Page 612 - Bird R.B. Transport phenomena

592 Chapter 19 Equations of Change for Multicomponent Systems

SOLUTION (a) Throughout this example, for brevity we omit the subscripts p, T indicating that these
quantities are held constant. First we write expressions for the intercepts as follows:

H =H- x (%f) ; H = H + xl§f) (19.3-10,11)
A B B
х
\° в/п \0Хв/п
in which H = H/(n + n ) = H/n. To verify the correctness of Eq. 19.3-10, we rewrite the ex-
A B
pression in terms of Я:
x
T3 _H l(dH
И
А - ТГ ~ ТГ I TZT I (19.3-1
n b
Now the expression H = (dH/dn ), implies that Я is a function of n and n , whereas
A A h A B
(дН/'дх ) implies that H is a function of x A and n. The relation between these kinds of deriva-
А п
tives is given by the chain rule of partial differentiation. To apply this rule we need the rela-
tion between the independent variables, which, in this problem, are
n A = (\- x )n; n B = x n (19.3-13,14)
B
B
Therefore we may write
дх ) \дп )„\дх ) \дп ) \дх в
в п
в п
А
в п
= Н (-п) + Н (+п) (19.3-15)
в
А
Substitution of this into Eq. 19.3-12 and use of Euler's theorem (H = n H A + n H ) then gives
B
A
B
an identity. This proves the validity of Eq. 19.3-10, and the correctness of Eq. 19.3-11 can be
proved similarly.
(b) One can also get H by using the definition in Eq. 19.3-7 and measuring the slope of the
A
curve of H versus n , holding n constant. One can also get H A by measuring the enthalpy of
B
A
mixing and using
H = n H + n H = n H + щН + AH (19.3-16)
A A B B A A в mi x
Often the enthalpy of mixing is neglected and the enthalpies of the pure substances are given
as H ~ C {T - T°) and a similar expression for H . This is a standard assumption for gas
A pA B
mixtures at low to moderate pressures.
Other methods for evaluating partial molar quantities may be found in current textbooks
on thermodynamics.
§19.4 USE OF THE EQUATIONS OF CHANGE FOR MIXTURES
The equations of change in §19.2 can be used to solve all the problems of Chapter 18, and
more difficult ones as well. Unless the problems are idealized or simplified, mixture
transport phenomena are quite complicated and usually numerical techniques are re-
quired. Here we solve a few introductory problems by way of illustration.

EXAMPLE 19.4-1 (a) Develop expressions for the mole fraction profile x (y) and the temperature profile T{y)
A
for the system pictured in Fig. 19.4-1, given the mole fractions and temperatures at both film
Simultaneous Heat and boundaries (y = 0 and у = 8). Here a hot condensable vapor, A, is diffusing at steady state
Mass Transport through a stagnant film of noncondensable gas, B, to a cold surface at у = 0, where A con-
denses. Assume ideal gas behavior and uniform pressure. Furthermore assume the physical

1 A. P. Colburn and T. B. Drew, Trans. Am. Inst. Chem. Engrs., 38,197-212 (1937).