Three-CSTR System

Frequency-domain indkatois that reaolt from Ziegler-Nichob settiags and 0.316 damping cmfiicicnt settings 

Bode plots of openloop and dosedloop BG /(l + BG ) for a thiee-CSTR system and PI oon-tiollers. 

Now we are ready to find the controller settings required to give various frequency-domain specifications with P, PI, and PID controllers. 

Gain margla Suppose we want to find the value of feedback controller gain Kf that gives a gain margin GM 2. We must find the value of that makes the Nyquist plot of 

Nyquist plots for a three-CSTR system with proportional controllers. 

---

Combine the three first-order ODEs describing the three-CSTR system of Sec. 3.2 into one third-order ODE in terms of Then solve for the response of to a unit step change in C 0 assuming all Jt s and t s are identical. 

Simulate the three-CSTR system on a digital computer with an on-off feedback controller. Assume the manipulated variable is limited to +1 mol of A/lt around the stcadystate value. Find the period of oscillation and the average value of for values of the load variable of 0.6 and 1. 

Examiik 9J. The isothermal three CSTR system is described by the three linear ODEs... 

A feedback controller is added to the three-CSTR system of Example 9.5. Now... 

Example 10.4. Consider again the three-CSTR system. We have already developed its doscdloop characteristic equation with a proportional controller [Equation (10.22)]. 

Root locus curves for a. three-CSTR system. 

Root locus results for three CSTR system... 

KU. Use the Routh stability criterion to find the ultimate gain of the closedloop three-CSTR system with a PI controller ... 

Figure lillb is a Nichols chart with two G B curves plotted on it. They are from the three-CSTR system with a proportional controller. 

Nichols chart with a three-CSTR system openloop Gmho plotted. 

The system of DEs (6.144) describes an initial value problem in nine dimensions, once we have chosen the initial values for the dimensionless temperatures j/ (0) and the dimensionless concentrations XAi(0) and x-s/JO) of the components A and B, respectively, in each of the three tanks numbered = 1, 2, 3 at the dimensionless starting time r = 0. We study various sets initial conditions for the problem (6.144) that lead to different steady-state output concentrations xb3 of the desired component B in the three CSTR system. 

The middle two plots show the dynamics of the reaction in the second tank. One steady state of tank 2 lies at (xAi(Tend), XBi(Tend), y Tend)) ss (0.33,0.67,1.28) and another at (xAi Tend), XBi Tend), y (Tend)) ss (0, 0.2,1.87). The latter gives the smaller yield of jg and results from the initial second tank conditions (x,i2(0), xb2(0), 2/2(0)) = (0.95,0,1.3) depicted in black. These two steady states are stable. There is another unstable steady state for this data, but our graphical method does obviously not allow us to find it because it is an unstable saddle-type steady state that will repel any profile that is near to it. It can be easily obtained from the steady-state equations, though. For a method to find all steady states of a three CSTR system, see Section 6.4.3. 

The actual number of steady states of our three CSTR system can be found by solving the steady-state equations of Section 6.4.3. 

Find estimates for the minimal amounts of the components A and B that are needed to prime the three CSTR system at start-up in order to obtain near zero output of component B in tank 3. (Most wasteful case scenario)... 

A feedback controller is added to the three-CSTR system of Example 7.5. Now C o is changed by the feedback controller to keep at its setpoint, which is the steady-state value of Ca3- The error signal is therefore just — Ca3 (the perturbation in Ca3). Find the transfer function of this closedloop system between the disturbance Cao and Ca3- List the values of poles, zeros, and steady-state gain when the feedback controller is ... 

EXAMPLE 8.2. A three-CSTR system has an openloop transfer function relating... 

FIGURE 11.14 Bode plots of openloop GmGc and closedloop GmGc + GmGc) for a three-CSTR system and PI controllers. 

A PI controller has two adjustable parameters, and therefore we should, theoretically, be able to set two frequency-domain specifications and find the values of T/ and Kc that satisfy them. We cannot make this choice of specifications completely arbitrary. For example, we cannot achieve a 45 phase margin and a gain margin of 2 with a PI controller in this three-CSTR system. A PI controller cannot reshape the Nyquist plot to make it pass through both the ( — y/l, 5 Jl) point and the (-0.5,0) point because of the loss of phase angle at low frequencies. 

---