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Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
1
1ยฉ Faith A. Morrison, Michigan Tech U.
Contining work with the linear driving force for mass transfer, i.e. mass transfer coefficients, ๐๐
2
Mass Transport โLawsโ
We have 2 Mass Transport โlawsโ
๐ ๐ ๐ฆ , ๐ฆ ,
Fickโs Law of Diffusion
Linear-Driving-Force Model
๐ ๐ฅ ๐ ๐ ๐๐ท ๐ป๐ฅ
Combine with microscopic species ๐ด mass balancePredicts flux ๐ and composition distributions, e.g. ๐ฅ ๐ฅ, ๐ฆ, ๐ง, ๐ก
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 ๐ , we can also predict a mass xfer coeff ๐ or ๐Diffusion coefficients are material properties (see tables)
Combine with macroscopic species ๐ด mass balancePredicts flux ๐ , 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, ๐พ , ๐พ
Use:
Use:
ยฉ Faith A. Morrison, Michigan Tech U.
SummaryTransport coefficient
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
2
3
Mass Transport โLawsโ
We have 2 Mass Transport โlawsโ
๐ ๐ ๐ฆ , ๐ฆ ,
Fickโs Law of Diffusion
Linear-Driving-Force Model
๐ ๐ฅ ๐ ๐ ๐๐ท ๐ป๐ฅ
Combine with microscopic species ๐ด mass balancePredicts flux ๐ and composition distributions, e.g. ๐ฅ ๐ฅ, ๐ฆ, ๐ง, ๐ก
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 ๐ , we can also predict a mass xfer coeff ๐ or ๐Diffusion coefficients are material properties (see tables)
Combine with macroscopic species ๐ด mass balancePredicts flux ๐ , 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, ๐พ , ๐พ
Transport coefficient
Use:
Use:
1
2
3
4
5
ยฉ Faith A. Morrison, Michigan Tech U.
4
Mass Transport โLawsโ
We now have 2 Mass Transport โlawsโ
๐ ๐ ๐ฆ , ๐ฆ ,
Fickโs Law of Diffusion
Linear-Driving-Force Model
๐ ๐ฅ ๐ ๐ ๐๐ท ๐ป๐ฅ
Combine with microscopic species ๐ด mass balancePredicts flux ๐ and composition distributions, e.g. ๐ฅ ๐ฅ, ๐ฆ, ๐ง, ๐ก
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 ๐ , we can also predict a mass xfer coeff ๐ or ๐Diffusion coefficients are material properties (see tables)
Combine with macroscopic species ๐ด mass balancePredicts flux ๐ , 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, ๐พ , ๐พ
Transport coefficient
Use:
Use:
1
2
3
4
5
1. Since we predict ๐ with Fickโs law, we can also predict a mass transfercoefficients ๐ or ๐
2. 1D Unsteady models can be solved (if good at math)
Mass Transport โLawsโ
ยฉ Faith A. Morrison, Michigan Tech U.
๐ท
๐
Mass transfer coefficients
3. Combine with macroscopic species ๐ด 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, ๐พ , ๐พ , to be used in modeling unit operations
Fickโs law of diffusion
Solutions are analogous to heat transfer
Relate ๐ and ๐ท
Remaining Topics to round out our understanding of mass transport:
We have 2 Mass Transport โlawsโ
Keep cranking at the list
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
3
Professor Faith A. Morrison
Department of Chemical EngineeringMichigan Technological University
5ยฉ Faith A. Morrison, Michigan Tech U.
CM3110 Transport IIPart II: Diffusion and Mass Transfer
Macroscopic Species ๐จ Mass Balances
Unsteady Macroscopic Species ๐ด Mass Balance
C.V.
3
Unsteady, Macroscopic, Species A Mass Balance
BSL2, p727
balance over time interval ฮ๐ก
ยฉ Faith A. Morrison, Michigan Tech U.6
C.V.
MOLES
Macroscopic control volume,
C.V.
Unsteady Macroscopic Species ๐ด Mass Balance
Keep track of:โขBulk convection of species ๐ด into and out of the C.V.โขMass transfer across control surfaces of species ๐ดโขMass of species ๐ด created by chemical reaction
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
4
Unsteady, Macroscopic, Species A Mass Balance
BSL2, p727
moles of ๐ด that flows intothe control volume
between t and ๐ก ฮ๐ก
moles of ๐ด that flows out of the control volume between t and
๐ก ฮ๐ก
balance over time interval ฮ๐ก
ยฉ Faith A. Morrison, Michigan Tech U.7
โณ , ฮ๐ก โณ , ฮ๐ก
๐ ๐
introduction of moles of ๐ดinto the C.V. by mass transfer
across the ๐ bounding control surface ๐ (C.S.)
๐ ๐ ฮ๐ก
๐ net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume
MOLES
Macroscopic control volume,
C.V.
๐ ๐
Unsteady Macroscopic Species ๐ด Mass Balance
C.S.
C.V.
moles of ๐ด that flows intothe control volume
between t and ๐ก ฮ๐ก
moles of ๐ด that flows out of the control volume between t and
๐ก ฮ๐ก
โณ , ฮ๐ก โณ , ฮ๐ก
๐ ๐
introduction of moles of ๐ดinto the C.V. by mass transfer
across the ๐ bounding control surface ๐ (C.S.)
๐ ๐ ฮ๐ก
๐ net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume
C.V.
ยฉ Faith A. Morrison, Michigan Tech U.8
๐๐๐ก
โณ , ฮโณ ๐ ๐ ๐ ๐
accumulation net flow in + production + introduction
ฮ is โoutโโ โinโC.S. = control surfaceC.V. = control volume
โณ , ๐ ๐ total moles of ๐ด in the C.V.
ฮโณ โ โณ , โ โณ , ,, bulk out
๐ net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume
๐ system volume
๐ ๐ ๐ ๐โ molar flux of ๐ด out
through the ๐ C.S.
MOLES
๐ โ ๐
Unsteady Macroscopic Species ๐ด Mass Balance
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
5
moles of ๐ด that flows intothe control volume
between t and ๐ก ฮ๐ก
moles of ๐ด that flows out of the control volume between t and
๐ก ฮ๐ก
โณ , ฮ๐ก โณ , ฮ๐ก
๐ ๐
introduction of moles of ๐ดinto the C.V. by mass transfer
across the ๐ bounding control surface ๐ (C.S.)
๐ ๐ ฮ๐ก
๐ net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume
C.V.
ยฉ Faith A. Morrison, Michigan Tech U.9
๐๐๐ก
โณ , ฮโณ ๐ ๐ ๐ ๐
accumulation net flow in + production + introduction
โณ , ๐ ๐ total moles of ๐ด in the C.V.
ฮโณ โ โณ , โ โณ , ,, bulk out
๐ net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume
๐ system volume
๐ ๐ ๐ ๐โ molar flux of ๐ด out
through the ๐ C.S.
MOLESUnsteady Macroscopic Species ๐ด Mass Balance
The choice of the โsystem,โ i.e. of the control volume, is an important first step.
(think โsourceโ and โsinkโ)
ยฉ Faith A. Morrison, Michigan Tech U.
Example 8: Bone dry air and liquid water (water volume 0.80 ๐๐๐ก๐๐๐ ) are introduced into a closed container (cross sectional area 150 ๐๐ ; total volume 19.2 ๐๐๐ก๐๐๐ ). Both air and water are at 25 ๐ถ, ~1.0 ๐๐ก๐ 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 10
air, then air + water
๐ป ๐
Unsteady Macroscopic Species ๐ด Mass Balance
closed container
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
6
ยฉ Faith A. Morrison, Michigan Tech U.11
What is our model?
(our assumptions)
Example: Bone dry air and liquid water (water volume 0.80 ๐๐๐ก๐๐๐ ) are introduced into a closed container (cross sectional area 150 ๐๐ ; total volume 19.2 ๐๐๐ก๐๐๐ ). Both air and water are at 25 ๐ถ 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
๐ป ๐
Unsteady Macroscopic Species ๐ด Mass Balance
closed container
Unsteady Macroscopic Species ๐ด Mass Balance
1. Control volume 2.
ยฉ Faith A. Morrison, Michigan Tech U.12
Solution:
Unsteady Macroscopic Species ๐ด Mass Balance
๐๐โ 1 ๐
๐ก 2.3 โ๐๐ข๐๐
Example: Bone dry air and liquid water (water volume 0.80 ๐๐๐ก๐๐๐ ) are introduced into a closed container (cross sectional area 150 ๐๐ ; total volume 19.2 ๐๐๐ก๐๐๐ ). Both air and water are at 25 ๐ถ 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
๐ป ๐
Unsteady Macroscopic Species ๐ด Mass Balance
closed container
(mass transfer coefficient for evaporating water)
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
7
ยฉ Faith A. Morrison, Michigan Tech U.
Example 9: 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 ๐๐ of surface area per ๐๐ volume of bed. What is the mass transfer coefficient when pure water flowing in (โsuperficial velocityโ5.0 ๐๐/๐ ) exits 62% saturated with benzoic acid?
13Cussler, p240, FAM v54 p63
2.0๐๐
1.0 10 ๐๐
Unsteady Macroscopic Species ๐ด Mass Balance
ยฉ Faith A. Morrison, Michigan Tech U.14
What is our model?
(our assumptions)
Unsteady Macroscopic Species ๐ด 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 ๐๐ of surface area per ๐๐ volume of bed. What is the mass transfer coefficient when pure water flowing in (โsuperficial velocityโ5.0 ๐๐/๐ ) exits 62% saturated with benzoic acid?
2.0๐๐
1.0 10 ๐๐
Unsteady Macroscopic Species ๐ด Mass Balance
1. Control volume 2.
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
8
(dissolving benzoic acid in packed bed, pseudo steady state)
ยฉ Faith A. Morrison, Michigan Tech U.15
Solution:
Unsteady Macroscopic Species ๐ด 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 ๐๐ of surface area per ๐๐ volume of bed. What is the mass transfer coefficient when pure water flowing in (โsuperficial velocityโ5.0 ๐๐/๐ ) exits 62% saturated with benzoic acid?
2.0๐๐
1.0 10 ๐๐
Unsteady Macroscopic Species ๐ด Mass Balance
๐๐ฃ๐๐ฟ
ln 1๐๐โ
๐ 2.1 10 ๐๐/๐
ยฉ Faith A. Morrison, Michigan Tech U.
Example 10: Height of a packed bed absorber
16BSL2 p742
Unsteady Macroscopic Species ๐ด 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?
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
9
ยฉ 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?
17BSL2 p742
Unsteady Macroscopic Species ๐ด Mass Balance
From lecture 8:
Example 10: Height of a packed bed absorber
ยฉ Faith A. Morrison, Michigan Tech U.18
Unsteady Macroscopic Species ๐ด Mass BalanceโGas Absorption
From lecture 8:
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
10
ยฉ Faith A. Morrison, Michigan Tech U.19
Unsteady Macroscopic Species ๐ด Mass BalanceโGas Absorption
From lecture 8:
ยฉ Faith A. Morrison, Michigan Tech U.20
Unsteady Macroscopic Species ๐ด Mass BalanceโGas Absorption
From lecture 8:
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
11
ยฉ Faith A. Morrison, Michigan Tech U.21
Unsteady Macroscopic Species ๐ด Mass BalanceโGas Absorption
Liquid in
Liquid out
Gas out
Gas in
moles of ๐ด that flows intothe control volume
between t and ๐ก ฮ๐ก
moles of ๐ด that flows out of the control volume between t and
๐ก ฮ๐ก
โณ , ฮ๐ก โณ , ฮ๐ก
๐ ๐
introduction of moles of ๐ดinto the C.V. by mass transfer
across the ๐ bounding control surface ๐ (C.S.)
๐ ๐ ฮ๐ก
๐ net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume
C.V.
accumulation net flow in + production + introduction
ฮ is โoutโโ โinโC.S. = control surfaceC.V. = control volume
โณ , ๐ ๐ total moles of ๐ด in the C.V.
ฮโณ โ โณ , โ โณ , ,, bulk out
๐ net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume
๐ system volume
๐ ๐ ๐ ๐โ molar flux of ๐ด out
through the ๐ C.S.
MOLES
๐ โ ๐
Unsteady Macroscopic Species ๐ด Mass Balance
? You try.
ยฉ Faith A. Morrison, Michigan Tech U.22
What is our model?
(our assumptions)
Unsteady Macroscopic Species ๐ด Mass Balance
1. Control volume 2.
Liquid in
Liquid out
Gas out
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
12
ยฉ Faith A. M
orrison, M
ichigan
Tech U.
23
Unsteady Macroscopic Species ๐ด Mass BalanceโGas Absorption
Liquid in
Liquid out
Gas out
๐ง๐ง ฮ๐ง
moles of ๐ด that flows intothe control volume between t and ๐ก ฮ๐ก
moles of ๐ด that flows out of the control volume between t and
๐ก ฮ๐ก
โณ , ฮ๐ก โณ , ฮ๐ก
๐ ๐
introduction of moles of ๐ดinto the C.V. by mass transfer
across the ๐ bounding control surface ๐ (C.S.)
๐ ๐ ฮ๐ก
๐ net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume
C.V.
accumulation net flow in + production + introduction
ฮ is โoutโโ โinโC.S. = control surfaceC.V. = control volume
โณ , ๐ ๐ total moles of ๐ด in the C.V.
ฮโณ โ โณ , โ โณ , ,, bulk out
๐ net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume
๐ system volume
๐ ๐ ๐ ๐โ molar flux of ๐ด out
through the ๐ C.S.
MOLES
๐ โ ๐
Unsteady Macroscopic Species ๐ด Mass Balance
๐ง
๐ง ฮ๐ง
๐งGas ๐ผ, ๐ด
Liquid๐ต, ๐ด
C.V.
๐
โณ
โณ
๐๐ฮ๐ง area for mass transfer
ยฉ Faith A. Morrison, Michigan Tech U.24
Unsteady Macroscopic Species ๐ด Mass BalanceโGas Absorption
โณ โก moles gas on an ๐ด-free basis
๐ด โก
๐ณ โก
๐ด ๐ฆ for dilute systems
๐ณ ๐ณโณ
โณ๐ด ๐ด
๐ด ๐ดโ
๐ณ ๐ณโ
โฆ
Operating line; from mass balance on the top of the column)
(result from ๐ , ๐ , ;
BSLโs equation 23.5โ21)
molar ratios
BSL2 p745
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
13
BSL2 p745 Bird, Stewart, and Lightfoot, Transport Phenomena, 2nd ed., Wiley, 2006. ยฉ Faith A. Morrison, Michigan Tech U.
25
Unsteady Macroscopic Species ๐ด Mass BalanceโGas Absorption
๐ดโ ๐ณโ
๐ณ
๐ด
๐ดโ
๐ด ๐ณ
(height of a packed bed absorber)
ยฉ Faith A. Morrison, Michigan Tech U.26
Solution:
Unsteady Macroscopic Species ๐ด Mass Balance
๐ฟโณ
๐ ๐ ๐
๐๐ด๐ด ๐ดโ
๐ด
๐ด
Answer: Perform numerical integration to obtain the column height ๐ฟ:
Need: Gas flow rate โณ , column cross sectional area ๐, data on thermodynamic equilibrium ๐ดโ ๐ณโ , desired separation (mole fractions top and bottom, and mass transfer coefficients ๐ ๐ and ๐ ๐.
Lecture2 11โ12: Macro Species A Mass Bal 4/12/2019
14
27
Mass Transport โLawsโ
We have 2 Mass Transport โlawsโ
๐ ๐ ๐ฆ , ๐ฆ ,
Fickโs Law of Diffusion
Linear-Driving-Force Model
๐ ๐ฅ ๐ ๐ ๐๐ท ๐ป๐ฅ
Combine with microscopic species ๐ด mass balancePredicts flux ๐ and composition distributions, e.g. ๐ฅ ๐ฅ, ๐ฆ, ๐ง, ๐ก
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 ๐ , we can also predict a mass xfer coeff ๐ or ๐Diffusion coefficients are material properties (see tables)
Combine with macroscopic species ๐ด mass balancePredicts flux ๐ , 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, ๐พ , ๐พ
Transport coefficient
Use:
Use:
1
2
3
4
5
ยฉ Faith A. Morrison, Michigan Tech U.
28
Mass Transport โLawsโ
We now have 2 Mass Transport โlawsโ
๐ ๐ ๐ฆ , ๐ฆ ,
Fickโs Law of Diffusion
Linear-Driving-Force Model
๐ ๐ฅ ๐ ๐ ๐๐ท ๐ป๐ฅ
Combine with microscopic species ๐ด mass balancePredicts flux ๐ and composition distributions, e.g. ๐ฅ ๐ฅ, ๐ฆ, ๐ง, ๐ก
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 ๐ , we can also predict a mass xfer coeff ๐ or ๐Diffusion coefficients are material properties (see tables)
Combine with macroscopic species ๐ด mass balancePredicts flux ๐ , 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, ๐พ , ๐พ
Transport coefficient
Use:
Use:
1
2
3
4
5
1. Since we predict ๐ with Fickโs law, we can also predict a mass transfercoefficients ๐ or ๐
2. 1D Unsteady models can be solved (if good at math)
Mass Transport โLawsโ
ยฉ Faith A. Morrison, Michigan Tech U.
๐ท
๐
Mass transfer coefficients
3. Combine with macroscopic species ๐ด 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, ๐พ , ๐พ , to be used in modeling unit operations
Fickโs law of diffusion
Solutions are analogous to heat transfer
Relate ๐ and ๐ท
Remaining Topics to round out our understanding of mass transport:
We have 2 Mass Transport โlawsโ
Model macroscopic processes; design units
Recommended