Figure 7.18 shows a liquid-level system in which two tanks have cross-sectional areas A1 and A2, respectively. The volume flow rate into tank 1 is qi. A pump is connected to the bottom….

## 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 *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 kg,

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

Conductive Heat Transfer (area = 1 × 10^{−}^{4} m^{2}, thickness = 0.1 m, and thermal conductivity = 401 W/(m・K)), and Convective Heat Transfer (area = 1 × 10^{−}^{4} m^{2} and heat transfer coefficient = 20 W/(m^{2}・K)). Assume that the heat flow rate is *q*_{0} = 400 J/s and the surrounding temperature is *T*_{0 }= 298 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*).