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.     

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.     

Closed form solution for unsteady convective diffusion in a fluid-saturated sparsely packed porous medium

Closed form solution is obtained for dispersion of a solute in a fluid-saturated sparsely packed porous medium using Brinkman model. The time-range of validity of the dispersion coefficient, which increases with time, is widened by using a generalised dispersion coefficient. The effect of porous parameter on the dispersion coefficient is clearly depicted. The dimensionless mean concentration distribution as a function of dimensionless axial distance and time for different values of physical parameters is shown. The corresponding results for the case of ‘pure convection’ is also reported.     

Thermosolutal convection in a fluid‐porous composite medium

In the present study analysis has been performed for thermosolutal convection in a fluid‐porous composite medium, consisting of a fluid‐saturated porous medium followed by an overlaying clean medium. The fluid‐porous composite medium is subjected to both a horizontal solutal and a thermal gradient. Top and bottom walls of the fluid‐porous composite medium are assumed to be impermeable and adiabatic. The Darcy‐Brinkman‐Forchheimer model is used to study the flow through the fluid‐porous composite medium. A single domain approach is taken into consideration for numerical simulation. The solution is done by control volume integration. A comprehensive analysis has been performed for various pertinent parameters to delineate their behavior. Results of the transport phenomenon have been provided in graphical and tabular form, for the complete understanding of the complex phenomenon. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21048     

A thermal resistance analysis on forced convection with viscous dissipation in a porous medium using entransy dissipation concept

The analytical solution of a two-equation model presented in an earlier study is examined. Heat transfer characterization is classified into two regimes which are dominated by fluid conduction or solid conduction and interstitial heat exchange, respectively by using the entransy dissipation concept. The computed pattern of variation of thermal resistance with shape factor S at a fixed Brinkman number for a low ratio of the fluid to solid effective thermal conductivities implies the occurrence of temperature gradient bifurcation as S decreases. Therefore, the thermal diffusion term in the fluid phase in the two-equation model is not negligible for both regimes.     

Unsteady thermocline degradation in a fluid-saturated,porous medium

A numerical model is developed to describe unsteady, three-dimensional, natural convective flows in a fluid-saturated, porous medium having a rectangular volume with impervious walls and finite heat transfer at the boundaries. The model is used to predict the transient decay of a thermocline in a packed bed during a period of stagnation in which there is zero net flow and no energy input into the bed. The computed results compare favorably with experimental data from a packed bed consisting of air and natural stone of mixed sizes and irregular shapes. The results show an upward shift in the position of maximum temperature along the vertical centerline of the bed and confirm the important influence of internal convection on the process. The results also demonstrate that a purely diffusive model would be incapable of a reliable predication. Further, the recognition of finite heat transfer rates at the boundaries is shown to be a significant factor in improving the predictive capability of the model.     

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.     

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.     

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.     

Convection in a fluid-saturated porous medium due to internal heat generation

Two-dimensional convective motions driven by uniformly distributed internal heat sources in a fluid-saturated porous medium are analyzed in the large Rayleigh number limit. An integral method is used to obtain a simple analytical representation of the flow field within a rectangular cavity whose vertical side walls are isothermal and whose horizontal boundaries are adiabatic. The results are in excellent agreement with numerical computations based on a finite-difference technique.     

Onset of convective stability in a fluid-saturated porous layer subjected to time-dependent heating

A theoretical analysis of thermal instability driven by buoyancy forces in transient temperature fields is conducted in an initially quiescent, fluid-saturated, horizontal porous layer. Darcy's law is used to explain characteristics of fluid motion and linear stability theory is employed to predict the onset of buoyancy-driven motion. Under the principle of exchange of stabilities, the stability analysis is performed on the basis of the linear amplification theory. The result predicts the critical condition of onset of buoyancy-driven convection, which is governed by the Darcy-Rayleigh number. The present stability criteria predict the experimental data quite well.     

Magnetic field and viscous dissipation effect on bioconvection in a permeable sphere embedded in a porous medium with a nanofluid containing gyrotactic micro‐organisms

Micro‐organisms play a vital role to understand the ecological system and therefore it is very important to understand the behavior of micro‐organism due to different parameters. In the present paper, we investigated the bioconvection about a permeable sphere with constant surface temperature embedded in a porous medium filled with a water‐based nanofluid containing gyrotactic micro‐organisms. The convection and movement of the micro‐organisms are constrained or assisted by thermophoresis, viscous dissipation effects, and an applied magnetic field. First, we have used the similarity transformation to simplify the governing equations. Then, we have solved the governing nonlinear partial differential equations numerically using a recent spectral relaxation method. The effects of the significant parameters on the local density of the gyrotactic micro‐organisms have been determined and discussed. It is observed that by introducing the magnetic field in the system, the skin friction, local nanoparticle Sherwood number, and the local density of the micro‐organism decrease, while the last two governing parameters show a positive response with increase in the viscous dissipation.     

MHD unsteady free convective flow through a porous medium

The two dimensional unsteady free convective flow through a porous medium bounded by an infinite vertical plate for an incompressible viscous and electrically conducting fluid is considered, when the flow is subjected to the action of a uniform transverse magnetic field. The magnetic Reynolds number is taken to be small, so that the induced magnetic field is negligible. Analytical expressions for the velocity field and temperature are given, and the influences of the various parametrs on the velocity field are discussed.     

Stability of a fluid-saturated porous medium heated from below by forced convection

A linear stability analysis determining the onset of convection in a bounded rectangular cavity containing a fluid-saturated porous medium is performed for insulated sidewalls, isothermal top wall, and bottom wall heated by forced convection. The nature of the bottom wall heating necessarily involves the Biot number, Bi. Numerical calculations of the critical Rayleigh number, R_c made over the range of Biot numbers 10⁻⁴?Bi?10⁴ for cavity aspect ratios 0?(a,b)?5 cover all effective bottom heating conditions from the constant heat flux global limit, R_c=27.096 found as Bi→0 to the isothermal global limit, R_c=4π² found as Bi→∞. Marginal stability boundaries, preferred cellular modes and disturbance temperature contours are displayed graphically.     

Nonlinear radiation on Maxwell fluid in a convective heat transfer with viscous dissipation and activation energy

The present analysis addresses linear and nonlinear radiation effects in hydrodynamic viscous Maxwell fluid flow on a unidirectional stretching surface through viscous dissipation. The relaxation effect is considered in the mathematical model, which elucidates mass transport mechanisms under binary chemical reaction and activation energy. Mathematical modeling contains nonlinear partial differential equations using boundary conditions. Appropriate transformations convert the partial differential equations into ordinary differential equations. Numerical solutions for regular differential equations are brought by Runge–Kutta–Fehlberg numerical quadrature and a shooting method with a tolerance level of 10⁻⁹. The influence of physical variables, such as Deborah relaxation number, rotation parameter, Biot number, activation energy parameter, reaction rate parameter, Eckert number, and Prandtl number are investigated. Increasing the Biot number improves the temperature region in the boundary layer. With high rotation, the increasing Deborah number enhances the fluid temperature substantially throughout the boundary layer.     

MHD free convective flow through a porous medium in a rotating fluid

In the paper, the velocity and temperature distributions of an electrically conducting liquid flowing past an infinite vertical porous plate through a porous medium in a rotating frame of reference are analysed for the case when a stron magnetic field is imposed in a direction perpendicular to the free stream and at an angle α to the vertical direction. The. influences of various parameters on the velocity field are discussed.     

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.