International Journal of Heat and Mass Transfer 103 (2016) 773–782

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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/(ms)] 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/(kgK)] g dimensionless value of r [–] m_ mass flow rate [kg/s] r entropy generation rate per unit volume [W/(Km3)] 2 2 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/(mK)] 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 3 1 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 wall [J/(K kg)] or [–] Acronyms s 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). Direct applica- frame of CIT and gas kinetic theory, as the theoretical foundation tion of previous interfacial entropy generation here will duplicate of second-law analysis. In Section 3, the analytical solutions and the entropy generation in the Knudsen layer. Thus in the present then the specific entropy generation formulations of heat convec- work, the interfacial entropy generation is derived as the difference tions in micro-pipe and micro-channel are presented. In Section 4, between entropy flux of gas at the wall and entropy flux in the we provide the results of second-law analysis of the two cases of solid wall. The former one is evaluated based on the velocity distri- þ microscale heat convection in Section 3, with the role of interfacial bution functions of incident gases (f ) and reflecting gases (f )at irreversibility to be illustrated and discussed. Concluding remarks the wall, which are related by the Maxwell gas-surface interaction are finally made in Section 5. model, as shown in Fig. 1(d). The obtained interfacial entropy Y. Guo, M. Wang / International Journal of Heat and Mass Transfer 103 (2016) 773–782 775

Fig. 1. Hydrodynamic modeling and thermodynamic irreversibility of microchannel gas flow in slip regime: (a) Modeling by Navier–Stokes equation with slip boundary s s condition, with also a comparison to the BTE solution in Knudsen layer (dashed line); (b) The total entropy generation including the bulk part rb and interfacial part ri ; (c) s s s Kinetic theory foundation of ri in previous work, as the difference between entropy flux of gases in the bulk region near the wall (Jg) and entropy flux in the solid wall (Jw), s ¼ s with Jg computed based on the Chapman–Enskog distribution function outside the Knudsen layer f g f CE; (d) Kinetic theory foundation of ri in the present work, as the s s s difference between entropy flux of gas at the wall (Js) and entropy flux in the solid wall (Jw), with Js computed based on the distribution functions of incident gas (f ) and þ reflecting gas (f ) which are related by the Maxwell gas-surface interaction model. generation in the present work is lower than that for rarefied gas ds q ¼r Js þ rs ; ð4Þ transport. The detailed derivation of bulk and interfacial entropy dt b generations is given below in the frame of CIT and kinetic theory gives rise to the expressions of entropy flux and entropy generation of gases separately. respectively:

2.1. Bulk entropy generation Js ¼ q=T; ð5Þ

The entropy generation in the bulk region is derived in the hi k l 2 l frame of CIT. The balance equations of mass and energy for heat rs ¼ ðrTÞ2 þ ru þðruÞT : ru ðr uÞ2: ð6Þ b 2 T 3 T and fluid flow are [39]: T For an incompressible flow (r u ¼ 0) considered in the present dq þ r ¼ q u 0 work, the bulk entropy generation Eq. (6) reduces to: dt ; ð1Þ de hi q ¼r q P : ru k l dt rs ¼ ðrTÞ2 þ ru þðruÞT : ru: ð7Þ b T2 T with e the specific internal energy of fluid, q and P the heat flux vec- tor and pressure tensor, governed by the Fourier’s law and Newton’s Eq. (7) evaluates the thermodynamic irreversibility induced by shear law respectively [39]: bulk heat conduction and viscous flow. q ¼krT hi 2.2. Interfacial entropy generation 2 ; ð2Þ P ¼ pI l ru þðruÞT þ lðr uÞI 3 The interfacial entropy generations induced by the velocity slip with I the unit tensor, the superscript ‘T’ in roman denoting the and temperature jump in microscale gas flow and heat transport transpose of a tensor, distinguished from the normal T in italics are derived based on the kinetic theory of gases. For mathematical denoting the temperature. Here the bulk viscosity of fluid has not simplicity and clear physical interpretation, the coupling effect is been considered. The fundamental relation in CIT is the Gibbs equa- assumed negligible between fluid flow and heat conduction, as to tion [17]: be separately discussed below in Sections 2.2.1 and 2.2.2, shown ds de p dq in Fig. 2(a) and (b). The assumption of negligible coupling between T ¼ : ð3Þ gas flow and heat conduction may be acceptable in the slip regime dt dt q2 dt where the non-equilibrium effect is moderate. Substitution of Eqs. (1) and (2) into Eq. (3), with an identifica- In the gas kinetic theory, the transport process is described by tion to the entropy balance equation [17]: BTE [40]: 776 Y. Guo, M. Wang / International Journal of Heat and Mass Transfer 103 (2016) 773–782

Fig. 2. Derivation of interfacial entropy generation in microscale gas flow and heat transport: (a) isothermal microscale gas flow with velocity slip; (b) microscale gas conduction with temperature jump. Steady states are considered for both cases.

@f @f @f 2.2.1. Interfacial entropy generation of velocity slip þ c þ F ¼ Cðf Þ; ð8Þ @t @x @c For the fully-developed micro-channel isothermal gas flow where f = f(x,c,t) is the velocity distribution function, with fdxdc shown in Fig. 2(a), the gas flow near the lower wall is taken to cal- denoting the probabilistic number of gas molecules within spatial culate the interfacial entropy generation. From Eq. (11), the distri- interval (x,x +dx) and molecular velocity interval (c,c +dc), and F bution function of gases near the wall reduces to: "# is the external force on per unit mass of gases. C(f) is the molecular 2 m @ux collision term, which is extremely complicated and usually f ¼ f 1 ðÞc u c l ; ð14Þ eq ð Þ2 x s y @y assumed the BGK relaxation approximation [41]. Thus Eq. (8) with- q kBT s out external force reduces to: where us is the gas slip speed at the wall and the local equilibrium @f @f f f eq distribution function becomes: þ c ¼ ; ð9Þ "# @t @x = 2 s 3 2 ðc u Þ þ c2 þ c2 ¼ q m x s y z : ð Þ with s denoting the molecular relaxation time, the equilibrium dis- f eq exp 15 m 2pkBT 2kBT=m tribution function feq being the local Maxwell–Boltzmann distribu- tion [40]: The main idea to derive the interfacial entropy generation has "# been proposed as the difference between the entropy fluxes at 3=2 ð Þ2 ¼ q m m ca ua ; ð Þ s s f eq exp 10 the gas side (Jg) and at the solid side (Jw) [29]: m 2pkBT 2kBT s ¼ s s ; ð Þ ri Jg Jw 16 where kB is the Boltzmann constant. Hereafter the Einstein’s sum- s ¼ = mation convention [39] is applied for elegance. Through a Chap- where Jw qn T, with qn the normal heat flux at the solid side and th man–Enskog expansion of f around feq within zero order, first assumed equal to the normal heat flux at the gas side [34]. For the order and higher order, different hydrodynamic equations are isothermal gas flow (qn = 0), Eq. (16) reduces to [29,44]: recovered respectively: Euler equation, Navier–Stokes equation Z s ¼ s ¼ : ð Þ and Burnett or super-Burnett equation [42]. In the present work, ri Jg kB cyf ln f dc 17 as the bulk fluid flow is described by the Navier–Stokes equations, the first-order expansion solution of Eq. (9) is adopted for further The distribution function of gases can be rewritten as analysis [43]: f ¼ f ð1 þ UÞ, with the global Maxwell–Boltzmann distribution 0 hi 2 3 3=2 2 2 @ q 2 m f eq @ua m f eq ub ¼ m mc ð Þð Þ ð Þð Þ f 0 exp . After some mathematic operations, ð Þ2 ca ua cb ub l @x ð Þ2 ca ua ca ua l @x m 2pkBT 2kB T 6 q kB T b 3q kBT b 7 f ¼ f 4 5: eq 2 3 the interfacial entropy generation Eq. (17) becomes [32–34]: m f eq ð Þ @T þ m f eq ð Þð Þð Þ @T Z ð Þ2 ca ua k @x ð Þ3 ca ua ca ua cb ub k @x q kB T a 5q kBT b 1 s ¼ U2 : ð Þ ð Þ r kB cyf dc 18 11 i 2 0 The entropy balance equation Eq. (4) can be derived from the In this work, according to the analysis at the beginning of Sec- BTE Eq. (9) by multiplying ‘ kBlnf’ on both sides and integrating tion 2, the interfacial entropy generation is instead derived from over the whole molecular velocity space [9]: s ¼ s s ¼ s s ri Js Jw Js with Js the entropy flux of gases at the wall: Z Z Z @ @ s kB f ln f dc þ kB cf ln f dc r ¼k c f ln f dc: ð19Þ @t @x i B y s s Z In Eq. (19), the distribution function of gases at the wall is: ¼kB ð1 þ ln f ÞCðf Þ dc: ð12Þ f ; for c < 0 ¼ y ; ð Þ f s þ 20 Therefore, the kinetic definitions of entropy density and entropy f ; for cy > 0 flux are obtained as: Z where the distribution function of reflecting gases f+ is related to qs ¼kB f ln f dc that of incident gases f through the gas-surface interaction model. Z ð13Þ The most common Maxwell model [44] is used here: the gas s molecules collide with the wall, and experience diffuse or specular J ¼kB ðc uÞf ln f dc: scatterings, the portion of them given by (1 s) and s respectively. Y. Guo, M. Wang / International Journal of Heat and Mass Transfer 103 (2016) 773–782 777

Thus the distribution function of reflecting gases is formulated as The deviation part (of distribution function) / in Eq. (24) rather [44]: than U in Eq. (18) is used in present work, since it makes the

þ analytical integration in Eq. (23) much simpler. The former is a per- ¼ ð ; Þþð Þ ð Þ: ð Þ f sf cx cy 1 s f 0 Tw 21 turbation from the local equilibrium distribution f eq (Eq.(15)) Substituting Eqs. (20) and (21) into Eq. (19), we get the full whereas the latter is a perturbation from the global equilibrium expression of interfacial entropy generation: distribution f 0. They are related through the identity Z Z Z f ¼ f ð1 þ /Þ¼f ð1 þ UÞ and a linearization of f around f 1 0 1 eq 0 eq 0 s ¼ ð þ UÞ ½ ð þ UÞ [33,34]: ri kB cyf 0 1 ln f 0 1 dcxdcydcz 1 1 1 Z Z Z ð Þ ð Þ 2 ð Þ 1 0 1 U ¼ dp 0 þ dT 0 mca 5 þ mcaua 0 þ ; ð Þ ð þ UÞ ½ ð þ UÞ : ð Þ / 27 cyf 0 1 s ln f 0 1 s dcxdcydcz 22 p T 2kBT 2 kBT 1 1 1 ð Þ ð Þ The integration in Eq. (22) is too complicated for a general case where the temperature and pressure jumps dT 0 , dp 0 are pertur- (arbitrary value of s). The fully diffuse wall (s = 0) is considered bations from the global equilibrium state at the gas–solid interface throughout the present work, such that Eq. (22) reduces to: and vanish for the present isothermal flow. In this way, Eq. (27) Z Z Z reduces to: 1 0 1 s ¼ mcxus ri kB cyf ln f dcxdcydcz U ¼ þ /: ð28Þ 1 1 1 Z Z Z kBT 1 0 1 : ð Þ Substitution of Eq. (28) into Eq. (18) produces the interfacial cyf 0 ln f 0 dcxdcydcz 23 1 1 1 entropy generation obtained in the previous work [33,34]: Similar to the analysis from Eq. (17) to Eq. (18), the following @ s l ux ¼ ð þ Þ r ¼ us : ð29Þ approximation is made (with f f eq 1 / ): i @ T y s

1 f ln f f ln f þð1 þ ln f Þf / þ f /2: ð24Þ The interfacial entropy generation Eq. (26) derived in the present eq eq eq eq 2 eq work is lower than the previous one in Eq. (29), although they have Substituting Eq. (24) into Eq. (23) and integrating over molecu- the same mathematical form. The reason has been elucidated in lar velocity space, we obtain the explicit expression of interfacial Fig. 1, where Eq. (29) signifies actually the entropy generation within entropy generation: the Knudsen layer while Eq. (26) is the entropy generation at the interface needed for second-law analysis. One should note that the l @u 2 Knudsen layer correction is sometimes taken into account for the rs ¼ k x ; ð25Þ i @ distribution function of gases near the wall [33], which will slightly pT y s improve the accuracy through a sacrifice of simplicity. Since an accu- where the molecular mean free path k hasq beenffiffiffiffiffiffiffiffi related to the rate analytical modeling of Knudsen layer remains still an open dynamic viscosity l through l ¼ 1 qck ¼ qk 2kBT [40]. With the question [7,42], the correction is not considered for the moment. 2 pm @ ¼ ux expression of slip speed in the Maxwell model: us k @y [4], s 2.2.2. Interfacial entropy generation of temperature jump Eq. (25) is slightly reformulated as: For the microscale gas heat conduction shown in Fig. 2 (b), the gas conduction near the wall is considered. From Eq. (11), the l @u s ¼ u x : ð26Þ distribution function of gases near the wall reduces to: ri s @ pT y s ()"# 2 2 2 k @T ðc þ c þ c Þ Eq. (26) implies that the interfacial entropy generation is pro- f ¼ f 1 þ c 1 x y z ; ð30Þ eq ð = Þ2 @y y 5k T =m portional to the gas slip speed (or velocity gradient) at the wall. q kBTs m s B s For gas flows in the continuum regime, the slip speed is negligibly with the equilibrium distribution function: small, with a nearly vanishing rs. It is thus reasonable to neglect i "# the entropy generation at the gas–solid interface and merely con- 3=2 c2 þ c2 þ c2 ¼ q m x y z : ð Þ sider that in the bulk region, as in the traditional second-law anal- f eq exp 31 m 2pkBTs 2kBTs=m ysis [15,16]. However, for gas flows in the slip regime where s appreciable slip speed exists, Eq. (26) gives a finite value of ri . In principle, the interfacial entropy generation of temperature The interfacial entropy generation is no longer negligible, but jump could be derived through similar procedures for that of may become comparable to or even dominant over that in the bulk velocity slip in Section 2.2.1. However, it is nontrivial to obtain the region, as to be demonstrated below. entropy flux in the non-equilibrium solid side near the interface

Fig. 3. Schematic of fully-developed microscale gas heat convections: (a) in circular micro-pipe; (b) in parallel micro-channel. For both cases, isoflux walls are considered. 778 Y. Guo, M. Wang / International Journal of Heat and Mass Transfer 103 (2016) 773–782

3.5 because of the distinction between kinetic theories of solids and Br=0.001 gases. Therefore the interfacial entropy generation is approximately 3.4 Br=0.005 estimated as a difference between entropy fluxes at the gas side and Br=0.01 at the solid side assuming the classical form (cf. Eq. (5)): 3.3 q q s ¼ s s ¼ n n : ð Þ ri Jw Js 32 3.2 Tw Ts A deeper microscopic exploration is still needed of the interfa-

s, b 3.1 N cial irreversibility of heat conduction across the gas–solid interface, 3 as a current important topic. To sum up, the expressions of interfacial entropy generations 2.9 induced by velocity slip and temperature jump are derived as Eqs. (26) and (32) respectively. Their dimension is slightly different 2.8 from the bulk entropy generation Eq. (7): the former denotes entropy generated in unit interfacial area while the latter denotes 2.7 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 the entropy generated in unit volume. The present Section 2 pro- Kn vides a theoretical ground for the second-law analysis of the two (a) cases of microscale heat convection introduced in next section.

1 3. Physical and mathematical models 0.9 Br=0.001 The microscale heat convection in parallel micro-channel and 0.8 Br=0.005 Br=0.01 circular micro-pipe in the near-continuum and slip regimes are 0.7 taken for a demonstration of second-law analysis in this section. The analytical solutions of the two classical cases shown in Fig. 3 0.6 are obtained as a first step, after which the specific formulations 0.5 s, iT of entropy generations are provided respectively. The following N assumptions are made [19,37,38]:(i) Steady-state incompressible 0.4 laminar gas flow; (ii) Both hydrodynamically and thermally fully- 0.3 developed; (iii) Neglected axial heat conduction; (iv) Constant properties. The analytical solutions of these two problems have 0.2 been indeed obtained in the classical work [37,38]. But slightly dif- 0.1 ferent trains of thought in the solution such as the used dimension- less parameters will lead to different expressions of the final result, 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 as shown in Ref. [19]. Therefore, the present work follows the gen- Kn eral lines in analytically solving the conventional heat convection (b) in any classical heat transfer textbook [45], with the microscale effect taken into account through the boundary conditions. 0.02 Microscale gas flow in the slip regime can be modeled by the Navier–Stokes equations supplemented with non-continuous 0.018 Br=0.001 boundary conditions [4]. The first-order velocity slip and tempera- 0.016 Br=0.005 ture jump boundary conditions at the fully-diffuse wall are respec- Br=0.01 tively [4]: 0.014 @ ¼ u ; ð Þ 0.012 us uw k @ 33 y s

s, iV 0.01

N 2c k @T T T ¼ ; ð34Þ 0.008 s w þ @ c 1 Pr y s 0.006 with c the specific heat ratio and Pr the Prandtl number. Zero wall

0.004 velocity (uw = 0) is considered. The rigorous condition for thermally fully-developed heat con- 0.002 vection is [45]: 0 @ T ðxÞTðy; xÞ 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 s ¼ 0; ð35Þ Kn @x TsðxÞTmðxÞ (c) with x denoting the axial coordinate of the channel or pipe, y (or r) the normal (or radial) coordinate. The cross-sectional mean temper- Fig. 4. Entropy generation number versus Kn for heat convection in micro-pipe: ature is defined based on the energy flow rate [19,45]: (a) bulk entropy generation number; (b) interfacial entropy generation number R induced by temperature jump; (c) interfacial entropy generation number induced qucpTdA by velocity slip. Pr = 0.7, c = 1.4, h = 1. Three different Brs are considered: ¼ Ac ; ð Þ w Tm _ 36 Br = 0.001 (black square-line), Br = 0.005 (blue circle-line), Br = 0.01(green mcp diamond-line). The dashed lines with symbols represent the bulk entropy _ ¼ generation number whereas the solid lines with symbols represent the interfacial where the mass flow rate of gas is m qumAc, with Ac and um being entropy generation numbers. (For interpretation of the references to color in this respectively the cross-sectional area and mass mean velocity. For an figure legend, the reader is referred to the web version of this article.) isoflux-wall case considered here, Eq. (35) reduces to [45]: Y. Guo, M. Wang / International Journal of Heat and Mass Transfer 103 (2016) 773–782 779

@T dT dT where the dimensionless temperature and coordinate are intro- ¼ s ¼ m : ð37Þ @x dx dx duced respectively as: h = kT/qR, g = r/R. The Brinkman number is defined as the ratio of viscous dissipation heat to the external heat ¼ 2 = 3.1. Heat convection in micro-pipe supply [37]: Br lum 2Rq. The dimensionless forms of symmetrical boundary conditions The velocity distribution is obtained through a solution of and temperature jump boundary conditions for Eq. (42) are: Navier–Stokes equation with the first-order slip boundary condi- dh tion Eq. (33) [4]: g ¼ 0; ¼ 0 dg : ð43Þ 1 þ 4Kn ðr=RÞ2 4c Kn u ¼ 2u ; ð38Þ g ¼ 1; h ¼ h ¼ h m 1 þ 8Kn s w c þ 1 Pr with the Knudsen number defined as Kn = k/2R. The temperature Solution of Eq. (42) with the boundary conditions Eq. (43) differential equation including the viscous dissipation heat, results in the analytical radial temperature distribution: becomes [45]: 4 Kn @T k @ @T ¼ c þ ð 2 Þþ ð 4Þ; ð Þ qc u ¼ r þ l U ; ð39Þ h hw þ B1 g 1 A1 1 g 44 p @x r @r @r q c 1 Pr where Uq is the dissipation function [39], and in the present case where the combinational parameters A1,B1 are defined as: reduces to: "# 16Brð1 þ 4KnÞ 1 @ 2 A 1 þ u 1 ð þ Þ2 ð þ Þ Uq ¼ : ð40Þ 1 8Kn 4 1 8Kn @r "# : ð45Þ 8Br 1 þ 4Kn The energy balance equation for a specific cross section of the B1 1 þ ð1 þ 8KnÞ2 1 þ 8Kn pipe is [19]: Z dT 1 It is trivial to validate that the present analytical solutions of m ¼ qP þ l U dA ; ð41Þ _ q velocity and temperature distributions are consistent with those dx mcp Ac in previous work [19,37]. Therefore, the velocity and temperature with the total mass flow rate of gas through the cross section fields are reliable for an accurate evaluation of entropy generation _ ¼ 2 m qumpR , the wetting perimeter P =2pR, and the differential in the next step. area element dA =2prdr. Substitution of Eqs. (37), (38), (40) and The source of entropy generation in microscale heat convection (41) into Eq. (39) gives rise to a dimensionless temperature differ- contains four parts: (i) viscous flow and (ii) heat conduction in the ential equation: bulk region; (iii) velocity slip and (iv) temperature jump at the "# gas–solid interface. Thus the total entropy generation is formu- d dh 1 þ 4Kn g2 8Br 32Brg3 g ¼ 4g 1 þ ; lated as: dg dg 1 þ 8Kn ð1 þ 8KnÞ2 ð1 þ 8KnÞ2 S ¼ S ; þ S ; þ S ; þ S ; ; ð46Þ ð42Þ gen gen H gen F gen V gen T

4.4 Br=0.001 previous result 4.2 Br=0.001 with interfacial irreversibility Br=0.005 previous result 4 Br=0.005 with interfacial irreversibility Br=0.01 previous result 3.8 Br=0.01 with interfacial irreversibility

3.6 s N 3.4

3.2

3

2.8

2.6 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Kn

Fig. 5. Total entropy generation number of heat convection in micro-pipe versus Kn. Pr = 0.7, c = 1.4, hw = 1. Three different Brs are considered: Br = 0.001 (black square-line), Br = 0.005 (blue circle-line), Br = 0.01(green diamond-line). The solid lines with symbols represent the present second-law analysis counting both the bulk and interfacial entropy generations, while the dashed lines with symbols represent the previous second-law analysis counting only the bulk entropy generation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 780 Y. Guo, M. Wang / International Journal of Heat and Mass Transfer 103 (2016) 773–782 with the subscripts ‘H’, ‘F’, ‘V’, ‘T’ represent ‘heat conduction’, ‘fluid 1.7 flow’, ‘velocity slip’ and ‘temperature jump’ respectively. Taking a volume element dV = Acdx across the pipe at (x, x +dx), each part 1.65 of the entropy generation is computed as below: Z Z Z Br=0.001 @ 2 Br=0.005 s k 2 k T 1.6 S ; ¼ r dV ¼ ðÞrT dV ¼ dx dA; ð47Þ Br=0.01 gen H H 2 2 @ V V T Ac T r Z Z Z 2 2 s, b 1.55 du du N ¼ s ¼ l ¼ l ; ð Þ Sgen;F rFdV dV dx dA 48 V V T dr Ac T dr Z Z 1.5 du du ¼ s R ¼ l R ¼ l ; ð Þ Sgen;V rVd us d Pdx us 49 R R pTs dr s pTs dr s 1.45 Z Z 1 1 1 1 ¼ s R ¼ R ¼ ; ð Þ Sgen;T rTd q d Pdxq 50 1.4 R R Ts Tw Ts Tw 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Kn R where = Pdx being the area of surface element along the pipe (a) wall. Substituting Eqs. (47)–(50) into Eq. (46), we obtain the total entropy generation in the volume element dV: 0.5 Z 2 Z 2 k @T l du l du 0.45 S ¼ dx dA þ dx dA þ Pdx u Br=0.001 gen 2 @r T dr pT s dr Ac TAc s s 0.4 Br=0.005 1 1 Br=0.01 þ Pdxq : ð51Þ 0.35 Ts Tw 0.3 A dimensionless entropy generation number is introduced as: 0.25 S S s, iT gen gen N Ns ¼ ¼ : ð52Þ k A dx pkdx 0.2 R2 c 0.15 Eq. (52) consists of two parts: Ns =Ns,b +Ns,i, with bulk part Ns,b representing the irreversibility induced by heat conduction and 0.1 fluid flow, interfacial part Ns,i the irreversibility induced by velocity 0.05 slip and temperature jump. When Ns,i is neglected, Eq. (52) reduces to the classical definition of entropy generation number in previ- 0 ous work [19].N is calculated through rectangular numerical 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 s,b Kn integration by putting the velocity distribution Eq. (38) and tem- perature distribution Eq. (44) into Eq. (51), whereas N is analyti- (b) s,i -3 x 10 cally obtained: 7 1 1 128Kn Br ; ¼ ; þ ; ¼ þ : ð Þ Ns i Ns iT Ns iV 2 2 53 6 hs hw ð1 þ 8KnÞ phs Br=0.001 Br=0.005 5 Br=0.01 3.2. Heat convection in micro-channel

4 The velocity distribution is obtained through a solution of s, iV

Navier–Stokes equation with the first-order slip boundary condi- N 3 tion Eq. (33) [4]:

3 1 þ 4Kn ðy=HÞ2 2 u ¼ u ; ð54Þ 2 m 1 þ 6Kn 1 with the Knudsen number defined as Kn = k/2H. The temperature differential equation in this case is [45]: 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 @T @2T du 2 Kn qc u ¼ k þ l : ð55Þ p @x @y2 dy (c) Through similar procedures in Section 3.1, the analytical solu- tion of dimensionless temperature distribution is obtained: Fig. 6. Entropy generation number versus Kn for heat convection in micro-channel: (a) bulk entropy generation number; (b) interfacial entropy generation number 4c Kn induced by temperature jump; (c) interfacial entropy generation number induced h ¼ h þ B ðY2 1ÞþA ð1 Y4Þ; ð56Þ w c þ 1 Pr 2 2 by velocity slip. Pr = 0.71, c = 1.4, hw = 1. Three different Brs are considered: Br = 0.001 (black square-line), Br = 0.005 (blue circle-line), Br = 0.01 (green dia- with the dimensionless temperature and coordinate respectively mond-line). The dashed lines with symbols represent the bulk entropy generation number whereas the solid lines with symbols represent the interfacial entropy introduced as h = kT/qH, Y = y/H, and Brinkman number: generation numbers. (For interpretation of the references to color in this figure ¼ 2 = Br lum 2Hq. The combinational parameters A2,B2 are defined as: legend, the reader is referred to the web version of this article.) Y. Guo, M. Wang / International Journal of Heat and Mass Transfer 103 (2016) 773–782 781

2.2 Br=0.001 previous result Br=0.001 with interfacial irreversibility 2.1 Br=0.005 previous result Br=0.005 with interfacial irreversibility 2 Br=0.01 previous result Br=0.01 with interfacial irreversibility 1.9

s 1.8 N

1.7

1.6

1.5

1.4 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Kn

Fig. 7. Total entropy generation number of heat convection in micro-channel versus Kn. Pr = 0.71, c = 1.4, hw = 1. Three different Brs are considered: Br = 0.001 (black square- line), Br = 0.005 (blue circle-line), Br = 0.01 (green diamond-line). The solid lines with symbols represent the present second-law analysis counting both the bulk and interfacial entropy generations, while the dashed lines with symbols represent the previous second-law analysis counting only the bulk entropy generation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

"# 18Brð1 þ 4KnÞ 1 number versus the Knudsen number is shown in Fig. 5. Three typ- A2 1 þ ð1 þ 6KnÞ2 8ð1 þ 6KnÞ ical Brinkman numbers Br = 0.001, 0.005, 0.01 are compared. "#: ð57Þ 3 6Br 1 þ 4Kn B2 1 þ 4.2. Heat convection in micro-channel 4 ð1 þ 6KnÞ2 1 þ 6Kn The bulk and interfacial, and total entropy generation numbers Again the velocity and temperature distribution solutions agree versus Knudsen number at three different Brinkman numbers are exactly with those in previous work [19,38], and are then used to shown in Figs. 6 and 7 respectively. calculate the entropy generation. Through similar procedures in The results in Figs. 4(a) and 6(a) show that the bulk entropy Section 3.1, the total entropy generation and the entropy genera- generation in microscale heat convection decreases with increas- tion numbers of heat convection in micro-channel are obtained. ing Kn, which may be explained by more flattened profiles and The total entropy generation in the volume element dV = Acdx with smaller gradients of velocity and temperature distributions. This unit width of channel (P = 2) is: trend is consistent with the commonly accepted knowledge in Z Z k @T 2 l du 2 l du much previous work [18–28]. However, with increasing Kn, the S ¼ dx dA þ dx dA þ Pdx u gen 2 @ s interfacial entropy generation increases due to temperature jump A T y A T dy pTs dy c c s and velocity slip, as shown in Figs. 4(b), (c) and 6(b), (c). The 1 1 þ Pdxq : ð58Þ discontinuous temperature and velocity profiles represent the Ts Tw non-equilibrium effect at the gas–solid interface. Thus larger The total entropy generation number is defined as: temperature jump and velocity slip at elevated Kn induce more ¼ Sgen ¼ Sgen ¼ þ irreversibility, resulting in more interfacial entropy generation. Ns k 2k Ns;b Ns;i. The bulk part Ns,b is also computed Ac dx dx H2 H The increase of interfacial entropy generation dominates over the through rectangular numerical integration by putting the velocity decrease of bulk entropy generation, giving rise to an increase of distribution Eq. (54) and temperature distribution Eq. (56) into total entropy generation when Kn increases. This is a totally Eq. (58). The interfacial part Ns,i is analytically obtained: different trend from previous classical result as compared in Figs. 5 and 7, and indicates the dominance of interfacial irre- 1 1 36Kn Br ; ¼ ; þ ; ¼ þ : ð Þ versibility in microscale heat convection. The evaluation of ther- Ns i Ns iT Ns iV 2 59 hs hw ð1 þ 6KnÞ phs modynamic performance of microscale system will be much distorted when the interfacial non-equilibrium effects are not taken into account. It is also seen that larger Br results in more bulk 4. Results and discussions entropy generation, and more interfacial entropy generation caused by velocity slip; in contrast, the interfacial entropy genera- 4.1. Heat convection in micro-pipe tion caused by temperature jump is independent of Br. Overall, the total entropy generation increases with Br. In terms of the separate The bulk entropy generation number, interfacial entropy gener- effect of temperature jump and velocity slip in the present cases, ation numbers are plotted versus Knudsen number in Fig. 4(a), (b) the former plays a main role and produces entropy generation and (c) respectively. The variation of total entropy generation about one order of magnitude larger than the latter. Nevertheless, 782 Y. Guo, M. Wang / International Journal of Heat and Mass Transfer 103 (2016) 773–782 the entropy generation caused by velocity slip will become larger [14] F. Vázquez, J. del Rio, Thermodynamic characterization of the diffusive at elevated Br as inferred from Eqs. (53) and (59). transport to wave propagation transition in heat conducting thin films, J. Appl. Phys. 112 (12) (2012) 123707. [15] A. Bejan, A study of entropy generation in fundamental convective heat 5. Conclusions transfer, J. Heat Transfer 101 (11) (1979) 718–725. [16] H. Herwig, The role of entropy generation in momentum and heat transfer, J. Heat Transfer 134 (3) (2012) 031003. The second-law analysis of microscale gas flow and heat trans- [17] S.R. De Groot, P. Mazur, Non-equilibrium Thermodynamics, Dover fer is investigated in a systematical way. The total entropy gener- Publications, New York, 1962. ation includes twofold: the bulk part from velocity and [18] M. Avci, O. Aydin, Second-law analysis of heat and fluid flow in microscale geometries, Int. J. Exergy 4 (3) (2007) 286–301. temperature gradients, and the interfacial part from velocity slip [19] K. Hooman, Entropy generation for microscale forced convection: Effects of and temperature jump. The bulk and interfacial entropy genera- different thermal boundary conditions, velocity slip, temperature jump tions are derived in the frame of classical irreversible thermody- viscous dissipation, and duct geometry, Int. Commun. Heat Mass Transfer 34 (8) (2007) 945–957. namic and kinetic theory of gases respectively. The former part [20] A. Arikoglu, I. Ozkol, G. Komurgoz, Effect of slip on entropy generation in a decreases with increasing Kn, as consistent with the conclusion single rotating disk in MHD flow, Appl. Energy 85 (12) (2008) 1225–1236. in previous work. However, the latter part usually ignored in pre- [21] M. Yari, Entropy generation analysis for Couette–Poiseuille flow through parallel-plates microchannel, Int. J. Exergy 6 (6) (2009) 809–825. vious second-law analysis, is proportional to the magnitude of [22] A. Sadeghi, M.H. Saidi, Second law analysis of slip flow forced convection velocity slip and temperature jump, and increases with increasing through a parallel plate microchannel, Nanoscale Microscale Thermophys. Kn. The increase of interfacial entropy generation may dominate Eng. 14 (4) (2010) 209–228. [23] G. Ibanez, A. Lopez, J. Pantoja, J. Moreira, J.A. Reyes, Optimum slip flow based over the decrease of bulk entropy generation, leading to an on the minimization of entropy generation in parallel plate microchannels, increase of total entropy generation at elevated Kn. Our theoretical Energy 50 (2013) 143–149. formalism is demonstrated by two classical cases of heat convec- [24] M. Shojaeian, A. Kosar, Convective heat transfer and entropy generation analysis on Newtonian and non-Newtonian fluid flows between parallel- tion in micro-pipe and micro-channel within slip regime. The plates under slip boundary conditions, Int. J. Heat Mass Transfer 70 (2014) results infer that neglecting the interfacial irreversibility in evalu- 664–673. ating the thermodynamic performance of microscale systems may [25] E. Ogedengbe, G. Naterer, M. Rosen, Slip-flow irreversibility of dissipative lead to a contrary decision. The present work mainly aims at clar- kinetic and internal energy exchange in microchannels, J. Micromech. Microeng. 16 (10) (2006) 2167–2176. ifying the physical nature and mathematical formulation of [26] K. Hooman, Heat transfer and entropy generation for forced convection entropy generation in microscale heat and fluid flow. More work through a microduct of rectangular cross-section: Effects of velocity slip, will be considered in the near future on more complicated situa- temperature jump, and duct geometry, Int. Commun. Heat Mass Transfer 35 (9) (2008) 1065–1068. tions, such as the isothermal boundary, axial heat conduction, [27] S. Chen, Z. Tian, Entropy generation analysis of thermal micro-Couette flows in other duct geometries and more realistic gas-surface interactions slip regime, Int. J. Therm. Sci. 49 (11) (2010) 2211–2221. beyond the fully diffuse walls. [28] L. Kuddusi, First and second law analysis of fully developed gaseous slip flow in trapezoidal silicon microchannels considering viscous dissipation effect, Int. J. Heat Mass Transfer 54 (1–3) (2011) 52–64. Acknowledgements [29] F. Sharipov, Onsager–Casimir reciprocity relations for open gaseous systems at arbitrary rarefaction: I. General theory for single gas, Phys. A: Stat. Mech. App. 203 (3) (1994) 437–456. This work is financially supported by the Key Basic Scientific [30] L. Waldmann, Non-equilibrium thermodynamics of boundary conditions, Z. Research Program (2013CB228301) and the Tsinghua University Naturforsch. A 22 (8) (1967) 1269–1280. Initiative Scientific Research Program. [31] D. Bedeaux, A. Albano, P. Mazur, Boundary conditions and non-equilibrium thermodynamics, Physica A 82 (3) (1976) 438–462. [32] H. Vestner, L. Waldmann, Boundary conditions for the distribution function References and for the moments of a gas at a moving wall, Z. Naturforsch. A 34 (3) (1979) 269–283. [1] H.S. Nalwa, Encyclopedia of Nanoscience and Nanotechnology, American [33] V. Roldughin, Non-equilibrium thermodynamics of boundary conditions for Scientific Publishers, California, 2004. slightly rarefied gases, J. Non-Equilib. Thermodyn. 16 (1) (1991) 13–26. [2] M. Gad-el-Hak, The MEMS Handbook: MEMS Introduction and Fundamentals, [34] V.M. Zhdanov, V.I. Roldugin, Non-equilibrium thermodynamics and kinetic Taylor & Francis, New York, 2006. theory of rarefied gases, Phys. Usp. 41 (4) (1998) 349–378. [3] M. Wang, Z. Li, Nonideal gas flow and heat transfer in micro- and nanochannels [35] F. Sharipov, Onsager–Casimir reciprocity relations for open gaseous systems at using the direct simulation Monte Carlo method, Phys. Rev. E 68 (4) (2003) arbitrary rarefraction: II. Application of the theory for single gas, Phys. A: Stat. 46704. Mech. App. 203 (3) (1994) 457–485. [4] G. Karniadakis, A. Beskok, N. Aluru, Microflows and Nanoflows: Fundamentals [36] F. Sharipov, Onsager–Casimir reciprocal relations based on the Boltzmann and Simulation, Springer, New York, 2005. equation and gas-surface interaction: Single gas, Phys. Rev. E 73 (2) (2006) [5] M. Wang, X.D. Lan, Z.X. Li, Analyses of gas flows in micro- and nanochannels, 026110. Int. J. Heat Mass Transfer 51 (13–14) (2008) 3630–3641. [37] O. Aydin, M. Avci, Heat and fluid flow characteristics of gases in micropipes, [6] G.A. Radtke, N.G. Hadjiconstantinou, Variance-reduced particle simulation of Int. J. Heat Mass Transfer 49 (9–10) (2006) 1723–1730. the Boltzmann transport equation in the relaxation-time approximation, Phys. [38] O. Aydin, M. Avci, Analysis of laminar heat transfer in micro-Poiseuille flow, Rev. E 79 (5) (2009) 056711. Int. J. Therm. Sci. 46 (1) (2007) 30–37. [7] F. Sharipov, Rarefied Gas Dynamics: Fundamentals for Research and Practice, [39] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Elsevier Pte Ltd., Singapore, 2004. Wiley-VCH, Weinheim, 2016. [40] S. Chapman, T.G. Cowling, The Mathematical Theory of Non-uniform Gases, [8] G. Chen, Nanoscale Energy Transport and Conversion: a Parallel Treatment of Cambridge University Press, Cambridge, 1953. Electrons, Molecules, Phonons, and Photons, Oxford University Press, New [41] P.L. Bhatnagar, E.P. Gross, M. Krook, A model for collision processes in gases. I. York, 2005. small amplitude in charged and neutral one-component systems, Phys. Rev. 94 [9] D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics, (3) (1954) 511–525. Springer, Heidelberg, 2010. [42] H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, [10] Y. Guo, M. Wang, Phonon hydrodynamics and its applications in nanoscale Springer-Verlag, Heidelberg, 2005. heat transport, Phys. Rep. 595 (2015) 1–44. [43] G.M. Kremer, An Introduction to the Boltzmann Equation and Transport [11] F. Sharipov, Heat transfer in the Knudsen layer, Phys. Rev. E 69 (6) (2004) Processes in Gases, Springer, Heidelberg, 2010. 061201. [44] C. Cercignani, The Boltzmann Equation and its Applications, Springer, New [12] V.A. Cimmelli, A. Sellitto, D. Jou, Nonequilibrium temperatures, heat waves, York, 1988. and nonlinear heat transport equations, Phys. Rev. B 81 (5) (2010) 054301. [45] T. Bergman, A. Lavine, F. Incorpera, D. DeWitt, Fundamentals of Heat and Mass [13] D. Jou, G. Lebon, M. Criado-Sancho, Variational principles for thermal transport Transfer, John Wiley & Sons Inc, Hoboken, 2011. in nanosystems with heat slip flow, Phys. Rev. E 82 (3) (2010) 031128.