Chemical engineering mathematical equations

Differential equations. 2. Chemical processes—Mathematical models. 3. Chemical engineering—Mathematics. I. Duong, D. Do. 

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). 

Optical designers and specialists in heat transfer calculations in the chemical engineering and mechanical engineering sciences are familiar with the mathematical construct known as The Equation of Radiative Transfer, although most chemists and spectroscopists are not. The Equation of Radiative Transfer states that, disregarding absorbance and scattering, in a lossless optical system... 

Most of you have probably been exposed to Laplace transforms in a mathematics course, but we will lead off this chapter with a brief review of some of the most important relationships. Then we will derive the Laplace transformations of commonly encountered functions. Next we will develop the idea of transfer functions by observing what happens to the differential equations describing a process when they are Laplace-transformed. Finally, we will apply these techniques to some chemical engineering systems. 

State variables appear very naturally in the differential equations describing chemical engineering systems because our mathematical models are based on a number of first-order differential equations component balances, energy equations, etc. If there are N such equations, they can be linearized (if necessary) and written in matrix form... 

Most standard chemical engineering tests on kinetics [see those of Car-berry (50), Smith (57), Froment and Bischoff (19), and Hill (52)], omitting such considerations, proceed directly to comprehensive treatment of the subject of parameter estimation in heterogeneous catalysis in terms of rate equations based on LHHW models for simple overall reactions, as discussed earlier. The data used consist of overall reaction velocities obtained under varying conditions of temperature, pressure, and concentrations of reacting species. There seems to be no presentation of a systematic method for initial consideration of the possible mechanisms to be modeled. Details of the methodology for discrimination and parameter estimation among models chosen have been discussed by Bart (55) from a mathematical standpoint. 

This is not the place for a treatise on the solution of differential equations, ordinary or partial. There are some excellent mathematical texts and some, such as Varma and Morbidelli s Mathematical Methods in Chemical Engineering (Oxford University Press, 1997), are specifically directed at the chemical engineer. What we shall try to do, however, is to explore some of the ad hoc methods that take advantage of peculiar features of particular problems and those that give partial solutions, as well as mentioning a few fall-traps for the unwary. 

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

A first major part of educating future engineers is to teach how to transform physical, chemical, and biological problems into mathematical equations, called modeling. The next step is to teach how to solve these equations or models numerically. 

In order to make design or operation decisions a process engineer uses a process model. A process model is a set of mathematical equations that allows one to predict the behavior of a chemical process system. Mathematical models can be fundamental, empirical, or (more often) a combination of the two. Fundamental models are based on known physical-chemical relationships, such as the conservation of mass and energy, as well as thermodynamic (phase equilibria, etc.) and transport phenomena and reaction kinetics. An empirical model is often a simple regression of dependent variables as a function of independent variables. In this section, we focus on the development of process models, while Section III focuses on their numerical solution. 

Thermodynamics and kinetics can surely be counted—along with transport phenomena, chemistry, unit operations, and advanced mathematics—as subjects that form the foundation of Chemical Engineering education and practice. Thermodynamics is of course a very old subject. For example, it was the same Rudolf Clausius, who in 1865 coined two immortal sentences (1) "The energy of the universe is constant" and (2) "The entropy of the universe tends to a maximum," that developed the famous Clausius-Clapeyron equation, one of the most basic physico-chemical relationships. Classical thermodynamics was largely complete in the 19th century, before even the basic structure of the atom was understood. 

Thomas K. Sherwood and C. E. Reed, Applied Mathematics in Chemical Engineering, McGraw-Hill, New York, 1939 William R. Marshall and Robert L. Pigford, The Application of Differential Equations to Chemical Engineering Problems, University of Delaware, Newark, 1947 A. B. Newman, Temperature Distribution in Internally Heated Cylinders, Trans. AlChE 24,44-53 (1930) T. B. Drew, Mathematical Attacks on Forced Convection Problems A Review, Trans. AlChE 26,26-79 (1931) Arvind Varma, Some Historical Notes on the Use of Mathematics in Chemical Engineering, pp. 353-387 in W. F. Furter, ed., A Century of Chemical Engineering [17]. 

Given mathematical expression, these laws lead to a network of equations from which a wide range of practical results and conclusions can be deduced. The universal applicability of this science is shown by the fact that it is employed alike by physicists, chemists, and engineers. The basic principles are always the same, but the applications differ. The chemical engineer must be able to cope with a wide variety of problems. Among the most important are the determination of heat and work requirements for physical and chemical processes, and the determination of equilibrium conditions for chemical reactions and for the transfer of chemical species between phases. 

A mathematical model consists, in principle, of a collection of equations that relate some inputs (explicative variables) to some outputs. Its goal is to reproduce the experimental behavior of a physical entity that exists on the real world (in chemical engineering we name these entities as processes). 

It is important to note that in using computer-aided models for batch distillation, the various assumptions of the model can have a significant impact on the accuracy of the results e.g., see the discussion of the effects of holdup above. Uncertainties in the physical and chemical parameters in the models can be addressed most effectively by a combination of sensitivity calculations using simulation tools, along with comparison to data. The mathematical treatment of stiffness in the model equations can also be very important, and there is often a substantial advantage in using simulation tools that take special account of this stiffness. (See the 7th edition of Perry s Chemical Engineers Handbook for a more detailed discussion of this aspect). 

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