Solution of Partial Differential Equations

One of the most frequently occurring partial differential equation in chemical engineering is the so-called parabolic type. This equation is used to describe time-dependent diffusion processes and fluid flow. Therefore, the numerical solution methods for this type of partial differential equation are important in heat transfer, molecular diffusion, and fluid flow. 

There are many numerical approaches one can use to approximate the solution to the initial and boundary value problem presented by a parabolic partial differential equation. However, our discussion will focus on three approaches an explicit finite difference method, an implicit finite difference method, and the so-called numerical method of lines. These approaches, as well as other numerical methods for aU types of partial differential equations, can be found in the literature [5,9,18,22,25,28-33]. 

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Lapidus, L. and Pinder, G. F., 1982. Numerical Solution of Partial Differential Equations in Science and Engineering, Wiley, New York. 

Partial Derivative The abbreviation z =f x, y) means that is a function of the two variables x and y. The derivative of z with respect to X, treating y as a constant, is called the partial derivative with respecd to x and is usually denoted as dz/dx or of x, y)/dx or simply/. Partial differentiation, hke full differentiation, is quite simple to apply. Conversely, the solution of partial differential equations is appreciably more difficult than that of differential equations. 

D.F. Hawken, J.J. Gottlieb, and J.S. Hansen, Review of Some Adaptive Node-Movement Techniques in Finite-Element and Finite-Difference Solutions of Partial Differential Equations, J. Comput. Phys. 95 (1991). 

Smith, G.D., 1985. Numerical Solution of Partial Differential Equations Finite Difference Methods, 3rd edition. Clarendon Press. 

Morton, K. and Mayers, D. (1994) Numerical Solution of Partial Differential Equations. An Introduction. Cambridge University Press Cambridge. 

Usually the finite difference method or the grid method is aimed at numerical solution of various problems in mathematical physics. Under such an approach the solution of partial differential equations amounts to solving systems of algebraic equations. 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

Thompson, ). F., Warsi, Z. U., Mastin, C. W., Boundary-fitted coordinate systems for numerical solution of partial differential equations, ). Comput. Phys. 

The Origin of Special Functions. The special functions of mathematical physics arise in the solution of partial differential equations governing the behaviour of certain physical quantities. Probably the most frequently occurring equation of this type in all physics is Laplace s equation... 

The single-column process (Figure 1) is similar to that of Jones et al. (1). This process is useful for bulk separations. It produces a high pressure product enriched in light components. Local equilibrium models of this process have been described by Turnock and Kadlec (2), Flores Fernandez and Kenney (3), and Hill (4). Various approaches were used including direct numerical solution of partial differential equations, use of a cell model, and use of the method of characteristics. Flores Fernandez and Kenney s work was reported to employ a cell model but no details were given. Equilibrium models predict... 

J. Crank and P. Nicolson, A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-conducting Type, Proc. Cambridge. Philos. Soc., 43,50-67 (1947). 

As mentioned earlier, calculations of diffusional rate processes are difficult as they involve the solution of partial differential equations. Even for processes which are clearly diffusional controlled, such as absorption, chemical engineers normally simplify the calculations by assuming equilibrium stages and may instead correct for possible deviations by using efficiency factors afterwards. Most commercial process design software, such as HYSYS, AspenPlus and ChemCAD, make the assumption of staged equilibrium processes. 

The independent variable in ordinary differential equations is time t. The partial differential equations includes the local coordinate z (height coordinate of fluidized bed) and the diameter dp of the particle population. An idea for the solution of partial differential equations is the discretization of the continuous domain. This means discretization of the height coordinate z and the diameter coordinate dp. In addition, the frequently used finite difference methods are applied, where the derivatives are replaced by central difference quotient based on the Taylor series. The idea of the Taylor series is the value of a function f(z) at z + Az can be expressed in terms of the value at z. 

Temperature profiles can be determined from the transient heat conduction equation or, in integral models, by assuming some functional form of the temperature profile a priori. With the former, numerical solution of partial differential equations is required. With the latter, the problem is reduced to a set of coupled ordinary differential equations, but numerical solution is still required. The following equations embody a simple heat transfer limited pyrolysis model for a noncharring polymer that is opaque to thermal radiation and has a density that does not depend on temperature. For simplicity, surface regression (which gives rise to convective terms) is not explicitly included. 

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