Page 530 - Modelling in Transport Phenomena A Conceptual Approach

510 APPENDIX A. MATHEMATICAL PRELIMINARIES

A.7.2 Numerical Methods

Numerical methods should be used when the equations to be solved are complex
and do not have direct analytical solutions. Various numerical methods have been
developed for solving Eq. (A.7-1). Some of the most convenient techniques to solve
the chemical engineering problems are summarized by Serghides (1982), Gjumbir
and Olujic (1984), and Tao (1988). The Newton-Ruphson method and the Secant
method are the two most widely used techniques and will be explained below.
One of the most important problems in the application of numerical techniques
is convergence. It can be promoted by finding a good starting value and/or a
suitable transformation of the variable, or the equation.
When using numerical methods, it is always important to use the engineering
common sense. The following advice of Tao (1989) should always be remembered
in the application of numerical techniques:

0 To err is digital, to catch the error is divine.

0 An ounce of theory is worth 100 lb of computer output.
0 Numerical methods are like political candidates; they’ll tell you anything you
want to hear.

A. 7.2.1 Newton-Raphson met hod
Expansion of the function f(z) by Taylor series around an estimate Zk gives

If we neglect the derivatives higher than the first order and let x = xk+l be the
value of x that makes f(x) = 0, then Q. (A.7-17) becomes

xk+l = xk - - (A. 7- 18)
f(xk
LI
dx Zk
The convergence is achieved when Izk+l - qJ < E, where E is a small number
determined by the desired accuracy.
The Newton-Raphson method proceeds as shown in Figure A.3. Note that the
method breaks down if (df/d~),~ = 0 at some point.

A.7.2.2 Secant method

Application of the Newton-Raphson method, Eq. (A.7-18), requires the evaluation
of the first derivative of the function. Unfortunately, this is not always easily found.
The secant method, on the other hand, requires only the evaluation of the function.