Theories for Mass Transfer Coefficients

Transcription

1 Mass Transfer Theories for Mass Transfer Coefficients Lecture 9, 5..7, r. K. Wegner

2 9. Basic Theories for Mass Transfer Coefficients Aim: To find a theory behind the empirical mass-transfer correlations (e.g. Tables 8.3- and -3 in Cussler s book) that is based on physics (first part of this class). Such models should connect mass transfer and fluid flow. We would like to predict the mass transfer coefficient k as a function of the diffusion coefficient and the fluid velocity v. First, we look at MTC s for fluid-fluid systems (9.) that are VERY important in industrial applications. In the following lecture we investigate models for simple fluid-solid interfaces (9.) that can be rather elaborate and detailed but have limited application in industry. Mass Transfer Basic Theories for Mass Transfer Coefficients 9-

3 9. Fluid-Fluid Interfaces, e.g. Falling film Gas bubbles in tank Liquid in packed tower Source: Cussler, Chapter.5. Source: Wikipedia, Blasensäule Source: Büchi Glas, Uster Mass Transfer Basic Theories for Mass Transfer Coefficients 9-3

4 Falling film Evaporation, Gas scrubbing From Table 8.3-: Gas bubbles in tank Aereation, Gas ab- and desorption, Bioreaktors Liquid in packed tower Extraction, istillation kz z v.69.5 kd d.3 3 g ρ / ρ ν 3 ν 3 k νg 3 v.5 a ν.67 ν.5 ( ad). 4 k ~.5, ~v.5 k ~ /3 k ~.5, ~v.67 Why is k ~ / or k ~ /3 or k ~ v.67?? Mass Transfer Basic Theories for Mass Transfer Coefficients 9-4

5 9.. The Film Theory (Nernst, 94) 9.. The Film Theory (Nernst, 94) Assumptions: - All action (fluid flow and mass transfer) occurs in a thin film at the interface - Bulk fluid (e.g. gas) FILM bulk fluid (e.g. liquid) - Steady-state flux across film Mass Transfer Basic Theories for Mass Transfer Coefficients 9-5

6 n ( c ) N c z k i This flux can be obtained also in terms of (for dilute concentrations) N j (c z i c ) Comparing equation (7.) and (7.) gives k (9.-) (9.-) (9.-3) Or by rewriting gives k Sh (9.-4) This simple theory gives k BUT all fluid characteristics (e.g. fluid velocity due to stirring) are in the unknown film thickness. Mass Transfer Basic Theories for Mass Transfer Coefficients 9-6

7 This simple theory provides the FRAMEWORK of most MTC s as follows: mass transfer characteristic other coefficient length Sh F system diffusion variables coefficient Applications: The film theory is used in some practical cases to determine the. Mass Transfer Basic Theories for Mass Transfer Coefficients 9-7

8 Example: CO is being scrubbed out of a gas by water flowing through a packed bed. Calculate the film thickness if.3-6 mol/(cm s) of CO are adsorbed when p CO atm, H 6 atm Solution: First find the interfacial concentration c i : i p H x H c and CO/HO.9-5 cm /s. c atm i 6atm 3 c 9.3 i c ( mol)/(8 cm ) mol/cm -4 3 Mass Transfer Basic Theories for Mass Transfer Coefficients 9-8

9 Calculate k from equation (9.-): N k(ci c ).3-6 mol/(cm s) k (9.3-4 mol/cm 3 - ) k.5-3 cm/s Now 5.9 cm / s 3 k.5 cm / s.76 cm Typically cm VERY IMPORTANT Mass Transfer Basic Theories for Mass Transfer Coefficients 9-9

10 9.. The Penetration Theory (Higbie, 935) Assumptions: Same as in Film-theory but the film is VERY thick iffusion is important in z-direction Convection is important in x-direction Equation (9.-): N k(ci c ) The solution to this problem was given before in the context of the semi-infinite slab (Cussler p. 46): ( ) N j v / π x (c c ) (9.-5) z max i where N is the flux and v max is the velocity of the liquid at the interface Mass Transfer Basic Theories for Mass Transfer Coefficients 9-

11 Note that this flux at the interface is valid at a specific x. To find the average flux, N (x) has to be averaged over the entire surface: LW N n dy dx z W L where L is the length of the film in x and W is its width in y. Since n does not vary in y, inserting 9.-5: L v max N c dx L π x N L max x L v π c N L v π L max c Mass Transfer Basic Theories for Mass Transfer Coefficients 9-

12 or v N c c π L ( ) max i so k vmax (9.-6) π L The L/v max is called contact time and is not known a priori in complex situations, as was in the film theory. Compare: k / (penetration theory) k (film theory) Mass Transfer Basic Theories for Mass Transfer Coefficients 9-

13 Mass Transfer Basic Theories for Mass Transfer Coefficients 9-3 These two theories bracket the experimental data (Table 8.3-) very well, almost too well to be accepted. Equation 9.-6 can be rewritten, assuming that the average velocity is v /3 v max (true for a laminar slit flow of a Newtonian fluid). Pe 6 Re Sc π Lv 6 L v 3 L L k π π Lv 6 ν ν π

14 9..3 The Surface Renewal Theory (ankwerts, 95) The success of the penetration theory with data despite its restrictive assumption motivated scientists and engineers to propose alternative and more realistic models leading, however, to the same variable dependencies at the end. Assumption: The thick film framework is replaced by TWO regions: interface and bulk. In the interfacial region mass transfer takes place according to penetration theory. Then elements of this region ARE EXCHANGE with the bulk region. This is the so-called surfacerenewal process. Mass Transfer Basic Theories for Mass Transfer Coefficients 9-4

15 The issue is how long the fluid elements stay in the interfacial region exposed to penetration. probabilityof asurface E(t)dt element to be at the surface for time t E(t) is the residence time distribution, RT, and E(t)dt The transfer of interfacial elements into the bulk is random and any surface element is equally likely to be withdrawn. Mass Transfer Basic Theories for Mass Transfer Coefficients 9-5

16 By definition the fraction of surface elements at time t is : [ τ] θ exp t / where τ is a characteristic constant equivalent to the average residence time of an element in the interfacial (surface) region. Now the fraction θ is also the sum of probabilities θ t E(t) dt Thus the residence time distribution of surface elements is: d dt t dθ dt τ t τ ( ) dt exp E(t)dt E(t)dt E( t) ) E t d dt Mass Transfer Basic Theories for Mass Transfer Coefficients 9-6 t

17 So, E(t) exp τ t τ The mass transfer coefficient at the interfacial region is obtained from the semi-infinite slab model (see eq and Cussler p. 46) as: n πt ( c ) N c z i The semi-infinite slab model is used here even though the interfacial region is not infinite. But if the surface is rapidly renewed and the τ is small, then the interfacial region appears as infinite. Mass Transfer Basic Theories for Mass Transfer Coefficients 9-7

18 Mass Transfer Basic Theories for Mass Transfer Coefficients 9-8 The average flux (over all surface elements) is: z,av dt n E(t) N [ ] ( ) dt c c t t exp i π τ τ ( ) [ ] dt t t exp c c i τ τ π ( ) ( ) τ τ π τ π i t erf c c

19 N τ ( c ),av i c ( c ) ( ) ( ) i c N,av πτ erf τ erf τ π τ Thus, As in the penetration theory, here k / τ k / Again the residence time τ is as unknown as the in film theory or the L/v max (contact time) in the penetration theory. The major contribution of the surface renewal theory is that it gives a more REALISTIC physical situation. This gives a better starting point for development of effective correlations and better models. Mass Transfer Basic Theories for Mass Transfer Coefficients 9-9

20 Summary: The Film Theory k The Penetration Theory Advantages Simple; good base for extension isadvantages Film thickness is unknown k v π L max Simplest including flow Contact time (L/v max ) usually unkown The Surface-Renewal Theory k / τ Similar math to penetration theory, but better physical picture Surface-renewal rate (τ) is unknown Mass Transfer Basic Theories for Mass Transfer Coefficients 9-

21 The dependencies of k on and v, like k ~ /, k ~ /3 or k ~ v.67, observed in the experimentally-based MTCs are typically not well reflected by the simple mass transfer models. These simple models for fluid-fluid interfaces pretend that fluid motion is incorporated in diffusion and everything is treated as a thin film or semi-infinite slab problem. In principle, these two extreme cases should bracket all possible geometries. Yet, especially the effect of flow (velocity) is usually not well reflected. One reason is that the simple theories assume a homogeneous system while real systems are heterogeneous with respect to concentration and flow (Schlünder, 977). Schlünder E.U., Chem. Eng. Sci. 3, Mass Transfer Basic Theories for Mass Transfer Coefficients 9-