Overall Shell Mass Balances I

1. Overall ShellMass Balances I

2. Outline 3. Molecular Diffusion in Gases • Molecular Diffusion in Liquids • Molecular Diffusion in Solids • Prediction of Diffusivities • Overall Shell Mass Balances • Concentration Profiles

3. Overall Shell Mass Balance Species entering and leaving the system by Molecular Transport + by Convective Transport Mass Generation by homogeneous chemical reaction Steady-State! * May also be expressed in terms of moles

4. Overall Shell Mass Balance * May also be expressed in terms of moles Common Boundary Conditions: • Concentration is specified at the surface. • The mass flux normal to a surface maybe given. • At solid- fluid interfaces, convection applies: NA = kc∆cA. • The rate of chemical reaction at the surface can be specified. ♪ At interfaces, concentration is not necessarily continuous.

5. Concentration Profiles I. Diffusion Through a Stagnant Gas Film

6. Concentration Profiles I. Diffusion Through a Stagnant Gas Film • Assumptions: • Steady-state • T and P are constants • Gas A and B are ideal • No dependence of vz on the radial coordinate At the gas-liquid interface,

7. Concentration Profiles I. Diffusion Through a Stagnant Gas Film Mass balance is done in this thin shell perpendicular to the direction of mass flow

8. Concentration Profiles I. Diffusion Through a Stagnant Gas Film Since B is stagnant,

9. Concentration Profiles I. Diffusion Through a Stagnant Gas Film Applying the mass balance, where S = cross-sectional area of the column

10. Concentration Profiles I. Diffusion Through a Stagnant Gas Film Dividing by SΔz and taking the limit as Δz 0, NA = constant

11. Concentration Profiles I. Diffusion Through a Stagnant Gas Film NA = constant But, Substituting,

12. Concentration Profiles I. Diffusion Through a Stagnant Gas Film For ideal gases, P = cRT and so at constant P and T, c = constant DAB for gases can be assumed independent of concentration

13. Concentration Profiles I. Diffusion Through a Stagnant Gas Film Integrating once, Integrating again,

14. Concentration Profiles I. Diffusion Through a Stagnant Gas Film Let C1 = -ln K1 and C2 = -lnK2, B.C. at z = z1, xA= xA1 at z = z2, xA= xA2

15. Concentration Profiles I. Diffusion Through a Stagnant Gas Film The molar flux then becomes OR in terms of the driving force ΔxA *, i.e. xA1> xA2 ǂ i.e. z2> z1

16. Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction Two Reaction Types: Homogeneous – occurs in the entire volume of the fluid - appears in the generation term Heterogeneous – occurs on a surface (catalyst) - appears in the boundary condition

17. Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction Reaction taking place 2A B Reactant A diffuses to the surface of the catalyst Reaction occurs on the surface Product B diffuses away from the surface

18. Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction Reaction taking place 2A B Assumptions: Isothermal A and B are ideal gases Reaction on the surface is instantaneous Uni-directional transport will be considered

19. Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction

20. Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction From stoichiometry,

21. Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction Substitution of NA into the differential equation Integration twice with respect to z, B.C. 1: at z = 0, xA= xA0 B.C. 2: at z = δ, xA= 0

22. Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction The final equation is And the molar flux of reactant through the film, *local rate of reaction per unit of catalytic surface

23. Concentration Profiles II. Diffusion With a Heterogeneous Chemical Reaction Reading Assignment See analogous problem Example 18.3-1 of Transport Phenomena by Bird, Stewart and Lightfoot

24. Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction Gas A dissolves in liquid Band diffuses into the liquid phase An irreversible 1st order homogeneous reaction takes place A + B AB Assumption: AB is negligible in the solution (pseudobinary assumption)

25. Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction first order rate constant for homogeneous decomposition of A S cross sectional area of the liquid

26. Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction Dividing by SΔz and taking the limit as Δz0,

27. Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction If concentration of A is small, then the total c is almost constant and Combining the two equations above

28. Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction Multiplying the above equation by gives an equation with dimensionless variables

29. Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction Thiele Modulus

30. Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction The general solution is

31. Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction Evaluating the constants, Reverting to the original variables,

32. Concentration Profiles III. Diffusion With a Homogeneous Chemical Reaction Quantities that might be asked for: 1. Average concentration in the liquid phase 2. Molar flux at the plane z = 0

33. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) Assumptions Velocity field is unaffected by diffusion A is slightly soluble in B Viscosity of the liquid is unaffected The penetration distance of A in B will be small compared to the film thickness.

34. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) Recall: The velocity of a falling film

35. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) * CA is a function of both x and z

36. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) Dividing by WΔxΔz and letting Δx  0 and Δz 0,

37. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) The expressions for , Transport of A along the z direction is mainly by convection (bulk motion) Recall:

38. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) The expressions for , Transport of A along the x direction is mainly by diffusion

39. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) Substituting the expressions for, Substituting the expressions vz,

40. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) Boundary conditions B.C. 1 B.C. 2 B.C. 3 BUT we can replace B.C. 3 with B.C. 3

41. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) or where

42. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption)

43. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) Reading Assignment See analogous problem Example 4.1-1 of Transport Phenomena by Bird, Stewart and Lightfoot

44. Concentration Profiles IV. Diffusion into a Falling Liquid Film (Gas Absorption) Quantities that might be asked for: 1. Total molar flow of A across the surface at x = 0