Imagine two nodes, which are connected with a certain unknown conductivity. One node has a fixed temperature of 30 degrees Celsius. The other node has a initial temperature of 80 degrees Celsius. This means that this node is cooling down. The node is after t seconds cooled down to a certain temperature but it hasnt reached Steady State yet. The tau time is determined from this cooling down.

How can I derive the heat capacity from the second node by using these known parameters?


If the temperature of the hot node is $T(t)$ then the temperature drop across the medium between them is:

$$ \Delta T(t) = T(t) - 30 $$

and the rate of heat flow is:

$$ \frac{dE}{dt} = k\Delta T(t) = k(T(t) - 30) \tag{1} $$

where $k$ is an unknown constant that describes the conductivity and geometry of the link between the nodes.

The heat content of the hot node is:

$$ E_h = C_hT $$

where $C_h$ is the heat capacity of the hot node, and therefore:

$$ \frac{dE_h}{dt} = C_h\frac{dT}{dt} $$

and combining this with equation (1) gives;

$$ \frac{dT}{dt} = \frac{k}{C_h}(T(t) - 30) $$

Which integrates to:

$$ T = 50\,e^{-tk/C_h} + 30 $$

By graphing $\log(T-30)$ against $t$ to will get a straight line of gradient $k/C_h$. However all you get is this ratio. You cannot determine $C_h$ unless you know $k$ and vice versa.


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