Mesoscale simulations of boiling curves and boiling

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ate boiling heat transfer matches well with simulated nucleate boiling heat transfer results .... which has been applied to a number of boiling and condensation.
International Journal of Heat and Mass Transfer 110 (2017) 319–329

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Mesoscale simulations of boiling curves and boiling hysteresis under constant wall temperature and constant heat flux conditions Chaoyang Zhang, Ping Cheng ⇑ School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai 200240, China

a r t i c l e

i n f o

Article history: Received 10 December 2016 Received in revised form 11 March 2017 Accepted 12 March 2017

Keywords: Pool boiling Surface wettability Saturated temperature Heating conditions Hysteresis Lattice Boltzmann method

a b s t r a c t Mesoscale simulations for pool boiling curves and boiling hysteresis on hydrophilic/hydrophobic surfaces, under constant wall temperature/constant wall heat flux conditions, are presented in this paper. It is found that simulated boiling curves in dimensionless form under these two different heating modes are identical in nucleate boiling and film boiling regimes, and they differ only in the transition boiling regime. Saturated temperatures have relatively small effects on boiling curves up to the fullydeveloped nucleate boiling regime, but have pronounced effects on critical heat flux and on film boiling. Boiling hysteresis between increasing heating and decreasing heating are also simulated numerically. It is confirmed numerically that boiling hysteresis exists in transition boiling regime for both hydrophilic and hydrophobic surfaces under controlled wall heat flux conditions. Under controlled wall temperature conditions, however, boiling hysteresis exists only on a hydrophobic surface during decreasing heat flux but no boiling hysteresis exists on a hydrophilic surface. Rohsenow’s classical correlation equation for nucleate boiling heat transfer matches well with simulated nucleate boiling heat transfer results for smooth horizontal superheated surfaces. Simulated critical heat fluxes are in qualitative agreement with those predicted by Zuber’s hydrodynamic model and by Kandlikar’s analytical model. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Pool boiling is a complex heat transfer process as it is affected by many factors, including heating modes (constant wall heat flux or constant wall temperature), thermal properties of vapor and liquid phases, size, orientation and surface properties of the heater (contact angle and surface microstructures), heating conditions (increasing or decreasing wall heat flux or wall temperature), and saturated temperatures. For high heat flux thermo-fluid systems using phase-change working media, boiling curve is the most important information for the design and safe operation. In 1934, Nukiyama [1] performed well-known experiments on pool boiling from horizontal wires in water under controlled heat flux conditions. When heat flux was increased to the maximum heat flux (which has been called the critical heat flux (CHF)), a temperature jump occurred after a slight increase in heat flux. He classified the heat transfer process into nucleate boiling regime, transition boiling regime and film boiling regime [1], and presented his data in terms of heat flux versus wall superheat which has since been called a boiling curve. Shortly after, Drew and Mul-

⇑ Corresponding author. E-mail address: [email protected] (P. Cheng). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.03.039 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

ler [2] studied boiling heat transfer in transition boiling regime under controlled temperature conditions. Subsequently, numerous researchers conducted experiments on different boiling regimes to study their heat transfer characteristics [3]. For example, Rohsenow [4] obtained a correlation equation for the nucleate boiling regime with different working fluids and surface conditions. Kutateladze [5] obtained a correlation equation for CHF based on a dimensionless analysis, and subsequently Zuber [6] proposed a CHF model based on a hydrodynamic instability analysis. Lienhard [7,8] refined the CHF model and studied heater size’s effects on critical heat flux. Kandlikar [9] performed a force balance analysis to study contact angle effects on CHF for smooth surfaces, and found that CHF decreases as the contact angle increases, which is in agreement qualitatively with pool boiling experiments [10]. Relatively less work has been published on transition boiling regime in comparison with other boiling regimes. Much of the transition boiling heat transfer data were obtained from quenching experiments [11]. One of the most interesting phenomena in transition boiling is the boiling hysteresis phenomena, i.e., boiling curves between increasing and decreasing wall temperatures (or wall heat flux) are different depending on wettability of the heater surface. Sakurai and Shiotsu [12] were probably the first to document the boiling hysteresis phenomena. Subsequently, Ramilison and Lienhard [13] carried out experiments to study hysteresis phe-

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nomena in transition boiling regime under controlled wall temperature conditions on surfaces with different advancing contact angles. They found that the heater surface with larger advancing contact angles had a larger boiling hysteresis than those of well-wetted surfaces. Based on the vapor volume fraction on the wall, Liaw and Dhir [14,15] obtained a correlation for transition boiling heat transfer on surfaces with different wettabilities under increasing and decreasing wall temperatures. However, Auracher and Marquardt [16,17] performed experiments with well-wetting fluids (such as FC-72, isopropanol and water) under controlled wall temperature conditions, and found that boiling hysteresis occurred only in transient boiling conditions. They claimed that no boiling hysteresis existed on clean surfaces (hydrophobic) under steady-state conditions. Recently, lattice Boltzmann methods have been successfully developed to study pool boiling phenomena numerically. Gong and Cheng [18,19] proposed an improved phase-change lattice Boltzmann model based on Hazi and Markus’s paper [20,21] with a simpler heat source term. Based on this improved model, Gong and Cheng [22] obtained boiling curves for smooth surfaces with different wettabilities under controlled wall temperature conditions numerically for the first time. Based on the modified version of this model, Zhang and Cheng [23] studied effects of subcooling and heater size on boiling curves under controlled wall heat flux conditions. Most recently, Gong and Cheng [24] investigated roughness effects on boiling curves under controlled wall temperature conditions numerically. Apart from Cheng and co-workers’ work [18,19,22–24], others [25–28] have also used similar methods to simulate bubble nucleation and boiling curves numerically. As a continuation of our previous work, we will use the improved version of the phase-change lattice Boltzmann method to simulate effects of the saturated temperature and boiling hysteresis on boiling curves in this paper. Simulated results for nucleate boiling regime are compared with Rohsenow’s correlation equation [4]. The good agreement between simulated results and existing correlation for nucleate boiling heat transfer suggests that the newly developed LB phase-change method is a promosing tool for studies of boiling heat transfer phenomena. Simulated critical heat fluxes are also compared with those predicted by Zuber’s hydrodynamic model [6] and by Kandlikar’s analytical model [9].

 are the mean value of time t and the estimated value  and u where q of time t + dt [23]. The evolution of temperature distribution gi(x, t) and its equilibrium distribution are given by

g i ðx þ ei dt; t þ dtÞ  g i ðx; tÞ ¼  " g eq i ðx; tÞ ¼ xi T 1 þ

ðg i ðx; tÞ  g eq i ðx; tÞÞ þ dt xi u

ei  U ðei  UÞ2 U 2 þ  2 c2s 2c4s 2cs

ð5Þ

Base on entropy balance equation and the thermodynamic relationship, Hazi and Markus [20] have derived the source term of the following form

u¼

1 qcv



@p @T



rU

ð6aÞ

q

In the above equation, (op/oT)q is evaluated from the equation of state and U is the real fluid velocity. When this source term was applied to the problem of vapor bubble rise from a superheated horizontal surface [20], they found that two unsymmetrical circulations exist in the flow field which is physically unrealistic. Subsequently, other source terms are proposed by different investigators. For example, Gong and Cheng [18] obtained the following source term for the phase-change lattice Boltzmann method

"

u¼T 1

1 qcv



@p @T

 #

rU

ð6bÞ

q

which has been applied to a number of boiling and condensation problems [22–24,30]. Recently, Li et al. [26] obtained the following source term

"

1 qcv



@p @T

 #

rUþ q



  1 k r  ðkrT Þ  r  rT qcv qc v ð6cÞ

The improved lattice Boltzmann phase-change LB model used in this paper is presented in details in our previous paper [23] and we will only be briefly mentioned here. Based on the Gong-Cheng phase change model [18,19], the density’s evolution equation in half-implicit scheme is given by

ð1Þ

where s is dimensionless relaxation time decided by the kinetic vis   eq  cous of the fluid with s = 3m + 0.5. f i x; t þ dt2 and Df i x; t þ dt2 are the corresponding equilibrium distribution and exact different term [29] given by

" #   2 dt ei  u ðei  uÞ u2 ¼ xi q 1 þ 2 þ x; t þ  2 2 cs 2c4s 2cs

ð2Þ

  dt eq ¼ f i ðqðx; tÞ; u þ DuÞ  f eq Df i x; t þ i ðqðx; tÞ; uÞ 2

ð3Þ

eq fi

#

2.2. The source term

2.1. Gong-Cheng’s improved phase-change LB model

   1 dt eq f i ðx þ ei dt; t þ dtÞ  f i ðx; tÞ ¼  f i ðx; tÞ  f i x; t þ 2 s   dt þ Df i x; t þ ; i ¼ 0; 1; . . . N 2

ð4Þ

where dimensionless relaxation sT = 3a + 0.5, with a being the thermal diffusivity. In the last term in Eq. (4), u is the source term which will be discuss in the next section.

u¼T 1

2. The computation model

1

sT

where c is the specific heat, k is the thermal conductivity. Markus and Hazi [21] as well as Tao and coworkers [27,28] also proposed similar source terms. After a careful derivation of the energy equation, we have derived the following exact expression for the source term.

"

  # 1 @p u¼T 1 rU qcv @T q    1 k þ r  ðkrT Þ  r  rT qcv qcp

ð6dÞ

where cp and cv are thermal specific heat at constant pressure and constant volume, respectively. It should be noted that Eq. (6b) or Eq. (6c) is just a special case of Eq. (6d): (i) Eq. (6d) reduces to Eq. (6c) if cp = cv, and (ii) Eq. (6d) reduces to Eq. (6b) if cp = cv and thermal conductivity as well as qcv is constant. The physical interpretation of the source terms given by Eq. (6d) are as follows: the first square blanket represents the source term owing to compressibility effects while the second square blanket represents the source terms due to property variations. In the following, it will be shown numerically that the second square blanket term in Eq. (6d) is also small in comparison with the first square blanket term if physical properties are not constant and cp and cv are not the same.

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321

(a) Heat source/sink term given by Eq.(6d)

(b)Heat source/sink term given by Eq.(6b)

(c) Heat source/sink term given by 2nd square blanket term of Eq.(6d) Fig. 1. Heat source terms during the nucleate boiling regime on a hydrophilic surface.

Fig. 1 shows that a vapor bubble has departed from the heater surface and rises up because of buoyancy force while two bubbles

are growing on the heater surface having a contact angle of 53°. As shown, black dotted curves represent bubbles’ interface, which is

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the average density of saturated liquid and vapor phases. The strength of the source term is indicated by different colours, with red, yellow and green indicating positive values (i.e., a heat source) and blue indicating negative values (i.e., a heat sink). Fig. 1(a) shows the source term u given by the exact expression of Eq. (6d). It can be seen from this figure that most of the computational domain is in light blue color with its strength in the order of 1010, indicating that the source term is vanishing in most of the computation domain. The strength of the source term near bubbles’ interface is in the order of 104, which is represented by dark blue color (a heat sink) near the upper interface and light green color (a heat source) near the lower interface. Inside the contact line of growing bubbles on the surface, the source term is positive with red colour (i.e., a heat source) as the contact line contracts when bubbles are rising up from the surface with cold liquid swapping the solid. Fig. 1(b) shows the strength of the first square blanket term of Eq. (6d) in the computational domain. We can see that color distributions in Fig. 1(a) and Fig. 1(b) are almost the same everywhere, indicating that the first square blanket term (representing the compressibility effects) of Eq. (6d) is the predominate one. Fig. 1 (c) shows the strength of second square blanket of the source term in Eq. (6d) which has mostly light blue color and in absence of red and yellow colors. Comparing scales of Fig. 1(b) and (c), it can be seen that the magnitude of the first term representing the compressibility effects is of the order of 104 and the second term representing variable property effects is of the order from 106 to 105. So, the second square blanket term in Eq. (6d), is indeed very small that have little effects on the total source term. Thus, the second blanket term in Eq. (6d), neglected by Gong and Cheng [18], are justified. 2.3. Computational domain, initial and boundary conditions As shown in Fig. 2, the 2D computational domain (x, y) is of rectangular shape with a length of 1200 lattices and a height of 600 lattices with the y-axis pointing upward. A heater with a length of LH = 300 lattices and a height of H = 40 lattices is placed at the center of the bottom boundary at y = 0. To convert the lattices units into real physical units, we use the capillary length as the reference length, which is widely used in boiling literature [3,31] and is defined as

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r

l0 ¼

gðql  qv Þ

ð7Þ

where r is the surface tension, ql and qv are densities of the saturated liquid and saturated vapor, respectively. In this paper, we set the gravity g = 0.00004 and choose water as the working fluid and its saturated temperature as the reference temperature, i.e., T0 = Tsat. The dimensionless time can be presented with the following form of [31]

t ¼ t

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi g ðql  qv Þ ¼ gt l0 r

ð8Þ

where t is the time in lattice units. The initial density in the whole computation domain is the saturated density of liquid phase determined by the PR equation of state at the initial temperature T0 with the saturated pressure corresponding to T0. On left and right sides of the computation domain, boundary conditions of uniform temperature and pressure are imposed with the non-equilibrium extrapolation scheme [32]. At the top of the computation domain, outlet boundary with constant pressure is set [33]. On the bottom of the heater at y = 0, control wall temperature or wall heat flux conditions are applied with time steps [23], while two sides of the heater are adiabatic. 3. Results and discussion Computations for pool boiling heat transfer from horizontal superheated surfaces were carried out with the phase-change LB model presented in Sections 2.1–2.3 for the following two heating modes: (i) For the case of controlled wall temperature conditions, Tb is given as the temperature at the bottom of heater surface at y = 0 with the method given by Zhang et al. [34], and the heat flux q is calculated from

P q¼

X ks ðT bþ1

 TbÞ

LH

ð9Þ

where ks is the conductivity of the solid heater, X is the top surface area of the heater and Tb+1 is the temperature of the upper layer lattice to heater’s bottom.

Fig. 2. Schematic of the computation domain.

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(ii) For the case of controlled wall heat flux conditions, 1-D Fourier heat conduction is used to specify the heat flux q at the bottom of the heater at y = 0. After computations have been carried out, simulation results are expressed in terms of the Jacob number Ja = cp,l(Tw-Tsat)/hfg, where cp,l and hfg are liquid’s thermal capacity and specific latent heat, respectively and Tw is the average temperature of the top heater surface (at y = H) obtained from the numerical solution. 3.1. Effects of heating modes on boiling curves

(a) Hydrophilic surfaces (θ = 53°)

(b) Hydrophobic surfaces (θ = 103°) Fig. 3. Effects of heating modes (controlled wall temperature and controlled wall heat flux) on saturated pool boiling curves for smooth horizontal superheated surfaces with LH = 12l0 and H = 1.5l0. (a) hydrophilic surfaces (h = 53°) and (b) hydrophobic surfaces (h = 103°).

In this section, we compare simulated boiling curves in terms of dimensionless heat flux versus Jakob number under controlled wall temperature conditions obtained by Gong and Cheng [22] and controlled wall heat flux conditions obtained by Zhang et al. [23] for a heater with LH = 12l0 and H = 1.5l0 at Tsat = 0.85Tc where l0 = 25 lattices. Fig. 3(a) shows simulated boiling curves for hydrophilic surfaces (h = 53°) under two heating modes while Fig. 3(b) shows the corresponding boiling curves for hydrophobic surfaces (h = 103°). It clearly shows that boiling curves for two different heating modes basically coincide at the natural convection, nucleate boiling and film boiling regimes. In particular, values of CHF are the same under these two different heating modes. The only difference between these two boiling curves is in the transition boiling regime: under controlled wall temperature conditions, as the wall temperature is increased beyond the CHF point (Ja = 0.28 for the hydrophilic surface and Ja = 0.21 for the hydrophobic surface), more vapor is gradually formed to become a vapor film. Thus, boiling heat flux decreases gradually as the wall temperature is increased to the minimum heat flux point where the stable film boiling regime begins to take place. However, for the controlled wall heat flux condition, as the wall heat flux is increased slightly beyond the CHF, nucleate boiling mode is suddenly changed to film boiling mode, and there is a sudden temperature jump during transition boiling regime under controlled heat flux process. Fig. 4 shows boiling patterns on a hydrophilic heater (h = 53°) with LH = 12l0 and H = 1.5l0 under controlled wall temperature conditions during transition boiling at Ja = 0.28 and Tsat = 0.85Tc at three different times (t⁄ = 1000, 1003 and 1006). It can be seen from this figure that vapor columns and vapor film co-exist on the top of the heater under controlled wall temperature heating conditions, which was also discussed by Dhir and Liaw [14]. As the wall superheat is increased further, the area covered by the vapor film increases to cover the whole heater surface, and at this point the heat flux decreases to the minimum heat flux point where the film boiling regime begins.

Fig. 4. Nucleate boiling and film boiling co-exist on a hydrophilic heater (h = 53°) with LH = 12l0 and H = 1.5l0 during transition boiling regime under controlled wall temperature conditions at Ja = 0.28 and Tsat = 0.85Tc.

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3.2. Effects of saturated temperature on boiling curves In this section, we investigate effects of saturated temperatures on boiling curves. For this purpose, we now simulate pool boiling under controlled wall heat flux conditions at a higher saturated temperature of Tsat = 0.9Tc, and compare boiling curves with those in Fig. 3 which was simulated for a heater with LH = 12l0 at Tsat = 0.85Tc where l0 = 25. For the same size of the heater, we choose LH = 14 l0 where l0 = 21.5 lattices at Tsat = 0.9Tc. Fig. 5(a) shows pool boiling curves at two different saturated temperatures of Tsat = 0.85Tc and Tsat = 0.9Tc on a hydrophilic surface (h = 53°) under controlled wall heat flux conditions while Fig. 5(b) shows a similar comparison on a hydrophobic surface (h = 103°). It can be seen from these figures that for both surfaces having the same contact angle, boiling curves for different saturated temperatures before the critical heat flux (i.e., in the nucleate boiling regime) nearly coincide with each other. However, the CHF is higher at a

Fig. 5. Effects of saturated temperature on saturated pool boiling curves for smooth horizontal superheated surfaces under controlled wall heat flux condition with LH = 12–14l0 and H = 1.5l0, (a) hydrophilic surfaces (h = 53°) and (b) hydrophobic surfaces (h = 103°).

smaller saturated temperature because of its larger latent heat and larger liquid-vapor density difference. In both the transition boiling and film boiling regimes, lower saturated temperature has a higher heat flux at the same degree of superheat for both hydrophilic and hydrophobic surfaces. 3.3. Boiling hysteresis under two different heating modes We now simulate boiling hysteresis phenomena on a hydrophilic and a hydrophobic surface under two heating modes of controlled wall heat flux and controlled wall temperature. After the wall heat flux (or wall temperature) is increased to the stable film boiling regime, the wall heat flux (or wall temperature) is set to decrease with the same time steps to simulate boiling hysteresis phenomena. For each step in the increasing and deceasing wall temperature (or heat flux), simulation results were averaged to obtain Ja and q for plotting boiling curves. Fig. 6 shows wettability effects on boiling hysteresis under controlled wall temperature conditions. Fig. 6(a) are two boiling curves for increasing and decreasing wall temperature on a hydrophilic surface (h = 53°) at Tsat = 0.85Tc (left) and Tsat = 0.9Tc (right), respectively. It can be seen that the boiling curve for decreasing wall temperature is exactly the same as the increasing wall temperature on the hydrophilic surface. Thus, no boiling hysteresis exists on a hydrophilic surface under controlled wall temperature heating conditions. Fig. 6(b) are the corresponding figures on hydrophobic surfaces (h = 103°) under controlled wall temperature conditions at Tsat = 0.85Tc (left) and Tsat = 0.9Tc (right), respectively. It can be seen from these figure that as the wall temperature decreases from film boiling, the heat flux continues to decrease and passes beyond the minimum heat flux along a lower transition boiling curve. Therefore, the boiling hysteresis exists on a hydrophobic surface under controlled wall temperature conditions. This simulated behavior is in agreement with previous experimental results [12–15]. Note that boiling hysteresis in transition boiling on a hydrophobic surface is due to the contact angle hysteresis between the advancing contact angle during the heating and cooling conditions [13]. On the hydrophobic surface, the advancing contact angle of the vapor bubble which is breaking up from the vapor film is larger than on a hydrophilic surface. As the wall temperature decreases from the film boiling regime, a larger advancing contact angle means less spreading of the liquid phase in contact with the surface. And the chance that nucleate boiling occurs is small with less wetted area on the surface. Therefore, it is harder for a hydrophobic surface to change from film boiling mode to nucleate boiling mode, resulting in a larger boiling hysteresis during the decreasing wall temperature process. While on a hydrophilic surface, it is easier for the surface getting wetted, the advancing contact angle during the increasing temperature and decreasing temperature has little difference. So, boiling hysteresis does not exist on the hydrophilic surface because its advancing contact angle is very small. Fig. 7(a) and (b) shows that transition boiling regimes for decreasing wall heat flux differ from those of increasing heat flux on hydrophilic surfaces (h = 53°) and hydrophobic surfaces (h = 103°) at Tsat = 0.85Tc and Tsat = 0.9Tc respectively under controlled wall heat flux conditions. Thus, boiling hysteresis exists on both hydrophilic and hydrophobic surfaces under controlled wall heat flux conditions. For comparison purposes, the boiling curve for increasing wall temperature presented in Fig. 6 (under controlled wall temperature conditions) are also plotted in Fig. 7 so that the minimum temperature point can be identified. When the heat flux begins to decrease from the film boiling regime, we can see from Fig. 7 that the wall temperature decreases along the film boiling curves until the point of minimum heat flux is reached. As the wall heat flux is reduced further, the wall temperature continues to decrease in the transition boiling regime with a smaller

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Fig. 6. Effects of wettability and saturated temperature on boiling hysteresis under controlled wall temperature conditions with LH = 12–14l0 and H = 1.5l0: (a) hydrophilic surfaces (h = 53°), and (b) hydrophobic surfaces (h = 103°).

slope to a point on the nucleate boiling regime, which is lower than the CHF point obtained under increasing heat flux conditions. With further reduction in heat flux, the nucleate boiling and natural convection curves nearly coincide with the curves under increasing heat flux conditions.

0.013 depending on surface conditions. The above correlation was obtained from experimental data with different fluids on heaters with different wettabilities. Eq. (11a) can also be re-written in the following forms

 Ja ¼ C sf

3.4. Comparison with existing correlation equations and analytic models 3.4.1. Nucleate boiling: comparison with Rohsenow’s correlation equation Based on a dimensionless analysis, Rohsenow [4] proposed the following correlation equation for nucleate boiling from smooth horizontal surfaces:

 Ja ¼ C sf

q ll hfg

1:7 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:33  cp;l ll r gðql  qv Þ kl

ð10aÞ

where ll and kl are liquid’s dynamic viscosity and conductivity, respectively, and Csf is a fitting constant in the range of 0.006–

q qbo

0:33 ðPrl Þ1:7

ð10bÞ

or

q ¼ qbo



Ja C sf

3

ðPr l Þ5:1

where qbo ¼ ll hfg =l0 ¼ ll hfg

ð10cÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gðql qv Þ

r

is the reference boiling heat

flux [24] and Prl is the Prandtl number of the liquid phase. Fig. 8 shows a comparison of our LB simulation results in the nucleate boiling regime with Rohsenow’s correlation equation given by Eq. (10c), where fluid properties were evaluated at saturated conditions (i.e., at Tsat = 0.85 Tc or 0.9Tc). It should be noted that the boiling curves under constant wall heat flux conditions

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Fig. 7. Effects of wettability and saturated temperature on boiling hysteresis under controlled wall heat flux conditions with LH = 12–14l0 and H = 1.5l0: (a) hydrophilic surfaces (h = 53°), and (b) hydrophobic surfaces (h = 103°).

in Fig. 8 are the same boiling curves in Fig. 5 except that the curves in Fig. 8 are expressed in terms of q/qbo. By matching our simulation results with Eq. (10c), we found that the fitting constant Csf = 0.0117 for the hydrophilic surface (h = 53°) and Csf = 0.0104 for the hydrophobic surface (h = 53°) fit well under saturated temperature of Tsat = 0.85Tc in Fig. 8(a) as well as at saturated temperature of Tsat = 0.9Tc in Fig. 8(b). Moreover, it is noted from Eq. (10c) that at the same value of Ja, the nucleate boiling heat transfer on a hydrophobic surface is higher than that on a hydrophilic surface because the fitting constant Csf = 0.0117 for the hydrophilic surface is larger than the Csf = 0.0104 for hydrophobic surface. This has been explained by Gong and Cheng [22] who showed numerically that both bubble departure diameter and bubble departure frequency from a hydrophobic surface are higher than those from a hydrophilic surface, and thus nucleate boiling heat flux from a hydrophobic surface is higher than those from a hydrophilic surface.

3.4.2. Critical heat flux: comparison with Zuber’s hydrodynamic model and Kandlikar’s analytical model In this section, we will compare our simulated critical heat flux with Zuber’s hydrodynamic model and Kandlikar’s analytical model. It should be noted that the value of CHF under controlled wall heat flux conditions and controlled wall temperature are the same according to Fig. 3. Our simulated results on the dimensionless critical heat flux (CHF), qCHF/qbo, versus contact angle at saturated temperatures of Tsat = 0.85Tc and Tsat = 0.9Tc, are plotted as square and circles symbols in Fig. 9(a) and (b), respectively. (i) Comparison with Zuber’s hydrodynamic model: Zuber’s well known model [6] on CHF is given by

 qmax;Zuber ¼ 0:131qv hfg

rðql  qv Þg q2v

1=4 ð11aÞ

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(a) Saturated temperature at Tsat = 0.85Tc

(a) Saturated temperature at Tsat = 0.85Tc

(b) Saturated temperature at Tsat = 0.9Tc Fig. 9. Effect of contact angle on CHF of saturated boiling at (a) Tsat = 0.85Tc and (b) Tsat = 0.9Tc.

(b) Saturated temperature at Tsat = 0.9Tc Fig. 8. Comparison of simulated boiling curves with Rohsenow’s correlation for nucleate boiling on smooth surfaces under controlled wall heat flux at (a) Tsat = 0.85Tc and (b) Tsat = 0.9Tc.

which can be non-dimensionalized with respect to qbo to give

pffiffiffiffiffiffiffiffiffiffiffiffiffi qmax;Zuber 0:0131 qv rl0 ¼ ¼ 0:0131Ar 1=2 qbo ll

ð11bÞ

where Ar is the Archimedes number given by Ar = gl30qv(ql-qv)/l2l = qvrl0/l2l , which is the ratio of the buoyancy force to the viscous force. Zuber’s hydrodynamic model is represented as a horizontal dashed line in Fig. 9(a) and (b). It can be seen that our simulated critical heat flux is slightly higher than Zuber’s hydrodynamic model [6] on a hydrophilic surface but is slightly lower than Zuber’s model [6] on hydrophobic surfaces. (ii) Comparison with Kandlikar’s model: The solid curves in Fig. 9 are the analytical model for CHF on a horizontal surface under constant wall temperature conditions given by Kandlikar [9], which is of the form

qmax qmax;Zuber

¼

 1=2 1 þ cos h 2 p þ ð1 þ cos hÞ 2:096 p 4

ð12aÞ

or in the following form

 1=2 qmax 1 þ cos h 2 p ¼ þ ð1 þ cos hÞ Ar 1=2 16 qbo p 4

ð12bÞ

It can be seen from Fig. 9(a) and (b) that both LB simulated results and Kandlikar’s analytical model [9] show that CHF decreases with increasing contact angle. For both Tsat = 0.85Tc and Tsat = 0.9Tc, CHF values predicted by Kandlikar’s model on hydrophilic surfaces at small contact angles are closer to LB simulated results than those on hydrophobic surfaces at large contact angles. This may be attributed to the following reasons: (i) Kandlikar’s model was derived based on the assumption that the bubble is of spherical shape, which is invalid on a hydrophobic surface; (ii) Kandlikar’s model assumes that the heater is of infinite size while numerical solutions are based on heater of finite size; (iii) numerical solution is a 2D simulation while Kandlikar’s model is 3D; (iv) The static contact angle in Kandlikar’s model is assumed to be constant. However, during the simulation process, as heater’s temperature increases, the surface tension decreases. It should be noted that the relationship between contact angle h and surface tension is cos h = (rsv  rsl)/rlv, i.e., the contact angle decreases as surface tension rlv decreases because the value of rlv is reduced as temperature increases but it has little effect on rsv and rsl [35]. The

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decrease of the realistic contact angle near the heating surface results in the deviate of CHF from Kandlikar’s model as the contact angle increases, which is also discussed in [3]. Moreover, Kandlikar’s model is based on the assumption that the departure bubble’s diameter Db being half of the instability wavelength from Zuber [6] which is independent of surface wettability. However, experimental and simulation results [18,36] have shown that the bubble departure diameter Db increases with the increase of contact angle. A larger bubble’s diameter means more heat flux is removed from the bubble’s interface. For this reason, it is reasonable that Kandlikar’s correlation on CHF on hydrophobic surface is lower than the actual value, which was also displayed in the comparison with Liaw and Dhir’s [37] experimental data [9].

4. Conclusions In this paper, effects of saturated temperature and heating modes on pool boiling curves as well as boiling hystereses under controlled wall heat flux and controlled wall temperature conditions are studied based on Gong-Cheng’s phase change lattice Boltzmann model. The following conclusions can be drawn from present simulation results: (1) An exact source term given by Eq. (7) is presented for phasechange lattice Boltzmann method. It is shown numerically that the simple source term given by Gong and Cheng [22] is accurate for numerical simulation of phase-change heat transfer phenomena. (2) Boiling curves in terms of heat flux versus Jacob number for controlled wall heat flux and controlled wall temperature conditions are almost the same in the nucleate boiling regime and film boiling regime. They differ only in the transition boiling regime: Under controlled wall temperature conditions, minimum heat flux point exists with the increasing wall temperature. Under controlled wall heat flux conditions, the temperature has a sudden jump during transition boiling regime from CHF to film boiling with a slight increase in heat flux. (3) The saturated temperature has no effect on heat transfer up to fully-developed nucleate boiling regime. The CHF as well as boiling heat transfer in transition boiling and film boiling regimes are higher at a higher saturated temperature. (4) Boiling hysteresis in transition boiling shows different behaviors under controlled wall heat flux and controlled wall temperature conditions. It exists on both hydrophilic and hydrophobic conditions under controlled wall heat flux conditions. Under the controlled wall temperature conditions, no boiling hysteresis exists on a hydrophilic surface because the boiling curve for decreasing superheat is nearly coincident with the boiling curve with increasing superheat. However, boiling hysteresis does exist on hydrophobic surfaces under controlled wall temperature conditions. (5) Rohsenow’s correlation equation matches well with LB simulated dimensionless boiling heat flux versus Jacob number in the nucleate boiling regime for nucleate boiling from smooth horizontal superheated surface. The variation of the fitting constant depends on the wettability of the smooth surface (6) Simulated critical heat fluxes agree with Zuber’s model reasonably well. Kandlikar’s analytical model for CHF agree well with the LB simulation results at small contact angles but its discrepancy becomes larger at larger contact angles. This probably can be attributed to a large number of assumptions made in Kandlikar’s analytical model.

Acknowledgements This research work was sponsored by National Natural Science Foundation of China through Grant Nos. 51420105009, and by Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51521004).

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