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高等輸送二 — 熱傳 Lecture 11 Simultaneous Heat and Mass Transfer 郭修伯 助理教授
Mathematical analogies ,[object Object],[object Object],[object Object],semiinfinite slab semiinfinite slab Thermal diffusivity Thermal conductivity
Mathematical analogies ,[object Object],Flat plate moved into an initially stagnant fluid viscosity plate velocity kinetic viscosity Confusing?
Mass flux Mass per volume Energy flux Not energy per volume? Momentum flux Not momentum per volume?
Interfacial mass flux: Interfacial energy flux: Interfacial momentum flux: Table 20.1-1
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.
Heat transfer from a spinning disc Imagine that a spinning metal disc electrically heated to 30C is immersed in 1000 cm 3  of an emulsion at 18C. The disc is 3 cm in diameter and is turning at 10 rpm. The emulsion’s kinetic viscosity is 0.082 cm 2 /sec. After an hour, the emulsion is at 21C. What is its thermal diffusivity? Energy balance: I.C., t = 0, T = T 0 α = ? Mass transfer away from a spinning disc:
[object Object],[object Object],[object Object],[object Object],[object Object],All caused by flow? a << bu a’ << b’u a’’ << b’’u Reynolds analogy
Reynolds analogy ,[object Object],[object Object],[object Object],[object Object]
The Chilton-Colburn analogy ,[object Object],[object Object],[object Object],[object Object]
The wet-bulb thermometer ,[object Object],[object Object],[object Object],[object Object],Coupling:
the Chilton-Colburn analogy =1 for gases Relative humidity =
Design of cooling towers Calculate the size of a tower required to cool a given amount of water:  Fig. 20.3.2 Fig. 20.3.1 Hot water in Cold water out z Cold air in Hot air out
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:
Coupling: X  Assuming,  the Chilton-Colburn analogy =1 for gases
integration or ... Fig 20.3-3
Fig 20.3-5 Fig 20.3-4 For  kc  values
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/m 2 .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 20.3-5, 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:
Thermal diffusion and effusion ,[object Object],Uniform salt solution Heated Cooled Soret, 1879 Dilute salt solution Concentrated salt solution For liquid, Soret coefficient For gas, Thermal diffusivity Heavier molecules usually will concentrate in the cooler region.
Experimental values:  Table 20.5-1 The temperature gradient effect disappears rapidly for dilute solution and is largest when solute and solvent concentrations are similar.
Thermal diffusion is studied in a two-bulb apparatus. Each bulb is 3 cm 3  in volume; the capillary is 1 cm long and has an area of 0.01 cm 2 . 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
Conclusions ,[object Object],[object Object],[object Object]

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Lecture 11

  • 1. 高等輸送二 — 熱傳 Lecture 11 Simultaneous Heat and Mass Transfer 郭修伯 助理教授
  • 2.
  • 3.
  • 4. Mass flux Mass per volume Energy flux Not energy per volume? Momentum flux Not momentum per volume?
  • 5. Interfacial mass flux: Interfacial energy flux: Interfacial momentum flux: Table 20.1-1
  • 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 cm 3 of an emulsion at 18C. The disc is 3 cm in diameter and is turning at 10 rpm. The emulsion’s kinetic viscosity is 0.082 cm 2 /sec. After an hour, the emulsion is at 21C. What is its thermal diffusivity? Energy balance: I.C., t = 0, T = T 0 α = ? Mass transfer away from a spinning disc:
  • 8.
  • 9.
  • 10.
  • 11.
  • 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. 20.3.2 Fig. 20.3.1 Hot water in Cold water out z Cold air in Hot air 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. Coupling: X Assuming, the Chilton-Colburn analogy =1 for gases
  • 16. integration or ... Fig 20.3-3
  • 17. Fig 20.3-5 Fig 20.3-4 For kc values
  • 18. 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/m 2 .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 20.3-5, 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.
  • 20. Experimental values: Table 20.5-1 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. Each bulb is 3 cm 3 in volume; the capillary is 1 cm long and has an area of 0.01 cm 2 . 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.