Practice Problems: Process Control

Question 1:
A process control engineer is designing a temperature control system for a CSTR (Continuously Stirred Tank Reactor). The reactor has a transfer function of G_p(s) = 3/(5s + 1), and a proportional controller with K_c = 2 is implemented. The transmitter has a gain of 0.5 mV/°C and the control valve has a gain of 4 °C/mV. What is the offset (steady-state error) when a unit step change in setpoint occurs?

(a) 0.077
(b) 0.083
(c) 0.125
(d) 0.000

Question 2:
A control engineer is tuning a PID controller for a heat exchanger using the Ziegler-Nichols ultimate gain method. During testing, the controller exhibits sustained oscillations at an ultimate gain (K_u) of 8.0 and an ultimate period (P_u) of 2.4 minutes. Using the Ziegler-Nichols tuning rules for a PID controller, what is the recommended integral time (τ_I) in minutes?

(a) 0.6
(b) 1.2
(c) 1.8
(d) 2.4

Question 3:
A chemical plant operator notices that a level control loop is oscillating. The process engineer analyzes the system and finds that the characteristic equation is s² + 4s + K = 0, where K is the loop gain. To ensure stability using the Routh-Hurwitz criterion, what is the maximum value of K?

(a) K <>
(b) K <>
(c) K > 0
(d) K > 4

Question 4:
A distillation column control engineer implements a cascade control system. The primary loop has a time constant of 15 minutes and the secondary loop has a time constant of 1.5 minutes. What is the approximate ratio of the primary to secondary time constants, and is cascade control appropriate?

(a) Ratio = 10; cascade control is appropriate
(b) Ratio = 5; cascade control is not appropriate
(c) Ratio = 10; cascade control is not appropriate
(d) Ratio = 15; cascade control is appropriate

Question 5:
A process control engineer is analyzing a first-order plus dead time (FOPDT) process with transfer function G(s) = (5e^-2s)/(10s + 1). Using the IMC (Internal Model Control) tuning method with a desired closed-loop time constant λ = θ (dead time), what is the proportional gain K_c for a PI controller?

(a) 0.5
(b) 1.0
(c) 2.0
(d) 4.0

Question 6:
A reactor temperature control system uses a thermocouple with a time constant of 10 seconds. The measurement exhibits first-order lag behavior. If a step change of 50°C occurs in the actual temperature, how long will it take for the thermocouple to register 63.2% of this change?

(a) 5 seconds
(b) 10 seconds
(c) 15 seconds
(d) 20 seconds

Question 7:
A process engineer is evaluating a control valve for a flow control loop. The valve has an equal percentage characteristic with a rangeability of 50:1 and an installed gain at 50% opening of 2.5 (gpm/%). What is the approximate installed gain at 25% opening if the valve follows ideal equal percentage behavior with α = 50?

(a) 0.35 gpm/%
(b) 0.71 gpm/%
(c) 1.25 gpm/%
(d) 3.54 gpm/%

Question 8:
A chemical engineer designs a feedforward control system for a heat exchanger where cooling water flow rate compensates for feed flow rate changes. The process gain from feed flow to outlet temperature is K_d = -3.5°C/(kg/s) and from cooling water to outlet temperature is K_m = 4.2°C/(kg/s). What is the required feedforward controller gain K_ff?

(a) -1.20
(b) -0.83
(c) 0.83
(d) 1.20

Question 9:
A process control engineer implements a PI controller with K_c = 3.0 and τ_I = 5 minutes for a liquid level control system. The controller output was at 45% before a step change in level error of +8% occurred. What will be the controller output 2.5 minutes after the step change (assuming the error remains constant)?

(a) 57%
(b) 69%
(c) 81%
(d) 93%

Question 10:
A control engineer analyzes a closed-loop system with open-loop transfer function G_OL(s) = 10/[(s+2)(s+5)]. Using the direct substitution method, determine if the system is stable and find the steady-state gain of the closed-loop system.

(a) Stable; K_ss = 0.5
(b) Stable; K_ss = 1.0
(c) Unstable; K_ss = undefined
(d) Stable; K_ss = 10

Question 11:
A reactor pressure control system has a second-order transfer function G(s) = 4/(s² + 2s + 4). A process engineer needs to determine the damping ratio (ζ) and natural frequency (ω_n) to assess system performance. What are these values?

(a) ζ = 0.5, ω_n = 2 rad/s
(b) ζ = 0.5, ω_n = 4 rad/s
(c) ζ = 1.0, ω_n = 2 rad/s
(d) ζ = 0.25, ω_n = 4 rad/s

Question 12:
A distillation column uses a ratio control scheme where reflux flow is maintained at 3.5 times the distillate flow. The distillate flow varies between 100 and 500 kg/h. If the ratio controller has a bias of 50 kg/h, what is the reflux flow when distillate flow is 300 kg/h?

(a) 1000 kg/h
(b) 1050 kg/h
(c) 1100 kg/h
(d) 1150 kg/h

Question 13:
A process engineer implements a split-range control system for reactor temperature using both cooling water (0-50% controller output) and steam (50-100% controller output). The controller output is currently at 65%. If the cooling water valve is air-to-close and the steam valve is air-to-open, what percentage is each valve open?

(a) Cooling: 0%, Steam: 30%
(b) Cooling: 100%, Steam: 30%
(c) Cooling: 0%, Steam: 15%
(d) Cooling: 70%, Steam: 30%

Question 14:
A chemical plant uses a Smith Predictor to control a process with significant dead time. The actual process has a transfer function G_p(s) = (2e^-5s)/(3s+1), but the model used in the Smith Predictor is G_m(s) = (2e^-4s)/(3s+1). What type of model mismatch exists?

(a) Gain mismatch only
(b) Time constant mismatch only
(c) Dead time mismatch only
(d) Both gain and dead time mismatch

Question 15:
A process engineer evaluates a control loop with a process transfer function G_p(s) = 5/(10s+1) and proportional controller K_c = 4. The measurement transmitter has negligible dynamics and unity gain. For a unit step disturbance entering at the same point as the manipulated variable, what is the offset?

(a) 0.048
(b) 0.050
(c) 0.200
(d) 0.952

Question 16:
A temperature control system for a jacketed reactor uses a PID controller. During commissioning, the engineer observes that the derivative action is amplifying high-frequency measurement noise. The derivative time is τ_D = 2 minutes. A derivative filter with time constant α = 0.1 is added. What is the effective corner frequency of this filter in rad/min?

(a) 0.5
(b) 5.0
(c) 10.0
(d) 50.0

Question 17:
A control engineer analyzes a multi-loop control system and constructs the relative gain array (RGA) for a 2×2 system. The steady-state gain matrix shows that when input 1 changes by 1 unit with output 2 constant, output 1 changes by 4 units. When input 2 changes by 1 unit with output 1 constant, output 1 changes by 1 unit. When both inputs change to keep output 1 constant with 1 unit change in input 1, input 2 must change by -2 units. What is λ₁₁ (the relative gain)?

(a) 0.50
(b) 0.67
(c) 1.33
(d) 2.00

Question 18:
A chemical engineer designs an override control system for a fired heater. The normal controller maintains outlet temperature at setpoint, but a high temperature override controller takes over when temperature exceeds 650°C. Both controllers manipulate fuel flow. What selector configuration is required, and what action should each controller have?

(a) Low selector; both reverse acting
(b) Low selector; normal reverse, override direct
(c) High selector; both direct acting
(d) High selector; normal direct, override reverse

Question 19:
A process control engineer tunes a flow controller using the Cohen-Coon method for a process with transfer function G(s) = (1.5e^-0.8s)/(4s+1). The Cohen-Coon tuning formula for PI controller gives K_c = (τ/K_pθ)[0.9 + θ/(12τ)]. What is the recommended controller gain K_c?

(a) 2.17
(b) 2.50
(c) 3.00
(d) 3.52

Question 20:
A distillation column uses a decoupler in a multivariable control system. The process gain from reflux (u₁) to top composition (y₁) is K₁₁ = 0.8, and from steam (u₂) to top composition is K₁₂ = -0.3. To decouple the effect of steam changes on top composition, what should be the decoupler gain D₁₂ in the feedforward path from steam controller to reflux valve?

(a) -2.67
(b) -0.375
(c) 0.375
(d) 2.67