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(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.

          (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.

(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 θ̅ ^∘ C (ambient temperature) and that at t = 0 it is
immersed in a bath of temperature θ̅ + ^∘ C, where is the bath temperature (which may
be constant or changing), measured from the ambient temperature.
Let us denote the instantaneous thermometer temperature by θ̅ + θ ^∘ C, so that θ is the change in
the temperature of the thermometer, satisfying the condition that θ(0) = 0.
The dynamics of this thermometer system can be characterized in terms of a thermal
resistance R (^∘ C/kcal/s) that resists the heat flow and a thermal capacitance C (kcal/^∘ C) that
stores heat.
Write a mathematical model for the system.
Thermometer
θ̅ + θ
θ̅ +
Bath
Figure 1 - Thin, glass-walled mercury thermometer system.

Added by Thomas F.

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