Model basis
This is a principle and training simulator, not a plant model. It demonstrates energy storage, forced circulation, heat-exchanger response and closed-loop control using a compact lumped-parameter model.
1. Physical background
Intuition
The vessel contains a finite thermal inventory. A constant process load adds heat. One or two pumps circulate water through CV-101 and HX-101, where an external cooling-water stream removes heat. If heat input exceeds heat removal, the vessel temperature rises; if removal is larger, it falls.
Causal chain
Pump state and valve opening → mass flow → heat-capacity rate and UA → heat removal → vessel-temperature response → controller error → valve demand.
Engineering interpretation
The large inventory, approximately 20.9 MJ/K, makes the temperature response slow. A pump trip removes forced cooling immediately, while temperature continues to change according to the stored-energy balance.
2. Governing equations
The vessel is perfectly mixed, with constant density and heat capacity. The implemented energy balance is:
ρ V cp · dT/dt = Q̇load − Q̇cool
Here ρ [kg/m³] is density, V [m³] is vessel volume, cp [J/(kg·K)] is specific heat, T [°C] is vessel temperature, and both heat rates are in W.
Each running pump contributes nominal mass flow, scaled linearly by the valve position xCV:
ṁ = npump · ṁnom · xCV
The heat exchanger uses a single-stream effectiveness approximation. The process-side heat-capacity rate is C = ṁcp. Conductance is scaled by flow and the user-selected HX capacity factor hf:
UA = UA0 · hf · (ṁ/ṁnom)0.8
ε = 1 − exp(−UA/C)
Q̇cool = max[0, ε C (T − Tcw,in)]
The 0.8 exponent reflects the Reynolds-number dependence of the Dittus–Boelter turbulent-tube correlation [1]. It is used here as an engineering scaling law, not as a full exchanger geometry calculation.
The displayed process return temperature is Treturn = T − Q̇cool/(ṁcp). The displayed cooling-water outlet assumes a fixed 30 m³/h utility flow.
3. Initial and boundary conditions
The reference case starts at 35 °C with P-101 running, P-102 stopped, a 200 kW heat load, clean-HX factor 1.0 and a 20 °C cooling-water inlet boundary. The controller is in AUTO with a 35 °C setpoint. Input Configuration loading treats the stored temperature as a new initial condition and restarts simulation time at zero.
Alarm thresholds are 45 °C (LAH-101) and 55 °C (LAHH-101). These are demonstration settings and are not derived from equipment design limits.
4. PID temperature control
TIC-101 acts in the cooling direction: a positive temperature error opens CV-101. The positional controller is:
u = b + Kpe + Ki∫e dt + Kdde/dt, e = T − Tsp
The output u is limited to [0,1]. Conditional integration provides anti-windup: when saturated, the integral is updated only if the change would move the output back toward the valid interval. In MAN, the valve is user-controlled and the error history tracks the process for a less abrupt return to AUTO. The formulation follows standard feedback-control practice [2].
5. Numerical solution method
The application advances the lumped energy balance with explicit Euler integration. Each rendered update is divided into 20 equal substeps. Real elapsed time is capped at 0.08 s per rendered frame before multiplication by the selected simulation-speed factor; this limits a single frame to 4 simulated seconds at 50× and therefore to 0.2 s per Euler substep.
The controller and energy balance are evaluated in every substep. The application has no scheduled discontinuity table: user actions define instantaneous state changes at the next browser frame. Calculations use JavaScript double-precision floating point. Non-finite values are trapped by the startup checks and IC validation.
6. Verification and numerical accuracy
Version 1.0.3 performs startup software checks for finite default calculations, the zero-flow analytical heating rate, and the reference Input Configuration schema. For zero flow and constant heat load, the model reduces to a constant derivative:
T(t) = T0 + Q̇loadt/(ρVcp)
Because the derivative is constant in that case, the Euler implementation reproduces the analytical temperature rise to floating-point precision. A default-case calculation also verifies finite flow, UA, effectiveness and cooling duty. These checks verify implementation consistency; they do not validate the model against experimental or plant data.
| Check | Basis | Acceptance |
|---|
| Zero-flow heating | Analytical constant-source solution | |error| < 10⁻¹⁰ °C over 60 s |
| Reference calculation | Finite positive flow, UA, ε and duty | All results finite and physically bounded |
| IC structure | Generated reference object through full validator | No missing, invalid or incompatible fields |
7. Model parameters and constants
| Symbol | Description | Value | Unit | Basis |
|---|
| ρ | Constant water density | 1000 | kg/m³ | Rounded demonstration value; not a property solver |
| cp | Constant water specific heat | 4180 | J/(kg·K) | Rounded near-ambient value [3] |
| V | Vessel liquid volume | 5.0 | m³ | Model assumption |
| ṁnom | One-pump nominal mass flow | 6.944 | kg/s | 25 m³/h with ρ = 1000 kg/m³ |
| UA0 | Nominal effective conductance | 13 333 | W/K | Model calibration parameter |
| Tcw,in | Cooling-water inlet | 20 | °C | Fixed boundary condition |
| Kp, Ki, Kd | PID gains | 0.10, 0.030, 0.000 | mixed | Demonstration tuning |
8. Model scope and limitations
Included
- Lumped vessel energy storage.
- Up to two parallel pumps with algebraic flow.
- Valve-position scaling, flow-dependent UA and ε–NTU cooling.
- PID control with output limiting and conditional anti-windup.
- Live P&ID, trends, scenarios, saved in-memory states and file-based IC.
Not represented
- Hydraulic pressure balance, pump curves, valve Cv, pipe inertia, cavitation or natural circulation.
- Cooling-side capacity limits, exchanger wall storage, fouling dynamics, phase change or temperature-dependent properties.
- Sensor lag/noise, actuator dynamics, dead time, interlocks or protection logic.
- Spatial temperature gradients, heat loss to ambient, plant-specific geometry or validated alarm limits.
Engineering-use disclaimer. Provided for engineering guidance, analysis and training. Results requiring safety, regulatory, contractual or operational significance shall be independently verified.
9. References
[1] F. P. Incropera, D. P. DeWitt, T. L. Bergman and A. S. Lavine, Fundamentals of Heat and Mass Transfer, 7th ed., Wiley, 2011. Heat-exchanger effectiveness–NTU method and turbulent internal-flow correlations.
[2] D. E. Seborg, T. F. Edgar, D. A. Mellichamp and F. J. Doyle III, Process Dynamics and Control, 3rd ed., Wiley, 2011. Feedback control, PID algorithms and integral windup.
[3] IAPWS, Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam, IAPWS R7-97(2012), 2012. Authoritative property formulation; the simulator uses rounded constant values rather than evaluating IF97.
Antranova model basis
[A1] Antranova Engineering AB, AntraSIM-0000 Cooling Demo System, software and embedded Theory & Method, version 1.0.3, 2026-09-19.