ML040300263
| ML040300263 | |
| Person / Time | |
|---|---|
| Site: | Beaver Valley |
| Issue date: | 01/07/2004 |
| From: | Sepelak B FirstEnergy Nuclear Operating Co |
| To: | Colburn T FirstEnergy Nuclear Operating Co, Office of Nuclear Reactor Regulation |
| References | |
| Download: ML040300263 (335) | |
Text
{{#Wiki_filter:,.<Io - SJ-41441-Memorandum i_ To: Tim Colburn From: Brian Sepelak Date: 01/07/04 Re: References Requested During 12/10/03 MAAP-DBA Meeting On December 10, 2003 a working level meeting was held with the NRC to discuss the Containment Conversion Pre-application Report submitted by Beaver Valley Power Station (BVPS) by letter L-03-188 dated November 24, 2003. During the meeting, Rich Lobel requested copies of References 10, 13 and 19 cited in Section 2.6 of the pre-application report. While these references are available in the public domain, they were readily available to BVPS and are being provided for the convenience of the staff. 1
0 AN EXPERIMENTAL AND THEORETICAL STUDY OF SIMULTANEOUS HEAT AND LASS TRANSFER APPLIED TO STEA11 DOUSING by EMIN IKULIC, Dipl.Ing., University of Sarajevo, Yugoslavia, N1.A.Sc., University of Waterloo A thesis presented to the University of Waterloo in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemical Engineering Waterloo, Ontario, 1976 O Emin Ruli', 1976
~~.. .0~
I _ --w----------- 1. 14-3 IOLRS :D8 CONTAINMENT SPRAY EXPERIMENTS FOR PRESSURE SUPPRESSION Susumu Kitani' ABSTRACT rennt vestsel . For the containvont spray cooling, the nresence A containment sprny mvsten is one of the nuclear Cif air in the containment vessel lowers the condtnst-safety 4cvices Installed in pressurized water reactors tion heat transfer rate. Furthermore, spray Cl% rate, to suppress the Inner pressure and to remove iodine spray drop size, and spray pattern will also affect It. vapor in the containment vessel In a postulated large Hanr fundamental researches on cooling the stean accident. atmosphere bV spray have been reported. ^evertheless. A subject of the Containment Spray Test is to the experimental investigation in a large vessel hns confirm the effectiveness cf pressure suppression. been performed to understand the gas-space rixinr and Cold water was Injected as spray droplets Into a tare the temperature distribution in the actual reactor vessel of 700 m. prepared in a given stean-air condi- containrent vessel. tion. Practically applied spray nozzles of hollow The parviken Full Scale Containeent experiments cone type were used at several heights by changing in Sweden (1) and the simtunted design basis testm nf the number of nozzles. the c~nrolinas Virr.InIa Tube Reactor ltnV.S.A. '2) The exoerimental results showed that the pressure evaluated the heat transfer in the spray cooling decreased mostly with flow rate of spray injected And qualitatively. independently of the position of no:zle height. Vitacht, Ltd. evaluated the heat transfer efi-clenc- of spray-cooling es a function of the pressure NNCL.%T1R: with a vensel of 42 ml t3.3 re In diameter and 6 m In height) (3). f - spray flux (Wzhr) Tokvo Shibaura Co. Ltd.made an experiment with J F - total flow rate (o;/hr) vessel .of 3 m3 (1.2 m In diameter and 3.L m high) and
'L- flow rate at i-th stage (=3thr) found that the theoretically calculared v*lue agreed h - average height of no:zle (in) well with the experimentnl reaults. (J'A) In their hi- height of noz:le at i-th stage (m) ePerimcntS the effect of gas-space mixine tnder I - spray injection ratio (hr-1) spraying was not evaluated.
T8 - average temperature In vapor phase (CC) In the present experiment a fairly larre vessel Tspl-temnperature of inlet vater ( C) was used for clarifyins the effei.tiveness of PX2R's Tsp2-temperature of water reaching bottom containment spray cooling. of containment vessel ( C) V - volume of containment above operating floor (tt ) TEST FACILITY V'- volunt of containment vessel (2&3) n - heat absorption efficiency (-) Fig. I shows a s*henatic dLngram of the contaln-ment oproy tesr apparatus. It consists of a onodel INTRODUCT10. In the safety analysis of a pressurized water reactor (P¶X7R), an accident of the primary coolant blow-dosin into the containment vessel (loss-of coolant accident or LtOCA) Is postulated. If the nostulated accident occurs, the inner pressure of the containment vessel will be built up due to the blown-down stear. Therefore, M.M.'s plants have the containment spray system to reduce the pressure by cooling the steam-air atmns,ihere. .4nother important role of the containment spray nrstem is to washout airborne fission products released from any perforated fuels into the contain-1 Japan Atonic Energy Research Institute, Tokal-nura, Ibaraki-ken, .lapan C'g Contributed by the Fuel Society of Japan for presentation at the lit Internationtl Conferenct on Liquid Atomizition and Spray Fig.1 E1odk Diogram of Cotaintimnt SpraY Test k Systerns held In Tokyo. Japan from August 27th to 31st. 1978. Apporatus
-3 55-
ff*d tentainnent vessel, spray systems, instruments for 2) Pressure measuring spveral experirental data and an air Steam duct 1 cleaning Ayster. Steam-air atmosphere Tine Mlodel Containnenr Vessel 3) Flow rate The Model Containment Vessel is 20 m high, 7 m in Steam under injection 1 diameter and 702 r3 in volume. It is designed for the Spray water condition of 5 kg/cm2G in pressure and 160 'C in water flowing on wall Xtcmnerature. *Ilic pressure test was carried out at 6.25 surface 12 kg /cm2 G and the leak was examined to be less than 0.1 Water circulated 7tlay Al. 3.5 kg!cm2 G. from the sump 1 it Is made of carbon steel plate ranged in thickness from 25 mm to 28 im. rXPERIESTL PROGRAM The inside surface of the vessel is clad with 2 mm thickness stainless steel. The containment spray test for pressure suppre-The wall of the containment vessel is covered ssion simulated te a UOCA condition of MIR was with 5 cm calcium silicate boards for the thermal perfnrmed, where heat transfer between spray droplets insulation (tha density of board is 0.22 g!cm3 and the and steamt-air atmosphere In the containment vessel was thermal conductivity 0.042 ccal/m.hr.C). investigated. Pressurization The heat absorption efficiency, n, is shown as Steam supply into the containment vessel can be folltws, sent from a boiler directly with flow rate of 6 ton/hr. Pressure relief valves are installed in the main r - (TP - T BP/(T - TBP) (1) ventilation loop header adjacent to the containment vessel and rupture disks are also set for the same object because of potential overpressurization and where 1 Is the gas temperature in the containment vacuum. vessel],,5p, the temperature of water in the spray
; S v~rsv systems header and Tsp2 the temperature of water above the
- The clevation, El.,of spray headers is taken from bottom of the containment vessel.
the bottom of the containment vessel. Single nozzle The initial conditions were set at 2.5 kg/cm2 C can be set at 18 m(10). The ring-shaped headers are for the atmosphere of the containment vessel and 40C located at 15 m(el), 12 m(P2) and 15 m(#3) on which for the water of spray tank. The distribution of 18 nozzles can be set respectively. spray nozzle on the spray headers was set according A hollow cone type nozzle (the same specification to the object of runs. witlvSPRAC0-1l73A) was used for the prejent work with The height of containment vessel of a 800 lWe water flow rate of 5S 1/min at 2.8 kg/cm-, spray angle class MWR's plant may be 80 m. however, the lower of 65 -67'. part of it consists of compartments for instrumen-f7 Air Cleaning System tation and radiation protection upto about 30 a high. Thus, there Is a large free volume above the highest 4., An air cleaning loop is provided to relieve the depleted air due to water vapor and supply fresh air floor, where the containment spray Is applied. with flow rate of 1700 m3/hr or 3400 M3/hr. The test condition for FWR is sibilated bv calculating the spray flux, f, defined by
"'-N f - I Fi hi/V - F h/V (2)
Data !1tiLuisition: A digital computer system Is I connected to the containment spray test apparatus for f - logging collected data, processing them and platting where V1 is the flow rate of solution issued from the i-th stage of spray header at the height hi above the the calculated results. highest floor or operating floor of the PWR's con-The system is to be able to file a large amount tainment vessel, and V is the free volume above the of analogue physical inputs such as temperature, operating floor. pressure, flow rate of fluid, etc., less than one On the other hand, the spray injection ratio. 1. second continuously. Heat transfer coefficient, kinetic viscosity, Is defined by Prandtle number, thermal conductivity and absorption efficiency of sprayed water are calculated to analyze 1* FiM'@ FI/V' (3) the e.xperimental data. lic'ar-Transfer-Data Measureent System: To measure heat where VI is the total volume of thc containment
-transfer data, following sensors are equipped in the vessel. The spray injection ratiO is introduced from containment spray test apparatus. the consideration that the steam cxistinj ont only
- 1) Temperature measurement by cnpper-constantan ther- in the spray region but also in non-spray region of mocouples compartments will work to retain the inside pressure of containment vessel.
Steam injected Table I shows the program of containment spray Steam-air Atmosphere 59 test for PMR's LOCA condition. For the model Spray water 58 containment vessel, V and V' are taken to be same. Spray header 6 Wall three positions RESULTS 6/position Water flowing on The steam-aIr atmosphere of 2.5 kg/cm2 G was made wall surfaces 12 by Introducing steam at flow rate of 6 ton/hr into Water in sump the model containment vessel having contained air 1 Steam Inlet 1 of 1 atm at the room temperature. After arriving at 2.5 kg/cn2G the supply of steam was stopped and the
-356-
Table 1 The Contoinment Spray Tests for Pressure Suppression Run No. umber of Nzzles on Header F Fh/V F/Vt Purpose _ 0 #1 #2 1#3 lm3/hr) (r/hhr) hir- J lI7 PWR containment spry demonstration PHS-l PHS-1 O 0 6 6 j O l 0 0 ~~1 20.9
- 2. 0.447
.473 3X152 test b transient Simulallon of PWRmethodflux i,_______ (r one subsystem flux simulated)
S 20.9 0.447 I -2 IPWR containment spray demonstration PHS-2 PHS-3 02 0 I 64
- l 12 0
0 X 0 0 23x 41.8 10 test (by steady state method) PWR containment spray demonstration 0.893 6xl t2 test by transient method (2 subsystems flux simulated) a O O 6 O 0.357 -2 Effect of spray flux and mixing PHS-4 _ _20.9 3x10 i within gas phase b O O 6 .62 8 i.(by trcnsient method) I PHS-50 6 0
.H'-5 6 41__8
'Simulation (bxtaIietdetod of PWR spray flux by 41.8O6 0.7156 PHS-5 O6 0.715 l6xlO transient method (2subsystems)
. _ ' I _ _ _ _
. Characteristic experiment of single PHS-6 O 0 0 3.48 0.0892 5xlO spray nozzle
_ . 8 i ~~~~~~~~~~~~~~~~~~~-2 PWR l 795 0.506 t.lxlO ( Temperature of Spray Solulion 40tJ vessel was left for about 20 min to decrease the temperature difference between ateem-air and the structure of the vessel. Then, spraying water was ejected into the vessel. Fig. 2 shows the experimental results for the pressure change over the time in run PHS-1 with a Al, curve obtained in calaulatlon of CO%`EMPT-LT (7). SPoy strt which Is popular for estimating temperature-pressure OLkA behavior of light water reactor's containment vessel 2.5-under the LOCA condition. In the calculation, the heat absorption efficiency Is taken as 100 %. It
.2.0 means that the temperature of spray droplets becomes ,,, 2 ^~~~~~kohn bye equal to that of rhe gas-phase In a period of falIlng.
The code includes the term of heat transfer between 1. (SprayFW&ole 21Mse/hr) thc structure of the containment vessel and the vapor atmosphere. In the calculation, the heat transfer coefficient reported by Uchida et al. was used (8). 01.0 Flowlq 2 Fig. 3 shows the pressure changu plotted with the computer after spray injection in all runs. In the figure, it is found that the presxure suppression S S S Sx S S 5 9 curves are classified roughly into thrce groups determined by rate of spray, regardless of the height of spray headers or the configuration of spray nozzles 0 20 40 60 80 100 120140 160 180 j on the conditinn tested. Tn other words, thc pressure Time (min) suppression Is closely related to spray Injection rare
*shown by F/V5 in Table 1. Fig.2 Pressure Change in PHS- I t
t I
-35 7-i i
i 41
------- 09102MO stW",
F1. UI~ WV I O.V 2.0 - 2 6 2 H PHS - -j #I 12 1 42_1 ID~~~~~I [i 1.0 r NX ~ 6m5.m
~ ~ 1
& 1.0 . PHS-4b 0.~~~~~~~~~.
PHS-3 ,r. -;5PHS-4a E 50 7. 5 m PHS-5 -~HS-te~~~~~~~~~~~~~~ \ 4. 0mr 0 60 120 180
- Non-Sprayed Region Time (min)
Fig. 3 Pressure Change in Pt-S Runs Sprayed Region 0 60 120 Fig. 4 shows the average tcmperaturc-profile Time (min) measured at the level in the containment vessel as a function of time after spray injection in run PHS- Fig.5 Temperature change in PHS-I .lb. The test was carried out with flow rate of water of 21 r3/hr from the hcader of 13 stage at the and PHS-6 elevation of 9 r, located nearly half of the vessel. FIg. 5 shows the average temperature-profile in It is noted that the temperature falls more runs PHS-l and PHS-6. In both cases, as the location speedily in lower level. The temperature at the level of spray no::les set at higher levels the temperature of 9.S u and ll n in non-spray region falls at the difference between spray region and non-spray region same rate as the temperature at the level of 7.5 a in becomes smaller In coa arlson with the results In spray region. However, the rate of temperature fall- run PHS-4b. This smooth decreasc of temperature in ing dccreases with higher elevation. These results the vessel is considered as the effect of convection may be related to the heat preservation by air mole- ln the gas spuce between spray region and non-spray cules in non-spray region. It Is interesting to see region by the drag forces of falling droplets. after the start of spray for a while and then that is Another process considered is that the vapori:ation kept constant. of droplets in the gas space of higher temperature may ocrure and the vapor move in the direction of the sprayed water on which the vapor condenses. As metntioned above, when the steam is condensed on water droplets, the pressure in the containment vessel decreases and not only the steam but also the air existing in non-spray region cove to spray region. Considering these phenomena. fission products, especially radloiodine vanor, existing in both region 0 C- of spray and non-spray in the containment vessel of ID PWR in the LOCA condition will have chance to be I- washed out. 0. CD References I.
- 1) Thoren, H-t., F.ricson. L.. Hleshtil, R., and I.- Brandt, W.; "Full-Scale Containment Experipents Performed in the !¶arviken Power Plant", ANS Fational Topical Neeting on Water Reactor Safety, Salt lake, Utah, March 26-28, 1973
- 2) Schmidt, R.C., Bingham, G.E. and Norberg, .1.A.;
"Sfmulated Design Basis Acr.ident Tests of the Carolinas Virginia Tube Reactor Containment --
Final Report", IN-1403 (1970)
- 3) Sagawa, N.; "An Experimental Study of Sprey lr 0 60 120 Cooling In Nuclear Reactor Centalners%, Journal of Time (min) Nuclear Science and Techuology, Vol. 5, pp. 419 s 426 (196B)
- 4) Nagasaka. H.; 1976 Fall 'Seeting of the Atomic Fig.4 Temperature in PHS-4b Eingery Society of Japan, B72 (1976)
-35 8-
bkd§kdbdfAMMNW-
- 5) Sag.awa, T., Nagas3ka, E. and Tobimatsu, T.; 1976 I Fall Heeting of tce Atvoic Encrgy Society of Japan, B;73 (1976)
- 6) Nagasaka. E., Tobiratsu. T.. Yanai, R. and Vagava, T.; 1977 Annual Meetirg of the Atomic Energy Society of Japan. t5Q (1977)
- 7) Argonne Code Center; "CONTEIT-LT rsers .Manual".
(1973)
- 8) Uchida, H., Oyana. A. and Togo, Y.; "Evaluatlon of Post-Incident Coaoling Systemq nf Light-Vater Poer Reactors", PruceedIng of the*Third Inter-national (onference on the Peacerul Utes of Atmic
- Energy Hold in Ceneva. 31 Augcst-9 Septcmfbcr 196' , Vol. 13 (0965) 1 ~~~~~~~~~~~~~~-3 59-
DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART I By P. N. ROINE, B.Sc.(Tech.), Ph.D., A.N .C.T., D.I.C. (ASSOCIATE ME.%iBER)* and G. A. HENWOOD*
SUMMARY
This paper describes experiments that were made to measure the drag on a sphere In a water stream and how this drag varied with the presence of neighbouring spheres. The spheres were J in. and J in. In diameter and the flow conditions gave Reynolds numbers In the range 10 to 1000. Increases of drag coefficient of almost tivo orders of magnitude are obtained when the sphere forms part of a regular packed array. Very large changes are associated with the surfaces of particle assemblies. Repulsive forces between particles have been found and also conditions that lead to a reduction In drag. the results are used to explain some of the features of fluidised beds. They Indicate the need to know more of the gas flow patterns before these results can be applied to bubble formation In gas fluidised beds. Introduction the flow section. The local water velocity was measured by When fluid is passed upwards through a bed of particles, 'introducing a fine stream of dye (about 0-016 in. diam.) the bed is said to be fluidised when the drag force equals the from a drawn-out glass tube bent at right angles near its effective weight of the particles and they are supported by end to discharge horizontally downstream. When the tank the stream. The drag on an individual particle will depend, was tapped sharply a small kink appeared in the dye stream amongst other things, on the closeness and geometrical and the progress of this was timed between scribe marks on arrangement of neighbouring particles for this will determine the tank walls. the local flow pattern. The object of this research was to The spheres were made of polythene, Xin. diameter, obtained
.examine a model consisting of regular spheres and to see as cast sheets of several attached spheres. The surface finish how the drag forces varied as fluid was.passed through was generally good except at the four points of attachment different geometrical arrangements. to other spheres. These rough points were removed by polishing. A small hole was drilled in the sphere and this was.
Apparatus plugged with solder to make the overall density just greater than that of water. The sphere was supported by a 4 in. A hydraulic system was used to measure drag forces as length of 0-007 in. Nichrome wire which was pushed into it. this simplifies much of the experimentation. It is well known that the drag on a body can be correlated with the product of For some experiments the sphere hung by its wire as a of its frontal area and the dynamic pressure by a coefficient pendulum as shown in Fig. 2(a). The wire was bent at its. (the coefficient of drag) which is a function of Reynolds top end and hooked over a tiny wire stirrup. Most of the number. Hence this investigation which is concerned with the wire was shrouded by a water-fillcd glass tube. The fluid geometrical arrangement of neighbouring bodies can be made independent of the nature of the fluid and of absolute size. There is no reason to suppose that the Reynolds number is a critical parameter in a fluidised bed. Systemns spanning several orders of magnitude of this group behave in a broadly similar manner and both liquid and gas supported beds may have the same Reynolds number. I in. spheres were used for convenience and drag forces could be measured at Reynolds numbers 25 to 100. This is roughly equivalent to 16 B.S.S. particles gently fluidised by air. In some different but related experiments observations were made on J in. spheres at Reynolds numbers around 400. The principal apparatus is shown in the photograph of Fig. 1. Distilled water was circulated through an open-topped Perspex tank 3 ft long with a flow cross section 6 in. x 6 in. The superficial velocity could be varied from 0-5 to l S ft/min. Th1is corresponds to Reynolds numbers from 32 to 96 with respect to a in. sphere in the stream. Calming sections at the two ends of the tank were arranged to give a substantially uniform velocity ovcr the centre two-thirds of
- Chemical Engincering Division, A.E.R.E., Harwell. Fig. I.-Photograph of apparatus-watertank TRANS. LNSTN CHEM. ENGRS, Vol. 39, 1961
44 ROWVE AND IIENNOOD. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART I I - / 'w.ter w.f sc in the glass shield and the drag force estimated from the dcflcction, weight, and length as before. A further modification is shown in Fig. 2(c). This is for measuring drag and lift forces simultaneously. It is similar to the force balance of Fig. 2(b) but the upstream-facing limb of the glass T is left open and the opposite end is drawn down to a narrow diameter and closed. The horizontal wire-support can pass freely through the restriction to allow measurement of drag forces as previously. When lift forces act perpendicularly to the plane of the sketch, the horizontal beam hinges about its downstream restriction. By viewing a
+in mirror below the transparent tank, the two deflections at right angles can be measured and the simultaneous drag and A lift forces can be estimated.
Observations were initially made on a solitary sphere in the tank and these were compared with the correlation of aoWater surfece other data in the same range of Reynolds number.' After
/-:--
_ _ making allowance for the drag on the short length of exposed wire, it was found that the drag using a downstream support was some 10% higher than previously reported.' This seems 5S f most probably to be a wall effect' but was considered small enough to be of no consequence when measuring the effect on drag of neighbouring particles. There was some variation from sphere to sphere (about a dozen in all were examined) but a given one gave reproducible drag values. All results are quoted as a ratio of the observed drag force to that when the same sphere was alone in the water tank (i.e. at infinite separation from any interfering sphere). Thus, B any variation in the drag characteristics from sphere to sphere was effectively eliminated. The other apparatus used was extremely simple and con-sisted of a Perspex tank 5 in. x 51 in. x 3 ft deep. This I was filled with water and 5 in. x 5 in. sheets of attached II polythene spheres were allowed to rise through it whilst their velocity was measured. II In all the experimental work observations were made in random order except for the selection of water velocity which I was not easy to set to a required value and was always tedious k__ -0 fio. to measure. Once adjusted, observations were made over a range of sphere configurations with frequent checks of the velocity. In almost all cases these observations were repeated C at least once with a fresh adjustment to the same water Fig. 2.-Sketch of apparatus-force balances velocity. Where appropriate, the pattern of observations formed a simple factorial experiment. stream caused deflection which was measured by sighting a Experimental Results-Part I cathetometer on one edge. At small angles the deflection is These results were all obtained with j in. diameter spheres proportional to drag and an absolute value can be calculated in the tank with horizontally flowing water just described. from the measured submerged weight of sphere and wire, the They are presented as a drag ratio, a = Fx/F,, varying with length of suspension, and the observed deflection. Typical a separation ratio, 6 = x/d. The particle diameter, d, was not values are, deflection = 0 05 cm, suspension length = 10 cm, varied as this was not practicable and the Reynolds number submerged weight = 0-08 g, and resulting drag force = 4 was varied by a factor of three simply by changing the water x 10-' g wt. The sensitivity of this method was about velocity. In this narrow range the results were found to be S x 10-6 g wt. The fluctuations in water temperature that independent of Reynolds number and it is believed that this were observed did not affect this sensitivity. An alternative is true over a wide range. McNown and Newlin' have shown method of suspension is shown in Fig. 2(b). This was particu- that this is at least approximately true for spheres within larly convenient when it was required to insert the sphere cylindrical boundaries. into some array of other spheres. It has the advantage of a Inspection of all the data suggests that the drag ratio varies support which is entirely downstream but it is heavier and inversely as the separation ratio for all the configurations hence its sensitivity is less. The horizontal support is counter- examined. At infinite separation the drag coefficient is unity weighted at its downstream end and is supported at its centre by definition so that the simplest equation that can be written of gravity in water by a hooked wire as previously. The wires is, are shrouded in a water-filled glass T drawn out where the wire protrudes about i in. downstream from the sphere. a = . + P 100 (I) The moving parts were weighed under water before assembly TRANS. LNSTNI CHEM. ENGRS, Vol. 39, 1961
F, ROWE AND HENWOOD. DRAG FORCES IN A HYDRAULIC N1ODEL OF A FLUIDISED BED-PART I 45 where K depends only on geometry. This equation does not with the convention that downstream distances are negative. meet the terminal condition when 6 = 0 so that it can only The error of a single observation about these curves is roughly apply above a certain minimum spacing. . 0-616 so that it is high when 6 is small. A practical limit The coefficient in equation (I) was found by averaging, to the drag ratio for near touching spheres appears to be i.e. 0-4 when the interfering particles is upstream and 0-8 when I n. it is downstream. Thus, equation (2) and (3) are not valid K = - Z (a - 1) with 6 less than 2. The effect of an interfering sphere in line is not large unless the separation is only a few sphere dia-and the resulting equation, where appropriate, has been meters. Nevertheless, it appears that a drag reduction could be drawn on the graph showing the results and is referred to as detected over 100 diameters away upstream but only about the "best hyperbola". 30 diameters away downstream. As would be expected, an Consideration of purely viscous and purely turbulent flow upstream obstruction has a greater effect on drag than a suggests that the relationship between a and 6 is more com- downstream one. plex than equation (I) in the flow regime that is of interest in At very low Reynolds numbers when two spheres of equal fluidisation. Nevertheless it will be seen that a simple hyper- diameter are in line in a fluid stream, the drag on each is bola fits the data reasonably well and provides a simple form equal.' At the Reynolds numbers relevant to fluidisation, the to include in any mathematical treatment of the consequences leading sphere develops a wake which affects the drag on a of particle interactions in a fluid stream. following one. The average drag obtained from equations (2) and (3) agrees within a few per cent with the drag calculated for each of a pair in purely streamline flow. Two spheres in line The drag on a sphere was observed whilst it remained stationary in the water stream and another sphere was placed Two spheres misaligned in line at different distances upstream and downstream. The The sphere was next observed whilst it remained stationary results are shown in Fig. 3. in the water stream and another, at a fixed radial distance When the interfering sphere is upstream of that observed, was placed around it in a horizontal plane. The results are the best hyperbola is, shown in Fig. 4 where the drag ratio is plotted against angular position. Curves have been drawn by hand to indicate the a = 1-00--O . .5 . (2) form of the variation but no attempt has been made to derive an equation which would obviously be fairly complex. and when the interfering sphere is downstream, As the previous experiments showed, the effect dies away rapidly with increasing separation. Comparing areas bounded a =1 00 + 0 * * * (3) by the curve and the horizontal line a = 1-0, it is seen that the general effect of a single near neighbour is to reduce the Obstructing spfteredownstream Obstructing sphere upstream N I-C Alex___~ * .m l Best hyberbol. a +151.r F.s For a singe observation
.7 Best hyberbol@ S . °
*;H.g s 1-0 _ _
- 0. I
§,~~~~~~
- ~~~~~~~~~
.4 . Re -32 X R-64
*Re - 96 F.seX (+_ IFlow
.2 t-)oe I (#)ye A
.I e1
-22 IS 14 -12 -10 -8 -3 -4 -2 0 2 4 6 8 10 12 14 16 IS 20 Z2 24 26 A -SEPARATION DISTANCE I
Fig. 3.-Graph of drag variation-two spheres in line TRANS. INSTN CHiEM. ENGRS, Vol. 39, 1961
I.
.1:
46 ROWE AND IIENWOOD. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART I 0.9i". I, i.
-V I
I 1 I go, I oRo . II 43* 135* c =96 j0 i 0.6 i ok~~~~~~~Jr i Flow 1lo 0-51 I S's. 0.41- e_ Li16 I 0.3 1- 2Z5. I 270 I 1 _ I I I etC 45. 90 -5A RIS 225' 270' 315'
* - ANGULAR OISPLACtMENT I
Fig. 4.-Groph of drag variction-two spheres misaligned I
-drag on a sphere. There is, however, a region adjacent to, Four sheets packed in this way were placed in the water tank but a little downstream of, the sphere where a drag increase so that they completely filled the cross section and all the
-occurs. The curve is, of course, symmetrical about the line fluid had to pass through the assembly. The drag on a single
-of flow. sphere at rest in the water stream was observed as this assembly was approached from the downstream and upstream direc-The lift and drag on adjacent spheres tions. In a second experiment one sphere was removed from A sphere was supported in the water stream in the manner the centre of the sheet facing the solitary sphere which was shown in Fig. 2(a) and its deflection normal to the line of moved up until it fitted into the vacant site.
flow observed as well as the usual parallel deflection. This
-was done by viewing in a mirror set below the transparent tank. Observations were made as a second sphere was placed adjacent to the first at various separation distances. The results are shown in Fig. S. By convention, lift and drag act at right angles. As the spheres approach, the drag increases ,., - : 19 1 by a maximum of about 15%. The lift increases sharply with
-approach and can be described approximately by the equa-0.0 tion, L 1/F. 151d (4)
. *6 19 l.96_
This equation is unreliable at separations less than + of a sphere diameter and the maximum lift observed was about .07 9O-6 of the drag on an isolated sphere. Although only large at separations of less than one diameter, lift was detectable
-with the interfering sphere almost 10 diameters away. This is evidence in support of the already suspected slight wall effect in this apparatus. a: *.,..s~r.e ls This lift force is a special case of the lateral dispersive forces arising from fluid viscosity and described by Bagnold.'
The drag on a sphere near an assembly The polythene spheres were cast as sheets of several attached i spheres arranged in square packing. The sheets were square and two opposite edges finished in complete spheres whilst I the other two edges %wereformed from half spheres as shown I in Fig. 6. When several sheets are set one on top of the other 0 I a 3 4 3 with alternate ones turned through 90°, they form an assembly d- SECPAATION DISTANCE of spheres in rhombohedral packing, the closest possible. Fig. 5.-Graph of drag variation-lift and drag on adjacent spheres TRANS. INSTN CHENM. ENGRS, Vol. 39,1961
mC111" I' ROWVE AND HENWOOD. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART I 47 The results for approach to a complete assembly are shown in Fig. 7. A hyperbola fits the data reasonably well at separa-tions greater than six sphere-diameters but closer than this 4 FH. the slope of the curve reverses. An upstream obstruction causes quite a large increase in drag which falls off as the lone sphere gains the shelter of one in the assembly. A downstream obstruction has little effect but reduces the drag. ahsttoct.. li*st-III 44.nItres,, hIrc.n *.nhIj .pts
.. I 3*0
- A ,- 3 It.
I
. is _ I 9 ~
Fig. 6.-Photograph of sheet of spheres and of a stack *,.t bY014614 . *. When there is a vacant site in the assembly the drag becomes very large indeed as the sphere approaches the hole, and __ __ _ _ _ _ _ I~__ _ _ __---- 25 20 IS 10 S 0 5 10 15 20 forces approaching one hundred times the drag of an isolated +/- -SEPARATION DISTAMCC sphere appear to be possible. Again the data can be repre-sented by a hyperbola and the constants are shown in Fig. 8. Fig. 7.-Graph of drag variatIon-single sphere near an assembly With an upstream obstruction the same curve fits the data at separations more than six diameters but when the obstruction particles and will be discussed later. Experimental observa-is downstream the forces are very different depending on tions with the sphere almost filling the vacant site are rather whether there is a vacant site or not. From the equations which unreliable because the sphere tends to foul its neighbours and are based on all the data, the drag with an upstream obstruc- precise measurement of position is difficult. tion is always greater than when it is downstream. It may Some of the above observations were repeated with the therefore be inferred that the terminal value is greater and thickness of the assembly increased from four up to twenty-this is particularly relevant to the behaviour of fluidised four layers without any detectable difference in drag forces.
* -1 0 O' I A-SEPARATION DISTANCE Fig. 8.-Graph of drag variation-sIngle sphere near a defect In an assembly TRANS. INSTN CHEM. ENGRS, Vol. 39, 1961
- -. 48 ROWE AND IIENWOOD. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED 13ED-PART I It is concluded from this that the effect is independent of the Experimental Results-Part 11 thickness of the obstructing assembly once it is greatcr than about four particle diameters. The results described in this part were all obtained by The greatly reduced drag as a sphere leaves an assembly allowing assemblies of polythene spheres to rise through a for the open stream arises from the change in local fluid stationary tank of distilled water. The assemblies were made velocity. In this investigation it is more meaningful to regard from cast sheets of i in. diameter spheres, each sheet 5 in.
the change as a geometrical effect which in turn controls the square (8 x 8 = 64 spheres per sheet) arranged in square fluid velocity. packing. Two opposite edges were formed by complete spheres and the other two by half spheres as shown in Fig. 6. The Perspex water tank was 3 ft deep and of 5 in. square in The drag on a sphere in an assembly and near a defect section. The clearance around the sheets of spheres was The previous experiment was repeated in a modified form therefore such that a hydraulic diameter for interstices at the so that the drag forces on a sphere adjacent to a vacant site edge was similar to one near the centre. It was intended that were measured as another sphere was brought up to the flow through a complete sheet as it rose piston-like through unoccupied place. Two adjacent spheres were removed from the tank would be essentially parallel and could be regarded the centre of one face of an assembly and one was replaced as typical of an infinite sheet.* by a sphere supported as in Fig. 2(c). By viewing in two directions at right angles the deflections due to lift and drag could be measured. Observations were made as a second The drag on spheres in a plane arrayfrom which spheres are missing sphere was moved into the remaining vacant site. Only the case of an upstream obstruction was examined. A single complete sheet of spheres was timed as it rose over the centre 12 in. section of the tank. Some spheres were then removed fromr the sheet and the experiment repeated. Spheres were always removed in a symmetrical pattern to ensure that the sheet remained horizontal in the tank and they were taken either from the centre or from the sides. 40 The observed average coefficient of drag per sphere was calculated from the known drag (the buoyancy force), the frontal area of a sphere, the fluid density, and the observed terminal velocity. _ gF CD=AipU2 . . . (5) I
= K. 12 . . . . (6)
Xt where t is the time to rise past the measured marks. A Rey-nolds number based on the diameter of a single sphere was also calculated, and this used to calculate the drag coefficient for a single sphere moving at the same velocity from Schiller and Naumann's correlation," 02.. 20 le. 011061 43 .I- -1 C'D = 24 (I + 0-15 ReO.6 87) . . (7)
- ~~~~~~~~~d which applies to the range of Reynolds number observed during these experiments (Re - 300). The ratio of these two drag coefficients, CDIC'D is a measure of the drag increase The results~ acsoniFi.9we lift andrgato caused by packing the spheres together. It is an average drag ratio, a.
The results for a single sheet are shown in Fig. 10 where Fig. 9 :,pofdrag variation-sphere in sonassembly near a defect the ratio of drag coefficients is plotted against the number of spheres removed from a sheet. The average drag per The results are shown in Fig. 9 where lift and drag ratios sphere does not appear to depend upon which spheres are are plotted against separation distance of the second sphere. removed but only on the number removed. This confirms the The drag ratio for a sphere in the assembly falls as its neigh- expectation of parallel flow and the equivalence of edge and bour leaves from an initial value of about 40 to a terminal centre spheres. The drag ratio for a complete sheet is 36-7 value of about 25, which is reached when the neighbour is and this is based on 30 independent measurements. The little more than one diameter away. As has already been standard deviation of a single observation is 41 0-6 but this stated, these limiting values are not necessarily very accurate error increases with the rate of rise until, with a single sphere, because of the difficulty of positioning a sphere in the assembly it is impossible to obtain a satisfactory drag value because whilst leaving it free to deflect in order to measure the drag. the terminal velocity is not achieved until the sphere has Nonetheless, the form of the variation is shown in Fig. 9. The risen through about half the depth of the tank. For all but a lift increases as a neighbour departs up to a terminal value single sphere this limitation was not reached. of 15 times that of the drag on an isolated sphere and this is achieved when the departing sphere is one-and-a-half diame- The drag force on a sphere in an array can be related to the pressure drop through the assembly and this compared with ters away. Its sense is such that neighbours tend to move accepted equations for flow through packed beds. The authors towards the path of the departing sphere. hope to publish a note on this topic in the near future. TRANS. LNSTN CHEM. EN'GRS, Vol. 39, 1961
ROWSE AND HENWVOOD. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART 1 49 As a further check on the absence of wall effect, a single 9fr observation was made in a 12 in. square tank in which a I ~~ ~~~~I large sheet built up from the standard 5 in. sheets was allowed 10*Iqts " lot Of, to rise. This gave a similar drag ratio. Thus, for a complete and regular array, the drag appears to be equally shared by all members. As spheres are removed, the average drag per sphere falls rapidly. The removal of 12 spheres (less than 20 V.) halves the drag ratio. Since the majority of spheres are still surrounded by neighbours as in a complete sheet, it follows that the reduced drag per sphere is not so much a changed local drag Al - CI 500 coefficient as a change in overall flow pattern and channelling through the hole. The drag is halved by a reduction in velocity of 1/o/2 and this would little more than double the velocity 20 of flow through a 20% hole. This is not a very great distortion of normal parallel streamlines and illustrates the point that small deviations from uniform flow may have large effects on the distribution of drag forces. 10 I z 4 5 4 r a 9 la "BEA Of LAYERS Fig. I I.-Graph-effect of layer thickness and packing on drag Increase t: This suggests a packing orientation cffectr although it must i be remembered that the stack now includes many half spheres in its upper and lower face. The experimental work reported in the previous section measured the drag on an individual sphere in the down-stream and upstream facing layers of a close-packed assembly. As explained, the actual terminal value is difficult to measure and it might be argued that this is an arbitrary concept anyway. As two spheres approach in a fluid stream, the repulsion caused by flow will eventually be replaced by a repulsion resulting from physical contact. In a case such as poSSLIC SP'tAtS ?(A S'tC OF 04 that represented by Fig. 8, the change from one form of repulsion to another might appear to be continuous as a Fig. IO.-Groph of drag voriation-plane array of spheres with members first approximation over a short range. In this way the best missing hyperbolas of Fig. 8 can be regarded as applying when the spheres arc lightly in contact and a terminal drag value can be defined by an appropriate selection of separation distance. The drag on spheres in multi-layered assemblies Arbitrarily, let'this terminal separation ratio, 6O, be 0-01, Several complete sheets of spheres. were stacked, tied which was about the limit of possible measurement. This together with fine wire, and allowed to rise through the tank. gives a "terminal" drag ratio of 91 for a sphere in a down-The sheets were stacked in either rhombohedral or cubic stream facing surface and 67 for one in an upstream facing packing and in the latter case small pins were necessary to surface. At small value of 6 the ratio of these two numbers hold the sheets in position. A stack of 10 layers was the is virtually independent of this arbitrarily chosen crit.ical naximrnum that could be accommodated without excessive distance. binding against the walls. Rcfcr again to Fig. II and in particular to the drag ratio The results are shown in Fig. 11 where the drag ratio is obtained for a four-laycr stack in close packing which is Plotted against number of layers for the two kinds of packing. analogous to the situation examined in Fig. 8. The value It is seen that close rhombohedral packing results in a drag obtained is about 74 which is intermediate between the two Coefficient almost four times that for open cubic packing. suggested values for drag on the spheres in the upper and With open packing, the average drag per sphere diminishes lower surfaces. This indicates that a sphere in the'upstream-as additional spheres shield one another. This effect, however, facing surface is typical of the interior whilst one in the is not felt above about 4 layers. With close packing the downstream facing surface is in a plane of high drag. 'Me Partial closing of interstices in one layer by spheres in the distribution of drag forces thus appears to be as shown in next irmcdiatcly increases the average drag which continues Fig. 12 giving a predicted average drag ratio of 73. This to increase with a stack up to about 4 layers thick. Further indicates that the arbitrarily chosen critical separation layers reduce the drag a little. A stack of 10 layers was turned distance is a fair estimate of contact conditions, on its side when it almost filled the cross section of the tank As the thickness of a stack grows, the average drag ratio and gave a drag ratio some25y%greater than previously. will tend to that of a typical interior sphere which appears to TRANS. INSTN CHENT. ENGRS, Vol. 39, 1961 4
50 ROWE AND HENWOOD. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART I be about 67. Thus the curve of Fig. 1i shows a maximum cases were included with spheres missing from the edges. before falling off asymptotically. A curve based on this and Stacks or 3, 5, 7, and 9 layers wcre examined. the distribution indicated by Fig. 12 is shown dotted in The results are shown in Fig. 14 where the drag ratio is Fig. 11. The data suggests that the difference bctwcen the plotted against the missing spheres expressed as a fraction of drag on the upper and lower layers is greater than the previous the total. They fall into two classes depending upon whether estimate but the absolute values are not greatly in error. the hole extends right through the stack or not. Although the Because of the greater drag on the downstream facing layer, it tends to separate from the stack unless they are all tied together. This is another fact of importance in interpreting fluidisation phenomena. 100l
,9I_ _
flow A.rqo or4 foror, OI
- at I
40a X.14CO tOfSeM9 %dts MSW.C Fig. 14.-Graph-effect of cavity size on average de ig per sphere L5 2ot-scatter is high [standard deviation of a single Iobservation of 0' drag ratio is + 5 (28 dlF)], there is a uniform tr-end for closed holes which can be described by, f 2; 4 NuOmEAtF UYERS a =68 5-82x (8). I Fig. 12.-Distribution of drag forces amongst layers where x is the fraction of spheres missing. 1?his applies to closed roughly spherical holes. In a few cases an open hole was formed. I'hat is to say, spheres were missing from the top and bottom. This arrange-ment produced a much more rapid reduction irn drag because of the streaming that can occur through the cent tre. Equation (8) has a necessary end condition. I rhen x = 7r[6 or 05, the spherical hole will just break the : surface of a containing cube and an open hole is former d so that the drag ratio will fall considerably. The results IFor assemblies l with spheres removed from the edges are sho' wn in Fig. 15
,0. . . . . . . . . .
i0o so 40
%NN ,.
Fig. 13.-Photogroph of sheet of spheres with members removed II The drag on spheres in multi-layered assemblies containing cavities II The previous experiment was repeated vith layers in close packing only and with spheres removed from some of the I
'4 a0400 . 044 a as6 04a a04 01 000 404 0 n 0 44 of-sheets. Spheres were removed symmetrically as shown in I - F6d44 4 OfSMO E00WS4I84 Fig. 13 and stacks were built in such a way that the void was roughly spherical and in the centre of the assembly. A few Fig. 15.-Craph-affect of edge cavities TRANS. INSTN CHEMI. EANGRS, Vol. 39, 1961
I - I ROWVE AND HENWOOD. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART I 51 and in general they lie in between the curves for open and 2. In a close assembly, the drag on a sphere is increased closed holes. When the pattern of spaces was such that by an order of magnitude. How many spheres constitute channels were formed, the drag ratio more nearly approached an assembly has not been determined but it is probably that for an open hole. quite a small number.
- 3. Adjacent spheres repel one another. (If fluidised The drag on linear chains of spheres particles achieve a degree of random motion, Bagnolds' The last experiment in this series was made with a chain dispersive forces originating from grain inertia will arise.)
of 6 attached spheres cut from one of the sheets. A sphere at 4. Upstream-facing surfaces strongly attract particles to one end was weighted with a plug of metal, and the assembly any defect. allowed to rise through the water tank whilst its velocity was 5. Downstream-facing surfaces strongly expel particles measured. An unweighted sphere was removed from the but this is a force that falls off very rapidly with distance. top end and the experiment repeated with 5 spheres. This 6. A sphere in a downstream-facing surface is subjected procedure was repeated for 4 and 3 spheres, the minimum to a higher drag than any other in a uniform packed array. number that would rise vertically without oscillation. At 7. The loss of a sphere from a downstream-facing surface each stage the drag force was measured by weighing the is accompanied by a reduction in drag on the remaining prepared chain under water. neighbours.
- 8. Small departures from parallel flow will have a large effect on drag distribution within any array of particles.
- 9. The drag on a sphere in an array varies fairly con-0.9 siderably with the packing arrangement.
- 10. The drag on spheres in an array seems to be in-o.3 \,F nFrom Fig. 3 sensitive to the size of any vcid in the assembly.
- 11. The drag ratio is insensitive to changes in Reynolds 0-7l number if not independent of it.
The liquid-like properties of a fluidised bed can be explained 91 O.1RI- in general terms by these conclusions. Lateral repulsive forces will cause the bed to spread and fill its container whilst the Q-S-Z .04 greatly reduced drag experienced by a particle which leaves z' the top will cause it to fall back and so maintain a stable and definite upper surface. i;k Stability of upper surface The stability of this surface can be considered in more detail. When the fluid velocity is just sufficient to support a 0*I particle in the interior of the bed, one on the surface will tend to be expelled since the drag here is higher than else-o where. If expelled, the particle will soon fall back as the I e drag falls off rapidly with distance from the surface but its NUMBEROFSPtHRESIN THECHAN departure will reduce the drag on those remaining which will Fig. 16.-Graph-effect of chain length on average drag per sphere tend to close the vacated site. Thus there is activity on the surface of a bed at the point of incipient fluidisation with numerous individual particles leaping a few diameters free The results are shown in Fig. 16 where the drag ratio is of the surface and falling back again. This can be seen quite plotted against the length of chain. The Reynolds number clearly especially when uniform spherical particles are based on a single sphere diameter was higher than previously fluidised by gas or liquid. and averaged about 3000. The average drag per sphere is Even if the fluid velocity is so great that occasional particles seen to reach a limiting value of about half that of a solitary do escape, there is still a fair chance that they will be re-sphere and this is reached once the sphere is 4 members long captured. This is because of the long-range shielding effect or more. Again it is seen that assemblies more than 3 or 4 of particles in line. Whenever particles align in a stream just f diameters thick approximate to infinite systems. Included in sufficient to support them, they will fall and "condense" back Fig. 16 is a value for a two-member chain obtained by to the surface from which they had previously escaped. Thus, averaging the limiting values previously observed and shown the surface has very great stability and a property somewhat in Fig. 3. This shows good agreement with the present results analogous to surface tension. It is interesting to note that this and, since it was obtained at a Reynolds number two orders surface stability stems from a repulsion between particles and of magnitude less than the others, supports the view that there therefore it may be misleading to pursue too far this analogy is no Reynolds number effect. with surface tension in liquids. It is only when spheres are in line that they reduce one Discussion another's drag, but this must be an unstable arrangement
- From all the foregoing evidence, the following facts appear because of the lateral repulsive forces that exist. The extreme to have an immediate bearing on fluidisation phenomena: case of open packing must be very rare and an approximation to close packing will be more usual. The repulsive forces will I. Spheres in line or approximately so shield one another keep the particles separated and physical contacts between so that the drag of each is reduced. This is quite a long- them will be limited so that the bulk of the bed has liquid-like range effect persisting for tens of sphere diameters. properties.
- TRANS. INSTN CHEM. ENGRS, Vol. 39,1961
I 52 ROWE AND HJENWOOD. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART Fig. 17.-Photograph of stationary tens-shaped void in a fluidised bed F - DRAG-fF[o0W 0 - - 0 a 0 I
-0 00 eao.0 0o ~o 0 a 0 C ..
'0, 11 aO 1
goO, I 0,0 R F-w Fig. 18.-Distribution of drag forces through a stationary void TRANS. INSTN CHEM. ENGRS, Vol. 39, 1961
ROWE AND HEINWOOD. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART I 53 Before the minimum fluidisation velocity is reached, the Formation of voids topmost layer of particles will be subjected to the greatest The formation of gas pockets, voids or "bubbles" in gas drag. They will therefore be the first to lift and under condi- fluidised systems is a puzzling feature that has so far defied tions of uniform flow, fluidisation will start from the top and any logical explanation. The foregoing assists in under-develop downwards as each layer lifts and allows the one standing the stability of such voids once formed. For example, below it to follow suit. This would not occur with square it has shown that the forces associated with a surface lead to packing where the leading sphere of each line carries the a form of surface tension and so maintain the stability of a greatest drag. V; downstream facing surface. An upstream-facing surface is The uniform fluidisation "from the top downwards" similarly stable for the drag forces increase enormously as a described above is a common feature of liquid fluidised particle approaches it. The surface is supported and main-systems but is rarely seen with a gas-supported bed. The tained by a flow of fluid through it at a rate far less than that reason for this is difficult to see but is probably associated needed to support an isolated particle. with non-uniform gas distribution. It has been shown that This can be illustrated by a simple experiment. Uniform the drag distribution is sensitive to departures from parallel spherical particles of 0-75 mm diameter were fluidised by flow. Drag is also sensitive to changes in packing arrange- both gas and liquid in a long tube the top of which was ment so that an initial small disturbance might grow and the closed by a porous plate. The fluid velocity was increased characteristically heterogeneous gas fluidised bed will de- until particles were transported and the material built up as velop. Because of the greater inertia of the fluid, liquid- cake against the upper closure. The under-surface of this supported systems will be less prone to channel formation cake was smooth and regular and the fluid velocity could and the maldistribution of drag forces that follows. now be reduced and the cake was still firmly supported. Fig. 19.-Photograph of "bubble" breaking tie surface ef ogas Iluidised bed TRANS. INSTN CHEMT. ENGRS, Vol. 39,1961
J . - 54 LROWE AND HENWVOOD. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART I Particles were not locked for the tube could be shaken but when spheres are in line that they reduce the drag on each eventually, at a rate roughly twice the minimum fluidisation other. Commonly drag forces vary inversely with separation velocity, they rained down in a uniform manner. The minimum distance. Large increases in drag are observed as a particle supporting velocity was quite clearly defined. The drag on a approaches the surface of an assembly. sphere at the bottom of the cake was thus a little less than on Some features of fluidised beds can be explained in terms one in the fluidised bed and the drag was fairly uniformly of the prevailing drag forces acting on its constituent distributed so that the particles eventually rained down and particles. Surfaces formed by particle assemblies normal to did not fall as a cake. This dynamic consolidation is in the direction of flow are shown to be stable and to possess a accord with the drag forces that have been reported. property loosely analogous to surface tension. Little light The stability of the lens-shaped stationary voids that has been thrown on the problem of bubble formation in occasionally occur in fluidised beds is thus explained. Fig. 17 gas fluidised beds but the investigation has pointed to the is a photograph of a stationary void against the transparent need to know something of the gas flow patterns that are wall of a bed of sand fluidised by air. A deliberately-long associated with bubbles. exposure shows that particles in the roof are more or less at rest whilst those in the floor are in a state of motion. (a few Acknowledgment particles are apparently stationary in the void but they are The authors are much indebted to Mr. J. B. Lewis for attached to the wall by static charges.) In the centre of the helpful discussion and suggestions throughout the research. void the drag is insufficient to support particles so that they fall. The lower surface is stable but maintained in a state of Symbols Used agitation and the upper surface is dynamically consolidated as a clear and definite interface. Fig. 18 illustrates the dis- A = frontal area of sphere. tribution of drag forces. Although its stability can be ex- CD = measured average drag coefficient. plained, the means by which such voids are initially formed is C'D = calculated drag coefficient for an isolated sphere. not clear. d = sphere diameter. Fig. 19 is a photograph of a free "bubble" just breaking F. = drag force with spheres separated by distance x (in the surface of a bed of sand fluidised by air. The bed, 12 in. weight units). wide but only I in. from back to front, is effectively two- = drag force at infinite separation (in weight units). dimensional. The void is seen to be fairly free of particles and g = gravitational acceleration. those that can be seen are mainly falling close to the wall K = a constant. where the gas velocity is low. The edges of the void are L = lift force (in weight units). definite and suggestive of the high drag force situations that Re = Reynolds number. arise at the surfaces of particle assemblies. The arched bubble- U = sphere velocity or nominal fluid velocity. roof implies dynamic consolidation and a horizontal com- x = separation distance or fraction of spheres missing. ponent of gas velocity. It does not break as a bubble in a = drag ratio = F3 /F, or ?D/C'D. liquid does but falls back as a coherent mass as the gas 6 = separation ratio = xd. from the void passes through it. Drag measurements give p = fluid density. some clues as to the nature of bubbles in fluidised beds butl little progress can be made until the gas flow associated with The above quantities may be expressed in any set of con-them is known. sistent units in which force and mass are not defined inde-The findings described by Fig. 14 are a little surprising. pendently. They show that the drag on spheres in an array is insensitive References to the presence of a void in the assembly. This conflicts Schiller, L. and Naumann, A. Z. Ver. dtsch. Ing., 1935, 77, 318. with the conclusions to be drawn from Fig. 10 which indicates ' Henwood, G. A. and Rowe, P. N. The Drag on a Half Inch channelling towards the hole in a single sheet. Bubbles in a Sphere in Water at Rc - 25 to 100, U.K.A.E.A. Memo. A.E.R.E., M.633, February. 1960. gas fluidised bed almost certainly disturb the gas flow in 3 McNown, J. S., Lee, H. M., McPherson, M. B., and Engez, adjacent material. Perhaps the important criterion of void- S. M. Influence of Boundary Proximity on the Drag of size in this context is the ratio of diameter to particle diameter Spheres in Proceedings of the 7th International Congressfor and by this standard the experimental voids were very Applied Mechanics, London, 1948. (London: The Institu-tion of Mechanical Engineers.) small. 4 McNown, J. S. and Newlin, J. 1. DragSpheres within Cylindrical Boundaries, in Proceedings of the Ist ANational Congress of Conclusions Applied Mechanics, 1951. (New York: American Society of Mechanical Engineers.) The drag on a sphere can vary widely depending on the ' Stimson, M. and Jeffery, G. B. Proc. roy. Soc., 1926, A.111, arrangement of neighbours. In a packed assembly, the drag 110. is increased by one or two orders of magnitude over that ' Bagnold, R. A. Proc. roy. Soc., 1954, A.225, 49. Graton, L. C and Fraser, H. J. J. Geol., 1935, 43, 77. on a sphere at the same nominal velocity but in isolation. Adjacent particles repel one another and it is generally only The manuscript of this paper was received on 5 April, 1960. TRANS. INSTN CHEM. ENGRS, Vol. 39, 1961 L
DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART II By P. N. ROWE, B.Sc., Ph.D., A.M.C.T., D.I.C. (AsSSOCIAT MEMBER)*
SUMMARY
A regular array of + Inch spherical particles was constructed and placed In a water stream. The drag on a sphere was measured as a function of the spacing between particles and expressed as a ratio of that on a sphere at a given spacing to one In Isolation at the same superficial fluid velocity. This drag ratio was found to vary Inversely as the separation distance. The results are applied to fluidised beds and it is concluded that small local changes In particle concentration are unstable as they would require a very large re- distribution of fluid velocity. The only exception to this Is when the particles are separated by tens of sphere diameters. On the other hand, the displacement of one sphere from Its lattice position makes very little difference to the drag forces. The results are also used to predict the minimum fluidisation velocity and this agrees quite weU with experi-mental observation. Introduction together as three layers with the le spacing, x, The particles in a fluidised bed are supported by the drag between them and te pitch of the centre layer displaced by forces produced by the relative flow of fluid upwards through half a unit. The assembly was placed in the water tank the bed. The drag force can be correlated with the product with the plane,, oftie~ayers inclined at 60° to the direction of of dynamic pressure and the projected area of the particle by flow. In this way e spheres lay in an hexagonal pattern in a coefficient, the coefficient of drag, which is a function of planes parallefto t e direction of flow. The assemblyjust filled Reynolds number, the particle shape, and the geometrical the cross-sec ion th-wvater tank and is shown in Fig. 1. arrangement and closeness or concentration of the particles. This paper describes an experimental determination of the effect of concentration on the drag acting on spheres arranged in a regular geometrical pattern in a fluid stream. The results are used to discuss the stability of fluidised beds and to pre-dict the minimum fluidisation velocity. Experimentation The apparatus has been described by Rowe and Henwood 1 and consisted of an open-topped Perspex tank 3 ft long and of 6 in. x 6 in. cross section. Water was circulated through the tank at a velocity of about I ft/min. so that the flow was streamline and the velocity was uniform over the centre section of the tank. A J in. diameter polythene sphere was Suspended in the stream. The sphere was supported on a horizontal wire which hung at its centre of gravity from a Vertical wire hooked at its upper and lower ends. The wire
'Was shrouded by an inverted glass T and a drag force on the sphere deflected the sphere and wire as a pendulum, the dis-placement being measured by a cathetometer. The submerged IWeight of the sphere and wire was about 0-1 g and a de-flection of about 0-02 cm occurred at a velocity of 1 ft/min.
This corresponds to a drag force of about 2 x 10-' gram weight at a Reynolds number of 64. A number of other i in. polythene spheres was drilled Fig. I.-Photograph of sphere assembly across a diameter and threaded on to 0-022 in. diameter Nichrome wire on which they were a tight sliding fit. Several Of these strings of beads were mounted parallel to one It was made so that the separation distance, x, could be another in a rectangular frame, the wire being kept taut by varied. tension springs. The beads in this "abacus" were arranged The centre sphere of the middle layer was omitted and its in a rectangular net with a constant distance, x, between ad- position occupied by the sphere mounted in the inverted jacent sphcrcs. Three of these frames were made and held glass T referred to above. In this way the drag on a sphere was measured as it was surrounded by a regular array of other
- Chemical Enginecring Division, A.E.R.E., Harwell. spheres at a constant and known separation distance. Pre-TRANS. INSTN CHEM. ENGRS, Vol. 39,1961
- 7 ROWE. D)RAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART II vious work' has shown that the drag on thrcc laycrs of close- work' it was thought that the variation might take the form of: packed sphcres approximates to the conditions obtaining in larger assemblies and so this arrangement gives a fair indi- =Aa K+ 1 00 (I) cation of the forces that will act on a sphere in a uniform large cloud of spheres in a fluid stream. and accordingly a line with the best estimate of K has been drawn which is, Procedure 0-68 a = l.O * (2) The experimental procedure was as follows. The water velocity was set to a given value and measured by timing the The standard deviation of an estimate from this equation is progress of an interrupted dye-strcam through the working about 40-11/ and it is seen to be a reasonably good descrip-section and the drag on the sphere alone was measured. The tion of the data. This assumption of a hyperbolic relationship sphere assembly, previously built to a chosen spacing, was between a and S is arbitrary but it does in fact fit the data well lowered into the tank, placed around the drag-measuring over the range examined. There is no reason to believe that sphere, and the drag measured again. The water velocity was equation (2) has theoretical significance and it is likely that changed and a further observation made after which the an analytically derived equation would be much more com-assembly was removed and the drag on the sphere alone plex. measured at the new velocity. The assembly was replaced and The value of the drag ratio when d = 0 01 is given as 69 the measurement repeated. In all, the measurements were by equation (2) and this is in good agreement with the pre-replicated three times at each of the water velocities, 0-5, viously determined value of 68.5.1 For reasons discussed in 10, 1-5 ft/min. The velocity was varied in random order as Ref. 1, 6 = 0-01 is taken to refer to the terminal condition. was the setting of sphere spacing but once a given spacing Over the range examined, there is no systematic variation had been built all measurements were made with the one with Reynolds number of the relationship between drag ratio setting. and separation distance and this agrees with previous findings. A supplementary experiment was also carried out. The In these experiments the sphere diameter, d, has not been sphere-assembly was built to a spacing 6 = 0-35, placed in varied but it is believed that the results are independent of the tank and the water velocity set to 1-5 ft/min. The drag- absolute size over the range of interest in fluidisation. The measuring sphere was placed in its normal position in the reason for this belief is that, with a great variety of shapes, lattice and the drag was recorded. It was then moved along a geometrical similarity is known to be preserved with respect principal lattice axis and the drag recorded at various dis- to drag forces.5 Variations in drag coefficient, boundary placement positions until it touched its neighbour. Observa- layer thickness, and other flow phenomena are usually tions were repeated with movement along another axis at functions only of Reynolds number. The application of this right angles and also with the sphere in its central position argument to systems of spheres has been developed in some whilst a neighbouring sphere was displaced similarly. Accurate detail by Richardson and Zaki.' It will be shown later that the positioning was difficult but no significant changes in drag results appear to apply to spheres two orders of magnitude were observed during these displacements. smaller than those on which drag measurements were made. This is convincing support for the hydraulic model. Results The Stability of Particles in a Fluidised Bed The results are shown in Fig. 2 where the ratio of the Consider an ideal fluidised bed of uniform spheres in which observed drag to that of a sphere in isolation, a = F/F, is the particles are arranged in an orderly rhombohedral pattern plotted against the separation ratio, 6 = x/d. From previous as was built in the water tank. To be supported in the fluid 9 1. 4 ?UtMI'? tnt, .unh
. "fM.m.1atf I.,,,
- ~~~~~.. -. A. 9i 00 I
~~~~LQ~~~~F.
I I 60-1.00~ ~ a
-4!
4 V 3 z I' - -
-0 I I 3 4 1 4 I 8 9 tO ,..
A' ! SEP9RAT£t RC* Fig. 2.-Variation of drag ratio with separation TRANS. IINSTN CHEM. ENGRS, Vol. 39, 1961
ROWE. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART I7 177 stream the submerged weight and drag must be equal: It follows from this that local small changes in concentration cannot persist: if they did the particles in this region would cease to fluidise. Very small local changes in concentration 8 d'3 . ps -PF) = a CD4 .. .2 PF 12 . (3) would require a very large redistribution of fluid velocity if they were to remain stable. Thus we might expect to find that Equation (3) must apply to each particle. The velocity, u, is the particles in a fluidised bed are uniformly distributed and that based on an empty cross section and CD is the coefficient superficially this appears to be the case for liquid fluidised of drag for a single sphere moving with a velocity, it, in an beds and for the dense phase of gas-supported systems. It infinite extent of fluid. Equation (3) is essentially a re-defini- would be virtually impossible to detect experimentally the tion of a and CD. small spacing changes that have been considered above but at Consider a disturbance of this orderly array. In a small least obvious concentration variations are not observed in region let the particle concentration fall by a small amount. fluidised beds with the exception of "bubble" formation. This That is to say, locally, 6 increases by an amount J6. To is not to say that concentration defects can never occur but accommodate this expansion, let the remaining particles in only that they cannot persist. the bed close up a little. If the region of expansion is small, Consideration of drag forces alone does not appear to changes in 6 elsewhere will be negligible. In the region where explain why bubbles form in some fluidised systems and not d has increased, the drag ratio, a will fall and spheres will be but in view of the foregoing argument it is signi-in others, supported only if the velocity increases to compensate. The ficant that bubbles involve an abrupt and discontinuous drag coefficient varies with velocity so that for continued change in particle concentration. Perhaps the reason for this stability equation (3) can be re-written: phenomenon lies with the flow responses that follow a small change in particle concentration. a CD U2- dga P. (4) Referring again to Fig. 2, it is seen that at large separations the drag ratio changes very little. Therefore, once the par-In the range of Reynolds number relevant to most fluidised ticles are well separated, their stability will be insensitive to systems, small changes in concentration. This corresponds to the so-0 7 (5) called "lean phase" fluidisation observed with gas-supported CD = 24 (1 + 015 Re ." ) . systems. From visual observation of 0-065 cm diameter (See Ref. 2.) spheres fluidised in air at high velocities, the stable lean phase '21. Between equations (2), (4), and (5), we can estimate the local concentration was seen to develop at a separation ratio, 6, change in velocity, Au, that is needed to support a particle of the order of 100. At this concentration the drag on a par- ,). S lb which has changed its separation from its neighbours by Ad. ticle will be within I % of its terminal value. This is seen to be: During the model-experiments it was shown that the axial 0 68 Ad drag on a sphere or on its neighbours does not change appre-1 gd . (pS -pr) ciably when it is displaced from its lattice position. Previous AUt' 18
- u . (l + 0 25 Re0.6s7) (0 68 + 6)2 work' has shown that there is a lateral repulsive force as The ratio of the velocity-increment to the spacing-increment, spheres approach closely so that fluid-supported spheres in a Auf/6 is not very sensitive to the absolute value of the spacing. rhombohedral lattice are in neutral or weakly stable equi-It increases by a factor of 4 as 6 decreases from 10 to zero. librium. This means that fluidised particles initially arranged The Reynolds number term is also not important. It varies in a regular pattern might be expected to decay slowly into a between I and 3 for much of the range of interest in fluidisa- more random assembly without great changes in the drag that of the forces occurring as long as the local particle concentration tion. The ratio is sensitive to density only when change. The conclusions about stability could there-solid approaches that of the liquid. The most important does not factors are the particle diameter and the fluid viscosity. The fore apply to a more disordered assembly of particles.
ratio has been evaluated for glass spheres fluidised in air and in water for the initial condition wher6 6 5 0, and the results The Minimum Fluidisation Velocity are shown in Table I. The ratios for air-supported systems are two orders of magnitude greater than for water but they are Consider a bed of uniform spheres arranged in uniform in proportion to the experimentally determined minimum close packing through which fluid is passing upwards. Fluidi-fluidisation velocities. Thus, in response to a disturbance of sation will start when the drag on the particles is equal to this kind, both systems are required to increase the local fluid their submerged weight. The drag on a particle in this situa-velocity by about the same factor if the particles are to remain tion has been shown' to be 68-5 times that of an isolated Supported. It is seen that a very small displacement requires an sphere at the same superficial velocity. Thus, the minimum enormous relative change in velocity to keep the particle fluidisation velocity can be calculated from the terminal falling supported. A change in separation ratio, Ad = 0-01 requires velocity for an isolated sphere in the same fluid. that the velocity approximately doubles in every case. This 'When a sphere is falling freely at its terminal velocity, the Separation corresponds to a local decrease in concentration drag force and its submerged weight are equal. The same is of 3%. true for a sphere that is fluidised so that the drag forces for the two situations can be equated: TABLE 1.-The Velocity Change Required by Particle Displacement (7) CD, . 4 d2. I PFUT = CD..r. 4 d2 . I PFU'n.r. Min. fluidising velocity Particle diam. Fluidising Au/A6 Um.r. d (cm) medium (cmls) (cm/s) (see Fig. 4) and therefore, 2-01 0-0l3 CDm.r.CDT = UIT11I m.r. * . (8) 0 0114 Water 0-0650 Water 40-7 o-60 = a 0 = 68-5 when the drag 0-0114 Air 179-0 I*O0 2 290-0 30 5 coefficients are measured at the same Reynolds numbers. 0-0650 Air TRAkNS. INSTN CHENM. ENGRS, Vol. 39, 1961
178 ROWVE. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART 11 Because of the complex relationship betwccn drag coefficient The minimum fluidisation velocity calculated in this way and velocity in the range of intcrest, the terminal velocity can- is shown in Table II for several systems and is also plotted not be calculated dircctly from first principles but it can be in Fig. 4. Some experimental values are also included for shown". ' that: comparison. In spite of the assumption of initially ideal pack-4 d3pFg(ps-pF) ing, the agreement is quite good. It compares favourably CDT .RC 27T=-3' . (9) with most published empirical correlations which is sur-prising in view of the premises involved in applying the Therefore the product of the drag coefficient and the square hydraulic model results to real fluidised beds. Fig. 5 illus-of the Reynolds number at the minimum fluidisation velocity trates that the method is not sensitive to the value chosen for will be 1168-5 of this. The relationship between drag coefficient ao. The work of Pinchbeck and Popper, and of van Herdens, and Reynolds number for a sphere is known from semi- Nobel, and van Krevelen' suggests a, - 70. empirical equations such as those of Schiller and Naumann. 2 Thus the term CD Re2 can be calculated and a graph of this product for spheres is shown as a function of Reynolds TAsLE Il.-Minimum Fluidisajion Velocities Ulrnf. number in Fig. 3. With this graph and equation (9), ReT (CD Re2)m. = .x (cm/s) d 4 dcPPg(Ps-Pr) Rem.r. Calcu- Ob-(cm) 3 ,u (from graph) lated served _= I1-b-Balloini in Water
§ - =_
0-0114 0-546 0-023 0-0202 0-0128 0-0650 101 3-5 0-538 0-597
.l- 0-300 9 940 94 3-13 3-87 1-20 492 000 1 030 8-58 10-12
_.!*.... ... ^sI ,.....-. V1 1 Ballozini in Air 0-0083 0-0097 1-44 2-30 0-061 0-096 0-961 1-29 0-823 0-853 0-0114 3-73 0-154 1-77 1-01 0-0273 51-2 1-74 8-33 7-01 0-0388 147 4-33 14-6 14-3 0-0650 691 14-3 28-8 30-5 X-1..- Copper Shot in Air 0-0083 4-32 0-175 2-76 2-84 0-0114 11-2 0-44 5-05 3-84 0-0400 484 11-1 36-3 40-2 0-0650 2 076 32-5 65-4 84-7 The spherical material on which measurements of minimum fluidisation velocity were made was classified by sieving, the diameters quoted being the average mesh size of adjacent screens. From this it follows that the material is not strictly of one size and deviation from single-sized particles increases with finer material. This means that the packing of the fine material will be more dense than the ideal that was assumed and hence a0 will be greater and the minimum fluidisation velocity will be less than that predicted. This is generally the case as Fig. 4 shows. With the large material, the sphere diameters are more nearly equal and the ideal packing becomes possible. However, as poured into the container, somewhat A, less dense packing will be achieved and a fluid velocity greater Fig. 3.-Varioaion of CD Re' with Re for spheres than that predicted will be required to fluidise the bed. Again Fig. 4 shows that this is in fact generally observed. (and therefore the terminal velocity) can be calculated for an / isolated sphere. If in equation (9) CDT is replaced by Conclusions Measurement of the drag on spheres arranged in a regular
. 68-5 CD.. array has shown that the fluid velocity required to support and Rer is replaced by Re0.r., the minimum fluidisation a particle is extremely sensitive to the separation between the velocity can be calculated in a similar manner. spheres. This means that in a fluidised bed small local changes For example, for a spherical copper particle of 0-040 cm of particle concentration are unstable because they require a diameter falling in air at 0C, the right hand side of equation very large change in velocity distribution. This applies whether (9) becomes 33,140. From Fig. 3 Rer = 202 which cor- the supporting fluid is a gas or a liquid, This is not so, how-responds to a terminal velocity of 666 cm/s. Substituting now ever, when the particles are separated by something in the the minimum fluidisation conditions, the right hand side of region of 100 diameters. At this concentration local changes equation (9) becomes 33 140/68-5 and Fig. 3 gives Re,,.r. can occur wvith little consequent adjustment of velocity. This
= 11 -1 corresponding to a minimum fluidisation velocity of condition corresponds to the lean phase fluidisation observed 36-3 cm/s. with gas supported systems. TRANS. INSTN CHEM. ENGRS, Vol. 39, 1961
PUB I I I ROWVE. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED BED-PART II 179 Cm.. -I. ,. II .. .
*P-U ISLK ta-(C.)
Fig. 4.-Minimum fluidisation velocity. Comparison of theory with experiment By considering the drag force that acts when the spheres are close packed and touching, it was shown that the minimum fluidisation velocity could be predicted. This agrees reasonably well with experimental measurements made with close sized spherical particles. The agreement is probably as good as is possible in a system where random variations in packing are bound to affect the result. The main value of this method of prediction is that it is based on first principles and a minimum of empirical coefficients. The empirical measurements are the S drag on a particle in close packing and the variation of drag _, _ _______ e coefficient with Reynolds number for an isolated sphere. The method can be applied to beds of non-spherical particles by applying similar corrections as are used when predicting the terminal falling velocities of irregularly shaped particles. Acknowledgment The author wishes to thank G. A. Henwood and D. L. Pyle (a vacation student of the Manchester College of Tech-nology) who carried out the experimental work and J. B. Lewis for helpful discussion. Symbols Used CD = drag coefficient = F 0 D . 04
-d2 . jjoPFU d = sphere diameter (cm or in.).
Fig. 5.-Minimum fluidisotion velocity. Effect of vaorying at F = drag force (in weight units) (g wt). TRANS. lNSTN CHEM. ENGRS, Vol. 39,1961
.4 .. , .. 180 HOWE. DRAG FORCES IN A HYDRAULIC MODEL OF A FLUIDISED lBED-PART 11 References Re = Reynolds number = dpFi Rowe, P. N. and Henwood, G. A. Trans. Instn chcm. Engrs, 1961, a = fluid velocity based on empty section or relative 39, 43. velocity of sphere and fluid (cm/s). 'Schiller, L. and Naumann, A. Z. Vcr. dtsch. Ing.. 1935, 77, 318. x = separation distance (cm or in.). ' Heywood, H. J. imp. Coll. chent. Engng Soc., 1948, 4, 17. a = drag ratio, F/FI, = CDICD.-
'Coulson, J. M. and Richardson, J. F. ChemicalEriginccring, Vol.
11, 1955, p. 489, (London: Pergamon Press Ltd.). ao = drag ratio when x = 0. 'Hoerner, S. F. Fluid Dynamic Drag, 1958 (Published by the 6 = separation ratio = x/d. author, 148 Busteed Drive, Midland Park, NJ., U.S.A.). p = fluid viscosity (g wt/cm s). Z Richardson, J. F. and Zaki, W. N. Trans.Instin ctem. Engrs, 1954, PF = fluid density (g wt/cme). 32,35.
- Pinchbeck, P. H. and Popper, F. Chemical Engineering Science, ps = solids density (g wt/cm'). 1956, 1, 57.
%van Heerden, D., Nobel, A. P. P. and van Krcvelen, D. W.
Subscripts Chemical EngineeringScience, 1951, 1, 63. toD = at infinite separation of particles, x = co. m.r. = at a velocity such that the bed is just fluidised. The manuscript of this paper was received on 3 November 1960, r - terminal falling conditions for a sphere in an infinite and the paper was presented at a symposium of the Institution in medium. London on 7 March 1960. TRANS. INSTIN CHEM. ENGRS, Vol. 39,1961
Vi ABSTRACT The containment of decompressed heavy water steam is a problem which has to be solved in the design of nuclear generating plants. In the CANDU type of reactor, this is done in large vacuum buildings into which large quantities of water are injected in the form of a spray. The condensation process from an air-steam mixture must take place in a relatively short period of time, and the pres-sure in the chamber must not rise above ambient pressure. The final objective of this project was to provide reli-able and scientific design bases for nuclear power plant vacuum buildings and these have been achieved through the following: i) The combined heat and mass transfer to a single water droplet moving freely in an air-stream mixture has been treated theoretically and experimentally. Three distinctive models of heat transfer within the droplet were developed. The ex-perimental results are in excellent agreement with the second model which allows for internal resistance and mixing in the droplet. ii) Two experimental techniques--"catch in cell" and "direct photo-graphy"--have been used to determine the droplet size distri-bution in a dousing chazber spray. iii) Findings of previous steps have been subsequently applied to I predict the results of existing dousing chamber tests. For this prediction a new numerical algorithm has been developed.
vii The agreement between the predictions and experiments is very good over a wide range of experimental results. This result supports the further use of combined heat and mass transfer theory in the design and simulation of full scale steam dousing systems.
viii CONTENTS Page ACKNOWLEDGEMENTS v ABSTRACT vi GENERAL INTRODUCTION I PART I SIMULTANEOUS MO4ENTUM, HEAT AND MASS TRANSFER APPLIED TO A SINGLE DROPLET
- 1. INTRODUCTION TO PART I 7
- 2. DROPLET MOVING FREELY (UP AND/OR DOWN) IN AN AIR-STEAM MIXTURE 9 2.1 The Solution Approach 9 2.2 Hydrodynamics of a Droplet 11 2.3 Gas Side Heat and Mass Transfer 15 2.4 Droplet Temperature Response with No Internal Resistance 24 2.5 Droplet Temperature Response with Internal Resistance and Mixing 25 2.6 Droplet Radius Change 28 2.7 Droplet Temperature Response with Internal Resistance and No Mixing 29 2.8 Thermodynamic and Transport Properties of Air, Water and Steam 34 2.9 Outline of the Numerical Algorithm, No Internal Resistance 35 2.10 Outline of the Numerical Algorithm with Internal Resistance and Mixing 36 2.11 Outline of the Numerical Algorithm with Internal Resistance and No Mixing 43
ix 2.12 Discussion of Droplet Temperature Response Predictions 45
- 3. THE SURVEY OF EXISTING EXPERIHENTAL RESULTS OF HEAT TRANSFER COEFFICIENT DURING CONDENSATION FROM PURE STEAM AND AIR/STEAM MIXTURE AND THEIR COMPARISON ^
WITH PREDICTIONS 50
- 4. EXPERIMENTAL APPARATUS 60 4.1 Air System 61 4.2 Steam System 61 4.3 Test Section 64 4.4 Droplet Producing Unit 64 4.5 Measurement Techniques 65 4.6 Experimental Procedure 68 4.7 Heat Conduction and Radiation Influence on the Droplet Temperature Response Measurement 70
- 5. EVALUATION AND DISCUSSION OF EXPERIMENTAL RESULTS 76
- 6. CONCLUSIONS AND RECOMMENDATIONS 88 PART II SIMULATION OF THE SPRAY SYSTEM AND DESIGN SYNTHESIS OF A DOUSING CHASTER
- 7. INTRODUCTION 92
- 8. EXPERIMENTAL 94 8.1 Dousing Chamber Tests 94 8.2 Droplet Size Distribution Measurements 90
- 9. THEORETICAL ANALYSIS 99 9.1 Problem Definition and Simplifying Assumptions 100 9.2 Basis of the Simulation 103
., 1~
I 9.3 Droplet Size Distribution 107 9.4 Equation of Motion Solution 108 9.5 Droplets Temperature Distribution 113 9.6 Heat Losses Simulation 119
- 10. THE NUMERICAL ALGORITHM OUTLINE 120
- 11. EXPERIMENTAL RESULTS AND DISCUSSION 124 11.1 Droplet Size Distribution 124 11.2 Dousing Tests Prediction 131
- 12. CONCLUSIONS AND RECOMMENDATIONS 140
- 13. FINAL
SUMMARY
142 REFERENCES 147 NOMENCLATURE 156
~~~~~~~~~~~~~~~~~~~
xi APPENDICES Page A. Some Analytical Solutions of the Macroscopic Momentum Equation for a Sphere 162 B. Reinhart's Correlations for Droplet Drag Coefficient 168 C. Evaluation of the Droplet Temperature Response with Internal Resistance and No Internal Mixing 170 D. Thermodynamics and Transport Properties of Water, Air, Steam and Their Mixture 175 E. The Computer Program for No Internal Resistance Solution of the Single Droplet Response 189 F. The Computer Program for Internal Resistance and Mixing Miodel of the Single Droplet Response 200 G. The Computer Program for Internal Resistance and No Mixing Solution of the Single Droplet Response 220 H. The Air Rotameter Calibration Curve 244 I. The Humidity Calibration Curve 247 J. Feed Water Thermocouple Calibration Curve 249 K. Thermistors and Recorder Calibration Curves 251 L. Data of Experimental Runs for Single Droplets 257 M. Dimensional Analysis for the Response Time Corelation 265 N. Dousing Chamber Simulation Algorithm 268
- 0. Droplet Size Distribution Measurements 284
xiv I Figure 35. Thermal Utilization Predictions Compared to Experimental Values 117 Figure 36. Different Diameter Droplets Temperature at the Different Positions in the Chamber 118 Figure 37. Comparison of Different Droplet Size Distributions Obtained by the Catch in Cell Technique 127 Figure 38. Droplet Size Distribution Obtained by the Direct Photography Technique 130 Figure 39. Dousing Tests #1, 1A and 1B 132 Figure 40. Dousing Tests #2A, 2B, 2C and 3A with a Single Nozzle 133 Figure 41. Dousing Tests #2D, 2E, 2F, 3D and 3E with Five Nozzles 134 Figure 42. Effects of Different Droplet Size Distributions
.on the Dousing Chamber Pressure Response 138 Figure Hi. The Air Rbtameter Calibration Curve 246 Figure Ii. The Humidity Calibration Curve 248 Figure J1. Feed Water Thermocouple Calibation Curve 250 Figure Ki. Thermistor '1 Calibration Curve 253 Figure K2. Recorder Calibration Curve for Thermistor #1 254 Figure K3. Thermistor #2 Calibration Curve 255 Figure K4. Recorder Calibration Curve for Thermistor #2 256 Figure 01. Photograph of Droplets Obtained Using the Catch in Cell Technique for 30 IGPM near the Spray Centre 285 Figure 02. Photograph of Droplets Cbtained Using the Catch in Cell Technique for 30 IGPII near the Edge of the Spray 286 Figure 03.. Photograph of Droplets Obtained Using the Catch in Cell Technique for 40 IGPM near the Spray Centre 287
xv Figure 04. Photograph of Droplets Obtained Using the Catch in Cell Technique for 40 IGPM near the Edge of the Spray 288 Figure 05. Photograph of Droplets Obtained Using the Catch in Cell Technique for 50 IGPM1 near the Spray Centre 289 Figure 06. Photograph of Droplets Obtained Using the Catch in Cell Technique for 50 IGPM near the Edge of the Spray 290 Figure 07. Droplet Size Distribution for Droplets Shown in Figure 01. 291 Figure 08. Droplet Size Distribution for Droplets Shown in Figure 02. 292 Figure 09. Droplet Size Distribution for Droplets Shown in Figure 03. 293 Figure 010. Droplet Size Distribution for Droplets Shown in Figure 04. 294 Figure 011. Droplet Size Distribution for Droplets Shown in Figure 05. 295 Figure 012. Droplet Size Distribution for Droplets Shown in Figure 06. 296 Figure 013. Photograph of Droplets Obtained Using the Direct Photography Technique for 30 IGPM near to the Edge of the Spray 297 Figure 014. Photograph of Droplets Obtained Using the Direct Photography Technique for 40 IGPHI near to the Edge of the Spray 298 Figure 015. Photograph of Droplets Obtained Using the Direct Photography Technique for 50 IGPM near to the Edge of the Spray 299 Figure 016. Droplet Size Distribution for Droplets Shown in Figure 014. 300 Figure 017. Droplet Size Distribution for Droplets Shown in Figure 015. 301
Ii xvi LIST OF TABLES Page Table I.1 Dimensions of the Needles 68 Table II.2 Droplet Temperature Response Model Fits 115 Table II.2 Upper Limit Droplet Size Distribution Parameter Predictions for the Catch in Oil Technique 125 Table II.3 Upper Limit Droplet Size Distribution Paramters Prediction for the Direct Photography Technique 129
GENERAL INTRODUCTION A provision for the remote possibility of released heavy water steam is a problem which has to be solved in the design of nuclear generating plants with heavy water cooled reactors. This steam could originate from the primary cycle shown in Figure 1, which illustrates a simple scheme of the primary and secondary cycles of a nuclear power generating plant. One of the ways to solve this problem is to entrap potential nuclear pollutants, con-tained in the steam, in vacuum buildings. Every reactor building is connected with the vacuum building by the relief duct (017 in Figure
- 2) which transfers the unwanted steam to it. A pressure rise in the vacuum building acuates the water cycle to inject a huge quantity of water (from the reservoir in the top of the vacuum building) in the form of a spray. The condensation process, in the presence of air as an inert gas, must take place in a relatively.short period of time (30 to 50 seconds) and the pressure in the chamber must not rise above ambient pressure. To fulfill this, it is desirable to be able to design and predict the performance of such a dousing chamber shown in Figure 3 on a firm scientific basis. Several questions re-lating to the steam dousing problem (an Atomic Energy of Canada--
AECL-term signifying the condensation of steam in a mixture with air on a falling water spray) have previously remained unanswered. Typical unanswered questions were: (a) How fast could free falling water..droplets pickup heat from an air/steam mixture in which the 1
2
,. '-.II. - 30.61I Vi fs 31SI.
XX Figure 1. Primary and Secondary Cycle of a Nuclear Power Plant.
. ;.I It Switchyard 2 Discharge Ducts
- 3 Power House Turbine Hall 5 Aominislralion Building 6 Service Wing 14 Screen House -
7 No 1 Reacior Building 15 Skimmer Wall B No 2 Reaclor Building 16 Reactor Auxiliary Bay No 3 Reactor Building 17 Peliel Duct 10 No 4 Reactor Building 18 Vacuum Building 11 Turoine Auxiliary Bay 19 D20 U09radIng To-e' 12 Intake 20 Water Treatment Building AuxIiiary Power Untils 21 Barge Unloading Dock Figure 2. Elements of a Nuclear Power Plant.
I vacuu" Building inie~nalsf - Pressue acluated -ale' olacement Sci 2 e C ieL system inlet headeC 3 Piessuf e actuated watel 5 lcmn system outlet header A Vacuum Charnoel 5 DistrC,,uo andI Soray Heacel 6 Erergen~y Water Stolage Tana 7 perimelef Wall 8 Basement 9 vacuum Ducts 100Mon Snd HOIS\ ces 1 1 Emergency Wale, Line 12Pressure Relief Valves 13 Sheli-n t Walls I APersonnel A.ilOCk 1F5PressuVe Relief Duct 16 ROOf/Wall Seal 1 7 Water lank Access HatctI 18 Basement ACce_5 RampHeader 1 9Vacuum PumpV Suction 20 Vacuum Duct Drain PrOe 21 Vacuum Duct Fill Pipe 22 ReactOf Building Pressure Relief LouvIes 23 Services Tunnel 24 EcurOpment Airlock Building and Relief Ducl 25 Pet mete, Wall M~~~~~~~~~n~~ta.1 ~Vacuum 26 Jib Crane and Relief Duct. Figure 3. Vacuum Building
- .i~ ~ ~ ~~~~~~~~~~~~~ -- W1 7~~~~~~
At Airview of ?ickeril ThNuclear PowerPat Figure 4.
4 concentration of the steam was changing with time and in which the droplets hada range of sizes and velocities? (b) What would be the effects of the previous variables on the heat transfer coefficients tvosprays? (c) Could the performance of a dousing* chamber be Pre-dicted with any accuracy by use ofbasic theories of heat.mass _and momentum transfer? It was therefore decided to study the problem in two parts, one concerning the transport processes associated with a single droplet and another one treating the spray problem as a design synthesis. Thus the thesis is presented in two parts: Part I. Simultaneous momentum, heat and mass transfer applied to a single droplet moving relative to an air/steam mixture; Part II. Simulation of a svray system and design synthesis of a dousing chamber. The main objectives of this research were as follows: i) to develop and/or apply theory of combined transport phenomena in the case of a single droplet moving relative to an air/ steam mixture, ii) to produce reliable experimental results and compare them with the, existing experimental evidence on heat and mass transfer rates especially in the case of.high concentrations of non-condensables 50% and over. iii) to use information obtained in steps i) and ii) to simulate the experimental dousing chamber situation, and to compare the simulation results with existing experimental data.
5 These steps will hopefully provide a sound basis for the simulation and the design of a real dousing system (Figure 4-- Air-view of Pickering nuclear generating plant). The optimal design of a full scale dousing system is not treated in this study but it is felt that a basis is now available for this to be done. Parts of this research have been reported in:
- 1. E. Kulic, E. Rhodes, G. Sullivan: Heat Transfer Rates obtained in Condensation on Droplets from Air/Steam Mixtures, 24th Annual Conference of the Canadian Society for Chemical Engineering, Ottawa, Oct. 1974.
- 2. E. Kulic, E. Rhodes, G. Sullivan: Heat Transfer Rate Predic-tions in Condensation on Droplets from Air/Steam Mixtures, The Canadian Journal of Cnemical Engineering, 53 (1975), pp. 252-258.
- 3. E. Kulic, E. Rhodes, G. Sullivan, K. McLean: Direct Contact Condensation from Air/Steam Mixtures on Falling Sprays, Heat Transfer Section Paper V.2.1, Vth-International CHISA Congress, Prague, August, 1975.
PART I SIMULTANEOUS MO"ENTUIM, HEAT AND MASS TRNSFER APPT IED TO A SINGLE DROPLET 6~~~~~~~~~~~~~~~~~~~~~~~~~~~
- 1. INTRODUCTION TO PART I Combined momentum, heat and mass transfer to droplets occurs in a number of technical processes such as spray drying [1],
spray cooling [2), flash drying, spray crystalization, cyclone evap-oration [3], combustion of liquid fuels [4], spray or void of fill cooling towers [5) and water sprays at the bottom of cooling towers [63, air conditioning units [7], direct contact condensers in ther-mal power generating plants [8], etc. In spite of the fact that one of these operations was used 5000 years ago (evaporative cooling [9]) and the relatively young process such as the combustion of liquid fuels is about 100 years old, and although intensive research has been undertaken for decades in the area, even the simple single droplet problem involving these three transport phenomena is far from being resolved and completely understood for the whole range of variables of practical interest. For an extensive review of the subject development and accompanying problems the readar is referred to survays of iedley [10) and Williams [11]. Although all these practical applications have many things in common it is necessary to investigate them separately in order to cover the range of variables of the snecific application. Thus, because there are no data supported by basic theory for the conden-sation of steam from a dilute air/steam mixture on moving droplets, the steam dousing problem has been approached in this specific fashion. In Chapter 2 the temperature response of a single droplet 7
8 moving freely in an infinite air/steam medium is investigated. Chapter 3 reviews existing experimental results and compares them with our predictions. The experimental apparatus and results are discussed in Chapters 4 and 5, respectively. e
9
- 2. DROPLET MOVING FREELY (?/OR DOW..) IN AN AIR/STEAM MIXTURE 2.1 The ADproach to Solving the Problem The problem can be stated as follows. It is necessary to predict the velocity, distance travelled and the temperature (as a function of time) of a droplet moving freely in an air/steam mixture of known temperature, pressure and humidity. In general, there are two approaches to solving this problem, which are:
i) the microscopic or the "point to point" treatment, ii) the macrosocpic balances treatment. In the first case it is necessary to write the go-vernirg differential equations of momentum, heat and mass transfer for the surrounding gaseous phase and solve them simultaneously (analytically or numeric-ally) with the momentum and energy equations for the droplet itself. This approach is the most rigorous one, and the solution will con-tain all the necessary information for every "point" of the system. Analytical solutions of this kind would be most convenient, kowever, there is presently no complete analytical solution available even for the momentum transfer to the droplet when the Reynolds Number exceeds
.f, 1 [10]. This lack of analytical solution is simply caused by the complexity of the problem itself. Despite the fact that numerical solutions of the basic equations have contributed immensely in many transport phenomena areas, in this particular problem there is still a lot of work to be done. Canadian workers,in particular, have made significant contributions towards solving the problem as is evident
'I
10 from recent references 112 to 20). A review of these contributions shows that they have been successful up to NRe - 400-500, or to the point when shedding of vortices from the rear of a falling drop takes place. The only exception in this respect is Hoffman and Ross's [17] treatment who used Gaierkin's error distribution solu-tion [21) for the momentum equation and integral-boundary layer formulation [22] for the energy equation, to extend the solution to N - 1000, which is the first solution (to the author's knowledge) Re going that far. The authors are aware of the fact that this exten-sion presents a problem since the originators of the method [21] express their doubts about it saying: "Additional study of separated. flows will be necessary before the application of the Galerkin's method in such systems can be definitely evaluated." The Reynolds numbers in this particular research program cover a relatively wide range (NRe 50 to - 4000) covered only in part by the existing analytical or approximate analytical solutions. Early in the present work the author was fortunate to be introduced to the Spalding and Patankar approximate boundary layer numerical methods [23] and serious consideration was given to their applica-tions in this study. However, it was realized that these methods were applicable only below the separation point Reynolds number. This left little doubt that the problem in question should be approached using a macrosconic balance treatment rv Zra nsvort rate equations.
11 2.2 Hydrodynamics of a Droplet The problem is to predict the velocity and position of a droplet injected into a moving air/steam mixture at any time of the moment of injection. A macroscopic momentum balance applied to a single free falling liquid droplet yields the following equation (neglecting buoyancy forces)- dvD - 7rd 2 () dt PMCD 8 NV - VM)1IVD - v.1 + Fb (1) where Fb = mg (2) The distance travelled by a droplet can be determined from t z =o vdt (3) In some cases it is possible to express the drag coefficient CD as a simple function of the Reynolds number NRe and obtain an analytical solution for the velocity and the distance travelled. This has been done in Appendix A, for the case of a solid sphere. However, the drag coefficients for water droplets are somewhat different than those for solid spheres. Some of the effects influencing the drag coefficient (not in order of a respective importance) are: mass transfer, acceleration, deceleration, free stream turbulence scale and intensity, particle rotation, internal circulation, roughness,
12 etc., and some of them influence spheres and droplets differently, resulting in different values of the drag coefficient. The most exhaustive review of these effects is presented in the six part paper by Torobin and Gauvin [26). The situation is complicated by the fact that some research findings on the subject contradict each other. For example, Ingebo's [27] results indicate a decrease in the drag coefficient due to acceleration rather than an increase found by most other investigators. Lunnon, Williams and others, as well as Torobin and Gauvin [24) suggest the increase in drag coeffi-cient values due to acceleration. Some drag effects cancel each other. For example an increase in wake turbulence caused by Reynolds number increase or by surface roughening seems to decrease the effect of the acceleration on the drag. When Torobin and Gauvin published their paper they dramatically underlined the necessity of further investigations. However, at the present time there is very little new information available. In the selection of the droplet drag coefficients to be t.,V, used over the whole range of Reynolds numbers, the technique proposed '% V~
.,"r*
by3aL and Shepard [Zi] was adopted to at first. This technique essentially uses the knoun value of the droplet terminal velocity to determine the steady state drag coefficient from the equation of mo-tion. The terminal velocities were determined from Best's equation: vRT 943 [1 - exp(- d/1.717) 1 147 - (4)
.I I\ F
-1~~~~~~~~~z Mv
. .. ~ 1s*.
*\9. 1.
rp
13 O: where:.
.. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ;
VRT - relative terminal velocity, [cm/s] 7I:. d - droplet diameter, [mm], which was obtained to approc.imate theex.perimental results of Gunn
- and Kinzer [28). Curve #11 in Figure 5 was obtained in this way, and afterwards approximated by three straight lines (#5, #6 and #7) in the same figure, as reported in [29]. The three straight lines i,,, I are
-0571 2 CD 14.098 N for 10<N < 10 (5)
D Re RRe
-0337 2 3 CD 4.61 N-Re for 10 < N Re < 10 (6)
C 0.0264 N 0 396 for N > 103 (7) D Re Re The drag coefficient values determined in this manner were then com-pared with Hughes and Gilliland [30] values, used by Hcllands [31] and Groeneweg [32), as shown in the same figure. The results of very thorough theoretical and experimental research by Reinhart [33,34) are summarized in Appendix B and repre-sented by curve 12 in Figure 5. Although the agreement between eouations (4) to.,(6) apnd Reinhart's resu3ts is ver; good over the whole range of Reynolds numbers it was later (in the dousing charmber simulation) decire' to use Reinhart's correlations, because of the thoroughness of the research and because of the extent of the Rey-nclds number range. Clearly enough, the equation of motion could only be solved using Reinhart's correlation by numerical methods.
1,2,4 - EQUATIONS (A-1, A-5, A- 14) 3 - C = (24/NRc) (+5 NI0 6 8 7 ) .1 5-7 EQUATIONS (5,6,7) 2 8 CD = 12 + 20/N R 0+ 66/ (1+175 (log0 (N 0 /280)) ) [31] 7-2 9 CD - 0O22 + 24/NRe (I+O 15 NR 06 ) [31]
'-3 10 EXPERIMENTAL EVIDENCE - PERRY [30]
11 LAPLE'S TECHNIQUE a BEST'S EQUATION A 12 REINHART [33, 34] I_ 5 r 8 N DROPLETS _z_ Figure 5. Drag Coefficients vs. Reynolds Nuunber 7J 9 _ 7- , 1-A l I l 1 I I I loo0 101 . 1l 10 0
15 For that purpcse two subroutines were written, one for the case of a droplet moving downwards and another one for the droplet moving upwards. Basically the equation of motion is solved for both cases using the fourth order Runge-Kutta technique, see Kuo [36]. The results of these calculations for the relative velocity a3nd the dis-tance travelled versus time, of a 1 mm droDlet moving upwards, sub-jected to different initial conditions (volume fraction of steam in a mixture 5%, total pressure 1 bar) are shown in Figures 6 and 7, respectively. The solid lines represent the conditions when the air/steam mixture (moving upwards also) velocity is smaller than the relative terminal velocity and the dashed lines represent the opposite situation. In any case the droplet reaches its relative terminal velocity of about 4 m/s. This is in agreement with the data of Gunn and Kinzer [28] and Foote and DuToit [37]. From Figure 7 one can see that when v. < vRT the droplet moves upwards initially then after it reaches its maximum height starts to fall down. How-ever, when fM > vRT the droplet is carried away by the air/steam mixture having higher velocity. 2.3 Gas Side Heat and Ilass Transfer The problem is to predict the rate of heat pick-up of a single free falling droplet in a mixture of steam and an inert gas (in this case air). A droplet of known diameter d. initially at uniform temperature T., is injected with a Velocity VD into an air/ steam mixture of knourm concentration C and temperature T*. Yne
16 CURVE DROPLE INITIAL MIXTURE N0 J VELOCITY rn!5 t VELOCITY m/s I 10 5 .- I-,.- 2 5 10 3 2 5 4 11 l 5 3 2 6 2 3 Figure 6. A Relative Velocity of Upward Moving Droplets C 1 1 VM < VRT Xs. -. VIA '-VRT 2
'I,. ~r iI 6
3 _______ e\ .A
I',I 17 i/7
~11 Figure 7. A Distance Travelled for Upward Moving Droplets II
/ I
/ l 3 /
I 1-7 / I I l F - / / I I OF1 I I // I
/
I I I I i 000,
'I .00 #I,
.-O 0-11 011 I 3 00 000,
-\II 0-1 I- Wool I 0-1 01-0.000,
.-II I
0 I
18 I' L injected droplet erperiences two unidirectional heat fluxes directed towards its surface; one caused by the temperature difference acting as a driving force between the surroundings and itself and usually called the sensible heat transfer component Q%; the second caused by the steam concentration difference in the bulk and at the droplet surface acting as a mass transfer driving force and called the latent heat transfer comnonent These components comprise the total heat QT1 which is exchanged between the surroundings and the droplet. The total heat (see Trybal [38)) can be determined from:
+% QL QS (8)
The sensible hea-t tr-nsfer flux may be expressed as: QS N4 h (T Ts) (9) Skelland [39) proposed the introduction of the Ackermann number N Ac to honour the early contributor in this area [40,41,42). The Acker-mann number 11A . -a (10) 1l- e allows for the mass transfer out of the gas phase and is the correc-tion factor to the heat transfer coefficient h, obtained without mass transfer to the droplet. In the case of mass transfer from a binary mix:ture vith one inert as (dr),te coefficient a can be represented as:
19 a = (NAIIA CPA + NSMSCpS) /hs N . ~.C* X , ,, X I
= N C/ (11)
.s csP' 'S The latent heat component can be expressed as QL = NSMS (12)
The mass flux of the steam from the mixture can be expressed as: Ns = F ln[(l - C5 i/C)/(l - CSM/CM)] Vl
= F ln[(l - PSi/P)/(l - PSM/PI )] (13)
Using equations (8) to (13) the total heat transfer to a droplet QT can be determined if the values of the heat and mass transfe$ccef-ficients are known. Heat transfer coefficients h can usually be calculated for most engineering situations by use of correlations involving the Nusselt number. However, the determination of the mass transfer coefficient F is not always at hand, mainly because there are fewer available experimentally based correlations for the Sherwood number, due to the complexities of the experimental and theoretical determinations. The situation is eased considerably by the discovery of transport phenomena analogies, especially the heat and mass transfer analogy which has proved useful in many engineer- .Fiz ing applications. The validity of this concept for a dilute (rich in an inert gas) air/steam ml::ture was confirmed some fifty years ago by Lewis (43], and reconfirmed by many investigators, e.g. for the
20 evaporation of liquids inside a tube [44], for the evaporation of water in film type of cooling towers [45], evaporation of droplets in spray drying [46,47) as well as for the evaporation of solid spheres in gas streams [48). A number of experimental results on the condensation of steam in the presence of an inert gas confirms the validity of the heat and mass transfer analogy, in the range of Reynolds numbers 20 to 2000, see Bobe and Malyshev [49], who studied the condensation of steam from air-gas mixtures on tubes. They in-cluded the experimental results of Semein [50) who studied the con-densation of steam from an air/steam mixture on cooling tower pack-ing producing liquid film flow, and of Schrodt and Gerhard [51] for the condensation of steam from a mixture with a non-condensing gas on vertical tubes in a bank. In a series of papers covering the last two decades Berman [52,53,54,55,56) (citing some of the papers) has treated the problem of combined heat and mass transfer for dif-ferent industrial applications (mainly condensers and cooling towers). He proposes the introduction of additional groups to correlate with the Nusselt and Sherwood numbers. Thus he suggests: NNu f(N PrINGr ,n , cPS /CPM) (14) 4 aid Nh ff(NRP IN cNr IgCE,RS/RM) (15) where:
21 j k ~~~P -P S n SM Si (16) is NSiS and CS S14/ M (17) In this manner it is possible to determine directly the corrected heat transfer coefficient (i.e. NAc hs) in the presence of mass transfer for many experimental situations. The same approach is used by Bobe and Solukhin [57]. The obvious limitation of this approach is the necessity of having at hand the correlation for the Nusselt and the Sherwood numbers in every particular case considered. However, in all the work reviewed above there were no reported ex-perimental data for the case of condensation on droplets in the presence of non-condensables. The literature search was encouraging enough, however, to allow the presumption of the validity of the analogy in our calculations of mass transfer coefficients. Thus for the situation involvting forced convection around a sphere or a drop-let of diameter d, the Nusselt number correlation is:
.n m
Nu Re Pr and the analogous equation for mass transfer is the Shenrood number correlation: NSh = A + BINRe IIISc The reported values of A in equations (18) and (19) are either 0 or
22 2 depending on the ranges of Reynolds number over which the experi-ments were performed. The only exceptions to this rule are Kramer's [58) values reported from 3.2 to 5 and reviewed by Keey and Glean [59] who reported only the value 5. Since these values for A cor-case of liquid-liqid-mass transfer, for the present respond to the .................
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
consideration the value A = 2 has been used. Thus equations (18) and (19) are then correct in the reported limit when NRe 0. The reported values of B vary from 0.085 [58] through 0.27 [48], 0.459 [60]), 0.6 [48,47] to 0.98 as reported by Ward et al [61). Theoreti-cal considerations of Hoffman and Ross [17) and Masliyah and Epstein [19] indicate that the exponents m and n in equations (18) and (19) depend on the value of the Reynolds number itself. This was not taken into account in the present study because the reported values cover relatively narrow ranges of Reynolds numbers. In most reported ' (e.g. 4,31,32,62,63) applicatoions of the correlation the values used are 1/2 and 1/3, respectively. The very thorough study of Ro7e et al [64) also reaffirms the applicability of the correlation. Some - of the correlations mentioned above are shown in Figure 8. Curve 1 represents the results of Ranz and Marshall [46) for the evaporation of droplets in a spray dryi.ng apparatus. Curve 2 represents Semeyin's [50] results for the case of condensation of steam from an air/steam mixture on the cooling tower packing of the film type. Curve 3 represents the results of Sher;;ood and Gilliland [44] for the evap-oration of liquids in a tube. Hoffman and Ross's [17] results on
'I~~~1
---- FIUGHMARK'S CORRELATIONS [48].
NNUOR N h- 2+0O6 NROl/z (NPr OR NSe.)13. FOR NRe C 450 NNU OR N Sh 2 +0-27 NR 0 '6 2 (Np,. OR NSC)"/3, FOR 450 < NRC<104
~
5. 6 Figure 8. Nusselt of!Sherwood Number vs. Reynolds Number 10 102 NRe
-- ---- .. -.-- -- . -- 7= ., ..
24 the evaporation of droplets are represented by curve 4. Curve 5 represents Bobe and H4alyshev's [49) results on the condensation of steam from air/steam mixtures on tubes. Berman's [45) results on the droplet evaporation are represented by curve 6. Hughmark's [48] correlations on the solid spheres heat and mass transfer are repre-sented by curves 7 and 8. All the curves for spheres and droplets are very close to each other. (;,rf.L The correlations proposed by Hughmark [481 have been used in this study, since they are based on a large number of experi-mental heat and mass transfer data. They satisfy the limintin value of NN = 2 for a stagnant fluid, see Lightfoot [65). 2.4 Droplet Temperature Response with No Internal Resistance Now when the heat transfer rate from the surrounding medium to the droplet is known it is possible to predict what effects this will have on the droplet temperature response. The simplest approach to the problem of predicting the temperature response of the droplet is to assume that the internal resistance to heat transfer of the droplet is negligible compared ti*the external gas phase resistance. (In normal convective heat transfer problems this is a perfectly valid assumption.) This assump-tion implies that there is no temperature gradient within the droplet. Thus the rate of heat transfer to the droplet is equal to the rate of heat pick-up of the droplet;
25 d (m Cp T) =QTA (20) Assuming negligible change in radius (see below for a discussion of this) and mass of the sphere, equation (20) may be rearranged to give: dT C(T T) + C AN ' (21) dt- 14 1 AA C1 = 6/p Cp d (22) C =N h (23)' C C1 C2 (24) If only a small temperature change is assumed, then all the quantities in equation (21) can be considered to be constants, except t and T.
- Hence, T = TM + Cs S +T NS S} exp (- Ct) (25) which represents the temperature-time response of a droplet over a small temperature change.
2.5 Droplet Temnerature Response with Partial Internal Resistance and Mixing Normal1yin convective heat transfer, the liquid side resistance in m as-ligid conacting is negligible. However, in com-bined heat and mass transfer, where the iatentJheat component is
26 larged the gas side resistance may be lowered enough so that the _uid side resistance becomes significant. It will be shown later that the internal resistance of the larger droplets may play a significant role in the droplet temperature response. The follow-ing theory is reviewed as a means of taking this into account. Both sensible and latent heat transfer contributions combine to provide an apparent vapour phase heat transfer coefficient h defined as: l app happ ' QT/(TM TS) (26) which thus includes the effects of sensible and latent heat trans-fer components (see equation (8)). This definition of an apparent heat transfer coefficient makes it possible to treat the droplet as a sphere suddenly exposed to convective heating or cooling 166,67). The temperature field in-side such a sphere is described by the following differential equa-tion. ate + a * - (27) TFr rar C with initial and boundary conditions: T(r,O) T. const , 0 < r < R. (28) T(Rt) f(R,t) , t > 0 (29) The heat balance across the dropler interface yields
27 Iitz h (T., - TS) = - k )TR (30) 7W.. and the symmetry condition may be expressed as: BT 0 aTr= at r =O (31) There is an additional condition imposed by the mass transfer caus-ing a droplet radius increase, according to dR NSMS dt - ~~~~~~~~~~~~(32) dt PD This problem is known in the literature as the Stefan problem. A number of situations where this problem arises is treated by Rubinsteinj[68]. Since there is no available analytical solution (i.e. with R and happ varying with time) of the differential equa-tion (27) with the conditions (28) to (32),an approximate numerical technique is accep d using constant coefficients over short inter-vals of time. Thus a dimensionless temperature profile solution for a sphere of constant diameter a short time after exposure is [66,67): T - TM n 2R (sin t- -in coslpn) sin(l r/R) 2 nl n((ln - i n*n cos n) rn ~ ep ( 2 Tm IM
-TTi n 2R nFo (33) where:
n - tha roots of the transcendental equation (34)
28 l- n cotin - NBE happR/k (34) The dimensionless droplet surface temperature is then (for r = R) n T - TX TM TM -Ti
= yi h-l in
'p (sinin in sin
-'i i
cos i)sin'p n n laexp-iP 2 N ) (35) The average temperature of the droplet T is determined by jR 4rT r2 dr (36) 4rR 3 0O which after integration yields n~co (sini, -Vn cos p) 22 T TM (TM -Ti) 13 3 p-sn sn c csn exp( - NF Fo (37) It should be emphasized that these equations are based on the assumip-tion that the apparent-liqat transfer coefficient is also constantl during the time increment. 2.6 The Droplet Diameter Change As it was pointed out the above derivations are valid for the case of a sphere of a constant radius, an assumption which is aot. quite true in this problem where steam is condensing on the dr-opjet. However, over sufficiently small droplet surface tempera-ture increases, the droplet size can be considered constant and for
29 repeated calculations over a succession of temperature (hence time) increments the mean droplet diameter can be adjusted according to the equation: Ld 2N H-At = - (38) (For the sake of completeness it will be Dointed out that the maxi-mum channe of a droplet diameter when fully utilized, i.e. heated A uniformly to the surrounding temperature, is less than 4% fox' XS < 0.5.) This unsteady state phenomenon representation by quasi-steady state conditions is adopted for all the correlations used for heat and mass transfer. In addition the steady state drag coef-ficients are assumed to be valid in the unsteady conditions provided the relevant non-dimensional numbers are calculated by use of the instantaneous values of the variables representing the condition of, .the droplet and its surroundings. This practice is exercised by many investigators, e.g. Frossling [69], El Wakiel et al [62,70), Burstein et al [4), Ross and Hoffman [12), Groenewegh[32], Domingos [71], etc. The validity of this assumption is verified experiment-ally prior to this study by El Wakeil et al [62) and Priem et al [72] who studied the evaporation of fuel droplets in air. 2.7 Droplet Temperature Response with Internal Resistance and No Mixingv Strictly speaking the solution presented in paragraph 2.6 can only be used in the first time interval, since the initial
30 condition Ti const for the subsequent time increments is not satisfied. To avoid this problem at the end of every time increment the average droplet temperature is determined and used in the next step as the uniform initial droplet temperature. Clearly, this
- approximation introduces higher temperature driving forces than exist in a solid sphere, or in other words, the solution represents
'. the response of a droplet which is periodically well mixed. This is the reason why the solution is called the internal mixing solu-tion, there being as many mixings as there are time increments in the solution. A limit to the above model is the model which is now introduced. Basically this solution does not allow for any inter-nal mixing, i.e. the droplet behaves like a solid sphere. At the end of every time increment the existing temperature profile within the sphere is determined, and used in the subsequent step as the initial temperature profile. The general solution of a temperature field in a sphere suddenly exposed to a convective heating or cooling, and having a temperature profile Ti - T(r) initially, is described by the follow-ing analytical solution [733: nco 2 ~ sin(+ r/R) e(r,t) - T(r,t) = I Tifi nsir An cos p r.R 2 R exp(- n nFo r r' f(r) Jo 1 snk/) r/R) dr sin(W (39)
31 where: T(r,t) - temperature at any point of a sphere. Assuming that the initial droplet temperature profile can be approximated by the polynomial i=n
-5 T (r) = I A.r (40) -1111,
..0.
i=O 1 .. r ; where n is the polynomial order, equation (39) can be solved. The function f 1 (r) in equation (39) can then be written f (r) = T - T.(r) (41) Introducing (40) and (41) into (39) and performing the necessary algebraic operations (see Appendix C) yields the following tempera-ture profile of a sphere when a seventh order polynomial for the initial temperature profile is used: n=r 2 sin(P r/R) F In sin lb cos - r n Fo n~l n n n n {M A0 )(sin On cos On) - Al R [2* sin ipn - 2 -2) cos - n
-A 3 n 2siAn n Tn 6co An ( n 3< 4n~ 6 A 2[3(2 -2si _* 2 R3 n 2 -6)
A2 2 3t~~)sn i 6) cos -A 3-
-~
4 4~~~~~~~~~~ sin '~ _ 2 %' + 24) cosn + 243- A 4 4 -l - H(5P 60~, + 120)
32 sin iv1 4n- 20(n + 120) cos pr A [r(64n 1204
+720) sin n - (% - 30
- + 360' -.720) cos An - 720) -
- A6 R6 [(7n6 - 210n + 2520'n2 _ 5040) sin n - In(*n - 42in +
- 6 Vn.
+ 840'p2 - 5040)
~n Cos~ I-n A7 4 7 7~'P'
['Pn(84n - 336ip 4 + 6720 2 +
- 40320) sin % - ( n -56'*n+ 1680in - 20160*n + 40320) cos In + + 40320)} (42)
During the numerical calculations it was found that it was necessary to use the seventh order polynomial in equation (40) to approximate the very steep profiles at r = R in the beginning of the process. Denoting the term in the curly bracket of equation (42) by I the surface temperature of the droplet can be written as: n2 ~ sin i~ (7) PNo 2x( O(R~t)
= op I-sini(- N n1I n s n sin Cn I F)
(43) The average temperature of the droplet can be determined from 3 R 2
'F(t) = 4rp? O 47T r 6 (r, t) dr (44)
33 Introducing (42) into (44) and performing the necessary algebraic operations the following is obtained: n-- 6 sin n - cos A (7 ) 2
=1 3 P - siniA cosep nnFo (45) n~lt ~n n n n If the initial temperature To of a sphere is uniform, then the solu-tions (42), (43) and (45) evolve into:
n=w sin 4 - p cos
- sin(P r/R) 6(rt) (TM - T°) i 2R -ns n n n exp(- t, 2 N )
n=1 *n " - sint % Cos n r n Fo (46) n=co 2 (sin i - i cos i ) sin t O(Rtt) = (TM - To) y 2Xn - 's n n1l *n *n 5 fln1Pn cos '~ ncos n e( 2N
*nFo
)
(47) n~o 6 (sin * -t cos t!n) 2 E)(t) = (TM-
) (M T0°)
) 3 Vn - sin in cos 9'n ep(8 n Fo Equations (46) to (48) are thus identical with equations (33), (35) and (37) used in the short time interval mixing model. This compari-son confirms the validity of the use of the polynomial function in the no internal mixing model.
2.7.1 Sumnary of Solutions The solutions using different models of the codbined heat and mass transfer problem have been presented above. The first cne is the case when the droplet is behaving as if it is perfectly
34 mixed, i.e. the internal temperature prcfilc is always uniform and there is no internal resistance to heat transfer. The second model represents the situation when the droplet is periodically perfectly mixed and therefore exhibits a partial internal resistance to heat transfer. The third model is based on the assumption that the drop-let is behaving as a solid sphere having the maximum possible inter-nal resistance to heat transfer depending on the thermal conductivity and the size of the droplet itself. Clearly, the real behaviour of a droplet must lie in the domain of these three models and it re- 4 mains to be determined experimentally which model best describes 4 the behaviour of the droplet. 2.8 Thermodynamic and Transport Properties of Air, Water, Steam and Air/Steam !fixtures In all the calculations reported in this thesis the thermo-dynamic and transport properties were determined from references [74,75,76,77 and 783. The thermodynamic properties of air were determined from the ideal gas law and the transport properties from [79,80). The transport properties of the air/steam mixture were s.. Z calculated using correlations given in [79] and [81]. Appendix D contains specific details for every property g calculated. 2g
35 2.9 Outline of the Numerical Algorithm used in the lo Internal Resistance Calculations Before the algorithm outline is started some pertinent assumptions (besides those already introduced) will be listed. i) Beat:transfer by radiation is assumed negligible in the range of conditions relevant to this work. This assumption has been adopted by previous workers [9,71,84]. (In higher temperature situations its contribution could be easily incorporated by use of the Spalding dimensionless group, see [12].) ii) The thermodynamic and physical properties of the air/steam mixture are considered uniform and evaluated at the arithmetic average temperature between droplet surface and surrounding medium temperature [81,85 and 86]. iii) The effects of natural convection are negligible compared to those of forced convection [12]. iv) The non-uniformity of heat and mass transfer around the sphere is taken into account by the use of the averaged heat and mass transfer coefficient correlations. v) The air/steam mixture is always saturated. Equation (21) was solved by the method of short interval integration. Firstly the difference between the temperature of the surroundings and the initial .droplet temperature (the total tempera-ture range through which the droplet passes) was divided into a large number of increments (typically 40) and the physical, thermo-dynamic and transport properties of the vapour phase at the mean film ii
36 temperatures (TX + T )/2 in each increment were calculated. In S addition the velocities, Reynolds numbers, convective heat and mass transfer coefficients were also calculated. Then the time taken for the droplet temperature to be raised through each increment was calculated from equation (21). Some results of these calculations are shown in Figures 9, 10, 11 and 12, where the mean temperature. responses of droplets injected at their terminal velocities and subjected to different saturated air/steam concentrations are shown as solid lines. The same calculations were performed using a fourth order Runge-Kutta technique [36) but there was no significant dif-ference in the results. The computer program developed for this purpose (short 4 interval integration) is given in Append-ix E. The Runge-Kutta pro-cedure computer program is not included since it does not contain anything essentially new over the previous program. 2.10 Outline of the Numerical Algorithm used in the Partial Internal Resistance and fixing Model A droplet of radius R and initial temperature T. was selected and assumed to be exposed to an air/steam mixture of known steam volume fraction x and temperature TM. From the gas phase data the bulk transport properties were calculated. A small in-crease in surface temperature AT5 was assumed. This enabled the calculation of the mean vapour film temperature and corresponding thermodynamic and transport properties followed by the heat and mass
PT = I BAR NO RESISTANCE SOLUTION Ti = 20 0C SOLUTION WITH PARTIAL INTERNAL RESISTANCE Ts 350 C Figure 9. Thcrmal lltillzation vs. TJ.me for x5 'X5 Xs= 0-5 VR = VRT . _--
- 11 ,-
/
/
/
/
/
/
/
f
--I
PT = I BAR SOLUTION WITHOUT INTERNAL RESISTANCE T1 = 201C ---- - SOLUTION WITH PARTIAL INTERNAL RESISTANCE .
.. 0--
,0 Ts = 64-990 C Figure 10. 'llermal Utilization vs. Time for X9 - 0.25 Xs= 0 25 R - VRT 1p 1-- -
.- -I /
N1 I if
/
/
/
/
-i /
/
/
/
l Ld I Li
.- -I I r-7 2 A I. 1 !n'
- .1 I - . -. n I r- I
-- It. . . `- ..... - -- --. -
- .7. . .... ---. . ... . - .-- ...- .. ... --- .., - - ---- - - ---- -- - --- ---I -
PT = I BAR NO RESISTANCE SOLUTION Ti 20 IC SOLUTION WITH PARTIAL INTERNAL RESISTANCE t a, 0 g~
-r - 53 99 0 C Figure 11. Thermal Utilization vs. Time for x S
°.15 0
X3 = 0-15
._0 VR ~ VRT
,* -0
¢7-I // /
/
/
- F /
/
I/
'1it/
C~E Ld
/
1-- /
.. . .. 10
PT = 1 BAR NO RESISTANCE SOLUTION
-20C -----. SOLUTION WITH PARTIAL INTERNAL RESISTANCE Ts =31*93 0C Figure 12. Thermal Utilization vs. Time for x - 0.05 XS -=°-5 . . _-
-7w V.RVRT
=v.
. 0 zi
'I /
1/
-j 1
'I
/
/
Ur
/
1't
/
/
I 0-V
-, . ~~~ ~~~~I
. V *.-
_ _ _ _ I -- ~ ~ -- ,
41 transfer coefficient using forms of equations (18) and (19). Thus an apparent heat transfer coefficient could be determined from equation (26). This heat transfer coefficient was then incorporated into the transcendental equation (34) to furnish a set of roots i1 to
*n The procedure for solving this transcendental equation was the "Half Interval Search" iterative technique [36). From equation (35) the time taken for the specified surface temperature increase was determined, using the same iterative technique. The average tempera-ture of the droplet was determined from equation (37). Finally the droplet diameter increase Ad was determined from equation (38) and the new radius was used for the next iterative step when the whole calculation was repeated using a new assumed surface temperature increase and a new uniform initial temperature profile equal to the mean temperature of the droplet calculated above. Figures 9, 10, 11 and 12 contain some typical predicted droplet temperature responses (dashed lines) taking into account the partial internal resistance.,
To see the influence of the number of mixings on the , 4 iteration steps Figure 13 is drawn. The solid line represents the case of - 35 mixings while the dashed line represents the situation with - 100 mixings. As it is expected the higher the number of mix-ing is this model goes closer to the previous one, and in the limit with an infinite number of mixings would probably coincide with it. All the calculations with this model were otherwise performed with a constant number of mixings (usually 35). Pfi U- 4 .
Figure 13. Influence of a Number of Mixings on Droplet Temperature Response DO SOLUTION WITH 35 INTERNAL MIXINGS
----- SOLUTION WITH 100 INTERNAL MIXINGS A-.-NO INTERNAL RESISTANCE SOWTION 80 ~d=I mm 80 ~ X'= 0-5 VR= VRT PT= I bar 60 40 20 1(?-3 lo -2 101 10°
_________ r si i~ ~~~~~~~
43 Appendix F contains the relevant computer program with all the corresponding subroutines included. 2.11 Outline of the Numerical Algorithm with Maximum Internal Resistance and No Mixing In the first time step, this algorithm is basically the same as the previous one, however in the second and subsequent time steps, the initial temperature profile is obtained by curve fitting the temperature profile obtained at the end of the previous time interval. For this purpose, the droplet radius is divided into 20 intervals beginning at r = 0 and ending at r = R, and the droplet temperature determined at each of these nodal points. A linear regression subroutine is introduced to determine the unknown poly-nomial coefficients A0 to A 7 (see equation (42)). This specifies the initial temperature profile for the second time step. In this time step (and in every subsequent one) the temperature at any point inside the droplet, at the surface and also average temperature are determined from equations (42), (43) and (45), respectively. The droplet diameter increase is determined in the same fashion as in the previous algorithm. Since computer time required for this algorithmwas exr-sive, onlya.fewicurves representing the thermal utilization as a function of time are shown in Figure.14 .:.-The maximum internal resistance solutions are represented by the dotted curves farthest to the right for each of the diameters considered. Thus the model
-X
PT =I bar ~- NO INTERNAL RESISTANCE SOLUTION Tv =20 0 C -"--SOLUTION WITH PARTIAL INTERNAL RESISTANCE AND INTERNAL MIXING s=0-5, -- SOLUTION WITH MAXIMUM INTERNAL RESISTANCE AND NO INTERNAL MIXING T 5 v 81V35 0 C Figure 14. Thermal Utilization Predictions Using Different VR : VRT= CON IQT %,I . Hodels
.00-- -
-I
/
/
/
/
/ /
/
/ /
i*)
/
6/
/
1/I
,/
/
7 1./ ID-Z' t0s] 10' lo,
- -- ~~~~~~
---.-.--- y---*
__ .- ._ - -~~~~~~~. .- ~- -.- . -. - _ _
45 .. predicts considerably slower response time, especially for the larger droplets. Since the droplet will probably experience some internal circulation the experimental results should fall into the region between the no internal resistance and no internal mixing solutions. The discussed solution procedure seems to be much simpler and more general than the one proposed by Banerjee and Crosbie [87 and 88J, since there is no need to introduce their assumptions: i) physical properties are independent of temperature ii) the heat transfer coefficient independent of the diameter of the sphere. All the necessary details and subroutines used in the maxi-mum internal resistance calculations can be found in Appendix G. 2.12 Discussion of Droplet Temperature Response Predictions By comparison of Figures 9 to 12 it can be seen that the predicted response of the droplet temperature is much faster for droplets in saturated mixtures of 0.5 volume fraction than for a mixture of 0.05 volume fraction. This is to be expected because the temperature and mass driving forces are greater at the higher steam concentrations. The time taken for the larger droplets to almost achieve the temperatures of the surroundings is of the order of seconds and since the terminal velocity of such droplets is of the order of 9 m/s the droplets may not be fully utilized for condensing steam, even in a very high dousing chamber. ; A
.$.A,' *
.t*A--t, I\
46 The effect of the internal resistance of the droplets is illustrated by comparing the solid, dashed and dotted response curves in Figures 9 to 12 and 14. The internal resistance is most saignificanr-;f_-the.-.ar gex4_dops. In addition, as the gas phase heat and mass transfer resistance increases, concentration increases, the relative that is, as the air I 111 importance of the droplet internal resistance is reduced (i.e. dashed and solid lines are closer to each other). Figure 15 illustrates the effect of the initial droplet velocity on droDlet temDerature response. The curve representing the fastest response is for a 1 mm droplet 1' injected at three times its relative terminal velocity. Wrnen the droplet is injected at its terminal velocity its response is considerably slower and the trend r ii continues for droplets entering at velocities less than the terminal velocity. These results might suggest P that in steam dousing the " 4-1Z_. A t, . _' . Lk, droplets should be injected with a high downwards velocity. However, since this would also shorten the effective lifetime of the droplets in a particular chauber, the optimum injection velocity may be quite low. This will be discussed further in part II of this study. Just for the sake of the completeness the effect of down-wards and upwards movement of the droplet is compared in Figure 16, for the case wihen the internal resistance of the droplet is taken into account. The peculiar shape of the thermal utilization curve in the case of upwards movement of the I, droplet can be explained as j 0O UP
Figure 15. Initial Velocity Influence on Response Times I I1-0
- -NO RESISTANCE SOLUTION
---- SOLUTION WITH RESISTANCE (PARTIAL) 0 d = 1mm VRD RT XA=0*5 I- VRD =
PT= I BAR Ti=20 OC VR O= Ts = 81-34 0C ) -1 I/' ILl H-I I 1-f-
) I r-
-4 I-
Figure 16. Comparison of Thermal Utilization for a Droplet Moving Upwards and Downwards 100o Xs 0-05 PM= I bar TI= 201C TM =32930 C d =Imm "Di= 11 rn/S Vmf = I M/s L* //- . t.
-- I ~~~~:'*;
~~ = I s0 14 (1) 6' A-.
t I f
' (1I
.i-I/
40 I 20 10-2 10- 10° I0- 10-I
/
49 follows: in the beginning, both the curves are close together. The downwards moving droplet has a decreasing velocity until it reaches its relative terminal velocity. However, the upwards moving droplet goes through the moment when vR = 0, and then accelerates towards the same relative terminal velocity as the other droplet.
',-V -z CC T*-
I2 ? I
) . .,N j _L-o' Li t.~b ~~
P. k ,
- -- 1
')
50
- 3. THE EXISTING EXPEPMIENNTAL EVIDENCE AND PREDICTIONS The purpose of this chapter is to compare the heat trans-fer coefficients predicted and confirmed experimentally in this study with those cited in the literature for various experimental situations. The vapour phase or "external" heat transfer coeffi-cient h is defined by equation (26) as:
app h Q /(T - T) (26) app T H S I The predicted values of heat transfer coefficients which were later indirectly confirmed by measuring thermal utilization for our system i (droplets and an air/steam mixture) are drawn in Figure 17. Thus I, the predicted average values of the apparent (with mass transfer) ; and convective (with no mass transfer) heat transfer coefficients for the droplets of various sizes (injetgd.at terminga veglocities) :i are presented over the range of concentrations 0.5 < xA < 1 which is il the practical range of interest in this research program. The most striking feature of the aDparent heat transfer coefficients is their rapid decline in'the more dilute end of the range. The enhancement !l lI of the heat transfer coefficient caused by the steam condensation is .! clearly illustrated by comDaring the two sets of curves. To see what influence the'different heat and mass transfer i} correlations might have on happ the predictions of Berman's [45) and Ranz and Marshall's [46) heat and mass transfer analogy correlations (curves 6 and 1 in Figure 8) for droplet evaporation are shown 1-1II iI
hchapp [w/m 20 C] 50.l O0 PURE CONVECTIVE HEAT COMBINED HEAT a MASS TRANSFER COEFFICIENT TRANSFER COEFFICIENT I'd P. II H iI 11 I So I en
> III C:
C. I-t P. I-A.
'I i-In
- a. H" :uC) 0 ('C 11 0)
Hi t-l a-. flrt 0(n rtt~ m~u x it r'i 9 zE-CD en I-A C-).
.'1
52 (Figure 17) by the dashed and by the dash-point lines for 0.1 and 6 [mm] droplets, respectively. They are very close to the predic-tions (solid lines in Figure 17) obtained when using Hughmalk's correlations represented by curves 7 and 8 in Figure 8. This was expected since the Barman's, Ranz and Marshall's and Hughmarl;'s correlations are very close together in Figure 8. 1 i Before comparing heat transfer coefficients arising in different experimental situations it is always necessary to care-fully define them. This has to be done in order to clarify which experimental values can be compared, and avoid possible confusion. The first basis of compariscn for the apparent heat trans-fer coefficient h is the interfacial heat transfer coefficient app hi predicted by the kinetic theory for pure steam condensation [89). __ 1 2 sa i R Tv T vv 2 (49)l hi = f A5 S v Clearly, a comparison with this heat transfer coefficient is possiblef only in the limit where an inert gas concentration tends to the value zero, and when the liquid internal resistance is negligible. Calcu-lated kinetic theory predictions of the heat transfer coefficients at the interface hi are represented by points 1 to 6 in Figure 18 , (on the pure steam line) for pressures 0.983, 0.167, 0.0689 and 0.0334 Bar, respectively [89]. These h. values were obtained using the value of the condensation coefficient f =1, see Mills gnd Seban [90). lll I~~~~~~~~~~~~~~~~~~~~~~~~I
a%s I 53 0 KINETIC THEORY PREDICTIONi USING EQUATION (1'J o,o DEFINING EQUATIO*N' (50) DEFINING EQUATION (2s) - r, - A ij '- '. I 2 rm - DEFINING EQUATION (50 a) - - 4k.-/ ( *1r -5, DEFIN1ING EQUATION (51 a) {;:, - I- I'(- 106 Figure 18. Comparison of Predicted and Experimental for Heat Transfer Coefficient f . t t Ln'.- I,.. .. ,-I -.
.I I a -, -.- -*vv t - . I
-t I- -.
Lf ~A
, _xl ; ;
.t.
b) A 105 Az t.( c: 1- ~- (J.-s . c'iJ 4:. E . Z*S 1N. I 14 7 i ) En-.4 ,-.
.4-; -r0 %l' .1J
*1I
54 The second basis of comparison could be the "internal" or "overall" heat transfer coefficient definition which hng been used in iie-ct-contact condensation of pure steam on water j t, ras.__c. ha a Q./(Tsat
~sat TL)
L (50). ; where: j ' Q - heat flux, l t 1 L -average temperature of the liquid. TL~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- An analogous definition of this heat transfer coefficient in our case for the saturated air/steam rioturf would be: all 'QT/(Ti Tav) (50a) Al Lekic's 189) pure steam experimental results for the "internal" heat transfer coefficients (defined by equation (50)) are represented by I points 5 to 9 for droplet diameters 0.1, 0.25, 0.5, 1 and 2 [mm], , respectively. Points 10 to 17 represent the same type of heat trans-fer coefficient for the pure steam condensation on water droplets (predicted by Brown [91)), for droplet diameters of 0.1, 0.3, 0.5, 1, 2, 3.175, 4.76 and 6.35 [mm], respectively. To distinguish bet-ween these "internal" heat transfer coefficients (equation (50)) and external ones (equation (49)) they are represented differently in Figure 18, and the latter are denoted by double circles, i.e. as C.
- '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I
, eNIKi
!lIIl
55 A third basis of comparison could be the heat transfer coefficieat us1La~lyfined for the case of "indirect" condpnqtion of pure steam on a liquid film flowing over some surface (tubes, walls, etc.). The following definition of this heat transfer coefficient is customary: hF =Q/(T T) (51) F ~sat W where: TW - wall temperature. Similarly, if the steam is condensing from a mixture on a liquid film ar. analogous definition applies:
.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.
hFM e QT/(Tsat TW) (51a) In order to distinguish among the three heat transfer coefficients defined by equations (26), (50a) and (51a) in the case of condensation from vapour-inert gas mixtures, the coefficients are represented in Figure 18 by solid, dashed and dotted lines, respectively. It should be noted that these coefficients were introduced and defined in order as their values decreased. The. highest values represent "external" heat transfer coefficients (equations (49) and (26)) followed by "internal" or "overall" ones used in cases of "direct" steam condensation (equations (50) and (50a)) and subsequently by thcse used for cases of "indirect" steam condensation (equations (51) and (51a)).
-1.
56 Now that the heat transfer coefficient definitions have been clarified, the corresponding experimental results (for mixtures) can be properly represented in Figure 18 and compared with our pre-dicted and experimental values for droplets. Curves 18, 19, 20 and 21 represent heat transfer coeffi-cients obtained for an air/steam mixture atatmosp-e~ricptes~sure condensing on a jet of water for a mixture (flowing parallel to the initial direction of the jet) with relative velocities 45, 36, 26 and 14 m/s, respectively (see Kutateladze, Figure 11.9, page 196, 192)). Curves 22, 23 and 24 represent the heat transfer coefficients h for the case of transverse flow past a horizontal pipe 153). Curve 25 illustrates the values obtained in reactor building coolers (fin-ned tubes with water flcwina insida) for rich mixtures of air and steam and for a frontal velocity 3.4 m/s 193]. The condensation of steam from an air/steam mixture on a vertical film is represented by curves 26 and 27 for wall temperatures of 55 and 10'C, respectively [94]. The condensation of steam from an air!steam mixture inside a pipe is represented by curves 28 and 29 obtained bv Ackermann [95). The only data available over practically the whole range of inert gas volume fractions are those of Renker [96], and these data are for the condensation of steam inside a pipe. Curves 30 and 31 in Figure 18 illustrate these data for the mixture velocities 25 and 5 m/s, respectively. It is now of interest to compare the results of this study with those of the other investigators cited above. The thicker
*1
57 parts (xA > 0.5) of the solid lines 32, 33 and 34 represent the h predictions of this studY using Hughmark's [48) correlations (for Nusselt and Sherwood nurbers) for droplet diameters 0.1. 2 and 6 rmn falling at their respective terminal velocities and for a total mixture pressure of ' Bar. These predictions have been confirmed experimentally in this study (see below). The thinner parts (xA < 0.5) of these lines illustrate the predictions of this work using the same approach (as for xA > 0.5). However, no experiments were performed in this region of steam concentrations. Our curve 32 for
.Lc-'-
a 0.1 mm droplet has the lowest internal resistance and t most t
.j.7a-I 2 rapidly to the kinetic theory prediction for the corresponding pressure of 1 Bar (point 1) as expected.
Pure convective heat transfer coefficients (no condensa-tion) as predicted by this study are represented by curves 35, 36 and 37 for droplet diameters 0.1, 2 and 6 mm, respectively, and fcr
- 4.
pure air (xA -+ 1) they are the same as the heat transfer coefficients calculated allowing for mass transfer (curves 32, 33 and 34). To be able to compare the heat transfer coefficients pre-dicted (and confirmed experimentally for xA > 0.5) in this study with the pure steam values obtained by Brown [91) and Ford and LekiE [97] our heat transfer coefficients have been recalculated using equation (50a). The results of these calculations are represented by lines 38, 39 and 40 in Figure 18. Point 5 represents the heat transfer coefficient for a 0.1 mm droplet obtained by Ford and Leki4c, while point 10 represents the value predicted by Brown. Our prediction
58 for a 0.1 mm droplet (based on the heat and mass transfer analogy extension in the region of very high steam concentrations) is represented by curve 38 in Figure 18 and tends to extrapolnte to somewhat higher values in the limit as xA apprnoaes zero. The same is true for the 2 mm droplet value represented by the point 9 (Ford and Lekic') and by the point 14 (Bromn) while our prediction is re-presented by curve 30. The same applies for a 6 [m=) droplet (com-pare points 15, 17 and curve 40). Brown's experiments were per-formed on sprays, while his theory was for single drops, thus there is some doubt about the comparability of his predictions and experi-ments. Ford and LekiE's experimental and theoretical results are i in good agreeuernt and they are for single droplets, and it is be-lieved that their results are more reliable. Comparison of our predictions (curves 38 and 39) with Ford and Lekic's results (points 5 and 9) shows reasonable agreement. Thus the final conclusion of this comparison is that the predictions of the heat and mass transfer analogy (for an air/steam mixture) applied to mixtures with inert gas concentrations of ° < xA < 0.5 tend to the corresponding values of the "external" heat transfer coefficients predicted bv the kinetic theory and the values of the "internal" heat transfer coefficient for the pure steam. wher. the heat transfer coefficients are carefully and properly defined. Finally it should be pointed out that the importance of the droplet internal resistance is illustrated once more in this l . figure. For a 0.1 mm droplet and xA > 0.5 the resistance to heat IL
59 transfer is practically all in the gas phase (compare curves 32 and 38), however for a 6 mm droplet the internal resistance is quite important (compare curves 34 and 40) almost as far as xA = 0.°5. To conclude this chapter it should be pointed out that_ this survey _,0 riental results is probably the most exhaustive
,,e3 collection of the data in the area of the condensation in the pres-ence of non-condensables. It is very useful to have these data together enabling the distinction between the different exoerimental situations1 and avoidingdp.posible confusion by improper use of the defining equations.
L Ij 60
- 4. EXPERI*ENTAL APPARATUS The purpose of the single drop experiments was to obtain measurements of thermal utilization, i.e. to measure as a function of time the percentage of the total possible heat pick-up of a drop-let. This is the same as the time dependence of the mean dimension-less droplet temperature (expressed as a percentage).
To make this measurement two possible techniques were at first considered. These were: i) the suspension of a droplet on a thermocouple or thermistor tip in an air/steam gas stream, ii) the free suspension of a droplet in a wind tunnel. The second approach (used by Garner and Kendrick 1101) successfully in mass transfer experiments) was abandoned because a) it was necessary to deal with the saturated air/steam mixture and problems of wet apparatus surfaces were unmanageable; for more details see Ford and Leki6 [90,97], b) it was practically impossible to measure the temperature res-ponse of such a droplet, since it did not stay in a fixed posi-tion in the tunnel, on the contrary it tended to oscillate and move in all the directions inside the wind tunnel cross sec-tional area, c) only the relative terminal velocity could be investigated whereas it was necessary to investigate relative velocities less than this value in this work. in Ct)
61 The first arrangement is much simpler and has been used by several previous investigators [69,62,12] to study the evaporation process. The general apparatus arrangement is shown in Figure 19. Figure 20 shows the air and steam lines together with air heater and calming chamber, while Figure 21 shows the General view of the apparatus, including the droplet producing unit, metering pump, cooling and feeding water lines, recorder and humidity meter. 4.1 Air Svstem Air was taken from the laboratory high pressure line. Its pressure was decreased to 20 psig by passing it through the regulat-ing valve RV and passing it through the rotameter P.1. The calibra-tion curve of this rotameter is in Appendix H. The calibration was performed with two American Aluminum case meters AL-600 and AL-350 Cfh. Both the calibration points are shown on the graph in Appendix H. From the rotameter R1 the air stream was split into three parts (1/4" PVC tubing used) to feed the air heater AlH. The electrical heater (GTE Sylvania Incorporated) was connected to a variac which varied the power to the heater over a range from zero to 500 watts. From the air heater the air was led to the calming chamber CCH. 4.2 Steam System In a similar fashion steam was taken from the low pressure steam line and led to the steam rotamater Ri.. This rotameter was previously calibrated by the supplier Schutte Koerting Co. (the
li WATE F Figure~19. Experimental Apparatus, kh-TE'LON TUBE WALL
-rT U' NEEDLE TUBE z WST
,I 1TERMISTOR .
It-i Ti THERMISTOR Uf) DEAD RI
~'DROPLET DETAIL "A' M Io tI 0~
GL %P LIP R2 RV B ky LEGEND i kw COOLING . V RI AIR ROT-AMAETER DPU I ,4 R2 STEAM ROTAMETER WATER OUT PCOOLIN_. RV REGULATING VALVE VWAER IN Ti TEMPERATURE -INDICATOR DETAIL "A" AH AIR HEATIER N SD C4 CCH CALMING CHAMBER Lb' DPU DROPLET PRODUCING UNIT SL - \VST WiATER SUPPLY TANK -I, TNiP METERING rPUNMP SL SAMPLING LINE CCH t3- GEAR AIND LEVER DS DA11PING SCREENS N NOZZI..E vvirrH A SCREEN 0 <iId U-, SD SUSPNDETD DROPLT I nI Pl PRESSURE INDICATOR TO DRAM
- .--. . __. __ -I , " I- _ _ __11 -.
- 771:. - ,
, .... ' 1. .. . .-- -- .. . - .
Figure 20. Side-view of Experimental Apparatus with Steam and Air Lines. Figure 21. Side-view of Experimental Apparatus with the Test Section _ . . __ _ . XQ_ _ _ _ . . A~.
64 supplier of the rotameter R1, also) for a steam pressure of 15 psig and temperature 212'F. The temperature indicator T1 in the rotameterts exit showed the temperature - 212'F during all the ex-perimental runs. The pressure indicator PI (range 0-30 psig) was used to determine the steam pressure before the rotameter. From the rotameter R2 the steam stream was also led to the calming cham-ber. The steam line tubing was heated by tape heaters and insul-ated carefully to prevent the steam condensation. 4.3 Test Section Air and steam were introduced through 3/4" pipes into the calming chamber CCH, which was made from a 20" long pipe with 10" OD. After being thoroughly mixed and calmed using a 80 mesh screen as shown in Figure 19, the stream was led through a 2" ID pipe to ! the nozzle especially designed toens5ure a flat velocity profile of mixture. 140 mesh screen .was provided at the end of the nozzle to decrease free stream turbulence [102). Two nozzles were used having 1" and 1-1/2" ID, respectively. 4.4 Droplet Producing Unit The droplet was produced in the droplet producing unit DPU. Essentially it consisted of a feed and a cooling water system. M1ore details can be found elsewhere [89). The feed water (deionized) was supplied through a very precise syringe metering pump M (model 341), a product of Sage Instruments a Division of Orion Re-search Inc. With a 50 cc syringe size the flow rate could be varied [I
65 I.: between 0.33 and'13 ml/min or nl/hr, depending on the gears selected. With a 10 pL syringe this could be reduced to 0.000092-0.0036 ml/min or m./hr, respectively. With the cooling water, the feed water could be cooled practically to the temperature of city supply water. A droplet was produced by starting and Stopping the syringe piston using the gear and lever arrangement. Droplets rang-ing in size from 1.3 to 4 =m could thus be produced. 4.5 Measurement Techniques The flow rates of air and steam were determined by the rotameters R and R2 respectively. The steam and water temperature were measured using iron-constantan thermocouples. The air/steam mixture velocity was determined by use of the continuity equation using measured flow rates. A pitot tube velocity measurement showed that the velocity profile just after the nozzle was flat. The humidity of the air/steam mixture was determined using a digital humidity and temperature sensing systegm c-tppl~j-eA 3yhumder " Scientific Corporation. This system uses the Brady Array Humidity Sensor [103] which is a semiconductor device which changes its re-sistivity as "water content" of the array increases. The digital readout is obtained just by exposing the sensor to the air/steam mix-ture. Although this instrument was extremely handy and simple to use it was also very tempramental even though AECL's experience with this instrument had previously been very satisfactory. The instrument
66 calibration was tested as follows: an air/steam mixture sample was sucked from the main air/steam line (about 5" before the exit nozzle) using the sample tap, provided. This mixture was passed through three containers filled with water cooled by city water inside glass cooling coils. Steam from the mixture in an intimate contact with water condensed, and the remaining air was fed to a precise gas meter to determine its flow rate. The bottles were weighed before and after the sample run. The difference in weight gave the amount of the steam condensed from the air/steam mixture and this allowed the calculation of the humidity of the cir/steam mixture. When the Brady Array System was working correctly the agreement between the two measurement techniques was very .good (within - + 5%). The calibration curve for the Brady Array instrument was supplied by the instrument manufacturer and it is given in Appendix The temperature measurement arrangements are shown schem-atically in Figure 22. The temperatures of the feed water and steam were measured in a normal manner therefore they are not dis-cussed In any detail. The calibration curve for feed water thermo-couple is given in Appendix J together with tabulated values for electromotive forces. For the temperature readings of the steam thermocouple tabulated values of electromotive forces were used. The -droplet temperature measurement deserves some comment since this measurement took some time to be developed. After several trials the following technique was adopted. Tiny thermistor wires
(IN Figtire 22. Temperature Measurements Schermal ivs - M. 11"ll-, I 'r 1 i ,'-" k: 1g.l
*71 .
- 1.
.i- -
i.,Y,! :. I. .."-:
- -1. .
II
68 (- 0.025 mm) were glued onto a teflon tube which was slid over the drop producing needle. Detail "A" in Figure 19 represents-the needle end (enlarged ten times) together with the 1.3 mm droplet (smallest diameter obtained experimentally) suspended at it. The thermistor bead is a spheroid with the longer axis length 0.3 mm. The ther-mistor is produced by Fenwal Electronics. Two needles were used. They were made of Popper and Sons Inc. aerospace stainless steel type 304 tubes as specified in the next table. Table I.1 Needle Gauge OD ID Wall Thiekness Ii mm inch l m inch l mM inch 1 30 0.3048 0.012 0.1524 0.006 0.0'762 0.003 2 26 0.4572 0.018 0.2413 0.0095 0.10795 0.00425 ji The corresponding thermistors were termed Nos. 1 and 2.' II For their calibration a liquid-in-glass thermometer serial No. SC2118, manufactured by Fisher, range -1 to 101 0C, graduation 0.1 0 C,was used. The thermistor calibration curves together with the corresponding calibrations of the recorder charts are given in Appendix K. 4.6 Experimental Procedure The experimental and measurement procedure was as follows. A previously decided air/steam mixture was fed to the nozzle and all
69 the parameters defining the mixture were measured and recorded. A -- droplet was formed on the needle tip and photographed. (The camera - used was Nikkormat FTN 35 mm with 55 mm Micro-Nikkor f 3.5 Lens and Braun 240 LS Electronic flash. Elements used f .32 exposure and - flash duration 1/1000 sec.) The films developed were subsequently - projected on a wall using a film strip projector. The diameter of r. the needle was known. This gave the reference dimension for the droplet determination. The droplets were prolate spheroids and the equivalent sphere diameter d was determined from (equal volumes of the spheroid and the sphere) d = (ab13 (52)1 where, a and b are the longer and shorter spheroid axes, respectively. v The estimated accuracy of the droplet diameter measuremenrt is + 0.1 mm. The high speed recorder was switched on and the droplet was exposed by moving it quickly into the air/steam mixture where it remained for about ten seconds. The recorded time intervals were 0.01. sec at a chart speed 5 in. per second. The chart paper used was Type 1895 Standard Kodak, 8 in. by 200 ft., specification 1. This type of paper showed the best recording for the specified elements on the recorder (0.01 sec time recording at 5 in. per second chart speed). Before proceeding to the evaluation and discussion of experimental results the possible influence of radiation, condensation V
70 and heat transfer on and through the needle with respect to the droplet temperature histories will be discussed. 4.7 Heat Conduction and Radiation Influence on the DroDlet Temperature Response Measurement As was mentioned earlier, radiation effects can be neg-lected safely in this range of temperatures (maximum surrounding temperature in the experiments was - 88'C and the lowest droplet temperature - 1200 and this difference decreases as the process goes on). Ranz [104) found that the radiation effects are negligible in the case of droplet evaporation at 260'C surrounding medium tempera-ture. Even in the case of liquid fuel burning, the radiant energy from the flame to the drop is less than 15% of the total heat trans-r fer [9). With respect to the possible effect of the needle on the temperature response, the first fact to be noted is that the whole wake region contribution to the total droplet heat and mass transfer is expected to be in the order of 10%, Hoffman and Ross [17]. Thus the needle presence at the droplet rear would be expected to have a relatively small influence on the total heat and mass transfer. However, due to the fact that the needles were made from stainless steel there existed the possibility that they might conduct a fair amount of heat away from the rapidly heating droplet. The hypodermic needle is represented as a semi-infinite rod with thermal insulation of its lateral surfaces (see Luikov [73),
71
- p. 203) at a uniform temperature equal to the initial surrounding medium temperature. For the most severe approximation of the real problem it can be assumed that:
i) the tip of the needle (droplet itself) experiences an infinite heat transfer coefficient with the surrounding medium bringing its temperature instantaneously to T. and ii) there is no heat transfer towards the lateral surface of the needle (covered by a teflon tube) from the surroundings. The differential equation describing the problem is
- at = ' ax2 ° <x < (53)-
with initial condition T(x,0) = = const (54)
,;: Is, and boundary conditions
.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
T(c,t) To (t) = O (55) The solution of this problem may be written as: T(x,t) - T01/2 TM - = erfc [x/2(att) J (56) M4 0 which can be written as: e = erfc tx/B)/'2(N F lO2) (57) where
72 B _c dc - ratio of the needle cross-sectional area to P 4 its perimeter N = - Fourier number. Fo B2 B To obtain some numerical values it will be assumed that the needle and its enclosed water represent the rod which has the physical properties of the needle which is made of stainless steel type 304. The outside diameter of the needle is 0.3 mm (0.012"). The physical j properties of this steel are x 16.6147 4.216 2 af - = 502.3*8027.08[ms cp 084.12*10 _0236O7 [m /SI Hence, for t 10 sec and B 0.3.10 /4 = 7.5-10 [m]
-6
=4.12-10 -10 7357 NF No ;5 2 7.5 .10 -10
; 7325.77, At the end of the needle x L = 5 cm i1§
= s io25 666.6 Introducing these values into (57) it yields e =erfc [666.6/2(7325.77)1/2) erfc(3.89) 1 - erf (3.89) 1 - 1= 0 and T(L,10) = To
73 This implies that the water on the other end of the needle does not "feel" what is happening to the droplet after the "droplet" is kept at its maximum possible temperature for 10 sec. By use of the theoretical models discussed earlier, the heat fluxes to the droplet surface have been predicted as a func-tion of time. Thus, the heat fluxes through the end of the rod may be compared xnith previously predicted fluxes to the droplet. Before this is done the areas of the rod end and the droplet will be com-pared. Areas of the rod end 2 ~2 -2 2 AR= d %/4 0.3 .3.14/4 = 7.0685-10 mM Surface area of the 1 mm droplet 2 2 2 ADl d n - 1 *3.14 = 3.14159 mM and the surface area of the 3 mm droplet AD3 = 32.3.14 = 28.27 mm2 The area of the rod end is - 2.2% of the 1 mm and - 0.25% of the 3 mm droplet total area. (The smallest droplet diameter achieved in the experiments was d = 1.3 mm.) To determine the heat flux through the rod end the follow-ing approach will be used: q - l~e a(0,T) (58)
74 Differentiating the solution for 6 wrt x (equation (56)) and intro-ducing it into (58) yields: q = - k(T - T ) exp(- x2 a (at)/2 (59) The heat flux entering the rod is found by taking x = 0 and is qO = - k(TM - TO)/(rat)12' (60) and the total heat flux entering in the time interval from t = 0 to t - t becomes 2 CI ' I qO dt - 2k(T 1 _ TQ).(t/wa) 1/ (61) Anticipating t = 0.2 sec and the 3 mm droplet being exposed to an air/steam mixture xS 0.5. After 0.2 sec the droplet temperature is 38C (used instead of V. Thus qO = 2.16.6(38-20)(0.2/3.14-4.12106)1f2
- 74285 [J/m 2 or the amount of heat entering the rod is
-2 -6 -3 QO = SO AR = 74285*7.0685-10 -10 = 5.25-10 3[J The total heat flux entering the 3 Tma droplet in the same time interval (taken from our predictions) is:
~ 4104 [J/M 2 J
75 and the amount of heat entering the 3 rmi droplet Q qDDAD 3 = 4.10 *28.27.10 = 1.13 [J] Thus the amount of heat conducted through the rod is less than 0.5 percent of the total heat entering the droplet. In a similar manner it can be shown that the amount of heat conducted by the rod is less than 3 percent of the total heat entering the 1 mm droplet at the same conditions (0.2 sec and xS = 0.5). Keeping in mind all the simplifications introduced in this treatment, i.e.: i) the sudden change of the rod and temperature; in the experi-ments the rod end was "protected" by the droplet itself and its temperature can only change as the droplet temperature changes, ii) the physical properties of steel; in the experimental situa-tion the needle is filled with water whose thermal conductivity is much lower than that of the steel, one can safely presame that the heat conduction along the needle can be neglected.
* '4;
.. w~~~~~~~~
76
- 5. EVALUATION AND DISCUSSION OF E)TERIMENTAL RESULTS Since the droplet temoerature-was recorded continuously,
+/-itwas -possible to produce smooth experimentalthermal utilization curves. Using the recorded experimental parameters (air/steam mixture ternerature. velocity, droplet diameter and initial droplet I
temperature) the thermal utilization of a droplet experiencing these parameters could also be predicted by use of the various computer algorithms discussed above. The experimental and predicted responses could then be plotted together.
- I..
Before we proceed to the evaluation and the discussion of ;v
- S I; 1 1:1 the experimental results the range of experimental variables will cr. 1.
Y be specified. These are: 40"L L .14 4I I - droplet diameter: d = 1.3 to 4 [mm) - air/steam mixture velocity: yR 108 to 691 cm/s if - steam concentration by volume: xS = 5 to 66% !i I! I! - water initial temperature: Twi 15.8 to 22.50 C. .i There were 90 runs altogether which covered the above range of variations. .I I For presentation purposes, only a few typical responses are selected I1 1;, and shown in Figures 23 to 27. However, all the experimental data I have been considered in the following discussion and are tabulated i in Table 1 of Appendix L. I It was pointed out in the paragraph 2.11 that the droplets would most likely experience some internal mixing and it was expected I-that the second model (internal resistance with periodic mixing) i I I
77 would probably provide a good approximation to the real situation. This intuitive prediction agrees very well with the experiments as one can see from Figures 23 to 28. Figure 23 represents the typical dimensionless-temperature response e for the run 3 (see Table I in Appendix L) and its com-parison with all the three theoretical models:
- no internal resistance (solid line), - internal resistance with periodic mixing (dashed line), - internal resistance and no mixing (dashed line).
From this figure (a typical one) it is clear that the experimental results are represented the best by the second model. In all the calculations around 35 mixings were used. To see the droplet diameter influence on the thermal utilization (keepIng steam concentration constant - 6%) Figure 24 was drawn. The dashed lines represent the thermal utilization pre-dictions using the second model (internal resistance with periodic *1** mixings) while the points represent the experimental results. The / I agreement between predictions and the experiments is very good for all the curves, which show a significant influence of the droplet diametar or. the droplet dimensionless temperature response. A similar graph of the approximately 15% steam concentra-tion datz is shown in Figure 25. This figure shows the same trend as the pravious onle vhich is that the bigger the droplet diameter, the slower its temperature change.
-,'I.-
S.
Figure 23. Comparison of Experimental Results with Different Models 0
-_NO INTERNAL RESISTANCE MODEL INTERNAL RESISTANCE MODEL WI' rH PARTIAL MIXING INTERNAL RESISTANCE MODEL WI TH NO INTERNAL MIXING d -29[mm] TM-=80° C y = 1-91 [m /s] Twi = 160C 03
/ )
r _ q
,j;J //~~~~~.
I. l . o [~~~~~~~~10 2-0 30
Figure 24. Comparison of Experimental Results with Predictions Using Internal Resistance Model with Partial Mixing for Different Droplet Sizes at xs Xu 6% 0 0
=4mm
--- INTERNAL RESISTANCE MODEL WITH PARTIAL MIXING I I I - I I- - - - __
I I I II
~~~~~~~~~~~~~~~I J,"-L, Ui
- ,oM. .-.t" ;
A a
- - -- ... . ~~
. I`
a -. - ii
'. , -7I
-,'.I .i ,I"I'..-i,,--
e:Fft .: .,--
, -". -, L -.,A 4'
-f -,... ,..jkj t r s 4.
v*~ 11.1 1,
.4 tj, .. : ;%;
Figure 25. Comparison of Experimental Results with Predictions Using Internal Resistance Model with Partial Mixing for Different Droplet Sizes at x 'u 15% O~ - g r___ _ ad~~~~~~~ - - Ar -. X ep i'~~d 3.7mm
---. INTERNAL RESISTANCE MODEL WITH PARTIAL MIXING X' -15 %
S 0 co 0 0o 2 4 6 4 6
81 To show how the thermal utilization of the droplet is affected bv the value of the steam concentration in an air/steam mixture. Figure 26 is plotted. In this figure the thermal utiliza-tion vs. time for the 2.5 mm droplet diameter for different steam concentrations (varying from 65% to 5%) is shown. Again the agree-ment between the experiment and the second model predictions is very good confirming the fact that the higher the steam concentra-tion is the "faster" the droplet temperature response. The influence of the mixture velocity on the droplet temnerature rezonse (droplet diameter - 2.8 [mn] and steam concen-tration - 35% by volume) is shown in Figure 27. Once more the agree-ment betw*'een the predictions and the experiments is very good, show-ing that the relative velocity is very important factor, i.e. the higher the relative velocity the faster the droplet temperature change. The agreement of the rest of the experimental runs (see Table I in Appendix L) with the predictions is good in the whole range of the variables. To obtain a measurement of the spread of the experimental data and Lhe accuracy of the predictions, two values of the thermal utilization, 50;% and 75% were chosen (on the theoretical curves) and the difference between the experimental values of thermal utili-zation 0= and the chosen predicted ones 0p are shown in Figure 28 plotted against the time taken to reach ep. From this figure it
'I I
Figure 26. Comparison of Experimental Results with Predictions' Using Internal Resistance Model with Partial Mixing - for Different Stenm Conqentrat:ions d X 2.5 mn 0~~~
.- - -Ix =5%
.~~~~~~~~~~~~~~~~~~~~~~~~~~~S
--- INTERNAL RESISTANCE MODEL WITH PARTIAL MIXING d 2 5 mm 00 U I I I I.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
, a -; V14: .. . . .
,%-W , Bu cket...? 2 111. 4 6 5 _DmM.5
-- fi{ ....-.. . - . .. .
-- L L .
7,
. . .1
-k,
. --- '.W.' I I R I.
ik f .-Jo
- A.
s ;.:i,. I
Figure 27. Comparison of Experimental Results with Predictions Using Internal Resistance Model with.Partial WU.xing for Different Velocities of the Air/Steam Mixture ' and d X 2.8 mm nnd xc by 35% a
-Q = 6*91 rn/S
[*68 m 4,
--- INTERNAL RESISTANCE MODEL WITH PARTIAL MIXING Xi-35 %
d 8 mm . . 2 t [sI _ i . .
)------ - + 20%
a S 3.0
- 0 9 .
0 . 9 - I 09 . S
*~ . 0
- en
**S
-. a . 0.- S- *- 0 -- -- - - 5 0
* *S 0 ~~~~~~~00
*0
* *
- S
-~~~~~~~~~~~
-) F - ~ - ~ -~ -~~20- %~-
Figure 28. Comparison of Predicted Values of Thermal Utilization (0 = 50% and O 0 75%) with Experimental REsalts 5 I* - - - -- - - - - - - - -- - - -- - - - - - - - - - -- - - - -- - _-_20 0
- )~ ~ ~ o _ , . * . 0 .5 50 Be.:. .- S*.. -
- _. A- . *-*- , .-. ....-..... - . - . -p=7-31 d
%)____k________ _._____ -_- _----- - -2
. ., '- - - I 2C : ;; ;) 4 5 6 7
_~~~~~ t -M Ar.i,. In
.~~---- J.tla[J Sl ,t.^.
--- - . . HEMMIANUMPIA1 85 can be seen that the maximum difference between the predictions and experiments is enclosed.by the + 20% thermal utilization lines for both the values of the thermal utilization (50% and 75%). A simple statistical analysis of these data gave a value for the stan-dard _eviati.onof.aout 5% for both chosen values of the thermal utilization.
To find out if the theoretical values of the thermal utilizations (50% and 75% chosen) differ significantly from the experimental average values e50 and e75 a simple student's t-test was performed [105]. From Table I (Appendix L) the average values of 6 and 675 are calculated as: '- 0 49.927 and 0 76.73.6 l
,50 75 The point estimates of the corresponding standard deviations are:
I:i S 6.0609 and s 3.968 50 75 When s is known t is then calculated from the formula 050 0 0 - 49.927 0.114 50 s An- 6,0609/iii0 and
= 075 875 76.716 75 4.0 90 s Fr. 3.96845//ii The value of t thus obtained is compared to the tabulated value for ta at the required confidence level (95% chosen) with n - 90 degrees
86 of freedom. Thus: ta (.95,90) = 1.99 The experimental value of the thermal utilization 650 is therefore not different from the predictions since t0 =0.114 < ta =1.99 however the 075 value is significantly different from the predicted one, since AJ. t =4.]02> t =1.99 75a This simple s tatistical a ies tat,there is 1/2nalqkpl some bias in 4 our predictions. It is felt that this bias lies in the number of mixings used in the calculations of the predicted values of the thermal utilization e The value 35 was used in all the calcula- X uions in spite of the fact that the bigger droplets would probably experience higher numbers of mixings. A trend of this kind could be observed when comparing the experimental results and the predic-tions. In Figure 24 the experimental results tend towards the no internal resistance model (infinite number of mixings) as the droplet diameter increases. This has been noticed in most of the other runs which can be seen from Figure 28, where the bigger droplets are represented by longer response times and more of the experimental values tend to be higher than the predicted ones, implying that they A
87 are "more" mixed. From Figure 13 one can see that the higher number of mixings (- 100 compared to - 40) gives bigger differences for higher values of thermal utilization. Thus, using the greater number of mixings for the bigger droplets the difference between experimental results and predictions would decrease and the "cause" of the bias in the predictions would disappear. However, it is not .'. - I...
\ possible to say what number of circulations or mixings should be used for any particular case. In any case our findings are in agreement with Harriot [106] who has found that the velocity of
! circulation increases with the diameter of the droplet.
- To conclude this section it can be said in general that the experimental results aigree close]y with the second model pre- -A..
'J -
dictions over the whole range of the experiments. The predicted t influence of all the important parameters (steam concentration, droplet diameter, air/steam mixture velocity) are confirmed experi-mentally.
88
- 6. CONCLUSIOMN AND RECO1ENDATIONS OF PART I From the material presented so far the following conclu-sions can be drawn:
i) The model with internal thermal resistance in a droplet and internal mixing (sometimes referred to as "partial internal mixing" or "second model") describes very well the effects of conbined momentum heat and mass transfer to a single drop-let since in a large majority of the experimental results correspond closely to the predictions of this model. ii) The predicted effects of all the parameters in the theoretical part of this study were confirmed experimentally: An increase of the droplet diameter (see Figure 24) increases the resDonse time. The highcr the concentration, the faster the droDlet temera-ture response (Figure 25). The higher the relative droplet velocity, the faster the droplet temDerature response (Figure 26). iii) Bigger droplets are experiencing more circulation (see Figures 24 and 28). iv) The resulrs..tend_.to..confir mthat it is correct to calculate the external (equation (49)) and internal (equation (50)) heat transfer coefficients by use of heat and mass transfer analog,2 and Hughnark's [48) or Ranz and Mtarshall's [46] correlations for Nusselt and Sherwood numbers (see Figure 18)
89
-J give practically the same excellent results over the whole range of the experimental Reynolds numbers (100 to 1200).
v) The resistance to heat transfer in this situation may occur-significantly in both phases. The internal resistance of the droplet is most'significant for fast moving large droplets in concentrated steam-air mixtures. The external heat transfer resistance is most significant for small, slow moving drop-lets in dilute mixtures. vi) The unsteady state temperature response calculations done by use of the quasi-steady state approach work out satisfactorily. vii) The analogy method of calculation may be extrapolated to the pure steam situation and predicts the "pure" steam heat trans-fer coefficients reasonably well (see Figure 18 and discussion of Chapter 3). Some possible suggestions and recommendations for future work are: i) To extend the range of experiments to the higher steam concen-trations, since very few runs were done in this range due to excessive condensation on the cold surfaces of the droplet producing unit. For this purpose the unit should be redesigned. ii) To modify the existing theoretical approach for the case of non-saturated air/steam mixtures when the droplet tends to reach the wet bulb mixture temperature rather than the dry bulb one.
90 iii) The apparatus could be used to study simultaneous heat and mass transfer inside the droplet by using different mixtures for the surrounding medium. iv) Instead of the Pass transfer towards the droplet the opposite process could be studied, i.e. droplet evaporation. The results of such work would be widely applicable to many opera-tions such as drying, air-conditioning, etc.
PART II SIMULATION OF TIlE SPRAY SYSTEM AND DESIGN SYNTHESIS OF A DOUSING CHAITBER 91
- 7. INTRODUCTION We are now in a position to attempt to solve a more comn-plex and realistic problem involving the combined momentum, heat and mass transfer in an experimental pilot plant, which was set up to provide a basis for the real dousing chamber design. The problem of the full size dousing chambers (see Figure 3) is even more com-plex due to the existence of the chamber internals and the fact that the air/steam mixture enters the building tangentially. There will be no attempts in this study regarding the possible simulation of the zeal dousing chamber.
The experimental Pilot plant was designed by AECL and the experimental results from this plant were provided by that companv (107). The droplet size distribution measurements reported below were initiated as a result of the findings of the first part of this work. The experiments were carried out at AECL and the results obtained so far were analysed at Waterloo. The drop size distribu-tion measurements are still in progress as an essential part of the general research project. For more details see K. McLean [1081 who is continuing this part of the work. The second part of this thesis is thus only concerned with the problem of the steam dousing simulation of the pilot plant. Chapter 8 will deal with the experimental part of the problem, while Chapter 9 will treat the theoretical approach to this problem. Chapter 10 will deal with the problem solution technique and the 92
93 corresponding numerical algorithm outline. Chapter 11 is devoted to the experimental results and their comparison with the theoretical predictions while Chapter 12 deals with the conclusions and possible suggestions and recommendations.
-'I
-Vi:
FZ
-4
- 8. ME2IMENITAL The following twc paragraphs will be devoted to short reviews of the pilot plant and droplet size distribution experiments.
More details can be found in (107] and [108). 8.1 Dousing Chamber Tests Figure 29 shows a schematic representation of the pilot plant experimental apparatus (as reported in [107)). Essentially, the plant consisted of a large vessel (56 [m3 ] and 5 [m] high) pump and motor unit and hot water header. The latter 1as used to provide live steam to the dousing chamber at a specified total pressure in the vessel. The pump and motor unit was used to provide the pres-cribed flow rata (and pressure drop) across the spray nozzles. The experiments were performed with one and subsequentlV idith five swirl nozzles tvDe TF48FC. produced by BETE FOG NOZZLES, INC., Greenfield, Massachusetts. The experimental procedure was as follows: i) The vessel was filled with air and its pressure recorded. ii) The steam was introduced into the vessel from the hot-water header until a desired total pressure was reached. iii) The pump was switched on and the water flow rate and tcmpera-ture were recorded. iv) The change of total pressure in the vessel (due to the steam condensation) was recorded as a function of time. 94
- w 95 PUMP AND MOTOR UNIT
.',f
'.Wp Figure 29. Pilot Plant Test Facility
96 Using this technique curves 1A to 3E (solid lines) shown in Figures 39, 40 and 4] were obtained. To provide an estimate of the heat losses from the vessel to the surrounding medium curve 1 (obtained with no water flow) in Figure 39 was recorded. In this case the condensation and the vessel pressure change was caused just by the cooling effect of the vessel, since the water pump and thus the nozzles were not used. The flow rates of water with a single nozzle were 45 IGPM and for the five nozzles case it was five times greater, i.e. 225 IGPH. 8.2 Droplet Size Distribution Measurements The experiments on the droplet size distribution of the particular nozzles used in the pilot plant tests are just a small part of an extensive study of droplet size distributions of the different spray producing equipment used by AECL such as spray plates and different spray nozzles. The experimental apparatus (see Figure 30) consists of a central adjustable height tower 45 feet high. In this manner it is possible to measure the droplet size distribution at different heights. On the top of the central water supply pipe is a flange to connect the cross arrangement with the outlets for the nozzles. There is a pump to provide the necessary water flow rates and the pressure drop across the nozzles. Preliminary review of drop size measurement techniques showed that the "catch in oi'" technique, used by Browm [991),
Figure 30. Experimental Apparatus for Droplet Size Distribution Measurement NOZZLE NOZZLES 1 CENTRAL PIPE -CENTRAL PIPE CATCH CATCH CATCH CELL CELL WATER CELL %0 IN STAND -4
- STAND GROUND LEVEL I \ LINEAR MOTOR
..or..V -. X,
-1
'I :111'1
;. - 4 i;
.1 i.
or. _ J
. ,/
i - 2
- e. ,
P I,
-: 'it ,- *.
98 Syhre (109), Kushnyrev [110) and Gelperin [111), had been found to be satisfactory and simple to deal with. After several oils for the cells were tried it was found that castor oil had the best char-acteristics for the intended purpose of catching the drops. High speed movies showed there was very little splashing when the drop-lets fell into the oil and the droplets were retained just at the interface and after a short time period would start to move slowly to the bottom of the shuttered catching box. This time period was long enough for shadow photographs of the drops to be taken. A special stand was designed for the purpose of mounting the catch cells [108]. On top of the stand could be mounted up to eight catch cells made of acrylic plates. Below each box was a special water and light proof compartment where the photographic plates (8" x 10") were placed. The tops of the cells could be exposed with sliding plates connected to a linear motor. These sliding plates were provided with slits which when slid over the box, ex-posed the oil to the spray for a while. Hence the following simple experimental procedure was as follows: i) The stand was placed in position beneath the spray and the boxes were loaded with the photographic plates and the castor oil layer. ii) The spray was formed by switching on the pump. iii) The linear motor was started moving the covering plates and opening the boxes for a while and allowing the droplets to
99 hit the oil and be suspended just beneath its surface. iv) The specially designed light source was placed over the boxes. This was essentially a long metal duct (8 feet) fit-ting over the box. The duct had an electronic flash built into the top. v) The flash was switched on exposing the plates to give a real size shadow (black and white) image of the droplets on the plates. The catch in cell technique worked quite satisfactorily in the range of flow rates (30 to 50 IGPM) with the BETE mozzle I placed 15 ft above the catch cells as the water distributor. How-ever, when the dousing plates were used very large droplets were. lot formed and the splashbig and the back splashing was so intensive that the reliability of the technique was questionable. To avoid this problem a direct photography technique is being developed (108] to produce an image of the moving droplets in air. In preliminary experiments, this technique has been tried out simultaneously with A "the catch in cell" technique. The development of this technique is still in progress and certain refinements such as a decrease of the depth of .the photographed droplet layer, light source distance, A, better image of smaller droplets, etc., will be necessary before it can be judged reliable. In spite of the necessity for the further improvements of this technique it is felt that the comparison of the results obtained by botn methods is useful, even at this stage of the development. This will be pursued in paragraph 11.2. I.
- 9. THEORETICAL ANALYSIS 9.1 Problem Definition and Sir'lifVinR Assuntions It was decided that the experimental pilot plant vessel could be represented by a chamber containing an air/steam mixture as shown in Figure 31. Thus the state of the air/stream mixture in the vessel is specified by its pressure PM, temperature TM and the cham-ber's volume V11. The parameters of the entering water are specified by its flow rate mWE and its temperature Tq Similarly the para-meters of the leaving water are t; L anrd TW L.
The problem can be stated briefly as follows: it is necessary to predict the pressure in the chamber as a function of time wher. the initial conditions of the air/steam mixture are known as well as the conditions of the entering and leaving dousing sater. To enable this, certain simplifying assurptions are necessary. These assumptions are:
- 1. The air/steam mixture in the vessel is perfectly mixed and al-ways saturated with steam.
- 2. The air/steam mixture velocity in the vessel is zero during the dousing process.
- 3. The thermodmnamic properties of the mixture components are determined from the ideal gas lwa; since the total pressure in the experiments was P 2 E ar. l St.-.... ', - . !-.
- 4. The spray immediately upon injectior. into the chamber breaks up into droplets of known size distribution whose defining para- ;*i I I')
Ik meters are specified in advance. 100
101 WATER IN WATER OUT Figure 31. Pilot Plant Dousing Chamber MNdel
102
- 5. The increase in a droplet diameter due to condensation was neglected, since its increase is less than 5% as found in Part I of this study.
- 6. Each droplet maintains its identity in the vessels, implying there is no:
i) shattering, whatever the cause may be (e.g., a) the distri-b~ution of aerodynamic pressures, see Hinze [112); b) waves and oscillations, see Levich and Krylov (in [113], p. 29°); l c) turbulence, see Hinze [112] and Brodkey [114]). ii) colliding which can cause: a) ccqiesence; b) separation and c) disintegration.
- 7. The void fraction of the system is very close to unity. and the amount of the droplet interaction can be neglected, thus each droplet is treated as a single one in an infinite medium for the purpose of its transport coefficient evaluations. fThe problem of interacting droplets is discussed by Yaron and Gal-or [115).
- 8. The dousing water supply source has constant temperature.
- 9. The flow rate of dousing water into the chamber remains constant.
- 10. Tpe volume of weater on the chamber floor is negligible as comn-pared to the volume of the vessel.
- 11. The pressure drop due to condensation of steam on the walls and o.bstructions of the chamber can be modelled from an experimental -z test (curve 1 in Figure 39 where tank pressure versus time was obtained without the use of a dousing spray). This effect can L thus be incorporated into the model in the form of heat losses to the surroundings.
103 9.2 Basis of the Simulation Consider the dousing system shown in Figure 30, consisting of a vessel filled with an air/steam mixture of known pressure, PMl, temperature, TM, and the steam volume fraction XSM corresponding to a saturation temperature T ,. The total pressure of the system will change due to the condensation of steam on the injected water drop-lets. The rate of pressure change in the vessel will depend on many factors governing the rates of heat and mass transfer from the gas phase to the liquid phase. For example, the local size, temperature and velocity distributions of droplets, the temperature nnd air concentration in the gas phase are all factors influencing the vessel pressure change. In order to utilize the information developed in Part I for a single droplet, to solve this complex dousing problem the following approach is adopted. A known initial droplet size distri-bution is discretized into M droplet diameters and the height of the chamber is divided into N increments. Applying the energy conserva-tion law (for a time interval At = t2 - t1) we can say that the water energy change (increase) AEW is equal to the air/steam mixture energy change (decrease) AEE. Allowing for energy losses to the surrounding QLS' we can write It2 AEM A w QLst (62)
ENIK 104 If the internal energy of the mixture in the chamber at the time t tI is Umi = C, 1 Ti kJ/kg (63) and at the time t = f2 kJ/kg (64) rL4 I-- Cv2 T21 t%"' then the energy change of the air/steam mixture can be written as AEM ' mtlauM1 - 'uV (65) The energy of all the water droplets in the chamber at the tire t = ti is i=M j=N (66) Hi = il j-l Ilih Ij
'.44,.'
and at the time t = t2 mi,;Jh I 1t i=M j=N (67) i=l j-l
-'p where, the corresponding enthalpies are hWii (68)
= Cp T i, j a a
ili Now, the energy of the droplets entering the system during the time
105 increment is M ~~~~t2 FE - ij, mW~ Jh 1 ~ (69) ial i,l *i 169 t and similarly the energy of water leaving the dousing system N I~~~t 2 EWL~~ ~ hW~ (70) i=l mWEiN iN i Thus the energy change of water phase due to heat pick-up from the air/steam mixture is E ?2 E Wi + EWL -EWE (71) In the last equation the effects of kinetic energy, potential energy and work are neglected. Hence, combining equations (62), (65) and (71) yields . W2 EWE 2I 2 . ml nSu.. e E142 - EW, + E1r - Els + QLS (72) or written in the same form as presented in 1116)
- 2 SuEW 1 ' = 'M~ul 2 + EW2 + EWML+ I1
'CS (72a)
CD (D (D (d (0 Q' Equations (72) or (72a) which basically represent a statement of the first law\ of thermodynamics for the dousing system can be used to determine state 2 of the air/steam mixture in the chamber a short
ENUUAF 106 period of time (0.5 sec say) after starting from the specified state 1. However, before we proceed to the description of the numerical algorithm itself it is necessary to detail how each term in these equations, (72) or (72a), can be determined. Term © represents the internal energy of the air/steam mixture and it is easily determined using equation (63) since the initial state of the air/steam mixture is specified. Terms Q and () represent the energy of all the droplets in the chamber at the beginning (specified) and: at the end of the time increment, respectively. When the droplet size distribution is specified (paragraph 9.3) and discretized the equation of motion solution (paragraph 9.4) enables us to determine the number of drop-lets in each height increment at the end of each increment. Finally, the temperatures of droplets (paragraph 9.5) combined with their numbers in each height increment allow us the determination of the energy of all the droplets in the chamber at the end of each time increment. Term Q is easily determined from the specified flow rate and the temperature of the entering water. The energy of water leaving the chamber represented by term © is determined by summing the energies of the droplets which left the last height increment (N = 100) during the time interval. The dousing chamber heat losses (term ( ) are determined from the experimental data on heat losses (paragraph 9.6).
107 Thus in effect the only unknown term in equation ( 7 2a) is Q representing the total energy of the air/steam mixture in the chamber at the end of the time increment. This term is deter-mined from equation (72a) by a trial and error procedure. For more details see Chapter 10 which describes the calculation algorithm. 9.3 Droplet Size Distribution To determine the total mass of anv droplet size at anv height increment in the system it is necessary to prescribe the initial droplet size distribution..Because none ol the emuations for droplet size distribution are derived on the basis of ayhjsical model of atomization the choice of which relation to use rests on its ability to represent the experimental data. A recent study [117], in that regard, showed that the uDDer limit function, proposed by Mugele and Evans 1118], and chi-souare distribution give good approxi-mations for drop size distributions. It can be shown that the other models[117] such as Nukiyama and Tanasawa, Rosin-Ramler, the log-probability function, which are often used to represent the experi-mental data on droplet size distributions, are in fact special cases of X-square and upper-limit distributions. Since the last two dis-tribution models are the most general, the other mcdels have not beer, considered. *The upper-limit function has been finally selected on the following grounds: i) a preliminary drop size analysis showed that for the nozzles used in the experimental tests [107] this function correlated local drop size distribution reasonably well;
108 ii) it is relatively simple to determine the defining parameters, see Mugele and Evans [118]; iii) it is the only drop size distribution allowing for a maximum droplet size. The upper-limit normalized volumetric drop size distribu-tion function is expressed as
. i dx 1/2 (x - x)x exp 6 I.nb ] (73) where: rb, 6 and xm, being the maximum droplet diameter, are distribution defining parameters, dv - is the normalized volume of droplets with diameters between x - d ared x + dx A normalized upper limit number distribution function could also be used to represent the experimental spray data, however the nurber distribution function gave worse approximations to the same data as shown in [117), and therefore only the normalized upper limit volume distribution function was used in the evaluation of our experimental data.
9.4 Equation of llotion Solution
'I Now when the droplet size distribution function is speci- W fied the problem of the droplet size distributions throughout the ....
I
.1 . IV -
I chamber can be solved by using the momentum equation for droplets. N.,u') - 1;
. .I
.11 The specified droplet size distribution is discretized into K droplet
.:'LI.II I.A' W
. t
.-ill
.I.I
..--Ifkt., I W.),II V"
109 diameters, the mass and the number of droplets of i-th diameter (i 1, ... , M) can be determined. By solving the equation of motion for every droplet diameter it is possible to determine the number of all droplets in the j-th (j = 1, ... , N) height increment in the vessel. In this manner the water droplet distribution throughout the chamber is specified in each time increment. The equation of motion for droplets is solved numerically using the fourth order Runge-Kutta technique [36] and Reinhart's [33] corre-lations for droplet drag coefficients. i The results of these computations are showm in Figures 32 to 34. Figure 32 represents the droplet velocity distribution j at different positions in a chamber, for a spray injected with the I! initial velocity 10 m/s in an air/steam system at P= 2.23 Bar, P 0.95 Bar and T = 98.250 C (initial state in experiments lA). Figure 33 shows the fall times of different diameter droplets neces- IjII sary to reach a specific position in a chamber for the same spray I'i as in Figure 32. Figure 34 represents the position of different diameter droplets (the same spray as in previous two figures) in a chamber for different times starting from time zero. From this figure it can be seen that all the droplets bigger than 1.2 [mm] reach the bottom of the chamber after only 1 [sec), while the 0.5 [mm) droplet does not reach the chamber bottom until 2.5 [sec] have elapsed. Thus the smaller droplets have a better chance of being fully utilized than the larger droplets.
ffi. h = 0 h =0-05H 9F-
=02H
=0-4H
=0 6H h =OSH h =H 71-H = HEIGHT OF CHAIMBER Figure 32. Veloci ty of Droplcts at Different Positions in the Dousing Cnamber (0(-A-)15
%J .
I I II I'. I I 0 ItR1 LE DI2 3 4 DROPLET DIAMIETER [mm]
II - ' IIII . ill 02 Figure 33. Times Taken by Different Droplets to Reach a Certain Position in the Chaziber 03 H = HEIGHT OF A CHAMIBER
%.-: I 0 ~vvp 41
.5 [se ]
0C6 0-7 i 08 II
.ji i
II il J! 0*9 ;1 1 il-
- i ..
.1.
- ii :
'i!
; I h=H . p I
0 I 2 3 4 ! i !i
;i DROPLET DIAMN;ETER [mm] :1
. Ill.
. I
. I'l 11
112 3 Figure 34. Positions of Different Droplets in the Chamber for Different Tines
-;4 1 2 -M"..I :1.
I: U) I H = HEIGHT OF CHAMBiER ,i (9 z I -J ILi h h 0 2 3 DROPLET DIAMETER [mmm]
113 9.5 Droplet Temperature Distributions In order to predict the temperature response of the water droplets in the same manner as was done in Part I of this study, it would be necessary to solve NIx N (>l = 10, N 100) times the single droplet problem in every time increment (0.5 sec). This is not feasible because it would require excessive computational time and a huge computer memory. To overcome this problem some other means of calculating droplet temperature responses have been considered. Tne "collocation method" [119) was tried, for a single drcplet response. This method was subsequently abandoned, since it did not
- offer any advantage over the procedure outlined earlier, because it was necessary to solve a system of algebraic equations at the colloca-tion points for each sphere for every time interval.
The striking similarity of all curves representing the individual dimensionless temperature responses (see Figures 9 to !2) suggested that they might be fitted by the equation 0 = exp(- A1 Z) (74) where A is an adjustable parameter. Simple dimensional analysis of certain important parameters (see Appendix M) gave the following expression for the dimensionless parameter Z SPsM vRo}1/3 The correlation (75), however, was found to be too simple
114 to enable the fitting of the thermal utilization curves reasonably well. Several other curve fits were tried (see Table II.1) sziTng the Waterloo Computer Center Library Subroutines GEPLSD or DPEN1N which are optional routines. Some of the curve fits were more successful than others. A dimensional f-ve parameter model of the typep 6 1 exp L A 1 Ro ( ] (76) gave the best agreement with the predictions. The results of the least squares estimates (for 1230 points) of parameters A 1 to A5 are: A1 = 2.3'7*!0 , A2 = 1.225449, A3 = 0.262816, A4 = 0.51S432 A3 = 0.102082 (77) To compare the correlation predictions with calculated droplet temperature histories, Figure 35 is drawn. This shaos that the correlation predictions (points) are in fairly good agreement. with the temperature responses over a wide range of air/steam mixture parameters. Now, using equation (76) and equalities (77) it is possible to calculate the temperature of the droplets in the dousing system at any time and position. This has been done and the results are illustrated in Figure 36 where, after a 0.5 sec exposure, typi-cal temperatures of different diameter droplets at different pos'- tions in the chamber are shomn., for the same spray used to produce Figures 32 to 34, after a 0.5 [sea] exposure.
1 115 TABLE II.1 Droplet Temperature Response Kodel Fits Model J
' 1 - exp (bIZ + b 2 )
1- exp (bZ 2 + b2 Z + b3 ) 3 2+ Z3 6=1- exp (bZ + b2 Z + bZ + b4) GEPLSD 6 =1 - exp (blZ 4 +b 2Z 3 +b 3Z 2 +b 4 z+b 5) 6 1 -exp (b Z 5 + bZ 4 + b Z 3 + b Z2 + b Z + b 1 (b Z + b 2 ) 6 =1- (b 1 Z 2 +bt 2 Z *b 3 ) e (b Z+b 2
=1- 2Z +Z bz+ GEPLSD e= 1- (b Z 4 + b 2 Z3 + b Z + b4 Z + b5 )
e =1- (bZ 5 + b Z4 + b Z 3 + b4 Z + b 5 Z + b6 ) 6 = 1 -b 1Z 6 = I - exp (bIZ + b 2) 6 = 1 - exp (- b Z ) 6 = 1 - exp (b Zb2 + b 3 ) 6 = I - b2 exp (- b 3 Z 4 ) DPE'LN' (Cont'd)
116 Table II.1 (Cont'd) Subroutine
~odel Used 9 =l1 - exp (b Z 2) + b 3 Zb 4 DPENLN 1 3~~~~
9 1 -exp (b Z+b Z + b 3 Z + b4 ) b4 b5 b6 b7
= b 1 -b2 exp (b p VR m t )
b2 b 3 b 4 b5 b6 9e= 1- exp (b p vR m t XS ) where P ,/m2 ], vR [m/s], t [s] and m [kg].
1 PT=1-5BAR, V 1=5m/s, D=05mm, Xs=0 2175 o CORRELATION 2 2 5-3 1b5 025 PREDICTIONS 3 2-5 1-0 2-5 0 4835 4 1[089 10.0 3-0 0 039 0 CP 0 0 II 0 0-% 0 I-I I-a I-Iii 0 10° 10I t ( sI Figtire 35. 7therrirln btlltzation lPVcciction1s Compared to
10 0 Tt 2.55 00 C I NI TI AL TE NMPE RA^TUIRE 2 18 OF SYSTEM4 90 - 80 - II h =0 8H 0 70 _ (f) h =O06H I-Ld
-J I 50k-CL 0
h=04H C) 50 LU a:: 40 h 2m i-L F- h =-1H - 30 h =005f E Twti = 240 C
-h =O INITIAL TEMPEFUATURE OF WATER DROPLETS H = CHAMBER HEIGHT 10 I I I II~~~~~~~~~
I a 0 I 2 3 DRO PLET D1 AMLAET ER [mm, , t4 IV _- 1 ....,z, 0'; r.:.-%r -,
119 The only quantity in equation (72a) left unspecified is the heat loss to the surrounding medium which will be dealt with in the next paragraph. 9.6 Heat Losses Simulation The heat losses to the surroundings QL were determined experimentally, see Figure 39, curve 1. These experimental results were fitted by P Pi exp(- kt) *(78) where: Pi - initial chamber pressure, k - adjustable constant, t - time. In this manner, the chamber pressure decrease due to heat losses was calculated rather than the corresponding heat losses. The least square estimate for the nonlinear model (77) of the parameter k is 0.002. Thus all the necessary terms in equation (72a) are speci-fied and we can proceed to the outline of the numerical algorithm
. ~~~~~~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~I
120
- 10. THE NUMIERICAL ALGORITHM! OUTLINE The only unknown term in equation (72a) is the term 0 which represents the total energy of the chamber after a certain time increment (0.5 sec used). This can therefore be determined from equation (72a). By a trial and error procedure the corresponding temperature and pressure of the air/steam mixture is determined and subsequently used as the initial state for the new time increment.
This implies the repetition of the procedure until the pressure change in the dousing chamber is satisfactorily decreased to a preselected value. This is the essenti-al step in the algoritnh, however before it can be taken, there are a number of steps involved as shown in the program contained in Appendix N. The algorithm entry data are: (i) chamber volume and he!.ght (ii) initial state of air/steam mixture in the chamber (iii) inlet water flow rate, temperature and velocity (iv) droplet size distribution parameters: xM, b and 6 (v) number of droplet sizes M and chamber height increments N ! (vi) a time increment. The program then proceeds through the following steps:
- 1. From the known droplet size distribution, the volume fraction distribution of droplets of 2 given mean diameter in each of the diameter intervals is calculated.
- 2. The nurbers of droplets of each-of the diameters in a unit volume of water is calculated.
_ _ _ _ _ _ _ _ _ _~~~~~~~~~~~ 121
- 3. The temperature of the system is determined from the partial pressure of steam present in the tank.
- 4. The density and heat capacity of water at the inlet conditions is determined.
- 5. The chamber is divided into a number of 5 cm height increments.
- 6. The velocity profile of each droplet size as it falls through the chamber is calculated by using a Runge-Kutta technique.
- 7. The time (see) for droplets of a given diameter to fall to a specific height increment in the chamber is calculated.
- 8. The energy (kJ) of the dousing water entering the chamber in a single time increment (0.5 see) is calculated.
- 9. The total residence time (sec) in the chamber of a droplet of given diameter is calculated.
- 10. The amount of air in the chamber at the start of test is calcu-lated.
- 11. The initial energy content (kJ) of the air/steam mixture is calculated.
- 12. Time is incremented by At (used value was 0.5 see).
- 13. At the end of the time increment the new local drop size distri-bution is calculated, i.e. the number of droplets of given diameters that are in the various height increments of the chamber.
- 14. If any droplets have left the chamber then the volume fraction of droplets of that particular diameter is calculated.
122
- 15. The local droplet temperature distribution is calculated. The new temperature (*C) of all droplets at all height increments of the chamber is.determined from the generalized correlation of thermal utilization (equation (77)).
- 16. Then using the temperatures of the droplets of given diameter the program then determines the energy (kJ) of all the droplets in the chamber at the end of the time increment. The density and heat capacity of each droplet is first determined from the droplet temperature and the total pressure of the system.
- 17. The energy (kJ) of any droplets that have left the chamber dur-ing the time increment is calculated. The value of volume fraction determined from step 14 is used if it is the first time for droplets of a given diameter to leave the chamber.
- 18. The energy reduction of the air/steam mixture is calculated by use of equation (72a).
- 19. The partial pressure of steam is determined by trial and error.
The procedure requires that the energy of air + energy of steam approximates the new energy of the mixture. Both energy of air and energy of steam are determined indirectly through functions of the partial pressure of steam.
- 20. A new system temperature is determined from the partial pressure of steam.
- 21. Total pressure of the-system is determined and printed out.
- 22. Steps 12 to 21 are repeated by advancing the time increment.
123 More details about every step can be found in the program given in Appendix N. I i
!i J
- 11. EXPERIMNTAL RESULTS AND DISCUSSION In this chapter the experimental results concerning the droplet size distributions will be treated firstly in paragraph 11.1.
The next paragraph 11.2 will deal with the problem of the dousing chamber test predictions on the basis of the information on droplet siza distributions supplied in paragraph 11.1. 11.1 Droplet Size Distribution Experimental runs were performed at three flow rates - 30,
- 40 and - 50 IGPM and the corresponding photographs (using the catch in cell technique) are shown in Figures 1 to 6 of Appendix 0 (two for each flow rate taken at different horizontal positions inside the spray). Each of these photographs is analysed on the departmental Quantimeter. The results of these analyses are shown in Figures 7 to 12 in the same Appendix corresponding to flow rates 30, 40 and 50 IGPM. The defining parameters x', b and 6 of the upper limit distri-buticn function were obtained by the least square fit by lineariza-tioa of the upper limit function model, see P. M. Reilly [121]. The least square fit estimates are represented by solid lines in Figures 7 to 12. The results of these calculations are shown in Table II.2.
From this table it can be seen that the maximum measured droplet size decreases as the flow rate through the nozzle innrease (corresponding to increasing pressure drops across the nozzle) which is expected. The curve-fitted values of the maximum droplet sizes 124 i
125 TABLE 11.2 Flow Run Number of Maximum Predicted Values Rate I Droplets Droplet IGPII Measured Diameter l Measuredn]_ . j 30.8 130 126 5.6 5.25 1.265 0.292 30.8 132 127 4.5 3.49 0.902 0.399 40.42 131 48 4.0 2.935 1.47 1.68 40.42 133 132 4.0 J5.295 2.172 1.008 50.05 134 143 3.7 4.629 0.677 j 1.569 50.05 136 181 3.7 3.917 0.822 00.779 differ from the measured ones, however, they correspond to the so-called maximum stable droplets (Mugele 1122), one of the originators of the upper limit distribution function theory). The dousing chamber tests were performed with 45 IGPM per nozzle, and therefore it was necessary to find some kind of average droplet size distribution for this flow rate. But before this was done it was also necessary to find an average droplet size distri-bution for one flow rate from the results obtained at different horizontal positions in the spray. Comparing the defining coeffi-cients in Table II.2 for the corresponding pairs of distributions, it can be seen that they differ considerably. A reasonable way to II
126 find an average distribution at the same flow rate is to take weighted values of the frequency distributions depending upon the amount of a liquid caught in the cells at the different positions in the spray. Thus for the 40 IGPN flow rate (Runs 131 and 133) the liquid volumes caught in the cells (during the same period of time) were - 15 mm3 and - 2646 mm3, respectively. Hence the sampled amount of water in the spray having distribution 131 (Figure 9 in Appendix 0) was negligible.and the distribution 133 (Figure 10 in Appendix 0, and circles 0 and the least square fitted curve 1 in Figure 39) is taken to be the representative one for this flow rate. Similarly, for the 50 IGPM flow rate the volumes caught in the cells were 707 and 1647 mm3 for distribution 134 and 136 (Figures 11 and 12 in Appendix 0), respectively. The weighted (30% and 70%) fre-quency distribution is shown in Figure 37 by dots a, while curve 2 represents the least square fitted line. The estimated parameters are: xm - 3.82 b = 0.9859 6 - 0.793 (79) To obtain a representative distribution for the 45 IGPM flow rate (dousing chamber experiments flow rate) the frequency distribution for 40 IGPII (Figure 10 in Appendix 0, and curve 1 in Figure 37) and the averaged frequency distribution for 50 IGPM (curve 2 in Figure
- 37) were weighted by a 50% to 50% correspondence, to produce the triangles A in Figure 37. The least square fitted curve to this data represented by curve 3 (in Figure 37) and its defining
Figure 37. Comparison of Different Droplet Size Distributions Obtained by the Catch in Cell Technique 0 CATCH IN CELL TECHNIQUE 0 40 IGPM O WEIGHTED AVERAGE TO 50 IGPM A WEIGHTED AVERAGE TO 40 & 50 IGPM A
- 1. LEAST SQUARE FIT TO 0 A
- 2. LEAST SQUARE FIT TO 0
- 3. LEAST SQUARE FIT TO A 2
-i l 2 3 4 5 d [mm]
128 parameters are: Xm = 3.847 [mm], b = 0.605, 6 = 0.886 (80) These are the defining parameters (obtained by using the catch in cell technique) for the upper limit distribution function to be used in the simulation of the dousing chamber runs. It was pointed out in paragraph 8.2 that a comparison of droplet size distributions obtained by the catch in cell technique and direct photography will certainly be interesting. For that purpose the direct photography method was attempted simultaneously with the experiments with the catch in cell technique. The photo-graphs of the spray in air at the flow rates 30, 40 and 50 IGPM are shown in Figures 13 to 15 (Appendix 0), respectively. These figures (13 to 15 of Appendix 0) represent the droplet images above the outer catch cell, i.e. they correspond to droplet sizes in cells represented by runs 132, 133, and 136 (Figures 2, 4 and 6 and distributions in Figures 8, 10 and 12 in Appendix 0). The photographs for the flow rates 40 and 50 IGPM had to be analysed manually, since the droplets did not appear as solid dots against the background, thus preventing analysis by the Quantimeter. The results of this analysis are shown in Figures 16 and 17 of AppendixO for 40 and 50 IGPM, respectively. The solid lines represent the least squares fit of the upper limit distribution function model. The parameter estimates are given in Table II.3.
129 TABLE II.3 _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Flow Rate Maximum Droplet i Predicted Values [IGPM) Diameter I Measured [mm] I X [m= I b 6 30.8 3.8 not analysed 40.42 3.5 3.31 0.7697 1.1071 50.05 3 x - 1.452 - 1.9978 The 50 IGPM case (Figure 17 of Appendix 0) is best fitted by the log-probability distribution function, which is a special case of the upper limit one. The volume log-probability distribution func-tion is defined as dv 6 1 exp ( 62Y) Zx_ . (81) with y- in (xFx) (82)
- X x~.b (83)
The representative droplet size distribution at 45 IGPM is obtained by weighting the frequency droplet size distributions at 40 IGPM (Figure 16 of Appendix 0) and 50 IGPHI (Figure 17 of Appendix 0) in the 50% to 50% correspondence. In this manner the droplet size distribution shown in Figure 38 by the solid line is
Figure 38. Droplet Size Distribution Obtained by the Direct Photography Technique 10 DIRECT PHOTOGRAPHY TECHNIQUE 0D WEIGTED AVERAGE Xm 4-334 (mm) LEAST SQUARE FIT b - 160588 x 8
- = 1 22337 x~~~~
4 4 _ / X~~~~~~~~~~~~~~~~~~~~~ 2 1 2 3 4
. d (mm)
131 obtained. The upper limit distribution function model fitted to these data is represented by the dashed line in Figure 38, and the defining parameters are xm = 4.334 [mm) b = 1.606 6 6 1.223 (84) Thus, the equalities (80) and (84) represent the upper limit distribution function defining parameters for the two differ-ent methods of droplet size distribution measurements, namely "the catch in cell" and "the direct photograph" methods, respectively. In the next paragraph it is shown how these parameters were used in the computer algorithm to predict the dousing chamber pressure charges as a function of time. 11.2 Dousing Tcst Predictions Dousing tests were undertaken [107] in the enclosed chamber of total volume 56 im3) (see Figure 29) provided firstly with single spray nozzles (experiments lA, 2A, 2B, 2C and 3A, Figures 39 and 40) and then a group of five spray nozzles (experiments 2D, 2E, 2F, 3D and 3E, Figures 39 and 41). The experimental results are represented by the solid lines while the triangles represent predic-tions using the droplet size distribution obtained using the catch in cell technique and the squares represent the predictions using the droplet size distribution obtained using the direct photography technique. From Figures 39 to 41 it can be seen that the predicted
A dmax-3-8 4 7(mm) 8=O0s886 a = 0-605 01 0 dma =4 5 (mm) 8 = 0.6656 a= 0-301 a dmax =4-334 (mm) 8= 1-223 a = 1606 1.1
-TEST #I[UNDOUSED a
AN M-i '1 ". a a 0 A TEST 4IA-DOUSED-SINGLE NOZZLE A T.;i = 24°C a u & TMi = 98-25°C C A a A 0 TEST : IB -DOUSED A
-FIVE NOZZLES. U A 0 a Ti =24°0C M A 0
0 0 0 Li\ 0 Tpji = 7-54°C l a c t? A A az a Figure 39. Dousing Tests #1, IA and 1B w
'Jj I I _ _ __
I I I I I __ __ I L I 10 20 *;)s 410 50 60 .70 so 90 I
'..., -V~
*1I ~';51 L
-^.t-
+ !+ji ;ME-: (:^)§*$;> ,J
SINGLE NOZZLE 135 -TEST No 2C Tyi = 176 ° F (800 C) 121 TMI =101.230 I.1 m0 ml 101
-TEST No 2B Twi. = 142° F (61° C) 0*9 TMI = 91.00 C
-TEST No 3A Twi= 159° F (70*6 0C)
; a0 0 TMi = 79.32 0 C A a-No 2A 1120 F (44 5 0 C)
TM i 77 920 C Figure 410. Dousing Tests #2A, 2B, 2C and 3A with a Single Nozzle 0 10 20 30 40 0 10 20 30 40 TIME (s)
FIVE NOZZLES TEST No. 2F TMI= 9795 0C TWA = 1600 F (71 0OC) TM i = 90 9°C
-TEST No 2E TMI = 92 570 C Twi = 1590 C (70- 60 C)
Twi = 1840 F (84-4°C) A
-TEST No 2D TMi = 84 19° C 0
Twi = 130 0 F (54-4 0 CC) 0 0 0 S 0
; 0 D
A S 32 a U A A U TEST No 3 D Twi = 153°F (67*3 0 C) TM; = 82-68 0C Figure 41. Dousing Tests #2D, 2E, 2F, 3D and 3E with Five 4-VI Nozzles
. M 20 30 40 0 10 20 30 40
".-"Bi1-- m l y., - i..
. .. . ..." , ."i
- 11. i
'!!. "C' I" 8 1-
.. I -
" 4.::, 114::
T;ijM.E , ( S),4 c<.. . I
135 results (triangles) show very good agreement for all the tests. The difference between the predicted and measured pressures at any time is within + 7% of the measured value for all the xuns except 1A, for times greater than 50 seconds. All the runs (except IA) were recorded for tine periods of from 10 to 40 seconds. The re-cording time length may be an important factor affecting the accur-acy of the predictions because of the following two phenomena which are bound to show increasing influences later in the dousing process. These phenomena are acting in the same direction, i.e. they both tend to increase the predicted pressure drop rate in the dousing chamber if they are not taken into account in the simulation algor-ithm. These phenomena are explained as follows: i) In Figure 38 the curve for the test #1 is shown, where a reduction in svstem pressure was obtained without the use of dousing water. The experimental curve shows that the steam condensation on the walls of the chamber contributes to the overall pressure drop at a rate of about 0.0021 [Bar/sec] for conditions similar to test #1. These conditions were always above 0.97 Bar, however, in the absence of other experimental information, this curve was used to estimate the heat losses in all the other dousing runs involving water sprays when the dousing chamber temperature and pressure changes were much more rapid which would mean lower actual heat losses to the surrounding medium than were allowed for.
___ M 0 136 ii) No attempt has been made to theoretically determine the effect of steam condensation on the water film which must inevitably be formed by droplet impigment on the walls and the obstructions of the chamber. In the initial time period of this film forma-tion this effect may not be so important, but as time progresses this effect probably becomes more pronounced as the walls be-come completely wetted. In the simulation runs however, it is presumed that all the water entering the dousing chamber flows in the form of spray which is certainly predicting a more rapid pressure drop in the chamber than would occur if part of the water flowed as a film on the walls and obstructions. Ing SPItP of hi4s i. can be~said~in~genczl that the simu-lation model predictioins fr uchja complex problem are highly satis-factory over the wide range of experimental conditions. The simulation model predictions using the droplet size distribution obtained from the direct photography technique predict slightly lower values (represented by squares in Figures 38 to 41) of dousing chamber pressures as compared to the values obtained by use of droplet size distribution obtained using the catch in cell technique. Further work on the direct photography technique will probably improve its reliability over the catch in cell technique. In any case the dousing chamber pressure drop predictions using the droplet size distribution obtained from the catch in cell tech-nique are conservative, if compared with the droplet size distri-bution obtained using the direct photography technique.
137 Finally, the sensitivity of the simulation model to the effects of droplet size distribution should be mentioned. To show the effects of the size distribution, the simulation has been run with three different imaginary droplet size distributions. The first droplet size distribution is represented by the parameters x 8.847 [lmm, b = 0.605 and 6 = 0.88 (85) which has the same "shape" as the one measured using the catch in cell technique but a larger maximum diameter x=. This droplet size distribution prediction (for the conditions of the run lA) are re-presented by circles in Figure 42. The second size distribution is a very small diameter spray, so small that all the droplets will be fully utilized in the chamber. This represents the heat balance on the chamber when all the water is utilized. The pressure in the chamber cannot be de-creased any faster than the pressure decrease represented by this curve, which is illustrated by squares in Figure 42. Finally, by a trial and error method a droplet size dis-tribution was selected which gave a very good fit to the experimental results (for the conditions of the run 1A). Its defining parameters are: xM = 4.5, b = 0.301, 6 = 0.66 (86) Figure 42 gives an indication of the sensitivity of the simulation to the different droplet size distributions. The
e Xm- 8 X 8 4 7 b: 0-605 8=0-88 . A Xm= 45 b= 0-301 B=0-66 a 5 FULLY UTILIZED SPRAY a v S U TEST No IA DOUSED a a SINGLE NOZZLE a
= 24° C U .
0 a S 0 a a A
. A A
a a a a a Figure. 42. Effects of Different Droplet Size Distributions co on the Dousing Chamber Pressure Response 20 30 40 !50. 70 80 90
-Iry A,1- I _ N
139 difference between simulated and predicted values can be very high. The heat balance curve predicts the pressure in the chamber 0.515 Bar after at t - 50 sec, while in the experiments at this time the pressure was 0.7 Bar. The droplet size distribution represented by circles (Figure 42) would predict a pressure of 0.81 Bar at this time. This sensitivity of the dousing chamber response curves to the droplet size distributions highlights the need for accurate drop size data. Obviously more effort needs to be put into the acquisition of such data to lend confidence to future scale-up and full size dousing chamber simulations.
- 12. CONCLUSIONS AND RECOMIENDATIONS OF PART II As demonstrated in Figures 39 to 41 the simulation model developed inthe second part of this report gives very good predic-tinnsofI the dousing chadberpressure droDs over the wide range of inlet water and dousing chamber conditions. The ability of the model to predict practically all the runs wit-in P fiw pornent is yery encouraging for the simulation of the real dousing chambers.
The correlation between thermal utililization and the initial droplet and gas phase variables (equation (76)) is a good practical way of getting around gigantic computer requirements of the dousing chamber simulation. The comparison between the correla-tion values of thermal utilization and the theoretically and experi-mentally confirmed values shows excellent agreement. At this stage of the project it can be concluded that the catch in cell technique of droplet size distribution measurements gives bigger average droplet sizes than the direct photography tech-nique. However, more experiments and some improvements in both tech-niques will give a firmer answer as to the question of the reliabil-ity of the techniques. The recommendations for future work are: i) modelling of the dousing chamber heat losses to the surround-ings; ii) modelling of the condensation on dousing chamber walls and obstructions; 140
141 iii) droplet size distribution measurement under the real condi-tions in a dousing chamber, which surely differ considerably from those in the pilot plant tests. Which of the measurement techniques will be more advantageous is not possible to assess at this stage of the project.
- 13. FINAL SUMIARY The following is a summary of the accomplishments of this project.
- 1. It has been demonstrated that the only practical way to solve the droplet combined heat and mass transfer problem for NRe over a wide range of values is by the macroscopic balance treat-ment involving the transport rate equations.
- 2. A survey of the literature on drag coefficients for droplets has led to the conclusion that the Reinhart's drag coefficient correlations appear to be the best.
- 3. Droplet velocity trajectories have been calculated using Reinhart's drag coefficient correlations for upwards and down-wards freely moving droplets. This could only be done numeric-ally. Solutions using Reinhart's drag coefficient correlations agree with existing data on droplet terminal velocities. These trajectories were the ones eventually used in the simulation of droplet temperature responses for free falling droplets (see below for summary).
- 4. Some analytical solutions for trajectories of solid spheres in free fall were either obtained from the literature or newly derived, however, it was decided in the end to use the "Reinhart's trajectories" instead of the solid sphere ones.
- 5. The sensible and latent heat transfer contributions to gas side heat transfer coefficients were discussed. It was shown that 142
143 mass transfer coefficients are very often not known. A detailed discussion of previous workers'use of the analogy led to the conclusion that its use in this work could be valid.
- 6. A comparison of the external heat and mass transfer analogy equations for different systems was enabled by collecting the correlations in Figure 8.
- 7. A model was developed in the form of an analytical solution based on the combined heat and mass transfer analogy to predict the temperature of a free falling droplet which is perfectly mixed (i.e. no internal resistance).
- 8. A model was developed in the form of a numerical solution based on the combined heat and mass transfer analogy to predict the temperature of a free falling droplet which is partially mixed.
This involves the combination of the external heat and mass transfer analogy with the internal conduction problem where the internal temperature profile is renewed periodically.
- 9. A third model was developed in the form of a combined analytical and numerical solution based on the combined heat and mass tran-sfer analogy to predict the temperature of a free falling drop-let which behaves as a solid sphere internally. This model describes the droplet having no internal mixing and the maximum possible internal droplet resistance.
- 10. It was shown that as the number of internal mixings increased the partial internal resistance model became assymptotic to the no internal resistance model (see Figure 13).
144
- 11. It was shown that in the limit as the number of internal mix-ings was decreased to zero and the initial temperature profile was uniform, the partial internal resistance model would become assymptotic to the no internal mixing model (see Chapter 2.7).
- 12. The maximum effect of a change of droplet size was investigated and although this was found to be small it was incorporated in the models.
- 13. All the necessary thermodynamics and physical properties for the air/water system were collected on one place (Appendix D).
- 14. Computer algorithms for each of the models were developed.
- 15. The effects of steam concentration, initial velocity and dia-meter, on the temperature response (thermal utilization) of free falling droplets were predicted. The possible range of effects on thermal utilization caused by internal mixing of varying degrees was shown.
- 16. Various ways of defining the heat transfer coefficient in the condensing process were carefully discussed.
- 17. The experimental results of many workers dealing with conden-sation from pure steam and also from air/steam mixtures were collected together in one place for comparison purposes.
- 18. It was shown that the predictions using the analogy theory extrapolate reasonably well to the pure steam measured and theoretical heat transfer coefficients.
- 19. An apparatus was built to enable the measurement of temperature responses of single droplets suspended in air/steam jets.
145
- 20. Exoerimentally measured mean temperature responses of droplets (using tiny thermistors glued on teflon tubing) were obtained.
- 21. The possible effects on the temperature responses of the needle used to create the droplets were shomn to be negligible.
- 22. Comparison of the experimental results and the theoretical models confirmed that a droplet appears to behave as if it is partially mixed internally.
- 23. The predictions of the partial internal mixing model and the results of 90 experiments showed standard deviations of 6% and 4% in the temperature response at 50 and 75% utilization respectively.
- 24. A slight trend in the data suggested that the larger droplets behaved as if they were experiencing more internal circulation than the smaller droplets.
- 25. It was concluded that the analogy theory could be applied with confidence to the problem of combined heat and mass transfer to droplets moving relative to an air/steam mixture provided that some internal resistance to heat transfer was allowed for.
- 26. Experimental data representing the size distribution of sprays produced by a typical dousing chamber nozzle were obtained by two methods, namely "the catch in cell" and the "direct photo-graphy" methods. The data were fitted by upper limit distri-bution functions.
- 27. A simulation model of a pilot plant dousing chamber was deve-loped based on the previously obtained knowledge of the thermal utilization of droplets and a step by step energy balance.
146
- 28. In order ta-make-the-use af-th mnodel feasi1be from tjhe point of view of computer time usage, a generalized dimensional correlation relating-the-thermna ut-ilzation. air/steam mixture pressure, initial droplet velonftfy m of droplet, steam con-centration and time was obtained by use of an optimm1 llr-ve fitting procedure.
- 29. The dousing chamber simulations and the experimental data supplied by Atomic Energy of Canada were compared and found to be in reasonable agreement.
- 30. The sensitivity of the simulation model to spray size distri-butions was illustrated and it was concluded that although the simulations of the pilot plant chamber were very good, it is expected that more data on droplet size distributions-ao.pee-tb-in the full-scale dousing chamber would be needed for scale-up purposes.
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/~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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100. E. Marshall and R. Meyder: On Condensation Mass Transfer in the Presence of Non-Condensables, Warme-und StoffUbertragung, 3 (1970), pp. 191-196. 101. F. H. Garner and P. Kendrick: Mass Transfer to Drops of Liquid Suspended in a Gas Stream, Part I: A Wind Tunnel for the Study of Individual Liquid Drops, Trans. Instn. Chem. Engrs. (London), 37 (1959), pp. 155-161. 102. H. L. Dryden and G. B. Schubauer: The Use of Damping Screens for the Reduction of Wind-Tunnel Turbulence, J. Aero. Sci., 14 (1947), pp. 221-228. 103. P. F. Bennewitz: The Brady Array - A New Bulk-Effect Humidity Sensor, Society of Autemative Engineers, Automobile EngineeringCD Meeting, Detroit, May 14-18, 1973. 104. W. E. Ranz: On the Evaporation of a Drop of Volatile Liquid in X High Temperature Surrounding, Trans. ASIME, 78 (1956), pp. 909-913. 105. P. M. Reilly: Introduction to Statistical Methods, University of Waterloo, Chemical Engineering Department, 1974. 106. P. Farriot: A Review of Mass Transfer Interfaces, Can. J. Chem. Eng., 40 (1962), pp. 60-69. 107. The Atomic Energy of Canada Ltd., Internal Publication. 108. K. McLean: U.A.Sc. Thesis, University of Waterloo, Chemical Engineering Department, in preparation. 109. H. Syhrc: Einige theoretisclhe Betrachtungen zur Kondensaticn an wasnertropfen (Some Theoretical Treatments of the Condensation on Hater Drcplets), Energietechnik, 11, No. 9 (1961), pp. 401-406.
155 110. V. I. Kushnyrev: Eksperinentalnoe issledovanie processa disper-govania zhidkosty primenitelno k smresitelnoy kondensacii (Experimental Investigation of the Disintegration Processes of the Liquids Applied to the Direct Contact Condensation), Trudy MEI Teploenergetika i mashinostroenie, vip. 104, 1972. 111. N. I. Gelperin, B. N. Basargin, V. S. Galustov, V. V. Shuvalov: Isledovanie dispersnosti raspyla centribezhno struynoi forsunki (Examination of Atomization by Centrifugal Nozzle), Himitscheskoe and neftnoe mashinostroenie, No. 11, 1972, pp. 15-16. 112. J. 0. Hinze: Fundamentals of the Hydrodynamic Mechanism of Splitting in Dispersion Processes, A.I.Ch.E. Journal, 1 (1955), pp. 289-295. 113. W. R. Sears, M. Van byke, editors: Annual Review of Fluid Dynamics, Vol. 1, Annual Review Inc., 1969. 114. R. .S. Brodkey: The Phenomena of Fluid Notions, Addison-Wesley, 1967. 115. I. Yaron, B. Gal-or: Convective Mass and Heat Transfer from Size Distributed Drops, Bubbles or Solid Particles, Int. J. Heat Mass Transfer, 14 (1971), pp. 727-737. 116. E. Kulic, E. Rhodes, G. Sullivan, K. McLean: Direct Contact Condensation from Air/Steam Mixtures on Falling Sprays, 5.- International CHISA Congress, Heat Transfer Section, Paper #E2.1, Prague, Aug. 25-29, 1975. 117. A. LekiC, R. Bajramovi6, J. D. Ford: Droplet Size Distribution: An Improved Method for Fitting Experimental Data, Submitted for publication in The Canadian Journal of Chemical Engineering. 118. R. A. Mugele and H. D. Evans: Droplet Size Distribution in Sprays, Ind. Eng. Ch., 43, No. 6 (1951), pp. 1317-1324. 119. W. E. Stewart: Solution of Transport Problems by Collocation Methods, A.I.Cn.E. Continuing Education Series #4 (1969). 120. R. A. Greenkorn and D. P. Kessler: Transfer Operations, McGraw-Hill, 1972. 121. P. M. Reilley: Engineering Statistics, Ch. E. 622, University of Waterloo, 1975. 122. R. A. Mugele: Maximum Stable Droplets in Dispersoids, A.I.Ch.E. Journal, 6, No. 1 (1960), pp. 3-8.
156 NUMENCLATURE a Ratio of heat fluxes caused by the concentration and temperature driving forces, defined by the equation (11) a1 Constant defined in Table 1, see Appendix B b ='Xm-X5 0 Skewness.parameter in upper limit equation (73) X50 A Area of a droplet m AB Constants in equations (18) and (19) A 1 ,,A2 A3 ,A 4 ,A 5 Constants in equation (81) C 1 , C2 ,C Constants defined by the equations (21), (22), and (23), respectively Molar concentrations of: an air/steam mixture; air CM'CAI'CSM' CSi in a mixture; steam in a mixture; steam at the inter-face, respectively 3 kg moles/mr C D S Drag coefficient of: a droplet, a disk and a D'C;'CD sphere, respectively Drag coefficient value at the maximum value d CDmax CD see Table 1 of Appendix B c Specific heat at constant volume, J/kg0 C v Specific heat at constant pressure of a droplet J/kg 0C pw CPACPS Molar heat capacities of air and steam, respectively J/kg mole 'K
157 d Droplet diameter m
- N2 Re/NFr Reinhart's dimensionless diameter
. 3p2 31 g1p2/
d Maximum value of d , see Table 1 of Appendix B max DAB Diffusivity of steam in air m2Is EMEW Energy content of mixture and water, respectively kJ Energy of water entering and leaving dousing chamber, EWE'EWL respectively kJ f Condensation coefficient f Wx Distribution function F Mass transfer coefficient kg moles/m2 s Fb Body force N g - 9.80665 Gravity acceleration m/s 2 ho 1haM'hF h. Heat transfer coefficients defined by equations (50) and (50a), (51) and (51a), respectively JIm2 sK h Convective or sensible heat transfer coefficient C 2., J/m s K h Apparent heat transfer coefficient defined by app equation (26), JIm 2 soK Enthalpies of air/steam mixture and water, respectively kJ/kg Mass flux defined by equation (17), kg/m s is
158 k Thermal conductivity of a droplet J/ms°K m = (1/6)ird 3pD Mass of a droplet kg mMmW Masses of air/steam mixture and water, respectively kg MAMS Molecular weights of air and steam, respectively kg/kg mole NA'N Molar fluxes of air and steam, respectively kg moles/mr s NAC Ackermann number defined by equation (10) NBi h appd/2k Biot number N at Fourier number FO R2 VR2 N d Froud number Fr dg ocL NGr .:,
- Grashof number NN =hkd Nusselt number NN kV N
Pr
=
k L c a Prandtl number 4o N = d VR Reynolds number N =- v -- DSchmidt number SC pDA DAB N Fd Sherwood number Sh CIPAB cdp N - 2 N /Nc Drop deformation number su7 2 Re
159 "weeW a H Weber number M Total pressure of a mixture N/m , Bar PAM, PSM Partial pressure of air and steam in a mixture, respectively N/M2 , Bar P Saturation pressure of steam N/m 2, Bar sat Si Partial pressure of steam at the interface N/M , Bar QTIQL9QS Total, latent and sensible heat fluxes to a droplet J/M2 s QLS Heat losses to surrounding flud, kJ r Distance from any point in a droplet to its centre m R Radius of a droplet m R Universal gas constant J/kg0 K R' R Nixture and steam constant, respectively, J/kg*K 4 3 S=4Re FrIN We=pjF 3 /gp4 Reinhart's system constant t Time s T TTT Initial, average gemperatures and a temperature at any time instant of droplet, respectively OC TM Initial temperature of air-steam mixture 0C T Saturation temperature of steam,. C sat Tw Wall temperature, °C WA Internal energy of mixture, kJ/kg
160 VD'VM Velocity of a droplet and the mi.:ture, respectively, [m/s] VDO Initial velocity of a droplet, [m/s] Relative, initial and terminal relative velocity, VRR V' RT [m/s5 Vg~vL Specific volumes of the vapour and water at saturation pressure m3/kg Volume fraction of droplets having diameter < x VR . e VR _f/pM/Pg Reinhart's dimensionless velocity x Diameter of particles x Average droplet diameter defined by equation (84) I i x Maximum stable droplet diameter m x5 0 Droplet diameter at V - 0.5 XAXs Volume fractions of air and steam in a mixture,
- respectively y Dimensionless function of x (equation (83))
z Distance travelled m PSMd1/3t z , (PS R) Dimensionless parameter used in equation (75) n1 -W rng Dimensionless numbers defined by equation (16) I.
161 Greek Letters a = k/pDCp Thermal diffusivity of a droplet m2/s 6 Size distribution parameter used in equation (73) and (82) A Difference T - Ti Thermal utilization TM - T. I Latent heat of water vaporization J/kg Dynamic viscosity of a mixture Ns/m 2 7? 3.14159 VI Kinematic viscosity m2 /s a Surface tension N/m P] 9'I Density of a droplet and mixture kg/m3 Dimensionless nuiber defined by equation (12) s
162 APPENDIX A SOIZE ANALYTICAL SOLUTIONS OF THE MACROSCOPIC MOMENTUM EQUATION FOR A SPHERE
-41 I.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,
163 In this approach are presented solutions obtained either from the literature or by the author for the relative velocity vR and position z of a sphere after it has been injected in an air/steam mixture with an initial velocity vRo. CASE 1. In the Stokes region (NRe < 0.1) the drag coefficient can be represented by C NRe/2< C (A.1) The solutions obtained are the same as those obtained by Lapple [24]. v'R g/A + (VRo - g/A) exp(- At) (A.2) Z = (g/A - v )t + (vR/A - 2/A - exp(- At)] (A.3) where A = 1814/pDd2 (A.4) CASE 2. In the region of 0.1 < NRe < 2 the standard solid sphere drag coefficient curve can be approximated by:
164 CD 24e ( .1 1 e] (A. 5) Thus G HDnDt2) VR, F exp(Dt/2) - (A. 6) and z -(2GDS2HDr)ZnF exp(Dc/2), z- (2G/DE+2U/DF) '
- (G/E +v4-VHt (A.7) where t=27 PM
, 4g-A
;D 2 =DI + 2D (A. 8)
(-O =1D =( D , 1 - 2Bv Ro - A-- D; E= - 2BD I (A. 9) F - -2BD 2 , G - D 1 (D - A) ; I - D2 (D + A) (A. 10) These expressions were not found in literature and were derived by the author by solving the equation of notion of the sphere. CASE 3. In the range of 2 < N < 500 the standard solid sphere Re drag coefficient curve can be represented 1'253 by:
165 C 1 /2 16/N~Re for 2 < NR < 10 (A.1) D CD D 11.5/N12 for 10 N< 500 (A.12)
~Re Re<
In this case, only an implicit relationship could be derived for the droplet velocity as a function of time t. This is given in equation (A.12). 2 l(J2/3 + 1/3 + vR21/3. 1 13 1 / 1/2)2 313113 2. L(J1/3 VR + J(i 1/3o1/ v R+ 3IJ1/3 l2 V1/2)2( 2/3
- tan 1
_r- t1/3 2(1/ 2 2(R
+ 1/2
-1/2 vRo) 1/3 131 1/3 vJ + (2vR +3i )(2v R (A.13)
C with I = g/J (A.14) J = gd PD/12p / pH/2 for 2 < NRe < 10 and (A.15) 1 J = gd3'2 p /8.625.pl/2p1/2 for 10 < NRe < 500 _ (A.16) . W In this case it was not possible to obtain an analytical solution for the distance travelled. M CASE 4. In the region where the Newton's law applies, i.e. 500 < NRe < 2.10, the drag coefficient is Re'
166 CD = 0.44 (A.17) an elegant solution can be obtained for both the velocity and the distance travelled tanh(Cgt) + CvR VR C[l + CvRo tanh(Cgt)] (A.18) where C =. 3 3 P /P gd (A.l9) The distance travelled depends upon the initial relative velocity: a) When the initial relative velocity of a drop vRo is smaller than its terminal relative velocity v RT, then: 1 £n cosh[Cg(t - C1 ) ] Z C2g cosh(- CClg) M (A.20) b) When the initial relative velocity of a drop VR is greater t shanit-erminal,.re-3ative v-,~~lo~iy vRT, then: 1 sinhECg(t - C1 )J Z C2g In sinhl- CClg - vM t (A.21) where g l1 + CvRO (A.22)
167 The relative terminal velocity i's in both the cases vRT = 1/C (A.23) It was first believed that derivations of the last two cases were new, however it has been recently discovered that a similar approach was used in [6] which confirms the validity of the solution. It is possible to obtain corresponding analytical solu-tions for a sphere initially moving upwards. The solutions are slightly different from those already derived for a downwards move-ment of a sphere until the moment when the droplet reaches its ma-xi-mum height. From this moment on the derived equations will hold.
'I
-X
168 APPENDIX B REINHART'S CORRELATIONS FOR DROPLET DRAG COEFFICIENT di
r .% f fY !:--l 3 APPENIDTC Bn
/ ; / g., .,
!,J T REINIIART'S DRAG COEFFICIENT FOR DROPLETS DOMAIN CD 3 f(NReS,Ap/P) UPPElR LIMIT OF VALIDITY d m f(Ap/p,S) AND/OR Nje - f(Ap/p,S)
I CD = 24/NRR d = 1.216/(AP/P b)"3 NR O.1 11a 24 0.687 * -1/3 -12 0.1311-2-.9 C5 -(1 +0.15N C d 70 'IO (10F S) N 974 (10 S).l S<c44.10 1 2 D NRe Re(T)R b OR ANY OTIIER VALID CORRELA-S>44 1012 TION FOR SOLID SPHERES d 30.8(a -1/3(10 -12) l N - 236(1G 1 2 )0 192 SPM Re R. aI~-200 -0.5315 -20.275 A-0.303038 12 CD ' 0.43(10-s 1 l) d 122(-) (10 S) NR N 2240(-E) (1012) Illb C12s-0. 05 dA- 0.535( 12 0.0367 NO.O -0.303 S>4112 S>44.10 CD 0.7(10 1S) d 300()P (10 1S) N Ie 1OAP) i (lo-2 0.224 -0.065 -0.447 0.114 -0.173 0.18
-IV V CD-0.0657d DO~O 6 S~d~pM) (AP (12 (10 - 5)
- d-N 16 A
0(AP.)
- -(A) (10
.) -12S) .
3 9
-0.173 (10-12 0
_ _ __ _ __ __ _ _ _ _ __ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _p i1 Re r iM 0.447 0.131 V CD-0.00335(~-") (10 S) d* -0.5 -0.238 0.232 D~~p .1 2224 (AP) (10-1 i-12 S) 1/6 NReaN, =4 4 7 0(A(.) (1o-12 ) I'AXIMUM~ C ~ O.7 -0.0232
)A 12 0.036 IX . C~~aa~~c 0.75( P) (.Lo- S)
Diannx PM Vt ~~CD C'D:na x+a I (d*-dTax) x NIY '!DYNAMIC STABILITY OF H[YDRODYNAMI[C STA"iLITY OF A DROPLET A DROPLET I
.. \X ^ '@Se~d~zE-,9:5 . ...
_ _ r~ .."rok ,%& .-;
170 APPENDIX C I EVALUATION OF THE DROPLET TE1PERATURE RESPONSE WITH INTERNAL RESISTANCE AND NO INTERNAL MIXING i i I I I I
*~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ii ii i
I. Ii I. i i I z i i I i
171 Equation (36) can be rewritten as (for the seventh order polynomial) f (r) = TM - (AO + Alr + A 2r2 + A3 r3 + ... + Ar7) (C.1) Introducing this into equation (34) it yields n=co 292 sin(bt r/R) 2 O(rt) = n - sin n cos n rR e P(-n Fo) nfR lrT (Ao + A r + A2r + A r + + A r7 10 r[ 14 ( 0 1 2 3 7
-sin(*nr/R) dr (C.2)
Considering just the integral in the last equation, one obtains R R I - TM J r sin(*nr/R) dr - Ao 0 r sin(mnr/R) dr -
- A1 Jf fR r
2 sin(*nr/R) dr + A2 f R r3 sin(*nr/R) dr + R
+ ... + A7 . r sin( nr/R) dr (C.3)
Denoting the integrals with T. and Ao to A7 by IM, I** I2 ...,7 I7
- the last integral can be rewritten as 7
I M =I IIi(C.4) iO i=o It is possible now to evaluate every single integral by the simple integration by parts. Thus
172 IM4 = TM r sin(+nr/R) dr (C.5) and I udv uv - vdu (C. 6) dv - sin(*, r/R) dr u = r or v - cos( 4,(ir/R) dv - dr and JR
'N~~~~p cos (O~nr/R) + f R. Cos(tpnIiR) arj 0
and 1j4~~~R R cos + -n Rsin(%r/R)] .11
*n
{R2 (in' -*.cos 0 I and finally I T (sin in - C (C.7) n Similarly, the second integral yields 10- AO 2 (sin n -ipn cos O) (C.8)
*n 1:
Ii 11
173 The next integral is SR2 It Al Jo r2 sin(*nr/R) dr Performing the integration by parts twice gives: 3~~~~~ Iim' 4 [ 2 in sin On - (V -2) cos n.-2J (C.9) Furthermore, the integral I2 is OR3 12 A2 o r sin(*nr/R) dr After the first integration is performed, the integral with lower power of r is obtained. Thus one can use the previous integration step in the evaluation of the higher power integral, to obtain I .A 2 2 2 2R44 [3(6 2 ) sin -n (At -6 co A2~~n n p)J (lO
*n Using the same technique the remaining integrals are evaluated 13 - f r sin(*nr/R) dr Vn~~
e<[4A (n ~6)sin* On~(n ~ 2n 2)cos An + 24) (C.ll) I4e A4 l r sin(nr/R) dr A4 - [54 - 60n + 120) sin in
+n
- ~(i'4 -2O0 2 +10
+12) o (C.12)
,n'n cosln
174 4 15 - A5 l: r sin(%Pr/R) dr . A- ' [*(6p - 12042 + 720) sin On 5 7 ~nfl lpn
- (n - 30n + 3604i - 720) cos in - 720) (C.13)
I - A6 I r7sin(4 r/R) dr =6 8 [(746 - 210t4, + 25204,2 0 np (6 42
- 5040) sin in - V(nn - 42*n + 840tn- 5040) cos n (C.14) and finally r8 sin(~nr/R) dr - A7 R 4 2
[n(8 - 3364k- + 67204,- 17 -A7fo
- 40320) sin ip - ( -8 6 + 1680<4 - 20160 2
_56 n n n ~ n
+ 40320) cos in + 40320) (C.15)
Introducing the values of integrals Im, Io, Il, ... , 17 into equation (34), equation (37) is obtained.
175 APPENDIX D OF WATER, THERMODYNAIICS AND TRANSPORT PROPERTIES AIR, STEAM AND THEIR MIXTURE I',, r L a- 13 t'J
*1
/
- ' 'I A1 L **
- q. -
1.' ,. it's, 1 I- V. II 19 I. _t>'s~~~t 7, " - 4.; r .e r- 1 Ar'j U
- 4
176 D.1 Saturation Pressure Steam saturation pressure Ps- f(T) is determined from Vesper's (76) correlation i-9. in p = = AiX " Bar) (D.l.1) SO where x - t 100 t - temperature in °C A0 = - 5.0982373-100 A5 = 2.4775634.10-1 AI = 7.2704899-10° A6 - - 8.6590250.10 2 A2 - - 3.0337268.100 A7 = 2.0153393.102 (D.1.2) A3 - 1.2567591.100' A8 =- 2.6934527-10-3 A4 = - 5.6086594.10-1 Ag = 1.5531799.10k4 Equation (D.1.1) is valid for 0 t < 374.15C. Accuracy of this equation is less than 0.02% for t 300°C and less than 0.01% for t > 300C, see Lekic 178). D.2 Saturation Tesoerature Steam saturation temperature Ts = f(P) is determined using Vesper's [76) correlation A.[in (1.01972 p))i C (D.2.1) i=0
177 where P - pressure in Bar 1 -~~~~~~~~4 A0 = 9.9092712,10 A6 = - 3.7393484*10 A1 2.7854242.101 A 3 - 1.7417752-10 5 1 7~~0-A - 2.2071712i0 (2 A2 - 2.7537565-10 ~~~~~~8 (D.2.2) 2 A3 = 2.1077805-10 1 = 1.5343731*10 6 A4 = 2.1296820.10 2 40 - - 4.2685685*10 7 A5 = 1.3283773.10 1 All = - 4.2924603-10 8 agree-Equation (D.2.1) is valid for 0.006108 < P < 221.2 Bar, and 0 ment with the table values is less than 0.06 C. D.3 Density and Specific Volume of Water Density of water is determined from p kg/m3 (D.3.1) 3 where v - specific volume of water, m /kg. equa-A specific volume of water is determined using the Tratz [77] tion 2 v S G H...K.T + (n - T) [L + (n - T) M N(q + ra + a2) (D.3.2) z T11 where: 2 D.3 w U - [kU2 + L(a _ MT)J1/ (D.3.3)
178 U - f - gT - hT (D.3.4) T t + 273.15 c' Y 647.3 (D. 3. 5) P P Ppc 221.286 (D.3.6) P c' Tc - critical pressure and temperature t - temperature in °C P - pressure in Bar. A = 4.17-10 1 M - 7.67662.110-1 g = 3.122199.108 H - 1.139706-104 N = 1.052358.10 11 h = 1.999850*105 K - 9.949927'10 5 f - 3.7.108 k - 1.72-100 n 1 L = 7.24165.105 = 6.537154*10 r 1.310268.101 Q - 1.362926.1016 q 6.25.101 z m 1.5108-10 5 1.500705-10 (D.3.7) Comparison of the values obtained using the Tratz equation with The 1967 IFC Formulation shows the differences less than 0.05% for t < 300 0 C and P < 400 Bar. D.4 The Density and the Specific Volume of the Saturated Steam The specific volume of the saturated steam is determined from the product P vII as proposed by Vesper [76) according to: i=8 3 s =I A.X, Bar m (D.4.1) si=O I ~kg
179 where t X a tS 100 0C t - saturation temperature in S AO = 1.2602044.100 A - 3.3488548-10 2 5 Al - 4.5905064.10-1 A6 - - 1.5087828.10 2 A7 3.1565330-10 3 (D.4.2) A2 - - 7.0419435*10 3 A 3 - 1.1476149-10 4 A8 - - 2.4890060104 I A4 - - 3.8647589-10 2 and (P v") m3/kg (D.4.3) PS and the density 1 kg/m3 (D.4.4) t P i D.5 The Density of and the Specific Volume of Saturated Water I..) The specific volume of saturated wate4was determined from Vesper [76] equation: i=9 Vsmas AjX. (D.5.1) i-b1 where xi
180 AO = 1.0001189-10 3 A5 - 5.2428875*10-4 Al = 5.3740533.10 6 A6 - - 2.8652450*10F4 A2 = - 2.67941471*10 5 A7 -9.0664926.10F5 (D.5.2) A3 - 3.0237718-10 4 A8 -- 1.5437041.10F5 A4 - - 5.5126569-10 4 Ag -l.0968357.10-6 D.6 The Enthalpy of Saturated Water i) The enthalpy of the saturated water is calculated using Vesper's [76] approximation of table values. i'9 h- 0 A.Xi kJ/kg (D.6.1) i= . where t 100 t - saturation temperature in oC S AO - 4.7554093.10 2 A ' 1.5159421.102 5 Al = 4.2490349.102 A6 - - 7.8226638.101 A2 e - 4.4519838.101 A7 - 2.3865162-101 (D.6.2) A3 = 1.2080923-102 A8 = - 3.9629482.100 A4 = - 1.7579053-10 Ag = 2.7663765.10-1
181 D.7 Saturated Steam Enthalpy . Vesper's [763 approximation of steam tables values will be used again i=6 h"= I A1 X kJ/kg (D.7.1) i=0 where t X- 100 t - saturation temperature in 0C 5 A* - 2.5006256ol03 A4 = 2.7974559-101 A1 - 1.8150666.102 A5 = - 9.4307509-10° A2 - 1.5749571,101 A6 - 1.0285952.100 A3 - - 4.3077814.10 D.8 Heat Capacity of Water Z9 The heat capacity of water is determined by differentiat-Ia ing Juza's expression for water enthalpy, Lekid [783: A + By + C 2 0.24927 + 0.021y) , -hl (D.8.1) where A = 4.1982594 - 0.1304363'X + 0.19097088-X2
- 0.07403256-X 3 + 0.02259985-X 4 (D.8.2)
182 B - 0.0307256 + 0.033444712-X - 0.024208803-X 2
+ 0.004837209.X 3 - Q.001100455 X4 (D.8.3) 5 -6 15 (D. 8. 4)
C- 0.032 + 0.003264-X +'0.26912-10 .x x .- t (D. 8.5) 10 y (D. 8. 6) t - temperature in 'C P - pressure in Dar D.9 Heat Capacity of Steam The heat capacity of steam is determined from the equation (see Lek1E [78)): X iA. r 1 - Ii t107724A + 5.1324.E(C - Sc} 1.82 T F9T L~~~ Dci - 7013 + 15 52(d'T - T T~~~~~~~
)DcDs 352-C 33 a 3
n rr 1Q where A1 -1.1698648.103 A3 - 7.3765806.101 (D. 9.2) A2 -' 8.0553613-10° A4 - 1.3026684 10I X T t +4 273.15 (D. 9.3) Tc 647.3
183 P P (D.9.4) a .P 221.287 C A - 4.7331.10 3 C - 1.55108 d - 1.26591 B - 2.9394510i3 (D.9. 5) C - 4.35507-106 D - 6.70126-10 E - 3.17362-10-5 I - 2.21287034.104 i D.10 Thermal Conductivity of Water water was deterxcined' by the II The thermal conductivity of tI I 1967 IFC Formulation [743. I BX + (P - p ) 2 C X k = AX IL + (P - P ) - im0 ino 3 (D.10. 1) insK where (D. 10. 2) Xt +273.15 273.15 t - temperature in 'C C0 1.6560-106 Bo0 - 9.4730-10 4 AO - - 0.92247 C = - 3.8929'10 6 A1 - 2.8395 B- M 2.5186-103 C 2.9323-10 6 B2 - 2.0012-10 3 2 A2 - 1.8007 B 5.1536104 C3 = - 7.1693-10 A3 = 0.52577 34 - -3 A4 0.07344
184 D.ll Thermal Conductivity of Steam The thermal conductivity of steam was determined from [74] 5t + 1.o4,1o07.t2 _ 4.511011.t3 k - 0.0176 + 5.87-10 S j (D.11.1) ms°K where t - temperature in 'C. D.12 Viscositv of Steam The viscosity of steam is determined using [74] Ps= 80.4 + 0.407 t - ps(1858. - 5.9*t)/100, 107. ES2 m (D.12.1) where t - temperature in 'C 3 p - density of steam in kg/mr. D.13 Viscosity of Air The viscosity of air was determined from 7 NS PA - 173.6 + 0.454't , 10 m (D.13.1) m where t - temperature in 'C. D.14 Viscosity of Air/Steam Mixture The viscosity of an air/steam mixture was determined following the procedure proposed by Bird et al [81]: i,
185 i-n x iv 1mix I _j_3 - (D.14.1)
£ Xjfij where xi, xj - mole fractions of the mixture components j~~~~~~~~~~~~~~~~~~~~
pi - viscosities of the mixture components
+ 1 (1 + -lI + p i ](D.14.2)a xi, M. - molecular weight of the mixture components. D D.15 Thermal Conductivity of Air The thermal conductivity of air was determined using the procedure outlined in Holland et al (79). It is assumed that air consists of 78% of nitrogen N 2 21% of oxygen and 1% of Argon, thus i=3 ~~~~~
A i=3 /35 0K
' (D.35.1) where rip HiM volume functions and molecular weights of N2 02 and Are respectively.
The thermal conductivity of the components was determined as
- k. =p 0 1 (1.32*CC + 508 - 104.6/X1 ) (D.15.2)
.c r
186 k2 e V02(1. 3 2-CV2 + 444.8 - 91.59/X2 ) (ID.35. 3) k3 2.5-o 3 CV3 (ID.15.4) 1 ' 28.02 , 2 = 32 , M3 ' kmol (I).15.5) The visco! sities o01 were determined 11 Voi c 8.175(4.58.Xi - 1.67) 5 1 8 .1 lo A/ (1).15.6) J 1873-10-5 °K1/6/[(kg/kmol) 11 (N/m2) ] 5
= 1.393.10 (I).15.7)
= 1.272-10O5 X = t + 273.15 t + 273.15 t + 273.15 1 126.1 2 154.4 3 151.2 (D.15.8)
The specific heat at constant volume Cvi was determined from 8 3 1 5 )/Mi , k (D.15.9) C - The specific heat at the constant pressure CPi is C = 27214 + 4.18(t + 273.15) , J/kmol°K CP2 = 34625 + 1.082(t + 273.15) - 78586.104 /(t + 273.15)2 Cp3 20808 , J/kmol°K (D.15.10)
187 D.16 The Thermal Conductivity of the Air/Steam Mixture The thermal conductivities of gas mixtures at low density may be estimated by a method analogous to that previously given for viscosity (see equations (D.13.1) and (D.13.2)): k ' nix x i-n I~in xik msQK(D1.1
.1 xiq The xi are mole fractions and the k are the thermal conductivities i~~~~~~~~
of the pure components. The coefficients qj are identical with those that appeared in the viscosity equation:
+
i- Ml F 1 2 (D.16.2) D.17 The DiffusivitV of Water Vapour Through Air The diffusivity of water vapour through air was determined from [79] (1 - Y )(TOK)l-8 ll-0 9 29 D w m2/s n (D.17.1)
+ + 3:Z L -6.7577~~~~~ 0.7794 +0.7503 where Y 0.78 , Y2 0.21 , = 0.01 y- volume fraction of water vapour in air.
w~
188 D.18 The Surface Tension of Water in Air The least square fit through the data of Rohensow and Hartnett [82] gave the following relations for surface tension a as a function of temperature in 'C a - 75.448 - 0.13071T - 3.5056-10 T (D.18.1) or a = 75.671 - 0.16131*T + 4.3348.10 4T - 5.2269*106 T3 (D.18.2) Since the differences of water surface tension in air and steam are negligible (see Vargaftik [83]) the water surface tension in air/ steam mixture can be calculated using eAitt~eh of qequations (D.18).
. ..- ~~~~~~~~~~~~~~~~i A,~
189 APPENDIX E THE COMPUTER PROGRAM FOR NO INTERNAL RESISTANCE SOLUTION OF THE SINGLE DROPLET RESPONSE 1'. I~~~~~~~ :;Xk~~~~ 1..~A
190 "i*'*IF A *I X* A 4~~~ A i A , K p~~~~~,A Pi,
- C C.Lw A 0*
A A A
- A
- AI A * *A A*
- A AA Or iA A *A A * *A . *
- At * * * *
- C C
- I-S Pt"t 0 6 k A M S L'L V L Tti L PPi dJiL E O(F SlIUlA7AN1E 0Lj 3 E.A x 35 C
- Ii~A~ F E t' F Iti; N. 1 I/S AP- I 1 IiE TO A 0 4LIP L E T .l J C &
C *A C i:iS IAr. I LUCITY t"YI IN1 E k LkESJIS I .C E I10 A H~E47 TRh:SFt.§ C * ~~~~Ifv A S9PLRE C
- ST--L"~~~~~II TS C
C C A *tA#* * **AA*A**** A* A * *A * * * *
- A A A* At**AA**A*I* it***** ,
C C A= THE &~PqAL #)IFF (SI VI TY #-I?/ SEC C AP iM F =IrE IMAL C(;ta)I.IC TlIV ITY OF MIA ILtJRE -- F I f-g j/..' OK s E.c C A ~!ir-P=I li~ R- AL CLi;I)JC II V IT' LIF SlEAr- -- P ILM J/m' u0 SEC C A Mi i V I E l -atL C('~"NLIC IV I TY uIF AIF -- FILM! .1/,1 (Up SEC C Lt - rltAT CAPA.CITY (IF A 1j,4c;o C CPt-.F - tlE T CAP'ACIIy UF --I YT UP E AlI F IL "' TL 'P RA I E J/t 0(lb( C CPP - HEAl CAt'ACIIY tuF S rFit m, KJKr UK C CPV - i4E.l CAP~CI1y LIF flh'1NL KJ/IKG OK C I)I&mF~oI.FF uSilvlIy OF lMixruki. -- F ILM kl?/SE L C Ol~ JE:%.:i-FRAl0-%E IN*Ct'.EASj. (C C F=MASS NA'NSF-E4 COEFIF]Clf.~:l KIMOL/.-.? SFC C G~mULA'~ ~'ASS VELL'l TY r-I soL / SEC C H=tiL AI I - ANSFE K C I EF FI L E N 1 J/r-2 Or, 5S.C C C KE - JnrF Igilt L-H OF I) A P. IF R C jJ - THE. O.,u~ F SIE AM/ A~I t4S T E AI-') R.AI I ts C FE- PFCLET Nti',Ei-? C -PP iP01IAL P1¶LbSUmE OF Tolf. SIF.AM 6ARA C PR-4 PriANiOIL t d C PT - TOTl4L P'cLSSL.F.. COF 1IHE MIXTIRiE f C P v - P I'I i) I)c I OF PrI P.v PY ( 1 (-lP) 6A C ~L t t H~a LI S .L'~ 3Ei OF f21l X iUItE -- t-U L K~ C t!EN! - Vt Y ?1j S f4WIe')'F.k C PMi- ':1 XIUKE GAS CUINSIANT- b0LKJ/G0 C H Mh - (It-N5IjY LIP f.IX1IRL - tULI' KG/H~3 C RUP - Dht-.jbJ7Y CIP 'WAP000; - FAILK KG/m3 C ROPtF=C)L-SlITY OF Slttr -- FJLP' KG/"%3 C pUv - o)LE.SlTY OIF KAIER K~G/rM3 C kimo=PrATIO O)F IHEFR."AL I)IFFU)SIVITY AND MOLE.CULAR OIMIJSIC"'J crr.rr. SC!4 SCHL1IUJT "U~I"&E C C 1F=TE'51110ATLIFE OF FILM O)C C C C I V - -, lIt.-- L)rt(II C. LL lI ~-P E P1AILI kEC C 11 - SIliw'.47101?- TP'.'EL-ATtp-;L COFRESP 'l~ ING 11) PPP 0C C usm .- NIjtSSFLT N-UI'rJLNL C vISf-'L - vJSCtjS~lY (.-I 'lIxl'i'i - 'L;K 10**7*raS/p2 C vjSPA - vISCOSIl Y OF Sltt,:i - huLK\ 1flA7*NS/m,3 C 'ISVti - VISCUSI I Y OF AIN - F;1.1L P, IflnA7
- i'S / v4 C v IS FvI SC t;S!I1IY (IF t'.lYTIibKL -- F IL.P l).**7 *% S C "
C vISf~vJSCL'bIlY C*F All' -- FI-]L 10.**7 *N- SFC/f-? C vjSfPf=,4IC0SIIY OF STEAM¶ -- FILM 10O.**7 A U vii -Ti~ lf.A VE-LOC ITY m/SLc C r4im-F~r'lILL CiJLlt-~ - .IGr~l (iUF !-;I X URE -- F ILN G K O C. C * * *A A A *A * * %A *A *A A A* Ak* A***A
- A
- A AA *A A A A A * * *
- AA ***A**A **
- A* A ' * * *k0 C *A* * ** ***** %*A I
- A A A * ~ t * *A ** AA**AAA*** AA **# * *A A t** %t it'
191 C 2 VPrE XP=i . 3 vknE Af'= I od". C --- - ------------- REAiD I,,IL,KP,PT t_5 FIt),AT
- (MJ 5, F 1 .9) 0Co ;) j= 1 P.
7 u'Ij qo J)l #FF 8 2 FCIR' A!r ( '~ F I 11 5 ) 9 If (vL~t)1PG; .0*. ) IG0t 7 43 10 READ) 2," (1),R(J). Vn'FXPTI, 10 11 GU it) 11) 12 113 ;eRE,' 2,.%(I),TZVPLXP,11,TO 13 LZS CUP:T I NUIE rv=r I 15 1 Hf. I l!=U . 16 17 P 1. I? = 0. . H AAPIP0 HAPS1IH=1 18 19 20 hSTLF=O. 21 22 H1APCHOJ=O. 23 A PC H r= O. 24 P I = ; . 1U 1 Q 25 viJ=vkrx)u Vt)=V'~Ft0. ! 26 C C. D)EN4SlTrY uF Ml~lA' UHE--'4ULh I -, C 27 It (Vwf X P. GI.) GOl IO 4"
?8 PPp=R (J ) *rT 29 CALL TLrItJ(l IP, IZ)
I 30 GLi Tn UI7 31 32 CALL I:Z:SS(IZIPPF) 33 Kt1J)=.4Pj'/P T 3L 35 9 mti= ,3 1:1, 7 / (4 (J)*l8. 2 + ( ISJ))2 i3 3b Nti -4ts= I~nl) o.ST . T (o4 : (P Z +Z7 3 . I ) C C vISCUSITY CrF -lTIIIRE--..jULK C 37 CALL SEK~vP(Tr.PV) .- 3* KLIPIP 1v I 39 a0 41 v= 1 v 1 2i2d.)3¶53&/SWt(
,FI / .1 +T 1.g+I).b222)*
S, I, 7 So I V S (1I.+STJ,-T(V1SP/VISV)*J .125q5)**2. 3z iI 2 I- k II 3.t .U. 1-<J5stJYl9((t
) I)^())i11) ,h())*Iv H(~fl Ii. 44 CALL v v(InC$(F111,VVOO) 4', l)= t'I'*§JI /*3 vh* U.**L
_ li6 t"Ea=1*'.(I )*2*1t,**-o) L7 C c [bIJ'4IaAL VELUCIIY VF MIX1LiE C lF (VRAP.G;l .0.) ttl 70 17 49 50 3 SlEAG^ IIA/lS0.8*7)/(f)(I)/1010.
. )i*1.5/RO}V
- * , *~:6t,Ail.si(~
192 53 kE~~~z.=-v:^ ~~r*u.t) Ivkll~ml.v~ 53 SS1 u V1 t (!p/ ( 2) c? . /; . ) ; b5t U h o)=~ n v41I v% Sb "t;^r1nr*V b I /w<T 57 VW1=VhtI1 56 GI) TO O 59 5 Vk1=9.'43*(1.-EXP(-(r)(1)/1,77)**1.1U7)) OV GU TO 1'! 61 17 CIJ:s.1 I lt) o2 voIT=VREXP b3 18 cUrTI ,uL bL r4 F cli=- T*r F( I )* I U0 0./vIS t oS o C I)&T I Ita bh P k I~% 1T 1I) I.) t (J ,PI ,TI, ~D VI NF b67 11 F 1? ri A 7 2 ,9 5 t V I ) ,F I U . 'i ! 1, 5 si k t J 1 ,F 1 0).S 2 X, 3 HP r =,F I U . ,2 x 6~~~~~~~~~~~~~~
~ 01 V I Pi I')
b9 UT=(IZ-I1)/I 70 iOT 1 = uT 71 tiL) TI0, 9 9 72 " J I+t IO C C: * *k: 73 K=0K C *it*s****t*~*A*Ai*'*t* A^9*****s*********j*t C 7 f vfIJULIl=1 V 75 IF (r.C;l . (J-l1 )) GO I0 12 76 G.; TC' IS 77 12. C .. TII r,Li ' 76 IF .G I.('SJ-7)) GO Lu 14 860 TOJGO13 61 14 tno I=6so 1 9 I .*(-3.1) b2 I Vti=,)l
- Ti 84 13 CLIN I I *JE I AS 1 V= I Vt l;T'I 8B I)
TVpo1V4 V')T/2. 87 ]F(1v.1;1.O.9999#lZ) G;O Li 34
*6 ~~Tf-(T/t1 V-rT/2. 3/2.
b'9 1F I7+ T 0T 2 90 CALL PZ AS(7F PLF) 91 CALL P/b;(1f MP?.Fm) 92 r(FfLFZf /PT 93 IF fMl=ViZF-'P1 94 C ALL S. rJ'V(1F ,PV) 95 CALL SEhPV TFMPVir ) 9tn KpF =e I/;V q7 KUPF6;=Vl/PVv C C ISCCUSiTr (OF !lIX1LlhL--FFILM1 C 98 C4LL VISASM(lF,lFvrCUPFVISM-F) ti9 CALL vI;ASMl(IFrlrk-Ft'loilviZ)rVISMFr-I) C C THERMAL COiO)UCT.IVI7Y UF MIXTUiRE -- FILM C 100 CALL At6AS*"( rFIF P.PXN?,CPUeCPAIAIMBMF-) 101 CAL.L. (7F',Fr,
- CPI1ŽMCPtd-., AtM3':F m .
vC -- C tSIpFL(Is;V1r8-Uf r~[YTLIRE -- hILM
193 102 ~~~~~CALL O IF A-SW (Fi F Fm ) 1 03 C AL L O ! I ruIF-C C GL.%!sII~ Y'F MI yIiJ04E....F..I L; 107 KuF,=IF I"A . IFLt2 73 I5) C C m I.T C A P1A C I IJF I x I WE -- F IL' C I 04 CALL CPr'Ah(PlF,:1Fm,CPPMi) 110 CPA=0.7.*CP:oflU.2IACI-'I2eO.01*CPAR C C C C SCH-a1UT NLI"ItER C C SCtV =, 1 I S*g:k4v. PiI 1I C PLCLC r ;frLl~HE
- 1 21 FPf i, 1E C
122 IF (RE;~-4-45. I, 7g,
- 1 23 7
- 127 1.1) *r~~~~~~~~~~~~~~~~Q CNM* Aq 132 hUS =? I O el 7 (.t2*g (S PFIN k
- I(1/3.)
1 31) + oS.!3.) C rnEAr ioA.,srr..w CIIEFFICIE!.T 1 32 9 r' G ;I' I-/. 13 3 (I (
- A! '4M-F .*tJSr,;/L()
-. g.'aLC Ss I k 4:.SF f'k COEFI- 1CIr f1
194 1 35 137 C C m ASs TIi;t.SF~v iFA1 C I 38 CALL PZA5(1V,1PL") la~ C ALL P ZAS ( IV¶,' Z V' I ~1 t.-~:- '4* LL(G ((1, -PZv,-/I- ) / (1 -FPP/P7 ) ) C C ti F AT OF VAP IZA IION C ILJ 2 CALL SFr.1(TF,hSEX) i3l C ALL t-k I" I1(1I1- f"~PttI' 14 5 CALL S1AI(1FPI,,,ISrI'.) 1 Lit) ~ CALL P';, J:.J ICT~,1NM C SENSISiLE HEAT T'RAtSFEq C 151 ACC'~ -?.A* A1 ~2".CfV I1b2 AC9l CA.-EXfP(-ACLA)) 153 C C (.AIENT i1EIT lhtNSSFER C 1 55 1LLM= i V A Pn IPJ A! I6 (J2U. C C fUlAL tiLAI INA:jSFfwN C 1 57 rGr.:IL0uOS' C C APPARENT HEAT T!kANSFE:R COJEFFICIENT 1 5b FlAPP=(i T/I( TZl-V4 0 1/2. 1 59 PIAPPa=i T.A~1~/ (1I Z-TI V'4+L1 lit C C rU1l NtJm8tEk C laO CALL AMOiA P T,IV ,Vj.Atj C C 7 IMCi :NFCCSSAlRY TO) TMPL0ATUlFF. IN~CREASE OT C 1b62 -CALL VV0LI)ES(tT, TV,,VV'li)) 1b3 ltc.iv= I. I JVLI') I bU CALL COV'1"? ( P IIIVCPV) lo b C,?=( tItt* 1 ?i 2*CPP*I 00./Il.-EXP (-Tr-A* Ih. U2*CPPA10IfOA(./H) 1 1o7 c C 1C,?C 169 1F-(AiPii1I
- I I 0. ) ; II 3 17 1 1'II 1 a. CI *!. I .of AL(JI A rd t; 3
C I 'f W:4 AL IiI JLIZAT 1U1.: 195 C 172 TmU 1=(1v-Il) / CtZ-11) C (. C KNAT I j i)F Tr~km AL DI FFUS IV ITY AND ) IF FUJSlIO. CLJEFFIC IENT C 173 A=A %A-'A F/ k1(1-rI'F *CPMF) 17 a C C APPAReEtT FIkA I TFANSFE.R C(.EFFICIENT--CHECK UP C 175 hAPCtilO'1ACPIV*(Tv-TvU)LD)-Il,00./A-?EA/(TZ..hv4L)r/2.)/THFTA C C AVLRAGE APPARENT I'EAJ TRANSE-R~ COEFFICIENT C
- 17eb HAPSTIP Af7PSF NA PP 177 htA PA vS='1 .51PSIP/K~
1 77
- 1 19 18')
11 HAPCMTtIAPCdi[4(HAPCIII)+HAPCH),THE TA/?-. 18? tIAPC rh=HAPCtHT/IHE Tu C C AVERAG3E SFN-S~hLE HEAT IRA.,SFFR. COEFFICIENT C hASrEP=,i5 TEP/K 1F5
- 113o 188 C
C Ei aHA;.jLEmL~t:I F ACTO)R I .,i
,-a, C -4I
,.;,4t EFS TF.P=-,A*AvS/HAVSTP 19 7 191 LF:fI,,, [="PA /PtV C
C h ALL S1hAM TO H 1M1IXUkE RATIO C -- A-122 h-AS=4jUc14I 0 *2 riShlt¶'=tPiAPAV I/ /hA.S
'19~41 19 d C
C I A V= I v
,.T1 C ;.195 P t'Ij 2br IV IA1V , 1:(ITm:l), THU T, l4EI U 19o 202 IL 1.i
- 7, 1'x b-' itt I= ,E: I AJ*1o)-I r F It!=, F.1 4. 7)
It 197 1 91i PKl1.?, 1 , 1',-APAVSi' ,t 11,7,rIAxV 't-APAV I:',.1i
*- 200 ?6 F ii.mA I (2X , I '.A ,t la . 7, 1 x , I F=I , FJI 7* ,Xt *4 EtN1 , 7, * , IxHA P
',F hj,7,Ix, EI APAS= F 1:. 7, 1 x ,IiAP:' , = I , 1). 7 IAC:'*Ij1(*L
..- 202 2 fI Fg L1I-
, I ( X, r= L AV TJ#
II EF S? T F~.='L r FP R -F
196 I I tE SLiPv I!LI.NLI b 1L I(; kAr TtI~ ' , / ) 207 Sb Ct. II'I)t-. 2019 PIenl.( a l /, I1jP.NntP, vI SP, v IS v, VI SM i 2ts 227 i (-( AT (,'x , - PP F I *.7, 1 x ,3. rI Z=, E 1 .7* 7 I X 1"IJP=, E Ili . 7, 1x ,5tV I S =, 1 1I :, . 7 w v zSv=, I". 7, I x, LH v I S . 7 El L¢=,tI C 21 0 PI%I',I2 ,k0 vk I L t s I ,F-F HS O 2 F 1JP~- *>2 f 9 H10 v E I 2ta"t£u.7. I x II m v K I 1 14. 7 I 1 H 1tE.mr; I ,j 7 I LX3 W 1= 1 U . 7 , I t,3 E F = I 4 . 7 I x, b H i S HHm =, E I 1. 7/ ) C 2 12 80 Cf) Ai I 4ti E 215 S3I'j 2 * -N LI C 2 15 S LI kR Ul I J Jt Z AISi I P.fS) C 2 1b x= T I O A.>5t> I3C!LI x,4 2 7 7 5 ,3u 1 4.) *A5 J13b 0 9n 14 ) 1 2~57;A. Ix)- 21 P S= E ;P P'S)I. O1S7 c 2 19 RE. T1)~ 220 NC) C 22 1 S LI mH I 1-IE IE P1 (I P 1'T I C C 2.22 X=tLtG11.n147?i'PPP) AZ3 ~~~~~1Z- ( ( ( t ( ( 1 ) O A 0 7 I17 1
?- .7d4tS /1 )0 0 )0. 74 1 Iku.od>e/ nsfiD. . I +5 .
213t3 /S I /l tOi!1) )*> 2 .2 7 1 3 tt 1/0 L x 2. )12 x-o 2I . 741 t2/ I 1nn7 ) t-3.7: 0
. t2 4/ . 7 5 3 l 7 )1
- t .27/ 8 tilU 72t24~iN c!. qt2. )1 *0 9 2 7 1 2t * /
2 24 Ozt Iijk p 22 5 e C 22b SUb ? i T i'IE vvU rS5( P I vvVOD ) C st***t*t*A**A**********************t***1******A****,* *t*,S,,********' *9 C 227 A=( T"4273. 15)/o(J7.3 22 6 S=p1 /22 1.2 ' 229 U=1 ()I.A (37ot!o0(*.-5l?21)q9.*A*A-1QQq.85/AI*tb) 23l v; J)4Sw d(T(I *72* *J+13*3 L. *lI .*tIV1S(5-I.50t070$5*A))) 92f, *0*.??94'1 17 2 31 e1 .wSIŽ3b* (e2.+S* ( 13.1O2b04S) ) /1 n.* IAt / (1.S10$/ I0II00, O+** 1 l ) 232 C=tu.i)5371b 4-A)**2 233 C = C ( 7. I I 5 / I 1)1l t tI u + 11. 7n7 ht2 1C ) 2S3 V' )t),. .4 1 7/ +C- 1 .39 71O-9.9 LIi27*A)/1 0 00. II-( 2 35 .H 1Et N ,, 23b EN-0} C C 237 S;UWjl:.ij1J jE StE PV(TZ, PV) C C 25ex Id It
197 21iou . ) A ~ .54udlbIbu/ 10 . )A K+ 1I.2h02Gu L 2 w0 6~E I. tit 2aI (, 2142 SU Ie~1mt C PV(1? ( PT , T vC v) 2'43 X= IV~/ I 0OU. 2135 Z *5 C~v=Cs-v(0 .U32+Z* (O .flf32hu 0 .klqI?* ZQ/ 1 00000,)) CPv=ZA y/ j 0(1. +CPV 25 0 CP I M. 25 1 252 E 7 tj;4 25 3 c*t. 1 C S L ~%Li~u 1,jF. CPP AP(PT, IF,C FP) C 2 55
- 25t, 25 7 c I-= Ii"* I
~h52 ** oo.9.9.6rh HEd)?~,6e,7 5 2b2
~SuC tl ES I .TA 2b3 C
26b 2b7 E.N.D C C AA A Al*LA
- 2b8 C RImI (I -IA Su: FI ')u(l'-
C 2b9 P. 7' )37o 5 /lIv. A 23~S..)x-7142$
- 270 271 IF C -6d,4 272 1 273 2 tE T OK "
27 4 C
198 275 T .E tVA (P, I.I;IIIA I, YL AIt4) C A AAAAA*AIA AA 'A*'A AItAAAA.AA AAA C 2/b xA=I.1./273.15 2 77 CALL PZI.S(I #PS) 27 VILAfim=( ( t 0. r i2 - .U t,3*k * - .3%,9) 4o ,6 (p-pS)/lfIuflIon(
- 26c? L NO C
C 2$3 St$N0U'J I I E VI ~A S~'(rIF, WF , ~PUF, VISMF) P h 5~ v IS v ~:1 7 5 . v -r. 4 5 j IF 29o E -N C C 291 SU~r'NL.1I I '%.- A%;3AS~' (IF , riF ,CP:.j2 CP)2, CPAARP, A-'!L;- ) 2911 30 ? CPc1`2272P.J.41$. 1~7A,( iF+i?73. 1b) 3 03 CPu?=!,Lo25.4 1.01402A E1F+273, 15)-7b8t8o. *I000O./(IF1273,15)*%2 305 Cv:-4 CP v2 -633' s 3 07 C VC44 31uiti-3I bleAp~ 306 V ISN2=6 1 75 &( u. be IkA2.'I.t 7 (b /8 .)IS I\./1. **II 3u9 V IS2.'I V, A ( '. 5'i*I kelE I f7)tA a5I/6. I/S 1 0211 .* A1 1 3 10 vi I 7b5(A . 5it TkA;;- I t, 7 ** 5 /9/S I A/ I 0. AA I 3 13 A'1_Ak:.5b V1SAk*CVAq 3 1 ( (I . 7~ a (1
=~0,1 A* )N)
A1 I) + .21A %k-2*A
*lb3 ~~~~~F 1I,2 t_0.:! 3b o 5Lr1 4(104
. C,'.2 1 ( . +Sflr (AA P*F /A-16V'~F I IJ 25 93 2
.317 A!- W'IF =-F*4vt-PF/I WF4LI.-,?F-) *F-II2L. +(I .- OF )*AM vT/(`*F 1?I . *--F 3 1m A 3 19E C
C 320 -q - I., - .'I; I I I 'i ;- 0 I F' A cZ- ( 7 F . -; F . Ii T :! ' - I
199 321 OIFM4F=(1 -RF ((F+273.15) **9/( (.-RF)*0.7i/0.7577+ lRF *u.21/0.77v4tf l*HF)*ul/0 7503) 322 HE I t2.8 323 Ed!) C CENI R Y 1.. .. ..
'4
/~~~~~~~~~~~
- - i
200 APPENDIX F THE COMPUTER PROGRAM FOR INTERNAL RESISTANCE AND MIXING MODEL OF THE SINGLE DROPLET RESPONSE
L.T FI v
- tt **~?,~ 201 CJI C A**t*4***A A****A A**A*AA tAA*A**t~t********A*
*C AIHIis 4-OtuISA~*MM S LI VL.S THE~ ;PUHLE", (IF SI "O LTI A14EMJS kr-OmENTUM, ,IFE'1 CI *111ASS TIIA'.S FL k oilkiI1.; fj-i*~hN~j IICpi OF ST F.A "iF UM SAIJIPAT F AIn-STEAm..
*fMIxrT.I-E fstmA Oka.i'IPLE T v.J Tt IkESlS IACI:C TO. tiEAT FLLtJN IN a DO~LPLE 1 C
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C DE~aS IT Y (IF mI XTULPhE-H..iULK C 50 IF(Vk.dXP.GI.0.) GO TO. abt 51 PPP~g(JJ *PT 52 CALL It.,-Pi(PbWI27) 53 514 (;cu-4TIC- '17 55 CALL PZAS(1L#PPI') 5b 57 Li 7 C ON'JT fi.e 58 59 C V IS C SI IY yOF I X7 UPE-HUL K C 60 CALL sEK~Pv(1L,PV) blI b2 vSIsID~J+o,1* (17*TZHJP* ( l15i. -5.9 A IZ)/I 1) 0. b)3 VJ 5V l346.LJ9JI 65 6t ._.. C C MASS OF A &PNILCT C 67 CALL vVt~hE5 (PI T I,VVOUJ) bb b9 70 71 p 0 I .jrT I I ( ,k iJ ),P I TI ,V ,VI NF 72 C C TERMIN~AL vELujC1ty OF MIXIURE C 73 IF(VrFXP.GT.0-) GO 10 17 .4
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A. 76 77 78 RE-ib=VRTAL)(1)ANqi'*iC1uou./v1SM6 79 80 £1 81 62 VIRj=Vi.T 1 83 8*Li "5 86 GO T11 c, Go If] lid .i 87 88 IqCD 18 CoNTI I4,f. HE mk~-=V ITI (lt rCI N IT N'j1J
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92 GE, It) lo 204 93 15 COtj I IatiE 97 It, ~CI IN11fLJ 0 1 ~~Kj=~3(I) 902 C C 103 P0 10k K= I J C C 1 04 IF(K.t;1.(hJ-I1)) GU TO 12 105 (;U rTo 1 3 ton ~ 12 C(IilI I(,. U 107 1 i; .1(-7) Kv.( GO I10 1l 109 GO if, 13 11 0 114 iHj=J)t~i~+9,/10.*R(K-31) 112 fJT=Tvll*-1 213 13 cfjr%II!mIc 115 119 ~~CALL PLAS(IF ;,jir) 121 CALLPZAS/I'rj~ 122 PFl'l=PLFM1PT I e3 CALL Sf.k'v(7~,Pv) 12a I 'CALL SF-id'V(7Fi,P~vfl) 2 ~~RuPFrzP I/IPVm C C VJSCVSITY OIF -mXvL-FL C 1 27 CALL VI3-ASm(1F,P4,N0LPF,V1s"¶F] 12b CALL vIStS,4( iFf0,Rf-t,frVPFI,VISMvFM) C C 7HFfr!MAL CONDDICTIVI TY OF~ MIXTU'RE -- FILM C 129 CALL AS164S"(IF,l<F,CFIJP,LP02,CPAK, A1~F 1 30 CALL A~A'i~IM C C DIFFUS1lIY-Lol MIXTUktc -- FILM C 131 CALL VFSlu r ~~i~ 1 32 CALL ,) JF ASMl1M.,kVM, VIF:4F,-) C C Dkr4SIli (IF P)XIUkE....Fl;im C 13 Hi A T C 1-tPA h1 I'1/4i &IIyII -- F1LI
- c 205 13 7 CALL L.PPAP(1'?F,11~,LPP) 33e CALL C:P;A~e(PZF:,, 1F',CPPm) 1 39 1a 0
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(. 1 L4 C C PRAIJDTL r40Jm1ER C 143 l44 C C HEY'jOLDS NUJ~bER C 1U 114 5
- 114b b~m~~m~F-1* IA (14(.-RF-M) *?8i. I Ammth=,R (.)J1 1 - ' 2+ c(I ()*I~28 W .%9 150 ,.:VRT* RJr/ 0i1M.$
I 151 152 IF MI~TTGrIU. ) Gtl TO bOO 153 15U ,U05 FEN~vH*0)(I) ArC.,r'F *I 1 U. 6o~ 'I SA 155
- 157 T(0LI)=rF TLI+ IMLIA 158 C
I .- C .( I C 159 tiC-,,v 1S -F/i.-ty1F"3F'.F/I('. ** 7 lbO 3c Nil= VI- I m R O/'WF M/1DIF-MF1M/ IfI. **7 X-, C PECLEI rj,'j~li,E C V 162
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C tNUSSELT rjtuMIJER C I b3 IF (REN-150. )7,7,h1 7 UjS.4=2,+It.t* (41!J**U.5)*(PRN**(1.3) lbS5 C C SkiERwviU0) 4U;MHER c 1 bb 167 1 68 (U 10 169 b jSt =,.+0 4 .?7A(RF-r1**U b?)*PiN**(.3
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17 5I% mmt¶3=K CJ) AI i. C?+(I .- P (J))*21h.g 206 1 7t, '. 4'* IIiA.~r~ (1*I.~-P )A? A. Qt C c -MASS je.itfJSFI:_k RATE C 179 CAL.L ,P/AS(TV#P7V) 16 1i4A=FA(ALUG( ( I.-PIV/PT MfI .- r'PP/P1 ) 1I CALL PZASL rvmpF'7?!) 182 TNmF*40; -31m/-I/ .PPPI C C HE.AI UF VAPORI'~ZATION' C 11i3 CALL SFKI(7F,1'SE:N) 1t0 CALL S.(F~tS,~ I1b7 CALL Pili:-III( IF.mlhPRIm) 186 ~~h vAP!-.=HSLr,~-milPI m C C SENS1IbLE H1EAT IRANSFER C 189 ACC=rfNA*1A020.ACPP/H 1 92 ACC:AC4/1 PC)1.APA 193 ACNA=ACL.'1/ (I.*-P. ~F(-ACCr.;j) C C LA1LNI HEAT 1kANJSFEh C 195 0)L~rivyAP11I.*1M20 I96 uJLM"=HVAvr';i r;,,Amo 1 o2u. C C 107AL HEAT TthM4SFEN C 1'97 JIJ*A 19b i=J415 C C APPAkEtil HEAT TIRA,,SFER Ci'EFF~ ICIErul C-1 99HPPt/(IT4)/) 200 HAPP?1=0iW/( 1Z-IV'v11+tT/L.) C C UiOT NOiMtEfk C 2 01 CALL AmDoA(pTjrV#VLA~tlb) 202 ,,AP'PJ/VA3/0O C C IPIE 4,007S L'IP1f.Af45CtftdDCAL EOlUATION A()x/IIl)o C 2 03 CALL TPA'NEO(5,ty) C C 11!-SE 1,rEli FUR OPU13LE1 1L--1PEkA1ir'L IN'Ci;ASE 01)-HALF Irj1EIkVAL C SEAI~Cil %1EhI ID C ** AA A * * ** A r LOO 2 G LI ~~LPSc = o LU)') i) 205 ltiA =O.OU1 V0I O 206 TiiLA = 7 i 207 LPtz1/U-O C C 7riEPr'*ltL tIFFUSI VI rY UF v.1. rr DRfO.1ILL
c 207 CALL C P li0a2UJ1"rIITVICPV) 2l 9 CALL. Vvk.hA~rS(F'TITV.' 4V O"l 211 211 ALF A= VLA: pi/ CP V/.qLIV/ 1 000. C
- 210 C C
SFJ..!A 0 215 SU.1H:o. i)U 110L1,!l 217 2 2171 t) 219 G;o TU 101 220 109 x(L)X CL-i )tPI 221 222 L ) )/ Y(L I ) 223 S5IEPA=(x(L)**2)*nUNJA 22j 2 1I-(STLPA.GT.l79.) GOu TO 107 225 S A=?. A IS A *t XP(-SlrLP A)
- 22t S UmMA = S ?AA + SA I F ( %,IS(3-tJA/l *O) T .AfS(CSA) GO TO 107 227 22ts 11 L0N 1p-oE
- 11) 7 F I H A r~¶,-L;A 230 231 IF(L.;1 .20) GO0 TO 112 232 GOc I 113 i*. 233 I112 X(L-)=\tL -1 )tP 235 I113 C(:11 I I NI.IF IL )) / X(L )
23b 239 241 IF(AtiS(Stl!'1/lO.**tO),GT.AloS(Stj)) Go To 11's 2'J2
- 2'J3 2L13 I I Li F I B'W=Th.IJI(...5JM2 24U 2 45 2071 tJrq+ I 24b 2u7 2'18 244 250 251 Dii leu L=1,21'O 252 IF(~r~.~?) G TO 115 253 X (L L)
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- 255 25b 25 7 25b S1LPX:(A(L)*s2)*FC)NX 259 r SfIk.T1S)GI) TO I117 2b0 S Li /1YxC TEP I 261 GI AhS(SX)) GO TO 117 2b2c IF ( A~iS (SUN.X/iI3A'o)
26bj 205 IF T)iJ Im i I2 , ?IO, I 208 26b 207 11 IF ( F 1r X ,IPo)1,(1,1 0, 1~1j3 A- h 2o3 IF (FT I Ht.T J?03,(or
- 01. 2h6 270 Flt ri f: I r4 x 271 GO TV, PG?
272 273 FlhA=F JX 27a GO TO 207 275 27b 209 f~r.MAI (ILatA IS WRUE tROUT) 277 GU hi 19$ 27b 2 10 c O~TN J1 E 2 7I 219 GU IC) 197 260 2 71 197~ 71r1IAZ IF`C v"1I *(, 1.0.) GO 710 IJ(7 263 NE~tL4 1 8an IF (r4L.L.E.3) G;U TO 1J06 2tsS 7=71AL'tItitIrtE TA 26bo 1U(fj1-..fE.t6) GO TO 4J13 26? GO lb. LI07 GUi 10 £10t 269 290 IF(!4L.(;E..20)) GO I0Q uIL
Ž91 I f (AhS I1/ I J-L.L1- I ~tT L *
- i.i v GO TO I107
?92 29a G (6VGE T0iTO"1 U) 295 I1F ( A $ S(I/G E L O3Gi
- l T o7. 3 296 GO IV 'JO-/0 297 15c (NT I;vuc 298 IF (r-jEL.f;E.c,0) LO~ 70 40j7 299 JF(AHS(I/rOLL2-I .).L1..oan j
- f. GO ILI '107 L
SUR~FACE TL'ajIUt OF AIEARt IN AIR~ C
.300 '106 SJF~7~-J '5TO~A *~(.)T*?/~O C SURFrACE TENSION GIROuP C
301 IF(VOkuP.GT.o,) GO'T0 505 3U2 CALL VELOOCJ 3 03 GO To £105 30a 505 Cuw1jIhjii 305 C~fLLL VEI i." 3ou GOu I( sIJ(5 307 4 07 C(ir-iTI4uL 30s If-(~~1.;TQ.)GO T0 zjt 309 ]F(vII.1P.EL;*.0.) (u Tu o'.( 31 u 3 11 C C AVCPAGL TLflPUE T1JRE OIF DIROhLE.1 C 312 313 Ii 315 SLIMA V=00. *, 31c 06U 51' L=It2utJ iII R,1 7
31 a X (L)=SN1.,L (Y L)) 209
-j1 4 320 152 AJ(L =A (L-1 VIP1 321 153 Cu.;. II i.:I.k
. .. 322 323 AIF(S L .= t I , (x (Lb. (' TL C (L IN L X%
32'4 S IFPA v*A L AV*2(SFt.A V)/( 325 32b S t AGO= II" AV15SAI 327 328 15 1 CUN I I .4LI 329 ISLS TAv=IZ-( IZ-r1)*SLMAv 33 ( TII TA V 331 I F (vM1 T . GT..0 GO I10 '4O9 3 3t2 GO To dUl 333 . 09 C(i%T I I J 334 Z=Z+vI1 rFiETA 335 ~J0 CU NT I' C C L)RUPLE F 01 A-LAIA JINCPEASL C .1A 33b 337 UM():(l )+OD c f, C THERMAL 01rILIZA11UJN i C 338 TMLI I=( 1Alv- i()/(TZ-TO) C RATIO UF 7H1~kMAL VIFFIJSIVITY AND DIFFUSIOI'a COEFFICIENT C 339 LI- 3au C C AVERA.GE APPAREN'T mEAT INAr.SFEQ COE~FICILNT f C
- 1. 3L 1 HA PS r= PS P+`tA PP 3Lj3 HAqPAV3=,4APSTIP/rK g'.APINI =AP1%Il+ (IAPP+1,* HA PPhi tHAPP) AT HE 1A/ b.
HAPA',L=mAPI%,/Tt1EIU LIj C c AVERAGE APPARE1N1 H1EA1 1HANSFER CUEFFICIEri'--CHECK UP C 3LJ6 HAPCH=Ir.1*Cf'V*(TAV-TAVIULrI)*IUOO,/ARFA/(TZ-Iv41)T/2. )/1,iE1A II HAP CH It =IiA~PCH ~T (H PC O HA C ) IETA/ 3L46 H A F C H0 = H-A PMC m1 / t L 350 I Av 0L1)= IAV C C AVERHArE SENS16Lt HEAT W1-ANSFER COEFFICIENT! I C I 351 iISTLP~t1S IEP 1H 352 nAVSTP=HS1EP/K 353 Hjll I r-,I. Ij ( n;C'Lj . *H I'+ HPA1T3* T IA / 6 354 HAV 1:4 =HIPi I/ THFITU 355 C
. I1 .
IS C ENHA'CE-iEfNT FACTOR C 356 E F ST CP h APA VS/H~A VS F 357 E.F j1NT=H-AP'AVI/tSA'VIIT C 17 0 ~~~~r-. 7L Cv f -yr n - t I aI A %# I 7 - 1 )
359 DLtSEPS= ( V~t IPP*T"F-I ** 3)/ W (I)**3) 210 3b0 q IJ
'lI 25, 1 ';,IAV O( I), I MD,1I"4fIT,TI4It.
3bI 25 F Ci,tiAI( 2x,3Hl V= E.I. 7, 1 utIt.%=l: 1E1 .7 ,Ix 5tl) (I)=,E I 7 I Yr5H,71 ,'t IE I. / , 1A Li MUl IU -. F 1 .7, x, &,4 I L I L= ,E1 4. 7 362 PI..121 , ' F, h I .1A,P, 'H AVS tAP A VI At 363 #t 1l(2X F ORMA rICA=7, E ,I 'J.I 7 1 I 'F= ,IEI7Jlx7 X E IIJ 7'El U7)xHAP 36 Q P.RI tT 21J , , rA VS~f ,iJ-AVI If-FS IEPEF :, F F NA'V 3tb5 24 IE Fmt aI .1(2 31x, I=l tF. I,ZE
,r('L~'S= 4I7,i 'I1 NEU7I ,FHAVS,,4JAPC 1P= ,El vI =E1 7 , I.XT , bAVI. 1F1)4)7,IX, 23 FUI.71ATseOx,5Htf/L=,F1L.7,lX, 'it,:'E~u.,Ix, t'~F,.7l I ' tF SI".b I # L .7u
='I, I i I Ef1 Nr= E I a . 710.2, l X, 'NF I, v= E F4.7 36h Pk I "J I 2 3,1 f (L. v V ,Z EN, N , 1; I ,HAPC I k 367 23 F (p41AlT (2( 5H I OLl:= F.I u . 7 t I x , I v Rf= I f E I Q 7 , Ix , ' Z = EI 7 I Y§ RF ts= ,
I EI 4 7 ,IA , '-v=t#1 3 , I , v DT= ', F I0 5 p I , PC TR=l F I Ol 3b8 'RlI" t 2,1, pt?!,q, scul, sm.q , 1.1SN %S , (IL AC-J , H,6Pcti 369 29 F LIE-0-A r ( ,eA , I PN= I,F I U.S, I X, ISCr,= ,fF1 0 .5, I X, SHN= I , F1I tl5 I x, 1.15!;= FI'F). 51X , Ix S= ', F I U ,I '{i 'F I 0. 2 , xACN@= I FFI0. 5,1XR r1APZCpi 2,F1o. I, /) 370 I 00 CON I Nur. I 371 GOU T 33 372 91 PP'1:41 55 373 55 Fll'1A4('0','l1Ti AjIN CANT Bt ACCOMPLISHEU #EE!'EEN THE LIf11TS 37U4 (;( TV 33 375 te;T C ., I :OE 37h PkJ'.;T 35 377 35 FL];tta1('0',1w.ATE4 SLJYFACE TEMPEFIA1URE IS GREATER 7hAN qr.V99% IU I TriE SUIJQVOIN) I1 I LmPLtA 1URE , / ) 37U3 33 L'J4rlN"Lt 379. 36v 27 FUE,.I1.(2~tt,~iPP~,El(*.7.1x,3t,1Z,( tlJ./,IXLjr1)p=,E1ls.7,1ts~VIsSP= '.7 381 90 .PHJ.I'. 2t5,W<1 ,ltEMIduIFI,.Jr,?1ApAvltrnAvp4 382 c~l FUtrAT(?xturl1rRILl*. ?,ix,5t1IkEt:,ofLJJ.7, 1X,3h01:,flb.7, I£, 'Li-i'll: C 383 90 Li'd11 tjE 384 S IiOp 3b5 E tN; C C **********i** * *t****A*A*********&*At4***i*tA*A.**.***4 386 SUI3HUU TI1 NE PZAS( t, P5) C C 38 7 38L8 PS=(((((((( (1.5531I/lu()OOO.*x-2.b93' 527/1 oon. ) *xŽ.U I 53 693/l ). )& k?033727 )~A *57 .I70Lit 9Q) *x-S.U7Ar 1) 369 PS=Ex'P(iS 3/1.01972 390 PEt. 7 Lli, ^
- 39) END C
39c SUlkoU I I vE TEPl ( PFPP TZ) C C 393 X=Al.(iG( I . 01 V47 2* PP )I. 3914 I =(( t (( (-Li r'92atifu3/ I vop)n. fi r) o, ".PS L /~1 nnn(f1t0 x1 I 13 431.5l1 I I1(()o)tf) )A Y .2( /I7 12/1 1 400v. )XI 7.7;1 7752/1 OflocI . )*.x-3 7i j: 2 iun, 4 / I O(Oh 1)\I )*X + I
- S; o3 7 7 3 1 1}on (i i Xt+ .I '9;?rI O;t) 42,1{77 1 ;*)!
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3qb 211 C c
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r-- 397 SIJbHlI rI N v vOJ.)S(PI,IV,vvO() t******t AA**t C **,**** ***,* * *A***AA**At ****** ******W******** C 398 A=(TV+273.5. )/o'h7.3 399 S=T?r c2 1I.206o tU=lUg.*hL,7000(IO(.-3122199. A*A-19Cq.h5/AA*h)
^=(ut;lT( I .7;*'* *U+1 3,o292oP. *IO .*-* 0 A (j9-1 .511l70t*A)))**0.2tV4 II7 6=1.0( P VJ*5(h,?.5*u3. Iu2f 8+5)))/1.**1I/( I-.5 IOh/A .2+A**tI 403 Ut(4 C=C* ( .la IIb/1 0JU0(J&.tO. ?.,?ho2I C** *)
vvub=0. 'i 7/e,4L-b- (II.397t~t)-9.4uY927* A)/ I)00vu. U (I h HE Trui i 407 ELiU C
**A ** ***A *** ****A****A**
C **** A *A r
£40da SUbRulJiot SE(PV(12,PV)
****** **L C **A***A**A*a*** *A***A********A******************'
C i409 x=T Z/ IOt). 4 I 1) PV=( ( ( ( ( (( (-eu.(o/¢Xl.
- X+3. 15t,533/ I 0 . )*x-1I.5of,7F,P. o*;'
I . Lueki 'j^ :i/ I O(t . ) `A-3 .k ol 75$h V / I o n . )A^x+t I . 1IU7t 1u,4/ I I10 )
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- t *** *A ***A*
£413 SUO R(JU TI flE CP V OU2(PIl9C[>/v '
i . C : X=TV/1 1I`). St 415 Y=PT/1 0(i. L=A**5 1417 Cfvl ./(Y+2. 30.cIŽ1*Y-a2 27 NI
£41 CIV=CPV*(0.(,0324L*(t).uU32?bu4u.Opc,91 *ZAZ/1 o0U0.))
1 419 2= . t 055 *tI .h372 u) *x-24 .20 dnI)
- x+33. u44 7 1 2l t3;). 7252
£420) C PV = Z*Y/ 1 t) 0. t CP v
£421 Z=( ((2.2s9Yi35*-I/.40325b) *A419.097")88)*X-13.43,33)*Y+u19.b?59L l 4 23 CpV=Z/1 to.+Cf-V k ETUli
£424: E NI)
C C k***
** A*A*
*** **A A *** *t** ** A
- A ** *** i********** t* s*A **t 425 su!iuIC-iE CPtA(PI,1KCPP)
L Cr*****l*s***************i******* C X=( IF+273.I5)/t,417*3 SlI / 2 ri!19 2;'b (~ 42h 42b L t3= I.tpZE0;20IJ ( I .5b I Oh-3/2. )*X ** s2/llt{.S.);t.e./01/*'
£429
£437 C= ( ?.o t)157Fz5 (:f 1. 7 2 21 4*i I1l._ x * *3 )/ I 6 O ) .- (I.2Z05'S7 o 1i5) / X**15 .
314t I)=22121%.7t)3*(b,*S4 (C- .53?4Av':b/X**33/I00fl.)AS**3) 4 31 C~P=(1109."nou-1J+X*(-1t.1107235+Y*(Z22.2V742- A*52. IOb73(i))) 4 32 HE TURN C,-..,II. C
. _ . .4_& O * 'A 4-~~~~~~~~~ t**i**t%***i*********
43 4 StJt3kOJI I r. St cI tI:,IA) 212 C A' s*A**stz*******X******i**t A 'A 'A*t~ C aI 3 Et 7 A=(((~ V2`b5 S 2* x-9 .a 3 e 7 5(o9 *x +2 17.9 7 4 5 5 9) x-4 3 . 07 7 I a )
- t 1 S.
U37 RE1IORrN 4 3 ff tf 1) C C 4 39 SLI~L Ll 1 1N I Iv PHj(,tINT PR I A) C C 4I 44(0 x =1/1OU. 44 1 E 1TA=(2(((((2.C 37b5/ I -3 96?9L 2 23.bol 2 x- 78 22v#-,lA Ix+159.'U?1) 175.791J53) 7- *xt 1201. 8u923)*X-4L.5 1 Qi3z *i x b. 173.Y*i 2-7. L4bbU9J'3/ 100(}.
£142 IF(T-35U.)2,2?,1 4uLI3 I FhtTA=Er;1A+0,.3nf73Cs*ELfP(.22555th*(l-350.))
L44It 2 LTU'f@. 445I 5 L C C *A*,*tA***1****ht*~**ft*t ************ 44 A~~iJt4 A;A(H T. A it A C ************** *******************i*t*.*. C 4u b CALL PAS(1 ,PS) 4 Q9 VL A1i= ( ( ( !..23?3- 0. (171 6w3* Y '6 36929 Y+ l. 1hS656 P-PS /I 00 0V U 450 YLAt-1= (VI.^: ( b . I S3o* x-20o fIl) x 25 1A ^) *X-'4. LI7 3 / ,00tt (. ) * ( P-p' 451 vL a:H=vLAb tU.b( S77-0 . 073La
- X-J .t)7)*x+2.tiS43c5) x-tu.9?227 4 52 RL 1 UR,'
4 53 E 0N C C 4 54 SU*3N(L'I;4F 1tt A ( ('im,I C I ANSC LNtI'IAL t!1tJA1(!tf SULUl 1 i -- 7A (X)NX/(XI-l.)=I.--IALF C JINlJRVAL SEARLM1 rML1HOV C . *** A****t** C 55SI t-P L IC I* 1 HEL *6 t Hv^-r _Z) 45 S ) I t:t_f-s I t) X I t 2 u 457 tzCI-"I-. £156 P1:3.1u1X 9ir53b 4 59 JF (LUr A ;(C).Lt. 0.Ct ) ) GU T(i 79 4bU 79 (L01 1 IU #IL £1O1 (uo 13 1=1,20 ,4o2 1F-(DAvS(C).L.l.%l.(COb) GU I(C 21 463 1F(r~I-1 . )2'i,Ž1 ,22 LI ' 20 A= ( - I) *pI+ )
- 0O ltl) 4 65 K= (c* 1-j1) AF'J1?2 L 6t) E S+1P ~)tt^#l~~t)rl-6 I)
Ut7?GO TO 2- £ti 21 Y:(eAI-1)*P]I2 469 (tCI 1o 25 470 22 F'1=3.1IJ1592b537 471 -A=(2*11-)*'1/P. 772 h=l *j1)1-t7. li~joo 147 3 t PS = u. ot, I t,( A ti 2 . PI-I li 7 L 2 F f' l I- ;,,. ( ,' I - r ( 1 ,_; I
4 75 F c= 0 TA N(6 ) -d/t - 213 4 76 7 K=r,+ I 4178 Y =(httl)/?. a79 t A=iTA'1 Y ) CX=Y/C a 1 f = r AI C x 1F ( F ) 1 2 , I 0 s II 12~~~~ (F t (f PS 3 , I10I(I1 11 IF (F-fF.P'b) i0, I1, 3 Li 874 3 IF(-*FA)5*,t),
£189 3S 1f (y
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14h F h=F GO TO 7 Li 935 6 A=y
£1bw 497 LI9Li F A=F 491 GO t 0 I 1 9 C' 9 FtJWM.A1(ILuHA IS TRUE kOUT)
GO lu 13 f 501 25 V-1 T 9 mY 49b 4o FUR--. 0Ox , 5HX L -F Iuia 15 . GO I0 13 4 99 10 C (,Ir4sTIN E u 0b 13 x t( I )=Y 50 1 RE 1 tw iJ E rv U) t C C C
, 5V2 SUk3hOL1M 1 NE vELDO _
C **
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C c ; NUM.EICAL SuLUTIJO OF THE E(QUATIUIN OF MU.ITION FOlR A I)ROPLET "OV k. D)%l1;:;AF~L)S hlhCENG--P.U i TA TECHNI' QUJlE C 5u3 C UJM^(i1. r. VhT,gfJ , 4?r, Ztrs,."cuT'5 , V I SMt (;. *I, ri ,NrJ, r1FE TA , VIT },Av'JL' 504; L)I EN4S I N l;(20)
- 50 5 50t6 UQ8 'h'=VIO 507 Z=Oj GO ;0 "10 II 5(7f 51 0 '1(19 IF('I,.L.F.2) GU TO 415 t
510 511 Z=ZI;K: .10 512 Gu Tu
*513 51 U 515 4 10 1 ft Vk 1:-vf?1)1USi I oi 45;2 45.3 4S I 1iF(vh-vn )U tiJ, 4 52, L452 5511 7,7 5~2 Lu5u Iz~
'I 53 If ( 1i2 'I-a
*51 1i Li 5 4 C f IT1 vd1jt 519 R E l)v :4) 1) JIt
- I' 1)sJ SYV I -
520 CALL CL'v4 .J\x[ EU,(.riCI) {e'~ 'be; ,sR~l?-fsltsvUi;'V~'**/(.*1(])*HtJV) '
'- 522 523 52';
525 CAL(. CDtEP-t1;ACUCLIF') tJ Li N,M
- l ()CIj* HFI
- IA* 2/ (Li '00I 1v .
526 1)= 3 00 Ui
- 529 5?2)
C ALL CO;KF I N ( IEci,C(P) a' 214 P !OO=3,)0(sb. *ktwmmj (VItLNO,%I PL IAT/U.)* /to( IJ *.-; JV) 53o h'E.2 = ( Vi, +C L I
- 7 HF TA./?e.) A)IJ( I) F, tj* 1 6 00{) ?. / s, I S :41 531 skE m2= ( vk iCrI *l 7 IE7A/ tJ.)1! 1 ) *Hiltit* 1IJI(Oe. /Vj Sr;fII
, 32 (.hK-.lI=(,^PhLiL*CC}(CLI t 533 LALL CUREI;(hLEICUCI;PJ PkhLlu0=3f(J01.1 *kkl'-* I.VP+CPt I* 1IiL Al/2. **2 (4 *1)t J )*if )v) 535 C r(ID;-i *Cj ci co 536 CALL L1;tI RE ,l~~k 53?
53?i CKl2=5 -Pq0L1AUACCUCL 539 h~f ( Vi.tCK2*1 nti 1 A ) *UL)(I ) *kh(lei4 *~ 1 0000./XV I SsIJ 54 0 CALL CW<i:r-"('E3,C0.:C'P) 5u I 54L3 CK3=C;-PlP100* CDC k CALL. Ct rE1,%(ktfd3,C1)CDk) 54 5 I 5116 Chi~t{(3Fi P9Ct.* .r-U/ C2 t) C (
- bE 1s *I) lH o.J 54 7 1F Zi:vti-~(tIts KUl2.*CKMl+.2.*Ch¢124C~t'3)*ItLfTA/12.
Vk'r'=vH4 549 Z?= VR ,1 1\ I-"f 550 V=VN ( CPCK42. *C^ 1+2. *Cr .2+C3)*TtlE1A/6. 551 Z3=VRVlCJF 552 Z=2+M(L+'J,*Z2tZ3)* IhElb/6. 553 GC' Trl DJ(1C 554 452 Vk=vH7 555 3F (rK-1) )t,18,1169 550, 557 Go OI =0. (f 559 5 ab9 IF(RIE.LE.2) GU 1 g75 560 TIME:=1jIEC F.I 561 7=1C 562 GC ii) L17s 563 u175 1IMtC=TImt bSu ZC=? 565 5bO Z=Z4(v*-v 1hfZ )+*T IL 5b7 £JJ C(l VT I NU. f~Eil up"~
- 569 f ND C
C C 570 SUbkuIjU1 I:rF VELUtIP C St * * * *
- t A A S * * * * * * *
- A* * * * %* t. *
- 4 *
- A
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- A A * *
- t *
- t t
- SkA * #* t * * * * *
- It
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C C NtOLfEh CAL StLUOT I(J (OF 7tE f.1itVATIuIN Of' M(,1 JteUN FLk A F)RjPij.L I 'vt (I C UIPi.-tALS kU'(;E--KU I TA I1:CI J!U C 571 I 'it.(Lr t- ,V.T. t,t Z* T.ti(!;I Z, vt.o, qiM , VIuS, , sJ'w OT ,VA I Re VI)Ii ,w.F 1, 572 ()J^'E 'SIClI'(?R ) 573 57u1 ' V 1OIl HY1Li 575 O 6t O, 41U=v 5 76 5,7 t9 O J f (r-C. L f .2 G T 41 5 57E V 11VFk 57 9 5EU G(O If) IJIo
215 582 583 b84 b56 CALL C(:;;E IN(HE ( rCi)LW') ( 5ffb 58t bI VA I k~v A I K-U. 1 5b9 bI IF (v 0 ).LI v IK )~ Go I*r 2lo 559U IF'('4.L1.1) Gu TO 316-591 GO To lb 592 lIb CONTIN~UE. 593 11- (J~*G Go1) 01) 202 59u Go Ti) 151 595 2 10 NP=ri %i+1 596 2ul CON I I r'E 597 Nb ('0 C1NT 59b 599 CKO=- ((;.tPPoI*COCLH)I oOU GO' 10) Ti 601 lb CtKL= G-PkLIL*CUCuR 602 17 WEI =(vRtcr(*0T 1l2 . )*0 ()AId* IJO0G./V1S!MF t603 CALL C1)LI;(PL,CD)CuK) bOb GO TO) 1it 607 CC1NTINL0E liii b09 lb 15 1F(NM.G1.II) G0 TO 18 boh I1i CK1= G-Fk0D.2Cr)C~w b613 17 19 CtALL C1IcL1t.(hF?11,CI)CL11) 609
- blU GI) 10 20
*blti 608 F (rs'.-mG1Io) GO 711 20 6 20 G1O 1C) 21 b61 2( CK'H1= G-PPLI)ACIOU~k 623 I. 1.1 CALL CvEf(~2C~k b1. PR(10=3Li000.* (IVI+CKI l*Uj/.*2(J.t()P IF( i.L I. I) GO TOI e1'2 627 GO TU 22 621i 629 IFL:.".0.0) ;r.' IC) 22 631) 631 Go rO 23 b32 2 i? CK2= G-PkL10ACDCVHI 633 e3 CALL C~iiI"'(~k'~2*CL%'C0",)
3'4 - !'i
.0 635 IF(1,:.LT.1) (tu 10 124*
01'i 636 63(s GO 10 2'J b3b 6362 639 C~rst2=-(L:+P~oU#Cr;CI1H) b ti U GL o ? . .11 6151 ?2 C~* -PUAUC' . r -it:-: I I
CALL c, ; ] ( L , I C~ 216 6145 F i4Di)= So)Jl;. *ku",:l- ( vi-+ctt 2*0)I) *?/ U. *D!( l )*PLiV ) 1F(r.L1.1) bu IL It 64 7 (;U To 2b h14h 12t) C'tait P rut 6149 lf ( !,. t r . o) ron i c 2b C e.3=- ((;+ Pil)
- CI)CI) )
b51 ;U 1LI 27 2b Cr,37= (;-(it00*C[CI 653 27 CALL Clh Ll'j(;4'i3,CbLsP) 65a 655 1F (N.Lr.I) Lu 1L, 12h 656 r;tLI tC, 26 657 12b CONtT1iUEc 658 IF (r:lt.GI.t)) GI' 111 26 b59 ttoO Gtj ri) 2YIL 28 CKO'3= (;-PRHLIUCI)CDR b62 29 CCONT I NlA 6b 3 II-('.LT.l) GU 10 130 GO TO 7 bSco 130 C)OTIfjlJE 66tb IF(r4M.GI.o) GO 10 7 bb7 /1=v R+v A I H bb8 GU I1i 51 7 Zl=vAll.-vp t 70 GO 10 51 67 1 51 V,,M:= V+ ((CI.O+?.*CI\tO-I2.*CIM?+CKM3)*UT/12. b72 3f (ii.L1. 1). (C, TO 132 6713 673 GI) 1l0 71 1 32 CUh IlNloit7 b7b IFv(:jFw.(o.o) GO TO 71 676 Z2=VWM4 VA Ik 677 GO 10 53 b7h 71 lcVI=V Ik-VIM 6o79 GLi 1f 5i S3 C0%.' I ;10IE b~il Vh=VY?4(Ct)+l~?. CK14I2. *CF;2tCK3)*G1/6t> b2 vweRS=VFw If.(V<.1.) 1;G( It 112 6813 GLI TO 1 I I 65 1 2 CU'T Jr*NUE Vk=A6S 1 V() 687 I II LU1i I Nt:O lF(Nr.LT.l) GO TO 13J 66d9 GO 10 72 bsn 134 CONTs1rPUL 691 ll (0N,.Gi. ') GO TO 7Ž2 7 b.) 69Y2 b.. 23=v+v Alk 6o93 GO Iii 35 72 Z3=vAlk-Pv b95 iGO To 35 35 C W-41 I r-LoL 697 ?=z4(zlt.*Z2tZ3)*uTr/o. 69 56 CUNT I 1 4to 689 IF (.sLI.1) (tGL 10 1jUt GO T1O 73 7011 1 4!5 CON't JMNE 7072 iF (-!4.6 T1.(;) GG TO 73 70'3 70*1 JUO 'h Ai 705 73 VO~vt .It-VP 788 " I I, i,':' 1 ', , v.t,
217 7u7 ==r I Lt +tT jv',;r.tN PN RII . I v)?C 'i C"s1l ,LC ^I ,( t 3,*1 EI,*FC 1,R -t1 iE?,P&3,^t 70L O. 5 l r Iu 1 02 FLU mI1a r ( X,xuh(KU=fF IL'.b, I X,4PLK 2=, 1 f 709 21 1A. 5 wI X # t 13^ L O= F5 . 1 11 I1 E = 3. 1 XI iRE3=, F C 710 7 11 2,fS.PliJl 3J,3.,r-, 1033 712 REIijkt: 713 f Itjl C IL A*******1** It****It *A A*********** A*A* C C 71 u ;****** C *t***********e*****,*****, C. C~tztiml 1,. 8aK, l ,;VE, VRI, ZI;4klm ,'; S ii 4 .VIl,,t4lNtio'-% T hETF A, VINFx VCSUftI 715 lSLRFTH0%fV 9fiC , .,2, uSE 1,vkrS 716 ,f1'MES I iJN U(23) 717 RE I =AbLS (RE I 3 71b S: ;Or.i'*iuhf 1 ,i*3k*1 0).ts* 2/G./V1IS~t **Ll 71 9 DELkN hf V - (P. () ST Ar = I i) 00 (kU(1)At
- A2 (FtJ1-h *%2 /V I S
- 3it 720 721 IOSTat= AR I 1*(8E 2<~tt 1-(0/ lt tl/
. *(-I ./3.) *(;*Illtl 12) )I .A IU1 ) ;
722 l)S1A2A=I0,UA(UrLhU/t A*(- LV S I A2 e-= 311, ( 0 EL h C!/ H .! ) i ! 723 72L 725 ( ) )*I* I. 1 10) r )=3, .i(Ul.rc)/tU l;)** (-U.U7)* (*G.** S kf.2A=I- 7,4. * (S*I 1).* t-A c t (_-V . I Q;2) 727 726 **(1 ) **I. 3 b RL 3 A=Ž22L(.* (IJEk(j/I . ) ta(-{. 3,I3) AS 10. 7c29 ht~~i4=71blJ.~ ~ ~ 6 3(5*I0t.**(-12) (-}3 .~~ltli~r* )** }.3l 730
.** . IbA 731 CtEQ=3 IAU. * (UELFtL/ktt15J**(-t. 73 )*(5*15 I.l;))**c2 HEr 5=tj4 7i) . * (iUt.L /i Or~t' A( - . ?3^ ) A
- 1. I0 - I* 0 232; 732 I ./t WItoA XS _ ,)2 . *(i (' / F U >Yb ) (-0 . 5 *1( I 0 . * -12)E
- 733 =t 73LJ ClIe;-AX=t).75* (t)LRLhti *A(-u v§ e,32 (S0 I O.** 12 )t. 03o i h)* S* . 25) **Q .2h2 q 735 S 1 L =3 . 273" 2 3 L(7 UL 0/
0AIJ; 73h vMAAXS=W'E5/I) MAXS WtSI AK=s40j~t riSTA3A~vL3A/DSTA3A 759 rS I A 3t=m 3t 15 TA 3ih] 7uo0 wS1A2A=PE2A/USTA2A 7 1 hS T At3=k.E 2i H/t) S t A2tA . 7 2 ViST lIS =0 . 1I)1Sh I 7 3 I)S i =0M.A XS 7 47u A=(! . ob5- t I1. /29 .qU ) %tALOG I t S** X . 5/ (t)ELk(E/kel-IS 10 . *(-3) 7LjL5 IF(,EI.LL.uL.S) GOT lI L1 7u1b 1F S GtT . 4u,.*If* J* 12 ) GUl I 0 2G 74~7 I FHE l . GT .k42A) GO It! 2S 714 A Co=(2n./HW~~~~5 )* * ** U bi,,7) Il.nl~*~ 7'u9 OSl A=(3. *hh l **CLkI (J . *l)fLhC) )* 0I5/ 1./ 3 S)
- 750 YS I A= ( U. *kRt I *Wl)F l H 01 ( . -ALi)* KUP-J: ) **(IX. / 3.)
751 GO TOf 210 752 20 CdT I Nut.L rO TO 2 6Zo r¢ 753 I F (F;7 I G I RLt2 b)
*' 75u4 755 3
.A 1 ELR/ (3. *l k1))*(1-/
Ul 75b hVS A(U 757 (;uL TO 210 -- 11 Iuu . 75h P? CotI
. -...Y *. L 7 sl a t. Tt- r
7b0 L D= 0,tL(SA IhS.**(1? )*u(j5 218 701 )DS ( 3. *kE I t *?LL. *At(L/ (t . *liLL .(n) ) (1I ./3. ) 762 r S 1A= (u .Ck IA lEL 1,(/3)(3. *I. /3. 7b3 GO IlI 21'.) 76t b C(('N I I';lILt 765 IF (K 1.GT.kt3t') GU t( 5ij 7b°C t V I I 0, A 5,/*(t I ;. C5) C 7bo 766 {,1) A= ( U. *Ft 1AIjELLi/3;A"Ci)AF<,slEi))w*(1./3. vSlA:(J.*f I Al)~2 C AtU3. h I-< L) ) 0 * -A1. .- 3 769 tLJ 1(121) 77U J CiJ=cU./rt 1 771 DS1A=(3.* 1**?*Cftadh(..h/(uj.*r)L Ot))t* (1./S.) 7 72 77 5 1F(uE1.Gr.PtJL) GO 7(' tO 777 ,USIA=(IET1/(U.5A *CDEtit-.L/ut-)**0(.3L.*(((*1(;.** (1- ))*(j11 )n (3 7 76 CD)=n. Ob5 o*LUS r;*t,A)~tt; Li *94.2; (S* I 0*t (-2 ) (-02 1~ 779 60 1C ?10 78 0 60 CLiNT I 'd'E 7 81 1 F 0-t. 1. G I. E5 ) G(I lCI 70u 7b2 K Sl 7A .ct.* (VLL RW/ t4t`*1*tO.2 b* (S A1 C. A* (-12) )s*0. 0t j 7 83 DSIA=HE 1/eSiTA 784 CV= l . 0 3 S* (C1. LfRL/i c )* o.477s tS* 1. *7At - I 2C)) *I (A 131') S IL 785 G;to Io 21
- 786 70 C rON I.Vt-7 87 t GI= U.
- DEL.; C.* S T 3 - 3iE I 1? t4io*
I (A (O)S T -0 W~StC) 7bbf Ge'l k b111=I e., i*O L I;t C :8 I A P-.3 *HE I ** *k Orlti
- A 789 PS Tr.=D5 I -(I /G;
/ GI Pk *I 790 1F ( AbS( USItf-;S I)/1S T).LT.).I)QJ 3) L;( T( 12 79 1 [I'S =os 1 '
792 GO I Es 11 793 12 CVCODI 'AX4A ADS T -I-1AXS) 7<#6 Sl1sJ
>~~~~;P 7 (U .L*XlL F*>SUh(I (
- 33. Mb I (A (L) S 7 N ) 1iA~ Y'S C )4, )iI.1 7945 bS1 A=0 I 79t, $SI A=r S I 797 21t0 C LN11tIJL 798 P rgt 799 END C
C o00 S11610'1,I tINE, VI SA5tA ( I F. F, R PLPVI S F) C 6 VII V 1 b PF = P0 . 4t('4 4 (7 ff I t *(5.6 5 .91 IF )/ 8l I v 802 vI5v= 1 73. t .L5J IF 803 f jI,F=R'3' AV1/5iFF1 (1.40(.-R? )s(1.iFsJ 12(vIIPF/VISvf )/(k.F25I3)' 606 F 1IJt l. 807 C C A - l b 0 ti %;uitjk'ifl I ',if. A MFSI i k (' I .1'F, CPN:' CP 2, CPiPiAfi-, 3 L * * *
- 0 tA * *** 0 I *** A i**
- 1* * *i- 1k j A * , A j * . -A A A.f 1.
i, f 809 ItP'LIP=0 0 17 5 r.JF7 1 I!0 0 L,o F /l).i7 *T r-t.I fF Tf *F7 F PI I 810 C/ 811 ,%MtLI=32. 81 2 V:14.tte3C1. 4J
219 8133 8 I LI IkA:?:(T&+273. 15)/151.1 81U 8131 17 131b e~ S 1 LJ2= .393/1O')(06lU 6131 SIAR=1 .27211t)'JUI0. 820 821 822 CVN2= (C PN2-6 3 iS 5 3 /wtl,; 823 824 C VAll=( LPAfR, b3 1 .)/~'W 825 637 630 E2E 829 AM'3LJ2=1S(Je0(I.32*CV-+404b-q1.59/TNU?.) A rib A~i= 2
- b AV IS5 A i4
- C V A k 631 f.
i I 132 F12L=Is 3 53 / S 1(1 40. h e22 *(.S i1 T (A 08 PF/A'.) vF ..1 93)'*2 6f33 834 I b35, 836 END C Suar)I.JILI II NE LIIF A S.( TF f1F,)IM ) 83 7 1339 C I. b43J C I1. C **h***~~~.***
***i****~~~~~ **A***********A*t***AL*Wi SENTRY
220 APPENDIX G THE COIMPUTER PROGRJII FOR INTERNAL RESISTANCE AND NO MIXING SOLUTION OF TEE SINGLE DROPLET RESPONSE
SJOt3 AATI-I,V.,t******khP-?9,P=50 221 C ****************
- C C
C *IHIS PROGldApmE SOLVES THE PR09LEM OF SIMULTANEOUS M0!~ENTIJMHEAT C *MASS TPANSFER DURING CONDENSATIUN OF STfAM FROM SATlIPATEDr AIR-S-C *MIXTTuRE ON A DROPLET iTH RF.SISTArjCE TO HEAT F-LOt IN A DROPLET C C C
*A NONUNIFO;4.t TE4PELPATURE (IF A DROPLET
*STEAUY STATE DRAG COEFFICIENTS FOR wATER DROPLETS USING REINHA.4 C *C0RRELAfrIt--INITIALLY DROPLET mOVES EITHER UPWAROS OR L)COvNAR)c C
- SI--UNITS C
- C ********* *************t*******************************
C ************** C C C A=IHERMAL DIFFUSIV[IY M2/SEC C AM$MF=TriEgI0L CONlDUC-IVITY OF MIXTURE -- FILM ,I/M Om SEC C AMbMP=THLK.AL CONDUCTIVITY nF STEAMA -- FILM J/I OK SEC C AM6MV=THE.RMAL CONDUCTIVITY OF AiR -- FILM J/M OK SEC C AMDA;THERMAL CONDUCTIVITY OF i4ATER J/M SEC OC C BI:='3lT tjt-lit ER C CDCI)R-1)'AG CuEFFICIENT OF A DROPLET C CDD -DRAG C tEFFICIENT OF A DISK c COSS -pRAG CuEFFICIENT OF A SOLID SPHERE C CKOCKICK2,CK3=flJ.N;E KuTTA'S COEFFICIENTS C CPA - HEAT CAPACITY OF AIR J/gF1OLOK C CPMF - HEAT CAPACITY OF- MIXTURE AT FILM TEMPERATURE J/KGOK
- C CPP - HEAl CAPACITY OF STEAM KJ/K.GOK C CPV=HEAT CAPACJTY OF vATTER KJ/KG OK C DD=kDOPLET LIIAMETAR INCREASE M C DIFMF=DIFFl.lSIVIlY UF MIXTURE -- FILM 82/SEC
- S C DT=TEMPE^ATLUREt IINCREASE OC C F=MASS TqAt.SFER COEFFICIENT KMOL/M2 SEC C FON=FPlURER NHUMHER C G=GRAVIIY ACCELLRATICN M?/SEC C H=HEAT TRANSFEk COEFFICIENT J/t42 OK SEC C HVAP=IiLAT OF VAPORIZAIION KJ/KG C KE - THE N.wmm.EP L1F DIAMEIERS C KF - THE NuIIbER OF STEAM/(A]R+STEAM) RATIOS C PEN - PECLET NUMBEAR C PPP - PARTIAL PRESSURE OF THE STEAM BAR C PRN - PRA;4: TL NUMBER C PT - TiTAL PRESSURE. OF THF MIXTURE 6AR
.. C Pv - PRODUCT OF PPP BY (1/ROP) BAR*M3/KG
.^
C O)L=LATENT HEAT THANSF-ER mJ/2 SEC L C OS=SE.,SIBLE hEAT TRANSFER J/M2 SEC C UT=t4St)L -- TOTAL hEAT TRANSFER J/142 SEC C. REmHRPEYNtULC1S NUMLLR OF mIxTuftE -- 1fULK C REN - PEYNOLDS NUI'IIEQ C RMB - rMIXTURE (;AS CONSTANT - I;ULK J/KGOK C RTtt!)N;?ATICj (IF THERMAL OIFFilSIVITY AND MOLECULAR DIFFUSION COEFF. C ROmB - ODEjSITY OF MIXTURFE - PULK KG/J3 C ROP - DENSITY OF VAPOUR - FIULK KG/M3 C RV - D)ENSITY OF WiATER KG/M3 C ROPF=t)ENSITY OF STEAM -- FILM KG/M3 C SCN - SCH4MIIOT NUmtER C SHN - SHEko5eOO NUCe; ER C SURFT-SU'FALE TENSION OF A DROPLET INAN AIR/STEAM MIXTuRE N/H C TF=TEmPERATUlRE OF FILM OC C TNA=NMASS TRANSFER RATE KMOL/M2 SEC C TV - iAlfER DROPLET 1EMPERATURE OC C IZ - SA1LURATION TEMPEkA7URE CORRESPONoING TO PPP OC
L USN - NJB.St.L I rNlumtEk c VI Std - VISCLiSITY OF MI.XTUOE - I3LILK 10**7*NS/m2 222 C V]Sr4F=VISCOSITT OF MIXTURNE -- FILM I0.t*7 *tN SEC/H2 c VISPI3 - VISCOSIIY (IF STEAM - FILLK 10*
- 7* NS /6I2 C VISPF=VISCOSITY OF STEAM -- FILM 10. **7 *N SEC/'12 C VISvI3 - VISCuSITY OF AIR - BULK 10**7 *NS/IM2 C vISVF=vIsCrf)51Y OF AIR -- FILM 10.**7 *N SEC/m2 C VO=VtLOClIY UF A DROPLET V/SEC C YINF=VELiOCI1Y OF SI)WROUNLUING MIXTURE H/SEC c VR=RELATIvE vELOCItY OF A DROPLET M./SEC C VkO=I0hITIAL RELATIVE V.LLJCITY I/SEC C VRT - TER.MINAL VELOCITY M/SEC C W=MULAR MASS VELOCITY KMOI./M2 SEC C wMMF=MlULLC(JLAR WEIGHT OF MIXTURE -- FILM KG/Kltl L C WMMF=MULECIJLAP V'LJGHT OF MIXTURE -- FILM KG/KMUL C Z=DJSTANCE TRAVELLED 8Y A DROPLET M C
C C 1 DOUBLE PRECISION iilYRDRTNtAOAIA2,A3#,AuASA6,A7 2 COMMON K,VRTI,'E,'VPKZRKHtl~tVISriG, NI, NMI,NN,THETA,VINF, VOUP, ISURFTDROVDVRO , VPRZ lETU,VRRS 3 DImF'4SIOW RN(20),TN(20),TtvP(20).RD(20) a DIMENSION 0(20)oR(20), L(200),Y(200) C -----------_---------- 5 VkTT=100. 6 VREXP:100. 7 VREXP=O. C--------------- _---- - - ---------- ------------- 8 READ 1,KE,KF 9 I FURr1AT7(21) 10 DO 6U 1=1,KE REAL 37,PT 37 FORMAT(F1IJ.5) 13 DU 90 J=1,KF 24 IF(VRExP.GT.0.) GO TO 43 15 REAo) 2i)(I),F(J),i0 11TH1 I6 2 FORMATC(5F10.5) 12Z 17 GO TO 4a 18 413 REAV aSD(I),TZVREXPTOTlTlH 19 45 FORMAT(hF]0.5) 20 44 CUNTINUE 21 IF(vRTr.GT.O.) GO TO 501 22 Vi00=PO. 23 VINF=l. IF(VOUP.GT.O.) GO 70 502 25 VDDOoN=1 1. 26 VD= Vl ) Pflt 27 VRO=Vlr4F+VD 28 GO TO 501 29 502 VRO=OAtiS(VOUP-VJtjF) 30 V[=YVIUP 31 501 CONTINUE 32 Tv-TI 33 TTHETU=O. 3a HAPP=OO. 35 HAPS P=UJ - 3b6 H APltjlO:. 37 Hn=o. 368 HSTEP=0, 39 HlNT=O. HAPCHO=:. f APCHT=U.
4£2 I IME=I.
£43 1 A VOL D= TO aa£ NI=O 223 1j5 NM=Q A6 NN=O
£17 ZRFK=O.
Li18 VRRk=O,
'Jq9 G=9 . 8Ohbb 50 P1=.3. 14 159 C
C DENSITY OF mIXTURE-6ULK C 51 IF(VREXP.GT.O.) GO' 10 46
- 52 PP=R (J) *Pl 53 CALL TEIPI(PPP,rZ) 5u1 GO TU U7
- 55 '46 CONTTNLIE CALL PZAS(tZPPP)
- 57 R (J ) = PPP/PT
- . 58 £47 CONTINJUF 59 R~b-83t4.7/(R(J)*18.02+tl.-R(J))*28.96) 60 ROMH=1OODOO.*Pr/(kMH*(TZ+273.15))
C C VISCOSITr OF MIXTUTL9RE-_ULK C i' 61 CALL SEKPV(TZ,PV) 62 ROP=p l /PV b3 VISP=50o.ii+O.£7*TZ-;;OP*(1858.-5.9*TZ)/1ooo. 6'J VISV=I73.o+O.u5i1*TZ 65 F112=(1.3536/Si t O(1.+0.6222)*(l.tSjw1t(VISP/VISV)*1.12593)**2 L; 66 F121=0.3536/SUIl t(1.tl.6071 )*(tt .SIRT(VISV/VISP)*O.dFYPl4)**2? 67 VTSMH=R(J)*VISP/(R(J)4t1,-R(J))*Fl12)+tl.-R(J))*VISV/(RtJ)*FI2I1 1-R(J)) 1 "' C C MASS OF A DR&PLET L t 68 CALL VVOl0)E5(PTT1,VVOD) 69 ROv=I ./VVO 70 I)M:HeJV*PlviO(1 ) **3/(b.*10.**9) .- s L 71 AREA=PI*0(I)**A*IO.**(-b) 72 PRINT I1 L(1 )#R(J)rPTTlIl),vINF
- 73 11 FORMAlT( ,2Xe5H¢O(I)=PFlO.5#2Xt5HR(J)=,FIO.5,2X,3HPT=,FIO.5t2xX34 l-,FlO,5,2xt3HV'O=#FlO,5,2x.'VlN4F=',FllO,5)
C ' r-
.I C TERMINAL VELUCITY OF MIXTURE #<?
C
-..t
_ 74 IF(VkEXP.GT.O.) GU TO 17 75 IF(D(I)-1.U)3,3,5 i 76 3 Fx1=4.625*S(JO(VlISMt*kF£,MB/10.**7)/(D(I)/1000.)**1.5/ROV
- i. 77 FK2=1.391 3*F-K1 78 VkI=(9.6065/FK1)**(2./3.)
- 79 REt.i-=3VRT*D(I)*'Cflrb*l10oo./VISM8 -V 80 IF(RE t3-1 0. )bh ,,JU
-. 81 4I %RT 1:(9.P~Oth/r'^2)*A (2,/3,) 1,s 82 REMii=HElr3*VRT1/vRT
- 83 VRT=VRT I
.t,
- . 8L GO TO h 5 VRT=9.u3*((.-EXIP(-(D(I)/1.77)**1.1147))
* *i 86 GO TO ISi 87 17 CONTIN4UE 88 VRT=VNExP 89 18 CONTINUE .... t 990 REMH=VPT*D(l)*1000./VISMt
ly JF1(VRTT.Ubl.U.J Lu IU 1 93 Go TO lb 224 9L 15 CU N TINU 95 VR=vRI q6 Vt0=VRT 97 Z=O. 98 lb COWT ITNUE 99 KI=30 100 DT=(TZ-1I)/hl 101 DT1=lD 102 KJ = I + 10 103 BDT=0.99 C C
- A I LI DO) 100 ,=1 ,tJ C **,t*** ********t******* **************.
C 105 IF(.K.;T.U(J-l1)) GO l) 12 10b GO tU 13 107 12 COTINIUE 106 IF(K.GT.(KJ-7)) GO TO 14 109 DT=UT1/5. 110 GU TO) 13 111 14 RDT=HD)T+Q./10.**(IK-3) 112 7VNr=t3[l*TZ 113 D I = TVf, - I V 11LI 13 C EO4TIt:UE 1 15 TV=TV4i.)I 116 TVM=TV-DT/P. 117 IF(1v.GT.0.Q99Q*1l) GO TO 34 118 TF=(TZ+lv-l1/2.)/2. 119 TFv.=(TZ+TVM-.)T/l.)/2. 120 CALL PZAS( IF ,fPZ7 ) 121 CALL PZAS(7FM,PZFm) 122 RF=P7F/P1 123 RF M=PZFf /tP T 1 24 CALL SEKPV(l, PV) 125 CALL SFKPV([FF!,PVM) 126 R)PF=PJ/PV 127 ROPFEmP1/PVm C C VISCOSITY OF AIXTURE-.FlL04 C* 128 CALL VISASM(TF RFR0PFVISMF) 129 CALL VISASM(TFMRFmvROPF M,V1SMFFm) C C IHERVAL Lt!NDUCTIVITY OF MIXT7LRE -- FILM C 130 CALL AeflAStl(TFFiCF',2,CPIJ2,CPARAMiBMF) 131 CALL Atl6ASP"(IFtti4F~wCl-N2F1,CP02M#CPAlmANBlMFe.) C C DIFFUSIVITY-OF HIXIURE -- FILM C 132 CALL UlDFASr( ,7FF,DJFN F) 133 CALL UI1FAIS(TF!,-RFPiFmFN) C .. C DENSI1Y O1F MIXTURE-FILM C 134 RME=#S31J.7/ RF*1 E.0()t( 1 .- If )*2fi,9b) 135 RiFm=831Lu.7/(RFMs*1 n.02+ (1.-kFMl)*28.9E,) 136 ROtiF=M100t)(i.*PT/U(q4M*(TF4273.15)) 137 ROME H=1 0000.*PT/ (RMFtM* (TFM+273. 15)
C HEAT CAPACIlY OF mIXTU5RE -- FILM 225 1 3 hl CALL CPPAkCPZF#,T~lCPP) 139 CALL CPPAx PZFi,TFm,CPPm) r 14Li CPA=0.71i*CPl~i2+(I.21*CPU2+0.fl1*CPAP C *CP02'i+l. 01* CPARP IU1 CPAM=O.i *P -(21
. 1 42 CPM'F=RE*CPP*1000.4(Is-PF)*CPA/28.qb
* '143 CPMFM=F#-1*CPPMi*OO(J+(1.-RF1/2l)*CPAM/28,.9b C NUM1ILR PkAFJDTL C
*PRN=CPrMF*V1SM~F/Amt8MF/10.**7 C
PRNM=C P MFM VlSs FM /AM ~ MP / 10 ***7 C NumbEk REYNULDS
- 1's7 C REN=VPl*O)(I)*ROMF*'10000./VISMF RE?;'I=VRl*O(1)*k5)?F~1*1U000./VISMFM
;4MMFkF*18.02+(1. 4 di*2&8.9b 1L16 kimFM=F~I*1b.02+(1.-¶V?1)*28e.96 150 wmMF3=P(J)*Id,02t(1 .-F(J) )*26,9b 151 152 NE~1 153 IF(VRTT.GT.0.) GO TO b~OO 15" 155 £505 REN=VkaI(1)*kHIMFiIOO00e/VISmF 156 RENM=VR*L)(1)t*HOMPM*10000,/VISMIFM 9 157 i~Vp*RO KH /y jHi-?K *: 1513 70OLDTHL1 LI+ THE TA 159 600 CONTINUE C
C SCHMIDT NUMBtR I- - C 160 LJOLJ SCN=VlSMF/ P.omF/DIFmF/1 0.**7
- 161 SC NM=VISHF H/h0MFMi/DIPF11/10.l**7 L C
C P.CLE T NIIMtILR C I. 162 PEN=RENtSCN 163 PENM=RENM*SCNM C .. II C NUSSELT NUMBER C 164 IF (REN-LJ50. )7r7r6 165 7 USN=2 6 40.6* (P.Er'*0.5)* (PRN** (1./3.)) 166 USN.9i=2.+0.b*(XENMi**0.')*(PRNM**(1./3.)) C C SHlERivOUP NUM!BER
- . C 167 1 bf 169 GO TO 9 170 8 USN=2.,o.27*UPFEN**0.b2)* (PRl~**(1./3,))
171 USNfl=2. +0.2 7* (PENM**0.62)* (PkNM4*A (./3.)) 172 173 SHNM=2.4o.27*(WENM~**0.bŽ?)*(SCCNM**(1./3.)) . W-1, C C HEAT TRANSFEk COEFFICIENT : . C 174 9 H1j0n0,*AmIJMF*tSN/D(I) . .* .FR 175 1M=1 0006 *AMbMIV- *USNM./D (I) C
".....T
C 176 r^M.HlR(J)*1M.Q2i( 1 .. R(J))*28.96 226 177 F=SHtj&N/PEt. 178 FM=SH'iM*l/PLNM C C MASS TR/t.SFER RATE C 179 CALL PZAS.(TVPZV) 180 TNA=F*(AL(JG((1.-PZV/iT)/(I .- PPP/PT))) 181 CALL PZAS(1IVM,P2ZVM) 182 1NA)=F* ( ALLUG(( I.-PZvt1/PT)/(1.-PPP/PT))) C C HEAT OF VAPURIZATION C 183 CALL SEKI(IF,HSEK) 18a CALL PHImjl(TFHPRIM) 185 HVAP=HSEK-hPRlt M 18b CALL SEI(TFM,t1SEK) 187 CALL PRIII(1FErHPR1',) 188 iVAPM=HSEK9-hPRlI C C SENS1iLE HEAT 1RANSFER C 189 ACC=TNA I8020.*CPP/th 190 ACN=ACC/(l.-ExP(-ACC)) 191 192 ACCM=1 NAM* I b020. *CPPM/tHM 193 ACNM=ACCL/( 1 .- EXP(-ACCM)) 19Y USM=ACNM*Hm*(TZ-lV+D7/4.) C C LATENT HEAT TRANSFER C 195 QL=HVAP*TNA*1802O0 196 r)LMF=HVAPM*TNAMA18020. C C TOTAL HEAT IkANSFER C 197 UrT =L+CS 198 Q M:=OLl+rJSM C C APPARENT iEIA1 TRANSFER COEFFICIENT C 199 HAPP=IQ1/(T7-TV4OT/2.) 200 HAPPMQ1m/( rz-TVM+1)7/4 .) C C B6OT tNUMBER C 201 CALL AM;DA(Pl,TV,VLAMF) 202 6I=HAPP*U,(I)/VLAfrti/20Uo. C THE HOOTS UF TRANSCENVEN:TAL E(UL'ARION TAN(X)+X/(B1-1.):0. C 203 CALL 1TRAN4EC(tI.Y) c *****,**'*********.*****~ *i******************t******.****.******t***t C 1IME TAhf.N Fuk D)ROPLEl 7EMPElRA7URE INJCREASE: VT--HALF INT~kVAL C C SEA.RCH oLt niuD 204 EPS=0.001 205 tHA=0.no0o1 206 ThH{Jrd1 207 IF(K.t;T.I ) GO lG 371 208 1EMPR=(TZ-TV)/(TZ-TO)
209 GO TO 372 210 371 CONTI 'LJl 227 211 TEMPP=TZ-TV 212 372 CONATINUE C C THLRMAL OIFFUSIVITY OF YATER DROPLET C ( 213 CALL CPVLD2(PT,TV,CPV) 21 4 CALL VVov0.5(Pt,1V,VVUD) 215 RUV=:./VVO') 216 ALFA=VLAmd/CPV/ROV/I 00(. C C FOURIER NUP18ER C 217 FONA=4.*ALFAATHA*(10**6)/(U(I)**2) 218 FU-Ji=3.*ALFA*1H3*( 10**b)/(D(I)**2) 219 SUMA=0. 220 SUMi=( . 221 DU 110 L=1,200 222 IF(L.GT.20) GO TO 109 223 X(L)=SNGL(Y(L)) 224 GO TQ' 108 225 109 X(L)=X(L-1)+Pl 2?6 108 CONTIIMJE 227 IF(K.GT.1) GO 10 310 228 AISA=(SIN(X(L))-X(L)*COS(X(L)))/(X(L)-SIN(X(L))*COS(X(L)))*(SIP. i 229 GO TO 311 230 310 CONTINOE 231 AIS=SSIN(X(L))/(X(L)-Slr4(X(L))*CUS(X(L)))/X(L) 232 I.. 233 S2A=AI*i,(I)*(2.*x(L)*SIN(X(L))-(A(L)**2-2.)*COS(X(L))-2.)/ 1 (2.*x (L)l 234 S3A=A?*I)(I)**2*(3.*(x(L)**2-2.)*SIN(X(L))-X(L)*(x(L)**2-6.)* ICOSx(LL)))/(g.*X(L)**2) 235 SI =A3**(T )**3*(Li.*X(L)*'(x(L**?6.L)*SIN(2(L)*)-t(x(L)**(-12.*X(L i 1**A2t2U.)*CLIS(X(L))+2a*)/(8.*X(L)**3) 236 237 SbA=A5*1)(I)**5*(X(L)t(6.*X(L)A**-120.tX(L)**2+720.)*SIN(X(L?))-( IL) i*b-30. *.x(L **,u+3hn.,tx(L )**2- 120 .)*COS (X M )-720. ) /t3?.*X(L9 238 S7A=A6*0(I )**b*( C 7.*x(L)**b-210.*X(L)**14+2520.*X(L)**2-501401-)* X L
- l. ISIN(X(Li)-X(L)*(X(L)**h-42.*X(L)**4+b4O,*X(L)**2-50ono.)*Acos 2/(b4.*X(L)*Ab) 7 239 S6A=A7*D(J)**7*(X(L)*(8.*X(L)**b-336.0*X(L)**u+b 20.*Y(L)**Ž2in l14o32o.)tSIN(X(L)()-(AL)**56.*X(LX()*btofbo8.*x(L)**14.201b0.*.X(
C. 2**2+40320.)*COS(X(L))4e0320.)/(128.*X(L)**7) 2440 SUSA=SI A-S2A-S3A"SJA-S5A-SbA-S7A-S8A 11 241 AISA=AJS*SUSA IL. 2412 311 CoU4 ITNUE 243 STEPA_(X(L)**2)*FONA 1 2q4 IF(STEPA.GI.175.) (; TO 107 245 SA=2.*AISA*EXP(-STEPA) . 246 SUMA=SLIM A +SA 2147 IF(A8S(SUMA/10.k*°).UT.ABS(SA)) Go TO 107 V 24 8 110 CONTIJUE .4L % 249 107 FTHA=1LMPR-SIJMA
,_- 250 DU III L=1,200 .S$
iI . 251 IF(L.GT.20) GO TO 112 ,. 252 x (LM) Ss;L ( L ?) 253 GO TO 113 - 254 255 113 CONh1 I'luE
S _ s ~~r t r'. sI.1 J '.v' us 257 AISP=(SIN(xL))-X(L)*COS(XL)) )/ X (L)SIN (xCLnCOS(x(L) )*(S1.(x IL) )/X EL)) 258 GO TO 313 259 312 CUNTINUE 2bO AIS:SJN(X(L))/(X(L)-SI1i(X(L))iCOSCX(L)))/X(L) 261 Sltl=(t1-1tl)*(S1N(X(L))-xCL)*CUS(X(L))) 262 S2H=A1*i)(] )*(2**X(L)*SIN(X(L)C-(X(L)**2-2.)*COS(X(L))-2.)/ 1(2.x CL)) 263 S3d=A2*tv(l)*s2*(3.*CXCL)*t?-2.)*SINCX(L))-XCL)*(X(L)**2-.),* lCOS(X(L)))/(C.*X(L)**2) 264 S48=A3*A(I )** 3 C.*x(L)* MXEL)*t2-6. )*SIN(X(L) )-(X(L)i*L4-12.*X(L) 1**242.)ACagSCCL))+214.)/C8.*XCL)**3)
- 1) *-2t* CL) **?+1?0. ) Clx CL)) )/C lb.* XL)*)
266 SbESAS*I)(I ) **S*C (L)*Cb.*X(L)*1f-2o.*x(L)**Ž+72o*)ts1,(xcL))c(xC iL *O30}. (1.3* +360.*x (L)**2-7?O.)*cO~s CX L) )7?o. )/(3? XL*5 2b7 S78=:Ae*V(1 Jt*haCC7 .*ZCL)**b-210*XCL)**14+?52 ?*X(L)**2-5tJrOd)* ISIN(X(L))-X(L)*(X(L)**h-42.*X(L)**14+840.*X(L)**2-5040.)*C(JS(X(L)) 2/(6LS.1XCL)**c,) 268 S86=A7*b( I)** 7*(X(L) A' 3 *x(L)**4+6720.AX(L)**2-1JO3?.U.)tS1N(X(L))-(XCL)**8i-56.*X(L)**b+1b8Oo.iX(L)**LI,2t01h0.*x(L) 2**2+40320.)*COS(X(L))+41)320.)/(128.*X(L)**7) 269 SUS5=SIO-si -S36-SUB-S5b-SbB-S7B-s88 270 AISt=AIS*SUS6 271 313 CONTINUE 272 STEPH=(X(L.)t*2)*FON8 273 IF(STLPS.(;.I~75.) (;0 10 IllU 274 SU=2.i*AJSF*txrP(-STEPb) 275 SUr48 SLIM9+ S 27b IF(ACS(SJ'it/1O** h).GT.A8S(S5)) GO I 0114 277 111 CONI.PUE 278 114 FIH;;1E-MPR-SUMB 279 N=O 280 207 14=1+l 26 1 xTh=(THA+TIHP)/2. 282 IF(XTeI.GT.0.9Q95*llI)GO T0 91 283 7EMf'=ILMPR 284 FUNX=ij.*ALFA*XTH*(10.**6)/(D(I)**2) 285 SUtx=(). 286 D0 120 L=1,200 287 IF(L.GT.20) GO TO 115 288 X(L)=SNGLCY(L)) 289 GO TU l1b
.290 115 X(L)=X(L-I)4PI 291 1lb CJtj TINlt
- 292 IF(K.GT.1) Gu TO 314 293 AISX=lCSIJ(N(L))-X(L)kCOS(X(L)))/(X(L)-SIN(X(L))*COS(X(L)))*(Slt(NX(
IL))/x(L)) 294 GE) TO 315 295 3114 C0O4T1q1JE 296 AIS=Slu(x(L))/(X(L)-SIN(x(L))*CUSCX(L)))/X(L) 297 slx=(1Z-AO)*(Sl:txCL))-x(LiiCOStxCL))) 298 srx:AI*O* C. 1(2. iX L)) 299 S33XA2*O(J )*A2*(3.*(x (L)*A2..2.)*S1W(X(L)).X(L)*CX(L)i*2.b.)* ICOSCX CL)) )/(LJ.'X(L)**2) 3300 S4 X=A 3 IA3*1)(1I)***3 (U. *X(L) (x (L)*s2-6. )*SIN(X(L)CX M(LA* a- 1 .*X(L) 30 lt*2t2Li.)ACt)(X(L))+2U,)/(!j*x(L)**3)i 301 S5x=AIA)(I)***I C(5.*(.L)A*4-oo.*X(L)**2+120.)*SIINCY L)) xCL)*AX(L 1 )**u-~2o.*xtL )**e+2Ut. )*CtJS(X(L)) )/(16.*X(L)*sJ) 302 S6X~Abi,)C1)*^jt t C IL *o3*(L*qojttJ*-t!)C S(()-2./3.(L*-'
303 S7X=Atb*I)( I)**6* (( 7.-*t (L**6-21 U.AX M**[J+2520. *X(L**2-50uO.)* I SI N( X(L ))-X (L)*C(L)M*.*X(L).*X (L) *+0. *X(L)**2-50L140.)COS(X((I 2/(',0.*x CL) **b) 3Q04 S8X=47*;( I)* A 7* Cx (L) (13.*XCL )t*b33b.O*x CL) **SJ+'72n.*tX L) **2-IL .2 .)A' N L)) ( * - 6.* L) * +I)0 * ( * - 0 6 .* 2*2+u01320. )*COS (x(L))4aU320.)/ (12H.*X L)**7)
. 305 SUSx~s1 x-.S2X-S3X-SL3X-S5X-SbX-S7X-S43X .kC. 306 A S X=AJIS *StS sX 307 315 CONTIN~UE 308 S TEPX= (X(L) **2) *FEOIjX 309 IF(sTEPx.GT.j75.) GO 70 117 310 SX=2.aAlsx*LxIP(-STEPX) 311 SU9x=SUMiA+Sk 312 IF(AIJS(StIfIX/10.**b).GTAt4S(Sx)) GO TO 117 313 120 CONTINUJE 3114 117 FTMx=TEmP-SIjmX 315 IF Cf THX)212,210,211 316 212 lF CFTHX*EPS)205,2I0u 210 317 211 IF(FrHX-P.PS)210,21O,203 318 203 IF(FTHx*FTHA)205#208,20b 319 205 THB:xTH 320 F7Hti:FTHX 321
- GOTO 207 322 20i, THA=X Ir
.323
- FTHIA=ITHx 324 GU TU 207
- 325 2 Ob PklrOl 209 326 209 FORM.AT(IUHA IS TRUE ROOT) 327 GO I( 198 328 210 CONTINUE 329 GU TO 197 330 iq8 XTh=A I '*
A... 331 197 THETA=xTH
\-* 332 ]F(VWIT.GT.0.) GO TO 407 333 NE=NE+ 1 334 IF(ME.LE.3) GO 10 406 335 T=lHE7UTtMf~A 33b IF (NE, Eti .) GO lO 41 3 337 IF(AliS(T/TOLP)-l.).ELTi.ool) GO TO 407 338 GO to aJOb 'I 339 413 CONitIUE 3340 IF(NE.GE.20) t;u TO 414 l341 IF(AHS(T/1OL!)-l.).LT.0.02 ) GO TO 407 -.
4~ 342 GO TO 406 343 414 CONTINUE 3414 IF(NE.GE.3O) GO 70 415 345 1F(A#iS(T/TOD-1.).LI.0.03 ) GO 10 107 i 3746 GO 10 uot)
'.347 115 CUNTItNJE
'- 348 IF('NE.GE.U0) GO TO u07 349 IFIARStT/TOLu-l.).LT.0.04 ) GO TO 07 C C SURFACE 1LNSION OF hATER IN AIR C 350 406 SURFT=(75.6-0. 15*TV-0.25*10.**C(-3)*TV**2)/1000. C C. SURFACE TENSION GROUP C . 1
. 351 IF(vDUP.(;T.O.) GO TO sOs
- -- 352 CALL VELUOU' 1:. l 353 GO Tl 1105 354 505 CONT I NUE :. i-I- 355 CALL vELl)UP ..1
35b GU 1OnUiS 357 4O7 CUNTINULE C 230 C AVLkA(;E TEMPERA1lUfE OF DRUPLET C 358 FONAV=I.*ALFA*1HE1AA(10.**6)/(D(I)**2) 359 SUMAV=O. 360 DU 151 L=1,20)0 31 IF(L.(;T,2U) G0 To 152 362 X(L)=SitiL(Y(L)) 363 GO IU 153 36U 152 X(L)=^v(L-1)+PI 365 153 CUONr1!4UE 366 IF(K.GT.1) Gu TO 320 367 AISAV=((Sl1f(X(L))-X(L)*ClIS(X(L)))**2)/(x(L)-SIN(x(L))*COS(x(L))) 368 GU TO) 321 369 320 CONTINUE 370 AIS=(SI.((X(L))-X(L)*COS(X(L)))/(X(L)-STN(X(L))*COS(X(L))) 371 SIAV=(TZ-AOJ)*(SIN(X(L))-X(L)*COS(X(L))) 372 S2AV=A I *t)(l)* (2.*X (L)*Sit.(x (L) )- OX (L)**2-2. )*COS (X (L) )-2. ) 1(2.*X(L)) 373 S3aV=A2*D)(I )**2*(3.*(X(L)**2-2.)*SIIJ(X(L) )-X(L)* (X(L)**2-6.)* 37U S4AV=A3*O(1)**I (4.$ (L.)* tX(L)**2-6.)*SIN(X(L))-(X(L)** -12.*X(L) 1**2424,)*COS(X(L ))t29.)/(O.*X(L)**3) 37S ~S5AV:AL*ol ()*tLJ ( C.'X (L)t*4-b0.*X (L)**?+1?0. )*SINX(XL))-X(L)*t(x ( 376 ~ 1)**Li-2O.*X(L)*t*?+12U.)*C(,3(X(L,) )/(1b.*zCL)**Us) 376 S6AV=AS*1)(l)**S*(XCL)*Cb.*xcL)**a-1?o.*X(L),*2.7ao)*sI'JcxcL))-(x lL)* C-30.*X(L)**U+3b(.iX(L)**2-120.)*COS(X(L))-720 .)/(32.*X(L)*?S 377 S7AV=Ab*I)(I)**6*b¶ ( 7 .*X(L)**o-210.*X(L)x*4+?520.*X(l)**2-50UO.)* IS/IbJ(X(L) )-x R)# Dr ( ) **b-a2.*x M **U+8dO.*X(L)*-2-5U D. )*COSOX (L) 378 S8AV=A7*1)(I)**7A(X(L)*(8.*x(L)*Ih-336.)*X(L)**u+672O.*X(L)**2-2**2+Yu32u.)*COS(X(L))+UU3?O.)1(128.*X(L)**7) 379 SUSAV=SA1 a v-S2A V-S3AV-SUAV-S5A4-S6AV...S7AV-.S8AV 380 AISAV=AIS*SU@SAV 361 321 CUNTr1N1JE 382 S7LPAV=(Y(L)**2)*FONAV. 383 IF(STEPAV.l;T.175.) GU tO ISa 38'J SAV=6. *AI$AV*E E'(-STEPAV)/X (L) **3 385 SUMAV=SUMAV+SAV 38b IF(ARS(SUN;AV/10.**6).GT.A6S(SAV)) GO TO 154 387 151 CUNTINuE 388 I Sa CONTINUE 389 IF(K.GT.1) GC TO 3hl 390 TAV=TZ-(T 1-103() *S1v 391 GU To 382 392 381 1AV=T?-SIJmAV 393 *382 CONTIN(UE 39LJ FONNra.*AL~.I; HtA*ClO.*it6),C(1 )*i2) 395 DE) 301 'M= 1, e 396 RD(14)=MAU(1)/40. 397 RNC(I)=SN(.LL(rD(m)) 398 SUMN=O. 399 DO 302 L=1,260 400 1F(L.GT.20) GO TO 303 L01 X(L)=S.JGL(Y(L)) 402 GO 1o 304 403 303 X(L)=X(L-I)PI1 4'04 304 CON11NUE 405 1F((K.GT.I) Go TG O 37 0 -i 406 ~ ~ A TS?4= (SINJ x(L )X (L)*C OX (L ))(DI /P )S 1 N(X (L 2. N;F)/t1)
* '407 GO TO 305 231
£108 300 CONNTI'd.F
'09 1H N ( F ) / X (L) 10 Slim=(TZ-AO)*(Slt(X(L))-x(L) 'COS(X(L)))
411 S2N=A1*IO(I)*(?.*X(L)*SIN(X(L))-C((L)**2-2.)*COS(X(L))-2.)/ 1(2.*X(L)).
'412 53N=A2*t)(1)* (s30*CX (L) A*2-2. )*SINCX CL) )-X(L)*(X CL) 't2-h.)
lC0So(X(L)))/(a.*X(L)**2)
'413 SLN=A3*II(])tA3*C~U.X(L)t(XCL)**2-h.)*SIJCX(L))-(XCL)**-lJ~2.*XC 1*A2+2u.)AC Os(X(L))+2u.)/(d.*X(L)**3)
'414 S5N=:A*0(1)A*i4*((5.*X(L)**4-bO.*X(L)**2+120.)*SIN(X(L))-X(L)*C 1)*hU-20.*X(L)**2+120.)*COS(x(L)))/(l6.*1(L)*2'4) 15 SbN=A5*I)(I )*A5A (X(L)
- h. *X (L)**1V12IJ.*XCL)**2+120.)*StNCXCL))-
IL)**o_30.*X(L)A*4+3cO.*X(L)**2-720.)*COSlX(L))-720.)/(32.*X(L) u16 S7N~Ab*i)(t)**fA(C7.*X~t.)**b-210.*X(L)*AU+2520.*XCL)ta?-50£10.)* 1S4CtX(l.))-X(L)*(XCL)**t-2.tXCL)**'4+b4Vt.*XCL)A*2-50U0.)*CCJS(XCI 2/Cb'4. ~X CL) *tth)
£417 St4=A7*r)(T)*A7*(X(L)*(8.*XCL.)**b-336.0*X(L)**'+672n.*x(L)**2-1450320.)*SIN(X(L))-(x(L)**8-5b.*x(L)**b+lb8O.*X(L)**L-20160.*x C 2**2+,'!320..)*C(IS(X(L))+4o3?o.)/128..*x(L)**7) 418 SUSN:=SIN-S2N-53N-SUN-SbN-SbN-S7N-S8N 419 AIS.N=A IS*SUSN
- '420 305 cONiINUE 421 SlEPw4=(x(L.)**?)*f(PN
- '422 IF(STEPN.GT.175.) GO 10306
'423 SN=2. *A1Slj*EX~ (-STEPN) 424 SU N=SUM-i +S4 S-
£425 IF(At3S(SuSt./1O.**6).GT.A tSCS )) G0 T0 303 .6 426 302 CON.TINJE
'427 30b CONTINUE
. ~'* '42 IFC(4.Gr.1) Gu TO 330
\~ '429 TN(M:)=TZ-( Tz-T )*SufIe
*'430 GO To 331 431 330 1N m )=IZ-SLIMN 4 32 331 CONTINUE 433 301 CLUNTIIN'JE 1i34 CALL TSiiFIT(NR , TNAO.A I,A2,AA3,ALu,5,A6,AA7)
- '435 AO=SNGL (AO) 436 AI=SwiGL(AI) t
£437 A2=SNGL (A2)
'438 A3=SNGL (A3) 439 A 4= S 6 L (A J
- '44O AS.sNGL(AS)
Ab=St4GL (Ab)
£442 A7=SNGL(A7) 443 DO 390 .'l=1,10 TNP(m)= A0 A1*?N(M)+A2*-RN(.C)**2+A3*N(M)**3+A4*RN(M)**U+A5*RN(Ci) 1+ A 6
- N (CA)*6+ A 7
- i (1C)* 7 445 390 CON r I 4vxE 14 4 6 THEf U=THETUtflETA IFtvRTT.GT.0.) GO TO u08 JF(V0)lP.E;J.0.) GO TO 408 44B IF(vRRS.Gr.0.) GO TO 'Ot0
- 45 0 N1=1 51 u08 CONTINLIE
. 452
, IF(vRr7,(;1.0.) GO 10 409 . . ,
*453 GO TO 410
'409 COiNTINUEE 455 Z=L+.YT*IHEIlA
- 456 410 CO.NTINUE C
C 457 DO=(2.*INA* IA.02/rtllv)*TtETA*10(0U. 232 458 D (I 0t1 C C THERMAL UTILIZA7ION C 459 THJTr=(lAV-IO)/(TZ-tr) C C RATIO OF TIERMAL DIFFUSIVITY ANt) DIFFUSION COEFFICIENT C 460 A=AM6hF/(RSOMF*CPMF) 461 RTMU=A/DlFMF C C AVERAGE APPARENT HEAl TRANSFER COEFFICIENT C 46b2 HAPSTPP=iAPSTPtHAPP 463 HAPAVN=HAPSTP/K Lj64 hAPINT=tiAPINT4 (tAPP04L.*HAPPM:+HAPP)*IHETA/b. 465 HAPAVI=HAPl.tl/ThETU 466 HAPPO=mAPP C C AVERAGE APPAkENI HEAT THA:4SFER COEFFICIENT--CHECK UP C 467 HAPCH=Dm*CPV*(TAV-TAVOLI))*1o00./AREA/(T2T-TV+DT/2.)/THETA 468 HAPCHl=^PAh'tll.(1APCh0+MAPICH)*ThEFA/2. 469 HAVCH0:=HAPCH 470 HAPCTR=HAPCHT/THETU 471 1AVOL.=l1AV C C AVERAGE SErtSIt3LE HIEAT TRANSFER COLFFICIENT C 472 hSlEP=HSfEP+M 473 hAVST P=HS1 .P/K 474 HINj=HINT4(h0$+.AHMtM)ihEIA/b. 475 HAVIfTT=HI;N1/THETU 476 HO=H C C ENHA.NCEMiENT FACTOR C 477 EFSlEP=HAPAVS/HAVSTP 478 EFINl=HPAAVI/thAViNT C 479 DLSTP=(TZ-TAv)/(TZ-To) 480 DLSRtIS=( vRT AP PAT1.ETlV**3)/(D(I)**3) 481 PRiNT 25,TV, TAV,1)tI) ,TMO,ThUT, TETU 482 25 FORIAT7(2X,3htV=,ElU.7. lCtIH7AV=,L14.7,lX,5HD(I)=,Elt.7,lt,5HPTrTP=. IEI 4 .7, I X , 5tI t'Ll= , F Q. I1, I X , bH IH LT L =#EItj. 7) 483 PRId47 2o,TljAFrtelnAPPIAI8APVSt1APAvI lE . 7, I tAPAVS =',E 1' ,7,IX 'HA AV I= ,LILi .7) 485 PRIrvi 24,HHAVS7P,1¶AVANllEFSTF.PEFINTFONAV 48~~~~~ oI2 f F S T4Et = I , ' I Li ' 7 t1 x .,I ,F I T' li I F.
=5 I li . . 7 , II FX 0
, A VsT= Y i E7 1 .7, ,
1IEfSTEP:',El1.7,IX,'EFPI:F.Il=tflJ.?,1XIFONAV:IE1*.7) 487 PRI N T 23,TOL,)V4,IRfr NjE ,)lt APC7R s88 23 FElkmAI (2X, ',IIJLL'=,E1L.7, IX, 'VRI',E1U.7, IX, 'Z:',E 14.7,1X, 'KCN=', 1E14.7, IX, ;sE:',I3,1X, 'I)1=',F10.b, iX HAPC1R',lF-lf.1 3 489 9 ~ 'N* (7,22)1?UTPTVRO Tl1ETlI
- 490 22 FOR;b;A(5Llb.b) 491 PRINT 29,iYN, SCN,SH"IYUSfJ,W4S,OL,ACN,HAf'CH 492 29 FOKM¶T?,SN:',F10.5,1X,6SCNIF10.5,1XISHN:6,f10.5I1XIUSNo=
I ',F1O.5,1X,x'.(S=:,FlO.2,1lx, 'uJL=:,F0.2, lX, 'ACt=',F 1O.5,IX, 'HAPCH=' I 2,F10. 1)
4193 100 CONTINUE
- 494 GU 10 33 233 495 91 PRI14T '5 496 55 FORmlA1 (IU I,' lTEAI I0rN CA rT bE ACCUMPLISHED HETwEEN THE LIMITS lT HA &11H6, W(s')
£197 Go 10 33.
_ 498 34 CONTIN(E.
£199 PkINT 35 500 35 Ft)R;MAT('0','w ATFR SURFACL. IEMPERATURE IS GREATER THAN 99,99Z UF ITHE SuRqOUN'VINtC TE.PF.RATUJRE',/)
501 33 CONTINUE 502 PRINT 21,PPP, TZoRNPVISPVISvvTISM 503 27 FO9PA1 (2AxUFlPPP=,EI£.7,lX,3HIZ=,E14£.7,IX,£UHR(P=,EIU.7,1X,5HVISF 114.7,IX,59-vISV=RL.72 1X,oF4VISM'R=,EI4.7/) 50U1 90 PRINT 2OVRIkE@M"DTEFINTHAPAVI.,HVINT 505 28 FORMAI(?xat1VRT=,ElU.7, IXShkEM=,EI4.7,1X,3hi)T=,EIU.7,lXIEFIt 1,E1l4.7, IX, 'HAPAVI=',El4. 7, 1X, HAVINT=',E1£.7///) C 506 80 CGN T I NUE 507 STOP 500 ENO I C C r 509 SUBROUTINaE PZAS(T,PS) C ******************1*****a*******,*************************** C I.. 510 %=1/100. 511 PS=((((((((1.5531,/10jo00.*X-2.693a527/lOo00)*X+?..0s3393/100.) 18.b59o25/100.)*X+2.£177563uL/1l.)*x-5.60hoS9l/10.)*X+l.256759l)*x S. 2033727)*X47.7270ObV9)*XS-.0/b71) 50-. 512 PS=EXP(P6)/l .ul972 513 RETURN 51U1 END C
- t. C *************************************************************
515 sUkoL rIN4E TLMP1(PPPTZ) C i. C 516 X=ALOG( 1.01972*PPP) 517 T?=((U(C(t(((-U.2924i603/10U00./lo0o0.*X-L.26R5685/J1nOOooo.)*Xt 134373 I/looC)00O.)*X42.2071712/100000.)*X-1.7u17752/100o00.)*X-3. 23484/10oo. )kX+1.32h3773/ 1000.) *<+2. I 2Qo82/1.00. )*X+2. 107780ci/10 1.. 3X+2.3753577)*X427.851J2£2)*Xt99.092712 518 RETURN 519 END C C S.. C 520 SUB3ROUTINE vvOD)E5(PT,.TVvvor ) C *A***A*A*************x***********,*** C 521 A=(TV+273.15)/b47.3 v 522 S=:P /2e?1 .28b 523 U-100.* ( s7uoono-3122199.*A*A-.1999.85/A**b) I-. 524 w= (LI+ SUR1 ( 1.72*U*IJt13rh2 926. *1o.**1)*( S-I .50)7i;5* A)))**fn. 2 9L 17 1
- ., 525 b~ln52sh*bU.+5*13.o~efs)/lo*^lo~l.510ff/1000oO.fA**l10 52b C=(0.65311541-A)v*2
- 527 C=C.* (7.2£a1 1IbS/l10OOU0.+0.767bb21*C**i.)
528 VVo0i=0.417/iutC-( 1 *i.3Y7Ub-9.9£49927*A)/100000, - 529 RETURN
c 234 C 531 SUHRUUTItNE SE'<PV(TZPV) C *****t*****a*******1t *********t*******A*********t***********t****,* C 532 X=TZ/100. 533 Pv:({(((-.90o/00.%31531 no.)*- ssRslo.* 1.3t8ffbiI/o0O.)*X-3.&047S53/lOO.)*X+1.1U7b1UQ/1Ooo.)*x-y.OLjlqu35/ 2000.)*X+4.5905064/10.)*Xt1.26020a4 534 RETURN 535 END C C ** **********tti** .*********t********* ** ** * *** ** ******
- 536 SUBURUTINE CPvO)?(PTTv,CPv)
C t***t**************************a***************************** C 537 X=Tv/1no. 538 Y=Pr/1oo. 53q 2=x**5 540 CPV=z./(Y+2. .1*O21*Y-o.24927 541 CPv=CPV*(0.0321i*((0.O032ou+0.O26912*Z*7/1ooo00.)) 542 Z=(( (-1. 100I5*X+43.837206)-X-2L4.2088O3)tX+33.uuLj7l2)*x+3o.72526 543 CPV=Z*Y/i1000.4CPV 544 Z=t((2.259Q85*x-7.40325b)*Xt1q.09708e)*X-13.04363)*X+J19.,62594 545 CPV=Z/lt). tCPv 5 Lb,E IJfUN 547 END C C 548 SUPROUT1NL CPPAk(PT,1F,CPP) C C 549 X=(TF+273.15)/o47.3 55 0 S=PT/bI.2Ot 551 b=1.o28,287*(I.5510b-S/2.)*X**1.82/1Q000.-5.098bhzU6/1o0./x**3.82 552 C=(2.010376*5*(17.72214*S-]1.*X**3)/10OO.-O.2O51b15)/x**15 553 D=22128.703*(b*S+(C-1..53298LQa/x**33/10oo. )*S**3) 554 CPP=(1lo9.ltbfl-UtX*(-16.11O723+X*(22l.29742-X*52.106736)))/647.3 555 RETURN 556 END C C * **t 557 SUbROlJTlINE SEKI(TEINTA) C *iA*******~***iA** k*****t**********fi*************+**********t**** C 558 X=1/100* 559 ENTA=(((Ol.UC'65952*X-Y.4307509)*X+27,974559)*X-4@N.077814)*X+15.7 19571)*X+181.5Ubbb)*X+250O.6256 5bO RETURN 561 END C C ****** ***.* *****************1*f****** 562 SUS~RUTIN'E PRIMII(T,ENTA) C t*.******t'*t't ***********************.t A***t************t*i*1**** C 563 X=T /I O 5b4 ENTA=((((((((2.7bb3765/li).*X-3.9b29482)*X+23.865162)*x-7 P.22bh36)
I lx+151.59u21) *X- I 5. 790S3? *X+120.bO923) *X-U.b 9b35) *x+u2s.1 3 2-7.4~i55 1 0q3/l1(1 . 5b5 IFUT-350. 32.2#1 235 50b 1 E~AE;A0303*XPO255*730) 567 2 FiETURN 5bt E NI C
'C C 569 SU6ROUTINE AM'~DA(P,T,VLAmB)
C C 570 X=1.,T/273. 15 571 CALL PZAS(T,PS) ' . 572 VLA ((i6(0.292923-0.O 71b93* Y)* X-0.32929* *+0. .1S5)**(-PS)/10 O0 OC 573 VLAm8=(VLANti3+(((5.1536*X-20.012)*X*25.1?3b)*X-9.473)/1O0000.)*(P-7 57L1 VLA~ld=VLAML4(((O.52577-1.073*X)x*x-.I.oo )*X+2.8395)*Y-..92247 . 5 575 RE TURN 576 END C C
', 577 C TRANSCENDOENTAL FQCUATION SOLU11ON -- TAN.(X)+X/(BI-1.)=o.~--H~ALF:
C INTERVAL SLAK~H METHUD C C 578 IMPLICIT REAL*S(A-HtO-Z)
- 579 D IMENS IJUNX (20) 580
- 581 582 IF(a)AiS(C).LT.0.005) GO TO 79
(-' 583 79 CON TI NWE 58u D0 13 1=1,,20 585 IF(u)A8S(C).LT.Q.005) GO TO 21 586 IF(t31-1 .)20,21 ,22 587 , I 588 589 . tl 590 GO TO 23 .4w. 591 21 Y=(2*1-1)*PI/2. . Z1. 592 GO TO 25 593 22 PI=3.1la15926537 594 A=(2*1-1 )*PI/29 595 B=IAPI-0.O00001
- 59b EPS=0.00C)1*(A+B)/2./(01-1.)
597 23 FA=07AN(A)-A/(1.-HI) l 598
- 599 60 0 7 K=K.l 601 Y =(A413)/2, 602 TA=DTA.N (Y) 603 Cx:Y/C 604 F=14+CX 605 IF(F) 12,10#,11
*_06 12 IF(F+EPS)3,10),ID 607 11 IF(F-EPS)10,10,3
! A* 608 3 IF(F*FA)b,E&,c b09 5 t3=Y 610 FB=F 6 11 GO 10 7
- , 612 6 .A=Y . . ,!7.1
. ,-.: . ;-11W,
- 613 FA=F
.. .o I
bi1l GO TO 7 615 H PRI1wT 9 b16 9 FuR maI(1 l-4HA IS 1RUFE 1OL'T) 236 617 GO TO 13 616 25 PRINT 9hqY b19 98 FORMAT(1Ux,5HX(L)=#Fj5.8) 620 GU TO 13 621 10 COJNTINUF 622 13 623 R'ETURN 624 END C C C *A****ii***t********************s*** 625 SL16ROUT INE vELOD0 C *****t**********.********,**t*******%******t C C NUME1ICAL S(JLUIlION OF THfE EWUATION llF MOTION FOR A DROPLET MOVING C DOWvJwAK0S HRL,'GE--KUTrA TECHNIQUE C 626 COMMON K,IVUT.NE,VRKZKROMH,VIShtIG,.NI,NM,NN, THETA,VINFiV[)iJP, 1SLIRFT,ROV,I),vROIVRZTHEIllVRRS 627 DIMENSIONJ D(20) 628 IF(K-1)da8,408,409 b29 u08 VR=VRO 630 Z=O. 631 GO TO 410 632 109 IF-(rJE.LE.2) GO TO 415 633 VR=VHRK 63" Z=ZRK 635 GO TO 410 636 41 5 VPRK=VR 637 ZRKZ 638 410 IF(VR0-vR1)U51,#452 9 53 639 451 IFc(V-Vpr) 454,J5is*,uiJ52 640 a53 IF(VH-vRT).1bZ,052,4J54L 641 454 CONTINUE 642 RElY 1 R~;ll(0 9ft 643 CALL CDREINj(xE0,C0CVR) 6a4 PROD=3ooO.*'Mvi?*VR**2/(4*.A(l)*ROv) 645 CKU=G-PROD*CDCDR 646 RE1_(VN+(:KoATHETA/2.)*u(cI)*p(,Mf*loooo./vIsM-647 R EMI( V C Al)
- I r IA/ u. ) *1)( I) i Ila G0 0 0./ vI S vi 648 CALL CIPREIN (kE1,C1DCDR) 649 PR O=30oto. *RuOmb *(V R+CK0*ThETA,2.) *2/(LI.*)(I)*RO) '
650 CK1=-Pt; 0*osCOh'. 651 CALL C0) EJN(AE-E1,CDCr)P) 652 PROD=3000. *RUAi* (Vk4CKUtTHF1'A/4. )**2/(t.*r)(I )*ROV) 653 RE2=(VP+CKlATHE1 A/2.)*A)(])*RoMf*looo../VISMd 654 RFM2=(VY+Cn*TtiE TA/'J.)*vtl)*Hk(~HA l(Joo./VlSMB 655 Cim1=G-pRiu0D*CDCDp 65b CALL CC0IPNJ(RE2,C)CDVR) 657 PROD1=30OC.*Ltiiit(VR'+CKs1*TIE.TA/2. )**2/ (U.*D(I)*ROV) 658 CK2=CG-PNIJ0*CiDCDk 659 CALL CDkEIN(t-EM2,Cf)CLt)R 660 661 CKM2=G-PROO*Cr)cR5.I 663 REM3=(v +CKPf2*THlA/P.)*L(])*ROMis*loooo./VISMH 664 CALL CLJEIN(kE3,Ci)CL15<) 665 PRUI)=3 jI:'. CC2 OCA*21 t V*+
- 7 ;4) (D,*D I ROV) 666 CK 3=G-1'?Ui)*CDC Dk'1
667 CALL COD(IN:(KLM3,lOCLP) 668 P9c1u=35{]),). *it§Mf3( vi.+cr^m2 *rHETA2 . ) **2/ ( ;1 . U1 1)
- RUv) 237 669 CKM3= ;-P t(iD*CUDCr 670 Zl=VR-vlrjF 671 VRM=VR+(CKtJ+2.*CK.4l+2.*CKM2+CKMi3)*THETA/12.
672 Z2=VNm-vlNF 673 VR=VR+(CKO+2.*CK1+2.*CK2+CK3)*THETA/6.
. ".. 67 4 Z3=vk-v,£NF 675 Z=Z+(Zl+u.*Z2+Z3)*THE7A/6.
676 GO T7 u05 677 452 VR=VRr 678 IF (K-i )14b8,lbaL69 679 468 TIME=0. 680 Z=O. 681 GO TO u170 682 469 IF(NE.LE.2) GO Tn 475 683 TIME= r1iEC 684 Z=ZC 685 GO TO 470 686 475 T[mEC=TIME 687 ZC=Z 688 470 TImE=tlmF+THETA i 689 Z=Z+ (VIk-VjNF)1*TlE 690 405 CONTIV4JE 691 RETuRN 692 END C C C .~As 693 SUHROllJTI ;E VE LlUP .,, I C ************** **** ****s************.****************.***I ! r% C C NUMERICAL SL9LUIIUN OF PIE EOUATlUN OF MOTION FOR A DROP C UPmARDS RUNGE--KUTTA TECI4NIULIE c 694 COMMON KsVRI#NFVSRKZRKROMIVISMH1G.NNMpNNI)TVAIR, lROV,0),VR',l,VW,Z.THEltJrRRS 695 DIMEN4SIOJ I)(20) 696 IF(K-1 )406,408,409 697 408 VR=VRO 698 Z=O. 699 GO 10 410 700 409 IF(NE.LE.2) GO TO 415 701 VR=VRHK - L 7702 *Z=ZRK i, - 703 GO TO 410 7011 415 VRHK=VR 705 ZRK=Z 706 410 COtN1 INJE 707 REO=Vk*' s(I ) I 0i03*100./VISMB ir 708 CALL Cl)UEi1I(kf.)ICDCO)C 709 PRkOD=OOU.*IUMU*VR**2/(4.*i)(l)*ROV)
- 1 7105 710 lF(VAIR-VHT)hO(,1, bO 711 61 VAIR=vAlR-0.001 I -
712 60 IF(VOO.LT.VAIR) GO 10 210 713 IF(N.LT.1) GO TO 116 7114 GO TO 16 1 16 CONTlttUE 716 IF(NIIGT.1) GO TO 201 717 GO TO 151 r 718 210 NM:=NsiY 719 201 CONTINUE
I'U CUU Il) 1h 721 151 CONTINU2 722 CKo=-(G+PqRUl*CDCDR) 238 723 GO TO 17 724 lb CKU= G-P;WODI)*CDCI)R 725 17 RE1(v;i+CKo*DT/2.)*li(I)*,umB*loooo./vlSiH 726 REMI=(Vq+CKo*o /n .)*i'(I )*'Oh8*IOoo./VlSl.m 727 CALL Cf)RLJN(RE1 CCt)VC) 728 PKOO=3(1G0.* (vR+CKO*UT/e. )**2/(4.*c)(c)*RoV) 729 IF(N.LI.1) Gb *0 118 730 GO TO 16 731 118 CN1TI 1LUE 732 IF(wi.GT.0) GO 1U 18 733 CK1=-(GiFRDIP*COCD[) 73, (;O TO 19 735 18 CKl= G-P~vl,*C0CuR 736 19 CALL. CVR.k1N(REMEM1C0CDR) 737 PROD=3(0t).*(VR+CKO*Dl/4.)**2/(4.*D(I)*Rv) 738 IF(N.L1.1)GO TO 120 739 GO TO 20 740 120 CONTINUE 74l1 IF(NM.(;T.O) GO TO1 20 742 CKM =- (G+PR(10*CJCIR l 743 GO ro 21 7u4 20 CKM1 G-P~frD*CL)CDk 745 21 RE2=(VP4CKI*0T/2.)*D( 3)*Ro*:F3*10000./visM6 746 REM2=(Vt +C. 1 *[jT//4. )*ED(1) *RrJMI* 1 000.o./VIS.MI 747 CALL C0HEIP:(kE2,Cl0CUR) 748 PRL)D-3i)0fJ.
- vk+CK1 *1)1/2. )**2/ (Y.*D (I) *ROV) 7Y9 IF(N.LT.1) Gu TO) 122 750 GO 70 22 751 122 CONTIN4UE 752 IF(rm.G1.o) GO TO 22 753 CK2=-(G+PROD*CDCDR) 75Y GO To 23 755 22 CK2= G-PR(oJ*CI)CUR 756 23 CALL COREIN(kEm2,2C)CpR) 757 PkOD=3000.*iu' eA(V'+CCKM1*DT/4.)**2/(4.*D(I)*ROV) 758 IF(-4.LT.1) GO 70 120 759 GO TO 24 760 124 CONTINUE 761 IF(NM.GT.0 GO TO 20 762 CKM2=-(GtPNOD*CI)CDR) 763 GL1 TO 25 764 24_ M2: C--PkUD*CDCUR 765 2S RE3=(VR+CI2*&T T)*D(I)*RUri.I31Q0000/vISM !
766 REM13=(ve+CKM2*OT/2.)*D(I )*RO1M*1OOO./VISMBi 767 CALL CDREIN(LE3,CVCDR) 768 PROD-3t(lo.*ks) e'tr (v. CK2+rT)**?/(.e*D3(I )*ROV) 769 IF(Nt.LT.1) GU TO 12b 770 GO TO 21 771 12o CONFINIJE 772 IF(14M.GtO) GO TI) 26 773 CK3= - (:;+PR(e*CoI)CR) 774 GO T) 27 775 26 CK3= G-PROI)*CUCDR 776 27 CALL CINE.1NtREP'3,COCDR) 777 PROD=3000. *RCIMB* t?+CK.>5e*112. )*21 (L.*( I *Rov. 778 IF(U4.L1.t) Gu TO 126 779 GO TO 26 780 128 CONTINUE 781 IF(titl.G1.o) GO TU 2. 782 CKM3=-: (G4 PR)i CDCfOH)
bU IiL e'
- 784 ?6 CKM3= G-Pkoji)ACI)COIJ 785 29 CUNTt JIJW 239 786 IF(IJ.LT.1) GU TO 130 787 GO T'.) 7 788 130 CON r I UJE
--. 789 IF(NM.rGT.n) GI) TO 7
(. 1790 Z1=VR+VAIW . 791 GO TO 51 792 7 Z1=VAIR-VR 793 GO TO S1 7941 51 VRM=vW+ (CsSO+2.*ChM142.tC1M2+CKh3)*TO/12. 795 IF(N.LT.l) GU TO 132 796 GO T0 71 797 132 CONTINUE 798 IF(t4.i.Gl.U) GO 10 71 799 Z2=VRM.+VAIR
- 800 GO TO b3 801 71 Z2=VA1R-VkM 802 GO TO 53 803 53 CONTlNUEE Vk=VR$(COt2.*CK1+2.*CK2+CK3)*DT/b.
805 ' YRS=VH 806 IF(VH.LT.O.) GO T0 112 807 GO 711IIt 808 112 CONrITNIE 809 vit=4tS (t 81c) 810 111 CONTINUE 811 IF(t.Lt.1) GO TO 134 8612 GO 10 72 813 1314 C(Jr4 r[NJE 814 1F(tim.(;7.0) GO TO 72 8 15 Z3=VH4 VA IR 816 GO TO 35 817 72 Z3=VAIR-VH 818 GO TO 35 819 35 CONTI NlIE 820 L=Z+(Z1+u.*Z2+zi)*01)/. 821 56 CUr4I ItNE 822 IF(rl.LI.1) GU/ TO) 1115 823 Go Tu 73 824 145 CONTINUE 825 IF(NMG.1.0) GO TO 73 826 Vl)=VR+ VA I R 827 GO TI) 41 828 73 VD=VA1R-VR 829 a1 CONT1NUE 830 T=lHt.TU+L)T 831 PRINT 1lJ2, CKIJCK#CK2,CK3,IEORF.I ,RE2,NER3,'iNNN L 832 102 FOURtAI(1xl44(:O=oFIO.5 l tHCKI= .FJO)51XxIOeCK2=,F15,[X 110.LS),xtdHmE's)=tF5.1FlXra>4RE1=,F5.1,1x^UJHRE2=,F5,1,tXtUtiRE, 21X,2HM,=,IS#Xl 2HN-=,13, IX3,4NN13) 833 PRf411 TlVNZVI),VNSZl,2,Z3CCICR,;fH 83 4 101 FORMAT27 15HVHRS=#,F5922X,3bIZI=,F5.2,2X.3HZ2=,F5.2,2X,3HZ3=,F5.2Z1X 2,F5. 1 1 x,3HNM=, 13) 835 RE TURN _ 836 END C C ** *****A*****
- A** **** A* **A*** * ***
837 SuF~eiouJIftJk cV)HEJN(RElCI)) C **AA*t4*A*************I*****
c 240 838 Ct!Sr TL},f.V, lop.E I,VFK, Zkt%, Th(It , v ISk3, t, N Nti, W4, THET A, VI NF, V1!UP, ISUqFIRl)V,'L)vROO'I#VhZ*TmETlJ,,VPRS 839 uImE*4SI;.. fD(20) BilO RE I =A US (< )E 84 1 S=:,(aj3e* SURFi I *F**3*10.**2$/G/VISMit*,*1 8.42 UTELAR =I2L-HNOM3 81Jq LDST~R1~1 .2I'r/ C!)f.L'd3/idili) ** ( 1 /3. ) 8146UiA50. 8YS U)STAZb3OO.A6(a)LL'L/I.M)** (-l.S3 )t (S*10** (-12) )**(n. 1111 8Uo9 DSTA21h).8*(ODLLNO/PIJMId)t (-O./3.U)*CS*1O.** (-12))t*(n .111J) 850 (- I 0. 2 (02)=974.*(S*10.** 851 E32A=23 .2 (Sa 1 L.** (- 8) )
- 3dl.
I92 852 RE3A:?22LJ0.s(fL /~&^*(o. (S*1O.** t(-12) )**o.3AM 853 PE3 =7L1i.*(ULLRU/iLlLUr0 ) *a (-0.303) * (S 1 . *(- 2n) o. 8 855 HE5UJJ~7O. * (DE.L<iU/KNfM) ** (-0.238)*i(S~lO.**(-1.2fl**o.232 856 D;M XS=2 .2 (I)EL./ RC) 1u ** (- 0.5)* (S* 1 I. *12))**( .2b.) 857 COMAX=0.75*I ELxU/koimk)**(-0.0232)*(s*1o.**(-12))**O.o36 858 k SIAP=3.273* ( (.)LR.D/ rOM)*S**o.S5) **o.2b2 859 Y:. A XS=RE5/ID;A x S 860 PSTALI=F4/0)3TA;:4 861 wSTA3A=Rh3A/t})STA3A 862 7ST A3t=Y3f'r/USTt3 863 .NS1A2A^=EdA/OSTA2A 8b1 wSTA28=RE2i/DSTA26 865 WTt-Rl=U.j1S7)SAR-866 DSt =th. XS 867 A~0.0fl5-( 1./?9.ba)*ALrGlocS**o.5/cDELpa/RoM5)*1O.**(-3)) 868 IF(REI.L.F.0.5) r-o To lo 869 Ius.GT.4'J.**1s) GO TlJ 20 870 1FU(tE 1.G(.t. E2A)60 TO 25 871 C(,(2u./RE1 )t(1.4t0 .150*HEIA*0.b87) 8721 ./3.) 8732 873 v!STA= Cu. *RE '1Aot.RU/(3.
- iC)R0Niv) )%*( /3.)
874 GO rW 210 875 20 CONTINOE 876 IF(P~l ,G.'rE2Ri) GO TO 26 877 -CD=(2a1./RE1)*(1,+0.150*9Lt**0.6b7) 878 DSTA=(3. Li' *CO*Rueb/(1J.tOEL,@(J))**(1 ./3.) 879 . S1A=(14.E1*OELNO/(3. Cl,*,oFd))*i (I./3.) 880 GO TU 210 881 25 CONTINUE 882 1F(REl.G7.IRE3A) GO TO 50 883 Co)o.u6*(S*10.tt (-12)) **v.05 884 U)STA=(3.keE I t2 *CD*NL(,mi'/(.*(.-EL RO) )** (1 /3.) 885 .. *.SI lt=.tkI*GDE-LR/(3.*CD*RO16) )**(1./3.) 886 GO 1JU 210 887 2b CUNTlNtUf. 888 JF(l.l.RE36) GO 70 50 889 cO=o.7'(S*1O.'* (-12)) (-0.05) 890 . US1A:(3.tNE1**?*CD~iRCI~ti/(;I.*I),LLNL1))**
................. (1/3.)
891 ,............... STA:(1j.tIE1 DELiW(2S3.*CV*RUn~i))**(I./3.) 892 GO TU 210 893 10 Co:1?U./WFI
- 894. . .............. (u . It.t))
) *(I./3.LS =(. tEI*2*1)*(13 895 .STAI(4.*kEEI*OLaU/(3.*CV*ROM))**t1.,/3.,)
896 GO 10 210 897 50 CONhTIN;UE 898 lF(RE1.(;l.RE4) GO( 1U 60
899 11I .214) 241 900 901 902 GU T0 210 903 bo CUNiT INUE 404 9 IF(iRl *r$T*p?5) rGo TO 70 qou 905 906 907 908 GO to 210 909 10 CONTINUE 910 11 Gku*ER*)S*33* IA2Ri,3(Ai (r.9T-DMiAXS)4COMAX) 911 GRR:=2*Eii*)7*23*E*2PO4* 912 OSTN:O)S1-GNt/GRPRIM 913 IF(AtiS((,)SI1N-0ST)/DST).L1.0.001) GO TO 12 91 a DSI=USTN 915 GO TO) 11 91b 12 CO=C~AX+A* (0STNvDMAXS) 917 WST=SIJR1(UJ.*i2ELNO*U~SN/(3.*ROM8B*(A*(DSTN-O)MAXS)+CDmAX)1) 918 DSTA=USTN 919 WIS IAZWS T 920 210 CONTINUE 921 C. R7uRN 922 END C 923 SU(4IRhUT I NEI SQF IT( XIsY IsAO, AI,,A2, A3,AJ,A 5rAhA7) C IMPLICIT P~LAI.*8I(A-?4,U-j)
- 925 11CC 10) C(10,10Q) R(20) k.. 92b N=20 927
- 928 Di) 1 I~l,N 929 930 931 932 A (14, )X I((I )3 933 A(5 I )=X1(I )**4 93a 935 A(8,I)=X1(I)**5 936 937 00 e J= ,M
- 938 8 fiI(,J)=A(J,I) 939
- 9LJo CALL MAMIP2u(A, tO,20),f320, 10),M,N,M, , ,10) 9Li 1 DUt b 1=1,m 9Li2 DO 6 =0 9 l3 6 C(1,j=)1j(iJ) 944 CALL MlAg-1P20(A. 10,20,Y,2olM,N,1,#v, 0))
93J5 CALL MIid4VND(C, 1C',~,0ET, I.HDIR, IC) 9ilb CALL :.IAMi'211(C, 1O,10,y.#U, l,t~MM,1q,p 1) 94 7 94 8 3 94 9 95 0 A02=Y (3,1 951 952 953 A3=Y (li, I). 9511 A b=Y'(,1 955
956 A7=Y U3,) 957 RL=0. 242 958 SS4=(u. 959 00 a Ij1,N 960 YF = 0 . 9b 1 D0 5 J= 1,'" 9b2 5 YF=FiYr(J,) * (Jul) 9b3 P(1)=Yl(l)-YF 9bLJ4 RU=IRJ4 k(]) 9b5 SSR=SS+R 1 I) *4t J) 966 P. I jT j o , I , I (l ) ,IF ,R (I) 967 10 F0PtltAl l0I,7L15.7) 9b8 U CONTINUE 9b9 RE TuH.N 970 ENO C C 971 SUBRoUTl ic VlSASM(TFVF ,RfjPFVISmF) C ********i***** ***t*******k******A*******t*** ***t**** 972 VISPF=80.4+u.407*F-kJPF *(1t58.-5.9*TF)/100O. 973 VjSVF=1 73.bi0.4S4I*TF 97 J F] I 0F=.353t,/S) T ( 1.4 o .6222)* (I .4SORT (VISPF/vISVF )* 1. t253)**2 975 9 756VI FJ21F~o.3b3b/btl'fl F21 =F=
- v J -/b -RlF F' (1..1.buI1).1.iSrviT(VIo~vF/vISF)*n.^f1id).**
( F + I.- f) I I I 4 SUR - V I ) Sv/V ISF. )
- FII 2IF 12rJt( .I a * *
-8) 976 VISMF~RF*VJSPF/(PF+(1.-RW)AFli2r)4(1.-RF)tVISVF/(,.F*FI21F.I.1;.eLF) 977 RETURN 978 END C'
C ***********s** h^a*******A*******A***************t*****************;I 979 SotiRCu1 I-NE AMP4AS?1 ItF ,tiF ,CPt.kCPO?,CPAR so.¶j)MF) C **^******+*******1.** *1**********wA*****t**s*'********t**t**~*t***** **t-;. 98 MWF0,o hA1SFo b5 -~7 I Ou1t,10 I .*F
+ IOQ/ I .*7*7TF* TF-0 ab I tTF* TF*l F/16 I 0 9B1 kNMN2=26.02 982 w~M02=32' 983 ,FM^k:39.9L .
98a rRN?=(TF+?73.15)/12b.1 985 TRU2=(TF+P73. lt)/154.u. 98b TRA K=( rF 1i?7. 15).11 51.2 987 SIN2=1 .b73/l000UU. , 988 .102=1.3V/ 3/I0O0fl0e 989 SIAk=1.27P711000(0. 990 CPNz272J14. 187* (tf+ ?13. 15) 991 CP0?=53Ip25.t1.ount ( IF+273. 3J) 78536,.*I0000./(TF+4?73.15)**2,2 992 CPAMz2O0U&. 993 CVN2=(CPN2-9315.)/vmtv2 99a Cvo)2=(CP'(j?-6315. )/v <W 995 CVAk=(CPAk-e3l5.)/ v$. 99b V I S2.J=e.5I7t* 4 . O7)I*U/I9 7 t(5. )/SIN2/lI.*tl 1 997 vISA2=d.61 5*(u.b h4(R-1 .h7)**(5./B.)/SIAR/1I).** 1 I 999 brN-vti=SNR* (l.32* I C'Vf.508!i.-104.ts/T Rr2) OOG A4"h0L=V ISO)2e I . 32*C V(.!2+4LIU6-19 1 .519/ l R02), 001 AMRAtA=t',5AVlIS4R'CVA.R 002 Al( IF= 7** AMB%.12+*.2 1 *,
*fiO2*
/3.)) AMRG?.0.0I I; I (N1t*~.3! taltl)(.h(is:*(./3.))+0.2 * (wM02**C1./3.))
20.01
- I(I** 10./3.))) '
003 F 1 L=0. 3516/SIicT (41
- t222C( I .4 SORTA(MPF/AfItiVE)f .1?2503)'A2 oO' 12ll0 =0. 353b/Strlt r t .4 I .btl7lI*U.*I+SIl AfiiVF/Alt4.PF) C.8bl )**2 1
005 AMtiF=RF A;PF //(kF+(I. -F)*F II2L)+ t 1 .- RF) /A4t4VF /(RF*FI2L+ I2) j , 006 RtETLt 007 INO
243 C C.1O0d SUi3RLJUT1NE r)IFASfM(TF#WFV1FMF) C 1010 HETut~'j 1011 END C
$ E.NT NY Fil
244 APPENDIX H THE AIR ROTAMETER CALIBRATION CURVE
245 The linear least square fit of the third order gave the values of the coefficients Y - - 0.49282 + 1.7852X - 0.28669X2 + 0.0638X3 where X - rota-ter reading Y - flow rate [t/sJ. The error variance was less than 1%. I. {. i LI
246 0 AL 600 0 AL 350 A STRAIGHT LINE FIT 4
^ THIRD POWER POLYNOMIAL FIT LU =3_.
LU 0 E~~~ 2 3X F'igure Ul. The Air Rotameter Calibration Curve 0 I 2 3 4 5 6 7
247 APPENDIX I THE HUMIDITY CALIBRATION CURVE
10024 WOO / ~~~~~~~~~~~~~~~~248 90 aE80p ; > 70 F<- F- 6 0 z LU
.30 20 10~~
Figure II. The Humidity Calibration Curve I 1 1 1 1 1 In tin I14
I! Ii I, 249 Ii i. APPENDIX J FEED WATER THERMOCOUPLE CALIBRATION CURVE t* I,
' . . 1~~~~~
250 0 TABULATED VALUES C CALU BRATED VALUES 3' I -J H-z uj I-- 0 2.M 0L
.v I
Figure Jl. Feed Water Thermocouple Calibation Curve 10 20 t230 40 60
251 APPENDIX K THERMISTOR AND RECORIDER CALIBRATION CURVES I.
252 The least square fit through the measured values gave the following values of coefficients: a) Thermistor 1 T - 140.15 - 0.24765R + 2.4901(R/100) - 0.11235(R/100)3
+ 0.0019093(R/100)4 , [IC]
where, R - thermistor resistance, (6aJ. The error variance is less than 1%. .4 T -0.12864 - 0.081275(CHP) + 0.022249(CEP)
- 4.2562,10- 4 (CEP) 3 + 2.9725*10-6(C?) 4 , c, where, CEP - percent of chart reading, %.
b) Thermistor 2 T - 143.77 - 0.21354.R + 1.7122(R/100) 2 - 0.067465(R/100) 3
+ 0.0010052(R/100)4 , [IC T -18.423 - 0.18619(CHP) + 0.22464*10 1(CHP) 2
- 0.34394*10-3 (CHP)3 + 0.21776*10-5 (CHP)4 4i
lo0 253 901- 0 MEASURED A FITTED 801-701-0-1 0 60W-w 50k 13-w wj .40 301-20 I: If ii Il'
- *10- . Figure R1. Thermistor #1 Calibration Curve
*I l l l l 2500-j li I
0~ 500 1000 1500 2000 2500 THERMISTOR RESISTANCE (S2)
254 901 80 70 U)
- 1-- 60 AI I
LiL
- 0 F-4 I z w 40 C) w- a MEASURED 0J 301 A, LEAST SQUARE FIT I
20 i3-10 Figure K2. Recorder Calibration Curve for Thermistor 40'
- 60 80- -:' 100
- j. roe
lUu 255 90_ 80j-
- MEASURED A LEAST SQUARE FIT 70 _
60 0 0 Enr M1U W-10 _ F-30 -
*201-10 _
Figure K3. Therristor V-2 Calibration Curve l l I l *~~~~R oo~~~~.. 0 500 1000 1500 2000 THFRMISTOR RESISTANCE (S)
100 256 90 0 MEASURED 80 A FITTED 0 0 CD IaJ 1= 50 I-w- llJ
/
30 20 10 Figure K4. Recorder Calibration Curve for Thermistor #2 A rT i0S-v-^ TV 60 FIN I V%) D10fE71- C' fcUAm-T QrA C-
257 APPENDIX L i DATA OF EXPERIENTAL RUNS FOR SINGLE DROPLETS
- 1 I
, 1i":
I I
;I
.1 I
f i I zII i.; i 1.i I i.i i k
'I1
.: i I!
0i
. -1 - 1 TABLE L.1 N~~~
2 U0 aw £ M,
" lThermal Utilization 0 [X]
0] j ", Predicted 10 20 30 40 50 60 70 75 80 90 95 1 2.6 47 15.8 191 5.5 13.5 23.5 33 43 52 66 70 74 89.5 96 2 2.9 47 16 191 6 17 33 57 57 67 75 78.5 83 93 98 - 4 3 2.9 47 16 191 6 12 25 51.5 51.5 63.5 73 77 81.5 91.5 - 4 1.9 47 16.2 191 8 21 34 55 55 64 71.5 75 79.5 91 96 5 2.5 65 17 140 6.5 18 28.5 48 48 67.5 73.5 78 80.5 89 94.5 6 1.4 66 17.5 140 5 18 30 57 57 66 74 78 81 92 97.5 7 2.5 31 16.9 258- 7 20 34 50.5 50.5 57 64 68 72 81 87 8 2.1 24 16.7 360 3.5 13 26 49.5 59.5 58 67 73 9 2.7 33 19.2 691 8 20 35 61 61 68.5 76 79 82.5 89 92 10 1.4 36 22.5 199 4 14 23 50.5 50.5 70 80 84 90 94 96 11 1.6 52 20 108 2 8 21 39 39 51 64 72 80 91.5 93.5 '.3
*.1 "I";%.
4 t, '. t -9.F , i%-? It - , I-.1-4.1'."..... Mu *1
TABLE L.1 (Cont'd) 10 20 30 40 50 60 70 75 80 90 95 12 1.9 52 18.7 108 6.5 21.5 37 51.5 58.5 68 75.5 78 81 87 93 13 1.7 53 19.8 108 8.5 20 34 .46.5 57 69 76.5 81 85.5 91.5 94 14 1.6 53 19.3 108 2 9 21 33 47 .60 70.5 75 79.5 87.5 92 15 1.7 53 19.5 108 6.5 18 35 47.5 57.5 69 77 80 84.5 90 92.5 16 2.9 17 18.1 376 5.5 13 22 33 51.5 52.5 68 71 77 88 94 17 2.5 18 18 376 2 6 11 18 34 57 72 77.5 82 90 83.5 18 3.1 19 19 376 * ,4 6 13.5 23 38 50 61.5 73 27.5 84.5 93.5 97 Id v* 19 2.6 19 18.7 376 d 4 8.5 16 28 43 57.5 77 83 86.5 94.5 98.5 20 2.4 18 18.5 376 as 10 23 41 51 60 68 73 77 81.5 91.5 97 21 3 13 18.5 405 5.5 16 27 41 54 65 75 80.5 86.5 95 98 U' 22 2.4 13 18 405 5 13 23 32.5 45 59 70.5 75 80.5 91.5 96.5 23 2.8 14 17.8 405 3.5 13.5 26 39 54.5 65 75 80 84 92.5 96.5 24 2.4 8 16.2 273 6.5 13.5 21.5 23 49 59 59 77 82.5 93.5 98.5 25 2.4 8 16.4 273 4.5 12 21.5 29.5 40.5 44 44 72 78.5 89.5 95 26 2.8 8 16.4 273 4.5 11.5 23 36 48.5 61.5 61.5 79 84.5 93.5 - L" nD _ _ * -. - r .--..- .. ll~ - -
<- - n
! I ** - -* 1-r TABLE L.1 (Cont'd) 10 20 30 40 50 60 70 75 80 90 95 27 1.3 8 16.9 2t3 5.5 16.5 27.5 39 49 59 69 74 78.5 87.5 94 28 3.0 8 16.8 273 5.5 14 22.5 41.5 53.5 65 75.5 79 84 - _ 29 1.5 8 17 273 7.5 20 30 42.5 53.5 63.5 73 77.5 82 91.5 26.5 30 1.3 7 17.2 349 6 17 17.5 36.5 46 55
- 65 70 74 86.5 92 31 1.3 6 16.8 349 5 17 26.5 36.5 47 57 68 73 78.5 89.5 95 32 1.3 6 16.8 349 5.5 16.5 27 39 48.5 58 68 72 77.5 87 92.5 33 2.3 6 16.2 349 4.3 5 14 22 32 43.5 56 67 73 80 89 94 ci a
34 2.9 6 17.8 349 5 14.5 25 36.5 46 58 78 73 78.5 - -
. i 35 1.8 6 16.5 349 9 20 31.5 43 53 62 71. 77 81 91 95.5 36 3.1 38 18.3 168 7 18.5 36 48 57 62 69 73.5 79 89 94.5 37 3.0 38 17.6 168 8.5 21.5 32 39 46 53.5 64 69 76 89 94 38 2.5 38 18.5 168 7 18 32 43 55 64 74.5 80 84.5 91.5 94 39 3.3 38 16.7 168 6 15.5 28.5 39.5 47 58 70 76 82.5 92 96 40 3 38 17.6 168 4.5 13 27 40.5 50 57 67 72 78 90.5 95.5 41 2.9 36 17.3 168 4.5 13.5 24 37.5 47.5 57 67.5 72.5 79 91.5 97 ON to a'
A .1
~~~
A ilL. .* .. *iS ... t
TABLE L.1 (Cont'd) 10 20 30 40 50 60 70 75 80 90 95 42 3.4 36 17.5 168 3.5 10 25 46 60 67.5 43 3.0 36 18.3 168 2.5 6.5 13 22 36 55 67 73 80 91 96 44 3.2 36 17.4 168 3.5 10 18.5 23 48 62 70.5 76.5 81.5 91.5 97 45 3.0 36 17.7 168 3.5 10 22 33 44 57.5 68.5 74 80 91.5 97.5 46 3.0 36 17.2 168 4 11.5 23 35 48.5 58.5 68.5 74.5 80 92.5 97.5 47 2.1 26 18.3 250 10 24 38 47 56 64.5 72.5 78 81.5 88.5 92.5 48 2.6 25 17.3 250 3.5 9 12.5 16.5 50 60 70 75 80.5 90.5 96
.*4 49 3.1 25 18.8 250 5 15 23.5 36 49.5 61 71.5 78.5 83.5 93 97 50 3.0 25 19.7 250 6 13 27 36.5 49 60.5 69.5 a)
Ci 73.5 78 87 92 51 2.5 25 18.2 250 6.5 20 36 48 60 70 78 80 82 91.5 96.5 52 2.8 25 16.7 250 3.5 11 23 32 49 60 70.5 78 13 93.5 97 53 3.0 25 18.9 250 3.5 13 24 36.5 50 61.5 70.5 76.5 81.5 91.5 96.5 54 2.9 25 17.5 250 3.5 9 21 33 59 45.5 73.5 79 84.5 93.5 97 55 2.9 25 17.6 250 5 14 22 35 47 57 69.5 75 80 92 96.5 56 2.6 25 17.2 250 3.5 9 17.5 29 4 58 67 73.5 79.5 91 95.5 Faj cry
TABLE L.1 (Cont'd) 10 20 30 40 50 60 70 75 80 90 95 57 2.6 25 16.6 250 5.5 12 24 36 48 59.5 72 78 83.5 94.5 98 58 3.2 14 20.1 341 3 75 17 28 43 58 70.5 77 81.5 88 - 59 3.7 14 18.7 341 5.5 14.5 28 41.5 53 66 77 82 85 90 - 60 3.2 14 19.3 341 2 6 14 27 40.5 53.5 67.5 73 78.5 88 _ 61 3.2 14 17.7 341 4.5 11 22.5 36 49 63.5 77 80.5 84 90 - 62 3.4 14 19 341 6 16 27.5 40 54 67.5 77.5 84 87 91.5 _ 63 3.0 14 17.7 341 as 5 10 23 36.5 47 61.5 73 78.5 84 90 94
'4 64 65 3.0 3.1 14 14 17.5 20.2 341 341 I.
0 7 6 17 17 29 20 42 42 53 52 67.5 6.1 78.5 70 83 74 87 78.5 91 87 91.5 66 3.9 14 20.6 341 4 11.5 24 39 53.5 - - _ - - - 67 3.5 6 19.7 351 7.5 18.5 27.5 38 49 60 60.5 75.5 81 - - 68 3.8 6 18.9 351 5 17.5 28.5 41 53.5 65 75 80.5 - - - 69 2.8 6 19.3 351 6.5 16.5 27 38.5 50.5. 61.5 72 77 81.5 88 - 70 2.5 6 18.8 351 7 17.5 29.5 41 52 62 71 76 80 86 - 71 2.4 6 18.6 351 6 17 29 41 51 63.5 73.5 78.5 82 88 - 0a g at 7> me ; ; , Mrp dub , ; jXJay
^;
giW~~~~~~~j.:G
,i.,,^,
TABLE L.1 (Cont'cd) 10 20 30 40 50 60 70 75 80 90 95 72 3.4 6 18.4 351 7 15 25 37.5 47 57.5 70 73.5 78 - 73 3.2 5 19.2 497 8.5 18.5 30 39.5 50.5 61.5 72 77 81 - 74 4.0 5 17.5 497 10 20.5 33 43 53 63.5 74.5 77 - - 75 3.3 5 16.8 497 9.5 20.5 33 44 55.5 68 77 79.5 - - 76 3.8 5 18.8 497 10 20.5 33 45 56 67 75.5 80 - - 77 3.9 5 17.7 497 12 23- 35 46 57 68 78.5 81.5 - - 78 3.5 5 17 497 6 21.5 30.5 41.5 51 60 69 73 77 s-a Vi 79 3.8 5 17.2 497 9 20.5 31 42.5 53 63 72 77 81 80 3.8 5 18.8 497 9 18.5 29.5 40 50 59 68.5 72 75.5 - 81 2.9 29 16.6 258 7 11.5 19 28.5 43 55 71 77 82 93.5 82 2.4 14 15.9 405 5 12 21 45.5 64 73 80 62.5 90.5 91.5 94 83 2.8 6.7 16.2 349 8.5 17 28.5 37 52.5 66 77.5 83 89 98 84 2.6 6.4 16.2 349 5.5 13 22 30.5 42.5 60.5 74 79 84.5 95.5 85 2.5 6.2 16.2 349 4.5 9.5 17 26 35 50 64 71 78 92.5 86 2.9 6.4 17.6 349 5 13 24 36 49 63 74 81 87 - CY% w-
TABLE L.1 (Cont'd) 10 20 30 40 50 60 70 75 80 90 95 87 1.5 5 19.2 1.76 5 11 20 26.5 35 37.5 60 67 73.5 86 93 88 2.5 5 19.6 1.76 E
-ri 10.5 20.5 31.5 42 59 71 84 88 91 - -
89 2.5 5 19 1.76 a 8.5 18 28 42 56.5 68 78.5 83 88 - - 90 2.5 5 19.1 1.76 9 17 29 44 54 64 75 82 89 - - 0o
-C- .... . I ..
P...
,I %,
"fit ,." 1:At., ") -
265 APPENDIX M DIMENSIONAL ANALYSIS FOR THE RESPONSE TIME CORRELATION
266 Let us assume that the response time will be a function of a steam saturation pressure PSM, initial relative velocity of a droplet VRo* mass of a droplet m and time t. The dimensional matrix in the MLT system of units is 1120] P V . m t M 1 0 1 0 L -1 1 0 0 (M.1) t -2 -1 0 1 We can construct square matrices from this dimensional matrix by deleting various rows or columns. The determinant formed from the last three columns in the dimensional matrix is: (o
~1 0
0 0 0~~~
-1 1
1V ) 1 0 1 O'~~~~~~~~~~~~~~~~~~~~~~~~~~~1 1 1 11 (1) 01 1 1 1
+ 0 1
0 10 0 l. 0
- Cl)(-l) + ° (M.2) Y Since the largest-order determinant, we can construct, is a third-order determinant, and the above third-order determinant is differ-ent from zero the rank of the dimensional matrix is 3. Since the rank is 3, the number of dimensionless products in a complete set is N mn - r 3 = 1.
Any product of the variables has the form
*iu
267 kI k2 k3 k4 Jr=P V M t (M. 3) or for the corresponding dimensions of v [M~lt2]k I k2 k3 k4 r 2] l[Lt [M (M.4) or rewritten r = M(k+k 3 ) L(-ki+k 2) t(-2k 1-k2+k4) (M.5) In order for v to be dimensionless the exponents of M, L, and t must all be zero k1 k3 0
- k1 + k2 ' 0 (M.6)
- 2k1 - k 2 + k 4 O Any solution of these equations results in a set of exponents for the dimensionless product v. Notice that the coefficients in each equa-tion are a row of numbers in the dimensional matrix. Equations (M.6) are a system of three equations in four unknowns and are undetermined, therefore, since there is an infinite set of solutions. For our pur-poses we shall assign a value 1 for k4, and solving for remaining unknowns k1 k = 1/3 k3 -- 1/3 (M.7) and (PV) 1 /t (p 11 t
= (1M.8) 1/
268 APPENDLX N DOUSING CHAMBER SIMULATION ALGORITM
*1 I-
269 $JOB WATFIV ****t***** ,KP=29, P=50 C ** ** ********** ****** *** *** *** ** ***** * ** **** *** * *** * **** *****+* **.
***************t*t******t*****************.**** ****** ******t *t****-
C THIS PROGRAM GENERATES PRESSURE DROPS IN NUCLEAR DOtUSING SrSTFmS C BY SOLVING THE DISCRETIZEO MEAT 6ALANCE EUUATION. C
*t*** ******** ***t***'****t************* ***t*** * ***t********** ***** t C
C INPUT PARAVETERS ARE: C UPPER-LIMIT DISTRIBUTION PARAMETERS C NUMBER OF DROPLET DIAMETER DISCOETIZATIONS C CHAMBER VOLUME AND HEIGHT C INITIAL PARTIAL PRESSURES OF STEAM AND AIR C INLET WATER VELUCITY,FLUORATE,AND TEMPEkATUPE C DRAG COEFICIENT CORRELA1IONt PARAMETERS C PRESSURE DROP TIME INCREMENT C C C I C LIST OF SYMBOLS AND ARRAYS __________-_--_____--W__--- -___ I SYMBOL U NI TS i; f_____ ii AMASS MASS OF WATER DROPLET KG CPv4F HEAT CAPACITY OF WATER DROPS LEAVING CHAMBER KJ/(KG*DEG.C) I i CPWO HEAT CAPACITY OF WATEP DROPS ENTERING CHAMBER KJ/(KG*DECG.C) CPWT HEAT CAPACITY OF-WATER DROPS WITHIN CHAM6ER KJ/(KG*L)EG.C) DCT TOTAL ENERGY ChANGE OF ALL DROPLETS i IN A TIME INCREMENT KJ DELAY TIME DELAY FACTOR SEC I DLCONT TOTAL ENERGY CHANGE OF ALL DROPLETS IN GIVEN TIME INCREMENT KJ I DNWL DENSITY OF hAIER DROPS LEAVING. CHAMBER KG/M*t3 i DNWO DENSITY OF IATER DROPS ENTERING CHAMBER KG/M**3 DNWT DENSITY OF '"ATER DROPS WITHIN i i CHAMRER KG/M**3 DPINC TOTAL NUMBER OF DROPLETS OF A SPEC-IFIC DIAMETER GENERATED IN A TIME INCREMENT
-ENAIR ENTHALPY OF AIR KJ/KG ENST ENTHALPY OF STEAM KJ/KG ENMIX ENERGY (IF AIR-STEAP. MIXTURE KJ i ENINA INTERNAL ENERGY OF AIR KJ/KG I1!
ENIN-M INTERNAL ENERGY OF AIR-STEAM MIXTUPE KJ/KG ENINST INTFRkAL ENFRGY OF STEAm KJ/KG i FRED FREQUENCY DISTRIBUTION CORRELATION FRN4EK FR OLD HEIGHT HEIGHT OF CHAMBER HTINC HEIGHT INCREMENT N TIME COUNTER NHI- NUMBER OF HIGHT INCREMENTS NI NC NUMBER OF DIAMETER DISCRETIZATIONS N55 PAIR PARTIAL PRESSURE OF AIR B&R I PAIROD INITIAL PARlIAL PRESSURE OF AIR BAR PAIRI INITIAL PARTIAL PRESSUlRE OF AIR PSI A 1!I PPP PARTIAL PRESSURE OF STEAM FA AP PPPO INITIAL PARTIAL PRESSt'WE Or STEAM
270 - PPP 1 INITIAL PARTIAL PRESSURE OF STEAM PSIA ps rI TOTAL PRESStPRE IN CHAM6ER PSIG PT TOTAL PRESSURE IN CHAMHER PTO INITIAL TOTAL PRESSURE BAR PT1 INITIAL TOTAL PRESSURE PSIA RAIR GAS CONSTANT FUR AIR BAR-H**3 DEG.K KG-MOLE RMWA .MOLECULAR WEIGHT OF AIR KG/K(;-M(LE ROMF DENSITY OF AIR/STAM MIXTURE KG/M**3 S SDEVAA PARAMETERS USED IN UPPER-LIMIT ANALYSIS TINIT INITIAL TEMPERATURE OF A WATER DROPLET IN A TIME INCREMENT DEG.C TKVOL CHAMBER VOLUME M**3 TPRESS LENGTH OF TIME INCREMENT SEC TTEST TOTAL TIME SEC TWA TO INLET WATER TEMPERATURE DEG.C TZ TEMPERATURE OF AIR-STEAM MIXTURE DEG.C TZO INITIAL TEMPERATURE OF AIR-STEAM MIXTURE DEG.C ULI MAXIMUM DROPLET SIZE IN SPRAY mm VISMF VISCOSITY OF AIR/STEAM MIXTURE 1 0**7*N*SEC/m VPRIM SPECIFIC VOLUME OF STEAM M**3/KG WAIR MASS OF AIR IN CHAMBER KG WTFLOW INLET WATER FLOWRATE Mm**3/SEC WTFON1 INLET NATER FL(1KRATE M**3/SEC ZINC NUMHER OF HEIGHT INCRFmENTS A DROP-LET OF A SPECIFIC DIAMETER FALLS CONT(I,N) ARRAY THROUGH IN A TIp.E INCREMENT ENERGY OF DROPLETS OF DIAMETER DtI) UNITS 4 WITHIN CHAMBER IN THE NTH TI.ME INCREMENT KJ DCI) DIAMETER OF ITH DROPLET MM DISNCI) VOLUME FRACTION OF DROPLETS WITH DIAMETER D(I) DROPS(I) NUMBER OF DROPS OF DIAMETER DtI) IN A UNIT VOLUME OF LIQUID DRPHTtI,J) NUMBER OF DRnPLETS OF DIAMETER D(I) OCCUPYING THE JTH HEIGHT INCREMENT FN(I )) ENERGY OF DROPLETS OF DIAMETER DCI) ENTERING CHAMBER KJ FOUT (I) ENERGY OF DROPLETS OF DIAMETER D(I) LEAVING CHAMRER KJ 1FLAG(I) CHECK TO DETERMINE IF DROPLET OF DIAMETER DtI) HAS REACHED OTTOM OF CHAMBER TIM(IJ) FALL TIME OF DROPLET OF DIAMETER D(1) TO THE JTH HEIGHT INCREMENT SEC TRES(I) RESIDECE TIME OF DROPLET OF DIAMETER D(I) IN CHAMBER VEL(I,J) VELOCITY OF DROPLET OF DIAMETER DtI) IN THE JTH HEIGHT INCREMENT M/SEC _3 WTPA(IJ) AVERAGE TEMPERATURE OF DROPLFT OF DIAMETER D(I) IN THE JTH HEIGHT INCREMENT C ** * * * **** *** * **** * * * * **** * *** * ** ** **** ** * * * .* *
* 'e * * * * * *4 * **
- 4 * * ** w* * * ** * * ** * . * * .* * * * * * ******* * . * *** ** * * *
- C I DIMENSION DISN(1O),r)DCHE10(lO) ,DROPsIIO),TINITI5J0t t
2 DIMENSION TIME(2000),DI5T(2000),VDROP(20OO)tlM(5th50),VEL (5,bSt 3 DIMEN4SION CONT (5t2)tFOUT( 0),F INtlO)OLIT(10) IS(500) IVlsA(2000) 4 DIMENSION ITRANS(q,200),VELTk(5,200),NS5(1O) 5 DIMENSION DRPtT(5,b50),ISE(10),TPES(1O0),Fe('LD(CO),IFLAG(IO) b DIMENSION W7PA(5,650),C(10),HH(1b0) 7 COMMON TZ,P1/AREAI / G,AREA,DMASS,POHB,VISMBA,SS,I)TAMr)ELPO, IDSTAR1,DSTA2A,DSTA28,DSTA3A,DSTA3B,DSTARL,RE2ARE2H,RE3ARE38,RE' 2RES,DMA XSrCDAXsWS TA P,J "A XS, %STARU V #HSTA 3 A, h~S T A38 , wST A 2A, hST A 26, 3hSTAR1,DSTSURFT/AREA2/TKVULPPPPENINMiwAIRPAIR.0uT77,T 8 100 FORMAT('11p9XSIMULATION OF PRESSURE DROP IN NUCLEAR DOUSING
- V I,'SYSTEMS")
9 101 FORMAT(' t9XWl*******s********s******************W**********l I0 102 FOPMAT('O',9XWINITIAL DROPSIZE DIST"IHUTION') I1 103 FORMAT(' ,9x,4*****************t**********l,//) I2 10a FORMAT'0',IIX,'UPPER-LlMIT PARAMETI.RS') 3 105 FORMAT(' 'lIIX,' ----------------------'). 4 106 FORMAT('O'1U1X, '1XMAX =',Fb.3,' MM') 'I I5 107 FORMAT(' ',14X,'DELTA=',Fb.4) I6 108 FORMAT(' '1,IJXIA =vF,3,//) I7 109 FORMAT(' ',llx,'DISCRETIZED INITIAL DROPSIZE OISTRIBUTION') I8 110 FORMAT ( ' ', 1 1X, ' ------------------ ) 1q III FORMAT('0',12X,'UROPSIZE VOLUME FRACTION NUMBER IN') 20 112 FORMA.T(' 212X,' (Mm) UNIT VOLUME') 2'1 113 FORMAT(' ',9X,F10.3,F13.4,FI6.4) 2'2 1114 FORMAT(' ',//,9X,' SYSTEM PARAMETERS') 2>3 115 FORMATW' ',9X,'*****************'J 2 14 116 FORMAT(1'O'11X,'TAtJK VOLUME -'1,F1o.3,' m**3') 2:5 117 FORmAT( .,11WXTANK HEIGHT =',F1O.3,' pil) 2'6 118 FORMAT(I' '1,11X'NITIAL STEAM PRESSURE ='1F10.4,' EAk') 2'7 119 FORMAT( ',1IX,'INIt1AL AIR PRESSUPF. =',F1O.U,' PAR') a '8 120 FORMAWt. ',IIXg'INITIAL SYSTEM PRFSSURE -',F10.U,' BAR') 2 >9 121 FOfMAT(' ',lXt'IINITIAL SYSTEM TEMPERATUR-=',F10.?,' DEG. CO) 50 122 FORMAT(' ',11x,'INLET hATER TEMPERATURE =1'F10.2,' DEG. C')
- 1I1 123 FORMAIC' ',11x,'1ILET %AIER VELOCITY =',F10.2,' M/SEC')
2Z 1 24 FORMAT' ',iIX,'lNLET WATER FLOWRATE =s,Fl0.uL H**3/SEC') 3 125 FORMAT('1',9x,'DYNAMIC RESPONSE OF SYSTEM') 354 126 FORMAT(W t,9X,'**************************') 55 1127 FORMAT('0',9X,' TIME SYSTEM STEAM AIR TOTAL
- 5 1 TOTAL TOTAL ')
b6 1 28 FORMAT(' ,aX, TEMP. PRESS. PRESS. PRESS. 1 PRESS. PRESS.') 7 129 FORMAT(' ,9X,' (SEC) (DEG C) (BAR) (BAR) (BAR) I (PSJA) (PSIG)') 58 1130 1 g.?MA~t# a -_____ O',9X'- ________ 3 p9 131 FORMAT('0') 10 132 FORHAT(' ',qXlF7.2,Fll.3,FlO.4tFlo.ZFlO.4,FlO.IFlO.3)
'1 133 FORMAT(' 1,11X,1NI1IAL v.ATER DFNSITY =',FlO.3,' KG/M**3')
2 1 34 FORMATC' '1,1XWINITIAL SYSTEM DENSITY =1,F10.3,1' G/M**3') . 3 135 FORMAIC' 1,IJX,'INITIAL SYSTEM VISCOSITY =',F10.3,' 10**7*N*S/M' 14'4 136 FGRMATt' ',IX.I'INITIAL SYSTEM SUHF TENSIN=',F10.5,' iN/m,//) H 15 137 FORMAT(' ',11X#'LNTERING "ATER DELAY TIME =',F10.3,# SEC',//) 14.6 138 FORNATlWI te=@F.,Xl-^F.,X n=l.3snl~f)=r.3. 15Xr WD=_F7,3,5X zD=IF7.3#5X,'D=IF7.3p$xt P=',F7.3)
'7 139 FORMAT('O',5X,'VELOCI7Y OF DROPLET OF DIA4METER PU) IN THE JI&H HE IGHT INCREMENT ') .l 46 1 40 F OR MAT( 5 X, - --- - -- - - - - ------ - - - ---- -- ----- _ ------- ----
1 ------ ---------- -- ----------------------------- ) 19 1141 FORMAT(3x, I3,5F14o5)
;0 t 42 FORMAW0T( WSX,'FALL TIME OF DROPLET OF VIAMETER rCI) TiTHE JTH IIGHT INCREMENT') THE 51 143 FORMAT(C'015WNUMBER OF DROPLETS OF DIAMETER D(I) OCCUJPYING THE
979 IJTH HEIGHT INCREeENTt) 52 145 FORMAT(0t5XAVERAGE TEMPERATURE OF DRUPLET OF DIAMETER 0(I) ITHE JTH HEIGHT INCREMENI' ) 53 1U4 FORMAT(3XI3,5FI4.2) 54 146 FURMAT(W h,llX,lUNGF-KlJTA TIME INCREMENT='1,Flo.a, SEC') 55 147 FORMAT(' ',9X,'DYNAMIC RESPONSE OF SYSTEM') C C 56 G=9.81 57 ICH=1 58 KK=1 59 RAIR=8314,7/28.96 60 RMWA=28.96 61 SUM=O. 62 TOTAL=o. 63 DELTIm=0.05 C C UPPER LIMIT PARAMETERS AND NUMBER OF DIAMETER DISCRETIZATIONS C ARE INPUTTED C 64 READSDEVAALUM,NINC 65 D114C=UM/NINC 66 OCHECK(t)=DINC 67 SINC=UM/500. 68 R=DCHECK(ICH)-SINC 69 DO I I=1,500 . 70 S(I)=(2*I-I)*SINC/2. C C DROPLET VOLUME DISTRIBUTION IS CALCULATED C 71 FREQ=((UM*SDEV)/(S(I)*(UM-S(I))*SoRT(3.14159)))*EXP(-(SDEv*( IALOG((AA*S(I))/(UM-S(I)))))**2) 7-2 FRDROP=FREU*SINC 73 TOTAL=TOTAL+FRDHOP C C DROPLET SIZE AND VOLUME DISTRIBUTIONS ARE DISCRETIZED C 74 C IF(S(I).LT.R) GO Tr) I 75 DISN(ICh)=T0TAL-SUM,
- 76. IF(ICH.E.,NINC) GO TO 1 77 ICH=ICH+1 78 DCHECK(ICH)=DCHECK(ICH-1)+DINC 79 R=DCHECK(ICH)-SINC 8O SUM=TOTAL 81 1 CONTINUE 82 DO s I=1.ICH 83 D(I)=(2*I-1)*DINC/2.
84 DISN(I)=DISN(I)/TOTAL C
.C DROPLET SIZE DISTRIBUTION IS CALCULATED C
85 4 DROPS(I)=DISt4(I)/(1.3333*3.14159*(D(I)/2.)**3) C SYSTEM PARAMETERS .ARE INPUTTED 86 READTKVOLHEIGHT 87 READTPRESS 88 T-0,0 89 READPTIPPPIPAIRI z; 90 READvTFLOiuVO,TIWATO 91 READDELAY 92 PTO=PT1/1i.504L 93 PPPo=PPPI/14.504L 94 PAIRO=PAIRI/14.50Ll 95 CALL TE'P1(PPP0rTZ0)'
273 96 PT-PTO C C INITIAL PHYSICAL PROPERTIES OF WATER DROPLElS ARE CALCULATED C 97 CALL PPisA1(1WAIOPT,CPWO,DNW0) 98 T0-h=FL0W/D^lhU 99 WTFLOw:-FE)wI*10.**9 100 PRINT100 101 PRINT101 102 PRINT102 103 PRINT103 104 PRIN7104 105 PRINt105 C C UPPER-LIM1T PARAMETERS ARE OU7PUTTEDl) C lOb PRINT1ObUM 107 PRIN7107,SDEV 108 PRINT108,AA c DISCREtIZED DISTRIBUTIONS ARE OUTPUTTED 109 PRINT109 110 PRINT11P ! 111 PRJNT111'Il 112 PRINT112 113 PRINT113,(D(J),DISN(J),DROPS(J),J=IICH) llU PRIN'll1' 115 PRIN71)5 C CHAMBER VOLUME AND HEIGHT ARE OUTPUTTED 116 PRINTltbt7KVOL 117 PRINT117,HE1GHT C C INITIAL CHAMBER PRESSURES ARE OUTPUTTED 118 PRINTll8,PPP0 119 PRINT1I9rPAIRO 120 PRlNT120,PTO C C INITIAL SYSTEM TEMPERATURE IS OUTPUTTED C INLET INLET hATER VELOCITYFLOwRATE#ANO TEMPERATURE ARE OUTPUTTEVl 2) C 121 PRINTI2,T20l 122 PRINTI22,TIATO 123 PRINT123VOl 124 PRINT12AWTF6W1X 125 P~lGP11-1a,69b 126 PPP=PPPO 127 XS=PPP/PT 128 HIN=HEJGHT/0.05 129 NHI~IFIX(H1N+0,5) 130 HTINC=HEIGHT/NH1 C C VELOCITY PROFILES ARE CALCtrLATEO USING RUNGE-KUTTA TECHNlOL'E C C DENSITY OF AIR/SlEAM MIXTURE 131 RMF-831 L,7/(XS*18. f24 (I .- XS)*'289,6) 132 RUMF=1000OO.*PTO/(RMF*(TZ0+273.15)) C C VISCOSITY OF AIR/STEAM MIXTURE C lug
133 CALL SEKPV(TZO#PV) 274 134 ROP=PT O/PV 135 VISP=8o.4+O.UO7*TZ)-ROP*(1858.-5.9*TZO)/10O0. 136 VISV=173.h0 .454*7Zo
.0, 137 FI12=0.3536/SU(RT(1.+0.e222)*(1.+SnRT(VISP/VISV)*1.12993)**2
- k. .138 FI2l=0.3536/S(IRT(1.+l.hU71)*(1.+S0RT(VISV/vISP)*0.8F781U)**2 139 VISMF=XS*VISP/(XSC(l.-XS)*Fll2)+(1.-XS)*VISV/(XS*FI?1+1.-XS)
C C SURFACE TENSION OF WATER IN AN AIR/STEAM MIXTURE C IIIO SURFT:(75.6-0.145*TWATO-0.25*10.**(-3.)*TWATO**2)/1000. VISMB=VISMF 142 ROV=DNWO 143 ROMB=ROMF I11113 PRINT 133,DNWO 145 PRINT 134,ROHH 146 PRINT 135,VISMB 147 PRINT 136,SURFT 148 PRINT 137,DELAY 149 PRINT 146,DELTIM PARAMETERS OF REINHARTIS DRAG CUEFFICIENTS CORRELATIONS 150 SS=ROMB*SURFT**3*10.**28/G/VISMB**4 151 DELRO=ROV-ROMS 152 DSTARl=1.216/(DELRO/ROMi)**(1./3.) 153 DSTA2A=70.0*(DELR0/RUd))**(1-./3.)*(SS*10.**(-12))**(-O.111) 154 DSTA2B=30.8*(DELRO)/ROMB)**(-1./3.)*(SS*10.**(-12))**( 0.111) 155 DSTA3A=122.*(DELR0/ROMB)**(-0.535)*(SS*10.**(-12))**(0.275) 156 DSTA38=300.*(D)ELRl/RCOm)**(0-.535)*(SS*10.**(-12)**tn(0.3h7) 157 DSTAR4=160.*(DELRO/ROMii)**(-0.447)*(SS*lO.**(-In ))**(0.1l43 158 RE2A=974.*(SS*10.**(-12))**(-0.1 9 2) 159 RE28=236.*(SS*10.W*(-12)) **0.1Q2 8 160 RE3A=2240.*(OELRO/H0m8)**(-0.303)*(SS*10.**(-12))**0.38 161 RE3q:=7160*(DLLRO/ROmd)**(-0.303)*(SS*10.**(-12))**0.08 162 RE4=3190.*(DLLRO(/ROMH)**(-0.173)*(SS*10.**(-12))t**0.1l 163 RE5=uu70.*(DELRO/RflMB)**(-0.238)*(SS*10.**(-12))**0.232 1615 DMAXS=224.*(DELRO/kUtm)**(-0.5)*(SS*10.**(-12))**(1./b.) 165 CDMAX=0.75*(DELRO/ROmd)**(-0.0232)*(SS*10.**(-12))**0.036 166 WSTAM=3.273*((DELRO/ROMt)*SS**0,25)**0.2b2 167 riMAXS=RES/DMAXS 168 KSTAR4=RE4/DSTAR4 169 WSTA3A=RE3A/DSTA3A 170 WSTA38=RE38/DSTA3B 171 NSTA2A=RE2A/DSTA2A 172 WSTA2B=RE28/DSTA26 173 WSTAHI=0. /DSTA.R 17Y DST=DMAXS 175 A=O.085-(1./29.54)*ALOGIO(SS**0.5/(DELRO/ROMi8)*10.**(-3)). 176 DIST (1I =U0. . 177 T. TI4E( ) = O. I ME ( I13:0. 178 VDRC - I ) =VO CM -- - - - - - ~ - -W - - - - - - - -- --- .- 179 DO 15 ID=10ICH 1& 0 VELDRP=VDROP(1) 181 LL-2000 182 IF(D(ID).GT..5) GO TO 37 183 DO 38 IT-2,LL 184 TIME(IT)=(IT-l)*DELTIM C C VELOCITY OF DROPLETS SMALLER THAN 0.5 MMTAKEN AS THE E -:M
* * ~_ @ ~_ _ ~_ _ ~e @~~ ~~ ~ ~ ~ ~ ~ ~ ~ I~la r
275 C 185 RSA=RAIR 186 ROSA=101300./HSA/293.15 187 IF(ROMF.GT.RuSA) GO 70 37 168 VRTSA(IT)=(((((l.bhb6/100O.*D(ID)-2.b781/1o0.)*D(ID)+2.36.12/lo.), 1(ID)-l.3806)*D(ID)+5,4506)*D(IO)r3.1682/10.) 189 YSA=0.L3*ALt1G10(POSA/RUMF)-0.* (ALOG1O(ROSA/ROMF))**2.5 190 VOROP(IT)=VRTSA(IT)*1o.**YSA*( .+0.0023*(1.1-ROMF/F(nSA)*(20.-TZO' 191 DIST(IT)=DIST(I1-1)4t(vDkOP(1T)+VDROP(I1-1))/2.)*DELTJH 192 IF(DIST(IT).GT.HEIGHT) GO TO 17 193 38 CONTINUE 194 IT=IT-1 195 GO 10 17 196 37 CONTINUE 197 DELTIM=.05 198 DMASS=(1.333*3.14159*(D(ID)/200D.)**3)*985.1 199 DO 16 IT-2.LL 200 TIME(IT)(JT1-1)*DELTIM 201 REN=(VELDRP*D(ID)*POMF*10000.)/VISMF 202 DIAM=D(ID) 203 RKVO=DV(VELDRP) 20'4 RKV1=DV(VELDRP+(RKVO/2.)*DELTIM) 1 205 RKV2=DV(VELDRP+(RKVI/2.)*DELTIM) 20b RKV3=DV(VELDRP4RKV2*DELTIM) 207 VDRDP(IT)=VELDRP*(DELTIm/b.)*(RKV0+2.*RKVI+2.*RKV24RKV3) 208 VELDRP=VDROP(ll) 209 RKZ0=DZ(VELDRP) 210 RKZ2=DZ(VELD0P+(RKZO/2.)*DELTIM) 211 RKZ2=DZ(VELDUP+(RKL2/2.)*DELIIM) 212 RKZ3=DZ(VELDRP+kKZ2*DELTIM) 213 DIST(IT)=DIST(IT-l)4(DELTIM/6.)*(RKZO+2.*RKZ4+'2.*RKZ2+RKZ3) 214 IF(DIST(IT).G1.HEIGkT) GO TO 17 215 16 CONTINUE 216 JT=1T-1 217 17 CONTINUE 218 ISTART=1 219 DO 18 IHK=INHI 220 Z-(IH*H7INC)-(HTINC/2.) 221 DO 20 IS=1STARTPIT 222 IF(DIST(IS3.GT.Z) GO TO 21 223 20 CONTINUE 224 IS-IS-1l 225 21 ISTART=IS-1 226 FACTOR=(Z-DIST(IS-1))/J(DIST(IS)-DIST(IS-1)) C C LOCAL VELOCITIES AND FALL TIMES ARE STORED 227 VEL(IDIH)=VDROP(IS-1)+FACTOR*(VfROP(IS)-VDROP(IS-1)) 228 TIH(1D, IH)=TJME(IS-1)+FACTOR*(I1ME(1S)-TI~iE(IS-1)) 229 18 CONTINUE 230 15 CO1TINUE
-- -- - --- _- -- - -------------- - - - - ---- _ ------- --- - n_--_-_-______
231 DO 200 K=:1NHI 232 NH(K)=K 233 200 CONTINUE 234 PRINT 138,iD(J)#J=1r1CH) 235 PR IT 139 236 PRINT 140 237 PRINT 1a1t(NH(J),(VEL(IJ),I=:,ICH),J=l1NHI) 238 PRINT 1 38#(D(J)rJ=1#ICH) 239 PRINT 1l2 240 PRINT 1 40 21i1 PRTNT 1U1 ah{l )(H.I I TTM(t I.n............l=1 =1
.lrHi ........ Nto T
242 IF(DELAY.LE.O.o) GO TO 210 276 243 PRINT 125 24 4 PRINT 126 245 PRINT 127
'- 246 PRINT 128 24 7 PRINT 129 248 PRIN1T 130 24J9 PRINT 131 250 210 CONTINUE C
C VARIAULES ARE INITIALIZED C 251 DO 'Io I=1,lCH 252 DO uo J=1,NHI 253 WTPA(IJ)=O. 254 40 DRPHTtl.J)=0. 255 TTEST=O. 256 TZ=TZO 257 N=1 258 DO 24 ID=IICH 259 IFLAG(ID)=O 260 ISECID)-l 4., C C CONTRIBUTION OF ENERGY OF WATER ENTERING SYSTEM TO SYSTEM PRESSll C DROP IS CALCULATED C
- 261 FIN(I):=(TPRESS*hTFLOW*CPwo*DNWO*DISNJ(Io)*TWATQ)/1o.*tq 262 NS5(11))=4HI+I 'S 263 CONTtJl),l)0.0
- 264 24 TRES(ID)=TIH(IDNHI) f., 265 WAIR=PAIhO*In.**5*TKVOL/(RAIR*(TzO,27315),) C C INITIAL ENERGY CONTENT OF MIXTURE IS CALCULATED C
- 266 CALL COEF C
C TIME IS INCREMENTED C 267 YPR=DELAY+1O.*TPRESS 268 41 T=TPRESS+T 269 IF(T.LE.DELAY) GO TO 62 2270 TTEST=TTEST+TPRESS 271 N=N+1-272 Do 90 1G=IICH 273 94 CONT(IG,2)=O.O C C MAXIMUM TIME IS SET 274 1 IF(T.GT.90.) GO TO 42 C C LOCAL DROPSIZE DISTRIBUTIONS ARE CALCULATED C C -- ------------ a ----------------------- __ ___ 275 DO 22 ID=1,ICH 276 DPINC=TPkESS*ITFLO)*DROPS(jD) 277 IF(IFLAG(ID).EO.l) GO TO 56 278 K1I1SECI!)) 279 MMO 280 DO 23 ITEST=KI,NHI 281 IFNTIM(IDITEST).GT.ITEST) GO TO 39 282 IF(ITEST.EU.NHI) GO TO 27 283 23 CONTINUE 284 39 CONTINUE PA C TM Tl ( TTti(Tn. TTFST-1I4TTM-(n. 7TFqT) )/P.
I /
- . . I - -- I * *I. IA
.4 V U UU 287 Mm=1 288 IF(ISE(ID).NE.1) GO I0 29 289 FROLD(IU)=1.0-((THID-11ES7)/(1IM(lD,1TEST)-TIM(IOITEST-1)))
290 ZINC=FkOLD(1D)+FLOAT(ITES7-2) 291 GO To 31 292 29 FRNEw11.-FRULV(ID) 293 FRNEW2=1.o-((TMID-TTEST)/(TItl(IDITEST)-TIM(IDITEST-1))) 2914 ZINC=FRNEhi1FRNEw2tFLOAI(lTLST-ISE(1C)-l) 295 GO TO 28 296 25 IF(ISE(ID).NE.1) GO TO 33 297 FROLD(ID)=(TTEST-TMlD)/(TIM(IOITEST)-TIM(IDITEST-1)) 298 ZINC=FROLD(ID)+FLOAT(ITEST-1) 299 GO TO 31 300 33 FRNEWl=1.-FRGLD(ID) 301 FRNEw2=(lTEST-TMID)/(TIMCID,ITEST)-lIm(lD.ITEST-1)) 302 ZINC=FRNEWI+FRNEP.2+FLOAT(ITEST-ISE(1U)) 303 28 DRPHT(ID,ISE(I)-l)-DRPHT(IDLISE(ID)-I)+DPINC*(FRNEW1Iz/NC) 304 FROLD(ID)=FRNE62 305 31 CONTINUE , 306 J4=ITEST-1-eMM 4 307 DRPHT(IOITEST-MM)=DPINC*(FROLD(ID)/ZINC) 308 VELTR( ID,N-1)=VEL(IDITEST-tMm) 309 GO TO 30 310 27 FRNEW2=(TTEST-(TIMCIDNHI)+.5*(TIM(IDNHI)-TIMCID,NHI-l))))/(TIM II; 2DNHX)-TIM(IDNHI-1)) 311 FRNEwl=-0.0 312 IF(ISE(ID).NE.1) FRNEWI:=.-FROLD(ID) 313 ZINC=FRIJEWI+FRNEi,2+FLOAT(IJHI-ISElID)+I) 314 1F(ISE(]O).EO.1) GO TO 34 315 DRPHT(ID,ISE(ID)-1)=DHPHT(DtISE(ID)-1)+DPl1NC*(FRNEw1/71NC) 316 34 OUT(ID)=(1TEST-(TIMt(ID,NHI)+,5*(TIM(IpNHI)-TItl(D, tJHJ-1 ))))/TP.i 3S 317 IFLAG(ID)=1 318 J4=NHI 319 30 K3=ISE(ID) 320 DO 35 IC=K3,JU 321 35 DRPHT(IlD,IC)=DPINC/ZTNC 322 ITRANS(ION-1)=ITEST-M 323 ISE(ID)=ITESTI1-Mm 324 22 CONTINUE C------------------------------m----___-__~-___________,________ miff . If~ 325 56 CONTINUE C-------_-----------------000 mmeeeeeemommmawoinmmme ___0 _________.__,_* 'o ! I 326 IF(T.GT.YPR) Ga TO 201 327 PRINT 138v(D(J)lJ=l,ICH) 328 PRINT 143 329 PRINT 140 330 PRINT 14U,(NH(J),(DRPHT(I,J),I=IICH),J=1,NHI) j i 331 201 CONTINUE 332 Do 47 II,11,ICH It C C- LOCAL DROPLET TEMPERATURE DIS7RIdUlIONS ARE CALCULATED C ^ 333 N2=1 'A' 334 7INIT7tJ1,h'2, 1)=7ltj~l (lD1 ,N2,2)=19ATO i... 335 AMASS=t 1.3333*3.14159* (D(ID1)/2. )**3.*DNWO)/1 O.**9. 336 V123=VO 337 NO=NH1+i 338 Do 1S 1H1=I,NO 33q IF(IHl.LE.I'fkANS(Ibl,N2))GO To 16 34 0 N2=N2t 1 3 11 TIIJIT(IO1,N 2,2)=l7TPA(IDltIHl-j) ti I
278 3a2 IF(N2.EO.(N55(ID1)-1)) GO TO 6 343 TINIT(ITlwN~2-1,1)=TINIT(IDl.N2-1,2) 344 6 CONTINUE 345 IF(IHI.EU.t4O) GO TO o4 346 IF(ti2.GT.(N1-)) Go TO 44 347 IF(N2.GT.N55(I01)) GO TO u6 348 V123=VELTR(IDIvN2-1) 349 IF(N2.NE.N55(1DI)) GO TO So 350 IF(TINIT(IDlp2vl).LT.TZ) NS5(ID1)=NS59(11)+1 351 50 CONTINUE 352 IF(TINIT(IOlIN2rl).LT.TZ) GO TO046 353 N55(IDI)=N2 354 48 T123=TIM(II)1,IHI)-(TPRESS*(N5S(ID1)-1)) 355 TINIT( IDlN2,1)=TlNIT(ID0,N55(ID1l,1) 356 GO TO 49
.357 46 CONTINUE 358 IF(N2.GE.N55(ID1)) GO TO 48 359 T123=TIM(ID1,IHI)-(TPRESS*(N2-1))
360 49 CONTINUE 361 CALL DIEM4P(PTV123,T123,AMASSXS,DLST) 362 WTPA(IDIIHl)=TIt4lT(IDllN2,l)+(TZ-TlNIT(IDIN2,1))*DLST 363 45 CONTINUE 36a 44 CONTINUE 365 IF(N55(ID1).LTNO) TINIT(ID1,NSS(ID1),1)=TINIT(I10,N55(lDl),2) 366 47 TINIT(IDl1N2,1)=NTPACIDl#IHl-l) C - --------- - - ---------------- ________________________.,. 367 98 OCT=O. C--------------- ------------------ _______--n-- ___,__________.. 368 IF(T.GT.YPR) GO TO 202 369 PRINT 138,(D(J),J=1,ICH) 370 PRINT 145 371 PRINT 140 372 PRINT 144,(NH(J),(WTPA(IJ),I=lICH),J=1tNHI) 373 202 CONTINUE 374 DO bl ID2=1,ICH 375 L1=NHI 376 IF(IFLAG(1D2) .NE. 1) L1=ITRANS(ID2,N-1) 377 378 379 FOUT(ID2)=0. DO 60 IH2=I,L1 TVI=wTPA(ID2,IH2) 4.4 380 CALL PPWAT(TVltPT,CPWTDNWT) 381 9' CONT(ID2,2)=CONT(ID2,2)t(DNIiT*CP"T*DRPHT(ID2,IH2)*(1.333*3.1r5 5D(ID2)/2.)**3)*]Vl)/10.**9 382 IF(IH2.LT.NHI) GO TO 60 C C CONTRIBUTION OF ENERGY OF "ATER LEAVING SYSTEM TO SYSTEM C DROP IS CALCULATED C 383 TV4=w7PA(ID2,Nel) 384 CALL PPAAT(TVYPTCPWFtDNWF) 385 FOUT(102)=(OUT(0)2)*TV4*DISN(ID2)*CPWF*DNWF*TPRESS*iTFLOW)/I 386 OUT(ID2)=I.0 387 60 CONTINUE C C TOTAL ENERGY CHANGE OF WATER IS CALCULATED r. (I 388 DLCONT=CONT(ID2t2)+FOI)T(ID2)-CONT(ID2,1)-FIN(ID2) 389 DCT=I)CT+DLCONT 390 CONT(ID2,1)=CONT(ID2,2) 391 61 CONTINUE C------- --------- --- --- -- ---- _______ ----- _--- _--_ C C A NEt4 ENERGY OF THE SYSTEM IS C4LCULATED
C 279 C 392 ENINM=ENINK-DCT C C A NEWi PARTIAL PRESSURE OF STEAM 15 CALCULATED C 393 CALL COEF 394 62 CONTINUE A NEW ENERGY OF THE SYSTEM 1S CALCULATED 395 CALL WALLS(PSIGTPRESS,WALEF) 396 PPP=PPP-wALEF 397 CALL TEMP1(PPPTZ) 398 CALL SPVOL(TZPPP,VPRIM) 399 CALL ENTHAL(TZENST) Li00 ENINSI=ENST-0.461415*(TZ+273.15) O101 ENINA=O.7O9*TZ i. PARTIAL PRESSURE OF AIR IS CORRECTED FOR TEMPER4TURE EFFECTS i; 4102 PAIR=PAIRO* l (TZ+273. 1S)/(TZO+273, 15)) I I A NEW TOTAL PRESSURE OF THE SYSTEM IS CALCULATED I I
.403 PT=PPP+PAIR I i
404 XS-PPP/PT ii
£105 ENINM=ENJINA*wAIR+ENINST*TKVOL/VPRIM
- 406 PSIG=PT*1u.50a-14,696 107 U~ PTHAR=PSIG/14,504.
*408 P=PT*14.504 I
DYNAMIC VARIABLES ARE OUlPUTTED
!1 409 IF(7'.LE.DELAY) GO TO 203 I, 410 IF(T.GT.YPR) GO TO 203 411 PRINT 117 R iri 412 PRINT126 l113 PRINT127 !1 i.
414 PRINT 128 il 415 PRINT129 416 PRINT130 I 417 PRINT131 :11,
- : I-1418 203 CONTINUE i .i 419 PRINT132,t, TZ.PPP.PAIRPTBARPPSIG 420 IF(TZ.LE.TWATO) GO T0 42 I
C t; C ENTIRE PROCESS IS REPEATED USING NEW VALUES OF DYNAMIC VARIAbLES C 421 GO TO 41 422 112 CONTINUE Y23 STOP 424 END C C 425 FUNCTION DV(VELDRP) C 426 COMMON /AREA1/ GrAREADMASSIOHMBVISMt3rASSDIAtlDELR4O, IDS1AR1,DSTA2AgDSTA2BDSTA3ADSTA3tiDSTAHR1RE2ARE2RFRF3A,36D'Edl 2RE5,nMAXSCOMAX,niSTAMWMAXSISTAk*4,hSTA3AeWSTA3E;,WlS.TA2A,w TA2bD* 3WsTAR1,I)ST SVliR 427 *RONF=ROMH6 ll
ztu 428 VISHF=9VSmB 429 RENDR=(DIJA*VELDRPRPOMF*10000.)/VISMF u30 SIGN=1. 431 CALL CDREIN(RENDRC) 432 AHEA=3.j1059*(DIAM/2000.)**2 433 DV=G-((SIGN4*C*AREA*IO"F*VELDRP**2)/(2.*DMASS)) 4311 RETURN 435 END C C 436 FUNCTION 0Z(VELORP) C C U437 DZ=VELDKP 438 RETURN 439 END C C *********************************************** 440 SUBROUTINE SEKPV(TZ,PV) C C 441 X=TZ/100. 442 PV=(((((((-2.489006/10000.*X+3.156533/1000.)*X-1.5OR7829/100Q.* l.3488548/100.)*X-3.,b475M9/100.)*X+1.1476149/1000J.)*)-7.0a1943 2000.)*X+4.5905064/10.)*X+1.2b02044 Y43 RETURN 444 END 445 SUBROUTINE PPirA1(TV#PTCPVDENS) C C U46 A=(TV+273.15)/647.3 447 S=PT/221.286 448 U=100.*(3700000.-3122199.*A*A-199Q.85/A**6) 449 W=(U+SoRT(1.72*U*U+1362926.*10.**10*(S-t.500705*A)))**0.29q£1177 450 B=1.052356*(62.5+S*(13.10266+S))/10.**11/(1.5108/100000.+A**1)' 451 C=(0.6537154-A)**2 ( 452 C=C*(7.241165/100000.+0.7676621*C**0) ;,,$ 453 VVOD_0,417/v'+C-E-(11.39706-9,9£19927*A)/100000. 454 DENS=I./VVOD u55 X=TV/100. 456 Y=PT/10O. 457 Z=X**5 458 CPV=:./(Y+2.)+0.021*Y-0.2a927 459 CPV=CPV*(0.032+Z*(0.0032Lu+0.026912*Z*Z/1O0000.)) 460 Z=(((-l.lOOUS5*Xt4.837208)*X-24.206883)*X+33.44712)*X+30.72526 '461 CPV=Z*Y/1000.+CPV 462 Z=(((2.259985*X-7.403256)*X+19.097088)*X-13.00363)*X+Ulq.8259 4 463 CPV=Z/100.+LPV 464 RETURN 065 END C 066 SUBROUTINE DTEMP(PTrVvTvAMXSPOLST) C *********i~******-*,i**************** ***************s C 467 P=PT*100000 468 V--(2.317,1O.**(9))*P**1.22509Ž*V**0.2b2P.1**1 A
u(0,51322) 281 469 IF(AtLS(D).GT.170.) GO TO 6 470 DLST=1-EXP(D) 471 GO TO 8 472 6 OLST=1.0 473 8 CONTINUE 47a RE7UH N - 475 END C C 476 SUBROUTINE lALLStPS1GT0I1) C ****************r**********t***************** t C 477 CZ-0.00200 478 W=(PSIG/l4.504)*(1-EXI(C*T)) £479 RETURN 48n0 END C C 481 SUBROUTINE TEMPI(PPPTZ) C C; J82 X=ALOG(l.01972*PPP) aB83 .T2=(( (U(( ( -£4.292Lib03/1000O./10OOt.*X-Ls.2b8%685/1 nn~n000.)*X+1. 1343731/1000000.)*X+2.2071712/1O00000.)*x-1.7al7752/lO000O.)*x 3.7. 23484/10000. )*X+1.3263773/1000. )*XI2.129b82/100. )*X+2.107780$/1o. 3X+2.37S3577)*X+27.854242)*X+99.092712 484 RETURN £485 END C *A***********t********************t******t*****1**t****.i 1 *****r** 486 SUBROUTINE ENTHAL(TPENTA) C ***********************):****t*t*********************~******,t*i*-*-*. C 467 X=T/100. 486 ENTAtt((((1.0285952*X-q.4307509)*Y127.974559)*X-uL3.07781)iJ)*X+15.' 19571)*X*181.50666)*X+2500.6256 489 RETURN q9Q EN*;D c 491 SUBROUTINE SPVOL(TCePPPPVPRIM)j C **fi*********t******************t************I*****tt**.**I****t****1 192 DI;MENSION A(q) 193 A(I)=1.2b0?0437 .94 A(2)=4.590S0b39/10. 95 A(3)=-7.04 194353/1o000. 96 A (L)=1 . 1l761490/10000. 97 A(S)=-3.6b£475685/100. 98 A(6)=3.346b5479/100. 99 A(7) =-1.50t87b2'17/100. 00 A(B)=3.15b53300/1000. 01 A(9)=-2 .48900596/10000j 02 PV:A(1) 03 X=lC/o00, 4 DO 10 I=1,6
)5 10 PV=PViA(1+1)*X**I )6 \'VPRIrM=PV/PPP )7 RETURsN
5 08 END 282 C C SI**********************t**************************************. 509 SUBROUTINE CGEf C C 510 COMMON4 /AREA2/TKVOLrPPPENMIXY,,AAIRPAIRO, TZOT 511 PPp=PPP+ 001 512 DO I I=1,90 513 PPP=PPP-. 001 5114 CALL TEMP1(PPP,TZ) 515 CALL ENTHAL(T.ZENTA) 516 ENINT=ENTA-0.4bljlS*(TZ+273.15) 517 CALL SPVDL(TZ,PPP,VP9Im) 518 PAIR=PAIRO*((TZ+273.15)/(TZO+273.15)) 519 ENINS=ENINT* TKVUL/VPRIH 520 ENINA=0.709*TZ 521 TENINT=ENINS+ENINA*AAIR 522 IF(T.LE.O.1) ENMIX=TENJINT 523 IF(TENINT.LE.ENMIX) GO TO 4 524 I CONTINUE 525 4 CONTINUE 526 RETURN 527 END C **************************************** **********************i 528 SUBROUTINE CDREIN(RE1,CD) C *********** ****t******t***** ******** ** ** ***** ******* *************t, C 529 COMMON /AREAI/ GAREA,DMASSDROMfVISMftDASS DIAMDDELHO, - IDSTAR1i STA2ADDSTA2BDSTA3ADSTA3,DSTAP4tPE2AREP3,RE3APE3P,;El 2PE5,Dt4AXSCL)MAX,45TAM2wMAXS.wSTAR4,NSTA3A,V-STA38,NSlA2AtW5,A.?s.4 3wSTARIDSTSURFT ,7 530 DSTARP=O.*DIAM*((ROCM**2)*G*10.**2/VISMB**2)**(l./ 3 .) 531 IF(REI.LE.0.5) GO TO 10 532 IF(SS.GT.LU.*10**12) GO TO 20 533 IF(tRE1.GT.RE2A)GO TO 25 534 CD:(24./REI)*(t.+0.150*R.l**0.6f87) 535 DSTA=(3.*REI**2*CD*RO.MB/(4.*n)ELRO))**(1./3.) 536 WSTA=(4,*REItDELRU/(3,*CD*RUMB))**(1,/3,) 537 538 539 20 GO TO 210 CONTINUE IF(RE.GT.RE28) GO TO 26 I 54 0 C0=(2U./REI)*(1.+0.150*RE1**0.687) 541 DSJA=(3.*REI**2*CD*RlMB/(11.*DELRO))**(1./3.) 542 WSTA=(4.*PEI*DELRO/(3.*CD*ROMB))**(1./3.. 543 GO TO 210 544 25 CONTINUE 545 IF(REI.GT.PE3A) GO TO 50 5416 CD=0,48*(S*100**(-12))**0.05 54i7 DSTA=(3.*REI**2*CD*ROHbI/(a.*DELRO))**(1./3.) 54 8 WSTA=(4,*REl*DELRO/(3.*CD*RODM))**(lS3.) 5419 GO TO 210 550 26 CONNT I f4 UE 551 IF(kE1.GT.RE3B) GO TO 50 552 CD=0.7*(SS*0IQ**(-12))**(-0.05) 553 DSTA=(3.*RE1**2*C1*POMF3/(a.*DELRo))**(I ./3.) 5514 'eSTA=(s.*REI*DELRO/(3.*CD*ROMS))**(l./3.) 555 GO TO 210 556 1'0 CD=24./RE1 557 DSTA=(3.*RE1**2*CD*R(j14/(14.*DELRO))**l(1./ 3 *) S58 hSTA=(4.*REtlDELRO/(3.*COl*ROMB))**(1./3.)
559 GO TO 210 283 560 50 CONTINUE 561 IF(REI.GTREI) GO TO 60 562 DSTA=(REI/(45* (DELRO/kOMB)t*0.3ee*(SS*10.**(-12) )**n.0323))**(I .i 11.29) 563 WSTA=DSTA**0.2Q*a.5*(DELRO/ROMR)**O.388*(SS*10.**(-12))**0. 0 3 2 3 564 CD=.06570*DSTA**0.42*t(DELRO/Rur;o)**o.2Zn* (SS*10.**(-12))**(-0.Oo' i
- 1 565 GO TO 210 566 60 CONTINUE 567 5b8 IF(RE1.GT.RES) GO TO 70 WSTA=19.9b* (DELRO/ROME)**0.262*(SS*10.** (-12))**0.0o65I 569 DSTA=REI/wSTA 570 CD=0.00335*(DELRO/ROM8)**0,477*(SS*10.**(-12))**(-0.131)*DSTA 571 GO TO 210 572 70 CONTINUE 573 11 GR=4.*DELRO*DST**3-3.*REI**2*RomE*(A*(DST-DMAXS)+COMAX) I,:
574 GRPRI i=12,iDLLRO*DST**2-3. *REI **2*ROMH*A I, 575 DSTN=DST-GR/GRPRIM 576 IF(ABS((DSTN-DSI)/DS1).LT.O.O01) GO TO 12 ,i 577 DST =DST N 578 GO TO II 579 12 CD=CDMAx+A*(DSTN-DMA XS) 580 wST7=SRT(4.*DELRO*DSTN/(3.*ROMB*(A*(DSTN-DMAXS)+CDMAX))) 581 DSTA=DSTN 582 kSTA=wST 583 210 CONTINUE 584 RETURN 585 END SENTRY
284 APPENDIX 0 DROPLET SIZE DISTRIBUTION HEASURSENTS
-.4.
'4
-I- !
..A1,.
0 i I..M 1 1'
- 265 II 4 b..- V; 7
I
'I i
I.1 .
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Figure 07. Droplet Size Distribution for Droplets Shown in Figure 01. 10 Xm = 5 252 mm, b = 0'292 8 = 1-265 FLOW RATE-30 IGPM, NEAR THE CENTER OF THE SPRAY 8 4 0 2 ED a 0-5 I 1P5
- 2 3 4 5 d (mm)
Figure 08. Droplet Size Distribution for Droplets Shown in Figure 02. to Xm 3.49 mm b= 0.399 = 0 902 FLOW RATE = 30 IGPM, NEAR TO THE EDGE OF THE SPRAY 4 o I /n '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 2 l 2d (mm) 3 4 . t... _
Figure 09. Droplet Size Distribution for Droplets Shown in Figure 03. 12-Xm=2 935(mm) b=1-683 8=-1473
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Figure 011. Droplet Size Distribution for Droplets Shown in Figure 05. iI Xm= 4-629 (mm) b= 1b569 8 = 0-677 8 FLOW. RATE - 50 IGPM, NEAR THE CENTER OF THE SPRAY 6 4 0 0 0
%0 nA 3 4 d (mm)
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Figure 012. Droplet Size Distribution for Droplets Shown in Figure 06. Xm = 3517 b = 0-779 = 0*822 8 FLOW RATE - 50 IGPM, NEAR THE EDGE OF THE SPRAY 6 0 I N. 0 0 2
%D ON 2 4 .Pj :1 ; .: .. d (mm)
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Figure 016. Droplet Size Distribution for Droplets Shown in Figure 014. 12 0 0 Xm= 3 31 mm b =0*7697 8= 1107 8 0 0 x 6 tix a
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0' I 2 3 4 d (mm)
Figure 017. Droplet Size Distribution for Droplets Shown in Figure 015. 10- . - 1452 mm 6 . 998 6 0 0 Mfi I e 2 3 4
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