CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > General Forums > System Analysis

1D heat flow through rod with changing thermal diffusivity

Register Blogs Community New Posts Updated Threads Search

Reply
 
LinkBack Thread Tools Search this Thread Display Modes
Old   December 2, 2014, 12:21
Wink 1D heat flow through rod with changing thermal diffusivity
  #1
New Member
 
Carl Johan
Join Date: Dec 2014
Posts: 1
Rep Power: 0
cjgn44 is on a distinguished road
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 ?
cjgn44 is offline   Reply With Quote

Old   December 7, 2014, 21:40
Default
  #2
Senior Member
 
Troy Snyder
Join Date: Jul 2009
Location: Akron, OH
Posts: 220
Rep Power: 19
tas38 is on a distinguished road
Quote:
Originally Posted by cjgn44 View Post
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 ?
You can solve the problem analytically by considering a two domain problem (each with constant conductivity) and match the temperatures and heat fluxes at the domain interface.

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.
tas38 is offline   Reply With Quote

Reply


Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are Off
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
Geothermal Reservoir 2D heat flow modelling phobo ANSYS 0 May 25, 2014 11:10
Transient Thermal: nonlinear heat equation vagantes ANSYS 0 May 2, 2013 06:38
Conjugate heat transfer, thermal boundary conditions Sarah Main CFD Forum 1 April 7, 2010 08:20
thermal diffusivity and specfic heat of NO2 intern CFX 0 May 6, 2002 12:07
Hot rod in transverse air flow Anders Larsson Main CFD Forum 3 April 19, 1999 10:28


All times are GMT -4. The time now is 08:53.