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Module 4, Lecture III 1/5/2021
1
CM3120: Module 4
Β© Faith A. Morrison, Michigan Tech U.1
Diffusion and Mass Transfer II
I. Mass transfer in distillation and absorptionA. Film modelB. Penetration model
II. Linear driving force model (mass transfer coefficient, πA. Review: no bulk convectionB. New: appreciable bulk convectionC. Predict mass transfer coefficientsD. Solve unsteady mass transfer problems
III. Macroscopic species A mass balancesIV. Dimensional analysis in mass transfer
A. Reviewβcompare to heatB. Engineering quantities of interestC. Data correlations for π (Sh or Nu correlations)
V. Overall mass transfer coefficients
CM3120: Module 4
Β© Faith A. Morrison, Michigan Tech U.2
Professor Faith A. Morrison
Department of Chemical EngineeringMichigan Technological University
www.chem.mtu.edu/~fmorriso/cm3120/cm3120.html
Module 4 Lecture III
Macroscopic species A mass balances
Module 4, Lecture III 1/5/2021
2
3Β© Faith A. Morrison, Michigan Tech U.
Contining work with the linear driving force for mass transfer, i.e. mass transfer coefficients, ππ
4
Mass Transport βLawsβ
We have 2 Mass Transport βlawsβ
π π π¦ , π¦ ,
Fickβs Law of Diffusion
Linear-Driving-Force Model
π π₯ π π ππ· π»π₯
Combine with microscopic species π΄ mass balancePredicts flux π and composition distributions, e.g. π₯ π₯, π¦, π§, π‘
1D Steady models can be solved1D Unsteady models can be solved (if good at math)2D steady and unsteady models can be solved by Comsol
Since we predict π , we can also predict a mass xfer coeff π or πDiffusion coefficients are material properties (see tables)
Combine with macroscopic species π΄ mass balancePredicts flux π , but not composition distributionsMay be used as a boundary condition in microscopic balancesMass-transfer-coefficients are not material properties Rather, they are determined experimentally and specific to the
situation (dimensional analysis and correlations) Facilitate combining resistances into overall mass xfer coeffs, πΎ ,πΎ
Use:
Use:
Β© Faith A. Morrison, Michigan Tech U.
SummaryTransport coefficient
Module 4, Lecture III 1/5/2021
3
5
Mass Transport βLawsβ
We have 2 Mass Transport βlawsβ
π π π¦ , π¦ ,
Fickβs Law of Diffusion
Linear-Driving-Force Model
π π₯ π π ππ· π»π₯
Combine with microscopic species π΄ mass balancePredicts flux π and composition distributions, e.g. π₯ π₯, π¦, π§, π‘
1D Steady models can be solved1D Unsteady models can be solved (if good at math)2D steady and unsteady models can be solved by Comsol
Since we predict π , we can also predict a mass xfer coeff π or πDiffusion coefficients are material properties (see tables)
Combine with macroscopic species π΄ mass balancePredicts flux π , but not composition distributionsMay be used as a boundary condition in microscopic balancesMass-transfer-coefficients are not material properties Rather, they are determined experimentally and specific to the
situation (dimensional analysis and correlations) Facilitate combining resistances into overall mass xfer coeffs, πΎ ,πΎ
Transport coefficient
Use:
Use:
1
2
3
4
5
Β© Faith A. Morrison, Michigan Tech U.
6
Mass Transport βLawsβ
We now have 2 Mass Transport βlawsβ
π π π¦ , π¦ ,
Fickβs Law of Diffusion
Linear-Driving-Force Model
π π₯ π π ππ· π»π₯
Combine with microscopic species π΄ mass balancePredicts flux π and composition distributions, e.g. π₯ π₯,π¦, π§, π‘
1D Steady models can be solved1D Unsteady models can be solved (if good at math)2D steady and unsteady models can be solved by Comsol
Since we predict π , we can also predict a mass xfer coeff π or πDiffusion coefficients are material properties (see tables)
Combine with macroscopic species π΄ mass balancePredicts flux π , but not composition distributionsMay be used as a boundary condition in microscopic balancesMass-transfer-coefficients are not material properties Rather, they are determined experimentally and specific to the
situation (dimensional analysis and correlations) Facilitate combining resistances into overall mass xfer coeffs, πΎ ,πΎ
Transport coefficient
Use:
Use:
1
2
3
4
5
1. Since we predict π with Fickβs law, we can also predict a mass transfer coefficients π or π
2. 1D Unsteady models can be solved (if good at math)
Mass Transport βLawsβ
Β© Faith A. Morrison, Michigan Tech U.
π·
π
Mass transfer coefficients
3. Combine with macroscopic species π΄ mass balance4. Are not material properties; rather, they are determined
experimentally and specific to the situation (dimensional analysis and correlations)
5. Facilitate combining resistances into overall mass transfer coefficients, πΎ ,πΎ , to be used in modeling unit operations
Fickβs law of diffusion
Solutions are analogous to heat transfer
Relate π and π·
Remaining Topics to round out our understanding of mass transport:
We have 2 Mass Transport βlawsβ
Keep cranking at the list
Module 4, Lecture III 1/5/2021
4
Professor Faith A. Morrison
Department of Chemical EngineeringMichigan Technological University
7Β© Faith A. Morrison, Michigan Tech U.
CM3110 Transport IIPart II: Diffusion and Mass Transfer
Macroscopic Species π¨ Mass Balances
Unsteady Macroscopic Species π΄Mass Balance
C.V.
3
Unsteady, Macroscopic, Species A Mass Balance
BSL2, p727
balance over time interval Ξπ‘
Β© Faith A. Morrison, Michigan Tech U.8
C.V.
MOLES
Macroscopic control volume,
C.V.
Unsteady Macroscopic Species π΄Mass Balance
Keep track of:β’Bulk convection of species π΄ into and out of the C.V.β’Mass transfer across control surfaces of species π΄β’Mass of species π΄ created by chemical reaction
Module 4, Lecture III 1/5/2021
5
Unsteady, Macroscopic, Species A Mass Balance
BSL2, p727
moles of π΄ that flows intothe control volume
between t and π‘ Ξπ‘
moles of π΄ that flows out of the control volume between t and
π‘ Ξπ‘
balance over time interval Ξπ‘
Β© Faith A. Morrison, Michigan Tech U.9
β³ , Ξπ‘ β³ , Ξπ‘
π π
introduction of moles of π΄into the C.V. by mass transfer
across the π bounding control surface π (C.S.)
π π Ξπ‘
π net rate of production of moles of π΄ in the C.V. by reaction, per unit volume
MOLES
Macroscopic control volume,
C.V.
π π
Unsteady Macroscopic Species π΄Mass Balance
C.S.
C.V.
moles of π΄ that flows intothe control volume between t and π‘ Ξπ‘
moles of π΄ that flows out of the control volume between t and
π‘ Ξπ‘
β³ , Ξπ‘ β³ , Ξπ‘
π π
introduction of moles of π΄into the C.V. by mass transfer across the π bounding control surface π (C.S.)
π π Ξπ‘
π net rate of production of moles of π΄ in the C.V. by reaction, per unit volume
C.V.
Β© Faith A. Morrison, Michigan Tech U.10
πππ‘
β³ , Ξβ³ π π π π
accumulation net flow in + production + introduction
Ξ is βoutββ βinβC.S. = control surfaceC.V. = control volume
β³ , π π total moles of π΄ in the C.V.
Ξβ³ β β³ , β β³ , ,, bulk out
π net rate of production of moles of π΄ in the C.V. by reaction, per unit volume
π system volume
π π π πβ molar flux of π΄ out
through the π C.S.
MOLES
π β π
Unsteady Macroscopic Species π΄Mass Balance
Module 4, Lecture III 1/5/2021
6
moles of π΄ that flows intothe control volume between t and π‘ Ξπ‘
moles of π΄ that flows out of the control volume between t and
π‘ Ξπ‘
β³ , Ξπ‘ β³ , Ξπ‘
π π
introduction of moles of π΄into the C.V. by mass transfer across the π bounding control surface π (C.S.)
π π Ξπ‘
π net rate of production of moles of π΄ in the C.V. by reaction, per unit volume
C.V.
Β© Faith A. Morrison, Michigan Tech U.11
πππ‘
β³ , Ξβ³ π π π π
accumulation net flow in + production + introduction
β³ , π π total moles of π΄ in the C.V.
Ξβ³ β β³ , β β³ , ,, bulk out
π net rate of production of moles of π΄ in the C.V. by reaction, per unit volume
π system volume
π π π πβ molar flux of π΄ out
through the π C.S.
MOLESUnsteady Macroscopic Species π΄Mass Balance
The choice of the βsystem,β i.e. of the control volume, is an important first step.
(think βsourceβ and βsinkβ)
Β© Faith A. Morrison, Michigan Tech U.
Example 9: Bone dry air and liquid water (water volume 0.80 πππ‘πππ ) are introduced into a closed container (cross sectional area 150 ππ ; total volume 19.2 πππ‘πππ ). Both air and water are at 25 πΆ, ~1.0 ππ‘π throughout this scenario. Three minutes after the air and water are placed in the closed container, the vapor is found to be 5.0% saturated with water vapor. What is the mass transfer coefficient for the water transferring from the liquid to the gas? How long will it take for the gas to become 90% saturated with water?
Cussler, p239, FAM v54 p59 12
air, then air + water
π» π
Unsteady Macroscopic Species π΄Mass Balance
closed container
Module 4, Lecture III 1/5/2021
7
Β© Faith A. Morrison, Michigan Tech U.13
What is our model?
(our assumptions)
Example: Bone dry air and liquid water (water volume 0.80 πππ‘πππ ) are introduced into a closed container (cross sectional area 150 ππ ; total volume 19.2 πππ‘πππ ). Both air and water are at 25 πΆ throughout this scenario. Three minutes after the air and water are placed in the closed container, the vapor is found to be 5.0% saturated with water vapor. What is the mass transfer coefficient for the water transferring from the liquid to the gas? How long will it take for the gas to become 90% saturated with water?
air, then air + water
π» π
Unsteady Macroscopic Species π΄ Mass Balance
closed container
Unsteady Macroscopic Species π΄Mass Balance
1. Control volume 2.
Β© Faith A. Morrison, Michigan Tech U.14
Solution:
Unsteady Macroscopic Species π΄Mass Balance
ππβ
1 π
π‘ 2.3 βππ’ππ
Example: Bone dry air and liquid water (water volume 0.80 πππ‘πππ ) are introduced into a closed container (cross sectional area 150 ππ ; total volume 19.2 πππ‘πππ ). Both air and water are at 25 πΆ throughout this scenario. Three minutes after the air and water are placed in the closed container, the vapor is found to be 5.0% saturated with water vapor. What is the mass transfer coefficient for the water transferring from the liquid to the gas? How long will it take for the gas to become 90% saturated with water?
air, then air + water
π» π
Unsteady Macroscopic Species π΄ Mass Balance
closed container
(mass transfer coefficient for evaporating water)
Module 4, Lecture III 1/5/2021
8
Β© Faith A. Morrison, Michigan Tech U.
Example 10: Flow through a packed bed of soluble spherical pellets.
liquid water
aqueous solution with
dissolved benzoic acid
Two-centimeter diameter spheres of benzoic acid (soluble in water) are packed into a bed as shown. The spheres have 23 ππ of surface area per ππ volume of bed. What is the mass transfer coefficient when pure water flowing in (βsuperficial velocityβ5.0 ππ/π ) exits 62% saturated with benzoic acid?
15Cussler, p240, FAM v54 p63
2.0ππ
1.0 10 ππ
Unsteady Macroscopic Species π΄Mass Balance
Β© Faith A. Morrison, Michigan Tech U.16
What is our model?
(our assumptions)
Unsteady Macroscopic Species π΄Mass Balance
Example: Flow through a packed bed of soluble spherical pellets.
liquid water
aqueous solution with
dissolved benzoic acid
Two centimeter diameter spheres of benzoic acid (soluble in water) are packed into a bed as shown. The spheres have 23 ππ of surface area per ππ volume of bed. What is the mass transfer coefficient when pure water flowing in (βsuperficial velocityβ5.0 ππ/π ) exits 62% saturated with benzoic acid?
2.0ππ
1.0 10 ππ
Unsteady Macroscopic Species π΄Mass Balance
1. Control volume 2.
Module 4, Lecture III 1/5/2021
9
(dissolving benzoic acid in packed bed, pseudo steady state)
Β© Faith A. Morrison, Michigan Tech U.17
Solution:
Unsteady Macroscopic Species π΄Mass Balance
Example: Flow through a packed bed of soluble spherical pellets.
liquid water
aqueous solution with
dissolved benzoic acid
Two centimeter diameter spheres of benzoic acid (soluble in water) are packed into a bed as shown. The spheres have 23 ππ of surface area per ππ volume of bed. What is the mass transfer coefficient when pure water flowing in (βsuperficial velocityβ5.0 ππ/π ) exits 62% saturated with benzoic acid?
2.0ππ
1.0 10 ππ
Unsteady Macroscopic Species π΄Mass Balance
ππ£ππΏ
ln 1ππβ
π 2.1 10 ππ/π
Β© Faith A. Morrison, Michigan Tech U.
Example 11: Height of a packed bed absorber
18BSL2 p742
Unsteady Macroscopic Species π΄Mass Balance
How can we use the linear driving force model for mass transfer to design a packed bed gas absorber to achieve a desired separation?
Module 4, Lecture III 1/5/2021
10
Β© Faith A. Morrison, Michigan Tech U.
How can we use the linear driving force model for mass transfer to design a packed bed gas absorber to achieve a desired separation?
19BSL2 p742
Unsteady Macroscopic Species π΄Mass Balance
From earlier lecture:
Example 11: Height of a packed bed absorber
Β© Faith A. Morrison, Michigan Tech U.20
Unsteady Macroscopic Species π΄Mass BalanceβGas Absorption
From earlier lecture:
Module 4, Lecture III 1/5/2021
11
Β© Faith A. Morrison, Michigan Tech U.21
Unsteady Macroscopic Species π΄Mass BalanceβGas Absorption
From earlier lecture:
Β© Faith A. Morrison, Michigan Tech U.22
Unsteady Macroscopic Species π΄Mass BalanceβGas Absorption
From earlier lecture:
Module 4, Lecture III 1/5/2021
12
Β© Faith A. Morrison, Michigan Tech U.23
Unsteady Macroscopic Species π΄Mass BalanceβGas Absorption
Liquid in
Liquid out
Gas out
Gas in
moles of π΄ that flows intothe control volume between t and π‘ Ξπ‘
moles of π΄ that flows out of the control volume between t and
π‘ Ξπ‘
β³ , Ξπ‘ β³ , Ξπ‘
π π
introduction of moles of π΄into the C.V. by mass transfer
across the π bounding control surface π (C.S.)
π π Ξπ‘
π net rate of production of moles of π΄ in the C.V. by reaction, per unit volume
C.V.
accumulation net flow in + production + introduction
Ξ is βoutββ βinβC.S. = control surfaceC.V. = control volume
β³ , π π total moles of π΄ in the C.V.
Ξβ³ β β³ , β β³ , ,, bulk out
π net rate of production of moles of π΄ in the C.V. by reaction, per unit volume
π system volume
π π π πβ molar flux of π΄ out
through the π C.S.
MOLES
π β π
Unsteady Macroscopic Species π΄ Mass Balance
? You try.
Β© Faith A. Morrison, Michigan Tech U.24
What is our model?
(our assumptions)
Unsteady Macroscopic Species π΄Mass Balance
1. Control volume 2.
Liquid in
Liquid out
Gas out
Module 4, Lecture III 1/5/2021
13
Β© Faith A. M
orrison, M
ichigan
Tech U.
25
Unsteady Macroscopic Species π΄Mass BalanceβGas Absorption
Liquid in
Liquid out
Gas out
π§π§ Ξπ§
moles of π΄ that flows intothe control volume between t and π‘ Ξπ‘
moles of π΄ that flows out of the control volume between t and
π‘ Ξπ‘
β³ , Ξπ‘ β³ , Ξπ‘
π π
introduction of moles of π΄into the C.V. by mass transfer across the π bounding control surface π (C.S.)
π π Ξπ‘
π net rate of production of moles of π΄ in the C.V. by reaction, per unit volume
C.V.
accumulation net flow in + production + introduction
Ξ is βoutββ βinβC.S. = control surfaceC.V. = control volume
β³ , π π total moles of π΄ in the C.V.
Ξβ³ β β³ , β β³ , ,, bulk out
π net rate of production of moles of π΄ in the C.V. by reaction, per unit volume
π system volume
π π π πβ molar flux of π΄ out
through the π C.S.
MOLES
π β π
Unsteady Macroscopic Species π΄ Mass Balance
π§
π§ Ξπ§
π§Gas πΌ,π΄
Liquidπ΅,π΄
C.V.
π
β³
β³
ππΞπ§ area for mass transfer
Β© Faith A. Morrison, Michigan Tech U.26
Unsteady Macroscopic Species π΄Mass BalanceβGas Absorption
β³ β‘ moles gas on an π΄ free basis
π΄ β‘
π³ β‘
π΄ π¦ for dilute systems
π³ π³β³
β³π΄ π΄
π΄ π΄β
π³ π³β
β¦
Operating line; from mass balance on the top of the column)
(result from π , π , ;
BSLβs equation 23.5β21)
molar ratios
BSL2 p745
Module 4, Lecture III 1/5/2021
14
BSL2 p745 Bird, Stewart, and Lightfoot, Transport Phenomena, 2nd ed., Wiley, 2006. Β© Faith A. Morrison, Michigan Tech U.
27
Unsteady Macroscopic Species π΄Mass BalanceβGas Absorption
π΄β π³β
π³
π΄
π΄β
π΄ π³
(height of a packed bed absorber)
Β© Faith A. Morrison, Michigan Tech U.28
Solution:
Unsteady Macroscopic Species π΄Mass Balance
πΏβ³
π π π
ππ΄π΄ π΄β
π΄
π΄
Answer: Perform numerical integration to obtain the column height πΏ:
Need: Gas flow rate β³ , column cross sectional area π, data on thermodynamic equilibrium π΄β π³β , desired separation (mole fractions top and bottom, and mass transfer coefficients π π and π π.
Module 4, Lecture III 1/5/2021
15
29
Mass Transport βLawsβ
We have 2 Mass Transport βlawsβ
π π π¦ , π¦ ,
Fickβs Law of Diffusion
Linear-Driving-Force Model
π π₯ π π ππ· π»π₯
Combine with microscopic species π΄ mass balancePredicts flux π and composition distributions, e.g. π₯ π₯, π¦, π§, π‘
1D Steady models can be solved1D Unsteady models can be solved (if good at math)2D steady and unsteady models can be solved by Comsol
Since we predict π , we can also predict a mass xfer coeff π or πDiffusion coefficients are material properties (see tables)
Combine with macroscopic species π΄ mass balancePredicts flux π , but not composition distributionsMay be used as a boundary condition in microscopic balancesMass-transfer-coefficients are not material properties Rather, they are determined experimentally and specific to the
situation (dimensional analysis and correlations) Facilitate combining resistances into overall mass xfer coeffs, πΎ ,πΎ
Transport coefficient
Use:
Use:
1
2
3
4
5
Β© Faith A. Morrison, Michigan Tech U.
30
Mass Transport βLawsβ
We now have 2 Mass Transport βlawsβ
π π π¦ , π¦ ,
Fickβs Law of Diffusion
Linear-Driving-Force Model
π π₯ π π ππ· π»π₯
Combine with microscopic species π΄ mass balancePredicts flux π and composition distributions, e.g. π₯ π₯,π¦, π§, π‘
1D Steady models can be solved1D Unsteady models can be solved (if good at math)2D steady and unsteady models can be solved by Comsol
Since we predict π , we can also predict a mass xfer coeff π or πDiffusion coefficients are material properties (see tables)
Combine with macroscopic species π΄ mass balancePredicts flux π , but not composition distributionsMay be used as a boundary condition in microscopic balancesMass-transfer-coefficients are not material properties Rather, they are determined experimentally and specific to the
situation (dimensional analysis and correlations) Facilitate combining resistances into overall mass xfer coeffs, πΎ ,πΎ
Transport coefficient
Use:
Use:
1
2
3
4
5
1. Since we predict π with Fickβs law, we can also predict a mass transfer coefficients π or π
2. 1D Unsteady models can be solved (if good at math)
Mass Transport βLawsβ
Β© Faith A. Morrison, Michigan Tech U.
π·
π
Mass transfer coefficients
3. Combine with macroscopic species π΄ mass balance4. Are not material properties; rather, they are determined
experimentally and specific to the situation (dimensional analysis and correlations)
5. Facilitate combining resistances into overall mass transfer coefficients, πΎ ,πΎ , to be used in modeling unit operations
Fickβs law of diffusion
Solutions are analogous to heat transfer
Relate π and π·
Remaining Topics to round out our understanding of mass transport:
We have 2 Mass Transport βlawsβ
Model macroscopic processes; design units