Page 327 - Introduction to chemical reaction engineering and kinetics

308 Chapter 12: Batch Reactors (BR)

To obtain the value of t for maximum Pr, consider equation 12.3-22 with fAl = 0,
fA2 = f*(t). Then the equation may be written as

Pr = Kf*(t)l(t + td) (12.3-32)

where K( = vccAoV) is a constant, and Pr is Pr(C). For maximum Pr,
1
dPr 0 + tddf,(tW) - f*(t) = o
dt= (t + CA2

or

(t + t,,y - f*(t) = 0

Equation 12.3-33 is solved for t, and the result is used in equation 12.3-32 or its equiv-
alent to obtain the maximum value of Pr.

Consider a liquid-phase, first-order reaction A + C, occurring in a reactor of volume V,
with a specified down-time, td. The reactor initially contains 5 moles of pure A. Determine
the reaction time which maximizes Pr(C), given kA = 0.021 mm’, and td = 30 min;
and calculate the maximum value of Pr(C).

SOLUTION

Since this is a constant-density system, equation 12.3-33 applies. To use this, we require
f*(t). From the rate law, and the material balance, equation 2.2-10,

dfA
(-rA) = ~ACA = kAcA,(l - fA) = -2 = cAo-
dt
From this.

g = k,(l - fA)

This integrates to
fA = 1 - &d (4

from which, in terms oft,
dfA - k +d (W
~-
dt A

Substituting the results from (A) and (B) in equation 12.3-33, we have

(t + t,)k,e- kit - (1 - e-kd) = 0 ((2