EFFECTS OF RAREFIED EFFECT, AXIAL CONDUCTION AND VISCOUS DISSIPATION ON CONVECTIVE HEAT TRANSFER IN 2D PARALLEL PLATE MICROCHANNEL OR NANOCHANNEL WITH WALLS AT UNIFORM TEMPERATURE

Experiments have shown that flow and heat transfer characters in microchannel or nanochannel are quite different from their well-known macroscale counterparts [1] because the effects of rarefied effect, axial conduction and viscous dissipation on heat transfer cannot be neglected. Significant efforts have been devoted to studying the separate effect of those three factors with non-slip boundary conditions. For example, Tso et al. [2-4] performed dimensional analyses and experiments to show the impact of Brinkman number on the microchannel flow. Their results stated that the Brinkman number has an important role in determining the flow transition point and the temperature distribution in spite of its relatively small values. A series of analytical solutions for heat transfer in one-dimensional microchannel when only axial conduction is considered has been extensively reported [5-12]. These may not exactly mimic convective heat transfer beacuse slip and joint effects of axial conduction, viscous dissipation and rarefied effects co-exist in the vicinity of the wall. In this paper, therefore, viscous dissipation and axial conduction terms are considered simultaneously in the process of deriving the analytical solution of the 2D energy equation with the first order velocity slip model and the temperature jump boundary conditions. We give the analytical expression of dimensionless 2D temperature profile via much simpler separation of variables and substitution approaches and apply it to examine the impacts of viscous dissipation, axial conduction and rarefied effect on heat transfer.


INTRODUCTION
Experiments have shown that flow and heat transfer characters in microchannel or nanochannel are quite different from their well-known macroscale counterparts [1] because the effects of rarefied effect, axial conduction and viscous dissipation on heat transfer cannot be neglected. Significant efforts have been devoted to studying the separate effect of those three factors with non-slip boundary conditions. For example, Tso et al. [2][3][4] performed dimensional analyses and experiments to show the impact of Brinkman number on the microchannel flow. Their results stated that the Brinkman number has an important role in determining the flow transition point and the temperature distribution in spite of its relatively small values. A series of analytical solutions for heat transfer in one-dimensional microchannel when only axial conduction is considered has been extensively reported [5][6][7][8][9][10][11][12].
These may not exactly mimic convective heat transfer beacuse slip and joint effects of axial conduction, viscous dissipation and rarefied effects co-exist in the vicinity of the wall. In this paper, therefore, viscous dissipation and axial conduction terms are considered simultaneously in the process of deriving the analytical solution of the 2D energy equation with the first order velocity slip model and the temperature jump boundary conditions. We give the analytical expression of dimensionless 2D temperature profile via much simpler separation of variables and substitution approaches and apply it to examine the impacts of viscous dissipation, axial conduction and rarefied effect on heat transfer.

Analytical solution of the 2D energy equation
Assuming that fluid property including density, specific heat, thermal conductivity and dynamic viscosity, are constants, the 2D energy equation for parallel plate microchannel or nanochannel including axial conduction and viscous dissipation, as well as boundary conditions can be established as where is the density, # is the specific heat at constant pressure, is the thermal conductivity, is the dynamic viscosity, & is the thermal accommodation coefficient, is the specific heat ratio, is the molecular free path, is the Prandtl number.
can be decomposed as an asymptotic temperature B and a transient term 1 . When ⟶ +∞, there is (3) and (4) can be rewritten as The asymptotic temperature B can be solved based on Eq. (5) and Eq. (6).
Eq. (7) can be transformed to the standard confluent hypergeometric equation [14], its solution is and g is B 1 . Therefore, the dimensionless temperature is Summation coefficients g are calculated via applying the Gram-Schmidt orthogonal approach in the literature [15].

Numerical simulation of forced convective heat transfer in a plane parallel channel
In order to validate the aforementioned analytical solution, a new extended heat transfer solver including viscous heat is developed based on the standard icoFOAM solver in OpenFOAM. The continuity, momentum and energy equations [16] for the incompressible fluid in the new solver is where is internal energy and is stress tensor. Because the linear Navier velocity slip and first-order temperature jump boundary conditions [17] are applied in the analytical solution of the 2D energy equation. Hence, these two slip boundary conditions shown as follows are also defined for the new solver in order to validate the correctness of the analytical solution. where <-is velocity of the fluid in the vicinity of the solid wall, ˜ is the velocity slip coefficient, is dynamic viscosity, is a unit diagonal matrix, is the normal vector of a boundary face of a mesh cell connecting with the solid wall. Subscripts and represent the central point of the boundary face and the mesh cell, respectively.

VALIDATION
In the test case, the length of the channel is 1.0 , and the half-height is 0.05 . The tangential momentum and thermal accommodation coefficient are set as = 1 and • = 1 , respectively. The thermal diffusivity is = 0.02 1 / , kinematic viscosity is = 0.014 1

RESULTS
Then the non-dimensional analytical method is used to examine the impacts of , and number on the temperature profile.   Fig. 2 (a), one can see that fluid asymptotic temperature is higher than constant wall temperature irrespective of whether 3g > <v˜˜ or 3g < <v˜˜. The asymptotic dimensionless bulk temperature of the fluid ~Ÿ at various viscous and rarefied conditions is shown in Fig. 2 (b) when considering axial conduction or not by setting = 10 or = 10 ± (the result is similar to one when = 10 and it is not shown here to save pages). ~Ÿ is a linear function of number. It roughly drops with increasing of and is independent on number. The critical length h , observed in Fig. 2(a), deserves further studies. The variation of h at various Br, Pe and Kn is illustrated in Fig. 3. Clearly, h reduces with larger Br and Pe and smaller Kn, corresponding to stronger viscous dissipation, weaker axial conduction and weaker rarified effect, respectively.

CONCLUSIONS
The analytical solution of the 2D energy equation with linear Navier velocity slip and first-order temperature jump boundary conditions for plane channel flow is performed, and summation coefficients in the analytical solution series are determined via using the Gram-Schmidt orthogonalization accompanied with Gauss-Legendre quadrature. The velocity and temperature profile obtained from the analytical solution approach for the case = 10, = −1, = 0.10 are in excellent agreement with numerical simulation results, which validates the accuracy of the analytical solution. The asymptotic fluid bulk temperature converges to a constant value higher than wall temperature if the viscosity dissipation is considered, and the asymptotic dimensionless bulk temperature of fluid increases linearly with number, drops with number and is roughly independent on number. If a fluid is used to cool the wall with uniform temperature, there is a critical point where the fluid bulk temperature reaches the same value as wall temperature and the Nusselt number becomes ill-defined at this point. This point moves much closer to the channel entrance at increasing | | or . On the contrary, the point moves towards the end of the channel with increasing .