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 C1 and C2, respectively. Assume that the temperatures of both objects are uniform, and they are T1 and T2, respectively. The

Question

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 [latex]C_{1}[/latex] and [latex]C_{2}[/latex], respectively. Assume that the temperatures of both objects are uniform, and they are [latex]T_{1}[/latex] and [latex]T_{2}[/latex], respectively. The heat flow rate into object 1 is [latex]q_{0}[/latex], and the temperature surrounding object 2 is [latex]T_{0}[/latex]. 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 [latex]R_{1}[/latex] and [latex]R_{2}[/latex], respectively.

a. Derive the differential equations relating the temperatures [latex]T_{1}, T_{2}[/latex], the input [latex]q_{0}[/latex], and the outside temperature [latex]T_{0}[/latex].

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

c. Build a Simulink block diagram based on the differential equations obtained in Part (a) and find the temperature outputs [latex]T_{1}(t)[/latex] and [latex]T_{2}(t)[/latex].

Accepted Answer

a. Similar to Example 7.8, the differential equations relating the temperatures [latex]T_{1}, T_{2}[/latex], the input [latex]q_{0}[/latex], and the outside temperature [latex]T_{0}[/latex] can easily be obtained as

[latex] \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} [/latex]

The equations can be rearranged as

[latex] \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}} ight) T_{2}=\frac{1}{R_{2}} T_{0} \end{aligned} [/latex]

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 [latex]q_{0}[/latex], and an Ideal Temperature source block is used to represent the surrounding temperature [latex]T_{0}[/latex].

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,

[latex] \begin{aligned} & C_{1}=C_{2}=m c=1 imes 447=447 \mathrm{~J} / \mathrm{K} \ & R_{1}=\frac{L}{k A}=\frac{0.1}{401 imes 1 imes 10^{-4}}=2.49 \mathrm{~K} \cdot \mathrm{s} / \mathrm{J} \ & R_{2}=\frac{1}{h A}=\frac{1}{20 imes 1 imes 10^{-4}}=500 \mathrm{~K} \cdot \mathrm{s} / \mathrm{J} \end{aligned} [/latex]

Run either simulation to generate Figure 7.45, showing the resulting temperature outputs [latex]T_{1}(t)[/latex] and [latex]T_{2}(t)[/latex] of the two adjacent objects.

Question 7.12: Temperature Dynamics of Two Adjacent Objects Figure 7.42 rep...

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