The room shown in Figure 7.34 has a heater with heat flow rate input of \(q_{0}\). The thermal capacitances of the heater and the room air are \(C_{1}\) and \(C_{2}\), respectively. The thermal resistances of the heater-air interface and the room wall-ambient air interface are \(R_{1}\) and \(R_{2}\), respectively. The temperatures of the heater and the room air are \(T_{1}\) and \(T_{2}\), respectively. The temperature outside the room is \(T_{0}\), which is assumed to be constant.

a. Derive the differential equations relating the temperatures \(T_{1}, T_{2}\), the input \(q_{0}\), and the outside temperature \(T_{0}\).

b. Using the differential equations obtained in Part (a), determine the state-space form of the system. Assume the temperatures \(T_{1}\) and \(T_{2}\) as the outputs.

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FIGURE 7.34 Problem 5. Room air T C T G Heater 90 R R To