Page 626 - Bird R.B. Transport phenomena

606 Chapter 19 Equations of Change for Multicomponent Systems

7. Discuss the similarities and differences between heat transfer and mass transfer.
8. Go through all the steps in converting Eq. 19.3-4 into Eq. 19.3-6. Why is the latter (approxi-
mate) result important?
9. Comment on the statement at the end of Example 19.4-1 that the rate of heat transfer is di-
rectly affected by simultaneous mass transfer, whereas the reverse is not true.

PROBLEMS 19A.1. Dehumidification of air (Fig. 19.4-1). For the system of Example 19.4-1, let the vapor be H O
2
and the stagnant gas be air. Assume the following conditions (which are representative in air
conditioning): (i) at z = 5, T = 80°F and x , = 0.018; (ii) at z = 0, T = 50°F.
H o
(a) For p = 1 atm, calculate the right side of Eq. 19.4-9.
(b) Compare the conductive and diffusive heat flux at z = 0. What is the physical significance
of your answer?
Answer: (a) 1.004
19B.1. Steady-state evaporation (Fig. 18.2-1). Rework the problem solved in §18.2, dealing with the
evaporation of liquid A into gas B, starting from Eq. 19.1-17.
(a) First obtain an expression for v*, using Eq. (M) of Table 17.8-1, as well as Fick's law in the
form of Eq. (D) of Table 17.8-2.
(b) Show that Eq. 19.1-17 then becomes the following nonlinear second-order differential
equation:

§d) 2 = 0
dz 2 l~x \dz,
A
(c) Solve this equation to get the mole fraction profile given in Eq. 18.2-11.

19B.2. Gas absorption with chemical reaction (Fig. 18.4-1). Rework the problem solved in §18.4, by
starting with Eq. 19.1-16. What assumptions do you have to make in order to get Eq. 18.4-4?
19B.3. Concentration-dependent diffusivity. A stationary liquid layer of В is bounded by planes
z = 0 (a solid wall) and z = b (a gas-liquid interface). At these planes the concentration of
A is c A0 and c Ab respectively. The diffusivity ЯЬ is a function of the concentration of A.
АВ
(a) Starting from Eq. 19.1-5 derive a differential equation for the steady-state concentration
distribution.
(b) Show that the concentration distribution is given by

0
= 7 (19B.3-1)
<db dc A
AB
(c) Show that the molar flux at the solid-liquid surface is

<3) (c )dc A (19B.3-2)
AB
A
(d) Now assume that the diffusivity can be expressed as a Taylor series in the concentration
®AB(C ) A = a A B [ l + fr(c A - c ) + p (c A - c ) 2 + • . •] (19B.3-3)
A
2
A
in which c A = \(c A0 + c ) and %b = ЯЬ (с ). Then, show that
АВ
Ab
А
AB
N * U = ^f Ьло ~ c )[\ + ± p (c - c f + • • •] (19B.3-4)
Ab 2 2 A0 Ab
(e) How does this result simplify if the diffusivity is a linear function of the concentration?