Mathematical Methods in Chemical Engineering

V. E. Jenson and G. V. Jeffreys, Mathematical Methods in Chemical Engineering, Academic Press, Inc., Orlando, Ela., 1990. 

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). 

References Bateman, H., Tables of Integral Transforms, vol. I, McGraw-Hill, New York (1954) Varma, A., and M. Morbidelli, Mathematical Methods in Chemical Engineering, Oxford, New York (1997). 

Matrix methods, in particular finding the rank of the matrix, can be used to find the number of independent reactions in a reaction set. If the stoichiometric numbers for the reactions and molecules are put in the form of a matrix, the rank of the matrix gives the number of independent reactions. See Amundson, N. R., Mathematical Methods in Chemical Engineering, Prentice-Hall, Englewood Cliffs, N.J. (1966, p. 50). 

Amundson, N. R. Mathematical Methods in Chemical Engineering Matrices and Their Application. Prentice-Hall, Englewood Cliffs, New Jersey (1966). 

Varma and Morbidelli, Mathematical Methods in Chemical Engineering... 

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. 

There are any number of good books on this subject, and three come to mind J. D. Murray, Asymptotic Analysis. Oxford Clarendon Press, 1974 C. C. Lin and L. A. Segal, Mathematics Applied to Deterministic Problems in the Natural Sciences. New York Macmillan, 1974 and R. E. O Malley. Introduction to Singular Perturbations. New York Academic Press. 1974. See also Varma and Morbidelli. Mathematical Methods in Chemical Engineering. New York Oxford University Press, 1997. 

N.R. Amundson, Mathematical Method in Chemical Engineering, Prentice-Hall, 1966 E. Kreysig, Advanced Engineering Mathematics, 2nd ed, John Wiley, 1967... 

Amundson, N. R. "Mathematical Methods in Chemical Engineering, Vol. I Matrices" Prentice-Hall Publishing Company, 1966. 

Since the last reaction is a linear combination of the first two (sum), it can be easily proved that the rank remains unchanged at 2. So to conclude, the concentrations of all components in this network can be expressed in terms of two, say H2 and Freon 12, and the first two reactions form an independent reaction set. In case of more complicated networks it may be difficult to determine the independent reactions by observation alone. In this case the Gauss-Jordan decomposition leads to a set of independent reactions (see, e.g., Amundson, Mathematical Methods in Chemical Engineering—Matrices and Their Application, Prentice-Hall International, New York, 1966). 

---