Module 3 Lecture III 3/1/2021
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CM3120: Module 3
Β© Faith A. Morrison, Michigan Tech U.1
Diffusion and Mass Transfer I
I. Introduction to diffusion/mass transferII. Classic diffusion and mass transferβQuick Start a): 1D EvaporationIII. Classic diffusion and mass transferβQuick Start b): 1D Radial dropletIV. Cycle back: Fickβs mass transport lawV. Microscopic species A mass balanceVI. Classic diffusion and mass transferβc): 1D Mass transfer with
chemical reaction
CM3120: Module 3
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 3 Lecture III
Quick Start 2:1D Radial Diffusion
Module 3 Lecture III 3/1/2021
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Β© Faith A. Morrison, Michigan Tech U.3
Continuingβ¦
Β© Faith A. Morrison, Michigan Tech U.BSL2, p547 4
Example 1: Water (40 πΆ, 1.0 ππ‘π) slowly and steadily evaporates into nitrogen (40 πΆ, 1.0 ππ‘π) from the bottom of a cylindrical tank as shown in the figure below. A stream of dry nitrogen flows slowly past the open tank. The mole fraction of water in the gas at the top opening of the tank is 0.02. What is the rate of water evaporation?
π§π§ π§ 0.3π
π§ π§ 1.0π
2π
π
π» π
0.25π
π§ 0
QUICK START
Last timeβ¦
Module 3 Lecture III 3/1/2021
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Β© Faith A. Morrison, Michigan Tech U.5
Example: Water (40 πΆ, 1.0 ππ‘π) slowly and steadily evaporates into nitrogen (40 πΆ , 1.0 ππ‘π) from the bottom of a cylindrical tank as shown in the figure below. A stream of dry nitrogen flows slowly past the open tank. The mole fraction of water in the gas at the top opening of the tank is 0.02. What is the rate of water evaporation?
π§π§ π§ 0.3π
π§ π§ 1.0π
2π
π
π» π
0.25π
π§ 0
QUICK START
Why does the water evaporate?
What limits the rate of evaporation?
What could be done to accelerate the evaporation?
What could be done to slow down the evaporation?
Interrogating the problem:
What is the driving physics?
Last timeβ¦
We sought to complete this problem.
The primary goal, however, was (is)
to grow our ability to troubleshoot
engineering problems.
We take note of the questions that were productive in leading us to the solution.
Β© Faith A. Morrison, Michigan Tech U.6
Last timeβ¦
The questions led us to
refine the problem; to frame it in a way that led
to the solution.
Example 1: Water (40 πΆ, 1.0 ππ‘π) slowly and steadily evaporates into nitrogen (40 πΆ, 1.0 ππ‘π) from the bottom of a cylindrical tank as shown in the figure below. A stream of dry nitrogen flows slowly past the open tank. The mole fraction of water in the gas at the top opening of the tank is 0.02. The geometry is as shown in the figure. What is water mole fraction as a function of vertical position in the tank? You may assume ideal gas properties. What is the rate of water evaporation?
π§π§ π§ 0.3π
π§ π§ 1.0π
2π
π
π» π
0.25π
π§ 0
QUICK START
Module 3 Lecture III 3/1/2021
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Β© Faith A. Morrison, Michigan Tech U.BSL2, p5477
Example 1
Seek concentration distributionβ microscopic species A mass balance.
Β© Faith A. Morrison, Michigan Tech U.BSL2, p5478
Example 1
First, we obtained the flux π ,and the concentration
distribution π₯ π§ .
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Β© Faith A. Morrison, Michigan Tech U.BSL2, p5479
Example 1
Then, we answered the question:
The rate of evaporation π΄ π , 3.9 10 mol/s.
π , πΆ ππ·π§ π§ ln1 π₯1 π₯
π΄ ππ·4
Β© Faith A. Morrison, Michigan Tech U.
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). What is the rate of evaporation and how does the water concentration vary in the gas?
π
π» π
BSL2, p550
2β
10
QUICK START
Ver 1
Letβs do another problem
The primary goals are β’ to grow our ability to troubleshoot engineering
problemsβ’ To explore βclassicβ mass transfer circumstances
and add them to our tool belt
Module 3 Lecture III 3/1/2021
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Β© Faith A. Morrison, Michigan Tech U.
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). What is the rate of evaporation and how does the water concentration vary in the gas?
π
π» π
BSL2, p550
2β
11
QUICK START
Letβs Interrogate the problem. Ver 1
Β© Faith A. Morrison, Michigan Tech U.
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). What is the rate of evaporation and how does the water concentration vary in the gas?
π
π» π
BSL2, p550
2β
12
QUICK START
Ver 1
Why does the water evaporate?
What limits the rate of evaporation?
What could be done to accelerate the evaporation?
What could be done to slow down the evaporation?
What is the driving physics?
Can we use any ideas from previous experience?
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Β© Faith A. Morrison, Michigan Tech U.
π
π
π» π
BSL2, p550
2β 2β
π and π» 0
13
QUICK START
Ver 2
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). The evaporation creates a film around the droplets through which the evaporating water diffuses. What is the rate of evaporation and how does the water concentration vary in the gas as a function distance from the droplet? You may assume ideal gas properties for air.
We are developing a model to address the questions of interest
π
π
π» π2β 2β
π and π» 0
Ver 2
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). The evaporation creates a film around the droplets through which the evaporating water diffuses. What is the rate of evaporation and how does the water concentration vary as a function distance from the droplet? You may assume ideal gas properties for air.
Β© Faith A. Morrison, Michigan Tech U.BSL2, p550 14
SOLVE
QUICK START
Module 3 Lecture III 3/1/2021
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π
π
π» π2β 2β
π and π» 0
Ver 2
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). The evaporation creates a film around the droplets through which the evaporating water diffuses. What is the rate of evaporation and how does the water concentration vary as a function distance from the droplet? You may assume ideal gas properties for air.
Β© Faith A. Morrison, Michigan Tech U.BSL2, p550 15
See hand slides
QUICK START
SOLUTION:
Β© Faith A. Morrison, Michigan Tech U.BSL2, p550 16
Solution:ππ¨ π1 π₯1 π₯
1 π₯1 π₯
ββ β
Ver 2
π
π
π» π2β 2β
π and π» 0
Ver 2
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). The evaporation creates a film around the droplets through which the evaporating water diffuses. What is the rate of evaporation and how does the water concentration vary as a function distance from the droplet? You may assume ideal gas properties for air.
QUICK START
Assumptions: β’ Steady diffusion; stagnant airβ’ Uniform film surrounds dropletβ’ Ideal gasβ’ Constant temperature and
pressureβ’ Constant π,π
Homework problem 3.14 do algebra; problem 3.8
plot this solution
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Β© Faith A. Morrison, Michigan Tech U.17
QUICK START
Solution:ππ¨ π
Ver 2
π
π
π» π2β 2β
π and π» 0
Ver 2
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). The evaporation creates a film around the droplets through which the evaporating water diffuses. What is the rate of evaporation and how does the water concentration vary as a function distance from the droplet? You may assume ideal gas properties for air.
Assumptions: β’ Uniform film surrounds dropletβ’ Ideal gasβ’ Constant temperature and
pressureWe can now explore these
assumptions, and modify, if needed, for more complex
problems.
Letβs Interrogate the problem.
Β© Faith A. Morrison, Michigan Tech U.BSL2, p550
π
π
π» π2β 2β
π and π» 0
18
QUICK START
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). The evaporation creates a film around the droplets through which the evaporating water diffuses. What is the rate of evaporation and how does the water concentration vary in the gas as a function distance from the droplet? You may assume ideal gas properties for air.
Film model of mass transfer
Assumptions: β’ Uniform filmβ’ Ideal gasβ’ Constant pressureβ’ Constant temperature
How good are these assumptions?
Are there energy issues?
Module 3 Lecture III 3/1/2021
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Β© Faith A. Morrison, Michigan Tech U.
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). The evaporation creates a film around the droplets through which the evaporating water diffuses. We can model the diffusion process as shown in the figure. The temperature in the film is not constant but varies as π π /π β π/β . What is the rate of evaporation and how does the water concentration vary as a function distance from the droplet?
π
π
π» π
BSL2, p550
2β 2β
π and π» 0
Note: notisothermal
19
QUICK START
Ver 3
Film model of mass transfer (more complex)
Ξπ» 0
Film model of mass transfer (more complex)
Β© Faith A. Morrison, Michigan Tech U.
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). The evaporation creates a film around the droplets through which the evaporating water diffuses. We can model the diffusion process as shown in the figure. The temperature in the film is not constant but varies as π π /π β π/β . What is the rate of evaporation and how does the water concentration vary as a function distance from the droplet?
π
π
π» π
BSL2, p550
2β 2β
π and π» 0
Note: notisothermal
20
QUICK START
Ver 3
What does changing temperature impact?
How do we modify the model?
Ξπ» 0
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Β© Faith A. Morrison, Michigan Tech U.BSL2, p550 21
Film model of mass transfer (more complex)
Where did we assume βisothermalβ in our previous solution?
Β© Faith A. Morrison, Michigan Tech U.BSL2, p550 22
Film model of mass transfer (more complex)
Where did we assume βisothermalβ in our previous solution?
When we integrated to obtain π₯ π .
Module 3 Lecture III 3/1/2021
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Film model of mass transfer (more complex)
Β© Faith A. Morrison, Michigan Tech U.
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). The evaporation creates a film around the droplets through which the evaporating water diffuses. We can model the diffusion process as shown in the figure. The temperature in the film is not constant but varies as π π /π β π/β . What is the rate of evaporation and how does the water concentration vary as a function distance from the droplet? You may assume ideal gas properties for air; you may assume that the diffusivity varies with temperature as follows:
π· π /π· , π/π /
π
π
π» π
BSL2, p550
2β 2β
π and π» 0
Note: notisothermal
23
HW 3.7 (stretch)
QUICK START
Ver 4
Β© Faith A. Morrison, Michigan Tech U.BSL2, p550 24
Solution:ππ¨ π
1 π₯1 π₯
1 π₯1 π₯
β / /
β / β /
Ver 4
Assumptions: β’ Uniform film surrounds dropletβ’ Ideal gasβ’ Temperature follows power lawβ’ Diffusivity follows power lawβ’ Pressure is constant
Film model of mass transfer (more complex)
Example 2: A water mist forms in an industrial printing operation. Spherical water droplets slowly and steadily evaporate into the air (mostly nitrogen). The evaporation creates a film around the droplets through which the evaporating water diffuses. We can model the diffusion process as shown in the figure. The temperature in the film is not constant but varies as π π /π β π/β . What is the water mole fraction in the film as a function of radial position? You may assume ideal gas properties for air; you may assume that the diffusivity varies with temperature as follows:
π· π /π· , π/π /
π
π
π» π2β 2β
π and π» 0
Note: notisothermal
HW3, problem 12 (stretch)
QUICK START
HW 3.7
Module 3 Lecture III 3/1/2021
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Introduction to Diffusion and Mass Transfer in Mixtures
Recurring Modeling Assumptions in Diffusion (βClassicsβ)β’ Near a liquid-gas interface, the region in the gas near the liquid is
a film where slow diffusion takes placeβ’ The vapor near the liquid-gas interface is often saturated (Raoultβs
law, π₯ πβ/π)β’ If component π΄ has no sink, flux π 0.β’ If π΄ diffuses through stagnant π΅, π 0.β’ If A is dilute in B, we can neglect the convection term (π π½β )β’ Because diffusion is slow, we can make a quasi-steady-state
assumptionβ’β’β’β’β’β’
Β© F
aith
A. M
orris
on, M
ichi
gan
Tech
U.
QUICK START
Β© Faith A. Morrison, Michigan Tech U.26
We have been performing a βQuick Start,β
Microscopic species A mass balanceβFive forms
π πππ΄ππ‘ π£ β βππ΄ β β ποΏ½Μ²οΏ½ ππ΄ππ· π» π π
π ππ₯π΄ππ‘ π£β β βπ₯π΄ β β οΏ½Μ²οΏ½π΄β π₯π΅π π΄ π₯π΄π π΅ππ·π΄π΅β2π₯π΄ π₯π΅π π΄ π₯π΄π π΅
πππ΄ππ‘ β β ππ΄ π π΄
In terms of mass flux and mass
concentrations
In terms of molar flux and molar
concentrations
In terms of combined molar flux and molar
concentrations
Weβll do a βQuick Startβ and get into some examples and return to the βwhyβ of it all a
bit later.
It turns out that there are many interesting and applicable problems we can address readily with thisform of the species mass balance.
Letβs jump in!
Microscopic species mass balance in terms of
combined molar flux π΅π¨
And have found the combined molar flux formulation useful.
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Β© Faith A. Morrison, Michigan Tech U.27
There are times it is not useful. We need to go back and discuss how/why/when this all works.
28 Β© Faith A. Morrison, Michigan Tech U.