DEVELOPMENT OF A POST-DRYOUT HEAT TRANSFER MODEL

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1 DEVELOPMENT OF A POST-DRYOUT HEAT TRANSFER MODEL Y.J. Wang a and C. Pan a,b,c a Institute of Nuclear Engineering and Science, b Department of Engineering and System Science, c Low Carbon Energy Research Center National Tsing Hua Uniersity, Hsinchu, Taiwan, 30013, R.O.C. dauw8017@gmail.com; cpan@ess.nthu.edu.tw ABSTRACT A one-dimensional mechanistic model for fully-deeloped post-dryout region is deeloped in the present work. This model is composed of four conseration equations, i.e. droplet momentum, droplet mass, apor mass, apor energy, and a set of closure relations accounting for arious heat transfer mechanisms. In particular, a new correlation for the interfacial heat transfer between the apor and liquid droplets is deeloped considering the Eckert number effect. The model proides the detail information including the droplet elocity, droplet radius, heat transfer rate and apor temperature depicting the mechanism in the post-dryout region. Comparison of the wall temperature predicted with the experimental data in the literature shows that the present model predicts the temperature rise and the degree of thermal non-equilibrium successfully. Oer 90% of 319 wall temperature data with the range of P= 30~140 bar, q " = 04~1837 kw/m and G= 380~5180 kg/m s can be predicted within 0% with the present model. KEYWORDS Post-dryout, liquid deficient film boiling, dispersed flow 1. INTRODUCTION Beyond the dryout point in a boiling tube with high apor quality, liquid no longer coers the tube wall and heat is primarily transferred to apor accompanying with dispersed droplets. This may cause a thermal non-equilibrium phenomenon in which the apor temperature exceeds saturation temperature and become superheated while the dispersed droplets still remain saturated. In this region, which is called the post-dryout region or liquid deficient region, the heat transfer rate is deteriorated and it will lead to a sudden rise of wall temperature which might damage the wall. Therefore, post-dryout heat transfer is of crucial importance for the safety of a nuclear power plant, or the design of a heat exchanger, etc. Numerous prediction methods hae been proposed in the literature. Empirical correlations are the simplest and the most straightforward methods that use the experimental data to fit the expressions of heat transfer coefficient [1, ]. This method has strong dependence on the data they used and is limited to the range of experimental data. Phenomenological models consider the thermal non-equilibrium phenomenon with empirical apor generation rate to calculate the apor superheat [3-5]. Howeer, the alidity of the empirical apor generation rate is still limited in certain cases. Corresponding author NURETH-16, Chicago, IL, August 30-September 4,

2 Another predicting method, mechanistic models, gie the detail structure and physical properties of the flow. Mechanistic models were first proposed by the seeral reports published at Massachusetts Institute of Technology in 1960s, and the most quoted model was proposed by Forslund & Rohsenow[6]. Their models were based on the use of conseration equations to describe the heat exchange between wall, apor and droplets, and consider the effect of heat transfer from wall to contacting droplets. Iloeje et al.[7] extended their research and gae a sophisticated description on droplet-wall interaction. Other researchers followed up with detailed analysis on this mechanism [8-10]. With the presence of dispersed droplets, the study from Varone and Rohsenow[11] proposed a model accounting for the influence of droplets, seeral studies focus on the hydrodynamic of droplets[1-14] and gie a close look of droplet behaior. The mechanistic models presented in the literature coer different degree of complexity ranging from onedimensional to multi-dimensional analysis and different degree of simplification on arious heat transfer mechanisms. The objectie of this paper is to deelop a one-dimensional mechanistic model for post-dryout heat transfer, with a wide range of flow conditions. Each heat transfer paths are considered with closure laws studied to modify, if needed, the one in the literature. With the conseration equations and specific heat transfer mechanism description, the thermal non-equilibrium phenomenon is fairly well depicted as well as the wall temperature. More studies are undergoing.. MODEL DESCRIPTION The present model is composed of four basic conseration equations, i.e. apor mass, apor energy, droplet mass, droplet momentum, and a set of closure laws. Seeral assumptions are made to simplify the calculation: (1) The pressure remains constant. () The temperature of droplets remains at saturation temperature. (3) Before the onset of dryout, apor and liquid are in thermal equilibrium. (4) The droplet diameter is uniform across the tube. (5) No droplet coalescence occurs..1. Conseration Equations Momentum balance of liquid droplets The droplets dispersed in the post-dryout region are accelerated by the buoyancy force and the drag force from the apor. du 3 u g( ) C ( u u ) d l d l D d (1) dz 8rd Where C D is the drag coefficient, which will be discussed in section.3 Mass balance of droplet NURETH-16, Chicago, IL, August 30-September 4,

3 The aerage droplet size is reduced by eaporation due to the effect of apor conection to droplet ( q d ), droplet-wall interaction ( q wd ), and radiation heat transfer from wall to droplet ( q rwd ) and from apor to droplet ( q rd ): dr 1 dr ( q q q q ) () dz u dt u r i d d d wd rwd rd d d(4 ) l l q d, q wd, q rd, q rwd are to be discussed in section.. Mass balance of apor The decrease in droplet mass contributes to the increase in apor quality. Thus, Where N is the droplet flux, Energy balance of apor dxa drd G Nl4rd (3) dz dz G 1 x N (4) 4 3 rd l 3 The heat diffused from the wall may be transferred to the liquid droplets through direct contact or radiation, and to the apor, through forced conection or radiation, to increase the apor temperature as well as the actual quality. Thus, D dxa D dt q" Dqwd " Dqrwd " DG il Cp ( T Tsat ) G Cp (5) 4 dz 4 dz Rearrangement of aboe equation leads to the following equation for apor temperature: dt 4 q" qwd " qrwd " dxa il Cp ( T Tsat ) (6) dz GDCp dz.. Models For Various Heat Transfer Mechanisms The present model considers all seen heat transfer paths[1]: wall to apor conection, wall and droplets interaction, apor to droplets conection, eaporation of droplets, wall to apor radiation, wall to droplets radiation and apor to droplets radiation Wall to apor conection NURETH-16, Chicago, IL, August 30-September 4,

4 The heat flux from the wall is transferred to apor and droplets. The contact area ratio between apor and the wall may be approximated by oid fraction. Hence, the relation between heat flux and the wall temperature can be expressed as q" q "+ q "+ q "+ h T T wd rwd rw w w (7) Consequently, the wall temperature can be calculated by the following equation: T w q" qwd " qrwd " qrw " T h w (8) Where q" is the total heat flux from the wall, qwd " is the heat flux between wall and droplets due to droplet-wall contact, qrwd " and q rw " are the heat flux transferred by radiation to the droplets and apor, respectiely. The following temperature dependent correlation from Heineman [15] is used in the present work for the heat transfer coefficient between wall and apor. h h w w / a k Gx D Cp L L for 6 60 D k D D / a k Gx D Cp L L for 60 D k D D (9) Where L is the distance from dryout point..1.. Wall and droplet interaction The heat transferred from the wall to droplets can be ealuated using Guo and Mishima s model [10]. They assumed that, as a droplet hits the wall, a apor base is generated at the interface and heat is transferred by film boiling mechanism and obtained the following time-aeraged heat flow rate to a single drop: 3 l hu dr qsd k( Tw Tsat ) trrd 9 k( Tw Ts) trrd 1/4 (10) Where t R is the droplet residence time, which is defined as the duration time that a single droplet has direct contact with the wall, and can be ealuated using the equation gien in [10]. The aerage dropletwall contact heat flux can then be calculated as: q wd " q sd m D 4 l r 3 3 d (11) Where the droplet deposition rate onto a heated wall, m D, can be ealuated by the following equation[10] NURETH-16, Chicago, IL, August 30-September 4,

5 .1.3. Vapor to droplet conection md l 0. Re l (1 ) (1) l D l It is found in the present study that the conection heat transfer between apor and droplets is the key mechanism in post-dryout region, which depicts the degree of thermal non-equilibrium and minate the accuracy of prediction. The heat transfer rate from apor to droplets per unit length tube is: Where h i is the interfacial heat transfer coefficient. q " h( T T ) (13) d i sat Lee and Ryley[16] proposed a model predicting h i of droplets in high temperature steam. Ban and Kim[17] modified the Lee-Ryley model to account for the effect of thermal non-equilibrium: k 0.5 1/3 l i ( 0.74 Red Pr f ) d i il h i (14) Where Re d is the droplet Reynolds number based on droplet diameter, relatie elocity between apor and droplet, apor density and mixture iscosity: ( u u )r (15) d d Red m Where the mixture iscosity is to be presented and discussed in section.3. Howeer, the calculation results in the present study shows that the slope of apor and wall temperature rise is too small, which indicates that the interfacial heat transfer rate needs to be modified. Applying the dimensional analysis, the Nusselt number can be expressed in the form of [18] Nu fn Re, Pr, Ec (16) Where Re, Pr are Reynolds number and Prandtl number, respectiely. Ec is the Eckert number, which accounts for the relation between kinetic energy and enthalpy, defined as: ( u ud) Ec Cp ( T T ) sat (17) In this study, the Ban and Kim correlation[17] is modified in the following way h k 0.5 1/3 n l i ( 0.74 Red Pr f Ec ) rd i il i (18) NURETH-16, Chicago, IL, August 30-September 4,

6 The effect of Ec on wall temperature is presented in Figure 1, which shows the strong effect of the exponent of Ec on the trend of temperature rise. The results implicate that the size of the droplets are extremely small, so it is necessary to consider the frictional effect on heat transfer. The iscous dissipation from droplet to apor at the interface reduces the total heat transfer rate from apor to droplets. Neglecting this phenomenon will lead to an oerestimation of interfacial heat transfer rate. The alue of n selected for predicting best-fitted wall temperature is 0.4. This alue is used for all the cases of this study Tw_exp n=0 n=0.1 n= n=0 n=0.1 n=0.4 T w (C) 400 T (C) z (m) z (m) Figure 1. (a) Comparison of predicted wall temperature based on different exponent (b) Comparison of predicted apor temperature based on different exponent.1.4. Radiation heat transfer The radiation heat transfer among the wall, apor and droplets in this paper is based on that proposed by Sun et al.[19] with the wall emissiity of 0.4 for Nimonic alloy at high temperature..3. Other Closure Relations Mixture iscosity The interactions among the droplets on the mixture iscosity should be considered. The present study apts the model proposed by Ishii and Zuber [0] for the mixture iscosity and, 1.5 ( 0.4 )/( ) ( d ) dm l l m l (19) dm Where dm is the maximum packing oid fraction, dm 0.6 ~1for dispersed flow. For the present study, 1 is used. dm Drag coefficient NURETH-16, Chicago, IL, August 30-September 4,

7 The size of droplets will decrease gradually while they flows wnstream, and the change in droplet size and elocity will affect the drag force applied on each droplet. The drag coefficient from Ishii and Zuber[0] is employed in the present work. Droplet break-up 4 C D=, for Re d<1 (Stokes regime) Re d 0.75 d (0a) 4(1 0.1Re ) C D=, for Red 1 (iscous regime) (0b) Re The diameter of a moing spherical droplet is goerned by the Weber number [1], We, which is a dimensionless number describing the relation between the inertia force and the surface tension: d ( u ud) rd We (1) According to Varone and Rohsenow[11], the critical Weber number in post-dryout region is Beyond this alue, the surface tension is no longer able to sustain the shape of a droplet and the droplet will breaks up due to the high alue of inertia force. The present study apts the Varone and Rohsenow s droplet break-up criteria for the modeling of droplet size and number density. If break-up occurs, 1 Wecr rd ( u ud) The droplet number density should be recalculated from aboe droplet size. ().4. Dryout Condition In order to sole for the four differential conseration equations, we need seeral boundary conditions for numerical integration. Void fraction It is found that the accuracy of oid fraction at the dryout location is important. The oid fraction directly affects the initial droplet elocity, u d, and hence affects the droplet behaior in the channel. For example, under the condition of high heat flux and low mass flux with high dryout quality (>0.9), different predicting correlation leads to different results in oid fraction and the calculated droplet elocities at dryout location, as shown in Table I. Although the difference among the three selected correlations is within 5%, the calculated uddo, demonstrates large discrepancy. This leads to significantly different wall and apor temperature distribution wnstream, as shown in Figure. Although the temperature is slightly under-prediction when the correlation proposed by Ahmad [] is used, the trend of the temperature rise is more reasonable, and this correlation is apted in the present study since it predicts the trend of temperature rise well. s u GD l ud l (3) NURETH-16, Chicago, IL, August 30-September 4,

8 Table I Comparison of Calculated Void Fraction and Droplet Velocity at the Dryout Point (Case 94of Becker et al[3]) Void fraction at dryout G(1 xdo) uddo, point with xdo We l (1 ) Ahmad[] Woldesemayat and (>17.5) Ghajar[4] Cioncolin and (>17.5) Thome[5] ( u ud) rd Tw_exp Ahmad Woldesemayat and Ghajar Cioncolin and Thome Ahmad Woldesemayat and Ghajar Cioncolin and Thome 600 # Tw (C) 400 T (C) z (m) z (m) Figure. (a) Wall temperature from different oid fraction at dryout point (b) Vapor temperature from different oid fraction at dryout point (Case 94 of Becker et al.[3]) Droplet size at dryout point The droplet diameter is one of the key parameter for post-dryout predictions which goerns the droplet number density and hence affects the total heat transfer rate. After comparing the calculation from seeral droplet models,the model of Kataoka et al.[6] is employed in the coupling computation with aboe model description: r d /3 3 ud 1/3 /3 xg l l ( ) ( ) ( ) (4) 3. SOLUTION METHOD The differential equations are soled using Runge-Kutta 4 integration technique to obtain the distributions of droplet size, droplet elocity, apor temperature, apor quality and wall temperature along the channel. The four differential in section.1 with boundary conditions in section.4 are soled with step size NURETH-16, Chicago, IL, August 30-September 4,

9 data from Bennett et al. [7] and Becker et al. [3] were compared and discussed in the next section. 4. RESULTS AND DISCUSSION The following results compare the difference between high and low degree of thermal non-equilibrium. Low mass flux will more likely to cause high degree of thermal non-equilibrium. As shown in Figure 3(a), for the case of high degree of thermal non-equilibrium, the apor temperature and wall temperature will rise continuously along the tube; howeer, for the case of low degree of thermal non-equilibrium, apor temperature remains nearly at saturation temperature and the wall temperature reaches an asymptotic alue shortly after the dryout location. In the aspect of relatie elocity, as shown in Figure 3(b), the relatie elocity between apor and droplet reaches equilibrium in a short distance beyond the dryout location for both cases. Howeer, for the case of high degree of thermal non-equilibrium, the elocity difference between apor and droplet is higher than the opposite case, which gie rise to the existence of dynamic non-equilibrium. The radius of droplets are much smaller for the case of low dryout quality, and the oerall heat transfer area are much more larger than that for the case of high dryout quality case Tw #315 Tw #333 T #315 T #333 x #315 x # (u -u d ) #315 (u -u d ) #333 r d #315 r d # x x x x10-5 T (C) x 0.6 u d -u g (m/s) x x x10-5.5x10-5 r d (m) x x (a) z-z 0 (m) (b) z-z 0 (m) Figure 3. (a) wall temperature and apor temperature change (b) relatie elocity of droplets and radius change for the case #315 and #333 from Becker et al.[3] #315: 30bar, G= kg/m s, x #333: 30bar, G=496.9 kg/m s, x x10-5 Figure 4 shows, under the same dryout quality and boiling number, the percentage of heat transfer rate from the apor to the droplets for the cases of high and low degree of thermal non-equilibrium, respectiely. For the case of low degree of thermal non-equilibrium, the heat transfer between apor and droplets is significant; while the opposite cases show less amount of heat is transferred to the droplet, which reduce to the thermal non-equilibrium phenomenon. As for the pressure effect, the result indicates that the heat transfer between apor and droplets is more effectie at high pressures when it is under high NURETH-16, Chicago, IL, August 30-September 4,

10 thermal non-equilibrium condition, and the eaporation of droplet will reduce the wall superheat. The calculated temperature, Figure 5, shows that the model successfully predicts the degree of thermal nonequilibrium and the trend of temperature change. 1.0 P=30bar, low degree of thermal non-equilibrium P=50bar, low degree of thermal non-equilibrium P=50bar, high degree of thermal non-equilibrium P=70bar, high degree of thermal non-equilibrium q d / q z-z0 (m) Figure 4. Heat transferred rate to the droplets in (a) Cases of low degree of thermal non-equilibrium #315: 30bar, x =0.366, G= kg/m s, Bo= from Becker et al.[3] #8: 50bar, x =0.316, G= kg/m s, Bo= (b) Cases of high degree of thermal non-equilibrium #307: 50bar, x =0.876, G=498.1 kg/m s, Bo= from Becker et al.[3] #77: 70bar, x =0.895, G=500.6 kg/m s, Bo= Tw (C) Tw #315 (Low P) T #315 (Low P) Tw #8 (High P) T #8 (High P) Tw exp #315 (Low P) Tw exp #8 (High P) Tw (C) (a) z-z0 (m) (b) Tw #307 (Low P) T #307 (Low P) 300 Tw #77 (High P) T #77 (High P) Tw exp #307 (Low P) Tw exp #77 (High P) z-z0 (m) NURETH-16, Chicago, IL, August 30-September 4,

11 Figure 5. (a) Vapor and wall temperature prediction in low degree of thermal non-equilibrium o #315: 30bar, T sat = C,x =0.366, G= kg/m s, Bo= from Becker et al.[3] o #8: 50bar, T sat = C, x =0.316, G= kg/m s, Bo= (b) Vapor and wall temperature prediction in high degree of thermal non-equilibrium o #307: 50bar, T sat = C, x =0.876, G=498.1 kg/m s, Bo= from Becker et al.[3] o #77: 70barT sat = C, x =0.895, G=500.6 kg/m s, Bo= Figure 6(a) shows the oerall comparison of the wall temperature prediction with the experimental data. The predicted results show fairly good agreement with the experimental data. Oer 90% of total 319 data can be predicted within 0% from the present model. The discrepancy between the predicted T w and experimental T w are majorly for the data taken in the short distance beyond the dryout location. The reason may be attributed to the premature temperature rise in the prediction as indicated in Figure 5(a) and Figure 6(b). The present model es not consider the transition boiling from annular flow regime to post-dryout regime in which the liquid film and droplets may still hae intense eaporation on the wall. As a consequence, this model can only be applied for the fully-deeloped post-dryout regime. As for the model being applied on the fully-deeloped post-dryout regime, the result shows a standard deiation of 6.3% and root-mean-square (RMS) error of 7.%. More study considering transition region is under deeloping now. Predicted Tw (C) % -0% bar 50bar 70bar bar 10bar 140bar Experimental Tw (C) Tw (C) Tw_exp #307 Tw #307 Tw_exp #89 Tw # z (m) Figure 6. (a) Oerall comparison of the predicted wall temperature with the experimental data. (b) The large discrepancy appears right beyond the dryout location. 5. SUMMARY AND CONCLUSION A one-dimensional mechanistic model of fully-deeloped post-dryout heat transfer is proposed and ealuated in this paper. A correlation has been deeloped for the interfacial heat transfer between apor and droplets, which modified the one by Ban and Kim[17] considering the contribution of Eckert number. This correlation indicates that in the post-dryout region, the iscous dissipation from droplet to apor at the interface reduces the total heat transfer rate from apor to droplets. Neglecting this phenomenon will lead to an oerestimation of interfacial heat transfer rate. NURETH-16, Chicago, IL, August 30-September 4,

12 319 experimental data points with the range of P= 30~140 bar, q = 04~1837 kw/m and G= 380~5180 kg/m s from Bennett et al.[7] and Becker et al.[3] are examined and compared with the present model predictions. The results show that the present model predicts fairly well in the wall temperature and the thermal non-equilibrium phenomenon in the fully-deeloped region, which gie the standard deiation of 6.3% and RMS of 7.%. The detail flowing information of apor and droplets in the post-dryout region is also shown in the article. Comparing the difference between high and low degree of thermal nonequilibrium, the results show that the heat transferred to the droplets is significant in low degree of thermal non-equilibrium, but poor in another. As for the pressure effect, the heat transfer between apor and droplets is more effectie at high pressure when it is in high thermal non-equilibrium condition, and opposite is the case in low thermal non-equilibrium condition. NOMENCLATURE q" Bo Boiling number ig l Cp specific heat, kj/ kgk D tube diameter, m G mass flux, kg/m s g graitational constant, m/s h heat transfer coefficient, W/m K i l latent heat of eaporation, kj/kg k thermal conductiity, W/mK N droplet flux, #/m s hk Nu Nusselt number D Pr Prandtl number = Cp k q heat flow rate, W q " heat flux, W/m ud Re Reynolds number = r aerage radius s slip ratio T temperature, K u elocity, m/s x quality Greek symbols oid fraction dynamic iscosity, Pa s density, kg/m 3 surface tension, N/m Subscript a actual c critical DO dryout d droplet f film temperature NURETH-16, Chicago, IL, August 30-September 4,

13 i l rwd rw rd sd d w wd w interfacial saturation liquid radiation between wall and droplets radiation between wall and apor radiation between apor and droplets single droplet apor apor to droplet wall wall to droplet wall to apor REFERENCES 1. R. S. Dougall and W. M. Rohsenow, "Film boiling on the inside of ertical tubes with upward flow of the fluid at low qualities," Cambridge, Mass.: Dept. of Mechanical Engineering, Massachusetts Institute of Technology,[1963] D. Groeneeld, "Post-dryout heat transfer: physical mechanisms and a surey of prediction methods," Nuclear Engineering and Design, 3(3), pp , (1975). 3. D. N. Plummer, "Post critical heat transfer to flowing liquid in a ertical tube," Cambridge, Mass.: Heat Transfer Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, D. Groeneeld and G. Delorme, "Prediction of thermal non-equilibrium in the postdryout regime," Nuclear Engineering and Design, 36(1), pp. 17-6, (1976). 5. P. Saha, "A nonequilibrium heat transfer model for dispersed droplet post-dryout regime," International Journal of Heat and Mass Transfer, 3(4), pp , (1980). 6. R. Forslund and W. M. Rohsenow, "Dispersed flow film boiling," Journal of Heat Transfer, 90(4), pp , (1968). 7. O. Iloeje, W. Rohsenow, and P. Griffith, "Three-Step Model of Dispersed Flow Heat Transfer (Post-CHF Vertical Flow)," ASME Paper, (1975). 8. E. Ganić and W. Rohsenow, "Dispersed flow heat transfer," International Journal of Heat and Mass Transfer, 0(8), pp , (1977). 9. R. A. Moose and E. Ganić, "On the calculation of wall temperatures in the post dryout heat transfer region," International Journal of Multiphase Flow, 8(5), pp , (198). 10. Y. Guo and K. Mishima, "A non-equilibrium mechanistic heat transfer model for postdryout dispersed flow regime," Experimental thermal and fluid science, 6(6), pp , (00). 11. A. F. Varone and W. M. Rohsenow, "Post dryout heat transfer prediction," Nuclear Engineering and Design, 95(pp. 3157, (1986). 1. M. Andreani and G. Yadigaroglu, "Prediction methods for dispersed flow film boiling," International Journal of Multiphase Flow, 0(pp. 1-51, (1994). 13. M. Andreani and G. Yadigaroglu, "A 3-D Eulerian-Lagrangian model of dispersed flow film boiling including a mechanistic description of the droplet spectrum eolution I. The thermal-hydraulic model," International Journal of Heat and Mass Transfer, 40(8), pp , (1997). 14. M. J. Meholic, "The deelopment of a non-equilibrium dispersed flow film boiling heat transfer modeling package," The Pennsylania State Uniersity, 011. NURETH-16, Chicago, IL, August 30-September 4,

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