I made a model for our seismometer can using actual data so that we know approximately what the time constant should be when we test it out. I used the appendix in Megan Kelley's report to make a relation for the temperature in terms of time.
so and
In our case, we will heat the can to a certain temerature and wait for it to cool on its own so
We know that where k is the k-factor of the insulation we are using, A is the area of the surface through which heat is flowing, is the change in temperature, d is the thickness of the insulation.
Therefore,
We can take the derivative of this to get
, or
We can guess the solution to be
where tau is the time constant, which we would like to find.
The boundary conditions are and . I assumed we would heat up the can to 40 celcius while the room temp is about 24. Plugging this into our equations,
, so
We can plug everything back into the derivative T'(t)
Equating the exponential terms on both sides, we can solve for tau
Plugging in the values that we have, m = 12.2 kg, c = 500 J/kg*k (stainless steel), d = 0.1 m, k = 0.26 W/(m^2*K), A = 2 m^2, we get that the time constant is 0.326hr. I have attached the plot that I made using these values. I would expect to see something similar to this when I actually do the test.
To set up the experiment, I removed the can (with Steve's help) and will place a few heating pads on the outside and wrap the whole thing in a few layers of insulation to make the total thickness 0.1m. Then, we will attach the heaters to a DC source and heat the can up to 40 celcius. We will wait for it to cool on its own and monitor the temperature to create a plot and find the experimental time constant. Later, we can use the heatng circuit we used for the PSL lab and modify the parts as needed to drive a few amps through the circuit. I calculated that we'd need about 6A to get the can to 50 celcius using the setup we used previously, but we could drive a smaller current by using a higher heater resistance. |