CHE2163: HEAT AND MASS TRANSFER
STEADY STATEDIFFUSION – BEYOND FICK WEEK 10
Learning Objectives, Readings, and Learn ChemE Videos
1. Fick’s Law with bulk motion – from the derivation of the general equation
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a. Incropera p792-794; Cengel p820-824; Welty p433-438
b. Learn ChemE video on “Deriving Differential Equations for
Mass Transfer”
2. Stagnant Film Diffusion (NB = 0)
a. Incropera p799-802; Cengel p825-827; Welty p452-458
b. Learn ChemE video on “Unimolecular Diffusion” and
“Unimolecular Diffusion Example”
3. (NA = -NB)
a. Incropera p805-808; Cengel p827-830; Welty p462-463
b. Learn ChemE video on “ ” and
“ Example”
By the end of this week, you should be able to:
Identify the best analytical approach for a particular problem involving mass diffusion (Fick, SFD, EMCD).
Solve these problems using appropriate equilibrium equations. Page 1
3.1 GENERAL MASS EQUATION (INCROPERA CH 14; WELTY CH 25)
For Heat Transfer, 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 – look at the file on “Derivation of general mass equation”; 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 (𝑗̅ 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
𝑗̅ =𝑦 (𝑗̅ +𝑗̅ )−𝐷
𝐴 𝐵𝐴 𝐵 𝐴𝐵𝑑𝑥
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
As previously discussed, Fick’s Law isn’t entirely accurate in practice because diffusive mass transfer will always involve
some (even very small) amount of bulk motion
This is very common in process equipment:
o Evaporation in a tank of a volatile solute in air
o Condensation of water vapour onto a cooling pipe
(a)Estimate the amount of ethanol lost to evaporation per day due to evaporation from a cylindrical tank, 2m in diameter, at a height of 3m. The liquid contains 5 % (mole) ethanol in water and the ambient temperature is 320K, and the saturated vapour pressure of ethanol is 5 kPa.
We have already solve this using Fick’s Law – but does it really apply?
3.3 STAGNANT FILM DIFFUSION
When a solute A is diffusing through a bulk vapour B, Fick’s law cannot easily be applied because the bulk vapour is not stationary
If both A and B are undergoing mass transfer, then the interaction must be taken into account when calculating mass transfer rates – this is very complex!
The “stagnant film” assumption states that while the bulk vapour B may be moving (i.e. by convective effects), there is
not net mass transfer of B, i.e. 𝑁 = 0; therefore, we can
ignore the mass transfer rate of the bulk vapour B
Under the stagnant film assumption, the rate of diffusive mass
transfer of solute A through stagnant phase B is:
𝐶𝐷𝐴𝐵 (𝑦𝐴2 − 𝑦𝐴1)
𝑃𝑡𝐷𝐴𝐵 (𝑃𝐴2 − 𝑃𝐴1) = − 𝑅𝑇∆𝑧 𝑃𝐵,𝑙𝑚
𝐶𝐷𝐴𝐵 (𝐶𝐴2 − 𝐶𝐴1) ∆𝑧 𝐶𝐵,𝑙𝑚
𝑗 =−𝜌𝐷𝐴𝐵(𝑤𝐴2−𝑤𝐴1) 𝐴 ∆𝑧 𝑤𝐵,𝑙𝑚
𝑗 =−𝜌𝐷𝐴𝐵(𝜌𝐴2−𝜌𝐴1) 𝐴 ∆𝑧 𝜌𝐵,𝑙𝑚
EXAMPLE: What is the rate of evaporation of ethanol from a tank (2m diameter; 3m height) containing 5% (mole) ethanol?
3.4 EQUIMOLAR COUNTERDIFFUSION
When a solute A is diffusing through a bulk vapour B, Fick’s law cannot easily be applied because the bulk vapour is not stationary
If both A and B are undergoing mass transfer, then the interaction must be taken into account when calculating mass transfer rates – this is very complex!
The “equimolar counterdiffusion” assumption states that the molar transfer rates of A and B are equal but with opposite
̇̇ directions, hence maintaining overall pressure, i.e. 𝑁 = −𝑁 ;
Under the equimolar counterdiffusion assumption, the rate of diffusive mass transfer of solute A through phase B is:
𝑗𝐴=− ∆𝑧 (𝑦𝐴2−𝑦𝐴1)
𝑗𝐴 = − ∆𝑧 (𝐶𝐴2 − 𝐶𝐴1)
𝑗𝐴 =−𝑅𝑇∆𝑧(𝑃𝐴2−𝑃𝐴1)
𝑗𝐴 = − 𝜌𝐷𝐴𝐵 (𝑤𝐴2 − 𝑤𝐴1) ∆𝑧
𝑗𝐴 = − 𝐷𝐴𝐵 (𝜌𝐴2 − 𝜌𝐴1) ∆𝑧
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