cm3120: module 4 - michigan technological university

Module 4, Lecture III 1/5/2021

1

CM3120: Module 4

© 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, 𝑘A. 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. Review—compare to heatB. Engineering quantities of interestC. Data correlations for 𝑘 (Sh or Nu correlations)

V. Overall mass transfer coefficients

CM3120: Module 4


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


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3© Faith A. Morrison, Michigan Tech U.

Contining work with the linear driving force for mass transfer, i.e. mass transfer coefficients, 𝒌𝒄

4

Mass Transport “Laws”

We have 2 Mass Transport “laws”

𝑁 𝑘 𝑦 , 𝑦 ,

Fick’s Law of Diffusion

Linear-Driving-Force Model

𝑁 𝑥 𝑁 𝑁 𝑐𝐷 𝛻𝑥

Combine with microscopic species 𝐴 mass balancePredicts flux 𝑁 and composition distributions, e.g. 𝑥 𝑥, 𝑦, 𝑧, 𝑡

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 𝑁 , we can also predict a mass xfer coeff 𝑘 or 𝑘Diffusion coefficients are material properties (see tables)

Combine with macroscopic species 𝐴 mass balancePredicts flux 𝑁 , 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, 𝐾 ,𝐾

Use:

Use:

© Faith A. Morrison, Michigan Tech U.

SummaryTransport coefficient


3

5



𝑁 𝑘 𝑦 , 𝑦 ,









Transport coefficient

Use:

Use:

1

2

3

4

5


6


We now have 2 Mass Transport “laws”

𝑁 𝑘 𝑦 , 𝑦 ,




Combine with microscopic species 𝐴 mass balancePredicts flux 𝑁 and composition distributions, e.g. 𝑥 𝑥,𝑦, 𝑧, 𝑡






Use:

Use:

1

2

3

4

5

1. Since we predict 𝑁 with Fick’s law, we can also predict a mass transfer coefficients 𝑘 or 𝑘

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



𝐷

𝑘

Mass transfer coefficients

3. Combine with macroscopic species 𝐴 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, 𝐾 ,𝐾 , to be used in modeling unit operations

Fick’s law of diffusion

Solutions are analogous to heat transfer

Relate 𝑘 and 𝐷

Remaining Topics to round out our understanding of mass transport:


Keep cranking at the list


4

Professor Faith A. Morrison

Department of Chemical EngineeringMichigan Technological University

7© Faith A. Morrison, Michigan Tech U.

CM3110 Transport IIPart II: Diffusion and Mass Transfer

Macroscopic Species 𝑨 Mass Balances

Unsteady Macroscopic Species 𝐴Mass Balance

C.V.

3

Unsteady, Macroscopic, Species A Mass Balance

BSL2, p727

balance over time interval Δ𝑡


C.V.

MOLES

Macroscopic control volume,

C.V.


Keep track of:•Bulk convection of species 𝐴 into and out of the C.V.•Mass transfer across control surfaces of species 𝐴•Mass of species 𝐴 created by chemical reaction


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Unsteady, Macroscopic, Species A Mass Balance

BSL2, p727

moles of 𝐴 that flows intothe control volume

between t and 𝑡 Δ𝑡

moles of 𝐴 that flows out of the control volume between t and

𝑡 Δ𝑡

balance over time interval Δ𝑡


ℳ , Δ𝑡 ℳ , Δ𝑡

𝑁 𝑆

introduction of moles of 𝐴into the C.V. by mass transfer

across the 𝑖 bounding control surface 𝑆 (C.S.)

𝑅 𝑉 Δ𝑡

𝑅 net rate of production of moles of 𝐴 in the C.V. by reaction, per unit volume

MOLES

Macroscopic control volume,

C.V.

𝑆 𝑆


C.S.

C.V.

moles of 𝐴 that flows intothe control volume between t and 𝑡 Δ𝑡


𝑡 Δ𝑡


𝑁 𝑆

introduction of moles of 𝐴into the C.V. by mass transfer across the 𝑖 bounding control surface 𝑆 (C.S.)

𝑅 𝑉 Δ𝑡


C.V.


𝑑𝑑𝑡

ℳ , Δℳ 𝑅 𝑉 𝑁 𝑆

accumulation net flow in + production + introduction

Δ is “out”‐ “in”C.S. = control surfaceC.V. = control volume

ℳ , 𝑐 𝑉 total moles of 𝐴 in the C.V.

Δℳ ∑ ℳ , ∑ ℳ , ,, bulk out


𝑉 system volume

𝑁 𝑘 𝑐 𝑐∗ molar flux of 𝐴 out

through the 𝑖 C.S.

MOLES

𝑆 ∑ 𝑆



6



𝑡 Δ𝑡


𝑁 𝑆


𝑅 𝑉 Δ𝑡


C.V.


𝑑𝑑𝑡

ℳ , Δℳ 𝑅 𝑉 𝑁 𝑆





𝑉 system volume



MOLESUnsteady Macroscopic Species 𝐴Mass Balance

The choice of the “system,” i.e. of the control volume, is an important first step.

(think “source” and “sink”)


Example 9: Bone dry air and liquid water (water volume 0.80 𝑙𝑖𝑡𝑒𝑟𝑠) are introduced into a closed container (cross sectional area 150 𝑐𝑚 ; total volume 19.2 𝑙𝑖𝑡𝑒𝑟𝑠). Both air and water are at 25 𝐶, ~1.0 𝑎𝑡𝑚 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

𝐻 𝑂


closed container


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What is our model?

(our assumptions)

Example: Bone dry air and liquid water (water volume 0.80 𝑙𝑖𝑡𝑒𝑟𝑠) are introduced into a closed container (cross sectional area 150 𝑐𝑚 ; total volume 19.2 𝑙𝑖𝑡𝑒𝑟𝑠). Both air and water are at 25 𝐶 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?


𝐻 𝑂

Unsteady Macroscopic Species 𝐴 Mass Balance

closed container


1. Control volume 2.


Solution:


𝑐𝑐∗

1 𝑒

𝑡 2.3 ℎ𝑜𝑢𝑟𝑠

Example: Bone dry air and liquid water (water volume 0.80 𝑙𝑖𝑡𝑒𝑟𝑠) are introduced into a closed container (cross sectional area 150 𝑐𝑚 ; total volume 19.2 𝑙𝑖𝑡𝑒𝑟𝑠). Both air and water are at 25 𝐶 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?


𝐻 𝑂


closed container

(mass transfer coefficient for evaporating water)


8


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 𝑐𝑚 of surface area per 𝑐𝑚 volume of bed. What is the mass transfer coefficient when pure water flowing in (“superficial velocity”5.0 𝑐𝑚/𝑠) exits 62% saturated with benzoic acid?

15Cussler, p240, FAM v54 p63

2.0𝑐𝑚

1.0 10 𝑐𝑚



What is our model?

(our assumptions)


Example: Flow through a packed bed of soluble spherical pellets.

liquid water



Two centimeter diameter spheres of benzoic acid (soluble in water) are packed into a bed as shown. The spheres have 23 𝑐𝑚 of surface area per 𝑐𝑚 volume of bed. What is the mass transfer coefficient when pure water flowing in (“superficial velocity”5.0 𝑐𝑚/𝑠) exits 62% saturated with benzoic acid?

2.0𝑐𝑚

1.0 10 𝑐𝑚




9

(dissolving benzoic acid in packed bed, pseudo steady state)


Solution:


Example: Flow through a packed bed of soluble spherical pellets.

liquid water



Two centimeter diameter spheres of benzoic acid (soluble in water) are packed into a bed as shown. The spheres have 23 𝑐𝑚 of surface area per 𝑐𝑚 volume of bed. What is the mass transfer coefficient when pure water flowing in (“superficial velocity”5.0 𝑐𝑚/𝑠) exits 62% saturated with benzoic acid?

2.0𝑐𝑚

1.0 10 𝑐𝑚


𝑘𝑣𝑎𝐿

ln 1𝑐𝑐∗

𝑘 2.1 10 𝑐𝑚/𝑠


Example 11: Height of a packed bed absorber

18BSL2 p742


How can we use the linear driving force model for mass transfer to design a packed bed gas absorber to achieve a desired separation?


10


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


From earlier lecture:

Example 11: Height of a packed bed absorber


Unsteady Macroscopic Species 𝐴Mass Balance—Gas Absorption



11








12



Liquid in

Liquid out

Gas out

Gas in



𝑡 Δ𝑡


𝑁 𝑆

introduction of moles of 𝐴into the C.V. by mass transfer

across the 𝑖 bounding control surface 𝑆 (C.S.)

𝑅 𝑉 Δ𝑡


C.V.






𝑉 system volume



MOLES

𝑆 ∑ 𝑆


? You try.


What is our model?

(our assumptions)



Liquid in

Liquid out

Gas out


13

© Faith A. M

orrison, M

ichigan

Tech U.

25


Liquid in

Liquid out

Gas out

𝑧𝑧 Δ𝑧



𝑡 Δ𝑡


𝑁 𝑆


𝑅 𝑉 Δ𝑡


C.V.






𝑉 system volume



MOLES

𝑆 ∑ 𝑆


𝑧

𝑧 Δ𝑧

𝑧Gas 𝐼,𝐴

Liquid𝐵,𝐴

C.V.

𝑁

ℳ

ℳ

𝑎𝑆Δ𝑧 area for mass transfer



ℳ ≡ moles gas on an 𝐴 free basis

𝒴 ≡

𝒳 ≡

𝒴 𝑦 for dilute systems

𝒳 𝒳ℳ

ℳ𝒴 𝒴

𝒴 𝒴∗

𝒳 𝒳∗

…

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

(result from 𝑁 , 𝑁 , ;

BSL’s equation 23.5−21)

molar ratios

BSL2 p745


14

BSL2 p745 Bird, Stewart, and Lightfoot, Transport Phenomena, 2nd ed., Wiley, 2006. © Faith A. Morrison, Michigan Tech U.

27


𝒴∗ 𝒳∗

𝒳

𝒴

𝒴∗

𝒴 𝒳

(height of a packed bed absorber)


Solution:


𝐿ℳ

𝑘 𝑎 𝑆

𝑑𝒴𝒴 𝒴∗

𝒴

𝒴

Answer: Perform numerical integration to obtain the column height 𝐿:

Need: Gas flow rate ℳ , column cross sectional area 𝑆, data on thermodynamic equilibrium 𝒴∗ 𝒳∗ , desired separation (mole fractions top and bottom, and mass transfer coefficients 𝑘 𝑎 and 𝑘 𝑎.


15

29



𝑁 𝑘 𝑦 , 𝑦 ,










Use:

Use:

1

2

3

4

5


30


We now have 2 Mass Transport “laws”

𝑁 𝑘 𝑦 , 𝑦 ,




Combine with microscopic species 𝐴 mass balancePredicts flux 𝑁 and composition distributions, e.g. 𝑥 𝑥,𝑦, 𝑧, 𝑡






Use:

Use:

1

2

3

4

5

1. Since we predict 𝑁 with Fick’s law, we can also predict a mass transfer coefficients 𝑘 or 𝑘

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



𝐷

𝑘

Mass transfer coefficients

3. Combine with macroscopic species 𝐴 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, 𝐾 ,𝐾 , to be used in modeling unit operations

Fick’s law of diffusion

Solutions are analogous to heat transfer

Relate 𝑘 and 𝐷

Remaining Topics to round out our understanding of mass transport:


Model macroscopic processes; design units

cm3120: module 4 - michigan technological university

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