1. 高等輸送二 — 質傳 Lecture 3 Dispersion
郭修伯 助理教授

2. Dispersion
Caused by the coupling of concentration differences and fluid flow
A form of mixing because of the flow
on a microscopic level, it involves diffusion of molecules
the microscopic dispersion is not understood in detail, but it takes place so rapidly that is is rarely the most important feature of the process.
Compared to diffusion
diffusion is caused by slow Brownian motion

3. Smoke pouring from a smokestack
Affected by wind, weather, and different amounts of smoke
Qualitatively similar to diffusion: the concentration profile is a Gaussian profile.
From our previous calculation on a sharp pulse of solute:
Dispersion coefficient (L2/t)
Speed of wind

4. Dispersion coefficient
Dimension [length2/time]
Independent of chemistry
Not a strong function of molecular weight or structure
carbon monoxide, styrene and smoke are of similar value
A strong function of position
Similar to diffusion coefficient
Unlike diffusion coefficient
Unlike diffusion coefficient
Dispersion
Diffusion
Fick’s law
Heat transfer
Although the mechanisms are very different, we will apply diffusion equations to study dispersion

5. Dispersion in Laminar Flow (Taylor Dispersion)
Solute pulse injected z
Steady solvent flow
Assumption:
the solutions are dilute
the laminar flow is unchanged by the pulse (i.e., the velocity varies only with radius)
mass transfer is by radial diffusion and axial convection. Other mechanisms are neglected.
Radial diffusion
Axial convection

6. A mass balance on the washer-shaped element
Radial diffusion
Axial convection
Boundary conditions:
Fick’s law
Laminar flow

7. Taylor, 1953; Aris, 1956
One solution
(1)
Average across the tube
(2)

8. An overall mass balance in terms of the average concentration:
Note: the radial variations of concentration are small relative to the axial ones
(i.e., )
(3)
Combine (1), (2) and (3)
Peclet number: relative importance of axial convection and radial diffusion

9. Dispersion coefficient is inversely proportional to diffusion coefficient !!

10. The analysis of chromatography
The walls of the tube are coated with a thin film of absorbent (the stationary phase). The injected solute is retarded by absorption in that thin layer. Because the solutes are absorbed to different degrees, they are washed out (the mobile phase) by the bed at different times.
Boundary conditions:
A mass balance on the tube’s contents
Radial diffusion
Axial convection
Axial diffusion

11. A mass balance on the absorbent Boundary conditions:
Thickness of the absorbent layer
Solving all these P.D.E.s together, the Golay equation gives:
Axial diffusion
Retardation in the absorbent layer
Axial convection + radial diffusion
We want to have good separation: reduce Ez v0 and R0

12. Dispersion in turbulent flow
The mass balance in such a flowing system:
By diffusion
By convection
By reaction
In turbulent flow, we assume that both velocity and concentration fluctuate:

13. Dispersion coefficient: How can we find its value?
Changes in reaction rate effected by the fluctuations
Fick’s law:
We define:
Dispersion coefficient: How can we find its value?
weak functions of the different chemical properties of various solute
must be determined by experiments

14. Dispersion coefficient, E
Dimension [length2/time]
Our experience:
We assume:
(i.e., Peclet number for dispersion~ constant)
roughly true from our experimental observation

15. Dispersion of plumes z, stack height x, wind directiony
x, wind direction
The dispersion coefficients in the y and z directions are commonly given as standard deviations  of the Gaussian profile:
For example, a slightly unstable plume, the dispersion parallel to the ground is:

16. At Reynolds numbers above 10,000,
Dispersion coefficients in a pipeline (containing air, water or other fluid)
At Reynolds numbers above 10,000,
the axial dispersion coefficient Ez is approximately:
the radial dispersion coefficient Er is approximately:

17. Dispersion in porous media
Flow through porous materials: filter cake, chromatographic column, reactor filled with solid catalyst. Dispersion coefficient in packed beds are presented as the sum of the contributions of diffusion and flow (Langer et al., 1978):
Common value ~ 2.0, especially for the dispersion of gases in beds of large particles. When the particles are smaller than 0.2 cm, the value of 2 rises.
The reciprocal of a tortuosity.
Common value ~ 0.7
Peclet number for dispersion
Peclet number for diffusion

18. (1) what is the dispersion coefficient at a flow of 1 cm/sec?
Example: A pulse of hydrogen cyanide in water is being dispersed by flow in a 1 cm pipe. Its diffusion coefficient is 1.5x10-5 cm2/sec.
(1) what is the dispersion coefficient at a flow of 1 cm/sec?
(2) what is it at 1 m/sec?
(1)
Reynolds number
Laminar flow: dispersion is caused by coupled radial diffusion and axial convection:
(2)
Reynolds number
Turbulent flow: dispersion is caused by coupled velocity and concentration fluctuation
Laminar flow > Turbulent flow > Diffusion

19. ~ 1% will contain mixed gases
Example: We have a 10 cm pipeline 3 km long for moving reagent gases at 500 cm/s from the wharf to the plant. If the pipeline is used for different gases, one after the other. How much will the gases mix?
We choose a coordinate system originally located at the initial interface between the gases but moving with the average gas velocity. The mass balance around this moving point gives: ( l >> d well-mixed in radial direction)
B.C.
The concentration change is significant when:
~ 1% will contain mixed gases