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Tutorial/HW Week #7 WWWR Chapters 24-25 ID Chapter 14PowerPoint Presentation

Tutorial/HW Week #7 WWWR Chapters 24-25 ID Chapter 14

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WWWR# 24.1, 24.12, 24.13, 24.15(d), 24.22.

To be discussed on March 11, 2014.

By either volunteer or class list.

Tutorial/HW Week #7WWWR Chapters 24-25ID Chapter 14Molecular Mass Transfer

- Molecular diffusion
- Mass transfer law components:
- Molecular concentration:
- Mole fraction:
(liquids,solids) , (gases)

- Velocity:
mass average velocity,

molar average velocity,

velocity of a particular species relative to mass/molar average is

the diffusion velocity.

- 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,

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

- As the total velocity,
- Or
- Which substituted, becomes

- Defining molar flux, N as flux relative to a fixed z,
- And finally,
- Or generalized,

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

Diffusion Coefficient

- Fick’s law proportionality/constant
- Similar to kinematic viscosity, n, and thermal diffusivity, a

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

- 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

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

- For gas mixtures with several components,
- with

- 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.

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

- As diffusivity changes with temperature, extrapolation of D simplified to Hayduk-Laudie eq.AB is by
- For diffusion of univalent salt in dilute solution, we use the Nernst equation

- Pore diffusivity simplified to Hayduk-Laudie eq.
- 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,

- But if Kn >1, then simplified to Hayduk-Laudie eq.
- If both Knudsen and molecular diffusion exist, then
- with
- For non-cylindrical pores, we estimate

Example 6 simplified to Hayduk-Laudie eq.

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

- F solids2 is the hydrodynamic hindrance factor, one equation is by Renkin,

Example 7 solids

Convective Mass Transfer solids

- Mass transfer between moving fluid with surface or another fluid
- Forced convection
- Free/natural convection
- Rate equation analogy to Newton’s cooling equation

Example 8 solids

Differential Equations solids

- Conservation of mass in a control volume:
- Or,
in – out + accumulation – reaction = 0

- For in – out, solids
- in x-dir,
- in y-dir,
- in z-dir,

- For accumulation,

- For reaction at rate solidsrA,
- Summing the terms and divide by DxDyDz,
- with control volume approaching 0,

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

- So by conservation of mass, general form:
- Written as substantial derivative,
- For species A,

- In molar terms, general form:
- For the mixture,
- And for stoichiometric reaction,

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