Modeling Kinetic Interphase Mass Transfer for Two-Phase Flow in Porous Media Including Fluid–Fluid Interfacial Area

Interphase mass transfer in porous media takes place across fluid–fluid interfaces. At the field scale, this is almost always a kinetic process and its rate is highly dependent on the amount of fluid–fluid interfacial area. Having no means to determine the interfacial area, modelers usually either neglect kinetics of mass transfer and assume local equilibrium between phases or they estimate interfacial area using lumped parameter approaches (in DNAPL pool dissolution) or a dual domain approach (for air sparging). However, none of these approaches include a physical determination of interfacial area or accounts for its role for interphase mass transfer. In this work, we propose a new formulation of two-phase flow with interphase mass transfer, which is based on thermodynamic principles. This approach comprises a mass balance for each component in each phase and a mass balance for specific interfacial area. The system is closed by a relationship among capillary pressure, interfacial area, and saturation. We compare our approach to an equilibrium model by showing simulation results for an air–water system. We show that the new approach is capable of modeling kinetic interphase mass exchange for a two-phase system and that mass transfer correlates with the specific interfacial area. By non-dimensionalization of the equations and variation of Peclet and Damköhler number, we make statements about when kinetic interphase mass transfer has to be taken into account by using the new physically based kinetic approach and when the equilibrium model is a reasonable simplification.     

Heat and Mass Transfer in MHD Micropolar Flow Over a Vertical Moving Porous Plate in a Porous Medium

An analysis is presented for the problem of free convection with mass transfer flow for a micropolar fluid via a porous medium bounded by a semi-infinite vertical porous plate in the presence of a transverse magnetic field. The plate moves with constant velocity in the longitudinal direction, and the free stream velocity follows an exponentially small perturbation law. A uniform magnetic field acts perpendicularly to the porous surface in which absorbs the micropolar fluid with a suction velocity varying with time. Numerical results of velocity distribution of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. Also, the results of the skin-friction coefficient, the couple stress coefficient, the rate of the heat and mass transfers at the wall are prepared with various values of fluid properties and flow conditions.     

Numerical simulation of fluid flow in a fractured porous medium

Various approaches to study the fluid flow in a fractured porous medium are discussed. Three approaches are compared by the example of axisymmetric flow in a fluid conducting layer. In the first approach, an elastic flow regime (Terzaghi’s model) is considered for a layer with homogenized properties. In the second approach, the problem is formulated in terms of two unknown functions of pressure averaged over fracture cracks and pores. In the framework of the third approach, a two-scale flow model in which the fluid flow through pores is limited by the size of each porous block is proposed.     

化学反应对竖直平板边界磁流体动力学微极流体滑流的影响

Heat and mass transfer effects on the unsteady flow of a micropolar fluid through a porous medium bounded by a semi-infinite vertical plate in a slip-flow regime are studied taking into account a homogeneous chemical reaction of the first order. A uniform magnetic field acts perpendicular to the porous surface absorb micropolar fluid with a suction velocity varying with time. The free stream velocity follows an exponentially increasing or decreasing small perturbation law. Using the approximate method, the expressions for the velocity microrotation, temperature, and concentration are obtained. Futher, the results of the skin friction coefficient, the couple stress coefficient, and the rate of heat and mass transfer at the wall are presented with various values of fluid properties and flow conditions.     

State space approach to unsteady magnetohydrodynamics natural convection heat and mass transfer through a porous medium saturated with a viscoelastic fluid

A technique of the state space approach and the inversion of the Laplace transform method are applied to dimensionless equations of an unsteady one-dimensional boundary-layer flow due to heat and mass transfer through a porous medium saturated with a viscoelastic fluid bounded by an infinite vertical plate in the presence of a uniform magnetic field is described. Complete analytical solutions for the temperature, concentration, velocity, and induced magnetic and electric fields are presented. The inversion of the Laplace transforms is carried out by using a numerical approach. The proposed method is used to solve two problems: boundary-layer flow in a viscoelastic fluid near a vertical wall subjected to the initial conditions of a stepwise temperature and concentration and viscoelastic fluid flow between two vertical walls. The solutions are found to be dependent on the governing parameters including the Prandtl number, the Schmidt number, the Grashof number, reaction rate coefficient, viscoelastic parameter, and permeability of the porous medium. Effects of these major parameters on the transport behavior are investigated methodically, and typical results are illustrated to reveal the tendency of the solutions. Representative results are presented for the velocity, temperature, concentration, and induced magnetic and electric field distributions, as well as the local skin-friction coefficient and the local Nusselt and Sherwood numbers.     

Upscaling LBM-TPM simulation approach of Darcy and non-Darcy fluid flow in deformable,heterogeneous porous media

This paper presents a numerical approach for the simulation of fluid flow through porous media by proposing a theoretical and numerical meso-to-macro multiscale framework, which combines the advantages of the lattice Boltzmann method (LBM) with the continuum Theory of Porous Media (TPM) to efficiently and accurately model fluid transport in heterogeneous porous media. In particular, LBM presents an alternative to experiments by studying the flow from a mesoscopic perspective, which in turn, allows the derivation of the material parameters needed for simulating the flow in the macroscopic TPM model. In this work, a meso-macro hierarchic upscaling scheme is applied to investigate the deformation-dependent intrinsic permeability properties and the Darcy/non-Darcy fluid flow regime. Concerning the mesoscale, the intrinsic permeability of the porous domain is computed by means of the LBM model at the first stage. Subsequently, deformation of the medium takes place in furtherance of determining the relation of the aforementioned deformation dependency. Thereupon, these findings are input into the TPM model in order to compute the primary unknown variables, where special focus is laid on the stability challenges in the compaction and near compaction states. With respect to the criteria of non-Darcy fluid flow, the conditions of its onset, i.e. the induced pressure gradient and mean fluid flow velocity, are computed as well using the LBM solver and conveyed afterwards to the macroscopic TPM model. Herein, the non-Darcy intrinsic permeability has been investigated in the TPM approach based on the Forchheimer equation. Simulations done on a synthetic porous micro-structure show that the combined framework proved to stand well between the two approaches.     

基于蒙特卡罗随机有限元方法的随机多孔介质内流体自然对流不确定性研究

为分析孔隙率不确定性对多孔介质方腔内自然对流换热的影响,发展了一种基于KL（Karhunen-Loeve展开）-蒙特卡罗随机有限元算法的随机多孔介质内自然对流不确定性分析数理模型及有限元数值模拟程序框架。通过K-L展开及基于拉丁抽样法生成多孔介质孔隙率随机实现,并耦合多孔介质自然对流有限元程序,进行随机多孔介质内自然对流传热数值模拟,得出了多孔介质内流场与温度场平均值与标准偏差,并分析了孔隙率不确定性条件下Da数对Nu数的影响。结果表明,孔隙率不确定性对多孔介质方腔内自然对流有重要影响。随机多孔介质内流场及温度场与确定性条件下的流场及温度场存在一定偏差,Nu数标准偏差随着Da的增大先增大后减小。     

Effect of variable porosity on laminar convection in a uniformly heated vertical porous channel

The present investigation is devoted to the study of fully developed mixed convective flow through a vertical porous channel. The lateral variations of porosity and thermal diffusivity in the bed near the wall, are approximated by exponential functions. The correlation between permeability and porosity is brought through Kozney-Carman approximation. The volume averaged one dimensional low speed momentum equation proposed by Vafai is employed for the analysis of the problem. Results are obtained for steady heating of ascending cold fluid and steady cooling of ascending hot fluid. For the above physical situations it is observed that the heat transfer rate, and ratio of friction factor increases with increase in porous parameter, whereas the ratio of mass flow rate decreases with increase in porous parameter. The velocity profiles exhibit hydrodynamic channelling and peak velocity shifts towards the wall for higher values of the porous parameter. For steady heating of ascending could fluid increase in Rayleigh number enhances the heat transfer rate, and mass flow rate, while it reduces the ratio of friction factor. An opposite trend is observed for the case of steady cooling of ascending hot fluid.     

Natural convection due to heat and mass transfer in a composite system

A numerical study is performed to analyse heat and mass transfer phenomena due to natural convection in a composite cavity containing a fluid layer overlying a porous layer saturated with the same fluid. The flow in the porous region is modelled using Brinkman–Forchheimer-extended Darcy model that includes both the effect of macroscopic shear (Brinkman effect) and flow inertia (Forchheimer effect). The vertical walls of the two-dimensional enclosure are isothermal whilst the horizontal walls are adiabatic. The two regions are coupled by equating the velocity and stress components at the interface. The resulting coupled equations in non-dimensional form are solved by an alternating direction implicit method by transforming them into parabolic form by the addition of false transient terms. The numerical results show that the amount of fluid penetration into the porous layer depends strongly upon the Darcy, thermal and solutal Rayleigh numbers. Average Nusselt number decreases while average Sherwood number increases with an increase of the Lewis number. The transfer of heat and mass on the heated wall near the interface depends strongly on the Darcy number. Received on 11 May 1998     

Numerical Simulation of Turbulent Flow and Heat Transfer in a Three-Dimensional Channel Coupled with Flow Through Porous Structures

This study investigates numerically the turbulent flow and heat transfer characteristics of a T-junction mixing, where a porous media flow is vertically discharged in a 3D fully developed channel flow. The fluid equations for the porous medium are solved in a pore structure level using an Speziale, Sarkar and Gatski turbulence model and validated with open literature data. Overall, two types of porous structures, consisted of square pores, are investigated over a wide range of Reynolds numbers: an in-line and a staggered pore structure arrangement. The flow patterns, including the reattachment length in the channel, the velocity field inside the porous medium as well as the fluctuation velocity at the interface, are found to be strongly affected by the velocity ratio between the transversely interacting flow streams. In addition, the heat transfer examination of the flow domain reveals that the temperature distribution in the porous structure is more uniform for the staggered array. The local heat transfer distributions inside the porous structure are also studied, and the general heat transfer rates are correlated in terms of area-averaged Nusselt number accounting for the effects of Reynolds number, velocity ratio as well as the geometrical arrangement of the porous structures.     

Conjugate model for heat and mass transfer of porous wall in the high temperature gas flow

Heat and mass transfer of a porous permeable wall in a high temperature gas dynamical flow is considered. Numerical simulation is conducted on the ground of the conjugate mathematical model which includes filtration and heat transfer equations in a porous body and boundary layer equations on its surface. Such an approach enables one to take into account complex interaction between heat and mass transfer in the gasdynamical flow and in the structure subjected to this flow. The main attention is given to the impact of the intraporous heat transfer intensity on the transpiration cooling efficiency. The project supported by the National Natural Science Foundation of China (19889209) and Russian Foundation for Basic Research (97-02-16943)     

Chemically reactive hydromagnetic flow of a second-grade fluid in a semi-porous channel

An analysis of a second-grade fluid in a semi-porous channel in the presence of a chemical reaction is carried out to study the effects of mass transfer and magnetohydrodynamics. The upper wall of the channel is porous, while the lower wall is impermeable. The basic governing flow equations are transformed into a set of nonlinear ordinary differential equations by means of a similarity transformation. An approximate analytical solution of nonlinear differential equations is constructed by using the homotopy analysis method. The features of the flow and concentration fields are analyzed for various problem parameters. Numerical values of the skin friction coefficient and the rate of mass transfer at the wall are found.     

Non-Darcy double-diffusive natural convection in axisymmetric fluid saturated porous cavities

Double-diffusive natural convection in a fluid saturated porous medium has been investigated using the finite element method. A generalised porous medium model is used to study both Darcy and non-Darcy flow regimes in an axisymmetric cavity. Results indicate that the Darcy number should be a separate parameter to understand flow characteristics in non-Darcy regime. The influence of porosity on heat and mass transfer is significant and the transport rates may differ by 25% or more, at higher Darcy and Rayleigh numbers. When compared with the Darcy and other specialised models of Brinkman and Forchheimer, the present generalised model predicts the least heat and mass transfer rates. It is also observed that an increase in radius ratio leads to higher Nusselt and Sherwood numbers along the inner wall.     

Fully-developed conjugate heat transfer in porous media with uniform heating

We propose a computational method for approximating the heat transfer coefficient of fully-developed flow in porous media. For a representative elementary volume of the porous medium we develop a transport model subject to periodic boundary conditions that describes incompressible fluid flow through a uniformly heated porous solid. The transport model uses a pair of pore-scale energy equations to describe conjugate heat transfer. With this approach, the effect of solid and fluid material properties, such as volumetric heat capacity and thermal conductivity, on the overall heat transfer coefficient can be investigated. To cope with geometrically complex domains we develop a numerical method for solving the transport equations on a Cartesian grid. The computational method provides a means for approximating the heat transfer coefficient of porous media where the heat generated in the solid varies “slowly” with respect to the space and time scales of the developing fluid. We validate the proposed method by computing the Nusselt number for fully developed laminar flow in tubes of rectangular cross section with uniform wall heat flux. Detailed results on the variation of the Nusselt number with system parameters are presented for two structured models of porous media: an inline and a staggered arrangement of square rods. For these configurations a comparison is made with literature on fully-developed flows with isothermal walls.     

Convective mass transfer across fluid interfaces in straight angular pores

Steady convective mass transfer to or from fluid interfaces in pores of angular cross-section is theoretically investigated. This situation is relevant to a variety of mass transport process in porous media, including the fate of residual non-aqueous phase liquid ganglia and gas bubbles. The model incorporates the essential physics of capillarity and solute mass transfer by convection and diffusion in corner fluid filaments. The geometry of the corner filaments, characterized by the fluid–fluid contact angle, the corner half-angle and the interface meniscus curvature, is accounted for. Boundary conditions of zero surface shear (‘perfect-slip’) and infinite surface shear (‘no-slip’) at the fluid–fluid interface are considered. The governing equations for laminar flow within the corner filament and convective diffusion to or from the fluid–fluid interface are solved using finite-element methods. Flow computations are verified by comparing the dimensionless resistance factor and hydraulic conductance of corner filaments against recent numerical solutions by Patzek and Kristensen (J. Colloid Interface Sci 236, 305–317 2001). Novel results are obtained for the average effluent concentration as a function of flow geometry and pore-scale Peclet number. These results are correlated to a characteristic corner length and local pore-scale Peclet number using empirical equations appropriate for implementation in pore network models. Finally, a previously published “2D-slit” approximation to the problem at hand is checked and found to be in considerable error.     

Poromechanics approach for modeling closed-cell porous materials with soft matrices

The main purpose of this paper is the formulation of a modeling framework for closed-cell porous materials by means of the nowadays classical approach of poromechanics. The cavities being saturated with a single fluid phase, Biot’s theory drastically simplifies since no fluid diffusion is present in this case. Nevertheless, the fluid contribution to the macroscopic behavior is still present and the fluid mass conservation controls the pore pressure that contributes to the total stress at the macro-scale. A constitutive law for the saturating fluid is introduced that encompasses both ideal gas and incompressible fluids as particular cases. The approach is built around the unified framework of continuum thermodynamics which is crucial in setting the convenient form for the state laws. As the finite strain range is of concern, the modeling strongly depends on the Eulerian porosity law adopted in the spatial configuration. Two model examples are discussed with parametric studies to show the effectiveness of the proposed approach.     

Phase-field modeling of hydraulic fracture

In this work a theoretical framework implementing the phase-field approach to fracture is used to couple the physics of flow through porous media and cracks with the mechanics of fracture. The main modeling challenge addressed in this work, which is a challenge for all diffuse crack representations, is on how to allow for the flow of fluid and the action of fluid pressure on the aggregate within the diffuse damage zone of the cracks. The theory is constructed by presenting the general physical balance laws and conducting a consistent thermodynamic analysis to constrain the constitutive relationships. Constitutive equations that reproduce the desired responses at the various limits of the phase-field parameter are proposed in order to capture Darcy-type flow in the intact porous medium and Stokes-type flow within open cracks. A finite element formulation for the solution of the governing model equations is presented and discussed. Finally, the theoretical and numerical model is shown to compare favorably to several important analytical solutions. More complex and interesting calculations are also presented to illustrate some of the advantageous features of the approach.     

MHD flow and mass transfer of chemically reactive upper convected Maxwell fluid past porous surface

The magnetohydrodynamic (MHD) flow and mass transfer of an electrically conducting upper convected Maxwell (UCM) fluid at a porous surface are studied in the presence of a chemically reactive species. The governing nonlinear partial differential equations along with the appropriate boundary conditions are transformed into nonlinear ordinary differential equations and numerically solved by the Keller-box method. The effects of various physical parameters on the flow and mass transfer characteristics are graphically presented and discussed. It is observed that the order of the chemical reaction is to increase the thickness of the diffusion boundary layer. Also, the mass transfer rate strongly depends on the Schmidt number and the reaction rate parameter. Furthermore, available results in the literature are obtained as a special case.     

Macroscopic Two-phase Flow in Porous Media Assuming the Diffuse-interface Model at Pore Level

The paper presents a model for two-phase flow, where liquid and gas are treated as one fluid with variable density. A one-component fluid and the diffuse-interface model for two-phase flow are assumed at pore level. The wetting properties of the fluid are described by the Cahn theory. Macroscopic equations are deduced in the framework of the Marle formalism. It is shown that two-phase flow in porous media can be described by the Cahn–Hilliard equation for the mass density. The concept of relative permeability is not needed. For non-neutral wetting, it is shown that a capillary pressure exists but that it is not a function of state. Two numerical illustrations are presented, one of them showing that the model is, at least in a simple steady-state situation, compatible with the generalized two-continuum model.     

多孔介质——自由流界面应力跳跃条件下流动特性解析解

对非对称多孔介质--自由流复合通道内多孔介质内部及多孔介质与自由流体界面处复杂质量、动量输运特性进行研究. 在多孔介质区采用Brinkman-extended Darcy模型并结合速度连续,剪切应力跳跃的界面条件对此复合通道内流体的传递现象进行求解,提出了考虑界面应力跳跃时非对称复合通道各区域流体运动速度及摩擦系数的解析式,分析了界面应力跳跃系数,达西数及无量纲多孔层偏心厚度对流体速度及摩擦系数的影响. 结果表明:改变界面性质可在一定条件下明显控制各区域流体速度分布;在达西数、多孔层偏心厚度一定情况下,界面应力系数的增大会使界面流速减小,而使流体摩擦系数增大,特别是界面应力系数小于0的情况下变化更明显,此时若不考虑界面应力系数则会造成较大误差. 当界面应力系数及多孔层偏心厚度均为较小负数值时,改变多孔层偏心厚度对界面速度的影响要大于改变界面应力系数的情况;而当界面应力系数及多孔层偏心厚度为较大正数值时,情况则相反. 较大达西数下,界面应力系数及多孔层偏心厚度对流体摩擦系数的影响均较大,继续减小达西数至一定程度时,界面应力系数对流体摩擦系数的影响可忽略不计而认为只与多孔层偏心厚度相关,且对较大多孔层偏心厚度更敏感.