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The applet at the bottom of this page uses a numerical approximation of the heat equation to determine the temperature in a thin, homogeneous bar.

The temperature range of the displayed temperature windows is between 0 and 200 (Kelvin). Draw an initial temperature in the bottom window (below the words initial temperature) by pressing the mouse button at the levels that you want the temperature to be at. Then press the calculate button. After that you can press the play button to see the temperature of the bar change over time. You can use the play, stop, and pause buttons as well as the scroll bar to control the playback of the changing temperature. Please feel free to experiment with changing the various parameters as well as the initial temperature.

This program allows you to draw an initial temperature and set the boundary conditions for the bar that this program tries to simulate. You can also change the various options to simulate different types of bars. The equation that I am using, where u(x,t) denotes the temperature at point x at time t and the derivatives should be partial derivatives with the one on the right being a second derivative, is: du(x,t)/dt = a^2 * d^2u(x,t)/dx^2

There are two types of boundary conditions for each side of the bar. These are keeping the end at the same temperature, which is constant, as the side of the bar (if "Use R" is not checked) and allowing free radiation (if "Use R" is checked). In the case of free radiation, the temperature change at the boundary point is equal to the radiation factor times the difference between the temperature at the point and the temperature on that side of the bar. The case where an end is insulated is just a special case of the free radiation condition where the radiation factor is equal to 0.

I have assumed that all of the variables that the user can set are constants. The following are the options that are visible in the applet. If the variable that the option represents is not unitless, the units of the variable are shown in square brackets. Also, if the variable is used in the above equation, then its representation is shown in parantheses.

- dx [m] = the distance in space between each point used in the calculation
- dt [s] = the amount of time that each frame represents
- Tend [s] = the time value of the last frame that is calculated
- Points = the number of points kept in memory for this model (should be at most 16,000,000)
- Length [m] = the length of the bar
- Thermal Diffusivity [m^2/s](a^2) = thermal conductivity / (density * specific heat)
- Stability Factor = a^2*dt/dx^2 (should be less than 0.5)
- End Time Factor = Tend*PI^2*a^2/L^2
- T [K] = The temperature at the specified side of the bar
- R = The radiation factor

Please note that if you type in a value that is not in the allowed range of numbers or is not a number, then the old value will be put back and an error dialog will pop up. This means that changing values may be a little more difficult than usual.

I have just one more thing to mention before the applet. Although the math may be difficult, all that the equation states in this case is that if the system is stable then the temparature will be smoothed out and turned into pretty much a straight line as time progresses. Anyway here's the applet.

This web page is maintained by Borys
Bradel. Last update: Sep. 24, 2001

Address:
http://www.eecg.utoronto.ca/~bradel/projects/thinrodtemp/index.html