Limiting Nusselt numbers for viscous dissipative flow between parallel plates kept at unequal temperatures

The limiting Nusselt number value of 4 for laminar convection through parallel plates kept at unequal temperatures in the absence of viscous dissipation is independent of the degree of asymmetry. Similarly, when the parallel plates are kept at equal temperatures, the limiting Nusselt number value of 17.5 when viscous dissipation is included is independent of the Brinkman number, which characterizes the viscous dissipation. This paper examines the dependence of limiting Nusselt numbers on the Brinkman number and the degree of asymmetry when the parallel plates are kept at unequal temperatures and viscous dissipation is included.     

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.     

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.     

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.     

Fully-developed heat transfer in annuli with viscous dissipation

For Newtonian concentric annular flows analytical solutions are obtained under imposed asymmetric constant wall heat fluxes as well as under imposed asymmetric constant wall temperatures, taking into account viscous dissipation and for fluid dynamic and thermally fully-developed conditions. Results for the special case of the heat flux ratio for identical wall temperatures and the critical Brinkman numbers marking changes of sign in wall heat fluxes are also derived.Equations are presented for the Nusselt numbers at the inner and outer walls, bulk temperature and normalised temperature distribution as a function of all relevant non-dimensional numbers. Given the complexity of the derived equations, simpler exact expressions are presented for the Nusselt numbers for ease of use, with their coefficients given in tables as a function of the radius ratio.     

First and second law analysis of fully developed gaseous slip flow in trapezoidal silicon microchannels considering viscous dissipation effect

Fully developed gaseous slip flow in trapezoidal silicon microchannels is studied. Friction factor, Nusselt number and entropy generation in the microchannel is obtained, effect of rarefaction, aspect ratio and viscous dissipation is explored and, the range of Brinkman number in which viscous dissipation effect is important and cannot be neglected is specified. The continuum approach with the velocity slip and temperature jump condition at the solid walls is applied to develop the mathematical model of problem in the trapezoidal microchannel. Transformation of trapezoidal geometry to a square provided ease of application of finite difference method in solution of the mathematical model. The effect of viscous dissipation is quantified by Brinkman number. The calculated Brinkman number for common engineering applications even with limiting operational and geometric conditions is found less than 0.005. It is observed that viscous effect for applications with Brinkman number less than 0.005 can be neglected. The region in which viscous dissipation effect cannot be neglected is specified as Br > 0.005. Decreasing effect of rarefaction and increasing effect of Brinkman number on irreversibility due to all sources, excluded axial conduction, is established. The dominant source of irreversibility in total irreversibility is specified as a function of Brinkman number.     

Effect of viscous dissipation and radiation on natural convection in a porous medium embedded within vertical annulus

The effect of viscous dissipation and thermal radiation on natural convection in a porous medium embedded within a vertical annular cylinder is investigated. The inner surface of the cylinder is maintained at an isothermal temperature $T_w$ and the outer surface is maintained at ambient temperature $T_{\infty }$. The fluid is assumed to obey the Darcy law. Finite element method is used to solve the partial differential equations governing the fluid flow and heat transfer behavior. The study is focused to investigate the combined effect of viscous dissipation and radiation. Results are presented for different values of the viscous dissipation parameter, radiation parameter, radius ratio, aspect ratio and Rayleigh number. It is observed that the viscous dissipation parameter reduces the average Nusselt number at hot surface. However, the average Nusselt number increases at the cold surface due to increased viscous dissipation parameter.     

Slip-flow heat transfer in microtubes with axial conduction and viscous dissipation – An extended Graetz problem

This study is an extension of the Graetz problem to include the rarefaction effect, viscous dissipation term and axial conduction with constant-wall-heat-flux thermal boundary condition. The energy equation is solved analytically by using general eigenfunction expansion. The temperature distribution and the local Nusselt number are determined in terms of confluent hypergeometric functions. The effects of the rarefaction, axial conduction and viscous dissipation on the local Nusselt number are discussed in terms of dimensionless parameters such as the Knudsen number, Peclet number and Brinkman number.     

Thermally developing forced convection in a porous medium: parallel plate channel with walls at uniform temperature, with axial conduction and viscous dissipation effects

A modified Graetz methodology is applied to investigate the thermal development of forced convection in a parallel plate channel filled by a saturated porous medium, with walls held at uniform temperature, and with the effects of axial conduction and viscous dissipation included. The Brinkman model is employed. The analysis leads to expressions for the local Nusselt number, as a function of the dimensionless longitudinal coordinate and other parameters (Darcy number, Péclet number, Brinkman number).     

Forced convective heat transfer in a microtube including rarefaction,viscous dissipation and axial conduction effects

Subsonic gas convective heat transfer in a microtube with a constant cross-sectional area and uniform wall temperature is investigated both analytically and numerically. First, the effect of rarefaction on heat transfer characteristics, at a distance from the inlet where Nu becomes constant, is analytically investigated for two cases: (i) including and (ii) neglecting the viscous dissipation effect. An exact solution for Nu in fully developed flow is presented for the case without viscous dissipation, while a closed-form solution for the asymptotic Nu is also provided for the case with viscous dissipation. Next, a numerical model is employed to investigate the simultaneous effects of rarefaction, viscous dissipation, and axial conduction for developing hydrodynamic and temperature conditions. The Nusselt number is substantially affected by viscous dissipation, rarefaction and axial conduction.     

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.     

Mixed convection with viscous dissipation and pressure work in a lid-driven square enclosure

Buoyant laminar flow in a square lid-driven enclosure is analysed. The vertical sides are kept isothermal at different temperatures, while the horizontal sides are insulated. Assisting mixed convection flow due to uniform motion of the top side is considered. The governing balance equations are solved numerically by employing a Galerkin finite element method. The effects of viscous dissipation and pressure work are taken into account. In order to investigate the influence of these effects, the Nusselt number is evaluated with respect to the heat fluxes at both vertical sides, for different values of the Rayleigh number and of the Péclet number based on the lid velocity. Two sample fluids are considered: a gas and a highly viscous liquid. In the framework of the Oberbeck–Boussinesq approximation, a comparison is made between three different energy balance models: (A) enthalpy formulation (pressure work and viscous dissipation are included); (B) internal-energy formulation (viscous dissipation is included); (C) both pressure work and viscous dissipation are neglected. It is shown that, in the absence of a lid motion, the three models yield substantially the same predictions. On the other hand, when the forced flow induced by the lid motion becomes sufficiently large, the three models yield discrepant results, thus implying that pressure work and viscous dissipation are not negligible. Moreover, it is shown that, in this case, model (A) yields unphysical results, while model (B) leads to reasonable predictions.     

Analytical solution to the problem of heat transfer in an MHD flow inside a channel with prescribed sinusoidal wall heat flux

An MHD laminar flow through a two dimensional channel subjected to a uniform magnetic field and heated at the walls of the conduit over the whole length with a sinusoidal heat flux of vanishing mean value or not, is studied analytically. General expressions of the temperature distribution and of the local and mean Nusselt numbers are obtained by using the technique of linear operators in the case of negligible Joule and viscous dissipation and by taking into account the axial conduction effect. The principal results show that an increase of the local Nusselt number with Hartmann number is observed, and, far from the inlet section, the average heat transfer between the fluid and the walls shows a significant improvement at all values of Hartmann number used when the frequency of the prescribed sinusoidal wall heat flux is increasing in the case of vanishing mean value of the heat flux and this is true especially at low Peclet numbers.     

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.     

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.     

Solution of the Graetz-Brinkman problem with the Laplace transform Galerkin method

The present study concentrates on the effects of viscous dissipation in laminar forced convection. A power law fluid rheology model is applied and the effect of heat conduction in the axial direction is considered negligible. The physical properties are considered constant. Assuming fully developed velocity profile, the development of the temperature profile and its asymptotic behavior are investigated. For the solution of the problem the Laplace transform Galerkin technique is used. The method allows for the most general boundary conditions. A detailed comparison with previously published results provides a verification of the numerical technique. An important feature of the approach is that derivatives and integrals with respect to the axial location can be obtained through the operational rules of the Laplace transformation and hence no numerical derivation or integration is needed. As an application of the numerical model, we focus on the natural cooling regime, when the viscous dissipation of energy is counter-balanced by keeping the wall temperature at the ambient value. We derive a correlation for the asymptotic behavior of the Nusselt number in the natural cooling regime. This correlation reproduces the known value for the Newtonian case and provides a convenient means to normalize the Nusselt number for a wide range of flow behavior indices.     

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.     

Natural convection flow of Cu–Water nanofluid in horizontal cylindrical annuli with inner triangular cylinder using lattice Boltzmann method

In this paper the lattice Boltzmann method is used to investigate the effect of nanoparticles on natural convection heat transfer in two-dimensional horizontal annulus. The study consists of an annular-shape enclosure, which is created between a heated triangular inner cylinder and a circular outer cylinder. The inner and outer surface temperatures were set as hot (T_h) and cold temperatures (T_c), respectively and assumed to be isotherms. The effect of nanoparticle volume fraction to the enhancement of heat transfer was examined at different Rayleigh numbers. Furthermore, the effect of vertical, horizontal, and diagonal eccentricities at various locations is examined at Ra = 10⁴. The result is presented in the form of streamlines, isotherms, and local and average Nusselt number. Results show that the Nusselt number and the maximum stream functions increase by augmentation of solid volume fraction. Average Nusselt number increases when the inner cylinder moves downward, but it decreases, when the location of inner cylinder changes horizontally.     

微通道内液体流动和传热研究进展

随着尺度的微细化,微通道内液体的流动和传热出现了不同于常规尺度的现象。液体流动的Re、传热Nu和摩擦常数C等都出现了新的变化规律。许多在常规尺度下不重要的因素如黏性耗散、轴向热传导和表面浸润性等都开始变的突出。研究流体在微通道的流动和传热规律,具有重要的现实意义。对微通道内液体的流动和传热研究进行了总结,尤其是对微通道内液体的黏性耗散进行了详细的分析。     

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.