
(a)
The pump resistance and the steady-state height.

Answer to Problem 7.32P
The pump resistance is
Explanation of Solution
Calculation:
The figure below shows the pump curve and the line for steady-state flow.
Figure-(1)
Write the expression of the two resistances in series.
Here, the outlet flow rate of the tank is
Write the expression of mass conservation.
Here, the outlet flow rate of the tank is
Write the expression of the height of the tank.
Here, the pump resistance is
Refer to Figure-(1) to obtain the pressure rise in the pump as
Substitute
Thus, the pump resistance is
Substitute 0 for
Substitute
Thus, the steady-state height is
(b)
The linearized dynamic model of the height deviation in the tank.

Answer to Problem 7.32P
The linearized dynamic model of the height deviation in the tank is
Explanation of Solution
Calculation:
Write the expression of the linear model.
Write the expression of mass conservation.
Here, the pump resistance is
Write the expression of small deviation in mass flow rates of the mass conservation.
Here, the area of the tank is
Write the expression of the outlet flow rate of the tank.
Write the expression of the deviation.
Here, the area of the tank is
Substitute
Substitute
Substitute
Thus, the linearized dynamic model of the height deviation in the tank is
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Chapter 7 Solutions
System Dynamics
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