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Chapter 25
Differential Equations of Mass Transfer
25.5 CLOSURE
The general differential equation for mass transfer was developed to describe the mass
balances associated with a diffusing component in a mixture. Special forms of the general
differential equation for mass transfer that apply to specific situations were presented.
Commonly encountered boundary conditions for molecular diffusion processes were also
listed. From this theoretical framework, a five-step method for mathematically modeling
processes involving molecular diffusion was proposed. Three examples illustrated how the
differential form of Fick’s equation presented in Chapter 24, and the general differential
equation for mass transfer presented in this chapter, are reduced to simple differential
equations that describe the molecular diffusion aspects of a specific process. The approaches
presented in this chapter serve as the basis for problem solving in Chapters 26 and 27.
PROBLEMS
25.1 Derive equation (25-11) for component A in terms of
molar units, starting with the control-volume expression for the
conservation of mass.
25.2
Show that the (25-11) may be written in the form
@rA
þð= # rA vÞ % DAB r2 rA ¼ rA
@t
25.3 The following sketch illustrates the gas diffusion in the
neighborhood of a catalytic surface. Hot gases of heavy hydrocarbons diffuse to the catalytic surface where they are cracked
into lighter compounds by the reaction: H ! 2L, the light
products diffuse back into the gas stream.
H
L
z=0
z=δ
a. Reduce the general differential equation for mass transfer to
write the specific differential equation that will describe this
steady-state transfer process if the catalyst is considered a
flat surface. List all of the assumptions you have made in
simplifying the general differential equation.
b. Determine the Fick’s law relationship in terms of only
compound H and insert it into the differential equation
you obtained in part (a).
25.4 A hemispherical droplet of liquid water, lying on a flat
surface, evaporates by molecular diffusion through still air
surrounding the droplet. The droplet initially has a radius R.
As the liquid water slowly evaporates, the droplet shrinks slowly
with time, but the flux of the water vapor is at a nominal steady
state. The temperature of the droplet and the surrounding still air
are kept constant. The air contains water vapor at an infinitely
long distance from the droplet’s surface.
a. After drawing a picture of the physical process, select a
coordinate system that will best describe this diffusion
process, list at least five reasonable assumptions for the
mass-transfer aspects of the water-evaporation process and
simplify the general differential equation for mass transfer
in terms of the flux NA.
b. What is the simplified differential form of Fick’s equation
for water vapor (species A)?
25.5 A large deep lake, which initially had a uniform oxygen
concentration of 1kg/m3, has its surface concentration suddenly
raised and maintained at 9 kg/m3 concentration level.
Reduce the general differential equation for mass transfer to
write the specific differential equation for
a. the transfer of oxygen into the lake without the presence of a
chemical reaction;
b. the transfer of oxygen into the lake that occurs with the
simultaneous disappearance of oxygen by a first-order
biological reaction.
25.6 The moisture in hot, humid, stagnant air surrounding a
cold-water pipeline continually diffuses to the cold surface
where it condenses. The condensed water forms a liquid film
around the pipe, and then continuously drops off the pipe to the
ground below. At a distance of 10 cm from the surface of the pipe,
the moisture content of the air is constant. Close to the pipe, the
moisture content approaches the vapor pressure of water evaluated at the temperature of the pipe.
a. Draw a picture of the physical system, select the coordinate
system that best describes the transfer process and state at
least five reasonable assumptions of the mass-transfer
aspects of the water condensation process.
b. What is the simplified form of the general differential
equation for mass transfer in terms of the flux of water
vapor, NA?
c. What is the simplified differential form of Fick’s equation
for water vapor, NA?
d. What is the simplified form of the general differential
equation for mass transfer in terms of the concentration
of water vapor, cA?
Problems
25.7 A liquid flows over a thin, flat sheet of a slightly soluble
solid. Over the region in which diffusion is occurring, the liquid
velocity may be assumed to be parallel to the plate and to be
given by v ¼ ay, where y is the vertical distance from the plate
and a is a constant. Show that the equation governing the mass
transfer, with certain simplifying assumptions, is
! 2
"
@ cA @ 2 cA
@cA
DAB
þ
¼ ay
@x2
@y2
@x
List the simplifying assumptions, and propose reasonable
boundary conditions.
25.8 Consider one of the cylindrical channels that run through
an isomerization catalyst as shown below. A catalyst coats the
inner walls of each channel. This catalyst promotes the isomerization of n-butane ðn $ C4 H10 Þ species A to isobutene ði $
C4 H10 Þ species B.
n $ C4 H10 ðgÞ ! i $ C4 H10 ðgÞ
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to explain the metabolic consumption of the oxygen to produce
carbon dioxide.
Use the general differential equation for mass transfer of oxygen
to write the specific differential equation that will describe the
diffusion of oxygen in the human tissue. What would be the form of
Flicks relationship written in terms of only the diffusing oxygen?
25.10 A fluidized coal reactor has been proposed for a new
power plant. If the coal can be assumed to be spherical, reduce
the general differential equation for mass transfer to obtain a
specific differential equation for describing the steady-state
diffusion of oxygen to the surface of the coal particle.
Determine the Fick’s law relationship for the flux of oxygen
from the surrounding air environment if
a. only carbon monoxide, CO, is produced at the surface of the
carbon particle;
b. only carbon dioxide, CO2, is produced at the surface of the
carbon particle.
If the reaction at the surface of the carbon particle is instantenous, give two boundary conditions that might be used in
solving the differential equation.
25.11 In the manufacture of semiconducting thin films, a thin
film of solid arsenic laid down on the surface of a silicon water by
the diffusion-limited chemical vapor deposition of arsine, AsH3.
2AsH3 ðgÞ ! 2AsðsÞ þ 3H2 ðgÞ
The gas phase above the channels contains mixture of A and B
maintained at a constant composition of 60 mol % n $ C4 H10
(A) and 40 mol % i $ C4 H10 (B). Gas phase species A diffuses
down a straight channel of diameter d ¼ 0:1 cm and length
L ¼ 2:0 cm. The base of each channel is sealed. This is rapid
reaction so that the production rate of B is diffusion limited.
The quiescent gas space in the channel consists of only species A
and B.
a. State three relevant assumptions for the mass transfer
process.
b. Based on your assumptions, simplify the general differential
equation for the mass transfer of species A, leaving the
equation in terms of the flux NA.
c. Using equations for the flux of A in your determined
equation, express the general differential equation in terms
of the concentration cA.
d. Specify relevant boundary conditions for the gas phase
concentration cA.
25.9 An early mass-transfer study of oxygen transport in
human tissue won a Nobel prize for August Krough. By considering a tissue cylinder surrounding each blood vessel, he
proposed the diffusion of oxygen away from the blood vessel into
the annular tissue was accompanied by a zero-order reaction, that
is, RA ¼ $m, where m is a constant. This reaction was necessary
The gas head space, 5 cm above the surface of the wafer, is
stagnant. Arsenic atoms deposited on the surface then diffuse
into the solid silicon to ‘‘dope’’ the wafer and impart semiconducting properties to the silicon, as shown in the figure below.Well mixed feed gas (constant composition).
Well-mixed feed gas (constant composition).
Diffuser screen
H2 (g)
AsH3 (g)
As, thin film
Si wafer
NA
The process temperature is 1050& C. The diffusion coefficient of
aresenic in silicon is 5 ' 10$13 cm/s at this temperature and the
maximum solubility of aresenic in silicon is 2 ' 1021
atoms/cm3 . The density of solid silicon is 5 ' 1022 atoms=
cm3 . As the diffusion coefficient is so small, the aresenic atoms
do not ‘‘penetrate’’ very far into the silicon solid, usually less
than a few microns. Consequently, a relatively thin silicon water
can be considered as a ‘‘semi-infinite’’ medium for diffusion.
a. State at least five reasonable assumptions for the mass
transfer of aresenic in this doping process.
b. What is the simplified form of the general differential
equation for the mass transfer of the aresenic concentration
within the silicon? Purpose reasonable boundary and initial
conditions.