overall shell mass balances i
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Overall Shell Mass Balances I. Outline. 3.Molecular Diffusion in Gases Molecular Diffusion in Liquids Molecular Diffusion in Solids Prediction of Diffusivities Overall Shell Mass Balances Concentration Profiles. Overall Shell Mass Balance . Species entering and leaving the system - PowerPoint PPT PresentationTRANSCRIPT
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 = kcโcA.4. The rate of chemical reaction at the surface can be specified.
โช 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
๐ ๐ด=โ๐๐ท๐ด๐ต๐๐ฅ๐ด
๐๐ง +๐ฅ๐ด(๐ ๐ด+๐๐ต)
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
๐ ๐ด=โ๐๐ท๐ด๐ต๐๐ฅ๐ด
๐๐ง +๐ฅ๐ด(๐ ๐ด+๐๐ต)
Since B is stagnant,
๐ ๐ด=โ๐๐ท ๐ด๐ต
(1โ๐ฅ๐ด)๐๐ฅ๐ด
๐๐ง
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
๐ ๐ด=โ๐๐ท ๐ด๐ต
(1โ๐ฅ๐ด)๐๐ฅ๐ด
๐๐ง
๐๐ ๐ด ว๐งโ๐๐ ๐ด ว๐ง+โ ๐ง=0
Applying the mass balance,
where S = cross-sectional area of the column
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
๐๐ ๐ด ว๐งโ๐๐ ๐ด ว๐ง+โ ๐ง=0
Dividing by Sฮz and taking the limit as ฮz 0,
โ๐๐ ๐ด
๐๐ง =0 NA = constant
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
โ๐๐ ๐ด
๐๐ง =0 NA = constant
๐ ๐ด=โ๐๐ท ๐ด๐ต
(1โ๐ฅ๐ด)๐๐ฅ๐ด
๐๐งBut,
Substituting,
๐๐๐ง ( ๐๐ท ๐ด๐ต
(1โ ๐ฅ๐ด )๐๐ฅ๐ด
๐๐ง )=0
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film๐๐๐ง ( ๐๐ท ๐ด๐ต
(1โ ๐ฅ๐ด )๐๐ฅ ๐ด
๐๐ง )=0For ideal gases, P = cRT and so at constant P and T, c = constantDAB for gases can be assumed independent of concentration
๐๐๐ง ( 1
(1โ ๐ฅ๐ด )๐๐ฅ ๐ด
๐๐ง )=0
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
๐๐๐ง ( 1
(1โ ๐ฅ๐ด )๐๐ฅ ๐ด
๐๐ง )=0Integrating once,
1(1โ๐ฅ๐ด )
๐๐ฅ๐ด
๐๐ง =๐ถ1
Integrating again,
โ ln (1โ ๐ฅ๐ด )=๐ถ1๐ง+๐ถ2
Concentration Profiles
I. Diffusion Through a Stagnant Gas Filmโ ln (1โ ๐ฅ๐ด )=๐ถ1๐ง+๐ถ2
Let C1 = -ln K1 and C2 = -ln K2,
1โ๐ฅ๐ด=๐พ 1๐ง๐พ 2
B.C.
at z = z1, xA = xA1
at z = z2, xA = xA2 ( 1โ๐ฅ๐ด
1โ ๐ฅ๐ด1 )=( 1โ๐ฅ๐ด2
1โ๐ฅ๐ด1 )๐งโ ๐ง 1๐ง 2โ ๐ง1
Concentration Profiles
I. Diffusion Through a Stagnant Gas Film
( 1โ๐ฅ๐ด
1โ ๐ฅ๐ด1 )=( 1โ๐ฅ๐ด2
1โ๐ฅ๐ด1 )๐งโ ๐ง 1๐ง 2โ ๐ง1
๐ ๐ด=โ๐๐ท ๐ด๐ต
(1โ๐ฅ๐ด)๐๐ฅ๐ด
๐๐ง๐ ๐ด=
๐๐ท๐ด๐ต
(๐ง 2โ ๐ง1 )ln (1โ ๐ฅ๐ด 2
1โ ๐ฅ๐ด1)
*, i.e. xA1> xA2ว i.e. z2> z1
๐ ๐ด=๐๐ท๐ด๐ต
( ๐ง2โ๐ง1)(๐ฅยฟยฟ๐ต)๐๐(๐ฅ๐ด1โ๐ฅ๐ด2)ยฟ
The molar flux then becomes
OR in terms of the driving force ฮxA
(๐ฅยฟยฟ๐ต)๐๐=๐ฅ๐ต 2โ๐ฅ๐ต1
ln (๐ฅ๐ต2
๐ฅ๐ต1)
ยฟ
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical ReactionTwo Reaction Types:
1. Homogeneous โ occurs in the entire volume of the fluid
- appears in the generation term
2. Heterogeneous โ 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
๐๐ ๐ด
๐๐ง =0
๐ ๐ด=โ๐๐ท๐ด๐ต๐๐ฅ๐ด
๐๐ง +๐ฅ๐ด(๐ ๐ด+๐ ๐ต)
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical Reaction
๐ ๐ด=โ๐๐ท ๐ด๐ต
1โ 12๐ฅ๐ด
๐๐ฅ๐ด
๐๐ง
From stoichiometry,
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical ReactionSubstitution of NA into the differential equation
๐๐๐ง (โ
๐๐ท๐ด๐ต
1โ 12๐ฅ๐ด
๐๐ฅ๐ด
๐๐ง )=0
Integration twice with respect to z,
โ2 ln(1โ 12 ๐ฅ๐ด)=๐ถ1 ๐ง+๐ถ2=โยฟ
B.C. 1: at z = 0, xA = xA0
B.C. 2: at z = ฮด, xA = 0
Concentration Profiles
II. Diffusion With a Heterogeneous Chemical ReactionThe final equation is
1โ 12๐ฅ๐ด=(1โ 1
2๐ฅ๐ด 0)
(1โ ๐ง๐ฟ )
And the molar flux of reactant through the film,
๐ ๐ด=2๐๐ท ๐ด๐ต
๐ฟ ln( 1
1โ 12๐ฅ๐ด0
)
*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
๐๐ ๐ด ว๐งโ๐๐ ๐ด ว๐ง+โ ๐งโ๐1โฒ โฒ โฒ๐ถ๐ด๐ โ ๐ง=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
๐๐ ๐ด ว๐งโ๐๐ ๐ด ว๐ง+โ ๐งโ๐1โฒ โฒ โฒ๐ถ๐ด๐ โ ๐ง=0
Dividing by Sฮz and taking the limit as ฮz 0,
๐๐ ๐ด
๐๐ง +๐1โฒ โฒ โฒ๐ถ๐ด=0
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction๐๐ ๐ด
๐๐ง +๐1โฒ โฒ โฒ๐ถ๐ด=0
If concentration of A is small, then the total c is almost constant and
๐ ๐ด=โ๐ท๐ด๐ต๐๐๐ด
๐๐งCombining the two equations above
๐ท ๐ด๐ต๐2๐๐ด
๐ ๐ง2โ๐1
โฒ โฒ โฒ๐ถ๐ด=0
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
๐ท ๐ด๐ต๐2๐๐ด
๐ ๐ง2โ๐1
โฒ โฒ โฒ๐ถ๐ด=0
Multiplying the above equation by gives an equation with dimensionless variables
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
๐ท ๐ด๐ต๐2๐๐ด
๐ ๐ง2โ๐1
โฒ โฒ โฒ๐ถ๐ด=0
๐2ฮ๐๐ 2
โ๐2ฮ=0
ฮ=๐๐ด
๐๐ด0,๐= ๐ง
๐ฟ ,๐=โ๐โฒ โฒ โฒ๐ฟ2/๐ท๐ด๐ต
Thiele Modulus
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
๐2ฮ๐๐ 2
โ๐2ฮ=0
The general solution is
ฮ=๐ถ1 cosh (๐๐ )+๐ถ2sinh (๐๐ )
Concentration Profiles
III. Diffusion With a Homogeneous Chemical Reaction
ฮ=๐ถ1 cosh (๐๐ )+๐ถ2sinh (๐๐ )
ฮ=cosh (๐ ) cosh (๐๐ )โsinh (๐ ) sinh (๐๐ )
cosh (๐ )=cosh [ฯ (1โฮถ )]cosh (๐ )
Evaluating the constants,
Reverting to the original variables, ๐ ๐ด
๐๐ด0=cosh [โ๐โฒ โฒ โฒ ๐ฟ2๐ท ๐ด๐ต
(1โ ๐ง๐ฟ )]
cosh (โ๐โฒ โฒ โฒ ๐ฟ2๐ท ๐ด๐ต)
Concentration Profiles
III. Diffusion With a Homogeneous Chemical ReactionQuantities that might be asked for:
1. Average concentration in the liquid phase
๐๐ด ,๐๐ฃ๐
๐ ๐ด0=โซ0
๐ฟ
(๐๐ด ยฟ๐ ๐ด0)๐๐ง
โซ0
๐ฟ
๐๐ง= tanh ๐๐
2. Molar flux at the plane z = 0
๐ ๐ด๐ง ว ๐ง=0=โ๐ท๐ด๐ต๐๐ ๐ด
๐๐ง ว๐ง=0=(๐๐ด0๐ท ๐ด๐ต
๐ฟ )๐ tanh ๐
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
๐ฃ ๐ง (๐ฅ )=๐ฃ๐๐๐ฅ [1โ( ๐ฅ๐ฟ )2]
๐ฃ ๐ง(๐ฅ )=(๐ ๐๐ฟ2 cos๐ผ2๐ )[1โ(๐ฅ๐ฟ )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 Wฮxฮz andletting ฮx 0 and ฮz 0,
๐๐๐ด๐ง
๐ ๐ง +๐๐ ๐ด๐ฅ
๐ ๐ฅ =0
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
๐๐๐ด๐ง
๐ ๐ง +๐๐ ๐ด๐ฅ
๐ ๐ฅ =0
๐ ๐ด๐ง=โ๐ท๐ด๐ต๐๐๐ด
๐๐ง +๐ฅ๐ด(๐ ๐ด ๐ง+๐๐ต ๐ง)
The expressions for ,
Transport of A along the z direction is mainly by convection (bulk motion)
๐ ๐ด๐ง โ๐๐ด๐ฃ๐=๐ ๐ด๐ฃ๐ง (๐ฅ)
๐ ๐ด= ๐ฝ ๐ดโ+๐๐ด๐ฃ๐Recall: ๐ฃ๐=๐๐๐๐๐ ๐๐ฃ๐๐๐๐๐๐ฃ๐๐๐๐๐๐ก๐ฆ
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
๐๐๐ด๐ง
๐ ๐ง +๐๐ ๐ด๐ฅ
๐ ๐ฅ =0
๐ ๐ด๐ฅ=โ๐ท ๐ด๐ต๐๐ ๐ด
๐๐ง +๐ฅ๐ด(๐ ๐ด ๐ฅ+๐๐ต๐ฅ)
The expressions for ,
๐ ๐ด๐ฅ โโ๐ท ๐ด๐ต๐๐ ๐ด
๐๐ง
Transport of A along the x direction is mainly by diffusion
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
๐๐๐ด๐ง
๐ ๐ง +๐๐ ๐ด๐ฅ
๐ ๐ฅ =0
Substituting the expressions for,
๐ฃ ๐ง(๐๐๐ด
๐ ๐ง )=๐ท ๐ด๐ต๐2๐ ๐ด
๐ ๐ฅ2
Substituting the expressions vz,
๐ฃ๐๐๐ฅ [1โ( ๐ฅ๐ฟ )2]( ๐๐ ๐ด
๐ ๐ง )=๐ท ๐ด๐ต๐2๐๐ด
๐ ๐ฅ2
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
๐ฃ๐๐๐ฅ [1โ( ๐ฅ๐ฟ )2]( ๐๐๐ด
๐ ๐ง )=๐ท ๐ด๐ต๐2๐๐ด
๐ ๐ฅ2
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)
๐ฃ๐๐๐ฅ [1โ( ๐ฅ๐ฟ )2]( ๐๐ ๐ด
๐ ๐ง )=๐ท ๐ด๐ต๐2๐๐ด
๐ ๐ฅ2
or
where
Concentration ProfilesIV. Diffusion into a Falling Liquid Film (Gas Absorption)
๐ ๐ด๐ฅ ว ๐ฅ=0=โ๐ท๐ด๐ต๐๐ ๐ด
๐ ๐ฅ ว๐ฅ=0=๐๐ด0โ ๐ท๐ด๐ต๐ฃ๐๐๐ฅ
๐ ๐ง
๐ ๐ด
๐๐ด0=1โ๐๐๐ ๐ฅ
โ 4๐ท ๐ด๐ต2 ๐ง
๐ฃ๐๐๐ฅ
=๐๐๐๐ ๐ฅ
โ 4๐ท๐ด๐ต2 ๐ง
๐ฃ๐๐๐ฅ
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)
๐ ๐ด=โซ0
๐
โซ0
๐ฟ
๐๐ด๐ฅ ว๐ฅ=0 ๐๐ง๐๐ฆ=๐ ๐๐ด 0โ ๐ท๐ด๐ต๐ฃ๐๐๐ฅ
๐ โซ0
๐ฟ 1โ๐ง
๐๐ง=๐๐ด0โ๐ท ๐ด๐ต๐ฃ๐๐๐ฅ
๐ ๐ฟ