ML20010E663
| ML20010E663 | |
| Person / Time | |
|---|---|
| Site: | Vermont Yankee  |
| Issue date: | 09/03/1981 |
| From: | Lellouche G, Zolotar B ELECTRIC POWER RESEARCH INSTITUTE |
| To: | |
| Shared Package | |
| ML20010E657 | List: |
| References | |
| NUDOCS 8109080112 | |
| Download: ML20010E663 (92) | |
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-) A Mechanistic Model j for Predicting Two-Phase Void Fraction for Water ' l in Vertical Tubes. Channels, and Rod Bundles by G. S. Lellouche
- B. A. Zolotar l
l 7 l i 8109000112 810903 PDR ADOCK 05000271 [ p PDR
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'A Nomenelature I Ah IC' "ft. 1
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i 1 = ratio of bubble area to bubble unit cell volund (V jc C = Distribution Coefficient with slip relation (TC ' ' o c = Constant in the slip relation C = Proportionality constant in bulk evaporation E coefficient ' W ,
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C = Proportionality :onstant in the Hancox-Nicol relation" 3 Co = Proportionality constant in the Dittus-Boelter relation c p
= Specific heat g i/ft.3 .2 for D = Diameter ft.
D, = Equivelent diameter (excludes wall effects) ft. D hy
= Hydraulic Diameter (includes wall effects) ft.
D = Heated rod diameter ft. "c E = Energy Generation terms for vapor, liquid BTU /ft.3-hr g,f F = Chen c? efficient Fg ,, = a Pg , ( 1 - 2) P, 2
#/ft.3 G = avgg p , (1 - a) v D g, #ft.2-hr g ,,
Go =Gg + Ge l h,hg,hg .hf ,h 3 = Enthalpy; general, vapor, liquid. vaporization, BTU /# saturation h = Evaporation enthalpy when liquid is not saturated STU/# gf h* = Film temperature enthalpy BTV/# h = Heat transfer coefficients:Hancox-Nicol, Thom, A,B,c,D tulk condensation, Dittus-Boelter BTU / F ft.2-hr- { ,, fi = Heat transfer coefficient j h, = Microconvective coefficient BTV/ F-ft.2-hr h = Macroconvection coefficient BTU / F-ft.2-hr 3 h = Pressure dependent portion of the Thom relation 3
o, . 9 [ s k g = Thermal conductivity, liquid, turbulent Bid / F-f t.-br J = Conversion constant Ja = Jacob number L (a,p) = Part of the slip correlation L = Length of heated section ft. . P = Power generation rate in fuel (or heater) rod Pe = Peclet number . Pr = Prandtl number p = Pressure #/ f t .2 P CR q" = Surface .1=at generation rate BTU / f t.2hr q"' = Volumetric heat generation rate BTV/ft.3hr q"out = Surface beat generation rate leading to liquid phase heating Re = Renolds number Rej = Renolds number of liquid phase S = Slip S t = Stanton Number S = Indicators in the vapor generation rate tern t c
= Condensation time of a bubble t
cps
= Collapse tii.e of a bubble .
O F T1 ,g,w.s= Temperature of liquid, vapor, wall, saturation ( T* = Film temperature T* = Temperature of subccoled liquid after bulk phase I condensation V = Volume ft.3 v = Velocity of vapor liquid ft./hr g,j X = Flow ouality = G g/Go X,q
= Equilibrium quality = gF /(G g
+F) g a = Void fraction
= 8g(h - hg )/(p h
# 3 g fg)
*4 (1 - a)f1P = Energy directly deposited in the bulk BUT/ f t.3-hr a
g ,,
= Vapor, liquid densities f/ft.3 pg = Vapor generation rate j ~ - _______ _ ___
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ABSTRACT A mechanistic model for flow boiling in vertical geometries is developed and is qualified against steady-state void formation data. Extensive testing versus rod bundle data is included. The model a110ws for 'subcooled void fonnation utilizing a modified Bancoff-Jones drift correlation. As established the model uses heat transfer and vapor generation correlations ivhich are continuous at over the full range of parameters but does not utilize flow regime maps. The correlations are qualified against tube, channel, and rod bundle voidage data. The qualification leads to excellent statistical agreement with the data: Void Fraction Number of Mean Error RMS Error Samoles Rod Bundles = -0.000210.0010 0.028 784 Tubes = -0.000710.0010 0.022 440 Channels = .002110.0018 0.051 776
'4 ,
The purpose of this report is to establish and qualify (by comparison with data) a mechanistic model for predicting two-phase vertical flow in heated rod arrays and other vertical geometries. Particularly important for the use of the model in nuclear reactor analysis is its qualification against rod bundle data. An earlier attempt at the establishment of such a model has been reported (1) and that model has been in use for some years. When the predictions of that model were recently examined statistically it was discovered that an approximately 5% negative bias (A a/a) in rod bunole void fraction existed (Figure 1). With this as the impetus the earlier model and its constitutive relations were re-examined and new correlations have been estcblished. In qualifying the new model a much larger data base has been utiiized consisting of a large number of additional rod bundle, rectangular channel and tube experiments along with the: mostly, channel experiments used in the first model's qualification. The present model has been optimized and the accuracy of the prediction can be seen in Figure 2. The authors have found overall comparisons such as those presented in Figures 1 and 2 to be invaluable in determing model biases and trends. They are generally much more useful than comparisons with individual experiments. This report consists of two parts and a number of appendicies. PART 1: Model description PART 2: Comparison with experimental data APPENDIX A: Development of the vapor generation form B: Choice of heat transfer coefficients C: Slip modeling D: Location of the Point of Net Vapor Generation E: Liquid Phase Superheating F: Comparisons with Individual Experiments
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t ! MODEL = 1 l l FRIGG DATR - REV CO AT LOW ALPHA
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PART 1: MODEL DESCRIPTION Conservation Equations Basically we establish a four equation model: Two mass, one energy, and a quasi-momentum (dr.ift-flux) relation.* Mass conservation aF 3G g g
+ = f9 (la) at az BF g 30 2
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+ = g (Ib) at az Energy conservation We write the energy equation in enthalpy form 3(h gF +hFl gi a(hggG +hG) gt 4 I ap 9,, + 9,, , + (2a)
+ , ___
3t az De J at In this form we neglect certain minor frictional components of the complete energy equation (see Ishii ). Here, the surface and volumetric heat sources are defined as (see ahead). q" =h 3 (T,- T 3) + h3 (T, - T1 ) (2b) I and q"' = (1 - a)n P (2c) where P is the total energy developed in the heating element and n defines the direct deposition fraction (e.Q. due to inelastic neutron scatter).
- The rela'tionship between the momentum equations and the drift-flux model
~
f is derived in ( ).
The convection coefficients are taken as Dittus-Boelter (h,) and Thom (h3 ) coe f ficients . Slip (quasi momentum balance) Starting with a drift flux relation vg =Co (avg + (1 - a)v g) + V gg (3a) we define the slip:
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v 1-a - 5a 9. . C + ov g gg/G g vg 1-C'c (3b) Constitutive Relations The constitutive relations for the model are developed in a very general form and include effects and phenomena that are not qualified in this report but which will hopefully be qualified in the future. At the end of this section a description of the model actually qualified is presented. I. Vapor Generation Relation We consider the term fg as being composed of three separate vapor sources: e Vapor formed at the wall due to wall superheat e Bulk boiling due to the direct deposition of energy (e.g. inelastic scatter of neutrons) e Flashing and condensation due to transient pressure effects.
The deriva?.fon of the individual terms is discussed in Appendix A the final relation is taken as: q"'-h[A (Ts T)l ( q" - q"out - h (T -T) c\yw l I t s g 4 33 1 g 1 h fg (1 + 8) , { 2 l b gg l 1 F dhg Fg dh s Pp /Gg dh g h + + + Gt dhs\ IE t (J dp dp j It dp If, (4a) sa 1 (hs -h) g g h f The coefficients 5 1,2 are zero until the bracketed terms become positive and then 51,2 = 1. If the bracket was positive and then becomes negative because of a change in power heat flux or power generation due to spatial or temperal effects the 5 1,2 coefficient remains equal to 1 until all of the vapor has condensed. II. Heat Transfer Relations The heating process is made up of two parts e Direct bulk heating and bulk condensation e Surface heating and condensation These processes are discussed in some detail in Appendix B; here we indicate only that the second follows from the following assumptions: l
d
' 1. The wall heat flux ( q" (z,t) ) is made up of micro-and macro-convective components. In a subcooled state micro-convection leads to vapor peneration and macro-convection leads to heating of the liquid. At sauration both lead to vaporgeneration. Superhecting of the liquid requires introduction of an evaporation coefficient (App. E).
- 2. The sum of these is taken as the sum of forced convection (i.e.
Dittus-Boelter) and boiling (i.e. Thom) correlations.
- 3. Part of the forced convection heat transfer produces vapor when the film temperature is hiph enough hence is included in the micro-convection term.
- 4. The vapor generation term is proportional to the difference between the micro-convection heat transfer rate and a condensation ~
term proportional to the bulk subcooling. The first three conditions lead to the following: micro macro 1 q" s h (T g -T)+h3 s (Tg -T)g (5)
= (hg + b)D,-T)+b g s o w -Tg+Ts -T) g (6)
\ l Hence, we may define: micro-convection: (h, + h gh ) P ,- Ts )
' T -T l Macro-convection: h (1 + ) ( T, - Tg )
twt L The fourth condition states that: f g1 = (h, + h hg)P -T)-h g s 3 U-T)s g
'. 1 hence, (see eq (4) above) that:
=
9" ~ 9" out (h, + 9D) U , - T,) Pa) 9" out = h3 (7, + Tg - 2Tg) (7b) The choice of heat transfer correlations, while key to the model development and application, selecting established forms and requiring as little adjustment of coefficients as possible. Here we have utilized: Forced Convection 0.8 0.4 Dittus-Boelter = h 3 =C3 Re g Pr g kg/Day (8a) and as developed by Weisman ( for rod bundles Cg (c) = 0.013 + 0.033c (8b) c s fraction of unit cell available for flow This modeling of the coefficient,taken from the literature. appears valid for heating outside tubes. For heating other g'eometries we use the literature values appropriate to those geometries ; e.g. 0.023 for rectangular geometries and tuces. Boiling Thom E h 3 =h[(p)(T,-T) s (Sa) 0 p/630 hg (p) = 193e (English units) (9b) Condensation near the Heater Surface 0 Hancox-Nicol s hg = C3Re .662 Prg k /D a (10a)
O As is pointed out in the modeling of the vaporgeneration term (Appendix B) the original analysis leading to a value for C A (ie C3 = 0.4 ) was based on the assumption that all the surface heat flux went to vapor generation (q"g = 0). In the present model this is not so, hence, the coefficient has been optimized against void data to give: CA = 0.2 for tubes and channels (10s)
= 0.1 for rod bundles (10c)
For rod bundles, the Nussult number in eq (10a) is based on the heated rod diameter (rather than the hydraulic diameter). III. The Distribution Coefficient In eq (3) the distribution coefficient is introduced. We use a modification of the Bancoff-Jones coefficient.
= (x1 + (1 - wy) a") /L(a) (gy) where a) c =c + (1 - :rg ) (og /cg ) .
5 1 = 1+e Re/10 (12b) o The relation for e o depends on the inlet Renolds number, For tubes . and rod bundles, data analysis (see Part II) indicates that 1/'O saturates at 1.25 (i .e. , Re >1.39 x 10s),
o The relation for r is found to be acceptably represented by r = (1 + 1.57 o /pg) / (1 - r ,) (12c) while the function L (a) satisfies L (a) = (1 - e~El *) / (1 - e-Cl ) (12d) C3=4p cr / (p (p er -p) (12e) A detailed discussion of the slip model is found in Appendix C. IV. The Drift Velocity The term vg g in eq (3a) represents the gravitational effect on the vapor velocity. This term is most important at low liquid flow rates and is not trivially modeled. Nonetheless for the range of flows and pmssures of interest at this time the term may be reasonably modeled as v gg (13a) (a) = v gg (1-a) 3/2 v e 99 c (13b) gt (U)
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V. The Qualified Model Only two of the correlations discussed above are excluded from the qual ification. They are the second and third parts of the vapor generation terms , eq (4a) (i .e. , r g and r g of Appendix A). These terms have little impact on the type of data which was used in the qualification process. However, rg , which describes direct energy deposition, is extremely im-portant to BWR transient response in roc drop calculations. Thus , although a bulk condensation coefficient is not recommended here, it is recommended that r be included utilizing the instantaneous condensation approxi. ation g in the surcooled regime and zero condensation in the post saturation regime. Implicit also in the qualification is the fact that departure from nucleat bo ling is not accounted for. Thus, although the correlation r g produces accurate void fractions up into the 90% range no indication of whether Df6 has occurred is producad by the model. yith these caveats the qua'ified model contains the follow correlations:
- 1. Vapor Generation rg =r g (eg A4, c, d) (14)
- 2. Heat Transfer a) Single Phase Subcooled for T,<Ts ; q" = hD (T,-Tg) (15a) for T,>Tsy, and T,<T D l q" = hD( T,-Tg ) + h B( T,-Ts ) , (15b) h from eq i3a, b)
B l l l
at T,= TO T defined by D q" = h D (T,7T g) + h B (TD-Ts) (15c) (15d) and gr = o - (hB+ hD)(TD-T3) = hA (Tg-Tg ) h fmm eq (10a-c) A b) Two-phase subcooled to saturated T, >TD , Tg<T s a" from eq (15b) rg = r g >o v fromeq(3a) g C,v ( ~
}
o 91 c) Two-phase superheated liquid q" from eq (15b) rg = (hD/2 + hB)(T,-Ts) + k gu N )E (Tg - Ts)/D,' /h g D, (15e) Nu)E from eq (E-8)
PART 2: Comparison with Experimental Data o The development of the model and its later qualification involved the use of comparisons with extensive experimental void fraction data. All of the cases to be presented here are based on steady-state measurements in electrically heated test sections. Comparisons with transient data will be carried out in the future. Experimental data were selected by the authors based on their applicability to water cooled nuclear reactors. The parameters of interest were flow rates, pressures, heat flux, inlet subcooling and geometry. However, a wide enough distribution of these parameters were selected so as to provide meaningful tests of the ability of the model to predict observed variations of void fractions under changed conditions. Some 1 experiments were rejected baced on the fact that their age cast doubt on their validity. Additionally, data at low pressures (Tess than 200 psi) l and with other fluids than water were not utilized. l l I
o The rod bundle geometry data which have been utilized are characterized in Table 1. they are obviously the most important part of the comparisons. In addition to representing nuclear reactor conditions, they also appear to be of high quality. Figure 3 presents the comparison of all of this data with the model. Tables 2 and 3 show the statistica; analysis which has been made. The model shows essentially a zero bias. The degree of agreement is well demonstrated by examining each of the six sets of rod bundle data. Figures 4, 5, 6, 7, 8 and 9, together with Table 4 demonstrate that the results are essentially statistically equivalent. This is particularly significant because:
- 1. The FRIGG BWR experiments (although of only 36 rods) were very close to the geometry of current BWR designs. The geometries of the other sets of data were significantly different.
- 2. The FRIGG FT-36C experiments, with an axially dependent power distribution, were not utilized in determining the basic model
" adjustment", hence tend to act as a " proof" of the model.
- 3. The CISE bundle experiments were carried out using a completely different technique (and at a different laboratory) than the FRIGG experiments.
i The CISE and FRIGG data show about a 0.01 difference in void fraction between the measurement techniques. While more CISE data will be desirable to see if this is statistically significant, t'.ere is always a potential a a bias inherent in one or both techniques (e.g. there is n potential of a bias due to finite valve closing times in the CISE experiments).
TABLE 1 Rod Bundle Experiments 5 E 5 = 8 8 i ? ? ? M C C C k # a C 0 O O O E E E -- . - E E G Number of heated rods 6 36 36 36 36 19 1 2 2 2 3 4 Type of rod array Circular ,, Circular Circular Circular Square Circular Rod diameter (ft) .0456 .0453 .0453 .0453 , .0656 Heated length (ft) 14.50 14.35 14.32 14.32 x 13.18 6 5 Flow Area (f t ) .03298 .1538 .1538 .1538 3"E .0312 8 N EO Hydraulic diameter 7 (ft) .1535 .1201 .1201 .1201 .0318 Axial heat distribution Uni form Uni form Uni form hon-uni form' Uniform Uniform 10 11 12 Radisl heat distribution Uniform Uniform Non-uni form Mon-uni form Uni form Non-uni form Measurement technique y-ray :-ray y-ray y-ray y-ray Valves Average pressure (psia) 585 723 800 725 929 6G1 0 0.705 1.105 1.366 Average flow rate 0.980 0.798 0.789 2\ (hr-f t / MBTU 0.123 0.157 0.191 0.154 0.128 Average heat flux 0.167 (hr-ftj Reference ; i
TisBLE 1 (cont.) Rod Bundle Experiments (footnotes)
- 1. One central heated rod with five surrounding rods
- 2. One central unheated rod surrounded by three rings of rods containing 6,12 and 18 rods respectively
- 3. A 6 by 6 square array of rods 4 A central heated rod surrounded by two rings of 6 and )? rods respectively
- 5. 0.0453 feet for FRIGG FT-6A case 2
- 6. 0.03299 ft for FRIGG FT-6A case
- 7. Hydraulic diameter defined as 4 x (Flow Area in ft 2)/(Total Surface Area per foot)
- 8. 0.1545 feet for FRIGG FT-6A case
- 9. Peak to Average = 1.18
- 10. Peak to Average = 1.180
- 11. Peak to Average = 1.097
- 12. Peak to Average = 1.141 l
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. 0l.7 . 0.9 FIGURE 3: Model vs. All Rod Bundle Data ti. o h l Vs F iI Ccd S. ic th '
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ATTACHMENT 1 aC = i i e= l 1
- n Average Error = c = I (ag - 4) t n=1 N
4 N - RMS Error = a = I n=1(a c n_am n . c}2 N-1 Model Bias with Uncertainty = c ! fR~- ) N = Number of data points in sample 1 4 _w-*-+we---- ww..-,,-_,.,,,... _ , , , _ . . _ _ _
.. . . . . . . . 0 TABLE 2 Rod Bundles, Model vs Data Statistical Analysis +
Average Error RMS Error Sample Size a, Range c o N 0.0 < am 1 0.1 .0027 .034 74 0.1 < am 1 0.2 .0043 .028 67 0.2 < am 1 0.3 .0019 .037 86 0.3 < am 5 0.4 .0010 .029 87 0.4 < am 1 0.5 .0027 .026 110 0.5 < a ,5 0.6 .0004 .024 119 0.6 < an 1 0.7 .0023 .026 104 0.7 < am 5 0.8 ,0041. .024 81 0.8 < om 1 0.9 .0060 .022 30 0.9 < a,1 1.0 .0336 .022 5 All a,* .0002 .028 784 Model Bias = .0002 .0010 l l + Attachment
- includes am 1 0.0 values l
) TABLE 3 i Rod Bundles, Model vs Data Error Distribution I a e-an Fraction in Range
< -0.15 0.0JO
-0.15 to .10 0.003
. -0.10 to -0.05 0.031 <
-0.05 to 0.00 0.488 0.00 to 0.05 0.440 0.05 to 0.10 0.034 0.10 to 0.15 0.004
> 0.15 0.000 l
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TABLE 4 Rod Bundles, Model vs Data Analysis of Experimental Sets RMS Error Sample Size Set Model Bias a N FRIGG FT-6 a FT-6A .00472.0024 .026 118 FRIGG FT-36A .0014:.0019 .025 190 FRIGG FT-36B .0002 .0031 .031 101 FRIGG FT-36C .0013:.0022 .028 157 FRIGG 36 Rod BWR .00172.0024 .032 182 All FRIGG Data .0004:.0011 .029 738 CISE Bundle Data .0098:.0029 .019 46 All Bundle Data .0002 .0010 .028 784 l 4 c-- -e +--<wa - - + - , ew--- . ~,,-.,.3,ee..,, , , . y. __ , , , ,
TABLE 5 Rectangular C5. "el Experiments e, E t ; u 2
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Heated length (ft) 2.250 4.200 4.167 3.310-4.950 2 Flow area (ft ) .0006- .0053 .0053 .0011-
.0007 .0016 Hydraulic diameter (ft) .0133- .0583 .0583 .0131-
.0147 .0184 Axial heat distribution Uniform Uni form Uniform Uniform Measurement technique y-ray y-ray y-ray y-ray Average pressure (psia) 1493 456 629 1598 MLB \
Average flow rate O.958 0.598 0.618 1.136 2 hr-ft )' Average heat flux "3 2 0.405 0.056 0.113 0.259 hr-ft ; Reference
- 1. Defined as in Table 1, footnote #7 t
TABLE 6 Round Tube Experiments IT-23 RT Heated length (ft) 3.281- 6.562 13.451 Tube diameter (ft) 0.0301 0.0295 i Axial Heat distribution Uniform Uniform Measurement technique Valve Valve Average pressure (psia) 650 589 MLB Average flow rate 2 1.023 1.166 hr-ft Averageheatflux[MBTU 0.242 0.236 ! hr-ft I \ l Reference l
In order to extend the range of conditions available for testing the model, comparisons have also been made to a number of void fraction measure-ments from heated rectangular channels and round tubes. Tables 5 and 6 show the characteristics of these experinents. Figure 10 and Tables 7 and 8 again show the statistical analysis which has been made on the channel data. Figures 11,12,13 and 14, and Table 9 provide the coinparisons for each of the different experimentors. It can be seen that there is some inconsistency between the measurements made by the various experimentors. In particular, the Martin data appears to be about 4% higher than the other data. This may be due to differences in experimental techniques. Howeve r, because of its relatively low statistical scatter and wide pressure range, the Martin data was particularly useful in formulating the pressure variation for the slip relationship. Figure 15 and Tables 10 and 11 show the statis-tical analysis for the CISE tube data. This data was important in formulat-ing the flow dependence in Co. It is also important to show that the model can follow the observed trends in void fraction with conditions. Table 12 shows the predictions of the model in different ranges of pressure and flow. The influence of these quantities appears to be reasonably well described by the model. l l
o21 - O
~
CP
~
X X o X X x O X X C XX X _ X g X g g X XX X O X X x X X
.o XX y-d X MX X C ~ #8 x Um X X>Wx X X X
CO 1 - X xV g X* 1- x A - X ' b' X CO k XX X fx vp - MxXX
- to XX NX 4 _ R; X #
X C x x n.$~hy * ! cu x x I Nf* x O
~
pY # &gx>* x y
- , & y
- / x xx
- x x O i l i I 8
1 Id$.RX_y 6 i i i i 8 l 4 l 6 I 0.6 0.9 1.0 C l 0.3 0l9 0.5 0.6 0.7 0.0 0.1 0.2 . RLPHA MERS. FIGURE 10 - Model vs. hilrectanaular CHANNEL DATR
TABLE 7 Rectangular Channels, Model vs Data Statistical Analysis Average Error RMS Error Sample Size um Range c o N 0.0 < om s 0.1 .0028 .037 257 0.1 < am 5 0.2 .0067 .050 105 0.2 < am 10.3 .0003 .054 102 0.3 < am 5 0.4 .0158 .057 96 0.4 < am s 0.5 .0070 .061 76 0.5 < am 5 0.6 .0245 .069 50 0.6 < am 5 0.7 .0141 .069 32 0.7 < am 5 0.8 .0181 .047 12 0.8 < am s 0.9 .0052 .050 7 0.9 < am 1 1.0 --- --- 0 All am* .0021 .051 776 Model Bias = .0021 .0018
- includes ams 0.0 values l
l l l l f L
TABLE 8 Rectangular Channels, Model vs Data Error Distribution 1 ae -g Fraction in Range
< -0.25 0.001
-0.25 to -0.20 0.001
-0.20 to -0.15 0.001
-0.15 to -0.10 0.025
-0.10 to -0.05 0.114
-0.05 to 0.00 0.420 i 0.00 to 0.05 0.309 0.05 to 0.10 0.102 0.10 to 0.15 0.016 0.15 to 0.20 . 0.007 0.20 to 0.25 0.004
> 0.25 0.000 t
. . - _ . .- -. ... - - . _ = - . - -.
U s6 o-CP x X
~
X o x x x o X X
~
o XX X X X x X
- X X X o x x x Nc k X
.4 x x x
u- x X X d x x cc - , um - y x ym 4 zo x t _ k :r' xx j a- g x x x x i _ x x CO xX
- X y x X X
X
< >f(
X o fX
. x?
o !" xy l t 4 l s l
~ ' I t l o : 1 l t 6 0l4 0.5 l
0 {. 6 0l7 0.8 0.9 1.0 0.0 0 l.1 0l2
. 0.3 .
t ALPHA MERS. FIGURE 11 - Model vs. MAURER CHANNEL DATA l
~
Ct e o c' o co o o o
.o u-a
-.3 c -
um - co x x I xx*x i l
.'.Je-
- 1 Co xx x x xW$ x cn M x '
- x o *
- x
- x f N x h' a
~
xN x x
* >k
- xx O E l x l
a.,-x 'x i i i i I i o I i I i I 6 1 6 I 8 I I i 0.0 0.1 0.2 0.i 3 0.9 0.5 0.6 0.7 0l8
. 0.9 1.0
! ALPHA MERS. i FIGURE 12 - Model vs. ST. PIERRE CHANNEL DATA
.m_ _..
T bb o - m o co o o
.o u-a x J X C -
x Um X X X X CO y XW X - 1 J
- r %
x* ! X x M X
- X o
_ x N
- X o
f
~
xx
- X X X o
p, o 2
-i i i t i 6 l t I l I I ' I l 1 0.0 O!! O!2 0.3 0.4 0.l5 0.6 0.7 0.i 8 0.9 1.0 ALPHA MEAS.
t t FIGURE 13 - Model vs. I CHRISTENSEN CHANNEL DATA
631
- o -
1-e,
. o . e o
- _ x 6 x
~ x g x
.o x* x do~ #5 x c - #4 Um x
_ x x x CO
. I -
x *xx 1, Mx* J_ x CC x
- xy, y x M
a
~~~ [X N
,x
'x x g x.27*
O _ x
'f x
[x
- gx E a
~
g fw# x _ v
/
--i
#* i
, i ; a g i t e I e ! I I
C [ 0.0 O!! 02 0.3 0.9 0.5 0.i 6 0.i 7 0.8 0.9 1.0 RLPHA MERS. FIGURE 14 - Model vs. MARTIN CHANNEL DATA
7 TABLE 9 Rectangular Channels, Model vs Data Analysis of Experimental Sets RMS Error Sample Size Set Model Bias a N Maurer .0253 .0068 .077 133 St. Pierre .0249:.0035 .038 115 Christensen .0070 .0030 .032 112 Martin .0208:.0019 039 416 All Channel Data .00'd +.0018 .051 776 l l l l l l 1 l [
V S:: o _ o o y% O x
/ < ,
.W h
o c4
*O x #
u- x _.i .x
- UD co r _ . x l A- '
J _ Co " - _ h cn o x .
.J ^
.x x
M
- x .
o
~
d
'Wx xX o
dix xxp.0 0.l1 6 1 i 6 t 1 l l 6 l l l 0.l 2 0.l3 0.l9 0.l5 0.l 6 0.7 0.8 0.9 1.0 l x4 x ALPHA MEAS. FIGURE 15 - Mocel vs. CISE TUBE DATA s . s _ _
4 TABLE 10 CISE Tube Data, Model vs Data Statistical Analysis Average Error RM3 Error Sample Size Qm Range c o N 0.0 < am s 0.1 .0153 .023 33 0.1 < am s 0.2 .0187 .023 38 0.2 < am s 0.3 .0120 .022 41 0.3 < am s 0.4 .0022 .015 43 0.4 < am s 0.5 .0026 .019 60 0.5 < am 5 0.6 .0050 .023 63 0.6 < am 5 0.7 .0116 .014 77 0.7 < am 5 0.8 .0100 .016 60 0.8 < am s 0.9 .0014 .019 13 0.9 < am s 1.0 --- --- 0 All am* .0007 .022 440 Model Bias = .0007 .0010
- includes ams 0.0 values
TABLE 11 CISE Tube Data, Model vs Data Error Distribution a e-am Fraction in Range
< -0.15 0
-0.15 to -0.10 .002 l -0.10 to -0.05 .016 i
-0.05 to 0.00 400 0.00 to 0.05 .580 i
0.05 to 0.10 .002
>0.10 0 i
i { 1 l
TABLE 13 Model vs Data Pressure and riow Range Comparisons Pressure Ranges in PSI 2 All Flow Range in MLB/HR- FT Configuration Data P s 700 700 s P s 1000 1000 s P s 1500 P 2 1500 1.0 s G G 3 1.0 i Bias Sample Blas Sample Blas Sample Blas Sample Blas Sample Blas Sample Blas Sample to Size to Size to Size to Size to Size to Size Size to i Rod Bundles .0002 784 .0027 189 .0000 479 .0051 140 --- 0 .0010 546 .0030 238 i t.0010 1.0023 1.0012 1.0025 t.0012 2.0018 1 Channels . 0021 776 .0192 184 .0026 43 .0137 187 .0072 378 .0011 510 .0042 266 t.0018 1.0027 1.0047 2.0044 t.0026 t.0024 t.0028 i Tubes .00G7 440 .0032 128 .0003 312 --- 0 --- 0 .0007 281 .0033 159 t.0010 t.0022 t. Doll .0013 t.0016 i I 1 1 i 1 I l s e
) -
e
.* i ,. ' . . 6 A Establishing the Vapor Generation Term As stated in the body of the paper, the term ig is considered as being composed of three separate vapor sources:
e Vapor formed at the wall due to wall superheat e Bulk boiling due to the direct deposition of energy (e.g. inelastic scatter of neutrons) e Flashing and condensation due to transient pressure effects.
- 1. Vapor generated at the wall .
In deriving a model for this term we assume the existance of a superheated boundry layer (B.L.) of liquid at the wall with j an average temperature: I l T* ' (Tg - T s)/2
>T s (AI)
That is, at the film temperature. We account for the process (assumed to be in dynamic equilibrium) of creation of voids, displacement of mass into ' ar. out of the B.L. and bulk, and due to condensation just cutside the B.L. Mix 109 is assumed to occur faster than the time scales of interest. The edge of the B.L. is assumed to be where gT = Ts , but its location anc~ the actual volume of the B.L. does not concern us. The characterization of the B.L. 'and the rest of the channel is made through T* and gT , except that liquid leaviril the B.L. it assumed to be at Ts* Consider the following sketch: i
l
< i o 9 - 9 out
' l M
' in I -g h,
/ i out
/ '
p'C e q" = hD(T,-T g)+hgU ,-Ts 9+ 9 * (See App. B) q"-q"out = boiling potential in the B.L. (See App. B) Min:
= Measures the mass of liquid returning to the B.L. from the bulk liquid due to a bubble leaving the B.L.
measures the mass of gas leaving the B.L. and entering the Mout: bulk liquid M: measures the mass of liquid leaving the B.L. and entering the c bulk due to condensation of bubbles at the B.L. (this is assumed to occur on the same time scale as other mass inter-changes). Balancing over the region we find: Mass: M in * "out + "c Energy: q" + M +Mh in hin " 9"out + "out h l If V is the rate of bubble formation, V g the rate of liquid mass g return, and V the rate cf liquid mass leaving the B.L. due to condensation in the bulk then: l l cogg V
- vapor mass leavino the B.L.
l (Pt -0 g)V g = excess mass of liquid leaving B.L. l due to creation of the voids in the B.L.
a(o -o)y = mass of liquid leaving B.L. due g g e to condensation at the B.L. Clearly, the net rate of vapor formation is given by: ap g (V g -V}* c g1 (A2) If all energy terms are measured from the local bulk liquid temperature (or enthalpy) we find; M h out out "D g(hg - hg )+ a(o g - go ) (h s -h) g, Vg M h t L (h g -hg) = 0
" "# Y in in Mhc
=
a (o, - og) (h s - h)V g c The energy term angg (h -h)V g c due to condensation in the bulk is not counted since it is contained in the equivalent Vg term.* There are 3 unknowns (V g . Vf , Vc) and only two equations . We assume the existance of a l heat transfer correlation, unspecified as yet, which accounts for near surface condensation and set l acg Vc h g
- hg + (# 1 ~#
a) (h s -h)t
=
h (T s
-T)
L ! , . (A3) r . g
*The vapor is assumed to be formed at the saturation temperature rather than the film temperature. Except for massive transients this should i. ave little impact.
The estimation of h (see A4d) is discussed in Appendix B. From eq (10) we find prog essively aogg V h fg (1 + 8) = q" - qout - "D g hfg sV c o t (hs -h) g 8
*Tg h fg (A4b)
I q" - q"out - (1 + 2S)E(T-T,)/(1+S)j s 4 I g1 3
- g (Y v -Y)*3 t 1I h
fg (1+8 ) l D e (A4c) 57=0 in the subcooled regime only up to the point where the bracket becomes >0, thereafter Sy = 1. Finally, set h = (1 + 28) ii / (1 + 8) (A4d) The term g out is defined in the standard way through the film temperature 9"out =hg(T*-T) (A5) hence tq" m q" - q"out =hgg (T - T*) +3 h D,-T) s (A6)
= (hg ^ @D w s This completed the derivation of y (the surface heat term).
- 2. kapor Generated due to direct energy deposition Since we do not account for superheating of the vapor due to direct
deposition we have a vapor mass increase rate ae VD = q"'/h g t =(1-a)nP/h gg h s h -h t gt g and the rate of bulk condensation (with condensation heat transfer coefficient h )* aogg Y *h c (T3 - Tg) /h gg The net rate of formation is then given by T g2 3 "#g,(Y ~Y ) ,
=S 2 (1-a) n P - h c (T -T),
s g (A7) C 1 h gg and S2=0 it the bracket term is negative, otherwise S2*1* The determination of the term h (A/V)c will be discussed in Appendix B.
- 3. Vapor Generation due to transient and spatial pressure effects Consider an expans:on o f the energy equation (eq 2A) leading to faF ah ah ah ah h
g -d + 3G ) + h 3 laF* g +3 3G+F E
) gy+G g gf + F gg 5
+ G L 8z' l
l f a(h,, - h,) F t 8("I - h s)0 1= _q 4 I 3P ! + + a
+q a 4 ._ __
l at az J at (A8) D,
- Note t:.at the condensation term proposcd by "cr.cox cn/ !!icol is .'. surf:ce condensation effect characterized by the surface area of the heat source.
For the bulk effect we must somehow deal with the surface area of the voids. This is disCJssed in App. B. l l l
where we have added and subtracted the term: ah F Bh3g G 3 t
+
at az tJsing eq (la,b) we can rewrite this as ah i + Fg at g Bh 3 Bh s a(hg - h3)FL B(hg-h3 )G g h fg 'g + G g az +F *0 1 + + - t at az at az
= R.H.S. (A9)
If we assume that h and h, are only pressure dependent (equalibrium thermal-dynamics) then the derivatives in the second to fifth terms on the left side of eq (A9) are only pressure dependent and can be replaced by terms like r dh g ap dh ap
~ 3 dp at dp az When gh
- h, the last two terms on the right side go to zero (neglecting superheat of theliquid' for the moment) r hance, the direct pressure effect of f gmust be
+ *0 1 g3 gt 9 1P g (A10) and the full vapor generation term is I
g" gl + bg2
- g3 4 I
,;5 (9" ~ 9"out - h (T -T) q"' - h 3
h 3 g De + S c JTT) 3 g fg (1 + 8) ) 2} h
+F g gh+G
~
h gg f0 + FgdN d
+ G t (All)
, , , . .-..._,y - , , . , - - - - - .
, , , , . -- y .-
It should be emphasized thatg r is a mass splitting relation. The energy equation (eq 2a) can however be split given the form of fg. Suppose a h ,,F9 , a_ G,,h-9 = E l 5t az g aheFt ,3Geht= E at az 1 Et+Eg =hq"+q"'+hh e l Then using eq (la) and eq (All) we have E g
=h g (rgy + f g2 ). +F g ) h+Gg f' and Eg follows directly.
Equation (A9) defines rg hence in the saturated regime, neglecting pressure effects for the moment, we must have 9" + 9"
- h fg f= g Equation (All) however states (setting S1,2 "1) hfg f= g (q" - q"out) + 9"'
hence,, q"outmust - O as Tg +T.s The definition of q"out (eq (7b) and eq (AS) shows however that 9"c ut fho (T, - T3 ) / 0
This fact implies either that the liquid superheats (Tt >Ts) or that we change the model setting q"out = 0 (accounting for pressure effects does not alter this conclusion). In fact we believe that the liquid phase does superheat and this aspect of the model is discussed in App. E. 1 e
'O i
l
Appendix B Choice of Heat Transfer Coefficients It has been the view of the authors that the " adjusting" of the model should be minimized. Fe have, within the structures of the postu-lated phenomenalogical basis of the model, chosen heat transfer coefficients from the literature. Only one such coefficient has been " adjusted" and the rationale for that is described below. The phenomena that are assumed to underly the processes of heating and ultimately vaporizing the liquid are: o forced convection heat transfer wall to boundry layer (B.L.) to bulk o Void formation in the film o Subcooled condensation just outside the B.L. o Direct heating of the liquid phase o Bulk condensation The model as established does not consider the variation of heat transfer coefficient with flow regime and no use of flow maps is allowed. he logic of the model allows various heating modes, however, which are discussed below.
- 1. Forced Convection Heat Transfer When the surface temperature is below saturation heat transfer is assumed to follow the well known relation
'q" ; T,'c Ts (B-1)
= hD (Ty -T) g and we assumeD h to be well represented by the Dittus-Boelter correlation.
The coefficient for this relation is usually taken as C g 2 0.023,
however, there is ample evidence ( , , ) that this value is dependent on the heating and flow geometries (inside tube, annullar,outside tube, etc.). We use the following data based values C g
= 0.023 - heating inside tubes
= 0.013 + 0.033c - heating outside tubes (B-2) c= flow fraction of unit cell Physically the heated surface will exhibit points of high free energy.
Whether these are cracks with associated higher internal temperatures or simply sharp edges, a new mechanism may be assummed to come into play when the film temperature T* > T3 . This mechanism will be referred to as the boiling heat flux and represented g by h p, 3 - T ) . h he Mal heat h h now taken as q" = ho (T, - Tg) + h a w s) bU We do not assume that we switch correlations (as some do .o.' example when ho ( T, - Tg ) = hg (T, - T )) s but that there are two dir.inct mechanisms operative. Eq (B-3) defines the surface heat flux under all conditions of flow and void fraction as long as T, > Ts . From a consideration of the data we believe g h h auegaQ Meled h he h (_) melah. h 3
=h g 0 (p) (T,- Ts) (B-4a) 0 pM30 (304) h, (p) = 193e (English units) l The reader will note the similarity to the Chen modeling where he takes l (for Tg= Ts )
l q" = (hg F + hmicro) (T,- T3 ) If we compare with our model, take F = 1 and hmicro = h g . Note that h micro is " (T, - T ) s'99 hense the comparison l l
- - , y - - . - _ -
is quite close. Indeed if we plot 3h / U, - Ts ) a h we have micro [(aT)*N Fig. B-1. St a the Chen relation was developed for pressuree below 500 pria (and most data was below 100 psi) and the Thom relation was optimized for pressures above 1000 psia it apears reasonable to assume the Thom relation has a more general pressure dependence validity. Further, the calculation fo.- the Chen coefficient (in Fig. B-1) does not include the S (Suppression) factor. There appears to us insufficient basis to the suggestion that voids stop forming at the wall simnly because the flow gets high. Examination of the data appears as well to show bubbles still forming in the film even during the annular flow which presumable forms the basis for assuming the existance of suppression in the first place.* In any event the final model predicts rather accurate void fractions well into the annular flow regime (see Part 2). While Eq (B-3) yields the total wall heat flux, we must still specify the separation into vaporization flux and bulk heating flux. When the B.L. l temperature T* > T. ying T, > T s) any microscopic void formation on the wall has a potenti ro. growth due to vaporization into it from the super-heated B.L. Thus, the vaporization flux is taken as q" = hg D, - T s) + h D w
~
=( hg + h)
D w
-T s The bulk phase heating flux follows as q" - q"y = hg (T* - Tg)
=
ho (T, - T g +T s -T) g (B-6) l It is important to notice that q" - q" O as T g +T. s This implies the liquid will superheat to some degree. This will be discussed in Appendix E. Note that the Reynolds number entering Dh is taken as the liquid Reynolds
- R. Duffey, Private Communication i
. A COMPARISON 0F THE TH0H AND CHEN
. MICRO-CONVECTION HEAT TR/.NSFER COEFFICIENTS
./ -' -
; g :: :
t . I j ..; ......I l . . L. . ! i i i ': '
; ? . , -
- l - + i.-V-}
s-l's t-- I * ' ( - < s -V .
} . [- [! i- p l .'-
- [i
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gy t-i . .i: . ::M :.i. -::t ..
!. ;4 *y8 .. ;
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= --
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p; ~
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- .-~ }/,::.
. t: .i i. - -
t : : -l ;. .rj
. ;:$ 4::- :i. ... : i.: .l . . .: il 1 - : i'.~ \ : .i .
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- g .' ' . : f "l * " - I-
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[ _ _. _. l......l.._.._ ._. ;_ .4, ._ . = ' ._I
! .1 :
l~ .i - - - -
- t . . ._l.-
'i -l l l 1 -l ! :
r : m.
, :I l. !-,._j ., . - i ! '
r / i . 1..- l -3 . ; 7. . y ..
. 1 . . . .., . . .
- 3. ..
t i
, . }: _ ..: m 9,1 .
.T- . = .
Ai! .-
. . I I . .
- l. ./._ _t. .
i
. ..- r -- . . _ _
.i I j
_bli ; f w g, _ . .., .y
' : f~i l._....:i : t. I r if: ! . t-
-l .: ._
..9
~
I i i,. i i I i: l '
- .l M i - [:
. j; ;nl i g ; i l l
~
e l N
' i
...i . _l
- I
.l. : i -
- i. l i-t t I -1 .
_ .= . . .
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-l a_._ _ _ . . .. :s d
. _.. i. . . . .._,i...<.. i - l ' I -
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___ ,t
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.....__.i_. i. . i.
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- s. :.; : .- *
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. . l' . i
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. . . . J. .
8 1 j :I
..i , 'I ! i ; l, ! l ' l 1 .i 1 -. i
. ,.. ! e .
j i ; . 3-gp
.1..___....
,,; _ _. ; _ . i- .. _. l
__ . . _d] : _ . _ _ _ .i.H_ . . . _ : -
- i -; .-
7 -- i . . .. . i ;
. p..d.I:u g .
I
- _t inn.
t
. .;. _. j -
. ]. . ._ . u g .
i-l
. . _ _ . ,,___..:_--- r . .:.__ --. : . - .
i b s *q :.va_I
! . . I t : , i t-i . ! 1 l ; } i ! ! i Fig. B1
In this report we do number (Re t ) hence hD + 0 as the void fraction + 1. not include heat transfer directly to the vapor.
- 2. Void Formation in the Film Although there is a potential for void formatinn when T, > Ts the sub-cooling of the bulk precludes significant voic formation until some later distance down the heated channel. Indeed there is a near wall condensation potential acting (just as there is a boiling potential) and we take this potential as:
q" = h (T -T) g (B-7) s and net void formation would depend on whether q" f q" (see eq A4c). The literature contains a number of potential candidates for h ; we have chosen the Hancox-Nicol (H & N) relation h =C Re*002 Pr Kg /0, (B-8) In this formulation H & N correlated the data based on circular tube exper-l iments and inlet conditions. We have expanded the formulation to local j conditions, a different voidase model and other geonetries. Because of l this we have optimized the coefficient (see Part 2): C = 0.2 for tubes l
= 0.1 for other geometries (B-9)
We have also defined the diameter entering the Nussait number to be the heated rod diameter ( = 0, for tubes). l l l
J
- 3. Direct Heating of the 1.iquid Phase The y and inelastic neutron scatter in the liquid comprise a form of bulk heat generation. We take the deposition to be related to the average bulk phase density hence q"' = (1 - o ) n 9 (B10a) where P is the total power generation in the rod and n accounts for the fraction of energy escaping the rod and the inelastic scattering phenomena.
We assume that for this form of energy deposition voids form directly (as microscopic bubbles). In other forms of bulk heating this may not be correct.
- 4. Bulk Condensation In Appendix A we have introduced a bulk condensation term q"' = h c (Ts -T) g (B10b) c This condensation term i; considered different from that leading to eqs B(7. 8 ) . That phenomena can be related to thelocation of the point of net vaporization and is not unrelated to surface condensation (the " surface" being the subcooled liquid interfacing the thermal boundry layer), while here we deal with the bulk void surface itself.
The following analysis starts with the single bubble model of Sursock and Duffey (_) and goes on to develop a formulation for HA/V)c (c.f. eq A-9 which does not contain sirgle bubble parameters. In the end we cannot utilize the result because of a lack of data necessary to evaluate the pro-portionality constant (at least wa are unaware of any such data). Thus, it is unclear that the f g term (eq A-7) is currently usablet in the subcooled flow regime unless or.e makes additional assumptions (instantaneous condensa-tion, i.e. , q"' heats tre liquid phase directly). Consider the condensation of a bubble. Following Sursock and Duffey we take for the individual condensing bubble
dR "
~
T (T, - Tt ) E ' kit (B-11) (o9hf9rfeog c hence assuming that the bubble condenses completely R (0) = 12pg PTCk [ AT }2
= 0 l
h I te t e (B-12) (9 f9 / Eq. (B-12hctually defines te . From thermodynamic considerations we can estimate the change of tem-perature of the liquid phase due to vapor condensation as 3,y . 4Iw ( O t )c 3/2 , (1 - a) o,cp (T * ~ I ) Y t t (B-13) Ogh fg We now make assumptions concerning initial bubble size which may be valid for energy deposition from a- rays and inelastic scatter which are probably not fully valid in other cases. We assume all initial bubble sizes are essentially the same and that the direct daposition void fraction is only a fraction of the local (cross section average) void fraction. In this case eq. (B-13) holds for many bubbles. ! Consider now l T*-T g t
= (Ts - T g) - (Ts - T *) > o i
If i is the bulk temperature that would exist W.aut direct bulk vaporization t then T,* - T must i be proportional to the amount of such voidage actually condensed. Hence we take (approximately)
-- - _ .=.
2 T*=T + h (T3 -T) d :: 1 1 c (B-14) G,cp hence T*-T = -T) 2 g L h(h)(T c s L p and we assume as well i h e (T3 -T) h e (Ts -T) 4 with these we may write
" IO Uc ) *
(I *)C t Ogh fg G z(T3 - T,) Vhe(h)c (B-15)
=YV c b(h)c From eq. B-12 we can determine the area per unit volume of a bubble (hence by I
the above assumptions of the mass of bubbles)as
) " "" (~ Y Y) ! h
) /3 hence i
, _ , - . , - . . . , . - _ . , . . , - - . . . - , , . - . _ _ . ~ . . - _ . . . , . . . . _ . . . _ , , _ . . . . - -_... _-, - , _ - . . . - . -
} (h) * (4n)3 (3 yy )2 h 2 c y3 (B-16) During conduction contrclied vapor condensation across a convective boundry layer the heat transfer coefficient can be represented as h c t
/d with d = t cp3 kg /c pgo and the tirae to collapse given by t
cps
=
R(0) / lvg - vg l. Using eq. (B12,15) d = !kg YV hc (f) c o jv - v l. pg g g hence h c Ltplv C c g -vlg 3
- 1/3 (B M v/ C (F YVkhAi ---,,,n-. .,-,. ,,--...,~.-n, , - . - ,..,_,,---,,-e., e,-4 ,, . . , . - . - - . , - - - - - - - - , - - - , <---,,-7 -n . - --,-- -
t If we multiply eq (B-16) by heand substitute the cube of eq (B-17) on the right hand side, we find after some cancellatf or,: h cp ot (Ts - T t) k g/D, (B-18) c(h=(362)DC c l3-1l c p h D g fg , where we have assumed V= D,2 L. Ns msuh can be can Mo ne fom h c( = Ce Ja l 5 - 1l zkg /D, L hence the condensation Nussult number can be taken as 2 0h c (h c n (Nu)c c Jal S - 1l z/L kg The final result is that bulk condensation is proportional to (T s-Tg)2 as found by Sursock and Duffey for the single bubble but all single bubble parameters have been eliminated. The analysis permits us to specify q"'c within a constant multiplier but the extant data on bulk condensation appears insufficient to estimate a value for this constant. g, .,,., ~ y. - . - - . - , -_.,y-. . , _ ._ , . , . , _ , , _ < . , , _ _
,. . - , . . , . . ..,,_,,,7 . , - . . - - _ _ , - . _ . . , - . . _ _ . , .
APPENDIX C Slip Modeling In the origional model established in 1972 (_) the slip relation was taken as a slight modification of the Bancoff-Jones model v g 1-a a = S y, x(a,p) - a (C-1) with x (a.p) = x1 + (1 - x 1) a M (C-2) x1 = 0.71 + 0.29 p/p i r(p) = 3.448275 - (1.875 X 10 5.85 X 10-7p )p i As was noted then this formulation has several desirable properties. 9 5 - 1 for p - p # CR 9 S - a fo r p - 0 at a= 1 i The rationale for these " desired" properties was: (1) for p y p CR differentiate between the phases hense S - 1 for all a , (2) since the vapor specific volume is infinite at p = 0 any voidage should lead to a = 1 and S-=. We commented that this implied that c was sonehow related to p/p as had been pointed out by Achmed. We also noted that when the bubble g g leaves the wall it starts with zero velocity, hence S should go to 0 as a - O for all p, but eq (C-1) did not exhibit this property. We suggested however l l that this could be achieved if x(a,p) was replaced by x(a,p)/L (a,p) where-1-e
-c 1(p)a l("*P)
- 1 - e -c 1(p) (C-3) l l
L
with C1 (p) = C PCR / (N(p ~ CR C = 4.4 There has been a great deal of work done on the subject of slip and we have attempted to synthesize a model which contains the features observed in experiment and is valid in vertical geometries. We start with a Zuber-Findley drift flux model v g = C, (avg + (1 -a) y,) + y ,(a) (C-4) The tems in eq (C-4) are cross section averages; if < > implies such an averaging then v a < a vg > /< a> g ae <a> a C, <a(a v g + (1 - a)vj)>
< av g 4 (1 - a) v><a>
j v g (a) = <a( 1 - a) ( v q - v,) >
<a>
Equation (C-4) can be rearcanged to yield 3 , g 1- a ~C + p,vg,(a) ~ (C-5) vg 1-Cmo , Gg _ Starting with eq (C-5) we must fit -11 of the physics into C, and vg (a). There are several conditions which a ratio of the phase velocities must meet:
1 C1 lim S (a,p) =1 p-p eg i C2 lim S (1.p) == p-o C3 S (a,p) finite except for C2 C4 lim S (a,p) = vg , (0)/vj>0 a- 0+ There are also a number of experimental observations 01 C, depends on mass flow j 02 Co depends on pressure 03 C, depends on void fraction 04 C, and vg , (a) depend on flow geometry (e.g. equivelent diameter) 05 v gg (a) = (1 - a) b These are not the only possible conditions or observations premoli et al (_) list 7 conditions on 5 relating to D,,G , op , and f a,none of which we would fully agree with. We have not listed any dependance of C, or v g on flow regime (flow maps) although various analyses have shown specific limits to which C, or v gj must tend if one had a fully developed invarient regime of a specific type. As with the heat transfer coefficients we aschew maps because of the jump phenomena associated with boundry crossing and because no extant flow map is--to our knowledge--demonstrably valid in any specific
situation. As is shown in Part 2 of this report flow maps are not needed to achieve excellent comparison with data. From the observations we may determine the characteristics of C, and v gj needed to satisfy the conditions C1 - C4 Condition C-1 lim S (a,p) =1 p-p eg from eq. C-5 this implies Co (1 - a) v , gP, / G, but, v g, is a measure of buoycncy henceg v - O as p - pCR lim Cg -1 for all a p -p ca Condition C-2 lim S (1,p) = a p-0 Since S (1-c,p) = C, + etc. 1-C o+Ce o - t and the numerator of S - O as c - 0 we must require 1 - C o
+ Co c - 0 as c - 0. Hence C2 also implies lim C - 1 o
a-1 and this yields S-0
e Expand eq (C-5) use 05 and L' Hopitals rule to yield lim S = lim 1
. b (1 -a)D'I vg /v, a-- 1 a -- I 1-d(1/Co ) 1+ d da
-}C where v g is the a independent part of v g, and vg is not differentiated here.
For C-2 to hold lim d 1/Co -- 1. Note that the second term is still in an a--I da indetermanent form (0/0) but if b > 1 then the first term dominates *, i.e. o ),y lim (1 + b(1 - a) b-1 y91j a--I Condition C-3 S (a,p) finite except for C2 implies 1
-> a 7-o and that the equality can only occur for a = 1, p = 0.
Condition C4 C-4 states simply that we are dealing with co-current flow and that the initial slip ratio is not zero but has the buoyancy value. It also implies that the vapor bubbles initially start rising in the superheated boundry layer. since C-4 also can be read as
- It actually doesn't matter if b = 1 the value for 5 is still infinite as a 1.
I
i lim S~ lim C 0
.- 0 0-0+ a4+
The four conditions on the slip ratio nay be replaced by equivelent conditions on Cg and vg ,: lim C, =1 for all a >0 pp (C6a) lim C, = 0 a -0 (C6b) dC lim da
" 1 a -1 p0 (C6c) lim b *> 1 0-1 (C6d) p-0 l (C6e) l Consider the following form for 1/Co.
g
= x 1(p)+(1- x (p) 1 a r(p) /L (a,p) (C-7)
If r(p) 5 0 and x,(p) 5 0, eq. (C-6a) is automatically satified if
.. l L(a, peg) = 1 and Eq. (C6b) is replaced by lim L(a,p)=0 (C-6b) a-0 Eq. (C-6c) implies L(1,p)r(p)(1-xy- a=1 = (C-8) lim 2 1
P-0 L (7,p) Consider the L(a.p) function. It is introduced to account for the obser-vations that bubbles formed near the wall tend to have slip values < 1, and conceptually at least a bubbse when it breaks off the wall should start of f with zero initial velocity (although we choose here to take the ac-celeration time to terminal velocity small, hence take vg , (0) as itsas-ymptotic. value). Also we know that these near wall effects are limited to small values of the average void fraction,hence that L (a) should have only a weak a dependance fora greater than a very few tenths. We choose the form for L (a) as
-c (p)e I~' -9)
L (a,p) : ' l 1 - e -c1(p) l which satisfies eq (C-6b'). Returning to eq (C-8) we see that eq (C-9) implies
- c (p)'
1 l c (p) e lim r(p) (1 - x 1) 1 p-0 g , ,-c 1 (p) l Now with the above definition of L (a,p) we can only have L (a, p p)=1 I (p c p ). We choose now for all a > 0 i f c = - p CR l l
p c (p) = p _ p) (C-10) CR This form eliminates the Cy term in the zero pressure limit and we have f t lim r(p) 11 - r y(p)'l= 1 (C-11) p-0 Quite generally since S S 0 we must require eq. (C-6e) hold for all p and this basically 1 - r (p) (1 - xy ) + S IP) ' c y(p) 3 g 1 - e -cy(p) for all p. Although the last term assists in satisfying the inequality we shall end up with a constant of proportionality of about 4 (See Part II) hence the final term is quite small; basically we require r (1 - x ) < 1 (C-12) The original determination of r by Jones led to 2 r(p) = 3.53125 - 0.1875 p p 000)+0.58594 1000) We adjusted this to the p = 0 limit of r(0) = 3.448275 in order to avoid negative slip below about 50 psi. Basically this r value is 1/0.29. If we extract the r(0) value and rescale to p Ca*
- I ""
- 2 r(p) = g1 1 p + 1.746 p
-0.1743[P \ CR / \ CR / ,
The simplicity of this form led to an examination of whether this poly-nomial could be replaced by one based on the thermodynamic variables. The final choice was
r(p) = h (1 + 1.57 pg / p, ) Although the initial slopes (around p = 0 ) are distinctly different the overall curves are quite close. Consider also a form for x 1 x (p) = K, + (1 - x,) (p/p CR This form is similar to that used in ( ) and was proposed by Jones. As above we replace p/p CR gi take (see Part II) x1 (p) = x , + (1 - x,) (p g/pg )N (C-13) with this choice r(p) = 1 1 + 1.57 p/p gg (C-14) 1 - x, . This does indeed satisfy eq (C-12) for all p. Indeed all conditions are satisfied except for eq (C-6d). The Drift Velocity Equation (C-4) is written as an equality. In fact, however, it is an identity and contains no infonnation as such. The term vg ,contains as i much information as all of eq (C-4) if one could but write a relation for l l it. Indeed this has been done a number of times ( , , .) for well chara-cterized flow regimes. Ishii for example has derived a number of analytical forms for annular flow which contain terms proportional to (1 - a)3 and (1 'a)2, but his definition for v , gappears to be different from ours. Although it is annoying,there are a number of ways that the identity between average vapor and liquid velocity can be written, Malnes discusses three, for our purposes the specific original form of v gj is not of im-t
portance since we have not proposed to try to determine analytic forms for C, or v gg but ultimately to fit the terms through the use of data. Thus, we apriori assume that eq (C-4) is an equality and attempt to fit C, and vg ,. We assume thatgv , is a measure of buoyancy in the low a limit and take
-k
'(p,- o g) cr 99 c "V gI (C-15) lim v =C 2 g1 a-0 ,
p,2 If we use the definition flow quality X a G /G g and o we remember that G o =G g +G j then we may write X X + g (1 - X) 99 hence using eq (C-5) we find X = apq fpj ( Co + ) (C-16) 1-Cao + Cnapg /p, As n - I we observe that X ~ 1+y (1) p,/Gg and since X must - I as a - I we have lim vg , = 0 a-1 and v,= g (1 - a)D with b > 0 Which along with condition C3 leads to eq (C-6d) and
b 51 Since the slip should not be bouyancy dependent as a -I we choose (finally) b = 3/2 (C-18 which is related to single bubble behavior. The coefficient C, is taken from the literature ( )as (C-19) C, = 1.41 The parameter r, is the only one that has not been determined
- id we use it to introduce mass M ow dependance.
It has been observed that gC has a limiting value of 2 in laminer flow in tubes;we introduce this observation throvgh an approximate relation 5 C-20 1/x,1 = 1 + e -Re/10 where Re is taken as the lnlet value using Go ans defining D as Dg. Numerical studies however show that for large values of Re kol becomes too large and we find it necessary to apply a cut off on Re such that "o " "I" "ol' #o2 (C-21) where x, = 0.8 for cylinders,annulli and bundles
= 0.71 for channels It is clear that this development will not satisfy all readersT The slip defined here is monotonic witn a and does not peak; the very low flow ' values are probably incorrect because vgg does not contain a strong diameter dependence (only tht ough Kg). However, the general quality of tne predictions are such as to make this slip model acceptable for its desired uses: relatively high flows and pressures above a very few hundred osi.
APPENDIX D Location of the Point of Net Vapor Generation (NVG) The succooled void formation model we establish and qualify is different from those in the literature and one of the adjustable parameters in the model is the condensation coefficient AC introduced in i g3 It is adjusted to secure a statistically good fit to the dati and basically C Adetermines the point of NVG and we believe deserves further discussion. This location (ZD ) is often referred to as the point of void departure or detachment and we shall use the tent.s interchangably. The location of Z Doccurs where Z g = Min {Z,,Zf o and Z,is the axial location where a is the first non-zero, and Zo is the axial location where 'g is first positive. In steady state where the fluid enters subcooled Zg = Z g; however, during a transient reduction in power generation could lead to a situation where i'g < 0 but a > 0 in this case Zg = Zg When 2 9
=Z o we can usually expect some condensation to occur. In the usual experimental situations where qP = 0 we t
l expect. Z g =Z.g Hence, by neglecting direct deposition and at steady state we have Zg defined as follows: when I' = 0
'q" (Zg )= q"out(2 D ) +hA (T s -T) g now using the definitions for q" (eq 2b) and q"out (eg AS) and first solving the energy equation up to the point of departure (NVG) we find 4
l hg(Zg ) = h g(0) + G30,
*o L
- , .- , -. . . . . . . _ ~ _
d from this we may determine the Nik temperature T,(ZD ) and two values for the wall temperature T,1 = (q"
- ho Tg+ h, T3 ) / D, + bg P T = T + 2q ' - 2(h + h ) (T -T) g s D 3 The correct value for Zg is where ,,1 =T g. This is in fact a fairly complex model for determining 3Z . Other authors have assumed that DZ is reached when the condensation tarm matches the total heat source, i.e.
when h c T 3
-T g(2))=q" 3 (D-2) and Zr is found from hg (ZD ) (i.e. from the energy equation). In this form the wall temperature (hence the wall superheat) does not appear at all. We believe this lack is a serious one since the wall superheat is a basic determinant for subcooled void formation. The statistical analyses of Part II shows that the h 3correlation of Hancox and Nicol (with a change in C ) predicts not only the point of detachment quite nicely but the subcooled modelg I allows an accurate prediction of the void fraction l in the rest of the subcooled regime as well.
If one were to try to compare the earlier correlations of Saha and Hancox and Nicol (H & N) with the present one then we should have to define a pseudo coefficient
~
hg +h g h* = hU + h^ j . h3+ @D . l The fact that h3 = T, - T sd es not impact here since S e h g multiplier must lie between 1 and 2 hence a simple rangt for h* can be determined. One can now craate a simple plot of the critical Stanton number at detachment l Since ha" U-T) w 3 T,1 is rea h sol M & N m a pa ha m e pa W n t
l s versus Paclet number. The definiticas imply ST = Nu/Pe
= h j
kPe/D i Where h is any of the correlations being considered. In this form I f i q" = h T s
-T(Z)f g D as for Saha but h* depends directly on the wall superheat. If we take the Saha data analysis ( ) and show it along with the H & N and the present correlation we have the presentation in Fig. D-1. While the Saha data display covers the same range for the Peclet numbers generally of interest the trend of the present correlation is significantly different. Because of the der ee of disagreement some further discussion appears warranted.
There are three points which appear of interest here, and they are: e The definition of the point of vapor generation (PVG) e The determination of the PVG from the experimental data e The sensitivity of the prediction of the Stanton number to the inferred liquid subcooling l 1. Defining the Point of Vapor Generation Basically all researchers appear to be agreed that there is a region of quite low void fraction followed by one of more rapidly increasing void fraction and that where one ends and the other begins defines the point of net VG. Unfortunately the data usually does not have such labels attached to it. The current model defines a point of VG which (viz. Fig. ) often exhibit long,lo.! void fraction t' ails. It is not clear then that the current model defines the same point of departure as other models do. l 2. Determining PVG from Exoerimental Data l A determination fo the PVG from the experimental data is necessary in
A COMPARISON OF . THREE HEAT TRANSFER CORRELATIONS FOR PREDICTING DETACHMENT
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order.to determine the liquid subcooling at that point. In order to demonstrate the difficutly of getting a reasonably clear estimate we have re-examined several sets of experimental data of void fraction versus heated length (Fig. 3-8). We have tried to determine the minimum and maximum values for the location of PVG that could reasonably be inferred from the data in order to place error bars on the subsequent detennination of the Stanton number. The results are shown in Fig. 2. The single symbol points are relatively cleanly determined as far as the published data is concerned while the others exhibit significant error ba rs . It is not clear that this presentation of data supports the S.Z. data range. Other data points due to Egan and Ferrell are also shcwn with extimates of the location of the PVG based on the steepest slope of void fraction versus distance and on a less steep projection (the circled crosses in Fig. 8).
- 3. The Prediction of the Stanton Number The procedure we have used in determining the St/Pe data space is quite straightforward. Thus if ATD a Ts -Tg (Zg) i 4T sub aT 3- T g(0) then S T e q"/G c ATg (17a) p Pe a GDcp /k and 'if q" is constant l
l ATg = AT sub - (17b) (ST . ATD ) 4Z /De Since(ST AT g ) is defined as q"/Gcp it is possible to secure ATg (hence ST) if Zgis known. Interestingly several experimenters have assumed l I
. THE EFFECT OF ERROR BARS OF THE LOCATION OF THE POINT OF NVG ON THE SAHN-ZUBER DATA DISPLAY
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ci-ri-
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= = -
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=- ._ l : ._ . -
a i
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.j .
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. r....=.=..=... . . = . . .. = . . = . . .a. _ n. . . F. =- .,
.3, n .. . . . , - . ] .
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. . . .'P l
'l .. . . .
..g .
i =l . . - j - w.i n = q=n=cj == =p=i= 1 = d= = I i i ; i i
$. ; .. u - [ d, . k. , ..M =k-;h r -
k! . (!. 9 i i j .
.l t
that the point of vapor generation is given by continuing the line of maximum slope of a versus Z, or a versus X,down to the abcissa; the intersection then defines the departure po' int. In Fig. 3-8 we show some of the Foglia and Ferrcli data along with what we feel are reasonable estimates of the point of void departure. In many cases the difference is not significant while in others (Fig. 8) it can learl to large changes in the calculated value of aTo. Using Fig. 8 as an example, we find the reported value of ATO to be 23 F (for each of the cases) while the calculated values range from 13-46 depending of which run is used and which intercept is considered " reasonable". Thus the Stanton number
' calculated from the ATgcan have considerable variation and exhibits, we believe, a wider scatter than is evident from the S-Z data display.
We believe in fact that the usual difinition of PVG may be counter-productive. The effects of tails may be quite significant (due to reactivity feedback effects) in application of models and their neglect may not be acceptable. Because of this we believe that the actual PVG is less important than the statistical agreement between data and pre-diction above some small value of void fraction (say 1 - 2%). The adequacy of the current model is established in this way. l l l i i l
Appendix E Bul k Phase Superheating The reader will note that although source terms may change with differing flow regimes they are continuous across flow regimes. This is true with the exception of the cross-over fron subcooled to !,aturated conditions. We have 0 as Tj T hence that there mest either be a jump in shown that q"out 3 f or we must allow the bulk phase to superhcat. The reader may su:pect tnat g had q"out been proportional to Ts -T g (instead of T* - T j) there would have been a smooth transition to saturated condition with no jump and no super-heating of the bulk phase. That conclusion would however have been incorrect. Approaching Saturation Conditions One of the difficulties (in our estimation) with earlier mechanistic models of flow boiling is that the source term for heating the liquid is usually pro-portional to T s -T.f In this situation we have as steady state (neglecting pressure effects): m dh G = 4h (Ts - Tg ) /D (E-1) l 1 dr. i and from eq (E-1) it is clear tha+ Tj T3 either when Gj -0 or when dh - 0). The first implies a = 1 and the second can h l h, (implying i d: occur only when 7 a. Neither of tiiese is physically acceptable. We avoid them here by taking the bulk source term non zero as Tg Ts and are faced with,the question of jumps or superheating. This problem and conclusion is not characteristic of this model, or even of one dimensional nodels, nor does it go away if we incidde pressure
,, _ . , , . , . ~ , - - ,-.
_ . . -~ -. , -
.-[ -
effects . It is fundamental . Thus if after Tg=T 3 g f=q /h fg then the bulk phase energy equation can be writte.n as Gg vhg = 4q /D but if g out 0 for gT -T Ren either GjorVh g-0 and we have the above s del emmt .* Superheating of the Liquid If the bulk phase is allowed to superheat then the derivation of i' implies that for Tg>T3 h fg (1 + #)-h ,g with h, > h s and that condensation ceases (h -0). In this case h q"out = (T,- Ts ) + Q sT -T) g (E-2) and the maximum degree of superhaat (occuring at Gg = 0) is T-T g s
=1 3 (T,- Ts ) (E-3) 0 This result is unacceptable cince T, - Ts can easily reach 20 - 40 F and
= as/q"/h and bulk superheats will hardly exceed a (very) few tenths of a degree. Thu phenomenon we have not introduced is the evaporation (or flashing) that will occur. Indeed since or is already generally large (in most cases) when g T-T s there will be a fairly large surface area for the superheated liquid to evaporate across.
- There is a mathematical out, if g out = (Ts- T )a g with 0 < + < 1 then Tg T at a finite value of Z.
s Indeed if G gdoes not change much Z s
=
(Ts - ](o)) CG pg with C a proportionality constant. There is no (1 - a)C evidence that a is < 1 indeed there is much to say it is E 1.
e e In a given volume the amount of additional voidage created by the super-heated liquid is Aa = (1 - a) Pj (hj-h)/Pg(h s g - hg ). we propose that the rate of evaporation is proportional to (E-5) (aa) pgg y = (aa) P gSv, hence for Tf a T s (E-6) f'g , 4 (q" - v h ,)/h g Dg q"out + Cg MPggg This leads to a maximum superheat of , (E-7) Tf-T =3 ls (T, - T )/(1 s +CG gfSe p /ho) and eqs(E-4,5,6) imply that the evaporation Nussult number is Nu)E =C E 'l Analysis of E-7 shows that the superheat still -%ith /q". Quisnti fication 0 107 B/hr ftf where T, - T = 200 F we however shows that even for q"= s
= 1 keeps the superheat find for small Pe number ( = 1000) that a value of C E 0.05 is sug-below b F. For reasonable values of q" and Pe a value of CE 0
ficiently large and we use this value in the model . It should be stated that we have not been able to find experimental data to support this approach to bulk evaporation and while the steady state void fraction and wall tem-it is not clear peratures are not meaningfully affected by the choice of C E how transient calculations will be affected. Such studies will be per-l formed in the future. It is clear that as the heated length is extended beyond the location of saturation (i.e., into very high void fraction) that the vapor will start to superheat. This effect will alter the form of r .g We do not consider this phenomenon at this time.
. . . , - . _ . .- _ . - - - - _ . . - - - . . - - _ -- -}}