Reference no: EM133644100

Modeling Dynamic Systems

The chemical processing plant is shown in Figure 1. The purpose of the device is to maintain the correct height of fluid in the tank. Fluid height is affected by the disturbance flow entering the tank at the top and by drainage through the outlet pipe at the bottom. The pump through a long thin inlet pipe provides replacement fluid. In the final system design the fluid height will be measured and a controller will determine the command signal to the pump motor. We wish to determine a model for the system shown in Figure 1 so that the controller can be properly designed.

For our preliminary analysis of this system, let us make the following assumptions:

• The dynamics of the motor are fast compared to the rest of the system and therefore the inductance of the motor windings can be neglected and only the winding resistance R_m, needs to be included. The motor's bearing losses and rotor inertia are also small and can be neglected.

• The pump's pressure-flow curves are shown in Figure 2 for various pump speeds. The pump is driven by the motor such that the motor torque T_m, and the pump pressure P_m are related by a constant n. The flow Q of the pump is also related to the speed ω of the motor by the same constant n. In other words, this is a positive displacement pump with no internal leakage or losses. The pump output pressure can be characterized by:

P_m = P₀ - KQ

where P₀ is the pressure drop across the pump at Q = 0 for a given motor torque and K is a proportionality constant.

• The inlet pipe has linear fluid inductance (inertia) I_L but no fluid resistance; therefore, the flow in the pipe Q_i is proportional to the pressure drop in the pipe P_i.
• The outlet pipe has only fluid resistance (and no fluid inductance) and is characterized by the following nonlinear equation:
P_b - P_a = R₁Q + R₂Q³ (2)
• The tank can be characterized as a linear fluid capacitor, C_T.

Question 1. Make a bond graph for each component and then combine them in order to generate a bond graph for this system.

Question 2. Augment the bond graph with causal strokes.

Question 3. Augment the graph to include the controller and comment on what type of signal the controller should produce.

Question 4. What is the order of this system without the controller? Determine a suitable minimal set of physical state variables to represent this system.

Question 5. Determine the state equations.

Question 6. Assume that the flow Q is small, such that the outlet pipe's behavior in Equation (2) can be simplified to P_b - P_a = R₁Q. Write the new state equations in state-space form.

Question 7. If the inlet pipe has resistance R_i will this affect the effective pump characteristics as seen by the tank? Explain by deriving a new set of state equations.

Question 8. Verify your explanation in Problem 7 by comparing the simulated response of the model with and without R_i (see system parameter values given below).

Question 9. For high flows, the cubic term of the outlet pipe resistance cannot be neglected. Compare the behavior of the system by simulating the linear (no cubic flow term) versus nonlinear (with cubic flow term) output pipe resistance. In both cases neglect the inlet pipe resistance R_i.

Attachment:- Modeling Dynamic Systems.rar