Dynamic simulation

Everything so far has been steady state: the plant running at constant conditions, with every accumulation term set to zero. Real plants are never quite steady. Feeds change, cooling water warms through the day, catalysts age, and operators start up, shut down and respond to upsets. Wren's operations team will run the new line, and she wants answers that steady-state models cannot give. How fast does the reflux drum level move when boil-up changes? How long do operators have if cooling water fails? Will the control scheme designed in CHE-374 actually hold the purity through a feed swing? These are questions for dynamic simulation.

From steady state to dynamics

A dynamic model restores the accumulation terms. Each vessel, tray and pipe with significant holdup gets differential equations for its mass, component and energy inventories. Flows are no longer fixed numbers but depend on pressure differences and valve openings. Controllers, with their tuning, measure and act. The model becomes a large system of differential and algebraic equations, integrated forward in time.

A dynamic model needs information a steady-state model does not. Vessel dimensions set the holdups. Valve sizes and characteristics set how flow responds to opening. Pump curves, line resistances and elevations set pressures. Controllers need their algorithms, tuning and ranges. Tray hydraulics determine how liquid holdup changes with flow. In practice, the steady-state model is the starting point: it is converted to dynamic mode by adding this equipment data, and it starts the dynamic run from the converged steady state. Aspen HYSYS and Aspen Plus Dynamics, gPROMS, UniSim and DWSIM all offer dynamic modes, and specialized operator training simulators are built on similar engines.

Numerical integration

The differential equations are integrated by numerical methods. The simplest, the explicit Euler method, steps forward using the current derivative. It is easy and often inaccurate unless the step is small. The fourth-order Runge-Kutta method, RK4, evaluates the derivative four times per step and is far more accurate for the same step size. Process models raise a further difficulty, stiffness: some parts of the model, such as pressure in a small gas space or the composition on a tray with little holdup, respond in seconds, while others, such as a large tank level, take hours. Explicit methods must use a step small enough for the fastest part, which can make them very slow. Implicit methods, such as backward differentiation formulas, remain stable with large steps and are the standard in dynamic simulators.

A reflux drum under level control

The first part of the code simulates the ester column reflux drum. Vapor from the column condenses into the drum; reflux returns to the column in a fixed ratio to the condensed flow, as in the CHE-374 control scheme; and the distillate valve is manipulated by a PI level controller with gain 1.67 and integral time 16 minutes, the tuning chosen in that course. At 5 minutes the boil-up steps up by 10 %, as it might when steam pressure rises. The model integrates the drum holdup and the controller's integral term together with RK4, using a step of three seconds.

# Dynamic simulation: reflux drum level under PI control after a boil-up step, and pressure rise on loss of cooling
# Part 1: drum holdup (kmol) with RK4; distillate valve on PI level control; time in minutes
A_drum = 6.0                 # kmol of liquid per meter of level (illustrative 1.2 m diameter horizontal drum, linearized)
Kc, Ti = 1.67, 16.0          # controller tuning from CHE-374 (% output per % level, minutes)
span_level, span_flow = 1.0, 80.0    # m for 0-100 % level; kmol/h for 0-100 % valve
V0, R = 89.4, 1.34           # boil-up (kmol/h) and reflux ratio, design point
D0 = V0 / (R + 1)
def run(step_frac, t_end=120.0, dt=0.05):
    h, integ, t = 0.5, 0.0, 0.0              # level m, integral of error, min
    sp = 0.5; out, peak, n = [], 0.5, 0
    def deriv(h, integ, t):
        V = V0 * (1 + (step_frac if t >= 5.0 else 0.0))
        Lr = R * D0 * (V / V0)                 # reflux set in ratio to the condensed vapor
        e = (h - sp) / span_level * 100        # % level error (direct acting: high level opens the valve)
        u = 100 * D0 / span_flow + Kc * (e + integ / Ti)
        u = max(0.0, min(100.0, u))
        D = u / 100 * span_flow
        dh = (V - Lr - D) / 60 / A_drum        # m/min
        return dh, e
    while t < t_end:
        k1 = deriv(h, integ, t); k2 = deriv(h + dt * k1[0] / 2, integ + dt * k1[1] / 2, t + dt / 2)
        k3 = deriv(h + dt * k2[0] / 2, integ + dt * k2[1] / 2, t + dt / 2); k4 = deriv(h + dt * k3[0], integ + dt * k3[1], t + dt)
        h += dt * (k1[0] + 2 * k2[0] + 2 * k3[0] + k4[0]) / 6
        integ += dt * (k1[1] + 2 * k2[1] + 2 * k3[1] + k4[1]) / 6
        t += dt; n += 1; peak = max(peak, h)
        if n % int(round(10 / dt)) == 0: out.append((round(t), h))
    return out, peak
out, peak = run(0.10)
print("boil-up +10 % at t = 5 min; reflux drum level (m), set point 0.50 m")
for t, h in out: print(f"   t = {t:4d} min   level {h:.3f} m")
print(f"peak level {peak:.3f} m ({(peak - 0.5) / span_level * 100:.1f} % of span above set point)")
print()
# Part 2: loss of cooling water; vapor keeps coming from the reboiler; gas space pressurizes (ideal gas)
Rg, T = 8.314, 350.0         # kJ/(kmol K), K
Vgas = 14.0                  # m3 of vapor space in the column top, overhead line and drum (illustrative)
P, t, dt = 101.3, 0.0, 0.5   # kPa, s
Vin = V0 / 3600              # kmol/s of vapor still arriving
Pset = 350.0                 # relief valve set pressure, kPa absolute (illustrative)
while P < Pset:
    P += Rg * T / Vgas * Vin * dt; t += dt
print(f"with no condensation the top pressure reaches the {Pset:.0f} kPa relief set point in about {t:.0f} s ({t / 60:.1f} min)")
print("a high-pressure trip of the reboiler steam at 250 kPa acts first if it works:")
P, t = 101.3, 0.0
while P < 250.0: P += Rg * T / Vgas * Vin * dt; t += dt
print(f"   trip reached after about {t:.0f} s, about {Pset - 250:.0f} kPa below the relief set point")

boil-up +10 % at t = 5 min; reflux drum level (m), set point 0.50 m
   t =   10 min   level 0.522 m
   t =   20 min   level 0.515 m
   t =   30 min   level 0.507 m
   t =   40 min   level 0.503 m
   t =   50 min   level 0.501 m
   t =   60 min   level 0.501 m
   t =   70 min   level 0.500 m
   t =   80 min   level 0.500 m
   t =   90 min   level 0.500 m
   t =  100 min   level 0.500 m
   t =  110 min   level 0.500 m
   t =  120 min   level 0.500 m
peak level 0.522 m (2.2 % of span above set point)

with no condensation the top pressure reaches the 350 kPa relief set point in about 48 s (0.8 min)
a high-pressure trip of the reboiler steam at 250 kPa acts first if it works:
   trip reached after about 29 s, about 100 kPa below the relief set point

The level rises to a peak of 0.522 m, 2.2 % of span above set point, then returns smoothly to 0.500 m over about an hour. That is the behavior wanted from averaging level control: the drum absorbs the disturbance, and the distillate flow changes gradually rather than passing the full step to the stripper and the decanter. If Theo raises the gain, the peak falls but the distillate swings more; the dynamic model lets the team see that trade-off before the plant exists.

Upsets and relief

Dynamic models are increasingly used to study abnormal situations, including those that set relief requirements. The second part of the code considers total loss of cooling water to the ester column condenser. The reboiler continues to boil up vapor, which no longer condenses, so the vapor space of the column top, overhead line and drum pressurizes. A simple ideal-gas model with 14 cubic meters of vapor space and the full design boil-up reaches the relief set pressure of 350 kPa absolute in about 48 seconds. A high-pressure trip on the reboiler steam, set at 250 kPa, is reached in about 29 seconds.

Ingrid points out what this simple model leaves out, and in which direction each omission errs. As the pressure rises, the boiling point at the bottom rises, the temperature difference across the reboiler falls, and boil-up drops: the real rise is slower. Some vapor condenses on cold metal and in the liquid on the trays at first: slower again. But the condenser may not lose all its cooling at once, and the vapor may be hotter than assumed. The simple model is conservative in most respects, and it shows two things clearly: operators have seconds, not minutes, so the protection must be automatic; and the steam trip is a real layer of protection, acting before relief. Relief sizing itself follows recognized practice, such as API Standard 521 and the requirements under OSHA's Process Safety Management standard, 29 CFR 1910.119, together with EPA's Risk Management Program rule where a listed chemical exceeds its threshold quantity, and for complex cases dynamic simulation of the relief scenario is accepted as a basis for the relief load. That work belongs to the contractor's relief specialists and is checked in the process hazard analysis.

Testing control strategies

The most common use of dynamic simulation in design is to test the control strategy. Theo builds the 17 loops from CHE-374 into the dynamic model of the ester column and decanter, and runs a set of disturbances: feed composition swings of ±5 %, a 10 % throughput change, a drop in cooling water temperature, and a step in steam pressure. For each he checks that product purity stays within specification and that no level or pressure reaches an alarm limit. Two findings come back to the control design. The tray 18 temperature controller, tuned for the original steam reboiler, is too aggressive with the vapor recompression system, whose compressor responds more slowly than a steam valve; it needs detuning. And a feed composition swing of 5 % takes about 40 minutes to show at the bottoms analyzer, confirming that the temperature control, not the analyzer, must carry the fast correction.

Operator training simulators

An operator training simulator connects a dynamic plant model to a copy of the plant's control system screens, so that operators can practice normal and emergency operations without risk. For a new line, it also lets operators learn the plant before it starts, and lets engineers check the control system configuration and operating procedures. Wren plans to have her operators train on a simulator of the new line for several weeks before commissioning, including the loss of cooling scenario and a recompressor trip. The model for the simulator comes from the same validated process model, tuned further when the plant runs.