This lecture was delivered by Hafiz Anees Rehman at Quaid-Awam-University, Nawab Shah Pakistan for Transport Phenomenon course. It includes: Diffusion, Stagnant, Gas, Liquid, Film,Law, Mass, Transfer, Molar, Concentrations

1. CONCENTRATION DISTRIBUTIONS IN
SOLIDS AND IN LAMINAR FLOW
1
/1
5
SHELL MASS BALANCES;
BOUNDARY CONDITIONS

2.  The diffusion problems are solved by making
mass balances for one or more chemical species
over a thin shell of solid or fluid.
 Having selected an appropriate system, the law of
conservation of mass of species A in a binary
system is written over the volume of the shell in
the form:
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(1)

3.  The conservation statement may, of course, be
expressed in terms of moles.
 The chemical species A may enter or leave the
system by diffusion (i.e., by molecular motion)
and also by virtue of the overall motion of the fluid
(i.e., by convection), both of these being included
in NA.
 In addition, species A may be produced or
consumed by homogeneous chemical
reactions.
What is happening?
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4.  After a balance is made on a shell of finite thickness
by means of Eq.(1), we then let the thickness
become infinitesimally small.
 As a result of this process a differential equation for
the mass (or molar) flux is generated.
 If, into this equation, we substitute the expression for
the mass (or molar) flux in terms of the
concentration gradient, we get a differential equation
for the concentration.
How Differential Equations are formed?
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5.  Integration of differential equation
 Constants of integration
 Using boundary conditions to calculate Constants
 Substitution of constants in integrated equations to
get General Solution
How To Solve the Differential Equations?
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6.  Liquid A is evaporating into gas B.
 We imagine there is some device that maintains the liquid
level at z =z1.
 Right at the liquid-gas interface, the gas-phase
concentration of A, expressed as mole fraction, is xA1.
 This is taken to be the gas-phase concentration of A
corresponding to equilibrium with the liquid at the interface.
 i.e. , provided hat A and B form an ideal gas
mixture and that the solubility of gas B in liquid A is
negligible.
System picture in TEXT.
p
p
x
vap
A
1A =
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7.  A stream of gas mixture A-B of concentration xA2 flows
slowly past the top of the tube, to maintain the mole
fraction of A at xA2 for z= z2.
 The entire system is kept at constant temperature and
pressure.
 Gases A and B are assumed to be ideal.
System picture in TEXT.
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8. 8/15

9.  A stream of gas mixture A-B of concentration xA2 flows
slowly past the top of the tube, to maintain the mole
fraction of A at xA2 for z= z2.
 The entire system is kept at constant temperature and
pressure.
 Gases A and B are assumed to be ideal.
Boundary Conditions.
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10.  When this evaporating system attains a steady state, there
is a net motion of A away from the interface and the
species B is stationary.
 Hence the molar flux of A is given by:
 A steady-state mass balance (in molar units) over an
increment Δz of the column states that the amount of A
entering at plane z equals the amount of A leaving at plane
z + Δz:
Fist step converting physical system into
mathematical model
(2)
(3)
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11.  Here S is the cross-sectional area of the column. Division
by SΔz and taking the limit as Δz0 gives:
 Substitution of Eq. (2) into Eq. (4) gives:
Further derivation:
(4)
(5)
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12.  For an ideal gas mixture the equation of state is
p=cRT, so that at constant temperature and pressure c
must be a constant.
 Furthermore, for gases DAB is very nearly independent of
the composition.
 Therefore, can be moved to the left of the derivative
operator to get
How c and DAB are constants?
(6)
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13.  Eq.(6) is a second-order differential equation for the
concentration profile expressed as mole fraction of A.
 Integration with respect to z give:
 A second integration then gives
Integrations . . . .
(7)
(6)
(8)
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14.  If we replace C1 by -In K1 and C2 by -In K2 ,Eq.(8) becomes:
 The two constants of integration, K1 and K2 may then be
determined from the boundary conditions
 When the constants have been obtained, we get finally
Boundary Conditions
(9)
(10)
(11)
(12)
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