Molecular diffusion in gasses (Transfer Process)

1. Molecular diffusion in gases
Part- I:
1. Introduction
2. Different types of molecular diffusion
in gases
Part-II:
3. Diffusion through varying cross
sectional area
4. Diffusion coefficient of gases
Part- III:
5. Experimental determination of
diffusion coefficient for gases
6. Multicomponent diffusion of gases
7. Mass Transfer Coefficient
Transfer Process
Presented by : Prakash Kumar
(Ph.D. Research Scholar, AGFE,
IIT-Kharagpur, India)
prakashfoodtech@gmail.com

2. Part-I
1. Introduction
2. Different types of molecular diffusion
in gases

3. 1. Introduction
 Mass Transfer: Transfer of mass as result of the species concentration difference in a system
or mixture. e.g., Distillation, absorption, drying, adsorption, membrane separation etc.
Gas A Gas B
 Modes of Mass transfer:
a. Diffusion: It occurs due to concentration, temperature or pressure gradient.
e.g., Fragrance of aerosols, dissipation of smoke, in cells, etc.
b. Convection: Mass transfer between a moving fluid and a surface, or between two
relatively immiscible moving fluids.
e.g., Evaporation, Distillation, etc.
c. Combination of both (diffusion & convection): In this there is simultaneous action of
diffusion and convection.
e.g., Mixing of water vapour with air during evaporation, plume of smoke, etc.

4.  Few terminologies:
1. Mass concentration (mass density): ρA of species A in a multi component mixture is
defined as the mass of A per unit volume of the mixture; kg/m³
2. Molar concentration: CA of species A is defined as the number of moles of species A per
unit volume of the mixture; mole/m³ or kg- mole/m³
3. Mass Fraction: m̽ A is defined as the ration of mass concentration of species A to the total
mass concentration of the mixture.
4. Mole Fraction: xA is defined as the ration of molar concentration of species A to the total
molar concentration of the mixture.
5. Velocities:
a. Mass average velocity ( vmass or v ): m̽ AvA + m̽ BvB =
b. Molar average velocity ( vmolar or v̽ ): xAvA + xBvB =
c. Mass diffusion velocity of component A (or vAd) = vA – vmass
d. Mass diffusion velocity of component A (or vAd) = vA – vmolar
ρA vA + ρBvB
ρA + ρB
CA vA + CBvB
CA + CB
(where, m̽ = mass fraction of species , xA = mole fraction of species A), vA and vB are absolute velocities of species A
and B, respectively)

5. 6. Flux in mass transfer: rate of mass transfer per unit area normal to the direction of flow
For species A of the multi-component mixture:
a. Absolute flux (NA)= CA vA (kg-mol A/s-m²); total flux of A relative to the stationary point
b. Bulk motion flux = CAv* or CA vmass ;bulk flux of A relative to the stationary point
c. Diffusion flux (J*A )= CA vAd ;diffusion flux relative to the moving fluid
Relationship between a, b, and c : Absolute Flux = diffusion flux + Bulk motion flux
 Fick’s Law: (considering binary mixture)
1. Fick’s first law: (one dimensional diffusion)
where,
JAZ = molar flux of species A in Z direction; kg-mol of A / m²-s
DAB = molecular diffusivity of the molecule A in B ; m²/s
CA = concentration of species A; mol/m³ or kg-mol/m³
Z = distance of diffusion; m
2. Fick’s second law:
(change in concentration of diffusant with time at any distance)
Adolf Eugen Fick
(1829-1901)

6. 2. Different types of molecular diffusion in gas
a. Equimolar counter diffusion:
J ̽A = - J ̽B
and, DAB= DBA
*
*
b. Diffusion plus convection:
N =C v*= NA +NB= 0
X1 2
e.g., Binary distillation
J ̽A = vAdCA
Mathematically the velocity of species A relative to the stationary point can be expressed as:
vA = vAd + v* (multiplying CA both side, we get following equation)
and, CAvA = vAdCA + v*CA (Absolute Flux = diffusion flux + Bulk motion flux)
or, NA = J ̽A + CAv* = (CA/C)(NA+NB) - DAB (dCA/dz)
or, NB = J ̽B + CBv* = (CB/C)(NA+NB) - DBA (dCB/dz)

7. c. Diffusion through stagnant non diffusing B:
e.g., evaporation of liquid acetone: NB=0 and pA2=0
Liquid
Acetone
NA = J ̽A + CAv* = (CA/C)(NA+NB) - DAB (dCA/dz)
Product: (xyz)
after putting rearranging:
NA = DAB P (pA1)
RT (Z2-Z1) PBM
tF = ρA (zF² - z0²) RT PBM
2MA DAB P (pA1-pA2)
Application of diffusion principle in developing ANN (Artificial Neural Network):
Source: http://dx.doi.org/10.1136/gutjnl-2019-320273

8. Part-II
3. Diffusion through varying cross
sectional area
4. Diffusion coefficient of gases

9. 3. Diffusion of A through Varying Cross-Sectional Area
In the above Steady State Diffusion, NA and J*A is taken as constant in integration. In
these the diffusion occur through constant area A through varying distance z.
𝑁𝐴 =
𝑁𝐴
𝐴
𝑁𝐴 is the kg moles of A diffusing per second, it is constant
A for Varying Area

10. a. Diffusion from sphere
For sphere
𝐴 = 4𝜋𝑟2
2
1 2
12
1
4 (1 )
4
A A AB A
A
A
A AB A A
A
BM
N N D dp
N
pA r RT dr
P
N D P p p
N
r RTr p


   


  
The mass diffusion of sphere can be
defined as:

11. In the above equation
• r2 is relatively large as compared to r1
• pA2=0 at the large distance from the sphere
• pA1 is the partial pressure of the A
Time of diffusion from r2 to r1
1 2
1
1
2
1
1 22 ( )
AB A A A
A
BM A
A p BM
f
AB A A
D P p p dr
N
RTr P M dt
r RT P
t
MD p p P



 


Equating with change of mass and integrating with proper limit

12. b. Diffusion through tapered tube
Rate of input
2
( ) A z
z r N
Rate of output
2
( ) A z z
z z r N 
  
Assuming there is no accumulation
in the system
2 2
( ) ( ) 0A z A z z
r N r N  
 
2 AB A
A
D dp
N r
RT dz
 

13. 1 2
1 2( )AB
A A A
D rr
N p p
RT L

 
Integrating with proper limit the diffusion flux can be found as
r is the local radius as described below
2 1
1 ( )
r r
r r z
L

 
2
1 0
2 1
1
A
A
p LA
Ap
AB
N RT dz
dp
r rD
r z
L

 
 
  
 
 

14. 4. Diffusion Coefficient for gases
1. Experimental determination of diffusion coefficients
This part will dealt in the forward slides.
2. Experimental Diffusivity Data
Various data has been tabulated by Perry and Green and Reid et al., the values
ranges from 0.05 x 10-4 m2/s for large molecule to 1.0 x 10-4 m/s for smaller
molecule as H2
.
3. Equation for Diffusion coefficient are given for mainly 3 given situation
a. Derivation uses mean free path
b. For non-polar molecules Lennard-Jones function is used for reasonable
solution
1
3
ABD u
1/2
7 3/2
2
.
1.8583*10 1 1
AB
AB D AB A B
T
D
P M M

 
  
  
7 1.75 1/2
1/3 1/3 2
1.00*10 (1/ 1/ )
[( ) ( ) ]
A B
AB
A B
T M M
D
P  



 

15. Part-III
5. Experimental determination of
diffusion coefficient for gases
6. Multicomponent diffusion of gases
7. Mass Transfer Coefficient

16. 5. Experimental determination of diffusion coefficient for gases
Assumptions
• Negligible capillary volume
• Each bulb is always at a uniform concentration
• Pseudo-steady state diffusion through the capillary
As the concentration in the bulbs change a little, a new steady state of diffusion is achieved.
Two different pure gases having equal pressure are filled in separate sections with a partition
valve in the capillary tube, allowed to diffuse for a time t (equimolar diffusion).
𝐽 𝐴
∗
= −𝐷𝐴𝐵
𝑑𝑐
𝑑𝑡
=
𝐷𝐴𝐵 𝐶2 − 𝐶1
𝐿
--1
Where 𝐶1, 𝐶2 = concentration of A in V1 and V2 at time t.
The parameters to be measured from the experiment :
 Initial pressure in the vessels.
 Partial pressure of one of the components in the vessel at the end of the experiment
 Time of experiment to be recorded

17. 𝑉1 + 𝑉2 𝐶 𝑎𝑣 = 𝑉1 𝐶1
0
+ 𝑉2 𝐶2
0
− −2
𝑉1 + 𝑉2 𝐶 𝑎𝑣 = 𝑉1 𝑐1 + 𝑉2 𝑐2 − −3 𝐶1 =
𝑉1 + 𝑉2 𝐶 𝑎𝑣 − 𝑉2 𝑐2
𝑉1
− −4
The average value 𝐶 𝑎𝑣 at equilibrium can be calculated by a material balance from
the starting compositions 𝑐1
0
and 𝑐2
0
is 0 at t=0.
Upon substitution of value of 𝑐1 in 1st equation we get
𝑉2
𝑑𝑐2
𝑑𝑡
= −
𝐷𝐴𝐵 𝐶2 −
𝑉1 + 𝑉2 𝐶 𝑎𝑣 − 𝑉2 𝑐2
𝑉1
𝐴
𝐿
𝐴𝐽 𝐴
∗
=
− 𝐷𝐴𝐵 𝐶2 − 𝐶1 𝐴
𝐿
= 𝑉2
𝑑𝑐2
𝑑𝑡
The rate of diffusion of pure A going to V2 is
equal to the rate of accumulation in V2 .

18. Rearranging and integrating between t = 0 and t = t, and concentration limits from
𝐶2
0
and 𝑐2 we get the final expression in the form as:
𝑐 𝑎𝑣 − 𝑐2
𝑐 𝑎𝑣 − 𝑐2
0 = exp
𝐷 𝐴𝐵 𝑉1+𝑉2
𝐿
𝐴
𝑉2 𝑉1
𝑡
Mutual diffusion coefficient can also be calculated in terms of pressure directly from the
equation below
−
𝑑 𝑃𝐴1 − 𝑃𝐴2
𝑑𝑡
=
𝐴 𝑥 𝐷𝐴𝐵 𝑃𝐴1 − 𝑃𝐴2
𝐿
(
1
𝑉1
+
1
𝑉2
Note: - Negative sign here shows the negative ingredient of flow.
𝐷𝐴𝐵 =
𝐿 𝑣1 𝑣2
𝐴 𝑥 𝑡 (𝑣2 + 𝑣1
ln
𝑝𝑡
(𝑝 𝐴1 − 𝑃𝐴2

19. 6. Multicomponent diffusion of gases
Case:- diffusion of A in a gas through a stagnant
non-diffusing mixture of several other gases B, C, D
and others, at a constant total pressure.
Hence 𝑁 𝐵 = 0, 𝑁𝐶 = 0, … . so on
The final equation for steady state diffusion is
𝑵 𝑨 =
𝑫 𝑨𝒎 𝑷
𝑹𝑻 𝒛 𝟐 − 𝒛 𝟏 𝑷𝒊𝑴
(𝑷 𝑨𝟏 − 𝑷 𝑨𝟐
Where𝑷𝒊𝑴 𝐢𝐬 𝐭𝐡𝐞𝐥𝐨𝐠 𝐦𝐞𝐚𝐧 𝐨𝐟𝑷𝒊𝟏 = 𝑷 − 𝑷 𝑨𝟏 and 𝑷𝒊𝟐 = 𝑷 − 𝑷 𝑨𝟐
𝑫 𝑨𝒎 =
𝟏
𝒙 𝑩
′
𝑫 𝑨𝑩
+
𝒙 𝑪
′
𝑫 𝑨𝑪
Where,
𝒙 𝑩
′
=
𝑿 𝑩
𝟏 − 𝒙 𝑨
𝒙 𝑪
′
=
𝑿 𝑪
𝟏 − 𝒙 𝑨

20. Different Type of Mass Transfer Coefficient
Diffusion A through non-diffusing B
Flux, NA Mass Transfer
Coefficient
Unit
𝑘 𝐺(𝑝 𝐴1 − 𝑝 𝐴2
𝑘 𝐺 =
𝐷𝐴𝐵 𝑃
𝑅𝑇𝛿𝑝 𝐵𝑀
𝑚𝑜𝑙
(𝑡𝑖𝑚𝑒 (𝑎𝑟𝑒𝑎 (∆𝑝 𝐴
𝑘 𝑦(𝑦 𝐴1 − 𝑦 𝐴2
𝑘 𝑦 =
𝐷𝐴𝐵 𝑃2
𝑅𝑇𝛿𝑝 𝐵𝑀
𝑚𝑜𝑙
(𝑡𝑖𝑚𝑒 (𝑎𝑟𝑒𝑎 (∆𝑦 𝐴
𝑘 𝑐(𝑐 𝐴1 − 𝑐 𝐴2
𝑘 𝑐 =
𝐷𝐴𝐵 𝑃
𝛿𝑝 𝐵𝑀
𝑚𝑜𝑙
(𝑡𝑖𝑚𝑒 (𝑎𝑟𝑒𝑎 (∆𝑐 𝐴

21. Diffusion A through non-diffusing B
Flux, NA Mass Transfer
Coefficient
Unit
𝑘′ 𝐺(𝑝 𝐴1 − 𝑝 𝐴2
𝑘′ 𝐺 =
𝐷𝐴𝐵
𝑅𝑇𝛿
𝑚𝑜𝑙
(𝑡𝑖𝑚𝑒 (𝑎𝑟𝑒𝑎 (∆𝑝 𝐴
𝑘′ 𝑦(𝑦 𝐴1 − 𝑦 𝐴2
𝑘′ 𝑦 =
𝐷𝐴𝐵 𝑃
𝑅𝑇𝛿
𝑚𝑜𝑙
(𝑡𝑖𝑚𝑒 (𝑎𝑟𝑒𝑎 (∆𝑦 𝐴
𝑘′ 𝑐(𝑐 𝐴1 − 𝑐 𝐴2
𝑘′ 𝑐 =
𝐷𝐴𝐵
𝛿
𝑚𝑜𝑙
(𝑡𝑖𝑚𝑒 (𝑎𝑟𝑒𝑎 (∆𝑐 𝐴
Conversion
𝑘 𝐺 𝑅𝑇 =
𝑅𝑇
𝑃
𝑘 𝑦 = 𝑘 𝑐 𝑘’ 𝑐 = 𝑘’ 𝐺 𝑅𝑇 =
𝑅𝑇
𝑃
𝑘’ 𝑦

22. THANK YOU
Suggestions are welcome…