Transient 1-D heat transfer 2

[chinmoy123]

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.

 

[espnloser]

Three methods I can think of---

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)

 

[25362]

With heat flowing perpendicular to the slab area in one direction,

[∂][θ]/[∂]t = [α] [∂]²[θ]/[∂]x²​

where t is time; [θ], temperature; x, the slab thickness, and [α] is the thermal diffusivity = k/[ρ]c_p

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.

 

[sailoday28]

chinmoy123 (Mechanical)
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://www.nzifst.org.nz/unitoperations/httrtheory4.htm

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

 

[athomas236]

sailday28

What an interesting site and in such an unexpected place.

athomas236

 

[rmw]

sailoday28,

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

 

[25362]

A Google search for "unsteady state heat transfer" can show us a variety of tutorials on the subject.