Page 268 - Introduction to chemical reaction engineering and kinetics

9.2 Gas-Liquid Systems 249

with boundary conditions given by equations 9.2-31 and -33. We solve equation 9.2-28
in dimensionless form by letting

A = c*Ic*i (9.2-37)

and

z = XlSe (9.2-38)

Then equation 9.2-36 becomes

(9.2-39)

where the Hatta number, Ha, is a dimensionless group defined by

Ha = S&IC~ID~~)~‘* = (DAek#*/kAe (n = 1) (9.2-40) /

if we use equation 9.2-7 to eliminate 6,. The solution of equation 9.2-39 gives the con-
centration profile for A through the liquid film in terms of A and z . The boundary con-
ditions 9.2-31 and -33, in terms of A and z , become

at z = 0, A=1 (9.2-31a)

atz = 1, A = CAbICAi = A,

The solution of equation 9.2-39 may be written as

A = Kl exp(Ha z) + K2 exp(-Ha z) (9.2-41)

and the integration constants evaluated from the boundary conditions are

-
K = 4 ed-Ha)
1 2 sinh (Ha)
K = dW - 4
2 2 sinh(Ha)

Elimination of Kl and K2 from equation 9.2-41 results in

A = A,sinh(Ha z) + sinh[Ha(l - z)] (9.2-42)
sinh (Ha)

To obtain the steady-state rate of transfer or flux of A into the liquid film, NA(z = 0),
we note that this flux is equal to the rate of diffusion at the gas-liquid interface:

NA(z = 0) = -DA, (9.2-43)