Revisited analysis of gas convection and heat transfer in micro channels: Influence of viscous stress power at wall on Nusselt number

https://doi.org/10.1016/j.ijthermalsci.2018.05.049Get rights and content

Highlights

  • Isothermally heated gas slipping flows in long plane micro-channels are considered.

  • Analytical solution of flow and heat transfer established from an asymptotic analysis.

  • Viscous stress power at the wall must be included in the total wall heat flux.

  • Wall Nusselt number tends to zero or to very small values for this flow type.

  • The very small values of the Nusselt number obtained in experiments are explained.

Abstract

This paper deals with the modeling of weakly rarefied and dilute gas flows in micro channels by the continuum approach, valid for Knudsen numbers smaller than about 0.1. It particularly focuses on the modeling of the associated heat transfer. The models proposed in the literature for the forced convection of gas flows in long micro channels between two infinite plates are more specifically discussed. The complete model for such flows is reminded after a compilation and a brief description of their possible applications in industries. The compatibility of the pressure work and viscous dissipation in the energy equation with the power of the viscous stress at the walls is discussed in detail. A dimensional analysis is proposed in the context of long micro channels. Analytical solutions for the velocity and temperature fields and for the Nusselt number are provided in the case of compressible micro-flows in isothermally heated flat plate channels, with pressure work and viscous dissipation included. The choice of an appropriate Nusselt number, including the power of the viscous stress at the wall, is particularly discussed. It is shown analytically and numerically, by solving the complete model for an isothermal wall micro-channel, that the Nusselt number tends to zero when the hydraulic diameter decreases, that is when the Reynolds number decreases and the Knudsen number increases. This could theoretically explain the very small values of the Nusselt number obtained in the experiments by Demsis et al. (2009, 2010).

Introduction

Due to the increasing development of Micro Electro Mechanical Systems (MEMS), the study of liquid or gas flows and heat transfer in ducts, heated or not, whose hydraulic diameter, Dh, is of the order of a few microns (say 1 to 100 μm), has given rise to a considerable amount of works over the past twenty years. A recent review by Kandlikar et al. [1] is dedicated to them. It is shown that monophasic liquid flows in micro channels have a behavior similar to that observed at the macroscopic scale and the classical continuum mechanics model can be used (Navier-Stokes equations with no slip boundary conditions).

However, for gas flows at the microscopic scale, specific phenomena are observed and require appropriate models [2,3]. A slightly rarefied flow regime close to the wall, generated by the interaction between the gas molecules and the wall atoms, must be taken into account at the microscopic scale whereas it is negligible at the macroscopic scale or for liquid micro-flows. More specifically, a Knudsen layer whose thickness is of the order of the mean free path of the gas molecules, λ, is formed closed to the wall. In this layer, the velocity magnitudes of the gas molecules considered individually are different at a fixed distance from the wall, due to their interactions with the wall. In other words, in this layer, the gas is in a state of local thermodynamic non-equilibrium which results in non-linear mean velocity profiles and relations between stress and strain rates. From the continuum mechanics point of view, at the micro channel scale, when the Knudsen number is such that 0.001<Kn=λ/Dh<0.1, these phenomena translate into a slip velocity and a temperature jump at the wall and, possibly, a gas flow driven by the tangential temperature gradient along the wall called “thermal creep” [4]. The consequences of these phenomena on the macroscopic quantities such as the mass flow rate, the friction factor, the bulk temperature and the wall heat flux can be significant [5] and must be taken into account in the modeling of the convective heat transfer in MEMS with gas flows. Indeed they may have antagonistic effects on the heat transfer.

Gas micro-flows, possibly with heat transfer, can be found in:

  • •micro heat exchangers for the cooling of electronic components or in chemistry [6,7],

  • •micro pumps and turbines, including the thermal transpiration-driven Knudsen pumps for vacuum pumping applications [[8], [9], [10]],

  • •micro-systems for the species separation in gas mixtures such as the method of gas separation by membranes [11],

  • •micro gas analyzers such as micro mass spectrometers and micro-chromatographs [12,13],

  • •supersonic gas flows in micro-nozzles to control the nano-satellite attitude or the boundary layers in aerodynamics [[14], [15], [16], [17], [18]],

  • •artificial lungs [19,20],

  • •pressure, flow rate and temperature micro-sensors in gas flows [21,22], etc.

This paper investigates the theoretical models available to simulate and analyze the slightly rarefied gas micro-flows with heat transfer, when Kn0.1. We focus on the modeling of forced convection of pure diluted gases in micro channels by a continuous approach based on the Navier-Stokes equations and first order slip and temperature jump boundary conditions. It appears that simplified models are often used in the literature for this flow type, but without relevant justification and with recurrent errors propagating from one paper to the other, particularly concerning the heat transfer analysis and the energy equation. Our purpose is to provide a consistent model for gas micro-flows and heat transfer and to compare with the vanishing values of the Nusselt number obtained in experiments [23,24].

In that aim, the characteristic length scales of the continuum description and the way of modeling the Knudsen layer are reminded in §2. The values of the slip and temperature jump coefficients are particularly discussed. The complete model for forced convection in heated micro channels is established and discussed in §3. A dimensional analysis is developed and the analytical solution of the temperature field given by a simplified asymptotic model for compressible gas convection in an isothermal micro-channel is established in §4. This solution is compared with the numerical solution of the full model obtained from finite volume simulations in §5. Furthermore, from the numerical simulations, the heat flux balances for slip and no slip flows and incompressible and compressible flows are analysed in details. The numerical method to solve the full model is presented in §5.1 and the analytical and numerical solutions are compared in §5.2. The heat flux balances and the very small values of the Nusselt number obtained in the experiments by Demsis et al. [23,24] are explained in §5.3.

Section snippets

Length scales of the continuum description and Knudsen layer modeling

The mean free path, λ, is the average distance traveled by the molecules between two successive collisions. It is the main scale to evaluate the rarefaction rate in a gas flow and the validity domain of the continuum description. In this paper, the most standard definition used for ideal gases is retained [2,5,[26], [27], [28]]:λ=μpπrT2=μρπ2rT=μπ2pρwhere r is the specific gas constant.

A scale analysis of the breakdown of the continuum description of gas flows was presented by Bird [25] and

Main physical phenomena and modeling issues

For flows in micro channels of large aspect ratio, L/Dh, and typical sizes 1μmDh10μm and 100μmL1mm, submitted to a moderate heating of the walls and to pressure variations between the inlet and the outlet of the channel of the order of 1 bar to a few bars, the conversion of the mechanical work of the viscous forces into internal energy is very important. It can also be shown that the Mach number, Ma=u¯/γrT, and the Brinkman number, Br=μu¯2/(kΔT), can reach or exceed the unit and the

Numerical method

An in-house finite volume code has been developed to solve the steady Navier-Stokes and energy equations with first-order slip boundary conditions (Eqs. (21), (22), (23), (24), (25), (26), (27), (28), (29), (30), (31), (32) in §3.5) on unstructured meshes. Details upon the discretization of the different terms can be found in Refs. [[64], [65], [66]]. Only a few points are reminded here. A second-order centered scheme is used for the diffusive and convective terms because the maximum Reynolds

Conclusion

The thermal aspects of the modeling of weakly rarefied gaseous flows (0.001Kn0.1) with first-order slip and temperature jump models have been discussed in details thanks to a dimensional analysis, an asymptotic analysis and numerical simulations. This model has been analyzed in the case of the forced convection of a cold gas flowing in long flat micro channels isothermally heated. The order of magnitude of the pressure work (PW) and viscous dissipation (VD) in the bulk flow and the order of

Nomenclature

Br
Brinkman number, Br=μu¯b2kΔTref
Cp,Cv
specific heat at constant pressure and volume respectively [J.kg1.K1]
CONV
integral mean of the convective term on a channel slice
d
mean molecule diameter [m]
Dh
hydraulic diameter, Dh=2H [m]
DIFF
integral mean of the diffusive term on a channel slice
ec
kinetic energy per mass unit, ec=v2/2 [J.kg1]
Ec
Eckert number, Ec=u¯b2CpΔTref
f
body force vector [N.m3]
g
gravity acceleration vector [m.s2]
h
enthalpy per mass unit [J.kg1 ]
h
heat transfer coefficient [W.m2.K1 ]

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