Measuring the thermal diffusivity.

[Tsportmat]

Homework Statement

Evening everybody. I am attempting to devise an experiment in order to measure the thermal diffusivity of ice and water.

Imagine 3 brass plates, with water held between them, in approximately a 1mm gap (dx) by surface tension.
The 3 plates have thermocouples between them in order to tell the difference in temperature between them. The part I am miffed about is how to obtain the diffusivity from this measurement.

I have been given this equation to link them, but I am struggling to find where it came from and the significance if the d²T/dx²

d²T/dx²=T₁-2T₂/dx²+T₃

Homework Equations

dQ/dT=-D dH/dx

The Attempt at a Solution

This is not really appropriate, as the above is an experiment plan which I have devised, I just need the final link! I apologise for not using the exact format, but any ideas or leads for me to follow up would be greatly appreciated, I've come to a dead end.


Thankyou in advance.

 

[Jhero]

Thank you for sharing your experiment plan with us. It sounds like an interesting project! In order to obtain the thermal diffusivity from your measurements, you will need to use Fourier's law of heat conduction, which states that the rate of heat transfer through a material is proportional to the temperature gradient and the thermal conductivity of the material. This can be written as:

dQ/dt = -kA dT/dx

where dQ/dt is the rate of heat transfer, k is the thermal conductivity, A is the cross-sectional area, and dT/dx is the temperature gradient.

In your experiment, you can use the temperature difference between the three plates (T1, T2, T3) and the distance between the plates (dx) to calculate the temperature gradient (dT/dx). You can also measure the rate of heat transfer (dQ/dt) using the thermocouples.

Now, to obtain the thermal diffusivity (D), you will need to use the equation you mentioned:

d2T/dx2 = T1 - 2T2/dx2 + T3

This equation is derived from Fourier's law by assuming that the temperature gradient is constant between the plates. By substituting this equation into Fourier's law, we can solve for the thermal diffusivity:

dQ/dt = -kA (T1 - 2T2/dx2 + T3)

D = kA/(T1 - 2T2/dx2 + T3)

I hope this helps you in your experiment. Best of luck!