Revisited analysis of gas convection and heat transfer in micro channels: Influence of viscous stress power at wall on Nusselt number
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, , 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 , 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 . 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]]:where 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, , and typical sizes and , 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, , and the Brinkman number, , 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 () 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 () and viscous dissipation () in the bulk flow and the order of
Nomenclature
- Brinkman number,
- specific heat at constant pressure and volume respectively []
- integral mean of the convective term on a channel slice
- d
- mean molecule diameter [m]
- hydraulic diameter, [m]
- integral mean of the diffusive term on a channel slice
- kinetic energy per mass unit, []
- Eckert number,
- body force vector []
- gravity acceleration vector []
- h
- enthalpy per mass unit [ ]
- h
- heat transfer coefficient [ ]
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