Free convection of composite porous medium in a vertical channel

A fully developed free convection flow of immiscible fluids in a vertical channel filled with a porous medium is analyzed in the presence of source/sink. The flow is modeled using the Darcy–Brinkman–Forchheimer equation model. The viscous and Darcy dissipation terms are included in the energy equation. The channel walls are maintained at two different constant temperatures. The transport properties of both fluids are assumed to be constant. Continuous conditions for velocity, temperature, shear stress, and heat flux of both fluids at the interface are employed. The resulting coupled nonlinear equations are solved analytically using regular perturbation method and numerically using finite difference method. The velocity and temperature profiles are obtained in terms of porous parameter, Grashof number, viscosity ratio, width ratio, conductivity ratio, and heat generation or heat absorption coefficient. It is found that the presence of porous matrix and heat absorption reduces the flow field. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20340     

Entropy Analysis of a Coupled‐Convection Flow through a Vertical Channel Partially Filled by a Porous Medium with Injection/Suction and Slip Boundary Conditions

Flow fields, thermal fields, and entropy generation have been investigated for fully developed mixed convection flow between two vertical porous plates. The vertical channel is partially filled by a porous medium, and channel walls are subjected to a constant injection velocity at the left wall and constant suction velocity at the right wall. The viscous dissipation effects and velocity slip for the longitudinal component of the velocity at the channel walls are also taken into account. The momentum and energy equations for the mixed convection problem in the vertical channel are solved by means of the perturbation series method, by taking perturbation parameter proportional to the Brinkman number. For the present problem, numerical solution is also obtained and compared with the analytical solution. The effects of various pertinent parameters on the velocity distribution, temperature distribution, entropy generation rate, and Bejan number are investigated and discussed graphically.     

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.     

On natural convection in enclosures filled with fluid-saturated porous media including viscous dissipation

Care needs to be taken when considering the viscous dissipation in the energy conservation formulation of the natural convection problem in fluid-saturated porous media. The unique energy formulation compatible with the First Law of Thermodynamics informs us that if the viscous dissipation term is taken into account, also the work of pressure forces term needs to be taken into account. In integral terms, the work of pressure forces must equal the energy dissipated by viscous effects, and the net energy generation in the overall domain must be zero. If only the (positive) viscous dissipation term is considered in the energy conservation equation, the domain behaves as a heat multiplier, with an heat output greater than the heat input. Only the energy formulation consistent with the First Law of Thermodynamics leads to the correct flow and temperature fields, as well as of the heat transfer parameters characterizing the involved porous device. Attention is given to the natural convection problem in a square enclosure filled with a fluid-saturated porous medium, using the Darcy Law to describe the fluid flow, but the main ideas and conclusions apply equally for any general natural or mixed convection heat transfer problem. It is also analyzed the validity of the Oberbeck–Boussinesq approximation when applied to natural convection problems in fluid-saturated porous media.     

Transient and non-Darcian effects on natural convection flow in a vertical channel partially filled with porous medium: Analysis with Forchheimer–Brinkman extended Darcy model

The present study addresses the transient as well as non-Darcian effects on laminar natural convection flow in a vertical channel partially filled with porous medium. Forchheimer–Brinkman extended Darcy model is assumed to simulate momentum transfer within the porous medium. Two regions are coupled by equating the velocity and shear stress in the case of momentum equation while matching of the temperature and heat flux is taken for thermal energy equation. Approximate solutions are obtained using perturbation technique. Variations in velocity field with Darcy number, Grashof number, kinematic viscosity ratio, distance of interface and variations in temperature distribution with thermal conductivity ratio, distance of interface are obtained and depicted graphically. The skin-friction and rate of heat transfer at the channel walls are also derived and the numerical values for various physical parameters are tabulated.     

Laminar convection in a vertical channel with viscous dissipation and buoyancy effects

The fully developed and laminar convection in a parallel-plate vertical channel is investigated by taking into account both viscous dissipation and buoyancy. Uniform and symmetric temperatures are prescribed at the channel walls. The velocity field is considered as parallel. A perturbation method is employed to solve the momentum balance equation and the energy balance equation. A comparison with the velocity and temperature profiles in the case of laminar forced convection with viscous dissipation is performed in order to point out the effect of buoyancy. The case of convective boundary conditions is also discussed.     

幂律流体饱和多孔介质通道中的粘性耗散效应

针对Darcy-Brinkman-Forchheimer流动模型,分析了幂律型非牛顿流体在填充多孔介质平板通道中强迫对流传热过程充分发展的黏性耗散效应,并比较了三个不同的黏性耗散项Darcy项、Al-Hadhrami项和Forchheimer项对流动传热率的影响。推导出了无量纲轴向流速分布和无量纲温度分布的计算表达式,并在恒热流边界条件下,利用经典Runge-Kutta法进行数值求解。模拟结果表明,布林克曼数Br、达西数Da、综合惯性参数F和幂律指数n等重要参数对无量纲温度分布有着较大的影响,同时发现不同的黏性耗散效应对流动传热特性也有着重要的影响。     

Adomian decomposition solution for propulsion of dissipative magnetic Jeffrey biofluid in a ciliated channel containing a porous medium with forced convection heat transfer

Physiological transport phenomena often feature ciliated internal walls. Heat, momentum, and multispecies mass transfer may arise and additionally non‐Newtonian biofluid characteristics are common in smaller vessels. Blood (containing hemoglobin) or other physiological fluids containing ionic constituents in the human body respond to magnetic body forces when subjected to external (extracorporeal) magnetic fields. Inspired by such applications, in the present work we have considered the forced convective flow of an electrically conducting viscoelastic physiological fluid through a ciliated channel under the action of a transverse magnetic field. The presence of deposits (fats, cholesterol, etc.) in the channel is mimicked with a Darcy porous medium drag force model. The effect of energy loss is simulated via the inclusion of viscous dissipation in the energy conservation (heat) equation. The velocity, temperature, and pressure distribution are computed in the form of infinite series constructed by Adomian decomposition method and numerically evaluated in a symbolic software (Mathematica). The influence of Hartmann number (magnetic parameter), Jeffrey first and second viscoelastic parameters, permeability parameter (modified Darcy number), and Brinkman number (viscous heating parameter) on velocity, temperature, pressure gradient, and bolus dynamics is visualized graphically.     

Assessment of heat transfer in large parallel plates with a narrow gap maintained at a constant temperature

Investigations are conducted on electromagnetohydrodynamic (EMHD) flow and heat transfer in a third-grade fluid flowing through large parallel plates, which are maintained at constant temperatures. The impact of convective heat transmission is disregarded since the space between the plates is small. The influence of viscous dissipation is considered. Despite being addressed for Newtonian fluids, the conduction problem with the viscous dissipation effect is not examined in third-grade fluids for EMHD flow and heat transfer behavior. The least-square method is adopted to solve nondimensional, nonlinear momentum and energy conservation equations to get the dimensionless velocity, temperature distribution, and heat flux. Temperature and heat flux are investigated in relation to the third-grade fluid parameter, the Hartmann number, the electric field parameter, and the Brinkman number. The findings show a rise in the Brinkman number dramatically increases heat transfer from both walls, necessitating cooling of both plates. The heat flow from both walls increases as the parameters of third-grade fluid increases.     

The Graetz–Brinkman problem in a plane-parallel channel with adiabatic-to-isothermal entrance

An analysis of laminar hydrodynamically developed forced convection in the thermal entrance region of a plane-parallel channel with isothermal walls is presented. The effect of viscous dissipation is taken into account in a self-consistent way, i.e. by assuming that this effect is important not only in the region downstream of the entrance section, but also in the region upstream. In the latter region, the channel is assumed to be thermally insulated. As a consequence, the thermal entrance temperature distribution is non-uniform. The temperature field and the local Nusselt number are evaluated.     

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 convection with viscous dissipation in the thermal entrance region of a circular duct with prescribed wall heat flux

The thermal entrance forced convection in a circular duct with a prescribed wall heat flux distribution is studied under the assumptions of a fully developed laminar flow and of a negligible axial heat conduction in the fluid, by taking into account the effect of viscous dissipation. The solution of the local energy balance equation is obtained analytically by employing the Laplace transform method. The effect of viscous dissipation is taken into account also in the region upstream of the entrance cross-section, by assuming an adiabatic preparation of the fluid. The latter hypothesis implies that the initial condition in the entrance cross-section is a non-uniform radial temperature distribution. Two special cases are investigated in detail: an axially uniform wall heat flux, a wall heat flux varying linearly in the axial direction.     

Forced convection in porous microchannels with viscous dissipation in local thermal non-equilibrium conditions

Fully developed, steady-state forced convection, in parallel-plate microchannels, filled with a porous medium saturated with rarefied gases at high temperatures, in local thermal non-equilibrium (LTNE) condition, is investigated for the first-order slip-flow regime (0 ≤ Kn ≤ 0.1). Both velocity and temperature jumps at the walls are accounted for. An analytic solution is proposed for the Darcy–extended Brinkman flow model with assigned uniform heat flux at the microchannel walls and viscous heat dissipation in the fluid phase. The solution for NTLE includes the shear work done by the slipping effects. A closed-form expression of the Nusselt number is derived. A validation analysis with respect to the case of channels filled with saturated porous medium is accomplished. The results show that the internal dissipation increases as the velocity slip increases. In addition, the heat dissipation strongly affects the fluid temperature profiles. The increases in velocity slip and temperature jump lead to decreases of temperature gradients in the fluid and solid along the sections. The heat transfer at channel walls is enhanced due to an increase in the bulk heat transfer.     

Comment on the paper I.A. Badruddin,Z.A. Zainal,Z.A. Khan,Z. Mallick “Effect of viscous dissipation and radiation on natural convection in a porous medium embedded within vertical annulus”, IJTS 46 (3) (2007) 221–227

In a recent paper [1] it is studied the steady natural convection heat transfer problem in a two-dimensional fluid-saturated porous medium embedded within a vertical annulus, whose inner and outer temperatures are uniform. Radiation heat transfer is taken into account, and the viscous dissipation effect is considered in the energy conservation equation. The present comment is related with two aspects of the paper, namely: (i) The radiation modeling of the problem, using the Rosseland approximation; and (ii) The energy formulation of the problem.     

Flow and heat transfer analysis of MHD third-grade fluid flow through a vertical microchannel subjected to entropy generation

This paper analyses the generation of entropy in an electrically conducting third-grade fluid through a vertical channel considering the variable thermal conductivity. Aspects of radiation, viscous dissipation, porous medium, Joule heating, convective boundary condition, and heat generation are studied. Nonlinear systems of ordinary differential equations are obtained via applying suitable dimensionless variables. After that, the system is solved with the aid of using the Runge–Kutta–Fehlberg method. The numerical solutions are used to characterize the irreversibility and irreversibility ratio. It is established that the entropy is enhanced with accelerating estimations of the third-grade material parameter, Brinkman number, magnetism, Biot number, porous parameter, and the impact is decelerated with elevating values of the radiation. The rate of heat transfer is higher for the Brinkman number, and a similar impact on drag force is noticed for magnetic and Grashof numbers. All the parameters on flow, temperature, fluid irreversibility and irreversibility ratio are discussed through graphical illustration.     

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.     

Thermal stability of a reactive viscous flow through a porous-saturated channel with convective boundary conditions

This study is devoted to investigate the thermal stability of a reactive viscous combustible fluid flowing steadily through a channel filled with a saturated porous medium. It is assumed that the system exchange heat with the ambient following Newton’s cooling law and the reaction is exothermic under Arrhenius kinetics, neglecting the consumption of the material. The Brinkman model is employed and analytical solutions are constructed for the governing nonlinear boundary-value problem using a perturbation technique together with a special type of Hermite-Padé approximants and important properties of the temperature field including bifurcations and thermal criticality are discussed.     

Pulsatile flow in heated porous channelEcoulement pulsatoire dans un canal poreux et chauffePulsierende strömung in einem beheizten porösen kanalПyльcaциoннoe тeчeниe в нaгpeвaeмoм пopиcтoм кaнaлe

The combined forced and free convection flow through a porous channel when a pulsatile pressure is applied across its ends is discussed. It is assumed that the ratio of the width of the channel to the length (δ) is small. Even in this physically realistic situation of a finite channel, the transverse velocity remains undisturbed and acquires its suction or blowing value at the channel walls. A salient feature of the investigation is the presence of steady streaming component in the higher approximations due to nonlinearity in the viscous dissipation heat. When the channel is horizontal, the axial velocity distribution obtained is exact and independent of δ. But the pressure and temperature distributions are exact and δ-dependent. On the other hand, if the channel is vertical, only the pressure distribution is obtained exactly. It is independent of the transverse coordinate but varies linearly with the axial coordinate. The velocity and temperature are obtained approximately to order O(δ). Also the shear stress and heat flux at the walls are discussed quantitatively.     

A comparative study of viscous dissipation effect on entropy generation in single-phase liquid flow in microchannels

The influence of viscous dissipation on entropy generation in fully developed forced convection for single-phase liquid flow in a circular microchannel under imposed uniform wall heat flux has been studied. In the first-law analysis, closed form solutions of the radial temperature profiles for the models with and without viscous dissipation term in the energy equation are obtained. In the second-law analysis, for different Brinkman number and dimensionless heat flux, the variations of dimensionless entropy generation and Bejan number as a function of the radial distance are investigated. The two models are compared by analyzing their relative deviations in dimensionless entropy generation and Bejan number. Comparisons are also performed for average dimensionless entropy generation and average Bejan number. Contribution of heat transfer irreversibility and fluid friction irreversibility to the deviations is analyzed and discussed. It is found that, under certain conditions, the effect of viscous dissipation on entropy generation in microchannel is significant and should not be neglected.     

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.