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Research Papers: Micro/Nanoscale Heat Transfer

# Modeling of Heat Transfer in Microchannel Gas Flow

[+] Author and Article Information
Tomasz Lewandowski

Institute of Fluid Flow Machinery, Polish Academy of Sciences, Fiszera 14, 80-952 Gdansk, Polandtomasz.lewandowski@imp.gda.pl

Tomasz Ochrymiuk

Institute of Fluid Flow Machinery, Polish Academy of Sciences, Fiszera 14, 80-952 Gdansk, Polandtomasz.ochrymiuk@imp.gda.pl

Justyna Czerwinska

Artorg Center, University of Bern, Stauffacherstrasse 78, CH-3014 Bern, Switzerlandjustyna.czerwinska@artorg.unibe.ch

J. Heat Transfer 133(2), 022401 (Nov 02, 2010) (15 pages) doi:10.1115/1.4002438 History: Received September 29, 2009; Revised July 30, 2010; Published November 02, 2010; Online November 02, 2010

## Abstract

Due to the existence of a velocity slip and temperature jump on the solid walls, the heat transfer in microchannels significantly differs from the one in the macroscale. In our research, we have focused on the pressure driven gas flows in a simple finite microchannel geometry, with an entrance and an outlet, for low Reynolds $(Re<200)$ and low Knudsen $(Kn<0.01)$ numbers. For such a regime, the slip induced phenomena are strongly connected with the viscous effects. As a result, heat transfer is also significantly altered. For the optimization of flow conditions, we have investigated various temperature gradient configurations, additionally changing Reynolds and Knudsen numbers. The entrance effects, slip flow, and temperature jump lead to complex relations between flow behavior and heat transfer. We have shown that slip effects are generally insignificant for flow behavior. However, two configuration setups (hot wall cold gas and cold wall hot gas) are affected by slip in distinguishably different ways. For the first one, which concerns turbomachinery, the mass flow rate can increase by about 1% in relation to the no-slip case, depending on the wall-gas temperature difference. Heat transfer is more significantly altered. The Nusselt number between slip and no-slip cases at the outlet of the microchannel is increased by about 10%.

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## Figures

Figure 1

Flow configuration geometry (L=1000 μm, H=300 μm). The pressure difference was applied between configuration inlet and outlet to drive the flow through the microchannel.

Figure 2

Pressure distribution for various Reynolds numbers and temperature difference. General characteristics of the pressure drop do not change significantly. Flow velocity was controlled by pressure difference between configuration inlet and outlet. Therefore, the values are a result of fixed boundary conditions. In all cases, Kn=0.008. (a) Pressure distribution for Re=0.55. (b) Pressure distribution for Re=120. (c) Pressure plots for Re=25 and two cases of hot gas cold wall and hot wall cold gas. Values are obtained along the microchannel: at the wall, at 1/4th of the channel width, and in the middle of the channel. Pressure values vary from the entrance of the channel up to 10% of its length.

Figure 3

Influence of the Knudsen and Reynolds numbers on the velocity profile in the microchannel. (a) Velocity at the inlet of the channel for three different Knudsen numbers and Re=25. The slip at the wall changes substantially, however, is still small in comparison to the average velocity. (b) Slip velocity along the wall for three different Reynolds numbers and Kn=0.008. The largest changes can be observed at the inlet of the channel up to 10% of channel length.

Figure 4

Velocity profiles at the inlet and outlet of the microchannel for various Reynolds numbers and the temperature differences. It can be noted that only the inlet is significantly influenced by the Reynolds number. The difference between gas and wall temperature changes the flow insignificantly. (a) Inlet velocity profiles. (b) Outlet velocity profiles.

Figure 5

Velocity in y-direction for Kn=0.008 and for the two Reynolds numbers. The changes across the channel in the relative velocity values are more pronounced for low Reynolds numbers. It is the best illustrated in normalized y-velocity profiles. (a) Velocity in y-direction for Re=0.55, vin,max=0.118 m/s, and vout,max=0.0114 m/s. (b) Velocity in y-direction for Re=120, vin,max=7.72 m/s, and vout,max=71.53 m/s. (c) Inlet y-velocity profiles for two Reynolds numbers. (d) Outlet y-velocity profiles for two Reynolds numbers.

Figure 6

Temperature distribution for Kn=0.008, two Reynolds numbers, and three different setups: no heating, cold gas hot wall, and hot gas cold wall. It is evident that the temperature field penetrates the microchannel much more extensively for a higher Reynolds number. (a) Tin=300 K, Tw=300 K, and Re=0.55. (b) Tin=300 K, Tw=450 K, and Re=0.55. (c) Tin=450 K, Tw=300 K, and Re=0.55. (d) Temperature distribution in the middle of the channel for Re=0.55. (e) Tin=300 K, Tw=300 K, and Re=120. (f) Tin=300 K, Tw=450 K, and Re=120. (g) Tin=450 K, Tw=300 K, and Re=120. (h) Temperature distribution in the middle of the channel for Re=120.

Figure 7

Temperature profiles for the inlet and outlet of the microchannel for two configurations: hot gas cold wall and cold gas hot wall. Reynolds number is Re=120. Knudsen number is Kn=0.008. (a) Inlet temperature across the channel width for various temperature differences (ΔT=Tin−Tw) for hot gas cold wall configuration. (b) Inlet temperature across the channel width for various temperature differences (ΔT=Tin−Tw) for cold gas hot wall configuration. (c) Outlet temperature across the channel width for hot gas cold wall configuration. (d) Outlet temperature across the channel width for cold gas hot wall configuration. It can be noted that for cold wall hot gas configuration and for a small temperature difference, the field does not penetrate the microchannel outlet the same way as for a large temperature difference.

Figure 8

Temperature field for Re=120 for hot gas cold wall (Tw=300 K) configuration and Kn=0.008. The inlet temperature is varied. Compared with Fig. 6, it can be noted that with small differences between the inlet and wall temperature, the inside of the microchannel gas is colder than the outside. For larger differences, the temperature field starts to penetrate the whole length of the channel and reaches the microchannel outlet. Simulations indicate that there is a temperature gradient for which the heat transfer between inlet and outlet regions can be blocked. (a) Tin=325 K. (b) Tin=350 K. (c) Tin=400 K. (d) Tin=450 K. (e) Temperature profile in the middle of the channel.

Figure 9

Temperature values for various simulation cases, Re=120 and Kn=0.008. The temperature is normalized by its average value. (a) Temperature jump at the wall along the microchannel. It is apparent that the largest changes are present in the vicinity of the entrance of the microchannel. (b) The temperature in the middle of the channel. Concerning hot wall cold gas, the temperature is larger than average temperature near the entrance. However, in the middle of the channel length, the same quantity is smaller than average.

Figure 10

Slip induced entrance effects. (a) The velocity slip along the microchannel wall for various temperature differences for Re=0.55 and Kn=0.008. (b) Velocity in the x-direction for cold gas hot wall (ΔT=−150 K). (c) Velocity in x-direction for hot gas cold wall (ΔT=150 K). The interaction between entrance geometry, velocity slip, and temperature jump creates changes in the x-direction of the velocity. Main stream velocity is still, however, sufficiently strong (large Reynolds number) to prevent flow separation.

Figure 11

Temperature difference between slip and no-slip simulations along the channel for Kn=0.008 and for two different Reynolds numbers: (a) Re=0.55 and (b) Re=120; for hot gas cold wall and cold gas hot wall configurations. The temperature is plotted on the wall, at 1/4th of the channel, and in the middle of the channel. For Re=0.55, the absolute value is taken. It can be noticed that for both cases temperature varies only slightly at the entrance of the microchannel. For the larger Reynolds number, the difference between the cold gas hot wall case and hot gas cold wall is much more pronounced and present all along the channel length.

Figure 12

Mass flow rate dependencies for different Reynolds and Knudsen numbers. (a) Mass flow rate as a function of Reynolds number for all calculated cases. Plotted line has equation: Q̇s=0.00001132Re kg/s+0.00001768 kg/s and depends on neither the temperature gradient nor the Knudsen number. (b) Mass flow rate as a function of the temperature difference between gas and wall (ΔT=Tin−Tw). It can be seen that for cold gas hot wall configuration, the mass flow rate is significantly affected by the temperature gradient. The opposite configuration remains unchanged. (c) Mass flow rate for Re=25 as a function of Knudsen number. The influence of the Knudsen number is insignificantly small. (d) Difference of the mass flow rate for slip and no-slip ((Q̇s−Q̇ns)/Q̇s) computations as a function of Reynolds number for three cases: no-temperature difference; hot gas cold wall and cold gas hot wall configurations. The latter one has the largest variation (triangles) in terms of applied temperature difference, which implies that slip effects are the most pronounced.

Figure 13

Nusselt number at the wall as a function of normalized channel length. (a) Nusselt number for hot wall cold gas and cold gas hot wall case. Additionally, the theoretical value for infinite channel is plotted (40). It can be seen that channel length has important role on Nusselt characteristics. (b) The difference between Nusselt at the wall for a slip and no-slip flow for two cases of Knudsen number. The rarefaction effects in Nusselt distribution are substantial especially for Kn=0.011.

Figure 14

Outlet velocity for Kn=0.008 and Re=120. (a) Outlet velocity for two accommodation coefficients. Line indicates solution with no-slip boundary conditions. Symbols refer to the solution with slip equations with two different accommodation coefficients σv=0.3 (circles) and σv=0.8 (squares). It can be noted that for considered cases, accommodation coefficient influences flow only near the walls and is the largest for σv=0.3. (b) Velocity profile for Kn=0.008 for three different grid sizes 48500, 27200, and 6800 elements. The solution does not demonstrate visible grid dependency for medium and fine meshes. Therefore, medium mesh was chosen for simulations.

Figure 15

Centerline (upper curve) and wall velocity (lower curve) in the function of Knudsen number for pressure driven flow in a channel; results of DSMC (34) compared with the Navier–Stokes slip flow approach (32); vertical line marks the boundary between slip and transition flow regime

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