Models for Mass Transfer at a Fluid-Fluid Interface

1. Models for Mass Transfer at a Fluid-Fluid Interface ~Of greater interest in separation process is mass transfer across an interface between a gas and a liquid or between two liquid phases. ~Such interface exist in absorption, distillation, extraction, and stripping. ~At fluid-fluid interfaces, turbulence may persist to the interface. ~The following theoretical models have been developed to describe mass transfer from a fluid to such an interface. Film Theory Penetration Theory Surface Renewal Theory Film-Penetration Theory

2. Film Theory Nernst postulated that the entire resistance to mass transfer in a given turbulent phase is in a thin, stagnant region of that phase at the interface, called a film, as shown in Figure 3.18 (a). This film is similar to the laminar sublayer that forms when a fluid flows in the turbulent regime parallel to a flat plate. ~no resistance to mass transfer in the gas phase due to pure A in the gas ~at the gas-liquid interface, equilibrium obeys the Henry’s law, cAi=HApA ~in the thin, stagnant liquid film of thickness , molecular diffusion only occurs with a driving force of cAi-cAb ~the film is assumed to be very thin, all of the diffusing A passes through the film and into the bulk liquid ~the bulk flow of A is neglected, the concentration is linear as in Figure 3.18 (a) nonvotatile pure component A Figure 3.18 (a)

3. diffusion flux total flux ~the liquid phase is dilute in A ~the bulk-flow effect is neglected ~the bulk-flow effect is not neglected experimental: n=0.5~0.75 theory: n=2/3(based on boundary layer)

4. Example 3.17 Sulfur dioxide is absorbed from air into water in a packed absorption tower. At a certain location in the tower, the mass transfer flux is 0.0270 kmol SO2/m2-h and the liquid-phase mole fraction are 0.0025 and 0.0003, respectively at the two-phase interface and in the bulk liquid. If the diffusivity of SO2 in water is 1.710-5 cm2/s, determine the mass transfer coefficient, kc, and the film thickness. Example 3.18 Sulfur dioxide is absorbed from air into water in a packed absorption tower. At a certain location in the tower, the mass transfer flux is 0.0270 kmol SO2/m2-h and the liquid-phase mole fraction are 0.0025 and 0.0003, respectively at the two-phase interface and in the bulk liquid. If the diffusivity of SO2 in water is 1.710-5 cm2/s, determine the contact time for Higbie’s penetration theory. Example 3.19 For the conditions of Example 3.17, estimate the fractional rate of surface renewal, s, for Danckwert’s theory and determine the residence time and probability distributions.

5. Penetration Theory Higbie provided a penetration theory for mass transfer from a fluid-fluid interface into a bulk liquid stream, as shown in Figure 18 (b). ~the stagnant-film concept is replaced by Boussinesq eddies. (1)move from the bulk to the interface (2)stay at the interface for a short, fixed period of time during which they remain static so that molecular diffusion takes place in a direction normal to the interface (3)leave the interface to mix with the bulk stream ~when an eddy moves to the interface, it replaces another static eddy. ~the eddies are intermittently static and moving Figure 3.18 (b)

6. unsteady-state diffusion B.C cA=cAb at t=0 for 0z cA=cAi at z=0 for t>0 cA=cAb at z= for t>0 tc=contact time of the static eddy at the interface during one cycle. average mass transfer flux ~the penetration theory is most useful when mass transfer involvesbubbles or droplets or flow over random packing. ~for bubbles, the contact time, tc, of the liquid surrounding the bubble is taken as the ratio of bubble diameter to bubble rise velocity. ~for a liquid spray, where no circulation of liquid occurs inside the droplets, the contact time is the total time for the droplets to fall through the gas. ~for a packed tower, where the liquid flows as a film over particles of random packing, mixing can be assumed to occur each time the liquid film passes from one piece of packing to another.

7. Surface Renewal Theory The penetration theory is not satisfying because the assumption of a constant contact time for all eddies that temporarily reside at the surface is not reasonable, especially for stirred tanks contactors with random packing, and bubble and spray columns where the bubbles and droplets over a wide range of sizes. Danckwerts suggested an improvement to the penetration theory that involves the replacement of the constant eddy contact time with the assumption of a residence-time distribution, wherein the probability of an eddy at the surface being replaced by a fresh eddy is independent of the age of the surface eddy. F(t) is the fraction of eddies with a contact time of less than t. {t}dt=the probability that a given surface eddy will have a residence time t.

8. The instantaneous mass transfer rate for an eddy with an age t in flux form as The integrated average rate is ~The more reasonable surface renewal theory predicts the same dependency on molecular diffusivity as the penetration theory. ~Unfortunately, s, the fractional rate of surface renewal, is as elusive a parameter as the constant contact time, tc.

9. Film-Penetration Theory Toor and Marchello combined feature of the film, penetration, and surface renewal theories to develop a film-penetration model. Their theory assumes that the entire resistance to mass transfer resides in a film of fixed thickness . Eddies move to and from the bulk fluid and this film. Age distributions for time spent in the film are of the Higbie or Danckwerts type. B.C cA=cAb at t=0 for 0z cA=cAi at z=0 for t>0 cA=cAb at z= for t>0 For small t, the solution is s is high surface renewal theory For large t, the solution is s is low film theory

10. Two-Film Theory and Overall Mass Transfer Coefficients Whitman suggested an extension of the film theory to two fluid films in series. ~Each film presents a resistance to mass transfer, but concentration in the two fluid at the interface are in equilibrium. ~There is no additional interfacial resistance to mass transfer. ~This concept has found extensive application in modeling of steady-state gas-liquid and liquid-liquid separation processes, when the fluid phases are in laminar or turbulent flow. ~The assumption of equilibrium at the interface is satisfactory unless mass transfer rates are very highorsurfactants accumulate at the interface.

11. cA*與pAb達平衡 pA*與cAb達平衡

12. Liquid phase: Gas phase: Taking into account the effect of bulk flow

13. Case of Large Driving Forces for Mass Transfer In a similar manner,

14. Example 3.20 Sulfur dioxide (A) is absorbed into water in a packed column. At a certain location, the bulk conditions are 50 C, 2 atm, yAb=0.085, and xAb=0.001. Equilibrium data for SO2 between air and water at 50 C are Experimental values of the mass transfer coefficient are as follows. Liquid phase: kc=0.18 m/h Gas phase: kg=0.040 kmol/h-m2-kPa Using mole fraction-driving forces, compute the mass transfer flux by: (a) Assuming an average Henry’s law constant and a negligible bulk-flow effect (b) Utilizing the actual curved equilibrium line and assuming a negligible bulk-flow effect (c) Utilizing the actual curved equilibrium line and taking into account the bulk-flow effect (d) Determine the relative magnitude of the two resistance and the values of the mole fractions at the interface from the results of part (c)