A STUDY OF NON-NEWTONIAN FLOW AND HEAT TRANSFER OVER A NON-ISOTHERMAL WEDGE USING THE HOMOTOPY ANALYSIS METHOD

The thermoconvective boundary layer flow of a generalized third-grade viscoelastic power-law non-Newtonian fluid over a porous wedge is studied theoretically. The free stream velocity, the surface temperature variations, and the injection velocity at the surface are assumed variables. A similarity transformation is applied to reduce the governing partial differential equations for mass, momentum, and energy conservation to dimensionless, nonlinear, coupled, ordinary differential equations. The homotopy analysis method (HAM) is employed to generate approximate analytical solutions for the transformed nonlinear equations under the prescribed boundary conditions. The HAM solutions, in comparison with numerical solutions (fourth-order Runge-Kutta shooting quadrature), admit excellent accuracy. The residual errors for dimensionless velocity and dimensionless temperature are also computed. The influence of the “power-law” index on flow characteristics is also studied. The mathematical model finds important applications in polymeric processing and biotechnological manufacture. HAM holds significant promise as an analytical tool for chemical engineering fluid dynamics researchers, providing a robust benchmark for conventional numerical methods.     

Spline method for solving continuous batch grinding and similarity equations

This paper investigates a numerical technique using cubic splines for solving the continuous batch grinding equation. The same approach is also used to solve the similarity equation in the case where the breakage function has a more complicated form than that for which analytical solutions are available. In the case of the similarity equation the spline coefficients are obtained from the solution of an algebraic generalized eigenvalue problem. Numerical solutions of the continuous equation for simplified selection and breakage functions are compared with those of analytical solutions in order to assess the validity of the numerical procedures. Experimental results from a limestone ore are used to test the validity of the simplified model. The spline method is also used to solve the continuous equation for a more complicated breakage function evaluated for a gold bearing ore body.     

Two Analytical Approaches for Solution of Population Balance Equations: Particle Breakage Process

Various particulate systems were modeled by the population balance equation (PBE). However, only few cases of analytical solutions for the breakage process do exist, with most solutions being valid for the batch stirred vessel. The analytical solutions of the PBE for particulate processes under the influence of particle breakage in batch and continuous processes were investigated. Such solutions are obtained from the integro‐differential PBE governing the particle size distribution density function by two analytical approaches: the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM). ADM generates an infinite series which converges uniformly to the exact solution of the problem, while HPM transforms a difficult problem into a simple one which can be easily handled. The results indicate that the two methods can avoid numerical stability problems which often characterize general numerical techniques in this area.     

Controlled Picard Method for Solving Nonlinear Fractional Reaction–Diffusion Models in Porous Catalysts

This paper discusses the diffusion and reaction behaviors of catalyst pellets in the fractional-order domain as well as the case of nth-order reactions. Two generic models are studied to calculate the concentration of reactant in a porous catalyst in the case of a spherical geometric pellet and a flat-plate particle with different examples. A controlled Picard analytical method is introduced to obtain an approximated solution for these systems in both linear and nonlinear cases. This method can cover a wider range of problems due to the extra auxiliary parameter, which enhances the convergence and is suitable for higher-order differential equations. Moreover, the exact solution in the linear fractional-order system is obtained using the Mittag–Leffler function where the conventional solution is a special case. For nonlinear models, the proposed method gives matched responses with the homotopy analysis method (HAM) solutions for different fractional orders. The effect of fractional-order parameter on the dimensionless concentration of the reactant in a porous catalyst is analyzed graphically for different cases of order reactions and Thiele moduli. Moreover, the proposed method has been applied numerically for different cases to predict and calculate the dual solutions of a nonlinear fractional model when the reaction order n?=??1.     

Dispersion of a Plume of Volatile Aerosol

A theoretical basis and numerical schemes are presented for simulating the dispersion of a plume of volatile aerosol. For special cases where analytical solutions are possible, excellent agreement is shown between analytical and numerical solutions. It is demonstrated that the error in the numerical simulation can be reduced to any desired level. Simulation results for volatile plume dispersion under realistic atmospheric conditions are also presented.     

An analytical–numerical method for solving a heap leaching problem of one or more solid reactants from porous pellets

In this paper we present an alternative method based on analytical and numerical solutions for solving the differential equations which describe heap leaching of one or more solid reactants from porous pellets. We propose to use analytical solutions for the differential equations which describe rate dissolutions along the pores and the surface of the particles under suitable regularity conditions. Moreover, we propose to use continuous and discontinuous solutions for the continuity partial differential equations describing balances. We comment on the efficient numerical solving of the remaining partial differential equations within the proposed numerical scheme. All of this, allows to obtain a numerical algorithm which is fast and accurate for the heap leaching problem. Also, we include particle size distributions on the proposed numerical methodology. This method applies to the case where the rate-controlling reagent is a component of the lixiviation solution only and not of the gas phase. The model includes the effects of particle scales, kinetic factors, heap scales and several operation variables. Finally, numerical experiments are presented.     

承压含水层地下水稳定流的有限层分析

有限层法是一种对空间某一方向进行数值离散,而在其余两方向采用连续函数的半数值半解析方法.该方法能有效地将三维问题简化为一维问题求解,从根本上解决了常用数值分析方法在模拟三维地下水运动时存在的计算工作量大、占用内存多、耗时大等缺点.文中基于有限层法的优点,推导了以伽辽金法结合贝塞耳函数为基础的层状非均质各向异性承压含水层的稳定流有限层方程,并编制了相应的计算程序.通过对2个经典算例的数值解与解析解对比分析,验证了该方法的正确性.     

Masked edge effects when measuring diffusion coefficients with the cup method

When diffusion coefficients are measured with the cup method, the exposed area is often less than the area of the sample. Because of this masking effect the measured diffusion coefficient is overestimated. The present article presents simple formulas based on two-dimensional analytical solutions to correct for this in cylindrical cup samples. Different cases of anisotropical diffusion are dealt with. The error in the analytical solutions is determined by numerical computer simulations.     

ON APPLICATION OF THE METHOD OF KINETIC INVARIANTS TO THE DESCRIPTION OF DISSOLUTION ACCOMPANIED BY A CHEMICAL REACTION

The dissolution, accompanied by chemical reaction, of monodisperse solid particles has been analysed. The resulting model, which accounts for the variation of mass transfer coefficient with the size of the dissolving particles, yields an approximate analytical form of a kinetic function. Rigorous numerical and approximate analytical solutions have been obtained for the governing system of nonlinear ordinary differential equations. The transient nature of the dissolution process as well as the accuracy of the analytical solution is brought out by the rigorous numerical solution. The analytical solution is fairly accurate for the major part of the range of operational times encountered in practice.     

明胶层中的扩散和反应的动力学研究 (Ⅰ)纯扩散和扩散伴随一级反应的数值模型

设计了两个数值模型,分别描述明胶层中的反应物的纯扩散和扩散伴随一般反应的动力学过程。用这些模型可以给出反应物在明胶层中不同时间下的空间分布。比之通常的分析解法,计算机化的数值模型更易求解,而且更具有解决复杂动力学过程的能力。为了验证这些数值模型的适用性,还同时与相应的分析解法作了对照。     

Population balance discretization for growth,attrition, aggregation,breakage and nucleation

This paper presents a new discretization method to solve one-dimensional population balance equations (PBE) for batch and unsteady/steady-state continuous perfectly mixed systems. The numerical technique is valid for any size change mechanism (i.e., growth, aggregation, attrition, breakage and nucleation occurring alone or in combination) and different discretization grids.The developed strategy is based on the moving pivot technique of Kumar and Ramkrishna and the cell-average method of Kumar et al. A novel contribution is proposed to numerically handle the growth and attrition terms, for which a new representation of the number density function within each size class is developed. This method allows describing the number particle fluxes through the class interfaces accurately by preserving two sectional population moments.By comparing the numerical particle size distributions with analytical solutions of one-dimensional PBEs (including different size change mechanisms and particle-size dependent kinetics), the accuracy of the proposed numerical method was proved.     

Solving population balance equations for two-component aggregation by a finite volume scheme

A conservative finite volume approach, originally proposed by Filbet and Laurençot [2004a. Numerical simulation of the Smoluchowski coagulation equation. SIAM Journal on Scientific Computing 25(6), 2004-2048] for the one-dimensional aggregation, is extended to simulate two-component aggregation. In order to apply the finite volume scheme, we reformulate the original integro-ordinary differential population balance equation for two-component aggregation problems into a partial differential equation of hyperbolic-type. Instead of using a fully discrete finite volume scheme and equidistant discretization of internal properties variables, we propose a semidiscrete upwind formulation and a geometric grid discretization of the internal variables. The resultant ordinary differential equations (ODEs) are then solved by using a standard adaptive ODEs-solver. Several numerical test cases for the one and two-components aggregation process are considered here. The numerical results are validated against available analytical solutions.     

Estimation of overall effectiveness factors for parallel catalytic reactions

Approximate analytical expressions of overall effectiveness factors for parallel catalytic reac-tions occurring in a spherical catalyst pellet with both internal and external mass transfer resistancesare obtained after taking the information given by asymptotic solutions valid for small and largevalues of Thiele moduli.This approximate procedure is feasible for power law as well asLangmuir-Hinshelwood kinetics and the agreement between the approximation and the numerical re-sults is good enough for practical purposes.Moreover,the method does not require any iterative ortrial-and-error computation,and can be conveniently applied in most practical calculations for reac-tors giving reliable results.     

Comparison of Analytical, Numerical, and Experimental Methods in Deriving Fracture Toughness Properties of Adhesives Using Bonded Double Lap Joint Specimens

Stress and fracture analysis of bonded double lap joint (DLJ) specimens have been investigated in this paper. Numerical and analytical methods have been used to obtain shear- and peel-stress distributions in the DLJ. The generalized analytical solution for the peel stress was calculated for various forms of the DLJ geometry and, by using crack closure integral (CCI) and by means of the J-integral approach, the analytical strain energy-release rate, G, was calculated. Experimental fracture tests have also been conducted to validate the results. The specimens were made of steel substrates bonded by an adhesive and loaded under tension. Specimens with cracks on both sides and at either end of the DLJ interface were tested to compare the fracture behavior for the two crack positions where tensile and compressive peel stresses exist. Tests confirmed that the substrates essentially behave elastically. Therefore, a linear elastic solution for the bonded region of the DLJ was developed. The fracture energy parameter, G, calculated from the elastic experimental compliance for different crack lengths, was compared with numerical and analytical calculations using the experimental fracture loads. The stresses from analytical analysis were also compared with those from the finite element results. The strain energy-release rate for fracture, G_f, for the adhesive has been shown to have no R-curve resistance, was relatively independent of crack length, and compared well with those obtained from numerical and analytical solutions. However, it was found that fracture energy for the crack starter in the position where the peel stress was tensile was about 20% lower than where the crack was positioned at the side, where the peel stress was found to be compressive.     

On the solution of population balances for nucleation, growth, aggregation and breakage processes

This work is concerned with the modeling and simulation of population balance equations (PBEs) for combined particulate processes. In this study a PBE with simultaneous nucleation, growth, aggregation and breakage processes is considered. In order to apply the finite volume schemes (FVS) a reformulation of the original PBE is introduced. This reformulation not only help us to treat the aggregation and breakage processes in a manner similar to the growth process in the FVS but also in deriving a stable numerical scheme. Two numerical methods are proposed for the numerical approximation of the resulting reformulated PBE. The first method combines a method of characteristics (MOC) for growth process with an FVS for aggregation and breakage processes. The second method purely uses a semidiscrete FVS for all processes. Both schemes use the same FVS for aggregation and breakage processes. The numerical results of the schemes are compared with each other and with the available analytical solutions. The numerical results were found to be in good agreement with analytical solutions.     

Mass transfer in a film flowing over semipermeable surfaces in microfiltration-desorption apparatuses

Desorption of dissolved water and gases during the purification of organic liquids in microfiltration-desorption apparatuses is simulated. The problem of desorption from a liquid film of variable thickness flowing over a microfiltration membrane is solved, and its numerical and approximate analytical solutions are presented. The problem of desorption from a film running down an impermeable surface in additional purification system is also solved, and its analytical solution is presented. The solutions are used to determine the generalized variables controlling the extent of desorption. The conditions ensuring high purification efficiencies are found.     

PROPAGATION OF REACTION FRONTS IN NONCATALYTIC NONISOTHERMAL GAS-SOLID FIXED BED REACTORS

This paper is devoted to the analysis of reaction fronts, in a noncatalytic nonisothermal gas-solid fixed bed reactor. A novel technique to obtain analytical approximate solutions by means of simple functions is presented. These analytical approximate solutions can be used to investigate the effect of various parameters in the reactor performance without resorting to numerical analysis. In addition, for adiabatic conditions and/or fast kinetics, the relationship between dependent variables can be investigated analytically. The technique is illustrated by analyzing the oxidation of ZnS in a fixed-bed reactor.     

Role of bond thickness on adhesive failure

An analytical, numerical and experimental program is described which establishes the basic fracture mechanics properties of an adhesive joint. A finite element analysis of a homogeneous finite tapered double cantilever beam is first presented and the results compared with elasticity and strength of materials solutions. Using analytical results developed in another paper, a finite bond line thickness correction factor is introduced to determine the crack tip stress intensity factor as a function of crack length. An experimental program is described wherein the crack tip stress intensity factor for the cantilever beam adhesive joint is measured by the compliance method and the results compared with those obtained by analytical and numerical methods. Finally, the critical value of the adhesive crack tip stress intensity factor is determined using the analytical and experimental techniques presented.     

An analytical solution for mass transfer in turbulent falling films with or without chemical reaction

A simple analytical solution is presented for mass transfer in an isothermal turbulent falling liquid film with or without chemical reaction. The exact solutions are compared with previous work for the cases of physical absorption and first-order reaction. New solutions are presented for a zero-order reaction, and generalized results are presented for first and pseudo nth-order reactions. The method of superposition is used to generate the first ten eigenvalues and corresponding values of the Sherwood number for several limiting situations. The analytical solution offers substantial advantages over previous finite-difference numerical solutions both in computation time and in describing the effects of operating variables.