[granzer]

Hello, I had recently written a Matlab code for solving the simple 2d heat conduction equation for steady-state temperature distribution in a square plate, using FVM. All the 4 edges of the square plate were set a fixed temperature. I am wondering if there are any experimental results showing steady-state temperature distribution in a square plate, for validation purposes?


Solution Ref: http://tutorial.math.lamar.edu/Class...placesEqn.aspx

[LuckyTran]

Maybe you can start over and describe your problem again?

A square plate with fixed temperatures... the same temperature? Then the plate is uniform temperature. What is there to even solve and what needs validating? Honestly I have no clue what you're asking for.

Initial conditions? Boundary conditions? Dimensions/material to establish a regime? Is this steady heat conduction or transient? If all four sides of the plate are the same temperature then the plate is a uniform temperature and there's nothing to even think about.

Unless there is some novelty in your setup, the answer is probably no. No one is going to spend their time testing something that has a known analytical solution. Temperature measurements are hard and expensive. Maybe you should try validating against the analytical solution instead of an experimental one.

[granzer]

Originally Posted by LuckyTran

Maybe you can start over and describe your problem again?

A square plate with fixed temperatures... the same temperature? Then the plate is uniform temperature. What is there to even solve and what needs validating? Honestly I have no clue what you're asking for.

Initial conditions? Boundary conditions? Dimensions/material to establish a regime? Is this steady heat conduction or transient? If all four sides of the plate are the same temperature then the plate is a uniform temperature and there's nothing to even think about.

Unless there is some novelty in your setup, the answer is probably no. No one is going to spend their time testing something that has a known analytical solution. Temperature measurements are hard and expensive. Maybe you should try validating against the analytical solution instead of an experimental one.

So sorry about the lack of clarity in my post. I coded a Gauss FVM based discretization to solve the transient 2D heat conduction equation without any source term. I used the central differencing method for the diffusion term and implicit trapezoidal rule for time integration. Each edge of the square can have different temperatures and the initial condition is set at room temperature. I solved the resulting system of linear equations using the successive over-relaxation method. I just wanted to know if there are any papers, maybe with different IC and BC, dealing with conduction in a square plate that gives the steady-state temperature distribution so I can compare the results.

[FMDenaro]

Originally Posted by granzer

So sorry about the lack of clarity in my post. I coded a Gauss FVM based discretization to solve the transient 2D heat conduction equation without any source term. I used the central differencing method for the diffusion term and implicit trapezoidal rule for time integration. Each edge of the square can have different temperatures and the initial condition is set at room temperature. I solved the resulting system of linear equations using the successive over-relaxation method. I just wanted to know if there are any papers, maybe with different IC and BC, dealing with conduction in a square plate that gives the steady-state temperature distribution so I can compare the results.

The steady state solution satisfies the Laplace equation with the BCs setting you prescribe, you can get many details on a lot of well known textbooks.


On the other hand, you can also build your own test-case, just assume the solution T(x,y) as any analytical function that satisfies the Laplace equation, use this function on the boundaries as either Dirichlet or Neumann BCs and then solve numerically. Thus, you can evaluate the discretization error on grids of several sizes.

[LuckyTran]

Don't bother looking for any experimental dataset. Just validate it against an analytical solution.

If properties are constant such that the equation is Linear then you can find exact analytical solutions (almost entirely) by hand. The steady state solution follows Laplace's equation. Even the entire transient solution can be solved exactly using eigenfunction expansion, Fourier transform,or Laplace transform approach + superposition. The simple square domain makes these analytical solutions accessible.

If non-linear, I even more highly doubt any experimental dataset exists.

[wangsen992]

With you problem set up, check solutions for laplace equation with eigenfunction expansion. You can have analytical solutions with no sweat.