ONSET OF DOUBLE-DIFFUSIVE CONVECTION IN A HORIZONTAL BRINKMAN CAVITY

This investigation reports on a linear stability analysis of the quiescent state within a horizontal porous cavity subject to vertical gradients of temperature and solute. The fluid motion is modeled using the Brinkman extension of Darcy's law, coupled with energy and species conservation equations. The horizontal boundaries are considered rigid-rigid, rigid-free, or free-free. Mixed thermal and solutal boundary conditions, of Dirichlet and Neumann types, are considered. The thresholds for monotonic and oscillatory convection instabilities are determined explicitly in terms of the governing parameters of the problem. The results for a viscous fluid and the Darcy porous medium emerge from the present analysis as limiting cases.     

Combined effects of internal heat generation and higher order chemical reaction on the non‐darcian forced convective flow of a viscous incompressible fluid with variable viscosity and thermal conductivity over a stretching surface embedded in a porous medium

In this paper, we study the combined effects of internal heat generation and higher order chemical reaction on a steady two‐dimensional non‐Darcian forced convective flow of a viscous incompressible fluid with variable dynamic viscosity and thermal conductivity in a fluid saturated porous medium passing over a linear stretching sheet. Using similarity transformations, the governing nonlinear‐coupled partial differential equations are made dimensionless and solved numerically for similarity solutions using very robust computer algebra software Maple 8. The non‐dimensional velocity, temperature and concentration distributions are presented graphically for various pertinent parameters such as relative temperature difference parameter, Darcy number, porosity parameter, reaction rate parameter and the order of the chemical reaction. The variations of Prandtl number and Schmidt number within the boundary layer are also displayed graphically when the fluid dynamic viscosity and thermal conductivity are temperature dependent. From the present numerical computations it is found that Prandtl number as well as Schmidt number must be taken as variables within the flow domain when the fluid's dynamic viscosity and thermal conductivity are variable. In the presence of internal heat generation, dynamic viscosity and thermal conductivity of the fluid are found to be higher than when it is absent. Increasing Darcy number reduces dynamic viscosity as well as thermal conductivity whereas increasing pore size reduces the Schmidt number and increases the Prandtl number within the boundary layer. For higher order reaction the rate of increase in mass transfer function is less compared to the rate of increase for the lower order reaction. © 2011 Canadian Society for Chemical Engineering     

Hydromagnetic non‐Darcy flow,heat and mass transfer over a stretching sheet in the presence of thermal radiation and Ohmic dissipation

A numerical analysis has been carried out to study magnetohydrodynamic boundary layer flow, heat and mass transfer characteristics on steady two‐dimensional flow of an electrically conducting fluid over a stretching sheet embedded in a non‐Darcy porous medium in the presence of thermal radiation and viscous dissipation. The governing partial differential equations are convected into a system of nonlinear ordinary differential equations by similarity transformation and are solved numerically by using the Successive linearisation method, together with the Chebyshev pseudo‐spectral collocation method. The effects of various parameters on the velocity, temperature, and concentration fields as well as on the skin‐friction coefficient are presented graphically and in tabular forms.     

Fully developed mixed convection of a binary fluid in a vertical porous channel

This paper reports an analytical and numerical study of mixed convection heat and mass transfer of a binary fluid in a vertical parallel plate channel filled with a porous medium. The thermal conditions applied on the walls of the system are uniform heat fluxes. Both the cases of double‐diffusion and Soret‐induced convection are considered. The governing equations for the porous medium rely on Darcy's model. The governing parameters for the problem are the Rayleigh number, Ra, Peclet number, Pe, Lewis number, Le, buoyancy ratio, φ, aspect ratio of the channel $A = L'/H'$ and the constant a (a = 0 for double diffusive convection and a = 1 for Soret induced convection). The resulting problem, in the limit of fully developed mixed convection, is solved analytically in closed form. A numerical solution of the full governing equations is demonstrated to be in good agreement with the analytical model. The temperature and velocity fields and the Nusselt and Sherwood numbers are obtained in terms of the governing parameters. The possible existence of reversed flows in the channel is discussed. © 2011 Canadian Society for Chemical Engineering     

Free convective heat and mass transfer in a doubly stratified non-Darcy micropolar fluid

The flow, heat and mass transfer characteristics of the free convection on a vertical plate with uniform and constant heat and mass fluxes in a doubly stratified micropolar fluid saturated non-Darcy porous medium are studied. The nonlinear governing equations and their associated boundary conditions are initially cast into dimensionless forms by pseudo-similarity variables. The resulting system of equations is then solved numerically using the Keller-box method. The numerical results are compared and found to be in good agreement with previously published results as special cases of the present investigation. The effects of the micropolar, Darcy, non-Darcy and stratification parameters on the dimensionless velocity, microrotation, wall temperature, wall concentration, local skin-friction coefficient and wall couple stress coefficient are presented graphically.     

Heterogeneous nanofluids: natural convection heat transfer enhancement

Convective heat transfer using different nanofluid types is investigated. The domain is differentially heated and nanofluids are treated as heterogeneous mixtures with weak solutal diffusivity and possible Soret separation. Owing to the pronounced Soret effect of these materials in combination with a considerable solutal expansion, the resulting solutal buoyancy forces could be significant and interact with the initial thermal convection. A modified formulation taking into account the thermal conductivity, viscosity versus nanofluids type and concentration and the spatial heterogeneous concentration induced by the Soret effect is presented. The obtained results, by solving numerically the full governing equations, are found to be in good agreement with the developed solution based on the scale analysis approach. The resulting convective flows are found to be dependent on the local particle concentration φ and the corresponding solutal to thermal buoyancy ratio N. The induced nanofluid heterogeneity showed a significant heat transfer modification. The heat transfer in natural convection increases with nanoparticle concentration but remains less than the enhancement previously underlined in forced convection case.     

NATURAL CONVECTION OF A BINARY MIXTURE IN A VERTICAL CLOSED ANNULUS

This article reports an analytical and numerical study of natural convection of a binary mixture within a vertical closed annulus. Neumann boundary conditions for temperature are applied to the vertical walls of the enclosure, while the short walls are insulated. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the vertical walls (double-diffusive convection, a = 0) or by temperature gradients (Soret effect, a = 1). The governing parameters for the problem are the thermal Rayleigh number R_T, Prandtl number Pr, Lewis number Le, buoyancy ratio ?, aspect ratio A, constant a, and curvature parameter η. An analytical solution, based on the assumption of parallel flow over a large portion of enclosure, is derived. Numerical confirmation of the analytical results is also presented.     

CHEMICAL REACTION ON NON-LINEAR BOUNDARY LAYER FLOW OVER A POROUS WEDGE WITH VARIABLE STREAM CONDITIONS

An analysis is presented to investigate the effects of variable viscosity and thermal stratification on non-Darcy MHD mixed convective heat and mass transfer of a viscous, incompressible, and electrically conducting fluid past a porous wedge in the presence of chemical reaction. The wall of the wedge is embedded in a uniform non-Darcian porous medium in order to allow for possible fluid wall suction or injection. The governing partial differential equations of the problem, subjected to their boundary conditions, are solved numerically by applying an efficient solution scheme for local nonsimilarity boundary layer analysis. Numerical calculations up to third-order level of truncation are carried out for different values of dimensionless parameters. The results are presented graphically, and the conclusion is drawn that the flow field and other quantities of physical interest are significantly influenced by these parameters. The results are compared with those known from the literature, and excellent agreement between the results is obtained.     

SORET AND DUFOUR EFFECTS IN A FREE CONVECTIVE DOUBLY STRATIFIED FLOW OVER A VERTICAL PLATE WITH CHEMICAL REACTION

An investigation was performed to study the influence of thermo-diffusion and diffusion-thermo effects in the transient, free convective flow of a viscous, incompressible, and doubly stratified fluid past an isothermal vertical plate in the presence of first-order chemical reaction. The governing boundary layer equations were solved numerically using an implicit finite difference scheme of the Crank-Nicolson type. The effects of the Soret number, Dufour number, thermal stratification parameter, mass stratification parameter, and chemical reaction parameter are analyzed and presented graphically. Also, the influence of the parameters on local as well the average skin-friction coefficient and the rate of heat and mass transfer are analyzed and discussed. The results are compared with particular solutions available in the literature. The present results are found to be in good agreement with the existing solution.     

Natural convection in a vertical porous cavity filled with a non‐newtonian binary fluid

Numerical and analytical study of natural convection in a vertical porous cavity filled with a non‐Newtonian binary fluid is presented. The density variation is taken into account by the Boussinesq approximation. A power‐law model is used to characterize the non‐Newtonian fluid behavior. Neumann boundary conditions for temperature are applied to the vertical walls of the enclosure, while the two horizontal ones are assumed impermeable and insulated. Both double‐diffusive convection (a = 0) and Soret‐induced convection (a = 1) are considered. Scale analysis is presented for the two extreme cases of heat‐driven and solute‐driven natural convection. For convection in a thin vertical layer (A ? 1), a semianalytical solution for the stream function, temperature, and solute fields, Nusselt and Sherwood numbers are obtained using a parallel flow approximation in the core region of the cavity and an integral form of the energy and constituent equations. Numerical results of the full governing equations show the effects of the governing parameters, namely the thermal Rayleigh number, R_T, the Lewis number, Le, the buoyancy ratio, φ, the power‐law index, n, and the integer number a. A good agreement between the analytical predictions and the numerical simulations is obtained. © 2012 American Institute of Chemical Engineers AIChE J, 58: 1704–1716, 2012     

A GENERALIZED DIFFERENTIAL TRANSFORM METHOD FOR COMBINED FREE AND FORCED CONVECTION FLOW ABOUT INCLINED SURFACES IN POROUS MEDIA

In this article, we apply the differential transform method (DTM) to obtain approximate analytical solutions of combined free and forced (mixed) convection about inclined surfaces (or wedges) in a saturated porous medium. Both aiding and opposing flows are considered. It is found that the parameter mixed convection from inclined surfaces in porous media is Gr/Re, where Gr is the local Grashof number and Re is the local Reynolds number. DTM solutions are obtained for mixed convection from an isothermal vertical flat plate as well as an inclined plate with constant heat flux having an inclination of 45°. Temperature and velocity profiles for these two cases at different values of Gr/Re are presented. The similarity transformations are applied to reduce the governing partial differential equations (PDEs) to a set of nonlinear coupled ordinary differential equations (ODEs) in dimensionless form. DTM is used to solve the nonlinear differential equations governing the problem in the form of series with easily computable terms. Thereafter a Padé approximant is applied to the solutions to increase the convergence of the given series. Excellent correlation between DTM-Padé and numerical quadrature (shooting) solutions is achieved. The DTM-Padé simulation is shown to be a robust benchmarking tool providing an excellent means of validation of numerical methods. The study has applications in geothermal energy systems, chemical engineering filtration systems, and packed beds.     

NATURAL CONVECTION OF A BINARY MIXTURE IN A VERTICAL CLOSED ANNULUS

This article reports an analytical and numerical study of natural convection of a binary mixture within a vertical closed annulus. Neumann boundary conditions for temperature are applied to the vertical walls of the enclosure, while the short walls are insulated. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the vertical walls (double-diffusive convection, a = 0) or by temperature gradients (Soret effect, a = 1). The governing parameters for the problem are the thermal Rayleigh number R_T, Prandtl number Pr, Lewis number Le, buoyancy ratio ϕ, aspect ratio A, constant a, and curvature parameter η. An analytical solution, based on the assumption of parallel flow over a large portion of enclosure, is derived. Numerical confirmation of the analytical results is also presented.     

Thermal Diffusion and MHD Effects on Combined Free‐Forced Convection and Mass Transfer of a Viscous Fluid Flow Through a Porous Medium with Heat Generation

The problem of thermal diffusion and magnetic field effects on combined free‐forced convection and mass transfer flow past a vertical porous flat plate, in the presence of heat generation is studied numerically. The governing momentum, energy and concentration equations are converted into a system of nonlinear ordinary differential equations by means of similarity transformations. The resulting system of coupled nonlinear ordinary differential equations is solved numerically by using the Shooting method. Numerical results are presented for velocity, temperature and concentration profiles within the boundary layer for different parameters of the problem including suction parameter, heat generation parameter, Soret number, Dufour number, magnetic parameter, etc. In addition, the effects of the pertinent parameters on the skin friction and the rates of heat and mass transfer are discussed numerically and illustrated graphically.     

GROUP THEORY AND DIFFERENTIAL TRANSFORM ANALYSIS OF MIXED CONVECTIVE HEAT AND MASS TRANSFER FROM A HORIZONTAL SURFACE WITH CHEMICAL REACTION EFFECTS

Viscous, laminar mixed convection boundary-layer flow over a horizontal plate, with chemical reaction, is considered. The governing equations are expressed in nondimensional form. Group theory is employed to determine the invariant solutions of these equations under a particular continuous one-parameter group. Series solutions of the transformed coupled system of equations are then generated for velocity, temperature, and concentration functions using the Differential Transform Method (DTM) with Padé approximants. The influence of thermal buoyancy parameter, species buoyancy parameter, chemical reaction parameter, order of chemical reaction, Prandtl number, and Schmidt number on the flow characteristics is evaluated in detail The obtained solutions are verified by comparison with the numerical shooting quadrature results. Applications of the study arise in sheet materials processing, bio-reactors, and catalytic systems in chemical engineering.     

MIXED CONVECTION BOUNDARY LAYER FLOW OF A MICROPOLAR FLUID ON A HORIZONTAL PLATE

An analysis is performed to study the heat transfer characteristics of laminar mixed convection boundary layer flow of a micropolar fluid past a semi-infinite horizontal fiat plate with variable surface heat flux. A nonsimilar mixed convection parameter £ and a pseudosimilarity variable v are introduced to cast the governing boundary layer equations into a system of dimensionless equations, which are solved numerically using the finite difference method. A single mixed convection parameter is used to cover the entire regime of mixed convection from the pure forced convection limit to the pure free convection limit. The effect of material parameters, the power-law variation of surface heat flux, nonsimilar mixed convection parameter and Prandtl number are considered. The micropolar fluids are observed to display drag reduction and reduced surface heat transfer rate when compared to Newtonian fluids. The effect of the buoyancy force results in the enhancements of friction factor, heat transfer rate and wall couple stress.     

Transient free convection flow over an isothemal vertical cylinder with temperature dependent viscosity

A numerical analysis is performed to study the influence of temperature-dependent viscosity and Prandtl number on unsteady laminar free convection flow over a vertical cylinder. The governing boundary layer equations are converted into a non-dimensional form and a Crank-Nicolson type of implicit finite-difference method is used to solve the governing non-linear set of equations. Numerical results are obtained and presented for different viscosityvariation parameters and Prandtl numbers. Transient effects of velocity and temperature are analyzed. The heat transfer characteristics against the viscosity-variation parameter are analyzed with the help of average skin-friction and Nusselt number and are shown graphically.     

Transient combined radiative and conductive heat transfer in plastics

By the addition of metal and oxide particles to plastics, thermal transport properties, heat capacity, and density of polymers can be varied systematically. Radiation effects in a particle filled with various fillers become more important and transient temperature responses including radiation can be significantly different from those by conduction alone. Transient combined conduction–radiation heat transfer is analyzed in a non‐gray layer of plastics, submitted to several thermal and radiative boundary conditions. The numerical method is an implicit finite difference procedure with nonuniform space and time increments. Coupling problems for the prescribed temperatures, prescribed radiative–conductive heat exchange laws, and mixed kind thermal boundary conditions are worked out for opaque as well as vitreous interfaces with specular reflections. Solutions are given to demonstrate the effect of different parameters, such as radiation–conduction parameter, radiation–convection parameter, and emissivity of the surfaces on temperature distribution and heat flux profiles across the layer. J. VINYL. ADDIT. TECHNOL., 11:28–37, 2005. © 2005 Society of Plastics Engineers     

NETWORK NUMERICAL SIMULATION OF HYDROMAGNETIC MARANGONI MIXED CONVECTION BOUNDARY LAYERS

The study of a steady coupled dissipative layer, known as the Mangaroni mixed convection boundary layer, in the presence of a magnetic field is presented. The mixed convection boundary layer is generated when in addition to Marangoni (thermocapillary) effects there are also buoyancy effects due to gravity and external pressure gradient effects. In the model considered the Marangoni coupling condition has been included in the boundary conditions at the interface. Similarity transformations are utilized to transform the governing partial differential conservation equations into nondimensional ordinary differential equations in a single independent space variable (η) and solved using the network simulation method (NSM) using an electronic circuit simulator, Pspice. NSM is founded on the classical thermoelectric analogy between thermal and electrical variables. A set of finite-differential equations, one for each control volume, was obtained by spatial discretization of the transformed equations. The solutions obtained are compared with earlier computations using other numerical techniques, showing excellent agreement. The influence of the Marangoni mixed parameter and Hartmann number on the velocity and temperature functions are studied in detail. The effectiveness of utilizing magnetic fields to control heat transfer in Marangoni convection boundary layers is identified. An increase in Hartmann hydromagnetic number (M) is found to strongly decelerate the flow but increase temperatures. An increase in Marangoni mixed convection parameter (λ) for the scenario opposing Marangoni flow (Γ > 0) considerably accelerates the flow but decreases temperatures in the boundary layer. Conversely, an increase in Marangoni mixed convection parameter (λ) for the case favorable to the Marangoni flow (Γ < 0) decelerates the flow but enhances temperatures in the boundary layer. Applications of the model include semiconductor crystal hydromagnetic heat transfer control.     

Effect of a Magnetic Field on a Micropolar Fluid Flow in the Vicinity of an Axisymmetric Stagnation Point on a Circular Cylinder

The effect of a magnetic field on a micropolar fluid flow in the vicinity of an axisymmetric stagnation point on a circular cylinder is studied numerically. The governing conservation equations of continuity, momentum and angular momentum are partial differential equations which are transformed into a system of ordinary differential equations by using the usual similarity transformations. The resulting system of coupled non‐linear ordinary differential equations is solved numerically by using the shooting method. The numerical results indicate the velocity, angular velocity and pressure distributions for different parameters of the problem including Reynolds number, magnetic parameter and dimensionless material properties, etc. In addition, the effect of the pertinent parameters on the local skin friction coefficient and the couple stress are discussed numerically and illustrated graphically.     

IMPACT OF THERMAL DISPERSION DURING FORCED CONVECTION CONDENSATION IN A THIN POROUS*/FLUID COMPOSITE SYSTEM

The effect of thermal dispersion during laminar forced convection filmwise condensation within a thin porous/fluid composite system is examined numerically. The model simulates two-dimensional condensation within a very permeable and highly conductive thin porous-layer coated surface. The local volume-averaging technique is utilized to establish the energy equation and to account for the thermal dispersion effect. The Darcy-Brinkman-Forchheimer model is employed to describe the flow field in the porous layer while classical boundary layer equations are used in the pure condensate region. The numerical results, which detail the dependence of the heat transfer rate and temperature field on the governing parameters (e.g., Reynolds number, Rayleigh number, Darcy number, Prandtl number, thermal dispersion coefficient, as well as porous coating thickness and thermal conductivity ratio), are calculated using a finite difference scheme. It is found that due to the better mixing of the thermal dispersion effect, the heat transfer rate is greatly increased and the effect becomes more pronounced as the Reynolds number increases. The results of this study provide valuable fundamental predictions of enhanced film condensation that can be used in a number of practical thermal engineering applications.