CHE2163: HEAT AND MASS TRANSFER
DIFFUSION – FICK’S LAW WEEK 9
Learning Objectives, Tasks, Readings, and Learn ChemE Videos
1. Fick’s Law – terms, applications and limitations
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a. Incropera p786-791; Cengel p801-805; Welty p399-406 b. Fill in student exercises – Diffusivity Mini-Quiz
c. Derivation of general mass equation (Moodle upload)
2. Calculate a diffusive flux using Fick’s Law, understanding the assumptions and limitations associated with its use
a. Incropera p786-790; Cengel p 801-802; Welty p 399-406
b. Fill in student exercises – Calculating Diffusive Flux
c. LearnChemE–“HydrogenDiffusionThroughPalladium”
3. Use Fick’s law and equilibrium data to solve steady state diffusion problems without any form of bulk motion
a. Learn ChemE video on “Deriving Molar Flux Equations” b. Examples and practice problems
By the end of this week, you should be able to:
Find molecular diffusivity constants and use these in combination with unit conversions and equilibrium equations to solve Fick’s Law problems
2.1 FICK’S LAW (INCROPERA P786-791; CENGEL P. 799-804; WELTY CH. 24)
Fick’s Law states that the rate of diffusive mass transfer of ̇ -1
solute A (𝑁diff,A in moles.s ) through phase B is equal to the product of the molecular diffusivity constant (DAB, in m2.s-1), the contact surface area (A, in m2) and the concentration gradient of A with respect to x (moles.m-4)
𝑁 =−𝐷𝐴 𝑑𝑖𝑓𝑓,𝐴 𝐴𝐵
In mass transfer, as in heat transfer, we often express this as a diffusive mass transfer flux (𝐽𝑑𝑖𝑓𝑓,𝐴 in mols.s-1m-2) rather than a rate:
𝐽 =−𝐷 𝑑𝑖𝑓𝑓,𝐴 𝐴𝐵 𝑑𝑥
Assumptions
o Steady state; therefore:
o Constant diffusivity; therefore:
o 1-dimensional mass transfer; therefore:
Note: what are the units of CA? Can you write similar equations for mass transfer rates if concentration units are in mass concentration? Mole fraction? Mass ratio? Partial pressure?
Let’s look in more detail at the terms in Fick’s Law
2.2 WHAT IS MOLECULAR DIFFUSIVITY?
Molecular diffusivity, DAB, is the diffusivity of A through B.
MOLECULAR DIFFUSIVITY IN GASES
(Treybal p31 – see also Cengel p802-803)
Typical Values: 1 x 10-5 m2/s
Theoretical prediction possible
Some values available in texts
For an ideal gas, DAB = DBA
Can extrapolate to different temperatures and pressures
f properties DAB DBA
(T in Kelvin)
Q. How will doubling pressure affect molecular diffusivity?
Q. How will doubling temperature from 25oC to 50oC affect molecular diffusivity?
MOLECULAR DIFFUSIVITY IN LIQUIDS
(Treybal p36 – see also Cengel p804)
Typical Values: 1 x 10-9 m2/s (varies with concentration)
Theoretical prediction possible but difficult
Values for some binary mixtures available in most texts
Can extrapolate to different temperatures
DAB fproperties T
DAB DBA
and account for different solute and solvent concentrations:
Q. If water temperature doubles from 15oC to 30oC, how will this affect the molecular diffusivity of O2 in water?
D D D AB ABB BAA
MOLECULAR DIFFUSIVITY IN SOLIDS
(Cengel p804)
Temperature & activation energy strongly affect molecular diffusion
Porosity and pore size also affect diffusion.
Important examples including diffusion of carbon into iron
(steel fabrication); doping silicon with boron to influence electrical properties
DAB 1 ks
DAB D0eQRT
Where D0 is a proportionality constant of units consistent with DAB, and is specific to every material.
DIFFUSIVITY MINI-QUIZ STUDENTS TO COMPLETE:
Is DAB dependent on temperature? Y/N
Is DAB dependent on pressure? Y/N
Is DAB dependent on nature of components?Y/N
Is DAB dependent on liquid concentration? Y/N
Is DAB dependent on liquid velocity? Y/N
2.3 WHY DOES MASS TRANSFER HAVE SO MANY EQUATIONS? AN EXAMPLE USING FICK’S LAW
Fick’s Law states that the rate of mass transfer of solute A (moles.s-1) through phase B is equal to the product of the molecular diffusivity constant DAB (m2.s-1) and the concentration gradient (moles.m-3.m-1)
ASSUMPTIONS
However, we could express the concentration gradient in a number of different units, particularly based on the concentration units in which we choose to work:
2.4 CALCULATING DIFFUSIVE FLUX – STUDENTS TO COMPLETE See Cengel p799-802; Welty p398-406 (full derivation – complex!)
Concentration Gradient
Fick’s Law Flux Equation
Units of driving
𝐽 = −𝐷 𝑑𝑖𝑓𝑓,𝐴 𝐴𝐵 𝑑𝑧
𝑗 =−𝐷 𝑑𝑖𝑓𝑓,𝐴 𝐴𝐵
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