TRANSPORT PHENOMENA
MASS TRANSPORT
Macroscopic Balances for Multicomponent Systems
Macroscopic Balances for
Multicomponent Systems
1. The Macroscopic Mass Balance
2. The Macroscopic Momentum and Angular
Momentum Balances
3. The Macroscopic Energy Balance
4. The Macroscopic Mechanical Energy Balance
Macroscopic Mass Balance
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Macroscopic Mass Balance
•
Macroscopic Momentum and Angular
Momentum Balances
The macroscopic statements of the laws of conservation of
momentum and angular momentum for a fluid mixture, with
gravity as the only external force, are
For most mass transfer processes these terms are so small
that they can be safely negleted.
Macroscopic Energy Balance
For a fluid mixture, the macroscopic statement of the law of
conservation of energy is
The Q term accounts for addition of energy to the system
as a result of mass transfer. It may be of considerable
importance, particularly if material is entering through the
bounding surface at a much higher or lower temperature
than that of the fluid inside the flow system, or if it reacts
chemically in the system.
Macroscopic Mechanical Energy Balance
The macroscopic statement of the law of conservation of
mechanical energy is
The additional term B0 accounts for the mechanical energy
transport across the mass transfer boundary.
Unsteady Operation of a Packed Column
EXAMPLE 23.6-2
There are many industrially important processes in which mass transfer takes
place between a fluid and a granular porous solid.
In this operation, a solution containing a single
solute A at mole fraction xA1 in a solvent B is
passed at a constant volumetric flow rate w/ρ
through a packed tower. The tower packing
consist of a granular solid capable of
adsorbing A from the solution. At the start of
the percolation, the interstices of the bed are
filled with pure liquid B, and the solid is free of
A. he percolating fluid displaces this solvent
evenly so that the solution concentration of A
is always uniform over any cross section.
Problem: Develop an expression for the
concentration of A in the column as a function
of time and of distance down the column.
A fixed-bed absorber
(a) pictorial representation
of equipment; (b) a typical
effluent curve.
Unsteady Operation of a Packed Column
EXAMPLE 23.6-2
Solution. We think of the two phases as being continuous and
existing side by side. We define the contact area per unit packed
volume of column as a.
Now, however, one of the phases is stationary,
and unsteady-state conditions prevail. Because of
this locally unsteady behavior, the macroscopic
mass balances are applied locally over a small
column increment of height Δz. We may use
macroscopic mass balance in molar units and the
assumption of dilute solutions to state that the
molar rate of flow of solvent, WB, is essentially
constant over the length of the column and the
time of operation.
Unsteady Operation of a Packed Column
We now proceed to use macroscopic mass balance to write the mass
conservation relations for species A in each phase for a column
increment of height Δz:
The symbols have the following meaning:
ε
– volume fraction of column occupied by the liquid
S
– cross-section area of column
cAs
– mole of adsorbed A per unit volume of the solid
phase
xA
– bulk mole fraction of A in the liquid phase
xA0
– interfacial mole fraction of A in the fluid phase,
assumed to be in
equilibrium with cAs
k0x
– fluid-phase mass transfer coefficient
Unsteady Operation of a Packed Column
For the fluid phase, in the column increment under consideration,
becomes
Here с is the total molar concentration of the liquid. Equation above
may be rewritten by the introduction of a modified time variable, defined
by
It may be seen that, for any position in the column, t' is the time
measured from the instant that the percolating solvent "front" has
reached the position in question. After substitution we get:
Unsteady Operation of a Packed Column
The two equations are to be solved simultaneously along with the
interphase equilibrium distribution, xA0 = mcAs, in which m is a constant.
The boundary conditions are:
Before solving these equations, it is convenient to rewrite them in terms
of the following dimensionless variables:
Unsteady Operation of a Packed Column
In terms of these variables, the differential equations and boundary
conditions take the form:
with the boundary conditions Y(ζ, 0) = 0 and X(0, τ) = 1.
The solution to above equations for these boundary conditions is: