Molecular diffusion in gases

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) [email protected]

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: 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. 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): ρ A of species A in a multi component mixture is defined as the mass of A per unit volume of the mixture; kg/m³ Molar concentration : C A 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 Mass Fraction : m̽ A is defined as the ration of mass concentration of species A to the total mass concentration of the mixture. Mole Fraction : x A is defined as the ration of molar concentration of species A to the total molar concentration of the mixture. Velocities: Mass average velocity ( v mass or v ): m̽ A v A + m̽ B v B = b. Molar average velocity ( v molar or v̽ ): x A v A + x B v B = c. Mass diffusion velocity of component A (or v Ad ) = v A – v mass d. Mass diffusion velocity of component A (or v Ad ) = v A – v molar ρ A v A + ρ B v B ρ A + ρ B C A v A + C B v B C A + C B (where, m̽ = mass fraction of species , x A = mole fraction of species A), v A and v B are absolute velocities of species A and B, respectively)

5. 6. Flux in mass transfer: r ate of mass transfer per unit area normal to the direction of flow For species A of the multi-component mixture: Absolute flux (N A )= C A v A (kg-mol A/s-m²); total flux of A relative to the stationary point Bulk motion flux = C A v * or C A v mass ;bulk flux of A relative to the stationary point Diffusion flux (J* A )= C A v Ad ;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, J AZ = molar flux of species A in Z direction; kg-mol of A / m²-s D AB = molecular diffusivity of the molecule A in B ; m²/s C A = concentration of species A; mol/m³ or kg-mol/m³ Z = distance of diffusion; m 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, D AB = D BA * * b. Diffusion plus convection: N =C v*= N A +N B = 0 X 1 2 e.g., Binary distillation J ̽ A = v Ad C A Mathematically the velocity of species A relative to the stationary point can be expressed as: v A = v Ad + v* (multiplying CA both side, we get following equation) and, C A v A = v Ad C A + v*C A (Absolute Flux = diffusion flux + Bulk motion flux) or, N A = J ̽ A + C A v * = (C A /C)(N A +N B ) - D AB ( dC A / dz ) or, N B = J ̽ B + C B v * = (C B /C)(N A +N B ) - D BA ( dC B / dz )

7. c. Diffusion through stagnant non diffusing B: e.g., evaporation of liquid acetone: N B =0 and p A2 =0 Liquid Acetone N A = J ̽ A + C A v * = (C A /C)(N A +N B ) - D AB ( dC A / dz ) Product: (xyz) after putting rearranging: N A = D AB P (p A1 ) RT (Z 2 -Z 1 ) P BM t F = ρ A (z F ² - z ²) RT P BM 2M A D AB P (p A1 -p A2 ) 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, N A and J* A is taken as constant in integration. In these the diffusion occur through constant area A through varying distance z.   A for Varying Area  

10. a. Diffusion from sphere For sphere   The mass diffusion of sphere can be defined as:

11. In the above equation r 2 is relatively large as compared to r 1 p A2 =0 at the large distance from the sphere p A1 is the partial pressure of the A Time of diffusion from r 2 to r 1 Equating with change of mass and integrating with proper limit

12. b. Diffusion through tapered tube Rate of input Rate of output Assuming there is no accumulation in the system

13. Integrating with proper limit the diffusion flux can be found as r is the local radius as described below

14. 4. Diffusion Coefficient for gases Experimental determination of diffusion coefficients This part will dealt in the forward slides. 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 m 2 /s for large molecule to 1.0 x 10 -4 m/s for smaller molecule as H 2 . Equation for Diffusion coefficient are given for mainly 3 given situation Derivation uses mean free path For non-polar molecules Lennard-Jones function is used for reasonable solution

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). --1   Where   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.       The average value at equilibrium can be calculated by a material balance from the starting compositions and is 0 at t=0.   Upon substitution of value of in 1 st equation we get       The rate of diffusion of pure A going to is equal to the rate of accumulation in  

18. Rearranging and integrating between t = 0 and t = t, and concentration limits from and we get the final expression in the form as:     Mutual diffusion coefficient can also be calculated in terms of pressure directly from the equation below   Note: - Negative sign here shows the negative ingredient of flow.  

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   The final equation for steady state diffusion is   Where and     Where,  

20. Different Type of Mass Transfer Coefficient Diffusion A through non-diffusing B Flux, N A Mass Transfer Coefficient Unit Diffusion A through non-diffusing B Flux, N A Mass Transfer Coefficient Unit

21. Diffusion A through non-diffusing B Flux, N A Mass Transfer Coefficient Unit Diffusion A through non-diffusing B Flux, N A Mass Transfer Coefficient Unit Conversion Conversion

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