NUMERICAL MODELING OF MHD CONVECTIVE HEAT AND MASS TRANSFER IN PRESENCE OF FIRST-ORDER CHEMICAL REACTION AND THERMAL RADIATION

An analysis was carried out numerically to study unsteady heat and mass transfer by free convection flow of a viscous, incompressible, electrically conducting Newtonian fluid along a vertical permeable plate under the action of transverse magnetic field taking into account thermal radiation as well as homogeneous chemical reaction of first order. The fluid considered here is an optically thin gray gas, absorbing-emitting radiation, but a non-scattering medium. The porous plate was subjected to a constant suction velocity with variable surface temperature and concentration. The dimensionless governing coupled, nonlinear boundary layer partial differential equations were solved by an efficient, accurate, extensively validated, and unconditionally stable finite difference scheme of the Crank-Nicolson type. The velocity, temperature, and concentration fields were studied for the effects of Hartmann number (M), radiation parameter (R), chemical reaction (K), and Schmidt number (Sc). The local skin friction, Nusselt number, and Sherwood number are also presented and analyzed graphically. It is found that velocity is reduced considerably with a rise in the magnetic body parameter (M), whereas the temperature and concentration are found to be markedly boosted with an increase in the magnetic body parameter (M). An increase in the conduction-radiation parameter (R) is found to escalate the local skin friction (τ), Nusselt number, and concentration, whereas an increase in the conduction-radiation parameter (R) is shown to exert the opposite effect on either velocity or temperature field. Similarly, the local skin friction and the Sherwood number are both considerably increased with an increase in the chemical reaction parameter. Possible applications of the present study include laminar magneto-aerodynamics, materials processing, and MHD propulsion thermo-fluid dynamics.     

DIFFUSION OF CHEMICALLY REACTIVE SPECIES IN BOUNDARY LAYER FLOW OVER A POROUS PLATE IN POROUS MEDIUM

Analysis was made to study the effect of diffusion of chemically reactive species in boundary layer flow of an incompressible fluid over a porous flat plate in porous medium. The first-order chemical reaction was considered and the reaction rate of the reactive species was taken in such a manner that it varied inversely along the plate. Self-similar equations were obtained using similarity transformations and were then solved by the shooting technique using the fourth-order Runge-Kutta method. This analysis revealed that at a fixed point, with increase of the permeability of the porous medium the velocity increases, but the concentration decreases. The suction reduces the thicknesses of momentum and concentration boundary layers but due to blowing the thicknesses become larger. With increase of both the Schmidt number and the reaction rate parameter, the reactive concentration profiles decrease. For a destructive chemical reaction, the concentration profiles exhibit negative value when the Schmidt number is large. It is very significant to note that in some constructive chemical reaction cases the mass absorption at the plate occurs for small Schmidt number.     

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     

EFFECTS OF HEAT GENERATION-ABSORPTION AND CHEMICAL REACTION ON MAGNETOHYDRODYNAMIC FLOW IN A PERMEABLE MOVING PLATE: NETWORK NUMERICAL STUDY

This study addresses the transient heat and mass transfer problem by laminar flow of a Newtonian, viscous electrically conducting, and heat generating/absorbing fluid on a continuously moving permeable surface. It was carried out with the well-known network simulation method. The effect of the parameters involved in the process were analyzed, so the influence of the parameters is represented graphically for several cases. The dimensionless parameters used in this study are: chemical reaction parameter, Prandtl number, Grashof number, absorption coefficient or heat generation, Schmidt number, and Hartmann number. The results presented are important because of the relevance of the conclusions for the steady and transient cases. In the study, the flow was considered laminar and two-dimensional. In addition, the transverse magnetic field is uniform and viscous dissipation effects, Hall effect, and Joule heating are omitted.     

UNSTEADY FLUID AND HEAT FLOW INDUCED BY A STRETCHING SHEET WITH MASS TRANSFER AND CHEMICAL REACTION

The effect of chemical reaction on the flow, heat, and mass transfer within a viscous fluid on an unsteady stretching sheet is examined. The stretching rate, temperature and concentration of the sheet, and the chemical reaction rate are assumed to vary with time. The time-dependent boundary layer equations governing the flow are reduced through a convenient similarity transformation to a set of ordinary differential equations, which are numerically solved by applying the fourth-order Runge-Kutta-Fehlberg scheme with the shooting technique. Results for the velocity, temperature, and concentration distributions as well as the wall temperature and concentration gradients are presented graphically for various values of the unsteadiness parameter A, Prandtl number Pr, Schmidt number Sc, and chemical reaction parameter γ.     

BOUNDARY LAYER FLOW AND DOUBLE DIFFUSION OVER AN UNSTEADY STRETCHING SURFACE WITH HALL EFFECT

The present investigation is concerned with the effect of Hall currents on boundary layer flow, and heat and mass transfer of an electrically conducting fluid over an unsteady stretching sheet in the presence of a strong magnetic field. The electron-atom collision frequency is assumed to be relatively high, so that the Hall effect is assumed to exist, while the induced magnetic field is neglected. The governing time-dependent boundary layer equations for momentum, thermal energy, and concentration are reduced using a similarity transformation to a set of coupled ordinary differential equations. The similarity ordinary differential equations are then solved numerically by the successive linearization method together with the Chebyshev pseudo-spectral collocation method. Effects of the Prandtl number, Pr, Schmidt number, Sc, magnetic field, M, Hall parameter, m, and the unsteadiness parameter, A, on the velocity, temperature, and concentration profiles as well as the local skin friction coefficient and the heat and mass transfer rates are depicted graphically and/or in tabular form. Favorable comparisons with previously published work on various special cases of the problem are also obtained.     

MHD MIXED-CONVECTION INTERACTION WITH THERMAL RADIATION AND nTH ORDER CHEMICAL REACTION PAST A VERTICAL POROUS PLATE EMBEDDED IN A POROUS MEDIUM

This article investigates the hydromagnetic mixed convection heat and mass transfer flow of an incompressible Boussinesq fluid past a vertical porous plate with constant heat flux in the presence of radiative heat transfer in an optically thin environment, viscous dissipation, and an nth order homogeneous chemical reaction between the fluid and the diffusing species. The dimensionless governing equations for this investigation are solved numerically by the fourth-order Runge-Kutta integration scheme along with shooting technique. Numerical data for the local skin-friction coefficient, the plate surface temperature, and the local Sherwood number have been tabulated for various values of parametric conditions. Graphical results for velocity, temperature, and concentration profiles based on the numerical solutions are presented and discussed.     

Thermal radiation effect on unsteady MHD free convection flow past a vertical plate with temperature‐dependent viscosity

This article investigates the influence of radiation and temperature‐dependent viscosity on the problem of unsteady MHD flow and heat transfer of an electrically conducting fluid past an infinite vertical porous plate taking into account the effect of viscous dissipation. The governing equations are converted into a system of nonlinear ordinary differential equations via a local similarity parameter which is taken as a function of time. The resulting system of coupled nonlinear ordinary differential equations is solved numerically using the fourth order Runge–Kutta integration scheme with the shooting method. The numerical results for the velocity and the temperature are displayed graphically showing the effects of various parameters. The results show that increasing the Eckert number and decreasing the viscosity of air leads to a rise in the velocity, while increasing in the magnetic or the radiation parameters is associated with a decrease in the velocity. Also, an increase in the Eckert number leads to an increase in the temperature, whereas an increase in radiation parameter leads to a decrease in the temperature.     

EFFECTS OF HALL CURRENT AND CHEMICAL REACTION ON OSCILLATORY MIXED CONVECTION-RADIATION OF A MICROPOLAR FLUID IN A ROTATING SYSTEM

This article is concerned with the analysis of the effects of thermal radiation on oscillatory mixed convection flow of a micropolar fluid in a rotating frame of reference in the presence of transverse magnetic field and Hall current. The influence of a first-order homogeneous chemical reaction and heat source effects is also analyzed. The governing partial differential equations with the appropriate boundary conditions are reduced to a set of ordinary differential equations using similarity transformations. The dimensionless governing equations for this investigation are solved analytically after using small perturbation approximation. The effects of various parameters on the velocity, temperature, and concentration fields as well as on skin-friction coefficient, Nusselt number, and Sherwood number with their amplitude and phase are discussed in detail. Numerical results are discussed with the help of graphs and tables. Present results are also compared with previously published work.     

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.     

Homotopy analysis of heat and mass transfer boundary layer flow through a non‐porous channel with chemical reaction and heat generation

This paper describes the two‐dimensional flow of an incompressible viscous fluid through a non‐porous channel with heat generation and a chemical reaction. Employing similarity transformations the governing non‐linear partial differential equations are solved both analytically and numerically. Analytically, we used the homotopy analysis method and numerically, we used the Matlab in‐built boundary value solver bvp4c. The effects of the Reynolds number Re, the Eckert number Ec, heat generation parameter δ, chemical reaction parameter γ, and the local Grashof number Gc on the velocity, temperature, and concentration fields are shown through tables and graphs and discussed.     

NATURAL CONVECTION HEAT AND MASS TRANSFER FROM A VERTICAL FLAT PLATE WITH VARIABLE WALL TEMPERATURE AND CONCENTRATION TO POWER-LAW FLUIDS WITH YIELD STRESS IN A POROUS MEDIUM

The problem of natural convection coupled heat and mass transfer of non-Newtonian power law fluids with yield stress from a vertical flat plate in a fluid-saturated porous medium is analyzed under boundary layer approximations. Similarity solutions are obtained for the general case of power law variations of the wall temperature and concentration. Velocity, temperature, and concentration profiles as well as local heat and mass transfer rates are presented and discussed for different values of the rheological parameters of a power-law fluid, the Lewis number, and the thermal and concentration buoyancy ratio.     

Modeling and numerical analysis of nanoliquid (titanium oxide,graphene oxide) flow viscous fluid with second order velocity slip and entropy generation

The prime objective of the present communication is to examine the entropy-optimized second order velocity slip Darcy–Forchheimer hybrid nanofluid flow of viscous material between two rotating disks.Electrical conducting flow is considered and saturated through Darcy–Forchheimer relation. Both the disks are rotating with different angular frequencies and stretches with different rates. Here graphene oxide and titanium dioxide are considered for hybrid nanoparticles and water as a continuous phase liquid. Joule heating, heat generation/absorption and viscous dissipation effects are incorporated in the mathematical modeling of energy expression. Furthermore, binary chemical reaction with activation energy is considered. The total entropy rate is calculated in the presence of heat transfer irreversibility, fluid friction irreversibility,Joule heating irreversibility, porosity irreversibility and chemical reaction irreversibility through thermodynamics second law. The nonlinear governing equations are first converted into ordinary differential equations through implementation of appropriate similarity transformations and then numerical solutions are calculated through Built-in-Shooting method. Characteristics of sundry flow variables on the entropy generation rate, velocity, concentration, Bejan number, temperature are discussed graphically for both graphene oxide and titanium dioxide hybrid nanoparticles. The engineering interest like skin friction coefficient and Nusselt number are computed numerically and presented through tables. It is noticed from the obtained results that entropy generation rate and Bejan number have similar effects versus diffusion parameter. Also entropy generation rate is more against the higher Brinkman number.     

考虑局部非热平衡的流体层流横掠多孔介质中恒热流平板的传热分析

对流体层流横掠多孔介质中恒热流加热的平板,应用Brinkman-Forchheime-extended Darcy流动模型和流体与多孔介质之间局部非热平衡理论建立守恒方程组,应用数量级分析和积分法,得出了速度边界层厚度、热边界层厚度、壁面黏性摩擦系数和对流传热系数、流体与多孔介质之间局部温差的计算公式。结果表明,速度边界层与光板时明显不同,其在平板前端迅速增长,之后越来越平坦,趋于一个恒定值;而热边界层则沿着流动方向不断增长,类似于光板时的情况;局部的表面对流传热系数在平板前端达最大值,之后逐渐减小,也类似于光板时的情况;多孔介质与流体间的局部温差在平板前端达最大值,之后呈现沿着流动方向逐渐减小的变化趋势。     

MHD flow and heat transfer of a micropolar fluid over a nonlinear stretching surface with variable surface heat flux and heat generation

An analysis has been carried out to study magnetohydrodynamic boundary layer flow and heat transfer of an electrically conducting micropolar fluid over a nonlinear stretching surface with variable wall heat flux in the presence of heat generation/absorption and a non‐uniform transverse magnetic field. The governing system of partial differential equations is first transformed into a system of ordinary differential equations using similarity transformation. The transformed equations are solved numerically. Results for the dimensionless velocity, micro‐rotation, and temperature profiles are displayed graphically delineating the effects of various parameters characterising the flow. The results show that the velocity profile decreases as the magnetic parameter and the velocity exponent increase, while it increases as the material parameter increases. The results show also that the temperature profile increases as the magnetic parameter, the velocity exponent, and the heat generation parameter increase. Furthermore, the temperature profile decreases as the material parameter, the heat absorption parameter, and the Prandtl number increase.     

The Role of Nonuniformity in Convective Heat and Mass Transfer through Porous Media,Part 2

Nonuniformities in porous materials can play a significant role in the convective and diffusive transport of fluid, heat, and mass. This study provides experimental results and corresponding numerical simulation results. The experiments report continuous data in transient test runs with measurements of temperature and velocity at distributed locations in the domain immediately downstream of the porous materials. An increasing degree of nonuniformity was found to produce a lower drying rate as well as an earlier onset of falling rate drying. The numerical model provides results of the effects of different nonhomogeneities such as distributed holes or distributed regions of varying permeability and moisture content. Comparisons between numerical and experimental test results indicate general agreement with differences with regard to details of the drying curve features. This provides a tool for studying the role of nonuniformity in fluid flow and heat and mass transfer in porous media.     

SLIP EFFECT ON DIFFUSION OF CHEMICALLY REACTIVE SPECIES IN BOUNDARY LAYER FLOW OVER A VERTICAL STRETCHING SHEET WITH SUCTION OR BLOWING

In the present investigation, we study the effects of slip boundary condition on the diffusion of chemically reactive species in steady boundary layer flow of viscous incompressible fluid over a vertical stretching sheet with suction or blowing. The first-order chemical reaction is considered and wall concentration varies linearly along the sheet. The self-similar equations are obtained using similarity transformations and are solved numerically using shooting method. Our study reveals that due to the increase of diffusion parameter and blowing, the velocity increases, and it decreases with suction, Schmidt number, and reaction rate parameter. Importantly, for increase of slip parameter, the boundary layer thickness increases. In contrast, the concentration at a point increases only for increasing slip and blowing, while it decreases for increase of all other parameters.     

THE EFFECTS OF VARIABLE FLUID PROPERTIES ON MHD MAXWELL FLUIDS OVER A STRETCHING SURFACE IN THE PRESENCE OF HEAT GENERATION/ABSORPTION

An analysis is performed to investigate the effects of variable viscosity and thermal conductivity on the two-dimensional steady flow of an electrically conducting, incompressible, upper-convected Maxwell fluid in the presence of a transverse magnetic field and heat generation or absorption. The governing system of partial differential equations is transformed into a system of coupled nonlinear ordinary differential equations, and is solved numerically. Velocity and temperature fields have been computed and shown graphically for various values of the physical parameters. The local skin-friction coefficient and the local Nusselt number have been tabulated. It is found that fluid velocity decreases with an increase in the viscosity parameter and the Deborah number. It is also observed that increasing the magnetic parameter leads to a fall in the velocity and a rise in the temperature. Furthermore, it is shown that the temperature increases due to increasing the values of the thermal conductivity parameter and the heat generation parameter, while it decreases with an increase of both the absolute value of the heat absorption parameter and the Prandtl number.     

非热平衡多孔介质内反应与传热传质耦合过程

采用局部热不平衡假设,对发生强吸热化学反应的多孔介质体系建立了反应与传热、传质耦合问题的数学模型,采用Ergun-Forchheimer-Brinkman方程描述多孔介质中的流体流动.运用交替方向隐式(ADI)方法对模型离散求解,并采用文献中的实验数据对模型进行验证.计算了不同条件下颗粒物料层内气体和固体骨架的温度场、产物气体浓度场以及固体转化率分布,以得到多孔介质体系内固有化学反应时的传热、传质规律.结果表明,不能忽略固体骨架与流体间的温度差.入口渗流速度、入口气体温度以及固体颗粒尺寸是影响系统反应特性的重要参数.研究结果对具有强吸热反应的固定床反应器的设计和运行具有一定的参考作用.     

Ionic mass transfer in parallel plate electrochemical cells

Experimental investigations have been made of ionic mass transfer in a parallel plate electrochemical cell under both laminar and turbulent flow. The results obtained in the laminar flow region were found to be well represented by a Leveque-type equation modified to include the cell aspect ratio as an additional parameter. The influence of decreased mass transfer at the edges of the electrodes due to changes in the velocity profile was found to be small. For the turbulent region, there is a correlation of the mass transfer coefficient with Reynolds number to an exponent of 0.875 and Schmidt number to exponent of 0.21. This is in accord with existing correlations for heat and mass transfer in similar geometries over the range studied.