ELOG - Time constant for heating the seismometer frame

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Message ID: 1952 Entry time: Mon Jun 29 11:13:21 2015

Author:	Megan
Type:	Misc
Category:	Seismometer
Subject:	Time constant for heating the seismometer frame

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.

$dQ = mc\cdot dT$

$\frac{dQ}{dt}=P=mc\frac{dT}{dt}$

$T(t)=\frac{1}{mc}\int P dt$

Where the value $P$ 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:

$T_{Al}(t)=\frac{1}{mc}\Big[P_{in}t-\int_{0}^{t}P_{out}dt\Big]$

$T_{Al}(t)=\frac{1}{mc}\Big[P_{in}t-\int_{0}^{t}KA_{side}d_{insul}(T_{Al}(t)-T_{lab})dt\Big]$

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).

$T'_{Al}(t)=A-BT_{Al}(t)$

where A and B are defined as follows: A=(1/mc)(P_in-KA_sided_insulT_lab), and B=(KA_sided_insul)/(mc). Solving the differential equation via Mathematica yields:

$T_{Al}(t)=\frac{A}{B}+C_1e^{-Bt}$

Therefore the time constant tau is just the reciprocal of B:

$\tau = \frac{1}{B}=\frac{mc}{KA_{side}d_{insul}}$

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:

$\tau=\frac{(10kg)(0.9\frac{J}{g\cdot K})(1000\frac{g}{kg})}{(58.121\frac{J}{K\cdot s\cdot m^2\cdot m})(0.75m^2)(0.0254m)}\approx\boxed{2.25\text{ hours}}$

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.

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