Solving partial differential equations of gas–solid reactions by orthogonal collocation

In this work the solution of the coupled partial differential equations for noncatalytic gas–solid reactions has been considered by orthogonal collocation. First of all, by an integral transformation and then by applying the orthogonal collocation method, these partial differential equations are converted to the ordinary differential equations. Then the equations are solved and the conversion–time profiles are obtained. The solution of the equations for volume reaction model, grain model and grain model with product layer resistance, modified grain model, random pore model, nucleation model and reaction of two gas with one solid has been presented in this work. The orthogonal collocation is a rapid method for solving of these equations and shows a good accuracy with respect to other solution techniques in the literature.     

反渗透膜元件的微分方程数学模型

在反渗透膜元件的理想结构模型基础上建立了反渗透膜元件运行的微分方程数学模型,在分析膜元件运行参数的基础上确认了微分方程的边界条件,从而使微分方程组具有惟一解。膜元件运行微分方程模型的建立为膜元件及膜系统运行精确模拟计算软件的编制提供了数学基础。     

Computation of the gradient and sensitivity coefficients in sum of squares minimization problems with differential equation models

In parameter estimation problems where the system model consists of differential equations, methods for minimizing a sum of squares of residuals objective function require derivatives of the residuals with respect to the parameters being estimated (sensitivity coefficients) or the gradient of the objective function (depending on the numerical optimization method). This paper considers two methods for generating such derivatives: (1) the adjoint equation — gradient formula; and (2) complimentary sensitivity coefficient differential equations. Particular attention is given to the consistency between the method used to solve the model equations and the proper formulation of the additional equations required by the two methods. Two example problems illustrate computational experience using a modified quasi-Newton method with the adjoint method used to generate gradients and applying a modified Gauss-Newton approach with the sensitivity coefficient equations to calculate both the Gauss-Newton matrix and the objective function gradient. Results indicate the superiority of the sensitivity coefficient approach. When comparing the computational effort required by the two methods and the results from the simple examples, it appears that the use of complimentary sensitivity coefficient equations is much more efficient than using only the gradient of the sum of squares function.     

Sensitivity analysis of initial value problems with mixed odes and algebraic equations

An efficient method is described for sensitivity analysis of nonlinear initial value problems, which may include algebraic equations as well as ordinary differential equations.The linearity of the sensitivity equations is utilized to solve them directly via the local Jacobian of the state equations. The method is implemented with the implicit integrator DASSL and is demonstrated on a stiff industrial reaction model.     

Chemical absorption of mercaptan in an aerosol operated loop reactor

A mechanistic mathematical model for the chemical absorption of mercaptan in sodium hypochlorite solution has been derived. In order to describe the process adequately, a semi-verified complex scheme of the involved kinetic reactions based on stopped-flow measurements with UV-detection has been implemented. The overall system of differential equations has been solved numerically. For some asymptotic cases, approximation formulae are given. The process has been carried out in an aerosol operated jet loop reactor which is characterized by high interfacial areas at low liquid flow rates. Fitting the model solution to the experimentally obtained conversion data enabled determination of the unknown hydrodynamic parameters. By means of a sensitivity analysis, the influences of the different parameters are discussed.     

Mathematical modeling of reverse osmosis systems

The present investigation pertains to modeling of seawater desalination system. A simulation model was developed and verified for a small-scale reverse osmosis system. The proposed model combines material balances on the feed tank, membrane module andproduct tank with membrane mass transfer models. Finally a comprehensive simulation model has been developed incorporating the effect of mass transfer inhibition The model is non-linear differential equation representing the feed concentration as a function of operating time and space. The solution of the simultaneous differential equations was obtained using the fourth order Runge-Kutta method, due to self starting and stability. The model was verified using the experimental data from the literature [17,24]. Parameter sensitivity was carried out to select the proper step size. The simulation was run for over 1000 11 enabling a prediction of operational performance at high overall system recoveries.     

APPLICATION OF A k-ε TURBULENCE MODEL TO AN ENCLOSED BUOYANCY DRIVEN RECIRCULATING FLOW

A numerical study is performed of the turbulent two-dimensional air flow in a square enclosure where the vertical walls are held at different constant temperatures and horizontal walls have linear temperature distributions. The equations solved are for continuity, mean momentum and mean thermal energy. The turbulent shear stresses and heat fluxes in these equations are prescribed using a three-dimensional turbulence model involving the solution of two extra differential equations for k, the turbulent kinetic energy, and ε, its rate of dissipation. Buoyancy effects on the turbulence structure are also accounted for. Results have been obtained in the range of Grashof numbers of 107-108. Moreover, various model constants were tested and a sensitivity study was carried out in order to determine the effect of these constants on the results. A comparison with experimental data is given: the agreement is good.     

Optimization and sensitivity analysis for multiresponse parameter estimation in systems of ordinary differential equations

Methodology for the simultaneous solution of ordinary differential equations (ODEs) and associated parametric sensitivity equations using the Decoupled Direct Method (DDM) is presented with respect to its applicability to multiresponse parameter estimation for systems described by nonlinear ordinary differential equations. The DDM is extended to provide second order sensitivity coefficients and incorporated in multiresponse parameter estimation algorithms utilizing a modified Newton scheme as well as a hybrid Newton/Gauss-Newton optimization algorithm. Significant improvements in performance are observed with use of both the second order sensitivities and hybrid optimization method. In this work, our extension of the DDM to evaluate second order sensitivities and development of new hybrid estimation techniques provide ways to minimize the well-known drawbacks normally associated with second-order optimization methods and expand the possibility of realizing their benefits, particularly for multiresponse parameter estimation in systems of ODEs.     

Solution and sensitivity analysis of differential-algebraic-Volterra equation systems

Step-response modeling of time-dependent systems leads to weakly singular Volterra equations for the interfacial states, which may be coupled to differential and algebraic equations for the states in an adjoining region. The numerical solution and parametric sensitivity analysis of the resulting system is a challenging problem because of the history-dependent nature of the Volterra states. This paper presents discretization strategies and a solver DAVES (Differential-Algebraic-Volterra Equation Solver) for doing these calculations for systems with regular and weakly singular Volterra kernels. Backward Difference Formulas (BDFs) are used for local discretization of the differential and Volterra operators, while piecewise Gaussian quadrature is employed for the past history terms of the Volterra equations. The solution strategies extend those used in the differential-algebraic solver DDASAC. The new integrator is demonstrated on various chemical engineering problems, including a differential-algebraic-Volterra system encountered in our data-based modeling of packed-tube reactors.     

A new mathematical solution for predicting char activation reactions

The differential conservation equations that describe typical gas-solid reactions, such as activation of coal chars, yield a set of coupled second-order partial differential equations. The solution of these coupled equations by exact analytical methods is impossible. In addition, an approximate or exact solution only provides predictions for either reaction- or diffusion-controlling cases. A new mathematical solution, the quantize method (QM), was applied to predict the gasification rates of coal char when both chemical reaction and diffusion through the porous char are present. Carbon conversion rates predicted by the QM were in closer agreement with the experimental data than those predicted by the random pore model and the simple particle model.     

A generalized two-environment model for micromixing in a continuous flow reactor—I. construction of the model

The conceptual frameworks of the two-environment model and the coalescence-redispersion model are unified in a generalized model for describing micromixing in a continuous flow reactor. The system equations of the model are then derived. The Monte Carlo simulation of the model is also given in order to make possible the solution of the derived integro-partial differential equations.     

Simulation of packed-bed separation processes using orthogonal collocation

The two-point boundary value problem resulting from the heat and material balance equations of a packed separation column are solved using polynomial approximation techniques. The model equations are based on the two-film theory of mass transfer. The resulting partial differential equations are first reduced to ordinary differential equations and then integrated using semi-implicit Runge-Kutta method of integration. Application of orthogonal collocation simplifies the solution of the two-point boundary value problem. For the examples studied, the algorithm is found to converge rapidly with respect to the number of collocation points used in the polynomial approximation.     

Application of the method of lines to the analysis of single fluid-solid reactions in porous solids

A mathematical model for the isothermal reaction between a fluid and a porous solid, in pelletgrain form, is studied. First, an analytical derivation of the time for total conversion is presented. Then the differential equations, given as a coupled pair of time-dependent and time-independent equations, are altered to a fully time-dependent form, without loss of accuracy, but with much greater amenability to an efficient numerical solution method. The solution is by the method of lines, whereby finite difference discretization in the spatial variable yields a stiff system of ordinary differential equations (ODEs), and the ODE initial value problem is solved with a modern general-purpose ODE solver. A further alteration to the equations overcomes the difficulty caused by the discontinuous ODEs for the solid variable. Results are given for two grain geometries and for a wide range of reaction moduli.     

Gear's procedure for the simultaneous solution of differential and algebraic equations with application to unsteady state distillation problems

The dynamic equations modeling a sieve plate at unsteady state are developed. Gear's procedure for the simultaneous solution of systems of stiff differential and algebraic equations is presented and demonstrated for the solution of unsteady state distillation problems. It is shown that the basic stage model can be modified by the addition of one variable and one equation such that Gear's procedures are readily applied. The proposed model and solution procedure is contrasted to recently published procedures. Numerical results are given for the solution of a problem involving an extractive distillation column at unsteady state.     

纳滤膜过滤多组分离子电解质溶液的模型与计算——Ⅰ理论模型

以固定电荷模型为基础,采用扩展的Nernst-Planck方程结合Gouy-Chapman(GC)理论和Donnan平衡模型,通过一系列假设和推导,得到多组分离子溶液透过纳滤膜时离子浓度在膜微孔内梯度微分方程和解膜微孔内梯度微分方程的有关参数和方程。     

DYNAMIC MATHEMATICAL MODEL OF A ROTARY DRYER: METHOD OF CHARACTERISTICS

A mathematical model for the dynamic behavior of a countercurrent rotary dryer has been developed and solved. The model consists of four hyperbolic partial differential equations with split boundary conditions. The equations are solved numerically using an algorithm based on the method of characteristics. The solution is stable and rapid. Sample results of a dryer simulation are presented.     

Cubic autocatalysis with Michaelis-Menten kinetics: semi-analytical solutions for the reaction-diffusion cell

Cubic-autocatalysis with Michaelis-Menten decay is considered in a one-dimensional reaction-diffusion cell. The boundaries of the reactor allow diffusion into the cell from external reservoirs, which contain fixed concentrations of the reactant and catalyst. The Galerkin method is used to obtain a semi-analytical model consisting of ordinary differential equations. This involves using trial functions to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. The semi-analytical model is then obtained from the governing partial differential equations by averaging. The semi-analytical model allows steady-state concentration profiles and bifurcation diagrams to be obtained as the solution to sets of transcendental equations. Singularity theory is then used to determine the regions of parameter space in which the four main types of bifurcation diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found by a local stability analysis of the semi-analytical model. An example of a stable limit-cycle is also considered. Comparison with numerical solutions of the governing partial differential equations shows that the semi-analytical solutions are very accurate.     

Estimating the parameters of chemical kinetics equations from the partial information about their solution

The inverse problem of identifying the parameters of sets of ordinary differential equations using experimental measurements of three functions that correspond to some components in the vector solution of a set is considered. A private case that is important for applications of chemical and biochemical kinetics when reduced equations linearly depend on the combinations of initial unknown parameters has been studied. An analysis and the numerical results are presented for two types of sets of chemical kinetics equations, such as the Lotka–Volterra model that describes the coexistence of a predator and a prey and the chemical kinetics equations that model enzyme catalysts reactions, including the Michaelis–Menten equations. The search for unknown parameters is confined to the problem of minimizing a quadratic function. In this case, the reduced differential equations of systems are used instead of their vector solutions, which are unknown in most cases. The cases of both stable and unstable search for unknown parameters are analyzed.     

热管式降膜吸收器的传热传质

对热管式降膜吸收顺溶液吸收传热传质并通过热管移出吸收热的过程进行了数值研究，根据所建立的数学模型，通过求解热管加热段外壁面溶液波动降膜的动态二维偏微分方程和热管传热方程，研究了膜雷诺数，低位余热温度，输出温度等因素对传热管质过程的影响，对进一步工作提出了新的见解。     

THE DRYING OF SOYBEAN SEEDS IN COUNTERCURRENT AND CONCURRENT MOVING BED DRYERS

The aim of this work was to analyze the simultaneous heat and mass transfer between air and soybean seeds in countercurrent and concurrent moving bed dryers by simulation. The technique chosen was based on modeling from mass and energy conservation equations for the fluid and particulate phases. The equilibrium, heat transfer and mass transfer equations were taken from specific studies. The equation representing drying kinetics was obtained by means of a thin-layer study, whereas the equilibrium equations was chosen from rival model discrimination, based on nonlinearity measures. Hence, the model parameters were defined by the respectives studies. The profiles for temperature and humidity of the fluid and the temperature and moisture of the seeds were obtained by numerical solution of the model. This model consisted of ordinary differential equations and the solution was obtained by a specific code. The simulated results indicated a significant.