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Overall Shell
Mass Balances I
Outline
3.
4.
5.
6.
Molecular Diffusion in Gases
Molecular Diffusion in Liquids
Molecular Diffusion in Solids
Prediction of Diffusivities
7. Overall Shell Mass Balances
1. Concentration Profiles
Overall Shell Mass Balance
Species entering and
leaving the system
by Molecular Transport +
by Convective Transport
* May also be expressed in terms of moles
Mass Generation
by homogeneous Steady-State!
chemical reaction
Overall Shell Mass Balance
* May also be expressed in terms of moles
Common Boundary Conditions:
1.
2.
3.
4.
Concentration is specified at the surface.
The mass flux normal to a surface maybe given.
At solid- fluid interfaces, convection applies: NA = kc∆cA.
The rate of chemical reaction at the surface can be specified.
♪ At interfaces, concentration is not necessarily continuous.
Concentration Profiles
I. Diffusion Through a
Stagnant Gas Film
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
Assumptions:
1.
2.
3.
4.
Steady-state
T and P are constants
Gas A and B are ideal
No dependence of vz on
the radial coordinate
At the gas-liquid interface,


 =

Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
Mass balance is done in this thin shell
perpendicular to the direction of mass flow

 = −
+  ( +  )

Concentration Profiles
I. Diffusion Through a Stagnant Gas Film

 = −
+  ( +  )

Since B is stagnant,
 
 = −
(1 −  ) 
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
 
 = −
(1 −  ) 
Applying the mass balance,
 ǀ −  ǀ+∆ = 0
where S = cross-sectional area of
the column
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
 ǀ −  ǀ+∆ = 0
Dividing by SΔz and taking the limit as Δz  0,

−
=0

NA = constant
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film

−
=0

NA = constant
 
But,  = −
(1 −  ) 
Substituting,


 
=0
1 −  
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film


 
=0
1 −  
For ideal gases, P = cRT and so at constant P and T, c = constant
DAB for gases can be assumed independent of concentration


1

=0
1 −  
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film


1

=0
1 −  
Integrating once,
1

= 1
1 −  
Integrating again,
− ln 1 −  = 1  + 2
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
− ln 1 −  = 1  + 2
Let C1 = -ln K1 and C2 = -ln K2,
1 −  = 1 2
B.C.
at z = z1,
at z = z2,
xA = xA1
xA = xA2
1 − 
1 − 2
=
1 − 1
1 − 1
−1
2 −1
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
1 − 
1 − 2
=
1 − 1
1 − 1
The molar flux then becomes
 
 = −
(1 −  ) 

−1
2 −1

1 − 2
=
ln(
)
2 − 1
1 − 1
OR in terms of the driving force ΔxA
*1 − 2 > 0, i.e. xA1> xA2
ǂ − 
i.e. z2> z1
2
1 > 0,

2 − 1
 =
(1 − 2 )
( ) =
(2 − 1 )( )

ln( 2 )
1
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical Reaction
Two Reaction Types:
1. Homogeneous – occurs
in the entire volume of
the fluid
- appears in the
generation term
2. Heterogeneous – occurs
on a surface (catalyst)
- appears in the
boundary condition
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical Reaction
Reaction taking place
2A  B
1. Reactant A diffuses to
the surface of the
catalyst
2. Reaction occurs on the
surface
3. Product B diffuses away
from the surface
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical Reaction
Reaction taking place
2A  B
Assumptions:
1. Isothermal
2. A and B are ideal gases
3. Reaction on the surface
is instantaneous
4. Uni-directional transport
will be considered
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical Reaction

=0


 = −
+  ( +  )

Concentration Profiles
II. Diffusion With a Heterogeneous Chemical Reaction
From stoichiometry,
 = −1/2
 
 = −
1
1 −  
2
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical Reaction
Substitution of NA into the differential equation

 
(−
)=0
1

1 −  
2
Integration twice with respect to z,
1
−2 ln 1 −  = 1  + 2 = −(2 ln 1 ) − (2 ln 2 )
2
B.C. 1: at z = 0,
B.C. 2: at z = δ,
xA = xA0
xA = 0
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical Reaction
The final equation is

1
1
(1− )
1 −  = (1 − 0 ) 
2
2
And the molar flux of reactant through the film,
2
1
 =
ln(
)
1

1 − 0
2
*local rate of reaction per unit of catalytic surface
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical Reaction
Reading Assignment
See analogous problem Example 18.3-1 of
Transport Phenomena by Bird, Stewart and Lightfoot
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
1. Gas A dissolves in liquid B and
diffuses into the liquid phase
2. An irreversible 1st order
homogeneous reaction takes
place
A + B  AB
Assumption:
AB is negligible in the solution
(pseudobinary assumption)
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
 ǀ −  ǀ+∆ − 1′′′  ∆ = 0
1′′′ first order rate constant
for homogeneous
decomposition of A
S cross sectional area of the liquid
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
 ǀ −  ǀ+∆ − 1′′′  ∆ = 0
Dividing by SΔz and taking the limit as Δz  0,

+ 1′′′  = 0

Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction

+ 1′′′  = 0

If concentration of A is small, then the total c is almost constant and

 = −

Combining the two equations above
2 
′′′

−

1  = 0
2

Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
2 
′′′

−

1  = 0
2

. . 1
. . 2
  = 0,
  = ,
 = 0

 = 0 
=0

Multiplying the above equation by
2
0 
gives an equation with dimensionless variables
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
2 
′′′

−

1  = 0
2

2Γ
2
−

Γ=0
2


Γ=
,
0

= ,

=
 ′′′ 2 /
Thiele Modulus
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
2Γ
2Γ = 0
−

 2
. . 1
  = 0,
. . 2
  = 1,
Γ=1
Γ
=0

The general solution is
Γ = 1 cosh  + 2 sinh 
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
Γ = 1 cosh  + 2 sinh 
Evaluating the constants,
cosh  cosh  − sinh  sinh 
cosh[ϕ 1 − ζ ]
Γ=
=
cosh 
cosh 
Reverting to the
original variables,

=
0
 ′′′ 2

cosh[
1− ]


 ′′′ 2
cosh(
)

Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
Quantities that might be asked for:
1. Average concentration in the liquid phase
,
=
0

( / )
0  0


0
tanh 
=

2. Molar flux at the plane z = 0
ǀ=0

0 
= −
ǀ=0 =
 tanh 


Concentration Profiles
IV. Diffusion into a Falling Liquid Film (Gas Absorption)
Assumptions
1. Velocity field is unaffected by
diffusion
2. A is slightly soluble in B
3. Viscosity of the liquid is unaffected
4. The penetration distance of A in B
will be small compared to the film
thickness.
Concentration Profiles
IV. Diffusion into a Falling Liquid Film (Gas Absorption)
Recall: The velocity of a falling film
 2 cos 
 () =
2
  = 
 2
1−( )

 2
1−( )

Concentration Profiles
IV. Diffusion into a Falling Liquid Film (Gas Absorption)
* CA is a function of both x and z
ǀ ∆ − ǀ+∆ ∆
+ǀ ∆ − ǀ+∆ ∆ = 0
Concentration Profiles
IV. Diffusion into a Falling Liquid Film (Gas Absorption)
ǀ ∆ − ǀ+∆ ∆
+ǀ ∆ − ǀ+∆ ∆ = 0
Dividing by WΔxΔz and
letting Δx  0 and Δz  0,
 
+
=0


Concentration Profiles
IV. Diffusion into a Falling Liquid Film (Gas Absorption)
 
+
=0


The expressions for  ,


= −
+  ( +  )

Transport of A along the z direction is mainly by convection (bulk motion)
Recall:
 = ∗ +  
 =   
 ≈   =   ()
Concentration Profiles
IV. Diffusion into a Falling Liquid Film (Gas Absorption)
 
+
=0


The expressions for  ,


= −
+  ( +  )

Transport of A along the x direction is mainly by diffusion


≈ −

Concentration Profiles
IV. Diffusion into a Falling Liquid Film (Gas Absorption)
 
+
=0


Substituting the expressions for   ,


 2 
= 

 2
Substituting the expressions vz,

 2
1−( )


 2 
= 

 2
Concentration Profiles
IV. Diffusion into a Falling Liquid Film (Gas Absorption)

 2
1−( )


 2 
= 

 2
Boundary conditions
B.C. 1   = 0,
B.C. 2   = 0,
B.C. 3   = ,
 = 0
 = 0


=0
BUT we can replace B.C. 3 with
B.C. 3   = ∞,  = 0
Concentration Profiles
IV. Diffusion into a Falling Liquid Film (Gas Absorption)

 2
1−( )


2
=1−
0

or
where
2 /
/ 4

exp − 2 
0

= 1 − 
0
erf  =

 2 
= 

 2

= 
2
4



2
0 exp(− ) 
∞
2
0 exp(− ) 
=

2
4


2 
2
exp(−

) 
 0
Concentration Profiles
IV. Diffusion into a Falling Liquid Film (Gas Absorption)

= 1 − 
0
ǀ=0

2
4


= 

2
4



 
= −
ǀ=0 = 0


Concentration Profiles
IV. Diffusion into a Falling Liquid Film (Gas Absorption)
Reading Assignment
See analogous problem Example 4.1-1 of Transport
Phenomena by Bird, Stewart and Lightfoot
Concentration Profiles
IV. Diffusion into a Falling Liquid Film (Gas Absorption)
Quantities that might be asked for:
1. Total molar flow of A across the surface at x = 0

 =
0
= 0
= 0

0
ǀ=0 
 

 


0
1


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