Page 528 - Elements of Chemical Reaction Engineering Ebook

498 Steady-State Nonisothermal Reactor Design Chap. 8

the heat-generated curve. At this point of tangency the slopes of the R(T) and
G(T) curves are equal. That is, for the heat removal curve we have

(8-74)

and for the heat-generated curve

Assuming that the reaction is irreversible, follows a power law model, and that
the concentrations of the reacting species are weak functions of temperature,
-rA = (Ae-EIRT) fn(C,)

then

(8-75)

Equating Equations (8-74) and (8-75) yields

E
-AH,,
Cpo(l + K) = - (-rA) - (8-76)
RT2 FAO
Finally, we divide Equation (8-67) by Equation (8-76) to obtain the fol-
lowing AT value for a CSTR operating at T = T, .

(8-77)

I 1
rf this AT,, is exceeded, transition lo the upper steady state will occur. For
many industrial reactions E/RT is typically between 16 and 24, and the reaction
temperatures may be 300 to 500 K. Consequently, this critical temperature dif-
ference AT,, will be somewhere around 15 to 30°C.

8.6.5 Steady-State Bifurcation Analysis

In reactor dynamics it is particularly important to find out if multiple sta-
tionary points exist or if sustained oscillations can arise. Bifurcation analysis is
aimed at locating the set of parameter values for which multiple steady states
will occur.23 We apply bifurcation analysis to learn whether or not multiple
steady states are possible. An outline of what is on the CD-ROM follows.

23V. Balakotaiah and D. Luss, in Chemicql Reaction Enginewing, Boston ACS Sympo-
sium Series 196 (Washington, D.C,: American Chemical Society, 1982), p. 65;
M. Golubitsky and B. L. Keyfitz, Siam. J. Marh. Anal., 11, 316 (1980); A. Uppal,
W. H. Ray, and A. B. Poore, Chem. Eng. Sci., 29, 967 (1974).