Tutorial/HW Week #7 WRF Chapters 22-23 ; WWWR Chapters 24-25 ID Chapter 14

1. Tutorial #7 WWWR# 24.1, 24.12, 24.13, 24.15(d), 24.22. To be discussed during the week 9-13 March, 2020. By either volunteer or class list. Homework #7 (self practice) WWWR #24.2 ID # 14.25. Tutorial/HW Week #7WRF Chapters 22-23; WWWR Chapters 24-25ID Chapter 14

2. Molecular Mass Transfer • Molecular diffusion • Mass transfer law components: • Molecular concentration: • Mole fraction: (liquids,solids) , (gases)

3. For gases, • Velocity: mass average velocity, molar average velocity, velocity of a particular species relative to mass/molar average is the diffusion velocity.

7. Flux: A vector quantity denoting amount of a particular species that passes per given time through a unit area normal to the vector, given by Fick’s First Law, for basic molecular diffusion or, in the z-direction, For a general relation in a non-isothermal, isobaric system,

8. Since mass is transferred by two means: • concentration differences • and convection differences from density differences • For binary system with constant Vz, • Thus, • Rearranging to

9. As the total velocity, • Or • Which substituted, becomes

10. Defining molar flux, N as flux relative to a fixed z, • And finally, • Or generalized,

12. Related molecular mass transfer • Defined in terms of chemical potential: • Nernst-Einstein relation

13. Diffusion Coefficient • Fick’s law proportionality/constant • Similar to kinematic viscosity, n, and thermal diffusivity, a

14. Gas mass diffusivity • Based on Kinetic Gas Theory • l = mean free path length, u = mean speed • Hirschfelder’s equation:

15. Lennard-Jones parameters s and e from tables, or from empirical relations • for binary systems, (non-polar,non-reacting) • Extrapolation of diffusivity up to 25 atmospheres

16. Binary gas-phase Lennard-Jones “collisional integral”

19. With no reliable s or e, we can use the Fuller correlation, • For binary gas with polar compounds, we calculate W by

20. where

21. and • For gas mixtures with several components, • with

23. Liquid mass diffusivity • No rigorous theories • Diffusion as molecules or ions • Eyring theory • Hydrodynamic theory • Stokes-Einstein equation • Equating both theories, we get Wilke-Chang eq.

26. For infinite dilution of non-electrolytes in water, W-C is simplified to Hayduk-Laudie eq. • Scheibel’s equation eliminates FB,

27. As diffusivity changes with temperature, extrapolation of DAB is by • For diffusion of univalent salt in dilute solution, we use the Nernst equation

28. Pore diffusivity • Diffusion of molecules within pores of porous solids • Knudsen diffusion for gases in cylindrical pores • Pore diameter smaller than mean free path, and density of gas is low • Knudsen number • From Kinetic Theory of Gases,

29. But if Kn >1, then • If both Knudsen and molecular diffusion exist, then • with • For non-cylindrical pores, we estimate

30. Example 6

34. Types of porous diffusion. Shaded areas represent nonporous solids

35. Hindered diffusion for solute in solvent-filled pores • A general model is • F1 and F2 are correction factors, function of pore diameter, • F1 is the stearic partition coefficient

36. F2 is the hydrodynamic hindrance factor, one equation is by Renkin,

37. Example 7

41. Convective Mass Transfer • Mass transfer between moving fluid with surface or another fluid • Forced convection • Free/natural convection • Rate equation analogy to Newton’s cooling equation

42. Example 8

44. Differential Equations • Conservation of mass in a control volume: • Or, in – out + accumulation – reaction = 0

45. For in – out, • in x-dir, • in y-dir, • in z-dir, • For accumulation,

46. For reaction at rate rA, • Summing the terms and divide by DxDyDz, • with control volume approaching 0,

47. We have the continuity equation for component A, written as general form: • For binary system, • but • and

48. So by conservation of mass, • Written as substantial derivative, • For species A,

49. In molar terms, • For the mixture, • And for stoichiometric reaction,