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Analytical solution for 1D heat flow problem

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Old   March 25, 2010, 11:25
Default Analytical solution for 1D heat flow problem
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Hi all,
I have a hollow cylinder with pipe wall and insulation of given thickness (dw,dt) . given is the inlet temperature, outside air temperature, air flow heat transfer coefficient and outside air heat transfer coefficient - all property data available.

I need the temperature distribution at a given cross section at any given length x ? T(x) = ?

Could you guys guide me to some analytical formulation for this?

Thanks,

CFDtoy
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Old   March 25, 2010, 12:34
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2d heat diffusion equation in cylindrical coordinates?
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Old   March 25, 2010, 12:58
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cool beans dude

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Old   March 25, 2010, 12:59
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Yyyyyyaaaaaaaaa
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Old   March 25, 2010, 13:32
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hi there,
i have a pipe thickness and then the insulation thickness and then the heat transfer coefficient outside and inside leading to an effective thermal resistance of the system. is there an analytical expression to compute the T(x) along the length of the hollow cylinder?

Thanks.

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2d heat diffusion equation in cylindrical coordinates?
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Old   March 26, 2010, 04:49
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Ok, so I assume you have a steady state heat conduction problem, so you can just use Laplace's equation.

The conduction constant you have given are imposed as boundaries to this equation. You can remove the theta-dependency, since T only depends on r and x.

So what you need is:
1. Laplace equation (wikipedia)
2. proper formulation of boundary conditions (Fourier?)
3. A book on how to solve partial differential equations

hint: the temperature profile for any x has the same shape in r.
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Old   January 17, 2014, 07:07
Default Problem with 1D problem
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Hi everyone!!!

I am trying to simulate a 1d heat transfer in a medium confined between two large walls (top and bottom wall). I created a 2d rectangle in icem cfd. For top and bottom lines I have given the boundary conditions as walls. But I am not clear what boundary condition I should use for left and right lines which are not supposed to be present. (I created them only for meshing)..


Pls reply.
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Old   January 17, 2014, 18:31
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Hi,

Impose adiabatic walls (symmetry condition).

Cheers,
Michujo.
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Old   January 17, 2014, 20:25
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>> I need the temperature distribution at a given cross section at any given length x ? T(x) = ?

This makes no sense. You're saying you need the distribution at a given cross-section, which means T(r), and at any given length, which means T(x). Therefore, this is a 2D (axisymmetric) problem T(r,x).

You are also implying in your original statement that there is flow in the pipe, but you provide _no_ information about the flow. Is it a developing boundary layer? Is it a fully developed flow? Is it laminar or turbulent? etc, etc. Depending on these conditions there may or may not be an analytic solution.

The suggestion by RBJ that this is a diffusion problem (Laplace equation in steady state) is incorrect in light of my comments above.

What I'm reading from your statement is that you're trying to solve a typical undergraduate (or graduate, depending on the school) heat transfer problem. It's always instructive, for you first of all, to begin with the mass, momentum and energy equations (Navier Stokes equations), do similarity analysis to eliminate terms that are relatively small, and get the final simplified equation set. Once you have that, the solution becomes obvious.

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Old   January 18, 2014, 04:47
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Quote:
Originally Posted by michujo View Post
Hi,

Impose adiabatic walls (symmetry condition).

Cheers,
Michujo.

HI !! thanks for your reply.. Actually I am dealing with radiation problem.. If it is conduction , we can give adiabatic. but for radiation problem, the wall may have some emissivity, absorbtivity problems right? So, I dont think, it s possible to give adiabatic condition. What do u say?
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Old   January 18, 2014, 21:11
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If it's a 1D problem between the two walls (meaning there is no variation of the dependent variable along the walls), then it means there are no gradients in that direction. The suggestion by michujo is correct (you can also apply periodic BC, but adiabatic is just fine). Also, you shouldn't bother putting too many grids in the second direction - just one cell should be sufficient

PS - I had earlier responded to CFDtoy, without realizing that it was a question sent out three yeas ago. This new question should have been posed under a new thread

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Old   January 20, 2014, 05:47
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Quote:
Originally Posted by adrin View Post
If it's a 1D problem between the two walls (meaning there is no variation of the dependent variable along the walls), then it means there are no gradients in that direction. The suggestion by michujo is correct (you can also apply periodic BC, but adiabatic is just fine). Also, you shouldn't bother putting too many grids in the second direction - just one cell should be sufficient

PS - I had earlier responded to CFDtoy, without realizing that it was a question sent out three yeas ago. This new question should have been posed under a new thread

Adrin
Thanks a lot adrin The periodic bc requires us to give mass flow rate or pressure gradient as input.. but in my case, the medium s nt moving and it is trapped... i dont hve either of these input values.. wat to do then?
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Old   January 20, 2014, 07:44
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Why not just use adiabatic as was already recommended (correctly)?
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Old   January 20, 2014, 14:17
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Quote:
Originally Posted by adrin View Post
Why not just use adiabatic as was already recommended (correctly)?
Thank u so much michujo and adrin The results I got is quite convincing ...

I also have one doubt.. If we have to specify a wall as isothermAL, we can just give the value of the temperature in boundary conditions>>edit>>thermal >>thermal conditions>>temperature....

Am i right??

And if we want to specify a medium (fluid /solid) as isothermal, how to give a specified temperature for that? Because in boundary conditions panel, for solid or fluid, we can only edit the name and nothing else.. plssssssssssss tell me how to specify a medium as isothermal... ?? Do we have to specify it in reference values section ?? or do we have to give it under cell zone conditions>>edit>>Fixed values>>Temperature ???

I tried it with fixed value once an as i am wrking in radiation heat transfer problems, i dont think I am right...

plsssssssssssss give me ur suggestions michujo and adrin... I badly need it...

Last edited by myaccount; January 21, 2014 at 07:46.
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