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Thermodynamic Analysis of Gas Flow and Heat Transfer in Microchannels

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Thermodynamic Analysis of Gas Flow and Heat Transfer in Microchannels International Journal of Heat and Mass Transfer 103 (2016) 773–782 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt Thermodynamic analysis of gas flow and heat transfer in microchannels ⇑ Yangyu Guo, Moran Wang Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics and CNMM, Tsinghua University, Beijing 100084, China article info abstract Article history: Thermodynamic analysis, especially the second-law analysis, has been applied in engineering design and Received 4 June 2016 optimization of microscale gas flow and heat transfer. However, following the traditional approaches may Accepted 25 July 2016 lead to decreased total entropy generation in some microscale systems. The present work reveals that the second-law analysis of microscale gas flow and heat transfer should include both the classical bulk entropy generation and the interfacial one which was usually missing in the previous studies. An increase Keywords: of total entropy generation will thus be obtained. Based on the kinetic theory of gases, the mathematical Entropy generation expression is provided for interfacial entropy generation, which shows proportional to the magnitude of Second-law analysis boundary velocity slip and temperature jump. Analyses of two classical cases demonstrate validity of the Microscale gas flow Microscale heat convection new formalism. For a high-Kn flow and heat transfer, the increase of interfacial transport irreversibility Thermodynamic optimization dominates. The present work may promote understanding of thermodynamics in microscale heat and fluid transport, and throw light on thermodynamic optimization of microscale processes and systems. Ó 2016 Elsevier Ltd. All rights reserved. 1. Introductions entropy generation decreased with the increase of Knudsen num- ber (Kn, defined as the ratio of mean free path to characteristic In recent years, with the rapid development of micro- and length) in microscale systems such as in the classical work [18,19]. nanofabrication and nanotechnology [1] and micro- However, the particular phenomena of velocity slip and tem- electromechanical systems (MEMS) [2], there are increasing inter- perature jump occur at the gas–solid interface in microscale gas ests and studies on microscale gas flow [3–7] and heat transport flow [4]. The non-continuous velocity and temperature profiles [8–11]. The thermodynamics at microscale has attached more come from the in-sufficient interactions between gas molecules and more attentions in theory, such as the non-equilibrium and solid walls. In other words, the gas-surface interaction near entropy [12–14], or in applications for optimization of thermal effi- the solid wall cannot reach a local equilibrium state as a funda- ciency of microsystems. The entropy generation minimization mental hypothesis in CIT [17]. Such a non-equilibrium effect, principle [15] or so-called the second-law analysis [16], originated non-doubtfully, will bring additional irreversibility and entropy in classical irreversible thermodynamics (CIT) [17], has been generation. Actually the entropy generation in rarefied gas systems extended from the conventional engineering field to microscale has been declared to consist of two parts [29]: one from the inter- systems. Hitherto the second-law analysis of microscale gas heat molecular interactions and the other from the gas-surface interac- convections in simple geometries (micro-channels, micro-pipes, tions, which are denoted as the bulk and the interfacial entropy micro-ducts, etc.) has been widely performed via either analytical generations, respectively hereafter. The study on interfacial approach [18–24] or numerical simulations [25–28]. All these entropy generation originated earlier in formulating fluid–solid works followed the traditional methodology: ‘‘computing the interfacial boundary conditions in the frame of non-equilibrium entropy generation after a resolution of velocity and temperature thermodynamics [30,31]. The boundary conditions were obtained field distributions” based on the entropy generation formula in as bilinear phenomenological flux-force relations from the non- terms of velocity and temperature gradients [15,16]. The size negative interfacial entropy generation restricted by the second effects at microscale were incorporated in obtaining the velocity law [30]. The kinetic theory foundations were also investigated and temperature distributions by solving the classical hydrody- rooted in linearized Boltzmann transport equation (BTE) [32–34], namic equations with velocity slip and temperature jump bound- where both the boundary conditions and the Onsager reciprocal ary conditions. A common conclusion was made that the total relations for kinetic coefficients [29,35,36] have been derived. These outstanding works laid a solid basis for interfacial boundary conditions from both thermodynamic and statistical physical per- ⇑ Corresponding author. spectives. But they mainly focus on the rarefied gas transport with E-mail address: [email protected] (M. Wang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.07.093 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved. 774 Y. Guo, M. Wang / International Journal of Heat and Mass Transfer 103 (2016) 773–782 Nomenclature c (cx, cy, cz) molecular velocity [m/s] Greek symbols u (ux) fluid velocity [m/s] k molecular mean free path [m] T thermodynamic temperature [K] l dynamic viscosity of fluid [kg/(mÁs)] x, y, r coordinate components [m] q mass density of fluid [kg/m3] Pr Prandtl number [–] c specific heat ratio [–] Kn Knudsen number [–] s molecular relaxation time [s] Br Brinkman number [–] h dimensionless temperature [–] cp specific heat capacity at constant pressure [J/(kgÁK)] g dimensionless value of r [–] m_ mass flow rate [kg/s] r entropy generation rate per unit volume [W/(KÁm3)] 2 À2 Ac cross-sectional area [m ] Uq dissipation function [s ] R radius of the micro-pipe [m] U, / deviational part of velocity distribution function [–] H half of the height of the micro-channel [m] a, b dummy index [–] k thermal conductivity of fluid [W/(mÁK)] P wetting perimeter [m] Subscripts Y dimensionless value of y [–] s variables of gas at the wall 2 q heat flux supply [W/m ] w variables of the wall A, B combinational parameters [–] m mean value of variables À3 À1 f molecular velocity distribution function [m (m/s) ] 1 parameters of micro-pipe 2 F external force on per unit mass of fluid [m/s ] 2 parameters of micro-channel kB Boltzmann constant [J/K] eq equilibrium state p thermodynamic pressure [Pa] b bulk region e specific internal energy [J/kg] i interface region s specific entropy or specularity parameter of Á wall [J/(K kg)] or [–] Acronyms s Á 2 J entropy flux [W/(K m )] CIT classical irreversible thermodynamics S entropy generation rate [W/K] gen BTE Boltzmann transport equation a finite Kn resulting from enhanced mean free path. Theoretical 2. Theoretical foundation of second-law analysis foundation of the entropy generation is seldom explored in micro- scale gas flow and heat transport, where the reduced system size The theoretical evaluation of thermodynamic irreversibility in contributes to the finite Kn. It will be shown that the interfacial microscale gas flow is intimately related to its hydrodynamic mod- entropy generation obtained in rarefied gas transport is not avail- eling, as is shown in Fig. 1. The microscale gas flow within the slip able directly to analysis of microscale gas flows due to a finite por- regime (0.001 6 Kn 6 0.1) is considered throughout the present tion of Knudsen layer (a thin layer with a thickness about one work, modeled by the Navier–Stokes equation (or Fourier’s law) mean free path near the wall) across the microchannel. On the with a velocity slip (or temperature jump) boundary condition other hand, the interfacial entropy generation has never been [4]. Although in principle the gas behaviors within the Knudsen incorporated into the second-law analysis of microscale gas heat layer should be described through a solution of BTE, the present convections in the previous work [18–28], to the authors’ best modeling has been proved to yield sufficiently accurate results knowledge. With the miniaturization of system and increase of within the slip regime [4,7]. Accordingly, the total entropy genera- Kn, the entropy generation at the interface may become compara- tion includes twofold shown in Fig. 1(b): the bulk part in the ble to or even dominant over that in the bulk region. The thermo- microchannel due to the fluid flow (or heat transport) and the dynamic performance of microscale system will thus be interfacial part at the wall induced by velocity slip (or temperature inadequately or erroneously evaluated when neglecting the inter- jump). The interfacial entropy generation has been calculated for facial irreversibility. It will be shown that the total entropy gener- rarefied gas transport based on kinetic theory of gases, as the dif- ation may increase with increasing Kn when both the bulk and ference between the entropy fluxes at the gas side and the solid interfacial entropy generations are counted. one [32–34]. The velocity distribution function of gases obtained Therefore, the present work aims at a systematic second-law by the Chapman–Enskog solution to BTE in the bulk region (outside analysis of microscale gas flow and heat transport with special the Knudsen layer) was used to evaluate the entropy flux at the gas attention on the role of interfacial entropy generation. Two classi- side. From the authors’ perspective, the obtained entropy genera- cal cases are taken for demonstration: heat convections in micro- tion is actually that within the Knudsen layer, as shown in Fig. 1 pipe [37] and in micro-channel [38]. The remainder of this article (c). As the Knudsen layer is modeled approximately by continuum is organized as: in Section 2, the mathematical expressions of bulk equation, the entropy generation within this layer has been and interfacial entropy generation are derived respectively in the accounted in the bulk part, as indicated in Fig. 1(b).
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