高等輸送二 — 質傳 Lecture 7 Fundamentals of Mass Transfer 郭修伯 助理教授

Understand diffusion? What is Diffusion? How to study diffusion? process by which molecules, ions, or other small particles spontaneously mix, moving from regions of relatively high concentration into regions of lower concentration How to study diffusion? Scientific description: By Fick’s law and a diffusion coefficient Engineering description: By a mass transfer coefficient

Based on empirical arguments, the amount transferred at the interface: Concentration difference Interfacial area Rate of mass transferred = k concentration in the bulk Flux at the interface (include both diffusion and convection) concentration at the interface mass transfer coefficient

c1 (t) k = 3.4 x 10-2 cm/sec t = 8.3 x 103 sec The vessel is isothermal at 25 C, so the water’s vapor pressure is 23.8 mmHg. The vessel has 0.8 liter of water with 150 cm2 of surface area in a total volume of 19.2 liters. After 3 min, the air is 5% saturated. What is the mass transfer coefficient? How long will it take to reach 90% saturation? c1 (t) k = 3.4 x 10-2 cm/sec t = 0, c1 = 0 t = 8.3 x 103 sec

2 cm-diameter spheres of benzoic acid are packed into a bed 0.2 cm-diameter spheres of benzoic acid are packed into a bed. The spheres have 23 cm2 surface area per 1 cm3 of bed. Pure water flowing at a superficial velocity of 5 cm/sec into the bed is 62% saturated with benzoic acid after it has passed through 100 cm of beds. What is the mass transfer coefficient? From mass balance: z = 0, c1 = 0 k is defined at the interface between the sphere’s face and liquid

Bromine is rapidly dissolved in water Bromine is rapidly dissolved in water. Its concentration is about half saturated in 3 min. What is the mass transfer coefficient? From mass balance: t = 0, c1 = 0

A bubble of oxygen originally 0 A bubble of oxygen originally 0.1 cm in diameter is injected into excess stirred water. After 7 min, the bubble is 0.054 cm in diameter. What is the mass transfer coefficient? Mass balance on the bubble itself c1 : the oxygen concentration in the bubble ~ constant (P/RT) c1sat : the oxygen concentration at saturation ~ constant t = 0, r = 0.05 cm k = 1.6 x 10-3 cm/sec

Mass transfer coefficient Concentration difference Interfacial area Rate of mass transferred = k Mass per unit volume, or? Local or average? Real contact area is? Partial pressure? Molar concentration? Mass concentration? Mole fraction?

A blood oxygenator A blood oxygenator is used to replace the human lungs during open-heart surgery. From published correlations of mass transfer coefficients, the mass transfer coefficient based on the oxygen concentration difference in the water is 3.3 x 10-3 cm/sec. The equation given by the oxygenator manufacturer is: Hypothetical oxygen partial pressure in equilibrium with water under the experimental conditions. Actual oxygen partial pressure in the gas The Henry’s law constant of oxygen in water at the experimental condition is: Mole fraction of the total oxygen in the water Find kp

Mass transfer coefficient The values of the coefficients are usually reported as correlations of dimensionless numbers Sherwood and Stanton numbers involve the mass transfer coefficient Schmidt, Lewis and Prandtl numbers involve different kind of diffusion Reynolds, Grashof and Peclet numbers describe flow (forced convection) (free convection)

Sherwood number The variation of Sherwood number with flow is complex because the flow has two physical origins: Forced convection the flow is caused by external stirring or pumping Free convection the fluid velocity is a result of the mass transfer itself. The mass transfer causes density gradients in the surrounding solution and then in turn cause flow.

Correlations Correlations for fluid-fluid interfaces Accuracy ~ 30% k ~ v0.7 k ~ D0.5-0.7 Correlations for fluid-solid interfaces much like the heat transfer equivalents accuracy ~ 10% or less k ~ v0.5 k ~ D2/3

A disc of benzoic acid 2.5 cm in diameter is spinning at 20 rpm and 25C. How fast will it dissolve in a large volume of water? How fast will it dissolve in a large volume of air? The diffusion coefficients are 1.0 x 10-5 cm2/sec in water and 0.233 cm2/sec in air. The solubility of benzoic acid in water is 0.003 g/cm3. Its equilibrium vapor pressure is 0.3 mmHg. The dissolution rate is: For water For air The flux in air is about 1/3 of that in water.

Air containing a water-soluble vapor is flowing up and water is flowing down in the experimental column. The water flow in the 0.07 cm-thick film is 3 cm/sec, the column diameter is 10 cm and the air is essentially well mixed right up to the interface. The diffusion coefficient in water of the absorbed vapor is 1.8 x 10-5 cm2/sec. How long a column is needed to reach a gas concentration in water that is 10% of saturation? From mass balance: absorption z = 0, c1 = 0 z = 4.8 cm

Mass transfer across interfaces From one well-mixed bulk phase into another different one solute flux relative to the interface overall mass transfer coefficient

Overall mass transfer coefficient Gas Liquid p10 p1i c1i c10 overall liquid-side mass transfer coefficient overall gas-side mass transfer coefficient

Estimate the overall liquid-side mass transfer coefficient at 25C for oxygen from water into air. For oxygen in air, the diffusion coefficient is 0.23 cm2/sec; for oxygen in water, the diffusion coefficient is 2.1 x 10-5 cm2/sec. The Henry’s law constant is 4.4 x 104 atm.

We are studying gas absorption into water at 2 We are studying gas absorption into water at 2.2 atm total pressure in a packed tower containing Berl saddles. For both ammonia and methane the mass transfer coefficient times the packing area per tower volume is 18 lb-mol/hr-ft3 for the gas side and 530 lb-mol/hr-ft3 for the liquid size. Their Henry’s law constants are different: 9.6 atm for ammonia and 41000 atm for methane. What is the overall gas-side mass transfer coefficient for each gas? For ammonia: For methane: