Averaging approach for microchannel heat sinks subject to the uniform wall temperature condition

The present paper is devoted to modeling methods for thermal analysis of microchannel heat sinks. The averaging approach presented in earlier works for the case of constant surface heat flux is extended to the problems subject to the uniform wall temperature condition. The solutions for velocity and temperature distributions are obtained by solving one-dimensional averaged governing equations without resorting to a two-dimensional direct numerical simulation. General solutions for both high-aspect-ratio and low-aspect-ratio microchannel heat sinks are presented. Asymptotic solutions in high-aspect-ratio and low-aspect-ratio limits are also given in explicit form. The solutions presented in the paper are validated by comparing them with the results of direct numerical simulation. The friction factors, Nusselt numbers and thermal resistances for microchannel heat sinks with a uniform base temperature are obtained from the presented solutions. The effects of the aspect ratio and the porosity on the friction factor and the Nusselt number are presented. Finally, characteristics of the thermal resistance of the microchannel heat sink are discussed.     

Analytical Study on Forced Convection of Nanofluids With Viscous Dissipation in Microchannels

This article presents an analytical study on forced convection of laminar fully developed flow of incompressible, constant-property nanofluids in microchannels. Closed-form solutions for the temperature distributions in the radial direction with the incorporation of viscous dissipation are obtained under isoflux boundary condition. The effects of the governing parameters, including modified Brinkman number, thermal conductivity ratio, and nanoparticle volume fraction of the nanofluids, on the temperature distributions are investigated and analyzed for both heating and cooling processes. The heat transfer performance characterized by the Nusselt number is investigated based on the effects induced by these parameters. In the comparison between the models with and without viscous dissipation, it is found that the thermal performance of a microchannel is overrated when viscous dissipation is excluded in the analysis. It is concluded that these governing parameters are intimately interrelated in the flow and thermal analyses of nanofluids in microchannels. The interrelationship of the viscous dissipation effect and the nanoparticle volume fraction is examined in a contour deviation map of Nusselt numbers between the model with and without considering the viscous dissipation.     

Constructal design and thermal analysis of microchannel heat sinks with multistage bifurcations in single-phase liquid flow

Based on constructal theory, five different cases with multistage bifurcations are designed as well as one case without bifurcations, and the corresponding laminar fluid flow and thermal performance have been investigated numerically. All laminar fluid flow and heat transfer results are obtained using computation fluid dynamics, and a uniform wall heat flux thermal boundary condition is applied all heated surfaces. The inlet velocity ranges from 0.66 m/s to 1.6 m/s with the corresponding Reynolds number ranging from 230 to 560. The pressure, velocity, temperature distributions and averaged Nusselt number are presented. The overall thermal resistances versus inlet Reynolds number or pumping power are evaluated and compared for the six microchannel heat sinks. Numerical results show that the thermal performance of the microchannel heat sink with multistage bifurcation flow is better than that of the corresponding straight microchannel heat sink. The heat sink with a long bifurcation length in the first stage (Case 1A) is superior. The usage of multistage bifurcated plates in microchannel heat sink can reduce the overall thermal resistance and make the temperature of the heated surface more uniform (Case 3). It is suggested that proper design of the multistage bifurcations could be employed to improve the overall thermal performance of microchannel heat sinks and the maximum number of stages of bifurcations is recommended to be two. The study complements and extends previous works.     

The effect of creep flow on two-dimensional isoflux microchannels

Microchannel convective heat transfer and friction loss characteristics are numerically evaluated for gaseous, two-dimensional, steady state, laminar, constant wall heat flux flows. The effects of Knudsen number, accommodation coefficients, second-order slip boundary conditions, creep flow, and hydrodynamically/thermally developing flow are considered. These effects are compared through the Poiseuille number and the Nusselt number. Numerical values for the Poiseuille and Nusselt numbers are obtained using a continuum based three-dimensional, unsteady, compressible computational fluid dynamics algorithm that has been modified with slip boundary conditions. To verify the numerical results, analytic solutions of the hydrodynamically and thermally fully developed momentum and energy equations have been derived subject to both first- and second-order slip velocity and temperature jump boundary conditions. The resulting velocity and temperature profiles are then utilized to obtain the microchannel Poiseuille and Nusselt numbers as a function of Knudsen number, first- and second-order velocity slip and temperature jump coefficients, Brinkman number, and the ratio of the thermal creep velocity to the mean velocity. Excellent agreement between the numerical and analytical data is demonstrated. Second-order slip terms and creep velocity are shown to have significant effects on microchannel Poiseuille and Nusselt numbers within the slip flow regime.     

The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer

The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer rates, as given by the Nusselt number, is numerically evaluated subject to constant wall heat flux (H2) and constant wall temperature (T) thermal boundary conditions. Numerical results are obtained using a continuum based, three-dimensional, compressible, unsteady computational fluid dynamics algorithm with slip velocity and temperature jump boundary conditions applied to the momentum and energy equations, respectively. For the limiting case of parallel plate channels, analytic solutions for the thermally and hydrodynamically fully developed momentum and energy equations are derived, subject to both first- and second-order slip velocity and temperature jump boundary conditions, from which analytic Nusselt number solutions are then obtained. Excellent agreement between the analytical and numerical results verifies the accuracy of the numerical algorithm, which is then employed to obtain three-dimensional rectangular channel and thermally/hydrodynamically developing Nusselt numbers. Nusselt number data are presented as functions of Knudsen number, Brinkman number, Peclet number, momentum and thermal accommodation coefficients, and aspect ratio. Rarefaction and viscous dissipation effects are shown to significantly affect the convective heat transfer rate in the slip flow regime.     

Thermally developing flow and heat transfer in rectangular microchannels of different aspect ratios

Laminar convective heat transfer in the entrance region of microchannels of rectangular cross-section is investigated under circumferentially uniform wall temperature and axially uniform wall heat flux thermal boundary conditions. Three-dimensional numerical simulations were performed for laminar thermally developing flow in microchannels of different aspect ratios. Based on the temperature and heat flux distributions obtained, both the local and average Nusselt numbers are presented graphically as a function of the dimensionless axial distance and channel aspect ratio. Generalized correlations, useful for the design and optimization of microchannel heat sinks and other microfluidic devices, are proposed for predicting Nusselt numbers. The proposed correlations are compared with other conventional correlations and with available experimental data, and show very good agreement.     

Fully developed electro-osmotic heat transfer in microchannels

Thermally fully developed, electro-osmotically generated convective transport has been analyzed for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. Such a flow is established not by an imposed pressure gradient, but by a voltage potential gradient along the length of the tube. The result is a combination of unique electro-osmotic velocity profiles and volumetric heating in the fluid due to the imposed voltage gradient. The exact solution for the fully developed, dimensionless temperature profile and corresponding Nusselt number have been determined analytically for both geometries and both thermal boundary conditions. The fully developed temperature profiles and Nusselt number are found to depend on the relative duct radius (ratio of the Debye length to duct radius or plate gap half-width) and the magnitude of the dimensionless volumetric source.     

Effect of substrate thickness and material on heat transfer in microchannel heat sinks

Heat and fluid flow in microchannels of size (200μm × 200 μm, 5 cm long) of different substrate thicknesses (t = 100 μm–1000 μm) and different MEMS (Microelectromechanical Systems) materials (Polyimide, Silica Glass, Quartz, Steel, Silicon, Copper) was studied to observe the effects of thermal conductivity and substrate thickness on convective heat transfer in laminar internal flows.The results of the model were first validated by the theoretical results recommended by standard forced convection problem with H1 (Constant heat flux boundary condition) condition before the results from the actual microchannel configurations were obtained. Thereafter, general Nusselt number results were obtained from the models of many microchannel configurations based on the commercial package COMSOL MULTIPHYSICS^® 3.4 and were discussed on both local and average basis.A general Nusselt number correlation for fully developed laminar flow was developed as a function of two dimensionless parameters, namely Bi, Biot number and relative conductivity k¹, to take the conduction effects of the solid substrate on heat transfer into account. It was also demonstrated when the commonly used assumption of constant heat flux boundary (H1) condition is applicable in heat and fluid flow analysis in microfluidic systems. For this, a new dimensionless parameter was employed. A value of 1.651 for this suggested dimensionless parameter (Bi^0.04k^1?0.24) corresponds to 95% of the Nusselt number associated with the constant heat flux boundary condition so that it could be set as a boundary for the applicability of constant heat flux boundary (H1) condition in microfluidic systems involving heat transfer.     

Analysis of the effect of viscous dissipation for laminar flow in stadium-shaped ducts

The fully-developed laminar forced convection of a Newtonian fluid in a duct with stadium-shaped cross section has been analyzed. The effect of viscous dissipation has been taken into account. Three different thermal boundary conditions have been considered: (T) uniform wall temperature distribution; (H1) axially uniform wall heat flux distribution with peripherally uniform wall temperature distribution; (H2) axially and peripherally uniform wall heat flux distribution. The adiabatic-wall boundary condition has also been analyzed as a special case of the H2 boundary condition. The velocity and temperature distributions in the fluid, as well as the Fanning friction factor and the Nusselt number, have been evaluated numerically, by employing a Galerkin finite element method. As expected, the numerical evaluation of the dimensionless temperature distribution and of the Nusselt number reveals that increasing discrepancies between the H1 and H2 boundary conditions exist if the stadium-shaped duct is gradually flattened.     

Laminar convective heat transfer in a microchannel with internal longitudinal fins

A numerical study was conducted to investigate the fluid flow and heat transfer characteristics of a square microchannel with four longitudinal internal fins. Three-dimensional numerical simulations were performed on the microchannel with variable fin height ratio in the presence of a hydrodynamically developed, thermally developing laminar flow. Constant heat flux boundary conditions were assumed on the external walls of the square microchannel. Results of the average local Nusselt number distribution along the channel length were obtained as a function of the fin height ratio. The analysis was carried out for different fin heights and flow parameters. Interesting observations that provide more physical insight on this passive enhancement technique, and the existence of an optimum fin height are brought out in the present study.     

Developing Laminar Flow and Heat Transfer in Annular-Sector Ducts

For annular-sector ducts, steady, laminar, and constant-property forced-convection flow and heat transfer in the entrance region have been analyzed numerically using a general, marching procedure. Two types of thermal boundary conditions have been considered: (1) uniform temperature both axially and peripherally (T boundary condition); (2) uniform axial heat flux with uniform peripherally temperature at any cross section (H1 boundary condition). Numerical analysis has been conducted in the following range of parameters: D_i/D_o = 0.00, 0.25, 0.50, apex angle of the sector 2 alpha = 18 degrees, 20 degrees, 24 degrees, 30 degrees, 40 degrees, and Pr = 0.707. The solutions of the developing Nusselt number and friction factor are presented as functions of nondimensional axial distance. Comparisons are made between the computed results and the analytical or numerical results available in the literature. For all cases compared, satisfactory agreement is obtained.     

Heat transfer in rectangular microchannels

Convection heat transfer in a rectangular microchannel is investigated. The flow is assumed to be fully developed both thermally and hydrodynamically. The H2-type boundary condition, constant axial and peripheral heat flux, is applied at the walls of the channel. Since the velocity profile for a rectangular channel is not known under the slip flow conditions, the momentum equation is first solved for velocity. The resulting velocity profile is then substituted into the energy equation. The integral transform technique is applied twice, once for velocity and once for temperature. The results show a similar behavior to previous studies on circular microtubes. The values of the Nusselt number are given for varying aspect ratios.     

Forced convection with viscous dissipation using a two-equation model in a channel filled by a porous medium

The effects of viscous dissipation on the temperature profiles for a fully developed forced convection flow between two parallel plates with a constant heat flux boundary condition are studied. A two-equation model that includes viscous dissipation in the fluid phase is solved analytically and exact solutions for the temperature fields are obtained. Based on the solutions, the effects of several parameters on the transverse temperature profiles and Nusselt number are studied. The solution reducing to two respective limiting situations of slug flow and clear fluid flow agrees with the literature. A comparison with a one-equation model is also presented.     

Analysis of laminar heat transfer in micro-Poiseuille flow

The present study examines laminar forced convective heat transfer of a Newtonian fluid in a microchannel between two parallel plates analytically. The viscous dissipation effect, the velocity slip and the temperature jump at the wall are included in the analysis. Both hydrodynamically and thermally fully developed flow case is examined. Either the hot wall or the cold wall case is considered for the two different thermal boundary conditions, namely the constant heat flux (CHF) and the constant wall temperature (CWT). The interactive effects of the Brinkman number and the Knudsen number on the Nusselt numbers are analytically determined. Different definitions of the Brinkman number based on the definition of the dimensionless temperature are discussed. It is disclosed that for the cases studied here, singularities for the Brinkman number-dependence of the Nusselt number are observed and they are discussed in view of the energy balance.     

Viscous dissipation effects on thermal transport characteristics of combined pressure and electroosmotically driven flow in microchannels

This study investigates the influence of viscous dissipation on thermal transport characteristics of the fully developed combined pressure and electroosmotically driven flow in parallel plate microchannels subject to uniform wall heat flux. Closed form expressions are obtained for the transverse distributions of electrical potential, velocity and temperature and also for Nusselt number. From the results it is realized that the Brinkman number has a significant effect on Nusselt number. Generally speaking, to increase Brinkman number is to decrease Nusselt number. Although the magnitude of Joule heating can affect Brinkman number dependency of Nusselt number, however the general trend remains unchanged. Depending on the value of flow parameters, a singularity may occur in Nusselt number values even in the absence of viscous heating, especially at great values of dimensionless Joule heating term. For a given value of Brinkman number, as dimensionless Debye–Huckel parameter increases, the effect of viscous heating increases. In this condition, as dimensionless Debye–Huckel parameter goes to infinity, the Nusselt number approaches zero, regardless of the magnitude of Joule heating. Furthermore, it is realized that the effect of Brinkman number on Nusselt number for pressure opposed flow is more notable than purely electroosmotic flow, while the opposite is true for pressure assisted flow.     

Heat and mass transfer analysis of chemically reactive tangent hyperbolic fluid in a microchannel

This study addresses the fully developed magnetohydrodynamic flow of non-Newtonian fluid in a microchannel using tangent hyperbolic fluid model. The physical situation has been modeled by accessing boundary layer theory along with the physical aspects of thermophoresis and Brownian motion. The heat and mass transport phenomena are depicted through graphical interpretations. The modeled equations are nondimensionalized using dimensionless variables. The obtained corresponding equations are solved by employing Runge–Kutta–Fehlberg scheme accompanied with shooting technique. The fluctuations in distinct entities of physical connotations, like, the Nusselt number, friction factor and Sherwood number are explored in this examination. A notable reduction in the concentration field of the tangent hyperbolic fluid has been obtained for a larger chemical reaction parameter. The result shows that non-Newtonian fluids exhibit higher Nusselt number than Newtonian fluids. Furthermore, a significant enhancement in Nusselt number has been attained through a rise in the power-law index and thermophoresis aspect.     

Fluid flow and heat transfer characteristics in rectangular microchannels

Experiments were conducted to investigate flow and heat transfer characteristics of water in rectangular microchannels. All tests were performed with deionized water. The flow rate, the pressures, and temperatures at the inlet and outlet were measured. The friction factor, heat flux, and Nusselt number were obtained. The friction factor in the microchannel is lower than the conventional value. That is only 20% to 30% of the convectional value. The critical Reynolds number below which the flow remains laminar in the microchannel is also lower than the conventional value. The Nusselt number in the microchannel is quite different from the conventional value. The Nusselt number for the microchannel is lower than the conventional value when the flow rate is small. As the flow rate through the microchannel is increased, the Nusselt number significantly increases and exceeds the value of Nusselt number for the fully developed flow in the conventional channel. The micro‐scale effect was exhibited. The Nusselt number is also affected by the heat flux. The Nusselt number remains the constant value when the flow rate is small. The Nusselt number increases with the increase in the heat flux when the flow rate is large. © 2008 Wiley Periodicals, Inc. Heat Trans Asian Res, 37(4): 197–207, 2008; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20206     

Fully developed mixed convection and flow reversal in a vertical rectangular duct with uniform wall heat flux

Combined forced and free flow in a vertical rectangular duct is investigated for laminar and fully developed regime. The velocity field, the temperature field, the friction factor and the Nusselt number are evaluated analytically by employing finite Fourier transforms. The thermal boundary condition considered is an axially uniform wall heat flux and a peripherally uniform wall temperature, i.e. an H1 boundary condition. The necessary and sufficient condition for the onset of flow reversal is determined either in the case of upward flow in a cooled duct or in the case of downward flow in a heated duct. The special case of free convection, i.e. the case of a purely buoyancy-driven flow, is discussed. The occurrence of effects of pre-heating or pre-cooling in the fluid is analysed. It is pointed out that although these effects occur in rectangular ducts, they are not present either in circular ducts or in parallel-plate channels.     

Heat flux and slip effects on liquid flow in a microchannel

In this study, the liquid flow with the slip boundary condition in a microchannel between two parallel plates with imposed heat flux was numerically investigated. The combined effect of pressure-driven flow and electro-osmosis was taken into account. Electric potential, liquid flow and thermal characteristics were determined using the Poisson–Boltzmann, the modified Navier–Stokes and the energy equations for a hydrodynamical and thermal steady fully developed laminar flow for an incompressible liquid. The results demonstrate the influence of the slip coefficient, the heat flux and the pressure difference on flow velocity, local temperature and Nusselt number. A comparison of the developed model results with those in a previous study was made.     

Laminar forced convection slip-flow in a micro-annulus between two concentric cylinders

Forced convection heat transfer in hydrodynamically and thermally fully developed flows of viscous dissipating gases in annular microducts between two concentric micro cylinders is analyzed analytically. The viscous dissipation effect, the velocity slip and the temperature jump at the wall are taken into consideration. Two different cases of the thermal boundary conditions are considered: uniform heat flux at the outer wall and adiabatic inner wall (Case A) and uniform heat flux at the inner wall and adiabatic outer wall (Case B). Solutions for the velocity and temperature distributions and the Nusselt number are obtained for different values of the aspect ratio, the Knudsen number and the Brinkman number. The analytical results obtained are compared with those available in the literature and an excellent agreement is observed.