Gas transfer rates If either phase concentration can not be predicted by Henry's law then there will be a transfer of mass across the interface until

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1 Gas transfer rates If either phase concentration can not be predicted by Henry's law then there will be a transfer of mass across the interface until equilibrium is reached. The mechanisms and rate expressions for this transfer process have been conceptualized in a variety of ways so that quantitative descriptions are possible. One common conceptualization method is the film theory.

2 Film Theory Developed by Nernst (1904) The simplest conceptualization of the gasliquid transfer process Near the interface, there exists a stagnant film. This stagnant film is hypothetical since we really don't know the details of the velocity profile near the interface.

3

4 In this film, bulk liquid is not moving. Therefore, transport is governed essentially by molecular diffusion. Fick's law describes flux through the film.

5 Molecular diffusion A random and spontaneous mixing process that moves the solute from regions of high concentration to regions of low concentration. Molecular diffusion is due to concentration gradients and the random motion of molecules. calm environment

6 Molecular diffusion Rates of simple one-dimensional microscale mass transfer processes in homogeneous fluids or solids are expressed in terms of Fick s first law of diffusion.

7 Molecular diffusion According to Fick s first law, the flux of component "c" across a chemical front per unit time is directly proportional to the size of the spatial gradient in concentration and a constant of proportionality (the diffusion constant D ) (cm 2 /sec) specific to solute Substance moves due to concentration differences. Area that is perpendicular to the flow.

8 Therefore, flux through the film is C J D X Typical units for flux are mol 2 or mg cm sec 2 cm sec

9 If the thickness of the stagnant film is given by d n then the gradient can be approximated by: C CbCi X dn C b and C i are concentrations in the bulk and at the interface, respectively.

10 Assume there are no reactions in the stagnant film. Therefore, no accumulation in the film. Assume that D = constant At steady-state: JD (CbC i) dn What is the value of C i?

11 C i is calculated by assuming that equilibrium is attained instantly at the interface. Nernst s model assumes that the other phase doesn't have a "film". Ci Cg H c (if the film side is liquid and the opposite side is the gas phase).

12 Two film model Nernst s model assumes that the other phase doesn't have a "film". This theory is the most widely accepted model. It makes it easier to visualize the concepts. It is usually applied for gas transfer by bubble formation in liquid. Therefore, we use the Lewis-Whitman (1923) two-film model as described below.

13 Two film model It assumes that there are two stagnant layers of film at the interface. The mass transfer across the films is by molecular diffusion. The bulks of gas and liquid are homogeneous with respect to solute.

14 uniform concentration and pressure because of turbulence uniform concentration because of turbulence Assuming that the mass transfer is in the X direction, in other words from bulk gas phase to bulk liquid phase.

15 Since the solute is diffusing from the gas phase into the liquid, there must be a concentration gradient in the direction of mass-transfer within each phase. The bulk gas concentration C g falls to C gi at the interface. Similarly, in the liquid phase, the concentration falls from C li at the interface to C l in the main body of liquid.

16 The bulk gas concentrations C g and C l are clearly not the equilibrium values since otherwise the mass transfer would not occur. the interface.

17 Now, the flux from the bulk gas to the interface is J g = k g (C gi -C g ) and the flux from the interface to the bulk liquid is J l = k l (C l -C li ) where k l Dl D g k d g n d n k l or k g = mass transfer coefficient for either the liquid or gas phase as (dimensions = Length/time).

18 The same assumptions apply to the two films as apply in the single Nernst film model. So at steady state, the flux from the bulk gas to the interface is equal to the the flux from the interface to the bulk liquid. Therefore, k g (C gi -C g ) = k l (C l -C li ) kl = kg J g = J l (Cgi C g ) (ClC li )

19 The difficulty is finding interface concentrations, C gi or C li. We cannot directly measure the interface concentrations. We want to be able to relate flux to bulk air and water concentrations.

20 Therefore, overall mass transfer coefficients are defined. These coefficients are based on the difference between the bulk concentration in one phase and the concentration that would be in equilibrium with the bulk concentration in the other phase.

21 C g 0,016 0,014 C g 0,012 C g C H C gi c li A Equilibrium line 0,01 C gi 0,008 0,006 C g * 0,004 B 0, C l C l C li C l * C l

22 In the previous graph, point A represents the two bulk phase concentrations and point B represents the concentrations at the interface.

23 Define: J * Kl ClCl C li is replaced with C l * J * Kg Cg Cg C gi is replaced with C g * K l = overall mass transfer coefficient based on liquid-phase concentration. K g = overall mass transfer coefficient based on gas-phase concentration.

24 K g,l have dimensions of Length/t. C l * = liquid phase concentration that would be in equilibrium with the bulk gas concentration. = C g /H c (typical dimensions are moles/m 3 ). C g * = gas phase concentration that would be in equilibrium with the bulk liquid concentration. = H c C l (typical dimensions are moles/m 3 ).

25 Expand the liquid-phase overall flux equation to include the interface liquid concentration. * J Kl C C C C l li li l

26 Then substitute C li Cgi H c and * C l C g H c to get: J Kl Cl C Cgi Cg li H c

27 In the steady-state, fluxes through all films must be equal. Let all these fluxes be equal to J. Cl and J g = k g (C gi -C g ) J l = k l (C l -C li ) On an individual film basis: Cli J k l Cgi Cg J k g

28 Since all J s are equal: J J J K l kl Hck g

29 This can be arranged to give: 1 Kl 1 1 kl H c kg H c : Contaminant specific k l and k g : Environment specific, such as mixing

30 A similar manipulation starting with the overall flux equation based on gas phase concentration will give: 1 Hc 1 K g k l kg

31 These last two equations can be viewed as "resistance" expressions where 1/K g or 1/K l represent total resistance to mass transfer based on gas or liquid phase concentration, respectively.

32 In fact, the total resistance to transfer is made up of three series resistances: i) liquid film, ii) interface iii) gas film. However, we assume instant equilibrium at the interface so there is no transfer limitation here. It should be noted that model selection (penetration, surface renewal or film) does not influence the outcome of this analysis.

33 The Two Special Cases - Single film control It is possible that one of the films exhibits relatively high resistance and therefore dominates the overall resistance to transfer. This, of course, depends on the relative magnitudes of k l, k g and H c. So the solubility of the gas and the hydrodynamic conditions which establish the film thickness determine if a film controls.

34 For highly soluble gases such as NH 3 and SO 2 (H c is low) 1 1 kl Hckg 1 Kl 1 Hk c g K l H c k g Mass transfer is controlled by gas film resistance.

35 In general, slightly soluble gases such as N 2 and O 2 (H c is high) Kl kl H c kg 1 Kl 1 1 kl Hckg 1 kl or Kl k l Mass transfer is controlled by liquid film resistance.

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