1. Principles of Diffusion and Mass Transfer Between Phases
Chapter 6
Principles of Diffusion and Mass Transfer Between Phases

2. 1.THEORY OF DIFFUSION
Diffusion is the movement, under the influence of a physical stimulus, of an individual component through a mixture
The most common cause of diffusion is a concentration gradient of the diffusing component.
E.g., The process of dissolution of ammonia into water: (1)A concentration gradient in the gas phase causes ammonia to diffuse to the gas-liquid interface; (2)Ammonia dissolves in the interface; (3)A gradient in the liquid phase causes ammonia to diffuse into the bulk liquid.

3. A concentration gradient tends to move the component in such a direction as to equalize concentrations and destroy the gradient.
If the two phases are in equilibrium with each other, diffusion, or mass transfer fluxes is equal to zero.
Other causes of diffusion: activity gradient (reverse osmosis); temperature gradient (thermal diffusion); application of an external force field (forced diffusion, e.g., centrifuge, etc).
Two kinds of diffusion caused by concentration gradient : molecular diffusion（分子扩散） and eddy diffusion（涡流扩散）. [e.g., diffusion process of ink in the stagnant or agitated water…]

4. Mass transfer driving forces
E.g., absorption or stripping process: Gas-liquid phases are not in equilibrium with each other.
Absorption column
Driving force
Driving forces:

5. Absorption column

6. Driving force
Absorption column
Driving forces:
Question: Can we use (p-c) or (y-x) as mass transfer driving force? Compare mass transfer driving forces with heat transfer driving force?

7. (1)Comparison of diffusion and heat transfer
Momentum transfer
Heat transfer
Mass transfer

8. (2)Diffusion quantities（扩散通量）
1.Velocity u, length/time.
2.Flux across a plane N, mol/area•time.
3.Flux relative to a plane of zero velocity J, mol/area•time.
4.Concentration c and molar density M, mol/volume (mole fraction may also be used).
5.Concentration gradient dc/db, where b is the length of the path perpendicular to the area across which diffusion is occurring.
Appropriate subscripts are used when needed.

9. (3)Velocities in diffusion（扩散速率）
Velocity without qualification refers to the velocity relative to the interface between the phases and is that apparent to an observer at rest with respect to the interface.

10. (4)Molal flow rate, velocity , and flux
Where
M=molar density of the mixture
N= total molar flux in a direction perpendicular to a stationary plane
u0= volumetric average velocity

11. For components A and B crossing a stationary plane, the molal fluxes are
By definition there is no net volumetric flow across the reference plane moving at the volume-average velocity u0, although in some cases there is a net molar flow or a net mass flow.
=Diffusion flux of component A in the mixture

12. =Diffusion flux of component B in the mixture
Fick’s first law of diffusion for a binary mixture（二元混合物）:
=diffusivity of component A in its mixture with component B, m2/s
=molar concentration gradient, mol/m4

13. (5)Relations between diffusivities
For ideal gases, and for diffusion of A and B in a gas at constant temperature and pressure,

14. For liquid with same mass density [kg/m3],
For no volume flow across the reference plane, the sum of the volumetric flows due to diffusion is zero. The volumetric flow rate is the molar flow rate times the molar volume M/ and
Substituting(代入) Eq.(17.13) into Eq.(17.14) gives

15. A common form of the diffusion equation gives the total flux of component A:
Where Dv=volumetric diffusivity, m2/h, cm2/s
For gases,
For liquid, if M =Constant,

16. (6)Interpretation of diffusion equations扩散方程
The vector nature of the fluxes and concentration gradients must be understood, since these quantities are characterized by directions and magnitudes.
The sign of the gradient is opposite to the direction of the diffusion flux, since diffusion is in the direction of lower concentrations. 物质A的浓度梯度在方向上的变化与扩散通量相反，表示扩散是沿物质浓度降低方向进行的

17. (7)Equimolal diffusion(等分子扩散)
Zero convective flow and equimolal counterdiffusion（ 等分子反扩散 ） of A and B, as occurs in the diffusive mixing of two gases and in the diffusion of A and B in the vapor phase for distillations that have constant molal overflow.

18. Fig.17.1(a) Component A and B diffusing at same molal (equimolal) rates in opposite directions [Like the case of diffusion of A and B in the vapor phase for distillations that have constant molal overflow].
Note that for equimolal diffusion, NA=JA.

19. Assuming a constant flux NA and zero total flux (N=0), integrating Eq
Assuming a constant flux NA and zero total flux (N=0), integrating Eq.(17.17) over a film thickness BT, or

20. The concentration gradient for A is linear in the film, and the gradient for B has the same magnitude but the opposite sign, as shown in Figure 17.1a.
[From Eq.(17.17),

21. (8)One-component mass transfer (one-way diffusion)
Fig.17.1(b) Component A diffusing, component B stationary with respect to interface. [Like the case of diffusion of solute A from gas phase into liquid phase in absorption process.]

22. (8)One-component mass transfer (one-way diffusion)
When only component A is being transferred, the total flux to or away from the interface N is the same as NA, and Eq.(17.17) becomes

23. Rearranging and integrating, we have
Equation(17.24) can be rearranged by using the logarithmic mean of 1-yA for easier comparison with Eq.(17.19) for equimolal diffusion. The logarithmic mean of 1-yA is

24. Combining Eqs(17.24) and (17.25) gives
[Comparing one-way diffusion in the Chinese textbook,

25. Comparing Eq. (17. 26) with Eq. (17
Comparing Eq.(17.26) with Eq.(17.19),the flux of component A for a given concentration difference is therefore greater for one-way diffusion than for equimolal diffusion.
[Example17.1.]

26. For diffusion in liquid,

27. 2.PREDICTION OF DIFFUSIVITIES
Diffusivities are physical properties of fluids.
Diffusivities are best estimated by experimental measurements, or from published correlations.
The factors influencing diffusivities are temperature, pressure, and compositions for a given fluid.
Fick’s first law of diffusion for a binary mixture:
(1)Diffusion in gases

28. Assume concentrations of component A in the two layers with distance of the molecular mean free path of a binary mixture are cA1 and cA2, respectively, the diffusion flux is
Therefore,
Where
=average molecular velocity, m/s

29. A more rigorous approach based on modern kinetic theory allows for the different sizes and velocities of the molecules and the mutual interactions as they approach one another [Eq.(17.28) for binary diffusion.]
The collision integral D decreases with increasing temperature, which makes DAB increase with more than the 1.5 power of the absolute temperature.
For diffusion in air,

30. In general, influence of concentrations for diffusion in gases can be neglected, and
[*Diffusion in small pores](自学)
Example [p.520] (自学)

31. (2)Diffusion in liquids
Diffusivities in liquids are generally 4 to 5 orders of magnitude smaller than in gases at atmospheric pressure.
Influence of pressure for diffusion in liquids can be neglected.
Diffusivities for dilute liquid solutions can be calculated from Eq.(17.31)[Empirical correlation].

32. Other empirical correlations for diffusivities:
For dilute aqueous solutions of non-electrolytes, using Eq.(17.32).（自学）
For dilute solutions of completely ionized univalent electrolytes,using Nernst equation(17.33). （自学）
(3)Schmidt number Sc
Sc is analogous to the Prandtl number.

33. For gases, Sc is independent of pressure when the ideal gas law applies, since the viscosity is independent of pressure, and the effects of pressure on  and Dv cancel. Temperature has only a slight effect on Sc because  and Dv both change with about T0.7~0.8.
For liquids, Sc decreases markedly with increasing temperature because of the decreasing viscosity and the increase in the diffusivity.
Unlike the case for binary gas mixtures the diffusion coefficient for a dilute solution of A and B is not the same for a dilute solution of B in A. i.e., DAB≠DBA [Comparing with eq.(17.15)?]

34. EXAMPLE [p.522]
From eq.(17.31), it is apparent that unlike the case for binary gas mixtures, the diffusion coefficient for a dilute solution of A and B is not the same for a dilute solution of B in A.
But, from EXAMPLE 17.3., the diffusivities of benzene in toluene and toluene in benzene have only a slight difference, and in this case the conclusion of Eq.(17.15) DAB=DBA is still effective

35. (4)Turbulent diffusion
In a turbulent stream the moving eddies transport matter from one location to another, just as they transport momentum and heat energy.
Where
=molal flux of A, relative to phase as a whole, caused by turbulent action
=eddy diffusivity
The total molal flux, relative to the entire phase, becomes

36. The eddy diffusivity is not a parameter of physical property, it depends on the fluid properties but also on the velocity and position in the flowing stream.