[Luyben] Process Mod.. - Student subdomain for University of Bath

NUMERICAL METHODS 97
T
Newton-Raphson convergence.
Generalizing Eq. (4.9), we get the recursive iteration algorithm :
where Tn + 1 = new guess of temperature
Tn = old guess of temperature
f, = value off,,, at T = T.
fi = value of the derivative off, df/dT, at T = T.
(4.10)
The technique requires the evaluation off ‘, the derivative of the function
f(r) with respect to temperature. In our bubblepoint example this can be
obtained analytically.
AT, = PC;: -
p = x &WT+BI) + (1 _ $ &‘z/T+Bz)
-xA,P, -(l -x)&P;
=
T2
(4.11)
If the function .were so complex that an analytical derivative could not be
obtained explicitly, an approximate derivative would have to be calculated
numerically: make a small change in temperature AT, evaluate fat T + AT and
use the approximation
(4.12)
A digital computer program using Eqs. (4.10) and (4.11) is given in Table
4.2. The problem is the same bubblepoint calculation for benzene-toluene performed
by interval-halving in Table 4.1, and the same initial guesses are made of
temperature. The results given in Table 4.2 show that the Newton-Raphson algorithm
is much more effective in this problem than interval halving: only 4 to 5