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Process Dynamics And Control 2nd Edition Dale E. Seborg, Thomas F. Edgar, Duncan A. Mellich - Solutions

What is the differential equation model of the series PID controller? Qualitatively describe its response to a step change in e(t).
Several linear transmitters have been installed and calibrated as follows:(a) Develop an expression for the output of each transmitter as a function of its input. Be sure to include appropriate units.(b) What is the gain of each transmitter? zero? span? Is the gain constant orvariable?
A process instrumentation diagram of a flash drum is shown in Figure. Steam is condensed in a steam coil to vaporize a portion of the liquid feed, and the liquid product is removed by a pump. There are control valves for the steam flow, vapor product, liquid product, feed flow, and the steam chest
Adapted from Luyben and Luyben (1997).(a) Calculate the gain of an orifice plate and differential pressure transmitter for flow rates of 10, 50, 75 and 90% of full scale,(b) Calculate the gains of linear, equal-percentage, and square root control valves at the same operating points assuming
Chilled ethylene glycol (sp gr = 1.11) is pumped through the shell side of a condenser and a control valve at a nominal flow rate of 200 gal/min. The total pressure drop over the entire system is constant. The pressure drop over the condenser is proportional to the square of the flow rate and is 30
A pneumatic control valve is used to adjust the flow rate of a petroleum fraction (sp gr = 0.9) that is used as fuel in a cracking furnace. A centrifugal pump is used to supply the fuel, and an orifice meter/differential pressure transmitter is used to monitor flow rate. The nominal fuel rate to
I. M. Appelpolscher has acquired considerable knowledge of pneumatic control valves over the years. In particular, he knows that the flow characteristic of a “quick-opening” valve is f = √ℓ which is one reason they also are called “square root” valves. Appelpolscher reasons that, if
A theoretical force balance for the control valve shown in Figure can be expressed asAssuming the second-order differential equation is linear, find values of the coefficients of the equation (in deviation variable form) and determine whether the valve dynamic behavior is overdamped orunderdamped.
Steady-state tests have been run on a neutralization tank, where pH is the controlled variable and w is the acid flow rate. The data follow the relationship shown in figure: You are asked to evaluate which type of valve (quick opening, linear, equal percentage) would best match the nonlinear gain
A thermocouple Located in a thermowell and connected to a temperature transmitter is used to measure the liquid temperature in a bioreactor. A steady-state calibration of this instrument (transducer) yields the following data: A process engineer runs a test on the reactor under controlled
An engineer sets the pressure in a supply tank using a very accurate manometer as a guide and then reads the output of a 20-psig pressure gauge attached to the tank as 10.2 psig. Sometime later she repeats the procedure and obtains values of 10.4 and 10.3 psig. What can she say about the
A process temperature sensor/transmitter in a fermentation reactor exhibits second-order dynamics with time constants of 1 s and 0.1 s. If the quantity being measured changes at a constant rate of 0.1 °C/s, what is the maximum error that this instrument combination will exhibit? ‘What is the
Consider the distillation column shown in Figure. It would be reasonable to control the liquid level in the reflux drum, hD, by manipulating either reflux flow rate, R, or distillate flow rate, D. How would the nominal value of the rel1u rat in (R/D) influence your choice? As a specific example,
A stirred-tank blending system with a bypass stream is shown in Figure. The control objective is o control the composition of a key component in the exit stream, x4. The chief disturbance variables are the mass fractions of the key component in the inlet streams, x1 and x2. Using the information
Suppose that the distillation column shown in figure has been designed to separate a methanol-water mixture that is .50% methanol (MeOH). This high purity column has a large number of trays and a nominal distillate composition of xD = 5 ppm of MeOH. Because a composition analyzer is not available,
It has been suggested that the capital cost for the distillation column in figure can be reduced by using a ?flooded condenser.? In the proposed design, the reflux drum would be eliminated, and the condensed vapor in the condenser would provide the liquid inventory for the reflux and distillate
The exit stream from a chemical reactor is sent to a storage tank, as shown in Figure. The exit stream from the storage tank serves as the feed stream to a separation process. The function of the intermediate storage tank is to ?damp? feed disturbances and to allow the separation process to
Consider the liquid storage system shown in figure. Only volumetric flow rates, q and q, can be manipulated. Determine the model degrees of freedom, NF, and the control degrees of freedom,NFC.
A double-pipe heat exchanger with a partial bypass for the cold stream is shown in Figure. The mass flow rate of the hot stream wh, and the bypass fraction, f, can be manipulated. Heat losses can be neglected.(a) Determine the model degrees of freedom, NF, arid the control degrees of freedom, NFC,
Consider the blending system of Exercise 102. Inlet how rate, w2, and the bypass fraction, f, can be manipulated. Determine the model degrees of freedom, NF, and the control degrees of freedom, NFC.
Nitrogen gas is used to purge a mixture of air and small amounts of a flammable vapor from a process vessel. The nitrogen is introduced continuously through an opening in the vessel, and the resulting mixture is vented to the atmosphere through another opening. The capacity of the vessel is 20,000
The probability of a particular type of sensor functioning properly is 0.99. Consequently, a triply redundant sensor system has been proposed for a critical measurement. Thus, three independent sensors will be installed, end the median of the three measurements will be used for the alarms and
Air samples from a process area are continuously drawn through a 1/4-in, diameter tube to an analytical instrument that is located 40 meters away. The tubing has an outside diameter of 6.35 mm and a wall thickness of 0.762 mm. The flow rate through the transfer line is 10 cm3/s (or ambient
The liquid feed stream for the gas-liquid separator (or flush drum) shown in Figure consists of a hydrocarbon mixture. Because the pressure in the vessel is significantly lower than the feed pressure, part of the liquid feed flashes to form a gas phase. The hydrocarbons are flammable and somewhat
The loss of the coolant to a process vessel can produce an unacceptably high pressure in the vessel. As a result, a pressure relief valve is used to reduce the pressure by releasing the vapor mixture to the atmosphere. But if the mixture is toxic or flammable, the release can be hazardous. For the
A temperature control system for a distillation column is shown in figure. The temperature T of a tray near the top of the column is controlled by adjusting the reflux flow rate R. Draw a block diagram for this feedback control system. You may assume that both feed flow rate F and feed composition
Consider the liquid-level PI control system shown in Figure with the following parameter values: A = 3 ft2, R = 1.0 min/ft2, Kv = 0.2 cfm/psi, Km = 4 mA/ft, Kc 4, and τ1 = 3 min. Suppose that the system is initially at the nominal steady state with a liquid level of 2 ft. If the set point is
Consider proportional-only control of the stirred-tank heater control system in Figure. The temperature transmitter has a span of 50°F and a zero of 55 °F. The nominal design conditions are T = 80 °F and T1 = 65 °F. The controller has a gain of 5 while the gains for the control valve and
It is desired to control the exit concentration of c3 of the liquid blending system shown in Figure. Using the information given below, do the following:(a) Draw a block diagram for the composition control scheme. (Use the symbols in the figure as much as possible.)(b) Derive an expression for each
A proportional-only feedback controller is implemented as shown in figure. We ordinarily think of the process and controller gains as being positive.(a) Give an example of a process whose gain (between the manipulated and controlled variables) is negative. Draw a schematic diagram of the process to
For a Liquid-level control system similar to that in figure, Appelpolscher has argued that integral control action is not required because the process acts as an Integrator (cf, Eq. 11-77). To evaluate his assertion, determine whether proportional-only Control will eliminate offset for step changes
A block diagram for internal model control, a control technique that is considered in Chapter, is shown in Figure. Transfer function Gp denotes the process model, while Gp,denotes the actual process transfer function. It has been assumed that Gv = Gm = 1 for simplicity, Derive closed-loop transfer
An electrically heated, stirred-tank system is shown in figure. Using the given information do the following: (a) Draw a block diagram for the case where T3 is the controlled variable and voltage signal V2 is the manipulated variable. Derive an expression for each transfer function. (b) Repeat part
The block diagram of a special feedback control system is shown in Figure. Derive art expression for the closed-loop transfer function, Y(s)/D(s).
A block diagram of a closed-loop system is shown in figure.(a) Derive a closed-loop transfer function for disturbance changes, Y(s)/Ds).(b) For the following transfer functions, what values of Kc will result in a stable closed-loopsystem?
A mixing process consists of a single stirred-tank instrumented as shown in figure. The concentration of a single species A in the feed stream varies. The controller attempts to compensate for this by varying the flow rate of pure A through the control valve. The transmitter dynamics are
A PI controller is to be used in a temperature control system for a bioreactor. For nominal conditions, it has been determined that the closed-loop system is stable when τ1 = 10 min and – 10 < Kc < 0. Would you expect these stability limits to change for any of the following instrumentation
A process is described by the transfer function Find the range of controller settings that yield stable closed-loop systems for (a) A proportional-only controller. (b) A proportional-integral controller. (c) What can you say about the effect of adding the integral mode on the stability of the
The block diagram of a feedback control system is shown in figure. Determine the values of Kc that result in a stable closed-loop system.
The question has been raised whether an open-loop unstable process can be stabilized with a proportional-only controller.(a) For the process and controller shown in figure a, find the range of Kc values that yield a stable response. (Note that τ is positive.)(b) Check the gain of Y(s)/Ysp (s)
For the liquid-level control system in Figure determine the numerical values of Kc and τ1; that result in a stable closed—loop system. The level transmitter has negligible dynamics, while the control valve has a time constant of 10s. The following numerical values are available:
As a newly hired engineer of the Ideal Gas Company, you are trying to make a reputation in the Process Control Group. However, this objective turns out to be a real challenge with I. M. Appelpolscher as your supervisor. At lunch one day, LM.A declares that a simple second-order process with a PI
It is desired to control the exit temperature T2 of the beat exchanger shown In Figure by adjusting the steam flow rate w3, unmeasured disturbances occur in inlet temperature T1. The dynamic behavior of the heat exchanger can be approximated by the transfer functions:
A process is described by the transfer functionwhich includes actuator and measurement dynamics. The process engineering group has the option of redesigning the process to eliminate the right-half plane zero. They need to determine whether this modification will yield a substantially better
A feedback control system has the open-loop transfer function, GOL(s) = 0.5Kce–3s / (10s + 1). Determine the values of Kc for which the closed-loop system is stable using two approaches:(a) An approximate analysis using the Routh stability criterion and a 1/1 Padé approximation for e–3s.(b) An
A process control system contains the following(a) Show how GOL(s) can be approximated by a FOPTD model;Find K, τ, and θ for the open-loop process transfer function(b) Use Routh stability methods and your FOPTD model to find the range of Kc values that will yield a stable closed-loop system.
A process has the transfer function, where K has a nominal-value of K = 1. PID controller settings are to be calculated using the Direct Synthesis approach with ?c = 5 min.Suppose that these controller constants are employed and that K changes unexpectedly from 1 to 1 + ?.(a) For what values of
The liquid level in a reboiler of a steam-heated distillation column is to be controlled by adjusting the control valve on the steam line, as shown in figure. The process transfer function has been empirically determined to be where H denotes the liquid level (in inches) and P is the steam pressure
A process has the transfer function, G(s) 2e–0.2t/ (s + 1). Compare the PI controller settings for the following design approaches: (a) Direct Synthesis method (τc = 0.2) (b) Direct Synthesis method (τc = 1.0) (c) ITAE performance index (disturbance) (d) ITAE performance index (set
A process including sensor and control valve can be modeled by:Show that the Direct Synthesis method can be used to derive Controller M in Table 12.1, by utilizing the time-delay approximation in Eq.12-24b.
A process stream is heated using a shell and tube heat exchanger. The exit temperature is controlled by adjusting the steam control valve shown in figure. During an open-loop experimental test, the steam pressure P was suddenly changed from 18 to 20 psig and the temperature data shown below were
Suggest a modification of the Direct Synthesis approach that will allow it to be applied to open-loop unstable processes.
A process including sensor and control valve can be modeled by a fourth-order transfer function:(a) Design PID controllers using two design methods:(i) A sccond-order-plus4ime-delay model using the model reduction approach proposed by Skogestad (Section 6.3) and the modified IMC tuning relation in
Consider the level control problem in Example 12.3. Attempt to reduce the set-point overshoot of the PI controller (?c = 15) by using set-point weighting (cf. Eq. 12-39). Which value of b gives the best results? Does the value of b affect the disturbance response?
Consider the PID controller of Example 12.3 for τc = 8. (a) Suppose that the PID controller is implemented as the series form in Table 12.2, rather than as the parallel form that was employed in Figure. Are the simulated closed-loop responses significantly different from the ones in
Consider the blending system shown in Figure. A feedback control system is used to reduce the effect of disturbances in feed composition, x1, n the controlled variable, product composition, x. Inlet flow rate, w2, can be manipulated. Do the following:(a) Draw a block diagram of the feedback control
A PID controller is used to control the temperature of a jacketed batch reactor by adjusting the flow rate of coolant to the jacket. The temperature controller has been tuned to provide satisfactory control at the nominal operating conditions. Would you anticipate that the temperature controller
Suppose that a process can be adequately modeled by the first-order-plus-time-delay model in Eq. 12-10. (a) Calculate PI controller settings using the Hagglund-Astrom tuning relations in Table 12.4. (b) Cohen and Coon (1953) reported the following tuning relations to PI controllers: These tuning
Consider the experimental step response data for the heat exchanger of Exercise 12.5. Determine the PI controller settings using the step response method and two controller tuning relations: (a) Direct Synthesis method with τc = τ/3 (b) Ziegler-Nichols settings in Table 12.6. Which
Consider the transfer function model in Eq. 12-10 with K = 2, τ = 5 and θ = 1. Compare the PID controller settings obtained from the Ziegler-Nichols and Tyreus-Luyben tuning relations in Table 12.6. Simulate the closed-loop responses for a unit step change in the set point.
IGC’s operations area personnel are experiencing problems with a particular feedback control loop on an interstage cooler. Appelpolscher has asked you to assess the situation and report back what remedies, if any, are available. The control loop is exhibiting an undesirable sustained oscillation
A problem has arisen in the level control loop for the flash separation unit shown in Figure. The level control loop had functioned in a satisfactory manner for a long period of time. However, the liquid level is gradually increasing with time even though the PI level controller output has
A heat transfer process has the following transfer function between a temperature T and an inlet flow rate q where the time constants have units of minutes:If the flow rate varies sinusoidally with au amplitude of 2 L/mm and a period of 0.5 mm, what is the amplitude of the temperature signal after
Using frequency response arguments, discuss how well e-Gs can be approximated by a two-term Taylor series expansion, that is, by 1 – θs. Compare your results with those given in the text for a UI Pad approximation.
A data acquisition system for environmental monitoring is used to record the temperature of an air stream as measured by a thermocouple. It shows an essentially sinusoidal variation after about 15 s. The maximum recorded temperature is 127 °F and the minimum is 119 °F at 1.8 cycles per mm. It is
A perfectly stirred tank is used to heat a flowing liquid. The dynamics of the system have been determined to be approximately as shown in figure. Where: P is the power applied to the heater Q is the heating rate of the system T is the actual temperature in the tank Tm is the measured
For each of the following transfer functions, develop both the amplitude and phase angle of the Bode plot. Find AR and ? for each transfer function at of values of w = 0.1,1, and 10.
A second-order process transfer function is given by(a) Find |G| and (b) Plot |G (jw)| vs. wτ and Note that you can obtain the wt plots by setting τ = 1 and reparameterizing τ0/τ accordingly.
Plot the Bode diagram (0.1 w 100) of the third- order transfer function, Find both the value of w that yields a ? 180o phase angle and the value of AR at that frequency.
Using MATLAB, plot the Bode diagram of the following transfer function:Repeat for the situation where the time-delay term is replaced by a 1/1 Padéapproximation.
Two thermocouples, one of them a known standard, are placed in an air stream whose temperature is varying sinusoidally. The temperature responses of the two thermocouples are recorded at a number of frequencies, with the phase angle between the two measured as shown below. The standard is known to
Exercise 5.19 considered whether a two-Lank liquid surge system provided better damping of step disturbances than a single-tank system with the same total volume. Reconsider this situation, this time with respect to sinusoidal disturbances; that is, determine which system better damps sinusoidal
For the process described in Exercise 6.5, plot the composite amplitude ratio and phase angle curves on a single Boric plot for each of the four cases of numerator dynamics. What can you conclude concerning the importance of the zero location for the amplitude and phase characteristics of this
Develop expressions for the amplitude ratio of each of the two forms of the PW controller:(a) The parallel controller transfer function of Eq.8-14.(b) The series controller transfer function of Eq.8-15.Put the results on a single plot along with asymptotic representations of each AR curve. You may
A process has the transfer function,With K = 10, ? = 2min, ? = 1min, Obtain the Bode plot for this transfer function. Do you need to approximate the denominator exponential term, or can your computer program handle it directly?
Appelpolscher has just left IGC’s Monday morning status meeting with Smarly, Quirk, and the operations personnel. Over the weekend, a major unit in the hydrogen production process, a centrifugal compressor supplying carbon monoxide to a reactor, failed. The operations people were able to switch
Plot a Nyquist diagram for the transfer function in Exercise 13.8. Repeat with the exponential term replaced by a 1/1 Padé approximation. Why are these two plots so different in appearance for large values of w?
Plot the Bode diagram for a composite transfer function consisting of the one in Exercise 13.8 multiplied by that of a parallel-form PID controller with Kc = 0.21, τ1 = 5, and to τ1 = 0.42. Repent using the same process transfer function but with a series PID controller with filter that employs
For the process described by the transfer function (a) Find two second-order-plus-time-delay models that approximate G(s) and are of the form One of the approximate models can be found by using the method discussed in Section 6.3; the other, by visual inspection of the frequency responses of G and
Obtain Bode plots for both the transfer function given as follows and a FOPTD approximation obtained using the method discussed in Section 6.3. What do you conclude about the accuracy of the approximation relative to the original transferfunction?
In Chapter we considered a stability analysis based on substituting s = jw into the characteristic equation Demonstrate that this approach is (or is not) identical to the Bode stability criterion oil.
A process that can be modeled as a time delay is controlled using a proportional feedback controller. The control valve and measurement device have negligible dynamics and steady-state gains of Kv = 0.5 and Km = 1, respectively. After a small set-point change is made, a sustained oscillation
A Bode diagram for a process, valve, and sensor is shown in figure.(a) Determine an approximate transfer function for this system.(b) Suppose that a proportional controller is used and that a value of Kc is selected so as to provide a gain margin of 1.7. What is the phasemargin?
Consider the storage tank with sight glass in Figure. The parameter values are R1 = 05 min/ft2, R2 = 2min/ft2, A1 = 10 ft2, Kv = 2.5cfm/mA, A2 = 0.8ft2, Km = 1.5mA/ft, and ?m = 0.5 min.(a) Suppose that R2 is decreased to 0.5 min/ft2. Compare the old and new values of the ultimate gain and the
A process (including valve and transmitter) has the approximate transfer function. G(s) = 2e–0.2s/(s + 1) with time constant and time delay in minutes. Determine Pt controller settings and the corresponding gain margins by two methods: (a) Direct synthesis (τc = 0.3 mm). (b) Phase margin
Consider the feedback control system in figure, and the following transfer functions: (a) Plot a Bode diagram for the open-loop transfer function. (b) Calculate the value of K that provides a phase margin of 30?. (c) What is the gain margin when Kc =10?
Frequency response data for a process are tabulated below. These results were obtained by introducing a sinusoidal change in the controller output (under manual control) and recording the measured response of the controlled variable. This procedure was repeated for various frequencies. (a) If the
For the process in Exercise 14.5, the measurement is to be filtered using a noise filter with transfer function GF(s). Would you expect this change to result in better or worse control system performance? Compare the ultimate gains and critical frequencies with and without the filter. Justify your
The block diagram of a conventional feedback control system contains the following transfer functions: (a) Plot the Bode diagram for the open-loop transfer function. (b) For what values of K is the system stable? (c) If Kc = 0.2, what is the phase margin? (d) What value of Kc will result in a
The dynamic behavior of the heat exchanger shown in figure can be described by the following transfer functionsThe valve lift x is measured in inches. Other symbols are defined in Figure.(a) Find the Ziegler-Nichols settings for a PT controller.(b) Calculate the corresponding gain and phasemargins.
Consider a standard feedback control system with the following transfer functions: (a) Plot the Bode diagram for the transfer function, G = GvGpGm. (b) Design a PI controller for this process and sketch the asymptotic Bode diagram for the open-loop transfer function, GOL = GcG. (c) Analyze the
Hot and cold liquids are mixed at the junction of two pipes. The temperature of the resulting mixture is to be controlled using a control valve on the hot stream. The dynamics of the mixing process, control valve, and temperature sensor/transmitter are negligible and the steady-state gain is
Consider the control problem of Exercise 14.10 and a PI controller with Kc = 5 and τ1 = 0.3mm. (a) Plot the Nyquist diagram for the open-loop system. (b) Determine the gain margin from the Nyquist plot.
Consider a process model,and a PI controller with the Hagglund-Astrom settings of Table 12.4. Use the robust stability limit of Eq. 14-35 to determine how robust this control system is to the assumed value of the process gain. Assume that the time constant and time delay do notchange.
For the process in Exercise 14.5, a PI controller is tie- signed so that the peak value of the complementary sensitivity function Is MT = 1.25, Use the robust stability limit of Eq. 14-35 to estimate the values of the time constant for which closed-loop stability is guaranteed. Assume that the
In ratio control, would the control loop gain for Method I (Fig) be Less variable if the ratio were defined as R = d/u instead of R = u/d? Justify youranswer.
Consider the ratio control scheme shown in Fig. Each flow rate is measured using an orifice plate and a differential pressure (D/P) transmitter. The electrical output signals horn the D/P transmitters are related to the flow rates by the expressions.Each transmitter output signal has a range of 4
It is desired to control liquid level h2 in the storage tank system shown in Fig by manipulating flow rate q3 Disturbance variable q1 can be measured. Use the information available to do the following: (a) Draw a block diagram for a feedforward feedback control system. (b) Derive an ideal
For the liquid storage system shown in Fig the control objective is to regulate liquid level h2 despite disturbances in flow rates, q1 and q4. Flow rate q2 can be manipulated. The two hand valves have the following flow-head relations: Do the following, assuming that the flow transmitters and the
The closed-loop system in Fig has the following transfer functions: (a) Design a feedforward controller based on a steady-state analysis. (b) Design a feedforward controller based on a dynamic analysis. (c) Design a feedback controller based on the IMC approach of Chapter 12 and tc = 2. (d)

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