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Patankar 1980 (Diffusion Convection Problems)

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Old   November 1, 2017, 17:44
Default Patankar 1980 (Diffusion Convection Problems)
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Harris
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Hello,

I'm a chemistry major and am very new to the CFD field and I'm having difficulties coping with the Partial differential concepts and derivations covered extensively in this course.

There are three problems I'm stuck on and have no clue on how to proceed with the derivations. I wish to better myself at this work, but unfortunately, there are no TAs or tutors for such a course and my professor expects us to already have sound foundation in fluid mechanics and PDE math.

One of the problems I'm working on today is Patankar's 1980 book Chapter 5, problem 2.

d/dx ( ρuϕ- Γdϕ/dx )=S

where ϕ=ϕ_0 @ x=0, and ϕ=ϕ_L @ x=L

(1) I need to find exact solution (exponential scheme) and compare it to a numerical solution.

Notes: I can at least do the integration over control volume and divergence theorem. But has no clue how the exact solution with the Peclet number is completely derived (the text merely states this is the exact solution). If someone can hint and show how, that would be nice. Maybe it can give me a clue on how to derive the (SL^2)/((ϕ_L - ϕ_0 ) )

I'm also advised to work on problems 5.4 and 5.5 in the chapter. Hopefully, someone can give me a nudge on these problems too, especially 5.4 (mass leakage through a porous duct).

I'm just too hopelessly ill -prepared to take this class. Thanks!
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Old   January 25, 2018, 07:56
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Rami Ben-Zvi
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Quote:
Originally Posted by hhandoko View Post
Hello,

I'm a chemistry major and am very new to the CFD field and I'm having difficulties coping with the Partial differential concepts and derivations covered extensively in this course.

There are three problems I'm stuck on and have no clue on how to proceed with the derivations. I wish to better myself at this work, but unfortunately, there are no TAs or tutors for such a course and my professor expects us to already have sound foundation in fluid mechanics and PDE math.

One of the problems I'm working on today is Patankar's 1980 book Chapter 5, problem 2.

d/dx ( ρuϕ- Γdϕ/dx )=S

where ϕ=ϕ_0 @ x=0, and ϕ=ϕ_L @ x=L

(1) I need to find exact solution (exponential scheme) and compare it to a numerical solution.

Notes: I can at least do the integration over control volume and divergence theorem. But has no clue how the exact solution with the Peclet number is completely derived (the text merely states this is the exact solution). If someone can hint and show how, that would be nice. Maybe it can give me a clue on how to derive the (SL^2)/((ϕ_L - ϕ_0 ) )

I'm also advised to work on problems 5.4 and 5.5 in the chapter. Hopefully, someone can give me a nudge on these problems too, especially 5.4 (mass leakage through a porous duct).

I'm just too hopelessly ill -prepared to take this class. Thanks!
Hi hhandoko,

I guess it may be too late, but I just noticed your message, and try to explain in simple terms how to solve this equation analytically. Please see the attached.
Regards,
Rami
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File Type: docx ADE.docx (46.2 KB, 48 views)
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