Transcript
Page 1: Contining work with the linear driving force for mass ...fmorriso/cm3120/lectures/lecturePart2_11-12_CM3120.pdf+ ยบ, we can also predict a mass xfercoeff๐‘˜ รฌor ๐‘˜ ร– Diffusion

Lecture2 11โ€12:  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, ๐’Œ๐’„

2

Mass Transport โ€œLawsโ€

We have 2 Mass Transport โ€œlawsโ€

๐‘ ๐‘˜ ๐‘ฆ , ๐‘ฆ ,

Fickโ€™s Law of Diffusion

Linear-Driving-Force Model

๐‘ ๐‘ฅ ๐‘ ๐‘ ๐‘๐ท ๐›ป๐‘ฅ

Combine with microscopic species ๐ด mass balancePredicts flux ๐‘ and composition distributions, e.g. ๐‘ฅ ๐‘ฅ, ๐‘ฆ, ๐‘ง, ๐‘ก

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 ๐‘˜Diffusion coefficients are material properties (see tables)

Combine with macroscopic species ๐ด 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, ๐พ , ๐พ

Use:

Use:

ยฉ Faith A. Morrison, Michigan Tech U.

SummaryTransport coefficient

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Lecture2 11โ€12:  Macro Species A Mass Bal 4/12/2019

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3

Mass Transport โ€œLawsโ€

We have 2 Mass Transport โ€œlawsโ€

๐‘ ๐‘˜ ๐‘ฆ , ๐‘ฆ ,

Fickโ€™s Law of Diffusion

Linear-Driving-Force Model

๐‘ ๐‘ฅ ๐‘ ๐‘ ๐‘๐ท ๐›ป๐‘ฅ

Combine with microscopic species ๐ด mass balancePredicts flux ๐‘ and composition distributions, e.g. ๐‘ฅ ๐‘ฅ, ๐‘ฆ, ๐‘ง, ๐‘ก

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 ๐‘˜Diffusion coefficients are material properties (see tables)

Combine with macroscopic species ๐ด 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, ๐พ , ๐พ

Transport coefficient

Use:

Use:

1

2

3

4

5

ยฉ Faith A. Morrison, Michigan Tech U.

4

Mass Transport โ€œLawsโ€

We now have 2 Mass Transport โ€œlawsโ€

๐‘ ๐‘˜ ๐‘ฆ , ๐‘ฆ ,

Fickโ€™s Law of Diffusion

Linear-Driving-Force Model

๐‘ ๐‘ฅ ๐‘ ๐‘ ๐‘๐ท ๐›ป๐‘ฅ

Combine with microscopic species ๐ด mass balancePredicts flux ๐‘ and composition distributions, e.g. ๐‘ฅ ๐‘ฅ, ๐‘ฆ, ๐‘ง, ๐‘ก

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 ๐‘˜Diffusion coefficients are material properties (see tables)

Combine with macroscopic species ๐ด 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, ๐พ , ๐พ

Transport coefficient

Use:

Use:

1

2

3

4

5

1. Since we predict ๐‘ with Fickโ€™s law, we can also predict a mass transfercoefficients ๐‘˜ or ๐‘˜

2. 1D Unsteady models can be solved (if good at math)

Mass Transport โ€œLawsโ€

ยฉ Faith A. Morrison, Michigan Tech U.

๐ท

๐‘˜

Mass transfer coefficients

3. Combine with macroscopic species ๐ด 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, ๐พ , ๐พ , to be used in modeling unit operations

Fickโ€™s law of diffusion

Solutions are analogous to heat transfer

Relate ๐‘˜ and ๐ท

Remaining Topics to round out our understanding of mass transport:

We have 2 Mass Transport โ€œlawsโ€

Keep cranking at the list

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Lecture2 11โ€12:  Macro Species A Mass Bal 4/12/2019

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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 ๐‘จ Mass Balances

Unsteady Macroscopic Species ๐ด Mass Balance

C.V.

3

Unsteady, Macroscopic, Species A Mass Balance

BSL2, p727

balance over time interval ฮ”๐‘ก

ยฉ Faith A. Morrison, Michigan Tech U.6

C.V.

MOLES

Macroscopic control volume, 

C.V.

Unsteady Macroscopic Species ๐ด Mass Balance

Keep track of:โ€ขBulk convection of species ๐ด into and out of the C.V.โ€ขMass transfer across control surfaces of species ๐ดโ€ขMass of species ๐ด created by chemical reaction

Page 4: Contining work with the linear driving force for mass ...fmorriso/cm3120/lectures/lecturePart2_11-12_CM3120.pdf+ ยบ, we can also predict a mass xfercoeff๐‘˜ รฌor ๐‘˜ ร– Diffusion

Lecture2 11โ€12:  Macro Species A Mass Bal 4/12/2019

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Unsteady, Macroscopic, Species A Mass Balance

BSL2, p727

moles of ๐ด that flows intothe control volume 

between t and ๐‘ก ฮ”๐‘ก

moles of ๐ด that flows out of the control volume between t and 

๐‘ก ฮ”๐‘ก

balance over time interval ฮ”๐‘ก

ยฉ Faith A. Morrison, Michigan Tech U.7

โ„ณ , ฮ”๐‘ก โ„ณ , ฮ”๐‘ก

๐‘ ๐‘†

introduction of moles of ๐ดinto the C.V. by mass transfer 

across the ๐‘– bounding control surface ๐‘† (C.S.)

๐‘… ๐‘‰ ฮ”๐‘ก

๐‘… net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume

MOLES

Macroscopic control volume, 

C.V.

๐‘† ๐‘†

Unsteady Macroscopic Species ๐ด Mass Balance

C.S.

C.V.

moles of ๐ด that flows intothe control volume 

between t and ๐‘ก ฮ”๐‘ก

moles of ๐ด that flows out of the control volume between t and 

๐‘ก ฮ”๐‘ก

โ„ณ , ฮ”๐‘ก โ„ณ , ฮ”๐‘ก

๐‘ ๐‘†

introduction of moles of ๐ดinto the C.V. by mass transfer 

across the ๐‘– bounding control surface ๐‘† (C.S.)

๐‘… ๐‘‰ ฮ”๐‘ก

๐‘… net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume

C.V.

ยฉ Faith A. Morrison, Michigan Tech U.8

๐‘‘๐‘‘๐‘ก

โ„ณ , ฮ”โ„ณ ๐‘… ๐‘‰ ๐‘ ๐‘†

accumulation  net flow in + production + introduction

ฮ” is โ€œoutโ€โ€ โ€œinโ€C.S. = control surfaceC.V. = control volume

โ„ณ , ๐‘ ๐‘‰ total moles of ๐ด in the C.V.

ฮ”โ„ณ โˆ‘ โ„ณ , โˆ‘ โ„ณ , ,, bulk out 

๐‘… net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume

๐‘‰ system volume

๐‘ ๐‘˜ ๐‘ ๐‘โˆ— molar flux of ๐ด out 

through the ๐‘– C.S.

MOLES

๐‘† โˆ‘ ๐‘†

Unsteady Macroscopic Species ๐ด Mass Balance

Page 5: Contining work with the linear driving force for mass ...fmorriso/cm3120/lectures/lecturePart2_11-12_CM3120.pdf+ ยบ, we can also predict a mass xfercoeff๐‘˜ รฌor ๐‘˜ ร– Diffusion

Lecture2 11โ€12:  Macro Species A Mass Bal 4/12/2019

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moles of ๐ด that flows intothe control volume 

between t and ๐‘ก ฮ”๐‘ก

moles of ๐ด that flows out of the control volume between t and 

๐‘ก ฮ”๐‘ก

โ„ณ , ฮ”๐‘ก โ„ณ , ฮ”๐‘ก

๐‘ ๐‘†

introduction of moles of ๐ดinto the C.V. by mass transfer 

across the ๐‘– bounding control surface ๐‘† (C.S.)

๐‘… ๐‘‰ ฮ”๐‘ก

๐‘… net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume

C.V.

ยฉ Faith A. Morrison, Michigan Tech U.9

๐‘‘๐‘‘๐‘ก

โ„ณ , ฮ”โ„ณ ๐‘… ๐‘‰ ๐‘ ๐‘†

accumulation  net flow in + production + introduction

โ„ณ , ๐‘ ๐‘‰ total moles of ๐ด in the C.V.

ฮ”โ„ณ โˆ‘ โ„ณ , โˆ‘ โ„ณ , ,, bulk out 

๐‘… net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume

๐‘‰ system volume

๐‘ ๐‘˜ ๐‘ ๐‘โˆ— molar flux of ๐ด out 

through the ๐‘– C.S.

MOLESUnsteady Macroscopic Species ๐ด Mass Balance

The choice of the โ€œsystem,โ€ i.e. of the control volume, is an important first step.

(think โ€œsourceโ€ and โ€œsinkโ€)

ยฉ Faith A. Morrison, Michigan Tech U.

Example 8: Bone dry air and liquid water (water volume 0.80 ๐‘™๐‘–๐‘ก๐‘’๐‘Ÿ๐‘ ) are introduced into a closed container (cross sectional area 150 ๐‘๐‘š ; total volume 19.2 ๐‘™๐‘–๐‘ก๐‘’๐‘Ÿ๐‘ ). Both air and water are at 25 ๐ถ, ~1.0 ๐‘Ž๐‘ก๐‘š 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

๐ป ๐‘‚

Unsteady Macroscopic Species ๐ด Mass Balance

closed container

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Lecture2 11โ€12:  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 ๐‘™๐‘–๐‘ก๐‘’๐‘Ÿ๐‘ ) are introduced into a closed container (cross sectional area 150 ๐‘๐‘š ; total volume 19.2 ๐‘™๐‘–๐‘ก๐‘’๐‘Ÿ๐‘ ). Both air and water are at 25 ๐ถ 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

๐ป ๐‘‚

Unsteady Macroscopic Species ๐ด Mass Balance

closed container

Unsteady Macroscopic Species ๐ด Mass Balance

1. Control volume 2.

ยฉ Faith A. Morrison, Michigan Tech U.12

Solution:

Unsteady Macroscopic Species ๐ด Mass Balance

๐‘๐‘โˆ— 1 ๐‘’

๐‘ก 2.3 โ„Ž๐‘œ๐‘ข๐‘Ÿ๐‘ 

Example: Bone dry air and liquid water (water volume 0.80 ๐‘™๐‘–๐‘ก๐‘’๐‘Ÿ๐‘ ) are introduced into a closed container (cross sectional area 150 ๐‘๐‘š ; total volume 19.2 ๐‘™๐‘–๐‘ก๐‘’๐‘Ÿ๐‘ ). Both air and water are at 25 ๐ถ 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

๐ป ๐‘‚

Unsteady Macroscopic Species ๐ด Mass Balance

closed container

(mass transfer coefficient for evaporating water)

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Lecture2 11โ€12:  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 ๐‘๐‘š of surface area per ๐‘๐‘š volume of bed. What is the mass transfer coefficient when pure water flowing in (โ€œsuperficial velocityโ€5.0 ๐‘๐‘š/๐‘ ) exits 62% saturated with benzoic acid?

13Cussler, p240, FAM v54 p63

2.0๐‘๐‘š

1.0 10 ๐‘๐‘š

Unsteady Macroscopic Species ๐ด Mass Balance

ยฉ Faith A. Morrison, Michigan Tech U.14

What is our model?

(our assumptions)

Unsteady Macroscopic Species ๐ด 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 ๐‘๐‘š of surface area per ๐‘๐‘š volume of bed. What is the mass transfer coefficient when pure water flowing in (โ€œsuperficial velocityโ€5.0 ๐‘๐‘š/๐‘ ) exits 62% saturated with benzoic acid?

2.0๐‘๐‘š

1.0 10 ๐‘๐‘š

Unsteady Macroscopic Species ๐ด Mass Balance

1. Control volume 2.

Page 8: Contining work with the linear driving force for mass ...fmorriso/cm3120/lectures/lecturePart2_11-12_CM3120.pdf+ ยบ, we can also predict a mass xfercoeff๐‘˜ รฌor ๐‘˜ ร– Diffusion

Lecture2 11โ€12:  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 ๐ด 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 ๐‘๐‘š of surface area per ๐‘๐‘š volume of bed. What is the mass transfer coefficient when pure water flowing in (โ€œsuperficial velocityโ€5.0 ๐‘๐‘š/๐‘ ) exits 62% saturated with benzoic acid?

2.0๐‘๐‘š

1.0 10 ๐‘๐‘š

Unsteady Macroscopic Species ๐ด Mass Balance

๐‘˜๐‘ฃ๐‘Ž๐ฟ

ln 1๐‘๐‘โˆ—

๐‘˜ 2.1 10 ๐‘๐‘š/๐‘ 

ยฉ Faith A. Morrison, Michigan Tech U.

Example 10: Height of a packed bed absorber

16BSL2 p742

Unsteady Macroscopic Species ๐ด 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?

Page 9: Contining work with the linear driving force for mass ...fmorriso/cm3120/lectures/lecturePart2_11-12_CM3120.pdf+ ยบ, we can also predict a mass xfercoeff๐‘˜ รฌor ๐‘˜ ร– Diffusion

Lecture2 11โ€12:  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 ๐ด Mass Balance

From lecture 8:

Example 10: Height of a packed bed absorber

ยฉ Faith A. Morrison, Michigan Tech U.18

Unsteady Macroscopic Species ๐ด Mass Balanceโ€”Gas Absorption

From lecture 8:

Page 10: Contining work with the linear driving force for mass ...fmorriso/cm3120/lectures/lecturePart2_11-12_CM3120.pdf+ ยบ, we can also predict a mass xfercoeff๐‘˜ รฌor ๐‘˜ ร– Diffusion

Lecture2 11โ€12:  Macro Species A Mass Bal 4/12/2019

10

ยฉ Faith A. Morrison, Michigan Tech U.19

Unsteady Macroscopic Species ๐ด Mass Balanceโ€”Gas Absorption

From lecture 8:

ยฉ Faith A. Morrison, Michigan Tech U.20

Unsteady Macroscopic Species ๐ด Mass Balanceโ€”Gas Absorption

From lecture 8:

Page 11: Contining work with the linear driving force for mass ...fmorriso/cm3120/lectures/lecturePart2_11-12_CM3120.pdf+ ยบ, we can also predict a mass xfercoeff๐‘˜ รฌor ๐‘˜ ร– Diffusion

Lecture2 11โ€12:  Macro Species A Mass Bal 4/12/2019

11

ยฉ Faith A. Morrison, Michigan Tech U.21

Unsteady Macroscopic Species ๐ด Mass Balanceโ€”Gas Absorption

Liquid in

Liquid out

Gas out

Gas in

moles of ๐ด that flows intothe control volume 

between t and ๐‘ก ฮ”๐‘ก

moles of ๐ด that flows out of the control volume between t and 

๐‘ก ฮ”๐‘ก

โ„ณ , ฮ”๐‘ก โ„ณ , ฮ”๐‘ก

๐‘ ๐‘†

introduction of moles of ๐ดinto the C.V. by mass transfer 

across the ๐‘– bounding control surface ๐‘† (C.S.)

๐‘… ๐‘‰ ฮ”๐‘ก

๐‘… net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume

C.V.

accumulation  net flow in + production + introduction

ฮ” is โ€œoutโ€โ€ โ€œinโ€C.S. = control surfaceC.V. = control volume

โ„ณ , ๐‘ ๐‘‰ total moles of ๐ด in the C.V.

ฮ”โ„ณ โˆ‘ โ„ณ , โˆ‘ โ„ณ , ,, bulk out 

๐‘… net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume

๐‘‰ system volume

๐‘ ๐‘˜ ๐‘ ๐‘โˆ— molar flux of ๐ด out 

through the ๐‘– C.S.

MOLES

๐‘† โˆ‘ ๐‘†

Unsteady Macroscopic Species ๐ด Mass Balance

? You try.

ยฉ Faith A. Morrison, Michigan Tech U.22

What is our model?

(our assumptions)

Unsteady Macroscopic Species ๐ด Mass Balance

1. Control volume 2.

Liquid in

Liquid out

Gas out

Page 12: Contining work with the linear driving force for mass ...fmorriso/cm3120/lectures/lecturePart2_11-12_CM3120.pdf+ ยบ, we can also predict a mass xfercoeff๐‘˜ รฌor ๐‘˜ ร– Diffusion

Lecture2 11โ€12:  Macro Species A Mass Bal 4/12/2019

12

ยฉ Faith A. M

orrison, M

ichigan

 Tech U.

23

Unsteady Macroscopic Species ๐ด Mass Balanceโ€”Gas Absorption

Liquid in

Liquid out

Gas out

๐‘ง๐‘ง ฮ”๐‘ง

moles of ๐ด that flows intothe control volume between t and ๐‘ก ฮ”๐‘ก

moles of ๐ด that flows out of the control volume between t and 

๐‘ก ฮ”๐‘ก

โ„ณ , ฮ”๐‘ก โ„ณ , ฮ”๐‘ก

๐‘ ๐‘†

introduction of moles of ๐ดinto the C.V. by mass transfer 

across the ๐‘– bounding control surface ๐‘† (C.S.)

๐‘… ๐‘‰ ฮ”๐‘ก

๐‘… net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume

C.V.

accumulation  net flow in + production + introduction

ฮ” is โ€œoutโ€โ€ โ€œinโ€C.S. = control surfaceC.V. = control volume

โ„ณ , ๐‘ ๐‘‰ total moles of ๐ด in the C.V.

ฮ”โ„ณ โˆ‘ โ„ณ , โˆ‘ โ„ณ , ,, bulk out 

๐‘… net rate of production of moles of ๐ด in the C.V. by reaction, per unit volume

๐‘‰ system volume

๐‘ ๐‘˜ ๐‘ ๐‘โˆ— molar flux of ๐ด out 

through the ๐‘– C.S.

MOLES

๐‘† โˆ‘ ๐‘†

Unsteady Macroscopic Species ๐ด Mass Balance

๐‘ง

๐‘ง ฮ”๐‘ง

๐‘งGas ๐ผ, ๐ด

Liquid๐ต, ๐ด

C.V.

๐‘

โ„ณ

โ„ณ

๐‘Ž๐‘†ฮ”๐‘ง area for mass transfer

ยฉ Faith A. Morrison, Michigan Tech U.24

Unsteady Macroscopic Species ๐ด Mass Balanceโ€”Gas Absorption

โ„ณ โ‰ก moles gas on an ๐ด-free basis

๐’ด โ‰ก

๐’ณ โ‰ก

๐’ด ๐‘ฆ for dilute systems

๐’ณ ๐’ณโ„ณ

โ„ณ๐’ด ๐’ด

๐’ด ๐’ดโˆ—

๐’ณ ๐’ณโˆ—

โ€ฆ

Operating line; from mass balance on the top of the column)

(result from ๐‘ , ๐‘ , ;

BSLโ€™s equation 23.5โ€21)

molar ratios

BSL2 p745

Page 13: Contining work with the linear driving force for mass ...fmorriso/cm3120/lectures/lecturePart2_11-12_CM3120.pdf+ ยบ, we can also predict a mass xfercoeff๐‘˜ รฌor ๐‘˜ ร– Diffusion

Lecture2 11โ€12:  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 ๐ด Mass Balanceโ€”Gas Absorption

๐’ดโˆ— ๐’ณโˆ—

๐’ณ

๐’ด

๐’ดโˆ—

๐’ด ๐’ณ

(height of a packed bed absorber)

ยฉ Faith A. Morrison, Michigan Tech U.26

Solution:

Unsteady Macroscopic Species ๐ด Mass Balance

๐ฟโ„ณ

๐‘˜ ๐‘Ž ๐‘†

๐‘‘๐’ด๐’ด ๐’ดโˆ—

๐’ด

๐’ด

Answer: Perform numerical integration to obtain the column height ๐ฟ:

Need: Gas flow rate โ„ณ , column cross sectional area ๐‘†, data on thermodynamic equilibrium ๐’ดโˆ— ๐’ณโˆ— , desired separation (mole fractions top and bottom, and mass transfer coefficients ๐‘˜ ๐‘Ž and ๐‘˜ ๐‘Ž.

Page 14: Contining work with the linear driving force for mass ...fmorriso/cm3120/lectures/lecturePart2_11-12_CM3120.pdf+ ยบ, we can also predict a mass xfercoeff๐‘˜ รฌor ๐‘˜ ร– Diffusion

Lecture2 11โ€12:  Macro Species A Mass Bal 4/12/2019

14

27

Mass Transport โ€œLawsโ€

We have 2 Mass Transport โ€œlawsโ€

๐‘ ๐‘˜ ๐‘ฆ , ๐‘ฆ ,

Fickโ€™s Law of Diffusion

Linear-Driving-Force Model

๐‘ ๐‘ฅ ๐‘ ๐‘ ๐‘๐ท ๐›ป๐‘ฅ

Combine with microscopic species ๐ด mass balancePredicts flux ๐‘ and composition distributions, e.g. ๐‘ฅ ๐‘ฅ, ๐‘ฆ, ๐‘ง, ๐‘ก

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 ๐‘˜Diffusion coefficients are material properties (see tables)

Combine with macroscopic species ๐ด 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, ๐พ , ๐พ

Transport coefficient

Use:

Use:

1

2

3

4

5

ยฉ Faith A. Morrison, Michigan Tech U.

28

Mass Transport โ€œLawsโ€

We now have 2 Mass Transport โ€œlawsโ€

๐‘ ๐‘˜ ๐‘ฆ , ๐‘ฆ ,

Fickโ€™s Law of Diffusion

Linear-Driving-Force Model

๐‘ ๐‘ฅ ๐‘ ๐‘ ๐‘๐ท ๐›ป๐‘ฅ

Combine with microscopic species ๐ด mass balancePredicts flux ๐‘ and composition distributions, e.g. ๐‘ฅ ๐‘ฅ, ๐‘ฆ, ๐‘ง, ๐‘ก

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 ๐‘˜Diffusion coefficients are material properties (see tables)

Combine with macroscopic species ๐ด 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, ๐พ , ๐พ

Transport coefficient

Use:

Use:

1

2

3

4

5

1. Since we predict ๐‘ with Fickโ€™s law, we can also predict a mass transfercoefficients ๐‘˜ or ๐‘˜

2. 1D Unsteady models can be solved (if good at math)

Mass Transport โ€œLawsโ€

ยฉ Faith A. Morrison, Michigan Tech U.

๐ท

๐‘˜

Mass transfer coefficients

3. Combine with macroscopic species ๐ด 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, ๐พ , ๐พ , to be used in modeling unit operations

Fickโ€™s law of diffusion

Solutions are analogous to heat transfer

Relate ๐‘˜ and ๐ท

Remaining Topics to round out our understanding of mass transport:

We have 2 Mass Transport โ€œlawsโ€

Model macroscopic processes; design units


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