NUMERICAL SIMULATION OF HEAT TRANSFER IN
RECTANGULAR MICROCHANNEL
Jun Yao*, Mayur K. Patel†, Yufeng Yao* and Peter J. Mason*
* Kingston University, Faculty of Engineering,
Roehampton Vale, Friars Avenue, London SW15 3DW, United Kingdom e-mail: k0429683@kingston.ac.uk
† University of Greenwich, School of Computing and Mathematical Sciences
30 Park Row, Greenwich, London SE10 9LS, United Kingdom
e-mail: m.k.patel@greenwich.ac.uk
Key words: Computational Fluid Dynamics, Heat Transfer, Microchannel Flow
Abstract.Numerical simulation has been conducted for the investigation of heat transfer in a rectangular, high–aspect ratio microchannel with heat sinks, similar to an experimental study. Water at ambient temperature is used as working fluid and the heating power of 180 W is introduced via electronic cartridges in the solids. Three channel heights of 0.3 mm, 0.6 mm and 1 mm are considered and the Reynolds number is in the range of 300 to 2360, based on the hydraulic diameter. The study is mainly focused on the Reynolds number and channel height effects on microchannel thermal performance. Validation study shows that Nusselt number variations agree with the theory and other predictions very well, but the numerical predicted Poiseuille number has shown some deficits compared to the theory, same as observed by other researchers. It is found that the scaling effect only appears at small channel height of 0.3 mm, for both the friction factors and the thermal resistance. While the predicted friction factor agrees reasonably well with an experimental-based correlation, the theoretical significantly under-predicts it. The thermal resistance tends to become smaller at small channel height, indicating that the heat transfer performance can be enhanced at small channel height.
1 INTRODUCTION
heat transfer parameter along the channel wall due to its three-dimensionality and micro-scale structures. As a consequence, numerical simulations have been increasingly adopted for the analysis of this type of flow problems (e.g. Gamrat et al. [5]).
resistance and substrate thickness. It was found that the thermal conducting resistance increases with increase in the thickness of substrate, and that both the convective and the conductive resistance are largely dependent on the channel geometry. Furthermore, the channel flow rate has a functional relationship with the capacitance resistance. Thus, the pressure loss could increase with an increase in the flow rate due to the nature of microchannels.
Following on from the experimental investigation of Gao et al.4 and the numerical study of Gamrat et al.5, this paper continues our recent study on numerical simulation of microchannel flow and conjugate heat transfer17, which couples fluid convection in a rectangular microchannel and heat conduction in the solids. Here we aim to extend the study to include the thermal characteristics. The main objectives of the present study are (1) to compare the numerical predictions with theoretical estimation and other numerical predictions and (2) to verify the Reynolds number and size effects on the deviation of heat transfer characteristics in terms of friction factor and thermal resistance from the classical corrections, which usually applies to macroscale flows.
2 NUMERICAL TECHNIQUE 2.1 Problem definition
Figure 1: Sketch of configuration with views of a plane at middle cross–section
0.1–1mm. Four electric cartridges are inserted inside two blocks in symmetry and are surrounded by insulting material. The length of the electric cartridges is slightly shorter than the microchannel length L. Comparing to curved entrance in Gao et al.’s experiment4, here we use rectangular entrances to achieve a uniform inlet velocity and minimise any possible distortions at the channel entrances. The same mass flow rate, at the two chamber inlets, is applied to ensure that the flow remains symmetric at the entrance of the channel. The heat losses from such an arrangement are mainly concentrated at the channel exit area as the heat sinks are approximately 25% shorter than the channel length.
2.2 Computational domain and boundary conditions
The computational domain has two components, fluid and solid parts. The fluid part consisting of two chambers, each having two inlets/outlets and are linked with the flow channel. The solid part, surrounding the channel and chambers, was embedded with four square shaped heat sinks. Based on this geometry, a multi-block structured mesh was generated and for each rectangular sub-domain, a “good quality” structured mesh was generated to achieve “good” numerical accuracy and efficiency in terms of CPU requirements. Across the domains of the fluid and solid parts, mesh lines were connected smoothly to ensure no numerical errors, due to interpolation, were introduced. A serious of grids was used to obtain grid–independent results.
Standard hydraulic boundary conditions of uniform inflow at the two chamber inlets were applied and the thermal boundary conditions for solving heat conduction equation were:
• The heat flux of electrical cartridge was kept constant at 180 W and the heat source was uniformly distributed over the fluid/solid interfaces;
• Water at ambient temperature (300K) and heat transfer coefficient of 10 Wm-2K-1; and
• Adiabatic wall conditions were prescribed for the remaining surfaces. 2.3 Numerical method and solution procedure
The commercial CFD package PHOENICS18 version 3.4 was used to carry out the numerical simulation which involves conjugate heat transfer. The governing equations for fluid flow were solved by the conventional finite volume methodology together with the pressure-velocity coupling technique, commonly referred to as the SIMPLE algorithm. The fluid (water) in the domain was assumed to be incompressible, having laminar flow characteristics with constant material properties. The buoyancy forces and radiation effects were neglected. Heat transfer, within the solid blocks was solved for by the heat conducting equation which provided the constraints for the flow boundary conditions.
that used by Gamrat et al.5 in their numerical study, where they found that the computed channel velocity profile was very similar for grid densities of 20, 30 and 40 the wall-normal direction. However, some differences did appear for mesh densities between 80 and 164 in the streamwise direction. Thus, the higher limit of 164 nodes was used in the present study. As the problem was “basically” two-dimensional, there were no significant influences from a spanwise resolution. A trial simulation of the laminar flow at a Reynolds number of 2166 has also been performed. The temperature distributions agree qualitatively with those reported by Gamrat et al.5 The predicted temperature has a value of 295.8K at the channel entrance, which is exactly the same value as that reported in [5], the temperature increases to around 301.7K at the channel exit, about 1.5% higher than that reported in [5].
3 RESULTS AND DISCUSSIONS
The numerical study by Gamrat et al.5 found that the characteristics of the flow and the heat transfer in the micro-channel had significant dependency on the Reynolds number, the Prandtl number and the geometry. Our previous study confirms these observations and in this paper, we extend further the study by concentrating on the issues of microchannel hydraulic performance, such that the relationship between the Reynolds number and Poiseuille number, and moreover, the friction factor and thermal resistance in the microchannel subject to the scaling and Reynolds number effects.
The hydraulic diameter (D ) length can be evaluated using the formulah
p A
h W
C
D = 4× , whereCA is the cross-sectional area and WP is the wetted perimeter, both measured at the
channel entrance. The range of Reynolds number considered varies between 300 to 2360, similar to that used in the experimental investigation of Gao et al.4 and the numerical study of Gamrat et al.5
3.1 Model validation against the theory
5 10 15 20 25 0 0.01 0.02 0.03 0.04 Dimensionless abscissa, x* N us se lt num be r, N u
Bejan and Sciubba (1992) Gamrat et al. (2005) Prediction (e=1mm) Prediction (e=0.3mm)
where φ is the heat flux. T is the temperature with subscripts ‘w’ stands for wall and ‘ f ’ for fluid. Parameterkis the local thermal conductivity. The dimensionless abscissa is defined as r e hR P D x x∗ = 1 .
The predicted local Nusselt number from the simulation can be evaluated via the heat transfer coefficient given by4
k hD Nu = h (2) where f w T T h −
= ϕ is heat transfer coefficient with
h wl P 2 = ϕ and h in out in f l x T T T T = +( − ) .
P is electric power in Watts. ‘w’ is the width of the channel, l is the heating resistant length, h subscripts ‘ in ’ represents the inlet and ‘ out ’ stands for the outlet.
Figure 2 presents comparisons of local Nusselt number variations along the axial direction of the channel for Re = 832, Pr = 0.7 and e = 1mm. It is evident that the present numerical model predictions are in excellent agreement with theory and predictions by Gamrat et al.5, for a uniform inlet profile.
Figure 2: Comparison of Nusselt number variations by theory and numerical predictions
10 30 50 70 0.002 0.006 0.01 0.014 0.018 Dimensionless coordinate, x+ P o is e u ille n u m b e r, P o
Predictions (e=1mm, Re=865) Predictions (e=0.3mm, Re=842) Shah and London (1978) Gamrat et al. (2005) e=1mm Gamrat et al. (2005) e=0.3mm
dimensionless coordinate. Mathematically, the Poiseuille number can be defined by the Fanning friction factor f and the Reynolds number as5
e o fR P = (3) where x D V p f h ave2 2×ρ ∆
= is related to the pressure drop over the hydrodynamically developing length x and diameter, Vaveis the averaged velocity in the channel.
Shah and London19 proposed an analytical formula based on laminar developing flows in two-dimensional channels as 2 5 ) ( 10 9 . 2 1 44 . 3 4 674 . 0 24 44 . 3 ) ( + − + + + + × + − + + = x x x x x Polam (4)
where the dimensionless coordinate is defined as
e hR D x x+ = .
Figure 3: Comparison of Poiseuille number variations by numerical predictions and the theory
Figure 3 gives presents a comparisons between the Poiseille number distributions and the dimensionless coordinate x+. Although similar trends are observed, the agreements are not as good as that in Figure 2 for the Nusselt number. The predictions by Gamrat et al.5 has a higher inaccuracy for smaller x+ than the current study results. Conversely, the current study results
0 0.1 0.2 0.3 0.4 0.5 0 500 1000 1500 2000 2500 Reynolds number, Re F ri c ti on f a c tor , f e=1mm e=0.6mm e=0.3mm White (1994) e=1mm White (1994) e=0.6mm White (1994) e=0.3mm Shen et al. (2005)
for these differences is not clear. It could be attributed to the boundary layer developed along the channel wall and also the entrance/exit effects, as discussed by Kandlikar and Grande22. Nevertheless, the present numerical predictions indicates that the there is a scaling effect from the entrance up until x+=0.015 after which the scaling effect is negligible. This finding agrees
that noted by Gamrat et al.5. 3.2 Friction factor
The laminar friction factor for fully developed flow in rectangular channels can be determined analytically, by a polynomial, which is a function of aspect ratio and the Reynolds number by24 Re / ) 9564 . 0 7012 . 1 9467 . 1 3553 . 1 1 ( 96 − α+ α2− α3+ α4 = the f (5) where 2 2 ) 1 / ( 1 ) / ( + + = e w e w α such that 0 ≤ α ≤ 1.
Shen et al.23 carried out an experimental study with a rectangular microchannel with rough walls and based on 140 sample measurement data, they proposed a correlation
4743 . 0 Re / 0922 . 4 = cor f (6)
Figure 4: Effects of channel heights on the variation of friction factor with the Reynolds number
Figure 4 presents the friction factor variations with the Reynolds number together with a
0 0.1 0.2 0.3 0.4 0.5 0 500 1000 1500 2000 2500 Reynolds number, Re T h er m al r esi st an ce, R t e=1mm e=0.6mm e=0.3mm
Fedorov and Viskanta (2000) Kawano et al. (1998)
predictions and that from theory, while the correlation of Shen et al.23 agrees well with the present predictions for larger channel heights (i.e. 1 mm and 0.6 mm). Numerical predictions show clear scaling effects, especially at smaller height (0.3 mm), while theoretical estimation totally neglected this effect. Kandlikar et al.8 also performed experiments to study the
relationship between wall roughness and friction factor using two capillary tubes of 1076 µm and 600 µm in diameters and higher roughness factors ranging from 0.00178 to 0.33. They concluded that channel wall roughness effect could be neglected for the larger tube diameters (>1 mm) where smooth wall condition can be used, whereas for small tube diameters (< 600 µm), the surface roughness increases with fraction factor.
3.3 Thermal resistance
Following the definition given by Shen et al.23, we evaluate the thermal resistance as
q T
Rt = ∆ m (7)
where the temperature deficit ∆Tm is calculated in the same way as that by Shen et al.23.
Figure 5 presents the thermal resistance variations against the Reynolds number. Although there are some differences between different channel heights, the differences are not as significant as for the other parameters (see Figures 2-4). Nevertheless, it is evident that the smaller channels do show smaller thermal resistance thus indicating better thermal performance. Due to the lack of published data available in the public domain, only numerical simulation of Fedorov and Viskanta14 and experimental measurement by Kawano et al.15, at lower Reynolds number are included, for qualitative comparison. Similar trend and magnitudes are evident.
4 CONCLUSIONS
We present a numerical study of heat transfer in a rectangular microchannel with heat sinks using a three-dimensional CFD model. Using similar geometry used by that reported by Gao et al.4 in their experimental study, our numerical study has successfully validated the test data.
Numerical predictions of Gramet et al.5 have also been verified. The current study has also included thermal characteristics, such as friction factor and thermal resistance.
In summary the following findings can be drawn based on the present study:
(1) The Nusselt number variation as a function of dimensionless abscissa agrees well with the theoretical estimation of Shah and London19. However, the Poiseuille number has shown clear difference compared to theory, similar to that reported by Gamrat et al.5 (2) The channel height has significant effect on the friction factor at smaller diameters
(i.e. 0.3 mm). Present numerical results compared reasonably well with the correlation proposed by Shen et al.23 which are based on experimental data, but theoretical
estimation under-predicts the values with no scaling effects considered.
(3) Channel heights have little influence on the thermal resistances compared to other parameters, but clearly the smaller the channel height, the smaller the thermal resistance. In the low Reynolds number regime, the study results show similar trends and magnitude range as those obtained by other researchers, both experimentally and numerically.
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