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

 

The room shown in Figure 7.34 has a heater with heat flow rate input of q0. The thermal capacitances of the heater and the room air are C1 and C2, respectively. The thermal resistances of the heater–air interface and the room wall–ambient air interface are Rand R2, respectively. The temperatures of the heater and the room air are T1 and T2, respectively. The temperature outside the room is T0, which is assumed to be constant.

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

b. Using the differential equations obtained in Part (a), determine the state-space form of the system. Assume the temperatures T1 and T2 as the outputs.

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

  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 equation relating the liquid height h and the volume flow rate qi at the inlet.

  Consider the single-tank liquid-level system shown in Figure 7.19, where the volume flow rate into the tank through a pipe is qi. The liquid leaves the tank through an orifice….

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

Figure 7.20 shows a hydraulic system of two interconnected tanks that have the same cross-sectional area of A. A pump is connected to tank 1. Assume that the relationship between the….