Transient 1-D heat transfer
Transient 1-D heat transfer
(OP)
Its almost 6 years since I have dealt with transient HT. So bear with me if the question is frivolous.
I have a 1-D sample with both sides being held at say 150C.
I want to find out the time taken for the mid plane to reach a temp (say 100 C)
How do I solve the follwoing transient 1-D HT equation
Density * Specific Heat *(del T/del t) = Kxx (del^2 T/del x^2)
Because one side deals with temperature vs time and the other side is temperature vs x (distance) how do I come up with a equation which tells me the time taken for the midplane of the plate to reach a certain temperature?
Any help will be appreciated.
I have a 1-D sample with both sides being held at say 150C.
I want to find out the time taken for the mid plane to reach a temp (say 100 C)
How do I solve the follwoing transient 1-D HT equation
Density * Specific Heat *(del T/del t) = Kxx (del^2 T/del x^2)
Because one side deals with temperature vs time and the other side is temperature vs x (distance) how do I come up with a equation which tells me the time taken for the midplane of the plate to reach a certain temperature?
Any help will be appreciated.





RE: Transient 1-D heat transfer
1) Heisler Charts : From what I remember from undergrad it usually dealt with convective Boundary conditions. I am sure it can be used for fixed temperature BCs
2) derive the exact solution : Will involve separation of variables and some math
example d^2T/dx^2 = 1/alpha *dT/dt
initial condition T(x,0)=Ti (initial condition of rod)
BC T(x=0,t)= Te (i.e temp at fixed condition at all times = 150)
since same BCs at end...the other BC at the midplane will be dT/dx (x=L/2,t) = 0
x=L/2=midplane of rod or beam
and using SOV and such you can derive an exact soluion for the temp at any time and at any location of the rod
for Reference Heat Conduction, Second Edition, Wiley, 1993
M. Necati Ozisik
3) finite difference --- my prof at grad school was an advocate of this......you can use excel or write a program. If your old school you will know fortran but try writing a Matlab code.
Peace
(I aint an expert or anything...so double check what I said...infact I need a job)
RE: Transient 1-D heat transfer
where t is time; θ, temperature; x, the slab thickness, and α is the thermal diffusivity = k/ρcp
Solutions for the case of heat flowing from both sides is generally found from the Gurney-Lurie diagram for heating or cooling a large slab.
See fig. 10.7 in Professor Aksel L. Lydersen's Fluid Flow and Heat Transfer (John Wiley & Sons). Chapter 10, Unsteady State Heat Transfer, provides examples.
RE: Transient 1-D heat transfer
The case you have brought up lends itself to exact solutions. And espnloser (Mechanical)and 25362 (Chemical) have both suggested using exact solutions.
An example of the type of charts being refered to is in
http:
The limitation with the exact solution is that the thermal diffusivity is assumed constant.This might be a good point for other to input on your original question.
Regards
RE: Transient 1-D heat transfer
What an interesting site and in such an unexpected place.
athomas236
RE: Transient 1-D heat transfer
That is an excellent site. The heat transfer pertains to more than just food processes. A star from me.
There was a poster a few weeks ago who wanted a primer on heat transfer. This would be a good place for him/her.
rmw
RE: Transient 1-D heat transfer
A Google search for "unsteady state heat transfer" can show us a variety of tutorials on the subject.