Question 7.12: Temperature Dynamics of Two Adjacent Objects Figure 7.42 rep...
Temperature Dynamics of Two Adjacent Objects
Figure 7.42 represents the temperature dynamics of two adjacent objects, in which 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 \left(\right. area =1 \times 10^{-4} \mathrm{~m}^{2}, thickness =0.1 \mathrm{~m}, and thermal conductivity \left.=401 \mathrm{~W} /(\mathrm{m} \cdot \mathrm{K})\right), and Convective Heat Transfer (area =1 \times 10^{-4} \mathrm{~m}^{2} and heat transfer coefficient =20 \mathrm{~W} / \left.\left(\mathrm{m}^{2} \cdot \mathrm{K}\right)\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. 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).
a. Similar to Example 7.8, the differential equations relating the temperatures T_{1}, T_{2}, the input q_{0}, and the outside temperature T_{0} can easily be obtained as
\begin{gathered} C_{1} \frac{\mathrm{d} T_{1}}{\mathrm{~d} t}=q_{0}-\frac{T_{1}-T_{2}}{R_{1}}, \\ C_{2} \frac{\mathrm{d} T_{2}}{\mathrm{~d} t}=\frac{T_{1}-T_{2}}{R_{1}}-\frac{T_{2}-T_{0}}{R_{2}} . \end{gathered}
The equations can be rearranged as
\begin{aligned} & C_{1} \frac{\mathrm{d} T_{1}}{\mathrm{~d} t}+\frac{1}{R_{1}} T_{1}-\frac{1}{R_{1}} T_{2}=q_{0}, \\ & C_{2} \frac{\mathrm{d} T_{2}}{\mathrm{~d} t}-\frac{1}{R_{1}} T_{1}+\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right) T_{2}=\frac{1}{R_{2}} T_{0} \end{aligned}
b. The Simscape block diagram corresponding to the physical system is shown in Figure 7.43. Two Thermal Mass blocks are used to represent objects 1 and 2. A Conductive Heat Transfer block and a Convective Heat Transfer block are used to represent the two modes of heat transfer involved, conduction between the objects and convection between object 2 and the air. An Ideal Heat Flow Source block is included to represent the heat flow rate input q_{0}, and an Ideal Temperature source block is used to represent the surrounding temperature T_{0}.
c. The Simulink block diagram built based on the differential equations obtained in Part (a) is shown in Figure 7.44. Based on the default values used in Simscape modeling, the following system parameters can be determined,
\begin{aligned} & C_{1}=C_{2}=m c=1 \times 447=447 \mathrm{~J} / \mathrm{K} \\ & R_{1}=\frac{L}{k A}=\frac{0.1}{401 \times 1 \times 10^{-4}}=2.49 \mathrm{~K} \cdot \mathrm{s} / \mathrm{J} \\ & R_{2}=\frac{1}{h A}=\frac{1}{20 \times 1 \times 10^{-4}}=500 \mathrm{~K} \cdot \mathrm{s} / \mathrm{J} \end{aligned}
Run either simulation to generate Figure 7.45, showing the resulting temperature outputs T_{1}(t) and T_{2}(t) of the two adjacent objects.
