Figure 7.46 represents the temperature dynamics of two adjacent objects, where the thermal capacitances of the objects are \(C_{1}\) and \(C_{2}\), respectively. Assume that the temperatures of both objects are uniform, and they are \(T_{1}\) and \(T_{2}\), respectively. The heat flow rate into object 1 is \(q_{0}\), and the temperature surrounding object 2 is \(T_{0}\). There are two modes of heat transfer involved, conduction between the objects and convection between object 2 and the air. The corresponding thermal resistances are \(R_{1}\) and \(R_{2}\), respectively.

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

b. Build a Simscape model of the physical system, and find the temperature outputs \(T_{1}(t)\) and \(T_{2}(t)\). Use default values for the blocks of Thermal Mass \((\) mass \(=1 \mathrm{~kg}\), specific heat \(=447 \mathrm{~J} \cdot \mathrm{K} / \mathrm{kg}\), and initial temperature \(=300 \mathrm{~K})\), Conductive Heat Transfer (area \(=1 \times 10^{-4} \mathrm{~m}^{2}\), thickness \(=0.1 \mathrm{~m}\), and thermal conductivity \(=401 \mathrm{~W} /(\mathrm{m} \cdot \mathrm{K}))\), and Convective Heat Transfer (area \(=1 \times 10^{-4} \mathrm{~m}^{2}\) and heat transfer coefficient \(=20 \mathrm{~W} /\left(\mathrm{m}^{2} \cdot \mathrm{K}\right)\) ). Assume that the heat flow rate is \(q_{0}=400 \mathrm{~J} / \mathrm{s}\) and the surrounding temperature is \(T_{0}=298 \mathrm{~K}\).

c. A Build a Simulink block diagram based on the differential equations obtained in Part (a), and find the temperature outputs \(T_{1}(t)\) and \(T_{2}(t)\).

Transcribed Image Text:

FIGURE 7.46 Problem 4. 90 A T J T2 C To R R