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1D heat flow through rod with changing thermal diffusivity |
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December 2, 2014, 12:21 |
1D heat flow through rod with changing thermal diffusivity
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#1 |
New Member
Carl Johan
Join Date: Dec 2014
Posts: 1
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Hello! I'm comparatively new to this whole world of CFD. I've looked at the examples in "12 Steps to navier stokes" and would say I understand that okay.
(http://lorenabarba.com/blog/cfd-pyth...navier-stokes/) I've now gone back to 1D, and I'm trying to do some computations on the 1D heat equation: http://en.wikipedia.org/wiki/Heat_equation My case is for example this: I've got a rod that's insulated around the sides such that the 1D case makes sense to use, the right side of the rod is connected to something such that the temperature is a constant T_1 and the left side a constant T_2. (similar to step 3 in the link) If the thermal diffusivity is constant throught the rod, then this problem is something I can find solved on the internet everywhere (using a finite difference scheme). But what if the thermal diffusivity is nonuniform, for instance, if the rod is splitted up into 2 equally big parts but with a different diffusivity for each part (ie, made of different materials). Can I then simply make the thermal diffusivity a function of x_i in the equation? Ie, write: du/dt = a(x)*(d^2 u/dx^2) instead of du/dt = a*(d^2 u/dx^2) If yes, why? if no, why not ? |
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December 7, 2014, 21:40 |
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#2 | |
Senior Member
Troy Snyder
Join Date: Jul 2009
Location: Akron, OH
Posts: 220
Rep Power: 19 |
Quote:
If you solve the problem numerically, the variable conductivity does not present a problem. You simply discretize the conductivity. For example, a finite difference discretization of the diffusion term would be: d/dx ( a*du/dx ) = ( (a*du/dx)_i+1 - (a*du/dx)_i ) / ( x_i+1 - x_i ) = ... a_i+1*( u_i+1 - +u_i ) / ( x_i+1 - x_i )^2 - a_i*( u_i - u_i-1 ) / ( ( x_i+1 - x_i)*( x_i - x_i-1 ) ) In the case of non-spatially varying conductivity and constant grid spacing, this reduces the second-order, central difference formula for second derivative. |
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