To determine the time constant of the system, a model of the heat flows through the system was made, starting with the basic ΔQ=mcΔT.
Where the value is the net power flowing through the system, or P_{in}-P_{out}. P_{in} is the power supplied by the heaters, and P_{out} is the power lost radiatively through the insulation. Pin is a known value, while Pout can be calculated via the K-factor, the thickness of the insulation, the area of a side, and the temperature difference between the two sides. Taking all this into account, the differential equation becomes:
This is just a differential equation where the temperatue of the frame (T_{al}(t)) is related to the integral of itself. The equation can be rearranged such that the temperature is related to the derivative of the temperature (as differential equations typically are).
where A and B are defined as follows: A=(1/mc)(P_{in}-KA_{side}d_{insul}T_{lab}), and B=(KA_{side}d_{insul})/(mc). Solving the differential equation via Mathematica yields:
Therefore the time constant tau is just the reciprocal of B:
The units of tau do work out to be time, given that the units of K are taken to be [E]/[A]/[T]/[t]/[d]. Plugging in numbers to get an estimation of the order of magnitude gives:
This value is an estimation so far; within the next couple days I will go measure the frame and get more accurate values. I will also recheck the calculation, because I think that the time constant should have some dependence on the power input to the system. |