Page 628 - Bird R.B. Transport phenomena

608 Chapter 19 Equations of Change for Multicomponent Systems

state the diffusion is exactly balanced by the chemical reaction. Show that the concentration
profile is

= Yr
7 c A0 г е -ь
2
2
in which R is the radius of the sphere, c is the molar solubility of A in B, and b = k"'R /^ ,
A0 AB
(b) Show by quasi-steady-state arguments how to calculate the gradual decrease in diameter
of the sphere as A dissolves and reacts. Show that the radius of the sphere is given by
2
= R 0 -2 b) <f - t 0 ) (19B.6-2)
in which R is the sphere radius at time f 0/ and p sp h is the density of the sphere.
o
19B.7. Various forms of the species continuity equation.
(a) In this chapter the species equation of continuity is given in three different forms: Eq.
19.1-7, Eq. (A) of Table 19.2-1, and Eq. (B) in Table 19.2-3. Show that these three equations are
equivalent.
(b) Show how to get Eq. 19.1-15 from Eq. 19.1-11.

19C.1. Alternate form of the binary diffusion equation. In the absence of chemical reactions, Eq.
19.1-17 can be written in terms of v rather than v* by using a different measure of concentra-
tion—namely, the logarithm of the mean molecular weight: 2

4: In M + (v • V In M) = <$ V 2 In M (19C.1-1)
ot AB
in which M = x M A + x M . (Caution: Solution is lengthy.)
B
A
B
Equation 19C.1-1 is difficult to solve even for the stagnant gas film of §18.2, because of
the variable mass density p that appears in the continuity equation (Eq. A of Table 19.2-3).
19D.1. Derivation of the equation of continuity. In §19.1 the species equation of continuity is de-
rived by making a mass balance on a small rectangular volume Ax Ay Az fixed in space.
(a) Repeat the derivation for an arbitrarily shaped volume element V with a sufficiently
smooth fixed boundary S. Show that the species mass balance can be written as
n
j j p dV = - J (n • )dS + j r dV (19D.1-1)
t A A A
V S V
Use the Gauss divergence theorem to convert the surface integral to a volume integral, and
then obtain Eq. 19.1-6.
(b) Repeat the derivation using a region of fluid contained within a surface, each point of
which is moving with local mass average velocity.
19D.2. Derivation of the equation of change for temperature for a multicomponent system. De-
rive Eq. (F) in Table 19.2-4 from Eq. (E). We suggest the following sequence of steps:
(a) Since the enthalpy is an extensive thermodynamic property, we can write

m 2 , m 3 ,..., m N ) = <о 2 , co 3 ,..., о^м) (19D.2-1)
H(m u mH(a> u
in which the m are the masses of the various species, m is the sum of the m , and the a) =
Q a a
m /m are the corresponding mass fractions. Both H and H are understood to be functions of T
Q
and p as well as of composition. Use the chain rule of partial differentiation to show that

С. H. Bedingfield, Jr., and Т. В. Drew, lnd. Eng. Chem., 42,1164-1173 (1950).
2