1. 高等輸送二 — 質傳 Lecture 2 Diffusion in concentrated solutions
郭修伯 助理教授

2. Dilute vs. concentrated (solution)
Any mass flux include both convection and diffusion (Maxwell, 1860).
Diffusion causes convection.
Dilute solution: convection caused by diffusion is small and can be neglected.
Concentrated solution: both convection and diffusion have to be considered.
Difference between Heat transfer and Mass transfer
Heat conduction can occur without convection
Diffusion and convection always occur together

3. Total mass transported
Mass transported by convection
Mass transported by diffusion
Total mass transported =
Convective reference velocity
???
Average solute velocity
w.r.t fixed coordinate
Local concentration

4. Convective reference velocity
How to choose its value ?
Our goal: choose va so that va is zero as frequently as possible!
The mass transfer reduced to “diffusion” only.
va can be:
the molar average velocity
good for ideal gases where the molar concentration is constant
the mass average velocity
good for constant-density liquid
the volume average velocity
good for constant-density liquid and for ideal gas
What are those?

5. The temperature and pressure are such that the diffusion coefficient is 0.1 cm2/sec. Find the molar average velocity v*, the mass average velocity v, the volume average velocity v0 at the average concentration in the system.
nitrogen
hydrogen
The volume in this system does not move, so v0 = 0. If the gases are ideal, the molar concentration is constant, so v* = 0.
Diffusion in the thin-film
For nitrogen at an average concentration of 0.5 c
For hydrogen at an average concentration of 0.5 c
Mass fraction of nitrogen
Mass fraction of hydrogen

6. Solute accumulated in volume Az
Fast evaporation by diffusion and convection l z
A mass balance on a differential volume A z gives:
Solute transported out at z + z
Solute transported in at z
Solute accumulated in volume Az =
Dividing A z
z  0

7. contribution of both diffusion and convection and it is constant.
s.s
choose volume average velocity v0
If the solvent vapor is stagnant
Total molar concentration

8. B.C.
Exponential concentration profile
n1 = constant
The diffusion flux is smallest at the bottom of the capillary and rises to a maximum value at the top of the capillary.

9. Concentrated solution
Dilute solution or
Linear concentration profile

10. Solute accumulated in volume Az
Fast Diffusion into a Semiinfinite slab z
l, >>
Fast evaporation by diffusion and convection
A mass balance on a differential volume A z gives:
Solute transported out at z + z
Solute transported in at z
Solute accumulated in volume Az =
Dividing A z
z  0

11. choose volume average velocity v0
independent of z =1
Continuity equation

12. Solvent gas is insoluble

13. B.C.

14. How important is the convection term?z l
The vapor pressure of benzene at 6ºC is about 37 mmHg
The vapor pressure of benzene at 60ºC is about 395 mmHg
Total flux
Diffusion + convection
Diffusion
6 ºC
2 %
60 ºC
40 %

15. General form of the mass balance equationx z y
Input rate through ABCD C G
Input rate through ADHE D
Input rate through ABFE H z B
Output rate through EFGH F y
Output rate through BCGF A E x
Output rate through CDHG
input - output + generation = accumulation

16. Diffusion and the convection term
General equation include the effects of chemical reaction, convection, and concentration-driven diffusion
Combine with Fick’s law

17. Membrane example Fresh water Salt water membrane x y
Find differential equations for calculating the drop in flux caused by the concentration polarization (i.e., by the salt accumulation near the membrane surface)
s.s. 2D
no reaction
Responsible to concentration polarization
Usually much smaller than the convection term

18. Spinning disc example
Solute from dissolving disc
flow
The dissolution rate is diffusion-controlled. Calculate the rate at which the disc dissolves.
s.s.
no reaction
Angularly symmetric
Angularly symmetric
Disc infinitely wide
Disc infinitely wide

19. B.C.
Levich, 1962

20. Reynolds number
Schmidt number
Independent of the disc radius: constant flux

21. Mass transfer versus Heat transfer
Diffusion-induced convection; chemical reaction only for mass transfer
Radiation only for heat transfer
J. Crank, (1975) The Mathematics of Diffusion, 2nd ed. Oxford: Clarendon Press
(include reactions)
H.S. Carslaw, and J. C. Jaeger (1986) The Conduction of Heat in Solids, 2nd ed. Oxford: Clarendon Press
(a more complete selection of boundary conditions)

22. Diffusion through a polymer film
Diaphragm（隔板） cell
Initial pressure zero
Measure the ethylene concentration in the upper compartment as a function of time.
Polymer film
Initial pressure one atmosphere
From mass balance and Fick’s law:
Boundary conditions:
l: film’s thickness H: Henry’s law coefficient

23. Mole balance on the top compartment:
Crank, (1975)
Mole balance on the top compartment:
t = 0, p = 0

24. Large time
From the intercept and the slope, we can obtain the equilibrium Henry’s law coefficient, H, and the diffusion coefficient, D, in a single experiment
Pressure
Time

25. A dissolving pill
Estimate the time required to produce a steady flux of drug pill in the gut (腸).
Assumption: the drug’s dissolution is controlled by diffusion into the stagnant contents of the gut.(i.e. ,
The dissolution is diffusion-controlled
The surrounding s are stagnant )
The mass balance on a spherical shell:
Diffusion control
Boundary conditions:

26. Crank, (1975)
Carslaw and Jaeger (1986)
The flux in a dilute solution:
The time to reach steady flux:
Time ~ 80h is much longer than the experimental result (i.e., 10 min)
Because: free convection driven by the density difference caused by the dissolution

27. Effective diffusion coefficients in a porous catalyst pellet
A porous catalyst pellet containing a dilute gaseous solution. Determine the effective diffusion of solute by dropping this pellet into a small, well stirred bath of a solvent gas and measuring how fast the solute appears in this bath.
A mass balance on a spherical pellet:
Diffusion control
Boundary conditions:
Bath concentration

28. A mass balance on the solute in the bath of volume VB
t = 0, C1 = 0
Carslaw and Jaeger (1986)
Crank, (1975)
Void fraction in the sphere