Numerical study of high heat flux pool boiling heat transfer

Transcription

1 Numerical study of high heat flux pool boiling heat transfer Ying He*, Masahiro Shoji** and Shigeo Maruyama*** *: Advanced Computer Center, The Institute of Physical and Chemical Research, -1, Hirosawa, Wako-shi, Saitama, Japan **: Department of Mechanical Engineering, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, Japan **: Engineering Research Institute, The University of Yokyo, Yayoi, Bunkyo-ku, Tokyo Japan Abstract A numerical simulation model is proposed based on a numerical macrolayer model [1]. In this model, the boiling curve was reproduced numerically by determining the macrolayer thickness. The change of critical heat flux dependent on the contact angle was estimated. In addition, the formation of nucleate site density was considered and the effect of surface roughness was investigated. Furthermore, the model was applied in the transient boiling process. The transient boiling curves were predicted though increasing power linearly and exponentially. The saturated liquid fluorinert FC-7 and water was used in the numerical calculation. Finally, this model was combined with three-dimensional transient heat conduction to investigate the spatial variations of wall temperature. Nomenclature A area A v area covered by vapor stems A w total area of heater * D c diameter of cavity g gravitational acceleration H heater thickness H fg latent heat of evaporation i, j, k coordinate indices in x, y, z directions L heater length N number of molecules per volume N s number of surface cavity q instantaneous heat flux q av time averaged heat flux q CHF critical heat flux q in input heat flux q α instantaneous heat flux component due to the increase of void fraction q δ instantaneous and average heat flux component due to the decay of the macrolayer thickness q m Gambill-Lienhard upper limit heat flux Q heat transfer rate r s radius of a vapor stem R ideal gas constant cavity mouth radius R ca t time t 0 exponential period in Fig. 11 T 0 initial averaged wall temperature T b temperature boundary condition at the bottom of the heater T rate wall temperature heating rate in transient boiling T w surface temperature v 1 volumetric growth rate of bubble w equivalent macrolayer thickness x, y, z rectangular coordinates α void fraction α 1 thermal diffusivity δ 0 initial macrolayer thickness δ instantaneous macrolayer thickness δ m macrolayer thickness corresponded to q m ε convergence coefficient m mass velocity of evaporated molecules t time-step size T act active wall superheat corresponding to a cavity Τ wall superheat λ thermal conductivity of water thermal conductivity of heater λ H 1

2 θ ρ l ρ v σ τ ξ contact angle density of liquid density of vapor surface tension bubble departure period volumetric ratio of the accompanying liquid to the moving bubble Subscripts 0 initial av average b bottom boundary cal calculated value h heater l liquid m maximum limit of evaporation s boiling surface v vapor w heated surface P present Superscript m, m+1 computational times n, n+1 time indices 1. Introduction Boiling is a complex and elusive process. For the past several decades, numerous investigations of modeling efforts on nucleate boiling have been carried out. According to the hypothesis on the heat transfer, these models may be divided into three major categories: bubble agitation models, vapor-liquid exchange models and latent heat and macro/microlayer evaporation models. The bubble agitation models are based on the assumption that process of bubble growth and release induce the surrounding liquid to the heat transfer process. The main action of the bubble is to agitate the liquid, but carry away little heat. These kinds of models consider the heat transfer as the turbulent forced convection. The pool boiling heat transfer is correlated as relationship between Nusselt number and Reynolds number [, 3]. The representative models are Rohsenow s model [4] and Zuber s microconvection model [5]. In the vapor-liquid exchange models, the movement of the vapor was postulated as microscopic pumps. When the bubble releases the from the heated surface, it pushes the superheated liquid from the near-wall region into the cooler ambient and draws the cold fluid to the surface, at this time strong convection heat transfer is carried out between cold and superheated liquid. However, the temperature boundary layer is usually much less than the diameter of bubble, so the heat carried by the boundary layer will be limited. This kind of model may be far from the reality [,3]. Recently, many researchers have shown that the evaporation of the liquid macro/microlayer plays an important role, especially at high heat flux regime. The liquid layer includes the microlayer underneath the bubble and the macrolayer at the base of coalescence bubble. On the other hand, the representative mechanisms responsible for the critical heat flux are Helmholtz-instability mechanism and the liquid macro/microlayer evaporation mechanism. Zuber [6,7] originally put forward a critical heat flux (CHF) theoretical model basing on Taylor instability and Helmholtz instability. The fundamental notion is that the vapor generated at the flat plate accumulates to form the large continuous columns. CHF takes place when the vapor-liquid interface of the escape passage becomes unstable due to Helmholtz instability. The equation is 1/ 4 π σg( ρ ) l ρ v qchf = ρ v H fg (1) 4 ρ v Bubble agitation mechanism and Helmholtz-instability mechanism can either explain most of heat transfer at the low heat flux regime or CHF, however, they can not account for the continuity of the pool boiling curve. At this point, macro/microlayer evaporation mechanism is reasonable because it can reproduce the pool boiling curve from the nucleate boiling to transition boiling. Therefore, in the recent decade, numerous experimental and modeling efforts on liquid layer evaporation mechanism were carried out. Two main different postulates on the evaporation of liquid layer can be divided: the macrolayer model and the liquid-solid contact model. Haramura and Katto [8] and Pan et al. [9] suggested an alternate CHF theory basing on the role of the macrolayer. These models retained the basic element of Zuber s model [6,7] that

3 hydrodynamic instabilities dictate the occurrence of CHF. However, they proposed that the controlling instabilities occur not at the walls of large vapor columns but rather at the walls of tiny vapor stems around active nucleate cavities that intersperse the liquid macrolayer on the heater surface itself [10]. Bhat et al. [11] put forward a theoretical model of macrolayer formation. It suggests that active site density may play a role in high-heat flux nucleate boiling and CHF. In the above models, the periodic supply of macrolayer liquid layer is considered to be the main mechanism of nucleate boiling and CHF. On the other hand, in some models, the evaporation of micro liquid layer and the increasing of dry area of individual bubbles are considered to be important in nucleate boiling heat transfer. Dhir and Liaw [1] showed this thought in their time-averaged model. In the model, the energy from the wall is conducted into liquid macro/micro layer surrounding the stems and is utilized in evaporation at the stationary liquid-vapor interface. Later, Nagai and Nishio [13] pointed out that the growth of the primary isolated dry area was the mechanism of critical heat flux according to their observations through a single crystal sapphire boiling surface. Zhao et al. [14] suggested the same viewpoint in their microlayer model. Although these models can explain critical heat flux and transition boiling fairly well, there appears to be some discrepancies among these models. Maruyama et al. [1] presented a numerical model basing on the macrolayer theory. The model retains the basic thought of Haramura and Katto s model [8] that the macrolayer forms underneath the coalesced bubble and dries out periodically. The difference is about the assumption of vapor stems. The model postulated that vapor stems are formed on the active cavity sites with a certain contact angle and the evaporation also occurs at the liquid-vapor interface. However, Haramura and Katto [8] assumed that the liquid-vapor interface was stationary and the entire surface heat flux contributes to macrolayer evaporation. In fact, this model can be considered to be the combination of the spatial-averaged and time-averaged model. Although the model by Maruyama et al. [1] combined the two kinds of viewpoints, there remain some problems that can be classified as: 1. The initial macrolayer thickness is determined by wall superheat, which has few experimental proofs.. The transition period between two vapor bubbles was not considered. 3. The situation when the wall temperature is not uniform was not considered. 4. The effect of surface roughness was not considered. On the other hand, recently, many researchers are trying to look for a valid numerical method to simulate boiling heat transfer. Pasamehmetoglu [15,16] developed a computer program to analyze low heat flux nucleate boiling. They simulated the bubble dynamics on the boiling surface by solving 3D transient conduction equation. The cylindrical control volumes with octagonal and square cross-sections in lateral and vertical directions respectively were used in the simulation. Each octagonal cell is assumed to contain a nucleation cavity and representing the footprint of a departing bubble. Bai and Fujita [17] performed a numerical simulation of a single bubble by combining hydrodynamics and heat transfer behavior. It was assumed that the evaporation heat was supplied from the superheated layer around the bubble by convection and from the meniscus part of the microlayer. Two-dimensional transient equations of mass, momentum and energy were solved by arbitrary Lagrangian-Eulerian finite element method (ALE-FEM). The growth process of an isolated bubble was simulated continuously. Son and Dhir [18] carried out the similar work associated with a single bubble. Lately, CML (Coupled Mapping Lattice) [19] and LGA (lattice gas automata) [0] method were also applied in boiling simulation. Although CML and LGA reproduced different boiling modes and full boiling curve, the methods can only simulate boiling phenomena qualitatively at present. In summary, more researchers have become to pay attention to the numerical simulation of boiling and tried to find more information which may be difficult to be found by experiments. Because there are still many arguments on macro/microlayer evaporation mechanism, we 3

4 tried to use the numerical methods to develop a macrolayer model and investigate the unity between the macrolayer and liquid-solid contact model. In this paper, we developed the model of Maruyama et al. [1] by reproducing the boiling curve numerically. Basing on their physical model, we used a simple numerical method and fewer empirical relations to determine the wall superheat from the heat flux. In addition, the formation of nucleation site density was included to investigate the effect of surface cavities. Furthermore, the model was applied in the transient boiling process. The transient boiling curves were predicted through increasing power linearly and exponentially. Finally, the partial numerical method was used to investigate the spatial variations of wall temperature.. The Developed Numerical Model.1 Governing equations In this model, the basic physical model and equations of Maruyama et al. [1] was used. Fig. 1 shows the schematic of the top and side views of a vapor bubble on a heated surface. In this model, the macrolayer containing vapor stems occupies the region immediately next to the wall. The vapor stems are formed on the active cavity sites. The most important feature of this model is the introduction of a liquid-vapor stem interface evaporation phenomenon, which means that the evaporation occurs not only at the vapor bubble-macrolayer interface but also at the stem interface. In this part of the study, we assumed the wall temperature was uniform and constant. The influence of temperature variation in the heater was not considered. From the heater surface heat is conducted into the macrolayer and is utilized in evaporation at the macrolayer-bubble interface. Therefore, the heat balance is written as dδ λ T ρ = () l H fg dt δ By the integration of equation (), the thickness of macrolayer can be obtained as 4 δ ( t) = δ 0 λ T ρ H l fg t (3) The rate of heat transfer from the liquid-vapor stem interface can be written as r1 Q = qda= qπrdr r0 (4) 1 δ δ = πr 0λ T log tanθ r0 tan θ δ m where A is the liquid-vapor stem interface area. Suppose that heat from the heated surface is conducted into the interface area and is applied in the evaporation at the stem-liquid interface. The evaporated heat just contributes to the increase of radius of the vapor stem. Therefore, the heat balance can be written as drs qda = ρl H fg A sinθ (5) dt where θ is defined as the contact angle in the model. The growth rate of vapor stems dr s /dt is then obtained from dr s 1 λ T δ = 1+ log (6) dt ρ H δ l fg δ m where δ m is the thickness corresponding to the maximum evaporation heat flux for saturated pool boiling. The introduction of δ m is to avoid the infinite heat flux at the liquid-vapor interface. The maximum evaporation heat flux can be obtained by considering the evaporation and condensation of molecules. The upper limit heat flux thus can be written as ρ ρ H H l v fg fg q = mh = T m fg (7) ρ ρ T πrt l v For saturated pool boiling of water at atmospheric pressure, the upper limit heat flux can be expressed as qm =786. T( MW/ m ) (8) Therefore, δ m can be written as λ T δ m = (9) q m The instantaneous heat flux can be obtained as dw q = ρl H fg dt dδ dα = ρl H fg ( 1 α) + ρl H fgδ (10) dt dt = q + q δ α sat sat

5 where the parameter w is defined as the equivalent thickness. It means the amount of liquid left on the surface. It can be expressed as w = δ ( 1 α ) (11) q δ and q α 1 are the heat fluxes related to the decay of macro-layer thickness and the growth of vapor stems, respectively. The macrolayer theory considers that the liquid supply occurs periodically and the departure period exactly corresponds to the sustained period of vapor bubble. Therefore, the averaged heat flux can be expressed as τ q = 1 qdt av τ (1) 0. Closure relationships With respect to the initial void fraction, Gaertner [1] showed that the diameter of the vapor stems had the relationship with the active site population for water as N 1 D = (13) A 9 where D is the diameter of the vapor stem, N/A is the active site population. In this simulation, the diameters of the vapor stems were controlled within 0.4mm according to the experiment []. Rajavanshi et al. [3] put forward an equation about initial macrolayer thickness according to Haramura and Katto s hypothesis [8], it is written as 0.4 ρ 0 = v ρv H fg σρv ρl ρl qav δ 0 (14) In this simulation, for a given heat flux, the initial macrolayer thickness was obtained by equation (13). In the transition boiling regime, the extrapolated values of the equation were employed as the initial macrolayer thickness. The bubble departure period is according to the equation of Katto and Yokoya [4], i.e. 1 ( ξρ ) 5 1 l + ρv 11 v1 5, = (15) ( ρ ρ ) τ = ξ 4π g l v 3.3 Numerical method to reproduce boiling curves As described in the previous, although the macrolayer model [1] provided a reasonable 5 assumption on the macrolayer liquid, the method to estimate boiling curve has a problem because it is difficult to determine the relationship between wall superheat and the thickness of macrolayer. Besides this, the boiling curves of saturated liquid except water are difficult to be predicted by their method. These problems can be solved well by the present method. Fig. shows the flow chart for predicting boiling curves. Because the heat flux is the function of wall superheat, from nucleate boiling regime to the critical heat flux, we can obtain the numerical solution of wall superheat by the bisection method. To be specific, initially, for a given heat flux, we obtained the initial macroalyer thickness by equation (14). Subsequently, we calculated the heat flux in a certain wall superheat by using the above macrolayer model. Finally, If the difference between the calculated and the given heat flux was less than a decided convergence coefficient, the wall superheat could be considered as the numerical solution of the given heat flux. The critical heat flux is determined by the following condition as qav qav 1 < ε (16) T T 1 where T 1 and T are the calculated wall superheat corresponded to q and q 1. In the transition boiling regime, we employed the extrapolated method to determine the boiling curve. Specifically, the initial macrolayer thickness and the wall superheat were gained through the extrapolated line of the obtained boiling curve in nucleate boiling regime. Afterwards, the heat flux can be calculated through the present model..4 Effect of Surface Roughness The action of the nucleate site density in the model of Maruyama [1] is vague because the effect of surface condition is not considered. Some researches have shown that nucleate site density might play a role in high heat-flux nucleate boiling and CHF [10,]. Therefore, it may be necessary to include the effect of surface roughness. So far, there're few studies to characterize the

6 site density quantitatively. Wang and Dhir [5] put forward a correlation about cavity densities and cavity diameters as the followings: 3 * Dc 6 * 5. Ns( sitescm / ) = Dc 6 * Dc 5.4 * D 5.8µ m c 3.5 D < 5.8µ m (17) * D 3.5µ m According to the above correlation, the cavity density will be so large that it is difficult to be simulated by the present computer. Hence, referring to the methods described by Shoji [19] and Sadasivan et al. [6], we employed the following method: 1. The number of surface cavities within 100-mm area is set to vary from 150 to By referring to Equation (16), we set that the cavity diameters were less than 7.0µm. Among them, 1~5% of the surface cavities was between 5.8~7.0µm, 9~0% was between 3.5~5.8 µm and 75~90% was less than 3.5µm. 3. The site spatial distribution and diameter are assigned randomly by satisfying the conditions in step. The simulated cavities were assigned and distributed in the 1 1 subdivisions. For the simplification, we assumed that there was at most one cavity in on mesh and only the biggest one was chosen to be the potential nucleation site. Consequently, a part of assigned cavities were selected to be the potential nucleation sites. Hence, the corresponded active wall superheat could be obtained from the following expression, σt sat T = (18) act ρ vh fg Rca As shown in Fig., when a certain power was input to the heater, some part of potential nucleate sites became activated and formed vapor stems instantaneously. Later, combining the macrolayer model described before, we could find the wall superheat corresponded to the input heat flux. 3. Application of the Model by Numerical Simulation 3.1 Simulation of transient boiling process The present model was applied in the transient boiling. Some assumptions were included as follows: 1. The macrolayer established immediately when c * c 6 the vapor mass departed from the surface. The motion pictures of Katto and Yokoya [7] showed that the liquid supply was initiated before the formation of vapor mass, however, the delay time of the new vapor bubble is much smaller than the bubble hovering period. Therefore, it is reasonable to make the assumption.. The initial thickness of the macrolayer was determined by the surface heat flux when the vapor mass taked off. In the transition-boiling regime, the initial thickness of macrolayer was determined by the extrapolated value of the obtained nucleate boiling curve. 3. The initial void fraction was not changed with wall temperature. Although the active site population increases with the wall superheat, the production of active site density and diameter of the vapor stem changes little with the wall superheat [1]. Therefore, the assumption may be reasonable. Besides the evaporation of macrolayer at heated surface, one-dimensional transient heat conduction within the heater was also considered. The heat conduction equation is T T = α 1 (19) t x Subjecting to the following initial and boundary conditions: t = 0, q q, T T s = 0 w = 0 (0) T x = 0, λ = q s (1) x T x = H, λ = q in () x Employing explicit FDM, we can obtain the instantaneous surface temperature according to the equations (18)-(0). 3. Spatial temperature distributions The three-dimensional transient heat conduction was considered in this part of study. Boiling process was included through boundary conditions. The transient heat conduction is described in Cartesian coordinates as T T T T w = α 1 + t + + x y z (3) ρc Subject to the following boundary and initial conditions:

7 T x=0 and x=l, for all y, z, t = 0 (4) x T y=0 and y=l, for all x, z, t = 0 (5) y T z=h, λ H = qb (6) Z T z=0, qs ( i, j) = λh (7) z The first four boundary conditions for the y and x directions are referred to as a representative rectangular portion of the total heater where the periphery acts as a symmetry line. The boundary condition of the bottom is shown in equation (6). At these two conditions, the heat generation term w/ρc in equation (3) is set to zero. Equation (7) shows the boundary condition the top of heater. Because the top of the heater is the boiling surface, the boiling effects are included through calculating the local surface heat flux of boiling. The local surface heat flux q s (i,j) can be calculated by equation (10). In addition, the initial condition is specified as T w ( i, j) = T0 (8) The initial temperature profile inside the heater is calculated by one-dimensional steady state heat conduction, i.e. k z T( i, j, k) = Tw( i, j) + qin (9) λh The fully implicit scheme was employed to the three-dimensional heat conduction problem. Although the calculation for the heat conduction is stable at any time, there is a limitation for the calculation of pool boiling heat transfer. The calculation is stable when the following condition is satisfied: t T ( t + t) (30) t T t + t 4. Results and Discussion 4.1 Boiling curves and the dependence of parameters The diameter of simulated area was 10mm. Fig. 3 shows the estimated boiling curve of water. We can see that the prediction in the high heat flux regime is in a good agreement with the experiments, however, it predicted higher in the low heat flux regime. This implies that this model 7 is more suitable for the high heat flux regime. The CHF predicted by the present simulation is 1.63MW/m at δ 0 =38µm, which is almost the same as that estimated by Katto and Yokoya [4]. The boiling curve of FC-7 was also calculated. The fluid is saturated at a temperature of 39K. The averaged bubble departure period of FC-7 is 30ms which is calculated from equation (14). The results are plotted in Fig. 4. We can see that except the lower critical heat flux, they are almost consistent with the experimental values carried out by Hohl et al. [8]. The predicted critical heat flux by the present model is 5 q = W/m at T=36K. The critical heat flux calculated by equation (1) is W/m. Therefore we believe that the predicted value is reasonable. The contact angle is a parameter that can be handled in the present simulation. In this model, the effect of contact angle is included and affects the growth of rate of vapor stem, dr s /dt. When the contact angle is changed from 5 o ~90 o, the CHF changes consequently. Fig. 5 provides the critical heat flux dependence on the contact angle. The vertical coordinate is the dimensionless CHF whose denominator is the value of Zuber's [7] correlation. In Fig. 5, other available experimental data are also plotted. As displayed in the graph, although the predicted curve gives a relatively smaller contact angle for the same CHF when θ<30 ο, the results indicate the same tendency as those by the available experiments Figure 6 shows the predicted boiling curves for different surface conditions. As displayed that, with the number of cavities increasing, the boiling curves moved the lower superheat region, whereas the CHF value didn't change so much. When the number of surface cavities was 600, for relatively rougher surface, the boiling curve shifted to the left of that with smoother surface. Fig.7 shows the variations of macrolayer thickness, void fraction and instantaneous heat flux when the average heat flux was 1.6MW/m. We can see that both of the macrolayer thickness and the void fraction when the surface cavities were 600 were larger than that without considering surface cavities. This means that the heat flux component q δ (related to the decay of macrolayer thickness) when Ns=600 is smaller than that without considering surface cavities,

8 whereas the heat flux component q α (related to the growth of vapor stems) when Ns=600 is larger (from Equation 10). Hence, at last the heat flux when Ns=600 does not change so much compared to the heat flux without considering surface cavities This indicates the same tendency of Bereson s experiments [9]. Thus, we can conclude that the reason that the roughness has little effect on the CHF is that the heat flux due to the evaporation of macrolayer thickness becomes smaller and the heat flux due to the growth of vapor stems becomes larger. 4. Results of transient heating The input heat flux was set to increase linearly and exponentially. The incipient boiling superheats of water and FC-7 were set at 10K and 15K respectively [30,31]. The simulated area was 10mm in diameter and the heater was copper with 10mm in thickness. The bubble departure periods of water and FC-7 are calculated by equation (15). Because the bubble departure periods don't change so much according to Huang [30], we employed the averaged values as 40ms and 30ms respectively. The averaged heat flux increases with time and reaches the peak value automatically. The wall temperature variation when the heating rate is zero was first investigated. The result is plotted in Fig. 8. In nucleate boiling regime (T w =1 o C), the variation is very small. With the wall superheat increasing, the fluctuation becomes larger, especially in transition boiling regime. This is because the macrolayer is considerably thin and is consumed completely within a short period. The transient boiling curves of water and FC-7 are plotted in Fig. 9 and Figure 10. From these two figures, we can see that the boiling curves change with the heating rates. For lower transients, boiling curves almost remain the same as the steady-state curve. Beyond the steady-state CHF, the nucleate boiling curves extend until the transient CHF is reached. For higher transient heating rate, the boiling curves deviate from the steady-state curve and the CHF becomes much higher. The experimental data by Hohl et al. [31] are also plotted in Fig. 10. It s found that the simulated results have the same tendency as the experimental data. In their experiment [31], the diameter of heater was 34mm, this may be the main reason that the CHF values are higher than the simulated results. The transient boiling curves of water with increasing heat input exponentially are plotted in Fig. 11(a). Referred to the experimets by Sakurai and Shiotsu [3], the exponential periods are varied from 5s~50ms. It shows the same characteristics as that shown in Fig.10. The change of transient CHF with time is plotted in Fig. 11(b). It can be seen that transient CHF generally increases the increase of heating rate. This tendency is the same as that by Sakurai and Shiotsu 's experiment on a thin wire [3]. However, the transient CHF begins to fluctuate when t 0 is less than 70ms. The reason may be because the formation of macrolayer becomes unsteady. We also tried to analyze the mechanism that the transient boiling curves change with heating rates. Fig.1 (a) shows the changes of macrolayer thickness, and Fig.1 (b) shows the changes of void fraction of point 1(steady state) and point (transient heating) in Fig. 9. We can see that the macrolayr thickness of point is thicker than that of point 1, whereas the instantaneous void fraction α of point is a little less than that in point 1. For this condition, from equation (10), we can duduce that the macrolayer thickness plays an important role in increasing transient CHF. 4.3 Spatial temperature distribution Fig. 13a shows the surface pattern at 0.5 and 30ms when the average heat flux was 1.57MW/m. The thickness of the heater was 1mm. Fig. 13b gives the temperature variations of five points plotted in Fig.8a within two bubble departure periods. We can see that, when vapor and liquid filled with the grids at the same time, the temperatures in these points were generally lower; when only vapor or liquid occupied the grids, the temperature were relatively higher. Fig.14 provides the local instantaneous temperatures on the whole surface at 0.5 and 30ms. As revealed that, the surface temperatures remained almost the same at the initial stage of the hovering period. The low temperature points are the place where liquid and vapor exist at the same time. With the time increasing, the areas that 8

9 liquid and vapor intersected became more, the areas that only liquid or vapor is filled with became less, thus, the places where the surface temperatures changed considerably became so more that the temperature in the whole surface varied greatly. 5. Summary and Conclusion In this study, through several numerical methods, the numerical macrolayer model [1] was developed significantly. The main conclusions can be drawn as follows: 1. The present model can explain steady state boiling quantitatively. (1). In this study, by using bisection method, the boiling curves of water and FC-7 are obtained. The simulated results for higher heat fluxes are in a good agreement with the available experimental. For lower heat flux, the predicted results are higher than the experimental results. (). Generally, the evaporation at the interface of vapor stem-liquid is the main contribution to the total heat flux. With the increasing of heat flux, the evaporation at the interface of macrolayer-liquid plays a more important role in the total heat flux. This is in an agreement with that of Pasamehmetoglu et al. [33] 's model and is consistent with the observations of Nagai and Nishio [13] 's observation in some degree. (3). CHF decreases with the increase of the contact angle, which agrees with the experimental results in qualitatively. (4). The boiling curve shifts to the left with the increase of surface roughness, whereas the CHF value only slightly changes with surface roughness. It is in an agreement with the experimental result of Bereson [8]. It is found that the little variation on the CHF value when the surface roughness increases is because the evaporation due to decay of macrolayer becomes decline and the evaporation due to growth of vapor stems becomes increased. (5). Through the investigation of D temperature distribution, it is shown that the low temperature point is generally in the place where vapor and liquid phase occupy at the same time, whereas in the place with which only vapor or liquid is filled, the temperature is generally higher. This also proves that the evaporation at the interface of vapor stem-liquid is the main part of the total heat flux. (6). It is revealed that the temperature fluctuation at the solid-liquid-vapor interface is significantly large. This suggests that the temperature variation at this interface should not be neglected.. The present model can explain transient boiling qualitatively. (1). The macrolayer model is extended to transient boiling. Combining the present developed model and one-dimensional transient heat conduction within the heater, transient boiling curves of water and FC-7 by increasing input heat linearly are obtained. The results are in an agreement with the experiment of Hohl et al [30]. (). Transient boiling curves are obtained by increasing input heat exponentially. They showed the same tendency as the experiments by Sukurai and Shiotsu [3]. (3). It is displayed that the macrolayer plays an important role in raising the transient CHF. 5. Reference [1] Maruyama, S., Shoji, M., and Shimizu, S., A numerical simulation of transition boiling heat transfer, Proceedings of the nd JSME-KSME Thermal Eng. Conf. pp (199). [] Xu, J. J., and Lu, Z. Q., Boiling Heat Transfer and Gas Liquid Two Phase Flow, p Nuclear Energy, Beijing (1993). [3] Carey, V. P., Pool boiling, In Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Process in Heat Transfer Equipment, Chap. 7, Hemisphere, Washington, D. C. (199). [4] Rohsenow, W. M., A method correlating heat transfer heat transfer data dor surface boing liquids, Trans. ASME, 84, p. 969 (196). [5] Zuber, N., Nucleate boiling. The region of isolated bubbles and the similarity with natural convection, Int. J. Heat Mass Transfer Vol. 6, (1963). [6] Zuber, N., Hydrodynamic Aspects of boiling heat transfer, AEC Report No.AECU

10 (1959). [7] Zuber, N., On the stability of boiling heat transfer, Trans. ASME, 80, p (1958). [8] Haramura, Y., and Katto, Y., A new hydrodynamic model of critical heat flux, applicable widely to both pool and forced convection boiling on submerged bodies in saturated liquids, Int. J. Heat Mass Transfer, Vol. 6 (), pp (1983). [9] Pan, C., Hwang, J. Y., and Lin, T. L., The mechanism of heat transfer in transition boiling, Int. J. Heat Mass Transfer, Vol.3, pp (1989). [10] Sadasivan, P., Unal, C. Nelson, R. A., Perspective: issues in CHF modeling the need for new experiments, Trans. ASME, J. Heat Transfer, Vol. 117, pp (1995). [11] Bhat, A. M., Prakash, R. and Saini, J. S., On the Mechanism of Macrolayer Formation in Nucleate Pool Boiling at High Heat Flux, Int. J. Heat Mass Transfer, Vol.6(5), pp (1983). [1] Dhir, V. K., and Liaw, P., Framework for Unified Model for Nucleate and Transition Pool Boiling, Trans. ASME, J. Heat Transfer, Vol. 111, No.3, pp (1989). [13] Nagai, N. and Nishio, S., Liquid-solid contact phenomena in boiling heat transfer ( proposal of concept of contact-line length density), Trans. JSME (B), Vol. 63, No.106, pp. 0-7 (1997). [14] Zhao, Y. H., Masuoka, T., and Tsuruta, T., Prediction of critical heat flux based on the microlayer model, Trans. JSME (B), Vol. 6, No. 598, pp (1996). [15] Pasamehmetoglu. K. O., and Nelson, R.A., A finite-control volume computer program for heater analysis of nucleation dynamics (HANDY), Los Alamos National Laboratory Report LA-UR (1991). [16] Pasamehmetoglu, K. O., Numerical modeling of a nucleate boiling surface, Numerical Heat Transfer: An Int. J. Computation and Methodology, Part A: Applications. Vol. 5, No.6, Jun., pp (1994). [17] Bai, Q. and Fijita, Y., Numerical simulation of the growth for a single bubble in nucleate boiling, Thermal Science and Engineering, Nol. 7 (4), pp (1999). [18] Son, G. and Dhir, V. K., Numerical simulation of a single bubble during partial nucleate boiling on a horizontal surface, Proc. of 11th IHTC, Vol., pp , Kyongju, Korea (1998). [19] Shoji, M., Boiling simulator: a simple theoretical model of boiling, Third Int. Conf. on Multiphase Flow, 0.-14, Lyon, France (1998). [0] Hashimoto, Y. and Ohashi, H., Digital boiling using cellular automata, Proc. High Performance Computing on Multiphase Flows, Tokyo, Japan (1997). [1] Gaertner, R. F., Photographic study of nucleate pool boiling on a horizontal surface, ASME, J. Heat Transfer, Vol. 87, pp.17-9 (1965). [] Shoji, M., Kuroki, H., Model of macrolayer formation in pool boiling, Proc. of 10th Int. Heat Transfer Conf., pp (1994). [3] Rajvanshi, A.K., Saini, J.S., and Prakashi, R., Investigation of macrolayer thickness in nucleate pool boiling at high heat flux, Int. J. Heat mass Transfer, Vol.35, No., pp (199). [4] Katto. Y., and Yokoya, S., Behavior of a vapor mass in saturated nucleate and transition pool boiling, Trans. JSME, Vol. 41, pp (1975). [5] Wang, C. H. and Dhir, V. K., Effect of surface wettability on active nucleation site density during pool boiling of water on a vertical surface, Trans. ASME, J. Heat Transfer, Vol. 115, pp (1993). [6] Sadasivan, P., Unal,C. Nelson, R. Nonlinear aspects of high heat flux nucleate boiling heat transfer, Trans. ASME, J. Heat Transfer, Vol. 117, pp (1995). [7] Katto, Y., Yokoya, S., and Yasunaka, M., 1970, Mechanism of boiling crisis and transion boiling in pool boiling, Proc. 4th Int. heat Transfer Conf., Paris, vol.5, pp. B3.. [8] Hohl, R., Auracher, H., and Blum, J., Identification of liquid-vapor fluctuations between nucleate and film boiling in natural convection, Convective Flow and Pool Boiling Conference, Section : Pool Boiling, Kloster Irsee (1997). [9] Berenson, P. J., Transition boiling heat transfer from a horizontal surface, M.I.T Heat Transfer Lab. Tech. Rep.17 (1960). 10

11 [30] Huang, Z. L., A study of steady transition boiling of water, Ph. D thesis, Faculty of Engineering, The University of Tokyo (1993). [31] Hohl, R., Auracher, H., Blum, J., and Marquardt, W., Pool boiling heat transfer experiments with controlled wall temperature transients, Proc. of the nd European Thermal Science and 14th UIT National Heat Transfer Conference, Vol. 3, pp (1996). [3] Sakurai, A. and Shiotsu, M., Transient pool boiling heat transfer, part : boiling heat transfer and burnout, ASME, J. Heat Transfer, Vol. 99, pp (1977). [33] Pasamehmetaglu, K. O., Chappidi, P.R., Unal C., and Nelson, R. A., Saturated pool nucleate boiling mechanism at high heat fluxes, Int. J. Heat Mass Transfer, Vol. 36, No. 15, pp (1993). [34] Kutateladze, S. S., On the transition to film boiling under natural convection, Kotloturbostroenie, Vol. 3, p. 10 (1948). [35] Shoji, M., Study of steady transition boiling of water: experimental verification of macrolayer evaporation model, Proc. Eng. Found. Conf. Pool External Flow Boiling, Publ. by ASME, pp.37-4 (199). [36] Maracy, M. and Winterton, R. H. S., Hysteresis and contact angle effects in transition pool boiling of water, Int. J. Heat Mass Transfer, Vol. 31, No.7, pp (1988). [37] Harne, E. and Diesselhorst, T., Hydrodynamic and surface effects on the peak heat flux in pool boiling, Proc. 6th Int. Heat Transfer Conf., Toronto, Vol. 1, pp (1978). [38] Liaw, S. and Dhir V. K., Effect of surface wettability on transition boiling heat transfer from a vertical surface, 8th Int. Heat Transfer Conf., San Francisco, Vol. 4, pp Hemisphere, Washington D. C. (1986). 11

12 Captions for The Figures Fig. 1. Schematic of Macrolayer Model Fig.. Flow chart to calculate wall superheat from heat flux Fig. 3. Simulated Boiling Curve of Water Fig. 4. Simulated Boiling Curve of Water Fig. 5. Prediction of Critical Heat Transfer Dependence on contact angle Fig. 6. Predicted Boiling Curves for Different Surface Conditions Fig. 7. Variations of δ, α and q with time when average heat flux q av =1.6MW/m (without considering cavities and N s =600) Fig. 8. Wall Temperature Variations in Different Boiling Regimes Fig. 9. Predicted transient boiling curves of water Fig. 10. Simulated transient boiling curves of FC-7 for different heating rates Fig. 11. Predicted results by exponential Heating increase (a) transient boiling curves (b) relationship between transient CHF and exponential period Fig. 1. Comparison of characteristic parameters for transient boiling and steady state (a) changes of macrolayer thickness (b) changes of void fraction Fig. 13. Instantaneous Surface Dry Pattern (b) Temporal Variations of Instantaneous Wall Temperatures For Different Positions (q=1.57mw/m, H=1mm) Fig. 14. Local Instantaneous Temperatures On the Heated Surface (q=1.57mw/m, H=1mm) 1

13 Vapor Mushroom Vapor Stem Macrolayer Vapor Mass Heater r 1 r 0 Vapor Stem θ Macrolayer δ H Heater Fig. 1. Schematic of Macrolayer Model 13

14 Assign Surface Cavities Input a value of heat fluxq in look for a certain T within the interval of T =5K~60K Some surface cavities become activated and form vapor stems immediately Calculate the initial macrolayer thickness by q in Using macrolayer model to calculate heat flux q cal Look for T in a smaller interval No q cal q q cal in < ε No N=50 Yes Break Yes Stop Fig.. Flow chat to calculate wall superheat from heat flux 14

15 Katto & Yokoya [4] 10 6 Kutateladze [34] 10 6 Heat Flux q av, W/m 10 5 Experiments Shoji et al. [35] Gaertner [1] Computational Results Wall Superheat T sat, K Fig. 3 Simulated Boiling Curve of Water [ ] 3 Simulated Value Experiments by Hohl et al. [8] Heat Flux q av, W/m Wall Superheat T sat, K Fig. 4 Simulated Boiling Curve of Fluorinert FC-7 15

16 q CHF /ρ v H fg [σg(ρ l ρ v )/ρ v ] 1/ Maracy & Winterton Heating [36] Maracy & Winterton Cooling [36] Harne & Diesselhorst [37] Liaw & Dhir Heating [38] Liaw & Dhir Cooling [38] Present Model Dhir & Liaw's Model [1] Zuber [7] Contact Angle θ, deg. Fig.5 Prediction of Critical Heat Transfer Dependence on contact angle Computational Result only considering vapor stems Heat Flux q av, W/m Number of Cavities N s 600(rougher) 600(smoother) Wall Superheat T, K Fig. 6 Predicted Boiling Curves for Different Surface Conditions 16

17 Macrolayer Thickness δ, mm δ (without considering cavities) α (without considering cavities) δ (N s =600) α (N s =600) [ ] Time t, ms Void Fraction α, % q, W/m 1 without considering cavities N s = Time t, ms Fig.7 Variations of d, a and q with time when average heat flux q av =1.6MW/m (without considering cavities and N s =600) 15 T w =1 o C Wall Temperature T w, o C T w =130 o C T w =136 o C Time t, ms Fig. 8. Wall Temperature Variations in Different Boiling Regimes 17

18 [ ] Heat Flux q av, W/m 0 10 Transient boiling curves for water 1 Steady State 5K/s 80K/s Wall Superheat T, K Fig. 9. Predicted transient boiling curves of water [ ] Heat Flux q av, W/m 1 Simulated results 77K/s(heating) K/s(heating) Steady State Hohl et al. [31] Steady State 4K/s(heating) 10K/s(heating) Wall Superheat T, K Fig. 10. Simulated transient boiling curves of FC-7 for different heating rates 18

19 Heat Flux q av, W/m 10 6 Steady State Q=Q 0 exp(t/t 0 ) t ms 1000ms 300ms 100ms 70ms 60ms 50ms (a) Wall Superheat T, K Transient CHF, W/m 10 7 Saturated Boiling P=0.101MPa Exponential period t 0, s (b) Fig. 11. Predicted results by exponential Heating increase (a) transient boiling curves (b) relationship between transient CHF and exponential period 19

20 δ, mm 0. 0 Time t, ms Steady state (point 1 in Fig. 9) Transient CHF at 80K/s ( point in Fig. 9) point in Fig (a) Void Fraction α, % Time t, ms Time t, 100 ms 100 Steady state (point 1 in Fig. 9) Transient CHF at 80K/s (point in Fig. 9) point in Fig Time t, ms (b) Fig. 1. Comparison of characteristic parameters for transient boiling and steady state (a) changes of macrolayer thickness (b) changes of void fraction 0

21 t =0.5ms t=30ms 140 Wall Temperature, K point a (1,5) point b (1,10) point 4 (11,4) point 8 (11,8) pont 19 (11,19) Time t, ms Fig. 13 (a) Instantaneous Surface Dry Pattern (b) Temporal Variations of Instantaneous Wall Temperatures For Different Positions (q=1.57mw/m, H=1mm) 1

22 t=0.5ms t=30ms Fig. 14. Local Instantaneous Temperatures On the Heated Surface (q=1.57mw/m, H=1mm)