Thermal Equilibrium in Transient Forced Convection Porous Channel Flow

The validity of the local thermal equilibrium assumption in the transient forced convection channel flow is investigated numerically. Axial conduction in both fluid and solid domains is included. It is found that five dimensionless parameters control the local thermal equilibrium assumption. These parameters are the thermal diffusivity ratio _R, the volumetric Nusselt number Nu, the dimensionless channel length _max, Peclet number Pe, and the solid to fluid total thermal capacity ratio C _R. The qualitative and quantitative aspects of the effects of these five parameters on the channel thermalization time are investigated.     

The Effect of Thermal Expansion on Porous Media Convection. Part 2: Thermal Convection Solution

The impact of thermal expansion and the corresponding non-Boussinesq effects on porous media convection are considered. The results show that the non-Boussinesq effects decouple from the rest, and solving the Boussinesq system separately is needed even when non-Boussinesq effects are being investigated. The thermal expansion is shown to have a lasting impact on the post-transient convection only for values of Rayleigh number substantially beyond the convection threshold, where the amplitude of convection is not small. In the neighbourhood of the convection threshold the thermal expansion has only a transient impact on the solution. It is also evident from the results that the neglect of the time derivative term in the extended Darcy equation might introduce a significant error when oscillatory effects are present.     

Transient Free Convection Fluid Flow in Domains Partially Filled with Porous Media

The problem of transient free convection in domains partly filled with porous substrates is investigated analytically using Laplace transformation technique. Four configurations are considered which are subject to an isothermal heating boundary condition. The Brinkman-extended Darcy model is adopted to describe the hydrodynamics behavior of the porous domain.     

Validation of Thermal Equilibrium Assumption in Transient Forced Convection Flow in Porous Channel

The validity of the local thermal equilibrium assumption in the transient forced convection channel flow is investigated analytically. Closed form expressions are presented for the temperatures of the fluid and solid domains and for the criterion which insures the validity of the local thermal equilibrium assumption. It is found that four dimensionless parameters control the local thermal equilibrium assumption. These parameters are the porosity , the volumetric Biot number Bi, the dimensionless channel length _max and the solid to fluid total thermal capacity ratio C _R. The qualitative and quantitative aspects of the effects of these four parameters on the channel thermal equilibrium relaxation time are investigated.     

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.     

The Effect of Thermal Expansion on Porous Media Convection Part 1: Thermal Expansion Solution

The effect of thermal expansion on porous media convection is investigated by isolating first the solution of thermal expansion in the absence of convection which allows to evaluate the leading order effects that need to be included in the convection problem that is solved later. A relaxation of the Boussinesq approximation is applied and the relevant time scales for the formulated problem are identified from the equations as well as from the derived analytical solutions. Particular attention is paid to the problem of waves propagation in porous media and a significant conceptual difference between the isothermal compression problem in flows in porous media and its non-isothermal counterpart is established. The contrast between these two distinct problems, in terms of the different time scales involved, is evident from the results. While the thermal expansion is identified as a transient phenomenon, its impact on the post-transient solutions is found to be sensitive to the symmetry of the particular temperature initial conditions that are applied.     

Backward Free Convection Boundary Layers in Porous Media

The well known steady free convection forward boundary layer (FBL) flows ascending over a heated upwards projecting semi-infinite flat plate embedded in a fluid saturated porous medium are compared in this paper to their less well known backward (BBL) counterparts descending over a cooled (also upwards projecting!) semi-infinite flat plate. The circumstance that the definite edge of the plate (x = 0) in the former case is a leading edge and in the latter one a trailing edge, leads to substantially different mathematical and physical features of the FBL and BBL flows, respectively. The paper considers under this aspect the case of similar flows corresponding to surface temperature distributions which are power-law functions of the distance x from the definite edge. For permeable plates the effect of an adequate lateral suction and injection of the fluid is also taken into account. The detailed investigation, however, is restricted to the particular values m = +1 and m = –1/3 of the power-law exponent m, where both FBL and BBL solutions are available in exact analytic form. For each of these values, both exponentially and algebraically decaying BBL solutions were found. In addition, the existence of an exact algebraic BBL solution valid for any value of m is reported.     

A Brief Introduction to Convection in Porous Media

Transport in Porous Media - This paper serves as a brief introduction to the longer introduction provided by the book by Nield and Bejan (NB). Attention is focussed on the modelling of the...     

Experimental Investigation and Modelling of Colloidal Release in Porous Media

Gas production from underground storage reservoirs is sometimes associated with solid particles eroded from the rock matrix. This phenomenon often called sand production can cause damage to the storage equipments, leading the operator to choke the wells and prevent them from producing at full capacity. Colloid release is often associated as a precursor of larger solid production. Indeed, in sandstone storage sites, clay release induced by the presence of condensed water associated with the gas production in the near-wellbore region can be a forecast of intergranular cement erosion. The objective of this work is twofold: firstly to experimentally investigate colloidal particle detachment through ionic strength reduction (absence of salinity of the condensed water) in porous media and secondly to determine its evolution with time and to model it. Laboratory experiments with model systems are developed to reproduce the particle generation and their transport in porous media. The model porous medium is a packed column of two powders: silicon carbide particles of 50 μm and silica particles of 0.5 μm (3% by weight) initially mixed together. Brine flows at different concentrations are imposed through the porous sample and, at very low salt concentration, colloid silica particles are massively released from the medium. Experimental evolutions of the particle concentration with time are compared to solutions of the advection–dispersion equation including first-order source terms for colloid release. The dispersion coefficients of the porous medium have been determined with tracer tests. The experimental results exhibit a different behaviour at short- and long-time intervals and a model has been built to predict the colloid production evolution with the introduction of two different time scales for the eroded rate. The model can be used in a core test to evaluate the amount of detachable fines and the rate of erosion.     

Thermal Analysis of Natural Convection Porous Fins

This work introduces a simple method of analysis to study the performance of porous fins in a natural convection environment. The method is based on using energy balance and Darcy’s model to formulate the heat transfer equation. The thermal performance of porous fins is then studied for three types of fins: long fin, finite-length fin with insulated tip and a finite-length fin with tip exposed to a known convection coefficient. It is found from the analysis that the effect of different design and operating parameters such as: Ra number, Da number, thermal conductivity ratio, Kr and length thickness ratio on the temperature distribution along the fin is grouped into one newly defined parameter called S_H. The effect of the variation of S_H on the porous fin thermal performance is established. The effect of varying the fin length and thermal conductivity ratio on the heat transfer rate from the fin is investigated and compared with that for a solid fin at certain conditions. It is found that the heat transfer rate from porous fin could exceed that of a solid fin. It is also found that increasing the fin length and effective thermal conductivity enhances the heat transfer from the fin up certain limit, where a further increase in these parameters adds no improvement to the fin performance. On Leave from Jordan University of Science and Technology, Irbid-Jordan     

Thermal Convection in a Non-darcy Hemispherical Porous Medium

A simple mathematical theory is proposed to investigate the development of the flow field which is the response of a fluid to the buoyancy force due to the existence of a temperature gradient in a hemispherical fluid-saturated porous medium, assuming the validity of the Brinkman model. The induced flow is assumed to be slow, and Stokes approximation is invoked. It is shown, at all times, the induced fluid motion occurs in the form of eddies on either side of the axis of symmetry. In the steady state, the behavior of the fluid motion on the free surface is similar to that of axial fluid flow.     

Thermal Capacity Effect on Transient Free Convection Adjacent to a Vertical Surface in a Porous Medium

In this paper we analyse how the presence of the thermal capacity of a vertical flat plate of finite thickness, which is embedded in a porous medium affects the transient free convection boundary-layer flow. At the time t = 0, the plate is suddenly loaded internally with a constant heat flux rate q, so that a transient boundary-layer flow is initiated adjacent to the plate. Initially, the transient effects due to the imposition of the uniform heat flux rate at the plate are confined to a thin fluid region near to the surface and are described by a small time solution. These effects continue to penetrate outwards and eventually evolve into a new steady state flow. Analytical solutions have been derived for these transient (small time) and steady state (large time) flow regimes, which are then matched by a numerical solution of the full boundary-layer equations. It has been found that the non-dimensional fluid temperature (or fluid velocity) profiles are reduced when the thermal capacity effects, described by a parameter Q ^*, are reduced. For small values of Q ^*, the approach of these profiles to their steady state values is monotonic. However, for large values of Q ^*, the temperature profiles are observed to locally exceed (pass through a maximum value) the final steady state values at certain distances from the plate. In general, the maxima in the temperature profiles increase in size as Q ^* increases and the time taken to approach the steady state solutions increases significantly.     

Radiative Effect on Natural Convection Flows in Porous Media

A regular two-parameter perturbation analysis based upon the boundary layer approximation is presented here to study the radiative effects of both first- and second-order resistances due to a solid matrix on the natural convection flows in porous media. Four different flows have been studied, those adjacent to an isothermal surface, a uniform heat flux surface, a plane plume and the flow generated from a horizontal line energy source on a vertical adiabatic surface. The first-order perturbation quantities are presented for all these flows. Numerical results for the four conditions with various radiation parameters are tabulated.     

Experimental and Numerical Study of the Onset of Transient Natural Convection in a Fractured Porous Medium

Transport in Porous Media - In this study, analysis of transient natural convection in a porous medium with a vertical fracture across it in a rectangular enclosure was performed experimentally and...     

Study of the Effect of Thermal Dispersion on Internal Natural Convection in Porous Media Using Fourier Series

Transport in Porous Media - Natural convection in a porous enclosure in the presence of thermal dispersion is investigated. The Fourier–Galerkin (FG) spectral element method is adapted to...     

A Two-Equation Analysis of Convection Heat Transfer in Porous Media

This paper presents a two-equation analysis on the convection heat transfer in porous media based on the modeling developed by Carbonell and Whitaker (1984). The porous system under consideration is bounded by two parallel walls and heated uniformly from one side surface. The Darcy flow is imposed and the fully developed heat transfer is assumed. General solutions, which take into account the additional convective and conductive terms, are obtained for the temperature fields and the Nusselt number. The detailed studies are presented for the porous systems characterized by consolidated and unconsolidated circular unit cells. The results show that, for the consolidated unit cell case, a prediction without the additional convective term overestimates the heat transfer, while for the unconsolidated unit cell case, this effect is negligible. The additional conductive terms are also examined and found to act conventionally as part of the conductive terms.     

Controlling Chaos in Porous Media Convection by Using Feedback Control

A method of using feedback control to promote or suppress the transition to chaos in porous media convection is demonstrated in this article. A feedback control suggested by Mahmud and Hashim (Transp Porous Media, doi:10.1007/s11242-009-9511-1, 2010) is used in the present article to provide a comparison between an analytical expression for the transition point to chaos and numerical results. In addition, it is shown that such a feedback control can be applied as an excellent practical means for controlling (suppressing or promoting) chaos by using a transformation made by Magyari (Transp Porous Media, doi:10.1007/s11242-009-9511-1, 2010). The latter shows that Mahmud and Hashim (Transp Porous Media, doi:10.1007/s11242-009-9511-1, 2010) model can be transformed into Vadasz-Olek’s model (Transp Porous Media 37(1):69–91, 1999a) through a simple transformation of variables implying that the main effect the feedback control has on the solution is equivalent to altering the initial conditions. The theoretical and practical significance of such an equivalent alteration of the initial conditions is presented and discussed.     

On the Nield-Kuznetsov Theory for Convection in Bidispersive Porous Media

We revisit the problem of thermal convection in a bidispersive porous medium, first addressed by Nield and Kuznetsov (Int. J. Heat Mass Transfer, 49: 3068–3074, 2006). We investigate the possibility of oscillatory convection by using a highly accurate Chebyshev tau numerical method. We also develop a nonlinear energy stability theory for the same problem. This yields a global stability threshold below which instabilities cannot arise. These thresholds together with the linear instability boundaries yield a zone where thermal instability may be found. The results and theory of Nield and Kuznetsov (Int. J. Heat Mass Transfer, 49: 3068–3074, 2006) are thus proven to be a highly important development in the modern theory of designer porous materials, cf. Nield and Bejan (Convection in Porous Media, Springer, New York, 2006), pp. 94–97. This work was supported in part by a Research Project Grant of the Leverhulme Trust—Grant Number F/00128/AK.     

Combined Free and Forced Convection in Vertical Channels of Porous Media

In this paper we investigate the combined free and forced convection of a fully developed Newtonian fluid within a vertical channel composed of porous media when viscous dissipation effects are taken into consideration. The flow is analysed in the region of a first critical Rayleigh number in order to interpret the multiple-valued solutions and discuss their validity. The governing fourth-order, ordinary differential equation, which contains the Darcy and the viscous dissipation terms, is solved analytically using perturbation techniques and numerically using D02HBF NAG Library. A detailed investigation of the governing O.D.E. is performed on both clear fluid and porous medium for various values of the viscous dissipation parameter, , when the wall temperature decreases linearly with height, and the pressure gradient is both above and below its hydrostatic value. Although mathematically the results in all cases show that there are two solution branches, producing four possible solutions, the study of the velocity and buoyancy profiles together with the Darcy effect indicate that only one of the two solutions at any value of the Rayleigh number appears to be physically acceptable. It is shown that the effect of the Darcy number decreases as the critical Rayleigh numbers increase.     

Experimental Investigation of Thermo-Oscillatory Convection

The thermal convection in an air column oscillating with a high frequency in a plane channel whose boundaries are isothermal and have different temperatures is investigated. The experiments were performed for various channel orientations and for a wide range of nondimensional governing parameters, i.e. the gravitational Rayleigh number and the thermo-oscillatory parameter. As follows from the experimental results, for relatively large oscillation amplitudes the latter parameter characterizes the average action of high-frequency oscillations on a non-isothermal incompressible fluid. The regions in which either the thermo-oscillatory or gravitational mechanism of thermal convection predominates are determined. The threshold of excitation of thermo-oscillatory convection under weightlessness conditions is found.