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Overall ShellMass Balances I
Outline
3. Molecular Diffusion in Gases 4. Molecular Diffusion in Liquids 5. Molecular Diffusion in Solids6. Prediction of Diffusivities
7. Overall Shell Mass Balances1. Concentration Profiles
Overall Shell Mass Balance
Species entering and leaving the system
by Molecular Transport +by Convective Transport
Mass Generationby homogeneous chemical reaction
* May also be expressed in terms of moles
Steady-State!
Overall Shell Mass Balance
* May also be expressed in terms of moles
Common Boundary Conditions:
1. Concentration is specified at the surface.2. The mass flux normal to a surface maybe given.3. At solid- fluid interfaces, convection applies: NA = kcβcA.4. The rate of chemical reaction at the surface can be specified.
βͺ At interfaces, concentration is not necessarily continuous.
Concentration Profiles
I. Diffusion Through a
Stagnant Gas Film
Concentration Profiles
I. Diffusion Through a Stagnant Gas FilmAssumptions:
1. Steady-state2. T and P are constants3. Gas A and B are ideal4. No dependence of vz on
the radial coordinate
At the gas-liquid interface,
Concentration Profiles
I. Diffusion Through a Stagnant Gas FilmMass balance is done in this thin shell
perpendicular to the direction of mass flow
π π΄=βππ·π΄π΅ππ₯π΄
ππ§ +π₯π΄(π π΄+ππ΅)
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
π π΄=βππ·π΄π΅ππ₯π΄
ππ§ +π₯π΄(π π΄+ππ΅)
Since B is stagnant,
π π΄=βππ· π΄π΅
(1βπ₯π΄)ππ₯π΄
ππ§
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
π π΄=βππ· π΄π΅
(1βπ₯π΄)ππ₯π΄
ππ§
ππ π΄ Ηπ§βππ π΄ Ηπ§+β π§=0
Applying the mass balance,
where S = cross-sectional area of the column
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
ππ π΄ Ηπ§βππ π΄ Ηπ§+β π§=0
Dividing by SΞz and taking the limit as Ξz 0,
βππ π΄
ππ§ =0 NA = constant
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
βππ π΄
ππ§ =0 NA = constant
π π΄=βππ· π΄π΅
(1βπ₯π΄)ππ₯π΄
ππ§But,
Substituting,
πππ§ ( ππ· π΄π΅
(1β π₯π΄ )ππ₯π΄
ππ§ )=0
Concentration Profiles
I. Diffusion Through a Stagnant Gas Filmπππ§ ( ππ· π΄π΅
(1β π₯π΄ )ππ₯ π΄
ππ§ )=0For ideal gases, P = cRT and so at constant P and T, c = constantDAB for gases can be assumed independent of concentration
πππ§ ( 1
(1β π₯π΄ )ππ₯ π΄
ππ§ )=0
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
πππ§ ( 1
(1β π₯π΄ )ππ₯ π΄
ππ§ )=0Integrating once,
1(1βπ₯π΄ )
ππ₯π΄
ππ§ =πΆ1
Integrating again,
β ln (1β π₯π΄ )=πΆ1π§+πΆ2
Concentration Profiles
I. Diffusion Through a Stagnant Gas Filmβ ln (1β π₯π΄ )=πΆ1π§+πΆ2
Let C1 = -ln K1 and C2 = -ln K2,
1βπ₯π΄=πΎ 1π§πΎ 2
B.C.
at z = z1, xA = xA1
at z = z2, xA = xA2 ( 1βπ₯π΄
1β π₯π΄1 )=( 1βπ₯π΄2
1βπ₯π΄1 )π§β π§ 1π§ 2β π§1
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
( 1βπ₯π΄
1β π₯π΄1 )=( 1βπ₯π΄2
1βπ₯π΄1 )π§β π§ 1π§ 2β π§1
π π΄=βππ· π΄π΅
(1βπ₯π΄)ππ₯π΄
ππ§π π΄=
ππ·π΄π΅
(π§ 2β π§1 )ln (1β π₯π΄ 2
1β π₯π΄1)
*, i.e. xA1> xA2Η i.e. z2> z1
π π΄=ππ·π΄π΅
( π§2βπ§1)(π₯ΒΏΒΏπ΅)ππ(π₯π΄1βπ₯π΄2)ΒΏ
The molar flux then becomes
OR in terms of the driving force ΞxA
(π₯ΒΏΒΏπ΅)ππ=π₯π΅ 2βπ₯π΅1
ln (π₯π΅2
π₯π΅1)
ΒΏ
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical ReactionTwo Reaction Types:
1. Homogeneous β occurs in the entire volume of the fluid
- appears in the generation term
2. Heterogeneous β occurs on a surface (catalyst)
- appears in the boundary condition
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical ReactionReaction 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 ReactionReaction taking place
2A B
Assumptions:
1. Isothermal2. A and B are ideal gases3. Reaction on the surface
is instantaneous4. Uni-directional transport
will be considered
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical Reaction
ππ π΄
ππ§ =0
π π΄=βππ·π΄π΅ππ₯π΄
ππ§ +π₯π΄(π π΄+π π΅)
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical Reaction
π π΄=βππ· π΄π΅
1β 12π₯π΄
ππ₯π΄
ππ§
From stoichiometry,
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical ReactionSubstitution of NA into the differential equation
πππ§ (β
ππ·π΄π΅
1β 12π₯π΄
ππ₯π΄
ππ§ )=0
Integration twice with respect to z,
β2 ln(1β 12 π₯π΄)=πΆ1 π§+πΆ2=βΒΏ
B.C. 1: at z = 0, xA = xA0
B.C. 2: at z = Ξ΄, xA = 0
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical ReactionThe final equation is
1β 12π₯π΄=(1β 1
2π₯π΄ 0)
(1β π§πΏ )
And the molar flux of reactant through the film,
π π΄=2ππ· π΄π΅
πΏ ln( 1
1β 12π₯π΄0
)
*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
ππ π΄ Ηπ§βππ π΄ Ηπ§+β π§βπ1β² β² β²πΆπ΄π β π§=0
first order rate constant for homogeneous decomposition of AS cross sectional area of the liquid
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
ππ π΄ Ηπ§βππ π΄ Ηπ§+β π§βπ1β² β² β²πΆπ΄π β π§=0
Dividing by SΞz and taking the limit as Ξz 0,
ππ π΄
ππ§ +π1β² β² β²πΆπ΄=0
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reactionππ π΄
ππ§ +π1β² β² β²πΆπ΄=0
If concentration of A is small, then the total c is almost constant and
π π΄=βπ·π΄π΅πππ΄
ππ§Combining the two equations above
π· π΄π΅π2ππ΄
π π§2βπ1
β² β² β²πΆπ΄=0
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
π· π΄π΅π2ππ΄
π π§2βπ1
β² β² β²πΆπ΄=0
Multiplying the above equation by gives an equation with dimensionless variables
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
π· π΄π΅π2ππ΄
π π§2βπ1
β² β² β²πΆπ΄=0
π2Ξππ 2
βπ2Ξ=0
Ξ=ππ΄
ππ΄0,π= π§
πΏ ,π=βπβ² β² β²πΏ2/π·π΄π΅
Thiele Modulus
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
π2Ξππ 2
βπ2Ξ=0
The general solution is
Ξ=πΆ1 cosh (ππ )+πΆ2sinh (ππ )
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
Ξ=πΆ1 cosh (ππ )+πΆ2sinh (ππ )
Ξ=cosh (π ) cosh (ππ )βsinh (π ) sinh (ππ )
cosh (π )=cosh [Ο (1βΞΆ )]cosh (π )
Evaluating the constants,
Reverting to the original variables, π π΄
ππ΄0=cosh [βπβ² β² β² πΏ2π· π΄π΅
(1β π§πΏ )]
cosh (βπβ² β² β² πΏ2π· π΄π΅)
Concentration Profiles
III. Diffusion With a Homogeneous Chemical ReactionQuantities that might be asked for:
1. Average concentration in the liquid phase
ππ΄ ,ππ£π
π π΄0=β«0
πΏ
(ππ΄ ΒΏπ π΄0)ππ§
β«0
πΏ
ππ§= tanh ππ
2. Molar flux at the plane z = 0
π π΄π§ Η π§=0=βπ·π΄π΅ππ π΄
ππ§ Ηπ§=0=(ππ΄0π· π΄π΅
πΏ )π tanh π
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 B3. Viscosity of the liquid is unaffected4. 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
π£ π§ (π₯ )=π£πππ₯ [1β( π₯πΏ )2]
π£ π§(π₯ )=(π ππΏ2 cosπΌ2π )[1β(π₯πΏ )2]
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
* CA is a function of both x and z
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
Dividing by WΞxΞz andletting Ξx 0 and Ξz 0,
πππ΄π§
π π§ +ππ π΄π₯
π π₯ =0
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
πππ΄π§
π π§ +ππ π΄π₯
π π₯ =0
π π΄π§=βπ·π΄π΅πππ΄
ππ§ +π₯π΄(π π΄ π§+ππ΅ π§)
The expressions for ,
Transport of A along the z direction is mainly by convection (bulk motion)
π π΄π§ βππ΄π£π=π π΄π£π§ (π₯)
π π΄= π½ π΄β+ππ΄π£πRecall: π£π=πππππ ππ£ππππππ£ππππππ‘π¦
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
πππ΄π§
π π§ +ππ π΄π₯
π π₯ =0
π π΄π₯=βπ· π΄π΅ππ π΄
ππ§ +π₯π΄(π π΄ π₯+ππ΅π₯)
The expressions for ,
π π΄π₯ ββπ· π΄π΅ππ π΄
ππ§
Transport of A along the x direction is mainly by diffusion
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
πππ΄π§
π π§ +ππ π΄π₯
π π₯ =0
Substituting the expressions for,
π£ π§(πππ΄
π π§ )=π· π΄π΅π2π π΄
π π₯2
Substituting the expressions vz,
π£πππ₯ [1β( π₯πΏ )2]( ππ π΄
π π§ )=π· π΄π΅π2ππ΄
π π₯2
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
π£πππ₯ [1β( π₯πΏ )2]( πππ΄
π π§ )=π· π΄π΅π2ππ΄
π π₯2
Boundary conditions B.C. 1B.C. 2B.C. 3
B.C. 3
BUT we can replace B.C. 3 with
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
π£πππ₯ [1β( π₯πΏ )2]( ππ π΄
π π§ )=π· π΄π΅π2ππ΄
π π₯2
or
where
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
π π΄π₯ Η π₯=0=βπ·π΄π΅ππ π΄
π π₯ Ηπ₯=0=ππ΄0β π·π΄π΅π£πππ₯
π π§
π π΄
ππ΄0=1βπππ π₯
β 4π· π΄π΅2 π§
π£πππ₯
=ππππ π₯
β 4π·π΄π΅2 π§
π£πππ₯
Concentration ProfilesIV. 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
Quantities that might be asked for:
1. Total molar flow of A across the surface at x = 0
IV. Diffusion into a Falling Liquid Film (Gas Absorption)
π π΄=β«0
π
β«0
πΏ
ππ΄π₯ Ηπ₯=0 ππ§ππ¦=π ππ΄ 0β π·π΄π΅π£πππ₯
π β«0
πΏ 1βπ§
ππ§=ππ΄0βπ· π΄π΅π£πππ₯
π πΏ