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A Coupled Wicking and Evaporation Model for Prediction of Pool Boiling Critical Heat Flux on Structured Surfaces H Purdue University Purdue e-Pubs CTRC Research Publications Cooling Technologies Research Center 2019 A Coupled Wicking and Evaporation Model for Prediction of Pool Boiling Critical Heat Flux on Structured Surfaces H. Hu Purdue University J. A. Weibel Purdue University, [email protected] S V. Garimella Purdue University, [email protected] Follow this and additional works at: https://docs.lib.purdue.edu/coolingpubs Hu, H.; Weibel, J. A.; and Garimella, S V., "A Coupled Wicking and Evaporation Model for Prediction of Pool Boiling Critical Heat Flux on Structured Surfaces" (2019). CTRC Research Publications. Paper 339. http://dx.doi.org/https://doi.org/10.1016/j.ijheatmasstransfer.2019.03.005 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. A Coupled Wicking and Evaporation Model for Prediction of Pool Boiling Critical Heat Flux on Structured Surfaces Han Hu, Justin A. Weibel1, and Suresh V. Garimella1 School of Mechanical Engineering, Cooling Technologies Research Center Purdue University, 585 Purdue Mall, West Lafayette, IN 47907 USA ABSTRACT. Boiling is an effective heat transfer mechanism that is central to a variety of industrial processes including electronic systems, power plants, and nuclear reactors. Micro-/nano- structured surfaces have been demonstrated to significantly enhance the critical heat flux (CHF) during pool boiling, but there is no consensus on how to predict the structure-induced CHF enhancement. In this study, we develop an analytical model that takes into consideration key mechanisms that govern CHF during pool boiling on structured surfaces, namely, capillary wicking and evaporation of the liquid layer underneath the bubble. The model extends existing wicking-based CHF theories by introducing the competing evaporation mechanism. The model reveals a wicking-limited regime where CHF increases monotonically with the wicking flux, and an evaporation-limited regime where additional increases in the wicking flux do not significantly affect CHF. The model predictions are shown to agree with experimental CHF data from the literature for boiling of water on surfaces structured with square micropillar arrays. A parametric study is performed for such micropillared surfaces and has identified the optimal structure based on the competition between wicking and evaporation, capillary pressure and viscous resistance, and conduction and liquid-vapor interfacial resistance. Keywords: pool boiling, critical heat flux, structured surfaces, wicking, evaporation 1 Corresponding author. E-mail: [email protected] (J.A. Weibel), [email protected] (S.V. Garimella) 1 Nomenclature D pillar diameter, m Greek letters d b bubble departure diameter, m fit fitting parameter difference in liquid and vapor d dry spot diameter, m dry lv density, kg/m3 * ddry critical dry spot diameter, m liquid film thickness, m 2 g gravitational acceleration, m /s c critical film thickness, m H pillar height, m porosity equilibrium contact angle at h enthalpy of vaporization, J/kg fg equ room temperature, ° evaporation heat transfer equilibrium contact angle at h lv coefficient, W/(m2·K) equ,b boiling point, ° receding contact angle at permeability of structures, m2 K rec,b boiling point, ° critical wavelength of K effective permeability, m2 B RT Rayleigh-Taylor instability, m 2 kl thermal conductivity, W/(m·K) dynamic viscosity, m /s 3 M molar mass, kg/mol l liquid density, kg/m 3 P pillar pitch, m s solid density, kg/m 3 P0 bulk liquid pressure, Pa v vapor density, kg/m 2 PC capillary pressure, Pa lv surface tension, J/m ˆ Pl liquid pressure, Pa accommodation coefficient Pv vapor pressure, Pa stress, Pa 2 3 qCHF critical heat flux, W/m e evaporation rate, m /s 2 3 qe evaporation heat flux, W/m g gravity-driven flow rate, m /s critical heat flux on flat surfaces, q wicking rate, m3/s CHF,f W/m2 w Structure-induced CHF characteristic wicking rate, q CHF,s enhancement, W/m2 w,ch m3/s 2 qw wicking heat flux, W/m e evaporation flux, m/s average evaporation flux from radial coordinate r e,avg the meniscus, m/s characteristic evaporation flux, r correction factor for wetted area area e,ch m/s rs roughness ratio of the side wall w wicking flux, m/s R wicking front location, m w,ch characteristic wicking flux, m/s * superhydrophilicity metric R radius of curvature, m 3 = PC KH , Pa·m 2 * modified superhydrophilicity Ru universal gas constant, J/(mol·K) 3 metric = PKHCB , Pa·m Tboil liquid boiling point, K Subscripts Ts surface temperature, K ch characteristic quantities Tv vapor temperature, K e evaporation ur radial flow velocity, m/s w wicking 1. Introduction Boiling is a key industrial transport process central to the operation of power plants, nuclear reactors, electronic systems, and other applications. While nucleate boiling offers the potential for high heat dissipation efficiency by taking advantage of the latent heat of vaporization, there exists a critical heat flux (CHF) limit above which an insulating vapor layer covers the heated surface. Above CHF, the heat dissipation efficiency is reduced by orders of magnitude compared to nucleate boiling. Therefore, it is of vital importance to accurately predict CHF to inform the design of surfaces that extend this limit. Over a half-century ago, Zuber [1] derived a hydrodynamics-based model that accurately captures the pool boiling CHF on flat surfaces [2-4]. Recent experimental studies [2, 3, 5-18] have shown that using micro-/nano-structured surfaces can enhance CHF well above Zuber’s limit. A variety of mechanisms have been proposed to explain the structure-induced CHF enhancement, including increased three-phase contact line length [11, 16], microconvection around the bubble [19], increased departure frequency of coalesced bubbles [3], and liquid rewetting via capillary wicking [2, 3, 5, 7, 8, 11, 13, 14]. Among these mechanisms, capillary wicking has been the most widely agreed upon, with strong support from experimental data [2, 5, 10, 17]. A number of CHF models have been developed [2, 8, 10, 17, 20, 21] based on this wicking mechanism, sharing a common formulation of qCHF=+ q CHF,f q w , where qCHF,f represents the critical heat flux predicted for a flat surface (i.e., without the capillary wicking) and qw represents the amount of CHF enhancement induced by capillary wicking. While is usually calculated with established CHF models based on hydrodynamics [1], the balance of forces acting on bubbles [22], or gravity- driven rewetting [8, 10], there is no consensus on how the wicking-induced CHF enhancement qw should be predicted. 3 Table 1 summarizes the empirical, analytical, and semi-analytical models available to predict CHF on structured surfaces. All symbols are defined in the source references, except that a generic symbol ( fit ) is used to represent the fitting parameters in the empirical and semi-analytical models. Many wicking-based CHF models rely on empirical relations that are developed by correlating CHF with experimentally-measured wicking properties [2, 17, 20]. Ahn et al. [17] characterized the wicking capability of structured surfaces by measuring the volume flux of a droplet wicked into a structured surface, and assumed that CHF increased monotonically with the measured wicked volume flux. Their results demonstrated wicking to be a key mechanism for CHF enhancement on structured surfaces. Rahman et al. [2] adopted the same method as Ahn et al. [17] to characterize the wicking capability of a broader range of structured surfaces, including biotemplated nanostructures, square micropillar arrays of different diameter, height, and pitch, and hierarchical structures. Their experimental data strongly supported a linear relationship between the CHF and the measured wicked volume flux. Kim et al. [20] measured the wicking distance of droplets spreading on structured surfaces and fitted the wicking distance as a square-root function of time. A wicking factor was defined using the coefficient of the fitted relationship, and CHF was reported to scale linearly with the wicking factor. Despite good agreement with experimental data, these correlations strongly depend on the experimentally measured wicking properties, limiting their applicability to the design of surfaces for enhanced CHF. Recently, efforts have been made to develop analytical and semi-analytical models (including fitted parameters) to predict the wicking-induced CHF enhancement [8, 10, 21]. Dhillon et al. [10] derived an analytical model for CHF based on rewetting of a dry spot on the boiling surface and predicted an optimal structure spacing resulting from the competition between capillary pressure and viscous resistance. Kim et al. [8] derived a model for CHF on structured surfaces assuming that CHF was governed by liquid-supply capability from the gravitational and microscopic pressure potentials including capillary and disjoining pressures. Li and Huang [21] used Kandlikar’s model [22] for qCHF,f and derived the wicking-induced enhancement qw as a linear function of the wicking front velocity in the surface structures. Their model was semi-analytical, with the coefficient of the linear relationship fitted using experimental data. We note that existing wicking-based CHF models have focused on developing wicking parameters to characterize the wicking behavior of liquid on structured surfaces, followed by the formulation of a relationship between CHF enhancement and the wicking parameters developed. A variety of standalone 4 wicking parameters have been proposed, among which Xiao et al. [23] and Allred et al. [24] have defined representative parameters for linear wicking and radial wicking, respectively. Xiao et al. [23] reported that the volumetric rate of linear wicking follows the Washburn-type square-root law; a propagation coefficient was introduced for the square-root relationship. Allred et al. [24] analyzed the volumetric rate of radial wicking and introduced a superhydrophilicity metric defined as the product of capillary pressure, flow permeability, and the height of the structure to characterize radial wicking behavior.
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