Surface temperature of a tile

In summary, the conversation discusses a MATLAB program for calculating surface temperature on a tile and finding the most stable and accurate differencing technique. The four methods being compared are accurate for small time steps, but Backwards/DuFort show parabolic behavior for larger time steps. The discussion also touches on the stability and accuracy of the Crank and Forward Euler methods, as well as potential issues with the implementation of the methods.
  • #1
mattyboson12
41
0
Hi, I have written a MATLAB program to calculate the surface temperature on a tile and I am trying to find the most stable/accurate differenceing technique. I have plotted the surface temperature (where its actual value is ~122.5) against the time step for the differencing techniques shown below. All of them are accurate for small time steps, and then Backwards/DuFort decrease/increase parabolically, is this due to second order accuracy? And why is crank unstable for large timesteps and fowards isn't?

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  • #2
Let's see the PDEs you're solving.

Chet
 
  • #3
Hi Chet

The program calculates the temperature using fourier's equation in one dimension: dU/dt=α*(d^2 U/d^2 x) . I'm guessing crank is the best method as it seems to have the best compromise between stability/accuracy compared to the others. I am just confused as to why the differencing methods form very distinct graphs ie. why the crank oscillates around the actual value whereas dufort flies off to infinity

kind regards
Matt
 
  • #4
Based on my experience, you should't be encountering these difficulties solving the 1D transient heat conduction equation using most of these methods. Backward differencing, in particular, should not give rise to the zigs and zags that you are seeing. Are you sure you are implementing these methods correctly? How are you solving for the temperatures at n+1 using your Backward Euler scheme? Are you solving a tri-diagonal matrix?

Chet
 
  • #5
I've double checked my code and all the methods seem to be okay. Yep, i have used the tri-diagonal matirx and then the built in function tdm() to solve the matrix

Matt
 
  • #6
Here's something you can check. The Backward Euler method should return the steady state solution after one time step if the time step is made extremely large. Does it? That's an indication of how stable the Backward Euler method is.

Chet
 

1. What factors affect the surface temperature of a tile?

There are several factors that can affect the surface temperature of a tile, including the type of material the tile is made of, the color of the tile, the amount of sunlight or heat it is exposed to, and the surrounding temperature and humidity.

2. Why does the surface temperature of a tile matter?

The surface temperature of a tile can affect the overall temperature and comfort of a room or outdoor space. It can also impact energy efficiency and heating and cooling costs.

3. How can I measure the surface temperature of a tile?

The surface temperature of a tile can be measured using a thermometer or infrared camera. It is important to take multiple readings in different areas of the tile to get an accurate average temperature.

4. Can the surface temperature of a tile change over time?

Yes, the surface temperature of a tile can change over time. Factors such as weather, exposure to sunlight, and the type of material can all impact the temperature of the tile.

5. Are there any safety concerns related to the surface temperature of a tile?

High surface temperatures of tiles in outdoor spaces, such as around a pool or in a patio, can be a safety concern. It is important to choose materials and colors that will not become too hot to touch in direct sunlight.

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