Mathematical modeling finite-difference methods

The analysis of polymer processing is reduced to the balance equations, mass or continuity, energy, momentum and species and to some constitutive equations such as viscosity models, thermal conductivity models, etc. Our main interest is to solve this coupled nonlinear system of equations as accurately as possible with the least amount of computational effort. In order to do this, we simplify the geometry, we apply boundary and initial conditions, we make some physical simplifications and finally we chose an appropriate constitutive equations for the problem. At the end, we will arrive at a mathematical formulation for the problem represented by a certain function, say / (x, T, p, u,...), valid for a domain V. Due to the fact that it is impossible to obtain an exact solution over the entire domain, we must introduce discretization, for example, a grid. The grid is just a domain partition, such as points for finite difference methods, or elements for finite elements. Independent of whether the domain is divided into elements or points, the solution of the problem is always reduced to a discreet solution of the problem variables at the points or nodal pointsinxxnodes. The choice of grid, i.e., type of element, number of points or nodes, directly affects the solution of the problem. 

The mathematical models of the reacting polydispersed particles usually have stiff ordinary differential equations. Stiffness arises from the effect of particle sizes on the thermal transients of the particles and from the strong temperature dependence of the reactions like combustion and devolatilization. The computation time for the numerical solution using commercially available stiff ODE solvers may take excessive time for some systems. A model that uses K discrete size cuts and N gas-solid reactions will have K(N + 1) differential equations. As an alternative to the numerical solution of these equations an iterative finite difference method was developed and tested on the pyrolysis model of polydispersed coal particles in a transport reactor. The resulting 160 differential equations were solved in less than 30 seconds on a CDC Cyber 73. This is compared to more than 10 hours on the same machine using a commercially available stiff solver which is based on Gear s method. 

For the solution of sophisticated mathematical models of adsorption cycles including complex multicomponent equilibrium and rate expressions, two numerical methods are popular. These are finite difference methods and orthogonal collocation. The former vary in the manner in which distance variables are discretized, ranging from simple backward difference stage models (akin to the plate theory of chromatography) to more involved schemes exhibiting little numerical dispersion. Collocation methods are often thought to be faster computationally, but oscillations in the polynomial trial function can be a problem. The choice of best method is often the preference of the user. 

The selected mathematical model is represented by a discretization method for approximating the differential equations by a system of algebraic equations for the variables at some set of discrete locations in space and time. Many different approaches are used in reactor engineering , but the most important of them are the simple finite difference methods (FDMs), the flrrx conservative finite volume methods (FVMs), and the accurate high order weighted residual methods (MWRs). 

Initial modelling on predicting velocity profiles in pultrusion dies was carried out by Gorthala et al. (1994). Here a two-dimensional mathematical model in cylindrical co-ordinates with a control-volume-based finite-difference method was developed for resin fiow, cure and heat transfer associated with the pultrusion process. Raper et al. 

To numerically solve equations of the above mathematical models, the general computational gas dynamics is adopted in the present work. The general differential equations (2.7) and (2.31) are then discretized by the control volume-based finite difference method, and the resulting set of algebraic equations is iteratively solved. The numerical solver for the general differential equations can be repeatedly appUed for each scale variable over a controlled volume mesh. This process must be conducted extremely carefully to avoid the influence of slight changes in the accuracy of discretization. 

The mathematical model described above involves non-linear, coupled, partial differential equations. The equations were solved using a Finite-Difference method. Details of this mathematical technique have been described elsewhere in the literature (8.9). Figure 2 shows a flowsheet for the numerical solution of these model equations. 

ABSTRACT In order to investigate the effect of coal particle size on gas desorption and diffusion law at constant temperature, the constant temperature dynamic coal particle gas adsorption and desorption experiment with different particle sizes was conducted in the coal gas adsorption and desorption experiment system. The results suggest that gas desorption laws of different particle size of coal samples show a good consistency at different pressures, and the cumulative desorption of gas coal particle is linear with time. For the same particle, the higher the initial pressure, the more the maximum gas desorption the smaller the coal particle is, the more quickly the gas desorption rate is at the same initial pressure. Then, the gas spherical flow mathematical model is built based on Darcy law and is analysed with finite difference method. At last, the gas spherical flow mathematical model is constructed with Visual Basic. The contrast between numerical simulation and experimental results shows that the gas flow in the coal particle internal micropore accords with Darcy s law. 

The radiation model of the radial methane flow in spherical coal grains is established and is calculated with finite difference method based on mass conservation law and Darcy s law and the mathematical model is non-dimensionalized. The coal particle gas diffusion equation is calculated with computer program developed with VB. 

Divide the spherical coal particle into units and using the finite difference method establish the mathematical equations and make them dimensionless, the purpose of which is to simplify the equations and decrease the number of conditions substitution for the model solution. By means of VB programming and calculation, finally the gas pressure distributions at the different time and the gas desorption amount in the coal particle are obtained. 

Rocha and Paixao [38] proposed a pseudo two-dimensional mathematical model for a vertical pneumatic dryer. Their model was based on the two-fluid approach. Axial and radial profiles were considered for gas and solid velocity, water content, porosity, temperatures, and pressure. The balance equations were solved numerically using a finite difference method, and the distributions of the flow field characteristics were presented. This model was not validated with experimental results. 

Zak and Frank [41] are pioneers who used discrete Markov Chain in finite state to describe a stochastic nature of supply and demand under centralized distribution of resources. The outcome shows that the size of system loss has close relations with functions of the system, the initial inventory level and allocation of resourees. To avoid material shortage and resources inactiveness, they designed mathematical models in different situations and algorithms applying brand and bound method to optimize the initial inventory and resource allocation. The goal of those measures was to minimize total system costs of losses. 

The study for predicting the velocity profiles in pultrusion by Gorthala et aL (1994) used a variable viscosity model. A comprehensive two-dimensional mathematical model in cylindrical coordinates was developed for resin flow, cure and heat transfer. A control-volume-based finite difference method (FDM) (Patankar method) was used for solving the governing equations. The use of artificial neural networks (ANNs) for pultrusion modelling in terms of the real process data and their potential for intelligent machine control was proposed by Wilcox and Wright (1998). Liu et al. (2000 Liu, 2001 Liu and Hillier, 1999) implemented a finite element/control volume... 

The solution of problem was carried out by the numerical finite-difference method briefly presented in [1], The calculating domain was covered by uniform grid with 76x46 mesh points. A presence of two-order elliptic operators in all equations of the mathematical model allow us to approximate each equation by implicit iterative finite-difference splitting-up scheme with stabilizing correction. The scheme in general form looks as follows ... 

Numerical simulations on the basis of the Poisson equation combined with the Nemst-Planck model offer another possibility of describing such systems. In the approaches by these groups, the finite-difference method was used to simultaneously solve the Nemst-Planck and the Poisson equations. Due to the complex mathematical procedure, the computing times are very large. So far, this approach has not yet been used for describing the response of thin membranes. Morf proposed a simpler finite-difference approach by combining the Poisson and Nemst-Planck equations in a stepwise way. The Poisson model was used to numerically evaluate the potential profile (Equation 22.23) after each iteration step of the concentration profile (Equations 22.21 and 22.22). The updated potential was then used in the next step ... 

Mathematical modeling in electrochemistry using finite element and finite difference methods... 

Computer simulation of the reactor kinetic hydrodynamic and transport characteristics reduces dependence on phenomenological representations and idealized models and provides visual representations of reactor performance. Modem quantitative representations of laminar and turbulent flows are combined with finite difference algorithms and other advanced mathematical methods to solve coupled nonlinear differential equations. The speed and reduced cost of computation, and the increased cost of laboratory experimentation, make the former increasingly usehil. 

A final note is in order. The finite-difference and finite-element techniques are entirely equivalent from a mathematical point of view. What is different about these are the conceptualization of the problem and the resulting computational techniques to be employed. One method is not better than the other, although in particular circumstances one may clearly be superior. The point is that a modeler and modeling systems should account for both methods as well as others not mentioned here. 

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