THIN FILM EVAPORATION IN MICROCHANNEL MEMBRANE FOR SOLAR VAPOR GENERATION

Transcription

1 Proceedings of the th Internationa Heat Transfer Conference, IHTC- August -,, Kyoto, Japan IHTC-9 THIN FILM EVAPORATION IN MICROCHANNEL MEMBRANE FOR SOLAR VAPOR GENERATION Ammar A. Asheghri, TieJun Zhang * Department of Mechanica and Materias Engineering, Masdar Institute of Science and Technoogy, P. O. Box, Abu Dhabi, UAE ABSTRACT Thin fim evaporation is investigated in microchanne membranes for soar vapor generation. Microchanne membranes provide a nove way to stimuate thin fim evaporation for capiary pumping and efficient vapor generation in compact devices. On the top of the membrane, there is a thin ayer that absorbs wide-range soar spectrum and converts soar energy to heat. The heat is conducted through the channes was into the iquid menisci that utimatey generates vapor. Due to capiarity, iquid is continuousy repenished from the bottom side of the membrane into the menisci at the top. High heat transfer rates occur in the thin fim region due to the sma therma resistance across the fim where iquid thickness is ess than a micron. The augmented Young-Lapace equation and the kinetic theory for mass transport are used to mode the evaporating thin fim in microchanne membranes. The effect of vapor pressure, channes spacing, and sip ength on the interfacia evaporative mass fux is incuded. The infuence of variabe wa temperature profie is discussed. A new mode is deveoped to account for the effect of rough sidewas. The proposed work aims to offer a basis for designing thin fim evaporation devices for power generation and therma desaination systems. KEY WORDS: Thin fim, Evaporation, Mass transfer, Nano/Micro, Soar vapor generation, Roughness. INTRODUCTION High-performance and ow-cost therma devices are enabing a wide range of soar energy appications, as summarized by the U.S. DOE SunShot Vision Study in. Such devices utiize the soar radiation in an energy-efficient way to generate vapor for therma power generation and therma desaination. Microchanne membranes are of significant interest for soar vapor generation, because the iquid can be driven by the meniscus capiary force in parae microchannes of the membranes. Therefore, no mechanica pump is required to suppy the evaporation process with fresh iquid, which further reduces the size, cost and maintenance burden of soar vapor generators. In these compact devices, the utiization of atent heat of vaporization offers high soar therma conversion efficiency. In fact, evaporation heat transfer performance becomes even more superior in the thin fim region because of its smaer therma resistance. Wang et a. [] modeed the evaporating thin fim in a microchanne based on the kinetic theory for mass transport across a iquid-vapor interface. Zhao et a. [] studied the effect of superheat and temperature dependent thermophysica properties on thin fim evaporation in microchannes. Yan et a. [] showed that considering ony momentum conservation underestimates the tota heat transfer rate compared with considering both momentum and energy conservation. Xiao et a. [] presented a nano-porous membrane for thin fim evaporation and achieved absoute iquid pressures of - kpa with about nm pore diameters. *Corresponding Author: tjzhang@masdar.ac.ae

2 IHTC-9 In this paper, thin fim evaporation is modeed in microchanne membranes for soar vapor generation. The effects of vapor pressure and characteristic spacing between the channes on the evaporating thin fim profies and evaporative mass fux are incuded. Constant and variabe wa temperature profies are compared. A new mode is deveoped to account for the effect of corrugated channes sidewas on the evaporation rate.. ANALYSIS A schematic of a microchanne membrane is shown in Fig. a. The top ayer of the membrane is made of an absorption ayer that absorbs wide-range soar spectrum and heat is generated due to the phototherma effect. Generated heat is conducted through the channes was into the iquid menisci that utimatey generates vapor. Liquid is continuousy repenished from the bottom side of the membrane into the menisci at the top due to capiary pressure. The shape of the evaporating interface, shown in Fig. b, is divided into three regions: () the adsorbed fim region where no evaporation occurs; () the thin fim region where high heat transfer rates occur due to the sma therma resistance across the sub-micron thick iquid fim; () the intrinsic meniscus region where capiary pressure is dominant. The heat input is assumed to be appied at the side was of the channes. Due to symmetry, ony haf of a channe is considered. The working fuid is water. Gravity is negected, constant thermophysica properties are considered, and uniform vapor pressure is assumed in this anaysis. (a) (b) Fig. (a) Schematic of a microchanne membrane (b) Extended meniscus in a microchanne membrane. Thin Fim Mode on Smooth Sidewas Fuid fow in the thin fim region is assumed to behave according to ubrication approximation where viscous terms dominate because of the ow Reynods number. The foowing anaysis is simiar to the anaysis of []. The momentum equation governing the iquid fow reduces to: dp d du ( ), dp () dx dy dy dy By imposing a no sip boundary condition at the wa and a no shear boundary condition at the iquid-vapor interface, the iquid veocity is obtained: dp y u ( y ) () dx The iquid mass fow rate is: m dx u dy The iquid pressure gradient is derived from Eqs. () and (): x ()

3 x dp m dx dx IHTC-9 The pressure difference between iquid and vapor at the iquid-vapor-interface is expressed by the augmented Young-Lapace equation [,,]: P P P P () The capiary pressure due to curvature P c is: P c K, iv i c d. K ( ) (6) The disjoining pressure due to intermoecuar interactions P d is: P d A (7) where A is the non-retarded Hamaker constant which accounts for the van der Waas interactions. In this anaysis, a constant vaue of.8 - J is assumed for the non-retarded Hamaker constant [6]. As shown by Eq. (), dp dy : therefore, P i = P and P iv =, the augmented Young-Lapace equation becomes:k Pv P Pc Pd (8) Substituting Eqs. (6) and (7) into Eq. (8) and differentiating with respect to x gives: dp. A ( ) (9) dx where d/dx = because is assumed to be constant. Substituting Eq. () into Eq. (9) and differentiating with respect to x gives the fourth-order non-inear differentia equation governing the iquid fim thickness: d A m dx ().. ( ) ( ) The interfacia mass fux coud be cacuated by equating the conduction heat fux across the thin fim and the evaporation heat fux on the interface: m k ( T T ) h () w The wa temperature T w is assumed to be given. The temperature at the iquid-vapor interface T v is unknown, therefore another equation is required to cacuate the interfacia mass fux. The kinetic theory for evaporation proposed by Schrage [7] and reviewed in [] provides a second reation between the interfacia mass fux and the interface temperature: ˆ M Pv Pv m () ˆ R T T v v where the vapor temperature Tv is the saturation temperature at the vapor pressure v fg () P v. The accommodation coefficient of water ˆ ranges between. and. and is taken to be. in this study [8]. P v is the pressure at which the vapor is in equiibrium with the iquid. Due to the effects of capiary and disjoining pressures, Pv is smaer than the saturation pressure P sat att v and is given by [7]: Pv ( Tv ) Psat ( Tv ) ( Pd Pc ) P T P T v ( v ) sat( v )exp () TvR / M Eqs. ()-() are soved simutaneousy to obtain the interfacia mass fux and the iquid-vapor interface temperature. Once the interfacia mass fux is obtained, Eq. () can be soved and the thin fim profie is obtained.. Thin Fim Mode on Corrugated Sidewas Scaop features, shown in Fig. b, are commony formed on the sides of channes during the etching process. Scaops are modeed as an array of circuar piars with a height h of nm, a height-to-diameter ratio h/d of, and a diameter-to-period ratio d/ of..

4 IHTC-9 Fig. Schematic diagram for thin fim evaporation. (a) Smooth sidewas. (b) Rough sidewas (scaops) To estimate the additiona viscous resistance caused by scaops, the one-dimensiona form of Brinkman s equation is used []. d u u dp () dy dx where ε is the porosity: d () and κ is the permeabiity which accounts for the voumetric drag induced by the roughness []:. n (6) where φ is the soid fraction: (7) Note that Eq. (6) is vaid for φ <., therefore in this work φ is assumed.96. By appying the no sip boundary condition at the sidewas and no shear boundary condition at the iquid-vapor interface, the iquid veocity is obtained: dp u Aexp y B exp y (8a) dx and dp dx exp exp exp A (8b) dp dx exp exp exp B (8c) Provided that scaops have the same hydrophiic materia of the microchanne membrane, the formation of these scaops adds an additiona spreading force that is negected in this work. Note that the no shear boundary condition is appied at the iquid-vapor interface, therefore δ represents the thin fim thickness and not the sidewa corrugations as in []. The iquid pressure gradient is obtained from Eq. () and Eq. (8) as: x dp m dx (9a) dx C where exp exp exp C (9b) exp Substituting Eq. (9) into Eq. (9) and differentiating with respect to x gives the fourth-order non-inear differentia equation governing the iquid fim thickness on rough sidewas:

5 IHTC-9 d A C m dx ().. ( ) ( ) Eq. () defines the new thin fim profie with considering scaops on the sidewas of the channes. Eqs. ()-() are sti vaid for finding the interfacia mass fux.. Soution Methods and Boundary Conditions The fourth order Runge-Kutta method was used to sove Eqs. () and () with MATLAB used as the simuation software. Water thermophysica properties were provided by the NIST REFPROP. The probem was treated as an initia vaue probem requiring four initia conditions for δ, dδ/dx, d δ/dx, and d δ/dx. By assuming no evaporation and zero curvature conditions at the adsorbed fim region, the disjoining pressure at the non-evaporating region is given by []: R Tw P v Tw Pd T w n Pv Psat T w () M Tv Psat T ( w ) Tv Note that Eq. () is obtained from Eqs. () and () by assuming = m and P c =. Once P d is obtained, δ coud be cacuated by Eq. (7). The initia condition for the profie first derivative was set to zero []. In this work, the vaue of δ was set to nm to ensure a non-zero mass fux. However, when comparing smooth and corrugated sidewas in section. a vaue of δ nm was set in both cases to avoid toerance errors at execution. The shooting method was used to provide the correct initia condition for the profie second derivative to satisfy the intrinsic meniscus curvature. That is an initia curvature shoud be iterativey determined to obtain the desired fina curvature. The desired fina curvature shoud satisfy the far-fied condition K = /W. The initia condition for the profie third derivative was set to zero. The thin fim region ength, L, is determined by using the foowing condition provided by []: P d x L Pd ( x ) () Eq. () provides the condition at which the disjoining pressure is sufficienty sma and the evaporating thin fim merges into the intrinsic meniscus region.. RESULTS AND DISCUSSION The effects of vapor pressure and channes spacing on the evaporation rate at constant wa temperature of K are discussed in sections. and., respectivey. The effect of surface wettabiity and sip ength on the evaporation rate is discussed in section.. Section. compares constant to variabe wa temperature profies whereas section. discusses the effect of roughness on evaporation rate and interfacia mass fux.. Effect of Vapor Pressure The variation of vapor pressure, at a channe width of µm and superheat of K, and its effect on evaporative mass fux is discussed in this section. Fig. a shows that decreasing the vapor pressure increases the fim thickness and resuts in a shorter extended meniscus region. The vertica ines in the figure indicate the point where thin fim merges in the intrinsic meniscus regime. Higher vapor pressures correspond to higher vapor temperatures and higher superheats. Thus, an increase in superheat reduces the ength of the thin fim region. This trend agrees with the one presented in []. In Fig. b, the maximum fim curvature continuousy increases with ower vapor pressures corresponding to ower radii of curvature. At ower vapor pressures more iquid moecues are repenished inside the thin fim region. This is expained by Fig. c where higher vaues for capiary pressure are obtained at ower vapor pressures. The iquid pressure is obtained from Eq. (8) and the iquid pressure drop is given by: P L P( x) P() L () Fig. d shows that the iquid pressure drop increases aong the x direction and approaches a maximum vaue. The decrease in the disjoining pressure and capiary pressure aong the x direction causes iquid to be pumped into the evaporating thin fim region. An absoute iquid pressure vaue cose to -7 kpa is obtained towards the

6 IHTC-9 start of the thin fim region at a vapor pressure of kpa and channe spacing of µm. Experimenta observations reported absoute iquid pressures of - kpa using isopropy acoho (IPA) in nano-porous membranes with pore diameters of nm []. The interfacia mass fux in given by: m q h fg ( T v ) () Fig. e shows that the interfacia mass fux, after reaching a peak, decreases aong the x direction. Increasing the vapor pressure resuts in reducing the evaporative mass fux. As the vapor pressure increases the iquid pressure gradient decreases and ess iquid is pumped into the evaporating thin fim region. Lower vapor pressures correspond to ower iquid-vapor interfacia temperatures, as evident in Fig. f, and thus higher evaporative heat fux and mass fux vaues.. x = kpa = kpa = kpa (a) (c) [nm] P c [kpa] = kpa = kpa = kpa 8 6 = kpa = kpa = kpa 6 8 (b) (d) K [m - ] P /L [kpa/m] x = kpa = kpa = kpa = kpa = kpa = kpa 9 9 m" [kg/m -s] T v [K] 8 8 = kpa 7 = kpa = kpa (e) (f) Fig. Effect of vapor pressure on extended meniscus profies (W = µm). (a) Fim thickness. (b) Curvature. (c) Capiary pressure. (d) Liquid pressure drop. (e) Interfacia mass fux. (f) Liquid-vapor interface temperature. 6

7 IHTC-9 The interfacia mass fux decreases with x whie the iquid-vapor interface temperature remains amost constant as shown in Fig. e and Fig. f, respectivey. At reativey high vapor pressures the mass fux and interface temperature gradients become smaer. It is concuded in this section that achieving ower vapor pressures improves evaporation and resuts in higher interfacia mass fuxes. However, this comes at the cost of demanding higher power input for the vacuum pump. W = m W = m W = m (a) [nm] W = m W = m W = m (b) P d [kpa] W = m W = m W = m W = m W = m W = m - P c [kpa] 8 6 P [kpa] (c) 6 8 (d) - W = m W = m W = m W = m W = m W = m m" [kg/m -s] h c [W/cm.K] 8 6 (e) 6 8 (f) 6 8 Fig. Effect of channe width on extended meniscus profies ( = kpa). (a) Fim thickness. (b) Disjoining pressure. (c) Capiary pressure. (d) Heat transfer coefficient. (e) Interfacia mass fux. (f) 7

8 IHTC-9. Effect of channe width The variation of spacing between the microchannes, under vapor pressure of kpa and wa superheat of K, and its effect on evaporative mass fux is discussed in this section. Fig. a shows that increasing the spacing between channes extends the meniscus and resuts in a smaer buk thin fim region. The vertica ines in the figure mark the point where thin fim merges in the intrinsic meniscus regime. Longer thin fim regions are obtained with increasing the spacing corresponding to higher interfacia mass fuxes as evident in Fig. e. The disjoining pressure, shown in Fig. b, starts at a vaue of kpa due to the very thin fim thickness (δ nm). Then, it decreases sharpy with the direction of x. Fig b aso shows that the disjoining pressure is not much affected by the channe width. Fig. c shows that increasing the width reduces the capiary pressure. This observation agrees with nanoporous membranes where capiary pressure is defined by the pore size []. When spacing is smaer, evaporation is suppressed by the high capiary pressure and the absoute iquid pressure decreases as shown in Fig. d. The decrease in disjoining pressure, at the beginning of the thin fim region, is baanced by an increase in capiary pressure. After the capiary pressure reaches a peak, both disjoining and capiary pressures decrease and the absoute iquid pressure increases according to the augmented Young-Lapace equation Eq. (8). The heat transfer coefficient across the fim, shown in Fig. f, is: h c k () The heat transfer coefficient across the thin fim decreases with x due to the increased fim thickness and therma resistance. Smaer spacing eads to thicker fims which resuts in ower interfacia mass fux and heat transfer coefficient vaues as evident in Fig. e and Fig. f, respectivey. It is concuded from this section that arger channe width yieds higher evaporation rates and higher interfacia mass fuxes at the cost of decreasing capiary pumping. However, after reaching a width of µm, the evaporation rate becomes ess dependent on channe width.. Effect of sip ength One way to mode the wettabiity of the membrane is to consider a sip condition at the wa. In this section, the effect of the sip ength on thin fim profies is discussed. The iquid sip veocity at wa is given by the foowing equation: du usip sip y (6) dy where β sip is the sip ength. By considering the above boundary condition Eq. () becomes: dp y y sip u (7) dx The differentia equation governing the iquid fim thickness is modified accordingy such that: d A sip m.. (8) dx ( ) ( ) Note that if β sip =, then Eq. (8) becomes equivaent to Eq. (). Pawsky et a. argued that hydrophobicity enhances sip []. Therefore, the sip ength coud be considered as a measure of the hydrophiicity or wettabiity of a certain surface. Zhao et a. used a sip ength of.7 nm to mode water evaporation on siica surfaces []. Bonaccurso et a. reported a vaue of 8-9 nm for the sip ength of water on smooth hydrophiic surfaces. The effect of sip ength on thin fim profies is expained by Fig. by considering a range of sip engths between - nm. The effect of sip ength on the fim ength is presented in Fig. a. Higher sip ength vaues increase the fim ength and reduce its thickness. Fig. b. and Fig. c. show that the iquid pressure drop and the capiary pressure decrease with increasing the sip ength, respectivey. The decrease in capiary pressure at higher sip engths resuts in increased evaporation rates as evident by Fig. d. 8

9 IHTC-9 [nm] = = nm = nm = nm P /L [kpa/m] 9 x = = nm = nm = nm = = nm = nm = nm = = nm = nm = nm P c [kpa] 8 6 m" [kg/m -s] 6 8 Fig. Effect of sip ength on extended meniscus profies. (a) Thin fim thickness. (b) Liquid pressure drop. (c) Capiary pressure. (d) Interfacia mass fux.. Effect of Variabe Wa Temperature In this section, the effect of a variabe wa temperature profie on interfacia mass fux is discussed and compared to the constant wa temperature profie. The anaysis is performed under a vapor pressure of kpa and a channe width of µm. A inear wa temperature profie is considered as foows: 6 T T x (9) w w where T w = T v + ΔT. Fig. 6a shows that the fim thickness profie is amost independent of the wa temperature profie. The iquid-vapor interface temperature trend at variabe wa temperature profie foows the same trend of the constant wa temperature as shown in Fig. 6b. However, at the variabe wa temperature, the iquid-vapor interface temperature increases sighty towards the end of the meniscus. The interfacia mass fux vaues at the constant wa temperature are sighty higher compared to the variabe wa temperature as shown in Fig. 6c. This is because the considered wa temperature decreases with x. The interfacia heat transfer coefficient is given by: hv q / T v Tv () The constant wa temperature assumption overestimates the interfacia heat transfer coefficient as evident in Fig. 6d. The figure aso shows that the interfacia heat transfer coefficient sighty increases towards the end of the meniscus due to the increase of the iquid-vapor interface temperature. The inear temperature profie is an approximation and we intend, in the next step, to consider a conjugate heat transfer case where the wa temperature is provided by couping the therma conduction equation with the kinetic theory for mass transport. 9

10 IHTC-9 Constant wa temperature Variabe wa temperature Constant wa temperature Variabe wa temperature [nm] T v [K] (a) (b) 7. Constant wa temperature Variabe wa temperature. x. Constant wa temperature Variabe wa temperature m" [kg/m -s] h v [W/cm.K]... (c) - (d) -. Fig. 6 Effect of wa temperature profie on extended meniscus profies. (a) Thin fim thickness. (b) Liquidvapor interface temperature. (c) Interfacia mass fux. (d) Interfacia heat transfer coefficient.. Effect of Roughness (i.e. Scaops) The formation of scaop features increases the roughness of the channes sidewas. Fig. 7a shows that roughness resuts in a arger buk fim thickness and a shorter fim ength compared to the smooth sidewas. The capiary pressure shown in Fig. 7b is higher in the case of corrugated sidewas. As a resut, the interfacia mass fux and heat transfer coefficient across the fim thickness become sighty smaer after considering the effect of scaop features as seen in Fig. 7c and Fig. 7d, respectivey. The size of scaops is assumed to be nm which is at the same order of the thin fim thickness. In Fig. 7e, the pressure drop for corrugated sidewas is arger compared to smooth ones because of the additiona viscous resistance induced by the scaops. Fig. 7f shows that the interfacia heat transfer coefficient is higher for smooth sidewas compared to the corrugated ones. This resut is justified by the reativey higher capiary pressure for corrugated sidewas as evident in Fig. 7b. High capiary pressures suppress evaporation and resut in ower interfacia mass fuxes as shown in Fig. 7c. It is worth mentioning that if the wetting behavior of the additiona scaops is considered, different resuts might be achieved.

11 IHTC-9 Smooth sidewas Rough sidewas Smooth sidewas Rough sidewas [nm] P c [kpa] (a) (b) Smooth sidewas Rough sidewas Smooth sidewas Rough sidewas 8 m" [kg/m -s] 6 h c [W/cm.K] (c) (d) 9 x 6 8. x Smooth sidewas Rough sidewas 7 P /L [kpa/m] 6 h v [W/cm.K].. (e) Smooth sidewas Rough sidewas (f). Fig. 7 Effect of scaops (roughness) on thin fim profies under kpa vapor pressure and µm channe width. (a) Fim thickness. (b) Capiary pressure. (c) Interfacia mass fux. (d) Heat transfer coefficient across the fim. (e) Liquid pressure drop. (f) Loca heat transfer coefficient.. CONCLUSIONS The evaporating thin fim is modeed in a microchanne membrane using the augmented Young-Lapace equation and the kinetic theory for mass transport. A new mode incuding the effect of viscous resistance is deveoped for rough sidewas. It is found that ower vapor pressures and arger channe spacing enhance evaporation and correspond to higher interfacia mass fux vaues. Higher evaporation rates coud aso be achieved by enhancing the hydrophiicity of the surface. Further, imposing a variabe inear wa temperature

12 IHTC-9 profie does not affect the thin fim thickness profie, but infuences the evaporation rate. The formation of scaop features on the sidewas of the channes decreases the fim ength as we as the interfacia mass fux and increases the pressure drop. ACKNOWLEDGMENT This work was supported by the Cooperative Agreement between the Masdar Institute of Science and Technoogy, UAE and the Massachusetts Institute of Technoogy (MIT), USA. The authors woud ike to thank Dr. Guanqiu Li, postdoctora research feow at Masdar Institute, for his support. NOMENCLATURE A Dispersion constant (J) m interfacia mass fux (kg/m s) d diameter (m) M moecuar weight (J/mo) h c conductive heat transfer coefficient (W/m K) P pressure (N/m ) h fg atent heat of evaporation (J/kg) q evaporative heat fux (W/m ) h v interfacia heat transfer coefficient (W/m K) R universa gas constant (J/mo K) k therma conductivity (W/mK) T temperature (K) K curvature (m - ) u veocity (m/s) pitch (m) W channe width (m) L thin fim ength (m) x,y x and y coordinates (m) Greek etters δ thickness of thin fim (m) ν kinematic viscosity (m /s) ɛ porosity (-) ρ density (kg/m ) κ permeabiity (m ) σ surface tension (N/m) μ dynamic viscosity (N s/m ) φ Soid fraction (-) Subscripts c capiary sat saturation d disjoining v vapor i interface w wa iquid REFERENCES [] Wang, H., Garimea, S.V., Murthy, J.Y., Characteristics of an evaporating thin fim in a microchanne, Int. J. Heat Mass Transf., (9-), pp. 9-9, (7). [] Zhao, J.J., Duan, Y.Y., Wang, X.D., Wang, B.X., Effects of superheat and temperature-dependent thermophysica properties on evaporating thin iquid fims in microchannes, Int. J. Heat Mass Transf., (-6), pp. 9-67, (). [] Yan, C., Pan, X., Lu, X., Mechanisms of Thin-Fim Evaporation Considering Momentum and Energy Conservation, Proc. of th ASME Int. Conf. on Micro/Nanoscae Heat Mass Transf., MNHMT-7, (). [] Xiao, R., Maroo, S.C., Wang, E.N., Negative pressures in nanoporous membranes for thin fim evaporation, Appied Physics Letters., (), pp., (). [] Wayner, P.C. Jr., Kao, Y.K., LaCroix, L.V., The interine heat transfer coefficient of an evaporating wetting fim, Int. J. Heat Mass Transf., 9(), pp. 87-9, (976). [6] Narayanan, S., Fedorov, A., Joshi, Y., Interfacia transport of evaporating water confined in nanopores, Langmuir, 7, pp , (). [7] Schrage, R.W., A Theoretica Study of Interface of Interface Mass Transfer, New York: Coumbia University Press, (9). [8] Carey, V.P., Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, New York: nd edition, Tayor and Francis, pp. -, (8). [9] Faghri, A., Heat Pipe Science and Technoogy, Washington, DC: Tayor & Francis, (99). [] Xiao, R., Enright, R., Wang, E.N., Prediction and optimization of iquid propagation in micropiar arrays, Langmuir, 6(9), pp. 7-7, ().

13 IHTC-9 [] Sangani, A. S., Acrivos, A., Sow fow past periodic arrays of cyinders with appication to heat transfer, Int. J. Mutiphase Fow, 8(), pp 9, (98). [] Hanchak, M.S., Vangsness, M.D., Gheorghiu, N., Ervin, J.S., Byrd, L.W., Jones, J.G., Thin Fim Evaporation Mode with Retarded Van Der Waas Interaction, Proc. of ASME Internationa Mechanica Engineering Congress and Exposition, IMECE-697, (). [] Xiao, R., Wetting and phase-change phenomena on micro/nanostructures for enhanced heat transfer, Ph.D. Thesis, Massachusetts Institute of Technoogy, (). [] Pawsky, J. L., Ojha, M., Chatterjee, A., Wayner, P. C. Jr., Review of the effects of surface topography, surface chemistry and fuid physics on evaporation at the contact ine, Chem. Eng. Comm., 96, pp , (8). [] Zhao, J.J., Duan, Y.Y., Wang, X.D., Wang, B.X., Effect of nanostructured roughness on evaporating thin fims in microchannes for Wenze and Cassie-Baxter states, Journa of Heat Transfer., (), pp. -, (). [6] Bonaccurso, E., Kapp, M., Butt, H. J., Hydrodynamic force measurements: Boundary sip of water on hydrophiic surfaces and eectrokinetic effects, Phys. Rev. Lett., 88, 76, ().