How to Solve 1-D Transient Heat Transfer for Constant Temperature Surfaces?

In summary, the author is trying to solve the 1-D PDE for a constant temperature wall. They are not sure how to get the temperature profile and need help.
  • #1
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I am a little confuse here. I am trying to solve the 1-D Transient Heat Transfer for a Constant Temp. Surfaces "wall".
The PDE is
[tex]\frac{\partial T}{\partial t}=\alpha \frac{\partial ^2 T}{\partial x^2}[/tex]
T(to)=200*x
I am not so sure about how I'm suppose to get the Temperature profile

any help?
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  • #2
I don't understand what you've written, what do you mean by "T(to)=200*x"? Is that the temperature profile at time t=0?

To solve the PDE, have you attempted the separation of variables technique?
 
  • #3
This is a very simple problem in the area of heat conduction.
There are several well written books in this area. However I have followed one of those best in the field. I will suggest
Özışık (Ozisik for non-UTF viewers), M. Necati, "Heat Conduction", John-Wiley, 1993

and inquirer meant the initial condition by "T(to)=200*x" here. Btw, boundary conditions were not supplied. Most probably solution is sought for an identical Dirichlet type boundary conditions.

I'm sure the reference will help.
 
  • #4
siddharth said:
I don't understand what you've written, what do you mean by "T(to)=200*x"? Is that the temperature profile at time t=0?

To solve the PDE, have you attempted the separation of variables technique?
Yes, T=T(x,t) T(x,0)=200*x for 0<=x<=.5L (symetric at 0.5L) T(0,t)=0 T(L,t)=0
No, not separation of variables..
The problem was that I was using the finite difference method to find the temperature profile in the wall at different times and I had two equations for the problem but I didn't knew which one was useful.

Here are the equations
[tex]T_m^{p+1}= Fo(T_{m+1}^p+T_{m-1}^p)+(1-2Fo)T_m^p[/tex]
Subscript are for spatial nodes.
Superscript are for the time nodes.

[tex]T_m^{n+1}-T_m^n=\frac{Fo}{2}(T_{m-1}^{n+1}-2T_m^{n+1}+T_{m+1}^{n+1}+T_{m-1}^{n}-2T_m^{n}+T_{m+1}^{n})[/tex]

I found that both equations are good for solving the pde using finite difference.

Thanks siddharth

PS This how should look, (axis are not labeled...:P)
 

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  • #5
OK. Both are different FD schemes of the problem.
The first one is explicit in time* so you shoul be aware of the CFL condition that ensures stability.

The other is the Crank-Nicholson type discretization which is stable but needs system solver. In this one you have to collect unknown values (i.e. the values for the new time step which are indexed as n+1 here) to one side and solve the resulting Ax=b type linear system. This is an implicit form(*)


* Explicit in time. Values for the new time step can be obtained by direct substitution of previously known values. Spatial derivatives are obtained from the n-th time step.

* Implicit form. Spatial derivatives involves unknown values of the n+1st time step.
 
  • #6
bilgealp said:
OK. Both are different FD schemes of the problem.
The first one is explicit in time* so you shoul be aware of the CFL condition that ensures stability.

The other is the Crank-Nicholson type discretization which is stable but needs system solver. In this one you have to collect unknown values (i.e. the values for the new time step which are indexed as n+1 here) to one side and solve the resulting Ax=b type linear system. This is an implicit form(*)


* Explicit in time. Values for the new time step can be obtained by direct substitution of previously known values. Spatial derivatives are obtained from the n-th time step.

* Implicit form. Spatial derivatives involves unknown values of the n+1st time step.

Thanks a lot for the help and info.
 
  • #7
My pleasure.
 

FAQ: How to Solve 1-D Transient Heat Transfer for Constant Temperature Surfaces?

What is 1-D transient heat transfer?

1-D transient heat transfer is the process by which heat is transferred in one dimension (usually along a solid object) over a period of time. This type of heat transfer occurs when there is a difference in temperature between two points along the object, causing heat to flow from the higher temperature point to the lower temperature point.

How is 1-D transient heat transfer different from 1-D steady-state heat transfer?

In 1-D steady-state heat transfer, the temperature at any point along the object remains constant over time. This means that the rate of heat transfer is also constant. In 1-D transient heat transfer, the temperature at any point changes over time, resulting in a varying rate of heat transfer.

What are the factors that affect 1-D transient heat transfer?

The factors that affect 1-D transient heat transfer include the thermal conductivity of the material, the temperature difference between the two points, the surface area of the object, and the time duration of the heat transfer process. Other factors may include the presence of any insulating materials or external factors such as wind or radiation.

How is 1-D transient heat transfer mathematically represented?

1-D transient heat transfer can be represented using the one-dimensional heat conduction equation, which is a partial differential equation that relates the temperature at a given point to the heat flux, thermal conductivity, and time. This equation can be solved numerically using various methods such as the finite difference method or the finite element method.

What are some real-world applications of 1-D transient heat transfer?

1-D transient heat transfer is an important concept in many engineering and scientific fields. It is used in the design of building materials and insulation, as well as in the analysis of heat exchangers, electronic devices, and thermal management systems. It is also relevant in fields such as meteorology and geology, where it is used to study the transfer of heat through the Earth's surface and atmosphere.

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