Derive the differential equations relating the temperatures T1, T2, the input q0, and the outside temperature T0.

 

Case study

Figure 7.46 represents the temperature dynamics of two adjacent objects, where 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 heat flow rate into object 1 is q0, and the temperature surrounding object 2 is T0. 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 R1 and R2, respectively.

a. Derive the differential equations relating the temperatures T1T2, the input q0, and the outside temperature T0.

b. Build a Simscape model of the physical system, and find the temperature outputs T1(t) and T2(t). Use default values for the blocks of Thermal Mass (mass = 1 kg,

specific heat = 447 J・K/kg, and initial temperature = 300 K),

Conductive Heat Transfer (area = 1 × 104 m2, thickness = 0.1 m, and thermal conductivity = 401 W/(m・K)), and Convective Heat Transfer (area = 1 × 104 m2 and heat transfer coefficient = 20 W/(m2・K)). Assume that the heat flow rate is q0 = 400 J/s and the surrounding temperature is T= 298 K.

c. Build a Simulink block diagram based on the differential equations obtained in Part (a), and find the temperature outputs T1(t) and T2(t).

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erive the differential equations in terms of the liquid heights h1 and h2.

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