Page 65 - The engineering of chemical reactions

Variable Density 49
To solve the batch-reactor mass-balance equation, we write

CA = $! = NAo(l - x) = cAo 1 - x
V0(1 +2x) 1 + 2 x
so that we have written CA as a function of X and constants.
For this problem the equation dCA/dt = -WA is not appropriate, and we must solve
the equation
dNA = v,(l + 2X)kC,4, 1 - x
- = -vkCA
dt 1+2x
Since dNA ‘= -NAP dX, this mass-balance equation on species A can be converted to
dx
- = +p - X)(1 +2x) = k(l _ x)
dt 1 + 2 x
This equation can be separated to yield
dX
- = k d t
1 - x
and integrated from X = 0 at t = 0 to give
X
dX 1 ’ dX
t= -=- -
s r(X) k s 1 - x
x=0 x=0

= --i ln(1 - X)
This equation can be solved for X(t),
X(t) = 1 - e-kr
Finally, substituting back into CA(X), we obtain

1 - x e-kt ,-kt
c , ( t ) = CA‘,-- = CAo - CAo
1 + x 1 + 2(1 - e-kt) - 3 - 2eekt
Next consider the preceding as an &h-order irreversible reaction
A-+mB, r=kC;
The mass balance is

dN.4
- = -VkC” A
dt
or
dX
- = kC;,‘(l - X)“[l + (m - 1)x]++’
dt
and this equation can be separated to yield

1 x [l + (m - l>X]n-l dX
t=-
kc;,’ s (1 - x>n
0
This equation can be solved for t(X) or t (CA) by partial fractions, but the solution is not
pahcuhrly simple to solve explicitly for X(t) or CA(r).