ML20069D275
| ML20069D275 | |
| Person / Time | |
|---|---|
| Site: | 05200003 |
| Issue date: | 03/09/1994 |
| From: | Viskanta R PURDUE UNIV., WEST LAFAYETTE, IN |
| To: | Khatibrahbar ENERGY RESEARCH, INC. |
| Shared Package | |
| ML20069D266 | List: |
| References | |
| NUDOCS 9406030087 | |
| Download: ML20069D275 (95) | |
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{{#Wiki_filter:) PURDUE UNIVERSITY l I l l RAYMOND VISKANTA W.F.M. Goss DISTINGUISHED PRortsSOR OF ENGINEERING Dr. Mohsen Khatib-Rahbar i Energy Research Inc. P.O. Box 2034 Rockville, MD 20847 1 Ref: NRC and NRC/ Westinghouse AP600 PCCS Scaling Meeting l Westinghouse Energy Center, Monroeville, PA I February 22-24,1994.
Dear Mohsen:
This is a follow-up to the discussion conceming the thermal-hydraulics scaling issues at the referenced meeting. I have spent considerable time in the Purdue University Potter Engineering Library and going through my files looking for up to-date information on turbulent i natural, forced and mixed convection heat trrnsfer relevant to extemal and intemal (annular i channel) heat and mass (evaporation) transfer. In this letter I will provide a brief summary on j my literature review findings and highlight my concems about Westinghouse's approach in predicting external and intemal thermal-hydraulics in the AP600 and the integral test facilities l that I expressed at the meeting. For the sake of clarity, I will keep my remarks separately. Findings of Review Literatum External Reasonably up-to-date discussion of mixed convection heat transfer is given by Chen and Armaly [1] and Pethukhov and Polyakov [2]. Correlations am available for horizontal and vertical plates, but experimental data under turbulent flow conditions are very limited. The data l are particularly meager for turbulent free and mixed convection under at high Rayleigh or l Grashof numbers for free convection and for high Richardson (Grashof over Reynolds number l squared) numbers under mixed convection conditions. For example, Siebers et al. [3] correlate their experimental data for turbulent free convection from a vertical plate by an equation Nu, = 0.098 Gr!/3 (T,/T )* (1) 9406030087 940429 PDR ADOCK 05200003 A PDR HEAT TRANSFER LABORATORY taas SCHOOL OF MECH ANICAL ENGINEERING . WEST LAFAYETTE. IN 47eo712ss tai 7:do4 se32
where x is the vertical distance, Grx is the local Grashof number based on the local coordinate x and T, and T., are the wall and free-stream temperatures, respectively. Their experimental data extend to Gr, = 1.86 x 1012. These values are higher than the range covered by Vliet (1969) to which Westinghouse referred (see Figure 1 of presentation by Dan Spencer). The data of Vliet were obtained under a constant heat flux boundary condition, and, therefore, he used a Grashof number (Grl) which is based on q, and not T -T., to define the Grashof number. Even if the temperature dependence of the thennophysical properties on temperature (i.e., factor (T,/I' )*34) is ignored, the coefficient of 0.098 differ, from the value of 0.13 used by Westinghouse (see Equation 1 in D. Spencer's notes). A number of mixed turbulent convection correlations are presented by Chen and Armaly [1, pp.14-24 to 14-26] and need not be discussed here in detail. Some of these correlations am purely numerical (theoreti:al) and others have received some experimental validation. In summary, the correlations proposed by Westinghouse (Equation 5 in D. Spencer's notes), is different from those published in the literature. What is probably more significant is that the local Nusselt number for a vertical plate is considerably higher than for a horizontal flat plate, and this suggests that Equation (5)in Spencer's notes can not be used for both orientations. In addition, there is a need to distinguish between the flows where buoyancy aids and where it opposes forced flow. Internal Heat transfer to fluids flowing in vertical tubes under conditions of mixed (forced and free) turbulent convection have shown to exhibit marked departures from the case of purely forced convection. Significant degradation or enhancement of heat transfer can occur, depending on the flow orientation and the degree of buoyancy influence. Several state-of-the-art discussions are available and include those of Pethukhov et al. [4], Aung [5], Gebhart et al. [6], Pethukhov and Polyakov [2], Jackson et al. [7], Cotton and Jackson [8] and Poskas and Vilemas [9]. Some of these reviews are very extensive and may include as many as a hundred references. The review by Aung [5] cites 87 references and the one by Jackson et al. [7] is probably the most complete and cites 99 references. Unfortunately, the extensive experimental data reported are for vertical pipes and not parallel plate channels and fluids other than air. For ascending flow in a uniformly heated vertical tube, an implicit expression for the Nusselt number ratio, Nu/Nur (Nusselt number for mixed turbulent convection Nu to Nusselt number for forced convection Nup) has been recommended by Cotton and Jackson [10] of the form
, , 0.46
= 1.0 i 8 x 104 B (2)
Nup Nup where the + sign applies to descending flow and the - sign to ascending flow. 'Ihe buoyancy parameter B is defined as B= 3A Re Pr0 *
, 4 In Eq.(3) Gr* is the Grashof number based on the wall heat flux q, (i.e., Gr* = g q D /kg), and Re is the Reynolds number based on the diameter D as the characteristic length. The Nusselt number for forced convection, Nup,is calculated from the Dittus-Boelter equation, Nup = 0.023 ReE8 PrEd (4)
II,quation (2) has been compared with a body of experimental data for turbulent mixed convection heat transfer to gases in tubes. He details can be found in Cotton and Jacksoa [10), Jackson et al. [7] and Poskas and Vilemas [9]. The buoyancy parameter B defined by Eq.(3) in the turbulent mixed convection regime is based on the consideration of the modified shear stress distributions which occur in the near-wall region in response to buoyancy forces. Both the buoyancy parameter B and the semi-empirical equation for the Nusselt number ratio, Nu/Nup, given by Eq.(2) are derived on the assumption of fully-developed hydrodynamic and thermal conditions; however, in ascending mixed convection flows, very long development lengths (-100 D) are found to occur both experimentally and numerically [10]. It should be noted that Eq.(2) gives a discontinuity in Nusselt number for heated upward flow when Gr*/Re .425Pr 3 E8
- 3 x 104 . The published results show that in ascending flow heat transfer may be either impaired with respect to forced convection (at moderate wall heat fluxes) or enhanced (at high wall heat fluxes); whereas, in descending flow, heat transfer is enhanced at all imposed heat fluxes. The results reported here raise technical issues concerning the validity of the computational mixed convection heat transfer model employed by Westinghouse to calculate turbulent mixed convection heat transfer in the annulus between the containment shell and the air baffle (see Equation 5 given in the notes of D.
Spencer). I was not successful in identifying any theoretical or experimental results for turbulent mixed convection heat transfer in an axisymmetrically heated vertical duct that is formed by one uniform temperature wall and one adiabatic wall. This physical situation would be quite close to the case of a dry containment wall. For laminar flow the experimental local convective heat transfer results in a two-dimensional axisymmetrically heated vertical channel where represented by an empirical correlation [11), Nux /Re'4 x = (0.296)" + (0.359 Rifd)" (5) where Rix =Gr x/Re 2 s the local Richardson number. The value of n = 2.0 gave the best correlation for the experimental data. He experimental local Reynolds number Rex was based on uowhich is the center line velocity at the inlet section of the duct. Note that according to Eq.(5) the relevant scaling parameters for local heat transfer along the heated vertical wall of a vertical parallel-plate channel are the local Reynolds number Re x and the local Richardson number Rix and not the Reynolds number based on the hydraulic diameter (Red) and the local Grashof number Grx as proposed by Westinghouse (see Equations 1, 2 and 5 of the notes distributed by D. Spencer). No turbulent mixed (combined) heat transfer in an asymmetrically heated vertical channel could be identified in the published literature.
l The experimental work of Kapoor and Jaluria [12] shows that the relevant heat transfer scaling parameter is the Richardson (Rir, = Gro/Reh) number based on the width of the slot D and the horizontal distance L to width D ratio, I./D. Different correlations were obtained for the ceiling, wall and ceilinF Pl us wall. This suggests to me that the correlation used by Westinghouse does not contain the relevant scaling parameters. The published results further show that the Nusselt number is not a monotonically decreasing function of the distance along , ceiling and wall as measured from the slot. This indicates to me that the correlation used by l Westinghouse would not capture the correct trends of the heat transfer coefficient variation along the dome and cylindrical part of the containment wall. Validity of Heat and Mass Transfer Analogy Relatively little research attention has been focused on the heat and mass transfer analogy which is being used by Westinghouse (see Equation 7 in notes provided by D. Spencer) to relate : the mass transfer coefficient to the heat transfer coefficient in the annular gap. Greiner and Winter [13] studied theoretically and experimentally the defects of heat and mass transfer analogy for laminar and turbulent forced convection along a horizontal flat plate and showed that significant defects in analogy exist when the normalized driving force of the evaporation process is relatively large. To eliminate the effect in the heat and mass transfer analogy a correction needs to be made in the film theory of Ackermann. This suggests that the heat, mass and momentum correction factors used by Westinghouse may not be appropriate. Technical Issues ! A number of technical issues were raised at the meeting concerning Westinghouse's I approach in modeling internal and external heat transfer in the AP600 containment. Here, I would like to repeat in writing some of the questions I raised at the meeting, provide relevant j literature citations and elaborate on my technical concems. j
- 1. The correlation used by Westinghouse for external mixed convection heat transfer l
[ Equations (1), (2), (5) and (6) in the notes of D.R. Spencer] are inconsistent in form with ) published correlations [1,12] (see also Equation 5 above). The relevant scaling parameters are the local Reynolds (Rex ) and the local Richardson (Rix) numbers, and the location of the coordinate axis x needs to be clearly defined for the correlations to be i applicable.
- 2. What are the relevant thermal-hydraulics scaling parameters for buoyancy (temperature and concentration difference) driven flow in the downcomer and the upcomer of the l PCCS? Westinghouse has made no attempt to identify the relevant scaling l (dimensionless) parameters for the physical system considered. For buoyancy driven flow the Reynolds number is not a relevant dimensionless parameter, because the velocity is not l known and hence the Reynolds number can't be specified a priori.
- 3. For internal mixed convection in the channel formed by the containment wall and the air baffle, the combined (free and forced convection) Nusselt number used by Westinghouse
[ Equations (3), (4), (5) and (6) in the notes of D.R. Spencer] is inconsistent in form with the published results [2,5] for internal mixed convection (see also Equation 2 given above). Equation (4) used by Westinghouse is for forced convection heat transfer along a vertical plate (i.e., when the effects of buoyancy are not important) and not for flow and heat transfer in a vertical, parallel plate channel after the two boundary layers on the
vertical walls have merged. I am afraid that Westinghouse is attempting to force their integral test data results to match correlations of highly questionable fann that have not been used previously in the published mixed convection heat transfer literature [2,5].
- 4. The heat and mass transfer analogy defined by Westinghouse [ Equations (7) and (8) in notes of D.R. Spencer] and its modification to account for evaporation / condensation are :
based on a laminar film theory (14] There is no evidence in the literature, at least that I could find, to suggest that these corrections are r.ppropriate for physical situations relevant to AP600 PCCS. I feel uncomfortable with the equations used by Westinghouse, because the corrections are of theoretical nature, have not been developed for turbulent flow and have not been validated for the problems being considered. I am concerned that the analogy (as it now stands) is being misapplied, and I do not have much confidence in its use as it now stands. My question is how can Westinghouse address my concems? Please do not hesitate to call me if you or NRC have any questions or I can provide any additionalinformation. Sincerely,
@w R. Vis ta RV/fb P.S. I am enclosing with this letter some of the papers that I have cited in my remarks. Some references are just too long to be copied in full.
LL& '
References
- 1. T.S. Chen and B.F. Annaly, " Mixed Convection in External Flow, in Handbook of Single-Phase Convection Heat Traufer, edited by S. Kakac, R.K. Shah and W. Aung, John Wiley and Sons, New York,1987 Section.14.
- 2. B.S. Pethukhov and AF. Polyakov. Heat Transfer in Turbulent Mixed Convection, Hemisphere Publishing Corp., New York,1988.
- 3. D.L. Siebers, RJ. Moffat and R.G. Schwind, " Experimental, Vatiable Pmpenies Natural Convection from a Large, Vertical, Flat Surface," J. Heat Transfer, Vol.107, pp.124 132,1985.
- 4. B.S. Pethukhov, AF. Polyakov and O.G. Martyenko, " Buoyancy Effect on Heat Transfer in Forced Channel Flow, in Heat Transfer- 1982, Hemisphere Publishing Corp., Washington,1982, Vol.
1, pp. 343-362.
- 5. W. Aung, " Mixed Convection in Internal Flows," in Handbook of Single-Phare Convection Heat Traufer, edited by S. Kakac, R.K. Shah and W. Aung, John Wiley and Sons, New York,1987, Section 15.
- 6. B. Gebhatt, Y. hluria, R.L. Mahajan and B. Sammakia Buoyancy-Induced Flows and Transport, Hemisphere Publishing Corp., Washington,1988, Ch. apter 10.
- 7. J.D. Jackson, M.A. Cotton and B.P. Axcell, " Studies of Mixed Convection in Vertical Tubes," Int.
J. Heat Fluid Flow, Vol.10, pp. 7-15,1989.
- 8. MA. Cotton and J.D. Jackson, "Venical Tube Air Flows in the Turbulent Mixed-Convection Regime Calculated by a Low-Reynolds-Number k-c Model," Int. J. Heat Mass Transfer Vol. 33, pp. 275-286,1990.
- 9. P. Postas and J. Vilemas, " Mixed Convection in Channels," in Power Engineering, " Academia,"
Vilenius (1993), Nr. 3, pp. 22-25 (in Russian). 10 M.A. Cotton and J.D. Jackson, " Comparison Between Theory and Experiment for Turbulent Flow of Air in a Vertical Tube with Interaction," in Mixed Convection Heat Transfer- 1987, edited by V. Prasad, I. Catton and P. Cheng, ASME, New York,1987, HTD-Vol. 84, pp. 43 50.
- 11. B.J. Beak M.B. Younger, BF. Annaly, and T.S. Chen, " Mixed Convection in a Two-Dimensional Axisymmetrically Heated Vertical Duct," in Mixed Convection Heat Tramfer - 1991, edited by D.W. Pepper, BF. Armaly, M.A. Ebadian and P.H. Oosthuizen, ASME, New York (1991), HTD-Vol 163, pp. 37-43.
- 12. K. Kapoor and Y. Jaluria, " Mixed Convective Heat Transfer Characteristics of a Downward Turning Buoyant Ceiling Jet," in Mixed Convection Heat Transfer- 1991, edited by D.W. Pepper, B.F. Armaly, M.A. Ebadian and P.H. Oosthuizen, ASME, New York (1991), HTD-Vol.163, pp.
9-17.
- 13. M. Greiner and E.R.F. Winter, " Analogy Defects of Mass, Heat, and Momentum Transfer in Evaporation Boundary Layer Flow," in Studies in Heat Transfer: A FestschrVtfor E.R. G. Eckert, edited by J.P. Hartnett, T.J. Irvine, Jr., E. Pfender and E.M. Sparmw, Hemisphere Publishing Corp., Washington,1979, pp. 41-53.
14 R.R. Bird. W.E. Stewart and E.N. Lightfoct, Transport Phenomena, John Wiley and Sons, New York (1960), pp. 659-666. I l
s i HANDBOOK OF SINGLE-PHASE CONVECTIVE HEAT TRANSFER i Edited by Sadik Kakag Department of Mechanical Enginering University of Miami Coral Gables. Florida
' Ramesh K. Shah Harrison Radiator Division General Motors Corporation Lockport. New York Win Aung National Science Foundation Washington. D.C.
A Wiley-interscience Publication JOHN WILEY & SONS Odchesser - Brisbane - Tomnio - Singapore New York
i I l t lt 1 15 ! l I MIXED CONVECTION IN i INTERNAL FLOW !, i Win Aung , National Science Foundation Washongton, D.C. 1 I 15.1 Introduction 15.2 Governing Equations and Parameters
;j 15.3 Laminar Mixed Convection in Vertical Ducts j' 15.3.1 Hydrocynamically and Thermally Fully Developed Flow in Venical Circular 6 Tubes 15.3.2 Thermally Developing Flow in Vertical Circutar Tubes l,l 15.3.3 Hycrodynamically and Thermally Developing Flow in Vertical Circular Tubes '
15.3.4 Hydrocynamically and Thermally Fully Develooed Flow in Vertical Annut 15.3.5 Thermatly Developing Flow in Vertical Annuli 15.3.6 Hycrodynamically and Thermally Fully Developed Flow between Venical Pars!>el Plates 15.3.7 Hydrodynamically and Thermally Developing Flow between Vertical Petallet r Plates 15.4 Laminar Mined Convection in Horizontal Ducts ' 15.4.1 Thermally and Hydrodynamically Fully Developed Flow in Honzontal Cir. cular Tubes 15 4.2 Thermally Developing Flow in Horizontal Circular Tubes 15 4.3 Flow en Circular. Concentnc Honzonta! Annuli 15.4.4 Thermally Fully Developed Flow between Horizontal Paratiel Plates 15.4.5 Thermally Developing Flow between Honzontal Paraliel Plates 15.4.6 Flow in Rectangular Honzental Channels 15.5 Transition from Laminar to Turbulent Flow 15.5.1 Transitional Upward Flow in vertical Circular Tubes 15.5.2 Transitional Flow in Horizontal Circular Tubes 15.6 Turbuler't Mined Convection in Ducts 15.6.1 Venical Ducts 15.6.2 Honzontal Tubes Nomenciatu o References 4 15 1
15 40 4 xt.D CONVECTION IN INTERN AL Flow indicauons are that for buoyanev aaded flos. the mecharusrn of transiuon a smular transfer in oo' to that m boundarulayer flow over a flat plate. Here, transmen conmts of the expenmental es appearance et regular oscillauons which gradually grow m extent and ampbtude until no the disturoance breaks mto fluctuaung monon that is charactensue of turbulent flow. For buoyanev-opposed flow transiuon consists of an asymmetne flow (resulung from an onpnally symmetnc fully developed flowl wiuch gives nse to reversed flow on one 15.6.1 Vertic side of the tube. The extent of the reversed flow increases m size as Grg/Re increases. From the quant leadmg to an eddymg now. The transiuon to an eddymg modon occurs suddenly. laminar nuxed ( The flow escillauons that accompany the transiuon process lead to fluctua6ons in the mercased vi wall temperatures, and these can in turn be used to indicate transmon. Using thn isince near wal approach. Hallman (12) found that for buoyanev assisted, hydrodynamically fully now, where the developed flow tat tube entrance) in a UHF vertical ctreular tube, the locadon of when natural c. transiuon depends on the heatmg rate and the flow rate. At a constant flow rate, an transfer is gene 5 increase in beanns rate causes the pomt of transidon to travel upstream (i.e.. down the The phenon tubeh lf the heating rate is held fixed, inercasing the flow rate moves the transition explained by a point downstream. Based on Hallman's data, the following correlation may be used to heated wall ext predict the locauon of transidon: direction of mo i the wall. Conse Gr3 Pr - 2664 Gr " (15.70s) taminanzation the onset of bu 15.5.2 Transitional Flow in Horizontal Circular Tubes in a heruontal circular tube, the transition to turbulent flow is strongly afected by the presence of secondarv flow. With a high initial turbulence level at the entrance to the tube. the onset of secondarv flow tends to suppress the turbulence, while at a low ininal where Gr= go turbulence level. secondary flow tends to increase the turbulence. Consequently, when a the initial turbulence level is low, the experimental critical Reynolds number Re,, defmed as the value where an intermittency begins to appear in the flow [43), decreases with Re Ra* but can be greater than 6000 at Re Ra* - 1.5 x 10'. At a high level of inidal turbulence such as that associated with a turbulence generator, Re, increases In tubes he with Re Ra* but is only about 1500 at Re Ra* - 1.5 x 10'. For Re Ra* < 5 x 10s sharp peaks sr with UHF tubes. Mori et al. [43) recommend the fellowing equadon when the initial [83), among o turbulence level is low: however, since Re,, start to incre. 3 Re,- , , (15.70b) i buoyanev-aide I the Grashof t in the above. Re.n is the critical Reynolds number without heating and can be more value at high than 6 umes larger than the high initial-intensity entical Reynolds number of ap- reponed orip proumately 2000 [80). Mori et al [43) obtained a value of Re,n - 7700. For high inibal supercrideal p f turbulence levels. Mori et al. [43] recommend. for UHF tubes. the UHF wal j becomes maxJ I Re, - 12S(Re Ra")"' (15.70c) the data shov l Herbert and 5 For turbul, the heat trar 15.6 TURBULENT MIXED CONVECTION IN DUCTS semiempirical For verdcal upward flow in turbulent mixed convection in heated tubes. fairly well-estabhshed entena for the onset of buoyancy induced impairment of heat transfer are *i available. No satisf actory correlating equation, however. is available yet for the beat
+
i.e s TURBULE.NT MtXE.D CONVECTION tN DUCTS 1fa41 . transition is similar transfer. In downflow, a satisfactory correlation now exists. For horizontal tubes. on comtsts of the expenmental esidence indicates that the effect of buoyancy is negligible in turbalent sad amphtude unul now, c of turbulent now. dow (resul6ng Irom tversed now on one 15.6.1 Vertical Ducts s Gr3/Re increr.ses. From the quanutadve information presented in Secs.15.3 and 15 4. it is clear that in
- curs suddenly. laminar mixed convec6on in a vertical duct. the heat transfer is improved (by virtue of
.d to Ductuadons m the increased velocity near the wall) for aided flow but is worsened in opposed Row msines. Using dus (since near-wall veloesties are reduced). He situauon is quite different in turtulent rodynamically fully now, where the beat transfer is sometimes less than the pure forced-convecuon value .be. the location of when natural convecuor, aids forced convec6on. and where in opposed flow the heat nstant flow rate an transfer is generally larger than the corresponding forced convection value. ream (i.e., down the The phenomenon of beat transfer impairment in vertical heated upflow em be aoves the transition explained by a two-layer model[81]. In this concept, the fluid in the layer close to the 6 n may be used to beated wall expenences a buoyancy force owing to the reduced density. Acting in the direction of motion. this force tends to decrease the shear stress in the layer away from the wall. Consequently, turbulence production is reduced across the tube, result ng in (15.702) laminanzadon. A simple approximate analysis [82] leads to the following criterinn for the onset of buoyancy-induced impairment of heat transfer: Gr 5 N .mgly affected by the Re sa
- the entrance to the while at a low inidal where Gr g(p, - E)Dl/(Er!). The integrated density E is defined as Consequently, when vnolds number Re,- 1 r E p dT (15.71a)
- flow [43). decreases T* - L *
- At a high level of rator. Re, increases In tubes heated at specified fluxes (UHF), the heat transfer impairment leads to r Re Ra* < $ x 10s sharp peaks in the local wall temperatures, which have been observed by Ackerman uon when the ini6al l83}. among others. The thermal impairment does not persist at higher buoyancy.
however, since the shear stress changes sign and energy inputs to the turbulent motion start to increase, as does the thermal performance of the tube. As a result, in (15.70b) buoyancy-sided flow, the heat transfer from the tube is impaired in the low ranges of the Grashof number. but recovers and may even exceed the pure forced-convection i i ng and can be more value at high Grashof numbers. Quantitative evidence of this behavior has been i olds number of ap- reported originally by Fewster [84] for upflow of carbon dioxide and water at superen6 cal pressures. Figure 15.2.2 shows the situation in which the temperatures of l 7700. For high inidal the UHF wall are below the pseudocritical temperature, the temperature at which c,, becomes maximum. It may be noted that the cmerion of Eq. (15.71) is supported by (15,70c) the data shown in Fig.15.22. Included in Fig.15.22 are the U%T upflow data of Herbert and Sterns (3). of which more will be said later in this sec6on. l For turbulent tmxed convecuon in buoyanev-opposed flow in verucal UHF tubes. the heat transfer is generally enhanced over that for pure forced convec6on. A semiempincal equanon for this 6tuaton has been developed by Jackson and Hall [E2l: I ed tubes, fairiv well- , g 3eein ; g it of he:.: transfer are
* - 1 + 2750 Re., (15.72) i Nu r i
, j able ves for the beat
\
l 4
---.___ _ - - - -l
N 22 uixts)( ovvtrTios is INTERN AL Flow , 10 pure forced corn
- DA7 A FROM FEWSTER l84j correlauon for b a DATA FROM HERBERT AND STERN l3)
Nu *'
. is
- Nu, , Thu equauon sh-
[+ $ ."; ,* b oyanev oppos 49 .. that buovancy ef. correlated by the
\ Nu G ,o4
= 15 Nu r 27
. Re ' ,
0.1 10-7 10-6 10-6 10-' 10-3 which gives valu g Eqs. (15.73) to O raw expenmenta Re;, results of [3] miti Figure 15.22. Average Nuswh number for buovanev.asusied turbulent now, normalaed with are shown in Fig Nuw.ch number for pure forced convection turoulent How, m a verueal tube. between Gr used made: 10 DATA FROM HALL l82) . DATA FROM HERBERT AND STERN [3]
- D = 19 mm
- Bmm 5mm i
Nu, 1.0 , , - u., _ , _ _,,n
.- . - - ~ - - . - -
, _ , _g
- EQUATION (15.72) 0.1 10-' 10-6 10-6 10-' 10-3 Gi 23 Re Figure 15.23. Average Nusselt number for buoyanev-opposed turbulent now, normalued with huswit number for pure forced convecuon turbulent now,in a verucal tube. j l
Trus equauon is compared with expenmental data for superentical-pressure waitr in !
- In the above the 4 15.23.
For vertical tubes at UWT. in tbc high Re range. Herbert and Sterns [3] have found - outlet of the sub ' p g - that buoyaney effe:ts on the heat transfer are negligible m aided turbulent mixed convection when Re exceeds a certain apparent critical value Re y . Their data were P" based on experiments with water, with Pr varymg from 1.8 to 2.2 and Gr, from l'h..ag - , l 2.0 x 10' to 2.6 x 10'. approximately. The value of Re, may be calculated from the .
- m W in d e r followmg equation: convection [86): l Re, - 3000 + 0.00027 Gr, Pr' (15.73) i-Thus, when Re is greater than Rey . the Nusselt number is given by the correlation for ei For opposed fio yI a
A l l
1S*43 TURBUL.ENT MtXE.D CONVEC" MON IN DUCTS - pure forced convention. When Re < Re,,, Herbert and Sterns (3) suggest the following correlation for buoyancy aided turbulent flow: (15.74) Nu - 8.5 x 10- 2(Gr. Pr)"'
*##
- This equadon should be used in the ranges Re - 4500 to 15.000. D - 0.0127 to 0.0254
- m. L - 0.254 to 3.30 m. Pr - 1.8 to 2.2; and Gr. - 3 x 10* to 30 x 10* For g T. s buoyancy opposed turbulent convecuen, the data of Herbert and Stern.13) indicaic that buoyancy effects may be neglected for Re > 15.000. For Re < 15.000, the data are correlated by the equation
,a, (15.75)
Nu - 0.56 Re"d'Pr ' n
; Re .
which gives values higher than those for pure forced convection for UWT cubes. In 10-3 Eqs. (15.73) to (15.75), all properties are evaluated at the film temperature. r Using the raw cxpenmental data given in {3], comparisons may be made between the Nu/Nu results of [3] w,th those of Fewster [84) and Jackson and Ha11[821. These compansons Af8 Sb0*U in Figs. !$.22 and 15.23. De agreement is very good. To effect conversion =. normahzed with between Gr used in 182) and Gr. used in [3}, the following approximations have been made: E
- pf ND STERN [3}
p, - D = - p.S(T. - Ti) T
. " p,0 . - T,,
s T
= p,0 . - T,
' 2 g( p. - 5) D 5 Gr -
pr J 10~3 gBo2 ( T. - T ) D'
~
2 p'. now. normshird with Gr +c. . 2 al. pressure water in in the above the overbar designates the arithmetic mean of the values at the mlet and outlet of the tube. For vertical tubes at UWT in the low Re range, the turbulent mixed convection Nu iterns [3} have found ted turbulent mixed is independent of Re in both aided and opposed flow. Testing w,th air in aided flow m e . Their data were the range Re - 385 to 4930. Brown and Gauvin (85] found that Nu may be predicted 7% in the range Gr - 5 x 10* to 1 x 10' by the following equation for pure free i 2.2 and Gr. from deulated from the to con$ection {B6): (15.76) Nu - 0.13( Gr3 Pr ) ' (15.73) For opposed flow in the range Re - 378 to 6900. Brown and Gauvm P51 shou that Nu by the correlauon for
,ir s I l HANDBOOK OF SINGLE-PHASE CONVECTIVE HEAT TRANSFER i I Edited by Sadik Kakag i Department of Mechanical Engineering University of Miami Coral Gables. Florida i Ramesh K. Shah Hamson Radiator Division General Motors Corporation Lockport. New York Win Aung National Science Foundation Washington. D.C. i l A Wilev-Interscience Publication JOHN WILEY & SONS New York - Cbchester - Brnbane - Toronto - Singapore i l
l I 1
)
l 14
- MIXED CONVECTION IN EXTERNAL FLOW T. S. Chen and B. F. Armaly University of Missoun- Rolla Rolla. Missouri e
1 14.1 introduction 14.2 Fundamentals of Mised Convecuon 14.3 Correladon Equauona for Nusselt Numbers ~ ^ ~ ~
- I 14.4 Correlations for Flat Pistes
, 14.5 Correlations for Continuous Moving Sheeta .
14.6 Correisuons for Vertical Cyllnoers in Longitudinal Flow l 14.7 Correlabons for Horizontal Cylinders 14.8 Corretauons for Spheres 14.9 Corretauons for Vertical Flat Plates in Cross Flow 14.10 Turbuknt Heat Transfer Corretabons for Flat Plates 14.11 instability and Transrbon 14.12 Conciusions Acknowkdgment Nomenclature Referonows 14 1
m__. . . . . -_. _ . . _. m . m . . _. - . . .__ . . _ . _ 14 24 - MixtD COWECTION IN ITiTJtWAL FLou . 3., , The correspondm; cortclanon for the avera;c Nusselt number can be expressed by 3
. (, -
N u, Ref ' .,* - 0.8N 1 + L f Gr*,, ' 3 ' "} (14.39) o 3 0,752 H :, Rej' ,
) . t y o p it should be emphutzed here that the above correlauons for vertical plates in crou How are hauted to fluids having Pr - 0.7 and that fluid prop: rues are evaluated at the 7 3o free stream temperature T,, Correla6on equauons for fluids with Prandtl numbers #
other than 0.7 can be developed only when the results from analyses and experiments 7.s become available. i P
.3 - >
14.10 TURBULENT HEAT TRANSFER CORRELATIONS FOR FLAT PLATES Reported studies on turbulent beat transfer in miaed convection are lacking in the literature. Turbulent mixed convection on vertical and horuontal flat plates under the g_ UWT condidon has been analyzed by Chen et al. [34.35] by employing a modined O o n ising. length model that accounts for the buoyaney. force effect. Their calculations for Pr - 0.7 yield local Nusseli numbers that converge to the bndt of pure forced convection. but underpredict by 20% the available results in the hmit of pure free 8 F6gure 14.20. Co conveeuen. The local Nusselt number for turbulent flow over a flat plate under UWT sim, vertical and i 5 for 0.5 s, Pr s 1.0 and 5 x 10 s Re, s 5 x 10* is given by [36) Nu,Ref'n - 0.0287 Pr* * (14.40) correlation equai verdeal plates, r The local Nusselt number for turbulent free conveedon along an isothermal verdcal flat sponding averag. plate for all Prandtl numbers and for Gr, to 103: is correlated by Churchill and Chu {37) as f 0.492 ' 'M -3'47 Nu,Gr,M - 0.15 Pr34 1+i
' Pr ' (14.41) with the same e For turbulcat free constetion over a beated horizontal flat plate facing upward, the Turbulent mi local Nusselt number for Gr,Pr > 5 x 10' is given by Fujii and Imura [38] as flow was analvz
,e number for a sqt -
Nu,Grl 3M - 0.13 Pr34 ' " (14.42) Equadon (14 42). obtained under netther the UWT nor the UHF condinon. was also eerihed later by Imura et al. for an isotbenna! borizontal flat plate (39). 3 j If Eq (14.12) is emploved to propose a correlation for the local Nusselt number in mixed convecuen. the resulting form is Equation (14 451 l Measurement Nu , Re'~ 8 G(Pr) ' Gr' , in . 1/- m I ng and H - l
. }+c (34.43) et al. [33J have f(Pr) , F(Pr) i Rejid ; , correlation equa 8
form as 1 where f(Pr) - 0.0287 Pr ' and G(Pr) - 0.15 Pr3M[1 + (0 492/Pr)'M*]-3'A' for a verdcal plate and 0.13 Pr34 for a borizontal flat plate. The analytical local Nusselt g"' numbers of Chen et al. [34.35] for Pr - 0.7 agree fairly well with the proposed t 0 1.
4
, , , , , in the range t s 600'C.
g The corre>; 1 = 2 7f,3 m expressed for a = 0 793 m W = 0 7 tair) 3 - e E
- R
- Y' #
- l
- x'I , , g_ys3 m
[2 - a = 2.524 m - with n - 4 pr. g' 6 Figure 14.! 7 . for a verdcal D
, D. :) location a w 0 -
1 7 - 1 = 1.637 m ~ (14.46). A con a = 2.814 m plate and the i i the two figuri correution eauston Y32=1+X32 numbers agret
, , , , , properties are 0 1 2 3 x = 7.os7(a/axg/RejM)l" (7,/7 )* 14.11 INS'l Figure 14.21. Companson between measured and correlated local N.tiselt numbers; turbulent air ocw across a vertical nat plate. UHF (33) u3yancy fort convecuon. 71 hence aber the i i
of laminar mix 3 induced by thi I I I mod:s. One of I " 2#
- in laminar mh
# = 3 030 m Pr = 0.7 (air) number Re,,,
the minimum
; 2 - become unstat o From tbc m q
a strong force g o . stabili:ing effe. p over a borizon I O force opposes "I
- inclined flat i Y' = 1 + X' destabihzing (
opposing buoy Dow along a vc ' opposing free : ,
, , i 0 The analyn )
0 g 2 over a heated. {' the endcal Res x= 5 653(1/N)(GrL/Ref)(7,/7.)" forPr-0.7$ ! Figure 14.22. Companson between measured and predseted average Nusseli numbers: turoulent 10' s Re. ' s 1 aar nos across a verucal mai plate. UHF [D)- , f Gr,.,/Re,'f,2 - 76, for water l much lower er , l 44.>n I l l
l -'aqui INSTABtLITY AND TRANSITION 14e27
' in the range p s Gr,,/Ref s 30, with Re, < 2 x 10', Gr*, < 2 x 10'3, and 40 s T, s 600'C.
The corresponding expression for the everage Nusselt number can be denved and expressed for Pr - 0.7 as Ru Rei"'
- v<
( L f Grn ' T. ' * *"
-1+ 5.653 (14.47) 0.031 i T. ;
( # i Re}.
- with n - 4 prosiding a better 6t than n - 3.2.
Figure 14.21 illustrates a compsnson between the measured local Nusselt number for a verucal flat plate (L - 2.954 m and # - 3.030 m)in cross flow of air [33l at three t x,:1 locations, as indicated in the 6gure, and the proposed correlation equadon 17 m ~ (14.46). A comparison between the mesured average Nusselt number over the entirr 4m plate and the correladon equadon (14.47)is shown in Fig.14.22. As can be seen from 4 the two 6gures. the proposed correladons for both the local and average Nusselt numbers agree well with the experimental results. In Eqs. (14.46) and (14.47) all fluid propertier - evaluated at the free-stream temperature T , 3 'i 14.11 INSTABILITY AND TRANSITION , umea turbant Buoyancy forces play a signi6 cant ro:e in afecting the laminar flow regime in mixed ij convection. Their presence may enhance or dim.nish the stability of laminu flow and hence alter the transport chuacteristics of the mixed convection regime. The instability of laminar mixed convection flow and its subsequent transidon to turbulent !!ow can be induced by the wave mode of mstability, by th vortex mode of instability, or by both modes. One of the major criteria for use in determining the incipience of the instaoility in laminar mixed conveedon flows is the reladonship between the critical Reyno'ds numoer Re,,, and the entical Grashof number Gr,,,, that is. the relationship between , the minimum Reynolds and Grashof numbers that will cause the laminu Row to become unstable. i From the analyses of wave instability by the linear theory,it has been found that for a strong forced flow sith weak buoyancy force, an aid 2ng buoyancy force has a , stabilizing effect on flow along a verdeal plate [40], but a destabilizing effect on flow i over a borizontal flat plate (41]. These trends are both reversed 3. hen the buoyancy I force opposes the forced flow. For the laminar mixed convection flow adjacent to an ' inclined flat plate, an increase in the inclination angle from the verdcal has a ' destabilizing effect for an aiding buoyancy force, but a stabilizing effect for an opposing buoyancy force (42]. For a strong free convection now with very weak forced flow along a vertical flat plate, an aiding free stream has a stabilizing effect, whereas an opposing free stream tends to destabilize the flow [43). The analysis of vonex instability by the linen theory for mixed convection flow 2 over a heated, isothermal, bonzontal flat plate has provided the relationships between the entical Revnolds number and the entical Grashof number as Gr s .,/Re!.'f - 0 447 for Pr - 0.7 and Gr,,,/Re!.'f - 0.434 for Pr - 7 in the Revnolds numoer range of ' *"5 '" * * "' 10' s Re,., s 10' [44}. On the other hand expenments on isothermal. heated. hori-2cntal flat plates provide relationships for the onset of vonex instability as Gr, ,/Refff - 192 for air [45] and Gr.,,/Re![f - 46 to 100. uith an avers;c value of
- 78. for water [46]. Thus. for a given Revnolds number. the linen theory predicts a much lower entical Grashof number than that observed in expenments. In another
.q w a . - n. . .. - .. .. , . . . - ,, 4 . ..
$$ . .$$ ?. *- .'Y :
*?? -
- .! - .~ . .
; :, g;.p- M d d d ftD? I h'2, 5&5 / !
RK5 i EU;!ANOT I?FICT ON ' CAT 2RAN!?IE IN ?O30II OEANNIl FI,0~lS l s. w I I.S.?etukhov a:d A.T.?olyakov 1 ~:stitu e f or Eigh Te:peratures , USSR Academy of Sciences, j f.:oscow, USSR
~
0.G .1:artynenko Eeat and r. ass Cransf er Institute , ESSR Acaier:y of Scienets, 3 !. inst, USSR thecretical studies on heat trantier and f A23 EA::: hv e ro d yn a t e r in the case cf botn laminar
.: anc t r ouler.: nixec convection n scooth The ; aper discusses the results of ex;er een;al and co=pu:stiona' and theo- . enannels e.re dis cussed. .te ca n a:tenti-retical studies :f rorted Jaminar and tur- on 2s 6:ven ental to the ;1ane and veHical caseschannels cf flows in orneri-a bu.ent fi:ws anc nea ansfer in chan-rounc cross-sectics srounc tubes,. 2ne se nels uncer the c:nditi:: of an escential -
bu:yancy eff ect. The cain attentie: is gi- cases are t?.e ciost :.nterts w e, .cr prae-tice and are the cess cc ;rchensively ve: to uprard a:1 downward turmulen; me-ti:: cf liquid i: hea ed vertical tubes. studied ones. The problems of flow stati-The 'coundaries and chara:ti.: cf ther- lity are c:ittec frs: the discussion;
- o;ravitational f orces effect on veloci- studies per:sining o othtr channel ec:fi-ty and tecperatu e distributics, friction gurations a:d considering additic:a1 e'-
rects on the flow, such as variability of re:1 stance and local heat transfer are physical properties, unsteady-sta e and
'8 deter =ined. *ielationships f or calculation of boundaries of the gravity force field others, are touched upon triefly. Anong 4 ! effect are generalized. the info:r.ation on the topic available " In the case of a turbulent flow, the
- to the aue:re, the cai: atten:io: is gi- !
bucyancy effect en the averaged flow and ven to the studies containing the cost
.h on fluctuating notion are analyzed. conclusive results and conducted during
.A the last decade. The review also deals t
~ with the results obtained by the authers
- 1. : -"If 02UCTION and reflects their scie:ti'io interests. '
Further on, incoc:ressible chemica;- I n the course of develerrent o' the lv homoge= ecus cecia w 11 ce consicered, , wnese cens;re variano :.s :ne :t:ction ; 1 cf-ve m ve neat ;ransfer inec:v. ne prob- of te--*-ere alone. .r.e assumptions 4 I 1ers ef f ereed and free conve::ler _f. tere mbi- adopted give t, hat :sotherral cc ;ressiti- l stu:? ei :iepeciently, wnile inet ned =f f ect was m.cuca2 Ay no: s .idied at lity cf tre redium with pressure varia-i- l aA. ; vas ena recen: y sna: it became on is neg;ec"ed, in contrast to thermal j c'"- --** m ---eme: ion of :orcec and expansion due to the te=;erature varia-free conveetion =ay be of essence. At the tion of the .ae cius. In ether e '-?- earlie r stages c. :ne ; o olu a . utiga- lowine tourr19eso. we snail assure that tfay,. tes sain es;nasas was laic :n laci- the f ollowing linear cepen ence c10 cs O nar :.xed convec:loL, rr.1 Ae :: turoulent is oiservec; fr ted flows tne cuoyancy e;;ect was con- . s_Hered .--eleysst. In fact, ::17 during g O , ,1 - p ( -T,)) 8 (1) ne last fifteen years it became clear that is turbulent liquid ' lows in tubes valid at re'.atively sea,1 di.,.,erences the ef'e: of "ree cc:vectics may be not c:1y perceptible, but, under certai: con- (0-T,g.an
, n, o e overwhelmin,,,e caj o.,.,,y,, of ca-I di-ions, ay govern the character o' flow ses sgdied, mixed co:.vect.on was descr.-
and heat transfer. The increased require-bed w..his -he framework c. Boussinesq j re:ts ic;osed c: the accuracy of determi- approximatics. i natio: cf temperature fields and heat ince ::e dens,.ty var,a .on due to
. I
'b a' pipelisItransp$
es# 'y'as~cu'arly ' such 'ht, d e' ds of rocket temperature variation alge is ec-sidered, I t e".Ihno20g we shall use along wit. ._e .ern buoyar-- l buildiLE _aerotter:co;;1cs and cary others, f orce s" th,e tez=s 'thercogra,vitatic:aa gave s.n !_petus _.o .,ur.her devel..
. % cent c,. ,ey
.orces" and thermogravitatio: , it:plyirs such investigations. the process developnent in a honogeneous
"'s review paper, the results nf gravitational field.
experimental as wel. as calculation and l l 343
. . . . . _ _ . _ _ - - - - - = -
- 2. IAMDIAR FLOW mass force effect is absent.
In Ref.1 the results are presented This problem in a more complete of investigations of laminar mixed con- statement was solved by Collins (12] vection in tubes, obtained before 1.M.2 The calculations were conducted for a In subsecuent years, a cumbe r a* p pers viscous-gravitational water flow in ve Were published, dealing with heat _.trans-tical tubes using complete two-dimensi fer s Min bau u hon zental T 2-10 nal equations of motion and energy wit. the boundary conditions tw=const and 13, 1 W 4 ./ ana vertigal_tJ1)es_f_11-lE.25-28,40J.2, a36 uin in inclined tubes T . qw=const. In Fig.1, the heat transfer calculation nsults for the case qw=co. ThTse atuale s yleiceo experimental as are shown. The calculations were condu well as calculation and theoretical data. ted f or the conditions realized in the The problems of hydrodynamics and experinents carried out by Schelle and heat transf er in the case of mixed con- Hanratty [ AIchI J. 1963, v.9 No.2 J vection in vertical channels are relati- on studying the stability viola, tion in vely simple. an ascending flow in a heated tube. Th A theoretical solution of the prob- se data are denoted by dots in Fig.1, lem of combined laminar forced and free the calculations being made up to a co: convection in vertical tubes for the responding value of I with the assigne-thermal stabilization region at q,=const Grg/Re. was obtained f or the first time by Ostro-umov and Hallman [11J. These and some JO other similar solutions yielded analyti- M, App, ron,ma!te; cal expressions f or velocity and tempera- h d,vd. l "#F # 2 ture distribution, friction resistance gg i NN- M"d86 and heat transf er. In the case of coin-ciding directions of forced and free con- l ' ' N" % ' vection near the wall, i.e. of the ascen- fI' I 7'd ding flow in heated tubes or the descen- ) ' FB estUn g, 2 ding flow in cooled ones, the velocity , 10 10 2 1p in the vicinity of the wall is increased with an increase of Gr number (liquid Fig.1. Calculation [12 J of heat trant flee rate being constant)Itand decreased a certain for in a viscous-gravitational in the core of the flow. i 1 value of the Grashof number, a concavity flow in vertical tubes [ appears near the axis, which increases ascending flow, heating; [ with the increasing Gr. In the opposite descending flow, heating; case, specifically, in a descending flow 1 - V - Grg/Re = 1904; in heated tubes the velocity near the 2 - * - 1176; wall is reduced,and in the midcle of the 3 - o - 800; tube is increased with the increasing Gr number. 4 - a - 400** ' Along with the analytical solutions 5 - V - 240; for stabilized heat transfer at qw=const, i approximate solutions were obtained f or 6 - s - 504. ! the starting length with the boundary ; 1 conditions of the first and second kind, Results of experimental and calcu ' and also numerical solutions of the prob- tion and theentical investigation of lem of heat transfer over the entire tu- heat transfer in an ascending flow in be length. Hucerical solutions on tne heated tubes at qw=const show that the basis of the finite-diff erence metnod local y nu=ber depends on the reduced wars obtained using various simplified length z/d) and the paramet systems of differential equations of en- (WRe) = =(gi(1/Fp(/M d qw G),[or RaA = gfd{ ergy and notion. Differential equations a ('dTb/dz )/16 V a) *; the buoyancy effe in tne boundary layer approximation f or on heat transfer n] ear the heat sourc solution of the problem on viscous-gra- essentially smaller than far from the vita tional flow (i.e. laminar liquid mo inlet; with an increase of Grq/Re the tion with an essential gravity field) in free convection effect on forced flow ! a tube were probacly used for the first propagates closer and closer to the hea time in [ 11 J . In snis pa per ascencing a urce; at a certain distance from the
- and descencing air flows were, studied at heat source the local Nusselt number large tempe rature diff erences, T i becomes staE111:ed. Stabilization is when the gas temperature assurea,/T n>5, large achieved at the values of Z which are values and the volu=etric expansion co- the smaller the larger Grg/Re. Lany of efficient was small. These studies yiel- l ded relatively small (witnin 205) beat transfer variations at the starting
- the values biguously of Gr /Rea interconnected. and For RaAaare unac uniform length as compared to the case when the ly heated round tube, (Gr /Re) g = 4RaA.
~ - ~ ~ ~ ~
M, f 1 l 4 ding liquid motion in a heated tube, the ) the enumerated specific f eatures of heat velocity gradient on the wall, according i l transf er behaviour are illustrated in to the data of f 12 J equals zero at Fig.l. Gr /Re z=,500 Variation of the Nu number in verti- nake independently while the analytical solutions of the coordi- j cal tubes in the case of coinciding for- for a stabilized flow give this value j ced and free convection near the wall is equal to 324. adequately described by the interpolati- The complex spatial flow character on equation suggested in [ 13 J: in the case of combinec acuon ox zorced and f ree convection in norizeaul onan-- Gr (2) nels maxes impossaole suiwi ~.1, i i w . ;- Nu/Nu,= (1 + d)0.27 , solution oI Ine prooAcc. Guiculasion and
~
theoretical stuales or mixec convecdon where Nuo is the Nusselt number in a la- m n ul: ental tubes was conucte=nuM g minar liquid flow not aff ected by mass the metnoe or small cas-um forces with the corresponding values of re spec t o solution f or curelv f nma the reduced length. convection /~ 1,3,28) allowing f or se_- For the parameter B(Z), the follo- 5cncazy : lows, in ug boundary 4ayeq ap-
. iu n e-wing relationships are chosen: proximanon ( y,q -%
p ,zz,c ,zbz G . m fenece m t 0 2nus , au M .E =u . m ou mate so-B = 5 42-1+ 312Z .2 5 at Z40.07 lution of the problem on heat transfer at 'Z > 0.07 in the tubes horizontally situated in B = 240 space and a relatively small thermogra-The equation (2) describes with the vitation eff ect was obtained. The follo-accuracy of 2 84 experimental data on air wing dependence was derived for local and water over the entire tube length, heat transfer at q,=const.: including the themal starting length. At Z > 0.07, negle cting the unity as com-pared to the second term in Eq.2, con- Nu/Nu0 = 1+Nu,' l.32 10'0(Gr /Re)[1-exph40r}
- 4 verts the latter into the interpolation Hall =an equation for stabilized heat sees 9 -9.1 10-6Ra(1-exp(-100Z))3cos& sin'*
transfer. 4 Iquation (2) is valid at Z < Zer. At (A) , Z=Zer, the laminar flow stability is vio- where @ is the angle over the tube cir- l
, the f ollowing cumf erence calculated from the upper ge- l lated. For Zer = h(j)#[13):
equation was chosen in neratriz; tp the angle between the axis , l andvertical;Ra=FI<irbtorilydesc11bes Equation 4Oatisf Zg = 12.9(Gr g/Re )~0 *8 (3) the experimental data on both vertical and horizontal tubes at 0.6<(Nu/Nuo )<1.5. ) The values of Z iose given in Fig.1 It is noteworthy that in the vertical (3) somewhat exceed b calculated (dots), which is due to the diff erent tubes from buoyancy (T=0) the Nu number depends on one parameter, viz. Grq/Re, while
. manner of their de te zuination. In the in the horizontal tubes (t[= % /2), on first case (Fig.1) they were determined anothe r, Raq from the beginning of the current lines Interaction of free and forced con-bending, in the se cond, Eq.3, from the vection in horizontal tubes leads to de-beginning of deviation of the local heat velopment of secondary flows in the pla-transf er values from the relationships ne normal to the tube axis. Secondary typical of a viscous-gravitational flow. flows form a pair of vortices covering Usually in the course of analysis the entire cross-section of the tube. In j of the calcu$ation and theoreticalofin.sta. the case of a heated wall, the liquid vestigation results, the bounda ascends along the lateral surfaces of the bility violation f or the case (8 /8z)< 0 tube and descends in the central part of is assigned to the parameters ma ing the it; simultaneously, the liquid moves velocity value on the tube axis become along the tube axis. Transverse circula-zero, and for the case (Bp/B z) > 0, to tion intensity increases with the incre-those making the velocity gradient on the the e xpe rime n-asing Ra number, wnile the vortices cen-wall becone zero. tres approach the wall and the ascending tal data given in However,d Fig.1 an those given flow is concentrated near the surface.
by the relationship (3) show that the This f act was used in some studies f or stability violation occurs much earlier approximate calculation of heat transfer than the time when the calculated velo- [ 9,10 ). Results of these calculations city values become the above-noted spe- adequately describe the experimental da-cific ones. Coreover calculations using ta on heat transfer far from the heat different approxicatIons yield diff erent source at Ra q> 106 results. Thus, in the case of a descen. 345
l l l Development of cecendnry circulati- Rab) the f ollo-ing equ3tions are ch sen , ons leads to violation in the velocity 9 l' and temperature distribution over the tu-be cross-section. Idazimu= velocity values RaU )= 5 103 :~1 at Z <l.7 10~3 and minimum temperature values (in the ca- 9 se of a heated wall) are displaced from A the channel axis toward the lower genera- Rab9
)= 1.8*10 +55 Z~1*7 atZ>1.710i trix. Then heat transfer near the lower and generatriz near the upper is essentially one it is increased,Just decreased. The results of calculation of [8(1)J as with the vertical tubes, the buoyancy of the starting boundaries of free con-eff ect on heat transf er near the heat so- vection effect on the perimeter-average urce is comparatively small and then ir- heat transfer are in a good agry ment with creases, and at a certain distance the the reduced dependenges for pq .
local Nusselt number becomes stabilized. At Z oo , Ragn =1.8 10 and Nu = o This character of the process development = 4.36. In this case, on the Easis of (5) was established as a result of numerous we obtain experimental and calculation and theore-tical studies f or homogeneous boundary A conditions. rigure 2 dis Nu = 4.36[1 + (Ra4 / 1.810 )A] 0.045(Sa) results of f 6 J_ plays on the the calculation circu=f erence- __ average number Nu compared to the experi- For 9symptotic values of Nu at mental data of [ 31J. The figure shows Ra o > 6*10', the following interpolation a good agreement of the calculation and dependence is suggested in [ 6 J: experimental data. It is also clear that stabilization of the perimeter-average 0.177 number Ilu occurs at considerably smaller __.Nu ,= 1.287 0( [Ra} values of Z as compared to stabilization without the mass f orce ef fest. In this ca-se, the constant value of Nu is establi- which is quite close to the relationship shed at the values of Z which are the sea- (Sa). 11er the larger Raq. Correspondence between the numerical solution results and the experimental da-24 a 0.5 10' 85 ' ta on perimeter-average heat transfer prompts the possibility of calculating 20 pg'. (p,577gy @w g4 g,s,.fe complex three-dimensional flows realized
- A' . s s with mixed laminar convection in horizon-16 calculation re-3 y 3
, 3m 3,g, tal tubes.
sults Unf ortunately,f on local heat trans er f or the bo-12
- e ,p undary conditions of the second kind are o e 5m not presented in the published papers and 8 -
,, m* the degree of their correspondence to the ry experimental results is not discussed.
4 - - O The results considered above were ob-tained under the thermal boundary condi-0 - a > a - tions homogeneous over the channel peri-10 " 0" 0" / meter. In practice , the cases are encoun-2* tered, when the condition of uniform tem-Z perature or heat flux distribution over
= d Re Pr circumference cannot be attained. ior ex-ample, heat flux incicent on the solar Fig.2. Comparison of the calculation re- collector tube is distributed over its sults of ( 6 J with the experimen- perimeter asymmetrically. These problems tal data of f 31 J. are close to the proolems of tec:perature The data on perimeter-average heat field control inside the channels. Such transfer are generalized in [ 31] by me- proble=s arise during laser radiation ans of the following equation:
propagation in thermohydrodynams.c light guides (gas lenses). In Refs.21,22, the case was conside-Tu/Nu o =.1 (5) red of a round horizontal tube witn the
+(Ra q /Ra(q )4]0.045 ' heat flux on the wall depencent on the angular coorcinat e where Nu o (Z) is the Nupgelt number value tha limit value
- in a viscous of Ra determined flow;fromRn[p)e th condition that 9, " 9+ = o(1 + a c o s $ + c s in t ) ,
Nu number in a viscous-gravitational flow differs from Nuo no more than by 50 For and at tne same time, indepenaent of sne 346
- ., .. _ ; ., . , c :. , .-
longitudinal coordinate. Such assignment ons , [ 8,11,38,3 9 J , temperature depen-of tne councary conditions cares it pos- dances of physical properties. In [35,3Q/, sible to deter =ine the velocity and tem- the data on lion-fiewtonian liquids are ob-perature proriles a11owing f or the mass tained, in ( 29,40,44-50 J, on channels f orce eff ect, which satisfy the pre-set with diff erent cross-section foms, and conditions, e.g. which are close to the in [ 43 ), on channels with cross-secti-axisym etric ones. ons variable over the length. In [ 23.J, the problem on heat tran-sf er with the following boundary conditi-
- 3. TURBUlIUT F1,0W ens was considered: a) constant heat flux cn the upper half of the tube the lower ene being thermoinsulated; b), constant It was assumed until recently that in c M"ve lopec tur culenT Ilow in tubes heat flux on the lower part of the tube' (He > T0* ), tne buoyancy effect on hydro-
.he ugper one being thermoinsulated. dynamics and heat transrer is m essen u-
.or these cases, .igure 3 shows the ependence of the perimeter-averare but mn hta mam a w e n t YTa_n ,.._____a invariable over the tube length IG number - ~ , :- m teate-that on the Grq number. In the case of heating D.
un der cedain concisions sne Ed effect the relationship E/Iluo is can ha e
- oniv essential out cocinant.
from above$y smaller than 1.5 over the en-considerab It was e.stablished experimentally and tire studied range , while with the heating theoretit ally that under the action of from below, the values of E/Iluereach 6 thermogralitational f orces variations and 12 f or Fr= 0.7 and 5, res pectively. are observed in the velocity and tempe-Tor the case of heating from below the ratu:v fields of the averaged flow, tur-data on heat transf er are described by bulent transf er characteristics, heat transfer and resistance, tne relationship ITE/I uo = f(Raq), while 'rhe system of equations of energy, with heating f rom above such dependence is not observed. e mentum and mass conservation for a ste-The waviness of curves exhibited in ady-state turbulent flow of an incompres-Fig.3 reflects the structure variation of sible liquid with constant physical pro-the secondary flow. 7/ith small values of perties (except density in the equation Gr the secondary flow is represented by f or buoyancy forces) in the absence of ) s ha,ir of vortices, while at larger Grq internal heat sources and energy dissi-values this simple flow pattern is trans. pation of an averaged flow is given by: 1 formed into a more complex one, composed of numerous vertices. . B (T) B BT 'T' k))*
, 9c p u g *og C A o (g) - P *p<"k
;- LS o -
/ (6) ,
Pr 5 B (uh 3 B(ug) 1 0( "~Uzggp) g
- iEi * %( D Ik
~ ~
gu. i
)
Nu ~ ;* / p/ Q #
}
Nu, j g . / f
/. -p(u[uh), (7) .
. 4 /^% / G/ - l
/ / 3(up) '
(k=1,2,3 ; i=1,2,3 ).
/,/Q7 g
= 0, (8)
*2 n,/,s f . .m . . , . . . . . . -
This system takea into account the tur-10' # # 10' ff 10' 10, bulent transfer of heat and momentum, f re spe ctively . J Olj Let us also write down the energy Fig.3. Perimeter-average Ausselt balance equation of fluctuating motion: number [ 23 J. , , B(u,) , , Unfortunat ely the restricted length of this paper does,not allow considerati- f"i"k> b ( ~ 59 "i> 81 + pDT=0 W en of the studies dealing with mixed lami-nar convection in the canditions cocpli- He re . oniv ceneratinn nf tu rbol e nt ener-av (first term) its dissication (third cated by the effect of additional factors and non-fiewtonian liquids [ 35,36 J. tn d nnn h o nyn n e se hva a antinn h acond In particular, works [ 32,33,37 J term) are taken into account. consider the effect of unsteady-state, -As can be seen from equations (6-9), [41,42), the energy dissipation effect, the buoyancy forces arrect nnt nnly +he [34), the effect of enemical conversi- average d, but also the fluctuating flow, 347
4 and. hence, the turbulent encan+ e and number, Ri = PrT Rf = g V dT/dx , writte heat transler. Inus. the a #m ? n' +he r=o- (dw/dx)2 6ravitation nn +W + " " " " + " - ' r is 6overned by two interconnected *#hets
- here in the gradient fom. Represent i On the one hand, the mogravita tional f or- In Ine universal c oo raan. 5 c. sy m ni o ces cnange the 11 elds of averaged veloci- turbulent flows near the wa114 tv ar= c errerature val ues , which entails variation of the en-hulan+ +*=a d r cha- 0#o dT*/dx*
ractersmtie=- na +% ^*%* M a d , +he rco- Ri = g = g rav ita t iona l fo r_q1 A_ dire c t ly_ arf e c t the moti on o'f tu rbule n t e l e- an + = a' +b li-quid. ennancing or weakeninglhc h turbu- Gr q lent transfer intensity. = 8 dT*/dx+
, (1 Ruoyancy lorces produce an eff ect, PrRe' (~& ) (dw'/dz')#
or ratner, ein prent *? acrs n-
- ne tu r-bulent transf er of ree entum and raat, and, whe re henne. on the turbulent Prnn 4+1 number gyq"d"h (see tne geophysical studies data of [31, Gr = ", Re = ud'Q#
523). stratirication or turbulent flows 4 M V has been studied in tne field of Ecophy- ' sics over several decades, and the results obtainea predeter=inea to a large extent Re* = d,ay]7{,=Re Y h, P tne approacn to studying of tne correspon-ding ;rocesses applied to engineering, + *
- which constitutes the subject of this re-w
=w/k/o ,
x=Y- f. view paper. The fundacental result nf the theory of tehperature-stratified _turbu- l'a W e
*he a" m b
- de n inaal lent flows conoists in cetemination of rameter. and in the form of (10) it is the enaracter of the gravitation field convenient for analyzing tra m istion effect on wrbulence. E m ca se n' = ta - local hydrodynacic and ther-n1 eharact-ble censity cistrioution, the vertical ristics or t ne wall flow enu=ed by the motion of turbulent elecents is accompani- buovency effect on turbulence. In the eii_Dy Inc va c %y wassed to counter tne homogeneous flows under s'tudy, the num A rc h i- p d i an inrces. Wnich leans to th" de- Ri Changes over the flow Cross-secticn cay of turbulent energy, and hence. of within a rather wide interval. The deg
- tyLbulent transler. In tne case of unsta- ,
of thermogravitation force effect on t b_le density als tribution, the Archime dian bulent wall flow is on the whole enara. i Iqrces operate on Ine vertical motion of Lerized oy Ine lii, number determined f.
- turbu le nt elements , causing an 4* crease the ne at Ilux censity, qw, and the f ri in the turbulent energy. Jealization of oF~st re s s en In e w a u . <r .. . 1.e . using -
th_ese ef fe eta 4a 9 0n iso tha rm a l rh m el derivative values of (10) on the wall: flows cay lead to essential qualitative cEanges .ui R ch = et-* a* h=+ +*ansfer as comparea To near 1ransIer in laminar
/
I cr 0 2 (1 0 =Re"Gr* ( & ) . flows under sicliar ccaditivus. He firs t I Ri* = EeI
' case occurs, ior example, in horizontal channels wnh bes eec utne r surma 2nd the seconc, witn neated bottou surface , Such deter:1 nation of the global Eicha.
or. in owner worcs , with upward and down- son number is the most justifiable wit. ward flows in vertscai neurea enannels. the themal boundary condition of the The results available on channel co' nd kind assi6ned on the surface. flows ~show that the eff ect of therc 1 tational zorces on turouient tr % yErravi-is 3.1. Ilorizontal Plane Channel I not scall as co= pared to that cn the ave-raced r iew. _De pending on the conHtion s, It appears interesting to esticat both effects are co= mensurable or one of first of all, the conditions which tri ' them is dominant. In order to establish ger the gravity force effect on the lo t5e enarac ter anc extent ef each factor neat transfer in the channel. This can l it is expenlent to cons 1Mr suen condit1- done on the basis of the problem solut ons wnich allow their u carate nnnlysis. under the assumption of a scall effect This possibility is pRvided in the case the Archimedian forces. Pro = the solut or a plane norizontal c?annel flow, where carried out in [ 61 ) using the equati the_seconcary i re e -c o n v c ; wive flows are absent. (9) for the relatively scall buoyancy
~ 7h* enraceter characterizing the ef-feet on the horizontal turbulent wall the following equations for heat l feet of density innomogeneity '"
- Era- flow,fer trans and f riction resistance in th l vity field on turbulence is the' 1(ichard son plane channel at Pr> 0 5 are obtained: l VieA M nN Ao be h ,gd 4. 0 b d d N4.
g M O w, k f d P#j i O4- )
4- , l
'g
. displaced downwards from the distributi-El on with indiff erent density stratifica-Nu = Nu,,1 e g St,Re,(4.51nnRe,-3.2+B) ,
, tien, demonstrating an increase in the Gr W $
1,= (&),(1 1 74 h), (13) o o 0
,,e
^e o 11 20
' " \ * *' 'e in which the upper signs refer to the ca- o o 3 a se of stable density distribution, and the o oa x", ,
lower signs, to the unstable ones; B=f(Pr) a 1 g St.*4# 44eo is the free term in the logarithmic law 000 3 I p,.jg y p h-f or the temperature profile (at Pr=0.7, 4 Bah). Fro = (12) it is clear that Riw does 15
, A*e e t 45 100 - 3 not completely define the heat transf er Ch a
,A A AA 02 8 a 3 6,1 55 75 variation in the channel with a tempera-
* * * * *
- j x 4 to 0 ture-stratified turbulent liquid flow. .
The ratio Nu/Nuo depends on the Pr and Be e s 'u ' * * " " '
- 5 32 5 I numbers. 10 g", *6 33 QS -
On the basis of tge solution obtai-ned, the ratio Grg/Grq ) (see ( 54 J) is ' y7 22 5 a8 6,5 55 < introduced as a parameter characterizing *9 5,3 120 l the degree of buoyancy effect on heat tra- s 10 4.4 240 , I nsfer and channel liquid flow, where Grf) 0 I X l determines the boundary of the 16-varia- 5 10 ' 2 5 10 2 2 5 tion of heat transf er and is given by p g gg ! versal coordinates near the hea- l 1 2 ted wall: 1-3 5 stable strati-Gr9 =1.3*10-4PrRe2*75[Re fication; 7-lb unstable strati-
* +1/8+2.h(Pr /3-l ' N fication; 11 - calculation usins lgHe+1.151g( T )+0.5Pr the Reichardt formula.
l (14) derived from (12). momentum tranaf er. In the case of stable Expe ri cen tn i n+"diae n* t he re ni and stratification, the curves W+(x+) are hydrodynamic characteristics under the shi.f ted upwards, demonstrating the momen-conditions or stable and unstable density tum trans fer decay and approaching the strati 1 cation witn stabili M V man veli :ity distributions typical of laminar values, neat transrer and rlow'(z/h > 30, flows. cnannel neignt) were conducted inf 5 2ne561. Ear 11er studie e nn . s en 4 pr, hTein3 made c1 =e boundarv-Inver finw =+r"cture
, Re to-scqne; ,
, ej 30 gg 1
[ 57-59] and the open tray / 60]. 3 +2 6.6 55 - I mencental Invest igations were made . l in /~ 53-D f Ior eleetric neating of_ one 5 6,1 55 ) of tne enannel surraces (qwi=const), the
~
^ I 2
otner surzace-TeiL6 aclacatic (qwp=0A~The ,- ~ results or velocity measur F*Etn h an hTr o o flow are presented in the universal enK o o dinates (g+, , , ,+ g)2 1
, , g, d,. .
Fig.4 f or diff erent degrees n! +ka *+$ k le anc unstacle censity stratification eff ect. O x+ Temperature distribution is of a similar 'O 25 50 75 10 0 l nature. I.t is noteworthy that the .. density Fig.5. Dependence of the turbulent Pran-stInirleation errect is noticeab1_t_as dtl number on the dimensionless early a_s at relatively large values of Be, Re c 3'104, even with tne enannel of mode- coordinate for the cases of neu-FIte c2;ensions (h=40mm). This ef fqct_ is tral (1), unstable (2) and stable (3) density stratification. developing beginning with the_ gow, core and with the increasing Gr70r e i.e. buoyancy eIIect degree, spreac over the Fi6ure 5 displays the data obtained g $$ y p reglans causer anc closer to the, wall. Witr % uc m = m e,u. -e r , Tae-buoy- ber g (PrT= Ag/0T) distribution gg over the ency errect is essentially incinased, which ein be seen Irom 21g.4 and the re- to the values 2x/h=0.5. An essential dif-lationship (14). Under the unsteady stra- f erence in the character of this relati-tification conditions , the curves are onship behaviour is observed with diffe-349
r' l rent density stratifications. In the ca- 63.64 J agree well with one another and ses of neutral and unstable stratificati- ~w itn Ine A. Alison relationship L b> J i ens, Pr is slightly decreased from the value close to unity near the wall to the g_pf )2 pg value of approximately 0.8 in the flow Pr.= (PrT )o G- 6cr) ' E#cr= y T-(15) core, the difference in PrT in these c7-ses being small. In the case of stable stratification, the relationship PrT(x+) ]The calculation curve plotted in Pig.6 displays a pronounced maximum at (2x/h)z corresp onds to (Pr )o=0.85 and Rfer=0.1. 0.2, reaching the value of 3.5. .W va- ( esAtdatathe obtained same time,by he atmos Pruitt62(pheric studi-lue of PrT being larger than unity at ) exhibit stable stratification in the wall region, a considerably smaller increase of PrT proves a stronger decay of heat transf er with the increasing Ri. than of momentum transf er in tLese condi- The results presented in Pigs.5,6 tions, demonstrate the necessity of a more com-4 prehensive studying of specific features Pg a- of turbulent momentum and heat transfer - o.2 in temperature-stratified wall flows, 2 d 13 o4
/ channels included.
40 g, - i ,_g f o 8 0-5 y i I.h 10s "0/l
- t . .
j p d k ,9 9e O D 10 ,; g ,, M 1_ i i ii -0.01 0 0.01 0.02 0.03 0,04 6 e u . .- u-.=*****m Fig.7. Heat transfer coefficient of the 6 , l J p [ iI heated wall as a function of the 4 , A density startification conditions g , t IRil and the degree of buoyancy force 2 'I#'**8 1 ~ **P'#i"'"**1 d***I z 10'3 2 4 6810 2 4 666i 2 4 681O' 2 - calculatien using [58). Fig.6. Dependence of the turbulent Prandtl number on the degree of buoyancy force eff ect with stable (1-6) and on heat P.isure transfer7 presents (Nu=q,2h/the data Tw-Tb f)53-56,, for unstable (7-9) density stratifica- close values of Re=(6+8).10 depending tion: 1-5,7,9 - laboratory studies ; on the global numoer usw. Over the entir 6 8 - atmospheric studies [ 62 ]; range of the conditions realizec in~the ivk -Oslaulati0n OSit6 the EllitOn e m"a:.cn ', . M n U.t Lin*,la"l Mt1Chy re lationshi 1 (R'ii= - 0.008) to the maximum stability ( 60); 4- [p;63 J,2,7,5-- E(55)j* 64 .3,9- (Eiw = v.un) or censity sim ulcation, a monotonous variation of the du number Experimental data on Pr? obtained is obse rvec. unstaois s.4 .iiicu61on
; both in Ine la coratory anc in natural at- (ul sv) Aeacs to intensification of heat 1, o its de-cospatrac Inve5U6a nons are given in transfer,
~
walAs stab 7s wu= - yie,.& a. u . .m w u a. .uu .acas nu=kct
^
tcriaruslau 2anina-liiii . La 2coui.. vi [ ZJ are given for t(c nea; . 4 auu.i-
. .' . 4Tunu...
- . . .W, T no the
. flow
- i6 v.all aus2cn vi i W iivw, c orr e s pon- witn a cons un1, value af Nu. Ixperimenta dDig to X' i'lU ano (2 did s 0.2_. u 1 the data are compared with the calculation daia .L i Lil 4 0.01.ru grouped near the results obtained using the relationship value tr7=v.op with sne accura _ty]E10% of [ 12). With unstable stratification, inoepanoenny oz me ouoyancy effect con- calculation and experisental results are dations (siac Ae or unstao Ae stratifica- in a good agreement, while in the case ;
Qgu . 1.b w4. ivu wr e t as thu ' - M ies). of the calculati-Evidently. me value inil =0.01 can be on stable stratification,he data slightly exceed t experimental i considered as tne initiation councary of ones. The latter fact may be attributa- l the buoyancy effect on the turculent tran- ble to the calculation dependence of ; s'ter processen. Ji m i6 1 esis diect [12] being obtained without taxing in-of unstable stratificat_ ion, the laborato- to account the considerable variation I ry oata / >> j and atmospheric observati- of Pr*. l one of W.Pruitt [62 J exhibit a weak thus, the data available demonstra- i tendency toward the decrease of"PrT. The te that the buoyancy Iorces cay e xe rt an i data or / 60j demonstrate a stronger de- e ss e nt i al inn uence cirectiv an *"*bu-crease of PrT and yield the minimum va- lence in the channels of limited dimen-lue PrT=0.3 at Ri= -0.2. With stable den- sinne- laanine to ensnees in +n nhule nt sity distribution. Pre incre ases with Ri . transfer of cocentum and heat. and hence Experimental laboratoiy data of f 55 mS9, in heat release. 350 1 l l
Q C K W W $ ~.$.3.jO f ! D & .? j N 1 ~-Y f l'D -;Y .f .: [ ;Y' . : ;fYj Figure 9 dis l the experimental datadistribution obtained in ({p 71 ays
) diameter, while the tangential component gy b 1/2 bon $urbulent ener- in tne horizontal diametral plane is di-rected upwards near the wall and down-horizontal and vertical (the angle coordinate @ is calculated 1ametral $(uD planes ) in the wares in the central part of the flow (Fig.10,A). This pattern of distrioution from the upper generatrix @
vely small ratio Grq/G;cq"/=at 12 and thewith relati-a f-g , D (6/wl).tf 8 (6/w,')# 4 -P- g W. 1 W.~ %
/ N. 1A
~
' I j 15 0 ^
1S l I 2 7'%%'!' " 10 a , , 10 4 %va"1
- r/r.$ .0 e-Ns B
.n 4' i / k U j 6 -.-m , y-J----%qM t
/
} j 10 20 40 60 80 100 5
y Fig.10. Determination of the boundaries
- q. o 5
}i =0 4.g i f existence of diff erent forms
%^ i of secondary flows [ 71J. Yelo-
$ m/F ~ 4,g 4,M P city vectors and provisional sche-0 r_
n yr mes of secondary flows: ) LD O 1,0 1# 0 0 I - Re =4 3 10", Gr=2 109; g
~ Fig.9. Comparison of measurements results Il-te=1.2 10", Grp5 10 .
g e u an ef ect of the velocity vector components cor-g g)=12; responds to the two-vortex structure of III- -Re=1.2 Re=4.3 g 10 , Gr /Gr 10 g , Gr /Gr{f t) secondary flows predicted in ( 72] and presented schematically in the same fi-g q =110. gure. It is similar to the secondary flow structure in the viscous-gravita-strong {pfluenceofbuoyancyforces tional flow Under a stron action of (Gr /Gr d 110). In the first case al- thermogravitation, Gr thokh0ce=rtainasymmetryinthev,ertical be seen from Fig.103,g/Gr q (gb50, as ca the radial veloci-distribution of b is observed, the diffe- ty component in the vertical diametral ) rences in turbulent energy distributions plane in the upper part of the tube is in the vertical and horizontal planes are directed upwards, and in the lower paz-t, negligige; in the second case, i.e. at just a s in the previous case , downwards. Grq/Grql =110, the distribution of b is At the same time, the tangential veloci- , I of a qualitatively different nature. In ty cos,ponent in the horizontal dia=etral the vicinity of the upper generatriz,@ =0, plane is absent while the radial compo-the turbulence decays under tne action of nent in this pla,ne is directed toward stable density stratification, and the va-lue of turbulent ener in the region near flows structure occurs at Grg/Grqthe 3 0+ centre the 35), corresponding to Gro /Re =(1.5+210 ler, wallthan(1-r/r in the<same o 0.2 here is much smal-region in the vici- (see the two cu2ve s dispIayed in Fig.10). nity of tne middle generatrix @ = Jt/2. These curves demonstrate variation of
' In the vicinity or the lower g,eneratrix, the radial velocity component values at
@ =0, the value of b near the wall is lar- the two points of the vertical diameter ger, than in the horizontal diametral pla- (4=0, R=0.48) and (4=K , R=0.48) g a no, which reflects the turbulence inten- function of the parameter Gra /Grq . The sirication under the conditions of unsta- radial velocity component vaIue on the ble density distribution in the lower por- plot is referred to the axial velocity tion of the tube. component on the channel axis.
It appears that exactly these speci- Specific features of the flow dis-fic features of turbulence variation over cussed above also determine the charac-perimeter under the action of buoyancy ter of heat transfer. With the combined entail changes in the secondary flows forced and free convection present in structure in a horizontal round tube. With horizontal tubes, heat transf er changes the effect of thermog j=tation being re- overlarger the tube latively small, Grg/Gr 30, the radial the the perimeter the jronger, ratio Grq/Gr . On the velocity compnent in t e vertical diamet- upper generatrix the Nu nu ber has the ral plane is directed along the entire minimum value, an,d is decreased with an 352
_/ T~ hl A'.'. - *
~f ^
~ ' ~
rease of Gr /GrqW ; near the lower ge- f er as compared to the momentum transfer.
@ .$strix, it hks the maximum value and The absence of similarity between them at h
~eresses with an increase of Grq/GrqM. stable density stratification is demons-Expericental investigation of local eat transf er over the length and perime-trated by the results given in Figs.5,6.
m The solution results satisfactorily des-ter durirg t..rbulent mixed convection in cribe the data of [ 67-71] on local heat 7' 68,69,75). orizontal round tubes is discussed in transfer under the two-vortex secondary
. . The experiments were con- flow struejure conditions, corresponding ducted with water and air agd comprised / ol k30 (see Fig.10), and are pre-the interval of values g10 S Ee 6 5104; toGrh,Gr sente specifically in Fig.ll.
0.7 < Fr < 8; 0.2 G Grg/Grq 4300. The experimental data obtained on lo-
' cal heat transfer near the upper and lo-Nu wer generatrix are generalized over the l
MO (, ly.? entire range of the determining parameters l specified earlier, by the follerting empi-g bm t--
. m. ~~
iN3 rical relationships: l 7 # r.t W ik t l Nu,/Nug = 1 + 0.03 5(Grg /Grq)0.43 (yg) Tig.ll. Distribu ion of the local Nu num- Nu,/Nu ,, = [1 +(Gr /Gr* g )3} 0.048, q (19)
'"'* ber along the hori: ental tube.
Expericental data [ 68 J: 1
?
1 -@ =0; 2 -@ = n * ' - calc u lati- in which Grq is de ter=ined usin relati- ["G on [ 77 J Rt=h.95 10s ' nship (17). Fr m equations (18)g(19) it , g# 4
- 3*g!103* foll ws t.at at
-C . cogravita tion efflarger valuest er ect para:e of Grq/Grgl the thy >=-
i Figure 11 shows 'the heat transfer va-
- 300, he at transfe r en the lowe,r genera-
~ 3 riation along the upper and lower genera- trix of the tube is by 405 higher than trices of the tube accorcing to the data in a forced turbulent flow, while on the of [68) f or the case of relatively small lower generatrix, the Nu,.o nu=ber is t
effect of thermogravitation forces. It smaller than Nu o almost twice. I can be seen in this figure that the buoy- The effect of thermogarvitation f or-8
~
ancy errect increases with,the increasing ces on the perimeter-averaged heat trans-distance from the heat soure, which is fer is exhibited to a lesser extent than 1 s due to the thercal boundary layer forma- that on the local heat transfer, and ap-tion. That is why in the initial thercal . para aj) essentially larger values of region the efrect of thermogravitation "YQ/Grq l
, which is proved b lafion results of [ 76,77 J. y the calcu-17
,, forces appears at much larger values of
& Grashof nu=ber tnan in the region remote l j from the heat source. The local Gr nu=ber 3.3. Vertical Tubes ;
a calculated using the para:eters in tne tube cross-section, where the relation In the first papers dealing with so-lution of the problem of turbulent flow y' (T,)4,o-(T,)e,3 = and heat transfer in vertical tubes in M 0.05 the gravity force field [ 78,79), the y 2 KT w )4=2-T] b thermogravitation effect only on the ave-l N* is satisfied is interpreted as the limit raged flow was taken into account, while I h n one at the Elven z/d, and is descrioed by tse rollowing depencence: turbulent transf er characteris tics were taken from the data on purely forced con - i m vection. .his approach does not allow a C - correct description, not only quantitati.- l SGrqU) .1 + ~200z[d exp(-z/d) (17) l N crN= 4
~
. ve , but also qualitative, of the charac-
,{
.a
. _ ter of dependence of the Nu number on the Gr number, which was established in later E$ The lines in Fig.11 represent the experimental investigations ( 80-88 J.
I* calculation results of [ 76,77) which b in the case of an ascending flow are in fair agreement with the experimen- Thus in a ,heate d tube , the calculations showed { 6 tal data. The solution is carried out using the equations of motion and heat that the Nu number increases monotonously with the increasing Gr at Re-idem, and d; transf er in the boundary layer approxica- decreases in the case o,f a descending g tion and the equations f or energy and flo w. This character of heat transfer va-p turbulence dissipation (the k-6 codel). riation qualitatively corresponds to that u , The equations describing turbulent heat in a lacinar flow. At the same time, the {i - transfer were not used. This, naturally, experimental data prove that in a turbu-cade it impossible to obtain the data on lent ascending flow in heated tubes *.he f, specific features of turbulent heat trans- Nu number first decreases with an increase m of Gr number, and then increases; in the
?-
}-
353
v case of a descending flow, it monotonous- consideration. Some information on flue ly increases. The noted differences in the tuating characteristics in an air flow heat transfer behaviour during turbulent was obtained in [ 82,85,91.92 J. Fig.12 and laminar flow make it possible to ex- displays the experimental investigatier pect a considerable variation of turbulent results on intensity of temperature flu transfer characteristics in the gravity tuation and axial velocity component, and also on turbulent tangential streer force field in the cases considered. obtainen in f 85J. It can be seen in Theoretical analysis of the flow and this figure that the intensity of the heat transfer characteristics under a axia A velocity component fluctuatione small gravity field effect on turbulent (Fig.12a) is decreased with an increase liquid flow in tubes conducted in [ 89, of Gro at He=const, provided that the 90 J, made it possible to establish that tempef ature fluctuation intensity (Fig. the said differences in the heat transfer 12b) first decreases and then increase: behaviour are due to the dominant effect with simultaneous displacement of the of buoyancy directly on turbulence at the initial stage of the process (at Pr > 0.5), maximum from the wall towards the tuce as ecspared to its effect on the averaged axis. Similar results on temperature f: flow. Under the assumption of heat trans- etuations were obtained in 4 82,91J. . f er variation only under the eff ect of is clear from Fig.12c tnat at Re = 5300 turbulent momentum transfer change due to the Reynolds stresses, decreasing with buo ancy forces, the limit value of the an incre e gf Grg, at Grq = 2.2 107 (Grg/Gr 23) approach zero over the e: Gr Inumber has been established for a tire fl w cross-section. ve tical round tube, which at Fr> 0.5 co-incides with the equation (14). T' l6r in the case of liquid motion in ver- f' la , l 20 tical tubes in the gravity f orue field, 20 i,
/
j,4 o M ' the ascending flow during liquid heating and the descending one during its cooling are practically equivalent; the ascending 10 G f"jym 10
/ ..
flow during cooling and the descending
*3 [al*d, .[
p 4
.c a.
flow during heating are also equivalent. c % .. According to the experimental data avai- M ,*4 s lable, we shall consider further on the 0 1 2 tgx' 0 2 1 , ascending and descending flow during the Fig.13. Velocity (a) and temperature (t ! dis trioutiou in an ascending ar ! liquid heating. descending air flows a)ascendir l 3.4. Ascending Flow With Liquid Heating motion Re 5 103 1 - Gr 2.20 !
/ =23B The data available on the theznogra- Grg/Frlie4=510-8;Grh,G ,
2 - 7.9 107, 1.8 10- 120 [6(v. vitation effect on turbulent transfer in 7 140cast [ 82 :j gpe,M/q= 'b. 3 - 8 10 , 1.8 10~7, flow 4 - free-convective
- 4) /g4 ~ Q vertical plate . Grg=2 1012 [ 9:
.}'
*s -
2- b) descending flow, 5 - Re=2.05 2- A 1.2 107 [ 66 J, Gr /FrRe c =1
'l '
4 1 c
' Grq=the 6- "-1/3* 1aw; 7*- Gr ~ 0 Ascending motion,1 - Re=9.81 8
U,V,/% /9) 0 42 44 05 OA .[ Grq= 1.2 10 , g,qf y,3,4. 2 .10~
- Grg/Grq = 18 [ 82 JJ 2 - free-
- 42. ^ A convective flow st a vertica 0
. q7
. - - plate, Grz = 2 10 [98].Desc 0 0,2 Q4 0$ OA1-RU 4 ding motign. 3 - (Grq/FrHe ) =
= 5.4 10- [ 93 J; 4 - the *-1 Fig.12. Distribution of turbulence charac- law; 5 - Gr 0.
teristics over the tube cross-seo-tion at Ee ~ 5300 [ 851: An essential change in the chara a -fluctuation intensity of the g gg f , g tubes under the effect of thermogravi n n ensity u-tua lent tan ential stresses; 1-tion, in particular, a decrease of tu bulent transfer under certain conditi Gr4=7 10 ; 2 - Gr4 = 1.5 107' is demonstrated by the velocity and t 3 - Grgal.8 107; 4 - Gr q= 2.2 10.7 perature distributions presented in t universal coordinates in Fig.13. Thie vertical tubes are very scarce. In fact. fig of [ure displays the experimental data 66 this problem was never given a systematic and descending air motion. l 1 f 354 , l
i ,. j. . _ . . ,. '
;t l
".g, f.y , " k Q ; v .- ,' '.. . ', c l
i ces. l With the values of Grq number (or ' 1ro /Gram parameter) being relatively smell, Results of calculation of the Nu thd experimental data are situated above number f or the case of an ascending flow the curves denoting the purely forced flow (Pr ~ 5+7) in heated tubes with constant (Grq-+- 0), which is due to a decrease in turbulent transf er. A qualitatively simi- 2uN I l a .isoon lar tendency of the velocity profile va- 2 I m w t ristion is exhibited by the data presen-ted in Fig.4 for the case of stable densi-g d fE
!*I L. N X
F " < ty stratification leading to the decay of 6 i Dh D# * ' turbulence. under While the the action Gr number increases of therkogravitation for- 4 e" "* .iea ces on the averaged flow, the velocity and *I!l
' ' ' ' b temperature profiles are gradually rearran- 2 Kf 2 4 660' 2 - 4 L 610' 2460 ged. The effect of buoyancy directly on the averaged flow is exhibited in the for- Fig.14 Theoretical calculation results mation of a maximum on the velocity profi- of heat transfer in an ascending le between the axis and the wall of the water flow in a vertical heated tube. Under these conditions, the flow is tube (curves) compared with the y characterized by large-scale velocity and experimental data (dots). Re=8 10.
temperature fluctuations. With further in- 83 ); 3 - ( 87 ); Re=104: 4- 80 ; eresseoftheGrknumber,theeffectof thermogravitatic *orces on the averaged flew becomes dominant, and finally, the Re=1.4'10j{::1 Re = 1.6*10 6-8- 80]; 8 7 ., .
- -[ 80 ); 2 - [
7
" free convection" Itgime is developed in the forced flow. In the velocity distribu- heat flux density on the wall are presen-tion, the maximum point between the axis ted in tig.14 as a function of the Gras-and the wall becomes more and more dis- hof number determined from the mean-mass tinct, and the profile is situated below liquid temperature gradient:
the velocity profile for a purely forced A flow (see Pig.13a), approaching the one sfr (dTb/dz) = Gr /4 Fe. typical of turbulent free-convective flow GrA= V2 4 along the vertical surface. It can be as-suced, that in this case the temperature The data refer to the region remote from distribution also approaches that at free the inlet (z/d=40 and 100). At small Grg turbulent convection, coinciding with the numbers the Nu number is independent or distribution in a purely forced flow (Gr., GrA and,is in a good agreement with the
~0), as can be seen in Fig.13b. known dependences f or purely forced con-The few data available on the veloci- vection. A further increase of GrA (Re ty and temperature fields reveal a number bt ing constant) leads to a decrease in exhibi- t)e Nu number due to a decreasing turbu-of tedspecific featuresofinheat in the character transfer, trans fer va.
riation. Consider the experimental data lut transfer under the conditions of sts.ble density stratification. Subsequent on heat transfer in conjunction with the increase of GrA above a certain value recently conducted calculation and theore , (which is the larger tse higner Re) the tical studies [ 77,94,95J and also with cependence enaracter is changed: the Nu the results of physical analysis and ge. number increases with Grg and is practi-neralization [ 8 9,90,96 J. cally incependent of Re. This portion of In [ 94], an attempt was made to ta- the curve corresponds to the dominant re-ke into account the effect of thermogravi- le of free convection in the flow fort:a-tation forces on turbulent momentum trans. tion. Tne calculation results satisfacto-fer. However, it didn't help to solve the rily describe the experimental cata. problem. In a more cecprehensive statement, A more adequate approach was develo- tne problem or turbulent riou and neat pad in f 95J. The system of averaged equ- transfer in rouno tuoen essentially af-ations of conservation, written in the bo. rected by buoyancy was solved numerically undary-layer approximation allowing for in [77). The case of an arbitrary tube thermogravitation forces, was solved nume. position with respect to the free-f all rically. The relationships f or the turbu. acceleration vector was considered (see lent momentum transfer coefficient and the above). turbulent Frandtl number, Pr , were found A generalization of the experimental from the analysis of approximate equations data on heat transfer in a water and air of turbulent energy balance and enthalpy flow for (t/d)>40 is displayed in Fig.15. fluctuations intensity, which has made it Expericental data over the entire range possible to account for the buoyancy effect of the determining para.neters variation, on turbulent momentum and heat transfer. from the flow in the absence of mass for-When the thermogravitation effect is ab. ces efrect to the flow with dominant mass sent (Gr 0). the dependences for VT and f orces eff ect ("f ree-convection
- regime)
PrT are er 4 arted into the known dependen-355
e 5 I are described b a cocplex enough equati-on obtained in 96 , we shall consider tions (small Pr numbers and relatively j a particular esse on)ly. small Re numbers) the ratio Nu(Gr) at
,4 4
, . Re=const one having may be a,conotonously increasir4 no point with the minimum va.
5t pe.10+ 2 -
- lue., Note that the discrepancy in the ex.
(20 pericental data available on heat transfe j m: Pr 0.7 to liquid metals, especially with small l +- G r- 0 turbulent values of Pr is probably due to different degrees o the free convee-4 (2p tion eff ect, which was not taken into ac-a-1 count in the analysis. 2 2
.Gr-0 -- . -._. / .3
,, Pr 6 i
I 0-g: 46810*2 4 68KP 2 4 686, 4
}/G rg /Prd 3.5. Descending Flow With Liquid Heating 68)
Lxperimental results of (66,80,83, show conotonously thatincreases in this case heat with thetransfe incres r Fig.15. Dependence of the t number on the sing buoyancy effect (De=const). In (89 parameter Gr o /PrRe . Air flow (Pr= it is shown that the heat transfer incre-
=0.7): 1 - et erimental data f 66, ase is caused by the turbulent transfer 82 2 - cal ulation usin the intensification under the unstable densi-formu'la of . ater f ow (pr' ty distribution conditions. Numerical so-eleu$a lutions of t*.e problem conducted in (77 n tefr la' 95 J provea this conclusion. The solution 0f [ 96
- results are i . air agreement with thi ex.
'"e b -6 and perimental data on heat transfer. Experi-S Ore /ir?values (Gr /PrDe ) > 10%300, heat tr0nsfer cental data on the increase of velocity is described ' .luctuations intensity near the cooled L by'tne following equation: surrace with an increase of buoyancy ef-A p,3/4 Grl /4 fect in an ascending air flow in a verti-gy: , o cal plane channel are obtainea in (92J.
1,333g(p,jg) 3,3(p,W-0.7) - (20) An increase in turoulent transfer is deconstrated by the scarce expericental data on velocity and temperature distribu. ( The calculations using (20) are denoted tiens [ 66,93 J, shown in Fig.13. Ixperi-in Fig.15 with dashea inclined lines. In mental data (blacx dots) are presented in this case , heat transf er considerably ex- the universal coordinates against the de-ceeds that in a forced flow in the absen- pencences (solid lines) characteristic of ce of cans forces effect (Re values being forced flow without the case forces effect the same). The values of U+ and T+ with the buoyancy Based on the analysis uonducted in effect are displaced downwards from the l ( 96), a conclusion can be drawn that if corresponding values at Gr + 0 which is it were not for the buoyan:y effect on similar to the results for horIsontal turbulenna, :he St number variation would flows under the unstable stratification practical., Jollow the dependences deno- conditions (see Fig.4). ted with da.1ed lines, Thus, in the case under study, which Previously we considered the data corresponds to unstable censity distribu-on the buoyancy,effect on turbulent flow tion, the turculent transfer is ennanced and heat transfer of li uids with modera- witn an increase of Gr, leading to the it is increasing ruiness rf velocity anc tecFe- l te shown Pr nu=bers. that for liquid In (86, 0,97J(Frul) metals this rature profiles and a corresponding in-effect is none the less important. The crease of resistance and heat transfer. turbulent cocentum and hdat transfer coef- With strong effact of thermogravita-ficients, calculated in [86] from the tion, a regime is developed, which can be measured velocity and temperature profi- defined as the regime Of "thermo les in a cercury flow, depend both on tional initiation of turbulence"gravita- . Then Heynolds ana Grashof nu=bers. The data ob- the flow and heat transf t:r characteris-tained in the same paper on heat transf er ties are mainly governed by the intensive demonstrate the character of the relation- turbulent transfer caused by turculence ship Nu(Gr) at Re=const similar to that generation due to the Archimedian forces given in Fig.14. A decre,ase in the Nu num- and not to the averaged flow. In equation I ber is due, as shown earlier, to a decre- (9), the second term is much larger than - ase in turbulent transf er under the acti- the first. The velocity and temperature ' profiles in the turbulent flow core are on of thermogravitation forces. Develop- given by the relationship corresponding ment of mixed convection in liquid metals, to the *-1/3" law, and denoted in Figal3 however, may obviously have its specific with dashed lines, llent transfer in this features. In particular, the analysis of [ 90) suggest s that under certain condi- case is given by the following dependence 356
5 9 7 . p ,s
- 4 ' p* , * = * * *
i,.
,,-l .' ,. ~- ' f y '. [ ' s* _'
(.~...' * '
* .-: ~ - ~
s' ' Q :^ ;. . .,
- Yn$.$. . , 2 i ^.
~ .
Ow -
. < ,. *' .y.
I the tube length. The experimental data ob- [ 99 J: tained in long tubes, demonstrating the 0 0 0.5Pr *5Gr,'25 appearance of heat transfer stabilization, Un = (21) speak in favour of it. E 1 + Pr 0*D Modern calculation techniques and computers ensure solution of complete Figure 16 presents the experimental three-dimensional equations of energy and data on the local heat transf er (z/d >40) motion for laminar flows,f thus providing
,,, , , a principle possibility o solution of va-Nu . rious problems on mixed laminar convecti-100 , F.e.X[tPr.5 [991 on. However, the region of mixed laminar 60 Tr*7,(21) convection existence is extremely limited
..t
-pe.3d,Pr 5 owing to the intensive thermal-hydrodyna-40 [ gg ) 4,(20 . mic flow disturbances realized in these conditions. This makes the necessity of (df systematic studying of stability violati-2% 4 55h 2 E 6 8 $0* 2 on of laminar channel flows essentially Fig.16. Dependence of the Nu number en aff ected by buoyancy f orces extremely the Gr number in a descending wa- important.
ter f18w in vertical heated tu- Further progress in studying of tur-bes: 12 - experimental data of bulent flow and heat transfer with combi- [ 80,83,88 J as ir= 5+6.4, Re= ned forcee and free convection requires systematic experimental investigations of
= (9.4+ 11.5) 203 and Pr= 3+6' the flow structure and regularities of Re =(2.2+4.6)*103 , re spe ctive ly. turbulent mo=entum and heat transfer.
Apart from studying the structure , it is obtained by several authors [ 80,83,88 J , necessary to obtain more comprehensive who considered the water flow (Fr=3+7) in experimental data on neat transfer and comparison with the calculation resu,lts resistance over a wide range of Gr, Re from the equation obtained 991 . The and Pr numbers, with different orienta-relationship suggested in [ in 99[J genera- tions of the system in the gravity force lizes the experimental data available on field. On the basis of experimental data water an air over the entire range of the and modern methods of the turbulence the-J determining parameters variation, viz. ory, adequately general calculation tech-
. from the conditions not affected by mass niques of heat transfer and resistance in forces to the regime of thermogravitatio- the case of turbulent mixed convection in tubes can be developed.
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Polyakov, A.F., Experimental Inves- Etude de in Structure . tigation of the Gravity Field Iffect l'Ecoulement et du Transfert de Cha-en Turbulent Air Flow in a Plane Ho- leur en Convection Mirte Dans un Tu-rizontal Asymmetrically Heated Chan- be de section Circulaire, Tnese de Docteur-Ingenieur, L'Universite de n pp.el,366-371, 197 9.Teplofiz. Vysok._Temo., vol. Paris, 17, 19b9.
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Number and Under Significant Influ- der the Effect of Thermogravitatio-in Turbu - nal Forces, J. Engng Phys., vol. 27 enosShear lent of Buoyancy Forces,15B-167, 68. pp. 1422-1423 1974 Flows.2, pp. Springer-Verlag Be rlin , 1980. Petukhov, leshov, V.A.,B.S,, Po Ayarov, Ku-Yu. A.F. ,
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Transfe r - VI, v,ol. 1, pt. 1, pp .160- ces, Proc. 5th Int. Heat Transfer lyd , Einsk 1980. Conf . Tokyo, vol. 3, lic 4.S, pp.16:
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Dubilier, I.G. , , Investigation o,f Ae-rodynsmics and Heat Transfer of a khter, Yu.L. , a,nd Euleshov, V.d.She-, Turbulent Stratified Flow in a Ho- Experimental Study of the Effects c: rizontal Plane Channel, in Heat and Thermogravitation Upon Turbulent Flc tass Transfer - VI, vol. 1, Pt. J, and Heat Transfer in Horizontal Pi-pp. 157-161 pes in Heat Transfer and Turbulent
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G. Buoy, ant convection, vol. 2, pp. 719-dy of Turbulence, in An aExperimental Stu-Density Strati- 727, Hemisphere Publ. Co. , Washing-ton, 1977. fied Shear Flow, J. Fluid Mech. , vol. 70. Petukhov, B.S., Polyakov, A.F., Tro-19, pp . 221-245, 1964. itsky, V.V., and Shekhter Yu.L., 58 Buoyancy Effects in a Arya, S.P.S.lat-Plate Horizontal F Boundary Layer Thermogravitational Pield,Effect on J. Fluid 1*ech., vol. 63, pt. 2, pp. , the Structure of a Turbulent Non-ise 321-34), 1975. thermal Flow in Horizontal Cylindri-
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zing of Dense Fluid in a turbulent pp. N -d3 liemiaphexT Publ. " Pipe Flow, J. Fluid Me ch., No. 8, vol.1, hington, $977. Was C o. ,khov pp. 514->44, 1960. 75. Petu B.S., and Polyakov, A.F.,
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197J. Soc., vol. 15, pp. 403-409, 1972.
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361
} ?. Y ' ' 1 Yh 7. .' f * [ . ! , ' -
L 3.2. Horizontal Round Tubes minimums are shifted dcwnwards. The devi-ations of velocity and temperature fields The buoyancy effect en the averaged W/W. flow in horizontal tubes exhibits itself 1,2 in the formation of secondary flows in the plane normal to the tube axis. A com- 1,0
' yC {~~ Wl"l [ -
T plex character of the averaged flow is demonstrated by the results of experimen-4 Ml l 6%k Da tal studies of a turbulent nonisothermal # l {- air flow stmeture in horizontal tubes 0'6 u1 88 mm [ 66 J and 144 mm [ 67-81) in dia- j {
. meter, at qw=const. g .2 i in order to determine the boundaries De 3 of initiation of the possible development g[ ,
. -4 of secoucary free-convective flows and l --S their character equations (6)-(8) were 0 Uf'l l r/r',0 solved in [ 72.f for the stabilized moti- 1,0 05 02 0 0.2 (16 1 6~#A on and heat transfer under the assumption U (T WE TJ of a relatively small thermogravitation effect. In this case, the gravity force 1,2 6
[ ---- field effect on turbulent transf er is neg-lected. Combined analysis of the soluti-ens of [ 61,72 J shows that in horizontal tubes with small Gr number values, ther-1.0 03 [. q
@'g"'l i
- f% _
megrnvitation forces affect primarily the a:6 I l g . i averaged ficw. The buoyancy eff ect on tur- l un l bulent transfer becomes noticeable at so- 0'4
^.2 "
mewha"., larger Gr number values. The appro- fjs l 4 3> l ximate solution of [ 72J made it possi* 0,2 -4 ble to obtain an equation defining the 0 boundary of a 15-deviation of the local 10 0.6 0.2 0 0.2 0.6 1,0 heat transfer (Nu numbers on the upper or Fig.8. Velocity (a) and temperature (b) lower generatrix) from its value during profiles in a horizontal tube /67/:
*purelya forced convection: 1 - Re=1 3 104, Grq=4.2 108 Grg/GrquI-100; 2 - Re=?
3,- g,N= 3 10-Sp,0.5 He.6{3,0.125+2.4(Fr-1). q 2
. Grq=5.2.lg4, Re=
Grq/Gr 8 7.7 10 , Gr o
=$; 4 - measurements in a Grq=g/Gr 24 W
(16) verticEl diametral plane; 5 - mea-Equation (16) is valid for the region far surements in a horizontal diamet-from the heat source at qw=consgand ral plane. calculations of
/Gr as an [72,74,7$,6-77).
Fr > 0.5. Using argument, it is the ratio Grko a0alyze the convenient buoyancy effect on turbulent flow and from those typical of purely f orced flow heat transfer in horizontal round chan~ are the larger the higher the Gr number nels. and the smaller the Re number y 1.e . g.ey l increase with the increating ur e/Grq The attempts to detemine the initi- . ation boundaries of the buoyancy effect on heat transfer were made in earlier stu- This data statement obtained inis[ also 66 J.justified At Grq/Gr by?}he =5 dies. In [ 73 J, diagrams are given f or temperature and velocity distributions , horizontal and vertical tubes, defining are in f air agreement with the solutions the boundaries between f orced and mixed of [ 72 part 74 of,the,76,77). turbulent convection. In this paper, the paper, aAs shown in the qualitatively simi- first said boundaries are obtained on the basis 'lar character of velocity and temperature of analysis of the data on the length- fields deformation under the effect of average heat tranaf er, that is why the buoyancy is observed also in a laminar boundary of initiation of the buoyancy ef- flow in the case of homogeneous themal feet on heat transfer corresponds to lar- boundary conditions. However, the qua nti-ger values of Gr number, than those fol- tative results and functional connections lowing from relationship (16). between the para =eters determining the va-The measurement re sults on velocity lue of therzogravitation f orces effect and temperature profiles obtained in [67J are, naturally, diff erent. Besides, the are shown in Fig.8. An ess ential asym =et- data cited in the pzevious section allow ry of velocity and temperature distribu- us to expect in the case of turbulent tiens in the vertical diametric plane it liquid flow In a horizontal tube in the noticeable. It should be stressed that presence of heat exenange with the wall, the prof the efIec t of gravity force field direc-Re-510{1edeformationisseenevenat. Velocity mazicatly andontemperature turbulent.e. 361
~"
m:etinette. tores pr ' G,,/ Re s 460 f m, f areioc0 Gr,e a 5:0
,, . ,a r . . . Gr e ,....0
\
, 9ee 1000. Gre/ Se e 26 0 6 - ' De/ A t e 4 0 0 s
9tet, Gr/Ree160 g
'1 .z 0
0001
.... .....n 001 01 1
,,g, x
rigure f Comparison of arithmetic mean Nusselt numbers with the results of Martineth and Boetter (adapted from Marner and k McMillan'*; with permission of the American Society of Mechanical
\ , ,,, ,
- y ,,
Engineers) Lawrence and Chato' and Collmsis.u have developed , , numerical models for predicting developing mixed convection 'O 06 cs it ie 20 taking attount of viscosity and density variations and using seG: marching solution procedures. Zeldin and Schmidt'* solved pj pg,e 2 Typical behavior of local Nusselt number near the point of the full chiptic equations governing the problem usmg an maximum velocity prt,Sie distortion (adapted from Marner and tierative method in order to avoid the use of marching pro- McMillan'*. with permission of the American Society of Mechanical cedures, which had been questioned for conditions of strong Engineers) Notation v Radial velocity fluctuation (m/s) V Radial velocity (m/s) a Radius of tube, d/2 (m) w Axial velocity Ductuation (m/s) C, Specific heat at constant pressure (1/kgK) W Axial velocity tm/s)
' r~ y Trans verse coordinat e measured from w all. a -r (m)
C, Integrated specific (heat, T.- r.Tn), C, d T (J/k gK) y* y M/r Axial coordinsic (m) C, Coefficient in 2 equation turbulence model s' Dimensionless axial coordinate, d . Tube diameter (m) d Re Pr g Acceleration due to gravity (m/s8 ) p Coefficient of volume expansion (K * ') G Mass velocity,4 M/ad8 (kg/m8s) Gr AT Wall to bulk temperature difference. T.- T. (K)
~
Grashof number, (p.-p.)d8g/p 2 e Rate of dissipation of1(m8/s8 ) Gr Grashof number based on h,(p.-/)d8 g/9 8 x Von Karman constant Gr* Grashof number based on heat aus, g#d'q/Ar8 i Thermal conductivity 4W/mK) Gz Gracta number, Re Pr d/: (= 1/ ') p Dynamic viscosity (kg/ms) h Heat transfer coefficient,9./(T.- T.)(W/ma K) v Kinematic viscosity, p/p (m8 /s) i Specific enthalpy (J/kg) p Density (kg/m8)
& Turbulent kinetic encrgy (m3/s8 ) g r, I Length .cale (m) A 8
L Tube length (m) Integrated density, (T. - T.)r.,p d T (L g'm ) M Mass flow rate (kg/s) a, Turbul:nt Prandtl number Nu Nusselt number, A4/2 1
, g, di D t Shear stress, N/m2 Nu.,,
Arithmetic mean Nusselt number 4 .(T,- HT, + T.))Subscripts 8 b Refers to fluid bulk conditions p Pressure (N/m ) c Refers to thermodynamic critical pressure Pe Peclet number, Re Pr F Forced conve: tion Pr Prandtl number. #C,/2 m Mean or mixing length M Prandtl number based on C, pC,/2 ' min Minimum q, Heat Aux at the wall (W/m ) 8 pc Refers to pseudocritical value r Radial coordinate (m) ref Reference value Ra Rayleigh number, Or Pr i Turbulent Ra* Rayleigh number based on heat aux. Gr* Pr w Refers to conditions at the tube wall Re Reynolds number, Gd/p r Based on axial length 7 Temperature (*C or K) 0 Refers to inlet conditions T* Temperature Ductuation (K) :o Refers to the fully den!oped condition int. J. Heat and Fluid Flow. Vol.10. No.1. March 1989 3
$rudres cf maad convscroon on wricaltubes ./ D .lockssn et al Se P g those predicted for forced convection were observed (Figure 31 N.o The approximate theoretical model of Martinelli and Boelter'
,, was used with remarkable success as a basis for correlating the data. The following equation, which is identical with the 5- theoretical result, apart from the slight adjustment of an indes from 0.75 o 0.84, correlates all their data and that of Watringer and Johnson to within 20%.
.\
[. Gr Pr d '" "8 k\.s Nu,,,, = l.75T, Gz,,, + 0.0722T 3 J G W' L , (l1
* ** r,and fa are functions which are derived from theory and
*** " G s, tabulated in Ref. l. No data mere produced by Maninelli er al.
! sonica ni i g er es t t.n,...., c,, ,, e, ( for the case where free convection opposed forced convection 3 ,
3 , , ,3 , g , (heated downward Dowl i 4 i io- . t i j . i o. Subsequent work on laminar vertical mixed convection has i a t : . to- . 3i . io- been carried out using uniformly heated tubes. Clark and i L e io . i n s . io$
' 4 i . s .' 5 0 ' .
Rosenhow, in experiments on water at high subcritical 6 s . io* pressure Dowing upwards in an electrically heated tube, produced Figure J Emperimental and predicted magnitudes of Nu,,, as a data which were used by Hallman3 for comparison with his function of G2, (adaoted from Martinelli er et"; with parmission of analytical predictions for the fully developed case. Funher the American institute of Chemical (ngeneers) experimental investigations were reported later by Hallman28 and Brown.8 These data are in good agreement with Hallman's buoyant innuences. Nevertheless, diMeulties in obtaining a analysis, which is failed well for 100< Gr*/Re < 10.000 by the solution were still encountered beyond a certain value of Gr/Re equation: corresponding to a reversed now in the central region of the , , , pip *- in the case of buoyancy sided convection, i.e. upward now Nu-095 - (2) Re in a heated pipe or dow nw ard now in a cooled one, the velocity in the vicinity of the wallincreases with wall to-Duid temperature Experimental results obtained by Kemeny and Somers 23 using water and oil are shown in Figure 4. However, direct difference (at constant flow rate) and decreases in the core,
%entually a concavity develops in the velocity profile in the comparison of these results with carher work cannot be made . ore nou, Associated with these changes there is an enhancement because the Nusselt numbers used by Kemeny and Somers were cf heat transfer coemeient. In practice, with sufficiently strong based on the inlet Guid temperature. More recent experiments heating or cooling, the velocity profile gradually becomes by Barozzi er ol.3' have been corr. pared with numerical solutions unstable at the point ofinflexion and now unsteadiness results by Collins and good agreement is reponed. Evidence of (see belo4 transition to turbulent now near the end of the heated section in the opposite case.i.e. downward heated now or upward was observed for Reynolds numbers of less than 1000.
cooled now, the velocity near the wallis reduced and there is Furthe data can be found in the papers by Scheele and an impairment of heat transfer. With strong enough heating Hanratty and Brown and Gauvin.28 However,these experi. (or cooling), the velocity gradient at the wall approaches zero ments were concerned primarily with transitional now, and an instability develo'ps there suddenly. The above discussion of esperimental studies of laminar The numerical solutions of Collins'8 illustrate the above behavtor. These calculations were carried out for conditions covered in the expenmental work of Scheele and other workers d' ~ t i p er,.eae o s s et e e , .i en the influence of heat transfer on stability oflaminar flow in %e oo,a n ae . is..aer no. te vertical tubes."*"-"It uns observed that as Reynolds number decreased, the critical value of Gr'/Re for onset of gradual see mais . a a.awner W
. - !.*e*
instability decreased asymptotically to approximately 42.5 for the esse of upward flow with uniform heat Aux. An increase in 1s _ . ;a?; 'y 1 '"1 ^- ' the critical value of Gr'/Fe *vith decrease in Reynolds number - ;;
' ^
was observed for the case of downward now with uniform heat , , , , , , flux. The highest critical vilue of Gr'/Re encountered was ici ios n' io- n' tr approximately 75 and this was for Reynolds numbers approaching cr */ ne zero. w'- Experimental studies of faminar mixed convection ~ ~ ~ * * ~ ne in verticaltubes f;
. Q',,
The early experimental work invalving mised convection in i t' -
*r .
vertical tubes was carried out using test netions in the form of
}- i i' 1" .- 8' double tube heat exchangers. Watriger and Johnson"in 1939
>oned esperiments in which wate? Dowing downward in a x was cooicd by an enternal 0ow. For that arrangement free i a,
- ' = '; ; 2 ;
ni w. a. a as a. convection aided forced convection. Martinelli er of." shonly ce */ a, afterwards reponed esperiments with upward now of water and cilin tubes having uniform wall;cmperatures. heated by means Figure 4 Plots of Nu, versus Gr'1te for ditterent vetues of G (adapted from Kemeny and Sorners% with permission of the ci stearn. Overall heat iransfer raies considerably in excess of American Society of Mechanical Engineers) 4 Int. J Heat and Fluid Flow. Vol 10. No 1. March 1989
, ,, u ,, . . . . . . . . . . . . . . , . . . . . ;
two case.s are the s.ame. l
- o. s..
. , , sssss n., htmples of buoyancy.induc2dimpritmint of heat c, , sus,r.q transfer h
$$e T in the earliest studies of turbulent mixed convection with k ' ," a n inu,.: atmospheric presure water and air,""*** there was little g g,,,,,. indication of the dramatic in0uences of buoyancy on heat
- 3"'W'* transfer that were later to become evident in the work on fluids near the critical point" and which have provided the l sac - f X H7 kW8C o Iseswe.
ncentive for considerable research work in recent years. i Dunng early development work on supercritical pressure o
\ <
c steam generators. Shitsman" and Ackerman" found severe localized impairment of heat transfer for upward now in heated
\ tubes at near.cntical ennditions (Figure 51. The effect was initially thought in be similar to film. boiling and was. in fact. l pren the name "pseudoboihng." There were. homeser. snme
*" surprising condnions uhere the wall temperatures were well below the pseudocritical value.' Under such condition, the fluid must have been in a liquid like state even in the wall.i.iyer i h region.
It became apparent that the effect was due to buoyancy and l
* - not a form of film boiling when experiments with upnow were compared with those for downnow at otherwise identical .
conditions. Investigations of this type were first reponed b) l Shitsman '8 and Jackson and Evans.Lutterodt"(Figure 61. 1 ((/ Similar compansons have since been made by a number of researchers, notably BourLe er of..". Fewster." Alferov er al..'* Watts and Chou." and Bogachev er al." { j si. The f.ict that the phenomenon oflocalized impairment due l to buoyancy is not restncted to Guids at high pressure, bus can
*' '3 )
occur in hquids and gases at normal pressure has subsequently Figure 5 Localded impairment of heat iransfer due to buoyancy become clear as a result of esperiments by Steiner." Kenning (adapted from Shitsman85 witn permission of Plenum Pubbshing er cl.." Hall and Price."" and Fewster." Kenning es ol. Corp.) found wall temperature peaks for upward flow of water at $ bar , (Figure 7)and Fewster made comparisons between upward and l mixed convection in vertical tubes has been restrieted to the downward now of atmospheric pressure water (Figures 8 and l case where free convection aids forced convection. An interesting 9). The case shown in Figure 9 is of special interest because l conclusion which has emerged from the present review is that the Reynolds number is below that normally associated with ! virtually no laminar Gow heat transfer measurements have been turbulent now. The measurements suggest that buoyancy. reported for conditions where free convection opposes forced induced turbulence is present the sudden fallin wall temperature convection (heated downward flow or cooled upward now). for upward Row being due to transition. For such flows the induence of buoyancy should be to impair Mised convection at low Reynolds number has also beert heat transfer relative to that for conditions of pure forced studied by Rouai." Buoyancy. induced transition to turbt. lent convection, provided the flow remains steady and laminar. However,as a result of the work of Scheele rf cl.on buoyancy. induced instability in laminar now. it is clear that the range of ,,g conditions availab!c for such capenments is likely to be rather t a'* H'o e e *t ose limited. 100 { ', y j
.. ise.u,.t Turbulent flow mixed convection so
, Whereas the efice:s :,f buoyancy on heat transier can be **" "*""" predicted readily for the larninar case, the situation is very 60 i
~
different with turbulent nows: the data for such conditions show some unespected trends (see reviews 3 **"). In configurations '"*h'* with forced and free convection aligned, local heat transfer so .-----~~~3...----~~~~ coefficients signincantly lower than those for forced now alone can result, in contrast, for downward now in heated tubes p.. i,. n,,,,,, , .rie.
,,..... 1 buoyancy forces cause a general enhancement of the turbulent H ddfusion properties of the now, with the result that wall n.
l temperature distributions are well behaved and heat transfer coefficients are higher than those for forced flow alona. Even. ' " ' '**. tually, as the free convection component becomes more and figure 6 Companson of upward and downward flow cases at near more dominant. heat transfer for upward flow also becomes entical conditions (from Jackson and Evans.Lutterodt**) int. J. Heat and Fluid Flow, vot.10. No.1. March 1989 5
Stucres of trused convection on verticaltubes:.I D. .lackson et at.
'5 0 -
8 ' 33 1 ** e,
' ' " I' Some further data which can be used for testing the validity
' l *U * * , of the criterion have been produced by Brassington and Cairns" for supercritical pressure helium. Buoyancy induced l u u, .'
wall temperature peaks occurred over a wide tage of reduced pressure but it was found that such effects were not present
.t.u.,
,o. n .u,.,
n u...
~ o" '. wo ,t n..,. (*)
x - - e . . > . u ,. . jf
,x,7&, -
se- s e a, ,e.aer ~e, e- - oe.a n e. SC ' ' O 31 60 60 s0 30 - figure 7 Buoyancy anduced smoairment with water at a pressure I ' of S bar (scapted from Kenning et st*t with permission of Harper 'to uone. eae e .arie. g _ and Row) 3/e now was clearly observed in upward now (Fig,e 10al whereas ' ' ' for downward flow turbulence existed througnout the heated * " ** section (Figure 105 4 in downward Gow a periodic Nusselt number vanation iFigure ll) suggested that small. buoyancy. * (b) n , , q ,, induced recirculatist cells might have been present. Wilkinson e,,n,w,: dr al." cxamined the conditions for now reversal at the pipe wall to - in a buoyancy opposec flow; otherwise the problem has received ~#**. ' little attention. " The exampics quoted above illustrate the importance of interactions between forced and free convection for Dow in so _ T ' ' " ' ' " , N. .". '. .". . . . , : vertical pipes. An explanation of the mechanism involved has f. ce.ane. '. been given by Hall and Jackson.s' They suggested that the n_ 5, dominant factor was the modification of the shear stress i distribution across the pipe, with consequential change in 't, unne. ene ee.ane. s ee turbulence production. The analysis of Hall and Jackson'* has j been extended by Jackson and Hall" to provide a general
,i .'o ,'o criterion for the onset of buoyancy effects applicable both for supercritical and normal pressure Guids. For innuences of (c) buoyancy on heat transfer coefficient to be less than 5% of the "' ""#'
forced convection value, the analysis yields a limit: '*"' " ' * ' G to . se -i i . toa R e,,', < 10 ' ' (3) ,,- uane.
~
. 5'
,- { ,o ane.
Cornparison of predicted criterion with experiment In order to test the validity of the criterion, experimental data so . l'-- - from the authors' programme of resesreh on heat transfer to I supercritical pressure carbon dioxide have been presented in voneleaeevane. 3 's terms of the parameter GJReF. Figures 12(a) and (b) show ' ' experimental data that are consistent with the criterion (the 3o [o are [o ordinate is the ratio of the observed Nusselt number to that for forced Gow in the absence of buoyancy effects but under n O tt.s/s (d) conditions that are otherwise identical). *" 25 s we. 8. Alferov el of." presented data for supercritical pressure water se -6I00 in terms of the ratio of calculated heat transfer coefficients for " ~
,/ W'"*-
forced and free convection (Figure 13). It can be demonstrated ",
. f c e.ane, with the aid of established correlation equations for free .....
and forced convection that A,,,Jh,,,,w is proportional to 50 - N ,- Gr/Re 3** Pr" raised to the power one third, Thus Ihe criterion I{. for negligible buoyancy suggested by the data of Alferov er al." ( gures 13 and 14), namely A,,,,,Jh,,,,> 3,can be re expressed n - We'* *a8 8' ah*- Gr 3/6 g4) - o' ,'o .'o .o Re ** Pr" < 2.4 10- s 8
/i ure F 8(s-v) Compenson of heat transfer for voward and down.
For 10* < Re < 10', this criterion gives effectively the same ward flow of water at atmospheric pressure m a tube of diameter classification of data as that of Jackson and Hall. 100mm (adaoied from Fewsier ') 6 .-.*v.- - < r*- - " = '-
c . ::sa which huopney effects can be neglected. s- .*.g.. . go . *
' ' * ::*\'t's wor:e=
A simple equation describing mixtd conv:ction hsat
~
n,,,,,,,, transfer in vertical tubes Although the theory of Hall and Jackson was developed in order to obtain a ersterion for the onset of heat transfer 30 - impairment for the case of upward now in a heated pipe,it is varie, eae es ane-also applicable to downward 00* where there is heat transfer 8
,a enhancement. It has been generalized ' to provide the following 20 -
8/8 , simple description of the manner in whi:h the ratio of buoyancy. 3 innuenced to buoyancy. free Nusselt numbers vanes with the 30 60 60 buoyancy parameter Or,,lRej 'Pr' 8 ftpure S Buoyancy induced transition to turbulent flow in water *** Nu 10*G r., at atmospheric pressure at a Reynolds number of 1500 (adapted - = = 11.- from Fewstera ) Nu.r Rej ' Pr,',' '- (6)
' ' t e 1 **
. - o Oe.aese.
' C' . e e, 15 1.. r .: , y,y,,
a,'If6 wee: 8e a st00 8 t(e c f sn* t t'en Mv' e aOsjaa 'II " t G'*' 6 Si *10, t ivrevieat serreise.ea a , or s 6 se
- = ,,, e O s i n . " "
o C o goo ooo o to,- i;; ..
.:awaer t er r e ret.oa " .
% t*
- e t he r* * =. . ,,,...<-
50 E t ,.
' ' ' ' f 10 i 10 : ?4 10 10 10 10 '*
m estes teagth 0 0 to 20 le so
'O'" e,s 2 9 e w/m? figure 11 Nusselt number variation in mised convectan heat e e es 3 mm ,, transfer to water (adaDied from House")
( t covient secreio,,on" k ,* hu,s 051 ao,*'" , { 10 - 10' - h
\ Lom.asr terreiotiea" " e
~, . 0 s o a.; u. -
;**p,,;.3./
- .w..e.q:g' f 4, f..e -
(b) ' 'I '8 '0 '
'O ' '
EV "* '
,, 2 i i i i F/ pure 10 Correlation of low flow data for water (adapted from -
C' '8' Rous "): (a) upflow and (b) downflow (8) to - under conditions where- '"
,$, @ 9150 t h ad 'i'"['
'i v 8 Me! ' < 2.4 10- 8 (5) i -
/
Bearing in mind that 6r. is somewhat less than Gr.. the bE agreement w th the criterion is estremely good. ' Although the main sources of data concerning the onset of ,,., ,',,, ,,,, ,',,, ,',., buoyancy effects have been investigations with superentical (b) pressure Guids, some work has also been done using water at pressures near to atmospheric'8 8' and with atmospheric pressure figure 72(a-6) Mised convection data for supercritical carbon air. These studies have further substantiated inequality (3) as dion.de (adapted from Fewster a
'): (a) upward flow- and (b) downward flow lat. J. Heat and Fluid Flow. Vol.10. No.1. March 1989 7
a Studies of mned convecroon on verticalrubes J D Jackson et at upward now with strong er.ough buoyancy induence a recovery in heat transfer is indicated. lcading eventually to enhancement
, a_
%ee at high values of the buoyancy parameter. Data for upward now of air" and mercury *' and for downward now of water *8
' ~
....ll.... ' are shown in Figures 16.17. and 18, respectively. These show ~
l s ., , behavior consistent with that indicated by Equaiion 6.
; .5,, .
]
3g Y. ', ', . ' s Correlation of data for inized convection ; i' ii - The theory referred to in the previous sectrons has prosed to I be useful for correlating data and has been employed when i presenting daia from the authors' program of research into
me heat transfer to Auids at supereritical pressure-see Figures j
- ) -
, "ef 12(a) and (b)(earlier). h is clear that for upward now (Figure l os i 1 6 6 e to 12a) the data do not correlate in the region where heat transfer , /ipwre 13 Correlation of data for upward flow of supercritical is impaired in terms of the parameters Nu/Nu r an) Cie,/ReF. I pressure fluids (adapted from Alferov el al", with permission of ***er. Ior downward 00w (Figure 12b) an excellent corre.
Scripta Technica Inc.) l lation is achieved throughout the mixed convection regime by j the equation ; an,me ~ . C r. '"8 e.o e'er Nu i + 2750 (7)
. 20 == Nu, g . i s ..
, Ref ,
N )( . 16 = = The form of the above equation was chosen such that Nu l 2
\ s. ** .' becomes independent of Re, for conditions where buoyancy is
, , dominant (i.e. tending towards pure free convectioni. l
'+\
\g* 9 - -,.*
Tt g i6o . Nu k erf te ,/ e in e r , i e i , Nef e s $ st3m 05 le JG Sc 10 0 16 0 - e Figure 74 Correlation of downward flow mined convection data - for supercrshcal pressure water (adapted from Alferov et st"; with e / permission of Senota Technica Inc.) P' L'o
- to - <. ---
O a0 - .
, . 0 61 i is <
9
- e a go ,
se 10** t et t e* ' ' ' ' 0 0 6 8 17 16
// pure 16 Effect of buovency on heat transfer to air at atmospheric U' erne. Ustie. l pressure for upward flow (adapted from Byrne and fliogu. with ;
, permission from the Councilof the institute of Mechanical Engineers) I sc ob Cr,/ n e ,P 5U S 20 -
st i ' ' ' "" ) 01 to voi es eree.t ee er rae tree 1 100
,, , , , , , , , , ,,,, 3 0 oli ee a l
j
// pure 15 Theoretical prediction of general features of mised t convection heat transfer in verticol tubes (adapted from Jackson **) )
e, i a,a## #.
- o-The negative sign refers to the buoyancy sided case and the II i q*/ ',,#
positive sign to the opposed case. it should be borne in mind f a, 4 , / ,,-#' , , that this espression was obtained using simpic physical theory a g /, h ,
"'*l',,
and empirical relationships for buoyancy free heat transfer (for details see Ref. 2g). In consequence it merely represents an
, o,e g' '
'M jj )
l
, , , , , ,3 attempt to predict gross trends as buoyancy effects begin to , _
modify the forced flow significantly, however it is surprisingly 5 ; */ n e8 consistent with observed behavior (see below). i __t t i i i With increase in the buoyancy pat.imeter, an impairment of 0 t i 3 6 5
- heat transfer is indicated for upward now and an enhancement for downnow. The curves representing Equation 6 for the two figur, ;7 H .i i,,nst,, with an ascending mercury flow in a bested wenical pipe (adapted from Buhr et #1". with permission of cases are shown in Figure 15. It is of interest to note that for the American society of Mechanical Engineers) i e !
- m. i w... . n...e now ve in w i o.~ oco _
4.eiiause uniseisising eine ensianiscuiciis os nea transler cue to ne ws.as i , busyancy effects for downward Anw in heated tubes. The cbserved trends are in allcases consulent with the simple m;/ict 200 -
- i es 6 a cf mixed convection in vertical tubes of Jackson and Hall and can be adequately described by expressions such as Equations 7 or 8. For upward now m heated tubes a significaat amount 100 of experimental data is also available. Whilst the trends are l
- ' broadly consistent with the model, the problem of correlating I A'l I data is more complicated, particularly becaur4 of complex now N '
development behavior.
'O - 15
,- The dimculty of correlating upward now data in terms of
'O '
- purely local parameters (Figure 21) led RouaiS' to examine S
, conditions at wall temperature peaks. Minimum heat transfer 20
,# f 0agl0 (Dasyttis:N was correlated by
' "" 8' r agiin Nu , Gr*
,, Nu r -R e ' * " Pr" *" 1 et 0: er os 1 s e figure 18 Effect of buoyancy on heat transfer to water at atmos. and the dntance to the first wall temperature peak by pheric pressure for downward flow (adapted from Jackson and ~
Fewster**; with permission of Harper and Row)
= 0 051 - -
(10) The contrast between the upward and downward now cases J -Re'*" Pt"*- , is interesting: for the former, buoyancy forces lead to localized 1 effects (wall temperature peaks), whereas for the latter they do not. For upward now impairment or enhancement of heat [ A r,,ng, aus 1 g,a,l= #;T I
**,.. l transfer can occur depending on the magnitude of the buoyancy *.
I force, whereas for dounward Dow only enhancement is found.
- Although the effects might 'eem to be anomalous when viewed _ M *3' '"' * "8" ' '* *'" ' '
without the aid of the model of Jackson and Ha!!. they can be *3" "'"" " " *' seen to follow a meaningful pattern when considered in the light i of the model. The general picture of mixed convection in vertical O/ apF :
- ~ - ~ ' '
tubes embodied in Equation 6 and illustrated in Figure 15 thus * 'do ... ,, .r. i c. - I proves to be surprisingly accurate. In parallel with work on supereritical pressure Guids, the Figure 19 correlation of mixed convection data for downward present authors and co workers have also studied mixed How of water (adapted from Jackson and Fewstar**; with permission convection to water and to air, both at atmospheric pressure of Harper and Row) (Jackson and Fewster*2 and Ancell and Hall"). Again '~ the theoretical guidelines have proved to be useful. Jackson and Fewster used as their correlating parameter the group -t ' Ci,/Re '"Frl8.The data of Jackson and Fewsier for down. 2 E, ' b 6m G/ *r *V P. ward now of water in a heated tube are shown in Figure 19 and A \ can be seen to be satisfactorily correlated. A correlation equation which fitted the data over the entire range of conditions \ was arrived at by adjusting the inden in order to make Nusselt
- number independent of Reynolds number when buoyancy 2 innuences become dominant. This equation is: , o* ,
Nu
~
4500Ci. 4" * . o '*
- U/ 'C " E
-= 1+ (8) i -.J Nu r . 2.W' Re8 '"Pi[8 t o'- So" n .io" Agure 20 cornpans n of d wnward flow corretaison weih data for Figure 19 also includes some early data for atmospheric pressure water." The agreement between the equation and # '" # ' *# * "d " * " **"""'"*"#"'*'*""
the data is quite reasonable when account is taken of the 'N'3 p uncertainties involved. particularly in the values of Nur used *' e es e for normalizing the data. -.
*" 2= ses
- 26 tea P Figure 20 shows the data of Ancell and Hall" for downward ""'
flow of air in a heated p*ipe plotted on the same basis. Some ,, measurements by Easby* of downward flow mixed convection to nitrogen at about 4 bar are also ofinterest. When compared with Equation 7 (the correlation equation developed for the "--t % supercritical pressure carbon dioxide data) they are found to
. be in fair agreement.This is also the case for the data of Brown p.,,,,...,,a and Gauvin," who found enhancement of heat transfer by up to 70% in downward now experiments on air. Further sources 5 0"g .. l,., l,., ,',..
of data on downward now mixed convection are Krasynkriva
.(. . . .:
er of.," Ikryannikov er of.." and Alferov er of." Thus it figure 21 Upward flow data for water at atmospheric pressure I can be seen that there is a significant body of information (from Rouar) 1 i int. J. Heat and Fluid flow. Vol.10, No.1. March 1989 9 I 1 l 1
. l Studios et must convecrosn on verticaltubes:J O .tockson en et.
Rouas also compared his data with a refinement of Hall and The usual practice is to set the turbulent Prandtl number to Jackson's model which accounts for the innuence of heat a constant value, a,=0 9. The different strategies adopted to transfer on the buoyant layer.**" An implicit expression for determine turbulent viscosity, v,. are vital to the success (or NufNurresults: lack ofit) experienced in the application of turbulence models N" Gr* Nu '2** i turbulent mixed c nvecti n:it will be seen below that the l
- - - l i R = 10' Nu, (11) simpler models fail to capture even the general trends of heat l
, Re* *" Pr* * \Nur / . transfer impairment and enhancement.
It should be noted inai this correlation gives a discon. smuity in Nusselt number for heated upward now when (i) Presenbed eddy di//usivity models Gr' ire i'" Pr' '- 3 x 10 *
- The " eddy diffusivity" approach presenbes turbulent viscosity as a function of postion in the 00m mithout any direct reference Numericalstudies of turbulent mixed corwection to local features of either the mean field or time.aseraged in verticaltubes turbulence field Tanaka er al. examined turbulent mixed The various applications of turbulence closuus to mixed convection tube Dows usmg a modi 0 cation of Reichardt's" i eddy diffusivity model. The wall temperature distnbutions for l convection are considered under the following headings:
a vertical heated tube computed by Tanaka er ol. were opposite
)
(il Presenbed eddy diffusivity models to observed behavior, showing heat transfer enhancement for ' tii) Mixing length models ascending now and impairment for descending flow, it was (iii) One equation transport models concluded that "the theory is not sufficient to estimate heat (iv) Two-equation transport models transfer coefficients." The work of Tanaka er of. is the first of (v) Higher order models a number of numerical studies to be reviewed in which the developing flow formulation of Equations 12 and 13 is not The categorization abose does not convey the complete picture employed. because of the absence of any reference to the now formula-tions adopted (whether fully developed or developing thermal. (ii) Mi'ing length mode /s hy draulic conditions are assumed to prevail). It is well esta blished in the Prandtl-Taylor mixing length hypothesis (MLH) it is that very long flow development lengths occur in regions of the postulated that r, may be expressed in terms of a mixing length, ascending mixed convection regime and therefore the validity I.. and the mean velocity gradient: of studies using even the most refined turbulence models cast in a fully developed formulation can be seriously compromised. OW The most general form of the mean now governing equations
,g* _,
g tr used in the works reviewed below are the tirne averaged momentum and energy equations written in the 'th;m shear . Numerous modifications to the original prescription of I. (or " boundary layer ) formulations. Thus m cylindrical polar (I = uy) have been proposed (see the discussion of Launder
* ' " and Spalding." for example). A modification that has found widespread application is due to van Driest"in which allowance Atomentum is made for the damping effect of a solid boundary upon Ii
- (rVW)
/
(W8)= i dp I / ~ / tW _\ - turbulence in the vicinity of the boundary: rr -- j g pg .p . )] N = 0 4; A * = 26.01 (17)
+ (1 - /l( T- T,,,)]6 (12) Malhotra and Hauptmann" implemented the van Driest where mixing length modelin a fully developed mean flow equation set.
Computed wall temperature distributions for heated ascending
-p for ascending now and descending now demonstrated the correct trends. indicating
- f. = heat transfer impairment for up0ow and enhancement for
+g for descending now downnow. Comparison 4 *i the wall temperature data of Encryy Jackson and Evans.Luttetodt for carbon dioxide at near.
critical conditions showed good agreement for descending now I b (rV7H b rh '.[-rf\-
/c (IPT)=
r ir '.\Pr$ir but poor quantitative agreement for ascending now IFigure 22). t /r (13)
/. although the highly variable Guid properties of the experiment represent a considerable added cornplication.
i The equations are written in the Boussinesq approximation: ' Walklate" tested four mining length models against the data density variations are neglected except m the body force term for tiented up0ow of air with uniform vall flux obtained by of the momentum equation where a imeanzed function of Carr er al" The data show signincant m0ucnces of buoyancy, temperature is employed. The models in categories (i) to (iv) demonstrating velocity profile inversion in three of the four test above make recourse to the concept of turbulent viscosity by case and a marked reduction in the level of Reynolds stress. which Reynolds stress appearing in the n% mentum equation The four mixir,g length models used by Walklate represent as related to the mean velocity gradient: various medincations of the van Driest formulation. Walklate __ /W used the developing now equations of the form given above. i thus eliminating an uncertainty present in the work of Tanaka j
~ 8* " 'i 7,- (14) er of. and Malhoira and Hauptmann." Comparison was made with the bulk parameter data (Stanton number and local Similarly, the turbulent heat Oux in the energy equation is friction coefficient) and profile measurements of Carr er ut.
l related to the mean temperature gradient: I Discrepancies between calculated and measured bulk parameters 1
-rT* = ; /T were large and computed velocity, temperature and Reynolds (15) stress profiles were in poor agreement with the experimental
- 8. Er data. l 1
to ini J Heat and Fluid Fica voi to No 1 Ma'ch 1989
. . . . . . --...v . .
'. '# #"' "I.5.,, constitutise equation for v,.
The turbulence quanuties mnzt
'*U
'S'~
often selected to form the scalesare the turbulent Lmelse energy, 1,and its raic of dissipation c.as evidenced by Shih's lucrature y ;,- - surveys.* The length scale is obtained as & #3 c, and the
"~ ~'~,,,.--W f Prandtl-Kolmogorov formul.a takes the follow n; form:
,/ o,de.... , vi.., wet, y, , gg sjg g9, ee w s.ag ne. e.o" * "
Walklate" tested three versions of the &-r model with developing flow formulations against data of Carr er al.:'* the first model examined was a standard "high Reynolds. number" so model (Launder and Spalding)in which the 1 and c transport o to 60 60 so 100 40 16 0 equations were solved for the region y* >.10, analytical wall a/o functions being employed to bridge the near wall region. The Fqure 22 Companson betwun calculations of Malhotra and other two models were variants of a partial low.Reynolds. Hauptmann'* and data of Jackson and Gans.Lutteroct" for number treatment m shich a damping term was applied to the supercritical pressure carbon dioside (adapted from Malhoira and espressron for v,(Equatmn 191following Jones and Launder?2 '8 Hauptmann'*; with permission of Harper and Row) The forms of the & and r equations were unaliered from the An early hybrid work by Hsu and Smith" combined an high. Rey nolds number version talthough t tr.ansport u.ts solved espression for near wall turbulent viscosity due to Deissler" ver the entire Dowl % alliate found that, as a group.the A -r with the mixing length model for the outer region. The m del 5 Performed better than the miaang length models in com. calculations indicated only enhancement of heat transfer for puting turbulent mised conveelson. Heat transfer calculated heated ascending flow, a result that is again contrary to using the partiall w Reynolds number models showed relatively observed behavior.The likely cause of this result is the Deissler g d agreement with the data of Carr er al.; poorer agreement near. wall formulation u hich, although strictly not a prf, scribed was evident when the high Reynolds. number model was apphed. eddy diffusivity model as defined earlier, nonetheless does not These results were supported by comparisons with the now correctly redect turbulence production. It should be added that pr Gle measurements of Carr er of.: the general picture to fully developed now was assumed in this early attempt to emerge was that improvement could be gained by the use of calculate turbulent mixed convection. partiall w.Reyn Ids. number modiGcations. However.no model yielded high quantitatise accuracy. Walklate made a single test of a partial 'ow Reynolds. number model against one of the (iii) One. equation transport models velocity proGles measured by Ancell and Hall" for descending Studies of turbulent mixed convection in vertical tubes have Cow but found that agreement with data was poorer than that been performed using turbulence models that employ the found in the comparisons made with the ascending now proGle turbulent viscosity concept but which form r, from turbulence data of Carr er of. quantities and include transpori effects. The basis for intro. Abdelmeguid and Spalding" combined a standard a-c duction of turbulence quantities into an equation for v,is the model with (unspeciGed) wall functions in a developing now
?randt!-Kolmogorov formula in which the square root of solution scheme. Computed results demonstrated the correct turbulent kinetic energy, k, forrns the velocity scale in the trends in mixed convection heat transfer i.e.for heated upflow, constitutive equation. Thus, impairment at low Grashof number was succeeded by enhance.
r, - & inl, ment at high Grashof nurnber and, for heated downDow, , (18) ' enhancement was found for all values of Grashof number Models in which a transport equation for one of the scales (Figure 24). There was no comparison with caperimental , (in practice the velocity scale via A transport)is employed are data for bull parameters; however, reasonable agreement was l known as "one. equation" models. Ascell and Hall" applied a obtained with the velocity and temperature proGle measure. variation of Wolfshtein's one-equation model to their esperi. ments of Buhr er of. for heated ascending flow of mercury. j mental data for heated descending air Dows. An important Thus, from the results of Walklate" and Abdelmeguid and l feature of the Wolfshtein modelis that it is appl. cable over the Spalding," it would appear that the A-e model offers an I entire now domain, including the viscosity a'Tected near wall improvement over simpler models in ihe calculation of turbulent l region and thus does not require the specification of " wall mined convection, although there are signiGeant discrepancies ! functions" to bridge that region. Atcell and Hall's calculations were qualitatively correct, showing enhancement of heat transfer l 3,, , with respect to forced convection. The predicted enhancement
- finewt 0' it' was, however. considerably less than that found experimentally h' s tinr e i r,'.: .io' .
(Figure 23). It was concluded that the one.cquation model does not offer any advantage over a mining length model in mixed 300 - convection calculations. Again there are discrepancies between . the thermal. hydraulic conditions of the experiment and the 6' ' , ao* modeled problem and a fully developed condition was assumed io , ' in the analysis (although rapid thermal. hydraulic development g, 00' _,,,,,, in descending mixed convection nows makes the assumption far less limiting than in ascending Dows where in many cases 4( ,o , , , , , it is wholly inappropriate). O o $ , e io i; a r . ,on 1 (iv) Two. equation transport models y ,,, "Two-equation" models have been applied with some success downward flow turbutent mined convection (adapted from Ancell to turbulent mixed convection. These incorporate transport l and Half **. with permission of Harper and Row) int. J. Heat and Fluid Flow. Vol.10. No.1. March 1989 11 l l
Studoes almind convsetoon nos vertoest tubw J O .lackson et al. 200 computed accurately over a development length of approni. to - in a u eae,a, vie. mately sixty tube diameters. Good agreement was also obtained n .. with the velocity profiles of Ancell and Hall" for descending
'O - now at high levels of buoyancy, but predictions were in less than complete accord with heat transfer data a feature that is to -
' crammed further below. Polyakov and Shindin" have recently u e ' >
published experiments 1 data st Swing heat transfer, velocity, s .m' too io' sod 1.)o* temperature and turbulence parameters in heated upward now, and the present authors intend to make comparisons with 3 ,, theoretical predictions as part of their continuing programme w i of research. u p in ceuce.ag m. W- It was noted above that the data of Ancell and Hall were not computed satisfactorily by Cotion and Jackson using the Launder and Sharma model: results presented in Refs. 69 and
85 show that the calculations return considerably higher levels u[ s.i:, io' to' '.8
' of heat transfer than the,se measured experimentally (Figure 26). Recent work by thz authors" indicates that improved 50 1 10
- agreement with data is obtained by the inclusion of an additional source term in the : equation proposed by Yap."
figure 24 Calculated vanation of Nu with Gr* for Re = 25.ooo Further work is in progress to investigate buoyancy induced (adapted liom Abdelmeguio and Spalding**; with permission of Cambndge University Press) recirculation using an elliptic solver developed by Huang and Leschziner." Two further low Reynolds. number 1-c studies are reviewed. Renz and Bellinghausen" used the Jones and Launder model [i 5*' i7 ( Se'a#' ser.ua ea e se r/. iff e I compute heat transfer to an ascending now of a refrigerant under the conditions of an experiment by Scheidt" carried out 58 ', "" near the thermodynamic critical point (and therefore with
'8 "j
, ~ ~ T,',l $ 'M highly variable Auid properties). The correct qualitative trends w . - - i tion i iii of wall temperature development were found, although there "g . - - i 5000 e eso were some significant quantitative discrepancies as shown in D '(
%^ r . 4. . : .s. "
Figure 27 Tanaka et oI." compared a slight variant of the Jones and Launder model against their data for heated upuow of nitrogen and found generally good agreement between 3' "h* measured and calculated Nusselt numbers. Comparison with g, . data was limited, however, and does not appear to include
" "gg N b e.,.q: , points in the vicinity of maximum heat transfer impairment.
Tanaka et of. used the fully developed formulation for their 8
" calculations which is unsuitable for application to Dows with
, , , , , , , , , high heat transfer impairment where long development lengths o a u a u 60 as 56 46 n occur.
/0 (v) Highet order models 7, pure 25 Compenson t.etwun calculations of Cotton and
' Jackson"" and data of Steiner" for ascending air flow (adapted Reynolds stress" or second mo. ment" turbulence models do from Cotton and Jackson **"; with permiss.on of the American not rely upon the turbulent visenstly concept, but instead Society of Mechanecal Enginnts) incorporate transport equations for second order velocity and temperature correlations. Launder" provides a comprehensive discussion of these models.The complexity of Reynolds stress between computational results and esperimental da:2. The models led to the development of truncated forms known as likely cause of these discrepancies lies in the treatment of the '
near.m all region: Hall and Jackson identified the importance
. [*[",[""'
of deviations from " universal" behavior in this region in determining levels of mixed convection heat transfer and m /
* /
,,__ l
M!klate founJ trat agreement with caperimental data was ~~* improsed by the adoption of a partial low.Reynolds number treatment.
**0 , g(,,, ,/,j,',"l". /
,,,,,,,,g ,,e, saa %u e. ,
Cotton and Jackson"" esamined the performance of a low Reynolds number & -c modeldue to Launder and Sharma" no -
/
(a minor re-optimization of the Jones and Launder"" model)
/ ~~~,
against a wide range of mixed convection data for air Dows. It * / **~
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was found that the fulllow Reynolds number treatment cast in - the developing flow formulation reproduced to good accuracy un i i . i > - both the experimental heat transfer and (where available) 8 05 to 15 to 35 H velocity Reynolds stress and temperature profile measurements Gr a to" of Carr et of.. Byrne and Ejiogu"and Steiner"(allascending figure 28 Comparison between ca'culations of Cotton and flow)and of Easby" (descending 0ow with moderate buoyancy Jackson" and data of Ancell and Hall" for descending air flow influenet). Figure 25 shows an esample of the results obtained (,.. drawn from Cotton and Jackson", with permissiun of the by Cotton and Jackson: the heat transfer data of Steiner are Amencan Society of Mechanical Enginnes) 12 Int. J. Heat and Fluid Flow. Vol 10. No 1. March 1989
.e... .
s
- impairment using local parameters. In contrast he.it transfer r t) " "*"'
levels in heaicd downflow merease monotonically with mereasmg
, busyancy and have been correlated suceci,sfull) in terms cf too -
e local viriables.
. Turbulent mixed conveedon may often be calculated accurately using turbulence modeling techniques provided that
*.. . 'I' a developing flow solution is used and that the turbulence model g _ **, allows for changes in both turbulence velocity and length scale.
" Low.Reynolds. number" models should be used. permitting
, solution up to the heated surface; the shear stress may vary t,. en..cr significantly close to the wall and wall functions based on uniform shear are not applicable.
too ev. ,,,e,,,,,, References 9 , i Martinclii. R. C. and thietier L. M. K. An.al>ue.i: predenon of supenmposed free and forced consecuon in .i sertgal pipe. Umrcesa r el Cahlhenna. l'uhlscarnun un Engenteruna. l%2. $. 23-58 2 Ostroumov. G. A. The mathemaucal theory of heat transfer in 6 I ' ' - circular, verucal tubes e ith combined forced and free consecuon 1 a 9 u (in Russiant Zhurnal TrajnirAcslui fi:Ai. 1950.20.740-757 8'"'
) Hallman. T. M. Combined forced and free laminar heat transfer
/< pure 27 Companson between caleviations of Flent and Bell ng. in verocal tubes with uniform internal heat pencraison. Trans.
housen" and data of Scheidt" for ascending refrigerant flow ASME.1956. 78.1831-1841 ' (idapted Isnm Rorna and Bellinghausen"; with permission of Harper 4 Hanratty. T. J., Rosen. E. M and Kabel. R. L. Effect of heat and R W transfer on flow field at low Reynolds numbers in serocal tubes. Industnal and Enouneenne Chemustry. 1958. 50. 5 E 5-$20 5 Brom n. W. G. The superposioon ornaivral and forced eonsection al teb,mc stress models'(ASMs). Reynolds stress models and ai som now rates in a venical tube. rerschung ../ d<= c,6,cre ASMs have found application in complex flows; however, as des Ingenaturwesens. 1960.26. VDI.Forschungsheit aSO shswn by Launder." the ASM formulation reduces to &-c 6 Monon, B. R. Laminar convecuan in uniformly heated verucal f:rm in thin shear flows. To and Humphrey'* applied low. pipes. J. Fluid Mech.,1960. s. 227-240 Reynolds-number &-c and ASM formulations to frec convection 7 Rosen. E M and Hanraity.T.J. Use or boundar> l:3er theory along a heated wenical plate but found only slight differences 80 Predici the effect of heat transfer on the laminar Dom field in between the performance of the two models. De Lemos and '" " "
"d(f,96!7.12- ""P"'"" '
4sonske" applied an A,SM to mercury mixed convection tube 8 Pigford. R. F. Nonisothermal now and heat transfer inside (
.cw and found quahtauve agreement with data, vertical tubes. Chemical Enginnenns Fragress Smos,um S<rics.
Finally, the direct interaction of buoyancy and turbulence 1955.51.79-92 via the fluctuating density. velocity correlation is considered. 9 Bradky. D. and Entwistle. A.G. Developed lammar now heat Abdelmeguid and Spalding" presented results neglecting transfer from air for variable physical properties. Inr. J. Heer buoyant production and reponed that tests indicated that Mas. Tro,u/cr, 1965. 8. 621-638 inclusion of such terms in the & and c equations had no 10 Mamn. W. J. and MeMillan, H. K. Combined free and fo cod significant effect upon their respits. The results of Cotton and laminar convection in a vertical tube with constant mall tem. Jackson" confirm this findinI, at least to the extent that the pusium. TransaSm.mer Tnins/cr. m032E-M buoyant production terms were found to exert at most a second Il Lawrence. W. T. and Chato. J. C. Heat transfer efIccis on the developmg laminar now inside vertical tubes. Trans. A5ME C. crderinfluence. Petukhov and Medvetskaya" adopted a some. j, u,,, 7,,,,g,,, g g66. 88. 214 222 what unusual twe equation turbulence modelin which equations 12 Coll ns. M. W. Combined convection bi verueal tubes. were formulated for turbulent kinetic energy and mean square Symposium on Heat and Mass Transfe % l.;ombmed Forced
, enthalpy Puctuations with buoyant production terms included and Natural Convection. Inst. Mech. Engrs., Manchesier.1971, but convettion and ditTusion neglected. Bulk parameter calcula. Paper Cl15/71 LiIns were consistent with experimental data but no comparison I) Collins. M. W. Heat transfer by laminar combined convecuon with profile measurements were made. Funher work is necessary in a venical tube-predictions for water. Proc. 6th int. Heat j in erder to clarify the importan,r of direct interaction terms. T nsfu n ' '
g g \comn'd 'F Ep ng 00 = th combined forced. free convection in an isothermal vertical tube. Trans. Conclus. tons ASME C. J. Hear Trem/cr. 1971. 94. 2il-223 15 . Schecie. G. F., Rosen. E. M., and Hanratiy T. J. Effect of natural convection on transition to turbulent now in venical in laminar mixed convection heat tra nsfer is enha nced in heated pipes. Canadsen Journal of CAemical Engineenne. 1960. 38. upward flow and impaircd in heated downward flow. The 67-73 prsblem is amenable to calculation, although transition to 16 Scheck. G. F. and Hanratty, T. J. Effect of natursi con <ecuan turbulent Row may occur earlier than for forced convection or on stability of now in a veneemt pipe. J. Fluid MerA 1962.14 free convection alone. 2aa-256 l 17 Scheele, G. F. and Greene. H. L. Laminar-turbulent transinon Turbulent mised convection heat transfer to moderate Prandtl for nonisoihermal pipe now. AJ.CA.E. Journel. 1966. 12. number fluids is dictated by changes in turbulent diffusion. In 737 740
' heated upward flow heat transfer is impaired with modest la Watzinger. A. and Johnson. D. G. Warmeubenragung von butyancy and enhanced with high buoyancy. It is not possible Wasser an Rohrwand bei senkrechter Str6mung im Obergangs.
i i int. J. Most and Fluid Flow. Vol.10. No.1. March 1989 13
/ -
/
3 f 4 100. Zhosla Hoffman T.W. an d i PollocE,J.I.,G., Combdmed Fo,rced and Natural Convection Heat Transfer to i Air in a Vertical Tube, Proc. 5th s Int. Heat Transf er Conf. , Tokyo. y t Vol. 4, jig 4. 4, pp. 144-14c , 1974.
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.EVIEW Studies of mixed convection in vertical tubes J. D. Jackson, M. A Cotton, and B. P. Axcell Department of Enginoring. University of Manchester, Manchester, UK Received 3 August 1988 and accepted for publication 6 December 1988 The early study of convective heat transfer considered the branches of forced and free convection independently with only passing reference to their possible interaction. In fact !
the two are extreme cases of the general condition of "mixrJ" or " combined" forced and I free convection where both mechanisms operate simultane.mly. The present contribution aims to provide an up-to date review of those works concemed with mixed convection heat transfer in vertical tubes. The review is divided into two sections, the first dealing with lammar flow and the second with turbulent flow; further subdivisions are made according to whether the work is theoretical or experimental. Comparisons between theory and experiment are made where possible, expressions defining the conditions for onset of buoyancy effects are presented and equations for determining heat transfer are given. The paper ends with some general comments and recommendations. The survey is stricted to fluids of moderate Prandtl number; mixed convection in houid metals can display very different characteristics which will be discussed in a future paper. Keywords: mixed convection; combined convection; buoyancy influenced flow; inter-action between forced and free convection; laminarization; thermogravitation [ ' troduction Laminar flovv mixed convection The term " mixed convection"is used to describe the process The effect of simultaneous buoyancy forces and externally-of heat transferin fluids where,due to va ristions of gravitational applied pressure forces on steady laminar now in a vertical pipe body force associated with non-uniformity of density within is amenable to calculation. The general result is that wan the the system, the flow field is signincantly modified from that now is in the upward direction past a heated suifae: (or which would prevail under conditions of uniform density. The downwards past a cooled surface) heat transfer is enhanced, processes involved are usually thought ofin terms of the concept whereas in the opposite cases heat transfer is impaired. These of Guid buoyancy and the effects are frequently referred to as effects of buoyancy on heat transfer, innuences are not the result of any change in thermal diffusivity but are instead a consequence of the distortion of the velocity in the early development of the subject of convective heat field and pattern of convection in the Guid. transfer, free and forced convection were studied separately and
.any interaction between the two was ignored. When the possibility of such interactions began to be investiga ted, attention Theoretical studies of isminst mixed convection was at first restricted to laminar and transitional flow conditions.
More recently it has become clear not only that measurable Work on the subject dates from 1942, when Martinelli and innuences of buoyancy can exist in fully turbulent flow, but Boelter' analysed a very simple fully developed now model. that under certain conditions buoyancy effects can m fact be Other fully developed solutions were subsequently reported by the dominant factor m determining heat transfer. Ostroumov,8 Hallman,8Hanratty et dt.,' Brown.8and Morton
- in this paper attention is focussed on rnized convection in . .
With time and the advent of digital computers the number of the simplify ng assumptions decreased and, in particular, venical tubes. Two thermal boundary conditions are of panicular mterest, namely uniform wall temperature and uniform wall attempts were made lo obtain developing flow solutions. Rosen heat Oux. The early expenmental work on mszed convection and Hanratty,' following earlier work by Pigford,' used the was carried out using test sections heated by means of saturated boundary layer integral method with power series for the steam, resulting in approximately uniform wall temperature. velocity and temperature proGles. Thus they reduced the Later workers have m the main utilized electrical heatmg problem to one of integrating a number of simultaneous le: ding to approximately uniform wall heat flux. non-linear differential equations. Numerical solutions, taking The effects of buoyancy on heat , transfer rates can be either account of the variations of all the physical properties, were is enhance the process or to impatr it depending on the flow obtained for upward now of air with uniform temperature by crientation (ascendmg or descending), the now conditions and Bradley and Entwistle.' Marner and McMi!!an also obtained ca heated length. A sounc understandmg of the processes numerical solutions for this boundary condition taking account ved as needed in order to take proper account of the effects of flow development. Their prejictions of arithmetic mean
- o. .,aoyancy on heat transfer in the design of thermal systems. Nusselt number are shown in Mgure 1. Figure 2 shows an interesting calculation oflocal Nusselt number for which the innuence of buoyancy increases following the thermal entry Edess repnnt requests to Professor Jackson, si the Depanment of development and then reduces funher downstream as the fluid En5mesnng. Univeruty of Manchester. Manchester M13 9PL (JK.
temperature approaches the mall temperature. C 1989 Butterworth Pubhshers 2 Int J Heat and Fluid Flow. Vol 10. No 1. March 1929
Studas ct truerd etnneran on netoct!tubss'./ D. .lackson st it. getnet ruischen laminarer und turbulenter Stromur.g. Forschung 40 Alferov. N. S Balunov, V. F.Zhukowdi A. V Rybin, R. A
' auf dem Gehirer des inurnierwescu. 1939.10.182-196 and Fokin. B. S. Features of heat transfer due to combined free 19 Manmelli. R. C Southwell. C. J., Alves. G Crais. H. L.
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Impairrnent of Pro)cct No 6627 surbulent forced convection heat transfer to superentical pressure 21 Hallman. T. M Espenmental study of combined forced and CO, caused by buoyancy forces. Repon N.E.2. Unnersity of free lanunar convection in a terticaliube. N.A.S.A. T.N. D Il04 Manchester.1968 1961 44 Bourke. P. J., Pulkng. D. J., Gill. L. E., and Denton. W. H. 22 Kemeny. G. A. and Somers. E. V. Combined free and forced-Forced convective heat transfer to turbuleni carbon dioxide in consecuve nom in vertical circular tubes-<speriments with the superentical region. Int. J. Hear Mass Tranger. 1970.13. mater and od. Trau. ASME C. J. Hess Traufer. 1962. 84, 1339.l)48 339-146 45 Fewster. J. Mined forced and free convective heat transfer to 23 Baro 2.n.G.$ Dumas. A and Collins M. W. Sharp entry and superentical pressure Guids Doming in vertical pipes. Ph.D. transioon effects for laminar combined conveenon of water in Thesis University of Manchester.1976 senical tubes. Int. J. Hrus and fluid flaw. 1984. 5. 235-241 46 Alferov. N. S. Balunov, B. F., and Rybin, R. A. The effect of 24 Scheele. G. F. and Hanratty. T. J. Effect of natural convection naiural convection on the heat iransfer of a single. phase flow mstabilittes on rates o(heat transfer at low Reynolds numbers. with sub-enucat and superentical presures. Trplofuda VysoliLA A J.Ch.E. Journal. 1963.9.181-185 Temperatur. 1976. 14. 1215-1221 25 Brow n. C. K. and Gauvin. W. H. Temperature profiles and 47 Alferov. N. S., Rybin, R. A., and Balunov, B F. Heat transfer Ductuations in combined free a nd forced convection Dows. Chem. Eng Sci. 1966.21,961-970 with turbulent water flow in a vertical tube under conditions of appreciable in0uence of free convection. Teplorarrecida.1969 26 Petuk hov. B 5. Turbulent Dom and heat transfer in pipes under 16.66-70 considerable effect of thermogravitational forces. Hrut Transfer 48 Watts. M. J. and Chou, C. T. Miaed convection heat transfer and Turbulens Buoyant Conwctuon. Semunar of Imernatuonal to supercntical pressure water. Proc. 7th International Heat Cenire for Hras and Mass Transfer.Duhrorna. Yugostaria.1976 Transfer Conference. Munich,1982. Paper MCl6 feds. Spalding. D. B. and Afgan N.). Hemisphere Publishing 49 Bogachev. V. A Yeroshenko, V. M., Sn>itina. O. F and Corporation. Washington D C.. 1977.701-717 Yashkin. L. A. Measurement of heat transfer to superentical 27 Hsu. Y. Y. and Smith. J. M. The effect of density vanation on helium an vertical tubes under forced and mixed convection heat transfer in the entical region. Trau. A$ME C. J. Hrar conditions. Cryogenics. 1985.25,198-20I Traufer, 1961.83.176-182 50 21 Siciner. A. A. On the reverse transition of a turbulent flow under Jackson. J. D. and Hall. W. B. Influences of buoyancy on heat the action of buoyancy forces.J. Tluid Af echo l971.47,503-512 transfer to fluids flowing in vertical tubes under turbulent 51 Kenning. D. B. R.. Shock, R. A. W., and Poon. J. Y. M. Local conditions. Turbulent Torred Cem arrion in Chanaris and Sundles reouction in heat transfer due to buoyancy effects in upmard Theory and Apphcations to Hear Exchangers and Nuclear turbulent now. Proc. $th lat. Heat Transfer Conference. Tokyo. Reennes. 2. Advanced Study Inssisuse Book (eds. Kaksc. S. and 1974. Paper NC 4.3 Spalding. D. B L 1979.613-640 52 29 Hall. W. B. and Price. P. H. Mized forced and free convection < Petukhov B.S. and Nolde. L. D. Heat transfer to mater nowing from a vertical plate to air. Proc. 4th Int. Heat Transfer in a vertical tube. Trpluencrperda. 1959.6.72-80 Conference. Pans.1970. Paper NC 3.3 30 Petukhov. B. S. and Sirigin. B. K. Espenmental investigation 53 Hall. W. B. and Price. P. H. Internetson betmeen a turbulent of heat transfer with viscous inertial-gravitational flow of a free convection layer and a downw ard forced flom Sy mposium liquid in verucal tubes. Teplu/i:da Fysnidh Temperatur.1968. Heat and Mass Transfer by Combined Forced and Natural Con. 6.931-937 vection. Inst. Mech. Engrs., Marchester.1971. Paper CII3.71 31 Herben. L. S. and Sterns. U. J. An expenmental investigation 54 Rousi.N. M. Innuences of buoyancy and imposed flom transients of heat transfer to water in film Gow. Constaan Journal of on turbulent convective heat transfer in a tube Ph.D. Thesis. Chemical Engenrerinti. 1968. 46, 401-412 University of Manchester.1987 32 Herbert. L. S. and Sterns. U. J. Heat transfer in vertical tubes- 55 Wilkinson. G. T Tsang. B. K..H., and Hoffman. T. W. Flow interactions of forced a nd free convection. Chemical Engistenne reversalin turbulent mised convection. Proc. 7th International Journef. 1972.4.46-52 Heat Transfer. Munich.1982. Paper MCl7
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la Ecken. E. R. G. and Diaguila A. J. Convective heat transfer heat transfer in the region of superentical state vanables of a for mised free and forced flom through tubes. Trans ASME. liquid. Ilrus Tranjer-Sneirs Rescurch. l973. 5. 49-52 l954 % . 497-504 58 Brassington. D. J. and Cairns. D. N. H. Measurements of forced 35 Hall. W. B ard Jackson, J. D. Heat transfer near the critical convective heat transfer to supercritical pressure helium. CEGB. point. Keynote Leoure. Proc. 6th International Heat Transfer CERL. Report R D/L/N/1975 i Conference.Toronio. Canada 1978 59 Byrne. J. E. and Ejiogu. E. Combined free and forced convection l 36 Jackson.J. D Hall. W. B Fewster.J Watson. A and Watts, heat transfer in a vertical pipe Symposium Heat and Mass 1 M. J. Heat uansfer to supercritical pressure Duids. UKAEA. Transfer by Combined Forced and Natural Convection. Inst. l Harwell. Report No. AERE R 8158,1973 I 37 shitsman M. E. Impairment of the he.t transmission at Mech. Engrt, Manchester.1971, Paper Cll8 71 j 60 Jackson.J. D Mixed forced arid free convection-the innuence ; supereniscal pressures. High Temperature. 1963.1.237-243 of buoyancy on heat transfer in veriscal pipes. Paper presented l 38 Ackerman. J. W. Pseudoboiling heat transfer to supereritical at Session V. Euromech-72 Conference. Salford University. I pressure water in smooth and ribbed tubes. Trans. ASME C.J. March 1976 Heat Tran.gfer, l970. 92. 490-49R I 61 Buhr. H. O., Horsten. E. A and Carr. A. D. The distortion of l 39 Vekhrev. V Baruhn. D and Kon'kov. A. S. A study of tu.bulent velocity and temperature profiles on hesung of mercury heat transfer in venical tulws at supercruical pr uures. in a vert cal pipe. Trum. ASME C. J. Hrur Trans/cr. 1974. 96. Trpl..wcrer:da. 1967, 14. 116-119 152-158 14 Int. J Heat and Fluid Flow. Vol.10. No 1. March 1989
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>414 AIA A 1.,1982. 20,436-437 w alklate. P. J. A comparative study of theorcucal models of 96 To W. M. and Humphrey, J. A. C. Numencal simulation of turbulence for the numencal predict on of be mdary layer flows. buoyani turbulent flom-l. Free convection alon8 a heated.
Ph.D. Thesis. Unisersny of Manchesect insutuie of Science and verocal. flat plaic. Inr.J.Heer Moss Transfer.1986.29.573-592 Technology.1976 97 De Lemos. M. J. S and Sesonske. A. Turbulence modelling m 76 Carr A. D Connor. M. A.. and Buhr. H. O. Velocity. combmed convecuon m mercury pipe now. /nr. J. Hear Mass temperature and turbulence measurements m air for pipe flow Transfer. 1985.28.1067-1088 wnh combined free and forced convection. Trans. ASME C.J. 98 Petukhov B. S. and Mediciskaya. N. V. Turbulent now and Heat Tron 4f er.1973. 95. 445-452 heat exchange in sertical pipes under conditions of strong 77 Deissler.R.G. Analysis of turbulens heat transfer, mass transfer ini'vence of upward forces. TiplofisiLa VysoleLk Temperatur. and friction in smooth tubes at high Prandtl and Schmidt 19M.16.778-786 numbers. NACA TN 3145.1954 99 Vlict. G. C. and Liu. C. K. An expenmental study of turbuleni 78 Wolfshtein. M. The velocity and temperature distribution in natural convection boundary layers. Trans. ASME C. J. Hrer one. dimensional flow wnh turbulence a ugmentation a nd preuvre Transfer. 1969. 91, 517-531 gradient. Int. J. Hrer Moss Transfer. 1969,12,301-318 Int J. Heat and Fluid Flow. Vol.10. No.1. March 1989 15
e Experimental, Variable Properties - D. L Siebefs Natural Convection From a Large, i S.--.-. e, . uvermore. Caut. Vert.ical, Flat Surface ! L Assoc. Mem. ASME " Natural convecnon heat transfer from a tertical, 3.02 m high by 2.93 m long, I electrically heat surface in air was studied. The air was at the ambient temperature V R F. M0ffatt and the atmospheriepressure. and the surface temperature was variedfrom 60 cuo 1 Dept or Mecn Engnneenng.
- 320 C. These conditions resultedin Grashofnumbers up to 2 x 10'2andsurface-to.
Stanfor0 Unuverssty, G ambient temperature ratios up to 2.7. Convective heat trartrfer coefficients were Stanfor0. Calst. Y measured at 103 locations on the surface by an energy balance. Boundary layer Mem.ASME mean temperature profiles were measured with a thermocouple. Results show that: Y (1) the turbulent natural convection heat transfer data are correlated by the ex. ~ t R. G. Schwind P'" " Nielsen Engineenng and Researen Inc.- **" Mountain view. Calit. Nu, = 0.098Gr f 7, 3L( T r o Mem. ASME when all properties are evaluated at T.; (2) varsable properties do not have a sogmficant effect on lamanor natural convection heat transfer; (3) the transition V Grashof number decreases with increasing temperature; and ( 4) the boundary layer
- mean temperaturue profiles for turbulent natural convection con be represented by a " universal" temperature profile.
Introduction An experimental study of the effects of variable properties vection u. =ases hm been examined experimentally and r on natural convection from a large, vertical, flat, electrically analytically by several aathors [4-7]. The general consensus is P heated surface has been conducted. The vertical surface was that the effects of property variations on laminar natural
- placed in air at the ambient temperature and atmospheric convection are smallin gases for the T./T. range studied te %
pressure and heated to temperatures between 60 and 520*C. date (0.25-4.0). Reference temperature methods are ! These conditions resulted in Gr, up to 2 x 10'2, and T./T. recommended by most authors to account for the small P ranging from 1.1 to 2.7. The objective of this paper is to variable properties effects. One author Hara (4), demon- I
. present the effects of variable properties noted in the ex- strated analytically that when all properties werc evaluated at L periment on the natural convection heat transfer from the T., the following relationship would account for variable k surface, on the location of transition from laminar to tur- properties effects in air bulent flow in the natural convection boundary layer, and on the natural convection turbulent boundary layer temperature Nu = Nu, profile. In addition, comparison of the turbulent boundary 1 -0.055 ( T* - T. )*.
T. (1) r layer temperature profiles to others in the literature indicate For those authors recommending a reference temperature j the existence of a " universal" temperature profile for tur- method. definitions of T,,, ranged from a film temperature - bulent natural convection. This work was part of a larger [6,7] Io that given in [5] effort to examine a range of heat transfer conditions, varying [ from natural convection, to mixed convection driven by T,,, = T* - 0.3 8(T* - T.) (2) orthogonal buoyant and inertia forces, to forced convection, in most reference temperature methods, #is evaluated at T.. all from the same vertical surface [1]. One exception is Clausing (7), who recommends that $ be i interest in variable properties effects on natural convection evaluated at T . This recomendations was based on ex-f stems from the desire to predict more accurately the heat loss perimental laminar heat transfer data taken at cryogenic , , by convection heat transfer from the external-type receivers temperatures, where T /T. ranged from i.0 to 2.6. ( { used in large-scale solar central receiver power plants. Variable properties effects on turbulent natural convection Convective heat loss from receivers can occur in a natural, in gases have also been examined by several authors [6-8). mixed, or forced convection mode. In the natural convection The numerical predictions of Siebers (8] for natural con-mode, the convective heat transfer from an external-type vection from a vertical surface in the T./T. range of 1.0 to receiver, essentially a large vertical, high temperature surface, 3.0 with T. at 20'C showed that evaluating the properties. l is complicated by variable properties effects as a result of the including 4, at T f did not correlate the turbulent natural high operating temperatures. There is little information in the convection heat transfer predictions in terms of Nu and Gr. I literature concerning variable properties effects on natural Pirovano et al. [6] correlated their turbulent natural con-convection, particularly in the turbulent flow regime. vection experimental results in air in the T /T. range of 1.0 to 1.5, with a T,,, heavily weighted toward T. and with # Literature Review evaluated at T. Pirovano et al. defined T,,t as The effects of variable properties on laminar natural T,,t - T. + 0.2(T. - T ) (3) convection in liquids have been reviewed in two recent articles [2, 3]. Vanable properties effects on laminar natural con- If Siebers [8] had used the recommendations of Pirovano et al. to correlate his numerical predictions of turbulent natural Contr buied by the Heat Transrer Division ror publicanon in the Jovam or convection heat transfer, the results, when expressed in terms Han Twarsa. Manuscnpt remved by the Heat Transrer Division May 2s. of Nu and Gr, would have agreed closely with the ex-19ss. perimental results of Pirovano et al. Clausing [7] used Tj with 1241Vol.107, FEBRUARY 1985 Transactionsof the ASME
w T I
,,,, ,,.uu . 7,.= An open-return, draw through wind tunnel,18 m long with
'- n am a 4.8 m wide by 4.3 m high inlet, was built for these ex-
- q. s . ,
s - periments (Fig.1). The inlet nozzle to the test section has a
- 1. . 5 ic. I two-dimensional 3:1 contraction. The heat transfer test
# O-
') surface,3.02 m high by 2.95 m long,is mounted in a sidewall c . l y .Ji _, of the test section. The test section is 4.3 m high,1.2 m wide, 1 bigiiM '
and 4.3 m long. The walls of the test section facing the heat l 9 h, transfer surface are w ater cooled to tne ambient temperature.
~
A small amount of boundary layer suction along the top of l : '.Z' .,7 ,,,, .* Po 9 7 .. Z.u. the test surface is used to ingest most of the hot, inner region
' ' ';*74',"'
. **"" * : ",'."j.".T*;"' ',";'
- of the boundary layer that leaves the top of the test suriace,,
thereby preventing stratification in the tunnel. The suction rp senernatic ot the apparsius. including the wind tunnel. the test rate was adjusted using velocity measurements and smoke , ecs, and the bouncarv inyer traverse (+ 's represent heet tranater Gow v sualization. The bottom leading edge is defined as the
* ** * ""'"'"*"" **" " horizontal line where heating starts. Below this line are an additional large temperature-dependent correction to another 6 cm of smooth unheated surface; the struts that arrelate the turbulent natural convection data obtained at support the boundary layer traverse 30 cm off of the test .xyogenic temperatures. He did note, however, that using T. surface extend through this unheated surface. Figure I shows 2s the reference temperature for evaluating the properties the boundary layer traverse centrally located oser the test ,1&nificantly reduced the size of the temperature-dependent surface. The traserse was used in some tests to take boundary arrection needed to correlate the data. Most basic heat layer mean velocity, temperature, and now angle profiles.
- ransfer textbooks generally recommend a T,,, gisen by The traverse could move a boundary layer probe sertically or ,
quation (2) or Tf with 6 esaluated either at T. or at T.,, to horizontally over the test surface, rotate a probe about an axis ~ account for variable properties effects in both laminar and normal to the test surface, or move a probe normal to the test I urbulent flow. surface. The struts supporting the traverse had no effect on The review of the literature shows that here are conflicting the vertical boundary layer flow in the natural consection . ccommendations on how to account for variable properties mode. I effects on turbulent natural convection heat transfer in gases. The test surface consisted of 21 horizontal strips of 304 I These recommendations range from using a T,,, weighted stainless steel foil stretched tightly over a slightly crowned toward T., given by equation (2), to a T,,, weighted toward surface of high-temperature insulating material 5.38 cm thick T., given by equation (3). Basic heat transfer textbooks (Fig.1). The strips were heated electrically using a c current. generally recommended a T,,, gisen by equation (2) or Tf to Each strip was made from a 14.43 cm wide by 0.13 mm thick account for variable properties effects on turbulent natural piece stainless steel foil. Along the entire length of both edges convection heat transfer in gases, but recent experimental and of each strip,0.32 cm of the 14.43 cm width was tightly folded numerical works [6-8) point to a T,,, weighted to*ard T. under, creating a strip with a net width of 13.79 cm. The 21 with S evaluated at T.. The differences in these various strips were mounted horizontally with a 0.64 cm wide gap recommendations are significant when there are large tem- between them to assure electric isolation of the strips from perature differences across the boundary layer. one another even with thermal expansion of the strips when heated. Folding the edges of the strips under was done for two Apparatus and Instrumentation reasons: first, it helped keep the strips flat on the surface (l] The following is a brief discussion of the apparatus and and, second, it provided additional electric power dissipation mstrumentation. Further details can be found in [1]. Much of (4 percent of the total strip power dissipation) along the gaps the appiratus was designed for the mixed. and forced- between strips to exactly compensate for the unheated gap convection experiments and is described here only to com. area (4 percent of the total area). pletely describe the environment for the natural convection Surface temperatures on the test surface were measured at ' expenments. 105 locations, each equipped with a rosette of three ther. Nomenclature layer (24), (q./oc, ) F3 (gS6) -' 3 n, = length scalar for the inner g = gravitationalconstant, m/s2 y = verticaldirection, m region of the boundary layer Gr, = Grashof number, gS(T. - t = distance normal to the wall, (24), (o2 /gS( T. - T. ))3 T )y /r2 3 cm gl = modified length scalar for the ha convec'ive heat transfer Greek inner region of the boundary coefficier.t. W/m:C a = thermaldiffusivity, m2 /s leyer, 1D = testindeniification number C = coefficient of volumetric k = thermalconductivity, W/mC k. CH a = exponent on the sariable expansion, K -' 9'(k")T,." 0 = dimensionless temperature, properties correction Nu, = Nusselt number, hy/k T. - T Subscripts Nuj = Nu ( T./ T. )a, n = 0.14 for ( T. - T. c = condaction heat transfer turbulent flow, n = 0.04 for 6 = Length scalar for the outer cp = co'istant propertiu solution laminar flow region of the boundary layer e = electric power Pr = Prandti number r/a 6, = thermal boundary layer /= film tempeinture q = heat flux, W/m2 thickness, r = radiation heat transfer T = temperature. *C ref = reference temperature To = temperature scalar for the w = wall or surface temperature outer region of the boundary *(TT-. -T* 0 T.)gg cm = ambient condition Journalcf Heat Transfer FEBRUARY 1985. Vol.1071125
?
mocouple jurictions arranged equally spaced on the cir, cumference of a 7.5.cm-dia circle. The three thermocouple Several checks were made on the uniformity of and'A the junctions of a rosette were connected electrically in parallel so measurement of the electric power dissipated (l]. One irn. ; that the average temperature at the three junction locations portant check confirmed the accuracy of the voltage and th f was sensed. The thermocouple rosettes were located un. current measurements used to determine q,. In this check,the } } derneath the stainless steel heating strips, electrically insulated resistivity of the stainless steel as a function ' of temp_ from the strips by a thin layer of mica. The 105 thermocouple determined itom the strip dimensions and ' .N from rosettes en the test surface formed a staggered array with a 10 voltage and current measurements made during actual vertical columns, as shown by the "+ 's" in Fig.1. Columns transfer tests was compared with direct measurements ofb.the <
), 3, 5, etc., had 11 measuring stations, open on each of resistivity of a sample of the 304 stainless steel heated in an i heating strips 1, 3, 5, etc.; Columns 2, 4, 6. etc., had 10 oven and with resistivities reported in the literature for 304 [j '
measuring stations, located on heating strips 2,4,6, etc. stainless steel. within 1 percent The independently measured resistivities agreed 4 An extensive program of analysis and bench testing was over the temperature range of the ex. periment. There was also similar good agreement between the
, [
conducted on prototypes of the electrically heated test sur. - face. The analysis and tests verified that the surface ther. measured resistivities and those reported in the literature. M mocouples installation would measure T, (in*C) within
- 1 The radiation heat transfer from the surface to the b tun percent, even though the thermocouples were not welded to wallswas q,, determined from a three tone, diffuse, gray
- the strips [1]. body radiation heat transfer model. The zones in the model The natural convection data were taken only on calm nights were (i) the tunnel walls surroundmg the test surface (which " .) t' since the tunnel was outside. A hatch in the roof were of thewater cooled), (ii) the local spot on the surface where A tunnel was open and the top. edge suction was operated to mimimize was being measured, and (iii) the remainder of the test sur.
face. Shape feetors between the zones were: 0, I, or an area {8 % stratification in the test section. Flow visualization with I
' ?
smoke showed that the air in the test section was essentially at ratio in the case of the tunnel wall to test surface shape factor. 5 tt rest. Except for the boundary layer on the heated test surface, The temperature boundary conditions for the three zones were *
*4 respectively (i) the average tunnel wall temperature deter. '
there was no discernible flow. The stratfication inmined the test from the measurement of the tunnel wall ternperature
,t!
section for the natural convection tests was small, ranging at s: from less than 1 *C from top to bottom of the test surface for 35 locations with thermocouples, (ii) the local surface ,( low. temperature tests to 10*C over the top half of the test temperature measurement on the test surface where h was - il surface at a T. of 320*C, (The bottom half showed no being measured, and (iii) the average test surface temperature .t stratification for any tests.) For tests where stratification measurement calculated from all 105 surface temperature I #' occurred over the top half of the test surface, the local T., at a measurements. The tunnel wall temperature rarely exceeded
- given elevation was used in the data reduction. 40*C at any spot, since it was water cooled. As a result, the ! g' 5,'t average tunnel wall temperature was always near ambient and Heat transfer coefficients were calculated at each of the 105 not an important factor in determining q, in equation (4). f'*
surface temperature measurement locations on the test "C surface on the basis of an energy balance The emittance of the stainless steel test surface was lo determined from measurements of the normal, spectral h = b "' "' (4) reflectance of the 304 stainless steelin the 2.0 to 25.0 micron
, 6811 Ali where ( T. - T. ) wavelength range at up to 70 wavelengths with a paraboloid B reflectometer. These measurements were integ".ted over the
{ q, = electrical power dissipated [. appropriate Planckian blackbody distribution for each q, = net radiation heat transfer from the front of the test temperature considered and subtracted from one to give the I i
"I"" '
a normal, total emittance. The hemispherical, total emittance li q, = conduction heat transfer through the back of the test was then obtained from the relationship that exists between mface J ti the hemispherical, total and normal, total emittances of 9 The data reduction was done on a minicomputer system which smooth metal surfaces found in (9]. These measurements of I.h acquired and then reduced the data on-line. For natural hemispherical, total emittance agreed very well with previous e t-convection tests, q, ranged from 14 percent for the lowest T.measurements of emittance for the same material by other P
' 11 cases to 44 percent of the electrical power dissipated for the researchers (10,11) and with the theory for smooth metal *C highest T. cases, while q, was typically 8 to 14 percent of the surfaces found in (9). Also, the measured temperature ' '
8 't. electrical power dissipated. dependence agreed with the temperature dependence given by I kh The electric power dissipated per unit area at a temperature the smooth metal surface theory. The measuerments show theCl . measurement point on a given heating strip, q,, was equal to emissivity is 0.13
- 0.01 at ambient temperatures and "
h the average electric power dissipated per unit area for that 0.22
- 0.01 at the peak temperatures of the experiment. No L strip. This resulted from the uniformity of the electric power detectable effect of material aging during testing was tecorded s "i t dissipation by each heating strip, which in turn resulted from for test conditions reported in this paper. Measurements of ' ;
k 'c ! the uniformity of temperature (i.e., uniformity of resistivity) the emissivity over the temperature range of the experiment ? on each horizontal heating strip for the natural convection were made on samples of the stainless steel from before, mode of heat transfer and the uniformity of the heating strip during, and after testing. . 7%. 5 ; thickness, t l percent. The average electric power dissipated "The conduction through the insulation on the back of the I ) i per unit area on a heating strip was determined from the ten surface q, was determined at each of the 105 surface . product of the measured RMS voltage drop across the length temperature measurement points by applying a one-of a heating strip and the measured RMS current through a dime.asional conduction model between two locations through h '/(i t
.c !
beating strip, divided by the surface area of the stri" 'in- the 5.38 cm depth on the it.sulation. The two locations were (i) ciuding the folded under edges) between the vcLage the surface temperature measurement point and (ii) a point - 8h I. U measurement points. The small effect of thermal expansion directly behind the surface temperature measurement point - D on the surface area of a heating strip, which caused a 1 where a thermocouple was located between the two layers of percent increase in surface area for the highest temperature insulation that made up the insulated back of the test surface. ,, [M 8 case, was taken into account. The uniformity of electric Variation of the thermal conductivity of the insulation with power dissipation for all the 105 surface temperature temperature was accounted for in the one-dimensional model. '-[ measurement points was
- 2.5 percent. A numerical, three-dimensional conduction analyis of the h # I' P insulation showed that the one-dimensional conduction model -
- i 126IVol.107, FEBRUARY 1985 r .-
Transactions of the ASME _ a
M uniform heat nux surface. The transition from lammar to . turbulent flow occurs at Reynolds numbers noted by the d r '< rM-5 '0
- b j y.
pr others in the literature. The forced convection baseline
'g boundary layer velocity and temperature profiles show similar good agreement with accepted forced convection profiles for 9 both laminar and turbulent flow on a flat, uniform heat flux i'
)
,d r g.o .o c,'s Ronge of surface.
3 C*g,Q n The natural convection baseline heat transfer results are j shown m Fig. 2. Two natural convection baseline heat
' transfer tests were taken with T., equal to 60*C and 128'C l and T. equal to 15'C and 18'C, respectively. The baseline q
, , y, !
g ; results are plotted in Nu, versus Gr, coordinates in Fig. 2 with
., 3,3 y ,,
>= ce us. is 4 Quid properties evaluated at a T,,, equal to T . (The choice of "g
3, l:- T,g for these low. temperature difference tests has sery little j 2 effect n the data in Fig. 2). The 21 points shown for each test d 10' 10' 10' 10* 10' 10' 10 are each the average of the five data points along a horizontal : Gr Y heating strip (see Fig.1). The RMS variation of the five data t convecuan tuteline ruutts-Nusselt number ursus points along a heating strip was 4 percent or less. j The solid line in the laminar flow region in Fig. 2 (Gr,
< 10') represents accepted constant properties natural l was accurate to within 1 percent for areas of the 3 m by 3 m convection et.rrelation for laminar flow on a vertical, uniform '
,est surtcc 8 cm away from its edges. The calculated one- heat flux surface in air [131 dunensional conduction loss through the insulation for the { Nu, = 0.404Grf' (5) , .hermocouple locations on the top and the bottom heating . trips, which were within 8 cm of an edge of the test surface in the turbulent flow region (Gr, > 10*). three lines are
- 10 of the 105 locations), was increased by 5 percent, based on shown. The two solid lines represent the range of correlations
- nc results of the numerical, three-dimensional conduction in the literature for turbulent natural convection from a i analysA vertical surface. The dashed line is a "best fit" line through I The values of h obtained by equation (4) are reliable within the data in the literature, represented by a6 percent for the baseline cases and the low-T. cases and N u , = 0.096G r "'
' (6) within cl0 percent for the hi2h-T. cases. The uncertainties cere calculated following the method of Kline and Me. Equation (6)is the correlation recommended by Churchill and .
Clintock [12), using observed values for the individual un. Chu (14] for turbulent natural convection from a vertical certainties in the input terms. The uncertainty analysis was an surface with the coefficient 0.096 evaluated for a Pr of 0.71. important part of the experimental planning process and The equation is based on their survey of the then current data served as the principal criterion fcr choosing among alter. (1975). The data at that time included only low temperature native measurement techniques and for setting the standards difference data with minor variable properties effects. for acceptable accuracy on the individual measurements. Figure 2 shows three main points. First, the agreement Boundary layer mean temperature profiles were taken with between baseline laminar data and the accepted constant a 0.052 mm type-K thermxouple probe [1]. Natural con. properties correlation is good. Second, the baseline turbulent vection velocity profiles, taken with a pressure probe, were data lie within the range of correlations appearing in the used in making a correction to the thermocouple readings. literature and agree very well with the "best fit" correlation The correction accounted for errors caused by radiation of Churchill and Chu. Third, the transition location for the between the thermocouple, the hot test surface, and the cold baseline data agrees with other data in the literature that show tunnel walls, as well as conduction heat transfer down the transition between a Gr, of 10' and 10*. This close thermciccuple stem. 'Ile model used for the radiation agreement between baseline data and the data in the hterature correction is similar to the three zone model used for the qualifies the apparatus and instrumentation. In addition, the radiation correction in equation (4), except that a fourth zone close agreement shows that the small horizontal gaps between has been added to the model to represent the thermocouple. the 21 heating strips have little, if any, effect on the results. The corrections made were small. The boundary layer mean The lack of effect was expected, since the total area of the temper:sure data are reliable within the larger of
- 2*C or e 4 gaps was small compared to the heated area of the test sur-percent based on the same method of uncertainty analysis that face, a small amount of additional heating to make up for the was used to determine the uncertainty of h. The measurement unheated gaps was provided along the gap edges by the folded of distance normal to the test surface for the boundary layer under strip edges, and the depth of the step created by the gap profiles was accurate to within
- 0.12 mm. (* 0.13 mm) was small compared to the boundary layer thickness, which was as large as 15 cm.
Apparatus and Instrumentation Qualification in addition to the baseline tests, a comparison was made To qualify the apparatus and instrumentation, baseline between the heat transfer coefficient as determined by (i.e., low wall to-ambient temperature difference) heat equation (4) at a given location on the surface and one which transfer tests were taken in both the forced and natural could be determined from a boundary layer temperature c nvection modes and compared with accepted Dat plate, profile meuured at that same location by the following uniform heat Dux, constant properties correlations appearing relationship in the literature. Also, baseline boundary layer profiles of h = de (7) temperature and velocity were taken for forced convection dr and compared with accepted profiles appearing in the This comparison was possible for 43 cases. These were cases hierature for now on a flat, uniform heat Oux surface. The which had boundary layer temperature profiles with tem-forced convection baseline results are presented in [1]. The perature measurements well into the viscous sublayer of the forced convection heat transfer results show excellent turbulent boundary layer or the inner region of the laminar agreement with the accepted forced-convection heat transfer boundary layer (# < 0.2), where equation (7)is valid. Two of correlations for both laminar and turbulent no.v on a flat. the cases were natural convection profiles, and 41 were forced Joumalcf HeatTransfer FEBRUARY 1985, Vol.107 /127
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Fig. 3 Effect of variable properties on natural convection from o verucal surf ace in air Fig. 4 Correlation of variable properties effects on natural convection J llVC8, D from a vertical surfsee in air t Thes or mixed convection profiles taken as part of the forced and boundary layer, the heat transfer coefficent is less than would mixed convection experiments also conducted with this ap- be predicted by the low-temperature difference constant.
- vection paratus The profiles were for the T. In the 200*C to 580*C properties correlation with all properties evaluated at T.,
range. The average ratio of the measurement of h from in the laminar region (Gr, < $ x 10')in Fig. 3, there is a j'~ O equation (4) to the measurement of h from equation (7) for small decrease in Nu, for a given Gr, as T,, increases. The Th'e teri I the 43 cases was 1.008 with a standard deviation of 14 percent. decrease in Nu, with increasing T. is most clearly shown by *The The closeness of this ratio to 1.0 is a check on the consistency the lower Gr, data point for each T. As T. increases. the tiansfe. of theindependently measured surface heat transfer data and lowest Gr, data point for each T,, lies progressively farther P boundary layer temperature profiles over the entire tem- below the uniform heat Oux correlation given by equation (5), ' d' perature range of the experiment. Other consistency checks the upper line in the laminar region. + D are presented in [1]. The decrease in Nu, with increasing T. in the laminar region is not totally due to variable properties effects as was {h ter Results and Discussion
- 0.356 i the case in turbulent flow. The decrease in Nu, is partly due to Heat Transfer. Variable properties effects on natural a change in the heat transfer boundary condition on the test bhe piict convection heat transfer from an electrically heated, vertical, surface as T. increases. At low temperatures, the surface has 3.02-m-high by 2.95 m-long surface were studied. The a uniform heat Dux, as described earlier. At high tem. I I
Ec e ambient temperature was approximately 20*C. Average peratures, the surface approaches a uniform temperature as a g), values of T, that were considered were 60,128,222,349,424, result of the radiation heat transfer from the surface. For the 477, and 520*C. These temperatures resulted in T,/T. ratios highest temperature case,520*C, the test surface is within 2 bf ects from t.1 to 2.7. Figure 3 shows the heat transfer results in percent of a uniform temperature surface in the laminar and transition regions. The correlation for uniform surface hpgot di terms of Nu, and Gr, with all properties, including #, evaluated at T. for each test. The 21 data points shown in temperature is the lower line in the laminar region in Fig. 3
, 4 4p
- The Fig. 3 for each surface temperature were obtained in the same [15]. A combination of variable properties effects and a tonvet manner as those for the baseline cases shown in Fig. 2. For all change in boundary condition are indicated by the fact that hevt tests except test ID $85, the test surface was at steady state the highest temperature laminar data point lies about 8 conditions. Test 585, which had to be stopped before steady percent below the uniform surface temperature relationship.
g I,, , ,g lj g state was reached, has a transient energy storage correction Unlike the situation in the laminar region, a change in 4g equal to -5 percent of electric power dissipated. This boundary condition does not occur in the turbulent region. A h ., arine correction accounted for energy being stored in the insulation. vertical surface with a turbulent natural convection now over p 'jdnpc The correction mas based on measurements of the insulation it is simultaneously a uniform temperature and a uniform heat F gier temperature taken over a 10 min interval and was made along flux surface (i.e., h is spatially uniform for a given T.). ; peran with the radiation and conduction corrections in equation (4). Churchill and Chu's uniform temperature and uniform heat In the turbulent region (Gr7 > 10) in Fig. 3, the data flux correlations based on "best fits" of data in the literature i4 t,7 Pirov. shows two important points. First, there is a small decrease in are only different by 2 percent for a Pr of 0.71. The difference p h Nu, for a given Gr, as T. increases, when properties are is well within the uncertainty of the data available. g. jperat evaluated at T.. Second, each data set for each temperature When all the properties are evaluated at T., the variable remains parallel to the low-temperature difference correlation properties effects noted in Fig. 3 for turbulent natural con.
! Ages given by equation (6) from Churchill and Chut the seco.:d vection in air are accounted for by the following relationship h%Ta' . prop point in most visible for test 585. This latter point means that Nu, remains dependent on Gr8 in the turbulent flow region T. [ 3Fer with increasing temperature clifference across the boundary Nu, = 0.098Grf3 ( T= ) -0 H (8) h yeth
. ,predi layer, or in other words, the heat transfer coefficient remains This equation is based on a "best fit" of the natural con- / b7, uniform in the turbulent region. Only the coefficient in vection data in Fig. 3, where the 1/3 power on Gr was 1 i equatior (6) is changing with increasing temperature. It assumed as a result of the lack of dependence of A ony(shown goor decreases from the 0.096 value for a small temperature dif- by each set of the heat transfer coefficient data for each T.). >
pop q ference across the boundary layer to a value of 0.08 for the The coefficient in equation (8) is 2 percent higher than that in h vall 520 C test, a decrease of 15 percent. This decresse does not the baseline equation, equation (6), and a temperature ratio imply that the turbulent natural convective heat transfer correction has been added to account for the effects of h abcm
- M Fc coefficient at a fixed location is lower for higher T.. It only variable properties. Figure 4 (where the Nusselt number, Nul, indicates that, as a result of property variations across the is defined so that it includes the temperature ratio correction
, yL [tiyes f,pera' 128 f Vol.107, FEBRUARY 1985 Transactionsof the ASME 1 {
- l l 0l l I
r' h g A comparison of recommendations for evaluating that methcd to h predicted by equation (8) lor three different
'. . le propertks effects on turbulent natural convection values of T,, (300,600, and 900'C) and a T of 20*C. The he % g.C). first two metnods listed [6. 7] are recent recommendations r,. L reported in tne literature based on experimental data. The g [6] 17] (16) (17] (18) next three [16.17 18) are recomendations reported in most Eq.(2) 9" g E4. 0) Tr Tr T,-
basic heat transler textbooks. All of the methods except 1/ T. t/Tf I / T. 1/ T, I / T. Clausing's, the second method, are reference temperature J(g g :,= h/h predicted by Eq. (8) methods. Clausmg evaluates all properties, including d, at T, r7c) and makes a large correction based on T,,/T, to account for 1.03 1.30 0.95 0.82 0.90 variable properties effects. The correction is gisen in Fig. I of
% .o.: i.17 0.90 0.88 0.71 0.65 0.87 0.82 l
l.06 1.10 [7 he constant properties Nusselt-Grashof number ( rin)n the definition of Nusselt number) shows that equation correlation to which all the recommended methods presented in Table I are applied is given by equation (8) without the r gaccounts for the small systematic decreases in Nu, with (T,/T.)"" term. The correlation represented by equation I" peasing temperature, which appeared in Fig. 3. The tur- (8) without the temperature ratio term is not significantly I data collapse to within 6 percent of the correlation different than the constant properties correlations used in by equation (8). This percentage is within the un- each of the references in Table 1. The only difference is in the gf ty band on the heat transfer data. coefficient (0.098 in equation (8)) in each correlation. Ap-fi ' e variable properties effects on turbulent natural con- plying all methods to one constant properties correlation f ~.pction could also hase been accounted for by evaluating all allows the different methods of handling variable properties
] : properties, except d in Gr,, at a T,,, defined as: to be compared, without introducing the small differences in b
T,,, = 0.3 7,,. + 0.7 T. the coefficient of the constant properties relationship that is f'- n (9) I . recommended in each work. Effectisely, this means that 1 : ' term B should still be evaluated at T.. Table 1 compares with relatise trend in the sariable properties Jq e correlation for the /ammar natural convection heat effects predicted b) each method for increasing T,, with that fer data, with all properties for air evaluated at T.,is predicted by equation (8). l3 -7 Table I shows that the first method, from Pirovano et al. f T,, F.i v, ' Nu, = aGr,"( )s -m (10) (the most closely related experiment to the work described in this paper), predicts substantially the same s ariable properties
' term a equals 0.404 for a uniform heat flux surface and effect as equation (8) for all temperatures. This reference
,h Q$6 for a uniform temperature surface, the values reported temperature rnethod uses a T,,, heavily weighted toward T.,
gthe literature. Equation (10) agrees with numerical given by equauon (3), with S evaluated at T,,. It is based on P' gdictions made by Siebers (8) for T,/T. < 3.0, with the an experiment with values of T. up to 150'C. The second l pc detailed laminar data for T,/T. < l.5 from the ex. method in the table, the method recommended by Clausing 9 fnents of Pirovano et al. and with various analyses for air based on data taken at cryogenic temperatures, predicts 30 q i Q]. Equation (10) demonstrates that variable porperties percent higher heat transfer at 300*C, but only 10 percent
' Tects on laminar natural convection heat transfer are small. higher at 900'C. This trend indicates that his method predicts 4e difference in h calculated from equation (10) with and a different sariable properties effect on turbulent natural l: [thout the T,/T. corrections at a T,/T. of 3.0 would only convection with increasing T, than equation (8) does. The last pyy d percent, three methods in Table I-the textbook methods-predict
, ibThe effects of variable properties on laminar natural progressively lower heat transfer coefficents with increasing g pouvection heat transfer can also be accurately accounted for T.. This trend is particulary true for the fourth method.
'. ':.ty cvaluating all properties except S in Gr, at T,, as noted by where d is esalauted at T, along with the rest of the
$ ,.5,, arrow and Gregg [5] and Pirovano et al. The term J should properties.
be evaluated at T., as in turbulent flow. The disagreement between the methods recommended in j[ ,The trends noted in the turbulent natural convection region this work and the last three methods is most likely explained
, _ new. No experiments were found in the literature for high- by the fact that these recommendations are based on forced
{' perature flows on a vertical surface with large temperature convection experience or on a laminar flow natural con-i hfferences across the boundary layer. The highest tem- vection analysis by Sparrow and Gregg [$]. No turbulent t Perature experiment with turbulent flow in gases was by natural consection data with significant variable properties p'
+
vano et al. with temperatures up to 150*C and T, /T. < effects are asailable. Recommendations based on forced convection experience should not be espected to work a priori i I h, o.The one experiment ss a turbulent boundary layer was that had large at cryogenic tem- temperature variations for turbulent natural convection. Similary, recommendations
- paratures. Clausing, who conducted this experiment, noted based on Sparrow and Gregg's lammar flow analysis should erent trends, not be expected to work a priori for turbulent natural con.
k ble I shows a comparison of the effects of variable vection since, first, the analysis was for laminar natural u'
. rues on turbulent natural convection heat transfer f rom convection heat transfer and, second, the variable properties
@crtical surface in air predicted by various recommended effects on which they based their conclusions in that analysis 4 pthods to account for those effects. In the table the effects- were only a few percent for the realistic gas models and eted by five recommendations appearing in the literature temperature ranges studied (330 K < T < 1000 K, T./T. <
h.,h7.16,17,18] are compared with the effects predicted by 3.0). For example, if an h for air predicted using T, as the ggthe method proposed in this paper. The present work reference temperature in the laminar heat transfer C 77 eBoommends evaluating all properties in the constant relationship, equanon (5),is compared to an h predicted with 1,. "Moperties heat transfer relationship at T. and making the the same relationship using Sparrow and Gregg's recom-I tll-to-ambient temperature ratio correction, (T /T.)d ", mended reference temperature, given by equation (2), there
'4.. ;pabornin ,t equation (8). would be less than a i percent difference for T at 600*C and ga Nor each of the five methods from the literature, the table T. at 20'C ( T,/ T. - 3.0). The reasons for the p 8tves the T,,, at which properties are evaluated, the tem- disagreement with the results of Clausing are not clear at this
, ute used to determine B, and the ratio of h predicted by point. One possibility is that variable properties effects on 4
alof HeatTransfer F EB RU ARY 1985, Vol.107 /129
l a e 7 l 3
' t2 ,
e or L d r j, i l b'o, ,(*,n("E m,nrnum n . Tref =Tr
- . a
- 4 -
, :.. . thform 7. S= 0.08 h(2/6 t) + 0.81 **
r : -
- F - ~ r sl y ' n * :. -
6 ' Y ; _Turovient
, i 9 \
[ l Tronsitiond ' h 06 3: 1 I
~8 y
n rns w ~ l 1.(C) r.(c) cr,(siO4) y(m) ro., ~
/ 2::!!:!ti :H i!! ::: 9 i u - !
e
- * ::l0 !!:: ?E != ::: c:
too t.25 *50 US 2.00 2.25 2.50 .?S 3 00 0.0 1:!!! Rt !!! Nt ::: g;;' i T*/T 7 i Fig. 5 Effect of welltemperature on the trs9sstion Grashof numt> erin 2# io" io' ' fiatural convection from a vertical surf ace in air [dj!d g d Fig.8 ary la yer on a vertical surf aceTemperature profHes in a tutt> turbulent natural convection are'different for different Guids ; in significantly different temperature ranges, Transition. Figure 5 shows the effect of T,, on the natural equal to 820*C. The corrections to the te.nperaturc # convection transition from laminar to turb'ilent flow for fixed measurements accounting for errors caused by radiation' heat T . The figure is a plot of the logia of the Grashof number transfer and conduction heat transfer (discussed earlier based on y versus T /T . The length y, is the location of small. ranging from + 10*C near the wall to -2*C at the l either the minimum h or the maximum h. The location of the outer edge of the boundary layer. In the following figures,the minimum h is the point where h begins to deviate from the uncertainty m :. =0.12 mm, results in a
- 15 percent un.
laminar h value. The location of the maximum h is the first ecrtatnty in tne nohdimensional distances used in the figures point at which h equals its fully turbulent value which is a for the point closest to the wall for each profile from this constant. The resolution in determining these focations is plus esperiment. This uncertainty is inversely proportional to the ' or minus the width of one heating strip. Uncertainty bands distance from the wall. s based on this resolution are shown in Fig. 5. The upper value The temperature profiles are shown in Fig. 6, which is a for Gr, at each T /T. corresponds to the location of plot of dimensionless temperature 8 versus distance normalto maximu,m h; the lower value corresponds to the minimum A the wall: disided by 6,. Also shown are turbulent bo location. The zone between the minimum and maximum h is layer temperature profiles from other works for whic defined here as the " transition zone"(the crosshatched data were availablearea).to plot. Two of the profiles shown were All properties in this figure are evaluated at Tf with the ex. taken in air by Cheesewright and two were taken in water by ception of $. which is evaiuated at T.,. Film ter.iperature was Chokouhmar.d [20). For Cheesewright's profiles. T. e used largely because transition starts in the laminar boundary 84.5 and 57.2*C. For Chokouhmand's profiles. T. equals , layer where one way to correlate heat transfer data is to 38.9 and 32.5'C. The high temperature profiles from this 7, evaluate properties at f T , as discussed earlier. The solid lines experiment compare well with the profiles for air and wate conne', ting the data points from this experiment are for visual lower temperatures taken by Cheesewright and Cho refen:nce only. mand, respectisely. The close comparison of Cheesewright's Figure 5 shows that T. has a significant effect on the Iow-temperatture profiles for air with the high. temperature stability of the boundary layer for a fixed T.,. As T./T,, profiles from this experiment indicates that there is very little increases, the Grashof number at which transition occurs, effect of variable properties on the turbulent natura Gr,,, decreases significantly up to a T./T. of 1.75. Fur. vection temperature profile shape. All of the profiles in Fig.6 . thermore, the size of the transition zone, in terms of the show a " viscous" sublayer or " linear" region near the i difference between Gr /6, < 0.2, and a " logarithmic" region in the outer region of at the minimum and maximum h locations, decreases up,to a T./T. of 1.75. This decrease in the boundary layer, x/3, > 1.0. The temperature distrib size is also true in terms of vertical distance. However, the in the " logarithmic" region is given by ratio of Grashof numbers at the minimum and maximum 4 8 = 0.081n(c/6,) + 0.81 (11) ' locations remains fixed at approximately 5.0. These results ' Equation (11)is represented agree closely with the results of Pirovano et al. for T./T., up .. l to 1.5. The very low temperature ratio case agrees with the Comparisons were also made with turbulent boundary layer results of Cheesewright (19]. temperature profiles from other experiments for which - tabular data was not readily available for plotting (21-23]. In ! Beyond a T./T., of 1.75, transition zone size and location ! ; in terms of Gr the outer region, the " logarithmic" region, the data from the believed due toappear a$oss fixed. This apparent of resolution in locating trend the is transition experiments of Warner and Arpaci [21), Vliet and Li and Fujii,123) agree very well with the data in Fig. 6. In the zone. By T /T. = 1.75, the transition zone has moved down inner region, f r t/6, < 0.6 the temperature Q, p to the first three heating strips on the test surface (see Fig.1) thesc same references agree m, terms of their shape, but there and is occurring over a very short distance ( = 1 strip). On the is more scatter than is indicated ,
,j basis of the resolution of the transition zone location, ac- proximately
- 40 percent at z/3, equal 10 0.1 The reasons for . j curate location of the transition zone is impossible when the scatter are not clear. One possible reason ,
T./T. exceeds 1.75. It is only clear that the transition zone effect. The experiments cover t ajPr nu does not move upward on the surface for T./T. > 1.75. The agreement of the turbulent profiles from the vanous q Temperature Profiles. Boundary layer mean temperature experiments strongly suggests the existence of a " universal . ;, profiles were taken for turbulent natural convection forExistence temperature a T. profile for turbulent natural convection. k3 q l of a universal profile has not been clearly 1301Vol.107, FEBRUARY 1985 , i j (7 Transactionsof the ASME j {
4- ' 0 The important points shown by Fig. 6 and equation (11) l d
. are: (i) there are no significant variable properties effects on i the temperature profile shape, (d) the temperature in the outer 10 #_ ,
# region appears logarithmically dependent on z. (h) 6, appears
. to be a good length scale for the outer region, and (iv) the c.s 6 = 0F W) .
profile appears to be independent of Pr in the outer region as suggested by George and Capp in their theory. This last point l appears true r the Pr range of 0.7 to 33 based on (21-23). 6 = t36-142 (Z/7h').y3 l
. ko.5 / The temperature profiles for air from both this experiment i
. / and Cheesewright's experiment (the same cases as shown in I 'M -
Fig. 6) are presented in Fig. 7 in terms of the inner region 1
! ,. .I:I ai c rdinate :/nl. The term glis a modified form of the inner !
wo _ :NI .., .o-us
- a I c.2 426 2s 2 32 io- n -
region length scale of George and Capp. The modification. l p'=l U ! N I 'E j $ ll" Hjl which is based on equstion (8), accounts for a small shift in a profile locations in these coordinates caused by variable 7 s properties effects. Also shown in this plot are the following 0 10 O equations D, 0 = 0.127(ugl) (15) N Q.7 Temperature profiles in a turt>ulent natural convection boun. g , g ,3g _ y ,4gf y a 3 (gg) 6er7l ayer on a vertical surf ace in air These equations are equations (12) and (13) with coefficients are citablished experimentally but has been theorized most modified to fit the data in Fig. 7. The coefficients in the a recently by George and Capp (24]. George and Capp proposed equations are only slightly different than those derived by i at dividing the boundary layer into an inner and outer region. George and Capp on the basis of the data available to them. m The inner region was assumed to be a constant heat flux layer The figure shows that the form of the equations for a y consisting of a "uscous" sublayer and a " buoyant" sublayer. " universal" temperature profile derived by George and Capp
.n. The " buoyant" sublayer is between the " viscous" and for the inner region of the boundary layer agree very well with l res " logarithmic" regions pointed out in Fig. 6. The form of the the data for air. The coefficients are Pr. dependent, though, u equations derived by George and Capp for the " viscous" and smee the water data of Chokouhmand, if shown, would be a " buoyant" sublayers for a uniform temperature surface are. significantly to the right of the air data.
respectively As a final note, equations (7) and (15) can be used to sa 0 C, (g/q,) (12) determine a heat transfer relationship for turbulent natural ito , p- , , y3 convection in air in a straightforward fashion arp -8 H
' In these equations, the definition of 6 is one minus that used Jar Nu, = 0.10Gr,"3 g/ T. (17) by George and Capp, and the terms iC , C2 , and C are T.
ere by f 3 hinctions of Pr. Based on the experimental data available, This heat transfer relationship and the heat transfer . als , George and Capp determined C , C 2 , and C to be 0.1,1.35, 3 relationship given by equation (8), determined from the in. l als and - 1.45, respectively, for air. dependently measured boundary layer temperature profiles ' nts ,' George and Capp theorized that the temperature profile in and surface heat transfer data, respectively, agree within 2 at the outer region of the boundary layer was dependent only on percent. The consistency of the independently measured
. n. g/6, where 6 is a length scale for the outer region of the botodary layer temperature profiles and the surface heat l t 's boundary layer. Their theory showed that there should be no transfer data is a very good check on the experimental I
- re Pr number dependence in the outer region. George and Capp technique and the 5ariable properties effects noted in this l ne . expressed the equation for the outer region as experiment.
T- T. , C ,z_ ) gg4) Conclusions 6 .i!. To An experimental study of variable properties natural of where 6 was undefined. No specific form for the equation was convection heat transfer in air was conducted on a large (3.02 on given by their theory. An empirical fit to data is required to m high by 2.95 m long), electrically heated, verticsl surface. obtain the equation for the outer region. The wall temperature was varied from 60 to $20*C. The g3 Figure 6 suggests a logarithmic form for the " universal" ambient temperature was approximately 20*C. Surface heat temperature profile in the outer region of the boundary layer. transfer coefficients were measured at 105 locations on the i Deviations of the data from equation (11) are within the surface. Boundary layer mean temperature profiles were also i er tmeertainty of the data available. Figure 6 also suggests 6, as a taken. Turbulent natural convection heat transfer results ; 3 , length scale and T. - T. as a scalar for temperature in the show that variable properties effects on turbulent natural In l outer region as a result of the close agreement of the various convection heat transfer can be correlated using the low. fk ;I sets of data over a large range of Pr. temperature difference correlation,if properties are evaluated i j Using 6, as an outer region length scale is contrary to the at T. and a small wall to-ambient temperature ratio I %) recommendation of George and Capp, who stated that since correction is added. This correlation is given by l m ' 4, had "no dynamical significance," it should not be used as a
' l re length scale for the outer or inner regions. It is worth noting Nu, = 0.00SGrf 3 l P~
3 however, that when the Boussinesq approximation is made. T. ) .o u j .'r the integral jo" ( T- T. )dz appears in the integral form of the The correlation fits the turbulent heat transfer data taken to j 't l momentum equation for natural convection. The integral within 26 percent. The lammar variable properties heat !
- appears in a source term in the integral momentum equation, transfer results show, as demonstrated by others, that the '
as a representing the buoyant force term in the differential laminar variable properties effects for gases are small. The momentum equation. The appearance of the integral suggests results also show that the location and extent of transition i strong dynamical significance for 6,, since dividing the in. from laminar to turbulent flow are significantly affected by , a tetral by T. - T. gives 6,. T, for a fixed T.. As T,,/T. changes from 1.1 to 1.75, the l E balof HeatTransfer FEBRUARY 1985, Vol.107 /131
e transition Gr decreases from about 7 x 10' to 3 x 108 when 5 Sparrow. E. M.. and Gregg, J. L. "The Vanable Fluid.Propeny Prg 4; the properties in Gr are evaluated at a film temperature. in N,atural Consecuon.",,ASME Trea,ssenoas. Vol. 80.1958 pp. 469-84 Boundary layer mean temperature profiles indicate that a p,,,, ,,,, 3 , y;, ,,,3,,, ;,,,,,, y,, ocon,,cno, g,,,,,u p".
" universal" temperature profile exists for turbulent natural Regime Turbulent Le Long D'Une Plaque Plane Venicale." Paper No. NC.3 I g 1970. Proceedmgr of rat WA Internanoast #rer Trey /er Cog /erene,, p *g convection. The " universal" profile for the inner region of versailles. France. pp.1-12. ,,
the boundary layer for air, which includes the effects of ' C '"" 8 ^ * "N" C"" *" " C '*"*"*'"""S*fa variable properties on the profile location, is given by the includmg Inn'uences of Variable Propernes.."'ASME d, Y,Jou% Taesna. s i.o 105.1983, pp. t38-i4L ' following equations p. 8 5,ebers. D. L.. " Natural Convecuon Heat Transfer From an E Viscous sublayer: 8 - 0.127(t/ ql) Receiver." Sandia Nanonal Laboraiones Repon No. SAND 78 8276. ia
% o Buoyant sublayer: 8 = 1.36- 1 A2(t/gl) . p3 Nanonal Laporziones. Livermore. Cahf. 1978.
9 Sieget. R., and Howell. J. R., I* k y,c,,..g,it 3, Yr,rk.1972, p.116. Thermel Aedserson Neer 7,* .$ The forms of the equations were taken from George and 10 Edu ards. D. K.. and Canon. i.. " Radiation Charactenstics of Rough ans'}r 4 Capp. The constants in the equations, which are Pr-Oxidued Metals." Advances in Thermophysical Propenies al Entr "' dependent, were determined from mean temperature profiles Tunpercurn is9-199. and Prusurn." ested by Serge Gratch. ASME.1963' " A
- taken in air. The outer region of the " universal" temperature 11 Edwards. D. K.. and deVolo. N. B.. "tJseful Approximanons for spectral g 6%
?
profile, based on temperature profiles from this work and and Total Emissmty of Smooth Bare Metals." Advancer sn TAccmop4 7s,,, b ternperature profiles taken in air and water from other works, Propertnes er Extreme Temperatures and Pressures, edited by Ser8t Grases' N. ASME.1965. pp.174-188. has a Iqstithmic form. 12 Ki ne. S. J.. and McClintock. F. A. "Desenbmg Uncenamnes in S,np (#i Sample Espenments." MerAenicalEngmeerms. Vol 75.1953. pp. 3-8. 13 Sparrom. E. M.. and Gregg, J. L.," Laminar Free Convecuon from a V, g$ AcknotwedgmentS I
. Jr tical Plate wnh L'niform Surface Heas Flua." ASME Transactions. Vol. 73, (
The authors would like to acknowledge the financial 1956 pp. 435 a40. 14 Churchill. S. W.. and Chu. H. H. S.. "Correlatin8 Equauons for Laminar k#IIII support of the Department of Energy, acting through Sandia and Turbulent Natural Convecuon From a Verucal Plate." //NT, Vot, ts, 3 '328 National Laboratories. Livermore. California. In particular, ']'(c[3 3,3,,,,,,s of Lammar Free. Convection Row and H,, i we would like to thank Dr. Robert Ga!!agher for his attentive sur overview of the program and Dr. John Kraabel for liis close Trander About a Flat Plate Parallel to the Direcuon of the Generstmg Body bel Force." NACA TR lill 1953. and effective technical monitoring. This work was supported 16 Kays. W. M.. and Crawford M. E., Convertwe Neet end Mess Transfer, g by the U.S. Department of Energy under Contract DE AC04- McGraw ew york (2d,ed d,m, g,, Tr mhion.13d ed.L McGraw-Hill Ne, 76DP00789. p8tt york.1954, p. 258. f pfd 18 Gebhart. B., hear Treq/cr. (2d ed.). McGraw. Hill. New York.1973.
- 19 Cheesewr sht. R.," Turbulent Natural Converuon From a Verucal Plane
- ggt References Surface." ASME Jovana or Haat TaAMsfla, Vol. 90,1968,pp.1-8. Din 4
I Siebers. D. L. "Espenmemal Mined Convection Heat Transfer from a 20 Chokouhmand. H. " Convection Naturelle Dans L' eau Le Long D'Une IE'f tarse. Verucal Surface m a Honzon al Row." Ph.D. theus. Departmem of Plaque Venicale Chaufee a Denane de Rua Constante." Ph.D. thesis. A 8qu Mechanical Engmeeting. Stanford University.1983; available as Sand a Na- L'Universne El Mane Cune. Paris, Rappon CEA.R-a867.1976. It Warner. C. Y., and Arpaci. V. S., "An Espenmeman invesusation of ( sional Laborsiones Report No.. SAND 83 8223. Sandia National Laboratones. # Livermore. Calif.,1983. Turbulent Naiural Converuon n Air si Low Pressure Along a Venical Heated Flat Plate," //NT. Vol. II.1968 pp. 397-406. 2 Carey. V. P. , and Mollendorf. J. C., " Variable Viscosity Effects in )l 22 Vliet. G. C., and Liu. C. K., "An Espenmental Study of Turbuleni Several Natural Convecuon Rows." I/NT. Vol.23.1990 pp. 95-l08. Natural Convecuon Boundary Layers." ASME Jovana or Haar TaAMsFaa. 3 Shaukazullah. H., and Gebhart. B.,"The Effect of Variable Properues on Vol 91,1969. pp. $17-531. " Larnmar Natural Convecuen Boundary. Layer Row Over a venicalisothennal Surface sn wmer." Numencel Near Trem/tr. Vol. 2.1979, pp. 215-232. 23 Fugli. T., "Espenmental Studies of Free Convecuon Heat Transfer." i
- ?
Butlerta /SME Vol 2, No. 8.1959 pp. 555-558. f 4 Hara. T., " Heat Transfer by Laminar Natural Convecuon About a Ver- ...t ucal Rat Plate with Lar8e Temperature Dirference." Butt. /SME. Yol.1. No. 3. 1958, pp. 231 234. 24 Georst. W. K.. and Capp. S. P.. " A Theory for Natural Convecuon Tur. bulen Boundarv Layers Next to Hesied Venical Surfaces." I/NT. Vol. 22. l su 1979. pp. 813-826. ' ca' I
. 13 3 t **
> vs Ik ag
~)
h a [ re a ar l P l 4 s. l 1 ti 0 L d b{ t1 s. t *M v
'f n: *c c
e o. I I h t er I
< Y 1321Vol.107, FEBRUARY 1985 Transactionsof the ASME
{
l l STUDIES IN , HEAT TRANSFER i A Festschrift for E. R. G. Eckert I-Edited by J. P. Hartnett University ofIllinois at Chicago Cnicle 1 T. F. Irvine, J r. State Universuy of New York at Stony Brook E. Pfender Universtry of Minnesota E. M. Sparrow University of Minnesota 1 Engr
/
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40 3.M.11ADON MAXIMILIAN GREINER nimdict was large at small values nf M. Indicating less silmulattim of embulent Ef)cAg g,y,wfyygg mixing al small injection rates. Analogy Defects of Mass, Heat, .
SUMMARY
and Momentum Transfer The correlation formula derived in the analysis above, in Evaporation Boundary. - Layer Flow f(r q) = Nw(r;q)eu/2ffwd se s ,, (1 - r) q-appests to represent the data Elven for the distribution of filn :ooling effectiveness
.m an insulated plane surface downstream of a point of lateral injection. Since it was thz solution appropriate to small blowing intensities, one might expect it to be unsatisfactory for large values of M, but since the Peclet number is then small, the maximum value of n is also relatively small and the formula remains adequate.
Of the four empirical constants N, q, A, and A appearing in the derived expression for n when w(r) is talten to be proportional to c ** sin nz, two are found to be independent of x/D while N and q are related to x/D by simple power laws. the q versus x/D relationship is, of course, the function that allows the ona dimensional Mathieu functions, with parametric q, to represent the two. dimensional effectiveness distribution that is required. NOMENCLATURE Althougi three values of M as well as two lateral injection angles are mentioned in Table 3, the matching of theory with data on Fig. 2 is given only for M = 0.5 C normalized driving force [= (xi. x,J/(I -x,,,)] and a = 35*. The ,ther five comparisons will.be published when they are ready, cf local skin friction coefficient together with calculations of local average values of rdz) for the six cases reported c, specific heat at constant pressure D binary diffusbn coefficient in [6[. F parimeter t'efined by Eq.(19) f(rl) modified stteam function, Eq.(15) REFERENCES M mass transfer parameter, Eq. (20) r5: mass transfer rate
- 3. M. Tribus and J. Klein, l orced Convecteen from Nonisothermal Surfaces. Ifeat Transfer N parameter defined by Eq. (21)
Symp., Univ. of Michigan, p. 261,1953. , local Nusselt number I r 2. J. W. Ramsey, R. J. Goldstein, and E. R. G. Eckert, A Model for Analysis of the tota! pressure P Temperature Distribution with Injection of a Heated let into an Iso *hermal llow, in #cer pn partial pressure of components Transfer 1970. Elsevier, Amsterdam,1970. Pr Frandt! number
. 3. M. Y. Jabbati amt R. 3. Gokisicin. Adiabatic Wall Temperature and IIcat Transrer Downstream of injecnon through Two Rows of Iloles ASME paper 77-GT-50. Gas Turbine heat transfer rate l af.
Conf., Philadelphia, Marcn 27-31.1977. qu heating rate
- 4. 3. P. Sellers, Jr., Gaseous Film Cooling with Multiple lajection Stations AIAA I vol.1, pp. R gas constant 2154-2356, 1963.
Re, local Reynolds number
- 5. I'. R. G. EckerI, I itni Conling with injection through Itoles AGARD, florence, Sept.1970.
- 6. R.1. Goldstein. I'. R. G. Eckert. V L Erieksen, anet J. W. Ramsey, Irlm e Cooling I:ollowing Sc Schmidt number injectinn throurli Intlined Circular Itoici, Isr. J. Technet, pp. 145-154,1970; afwi Univ. of Sh, local Sherwood number Minneinta Rept. IITL.91,1969. T temperature, K 7.t.BahnLcaml12 I este, ThfJew ..f Fentrri.mt p. 293. Ih ver. M w York, l94s. 88 vth* city ctmystment parallel to plate R. F. L Ince. Tables of the 1.thptic-Cylinder t:ut ctions, hve. R. Soc. Elinheerste, vol. 52, pp.
u velocity component normal to plate 355-391,1931-1932. We dimensionless mass fraction, Eq. (17) w mass fraction v
42 M. r;HI INi;ll ANil E. H. B . wlHit M AN AR.OGY Dr.ITCr3 43 x mole fraction O.,
' -l y coordinate distance along plate g l 68 ~
y cocedinate distance normal to plate ,, ,- a local heat transfer coefficiens. Eq. (7) ll A j
$ local mass transfer coefficient, Eq. (8) = t / y - 6 .--
j [l p 4
~
n similarity coordinate, Eq. (14) . . - . d tempera ture, *C b 1 / 0 A dimensionicss tempera ture, Eq. (17) thermal conductivity Pi= ; //
/p,,const [ N u. 0 g , n,) _
, I Q.11:1 g
tii,.Ilil p dynamic viscosity K "-NN [ 0 cons!N
< < ' < < - < < - - - - - a r kinematic viscosity sf p mass density V(Ibrms flat plate (kmesceri Ig;d fim I4, . fin) r dew point temperature 7,,, shear stress at wall Figure 1 Physical model of boundary-tsyer flow with evaporation, y normalized fluid propelty, Eq.(18) 4 stream function Eq.(16) w normalized veincity Eq.(17) PIIYSICAL MODEL Subscripts De Problem under investigation is illustrated in Fig.1. A heated flat porous plate is covered by a quiescent thin liquid film (index I) of constant surface temperature.
Fluid properties without any subscript are related to the mixture. The plate is subjected to parallel flow at zero incidence of a gas (index 2), which 0 vanishing mass transfer may contain vapor of the evaporating liquid. He pertinent flow, wall,and boundary-I evaporating substance layer parameters for momentum, heat, and mass transfer are given in the figure. 2 free stream gas (air) m fluid composition at xi. = 0.5 (xi. + xi ) and T. = 0.5 (T. + T ) w liquid surface (r = 0) GOAL OF Tile INVESTIGATION
= free stream (y * =)
differentiation with respect to n if the following conditions are met Ps. - Pi
- O P, *O T - T.
- O (1)
INTRODUCTION then with uncoupled transfer mechanisms the well-known Pohlhausen and Blasius l De analogy between heat and mass transfer processes provides a useful tool for the ' *" "* *" * "" * "W'*" I * "* * '"'* under laminar flow conditions wi ".th the equations' ' * " . " * * * ' ' * " ' ' * " I solution of mass transfer prob! cms. De basis for this engineering technique is the similarity relations first derived by Schmidt [1] and Nusselt 12], which allow ' ' , a.x = 0'332 Ref8Pr "' (2) application of heat transfer solutions to mass transfer problems and vice versa. A Ackermann l3l, however, demonstrated that the application of analogy yields correct results only if the boundary conditions are similar and the mechanisms of Sh . = g,,7 = 0.332 Ref' Sc"' (3) heat, man, and momentum transfer are uncoupled. In the converse case, deviations from the analogy relations arise, which have been estimated by Ackermann (film 0.664 theory) l31. Eckert and lieblein (Kirmin.Pohlhausen integral technique) l4), and 9" Ref' Schlander (penetration theory) 15l.
* * " '* F " E'9"* "* ' * *
De objective of this contribution is to verify analytically and experimentaHy .' the applicability of the approximate solutions to heat and mass transfer problems in a.x (5) evaporation processes. s.* " T = 0.0295 Re" Pr"
44 M. tilt 1:lNI'It ANI) I" 14. 5. wir4TI C2 ANQt.OGY f)F.FEcTS 45 Sh ,. 0.02% Itc7 Sc"' (6) y _ g J" Pp di- (14) - yex;^ p. If tir conditions under Eqs. (1) are not satisfied, then the above analogy relations, I!qs. (2)-(6), are no longer valid. Dese so<alled analogy defects are the specific and the dimensionless -tream function f(n).given by - object of this investigation. p (IS)- f(n) = dv x u ANALYSIS ne continuity equation is satisfied by t5e stream function & in terms of 34 34 De analysis is ecstricted to laminary boundary-layer flow. To determine the heat pu = p. p pu=-p.p (16) and ma2s transfer and friction coefGcients a,$, and cf denned by Normalizing the velocity, the mass fraction, and the temperature d = a ( T, - T ) = - A (7) u_ w, - w,, T-T, g_ ,_ 3 _ T - T,, g 7)
- u. w -w, e
" # and the physical properties of the mixture rir, = R , T, (p . - p, ) = (8)
R, T, p - p,, \By ), r, = cf $ p., an'. = p. a
# = P-- #"~#---
cx =A-1 (18)
. (9),
D cp _ c,, - c ,, e,,a - the temperature. p:utial Iwetstue, amt velocily gradients at the wall must be wo =D_ y, = cp. c,. computed by numerical solution of the following set of boundary-layer equations- and employing the dimensionless abbreviations which teke into account temperature- and concentration-dependent properties: Continuity: V F=2 Wq (19) J* 3(PH) 4 3(90) = 0 (10) At DJ p.v. M=2 VRe, _ Momentum: p.u- (20) au B h's au a f au\ (11) N = fofofo$n (w, - wi.) p (23) pugg 4 pu gy gy (p g,,; 4 Diffusion: the following system of ordinary differential equations results ultimately from the ! similarity transformation: aw, Bw, 3 Bw
, DH p + #" p = a- pD %.,\ . .
(12) [,, p, j g' + f a(F - M) = 0 (22) l Energy: Ifa fa to We'I* + We' Sc (F - M) = 0 (23) l i BT BT 3 BT + pD(c,, - c,2) BT Dw, if, fx O'l' + 0' I'r y, (F - M + N) = 0 (24) l pur,37 + pur,gy gy gy 3v B i-(lJ) In these equations the primes denote differentiation with respect to the similarity variable n.
'lla thermal 41iffmion anil dilInsion thermal effects as well as the cirect of De transformed boundary conditions are aerodynamic heating have been neglected in Eqs. (12) and (13). Equations (1I)-(13)
I are reduced to a set of ordinary simultaneous differential equations I,y a simibrity n = 0: J' = 0 W=0 0=0 transformation, introducing the variable (25) ! y. 7 -2 h*, = 1 0-l
46 c4. <;to laut ANei t . n. t . wia it it unuu:v titttels 47 Formal integration of ligs. (22)-(24) using the imundary conditions resuhs in o set to of integral equations for the determination of the velocity, ccmcentration, and ,_' \ , temprature fields. 69
\ ,cos -e.
kq ee
\
. se sp 89
-Mp e , en te.e,a,es. en 03.o se e.sn.f.ee
( l /f,f,) c n p l ( l /f,c,)(F Af) dyldy e7 .a -w . we e. g, k waae = h >C04 -- k i . CC's -*" - (1/f,f,) esp l- (l/f,,9,)(F- Af)Juldn
" } \ "
\\ \ _
\ U \1 1 j,
(1/fpfato) eRP [- (Sc /fafafo)(F- Af)dn]dn ti- ,.i
\ \\ k Uk W= *
(27) l I L k (llfafato) exp [ (Sc lf,W,fo)(F - M)dn]dn 82- *
*"j'
\ '\ ,
'(llf,PA) exp l "(Pr fc/f,px)(F - M + N)dnlln ou l I _ I c6 os so ca os es os to e2 c4 es ce io 02 c' 0=
(1/f,PA) exp l-y (Pr fc/p,px)(F- M + N)dqldn (28) c.Q ~ e Figure 2 Normalized trsmfer coefficients as a function of the dirnensionless driving force for the evaporation of severalliquids eith different molecular weights. This system of integral equations can be solved according to an iteration procedure used by Piercy and Preston [6] and extended by Schuh l7l to include mass transfer. The thermal and transport properties of the inert gas (air) and the vapor of the evaporating substances are taken from data collections and curve. fitted, so that the
\ y ce
\ l l t'w -*e 1
\
temprature and concentration-dependent physical properties of the mixtures caa bc y l
- 4y- y
\,_\ (
tomputed by customary mixing rules l8l, (l n. .,,,i
} 1, k .
i gy. a . Co "*** N \ ANALYTICAL RESULTS g.8 _ . Computations were performed for the evaporation of water, methanol, toluene, and carbon tetrachloride into air. These four substances cover a range of molecular
\
es-K \
\, l
- waa ^ ;
l k g3 weights commonly encountered in chemical engineering processes. 2,N * \ ue results of tic numerical computations for the evaporation of these four
'g!
,H - *,J
\ I. k k \
[ liqui.Is are displayed in Figs. 2-4. The transfer coefficients 0,o,and cf are normalized with 0., o., and ef n calculated from liqs. (2) aml(3), the lluid prope-ties being taken I or- ' ' * " " * * '
\ \
at xi., and T,,,, and Fq. (4), using r e and T., as the reference state. The transfer *T " , coefficients thus normalized are plotted as a function of the normalized driving force "O '2 0' os os in J os oa os e or os 06 os _m of the evaporation process, C= (xi,, -xi )/(I -xi ). Deviations from unity repre. ** N sent the analogy defects.
. Pleave 3 ormeured transfer coefficients as a function of the darnensionless driving force at De results, shown in Fig. 2, were obtained for zero mole fraction of the vapor m. ,,g,,,,,,, ,
the free stream, x, = 0, and a free-stream temperature 0 of 100 C. In chemical - engineering processes such as drying and humidification, a wide range of 0 and xi may occur. Therefore the influence of these parameters on t'ie transfer coefficients was studied. The results, depicted in Figs. 3 and 4 for the systems
43 St. e;tti INI Bt ANil 9.. M. 0, WIN il 94 AkiAIDg;Y til l 9 m 49 e .4 in ll/(8 - C))
- a. " d en,. p in lInl - Ol}- 1 (30)
\ I=*-h*- h I
c
- "w.o l pr, M p c,. p c f ( f s= -
(n-p \ ( \_w... A ( .\ Sc. R i T. A.,c,. i __ l h % %
,.m T A = 11 + 0.2 (D.p/p R T.)In(1/I-C)l i g, wo .c _
De results obtained with these improved relations, curves 4, are in excellent
.\ \ l l ca \, agreement with the numerical solutions.
" ml w l % -s*
~ V' l%
(ti.ge
, % ( ( Q % EXPERIMENTS
.eTte=- For the investigation of the analogy defects in turbulent boundary-layer flow, an experimental facility including laminar investigation capabilities has been built.De 8
yp \ q core of this experimental setup is the test plate depicted in Fig. 6. This flat plate is e or et es os io na os Os os in c2 na os ce to installed horizontally in the test section of an open-loop laminar wind tunnel, in
- c. Q -
which the free-stream velocity and temperature can be adjusted continuously from 0 Figure d Normalized transfer coefficients as a funcilon of the dimensionless drMng force at to 20 m/s and from ambient temperature to 150*C. The degree of turbulence in the Mferent vapor convents of the free stream. free stream is less than 0.11 The evaporation area of the flat plate consists of six sections composed of wattr. air and toluene air for d = 20 and 400*C and xi = 0 and 0.6, show that porous, hydraulically smooth sintered plate segments, which are fabricated from the behavior of all the normalized transfer coefficients as a function of C is only sherrit nickel powder. Their mean surface roughness is 2 pm (CLA), measured with slightiy affected by these free-stream parameters. a 3-pm stylus. The excellent wettability features of the sintered plates serve for the generation of a very thin and continuous quiescent liquid film.
" .< < r e APPROXIM ATE SOLUTIONS , 7 i l' Q r )
j / j / / / / In I-ig. 5, results obtained with approximate theories are compared with the exact BU < , numerical predictium of boundary-layer theory for the evaporation into air of several substances with different molecular weights. D.
,c
[ '
/
G
" ~
D e[f G N Again, the coefficients #a and noare computed by using the Pohlhausen kb M a3 th \ F**'* 3
' N FC't a"]_
relations. Eqs. (2) and (3), the fluid properties now being taken at xi and T.. _ Curves I represent the approximate theory of Ackermann [3], which was confirmed i N Q by Colburn and Drew [9]. The results of this tl.cory, which is commonly called \ ;\ h\ s ( %% F film theory 110), . agree best with the exact numerical boundary layer solutions for j*'6 \ Q'N ',\ \\\ \ N ' systems with almost constant density. The same holds true for the theory \ \ E h \ ' introduced by Eckert and ljeblein [4l, curves 2, and the penetration theory [5], o< ege,.i rtds 's E \ \ ' curves 5. Ackermatm {3] extended the film theory by a factor A. This factor takes g'$ V}f e-fu]
\
'(4 h a . sao c
%h into accotml the influence of the " diffusing masses ** but results in an overcorrec- o2-cao. -*,,gl
' g"[y\Qh tion, curves 3. The deviation can be eliminated by a simple variation of Acker.
mann's original factor A, leading to 00 S' a
*7 e i k3 % I I %
02 01 06 08 to 02 at 06 Ce to 02 04 06 06 to E (I - x,,,) = A I In
' ^
(29)
- p. C 1-C I%ure 5 Coriparison of the results of approximate theories with the numerical solutions of the boundary-layer equationes.
i. 52 M. CH6.INI R ANil 1. H. I . wtNil;R AN ALOGY 81EI'LCTS 53 ! 16 --- (MHhonet. Asl ly.0l t ig iommer boundizy toy"
- - 16 -l Methanol- An] '.Y*pr
* ./
9378 p
- 961 usar IksnerEel solulen , L' Itsbulent boisdory leg ,y g 7
- 0,,8 16 *C 1 # 8- .
*>j p _ Sl a t i 2 *C , -#FJ 3
~
u lx,, 0l o s .n [ 10 --
- 's a 1 g, 945 e p a 961 mbor filsa theory =.
e I i
- 4 s v s 7 m/s D 3 , (e 8 *C u Im
/
- 7 s u,, s 10 m/s flo -30 t t e -14 *C ,g *[, S,o , *-
08 i
- y,'
n 10 s u,,s 17 m/s 12 , - I #" ' ' 06 I N - o 4 s u, s 7 rri/s 0 01 02 03 04 05 06 r. ' 08 ,4
'
- 7 s u,, s 10 m/s c = " -' i" 10
'#"* 'T 4 o 3g 3 ,_3 37 ,fs
- l l Figure 7 Mass transfes as a function of the driving fosa of r vaporation for laminar tmundary4ayer n,.w.1,ag=.rasi..n was into n >iss air. 0 01 02 03 04 05 % 07 na C=
I ' " of approximately -20*C before entering the wind tunnel are displayed in Fig. 8. Since these data, obtained for evaporation in laminar boundary layers, are in good Figure 9 Mass transfer as a function of the driving force of evaporation for surbulent boundary-layer now. tvaporanon was into dry sis. agreement with the corresponding theoretical predictions, the experimental data measured under turbulent flow conditions (Fig. 9) should also be very reliable. The results are restricted to mass transfer in constant-density boundary. layer flow, liccause the molecular weights of methanol and air are nearly equal and the REFERENCES i temperature differences across the boundary layer have been snull. lhe experi- ,, 3 Schmidi, verdunstung und warmeubergang. Gesund Ing. vol. 52, pp. 525-529,1929; mental data are reproduced quite well by the film theory. For the case of mass ,,p,io,,o 3, f ,, j ff,,, 3,,,, p,,,f,,, ,ol. 3 9, pp. 3-8,1976. it.msfer in boundary layers with large density gradients, however, the accuracy of 2. W. Nuswit, Warmcubergang, Diffusion und Verdunstung,2. Angew. Afash. Afech., vol 10, pp. the film lhcory is still an open question. 105-121,1930.
- 3. G. Ackermann, Waruwbbcrgang und molckulare Storfubertsagung im glekhen ield bei grossen lemperatur- und Partsaldruckdarterenzen, l'DI foru-tmngsh., vol. 382, pr I-16, 1937-16 y j
- 4. E. Eckert and V. l_acbicin, Besechnung des Stoffubergangs an einer ebenen,langs angestrom-15 ten Obernathe bei grossem Teildruckgefalle, Forsch. Ingenwurwes. vol. 16, pp. 3 3-4 2, tomnar bmsidory lays ,
i949, l' - ' 5. E. U. Schlundes, Stoffubcigang bcs Verdunstungs- und Absosplionsvorgang;ca an einer j isn , n68, , ik-,w s M n y - - " - ru".c - < r-*-.-l36.ca<8"'2 "64.
,I q y. 6. N. A. V. hercy and 1.11. Prem.n, A Simpic Solution of the lias riate Probicm of S6m S_ , , ,
I rkiion and inc.a isanoce, nui.a na, me. 21, pp. 995-suus.1936 p,, - , _ v
,I
- 2-*" 7. II. Schuh, 'I hc soluiion or the l ammar-HouvJarplayer I;quation for the f lat Plate for
/A ' i IVI.
Velocity and lemperature l'icids for Variatie Physical Properties and for the thflusaon gy g ar r~ a
. As u s ? m/s 7 s u s10 m/s Field at livh Concentration, NACA Tecit Memo 1275, pp.1-19,1950; nanslated from
- Fwh /kr., no. 1980,1944.
I fl - a 105 u 517 m/s
- 8. VDI.Wermcarlas. VI)l.Verlag Dusseldost,1974.
p' g g 09 ~
- 9. A. P. Colburn am! T. Lt. Drew, 7he Condensation of Miaed vapors. Tiens. AJCAF, vul 33 O 01 02 03 04 05 06 07 08 pp, g97-212,1931
~ ** - 10. R. B. Dard. w. ii. Stewart, and 1. N. Lightfoot. Trsasporr I'larnonwas. Wiley, New Yos k.
- C.
I *% 1960. ll. re aer. Aa swereue W M Munstung an du langs agn@nuen &nen he Figuss s Mass transfer as e function of the driving foras of evaporation for laminar boundary 4syer n en am und tus u enw uomunswentsh, docM thesis, MM UnWiM finw. I:vaporation was into dry air. Minnchen,1972.
4 HTD.Vol. 84 l l l Mixed Convection Heat Transfer - 1987 presented at THE WINTER ANNUAL MEETING OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS BOSTON, MASSACHUSETTS DECEMBER 13-18,1987 sponsored by THE HEAT TRANSFER DIVISION, ASME edited by V.PRASAD COLUMBIA UNIVERSITY 1.CATTON UNIVERSITY OF CALIFORNIA, LOS ANGELES P.CHENG UNIVERSITY OF HAWAll THE AMERICAN SOCIETY OF MECH ANIC AL ENGINEERS United Engineering Center 345 Eas: 47th Streat New York, N.Y.10017 ENGR
/ o i
s e-, .
4 COMPARISON BETWEEN THEORY AND EXPERIMENT FOR TURBULENT FLOW OF AIR IN A VERTICAL TUBE WITH INTERACTION BETWEEN FREE AND FORCED CONVECTION M. A. Cotton and J. D. Jackson Nuclear Engineereg Laboratories University of Manchester Manchester, United Or.pdom ABSTRACT N Risselt utsaber, 4D/1 (T -T m, Neaalt number for forced coNv)ection He a t transfer to turbulent flow in a vertical tube p pressure under conditions of mixed convection can exhibit marked P rate of production of k by mean flow shear d spar tures from the behaviour found in forced Pr Prandt1 ntsaber, c pp/1 cozvection : in ascending flow heat transfer may be 4 vall heat flux either impaired with respect to forced convection r, z radial, axial cylindrical polar coordinates levels (at moderate heat loadings) or enhanced (at high R tube internal radius test loadings), whe reas , in descending flow, heat Ra Reynolds ntsaber, p WbD/u tressfer is enhanced at all heat loadings. The present Re g turbulent Raynolds ntsaber centribution reviews earlier comparisons between T temperature (time-averaged) j thioretical formulations and experimental data for To reference temperature (upstream of the start I turbulent mixed convection and report s numerical of heating) I racults obtained using a low-Reynolds- v, w velocity components in r, z-directions number k - e turbulence model. Direct comparison with (fluctuating) ] i experimental data for t urbulent mixed convection heat l trtnsfer to air is made and t he general picture to Y W/(r /p) ) sestge is one of satisf actory computation of measured bulk heat transfer. ]n addition, approximate V, V velocity components in r, 2-directions 1 40mpa risons are ande with data obtained over vide (time-averaged) ranges of Reynolds and Crashof numbers using a buoyancy y normal distance f rom wall, R-r pirameter which combines these groups, y* y it ,lpiv bOK!<IATURE Creek letters A+ van Driest constant g coefficient of voltsee expansion l 8 buoyancy parameter e rate of dissipation of k specific heat capacity at constant pressure ? nodified dissipation variable j c' C, ,C g c2 e sta to in pr duction, sink terms c von Karman constant I of -equation 1 thermal conductivity 1 C constant in constitutive equation of u dynamic viscosity W k - e model v kinematic viscosity, 9/p D tube internal diameter p density correspond 1og to temperature T l D ' additional' dissipation term in k-equation o' fluctuating temperature-dependent dens $ty f*2 function in sink ters of c-equation og turbulent Pr andtl number ) i f f unction in constitutive equation of o,o turbulent Prandtl ataber for diffusion of I W k 8 k - e model k, c g magnitude of acceleration due to gravity g, component of acceleration due to gravity in t shear stress
-direc tio n Cr Crashof number based on wall heat flux,gg(D'/1v Subscripts Superscripts k turbulent kitatic energy b bulk -
time-averaged 1, mixing length t turbulent ' fluctuating w well 43
1NTR ODUCTION and in the nmerical study discussed below. This limits t he usef ulness of t he semi-e9parical equation in in situations where the mechanisms of f orced and ascending flow to providing a guide to the magnitude of free convection operate simul t aneously t he heat impairment for specified conditions; in descending transfer mode is termed ' mixed' or ' combined' flow, by contrast, developnent leneths are short and convection. Turbulent flow in vertical tubes under the equation currelates data well. It la nonetheless this heat transfer regime reveals complex t he rmo-f luid evident from the following comparisons of data and the d e v elo pmen t which prod uc e s bulk effects markedly k - c turbulence model t ha t the buoyancy parameter, B, different from those t ha t would r e s ul t if tne influence is a primary correlating paraceter for flow in vertical of free convection were combined with t ha t of forced tubes under conditions of turbulent aimed convection. convection in a simple ' additive' manner. The mixed convection regime is thus of considerable intrinsic A R EV IEW OF T Hy AP PL IC AT IO N OF TURBULE CE MOELS 70 physical interest, and, because of the possible impact TURBULEb7 PIXED CONVECT ION IU VERT ICAL TUBES that significantly modified levels of heat transfer may have upon t he pe r f ormance of thermal engineering The present neerical results are obtained using a systems, the problem is also of practical i=portance. ' low-pa yno ld s-n um be r ' two-equa tion (k - c ) turbulence hvastigators have obtained ex pe r im en t al model which is applicable over the entire flow domain, results for turbulent mixed convection heat transfer in including the viscosity-af fected near-vall region where a n um be r of fluids under a wide range of conditions turbulent Feynolds number, Re t, is small. The ( r e vie wed by Jackson and Hall [11). The behaviour par ticular f ormulation adopted here is that of launder observed experimentally is that, f or heated ascendinR and Sharma [8), a minor re-optimization of the flow, heat transfer (quantified in terms of Naselt pioneering model of this kind oue to Jones and ta und e r o m be r) may be either impaired with r e s pe c t to forced [ 9,10 ) . The selection of model t ype was guided by convection (at moderate heat load ing s) or enhanced (at consideration of the importance of sodifications which high heat loading s); in heated deseanding flow heat occ ur to the turbulent shear stress distribution in the transfer is always enhanced by buoyancy effects. The ne a r-v all region or mixed convection flows, identified difference between these trends and those found in by Jackson and Hall [1], and by trte experience sained laminar mixed convection should be noted : the general by other workers in applying cifferent forms of re sul t s found for the laminar case (in which t u rbule nc e closure to t he problem. Earlie r re s ul t s dif f usivities are purely solecular properties) are that obtained using turbulence models of varying complexity hast transfer is enhanced in heated upflow and lepaired are reviewed below and related to t he theoretical in heated downflow. h t he present work comprison is asemptions implicit in the models. The turbulence ecde with a body of expe rimen tal data fo r t ur bul en t models discussed are of the 'Boussinesq viscosity' or mixed convection heat transfer to gases . Carr , Conno r ' turbulent viscosity' type in which Paynolds stress is and Buhr [2), Pyrne and tjiogu [3] and Steiner [4] have related to the mean ve?ccity gradient via turbulent
- tade measurements for ascending air flows, while Axcell viscosity,pg:
sad Hall [5} obtained data for a descending air flow cod Easby [6] examined a descending nitrogen flow. Jackson and Hall [1] proposed a p'E = yt irE (3) phenomenunological model of he at transfer in the t ur bul e n t mix ed convection regime based upon consideration of the modified turbulent s he ar stress Fddy dif f usivity models distributions which occur in the near-wall region in re s pon se to buoyancy forces. A slightly revised form The simplest approach to closure of the mean flow of the earlier model based upon Crashof neber equation set relies upon pr e s cri p t ion of t ur bulent appropriate to t he unifot1n wall heat flux thermal viscosity without direct reference to spatial b ound a ry condition is employed in the present work. variations of the mean or turbulence fields. Podels in The analysis leads to a ' buoyancy parameter', B, by which this approach is adopted are k.nown as ' eddy which the importance of mixed convection effects may be diffusivity' models (note this terlainology introduces chsrecterised: some ambiguity and care should be taken to distinguish these models from ti e wider class of Ibu ssinesq viscosity models of which they f o rts part; a better Cr B= (1) description eight be ' prescribed eddy diffusivity' Ra 3.425 Pr .8 0 m od el s) , Re ichard t [11] pro po sed an oddy diffusivity Further analysis, in which recourse to espirical model in which the ratio of turbulent to molec ular hsat transfer and flow resistance correlations is made. viscosity la prescribed as a function of the universal leads to an eq ua tion for the ratio of Posselt ne ber distance coordinate, y'. Tanaka et al. [12] exaatned for mixed convection to that for forced convection as a turbulent mixed convection tube fluws usinj a function of the buoyancy parameters modification of Reichardt's expression in which y was replaced by the integral with re s pec t to y w Bx 10'8 of /Elo~/W from the wall to position y. In fset, p * [1 2 (2) neither k ic ha rd t's original f ollnula tion nor Tanska's o ( N/ N,)2 } 0.4 f, modification correctly reflect turbulence production by sheer which is given as the product of local Re ynold s where t he + sign .pplies to descending flow and the stress and velocity gradient: sign to ascending flow. The buoyancy pa r ame ter (equation (1)) and semi-empirical equation (equation (2)) are derived upon p - p's E (4) t he asstssption of a f ully-developed condition, however, 33 in ascending sized convection flows, very lon4 Tanaka et al. found that computed wall temparature dsvelopment lengths (of the order of 100 tube dissetets distributions for a vetrica3 heated tube were opposite in some essee) are found to occur both experimentally to observed behaviour , exhibiting heat transfer 44
epa""""
. l l
l l enhancement for ascending flow and impairment for rne and t wo-e eus tio n t r anspo r t models descending flow. Fixing leneth models Transoort models of turbulence in which recourse is made to the turbulent viscosity concept (but not to the remainder of the mixing length formulation) have Frandtl's sixing length hypothesis provides the been applied to turbulent mixed convection in vertical f rameerk for a group of turbulence models in which, by tubes. nese models employ alternative means of contrast with eddy diffusivity models, local mean de t e rminine characteristic length and velocity scales velocity grsdient is incorporated in a formula for u timing length models are significant on t he ir b w.n to determine u De models have two distinguishing features f irIr.ly, the cha r ac teris tic scales are serits, finding vide application in tur bulent flow deterwined by reference to turbulence quantities, as calcula tion; t he y also have an important bearing on opposed to mean field cuantities. Secondly, t ransport more detailed models which includ e equations for .the eqrations are introduced for the relevant quantities t rans por t of turbulence quantities. This connection is and y, i s f o rmed , at least in part in the case of one-two-fold: firstly, consideration of turbulent kinetic equation models, from differentially de te rmined (as energy transport reveals that, under the assumption of o ppo sed to algebraically prescribed) scales. "hus local equilibriise between production and dissipation of local equilibrium is not assumed in the manner of t urbulent kinetic energy, the mixing length formulation mixing length models . correc tly reflec t s t urbulence production (Townssnd 'odels in which a t r ans po r t equation for one [13]). Secondly, transport models rely in part u po n of t he scales ( in practice the velocity scale via k-mixing length re sult s for the determination of model t ran s po r t) is emoloyed are te rmed 'o n e-e q ua t io n' constants (launder and Spalding [14]). models. Axc ell and Pall [$) applied a variant of b t he mixing length formulation it is Wol f s h t e in' s (19) one. equation k-t model to their data po s t ula ted that u, appearing in equation (3) may be for heated descending flow of air. An t icipa ting the expressed in terms of a sizing length, 1 ,, and mean la und e r and Sharma [8] sodel employed in the current velocity gradient as follows: s t ud ie s , it s hould be noted that Wolfshtein's model is a pplic able over the entire flow domain, including the 2 g viscosity-offected near-wall region. The model may ug =p 1, (5) therefore be classed as a low-Reynolds-number model, in common with the van Driest mixing length model. Axc ell and Ihll's calculations were qualitatively correct, Prandtl's original sugg estion was that i should s howing enhancement of heat transfer with respect to be proportional to distance from t he wall. keerous forced convection. We computed enhancement was, propo s als for alternativa prescriptions have been however, considerably less than that founo advanced one a uch modification of Frandtl's linear e x pe rimen t ally . sodel which has found widespread application is due to highe r order models that have been applied van Drie st [15). The van Driest approach makes with some success to turDulent mixed convection data allowance for the effect of a solid boundary upon are 'two-equation' modela. De se models incorpo rate turbulence by incor pora tion of an exponential damping t rans po r t effects on both the velocity and lengt h term in the expression for mixing length: scales forming the De turbulence quantities constitutive most of ten equation selected fortou$r. f m the basis of the model are the turbulent kinetic energy, k. 1, a ey [1 - exp ( y*/A*)] (6) and its rate of dissipation, t . Walkla c e (18 ), whose l mixing length results were discussed above, tr.s t e d 1 three versions of the k - e model against Runs kl0 and j Pelhotra and Pauptmann ll6) applied the van Driest K13 of Carr et al. [2]. De first model tested i model to turbulent mixed convection with the re s ul t consisted of a standard 'high-Peynolds-nts:iber' model in t ha t computed wall temperature distributions for heated which the t r an s po r t equations for k and c were solved upflow and downflow demonstrated the correct t re nd s , over t he region y' a 30, analytical ' wall functions' indica ting heat t ran s f e r impairmen t for upflow and being employed to bridge the near-vall region and thue enhancement for downflow. 'iowe v e r , comparison made provide boundary cunditions for the transport ecuations l with the wall temperature data of Ja ckson and Br an s lu terodt [17] for carbon dioxide at nea r-cri tical at [a 30. Be remaining two models were variants of ! conditions showed good agreement for downward flow but a hybrid partial l ow-F= yncid s-n umbe r treatment, in l which a damping ters was applied to the expression a marked absence of quantitative agreement for upward for u , is11owing Jones and teunder [13,14), but the flow, I fora *of the k and t equations was unaltered from the l Walklate [16] tested four mixing length models hig h-Re ynold s-n tas be r form (althoug h k transport was i against data for heated upflow of air with uniform wall heat flux obtained by Carr et al. [2]. solved over the entire flow). Cart et al. Ualklate found that, as a g roup , the numbered their testa NIO to N13 in order of increasing k - t models performed better t han the mixing le'ng t h i l Crashof nta.ber and, of these, Walk.lete made comparisons models in computing turbulent mixed convection. He a t with the lowest and highest Crashof number tests, Runs I transfer computed using the partial low-Raynolds-number bl0 and N13. models showed good agreement with the data, however { j The four models used by Walk.l a te represent poorer agreement was evident when t he hig h-Ra ynold e- I various modifications of t he van Drie st formulation, ntsiber model was applied. however a detailed de sc ription of t he models is not Turther encouraging results using a given here since the results indicate that all perform k - e model have been obtained by Abdelseguid and similarly. Comparison made with the heat transfer data Spalding (20] who combined a high-Reynolds-number model l of Carr et al. revealed large discrepancies between with wall functions. Computed results demonstrated the calculated and measured parameters. correct trends in mixed convection heat transfer, i.e. for heated upflow, impairment at low and moderate Cr a s ho f n tsa be r was followed by e nhanc eme nt at highe r values, vnereas for downflow enhancement was found for 45
._ . . - _ . . - - - . = _ . - - < ~ _ ,.. . __ _ _ _ . . . - all values of Cr a sho f ne ber. to comparison with (iii) A further important feature of the experimental beat transfer data was nada by Abdelseguid turbulence model is that it is of the low-and Spalding, however their calculationa yielded levels 'e ynold s-n e be r typet turbulence transport of impairment and enhancement lower than those evident equations are solved over the entire flow in available esta. domain, thus rendering empirical wall Two recent numerical studies of turbulent functions and associated assumptions of near-aized cunvection in vertical tubes using low-Raynolds- wall universality redundant. num ber k - c models are par ticul arly relevant to the present work. h the first of these lie ns and Complete formulation of the mean flow and turbulence tellinghausen [21] used the Jones and taunder model to model (launder and sharua) ecuations conpute heat transfer to an ascending flow of a refrigerant under the conditions of an experiment by The mean flow equations are written in the ' thin Sc heid t [22) carried out near t he thermodynamic s he a r' (or ' boundary laye r' ) and Poussinesq critical point. The correct qualitative trends of wall approximations. Turbulent viscosity, w g , appearing in temperature oevelopment were found, although there were t he homentum and Energy Equa tions is evaluated as a some significant q uan tit a t iv e discrepancies (c.f. function of turbulent kinetic energy, k, and the Falhotra and Hauptmann's [16] a1xing length results for modified dissipation variable, 7. (to which the naar-critical point data discussed a t,ov e ) . It should boundary condition 7=0 at y = 0 applies). The be noted. however, t ha t the highly variable properties turbulent kinetic energy and dissipation rate are of t he near-critical point fluid add a further determined f rom transport equations. complication to the problem and make this an especially seriegent test of t he model. In support of this Continuity observation, it is foteresting to examine the work of Tanaka et al. [23] woo compared a slight variant of the 18(TV)+d."=0 I 3' #8 (7) Jones and launder model against their data for heateo upilow of nitrogen and found generally gotid agreement .'tnentua between measured and calculated tusselt number. In setempting to form an overall picture of 2 the relative performance of the turbulence models 13-r 3r (p rVW) + as S- (pV ) =dsN +r 13rb [r (u + 9t) 3r El reviewed above, it is necessary to bear in sind that such a picture is not wholly complete because of sparse comparison with (different) data and the various + [] - B (T - T*)) og 8 thermal-hydraulic formulations adopted (Axcell and Hall [5), Tanaka et al. [121, talhotra and Hauptmann [16) (8) and Tanaka et al. 123) assumed that a f ully-developed, or ' qua s i-d ev elo pe d ' [12), condition prevailed). where icnetheless, t he results of previous studies acrongly indicate that k - c formulations offer an improvement over simpler models in the calculation of turbulent g for ascending flow ga= { +g for descending flow sized convection. b applying k - c models particular attention should be given to the treatment of the near-wall region; Jackson and Hall [1] identified the En e rg y importance of deviations from ' universal' behaviour in determining mixed convection heat transfer and Walklate [18) found t ha t agreement with experiment was improveo 1 (prVT)+h(PVT)= h [r (h + y ) } (9) by the adoption of a partial low-Re ynold s-num be r t treatment. Consitutive soustion PRES DT AMLYS IS: LOW-R FY tc01DS-MJPB ER TWO-E00AT IOf' TUR5ULEICE KIDEL 2 y -Cf I b (10) Calculations have been pe rformed using the low- WWf Reynolds-number model of launder and Sharma ;8), details of which are given below, h a physical sense k-transport t he f o11sula tio n of the mean equations and the , t urbulence model possess three characteristics which 1 3 (prVk) + 8.- (pWk) = yt(U) are important to the accurate computation of turbulent r at as j sized convection heat transfers 1
+d3 y *) I ]
(1) The formulation is for developing thermal- ' U [r(v + 'k hydraulic conditions, a feature which is necessary if one alas to resolve the , (p , p' ) significant development effects occuring over long lengths in ascending turbulent mixed (11) convection, where (11) The turbulence model is of the two-equation class. thus transport effects on both the length and velocity scales are permitted and "N D* ={ 29(3k 2 (I+' } the implict assumption uf local balance I /3y)2 + (7 > 2) between production and dissipation of turbulent kinetic energy is absent from the The special form for D adopted in the region model (c.f. a1xing length models). ys* 2 is used because of cobvergence difficulties 46
quer=""' experienced when the standard form is retained in this a/D = 106 and z/D = 207) ano there is evidence that, region, even at such high length-t o-dia16e ter ra tios , a fully-developed conditgon has not 10-6 been attained over the (:aximm impatraent 7 - transoort range 2.5 x 10~ < 8< x occurs at P . 2.75 x 10~g) . b the case of descendine
* . . . flow a f ully-d eveloped conoition is attained by h (p t d) + (o d) = C g{w (@ s/D = 30. The experimental data show on Figure 1 were nbtaineo under a wide rense of conditions: Cart et al
- 3) u t ip [2] made pensurements on agr for 5000 < Ra < 5400 and
+ 7 g [ r (u + -) gl 1.06 x 10 < Cr < 2.22 x 10 at s/D = 103.45; the data e of Steiner (4) are for air in the ryges g2 2n g 3 g.2 5000 < pe s 14900 and 9.3 x 10 7 < Cr < 2.17 x 10 at
-C I 22T* c (7) 3r s/D . 60 and Easby's (6) descendinr flog data are for nigrosen at 2100 < Re < 8300, 1.4 x 10 < Cr < 6.8 x (12) 10 and /D . 83 - 159. b all cases. B is evaluated at the experimental conditions ouoted by t he authors constant s and Functions (takinx Pr = 0.7 in t he presentation of Steiner's and Ea s by 's data) and experimentally-cetermined Naselt k
2 number is normalized with respect to the Dittus-Poelter C = 0.09; f = exp (-3.4/[1 + Ra M h k *T g g enuations vt ok= 1.0 ; e, = 1.3 ; (13) N, = 0.023 Re U*U Pr0 *' (15) C,, = 1.44; Cg = 1.92; f2 = 1.0-0.3 exp (-Pe g2) D.e a sc ending flow dats may be considered as a grous since all are for large : D and therefore realistic comparison with the present res ult s for Turbulent Frandel nm ber s/D > 103.4 5 may be made. No points emerge: firstly, agreement between present n merical results and the experimental data le seen to be close and, secondly, c g= 0.9 (14) weight is given to the statement sade in the In t rod uc t ion that the buoyancy parameter is a primary Naerical solutions of the governing eaustions are correlating parameter fur ib ' N, since the data are obtained using a finite volume / finite dif f erence scheme obtained uver wide ranges of Raynold s number and following Iaschsiner [241 and the discretired parabolic Crashof cusber. Unfortunately, none of the three sets equations are solved using a ' marching' solution of experimental data include measurements for fully-procedure. N11 details of t he neerical scheme are developed forced convection heat transfer and an given in Raf. [7 ] . uncertainity in the comparison is consequently introduced by the use of the Dittus-Poetter equation: RESULTS Arc DISCUSSI0F t his uncertainty may be partly cuantified by the cbservation that computed N,at Be = 5000, Pr = 0.699 Attention in the present work is focussed upon the is u.3% lower than that yielded by equation (15). It modification of heat transfer in the mixed convection is also worth noting t ha t direct simulation of Runs regime c om bined with some consideration of mean b! 0, i12 and N13 of Carr et al yielded discrepancies velocity profile distortion. A companion oa per by the between computed values of Naselt numoer and those authors {25] nakes detailed comparison between measured deduced from the experimental measurements of 82, 12% and computed mean flow and turbulence profiles. and 11% respectively (computed values being lower). Comparison with experimental data is made at two A plot of the revised form of tne Jackson and Hall levelas at the first level direct comparison with data semi-empirical soustion, equation (2). is also shown in is made in which the experimental conditions are Figure 1. Agreement between the equation, the present precisely replicated. At the second level experimental numerical resalts and the experimental data is fairly c onditions are not reproduced exactly, but instead close in general, although the semi-espirical equation approximate comparisone with data are made by casting would appear to und e r-pred ic t the degree of maximum both in the form of tusselt nunber normalised to the impairment by a margin of approximately 20%. appropriate forced convection value and then plotting Figure 2 shows a second approximate comparison, in N/W, as a function of the buoyancy pa r ame t e r , B this case made with experimental measurements of heat (equation (1)). transfer obtained at low s/ D. Pyrne and Ej iogu [3] Figure I shows such an approximate comparison with a hgated a scending flow for l made meagurements on data. The trends of he at transfer impairment (with 1.6 10 < Re < 1.4 x 10 and 3.15 x 10gir< Gr ( 6.72 i maximum impairment of over 50%) and enhanc ement x 10 I and A.xcell and Hall (5) mede seasurements on the I discussed above are clearly evident. Present re s ult s obtained using the launder and Sharma low-Re ynold s- same experimental t he ranges rig <for aBaheated 2 x 10 < 1.3oescerging x 10 andflow overx 1.39 number k - c model are for sa = 5000 and Pr = 0.699 1031 < Cr < 5.03 x 1011 Bot h sets of data were obtained at sa = 5.5. Figure 2 shows costputed N/N 8 is varieg 4.4 x 10 <byCr increasing Crashof
< 9.0 x 108 nunber and Neseltovernumber the range is for ascending flow at s/D = 10, 20 and 50 (Re = 5000,,
normalised to t he computed v alue for f ully-d eveloped Pr = 0.699) and t he f ully-develo ped descending flow forced convection at the given Reynolds and Prandtl curve reproduced from Figure 1. Ne ascending flow n usber s . In the main, the points plottee for ascending curve s demonstrate marked development effects. The flow represent Wasalt number at an axial position distortions apparen t in these curves arise in s/D = 103.45 (the location at which Carr et al (2] made consequence of local recoverlas in Neselt number. their seasurements), although some points in the region similar to those apoorent in Figure 3 (see below). A of maximum impatruent are for higher t/D (between Frandel number of 0.7 was taken in processing the 47
a i 4 I l experimental data a ppe aring on ngure 2 and measured = l1 - exp (- y*/A*)). Neselt number is again normalized to the Dittus-Bositer equation. The results of direct simulations of Steiner's [4) (0.14 - 0.08 (1 - {}2 -0.06(1-{}*) (16) sxperimente un ascending air flows are shown in To the authors' knowledge, St einer's resul t e The profile is .;omputed accurately by the launder Figure 3. represent the only set of data for ascending mixed and Sharma model whereas the mixing length model convection heat transfer for air in which Fusselt incorrectly ytelds an of f-centreline velocity maximum, number development is measured. Agreement between the a f inding in confirmation of Walklate's earlier study corputed values and experimental points is acceptably [18l. close for all four cases over the f ull range of axial h conclusion. the present results and those of position. ( No t e s h Sa f . [4] Steiner does not mark other workers reviewed above indicate that the low-values on t he s/D axis but it seems likely from the Re ynold s-n um be r two-equation turbulence model of tsut of the pa pe r that this extends f rom s/D = 0 to launder and Sha rma (8) offers significant advantages 3/D
- 60). It is in tere s ting to observe the local over stapler mod els in the calculation of turbulent recovery in Neselt number occuring in the calculations aired convection heat t r an s f er to air in vertical End for which there is evidence in Steiner's data. tubes. There remains some uncertainty regarding t he Figures 4, 5 and 6 show direct comparisons between applicability of the model to de scending aired present calculations and the data of Byrne and Ej iogu convection flows at hign heat transfer enhancement (3) for ascending flow and Ax cell and mil 15) and levels and this question together with application of Earby [6] for the descending flow case. The resuls t he mod el to other fluids (liquid metals and water) s hown in Figure s 4 and 6 wo uld appear to indicate form the subjects of continuing investigations, estisfactory computation of the experimental data, although, in examining Easby's data, chia remark should ACK NOW1.FDC E P'UTS be qualified by t he observation that the data exhibit large scatter in relation to the relatively low The aut ho rs woul d ;1ke to thank Pro f e s so r B.E.
enhancement levels. Comparison with Axcell and Hall's launde r of t he tiniversity of Nnc he s ter Institute of data at generally highe r levels of enhancement Science and Technology f or his help and advice in the ( Figure 5) reveals by contrast computed Naselt number course of t he theoretical study reported here and also enhancement consistently higher an the measured Pt s e 1.. Yu for her assistance in pe r f o rming computer points. The cause of this discrepancy is not presently runs for some of the cases shown. known with enrtainty, altMugh, without prejudice, two clear possibilities may be identified a firstly, it may R EFERDCES be the case that the turbulence model per fo rms less tecurately where t u rbulen t diffusivity is increased; 1. Jackson. J.D. and mil, V.B. alternatively, experimental inaccuracies may be the "Influnces of buoyanew on heat transfer to fluido primary source of error. A third interesting in vertical tubes under tubulent flow conditions" hypo the sis is that descending mixed convection flows in " Turbulent Forced Convection in Channels and cty. under certain circumstances, exhibit an elliptic Pundles. Tneory and Applications to hat flow structure, t he reby rendering inappr opria te the Fxchangers and tuclear Reactors" Eds. S. Kakac and praeont parabolic f o rmula tion of the problem. This D.B. Spalding, Netsphere, te w Yo r k , 1979. qu3stion is currently under investigation in order that 2. Ca rr , A. D., Connor , ). A. and fuhr , H.O. tha source of the discrepancy might be identified with " Velocity, tesoerature and turbulence measurements greater certainty. in air for pipe flow with combined free and forced Raturning to the ascending flow data of Carr et al convection". Tr an s . AS >T C , J. He a t Tr an s f e r , 95, (2), Figure 7 shows the velocity profile measurement s pp 445-452 (1973). ,for Run F13 together with the present computed 3. Pyrne , J.F. and Ej iogu, E. points. De marked distortion of the profile sessured " Combined free and f roced convection heat transfer by Cart et al is captured to good accuracy by the model in a vertical pipe". Paper C116/71, I. itch. E. computations. Presentation of the same data in W' - y+ Symposius on Heat and has Transfer by Combined coordinate s (Figure 8) serves to indicate the Forced and th tural Convection, Fanchester,1971. pronounced departure from near-wall ' univ er s tlit y' 4 Steiner, A. evident under conditions of t urbulen t mixed "(h the reverese transition of a turbulent flow convection. nus, any assumptions of universality made under t he action of buoyancy forces". J. Fluid in order to construct well functions for use with Meh., 47, pp 503-512 (1971),
'high-Raynold s-numbe r ' turbulence models applied to 5. Axcell, B.P. and Fall, ll.B.
cixed convection are clearly highly questionable. " Fixed convection to air in a vertical pipe". ligure 9 provides an illustration of the Paper K-7, Proc. 6th h te rna tional hat Tr ans f er inadequacy of mixing length modele for the calculation Conference, Toronto,1978. af turbulent mixed convection flows. The experimental 6. Tasby, J.P. points shown are the velocity profile measurements of "h e effect of buoyancy on flow and heat transfer kun N10 of Carr at al for which profile inversion was for a gas passing down a vertical pipe at low cot found. We computed curves are obtained using the turbulent Reynolds nusbers" . ht. J. Hi s t tas s launder and Sharma low-Raynolds-number k - e model u:ed Transler, 21, pp 791 - 801, (1978). t hro ug hout this study and a low-Raynolda-number mixing 7. Cc t t on , P. A . length model obtained as the product of the van Driest "heoretical s tudie s of mixed convection in damping function and the Nikur adse distribution of vertical t ube s" . Ph.D. Thesis. University of mixing length for pipe flow: ta nche s t e r , 1987.
- 8. launder , B.E. and Sharna. B. I.
" Application of the energy-dissipation model of turbulence to the calcula tion of flow nest a spinning dise". latt. Ra a t Pa s s Tr ans f er , 1, pp 131-138, (1974).
48
. 1 l
l
- 9. Jones, W.P. and la under, 3.E.
"The prediction of laminarizatian with a two- to. ~. 38, Os [.,1. g.T$Irur f ecuation model of turbulence". . h t . J. Is e t Pass ... j Transfer,15, pp 301-314. (1972), is .
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l 10. Jonee, W.P. and launder, 3.E. . e.i. win.a m "The calculation of low-Paynolds-number phenomena ee .
. sata w asses ist with a tuo-equation model of turbulence". h t . J.
tera t tesa Transfer,16, pp !!!9-1130 (1973). '
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y, w ...\\ suusee ea
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, .. g'y, -
in vertical circular tubes - criteria of the .. effects. velocity and temperature profiles and < r ever se transition from turbulent ta laminar Figure 1 Heat transfer impairment and flow", Int . J. Hast tese Transfer, 16 pp 1267- f tnhancement in ascencing and descencing turbulent mixed convection.
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" Equilibrium layers and well turbulence" J. Fluid tech.,11, pp 97-120 (1961).
14 Launder, f.E. and Spalding, D. B. " ' f *3,, *. *
*1actures in mathenatical models of turbulence ~. " -
Acadesic Press. London,1972. - . - a nam
- 15. van Drisat. L.R. " '
. e.ia y e.a. us anses esi
" Oi turbulent flow near a wall", J. Aero. Sc i . , * **" * * ** "a ma m 23, pp 1007-10!! and 1036 (1956).
- 16. Psihotra, A. and louptmann, E.C. .
" hat trar.a f er to a supercritical fluid during
- n -
l turbulent, vertical flow in a circular duct" l Paper presented at the hternational Centre for , , Heat and Pa ss TY ans f e r Symposium on Turbulent . Suoyant flow in Ducts Dubrovnik,1976. ,, , 17 Jackson, J.D. and Evans lutterodt K.0.J. .
*
- g.mo
" hfluence of buoyancy on heat ransfer to ,, .
g,su
- superc ritical pressure carbon dioxide in a g e nas **7.
vertical pipe" Report PE2, ibclear Engineering de ,m e s'sai n u labortories, University of innchester,1968. arat we s t ud y of theoretical models of gu n 2 o e o a ans turbulence for. the numerical prediction of boundary layer flows Ph.D. Thesis, UPIST 1976. turbulent mixed convection.
- 19. Dolfshtein, W.
"The velocity and temperature distribution in one-dimensional flow with turbulence augmentation and "'
pressure gradient", ht. J. hat Pe s e Tr ans f e r , ' n= m .. ,, p , g. ,, g* 12, pp 301-318 (1969).
- 20. Abdelseguid, A.P. and Spalding D.B.
54 < I i* l
-"~
$itse lN!
im s E, s eal
- Turbulent flow and heat transfer in pipes with . - tm em im g 44 buoyancy ef fects", J. Fluid hech., 94, pp 383-400 g (1979). .c
- 21. Dens. U. and Bellinghausen, R. 4a < a
*Hrat transfer in a vertical pipe at supercritical f, g
**. .A pre s s ure*, Pr oc . 8th hternational Ha s t Tr ans fe r a g g g .... D *" ** r" '"" #"* 8-~ s~ .n Conf erence, San Francisco,1986. = \
- 22. Sc he id t , F. 3 \*
Dissertation, thiversity of Hannover,1983.
- 23. Tanaka, H., istuyama, S. and Hatano, S.
l36' h\y ,
" Combined f o rced and natural convection heat .a transfer for upward flow in a uniformly heated N *Nd-I P ' ' -
ver tical pipe", h t . J. He a t Pass Transfer, 30, pp 165-175 (1987). 8' 24 la schriner, t'.A. 6
*An introduction and guide to the computer code PASSABLE", UP:ST report, 1982.
I' m I g'g ,
- 25. Q>tton, P. A. and Jackson, J.D* Figure 3 Nusselt number development in
- Calcula tion of turbulent sized convection te a ascending turbulent mixed convection:
vertical tube using a low-Reynolds- direct conparison with the data of number k - e turbulence model", 6th Synoositas on 5te ner Turbulent Shear Flows, Toulouse, September 1987. 49
a su ., ..m.. . e ,.. m i i
" '9 g ---
-~~
3 , 6* - Wb h< e o Y ' "
" 1~ 0 A / .. //
fy i l l ii . ' " . 3 " k-E MODEL : n /f1 ___.2/D 0 06
// i i z/D : 103.45
// l i e DATA 0F fp i i CARR ET AL [2]
0& I, , RUN N13 z/D :103.45 gy [ Figure 4 Ascending flow: direct coriparison with ' the cata of Byrne and Ejioqu. I g0' ' Y/R
- 0. 0 01 0.2 0.3 0.4 0.5 0. 6 0.7 08 0.9 1.0 Figure 7 Velocity profile in ascending flow (Carr et al. Run hl3).
n .iJia "** ,,.
.= t MOON 1,
. GafA CF anstLL 4 0 MALL 151 l ! l l p#
l ',&% 'd ....
'l, l .
u . l ./ ye l ln x u . i, I f/ i l!! u . nig,o *
/ f/ Il ll u -
ya -
- . . noon
* ..., l
....,,.. ~ u .. . , . -
g , i -
.d! Gl#N N1) l asi se it s ...,.
ai 3
/ 4 . l i
_. . w'e .ss.6 s i ,.5 ,' Figure 5 Descending flow: direct comarison s,
#l ll , , , , , , , ,, , , , , , , , ,
j with the cata of Ancell anc riall. ,. Figure 8 Velocity profile in ascendino flow plottee in ' universal' coordinates (Carr et ! al, Run h13). gg. .3L <~~ I
*e ,
t ""
- k. w t -t M00tL 11
.- l
\
a CATA Of EA18Y lol l4
- g, /. [ N ! !
ll / IN w l
- I'
- 1. 4 l N, f/ e 04,4 O N Cts (IMOING no, il i (Ana. (.,f.A L.
i,i
-.- anon i
.... f
. . . #. l' -WlEW*" i El f f ' t u'
- I l i se --
f j l
' a' .%, a ",, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ..L81 L1 to see l
Figure 9- Velocity proffle in ascending flow, Figure 6 Descending flow: direct comarison comarison with a m,xing length model ; with the cate of Easby. (Carr et al, Run N10). ! 50 I
HTD-Vol.163 MIXED , CONVECTION HEAT TRANSFER 1991 - presented at THE 28TH NATIONAL HEAT TRANSFER CONFERENCE MINNEAPOLIS, MINNESOTA JULY 28-31,1991 sponsored by THE HEAT TRANSFER DIVISION, ASME edited by DARRELL W. PEPPER ADVANCED PROJECTS RESEARCH INC. BASSEM F. ARMALY UNIVERSITY OF MISSOURI-ROLLA
% [- .,-.
M. All EBADIAN FLORIDA INTERNATIONAL UNIVERSITY /' 6 ' ' 0 a P. H. OOSTHUIZEN QUEEN'S UNIVERSITY THE AMERICAN SOCIETY OF M E C H A NIC A L E N GIN E E R S United Engineering Center *O*O+ 345 East 47th Street *O*O+ New York, N.Y.10017
e HTD. Val.163, Mixed Crnvictitn Hxt Transfer ASME 1991 l l l MIXED CONVECTION IN A TWO DIMENSIONAL ASYMMETRICALLY HEATED VERTICAL DUCT l B.J. Beek Departrnent of Precision Mechanical Engineering Chonbuk National University Chonju, Korea M. B. Younger, B. F. Armaly, and T. S. Chen Department of Mechanical and Aerospace Engineering and Engineering Mechanics University of Missouri-Rolla Rolla, Missouri ABSTRACT ua = centerline streamwise velocity at the duct inlet v = transverse velocity component Measurements and predictions of mixed convection air flow W = width of duct in a vertical two-dimensional duct that is formed by one adiab- a = axial coordinate erature wall are reported. x stic walldistributions Velocity and one uniform and walltemheat flux were measured by P = [enetration depth of the reversed now as measured om the exit section using a laser-Doppler Velocimeter (LDV) and a Wollaston y = transverse coordinate (measured from the adiabatic wall) Interferometer (WI), respectively, for different buoyancy force z = spanwise coordinate effects and for various mter-wall spacings. Reversed flows a = thermaldiffusivity were observed adjacent to the adiabatic wall when the inlet air p = volumetriccoefficientof thermalexpansion Dow was restricted to a level below the natural convective value p (starved flow conditions). Velocity distributions skew toward # =
= density)/(Tg (T To -To),dimensionless temperature the bested wall as a result of the asymmetric heating, and the v = kinemauenscosity occurrence of flow reversal diminahes as the buoyancy force ( = Grb /(Reb 2),buoyancyparameter decreases. The occurrence of reversed flow adjacent to the adiabatic wall influenced significantly the velocity distribution in the duct, but it had only small effect on the beat transfer INTRODUCTION from the beated wall. Numerical predictions of velocity distri-butions and beat transfer con: pare well with measured values. Convecuve beat transfer between two parallel lates is of considerable interest in many engineering designs ' with electronic cooling and best exchangers. Tbc beat er in NOhENCLATURE that geometry can take place via forced convection (Mercer et al.,1967 ), free convection (Sparrow and Azevedo,1985), or b = spacing between walls mixed forced and free convection Cebeci et al.,1982) with g_ = gravitano leration either symmetric (Beckett and Fri(end,1982 and Wirtz and Grb = g4 -T , Grashof number based on b Stutrman 1982) or asymmetrie (Aunt 1972 and Quintiere and G rx = 1pt T , Grashof number based on x Mueller,'1973) beating condition. fn addition, the geometry bx =q - To), local best transfer coefficient at the can be horizontal Kamotani and Ostneh.1976 and Emery and heate wall Gessner,1976), i(nclined (Ou et al.,1976 and Azevedo and k = thermalconductivity Sparrow,1985) or vertical (Bododia and Osterle,1962), and the L = length of duct flow and beat transfer phenomena can be either laminar or Nux = hx/t, local Nusselt number at the bested wall turbulent.
P = pressure Pr - Frandt! number The problem of lammar mixed convection in vertical ducts qw = wall heat Dux at the heated wall has been studied extensively (Habchi and Acharya,1986) for Red = u obfy,Reynolds number based on b either symmetrically or asymmetrically beated condition. The Rex= u x/v,Reynolds o number based on r study of mixed forced and free convection with reversed now S = b/L dimensionless spacing between wall: rel pons in vertical ducts, has received relatively little attention. T = Duld temperature Flow revenals have been observed inside two dimensional To = inlet air teroperature vertical ducts of finite length with asymmetrical beating condi-T1 = temperature of heated wall (at y=b) tions (two walls t uniform but different temperatures) m natu. T2 = temperature of unheated (adiabatic) wall (at y = 0) and in mixed convection u = streamwise velocity component ral (Baekconvection et al.195 )).(iparrow et for The criteria al.1984) developing a reverse Do un = centerline streamwise velocity at the duct inlet for region in such flows and the extent of that reverse Dow region natural convectum are important for design considerations. Such criteria have 37
AN A1.OGY pt's I CIS 58 To achieve controlled film temperatures, extending from the adiabatic to close
. to the boiling temperature of the evaporating fluid, the plate sections are heated
~g i; individually by electrical heaters embedded in solid nickel plates soldered to the
_ _ r S$ E lower side of the porous plate segments. Each of the porous plate segments is NM " SS S .. supplied separately with degassed liquid. With the aid of a specially developed liquid 2% Y5' .* E supply system in combination with an optical observation device, a very thin liquid l y3 5-
- film with a constant thickness of less than 2pm can be maintained on the surface y 3 $
_ ]I 4 , I 8 h"h. 83 *d li
- 23M S$ 5'5 a g 4; F fp .heM e f each porous plate segment. This is accomplished at any local evaporation rate.
The very thin film is a prerequisite for the achievement of high surface temperatures with exclusion of boiling. In the present case, heat fluxes of the order 6 M
* " d- M f -E-I s b S 6 8 $ S o 5 of 50 kW/m 2are to be transferred from the heated plate to the film surface,
-Z - N 89 e
- b e
- C2 ; Cf C 3 causing temperature gradients of approximately 25*C per 0.1 mm. Since the nuximum temperature drop across a film 2 pm thick amounts to 0.5*C, the surface S j4 j { 7 temperature to be determined exactly for the evaluation of the mass transfer N 4 4 .
coefficient can be measured within a few tenths of a degree with miniature g- l i. thermocouples soldered into grooves on both sides of each section of the test plate. p Nl a .
- l .
d 2 l* j h Tin temperature of the su9 port frame and the bottom of the flat plate as well as l
-! the leading edE e is controlled by liquid bath thermostats.
hi , i I N - Velocity and thermal boundary layers origmating at the leadmg edge of the test l" ! ! jl
= & h, !
I J l
, ptte are sucked off a sho,' distance upstream of the evaporation area, causing appiWma:e coincidence of the lines of origin of the three boundary layers. For the
& 4 l j
-4 m . M > gl i i ~ experiments with turbulent boundary layers, a trip wire is installed across the flat
% plate 6 mn. downstream of the suction slot, ensuring the existence of fully g p"lg developed turbulent boundary layers obeying the $ power relation along the e-(I h~ 4 p3 f
_ .H /l' i 9., g measurement area of the dry flat plate. A detailed description of the experimental jN .p 2 [ #~ { facility, the test procedure, and the data reduction is given by Greiner llI). n% I G 50 g [ f
~
EXPERIMENTAL RESULTS
-{ Le first series of experiments conshh.J d heat transfer rneasurements in lanunar 1 j , .
~ y ,
' #MM / .- ,[ j and turbulent boundary layer flow with the flat plate kept dry. The results (not
\\ -
j f g shown here) are in very good agreement with the predictions of Eqs (2) and (5). After the f acihty was checked out, experiments in which methanol was evaposated att 'y h- ; y into slightly moist air under laminar flow conditions were performed. The reduced N m , i 3: 3 mass transfer coefficients are displayed in Fig. 7 as a function of the drivmg force m 4 g2
/ ,
4 f j 7 . 1 C They deviate from the exact numerical solution, especially at low interphase _g a temperatures. This is due to absorption of water vapor by the methanol film. The 4
.h Q .h x - i- , j methanol vapor pressure from the methanol water mixture is lower than that of j
h< j
~ '
7 P g a pure methanol. Consequently, the rate of evaporation is reduced. The vapor pressure lowering, however, was not taken into account in the data reduction process. g g {~Mklt j Although in the experiments the air was quite dry, with values of the dew-point temperature r between - 5 and + 2*C, a deviation of 30-40% arises in the region
- where the analogy should be most applicable.
Results of experiments for which the air was dried to a dew-point temperature sa
l
\
be:n established for the asymmetrically bested duct with uni. The two-dimensional duct was formed by two plates that form wall temperatures (Back et al.1990). The objective of were placed across the width of the air tunnel. Each plate was this investigation is to extend that experimental and numerical instrumented with twelve copper-constantan thermocouples. study to a two dimensional asymmetncally beated duct having Each thennoccuple was insened into a small bole on the back i one of its walls maintained at a uniform temperature while the of the plate, and its measuring junction was 1 mm below the l other wall is adiabatic. Heat transfer and velocity distnbution test surface. Electncal energy was supplied to the beated alu- ' measurements and predictions are presented for various heat- minum plate via six beating pads that were attached to the back ing levels, spacmgs between the duct walls, and inlet flow veloc- side of the plate. De power input to each of the six beating iues. pads was controlled by mdividual theostats to maintain a uni-form temperature over the entire beated test surface. The back side of the beated plate was insulated to minimin energy loss EXPERBENTALFACI1JI'IES AND PROCEDURE from that side. The beated plate could be maintained at uni-form and constant temperature to within 0.20C by controlling De experim. ental investigation was performed in a low tur- the voltage across the individual heaters. De other plate that bulence, open ctreuit, vertically aligned air tunnel. A schematic formed the two dimensional duct was made from plenglass and di+ gram of the air tunnelis shown in Fig.1. De tunnel had a its back was insulated with styrofoam board to simulate an smooth converging nozzle with a contraction ratio of 18:1, a adiabatic surface. Thus, the electrically beated and the insu-straight test section and a smooth diverging diffuser. De two lated (adiabatic) plates formed an asymmetrically bested duct dimensional duct was constructed in the test section of the air which was 27.62 cm wide and 106.68 cm long, with adjustable tunnel De test section of the air tunnel had two glass win- wall spacing between 0.95 and 437 cm. dows that extended along its length to facilitate interferometric measurements of the wall beat tius and the velocity measure. Velocity measurements were made using a, single channel menu by the laser-Doppler velocimeter. Plastic boneycomb laser Doppler velocimeter (LDV) operaung in a backward and several wire screens were used in the front section of the scattering mode and utilizing a counter for processing the tunnel to straighten the Dow and to reduce the turbulence level Doppler signals. The counter mterfaced with a microcomputer A single speed fan and a mixmg chamber were attached to the for rapid data acquisition and reduction. Glycerin parueles, tunnel entrance secuan to provide a forced rJr Gow through two to five microns in diameter, were mixed with the inlet the test section at a controlled rate. Forced flow velocities forced air flow to provide the scattering centers for the velocity from 0.03 to 13 m/s could be provided through the inlet section measurements. Flow visualizations were carried out when of the duet by controlling the pressure drop across the fan and needed by using a 15-watt collimated white light beam. 2.5 cm the inlet mixing chamber. in diameter, and Glycerin panicles as scattering centers. Heat flux measurements at the beated wall were performed by using a Wollaston Interferometer (WI) which provided a 15.24 cm diameter beam of parallel light for interferometric measure-ments. This type of interferometer and its use for measuring and deducing the neat flux from the interferogram were l described in detail by Semas et al. (1972) and by Younger (1987). Each interferogram typically had 9 fringes terminatmg ou'imr
\ -
/ at the beated wall from which fringe displacements could be measured. Dese fringes usually spanned a length of about 9 em of the bested wall. Heat flux measurements from each interferogram were averaged over the 9 cm length, and the average magnitude was considered as occurring at the mid-point of the mterferogram. The duct was aligned with the beam of the interferometer and was moved vertically up or domi to make beat flux measurements at different axiallocations of the clam wwso- -- - .-
duct. All measurements, velocities and wall beat fluxes, were made after the system had reached steady state conditions. Under these conditions the temperatures of the beated and the "0 '" adiabatic walls varied by less than 0.50 C during a four bour period. Tbc beat aux and the velocity measurements were not um 121 c' made simultaneous . De air pro rties were calculated at a w s== - , mean temperature, o + T1)/2, t is based on the inlet air temperature to the uct and the temperature of the bested wall. Flow visualizations were used to determine the onset of 7 flow reversal and the depth of its penetration. J ( NUMERICALANALYSIS Na* ,
- 3. Predictions of the flow and temperature fields in the experimental geometry were obtained by numencally solving ,
the governing elliptic partial differential equations. De flow 1 was assuned to be steady, twcslimensional, laminar, and the 8"*"ar - properties were considered constant (evaluated at the average sama cw n em temper,aturet (Tge+To govern)/2). By utilizing the ,Boussinesq l
, appronmation, mg conservauon equanons can be ,
hsted as follows : i N As==W7 atcm m. l au av
- + - -0 (1)
#x ay Fig 1 Schematic diagram of air nmnel 38 1
4
r , 3(uu) a(uv) 1 aP ax ay p ax ax ay a AT = 29.0' C o AT = 35.0' C v AT = 40.5 C
+ 9# (T-To) (2) (2) e AT = 46.1 C
" - analytical result m s,7 3 4. p m. m . . . . . .. u -
AfuT) a(vT) a'T g
+
,y a( p + a'T ) (4) u
. ... a.: ...
Gr xx 10-8 The applicable boundary conditions are given by Fig. 2 Heat transfer from a vertical, heated, constant y = 0: u = v = 0,aT/sy = Oor T = T (measuredvalues)(5) 2 temperature plate by natural convection y - b : u = v = 0, T = T1 (6) x = 0 : u = uo y = 0, T = To (7) presented in Fig. 3 as a function of the axial distance x. Measurements at axial location of x = 38. 55 and 69 cm, for heating conditions of (Ti . T i = 24,30,38,460C, and far x = L : s ' u/a x8 = 0, a vla x = 0, s ' Tla x' = 0 (8) duct wall spacings of b = t195,3.71,3.18, and 4.37 cm, %rc ne numerical solution of the above equations was obtained made to cover a wide ra. ige of the laminar natural convection domain in this geometry, M < (b/L)Grb.Pr < 10,000. The by using a finite-difference scheme, embodied in the computer code TEACH using the SIMPLE algorithm. The grid distribu- results in Fig. 3a and 3o a e tbr a duct spacmg of b = 1.91 cm, while those m Fig. 3c are lor a duct spacing of b = 4.37 cm. tion in the calculation domain was non uniform m both the The temperature differen , (T - T4 is 240C in Fig. 3a and longitudinal and the cross-Dow coordinate directions. A large 460C in Fig. 3b and 3c. Ead. 1miihenel point in these fig-number of grid points were placed in the regions where steep variations of velocities were expected. Some of that informa- ures represents thetaaveraf.e ';oniax.al a single Nollaston Interfe tion was deduced from the experimental observations and also ometer interferogram en at a given location. The dotted line on these Fgures represents the predictions for a from preliminary calculations using an equally spaced numeri-cal gnd. It was established from compansons of predicted single uniformi, Lted vertical plate under natural convection conditions (Ostrach,1953), and the solid line represents the velocity distributions using different grid density at different predictions for the asymmetrically heated duct with conditions axiallocations that, for the range of parameters mmmed in this study, a grid density of Nx xN corresponding to the expenmental geometry. providing a grid-independent soluti5n=. 90 x 60 is suf5cient for De heat transfer coefScient results in Figs. 3a 3e indicate a good agreement between the predicted and measured values. It is also clear from the figure that for some thermal conditions, DISCUSSION OF RESULTS Fig. 3a, the local heat transfer coefficient at the heated wall in he two-dimensional nature of the flow was verified by the vertical duet could be smaller than the local beat transfer measuring the streamwise velocity distribution at the midplane coefficient at a single uniformly heated vertical plate under the b/2, across the width of the tunnel same thermal condition. For these thermal conditions, the between the two walls, y = ion for isothermal Dow conditions. thermal boundary layer adjacent to the heated wall becomes in the spanwise, z, direct nese results reveal a constant velocity across 80% of the relatively large m companson to the wall spacint,, and beyond duct's width, thus confirming the two dimensional nature of the some pomt along the fen Dow any from the side walls. All reported streamwise velocity becomes relatively high,gth causing thermalofsaturation. the duct the In that bulk tem distributions in the transverse 'y" direction were measured region the heat transfer coefficient decreases at a rate faster than what occurs for the sinr.le uniformly heated plate. The along the midplane (spanwise direction, z = 0) of the duct's < adiabatic wall that forms the duct also helps to form a chimney width. The inlet veloc:ty prc'le was measured and was found to be uniform over 90% of tl.s cross sectiott nis justi5ed the effect that increases the velocity adjacent to the heated plate as use of the-measured centerline inlet velocity as the inlet condi- compared to that of the single umformly heated plate, and thus tions in the numerical model, he uncertamty associated with causes an increase in the heat transfer coefficient. His can be temperature measurements was determined to be 0.20C and seen by comparing the results of Fig. 3a with those in Fig. 3b. in that case an increase in wall temperature increases the chim-with the velocity measurements I percent. Similarly, in order ney effects and thus increases 'se heat transfer rate as com-to establish the accuracy level of the Wollaston Interferometer pared with the single uniformly beated plate results. Similarly, for measurin6 the loca, surface heat Dux, local beat transfer the results in Fig. 3b and Fig. 3c illustrate that an increase m rate from a smgle vertical heated plate was measured under ntal and analytical the duct wall spacing decreases the chimney effect and thus natural results, Nuxconvective
= 0.508Pr condig( . The expex '0.952 + Pr ) 4Gr W are the decreases com-heat transfer rate. It should be noted that if the pared in Fig. 2 in terms of Nux vs. Grx . Rese results indicate adiabvic wall is removed sufficiently away from the heated that the present Wollaston Interferometer setup could be used plate, s.e. if the wall spacing of the duct is mereased, then the to measure the local surface heat Dux to within an accuracy duct geometry becomes ecjuivalent to the vertical uniformly heated plate case. This lim 2 ting flat plate condition a ears to level of1.5% <
occur when (b/x)GrbP r > 7000. For cases when( duct De measured and predicted local beat transfer coefficient, 200 in Fig. 4, the heat transfer from the heated wall of br, for natural convection at the bested wall of the duct are, could be smaller than the single uniformly heated vertical flat 39
10 set -- - w
- 7. 6 s, .... -
- i. .'
5 .'~* a b = 0.95 cm 2 A b = 1.91 cm o - it . o b = 3.*8 cm
. e b = 437 cm 0 0.2 0.4 0.6 0.3
- single beated plate aM Fig.3a aT = 24' C,uo = 031 m/s and b = 1.91 cm ir , , , , , ,
10 (b!x)GrbP r a Fig.4 Nusselt number for the bested wall in natural p convection 6 . ,s
.E. '.'
E 4 . j - . . . . . , 2 - O 0 g , s ,. ,. 0.2 0.4 ' 0.6 0.s. y -
*W
% pure free convection -
Fig. 3b aT = 46' C,n =o0.41 m/s and b = 1.91 cm
$ pure forced convection to correlation equation with N = 2.0 redicted results a = 1.91 cm 7 . o b = 3.18 cm 6 's, 1 .
.r, , ,, ,,
E, . \ ..., Grx/Re x2 I " ~ o-- Fig.5 Correlated mixed convection local Nusselt 2 - number for the beated wallin a vertical duct. 0 0 0.2 a4 as as duct is smaller than the duct wall spacing. De results for this regime can be e rrelated by the followmg relation. Fig. 3c aT = 46' C, 031 m/s and b = 437 cm (o experimental results,- predicted result, Nux/Rex 0.5 = ((0.296)N +(0359(Grx/Re x )0.25)Njl/N (9)
- single verticalplate result) ne value of N = 2.0 seems to give the best correlation for Fig.3 Heat transfer coeffi-fent along the heated wall of the data. His form is similar to the one proposed by Chen et the duct al. (1986) for mixed convection adjacent to beated flat plate where the value of N for that case was 3. Note that the r-imental R olds number was calculated based on o u whi is plate values along the length of the duct due to thermal satura* the center e velocity at the inlet section of the duct.
titn and boundary layer interaction with the adiabatic wall. Measurements of beat transfer in the starved convection Revened flow from the exit section of the duct was not regime, where the centerline velocity at the inlet section of the observed in the experimental geometry for the experimental duct was lower than the induced value natural convection, conditions in natural convection flow. are presented in Fig. 6. The beat trans er coef5cient is pre-sented in that figure as a function of the flow restriction ratio De measured beat transfer results for the beated wall of the uonM . For some of the starved Dow meditions, the asymmetric duct under heanng (bu cy forces) accelerates the Dow at some point of N /Rex N'vs.ed Or convectio in Fig. conditions 5, and ey are com are ared resented in terms along the du en to a level bij;ber than what can be pro-with e natural and orced convection heat transfer ts vided by the fixed nlet Dow condiuon. Dus additional mass for a single vertical uniformly bested plate. De results in this Dow rate must be provided to that section of the duct via a figure were generated by using the asymmetricall bested duct reversed flow from the top section (normally an exit section) of with ull spacings of b = 3.18 and 1.91 cm and e inlet velo- the duct to satisfy the conservation of mass and momentum. dty was larger than what would develop under naturally con- The reverse Dow starts at the exit section of the duct and Dows vective conditions, Le. not under a starved flow condition, and downward adiscent to the cooler adiabatic wall. De penetra-where the boundary layer adjacent to the heated wall of the tion depth of that reversed Dow depends on the level of the 40
w *
\
l 8 - g ,, 7 a x = 32.1 cm .
. o x = 66.7 cm u .
b = 4.37 cm ^ G, - u
,' no reverse flow :
o x = 16 cm A X = 44 cm reverke OoW exisu but did not penetrate to 8 this location
*pe" [cti n revene flow exists and penetrate below this location I
'. u u a u . o La e a u u u a uon
/u t (m/s)
Fig.6 Heat transfer coefficient for the bested wall in Fig.7 Axlal velocity, distributions in the duct for starved convecuan flow, AT=250C, uo =0.15 m/s, the starved cotrvection flow regime.aT = 300C and b = 3.17 cm starved flow condition as dictated by the geometry, hunting _ , level and the fixed inlet velocity. De results presented in Fig.
- 6 correspond to measurements at an axiallocation of x = 32.1 ,
and 66.7 cm when the duct acing is b = 4.37 cm and the tem. u . perature diHerence is (T t o) - 300C. De measurements in . that Egure represent starve now conditions with and without a reversed Gow region. For the case when a reverse flow occurs o at the exit section of tbc duct, measurements of beat transfer p inside and outside the reversed Dow region are identified and A presented in the figure. De data indicate that reverse flow will a , occur when the starved flow satis 5es the conditions of n /uon< o x = 16 cm 0.5. From the available but limited set of experimental data,it . A x = 44 cm also appears that the beat transfer from the heated wall e
- o x = 74 cm depenas more strongly on the level of starved Gow conditions -prediction (ugu )when n the reverse Dow occurs, Le,when u /un n < 0.5. ,
Flow visualizations were carried out to determine the occur- L.: . u u u u u rence of the reversed Dow and its penetration depth under u (m's) starved flow conditions. De results show that reverse flow will occur in the duct when u /uo n < 0.5. The penetration depth F g. 8 Axial velocity distributions in the duct for starved increases rapidly as the starved flow condiuon increases, and starved convection flow, aT = 400C, uo= 0.15 m/s, and reaches its maximum value for the experimental geometry at b = 3.17 cm about 0.7 - 0.8 meters (70 - 80% of the duct's beight) when u /uo n < 0.2, where it remains approximately, constant as the level increases for a fixed geometry and inlet velocity, a starved degree of starved Dow condicon contmues,to mcrease. When Dow regime could develop in the duct, causm' g a reversed Dow the penetrauon depth ceases to increase with increasing buoy- to occur along the cooler adiabatic wall of tbc duct. ancy,u /un n < 0.2, the transition fromInmmar to turbulent flow starts to occur to balance the momentum and mus conserva- ne effects of duct wall spacing on the velocity distributions tion requirements. under natural convection conditions are shown in Figs. 9 and 10 where the beated wall temperature was maintained constant at Measurements of the velocity distribution m. the asymmetn- 250C and the wall spacing was changed from b = 3.17 cm in cally beated duct were performed using one component LDV pig,9 to b = 1.91 cm in Fig.10. When the duct wall, spacing is system and they were not done simultaneously with the pre- decreased from b = 3.17 cm in Fig. 9 to b = 1.91 cm in Fig.10. viously reported beat transfer measurements. ney were, how- the naturally induced velocity at the center of the inlet section ever performed on the same experimental duct. Measuremenu of the duct mereased from uo= 0.25 m/s to ue= 0.40 m/s due to and predictions of velocity distributions in the mixed and the chimney effects. The increase in velocity reduces the rela-starved Gow regimes are presented in Figs. 7 throu 11. The tive influence of the buoyancy force on the velocity distribu. numerically predicted values, which are presented solid lines tion, and thus the skewness of the velocity distribunon toward in these figures, are in very good agreement wi measured the heated wall decreases as observed in Figs. 9 and 10. When values. Measurements and predictions are presented at differ- the inlet velocity is increased by forced Dow to a value of au = ent axial locations, for different inlet velocities and different 0.9 m/s while holding the other conditions of Fig.10 constant, heating (buoyancy force) levels to explore the effects of the the buoyancy force effects decreases even further, and the various parameters on the final results. ne characteristic Dow skewness in the velocity distribution diminiibes further as behaviors consist of a skewed velodry distribution toward the shown in Fig.11. It is important to note that by utilizing the heated wall, with the level of skewness increasm' g as t!'e level of u, the numeri-adiabatic boundary condition, aT/sy = 0 at y = distribution that cal code provided predictions for the velocity buoyancy force increases, as shon in Figs. 7 and 8. De differ-ence between the conditions of Figs. 7 and 8 is that the temper- agree favorably with measured values when there is no reverse ' sture difference was increased from 25 to 400C while maintain- Dow in the de1 te. in the natural or mixed convection regune ing everything else constant, no = 0.15 m/s and b = 3.17 cm. (Fig. 9 Fig. s1). But for the starved Dow conditions, where ( Rese two Sgures also show Ihat as the buoyancy (heating) reverse flow appears in the duct, the use, of the adiabatic 41
l 8 _ significantly the magnitude of the velocity in the reverse flow as shown in Fig.12 where a sample of the results from Fig. 7 are
- 7 , . presented and compared for the two stated boundary condi-i l
,' tions.
l Measurements of the temperature distribution inside the I u duct were not performed but they were redicted the
- numerical code and a sample is presented in ig.13. Th fluid i, temperature and the adiabatic wall temperature increase as the
... axial distance increases, because of the added energy from the bested wall. The reversed flow could not be detected from the
, ,. o x = 22 cm measurements of the temperature distribution.
A x=51c:n o x - 74 cm e
- Prediction = 'C ,
'. ... u ..: ... u u ua = 0.15 m/s b = 3.18 cm
,g3)
Fig. 9 Axial velocity distributions in the duct for natural con- ... ]',Y, ef results , vection flow AT = 250C, ua= 0.25 m/s, and b = 3.17 cm E* a x = 74 cm predicted results s
.. m = 44 cm 8d *
/, . - x = 44 cm
[j[ ( // [( .--- x = 74 cm
- . - x = 74 cm (using adiabtic wall) g ,, \N f~
(using measured wall temperature) 4^ *
'... . ... ... u ...
,,, u(m/s) o x = 22 cm Fig.12 Sensitivity of numerical solution to boundary condi-a x = 51 cm tions in the starved flow regime u o x = 74 cm
-prediction a
'. u ... u ...
u (m/s) x = 21 cm / x = 31 cm / Fig.10 Axial velocity distributions in the duct for natural con- ,', x = 51 cm vection flow aT = 250C, u o= 0.40 m/s, and b = 1.91 cm x = 74 cm s e , u
/
/
" o x = 22 cm
, ... ,9 aA a x=51cm '
o x = 74 cm " * " " *
** -prediction .
y/b Fig.13 Dimensionless temperature distributions in tbc duct u for starved Dow, AT= 460C, u o= 0.41 m/s, and b = 1.91 cm
'. u u ..: u ..- CONCLUSION u(m/s)
In the present study, velocity and wall beat Dux measure. Fig.11 Axial velocity distributions in the duct for mixed con- ments are reported for mixed convection air flow in a vertical vection flow aT = 250C, u = o 0.90 nVs, and b = 1.91 cm two dimensional duct that is formed by one adiabatic wall and one uniformly heated wall. Results are reported for different boundary, condition in the numerical code produces results that buoyancy force effects by varying the heated wall temperature, deviate u ' icantly from measured data to the reversed flow inlet air veloci and wall a The numerically predicted region. r that reason measured adiabatic wall temperatures velocity distri tion and all est flux compare well with were used as the boun condition in the numerical code for measured-values. The velocity distributions are found to skew that Dow domain (Fig. 7 Fig. 8). 'Ite velocity in the reversed toward the heated wall as the buoyancy force increases. A Dow region is normally very small and its magnitude is very starved Dow regime could develop in the duct, at high buoyancy sinsitive to the wall temperatures. Thus amall deviations from levels, causing a reverse Dow to occur along the cooler adiab-the adiabatic condition in the experimental apparatus affect stic wall of the duct. For the natural convection case a reduc-42
i tion in the duct wall spacmg, whUe keeping au the other condi- Ou, J., Cheng, K. C.. and Lin.R.,1976, ' Combined Free and tions constant. wtll produce an increase to the naturally induced Forced L.aminar Convecuon m inclined Rectangular Chan-inlet velocity due to the chimney effect. His causes the buoy- nels,' httInstional Joumal of Heat and Mass Trnnsfer. Vol ancy force effects to decrease and decreases the skewness cf 19, pp. 277 283. the velocity distnbutions. For tbc mixed convection case, a Quintiere, J., and Mueller, W. K.,1973,'An Analysis of simple correlation was developed for predicting with reason- Ltminar Free and forced Convection Between Finite Vertical able accuracy the local mixed convecuon Nusselt number as a Parallel Plates," Journal of Hgat Trnntfer. Vol. 95, pp. 53-59, function of the Grasbof and Reynolds numbers. Measurements Sernas, V., Fletcher, L S., and Jones, J. A,1972, 'An Inter-of the wall beat Dux show that the beat transfer rate is sensitive ferotnetric Heat Flux Measurmg Device,
- I. S. A Transne-to the level of starved flow condition, ugun, in the region liant Vol.11, No. 4, pp. 346-357.
where ugun < 0.5. Reverse flow will occur m the duet adja- Sparrow, E. M., Chrysler, G. M., and Azevedo, L F. A., cent to the adiabatic wall when uguo < 0.5. De penetrauon 1984, ' Observed Flow Reversals and Measured Predicted depth of the reverse flow will increase rapidly as the buoyancy Nusselt Numbers for Natural Convection in a One-Sided levelincreases through the decrease of the starved flow par. Heated Vertical Channel," Journal of Hent Transfer. Vol. ameter, ugu , but it reaches a limiting value when ugun< 106, pp. 325-332. 0.2. Higher Euoyacey levels will not cause an increase m the Sparrow, E. M., and Azevedo, L F. A.,1985, ' Vertical-penetration depth but will cause a transition from laminar to Channel Natural Convection Spanning Between The FuUy-mrbulent flow. Developed limit and the Single Plate Boundary-Layer Limit," International Journal of Heat and Tramfer. Vul. 28, pp. 1847 1859, ACKNOWLEDGEMENT Wirtz, R. A, and Stutzman, R. J.,1982,
- Experiments on Free Convection Between Vertical Plates with Symmetric De study reported in this paper was supported in part by Heating
- Journal of He at Transfer. Vol 104, pp. 501507.
the National Science Foundation under the grants NST Younger, M. B.,1987, " Experimental Measurements of MEA 8300785 and NSF CTS-8923010. Natural and Mixed Convecuve Heat Transfer in an Asymme-tncaUy Heated Parallel-Plate Channel," M S. Thesit Mechan. ient Ennneerine Derartr-ent. University of Missoun.Rotta. REFERENCES Aung, W.,1972,' Fully Developed I aminar Free Convection Between Vertical Plates Heated Asymmetrically,' hlCUll: gional Journal of Heat and Mass Tramfer. Vol.15, pp. 1577 1580. Azevedo, L F. A, and Sparrow, E. M,1985, ' Natural Con-vection in Open Ended inclined Channels,' Joumal of Hent Transfer. Vol 107, pp. 893 901. Baek, B. J., Palaski, D. A., Armaly, B. F, and Chen T. S., 1990,
- Mixed Convection in an Asymmetrically Heated Vertical Parauel Plate Duct Flow,' Proceedines in 9th Inter-l national Hent Transfer Conference .Vol.2, pp. 369 374. 1 Beckett, P. M., and Friend,1. E.,1982, " Combined Natural and Forced Convection Between Parauel Walls: Developing -
Flow at Higher Rayleigh Numbers.* Intemational Journal of l Hent and M ass Transfer. Vol 27, pp. 611-621. Bodota, J. R. and Osterle, J. F,1962,"The Development of 1 Free Convection Between Heated Vertical Plates," Joumal of l litat Trnnsfer. Vol. 84, pp. 40 44 Cebect, T., Khattab, A. A., and L.amont, R,1982, 'Com- i bined Natural and Forced Convection in Vertical Ducts,' i Heat Trsefer 82. Proceedines of the 7th Intemational Heat Trnmfer Conference. Vol3,pp. 419-424 l' Chen. T. S., Armaly, B. F, and Ramaebandran, N.,1986,
' Correlations of Mixed Convection Flows on Vertical, loclined, and Horizontal Flat Plates,
- Joumal of Heat Tram-ist Vol 108, pp. 835-840.
Emery, A. F., and Gessner, F. B.,1976, 'The Numerical Predicuon of Turbulent Flow and Heat Transfer in the Entrance Region of a Parallel Plate Duet,' Journal of Heat Itansfer. Vol 98, pp. 594-600. Habchi, S, and Acharya, S.,1986,'Imm nnr Mixed Convec-tion in a Symmetrically or Asymmetricauy Heated Vertical Channel,
- Numerical Heat Tramf er. Vol. 9. pp. 605-618.
Kamotani, Y., and Ostrach, S.,1976, *Etteet of Thermal Instability on Hermally Developing i aminar Channel Flow,' Joumn! of Hent Transfer. Vol. 98, pp. 62-66. Mercer, W. E., Pearce, W. M., and Hitchcock, J. E,1967,
- Laminar Forced Convection in the Entrance Region Between Parauel Flat Plates,' Joumal of Heat Transfer. Vol 89, pp. 251256.
Ostrach, S,1953,"An Analysis of I ammar Free Convection Flow and Heat Transfer about a Flat Plate Parallel to the Generating Body Force,' NACA Rept.1111, Washington, D.C. 43
HTD-Vol.163 ) l l l l MIXED CONVECTION i HEAT TRANSFER 1991 - presented at THE 28TH NATIONAL HEAT TRANSFER CONFERENCE MINNEAPOLIS, MINNESOTA JULY 28-31,1991 sponsored by THE HEAT TRANSFER DIVISION, ASME edited by DARRELL W. PEPPER ADVANCED PROJECTS RESEARCH INC. BASSEM F. ARMALY UNIVERSITY OF MISSOURI-ROLLA , 5El..'a M. All EBADIAN , - FLORIDA INTERNATIONAL UNIVERSITY d' ni . . . , , ;L P. H. OOSTHUlZEN QUEEN'S UNIVERSITY
/ 9:
THE AM ERIC AN SOCIETY OF M E C H ANIC AL E N GIN E E R S United Engineering Center *O*O* 345 East 47th Street *O*O+ New York, N.Y.10017
- a HTD.Vol.163, Mixed Convection Heat Transfer ASME 1991 MIXED CONVECTIVE HEAT TRANSFER CHARACTERISTICS OF A DOWNWARD TURNING BUOYANT CEILING JET Kamlesh Kapoor and Yogesh Jaluris Department of Mechanical and Aerospace Engineering Rutgers, State University of New Jersey New Brunswick, New Jersey ABSTRACT Nup Nusselt number based on D,hD/k aNup difference between the maximum and the minimum value An experimental investigation has been carried out on the of Nup at the corner mixed convective heat transfer characteristics of a buoyant ceili:3 jet turning downward at a corner. Such flows are q local heat transfer flux to the surface frequently encountered in buoyancy driven transport in enclosed Q total net heat transfer to the isothermal surface from the regions, such as those associated with thermal energy storage heated jet flow pt:blems and enclosure firet However,very little work has been doIe on the basic heat transfer mechanisms in such flows, Ogg total thermal energy input by the ceiling jet, particularly on the local heat flux distributions over the ceiling poUACo o p(To T.) sid the vertical wall near the corner. In this study, a two. Re Reynolds number,UO DN dimensional horizontal jet of heated air is discharged adjacent to Richardson number, Rl = Or/Re 2 the underside of an isothermal horizontal plate whose other end is Ri attached to an isothermal vertical plate, making a right angle To discharge temperature of the ceiling jet corner with it. The distarice between the jet discharge and the T, temperature of the isothermal surface corner could be varied Extensive heat flux measurements were carried out for different inflow conditions of the jet. The T,,, temperature of the surroundings variation of the local heat transfer rate along the ceiling and the vertical wall shows a minimum followed by a recovery, as the Un discharge velocity of the ceiling jet f1;w turns at the corner. The, average Nusselt numbers for the W width of the horizontal and vertical plates horizontal surface and the vertical walls arg obtained as functions x horizontal coordinate distance, measured from the slot cf the mixed convection parameter Or/Re which is also known as the Richardson number Ri of the discharged ceiling jet. The x distance along the two surfaces from the slot t:tal heat transfer rate to the isothermal ceiling and the vertical wall by the jet flow is siso obtained. The study brings out several y vertical coordinate distance measured downward from basic considerations in the thermal transport from a buoyant jet thesfor t) the isothermal surface over which it flows, including the x transverse coordinate distance e.ffects of opposing buoyancy for the vertical wall Greek symbols NOMENCl>.TURE # coefficient of thermal expansion of the fluid cross sectional area of the slot through which the heated v kinematic viscosity of the fluid A, air jet is discharged penetration distance of the jet flow, measured downward
#p Cp specific heat of the sluid at constant pressure from the slot D width of the slot for jet discharge to density of air at the jet discharge g magnitude of gravitational acceleration i dimensionless temperature, # =(T.T.)/T o T.)
32 Or Grashof number,gA(TgT.)D /r h local heat transfer coefficient, q/(To T,) thermal conductivity of air Buoyant jets are of considerable importance in many k practical problems. For example, heat rejection to the atmosphere L horizontal distance between ceiling jet discharge and and to water reservoirs, heat extraction and inermal energy
**'**' storage involve buoyant jet flows. Similarly, at the early stages of 9 .
A
f .
. j l
1 1 I fire growth in an enclosure, the ceiling jet driven by the fire The flow and heat transfer due to a horizontal buoyant plume turns downward at the corners and generates a wall flow ceiling jet has also been considered by severalinvestigators. Most with opposing thermal buoyancy. The flow and heat transfer of these studies have been concerned with enclosure fires. Alpert characteristics of vertical buoyant jets have been considered in (1975) studied the flow generated by the fire plurne driven ceiling the literature. But very little work has been reported on the flow je t. The characteristics of a ceiling jet were investigated by i of a downward turning buoyant ceiling jet. This is the problem Cooper (1986,1987) He developed analytical estirnaies to predict considered in the present paper, which reports a detailed the spread and entrainment into the ceiling jet. Alpert (1971) and esperimental investigation on the heat transfer characteristics of You and Faeth (1978) have measured steady state velocity and such flowt Interest lics mainly in the transport processes in the temperature profiles in ceiling jett Transient ceiling jet vicinity of the corner and in the negatively buoyant flow that characteristics were investigated by Veldman et al (1975) and arises af ter the turn. Motevalli et al. (1987) Transient ternperature and velocity distributions were measured in the ceiling jet flow caused by a Laminar buoyant jets have received considerable attention fire plume impinging on a ceiling. Cooper (1982,1969), in a in the literature, as reviewed by Jaluria (1986) Vertical jets with a theoreticalinvestigation. obtained the heat transfer from a ceiling small amount of thermal buoyancy are considered in many jet to the horizontal boundary. He developed an algorithm to investigations. However, turbulent buoyant jets have been studied predict the thermal response of the ceiling, for a variety of more extensively because nf their practical relevance, see the realistic constructions, to different fire scenario, reviews by Chen and Rodi (1979) and List (1982) Goldman and Jaluria (1966) carried out a detailed czperimental study of It is seen from the above review of the relevant literature I vertically discharged, negatively buoyant, wall jets. They obtained that, although many studies have been carried out on vertical and the flow characteristics of a two.dirnensional wall jet, with the l ceiling jets, no effort has been made to study the heat transfer buoyancy force acting in the direction opposite to that of the characteristics of a downward turning buoyant ceiling jet, flow, in an isotherrnal environment An insulated surface was particularly near the corner. In this paper, an experimental study employed to simulate the wall The penetration distance i . is carried out on the heat transfer behavior of a horizontal two-which characterizes the downward penetration of the flow, w[s dirnensional buoyant ceihng jet as it turns downward at the obtained and related to the inflow conditiops of the jet in terms corner. The results presented here consist largely of temperature i of the mixed convection parameter Gr/Re , defined later. This and local heat transfer data. Detailed velocity measurements were ! paramgter is also of ten known as the Richardson number Pi
- also undertaken but are not presented here for conciseness. The Gr/Re , where Ri is defined in terms of the discharge velocity
- ternperature and slot width, as defined in the nomenclature. The flow characteristics. including entrainment, will be the subject of 1 an accompanying paper (Kapoor and Jaluria,1991b) '
penettgtion distance 6 was found to decrease with an increase in 1 Gr/Re , indicating the,effect of increasing therrnal buoyancy. In this paper, the jet penetration distance and the thermal field were measured. The local heat transfer flus to the The fluid entrainment into 1 was found to increase with Gr/Re,the flow was also measured and horizontal and vertiegl plates was measured extensively over the over the investigated rgnge of range of 0.02 s Gr/Re s 04 From these heat flux rnessurements. l 0.01 to 0.11 Jaluria and Kapoor (1988) extended this Gr/Re range the heat transfer from the ceiling jet to the ceiling and to the ! to Oli and found that very large flow rates are indeed entrained ' into the jet flow due to the large horizontal spread of the flow vertical wall is obtaiced. The distance between the jet dischstge and the corner, l., has been varied in the czperiments to ! particularly near the stagnation region. Kapoor and Jaluria (1989) determine the effect of the parameter UD on the heat transfer have reported a detailed experimental investigation on the heat ) rates. Results are presented for three values of UD. The basic ' transfer characteristics of such two-dimensional negatively heat transfer characteristics of the downward turning jet are buovant wall jets in isothermal media. A heated jet was i emphasized and some relevant aspects of the resulting flow are discharged vertically downward adjacent to a water cooled, also reportect The heat transfer in the vicinity of the corner is isothermal, aluminum plate. The local heat transfer flux was important in the simulation of these flows. Similarly, the measured over the plate various values of the mixed convection parameter Gr/Re{or resulting negatively buoyant flow is important in the modeling of
, ranging f rom 0.05 to L0i It was enclosure fires to study the changing environment in a room.
found that the total heat transfer rate from the jeg to the l isothermal surface decreases with an increase in Gr/Re due to EXPERIMENTAL ARRANGEMENT the decrease in the penetration distance and the consequent reduction in the heat transfer area with increasing buoyancy Figure 1 shows a sketch of the experimental arrangement. effects The effect of the wall temperature on the jet penetration A blower sends ambient air, over a wide range of flow rates, distance was also determined and it was found that the through a heated copper tube. The blower is fixed at the bottom penetration distance decreases with an increase in the wall of a vertical duct which is 135 rn high and 03m x 03m in cross j tempera ture. This was due to the decrease in the beat transfer section. To ensure uniform flow and a low level of turbulence in i l rate which results in a higher buoyancy level in the jet flow and, the duct, a honeycomb section and three very fine screens are ' hence, smaller penetration distance- provided at the entrance. Two very fine screens, which are 03m apart from each other, are provided in the central portion of the The penetration of vertical buoyant jets into thermally duct. This arrangement provides a very low level of turbulence. stratified media has also been investigated by many workers. with intensities usually less than 03% in the duct and a fairly Jaluria (1982) obtained a numerical solution to the problem of uniform flow, with a variation of less than 5% in the velocity is penetration of a laminar vertical plane jet into a thermally obtained across the duct width. The copper tube, which is 5 cm in stratified environment. Kapoor and Jaluria (1991a) have carried diameter and I m in length,is heated by means of three fiberglass
. out an experimental investigation on the flow and therrnal insulated heaters which are wrapped around it. The electric ji characteristics of a plane turbulent wall jet in a two-layer energy input to each of the heaters is varied by means of power thermally stable environment The penetration distance in in the controllers. A two-dimensional diffuser, whose width could be stably stratified medium was found to be larger than thft in an varied,is fired to the copper tube. Several diffuser designs were
- , isotherinal medium at the lower temperature level Again, this considered to ensure uniform two. dimensional flow at the exit 3 behavior is expected from the increased opposing buoyancy effect Velocity and temperature rnessurements across the diffuser exit 6
in the latter case. The mass flow rate penetrating downward showed fairly uniform distributions The variations in the h across the interface between the two layers was estimated and temperature and velocity were less than about 3*C and 2 em/s, l compared with the jet inlet mass flow rate. The heat transfer to generally giving rise to values that were less than 2 and 5 percent p the isothermal surface was also measured for several wall of the mean values, respectively. The copper tube was provided temperatures and jet dischstge conditionk with an insulation jacket. which consists of an inoer layrt of 10 l
1 i . I, Eg' S .CT80s. 1 r s ( t 5
-r ; re.,se...u. -
l
-E-8 35 = . . ~,,,,,,,,_n,,
-- a _ . _
'-- .y a=== y- ,.;-.. l .. m .. W i 3
_.m.< l} r
=
, ,, m g - igtu... tate
- I
-t 3 etste sesotavros sacs., . . _ l q
. _...,.;_ .. . ,s -c1
( ... : - I j 7
== :
\j '
'5- j j
..,. . s
.m . ..
--- i. ,
- i. --
1.._ .__,m_. e.1, .
- - - - I :
- e . s. ~ i; g' g i ins Fig. L Experimental arrangement for the study of the mixed _b ' '* I t.3 4 ~}
convection flow of a heated.two. dimensional ceiling jet i ~I sees witw discharged horizontally into an enclosure. mm., - <-.s.-a n .
** 3 As.-
raont vitw r -- , wv e.m?fraL J.T - se . vo . ne.e n.,,,,,,, ue . i s. Fig 1 Sketch of the experimental arrangement used for y,,, ,,,, simulating isothermal horizontal and vertical surf acet 36 s e 4 g
$2 U [Q- .s
.__ 'CW W -'
.S$
is
.49 e .es
. =
.es e -
** *.s e.a e. e.< e.s e.a e.,
8 se c.i , Fig 1 Variation of the penetration distance s with the mixed Fig,4. Emperimentally obtained isothernes for a downward P turning buoyant ceiling jet at Gr/Re 2 ,g3gy convection parameter Gr/Re2,
5 1 l l l i i fiberglass tape and outer layer of glass wool rnaximum temperature difference measured between any two , embedded thermocoup!es on the two plates was always less than The discharge temperature of the jet To could be varied up LO*C, giving an error of less than 3% in the plate temperature to 150*C and the discharge velocity could be adjusted between 03 datt
' and 2.5 m/t Thus, a fairly wide range of the governing parameters, Re and Gr, based on the inlet conditions, could be The heat transfer from the hot jet to the ceiling and obtained with the present experimental arrangement. The vertical plate was measured by using microfoil heat flow sensors maximum values for the Grashof number Gr and the Reynolds (RdF type 20472 3) Each of the heat flux sensors was 15mmmomm l nutnber Re obtained in the present experiments, were 3500 and in surf ace area and 03mm in thickness, and could easily be 6
1 10, respectively. The discharge velocity of jet was obtained by attached to the plate. The typical distance between two heat flux measuring the velocity distribution at the crit of the diffuser. A sensors along the ceiling and vertical plate was approximately 5.0 calibrated DANTEC hot wire sensor was kept horizontally at the em except near the corner where the sensors were placed typically exts of the diffuser to measure the velocity distribution there. I cm from each other. This arrangement provided detailed The hot wire was calibrated by using a special calibration facility distributions of the heat transfer flux along the ceiling and the which was designed to calibrate the sensor at small velocity levels, vertical plate. The electric output (in millivolts) from the heat ranging from 0 to 0.5 m/s, and for a desired flow directiort The flux sensor was converted into heat flux (in W/m,) with the help details of the calibration rcethod and of the velocity of individual calibration curves supplied by the manufacturer and measurements are given by Tewari and Jaluria (1990) verified in the laboratory by employing surfaces with known heat flux inputs. The accuracy of the measured heat flux was The discharge temperature of the jet was measured using a estimated to be very high, with an estimated error of less than 3% rake of itve thermocouples located across the diffuser. The in the present experiments. The fluctuations in the heat flux were average of five temperatures was then taken as the jet discharge found to be less than 10% in most of the cases and may be temperature T.o As mentioned earlier, the temperature attributed to the turbulent flow over the surfaces The output of distribution across the diff,iser exit was very unifortrL A the heat flux sensors were constantly monitored on a strip chart maxirr recorder and all the heat flux data were collected by using a Gr/Re(um variation employed of 3.0*C in the presentwas enessuredFor experiments. over the range further derailsof Keithley data acquisition system. Thus, the errors in the reported on this system, see Goldman and Jaluria (1986) and Kapoor and Nusselt and Richardson numbers were estirnated to be less than Jaluria (1989) 5% and 3%, rer,pectively. The temperatures were measured by copper-constant thermocouplet Considerable care had to be The heated, two-dirnensional jet of air, was discharged exercised to obtain accurate and repeatable data. In general, the horizontally adjacent to the isothermal surfaces, as shown in Figr repeatability was very high, being within 5% for most
- 1. A water cooled aluminum plate represents the ceilitig of an messurements reported here.
enclosure. Another water cooled vertical aluminum plate was fixed to Ibe ceiling plate, thus forming a 90* corner, The RESULTS AND DISCUSSIONS horizontal distance L between the corner and the jet discharge could be varied from about Ohm to 10m by moving the jet slot The height D of the two. dimensional, wide. slot along the ceiling plate. The width of the jet could also be varied discharging the jet is generally taken as the characteristic in the present caperiment, most of the measurements were taken dimension for a jet in an extensive environment (Jaluria.1960; at D = a065m and 0.0508m. Thus, this arrangement allowed a Gebhart et at 1988) The dimensionless governing parameters in change in IJD from 10 to approximately 2Q the present experiment are the Reynolds number Re, the Granhof The sides of the flow region were kept partially open in number Gr and the Richardson number R1, which js also of ten known as the mixed convection parameter, Gr/Re . The heat order to avoid thermal stratification. Extensive data were taken transfer at the ceiling and vertical wall may be presented in terms without any side walls, and it was found that the jet flow was of the Nusselt number Nugy These dimensionless quantities are confined about 4D from to athe maximum value vertical wall of about throughout the 2D from range the ceilingdefined of Gr/Re as follows-and of. Based on these experimental observations, as shown in Fig.1, the UD g#(T .T )D3 sides were closed up to 2D and 4D from the ceiling and vertical Re = 2- , Or= "" wall, respectively,in order to avoid entrainment close to the wall (1) r y2 flows and, thus, maintain the two4imensionality of the ficw. However, the results near the midplane were found to be essentially independent of the side walls. Thus, the results gA(T T*)D presented here are not affected by the edge flow or the side walls. RI . Nup = D (2) All esperimental results reported in this paper are with the k sidewalls partially closed, as outlined above, and closely represent IJ*2 physical two-dimensional flows encountered in practical situations where U and T are such as enclosure fires. g the velocity and temperature at the discharge,of the jet, g the magnitude of the gravitational acceleration,# the coefficient of thermal expansion of the fluid, h The water cooled aluminum ceiling and vertical plates are the local convective heat transfer coefficient, and v the kinematic identical in desigrL As shown in Fig. 2, four rectangular copper viscosity of air at the discharge temperature T Thus, b is given t tubes, each 2.5 cmxt25 cm in cross section, were fixed along the by b = q/(ToT 3) where q is the measured localTiest fluz and T is length of the plate. It was made sure that the surface contact the surface temperature. s between the aluminum plate and copper tubes was excellent. Water from an outside source enters at the top of the vertical The data consist mainly of the temperature distributions in plate and the water coming out of the arrangement for this plate the jet flow and of the local heat flux measurements along the was allowed to enter at one end of the ceiling plate. Water ceiling and the vertical wall for differect values of the jet coming out of the other end of the ceiling plate was allowed to discharge conditions such as temperature and flow rates The
, drain into a sink. The temperature of tre water entering the results are presented for Lp = 10,111 and 165 and the mixed
;lj. vertical plate was maintained at a desired value by mixing hot and convection parameter Gr/Re or Ri was varied from 0.02 to 0.65 in cold water streamt Twelve thermocouples were embedded in the the present experiments Some of the typical experimental results
! ;- plate to monitor plate temperatures. For further details on the obtained in this study are presented here. ll plate assembly, see Kapoor and Jaluria (1989) A fairly uniform I temperature distribution was found over the plate surface. The ' 12 i
,h V
The buoyant horizontal ceiling jet, af ter losing some of its As mentioned earlier, the local heat transfer rate from the rnementum and thermal energy to the ceiling, reaches the corner jet to the isothermal ceiling and vertical wall was measured by and turns downward along the vertical wall and behaves like a , using heat flux meters. These heat flux gages were fixed at about eegatively buoyant wall jet, as studied by Goldmaa and J,aluria 5.0 cm apart from each other along the midplane of the ceiling (1966) The hot jet flow penetrates to a finite distancepi in the and vertical plates. In order to measure detailed heat flux enclosure, due to the opposing buoyancy. It then turns upward distributions at the corner, the sensors were fixed more closely (up and flows out of the enclosure. In this process, the jet flow to 1 cm) at the ceiling and vertical plate near the corner. The entrains significant amounts of fluid from the ambieni medium. temperature of the ceiling and vertical plates was kept the same resulting in a much larger flow rate out of the enclosure, as and close to the ambient temperature (within 2LO*C)2 Extensive coropared to the discharged flow rate. heat transfer data has been obtained for 0.02 s Gr/Rc s 045 and UD = 10,13.1 anc,10. Only a few of the typical results are shown The penetration distance sp, which represents the here for concisenest downward penetration distance of the thermal effects in the flow, is defined as the vertical distance at which a sharp increase in the igure 5 shows the variation of measured best flux qfin temperature level is seen as one proceeds from bottom to the top watts /m ) along the ceiling and the vertical plate for Gr/Re = of the enclosure (Kapoor and Jaluria,1989) The variation of the 0.0435 and UD = 10. It is seen from the figure that the local heat with the mixed convection parameter transfer rate decreases along the ceiling plate and reaches its penettgtion distance # 3 Gr/Re for L/D = 141.ll,10 and 20.5, is shown in Fig, 3. The lowest value just before the corner. It is seen to increase sharply ceiling and the vertical wall temperatures are kept the same and as the jet flow turns downward at the corner and reaches its are very close to the ambient temperature (within i 1.0*Cl it is maximum value just below the corner on the vertical wall The seen from the figure thatmixed the penetration, local heat transfer flux is then seen to decrease gradually along with an increase in the convectioninparameter general, decreasef. Gr/Re the vertical wall and to become almost acto further downstreant These results are similar to those s a negatively buoyant vertical The penetration distance i has also been shown in the figure. Jet.The results for a vertical jet an also shown in the figure. The As implied by isotherms in Eig. 4, the jet flow separates from the penetration distance was found to be smaller for the higher value ceiling plate and reattaches to the vertical wall at the corner. Fig. of IJD at a given Gr/Re2 . It was found that the velocity level 5 further confirms the existence of a small recirculation zone in decreases at a much f aster rate than the thermal energy along the the corner. ceiling plate as the flow approaches the cornet. Hence the jet flow has, relatively, smaller momentum, while retaining a larger Figure 6 shows the variation of the local Husselt number amount of its thermal energy after it turns downward at the Nu n ng the ceiling and the vertical plate for four values of corner. This results in a smaller penetration distance for the Gr/R and for L/D = 10. The local Nusselt number is defined as higher value of L/D. The velocity data are not presented here for Nu = hD/k, where h = q .T,) The basic trends are similar to brevity. But the flow field measurements agreed closelv with the those observed in Fis 4. he dimensionless local heat transfer. results from the temperature and heat flux data (Kapoor and rates, in terms of the Nusse(t number, are seen to be higher for Jaluria,1991b) the smslier values of Gr/Re . It should be mentioned here that the measured vplue of local heat flux q is smaller at the lower Appropriate correlations may be obtained to calculate the value of Gt/Re . Higher values of Nun are obtained because of non-dipensional penetration distanceP3 /D as a function of still smaller value of (TgT,) It is seen from the figure that, in Gr/Re for various values of L/D. A general correlating equation general, the local heat transfer rate first decreases along the to calectate sp /D, incorporating both IJD and Gr/Re' as input ceiling plate, then increases at the corner and again decreases parameters, is along the vertical plate, becoming altnost zero at some location downstream. The penetrations distance 3P has also been marked along the ceiling and the wall 2 on the figure. is obviously The due to the decrease increasing in Nuhoundary layer thickness o b = 13D7 (UD)D(Gr/Re )el (3) D flow (Jaluria,1980) . Other correlating equations for different thermal conditions such it is seen in Fig. 6 that the recovery in the local begt as the vertical negatively buoyant jet have been obtained earlier transfer rate at the corner depends upon the value of Gr/Re . (Goldman and Jaluria,1986) The recovery in the locaj heat flux was found to be larger for the smaller values of Gr/Re . Figure 7 shows variation of this heat In order to understand the basic nature of the resulting transfer recovery, expressed in terms of ANun defined as the transport processes, detailed measurements of the thermal field difference between the rnaximum an the minimum values of the across the enclosure were carried out. The thermal field was local Nu n ta the corner, with Gr/Re It is seen from Fis 7 that mapped very c!osely, by using a rake of thermocouples ubich was decreases sharply up to a placed at the midpoint of the depth (z direction) of the enclosure, the heat transfgr recovery factor, ANu,d by a gradual decreas From these temperature measurements, the corresponding value the valueof Gr/Re = 0)01 of Gr/Re This is followe is increased to Q431 This is an espected isotherms were determined by int rpolation. Figure 4 shows a result because, at the lower value of Gr/Re2 , the ceiling jet flow typical set of isotherrns for Gr/Re = 0301 It is seen from this has relatively larger momentum before it turns downward at the figure that the horizontal buoyant ceiling jet loses thermal energy corner. Therefore, it separates from the ceiling plate at s' larger as it moves along the ceiling and turns down at the corner along distance from the corner and teattaches itself to the vertical well the vertical wall As expected, the jet flow loses thermal energy at a larger distance bel w the corner. This suggests that at the as it penetrates downward to a finite distance i , as discussed smaller value of Gr/Re , the size of recirculation region at th earlier, and then rises upward due to thermal Suoyancy and corner is larger than that foi.nd at thg higher values of Gr/Re . finally escapes out of the enclosure. It is seen from the isotherms Hence, at the smaller value of Gr/Re , the separated ceiling jet that the ceiling jet flow starts turning downward before reaching flow has to travel a larger distance at the corner before it the corner. This shows that the ceiling jet flow separates from reattaches itself to the vertical well, resulting in a higher heat the ceiling some distance before the corner and reattaches to the transfer recovery f actor. vertical wall downstream. This indicates the presence of a small recirculation tone in the corner. This conclusion was also The variation of the average Nusselt number En for the confirmed by the velocity profiles measured near the ceiling and ceiling plate with Gr/Re 2for UD=10,111 and 165 is shown in Fig. vert. cal plates The flow separation at the corner is also & As defined earlier, E 4D'k where E=O# T, T,).W.L confirmed by the corresponding heat flux measurements on the Here, W and L are the wi th and the length di ceiling plate, ceiling and vertical walls. 13 l (
u. i se a __ _ xa3 ~.y .,
,u. . . . . . .
..u. N ..-
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cel.m E Fig. 7. Variation of heat ansfer recovery factor AN"D at ebe Rg & Vari. in of ee awnge hit numkr E D *M' corner with Gr/R me IJD - 10g Gr/Re for the ceiling plate at different v. lues al I>Il I
. -- _- - -- e a - -- -
l respectively. The total net heat transfer rate to the ceiling and O, with was obtained by integrating the measured heat flux over Gr/Re 2 Figure for L/D 11 shows
= 10. the In this variation figure of O,,;i; O is nonJ 5nensionafz'ed by O'Ne.'rNth th of the ceiling plate. It is seen from thisthe figure that thfenergy input by the ceiling jet Ogg, which is total thermal average Nusselt number decreases with an increase in Gr/Re . calculated as follows This is due to the fact that, while the net heat transfer to the ceiling plate O increases with Gr/Re , the jet temperature Ogg = p,U,A,Cp(T, T,) (11) difference (T,'l');Neresses more rapidly thanThis 0,gdence a increases with an increase results in a lower average heat trans(er coefficient an It is seen 2 from this figure resu that 0,,;ift"because at the higher value looer value of ED at higher Gr/Re . Figure 8 shows that the in Gr/Re This is an capected average Nusselt number was also found to be larger for the lower of Gr/RcI, the ceiling jet has a relatively larger amount of value of UD. As discussed earlier, the definitions of ED and T1 thermal energy which results in larger heat transfer rate to the are ceiling plate. However, the net heat transfer to the verticpl wall was found to decrease gradually with an increase in Og/Re . This O. is due to the fact that at a higher value of Gr/Re , the flow E=0 p k where d'"'
b =( T, T,). W,. L (4) penetration 4 is smaller. Thus, the thermal transport area is smaller on thE vertical mall Sirnilar results were found in the case of a negatively bunyant vertical jet. Figure 11 also shows the
- eD total heat transfer to the ceiling and vertical wait 0,,,,g is the UD *OL h),y,L
. 0) sum of O,,;i;,, and O.,ii, as discussed earlier.
This equation can also be wntten as The effect of UD on the total heat transfer is shown in Fig.12. It is seen from this figure that the total heat transfer rate is larger at the higher value of UD. This is due to the g D,O-einne g corresponding surf ace area of the ceiling plate being larger, which results in greater heat transfer. k(T,-T,) W.(L/D) The inulis presented here indicate the basic heat transfer As seen later in this paper,0 g was found to be la er at the higher value of UD, the c,orr'c"shonding value of uD calculatcJ from Eqn.(5) was lower due to presence of (IJD)in the downward to give rise to a vertical, negatively buoyant wall jet. ! The flow over the wall is very similar to that of a heated, denominator. downward, jet discharged adjacent to the wait However, the Figure 9 shows variation of the average Nusselt number traosPort over the ceiling. or the , horizontal boundary, N7 for the vertical plate. In this case the average heat transfer substantially affects the conditions obt,sined at the corner and ) coefficient b = 0 thus the wall jet. The overall transport is determined by the inlet j Here, O
- is the net heat transfer from thh(t/ Tflo-T'w).Westo Ehe verticaI all up to theconditjons. characterited by the raised convection parameter Gr/Re , and by the horizontal distance L/D over which the heated corresponding penetration distance s . The basic trends of the oults are similar to those discussed P in the case of the ceiling let interacts with the ceiling, The observed trends are found to agree with earlier work on ceiling jets and on wall jets and with pla te, the underlying physical mechanisms The average Nu:selt number Nuo for bo:b the ceiling and CONCLUSIONS the vertical wall can be expressed as A detailed experimental study has been carried out to O +O*# investigate the basic heat transfer characteristics of a heated,
, hD where (7) k It =(T#"'.We(LH d-T,) p) horizontal, ceiling jet which turns downward at a corner to become a wall jet with opposing thermal buoyancy. Such buoyant ceiling jets are frequently enc untered in many practical
' The variation of the average Nusselt a mber k for both the ceiling and the vertical wall with Gr/Re is shown in Fig.10. The Problems, such as enclosure fires and thermal, energy storage results show the average heat transfer rate from the jet flow to systems A heated twodirnensional jet of air is discharged the ceiling and the vertical wall The basic trends, as expected, horizontally adjacent to the underside of an tsothermal ceiling are a combination of individual results for the ceiling and the plate. An isothermal vertical plate is fixed to the other end of rettical wall, as shown in Figt 8 and 9. respectively. this horizontal plate, making a right angle corner. The distance between the corner and location of jet discharge was varied, along All these results may also be expressed in terms of with the width of slot through which the jet is discharged and the
& My M h wm M & p h Mu#
correlgting equations to indicate the dependence of N7p on ceiling jet turns d wnward at the corner and then penetrates Gr/Re and UD. There is also a weak additional dependence on along the vertical well to a finite distance as a negatively buoyant Re- However, for s3ch turbulent mixed convection flows, the jet. In this study, the heat transfer characteristics of the resulting dependence on Gr/Re is much stronger than that on either Gr or Re and the results may be expressed fairly accurately in II
- is reported. The following are the rnajor conclusions of the study:
terms of thepined convection parameter. Thus, over the ranges 0.03 < Gr/Re < 045 and 10 < UD < 20, the experimental results are well correlated by the equationt Th ,gg; g M W N m n eggg n & W W & 2 the jet flow separates frorn the ceiling just before the corner and (E )Dceiling = 25Q47(Gr/Rc )e(gp)@ (8) teattaches itself to the vertical wall at some distance below the (EIsD) wait = 82.77(Gr/Rc 2
)e(gg)e (9) corner. These results indicate the existence of a small recirculation zone at the corner.
2 (EuD) ceiling + wall = 6135(Gr/Re )"(UD)* (10) 7 g g The correlation coefficients for these equations are larger than of the corner. This value recovers downstream, attaining a a95, indicating a fairly close approximation of the data with these maxirnum just below the corner and then decaying downstrearn. equations 15
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