A study of boiling on surfaces with temperature-dependent wettability by lattice Boltzmann method

International Journal of Heat and Mass Transfer 122 (2018) 775–784

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A study of boiling on surfaces with temperature-dependent wettability by lattice Boltzmann method Lei Zhang a,b, Tao Wang a,b,⇑, Yuyan Jiang a,b,⇑, SeolHa Kim a, Chaohong Guo a,b a b

Institute of Engineering Thermophysics, Chinese Academy of Sciences, No. 11, Beisihuanxi Road, Beijing 100190, People’s Republic of China University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 30 September 2017 Received in revised form 5 February 2018 Accepted 7 February 2018

Keywords: Temperature-dependent wettability Boiling heat transfer Lattice Boltzmann method Smart control of heat transfer

a b s t r a c t Effects of surface wettability have been the focus of boiling heat transfer research in recent years, due to its important role on boiling performance. It is reported that hydrophobic surface has higher boiling heat transfer coefficient while hydrophilic surface has higher critical heat flux. In this study, a surface with temperature-dependent wettability was proposed to take advantages of both hydrophilic and hydrophobic surfaces. A hybrid thermal lattice Boltzmann model with an improved forcing scheme was used to simulate and evaluate the effects of wettability control. First, single bubble dynamics on hydrophilic and hydrophobic surfaces were depicted to analyze the heat transfer features on both surfaces. Second, boiling curves for each condition have been obtained under stepwise heat flux control condition, and the controlled wettability surface shows higher boiling performance than both hydrophobic and hydrophilic surfaces. In addition, we found an optimal relation between temperature and surface wettability for heat transfer rate, and it is evaluated through the parameter test of the temperature-wettability relation. This research may provide a potential way of controlling surface wettability to improve boiling performance and also offer conceptual design of enhanced boiling surface. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Boiling heat transfer (BHT) is one of the most effective heat transfer modes in energy conversion and cooling system. It has been widely used in electric power generating and thermal management of electric components. Over the past century, many researchers intended to enhance the heat transfer coefficient (HTC) and the critical heat flux (CHF) to improve the performance of BHT. Recently, the effect of wettability has arisen much attention, for that it influences the volume of trapped vapor/air in a cavity, bubble departure diameter, bubble departure frequency, incipience superheat and liquid supply, which play important roles on boiling performance [1,2]. There were many reports about wettability effects on boiling performance. Takata et al. [3] used UV light to change the wettability of TiO2 coated surface, they found that the CHF doubled compared to non-coated surface but the temperature at the onset of nucleate boiling increased about 100 K. They also reported that there was no nucleate boiling regime on a super water-repellent surface. The stable film boiling occurred in very small superheating ⇑ Corresponding authors at: Institute of Engineering Thermophysics, Chinese Academy of Sciences, No. 11, Beisihuanxi Road, Beijing 100190, People’s Republic of China. E-mail addresses: [email protected] (T. Wang), [email protected] (Y. Jiang). https://doi.org/10.1016/j.ijheatmasstransfer.2018.02.026 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

[4]. Then, Forrest et al. [5] changed the surface wettability through application of nanoparticle thin-film coatings and found that hydrophobic surface coated with fluorosilane can enhance HTC by about 100%, but gives lower CHF than hydrophilic surface. After this, Bourdon et al. [6] studied the influence of wettability in terms of surface temperature quantitatively and showed that grafted surfaces which has higher contact angle enabled an easier boiling, they also reported that wettability can be of primary importance and can control the position of incipient boiling even with a certain amount of roughness. Hsu et al. [7] investigated the effects of surface wettability on CHF under various wettability from superhydrophilic to superhydrophobic and found that CHF values increased with the decrease of surface contact angle. They also reported that size of growth bubbles increases with increasing contact angle. Most recently, Girard and Kim [8] conducted pool boiling experiments and verified that boiling heat transfer coefficient increase with contact angle increase. Besides, You et al. [9] and Bang et al. [10] reported a big enhancement in CHF with nanofluid and Kim et al. [11], Coursey et al. [12] attributed the CHF enhancement to the nanoparticle deposition which increased wettability of the heating surface. According to the above literatures, generally, it is showed that hydrophobic surface can promote bubble nucleation and has a higher HTC than hydrophilic surface, while hydrophilic surface can induce liquid supply to the dry area of a heater and delaying

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Nomenclature a, b c e f F Fb Fm Fads g hfg JaR Ja LH l0 p q s t⁄ t0 T

v

wa

parameter in P-R equation of state lattice speed (m  s1 ) lattice velocity vector (m  s1 ) distribution function body force vector (N) gravitational force (N) interaction force (N) fluid-solid interaction force (N) gravitational acceleration (m  s2 ) 1 latent heat (J  kg ) Jacob number calculated by Rohsenow’s equation Jacob number length of heater (m) characteristic length (m) pressure (Pa) heat flux (W  m2 ) entropy (J  K1 ) dimensionless time characteristic time (s) temperature (K) velocity vector (m  s1 ) weight coefficient

CHF. Based on those features, researchers considered taking advantages of both hydrophilic and hydrophobic surfaces. One method is using surface with mixed wettability, such as the work done by Betz et al. [13] and Jo et al. [14], they both found an increase of boiling heat transfer but their conclusions on CHF were not consistent. Different from their ideas, Bertossi et al. [15] considered changing the wettability during the process of boiling. They designed specific surfaces called ‘‘switchable surfaces” with polymer coating. The polymer showed wetting transition from hydrophilic to hydrophobic when temperature exceed a specific value. Since hydrophobic surface promotes bubble nucleation and hydrophilic surface enhances bubble detachment, the switchable surface increased boiling heat transfer in the nucleate boiling regime. Recently, Kim et al. [16] found that the wettability of TiO2coated surface (TCS) could change with temperature under high pressure conditions. They reported this kind of surface had a higher HTC in low-wall-superheat region due to its hydrophobicity and higher CHF because its better wettability in high-wall-superheat. In this study, we proposed surface whose wettability can be changed with temperature, it has large contact angle at low superheat which can promote bubble nucleation and turns to hydrophilic at higher superheat to increase CHF. Four relations between temperature and wettability were designed and their boiling performance were obtained to investigate whether there is an optimal one. Numerical method is chosen to investigate various cases and get more details to provide guidance on controlling surface wettability to improve boiling performance. Recently, the lattice Boltzmann method has received great attention and shows promising future in phase change field, models like phase field lattice Boltzmann model [17–19] and pseudopotential model [20–22] were used to study boiling phenomena from nucleate boiling to film boiling. There are several advantages of phase change pseudopotential model. First, the state of vapor or liquid at each lattice is determined by the equation of state, so there is no need to track the interface between liquid and vapor explicitly. Secondly, the nucleation process could be simulated automatically, there is no need to initialize a small bubble at the beginning, so we can get the entire ebullition cycle. Moreover, the effects of surface wettability can be easily implemented [2]. Until now, there are plenty of researches done with LBM in pool

x

co-ordinates (m)

Greek symbol acentric factor W effective mass k thermal conductivity (W  m1  K1 ) r surface tension (N m1 ) t kinematic viscosity (m2  s1 ) s relaxation time q density (kg  m3 ) h contact angle (°) l dynamic viscosity (Pa  s)

x

Subscripts or superscripts ave average c critical eq equilibrium l liquid t time v vapor sat saturation

boiling field, especially in studying the effects of wettability [2] or designing the patterns of mixed wettability [23–26]. Different with previous models, Li et al. [21] introduced a hybrid thermal lattice Boltzmann model to reduce the spurious term caused by the forcing-term. All terms in the energy equation instead of only the source term are calculated with finite-difference methods in hybrid thermal scheme. In this study, this hybrid thermal lattice Boltzmann method was adopted to investigate the wettability effects on boiling performance.

2. Model description A brief introduction of hybrid thermal lattice Boltzmann method will be described in this part. As same as DoubleDistribution-Function (DDF) approach, the flow field is solved by density distribution function, but instead of LB approach, the temperature field in hybrid model is solved with conventional numerical methods, such as the finite-difference or finite-volume method [27]. For the density distribution function, a multiple-relaxationtime (MRT) collision operator is adopted here, for it has a better numerical stability and lower spurious velocity. 2.1. MRT pseudopotential lattice Boltzmann method for flow field The standard LB equation with a MRT collision operator can be expressed as follows:

f a ðx þ ea dt ; t þ dt Þ ¼ f a ðx; tÞ  ðM 1 KMÞab ðf b  f b Þ þ dt F 0a eq

ð1Þ

where f a ðx; tÞ is the density distribution function in a th lattice eq direction, ea is the discrete velocity, dt is the time step, f b is the equilibrium density distribution function in b th lattice direction, F 0a is the forcing term, M is an orthogonal transformation matrix and can be found in [28], K is a diagonal matrix given by (D2Q9 lattice) 1 1 1 1 1 1 1 1 K ¼ diagðs1 q ; se ; sf ; sj ; sq ; sj ; sq ; sv ; sv Þ

ð2Þ

Using transformation matrix M, the right side of Eq. (1) can be rewritten as:

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 K m ¼ m  Kðm  meq Þ þ dt I  S 2

ð3Þ

eq

where m ¼ Mf , meq ¼ Mf , I is the unit tensor, S is the forcing term   and I  K2 S ¼ MF 0 . For the D2Q9 lattice, the equilibria can be given by

m ¼ qð1; 2 þ 3jv j ; 1  3jv j ; v x ; v x ; v y ;v y ; v 2

eq

2

2 x

v vxvyÞ 2 y;

T

ð4Þ

where jv j2 ¼ v 2x þ v 2y . After collision step, the streaming process can be expressed as 

f a ðx þ ea dt ; t þ dt Þ ¼ f a ðx; tÞ 

1

ð5Þ

8 > < ð0; 0Þ ea ¼ ð1; 0Þc; ð1; 0Þc > : ð1; 1Þc

a¼0 a¼14 a¼58

ð6Þ

a

X dt f a ; qv ¼ ea f a þ F 2 a

ð7Þ

where F is the total force acting on the system, which contains interaction force between particles and gravity force and/or other external forces. The interaction force is used to mimic the molecular interactions that cause phase separation and given by [29]

X F m ¼ GWðxÞ wa Wðx þ ea Þea

ð8Þ

a

WðxÞ is called ‘‘effective mass” and defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi WðxÞ ¼ 2ðpEOS  qc2s Þ=Gc2 , in which pEOS is obtained from the equawhere

tion of state. The Peng-Robinson (P-R) equation of state was chosen here, because it is more accurate and popular for real fluids (especially for water and ammonia). The P-R equation of state is given by

pEOS ¼

where Gw reflects the interaction strength which can be tuned to obtain different contact angles, xa ¼ wa =3, Sðx þ ea Þ ¼ uðxÞsðx þ ea Þ, in which sðx þ ea Þ is an indicator function that equals 0 for fluid and 1 for solid phase, uðxÞ can be set to WðxÞ.

qRT auðTÞq2  1  bq 1 þ 2bq  b2 q2

2.2. Energy equation for temperature field The temperature equation is derived on the local balance law for entropy, which is given by (neglecting the viscous heat dissipation)

ð9Þ

pffiffiffiffiffiffiffiffiffiffi 2 where uðTÞ ¼ ½1 þ ð0:37464 þ 1:54226x  0:26992x Þð1  T=T c Þ ,

Ds ¼ r  ðkrTÞ Dt

with x being the acentric factor, and a ¼ 0:45724R2 T 2c =pc , b ¼ 0:0778RT c =pc , where T c and pc represent the critical temperature and critical pressure, respectively. In the present work, we chose a = 3/49, b = 2/21 and R = 1. The acentric factor x was chosen as 0.344 for water. The gravity effect is added through buoyant force given by

F b ¼ ðq  qav e Þg

ð10Þ

where qav e is the averaged fluid density of the computational domain and g is the gravitational acceleration. The forces are incorporated via an improved forcing scheme and the forcing term S is given by

0

3

7 6 6v  F þ 2 cjF m j2 6 W dt ðse 0:5Þ 7 7 6 2 7 6 7 6 6v  F  2 cjF m j W dt ðsf 0:5Þ 7 6 7 6 7 6 Fx 7 6 S¼6 7 F x 7 6 7 6 7 6 F y 7 6 7 6 F y 7 6 7 6 4 2ðv x F x  v y F y Þ 5

v xFy þ v yFx

ð11Þ

ð13Þ

where s is the entropy and k is the thermal conductivity. Substituting the following thermodynamic relation into Eq. (13)

Tds ¼ cv dT þ T



 @pEOS 1 d @T q q

ð14Þ

one can obtain temperature equation

qcv


 DT @pEOS rv ¼ r  ðkrTÞ  T Dt @T q

Using relation

DðÞ Dt

@ t T ¼ v  rT þ

ð15Þ

¼ @ t ðÞ þ v  rðÞ. We can get

1 T r  ðkrTÞ  qc v qc v


 @pEOS rv @T q

ð16Þ

For simplicity, we use KðTÞ to represent the right-hand side of Eq. (16). The forth-order Runge-Kutta scheme is adopted to solve the temperature equation

T tþdt ¼ T t þ

2

2

ð12Þ

a

qT

The macroscopic density and velocity can be calculated via

q¼

X F ads ¼ Gw WðxÞ xa Sðx þ ea Þea



where f ¼ M m . For the D2Q9 lattice, discrete velocity vectors ea are

X

where c is a parameter used to tune the mechanical stability condition, jF m j2 ¼ F 2m;x þ F 2m;y , and F is the total force, v is the macroscopic velocity. The wetting condition is added through a fluid-solid interaction force given by

dt ðh1 þ 2h2 þ 2h3 þ h4 Þ 6

h1 ¼ KðT t Þ,

where

  h2 ¼ K T t þ d2t h1 ,

ð17Þ   h3 ¼ K T t þ d2t h2 ,

t

h4 ¼ KðT þ dt h3 Þ. In summary, Eq. (3) the pseudopotential LB model and Eq. (16) the temperature equation which are solved with finite-difference method are the most important parts of the hybrid thermal LB model. 3. Code validation and computational setup 3.1. Code validation To validate the present model, two representative problems are simulated. The first one is a benchmark thin film evaporation problem which was also used by other researchers [30,31]. The computational domain is illustrated as Fig. 1(a), where constant heat flux is added on the bottom wall and periodic boundary condition is applied on the right and left sides, while the top side is pressure outlet to close the system. If the input heat flux is small enough, no vapor bubble will emerge and liquid could evaporate into vapor through the interface directly. For a given heat flux, there is a stable stream mass flow rate which given by

m00 ¼ q=hfg where hfg is the latent heat of the phase change at 0.86 T c .

ð18Þ

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Fig. 1. Thin film evaporation: (a) Schematic of computational domain. (b) Comparison of LBM results and analytical results.

It can be seen from Fig. 1(b) that the LB simulation results are in good agreement with analytical results. For the second problem, we performed LB simulation of bubbles departure diameter with different wettability. Fritz [32] investigated bubble departure diameter and proposed a semi-empirical correlation for bubble departure diameter from a horizontal heating surface, given by

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

D ¼ 0:0208h

r

ð19Þ

gðql  qv Þ

where h is the contact angle measured in degrees. As shown in Fig. 2, our simulation results for bubbles departure diameter agree quite well with Eq. (19). 3.2. Computational setup In our simulation, variables are showed in a dimensionless form. The reference length l0 , reference velocity u0 and reference time t0 are given by

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l0 ¼

r

gðql  qv Þ

;

u0 ¼

qffiffiffiffiffiffi gl0 ;

t0 ¼ l0 =u0

ð20Þ

Fig. 2. Comparison of LBM results and Fritz’s semi-empirical correlation in terms of bubbles departure diameter.

where r is the surface tension, ql and qv are densities of saturated liquid and saturated vapor, respectively. Those characteristic values are widely used in previous boiling literatures [2,33,34]. As shown in Fig. 3, the computational domain is a 2D rectangular with a heating plate located at the central part of the bottom wall. After grid dependency checking, we chose a 600  1200 lattice size which could ensure accuracy and minimize computation time. The size of heated plate is LH in length and H in thickness. Two values of the heater size LH were chosen for calculation. The smaller one is 80 lattices which was used to ensure that a single bubble was emerged for the study of bubble dynamics and the bigger one was used to get the boiling curves. According to Zhang’s study about the heater size effects on boiling curves [33], if the ratio between heater length and capillary length LH =l0 is larger than 12, boiling curves are independent of heater size. Therefore, LH ¼ 300 lattices (>15 l0 ) was chosen for the bigger heater to eliminate the influence of heater size. The left and right sides of heater are adiabatic and a uniform wall heat flux q is specified on the bottom of the heater. Periodic boundary conditions are imposed on the left and right boundaries. At the top of the computation domain, outlet boundary with constant pressure is set. Initially, the computational domain is filled with saturated water at T s ¼ 0:86T c . The corresponding variables are represented in lattice units: saturated liquid and vapor density

Fig. 3. Schematic of computational domain and boundary conditions.

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4. Results and discussions In this part, the above-mentioned hybrid thermal lattice Boltzmann model was used to study single bubble dynamics on hydrophilic and hydrophobic surfaces. Boiling curves from onset of nucleate boiling until CHF point were obtained numerically. The boiling performance of temperature-dependent wetting surface was illustrated, and different relations between temperature and surface wettability were compared to study the effects of transition point. 4.1. Single bubble dynamics on hydrophilic and hydrophobic surfaces

Fig. 4. Controlled wall heat flux: Heat flux rise with numerical steps increase.

ql ¼ 6:5, qv ¼ 0:38, kinetic viscosity tl ¼ 0:04, tv ¼ 0:1732. The specific latent heat is calculated using the theoretical model proposed by Gong and Cheng [35], and dependent on the parameters in equation of state and temperature. In our case, the value is hfg ¼ 0:5702, and thermal conductivity is chosen as kl ¼ 0:48. To obtain boiling curves, controlled wall heat flux mode was adopted. Similar with experimental procedure, heat flux was increased slowly to avoid large temperature fluctuation on heater surface, as shown in Fig. 4. After a new heat flux was imposed, it would keep constant for a period of 125 t 0 to make the flow field and temperature field reach a steady state. Simulation results during one constant step were averaged to represent one state.

Fig. 5 shows the cyclic growth and release of vapor bubbles on a hydrophilic surface (h = 45.46°) at dimensionless superheat Ja ¼ cp;l ðT w  T s Þ=hfg ¼ 0:18. As shown in the figure, at t ¼ t=t0 ¼ 31:08 one bubble departed and the bulk liquid flooded in and contact with the heating surface. Then transient heat conduction from heater to liquid happened, but no bubble emerged at this period (from t  ¼ 31:08 to t ¼ 44:90). At the end of waiting period, one bubble occurred and a new cycle began. During the early stage of bubble growth (t  ¼ 51:81 to t ¼ 55:26), the size of bubble was small and surface tension was in domain, so the bubble expanded spherically until it got to a critical size when the lifting force and the holding force were in balance. Beyond this, the bubble continued to grow and the triple lines started to retreat (as shown at t ¼ 62:17) until the bubble departed from the heating surface smoothly. Since the liquid supply on hydrophilic surface is fast, the expansion of the bubble base is restricted and the CHF is delayed [36], but natural convection heat transfer during waiting period is low compared to boiling heat transfer, therefore, hydrophilic surface has lower HTC than hydrophobic surface [37].

Fig. 5. Simulation of single bubble dynamics in a typical ebullition cycle on hydrophilic surface at Ja = 0.18.

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Fig. 6. Simulation of single bubble dynamics in a typical ebullition cycle on hydrophobic surface at Ja = 0.18.

Fig. 6 shows the ebullition cycle on a hydrophobic surface (h = 91°) at Ja = 0.18. Different with the bubble dynamics on hydrophilic surface, a seed bubble was left on the surface after the prior bubble’s departure, as shown at t  ¼ 17:27 or t ¼ 51:81. When the bubble grew to a certain size, the bubble interface was stretched vertically (as shown at t  ¼ 41:45), and then a necking phenomenon happened accompanied by triple lines retreating. After that, the bubble was broken at the thinnest part of the neck and only a part of the bubble departed from the surface and the other part stayed on the surface as a nucleus of the next bubble’s circulation. Nam [38] also reported bubble interface necking phe-

Fig. 7. Effects of wettability on boiling curves under controlled wall heat flux condition.

nomenon and a seed bubble remained on hydrophobic surface. They attributed the reason to the change of forces direction which opposes the shrinking of bubble base. But no waiting period may be the directly reason which makes hydrophobic surface have better boiling heat transfer in the low heat flux regime, for boiling heat transfer is higher than natural convection [37]. 4.2. Multi bubbles dynamics and boiling curves on hydrophilic and hydrophobic surfaces In this section, the larger heating plate described before was used to obtain boiling curves. Fig. 7 shows a comparison of boiling curves on hydrophilic surface with hydrophobic surface. We can see that hydrophobic surface has a lower onset of nucleate boiling (ONB) temperature than hydrophilic surface, and the boiling curve of hydrophobic surface is in the left side of hydrophilic surface. This indicates that hydrophobic surface is easier for bubble nucleation and has a higher heat transfer coefficient in the nucleate boiling regime. For ease of analysis, we divided the nucleate boiling regime into three stages, represented by blue,1 yellow and red shades in Fig. 7, respectively. As can be seen from the figure, boiling curves of the second stage have the largest slope. This is because as the wall superheat increase more bubbles emerge as shown in Fig. 8(b) and those bubbles have no strong interactions among adjacent bubbles and can depart separately. Based on heat flux partitioning model, nucleate boiling heat transfer consists of three kinds of heat transfer mechanisms: convective heat flux, quenching heat flux and evaporative heat flux. As the nucleation sites increase, both quenching heat 1 For interpretation of color in Fig. 7, the reader is referred to the web version of this article.

L. Zhang et al. / International Journal of Heat and Mass Transfer 122 (2018) 775–784

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Fig. 8. Effects of wettability on multi-bubble dynamics on smooth surface at different stages under controlled wall heat flux.

flux and evaporative heat flux increase sharply [39], and those mechanisms of heat flux make the major contributions to the heat transfer increase at the second stage. But at the first stage, only one or two bubbles emerge as shown in Fig. 8(a) because of low superheat. Convective heat flux is the main mechanisms of heat transfer at this stage due to inactive bubble activity, so the total heat transfer rate is lower than second stage. At the third stage, the boiling heat transfer still strong but the increase rate of heat transfer coefficient is decreased because the wall superheat is too high and excessive bubbles emerge, they have strong interactions with their adjacent bubbles and merge with each other on the surface to form large bubbles which cause partial dryout at different locations on the surface, which would reduce the boiling heat transfer (BHT). Jo et al. [37] also found this phenomenon and they attributed the decline in BHT in the high heat flux regime to excessive bubble generation that interfered with surface cooling via rewetting. However, as discussion of Fig. 5, liquid supply on hydrophilic surface is strong and the bubble base expansion is slow, which delayed the formation of large dry patch caused by bubble merging and made higher CHF. In conclusion, hydrophobic surface is easier for bubble emerge and has higher boiling heat transfer rate in low heat flux regime, but hydrophilic surface has higher critical heat flux. Both hydrophilic and hydrophobic surface have best boiling performance at the second stage and slightly decrease in the third stage.

homogenous surface and without structures. Nevertheless, low threshold of critical heat flux limits the usage of hydrophobic coated surface. Hence, we put forward a new surface with changeable wettability which can realize better boiling heat transfer with a higher critical heat flux. As the first step, a wettability versus temperature relation was set as shown in Fig. 9, represented by up-faced triangles. The wettability starts to change with wall temperature increasing after onset of boiling point to maintain the advantage of hydrophobic surface, then the surface wettability

4.3. Boiling performance on temperature-dependent wetting surface Thus far, we can confirm that hydrophobic surface has higher HTC than hydrophilic surface when the surface is treated as

Fig. 9. Comparison of temperature-dependent wetting surface with hydrophilic/ hydrophobic surfaces in terms of boiling curves under controlled wall heat flux.

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turns better as the wall superheat increase to delay the critical heat flux. Numerical results of boiling curves are shown in Fig. 9, which compares the temperature-dependent (hereinafter referred to as T-dependent) wetting surface with hydrophobic and hydrophilic surfaces. The results of T-dependent wetting surface are represented by the hollow circles with a cross inside. As can be seen in the figure, at the early stage of nucleate boiling, the boiling curve of the T-dependent wetting surface is overlapped with the hydrophobic surface. It was because the temperature increment at this stage was small, and the change of surface wettability is small. In this way, we can maintain the advantages of hydrophobic surface which has higher heat transfer rate. Just after the nucleate boiling entered the aforementioned third stage, the boiling curve of T-dependent wetting surface separated with hydrophobic surface to obtain higher heat transfer rate. As the surface wettability becomes better with temperature increase gradually, the critical heat flux was delayed step by step. Besides, the boiling curve of the T-dependent wetting surface is on the left side of the hydrophilic surface, which means at the same heat flux T-dependent wetting surface has lower wall superheat and higher heat transfer rate. So the T-dependent wetting surface could take advantages of both hydrophilic and hydrophobic surfaces, moreover, the wettability transition only related to temperature and does not need additional external control. The transition range was set between 91° and 45.46° because surface with a contact angle among this range is easier to obtain in practice, but with the development of technology and emergence of new materials this range can be expanded. As shown in Fig. 10, a larger transition range was chosen and surface with temperature-dependent wettability could take advantages of both hydrophobic surface which has an earlier onset of nucleate boiling and hydrophilic surface which has higher critical heat flux. We next turn our attention to boiling heat transfer performance under different relations between temperature and wettability. Four kinds of relation with different contact angle under same wall temperature were designed and their boiling performance were obtained as shown in Fig. 11(a) and (b). For case1, the surface wettability starts to change linearly at the very beginning, in Fig. 11(b) we can see that the onset of nucleate boiling point is delayed because as the wall superheat increase the wettability of surface becomes better, as analyzed in Section 4.2, hydrophilic surface needs higher wall superheat to make bubble nucleation. Although the heat transfer rate on surface in case1 is still higher than hydrophilic surface, it is obviously lower than hydrophobic surface or

Fig. 10. Boiling performance on surface with temperature-dependent wettability with larger transition range under controlled wall heat flux.

case 2, 3 and 4. This indicates that this kind of trend can’t take the advantage of hydrophobic surface, so, in case 2, 3 and 4, the surface wettability starts to change after the onset of nucleate boiling. Among those three cases, case 4 has the smallest contact angle while case 2 has the biggest one under same wall temperature as illustrated in Fig. 11(a). But the boiling curve shows that case 3 has the best performance. An enlarged plot of these three cases is made as an insert plot in Fig. 11(b) to show the results more clearly. At the early stage of nucleate boiling, case 2 has slightly advantage of boiling heat transfer over others because it has the largest contact angle among these three cases, but as the heat flux increase, too much bubble emerged and it stepped into the aforementioned third stage of nucleate boiling. Since it has the largest contact angle among these three cases, it has the lowest critical heat flux. And for case 4, the wettability changes too fast, so the surface transform from high contact angle to low contact angle too early and could not take advantage of high heat transfer rate of high contact angle. This indicates that there is an optimal relation between temperature and surface wettability which can make the plain surface has best heat transfer rate under different wall superheat. 4.4. Analysis with empirical correlations To determine the optimal relation between wall temperature and surface wettability. We resorted to analytical analysis. Two

Fig. 11. Comparison of boiling curves on smooth surface with temperaturedependent wettability under different cases: (a) contact angel transition trends and (b) boiling curves under controlled wall heat flux.

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widely used correlations are chosen, one is Rohsenow’s correlation equation [40] to predict nucleate boiling performance, and the other one is Kandlikar’s equation [36] to predict the critical heat flux under different contact angle. Based on dimensionless analysis, Rohsenow proposed a correlation for nucleate boiling, which given by


JaR ¼ C sf

q ll hfg

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

ð21Þ

where q is the input heat flux and kl , ll are liquid’s conductivity and viscosity, respectively; C sf is a fitting constant depending on the surface conditions, which contains roughness and wettability. Fig. 12 shows a comparison of the LB simulation results with Rohsenow’s equation in the nucleate boiling regime, where fluid properties were evaluated at saturated conditions. We found that the fitting constant in Eq. (21) C sf ¼ 0:017 for hydrophilic surface and C sf ¼ 0:015 for hydrophobic surface match the simulation results well. Moreover, it is noted from Eq. (21) that at same heat flux q, the dimensionless wall superheat Ja of hydrophilic surface is higher than that of hydrophobic surface because the fitting constant C sf ¼ 0:015 for hydrophobic surface is smaller than C sf ¼ 0:017 for hydrophilic surface. It indicates hydrophobic surface has better nucleate boiling heat transfer rate than hydrophilic surface, which is consistent with our previous analysis. Similar comparison of LB results with Rohsenow’s correlation equation was also made by Zhang and Cheng [34]. Kandlikar’s relation is obtained through analyzing the forces which parallel to the heater surface, including the force pulling the bubble interface spreading on the heating surface and the force holding the bubble. This equation is widely used to predict the critical heat flux at different contact angle, given by


 1=2 1 1 þ cosh 2 4 qc ¼ hfg q2v þ ð1 þ coshÞ  ½rgðql  qv Þ1=4 16 p p ð22Þ where h is contact angle. Through Eq. (22) we can get the critical heat flux qc under specific contact angle, furthermore through changing C sf in Eq. (21) we can get the boiling curves under different contact angles. Combining those two correlations, we can predict the boiling curves from nucleate boiling to critical heat flux under different contact angle. Substituting Eq. (22) into Eq. (21), we can get:

Fig. 13. Schematic of the ideal boiling curve for temperature-dependent wetting surface to take advantages of both hydrophobic and hydrophilic surfaces.


JaR ¼ C sf

qc ll hfg

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

ð23Þ

Since C sf is related to contact angle, Eq. (23) can be rewritten in the following form

JaR ¼ Af ðhÞ

ð24Þ 1

where

A¼

q2v ½rgðql qv Þ1=4 ll

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi r

gðql qv Þ

!0:33



cp;l ll kl

1:7

and

f ðhÞ ¼

0:165    h 0:33 2 4 C sf ðhÞ 1þcos is a function of contact angle. p þ p ð1 þ cos hÞ 16 Then an ideal relation between dimensionless wall superheat Ja and contact angle h can be obtained. Fig. 13 is a schematic plot illustrated boiling curves of different contact angles. Points C–G are predicted critical heat flux using Eq. (22). Under ideal situation, initially the surface wettability should maintain constant and then changes from a high contact angle to a lower one as described in Eq. (24) just before it gets the threshold of critical heat flux, as a result, we can delay critical heat flux step by step and maintain the highest heat transfer coefficient at different wall superheat. This trend is illustrated in Fig. 13 as curve A—B—C—D—E—F—G and it can guarantee that the heat transfer coefficient is the highest on plain surface without structures at different heat flux. But in real situation, there is a delay between surface wettability transition and boiling patterns change. Besides, as analyzed in Section 4.2, near the threshold of critical heat flux, excessive bubbles emerge and coalesce with each other and prevent liquid to rewetting the surface, so the heat transfer coefficient starts to decrease or maintain. Hence, the contact angle should change before the threshold of critical heat flux, and the boiling curve is illustrated in Fig. 13 as A—B—C0 —D0 —E0 —F0 —G0 . 5. Conclusions In this paper, the boiling performance on surface with temperature-dependent wettability was investigated. The boiling performance on hydrophilic and hydrophobic as well as temperature-dependent wetting surfaces were evaluated by hybrid thermal lattice Boltzmann method. The following findings can be drawn from present study:

Fig. 12. Comparison of numerical obtained boiling curves with Rohsenow’s correlation for nucleate boiling regime on smooth surfaces with different wettability.

(1) Numerical results of single bubble dynamics obtained by hybrid thermal LB model are consistent with previous experimental results. The necking phenomenon on hydrophobic

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surface and existence of waiting period on hydrophilic surface are captured. The hybrid thermal LB model is verified suitable for study of wettability effects. (2) In terms of heat transfer performance, wettability consideration is reflected and boiling curves obtained by hybrid thermal LB model on plain hydrophilic and hydrophobic surface fit well with Rohsenow’s correlation. The fitting constant depends on the wettability of the surface when there are no roughness effects. (3) The temperature-dependent wetting surface can take advantages of both hydrophobic and hydrophilic surfaces, and has similar heat transfer rate as hydrophobic surface while maintains the merits of hydrophilic surface which has high threshold of critical heat flux. Moreover, the transition of surface wettability is completely passive and doesn’t need additional external control. (4) If the initial wettability is selected, through contact angle control, we can obtain highest heat transfer rate on smooth surface under different wall superheat. In this study, a relation between temperature and surface wettability is deduced to provide a potential design of T-dependent wetting surface.

Acknowledgements The authors gratefully acknowledge the financial support provided by the Natural Science Fund of China (NSFC Grant Nos. 51376179 and 51476167). This work was also supported by ChineseJapanese Research Cooperative Program (Grant No. 2016YFE0118100).

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