SIMILARITY SOLUTION FOR UNSTEADY HEAT AND MASS TRANSFER FROM A STRETCHING SURFACE EMBEDDED IN A POROUS MEDIUM WITH SUCTION/INJECTION AND CHEMICAL REACTION EFFECTS

An analysis is presented to investigate the effects of chemical reaction on unsteady free convective heat and mass transfer on a stretching surface in a porous medium. The governing partial differential equations have been transformed by a similarity transformation into a system of ordinary differential equations, which are solved numerically using an efficient tri-diagonal implicit finite-difference method. The results obtained show that the flow field is influenced appreciably by the presence of unsteadiness parameter, chemical reaction parameter, permeability parameter, and suction/injection parameter.     

UNSTEADY MHD BOUNDARY LAYER FLOW WITH DIFFUSION AND FIRST-ORDER CHEMICAL REACTION OVER A PERMEABLE STRETCHING SHEET WITH SUCTION OR BLOWING

In this study, unsteady MHD boundary layer flow with diffusion of chemically reactive species undergoing first-order chemical reaction over a permeable stretching sheet with suction or blowing and also with power-law variation in wall concentration is investigated. Using similarity transformation, the governing partial differential equations are converted into nonlinear self-similar ordinary differential equations. The transformed equations are then solved by the finite difference method using the quasi-linearization technique. Due to the increase in the unsteadiness parameter, the velocity initially decreases, but after a certain point it increases. A similar effect is also observed in case of concentration distribution. The increase in magnetic parameter causes a decrease in velocity and an increase in concentration. For increasing strength of applied suction both momentum and concentration boundary layer thicknesses decrease. On the other hand, applied blowing has reverse effects. Moreover, the mass transfer from the sheet is enhanced with increasing values of Schmidt number, reaction rate parameter, and also power-law exponent (related to wall concentration distribution). For high negative values of the power-law exponent, mass absorption at the sheet occurs. Moreover, due to increase of unsteadiness, this mass absorption is prevented.     

Magnetohydrodynamic slip flow and diffusion of a reactive solute past a permeable flat plate with suction/injection

The magnetohydrodynamic (MHD) boundary layer slip flow and solute transfer over a porous plate in the presence of a chemical reaction are investigated. The governing equations were transformed into self-similar ordinary differential equations by adopting the similarity transformation technique. Then the numerical solutions are obtained by a shooting technique using the fourth order Runge-Kutta method. The study reveals that due to the increase in the boundary slip, the concentration decreases and the velocity increases. On the other hand, with an increase in the magnetic field and mass suction, both boundary layer thicknesses decreased. As the Schmidt number and the reaction rate parameter increases, the concentration decreases and the mass transfer increases.     

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.     

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.     

HEAT TRANSFER DUE TO MHD SLIP FLOW OF A SECOND-GRADE LIQUID OVER A STRETCHING SHEET THROUGH A POROUS MEDIUM WITH NONUNIFORM HEAT SOURCE/SINK

An analysis is carried out to study the heat transfer characteristics of a second-grade non-Newtonian liquid due to a stretching sheet through a porous medium under the influence of external magnetic field. The stretching sheet is assumed to be impermeable. Partial slip condition is used to study the flow behavior of the liquid. The effects of viscous dissipation, nonuniform heat source/sink on the heat transfer are addressed. The nonlinear partial differential equations governing momentum and heat transfer in the boundary layer are converted into nonlinear ordinary differential equations using similarity transformation. Analytical solutions are obtained for the resulting boundary value problems in the case of two types of boundary heating, namely, constant surface temperature (CST) and prescribed surface temperature (PST). The effects of slip parameter, second-grade liquid parameter, combined (magnetic and porous) parameter, Prandtl number, Eckert number, and nonuniform heat source/sink parameters on the heat transfer are shown in several plots. Analytical expressions for the wall frictional drag coefficient and wall temperature gradient are obtained.     

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.     

Chemically reactive solute transfer in a boundary layer slip flow along a stretching cylinder

This paper presents the distribution of a solute undergoing a first order chemical reaction in an axisymmetric laminar boundary layer flow along a stretching cylinder. Velocity slip condition at the boundary is used instead of no-slip condition. Similarity transformations are used to convert the partial differential equations corresponding to momentum and concentration into highly nonlinear ordinary differential equations. Numerical solutions of these equations are obtained by the shooting method. The velocity decreases with increasing slip parameter. The skin friction as well as the mass transfer rate at the surface is larger for a cylinder than for a flat plate.     

A process control algorithm for reaction-diffusion minimum time heating cycles for binder removal from green bodies

Minimum time heating cycles have been simulated for binder removal from ceramic green bodies where the product of binder decomposition exits the green body via diffusion. The model consists of the reaction-diffusion partial differential equation, ordinary differential equations describing the reaction kinetics and a heating function, and algebraic equations, one of which imposes a constraint on concentration or pressure to avoid failure of the green body. Two solution approaches were compared: an earlier approximate method based on the pseudo-steady state assumption combined with a variational calculus algorithm and a new approach based on the finite element method combined with a process control algorithm. The agreement between the two solution strategies reinforces the validity of the pseudo-steady state approximation and the utility of the process control methodology. The latter, which was also applied to problems in which no approximate solution was obtainable, is thus a general method for obtaining minimum time heating cycles.     

Mass transfer simulation of biodiesel synthesis in microreactors

A coupled nonlinear mathematical model for the mass transfer of the species involved in the transesterification reaction between soybean oil and methanol in a parallel plates geometry microreactor is presented. The set of partial differential equations that governs the concentration profile of these species were obtained from the general mass balance equation for the case of isothermal flow and steady state with constant physical properties. The velocity profile was obtained from the Navier-Stokes equations assuming fully developed stratified laminar flow for two immiscible Newtonian fluids, with a plane interface between them, based on experimental observation of this flow pattern. The second order kinetic equations for the species were developed assuming homogeneous and reversible chemical reactions and these equations were written as source terms in the main equations. The mathematical model was solved using the hybrid method known as Generalized Integral Transform Technique (GITT). The simulation results were critically compared with those obtained by using the COMSOL multiphysics platform, showing a good agreement between the hybrid and fully numerical simulations. The effects of governing parameters such as residence time, temperature and microreactor dimensions were investigated. It was observed that higher triglycerides conversion rates occurred at higher temperatures and residence times and lower microreactor depths.     

HYDROMAGNETIC BOUNDARY LAYER FLOW AND HEAT TRANSFER IN VISCOELASTIC FLUID OVER A CONTINUOUSLY MOVING PERMEABLE STRETCHING SURFACE WITH NONUNIFORM HEAT SOURCE/SINK EMBEDDED IN FLUID-SATURATED POROUS MEDIUM

Analytical study for the problem of flow and heat transfer of electrically conducting viscoelastic fluid over a continuously moving permeable stretching surface with nonuniform heat source/sink in a fluid-saturated porous medium has been undertaken. The momentum and thermal boundary layer equations, which are partial differential equations, are converted into ordinary differential equations, by using suitable similarity transformation. The resulting nonlinear ordinary differential equations of momentum are solved analytically assuming exponential solution, and similarly thermal boundary layer equations are solved exactly by using power series method, with the solution obtained in terms of Kummer's function. The results are shown with graphs and tables. The effect of various physical parameters like viscoelastic parameter, porosity parameter, Eckert number, space, and temperature-dependent heat source/sink parameters enhances the temperature profile, whereas increasing the values of the suction parameter and Prandtl number decreases the temperature profile. The results have technological applications in liquid-based system involving stretchable materials.     

强制浓度振荡下的反应动力学参数特征函数估计法

研究了特征函数法(Karhunen-Lo(?)ve方法)在降低常微分方程组维数方面的可能性,并以CO氧化反应系统为例,说明该法的应用步骤.模拟结果证实了此方法的有效性.     

基于B-Z化学振荡反应过程的简化与软件分析

以Belousov—Zhabotinskii反应的Oregonator模型为基础,依据FNK机理,对振荡过程进行了简化,给出了分步基元反应的动力学参数和非线性的常微分方程,借助数学软件Matlab和Maple对方程进行了图解,并与实验数据进行了对比,该文提出的简化分析模型能较好地预测组分的初始浓度和振荡频率。     

PET固相缩聚反应器动态模型

PET固相缩聚过程涉及反应固体颗粒和反应器床层两个空间尺度的传质,是一个复杂的多维多相对象。以反应过程控制为目标,在已有的反应动力学模型基础上,同时考虑可逆化学反应和小分子产物内外扩散两个控制因素,并借用固定床拟均相模型的建模思想,将小分子产物在颗粒内外扩散形成的浓度梯度用一个有效系数来表示,从而建立了简化的PET固相缩聚反应器一维动态模型。在相关文献的实验条件下求解分布参数模型的数值解,并进行启动过程仿真和过程动态分析,计算反应器出口质量指标,与文献曲线和数据对照,验证了模型的正确性。     

Free convective flow over a vertical plate in a doubly stratified medium with electrophoresis, heat source/sink and chemical reaction effects

We analyzed the problem of unsteady, incompressible free convective doubly stratified flow past a semiinfinite vertical plate with the influence of electrophoresis, heat source/sink and chemical reaction. The partial differential equations governing the flow are solved by employing an implicit finite difference scheme of Crank-Nicolson type. The effect of heat generation and absorption in stratified and unstratified flow are examined and hence the influence of stratification on velocity, temperature and concentration are investigated and presented graphically. Further, the impact of the electrophoresis on particle concentration in the presence of generative and destructive reaction is analyzed. As well, the effects of the physical parameters on local and average values of skin friction, Nusselt number and Sherwood number are also investigated and illustrated graphically. The particular solutions of the present results are compared with the existing solution in literature and are found to be in good agreement.     

A technique for analysing integral rate data in non-ideal flow reactors to estimate unknown parameters

A technique is described for analysing integral rate data taken from a non-ideal flow chemical reactor characterised by the axial dispersion model, using a weighted residual approach. This approach avoids the formidable computational difficulties involved with the solution of the boundary value problem associated with the model differential equations. The measured concentration profile is approximated by a series of orthogonal polynomials. Model parameters are estimated by specifying that the fitting polynomial satisfy certain moment relations derived from the mass balance equation and boundary conditions. If the form of the rate equation is known, a priori, the estimation of the rate constant and the dispersion coefficient becomes explicit. In the more general case, in which the reaction order is also unknown, the estimation problem reduces to a uni-dimensional search for the reaction order parameter (or the adsorption parameter in the case of complex kinetics). The method is applied to both linear and non-linear systems and conclusions are drawn regarding the sensitivity of the chemical kinetic and transport parameters to experimental error.     

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.     

Sequential experimental design procedures for precise parameter estimation in ordinary differential equations

In this paper, it is shown how an experimental program for precise parameter estimation can be designed sequentially for the case that the mathematical model is given in the form of a set of ordinary differential equations. Two strategies are proposed. The first aims at minimizing the volume of the joint confidence region associated with the parameter estimates. The second attempts to alter as much as possible the shape towards a spherical region, by shortening the length of the longest principal axis of the confidence region to the maximum extent. The application of both criteria is illustrated by means of examples, representative for real problems in chemical reaction engineering. The techniques are easily applicable with our present day computing facilities. Qualitative indications are derived concerning the question when the use of an experimental design will result in an appreciable gain in significance for the parameter estimates.     

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.     

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