Partial differential equations chemical engineering

References Brown, J. W., and R. V. Churchill, Fourier Series and Boundary Value Problems, 6th ed., McGraw-Hill, New York (2000) Churchill, R. V, Operational Mathematics, 3d ed., McGraw-Hill, New York (1972) Davies, B., Integral Transforms and Their Applications, 3d ed., Springer (2002) Duffy, D. G., Transform Methods for Solving Partial Differential Equations, Chapman Hall/CRC, New York (2004) Varma, A., and M. Morbidelli, Mathematical Methods in Chemical Engineering, Oxford, New York (1997). 

Advanced CFD simulations (both in terms of numbers of grid points and partial differential equations) therefore require increasing amounts of computer memory and CPU-time. Chemical engineers increasingly get familiar with the idea of exploiting CFD, though still mostly of the RANS-type. Gradually, the... 

Thus to describe a chemical reactor (or almost any process unit in chemical engineering) we have to solve these partial differential equations with appropriate boundary and... 

Amundson, N. R. and Aris, R. (1972) Mathematical Methods in Chemical Engineering First Order Partial Differential Equations with Applications. Prentice-Hall, Englewood Cliffs, NJ. 

CHEMICAL ENGINEERING SYSTEMS BY THE FABRICATION, SOLUTION, AND PRESENTATION OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS COPIOUSLY ILLUSTRATED BY EXAMPLES FROM THE PAPERS OF RUTHERFORD ARIS... 

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. 

Subramanian, V.R., White, R.E. Semianalytical method of lines for solving elliptic partial differential equations. Chemical Engineering Science 59(4), 781-788 (2004)... 

Orthogonal collocation in the chemical engineering literature refers to the family of collocation methods with discretization grids associated to Gaussian quadrature methods [34, 204]. Spectral collocation methods for partial differential equations with an arbitrary distribution of collocation points are sometimes termed pseudo spectral methods [22]. 

These are systems where the state variables are varying in one or more directions of the space coordinates. The simplest chemical reaction engineering example is the plug flow reactor. These systems are described at steady state either by an ordinary differential equation (where the variation of the state variables is only in one direction of the space coordinates, i.e. one dimensional models, and the independent variable is this space direction), or partial differential equations (when the variation of the state variables is in more than one direction of the space coordinates, i.e. two dimensional models, and the independent variables are these space directions). The ordinary differential equations of the steady state of the one-dimensional distributed model can be either initial value differential equations (e.g. plug flow models) or two-point boundary value differential equations (e.g. models with superimposed axial dispersion). The equations describing the unsteady state of distributed models are invariably partial difierential equations. 

U. Nowak. U. Nieken. G. Eigenberger. Fully. Adaptive. Algorithm for Parabolic Partial Differential Equations in one Space Dimension. Computers and Chemical Engineering 20 5 (1996) 447... 

We have demonstrated the application of the finite integral transform to a number of parabolic partial differential equations. These are important because they represent the broadest class of time-dependent PDEs dealt with by chemical engineers. Now we wish to illustrate its versatility by application to elliptic differential equations, which are typical of steady-state diffusional processes (heat, mass, momentum), in this case for two spatial variables. We have emphasized the parabolic PDEs, relative to the elliptic ones, because many texts and much mathematical research has focussed too long on elliptic PDEs. 

In this chapter, we will present several alternatives, including polynomial approximations, singular perturbation methods, finite difference solutions and orthogonal collocation techniques. To successfully apply the polynomial approximation, it is useful to know something about the behavior of the exact solution. Next, we illustrate how perturbation methods, similar in scope to Chapter 6, can be applied to partial differential equations. Finally, finite difference and orthogonal collocation techniques are discussed since these are becoming standardized for many classic chemical engineering problems. 

Smit, J., van Sint Annaland, M. and Kuipers, J.A.M. (2005) Grid adaptation with weno schemes for non-uniform grids to solve convection-dominated partial differential equations. Chemical Engineering Science, 60 (10), 2609-2619. 

Partial differential equations (PDEs) of parabolic type arise from the modeling of a whole variety of systems in chemical engineering. Examples are dynamic models for fixed bed reactors, absorption columns, adsorbers, as well as the simulation of catalyst pellets and membrane reactors. Quite often spatially one dimensional models are sufficient to study the interesting phenomenon. 

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