The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

Forced convection gaseous slip flow in circular porous micro-channels

Laminar forced convection of gaseous slip flow in a circular micro-channel filled with porous media under local thermal equilibrium condition is studied numerically using the finite difference technique. Hydrodynamically fully developed flow is considered and the Darcy–Brinkman–Forchheimer model is used to model the flow inside the porous domain. The present study reports the effect of several operating parameters (Knudsen number (Kn), Darcy number (Da), Forchhiemer number (Γ), and modified Reynolds number ) on the velocity slip and temperature jump at the wall. Results are given in terms of the velocity distribution, temperature distribution, skin friction , and the Nusselt number (Nu). It is found that the skin friction is increased by (1) decreasing Knudsen number, (2) increasing Darcy number, and (3) decreasing Forchheimer number. Heat transfer is found to (1) decrease as the Knudsen number, or Forchheimer number increase, (2) increase as the Peclet number or Darcy number increase.     

Analysis of Momentum Transfer in a Lid-Driven Cavity Containing a Brinkman–Forchheimer Medium

The CE/SE (the space-time conservation element and solution element method) scheme with the second-order accuracy has been proposed. And the pretreatment method has been introduced to convert the parabolic equations to the hyperbolic equations, which are accurately solved by the CE/SE method. The lid-driven rectangular cavity containing a porous Brinkman–Forchheimer medium is studied in this numerical investigation. The Brinkman–Forchheimer equation is used such that both the inertial and viscous effects are incorporated. The governing equations are solved by the improved CE/SE approach. The characteristics of the flow are analyzed with emphasis on the influence of the Darcy number and the cavity depth. It is found that the porous medium effect decreases both the strength and the number of eddies, especially for deep cavities.     

Forced convection of gaseous slip-flow in porous micro-channels under Local Thermal Non-Equilibrium conditions

Steady laminar forced convection gaseous slip-flow through parallel-plates micro-channel filled with porous medium under Local Thermal Non-Equilibrium (LTNE) condition is studied numerically. We consider incompressible Newtonian gas flow, which is hydrodynamically fully developed while thermally is developing. The Darcy–Brinkman–Forchheimer model embedded in the Navier–Stokes equations is used to model the flow within the porous domain. The present study reports the effect of several operating parameters on velocity slip and temperature jump at the wall. Mainly, the current study demonstrates the effects of: Knudsen number (Kn), Darcy number (Da), Forchheimer number (Γ), Peclet number (Pe), Biot number (Bi), and effective thermal conductivity ratio (K _R) on velocity slip and temperature jump at the wall. Results are given in terms of skin friction (C _f Re ^*) and Nusselt number (Nu). It is found that the skin friction: (1) increases as Darcy number increases; (2) decreases as Forchheimer number or Knudsen number increases. Heat transfer is found to (1) decreases as the Knudsen number, Forchheimer number, or K _R increases; (2) increases as the Peclet number, Darcy number, or Biot number increases.     

Numerical Study of Heat Transfer in a Rectangular Channel with Porous Covering Obstacles

A numerical study is performed to analyze steady laminar forced convection in a channel in which discrete heat sources covered with porous material are placed on the bottom wall. Hydrodynamic and heat transfer results are reported. The flow in the porous medium is modeled using the Darcy–Brinkman–Forchheimer model. A computer program based on control volume method with appropriate averaging for diffusion coefficient is developed to solve the coupling between solid, fluid, and porous region. The effects of parameters such as Reynolds number, Prandtl number, inertia coefficient, and thermal conductivity ratio are considered. The results reveal that the porous cover with high thermal conductivity enhances the heat transfer from the solid blocks significantly and decreases the maximum temperature on the heated solid blocks. The mean Nusselt number increases with increase of Reynolds number and Prandtl number, and decrease of inertia coefficient. The pressure drop along the channel increases rapidly with the increase of Reynolds number.     

On forced convection in channels partially filled with porous substrates

 This paper numerically simulates the forced convection flow in the developing region of a parallel-plate channel partially filled with two porous substrates of equal thickness deposited at the inner walls of the channel. The major objective of the present work is to investigate the impact of several operating and design parameters on the thermal performance of the channel under consideration. The physical problem is simulated by using Darcy–Brinkman–Forchheimer model. For a prescribed amount of porous material, the current investigation discusses the comparison between inserting this entire amount at one side of the channel and inserting half of this amount at each side of the channel. Received on 25 May 2000 / Published online: 29 November 2001     

Examination of the Thermal Equilibrium Assumption in Transient Natural Convection Flow in Porous Channel

The local thermal equilibrium assumption in the transient natural convection channel flow is investigated numerically. The Darcy–Brinkman–Forchheimer model is used to model the flow inside the porous domain. The effect of different parameters on the validity of the local thermal equilibrium assumption is examined. It is found that the volumetric Nusselt number has the most significant effect on the local thermal equilibrium assumption.     

A theoretical analysis of forced convection in a porous-saturated circular tube: Brinkman–Forchheimer model

Fully developed forced convection inside a circular tube filled with saturated porous medium and with uniform heat flux at the wall is investigated on the basis of a Brinkman–Forchheimer model. The matched asymptotic expansion method is applied at small Darcy numbers. For large Darcy numbers, the solution for the Brinkman–Forchheimer momentum equation is found in terms of an asymptotic expansion. Once the velocity distribution is determined, the energy equation is solved using the same asymptotic technique. The results for the two limiting cases of clear fluid and Darcy flow conditions show good agreement with those available in the literature.     

The Horton–Rogers–Lapwood Problem Revisited: The Effect of Pressure Work

The problem of the onset of convective roll instabilities in a horizontal porous layer with isothermal boundaries at unequal temperatures, well known as the Horton–Rogers–Lapwood problem, is revisited including the effect of pressure work and viscous dissipation in the local energy balance. A linear stability analysis of rolls disturbances is performed. The analysis shows that, while the contribution of viscous dissipation is ineffective, the contribution of the pressure work may be important. The condition of marginal stability is investigated by adopting two solution procedures: method of weighted residuals and explicit Runge–Kutta method. The pressure work term in the energy balance yields an increase of the value of the Darcy–Rayleigh number at marginal stability. In other words, the effect of pressure work is a stabilizing one. Furthermore, while the critical value of the Darcy– Rayleigh number may be considerably affected by the pressure work contribution, the critical value of the wave number is affected only in rather extreme cases, i.e. for very high values of the Gebhart number. A nonlinear stability analysis is also performed pointing out that the joint effects of viscous dissipation and pressure work result in a reduction of the excess Nusselt number due to convection, when the Darcy–Rayleigh number is replaced by the superadiabatic Darcy–Rayleigh number.     

Mixed Convection in a Vertical Porous Channel

A numerical study of mixed convection in a vertical channel filled with a porous medium including the effect of inertial forces is studied by taking into account the effect of viscous and Darcy dissipations. The flow is modeled using the Brinkman–Forchheimer-extended Darcy equations. The two boundaries are considered as isothermal–isothermal, isoflux–isothermal and isothermal–isoflux for the left and right walls of the channel and kept either at equal or at different temperatures. The governing equations are solved numerically by finite difference method with Southwell–Over–Relaxation technique for extended Darcy model and analytically using perturbation series method for Darcian model. The velocity and temperature fields are obtained for various porous parameter, inertia effect, product of Brinkman number and Grashof number and the ratio of Grashof number and Reynolds number for equal and different wall temperatures. Nusselt number at the walls is also determined for three types of thermal boundary conditions. The viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. The Darcy and inertial drag terms suppress the flow. It is found that analytical and numerical solutions agree very well for the Darcian model. An erratum to this article is available at .     

Convective Roll Instabilities of Vertical Throughflow with Viscous Dissipation in a Horizontal Porous Layer

The vertical throughflow with viscous dissipation in a horizontal porous layer is studied. The horizontal plane boundaries are assumed to be isothermal with unequal temperatures and bottom heating. A basic stationary solution of the governing equations with a uniform vertical velocity field (throughflow) is determined. The temperature field in the basic solution depends only on the vertical coordinate. Departures from the linear heat conduction profile are displayed by the temperature distribution due to the forced convection effect and to the viscous dissipation effect. A linear stability analysis of the basic solution is carried out in order to determine the conditions for the onset of convective rolls. The critical values of the wave number and of the Darcy–Rayleigh number are determined numerically by the fourth-order Runge–Kutta method. It is shown that, although generally weak, the effect of viscous dissipation yields an increase of the critical value of the Darcy–Rayleigh number for downward throughflow and a decrease in the case of upward throughflow. Finally, the limiting case of a vanishing boundary temperature difference is discussed.     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

The Use of Porous Fins for Heat Transfer Augmentation in Parallel-Plate Channels

In this study, a steady, fully developed laminar forced convection heat augmentation via porous fins in isothermal parallel-plate duct is numerically investigated. High-thermal conductivity porous fins are attached to the inner walls of two parallel-plate channels to enhance the heat transfer characteristics of the flow under consideration. The Darcy–Brinkman–Forchheimer model is used to model the flow inside the porous fins. This study reports the effect of several operating parameters on the flow hydrodynamics and thermal characteristics. This study demonstrates, mainly, the effects of porous fin thickness, Darcy number, thermal conductivity ratio, Reynolds number, and microscopic inertial coefficient on the thermal performance of the present flow. It is found that the highest Nusselt number is achieved at fully filled porous duct which requires the highest pumping pressure. The results show that using porous fins requires less pumping pressure with comparable high heat augmentation weight against fully filled porous duct. It is found that higher Nusselt numbers are achieved by increasing the microscopic inertial coefficient (A), the Reynolds number (Re), and the thermal conductivity of the porous substrate k ₂. The results show that heat transfer can be enhanced (1) with the use of high thermal conductivity fins, (2) by decreasing the Darcy number, and (3) by increasing microscopic inertial coefficient.     

Flow near the permeable boundary of a porous medium: An experimental investigation using LDA

 Fluid flow at the interface of a porous medium and an open channel is the governing phenomenon in a number of processes of industrial importance. Traditionally, this has been modeled by applying the Brinkman’s modification of Darcy’s law to obtain the velocity profile in terms of an additional parameter known as the “apparent viscosity” or the “slip coefficient”. To test this ad hoc approach, a detailed experimental investigation of the flow was conducted using Laser Doppler Anemometry (LDA) in the close vicinity of the permeable boundary of a porous medium. The porous medium used in the experiments consisted of a network of continuous glass strands woven together in a random fashion. A Hele–Shaw cell was partially filled with a fibrous preform such that an open channel flow is coupled with the Darcy flow inside the preform through the permeable interface of the preform. The open channel portion of the Hele–Shaw cell also acts as an ideal porous medium of known in-plane permeability which is much higher than the permeability of the fibrous porous medium. A viscous fluid is injected at a constant flow rate through the above arrangement and a saturated and steady flow is established through the cell. Using LDA, steady state velocity profiles are accurately measured by traversing across the cell in the direction perpendicular to the flow. A series of experiments were conducted in which fluid viscosity, flow rate, solid volume fraction of the porous medium and depth of the Hele–Shaw cell were varied. For each and every case in which the conditions for Hele–Shaw approximation were valid, the depth of the boundary layer zone or the screening length inside the fibrous preform was found to be of the order of the channel depth. This is much larger as compared to the Brinkman’s prediction of the screening length which is of the order of √K, where K is the permeability of the fibrous porous medium. Based on this finding, we modified the boundary condition in the Brinkman’s solution and found that the velocity profile results compared well with the experimental data for the planar geometry and the fibrous preforms for volume fractions of 7%, 14% and 21% for Hele–Shaw cell depths of 1.6 and 3.175 mm. For a cell depth of 4.8 cm, in which the Hele–Shaw approximation was not valid, the boundary layer thickness or the screening length was found to be less than the mold or channel depth but was still much larger than the Brinkman’s prediction. Received: 10 May 1996 / Accepted: 26 August 1996     

Numerical Simulation of Two-Phase Inertial Flow in Heterogeneous Porous Media

In this study, non-Darcy inertial two-phase incompressible and non-stationary flow in heterogeneous porous media is analyzed using numerical simulations. For the purpose, a 3D numerical tool was fully developed using a finite volume formulation, although for clarity, results are presented in 1D and 2D configurations only. Since a formalized theoretical model confirmed by experimental data is still lacking, our study is based on the widely used generalized Darcy–Forchheimer model. First, a validation is performed by comparing numerical results of the saturation front kinetics with a semi-analytical solution inspired from the Buckley–Leverett model extended to take into account inertia. Second, we highlight the importance of inertial terms on the evolution of saturation fronts as a function of a suitable Reynolds number. Saturation fields are shown to have a structure markedly different from the classical case without inertia, especially for heterogeneous media, thereby, emphasizing the necessity of a more complete model than the classical generalized Darcy’s one when inertial effects are not negligible.     

Polynomial Filtration Laws for Low Reynolds Number Flows Through Porous Media

In this study, we use the method of homogenization to develop a filtration law in porous media that includes the effects of inertia at finite Reynolds numbers. The result is much different than the empirically observed quadratic Forchheimer equation. First, the correction to Darcy’s law is initially cubic (not quadratic) for isotropic media. This is consistent with several other authors (Mei and Auriault, J Fluid Mech 222:647–663, 1991; Wodié and Levy, CR Acad Sci Paris t.312:157–161, 1991; Couland et al. J Fluid Mech 190:393–407, 1988; Rojas and Koplik, Phys Rev 58:4776–4782, 1988) who have solved the Navier–Stokes equations analytically and numerically. Second, the resulting filtration model is an infinite series polynomial in velocity, instead of a single corrective term to Darcy’s law. Although the model is only valid up to the local Reynolds number, at the most, of order 1, the findings are important from a fundamental perspective because it shows that the often-used quadratic Forchheimer equation is not a universal law for laminar flow, but rather an empirical one that is useful in a limited range of velocities. Moreover, as stated by Mei and Auriault (J Fluid Mech 222:647–663, 1991) and Barree and Conway (SPE Annual technical conference and exhibition, 2004), even if the quadratic model were valid at moderate Reynolds numbers in the laminar flow regime, then the permeability extrapolated on a Forchheimer plot would not be the intrinsic Darcy permeability. A major contribution of this study is that the coefficients of the polynomial law can be derived a priori, by solving sequential Stokes problems. In each case, the solution to the Stokes problem is used to calculate a coefficient in the polynomial, and the velocity field is an input of the forcing function, F, to subsequent problems. While numerical solutions must be utilized to compute each coefficient in the polynomial, these problems are much simpler and robust than solving the full Navier–Stokes equations.     

Resolution of a Paradox Involving Viscous Dissipation and Nonlinear Drag in a Porous Medium

The modelling of viscous dissipation in a porous medium saturated by an incompressible fluid is discussed, for the case of Darcy, Forchheimer and Brinkman models. An apparent paradox relating to the effect of inertial effects on viscous dissipation is resolved, and some wider aspects of resistance to flow (concerning quadratic drag and cubic drag) in a porous medium are discussed. Criteria are given for the importance or otherwise of viscous dissipation in various situations.     

Mixed convection with viscous dissipation in an inclined porous channel with isoflux impermeable walls

Combined forced and free convection flow in a fluid saturated inclined plane channel is investigated by taking into account the effect of viscous dissipation. Steady parallel flow is considered assuming that the temperature gradient in the parallel flow direction is constant, and the channel walls are subject to uniform symmetric heat fluxes. Two possible formulations of the Darcy–Boussinesq scheme are considered, based on two different choices of the reference temperature for modelling buoyancy. The first choice is a constant temperature, while the second is a streamwise changing temperature. It is shown that both approaches substantially agree in the formulation of the balance equations for the range of values of the Darcy–Rayleigh number such that viscous dissipation is important. The boundary value problem is solved analytically for any tilt angle, revealing that it admits dual solutions for assigned values of the governing parameters. The rather important effect of viscous dissipation in the special case of adiabatic channel walls is outlined. E. Magyari is on leave from Institute of Building Technology, ETH—Zürich     

Estimating the Hydraulic Conductivity of Two-Dimensional Fracture Networks Using Network Geometric Properties

Based on a modified Darcy–Brinkman–Maxwell model, a linear stability analysis of a Maxwell fluid in a horizontal porous layer heated from below by a constant flux is carried out. The non-oscillatory instability and oscillatory instability with different hydrodynamic boundaries such as rigid and free surfaces at the bottom are studied. Compared with the rigid surface cases, onset of fluid motion due to non-oscillatory instability and oscillatory instability is found to occur both more easily for the system with a free bottom surface. The critical Rayleigh number for onset of fluid motion due to non-oscillatory instability is lower with a constant flux heating bottom than with an isothermal heating bottom, but the result due to oscillatory instability is in contrast. The effects of the Darcy number, the relaxation time, and the Prandtl number on the critical Rayleigh number are also discussed.     

Analysis of hydrodynamic and thermal dispersion in porous media by means of a local approach

A pore scale analysis is implemented in this numerical study to investigate the behavior of microscopic inertia and thermal dispersion in a porous medium with a periodic structure. The macroscopic characteristics of the transport phenomena are evaluated with an averaging technique of the controlling variables at a pore scale level in an elementary cell of the porous structure. The Darcy–Forchheimer model describes the fluid motion through the porous medium while the continuity and Navier–Stokes equations are applied within the unit cell. An average energy equation is employed for the thermal part of the porous medium. The macroscopic pressure loss is computed in order to evaluate the dominant microscopic inertial effects. Local fluctuations of velocity and temperature at the pore scale are instrumental in the quantification of the thermal dispersion through the total effective thermal diffusivity. The numerical results demonstrate that microscopic inertia contributes significantly to the magnitude of the macroscopic pressure loss, in some instances with as much as 70%. Depending on the nature of the porous medium, the thermal dispersion may have a marked bearing on the heat transfer, particularly in the streamwise direction for a highly conducting fluid and certain values of the Peclet number.