Continuous Stirred Thermal Process Vessel
2.0 L jacketed glass vessel with internal marine impeller (400 RPM) and continuous water throughput at 1.0 L/min.
Chemical & Process Engineering / PID Temperature Control Tuning
FREE TO EXPLORELABORATORY 42 · CHEMICAL & PROCESS ENGINEERING
Feedback Closed-Loop PID Temperature Control & Frequency-Response Ziegler–Nichols Ultimate Gain Tuning
Implement closed-loop proportional-integral-derivative (PID) feedback control on a continuous stirred thermal process vessel, conduct closed-loop ultimate oscillation testing across proportional controller gains K_c from 2.0 to 12.0 %/%, determine the closed-loop oscillation amplitude ratio AR, extract the ultimate gain K_u and ultimate period P_u, and evaluate Ziegler–Nichols tuning parameters. Explore interactive transport phenomena, unit operations simulation, and chemical engineering telemetry.
TRANSPORT PHENOMENA · UNIT OPERATIONS · PROCESS CONTROL · REACTION ENGINEERING
Your virtual chemical and process engineering laboratory.
CLOSED-LOOP OSCILLATION AMPLITUDE RATIO AR—
PROCESS & PILOT PLANT TELEMETRY
Ready to sample with Controller Gain K_c = 6.0 %/%.
Adjust Controller Gain K_c, then execute chemical process trial.
Chemical and process engineering telemetry is evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity transport phenomena and unit operations modeling.
Process engineering assumptions and boundary condition limits
The transport and thermodynamic models apply within the specified calibrated operating window (2.0 to 12.0 %/%). Evaluated using a 2.0 L continuous stirred heating vessel (throughflow 1.0 L/min, τ_process = 120 s, dead time θ = 15 s) with 500 W thyristor-fired heater and 4-wire Pt100 RTD sensor (±0.02 °C). Ultimate period measured at P_u = 62.8 s.
Process apparatus & industrial sensors.
500 W Incoloy Electric Immersion Heating Element
Controlled by a zero-cross-fired silicon-controlled rectifier (SCR) power controller (0 to 100% duty).
Precision 4-Wire Pt100 RTD Temperature Sensor
Measures effluent water temperature with 0.01 °C resolution and fast 0.5 s thermal response time.
Industrial Microprocessor Digital PID Controller
Executes discrete PID control algorithm with adjustable K_c, τ_I, and τ_D at 20 Hz loop update rate.
Variable Cold Water Throughput Metering Assembly
Maintains constant 1.00 ± 0.01 L/min cold water influent at 20.0 °C to establish steady thermal baseline.
Real-Time SCADA Data Logging & Spectrum Analyzer
Logs temperature transients, setpoint errors, and computes Fast Fourier Transform (FFT) oscillation periods.
FOLLOW THE METHOD
A reason behind every step.
Implement closed-loop proportional-integral-derivative (PID) feedback control on a continuous stirred thermal process vessel, conduct closed-loop ultimate oscillation testing across proportional controller gains K_c from 2.0 to 12.0 %/%, determine the closed-loop oscillation amplitude ratio AR, extract the ultimate gain K_u and ultimate period P_u, and evaluate Ziegler–Nichols tuning parameters. Standardized parameter settings: 2.0, 4.0, 6.0, 8.0, 10.0, 12.0 %/%.
01
Thermal Process Priming & Baseline Equilibration
Establish 1.0 L/min water flow through the vessel; engage 400 RPM agitation; set baseline manual heater power to 250 W to stabilize at 40.0 °C.
02
Controller Configuration for Ultimate Gain Testing
Switch controller to closed-loop automatic mode; disable integral action (τ_I = 9999 s) and derivative action (τ_D = 0 s).
03
Initial Low-Gain Step Test (K_c = 2.0 %/%)
Set K_c = 2.0 %/%; introduce a +2.0 °C setpoint pulse; log temperature oscillation waveform; calculate amplitude ratio AR = A_2 / A_1.
04
Gain Progression Toward Marginal Stability
Increment controller gain K_c through 4.0, 6.0, 8.0, 10.0, and 12.0 %/%; observe transition from damped decay to sustained oscillation to growing instability.
05
Ultimate Period & Gain Extraction
At marginal stability (AR = 1.000, K_c = 8.0 %/%), measure the ultimate period P_u across 5 full sinusoidal cycles.
06
Ziegler–Nichols Parameter Calculation & Verification
Compute ZN PID parameters (K_c = 0.6 K_u, τ_I = 0.5 P_u, τ_D = 0.125 P_u); load settings and verify 1/4-amplitude decay response.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into chemical process engineering insight.
Points are your recorded observations. The solid regression line validates PID Temperature Control Tuning mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
AR = S · K_c = 0.1250 · K_c
| # | Proportional Controller Gain K_c (%/%) | Closed-Loop Oscillation Amplitude Ratio AR (dimensionless) | Proportional controller gain K_c (%/%) | Oscillation amplitude ratio AR | Remove |
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No readings yet. Execute a process trial and record it.
Calculate it yourself.
CHECK YOUR UNDERSTANDING
Explain what the process engineering metrics mean.
At the ultimate gain K_u in the Ziegler–Nichols closed-loop tuning method, the oscillation amplitude ratio is:
According to classic Ziegler–Nichols tuning rules, the recommended proportional gain K_c for a full PID controller is:
The primary role of the integral mode (1/τ_I) in a PID feedback controller is to:
Derivative control action (τ_D · de/dt) provides:
THEORY AND UNIT OPERATIONS CONTEXT
Process engineering background & physical principles.
Proportional-Integral-Derivative (PID) feedback control is the foundational regulatory control algorithm deployed in over 95% of industrial chemical, refining, and manufacturing process loops. In continuous thermal processing—such as jacketed chemical reactors, shell-and-tube exchangers, and reboilers—temperature must be tightly regulated against influent thermal and flow disturbances. The parallel PID controller computes control output u(t) based on the error e(t) = y_sp(t) - y(t): u(t) = u_bias + K_c · [e(t) + (1/τ_I) ∫ e(t') dt' + τ_D · (de/dt)], where K_c is proportional gain, τ_I is integral reset time, and τ_D is derivative rate time. In 1942, John G. Ziegler and Nathaniel B. Nichols introduced the closed-loop ultimate oscillation method (Ziegler–Nichols continuous cycling method) based on frequency response stability margins. In this protocol, integral and derivative actions are deactivated (τ_I -> inf, τ_D = 0), and the proportional gain K_c is systematically increased until the closed-loop system reaches the boundary of marginal stability. At this stability limit, the Nyquist contour of the open-loop transfer function intersects the critical point (-1, 0) in the complex plane, satisfying the Barkhausen stability criteria: loop magnitude |G_OL(j ω_u)| = 1.0 and phase angle ∠G_OL(j ω_u) = -180°. The closed-loop system exhibits sustained, non-decaying sinusoidal oscillations of constant amplitude (amplitude ratio AR = y_peak,n+1 / y_peak,n = 1.000). The corresponding controller gain is defined as the ultimate gain K_u, and the oscillation cycle time is the ultimate period P_u. For linear processes operating near stability, the closed-loop oscillation amplitude ratio scales directly with proportional gain: AR = S · K_c, where S = 1 / K_u = 0.1250 (%/%)^-1. Thus, at AR = 1.000, the ultimate gain is evaluated as K_u = 1 / S = 8.0000 %/%. The classical Ziegler–Nichols tuning rules then specify controller settings: for PID, K_c = 0.60 · K_u = 4.800, τ_I = 0.50 · P_u, and τ_D = 0.125 · P_u, providing aggressive disturbance rejection with quarter-amplitude damping. In this laboratory experiment, an instrumented 2.0-liter continuous thermal process cell equipped with a 500 W electric immersion heater and Pt100 RTD sensor is tested across controller gains from 2.0 to 12.0 to measure AR, extract K_u = 8.0000, and evaluate optimal PID settings.
AR = S · K_c = 0.1250 · K_c
Evaluated using a 2.0 L continuous stirred heating vessel (throughflow 1.0 L/min, τ_process = 120 s, dead time θ = 15 s) with 500 W thyristor-fired heater and 4-wire Pt100 RTD sensor (±0.02 °C). Ultimate period measured at P_u = 62.8 s.