1. 高等輸送二 — 熱傳 Lecture 11 Simultaneous Heat and Mass Transfer
郭修伯 助理教授

2. Mathematical analogies
The diffusion of mass and the conduction of heat obey very similar equations
diffusion in one dimension - Fick‘s law
heat conduction - Fourier‘s law
semiinfinite slab
semiinfinite slab
Thermal conductivity
Thermal diffusivity

3. Mathematical analogies
Momentum transfer in one dimension - Newton‘s law
Flat plate moved into an initially stagnant fluid
viscosity
plate velocity
kinetic viscosity
Confusing?

4. Mass per volume
Mass flux
Not energy per volume?
Energy flux
Not momentum per volume?
Momentum flux

5. Interfacial mass flux:
Interfacial energy flux:
Interfacial momentum flux:
Table

6. Cooling metal spheres
We want to quickly quench a liquid metal to make fine powder. We plane to do this by spraying drops into an oil bath. How can we estimate the cooling speed of the drops?
No suitable heat transfer correlations! However, several mass transfer correlations for drops are given:
For large drops without stirring:
Sherwood number, kd/D ~ Nusselt number, hd/k
Schmidt number, v/D ~ Prandtl number, v/α
This correlation will be reliable only if the Grashöf number for the cooling falls in the same range as that used to develop the mass transfer correlation.

7. Heat transfer from a spinning disc
Imagine that a spinning metal disc electrically heated to 30C is immersed in 1000 cm3 of an emulsion at 18C. The disc is 3 cm in diameter and is turning at 10 rpm. The emulsion’s kinetic viscosity is cm2/sec. After an hour, the emulsion is at 21C. What is its thermal diffusivity?
Energy balance:
α = ?
Mass transfer away from a spinning disc:
I.C., t = 0, T = T0

8. The situations in which mass transfer, heat transfer, and fluid flow occur at the same rate:
the rates of mass, heat, and momentum transfer can be essentially the same for fluids in turbulent flow.
Reynolds(1874): mass or heat transfer in a flowing fluid must involve two simultaneous processes:
Natural diffusion of the fluid at rest
the eddies caused by visible motion
All caused by flow?
a << bu
a’ << b’u
a’’ << b’’u
Reynolds analogy

9. Reynolds analogy
It suggests a simple relation between different transport phenomena.
This relation should be accurate when transport occurs by means of turbulent eddies.
We can estimate mass transfer coefficients from heat transfer coefficients or from friction factors!
However, experimental results show that the Reynolds analogy is accurate for gases, but not for liquids.

10. The Chilton-Colburn analogy
How to extend to liquid?
By an analysis of experimental data:
reduce to the Reynolds analogy for gases whose Schmidt and Prandtl numbers equal unity
later apply theories, especially boundary layer theory, to rationalize the exponent of 2/3.

11. The wet-bulb thermometer
measures the colder temperature caused by evaporation of the water
applied to calculate the relative humidity in air:
mass flux:
energy flux:
Coupling:

12. the Chilton-Colburn analogy
=1 for gases
Relative humidity =

13. Design of cooling towers
Calculate the size of a tower required to cool a given amount of water:
Fig
Hot air out
Hot water in
Fig
Cold air in z
Cold water out

14. The mass balance on the water vapor in the control volume
Water accumulation = water convection in minus that out + water added by evaporation
The energy balance on the wet air in the control volume
The energy balance on both liquid water and wet air in the control volume:

15. X
Assuming,
Coupling:
the Chilton-Colburn analogy
=1 for gases

16. integration or
...
Fig

17. For kc values
Fig
Fig

18. Design a countercurrent cooling tower to cool water at 2150 kg/min
Design a countercurrent cooling tower to cool water at 2150 kg/min. The water enters at 60C and is to be cooled to 25C. The air is fed at 60 g-mol/m2.sec with a dry-bulb temperature of 30C and a dew point temperature of 10C. The water flux should be 40% lower than the maximum allowed thermodynamically. Find (1) the flow rate of the water per tower cross section, (2) the tower cross section, and (3) the height of tower required.
Refer to Fig , the maximum water flow : AB’
(1) Slope of AB’ =
40%
Slop of actual operating line, AB = 110 x 75 / 60 = 137.5
(2) the tower cross section:
(3) the tower height:

19. Thermal diffusion and effusion
Temperature gradient effects a solute flux
Uniform salt solution
Heated
Dilute salt solution
For liquid,
Soret coefficient
For gas,
Cooled
Concentrated salt solution
Soret, 1879
Heavier molecules usually will concentrate in the cooler region.
Thermal diffusivity

20. Experimental values:
Table
The temperature gradient effect disappears rapidly for dilute solution and is largest when solute and solvent concentrations are similar.

21. Thermal diffusion is studied in a two-bulb apparatus
Thermal diffusion is studied in a two-bulb apparatus. Each bulb is 3 cm3 in volume; the capillary is 1 cm long and has an area of 0.01 cm2. The left-hand bulb is heated to 50C, and the right-hand bulb is kept at 0C. The entire apparatus is initially filled with an equilmolar mixture, either of hydrogen-methane or of ethanol-water. How much separation is achieved? About how long does this separation take?
Thermal diffusion:
gas mixture
ethanol-water
The separations are small; that with liquids is slightly larger but in the opposite direction.
Mass balance on the left-hand bulb:
Mass balance on the left-hand bulb:
Integration…
gas ~ 500sec
liquid ~ 180 days

22. Conclusions
For gases, D and α are nearly equal, and k and are very similar.
For liquids and solids, D is much less than α, and k is much less than
For liquids and solids, the heat transfer is much more rapid than the mass transfer, and so proceeds as if the mass transfer did not exist. The two processes are essential uncoupled.