Heat transfer with laminar forced convection in a porous channel exposed to a thermal asymmetry

The effect of thermal asymmetry on laminar forced convection heat transfer in a plane porous channel with Darcy dissipation has been investigated numerically. The parallel plates making the channel boundaries were kept at constant, but different temperatures. The thermal asymmetry thus imposed on the system, results in an asymmetric temperature field and different heat fluxes across the channel boundaries. Depending on Darcy, Peclét and Reynolds number, the thermal asymmetry may lead to a reversal of the heat flux at a certain position along the flow at least at one of the channel walls. The corresponding Nusselt numbers become zero and might experience discontinuities thereby jumping from infinite positive to infinite negative, or vice versa. This feature is observed not only in the region of thermal development, but also in the fully developed region. In the fully developed region, an analytical expressions for the Nusselt numbers were obtained. From these expressions, analytical equations were deduced for the calculations of the axial positions along the channel where the Nusselt numbers become zero, or experiences discontinuity.     

Heat Transfer of Power-Law Fluids in Plane Couette–Poiseuille Flows with Viscous Dissipation

Abstract

Analytical expressions for the velocity and temperature profiles, bulk temperature and Nusselt numbers, in a fully-developed laminar Couette–Poiseuille flow between parallel plates of a power-law fluid with constant, and distinct, wall heat fluxes, in the presence of viscous dissipation are deduced and presented. Both favorable and adverse pressure gradient cases were analyzed. The walls’ shear stresses ratio, which arises naturally when the dimensionless hydrodynamic solution is obtained, together with the fluid power-law index Brinkman number and the walls’ heat fluxes ratio are the independent variables in the heat transfer solutions. With the exception of Newtonian fluids, there are in general two distinct analytical solutions, one for positive and another for negative values of the walls’ shear stresses ratio. The existence of singular points are also observed, where for a given value of the power-law index, there are values of the walls’ shear stresses ratio for which the Nusselt number becomes independent of the Brinkman number. It was also found that in a Couette–Poiseuille flow, for each value of the power-law index there exists a certain negative value of the walls’ shear stresses ratio that makes the Nusselt numbers at both walls identically zero.     

Natural convection heat and mass transfer of non-Newtonian power law fluids with yield stress in porous media from a vertical plate with variable wall heat and mass fluxes

This work studies the heat and mass transfer by natural convection from a vertical plate with variable wall heat and mass fluxes in a porous medium saturated with a non-Newtonian power law fluid with yield stress for the general case of power law variations in wall heat and mass fluxes. The governing equations are transformed into a dimensionless form by the similarity transformation and then solved by a cubic spline collocation method. Results are presented for velocity, temperature, and concentration profiles, as well as the Nusselt and Sherwood numbers for various parameters of the power law fluid with yield stress in porous media. The existence of threshold pressure gradient in the power law fluids tends to decrease the fluid velocity and the local Nusselt and Sherwood numbers. An increase in the power law exponent increases the local Nusselt and Sherwood numbers.     

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.     

Conjugated effect of joule heating and magneto-hydrodynamic on double-diffusive mixed convection in a horizontal channel with an open cavity

A finite element analysis is performed on the conjugated effect of joule heating and magneto-hydrodynamic on double-diffusive mixed convection in a horizontal channel with an open cavity. Homogeneous flows are imposed throughout the channel. Consistent high temperatures and concentrations are imposed at the bottom wall of the cavity. The other sides of the cavity along with the channel walls are considered as adiabatic. The effects of the various parameters (Richardson number, Hartmann number, joule heating, buoyancy ratio and Lewis number) on the contours of streamline, temperature, concentration and density have been depicted. Moreover, the average Nusselt and Sherwood numbers as well as bulk temperature is presented for the aforementioned parameters. The results show that the aforesaid parameters have noticeable effect on the flow pattern and heat and mass transfer.     

Heat Transfer of Bingham Fluids in an Annular Duct with Viscous Dissipation

ABSTRACT

Analytical expressions for the velocity and temperature profiles in a fully-developed laminar Poiseuille flow through a concentric annular duct of a Bingham fluid with constant wall heat flux at the inner and outer wall, in the presence of viscous dissipation are deduced and presented. It is found that the proportion of the heat generated by viscous dissipation near the outer wall increases with an increase of the dimensionless flow parameter, and a decrease of the duct radius ratio. The Nusselt numbers are first calculated based on a single bulk temperature for the entire duct cross section. The possibility of performing calculations of the relevant parameters discussed in this work is available via the Supplementary Material as an Excel file. Also in this work a new approach is employed, where two different bulk temperatures are used, one for each side of the radial location in the temperature profile whose derivative is zero. With this new approach the Nusselt number behavior is free of either unphysical discontinuities or negative values. As a consequence, the Nusselt number values better reflect the actual heat transfer coefficient at the walls and are more comparable with the heat transfer inside ducts when the temperature profile is symmetric.     

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.     

Micro-scale thermo-fluidic transport in two immiscible liquid layers subject to combined electroosmotic and pressure-driven transport

A theoretical analysis is presented for fully developed convective heat transfer in two immiscible fluid layers confined within parallel plate microchannels subject to combined effects of axial pressure gradients and imposed electrical fields. Assuming desperate zeta potentials at the interfaces thus formed, closed-form expressions are derived for the velocity and temperature distributions under fully developed conditions, with uniform wall heat flux boundary conditions. For the heat transfer analysis, the viscous dissipation effects are neglected as compared to the Joule heating effects. Results are subsequently obtained for different ranges of the ratios of various electrical properties of the two fluid layers and various relative strengths of the ratios of the electrical fields and the imposed pressure gradients. These results demonstrate the effects of the applied electric fields and pressure gradients, presence of external heat source or sink and interfacial positions on the temperature distributions in the two layers and the corresponding Nusselt numbers.     

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.     

Experimental Study on Fluid Flow and Heat Transfer from a Rectangular Prism Approaching the Wall of a Wind Tunnel

Experimental investigations in fluid flow and heat transfer have been carried out to study the effect of wall proximity due to flow separation around rectangular prisms. Experiments have been carried out for the Reynolds number 2.6 × 10⁴, blockage ratios are 0.1, 0.2, 0.3, and 0.4, aspect ratios (d/c) are 1.5, 1.33, 0.667, and 0.333, with different height‐ratios and various angles of attack. The static pressure distribution has been measured on all faces of the rectangular prisms. The results have been presented in the form of pressure coefficient, drag coefficient for various height‐ratios and blockage ratios. The pressure distribution shows positive values on the front face whereas on the rear face negative values of the pressure coefficient have been observed. The drag coefficient decreases with the increase in angle of attack as the height‐ratio decreases. The heat transfer experiments have been carried out under constant heat flux conditions. Heat transfer coefficients are determined from the measured wall temperature and ambient temperature and presented in the form of a Nusselt number. Both local and average Nusselt numbers have been presented for various height‐ratios. The variation of the local Nusselt number has been shown with nondimensional distance for different angles of attack and blockage ratios. The variation of the average Nusselt number has also been shown with different angles of attack for blockage ratios. The local as well as average Nusselt number decreases as the height‐ratio decreases for all nondimensional distances and angles of attack, respectively, for rectangular prisms. Empirical correlations for the average Nusselt number have been presented for a rectangular prism as a function of the Reynolds number, Prandtl number and relevant nondimensional parameters.     

Axial conduction in laminar pipe flows with nonlinear wall heat fluxes

A numerical analysis to determine the heat-transfer parameters of a fluid flow rejecting heat to the surrounding medium by convection and radiation is developed. The influence of axial conduction is included and the velocity profile is taken as nonuniform in the transverse direction. Use of a transformation eliminates the required boundary conditions at infinity. Approximate numerical techniques are employed to solve the nonlinear conjugate problem. As Péclet number increases, the temperature fields simplify to those where axial conduction is excluded. The computed results indicate that the effects of axial conduction are strongly altered by the parameters responsible for the convection and radiation. Bulk fluid temperatures, wall heat fluxes and Nusselt numbers are plotted against Graetz numbers. Critical Péclet numbers for a variety of cooling conditions are presented using the bulk fluid temperature as a reference.     

Natural convection heat transfer of water in a horizontal gap with downward-facing circular heated surface

An experimental study of natural convection heat transfer from a downward-facing horizontal circular heated surface in a water gap has been carried out. The results were correlated in different forms of Nusselt number vs Rayleigh number according to different independent variables. The effects of different characteristic length and temperature were discussed and the gap size is the preferred characteristic length, the average fluid temperature between bulk temperature and the saturated temperature is the preferred film temperature. For the estimation of the natural convection heat transfer under the present conditions, empirical correlations in which Nusselt number is expressed as a function of Rayleigh number, or Rayleigh and Prandtl numbers both, may be used. However, the best accuracy is provided by an empirical correlation which expresses the Nusselt number as a function of the Rayleigh and Prandtl numbers, as well as the gap width-to-heated surface diameter ratio, the dimensionless temperature. Artificial neural networks have been trained successfully for analyzing the influences of the gap width-to-heated surface diameter ratio and the wall temperature difference between the temperature of wall and ambient fluid on natural convection heat transfer based on the experimental data in the present study. The results show that the Nusselt number will increase by increasing the gap ratio and decrease by increasing the wall temperature difference.     

双分散多孔介质圆管通道中骨架发热对强迫对流传热的影响

多孔介质填塞是传热强化的有效方法之一。与单分散(普通)多孔介质相比,双分散多孔介质含有多孔骨架相和裂纹相,流体充满裂纹相(大孔隙)和多孔骨架相内小孔隙,因此具有更大的比表面积。基于双速度-双温度模型,分析了恒热流边界条件下双分散多孔介质圆管通道的强迫对流传热,并推导了两相的无量纲温度以及Nu的解析解。参数分析表明,多孔骨架发热会在通道壁-双分散多孔介质交界面上出现热流(温度梯度)分岔现象,并从数学和传热学角度阐释了其发生机理。在骨架吸热情形下,单分散和双分散多孔介质通道的Nu均呈现不连续特征,而对于双分散多孔介质通道,当有效导热系数比较大时,这种不连续特征仅出现在骨架生热情形。并且分析了Nu的渐近行为。     

Asymmetric wall heat fluxes boundary conditions in double-pass parallel-plate heat exchangers

Boundary conditions for asymmetric wall heat fluxes in double-pass parallel-plate laminar countercurrent operations are analyzed theoretically in this work by using an eigenfunction expansion in terms of power series for the homogeneous part and an asymptotic solution for the non-homogeneous part. Effects of variable ratio of heat fluxes on both sides and impermeable-sheet location are also studied. Quantitative and qualitative interpretations of theoretical predictions are utilized to investigate of heat-transfer efficiency enhancement of the double-pass model under consideration as compared to those in the single-pass operations without an impermeable sheet inserted. Results are presented in terms of Nusselt number and device performance improvement. The influence of the impermeable-sheet location on the heat-transfer efficiency enhancement as well as on the power consumption increment in double-pass operations has also been delineated.     

A numerical study of single-phase convective heat transfer in microtubes for slip flow

The steady-state convective heat transfer for laminar, two-dimensional, incompressible rarefied gas flow in the thermal entrance region of a tube under constant wall temperature, constant wall heat flux, and linear variation of wall temperature boundary conditions are investigated by the finite-volume finite difference scheme with slip flow and temperature jump conditions. Viscous heating is also included, and the solutions are compared with theoretical results where viscous heating has been neglected. For these three boundary conditions for a given Brinkman number, viscous effects are presented in the thermal entrance region along the channel. The effects of Knudsen and Brinkman numbers on Nusselt number are presented in graphical and tabular forms in the thermal entrance region and under fully developed conditions.     

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.     

An analytical solution for thermally fully developed combined pressure – electroosmotically driven flow in microchannels

An analytical solution is presented to study the heat transfer characteristics of the combined pressure – electroosmotically driven flow in planar microchannels. The physical model includes the Joule heating effect to predict the convective heat transfer coefficient in two dimensional microchannels. The velocity field, which is a function of external electrical field, electroosmotic mobility, fluid viscosity and the pressure gradient, is obtained by solving the hydrodynamically fully-developed laminar Navier–Stokes equations considering the electrokinetic body force for low wall zeta potentials. Then, assuming a thermally fully-developed flow, the temperature distribution and the Nusselt number is obtained for a constant wall heat flux boundary condition. The fully-developed temperature profile and the Nusselt number depend on velocity field, channel height, solid/liquid interface properties and the imposed wall heat flux. A parametric study is presented to evaluate the significance of various parameters and in each case, the maximum heat transfer rate is obtained.     

Recycle effect on heat transfer enhancement in double-pass heat exchangers under asymmetric isotherm conditions

The mathematical formulation of a heat transfer flow problem in the recycling parallel-plate heat exchanger under asymmetric wall temperatures was developed theoretically with ignoring axial conduction, and the analytical solution was obtained using a superposition principle and an orthogonal expansion technique in extended power series. The influences of the design parameters of the impermeable-sheet position (∆), and the operating parameters of the Graetz number (Gz), wall temperature ratio (σ) and recycle ratio (R) are examined. Significant heat transfer improvement is obtainable by employing double-pass devices instead of using single-pass ones for a larger Graetz number system. A technical feasibility of the new double-pass device was investigated in terms of the Nusselt number and device performance improvement under the effect of wall temperature ratios.     

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.     

Viscous dissipation effects on parallel plates with constant heat flux boundary conditions

This paper investigates basic analytical expressions for Nusselt number with the effect of viscous dissipation on the heat transfer between infinite fixed parallel plates, where the focus is on hydro-dynamically and thermally fully developed flow of a Newtonian fluid with constant properties, neglecting the axial heat conduction. Thermal boundary conditions considered are: both the plates kept at different constant heat fluxes, both the plates kept at equal constant heat fluxes, and one plate insulated. From the analysis, new expressions for Nusselt numbers have been found, as a function of various definitions of the Brinkman number.