Diffusion Lecture Notes

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DIFFUSION

CHEMICAL ENGINEERING REVIEW

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PRINCIPLES OF DIFFUSION

MASS TRANSFER

 Is the net movement of a component in a mixture from one location to another location where the component exists at a different concentration

 The transfer takes place between the two phases across an interface

 Mass transfer occurs by two basic mechanisms: (1) molecular diffusion and (2) eddy diffusion

MOLECULAR DIFFUSION

 Transfer or movement of individual molecules through a fluid by means of the random, individual movements of the molecules

 In a binary mixture, molecular diffusion occurs because of one or more different potentials or driving forces

1. Concentration gradient (ORDINARY DIFFFUSION) 2. Pressure gradient (PRESSURE DIFFUSION) 3. Temperature (THERMAL DIFFUSION)

4. External force fields (FORCED DIFFUSION) as in centrifuge 5. Activity gradient as in reverse osmosis

 Molecular diffusion occurs in solids and in fluids that are stagnant or in laminar or turbulent motion

EDDY (TURBULENT) DIFFUSION

 Takes place in fluid phases by physical mixing and by the eddies of turbulent flow

FICK’S LAW OF DIFFUSION AT STEADY STATE Fick’s Law of Diffusion

 Gives the rate of mass transfer by molecular diffusion perpendicular to an relative to a stationary surface which is at fixed distance from the interface

 Fick’s First Law of Molecular Diffusion is proportionality between a flux and a gradient. For a

binary mixture of A and B

𝑟𝑎𝑡𝑒 𝑜𝑓 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 = 𝑑𝑟𝑖𝑣𝑖𝑛𝑔 𝑓𝑜𝑟𝑐𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝜓𝑧 = −𝛿

𝑑Γ 𝑑𝑧

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𝐽𝐴𝑥 = 𝑁𝐴 𝐴 = −𝐷𝐴𝐵 𝑑𝐶𝐴 𝑑𝑍 and 𝐽𝐵𝑥 = 𝑁𝐵 𝐴 = −𝐷𝐵𝐴 𝑑𝐶𝐵 𝑑𝑍 where:

𝐽𝐴𝑥 - is the molar flux of A by ordinary molecular diffusion relative to the molar average velocity of the mixture in the positive z direction

𝐷𝐴𝐵 - mutual diffusion coefficient of A and B 𝑑𝐶𝐴

𝑑𝑥

⁄ - concentration gradient of A, which is negative in the direction of ordinary molecular diffusion 𝑁𝐴 - moles of A diffusing per unit time

𝐴 - cross sectional area 𝐶𝐴 - concentration of A

𝑍 - distance of diffusion

Fick’s Law of diffusion is based on the following observations:

1. Mass transfer by ordinary molecular diffusion occurs because of a concentration difference or gradient; that is, a species diffuses in the direction of decreasing concentration

2. The mass transfer rate is proportional to the area normal to the direction of mass transfer and not to the volume of the mixture. Thus, the rate can be expressed as a flux

3. Mass transfer stops when the concentration is uniform

PREDICTION OF DIFFUSIVITY For Gases:

1. Chapman and Enskog Equation 𝐷𝐴𝐵 = 1.8583 𝑥 10−7 𝑇3/2 𝑃𝜎𝐴𝐵2Ω𝐷,𝐴𝐵 ( 1 𝑀𝐴 + 1 𝑀𝐵 ) 1/2 Where: 𝐷𝐴𝐵 – diffusivity, m2/s 𝑇 - temperature, K

𝑃 – absolute pressure, atm

𝑀𝐴, 𝑀𝐵 - molecular weights of A and B, in kg/kmol 𝜎𝐴𝐵 - average collision diameter, Å

𝛺𝐷,𝐴𝐵 - collision integral based on the Lennard-Jones potential

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2. Fuller Method (Geankoplis) 𝐷𝐴𝐵= 1 𝑥 10−7 𝑇1.75 𝑃[(∑ 𝑢𝐴)1/3+ (∑ 𝑢𝐵)1/3]2 ( 1 𝑀𝐴 + 1 𝑀𝐵 ) 1/2 Where:

∑ 𝑢𝑖 - sum of structural volume increments (table 6.2-2, Geankoplis) 3. Schmidt Number 𝑆𝑐 = 𝜇 𝜌𝐷𝐴𝐵 For liquids: 1. Stokes-Einstein Equation 𝐷𝐴𝐵 = 7.32 𝑥 10−16_𝑇 𝑟𝑜𝜇 Where: 𝐷𝐴𝐵 - diffusivity, cm2/s 𝑇 - absolute temperature, K 𝑟𝑜 - molecular radius, cm 𝜇 - viscosity, cP 2. Wilke-Chang Equation 𝐷𝐴𝐵 = 7.4 𝑥 10−8 (𝜓𝐵𝑀𝐵)1/2𝑇 𝜇𝑉𝐴0.6 Where:

𝑉𝐴 - molar volume of solute as liquid at its normal boiling point, cm3/mol

𝜓𝐵- association parameter for solvent (water = 2.6; methanol = 1.9; ethanol = 1.5; benzene = 1.0; heptanes = 1.0)

3. Nernst Equation (for dilute solutions of completely ionized univalent electrolytes 𝐷𝐴𝐵 = 2𝑅𝑇 (1 𝜆+0 + 1 𝜆0−) 𝐹𝑎 2 Where:

𝐹𝑎 - Faraday constant, 96,500 coul/gequiv 𝑅 - gas constant, 8.314 J/K·gmol

𝜆0+, 𝜆−0 - limiting (zero concentration) ionic conductances, A/cm2·(V/cm)·(gequiv/cm3), Table 17.1 Unit Operations by McCabe and Smith

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MOLECULAR DIFFUSION IN GASES

1. EQUIMOLAR COUNTER-DIFFUSION

𝐽

∗ 𝐴

= −𝐽

∗𝐵

𝐽

_𝐴∗

=

𝑁

𝐴

= −𝐷

𝐴𝐵

𝑑𝐶

𝐴

𝑑𝑍

𝐶

𝐴

=

𝑛

_𝐴

𝑉

=

𝑝

_𝐴

𝑅𝑇

𝐽

𝐴∗

= −

𝐷

_𝐴𝐵

𝑅𝑇

𝑑𝑝

_𝐴

𝑑𝑍

Where:

𝐽

_𝐴∗

- diffusion flux relative to the moving fluid

2. GENERAL CASE FOR DIFFUSION OF GASES A AND B PLUS CONVECTION

In terms of velocity of diffusion of A to the right

𝐽

_𝐴∗

= 𝑢

_𝐴𝑑

𝐶

_𝐴

If the whole fluid is moving in bulk or convective to the right

𝑢

_𝐴

= 𝑢

_𝐴𝑑

+ 𝑢

_𝑀

𝐶

_𝐴

𝑢

_𝐴

= 𝐶

_𝐴

𝑢

_𝐴𝑑

+ 𝐶

_𝐴

𝑢

_𝑀

𝐽

𝐴

= 𝐽

𝐴∗

+ 𝐶

𝐴

𝑢

𝑀

Let 𝑁 = total convective flux of the whole stream, relative to the stationary point, then

𝐽 = 𝐶𝑢

_𝑀

= 𝐽

_𝐴

+ 𝐽

_𝐵

𝑢

_𝑀

=

𝐽

𝐴

+ 𝐽

𝐵

𝐶

𝐽

_𝐴

= 𝐽

_𝐴∗

+

𝐶

𝐴

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𝐽

𝐴

= −𝐷

𝐴𝐵

𝑑𝐶

_𝐴

𝑑𝑍

+

𝐶

_𝐴

𝐶

(𝐽

𝐴

+ 𝐽

𝐵

)

For equimolar counter-diffusion,

𝐽

_𝐴

= −𝐽

_𝐵

𝐽

𝐴

=

𝑁

_𝐴

𝐴

= −𝐷

𝐴𝐵

𝑑𝐶

_𝐴

𝑑𝑍

= −

𝐷

_𝐴𝐵

𝑅𝑇

𝑑𝑝

_𝐴

𝑑𝑍

3. SPECIAL CASE FOR A DIFFUSING THROUGH STAGNANT, NON-DIFFUSING B

In this case,

𝐽

_𝐵

= 0

𝐽

𝐴

= −𝐷

𝐴𝐵

𝑑𝐶

_𝐴

𝑑𝑍

+

𝐶

_𝐴

𝐶

𝐽

𝐴

𝐽

_𝐴

(1 −

𝐶

𝐴

𝐶

) = −𝐷

𝐴𝐵

𝑑𝐶

_𝐴

𝑑𝑍

𝐽

𝐴

(

𝐶 − 𝐶

_𝐴

𝐶

) = −𝐷

𝐴𝐵

𝑑𝐶

_𝐴

𝑑𝑍

𝐽

𝐴

[

𝑃

𝑅𝑇 −

𝑝

_𝐴

𝑅𝑇

𝑃

𝑅𝑇

] = −

𝐷

𝐴𝐵

𝑅𝑇

𝑑𝑝

_𝐴

𝑑𝑍

(7)

𝐽

𝐴

(

𝑃 − 𝑝

_𝐴

𝑃

) = −

𝐷

_𝐴𝐵

𝑅𝑇

𝑑𝑝

_𝐴

𝑑𝑍

𝐽

_𝐴

=

𝑁

𝐴

= −

𝐷

𝐴𝐵

𝑃

𝑅𝑇

(

1 𝑃 − 𝑝

_𝐴

)

𝑑𝑝

𝐴

𝑑𝑍

MOLECULAR DIFFUSION IN LIQUIDS

4. EQUIMOLAR COUNTER-DIFFUSION

𝐽

𝐴

=

𝑁

𝐴

= −𝐷

𝐴𝐵

𝑑𝐶

𝐴

𝑑𝑧

𝐶

_𝐴

= 𝐶

_𝑎𝑣

𝑥

_𝐴

𝐶

_𝑎𝑣

= (

𝜌

𝑀

)

𝑎𝑣

=

1

2 (

𝜌

1

𝑀

1

+

𝜌

2

𝑀

2

)

Where:

𝐶

_𝑎𝑣

- average total concentration of liquids A and B, mol/vol

𝑥

_𝐴

- mol fraction of A

5. DIFFUSION OF A THROUGH A NON-DIFFUSING B

𝐽

_𝐴

= −𝐷

_𝐴𝐵

𝑑𝐶

𝐴

𝑑𝑧

+

𝐶

𝐴

𝐶

_𝑎𝑣

(𝐽

𝐴

+ 𝐽

𝐵

)

𝐽

𝐵

= 0

𝐽

_𝐴

=

𝑁

𝐴

= −𝐷

𝐴𝐵

𝑑𝐶

𝐴

𝑑𝑧

+

𝐶

𝐴

𝐶

𝑎𝑣

𝐽

_𝐴

𝐽

_𝐴

(1 +

𝐶

𝐴

𝐶

_𝑎𝑣

) = −𝐷

𝐴𝐵

𝑑𝐶

𝐴

𝑑𝑧

𝐶

_𝐴

= 𝐶

_𝑎𝑣

𝑥

_𝐴

𝑑𝐶

𝐴

= 𝐶

𝑎𝑣

𝑑𝑥

𝐴

𝐽

_𝐴

(1 + 𝑥

𝐴

) = −𝐷

𝐴𝐵

𝐶

𝑎𝑣

𝑑𝑥

𝐴

𝑑𝑧

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𝐽

_𝐴

∫ 𝑑𝑧

𝑍2 𝑍1

= −𝐷

_𝐴𝐵

𝐶

_𝑎𝑣

∫

𝑑𝑥

𝐴

1 + 𝑥

_𝐴 𝑥𝐴2 𝑥𝐴1

𝐽

_𝐴

(𝑍

₂

− 𝑍

₁

) = −𝐷

_𝐴𝐵

𝐶

_𝑎𝑣

ln

1 + 𝑥

𝐴2

1 + 𝑥

_𝐴1

𝐽

_𝐴

=

𝑁

𝐴

=

−𝐷

𝐴𝐵

𝐶

𝑎𝑣

(𝑍

2

− 𝑍

1

)

ln

1 + 𝑥

𝐴2

1 + 𝑥

_𝐴1

𝑥

_𝑙𝑚

=

(𝑥

𝐴1

− 𝑥

𝐵1

) − (𝑥

𝐴2

− 𝑥

𝐵2

)

ln

𝑥

_𝑥

𝐴1

− 𝑥

𝐵1 𝐴2

− 𝑥

𝐵2

For dilute solution,

𝑥

_𝐴

≪≪ 𝑥

_𝐵

𝑥

𝑙𝑚

=

(0 − 𝑥

_𝐵1

) − (0 − 𝑥

_𝐵2

)

ln

0 − 𝑥

_{0 − 𝑥}

𝐵1 𝐵2

𝑥

_𝑙𝑚

=

𝑥

𝐵2

− 𝑥

𝐵1

ln

𝑥

_𝑥

𝐵2 𝐵1

ln

𝑥

𝐵2

𝑥

𝐵1

=

𝑥

𝐵2

− 𝑥

𝐵1

𝑥

𝑙𝑚

𝐽

𝐴

=

𝑁

_𝐴

𝐴

=

−𝐷

_𝐴𝐵

𝐶

_𝑎𝑣

(𝑍

2

− 𝑍

1

)

ln

1 + 𝑥

𝐴2

1 + 𝑥

𝐴1

𝐽

_𝐴

=

𝑁

𝐴

=

−𝐷

_𝐴𝐵

𝐶

_𝑎𝑣

(𝑍

₂

− 𝑍

₁

)

ln

𝑥

_𝐵2

𝑥

_𝐵1

𝐽

_𝐴

=

𝑁

𝐴

=

−𝐷

𝐴𝐵

𝐶

𝑎𝑣

(𝑍

2

− 𝑍

1

)

(

𝑥

𝐵2

− 𝑥

𝐵1

𝑥

_𝑙𝑚

)

For dilute solutions,

𝑥

𝑙𝑚

= 1

𝐽

𝐴

=

𝑁

_𝐴

𝐴

= −

𝐷

_𝐴𝐵

𝐶

_𝑎𝑣

(𝑍

2

− 𝑍

1

)

(𝑥

𝐵2

− 𝑥

𝐵1

)

𝐽

_𝐴

=

𝑁

𝐴

= −

𝐷

_𝐴𝐵

𝐶

_𝑎𝑣

(𝑍

₂

− 𝑍

₁

)

(𝑥

𝐴2

− 𝑥

𝐴1

)

(9)

𝐽

𝐴

=

𝑁

_𝐴

𝐴

= −𝐷

𝐴𝐵

𝐶

𝑎𝑣

(𝑥

_𝐴2

− 𝑥

_𝐴1

)

(𝑍

2

− 𝑍

1

)

= −𝐷

𝐴𝐵

(𝐶

_𝐴2

− 𝐶

_𝐴1

)

(𝑍

2

− 𝑍

1

)

MOLECULAR DIFFUSION IN SOLIDS

𝑱

_𝑨

=

𝑵

𝑨

= −𝑫

𝑨𝑩

𝒅𝑪

_𝑨

𝒅𝒛

6. DIFFUSION EQUATIONS USING GAS SOLUBILITY IN A SOLID SURFACE

𝐶

_𝐴

=

𝑆𝑝

𝐴

𝑉̅

Where: 𝑉̅ is the molar volume of gas at STP, 22.4 m

3

_{/kmol; 𝑆 – solubility of solute}

gas in a solid

7. DIFFUSION IN SOLIDS USING PERMEABILITY EQUATIONS

𝑃

𝑀

= 𝐷

𝐴𝐵

𝑆

Where: 𝑃

𝑀

- solute gas permeability, or the volume of solute gas at STP diffusing per

second per unit cross sectional area through a solid under a pressure difference of 1

atm pressure

8. DIFFUSION IN POROUS SOLIDS THAT DEPENDS ON STRUCTURE

𝐽

_𝐴

=

𝑁

𝐴

=

𝜀𝐷

𝐴𝐵

(𝐶

𝐴1

− 𝐶

𝐴2

)

𝜏(𝑍

₂

− 𝑍

₁

)

Where:

𝜀 - open void fraction; 𝜏 -

tortousity (factor that corrects for

the path longer than the (𝑍

2

− 𝑍

1

)

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MASS TRANSFER COEFFICIENTS

MASS TRANSFER COEFFICIENT

It is defined as the rate of mass transfer per unit area per unit concentration difference

and is usually based on equal molal flows

𝑘

_𝐶

=

𝐽

𝐴

𝐶

_𝐴𝑖

− 𝐶

_𝐴

𝑘

_𝑦

=

𝐽

𝐴

𝑦

_𝐴𝑖

− 𝑦

_𝐴

𝑘

𝐶

= [

𝐷

_𝐴𝐵

(𝐶

_𝐴𝑖

− 𝐶

_𝐴

)

𝑍

2

− 𝑍

1

] (

1 𝐶

𝐴𝑖

− 𝐶

𝐴

) =

𝐷

𝐴𝐵

∆𝑍

Sherwood Number

𝑆ℎ =

𝑘

𝐶

𝐷

_𝐴𝐵

Reynold’s Number

𝑅𝑒 =

𝐷𝑢𝜌

𝜇

=

𝐷𝐺

𝜇

Graetz Number

𝐺𝑧 =

𝑚̇

𝐷

_𝐴𝐵

𝐿𝜌

Schmidt Number

𝑆𝑐 =

𝜇

𝜌𝐷

_𝐴𝐵

Peclet Number

𝑃𝑒 = 𝑅𝑒 𝑥 𝑆𝑐

MASS TRANSFER WITH FLOW INSIDE PIPES

- Prediction of the internal mass-transfer resistance for separation processes using