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 evaluation of secondary effects on microchannel frictional and convective heat transfer characteristics

The frictional and convective heat transfer characteristics of rarified flows in rectangular microchannels, with either isoflux or isothermal boundary conditions, are evaluated subject to second-order slip boundary conditions, creep flow, viscous dissipation, and axial conduction effects. Numerical results are obtained using a continuum based, three-dimensional, compressible, unsteady computational fluid dynamics algorithm with first- and second-order slip velocity and temperature jump boundary conditions applied to the momentum and energy equations, respectively. The results, reported in the form of Poiseuille and Nusselt numbers, are found to be significant functions of aspect ratio, Knudsen number, slip model parameters, Brinkman number, and Peclet number.     

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.     

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.     

Heat and fluid flow characteristics of gases in micropipes

In this study, laminar forced convective heat transfer of a Newtonian fluid in a micropipe is analyzed by taking the viscous dissipation effect, the velocity slip and the temperature jump at the wall into account. Hydrodynamically and thermally fully developed flow case is examined. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). Either wall heating (the fluid is heated) case or wall cooling (the fluid is cooled) case is examined. The Nusselt numbers are analytically determined as a function of the Brinkman number and the Knudsen number. 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.     

Effects of viscous heating,fluid property variation,velocity slip,and temperature jump on convection through parallel plate and circular microchannels

Theoretical results, based on perturbation techniques, are presented for fully developed, both hydrodynamically and thermally, forced convection in both parallel plate and circular microchannels subject to isoflux wall boundary condition. Scaling effects, including variable property, viscous dissipation, velocity slip, and temperature jump, are studied for flow of both gases and liquids. The interactions between the aforementioned effects, due to temperature-dependence of the fluid properties that couples the momentum and thermal energy equation, are also examined in detail.     

Analysis of Gaseous Flow Between Parallel Plates by Second‐ Order Velocity Slip and Temperature Jump Boundary Conditions

In this study the momentum and energy equations are solved to analyze the flow between two parallel plates by employing second‐order velocity slip and temperature jump conditions. The flow is considered to be laminar, incompressible, hydrodynamically/thermally fully developed, and steady state. In addition to the isoflux condition, viscous dissipation is included in the analysis. Closed form expressions for the temperature field and Nusselt number are obtained as a function of the Knudsen number and Brinkman number. The Nusselt number obtained by employing the second‐order model is found to be lower compared to the continuum value and agrees well with the other theoretical models. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21116     

Effects of viscous dissipation on the heat transfer in a forced pipe flow. Part 2: Thermally developing flow

In this part of the study, consideration is given to thermally developing laminar forced convection in a pipe including viscous dissipation. The axial heat conduction in the fluid is neglected. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). Both the wall heating (the fluid is heated) case and the wall cooling (the fluid is cooled) case are considered. The distributions for the developing temperature and local Nusselt number in the entrance region are obtained. Results show that the temperature profiles and local Nusselt number are influenced by the Brinkman number (Br) and the thermal boundary condition used for the wall. Significant viscous dissipation effects have been observed for large Br.     

Extended Graetz problem including streamwise conduction and viscous dissipation in microchannel

Extended Graetz problem in microchannel is analyzed by using eigenfunction expansion to solve the energy equation. The hydrodynamically developed flow is assumed to enter the microchannel with uniform temperature or uniform heat flux boundary condition. The effects of velocity and temperature jump boundary condition on the microchannel wall, streamwise conduction and viscous dissipation are all included. From the temperature field obtained, the local Nusselt number distributions are shown as the dimensionless parameters (Peclet number, Knudsen number, Brinkman number) vary. The fully developed Nusselt number for each boundary condition is obtained also in terms of these parameters.     

Slip-flow and heat transfer of gaseous flows in the entrance of a wavy microchannel

The present work investigates the developing fluid flow and heat transfer through a wavy microchannel with numerical methods. Governing equations including continuity, momentum and energy with the velocity slip and temperature jump conditions at the solid walls are discretized using the finite-volume method and solved by SIMPLE algorithm in curvilinear coordinate. The effects of creep flow and viscous dissipation are assumed. The numerical results are obtained for various Knudsen numbers. The results show that Knudsen number has declining effect on both the C_f.Re and Nusselt number on the undeveloped fluid flow. Significant viscous dissipation effects have been observed for large Knudsen number. Also, viscous dissipation causes a singular point in Nusselt profiles.     

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.     

Convective heat transfer in pressure-driven nitrogen slip flows in long microchannels: The effects of pressure work and viscous dissipation

An analytical and numerical study is carried out to examine the convective heat transfer in two-dimensional pressure-driven nitrogen slip flows in long microchannels, whose length-to-height ratios are above 500. The momentum and the energy equations are solved, where variable properties, rarefaction that involves velocity slip, thermal creep and temperature jump, pressure work, and viscous dissipation are all taken into account. Nitrogen is assumed to be a perfect gas. The effects of pressure work and viscous dissipation, which are particularly significant for long microchannels, are examined by analyzing the uniform wall temperature and the uniform wall heat flux cases. It is found that the degree of rarefaction, which is characterized by the Knudsen number, is the key factor that determines the relative importance of pressure work and viscous dissipation. It is demonstrated that, for perfect gases, rarefaction promotes the conversion of internal energy to mechanical energy. Specifically, regardless of the fluid field development, pressure work and viscous dissipation cancel out in the absence of rarefaction, while pressure work is greater than viscous dissipation with rarefaction and its dominance increases as the Knudsen number increases. It is shown that the combination of pressure work and viscous dissipation makes a significant impact on the Nusselt number in both the continuum and the rarefaction cases. Therefore, it is concluded that for convective heat transfer in internal gas flows, both pressure work and viscous dissipation need to be considered in analysis.     

Scaling effects for flow in micro-channels: Variable property,viscous heating,velocity slip,and temperature jump

This paper reports closed form solutions, based on perturbation techniques, for fully developed, both hydrodynamically and thermally, slip-flow forced convection in both parallel plate and circular microchannels subject to isothermal wall boundary condition. Scaling effects, including variable property, viscous dissipation, velocity slip, and temperature jump are studied in detail. The results are not only applicable to gaseous flow in the slip-flow regime but also can be used for no-slip liquid flow in microchannels.     

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.     

Effect of viscous heating on heat transfer performance in microchannel slip flow region

Based on the superposition principle, an analytical solution for steady convective heat transfer in a two-dimensional microchannel in the slip flow region is obtained, including the effects of velocity slip and temperature jump at the wall, which are the main characteristics of flow in the slip flow region, and viscous heating effects in the calculations. The cases of constant heat flux boundary conditions and one wall as adiabatic and the other wall at constant heat flux input are studied. The solution method is verified for the cases where micro-scale effects are neglected. The effects of viscous heating on the temperature profiles and on the heat transfer performance are analyzed in detail. It is concluded that the effect of viscous heating, like an internal energy source, heats the fluid along the flow direction and severely distorts the temperature profiles. The effects of key parameters, such as the Brinkman and Knudsen numbers, on the Nusselt number, which expresses the heat transfer performance are investigated.     

Microtube gas flows with second-order slip flow and temperature jump boundary conditions

Second-order slip flow and temperature jump boundary conditions are applied to solve the momentum and energy equations in a microtube for an isoflux thermal boundary condition. The flow is assumed to be hydrodynamically fully developed, and the thermal field is either fully developed or developing from the tube entrance. In general, second-order boundary conditions assuming an effective mean free path model predict a lower slip velocity than a first-order model assuming a hard sphere mean free path model. Heat transfer effects associated with rarefied flow are reduced for the second-order model. The effect of the second-order terms is most significant at the upper limit of the slip regime. For airflow at standard conditions, the maximum second-order change to the Nusselt number is on the order of 15%. The second-order effect is also more significant in the entrance region of the tube. Nusselt numbers are found to increase relative to their no-slip values when temperature jump effects are small. In cases where slip and temperature jump effects are of the same order, or where temperature jump effects dominate, the Nusselt number decreases when compared to traditional no-slip conditions.     

Similarity solutions for flows and heat transfer in microchannels between two parallel plates

In this paper we give analytical similarity solutions of the Navier–Stokes equations coupled with energy equation of Newtonian fluid in a microchannel between two parallel plates taking into account the effects of viscous dissipation, the velocity slip and the temperature jump at the wall. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). We provide new similarity transformations for the governing equations and derive the expressions of Poiseuille number (Po) and Nusselt number (Nu). Then, the homotopy analysis method (HAM) is employed to solve the nonlinear differential equations with related boundary conditions. Both the dimensionless analytical expressions of velocity and temperature are obtained. The rarefaction effects on velocity distribution and flow friction are exhibited. The interactive effects of the Brinkman number (Br) and the Knudsen number (Kn) on Nu are analytically studied for both the CHF and CWT cases.     

Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation

Graetz problem inside the microtube is revisited considering rarefaction effect, viscous dissipation term and axial conduction in the fluid for uniform wall temperature boundary condition in the slip flow regime. The flow is assumed to be hydrodynamically fully developed, thermally developing, and the velocity profile is solved analytically. The temperature field is determined by the numerical solution of the energy equation. The rarefaction effect is imposed to the problem via velocity-slip and temperature jump boundary conditions. The local and fully developed Nu numbers are obtained in terms of dimensionless parameters; Pe, Kn, Br, κ. Fully developed Nu numbers and the thermal entrance length are found to increase by the presence of the finite axial conduction.     

Effects of viscous dissipation on forced convective heat transfer in a channel embedded in a power-law fluid saturated porous medium

The convective heat transfer analysis in a channel embedded in a power-law fluid saturated porous medium subject to uniform heat flux is presented and compared with a Newtonian fluid concerning the effects of viscous dissipation. Governing momentum and energy equations for non-Newtonian fluids which accounts for the viscous dissipation effects are solved numerically. The temperature profiles of the non-Newtonian fluids are found to relate closely to the velocity profiles. When viscous dissipation is taken account of, Nusselt numbers for non-Newtonian fluid are found to deviate more from Newtonian fluid with increasing Brinkman number for a certain range of the Darcy number.     

Effects of entrance region transport processes on free convection slip flow in vertical microchannels with isothermally heated walls

Natural convection gaseous slip flows in vertical microchannels with isothermal wall conditions are numerically investigated, in order to analyze the influence of the entrance (developing) region on the overall heat transfer characteristics. A long channel aspect ratio is considered, so as to achieve both hydrodynamically and thermally fully developed conditions at the channel exit. In other words, the flow-field within the microchannel consists of both developing and fully developed regimes. A wide range of Rayleigh number is covered, so that the cases of very short and relatively large entrance lengths can be analyzed in the same unified mathematical framework. With first order velocity and temperature jump conditions at the microchannel walls, local and average Nusselt number values are computed, by invoking the Navier Stokes equation and the energy conservation equation. It is recognized that the micro-scale effects, being associated with the velocity slip and temperature jump conditions, exhibit enhancements in the rate of heat transfer, as compared to the similar macro-scale geometries. The relative enhancements in the average Nusselt number become more prominent for higher values of Knudsen number, whereas this augmentation effect is found to be somewhat arrested at higher values of Rayleigh number. Contrasting features in the heat transfer rate predictions with and without the considerations of the entrance region effects are also carefully noted.