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,
![T(t)=\frac{1}{mc}\int_{0}^{t}\frac{kA}{d}[T_{lab}-T(t')]dt'=\frac{kA}{mcd}(T_{lab}t-\int_{0}^{t}T(t')dt')](https://latex.codecogs.com/gif.latex?T%28t%29%3D%5Cfrac%7B1%7D%7Bmc%7D%5Cint_%7B0%7D%5E%7Bt%7D%5Cfrac%7BkA%7D%7Bd%7D%5BT_%7Blab%7D-T%28t%27%29%5Ddt%27%3D%5Cfrac%7BkA%7D%7Bmcd%7D%28T_%7Blab%7Dt-%5Cint_%7B0%7D%5E%7Bt%7DT%28t%27%29dt%27%29)
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)
![T'(t)=-\frac{16}{\tau}e^{-t/\tau}=B-C[16e^{-t/\tau}+24]](https://latex.codecogs.com/gif.latex?T%27%28t%29%3D-%5Cfrac%7B16%7D%7B%5Ctau%7De%5E%7B-t/%5Ctau%7D%3DB-C%5B16e%5E%7B-t/%5Ctau%7D+24%5D)
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.
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