Heat rate distribution in converging and diverging microchannel in presence of conjugate effect
Introduction
Flow in tapering passage is encountered in nozzle, diffuser, venturi-meter, rotameter and various other engineering devices. At the micro-scale, tapering passage is employed in valve-less micro-pump, micro-nozzles, micro-mixers, micro-devices for separating proteins, and reducing boiling flow instability making study of such flow passages relevant from both practical and fundamental viewpoints. For example, from a fundamental viewpoint, one struggles to define a characteristic length scale for such passages. Despite a large number of interesting applications, previous studies have mainly focused on flow and heat transfer aspect in straight passages, and comparatively less information exists on diverging/converging passages. Here we attempt to partially overcome this lacuna. In this study, the effect of variation in cross-sectional area and wall heat conduction on heat flux re-distribution is examined in diverging/converging passages through detailed three-dimensional numerical simulations.
Studies involving diverging and converging microchannels in applications are available in the literature. The applications include particle separation using electrophoretic separation technique by Xuan et al. [16]; stretching of DNA molecules using thermo-electrophoresis and convection by Hsieh et al. [9]. In both these studies, a converging–diverging microchannel was employed in series. The microchannels were heated in the latter study. In a numerical study, Yong and Teo [18] showed increase in mixing and heat transfer rate by employing converging–diverging microchannel. Further advantage of employing diverging microchannel is evident from the work of Fu and Pan [8]. These authors showed enhanced chemical reaction because of better diffusion mixing in a diverging microchannel. Kates and Ren [10] also employed a diverging microchannel. Their interest was in demonstrating isoelectric focusing application in presence of a temperature dependent pH gradient in a diverging microchannel. Studies have also reported reduction in instabilities during boiling in a diverging microchannel [1], [11]. Duryodhan et al. [6] employed diverging microchannel in an innovative manner. They demonstrated both experimentally and numerically that it is possible to achieve a constant wall temperature, over a large parameter range, using diverging microchannel. In another study, Duryodhan et al. [3] showed that the characteristic length scale for diverging microchannel can be taken as the hydraulic diameter at L/3 from the inlet (where L is the length of the microchannel); similarly, the characteristic length scale for converging microchannel can be taken as the hydraulic diameter at L/3.6 from the exit [4]. These locations were interestingly found to be independent of all governing parameters (microchannel angle, length, hydraulic diameter, flow rate). These applications exhibit the importance of studying heating/cooling in diverging and converging microchannels.
Heat transfer in microchannel can be severely affected by axial conduction in solid, in the direction opposite to the fluid flow. Similarly, axial conduction in fluid becomes substantial at low values of Reynolds number. Wall conduction number (M; defined below through Eq. (12)) and Peclet number (Pe) respectively govern the effect of conduction in solid and fluid. Typically, for commonly used fluids (such as water) the value of Peclet number is more than the critical value of 50; therefore, conduction through fluid can be neglected [15]. However, the wall conduction effect through solid is relevant at microscale because of the large thermal conductivity ratio and thickness ratio of substrate to fluid in microchannels. The wall conduction cannot be ignored for M > 0.01 [12]. Study of conjugate heat transfer in uniform cross section microchannel is available in the literature [13], [2], [7]. Studies of wall conduction in uniform cross section microchannel have reported distortion in the supplied boundary condition owing to conduction in the wall. Reduction in the average Nusselt number because of increased heat distribution at larger solid-to-fluid thermal conductivity ratios has also been identified [13]. In case of gas flow through microtube, wall conduction results in the under-prediction of Nusselt number [17]. Further, Moharana and Khandekar [14] suggested the presence of an optimum aspect ratio at which the wall conduction dominates and results in a minimum Nusselt number. This indicates that the aspect ratio is also an important parameter affecting the conjugate heat transfer. However, study of conjugate heat transfer in non-uniform/varying cross section microchannel has not received attention. In the case of diverging/converging microchannel, the surface area, perimeter, and aspect ratio (and hence the heat flux on the inner wall) varies in the flow direction; which makes the problem interesting and non-trivial. This wall conduction leads to a substantial redistribution of the heat flux which could be of much importance for design optimization of the above mentioned microdevices. Obtaining a better understanding of these issues provided the motivation for undertaking this work.
The objectives of the present work are three-fold:
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Understand the effect of wall conduction on heat rate distribution in diverging and converging microchannels.
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Examine the use of wall conduction number in non-uniform cross sectional passages.
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Identify combination of parameters, which leads to large temperature gradient and/or heat flux gradient in diverging/ converging microchannels.
Three-dimensional numerical simulations of liquid flow through heated diverging and converging microchannels are performed towards this end. Diverging and converging microchannels with different geometrical configurations: angle (θ = 1–8°), depth (H = 86–200 μm), length (10–30 mm), and solid-to-fluid thickness ratio (ts/tf = 1.5–4.0) have been employed in the simulations. Microchannels with varying solid-to-fluid thermal conductivity ratios (ks/kf = 27–646) are also studied. Simulations are carried out for mass flow rate ( = 3.3 × 10−5–8.3 × 10−5 kg/s) and heat flux (q″ = 2.4–9.8 W/cm2) conditions. Heat rate distribution for all the simulated cases are discussed. Comparative study of wall conduction in diverging and converging microchannels is carried out. Difference between the two approaches (i.e. heat flux distribution versus heat rate distribution) used for analyzing the conjugate effect has been explained.
Highlights from the present work are as follows:
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Wall conduction and area variation are responsible for re-distribution of heat flux in diverging/converging microchannel.
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Non-dimensional heat rate distribution proposed as a parameter to analyse wall conduction in varying cross section microchannel.
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Wall conduction reduces substantially with increase in divergence angle but depends only weakly on convergence angle.
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Wall conduction larger in converging microchannel as compared to diverging microchannel.
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The surface temperature gradient versus heat rate variation exhibits two different zones demarcating diverging and converging microchannels. Diverging microchannel exhibits lesser surface temperature gradient compared to converging microchannel.
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Standard deviation in heat rate distribution varies linearly with wall conduction number.
Section snippets
Methodology
The geometry of the microchannels considered in this study are shown in Table 1. The mesh for each microchannel is generated using Gambit software. The mesh element is chosen to be quadratic–hexahedral in both solid and fluid zones. The three-dimensional model of diverging microchannel employed in the simulation is shown in Fig. 1. The figure also shows the geometrical configurations and boundary conditions employed in the simulations.
Results
This section discusses the simulation results for the different cases (angle, depth, length, solid-to-fluid thickness and conductivity ratio) shown in Table 1. Simulations are performed for the range of mass flow rates ( = 3.3–8.3 × 10−5 kg/s) and heat flux (q″ = 2.4–9.8 W/cm2). Non-dimensional heat rate distribution (Eq. (10)) at the inner walls of diverging and converging microchannels are plotted along the streamwise direction and discussed to understand the effect of different geometrical,
Discussion
This section presents the summary of parametric study performed in Section 3. Surface temperature gradient (ΔT′s = (Tso − Tsi)/L), percentage of standard deviation in and the wall conduction number (M) are utilized to evaluate the effect of different geometrical, thermo-physical and flow parameters on wall conduction as shown in Table 3. Further, wall conduction number (M) is correlated with for each case of diverging and converging microchannels.
Table 3 shows that for M greater than critical
Comparison between heat rate and heat flux variation
Literature suggests that the study of conjugate heat transfer, typically involves the analysis of heat flux distribution. However, in present work variation of heat rate distribution along the streamwise direction has been analysed to understand the wall conduction. Intention was to isolate the effect of area variation from heat flux redistribution. This section explains the difference between two approaches i.e. heat flux and heat rate distribution. For this, heat flux distribution is
Conclusions
Study of conjugate heat transfer in diverging and converging microchannels has been presented in this paper. The energy equation has been solved in both the fluid and solid domains, with appropriate coupling between them. The variation in thermo-physical properties of fluid with temperature has also been accounted for in the simulations. The numerical solution is validated against experimental data. Three-dimensional numerical simulations are carried out on diverging and converging
Acknowledgments
We are grateful to Mr. Abhimanyu Singh for his help in carrying out numerical simulations.
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