(4 Marks)
Mathematical modeling of a thermal system - Thermometer system.
Consider the thin, glass-walled mercury thermometer system shown in Figure 1. Assume that
the thermometer is at a uniform temperature $\bar{\theta}$ $^\circ C$ (ambient temperature) and that at $t = 0$ it is
immersed in a bath of temperature $\bar{\theta} + \theta_b$ $^\circ C$, where $\theta_b$ is the bath temperature (which may
be constant or changing), measured from the ambient temperature.
Let us denote the instantaneous thermometer temperature by $\bar{\theta} + \theta$ $^\circ C$, so that $\theta$ is the change in
the temperature of the thermometer, satisfying the condition that $\theta(0) = 0$.
The dynamics of this thermometer system can be characterized in terms of a thermal
resistance R ($^\circ C$/kcal/s) that resists the heat flow and a thermal capacitance C (kcal/$^\circ C$) that
stores heat.
Write a mathematical model for the system.
Thermometer
$\bar{\theta} + \theta$
$\bar{\theta} + \theta_b$
Bath
Figure 1 - Thin, glass-walled mercury thermometer system.
