L8 Diffusion 2 Fall 2009

# L8 Diffusion 2 Fall 2009 - Diffusion II Solving Diffusion...

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Diffusion II Solving Diffusion Problems Oct. 27, 2009

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Announcements Diffusion Exam next Thursday, Nov. 5 Diffusion Homework (5) Due Tuesday, Nov. 3 work through the examples in the textbook
Fick’s Laws of Diffusion x J x c D = t c x c D = 2 2 1 st law: Concentration gradient gives us direction and magnitude of flux 2 nd Law: Curvature of the concentration profile at x gives us the change of concentration in the next small bit of time, at x.

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See the handout on types of boundary conditions and solutions 1) Steady-State 2) Constant source, semi-infinite solid 3) Limited source, semi-infinite solid There are also graphs for solving different geometries.
Example: purifying gas through a membrane 5 mm thick sheet of Pd cross sectional Area = 0.2 m 2 c(H, high) = 0.3 kg/m 3 C(L, low) = 0 kg/m 3 D = 1.0 x 10 -8 m 2 /s What mass of hydrogen is purified each hour? Higher pressure gas mixture, Higher partial pressure of H 2 lower partial pressure of pure H 2 gas

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Draw a picture Identify what is known Identify the boundary conditions and initial conditions. What case are we dealing with? Identify what we need to find Write down symbolic relationships between what we have and what we need to find Last step should be “plugging in” numbers Problem-Solving Strategy
The words, “ steady-state diffusional membrane for purifying hydrogen” tell us that there will be no changes in concentration over time . This means that dc/dt = 0 If dc/dt = 0, then we know that the curvature of the concentration profile is also zero . This means that there will be a linear concentration profile. This means that the concentration gradient is constant in the membrane. This means that the flux is constant everywhere in the membrane. If the concentration profile is linear, than either two points or one point and a slope will determine the line. How to interpret the problem wording

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Pd membrane 0 5 mm x High pressure side Low pressure side Constant flux of H through the membrane C high = 0.3 kg/m 3 C low = ??? kg/m 3 D (H in Pd) = 1.0 x 10 -8 m 2 /s How many g(H)/h? ... “steady-state diffusional membrane for purifying hydrogen” A possible diagram for this situation
() ( ) 1 2 1 2 x x x c x c D x c D x c D J x = Δ Δ = =

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() ( ) 1 2 1 2 x x x c x c D x c D x c D J x = Δ Δ = = thickness membrane high low x c c D J =
() ( ) 1 2 1 2 x x x c x c D x c D x c D J x = Δ Δ = = thickness membrane high low x c c D J = () ( ) 5mm 3 . 0 10 0 . 1 3 2 8 m kg low s m x c J × = The book should have specified c low . That is a mistake. They use c low = 0 3 m kg

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() ( ) 1 2 1 2 x x x c x c D x c D x c D J x = Δ Δ = = thickness membrane high low x c c D J = () ( ) 5mm 3 . 0 10
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L8 Diffusion 2 Fall 2009 - Diffusion II Solving Diffusion...

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