Page 94 - Introduction to chemical reaction engineering and kinetics

76 Chapter 4: Development of the Rate Law for a Simple System

M Ail = kAd&,,-,lf (4.3-4)

where tis the reaction time in a BR or PFR, or the mean residence time in a CSTR.

4.3.5.1 BR or PFR (Isothermal, Constant Density)
For an &h-order isothermal, constant-density reaction in a BR or PFR (n # l), equa-
tion 3.4-9 can be rearranged to obtain cA/cA~ explicitly:

l-n - &id” = (n - l)k,t (3.4-9)
CA (n + 1)
= (n - l)MA,/c~~’ (3.4-9a)

(note that f = t here). From equation 3.4-9a,

= [l + (n - l)MA,I1’@“) (4.3-5)
CA/CA0 (n + 1)
For a first-order reaction (n = l), from equation 3.4-10,

= eXp(- kAt) = eXp(-MA,) (n = 1) (4.3-6)
cAicAo
The resulting expressions for cA/cAO for several values of n are given in the second
column in Table 4.1. Results are given for n = 0 and n = 3, although single-phase re-
actions of the type (A) are not known for these orders.
In Figure 4.3, CA/CA* is plotted as a function of MA,, for the values of n given in Table
4.1. For these values of II, Figure 4.3 summarizes how CA depends on the parameters
and f for any reaction of type (A). From the value of CA/CA~ obtained from
kA, cAo,
the figure, CA can be calculated for specified values of the parameters. For a given n,
CA/CA~ decreases as MA,, increases; if kA and cAo are fixed, increasing MA,, corresponds

Table 4.1 Comparison of expressionsa for CA/CA~ 5 1 - f~
I CA/CA~ = 1 - fA
Order(n) 1 BR or PFR I CSTR
0 = 1 -MAO; MAO 5 1 = ~-MAO; MAO 5 1
= 0; MAO 2 1 = 0; MAO 2 1

112 = (1 - h’f~,,#)~; ki~l/z 5 2
= 0; MA~/z 2 2

1 1 = exP(--Mid 1 = (1 + MAI)-’
312 = (1 + MA~,#)-~ from solution of cubic equation [in (cA/cA~)~'~]:
MA~~(CAICA,)~" + (CAICA~) - 1 = 0
= (1 + ‘tit’f~2)“~ - 1
2 = (1 + MAZ)-~
~MAZ
3 = (1 + 2it’f~3)-~‘~ from solution of cubic equation:
MA~(cA/cA,)~ + (CA/CA~) - 1 = 0

“For reaction A + products; (-TA) = kAcz; MA” = kAcAo“-l t; isothermal, constant-density conditions;
from equations 4.3-5, -6, and -9.