CHE2163: HEAT AND MASS TRANSFER
DIFFUSION – FICK’S LAW
Learning Objectives, Tasks, Readings, and Learn ChemE Videos
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1. Fick’s Law – terms, applications and limitations
a. Incropera p786-791; Cengel p801-805; WWWR p399-406
b. Fill in student exercises on p9 and 10
c. Derivation from 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; WWWR p 399-406
b. Fill in the blanks – Section 1.5
c. LearnChemE – “Hydrogen Diffusion Through Palladium”
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 mass transfer flux equations” as per wk 7, but focus on second half
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
FICK’S LAW (incropera p786-791; Cengel p. 799-804; WWWR Ch. 24)
· Fick’s Law states that the rate of diffusive mass transfer of solute A (diff,A in moles.s-1) 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)
· In mass transfer, as in heat transfer, we often express this as a diffusive mass transfer flux ( in mols.s-1m-2) rather than a rate:
· Assumptions
· Steady state; therefore:
· Constant diffusivity; therefore:
· 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
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
(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
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?
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
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
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:
Calculating diffusive flux – STUDENTS TO COMPLETE
See Cengel p799-802; WWWR p398-406 (full derivation – complex!)
Concentration
Fick’s Law
Flux Equation
Units of driving
Concentration gradients and Equilibrium relationships (Incropera p795-798; Cengel p807-810; WWWR p438-439)
Concentration gradients can be determined using equilibrium relationships: (these will form the boundary conditions of many problems) For example:
1. Concentration at gas-liquid interface for a dilute solution:
Henry’s law (low solubility)
xA (interface) =pA (interface) /H
H= Henry’s Law constant
2. Partial pressure at gas-liquid boundary for an ideal liquid mixture (could even be pure liquid)
Raoult’s law
pA(interface) =xA(interface) PA,sat
PA,sat = saturation vapour pressure of A
3. Concentrations at gas- solid boundary:
solubility relationship
CA (interface) = S pA (interface)
S = solubility ( kmol/m3kPa)
pA = partial pressure (Pa)
3.1 GENERAL MASS EQUATION (incropera ch 14; WWWR CH 25)
· As in Heat Transfer Study Guide p. 29-35, there is a differential form of the “General Mass Equation” from which analytical solutions to specific problems (defined by transfer modes, geometry, media, assumptions) can be derived
· Remember that all of the equations you encountered in Heat Transfer (1-D steady-state conduction through a cylinder; 1-D transient conduction, etc.) can all be derived from the general form of the heat equation, usually by making many assumptions (e.g. steady state, 1D-transfer, constant property data, etc)
· The same can be said for Mass Transfer; equations for the same problems (1-D conduction through a cylinder; 1-D transient conduction, etc.) can be derived from the general form of the mass equation
· As a student exercise before the lecture, go to Moodle – Learning Resources – Mass Transfer, and look at the file on “general mass and heat equations”; we will not go through derivation in class!
· To save us all the hassle of derivations, the textbooks have lists of equations for certain situations!
· No derivations required in this unit, but it is important to know that each MT or HT equation comes with its own set of assumptions based on these derivations fodder for critical analysis of your solutions!
3.2 steady-state diffusion with bulk motion (cengel p825-830)
· So far we have only considered molecular diffusion in a stationary medium, i.e. a situation in which diffusion happens in the absence of convection
· In reality, molecular diffusion usually occurs in combination with at least a small degree of bulk fluid movement, or convection.
· The bulk fluid movement could be due to external factors related to the medium itself (e.g. natural convection) but is also a direct reaction to the molecular diffusion itself
· Fick’s Law states that the rate of diffusive mass transfer of solute A in the presence of bulk fluid movement ( A in moles.s-1m-2) through phase B is equal to the Fickian diffusive term as described in Section 2.1, plus a convective term to account for the bulk motion of fluid
Where is the molar-average velocity
· The molar-average velocity is difficult to measure or calculate, and so in practical situations, the original equation for Fick’s law (Section 2.1) is used, but the DAB may be corrected to account for the effect of bulk motion. Therefore, while you do NOT need to USE the equation above in any of your calculations, you need to know about the assumptions underlying the diffusivity data.
· For a full derivation, see Cengel Chapter 14-8 or Welty p402-407
properties
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