Page 567 - Bird R.B. Transport phenomena

§18.2 Diffusion Through a Stagnant Gas Film 547

Substitution of Eq. 18.2-1 into Eq. 18.2-3 gives
xA .
dz \1 - x dz Q ( 1 8 2 4 )
A
For an ideal gas mixture the equation of state is p = cRT, so that at constant temperature
and pressure с must be a constant. Furthermore, for gases 4L is very nearly indepen-
AB
dent of the composition. Therefore, c% can be moved to the left of the derivative opera-
AB
tor to get
d ( 1 dx
A = 0 (18.2-5)
dz \1 - x dz
A
This is a second-order differential equation for the concentration profile expressed as
mole fraction of A. Integration with respect to z gives

(18.2-6)
1 - x dz
A
A second integration then gives
-ln(l - x ) = Qz + C 2 (18.2-7)
A
If we replace Q by In К and C by In K , Eq. 18.2-7 becomes
-
-
г
2
2
l-x A = K\K 2 (18.2-8)
The two constants of integration, K and K , may then be determined from the boundary
} 2
conditions
B.C. 1: at z = z u x = x M (18.2-9)
A
B.C. 2: at z = z , x = x A2 (18.2-10)
2
A
When the constants have been obtained, we get finally
~~ (18.2-11)

The profiles for gas В are obtained by using x = 1 — x . The concentration profiles are
A
B
shown in Fig. 18.2-1. It can be seen there that the slope dx /dz is not constant although
A
N Az is; this could have been anticipated from Eq. 18.2-1.
Once the concentration profiles are known, we can get average values and mass
fluxes at surfaces. For example, the average concentration of В in the region between z x
and z is obtained as follows:
2
\\x /x )dz \\x B2/x mYd£ c
m
B
•*B,avg _ J z, _ JQ _ \X B2/X m)-
(18.2-12)
о
Z
Z
in which С = ( z ~ \)/( 2 ~~ z^ is a dimensionless length variable. This average may be
rewritten as
• - Хв2 ~ Хт (18.2-13)
Лв '^ Ых /х )
В2 В1
That is, the average value of x is the logarithmic mean, (x ) , of the terminal concen-
B
B ln
trations.