USE CASE

One tank, three steady states, and a runaway.

Put an exothermic reaction in a cooled stirred tank and the simplest reactor in chemistry turns into a nonlinear-dynamics showcase. The heat it makes rises steeply with temperature while the heat you remove rises only linearly — and where those curves cross can give one operating point or three, sudden ignition and extinction, self-sustained oscillations, or a thermal runaway. This program maps all of it from the coupled mass and energy balances, then shows how feedback control holds the reactor where it otherwise can't sit.

An S-shaped steady-state curve of reactor temperature against coolant temperature, with stable lower and upper branches, an unstable middle branch, and two fold points marking ignition and extinction

steady states (one tank)

5.1 K

ignition/extinction hysteresis

~134 K

limit-cycle oscillation

+20 K

feed kick that runs away

90.1%

design conversion

design notebooks

THE MULTIPLICITY

Heat made vs heat removed.

The reaction rate climbs exponentially with temperature, so the heat it generates is an S-shaped curve; the cooling jacket removes heat along a straight line. Where the S crosses the line is a steady state — and an S can cross a line in three places. The middle crossing is unstable, the outer two are stable, and as you slide the coolant temperature the stable states appear and vanish at fold points. That's the van Heerden picture, and it's why the same reactor can sit cold and barely reacting or hot and nearly complete, depending only on its history.

The van Heerden diagram — an S-shaped heat-generation curve crossing a straight heat-removal line at one or three points — The van Heerden diagram: the exponential heat-generation S-curve against the linear heat-removal line. One intersection means a single operating point; three means multiplicity. Tilting or shifting the removal line — changing coolant temperature or flow — is how the steady states are born and destroyed.

The steady-state S-curve of reactor temperature versus coolant temperature, with stable branches, an unstable saddle branch, and ignition/extinction fold points — The multiplicity S-curve, traced by continuation and classified by stability: a cold stable branch, a hot stable branch, and an unstable saddle between them. The fold points are ignition (303.2 K) and extinction (298.1 K) — a 5.1 K window where the reactor's state depends on where it came from.

THE DYNAMICS

It can oscillate, and it can run away.

Multiplicity isn't the only surprise. Around the unstable middle state the reactor can settle into a self-sustained limit cycle — temperature swinging more than a hundred degrees, forever, with no changing input — a Hopf bifurcation. And push the feed too hard and the heat generation outruns the cooling entirely: a +10 K disturbance is absorbed, but a +20 K one ignites a thermal runaway to over 460 K. The program draws the limit-cycle orbit in the phase plane and maps the runaway boundary — the safety envelope a real reactor has to stay inside.

A closed limit-cycle orbit in the concentration-temperature phase plane, with transient trajectories spiraling onto it from inside and outside — The Hopf limit cycle: a closed orbit in the concentration-temperature plane that the reactor circles forever, with transients spiraling onto it from both sides around the unstable steady state at its centre. Self-sustained oscillation from a constant feed — the reactor as a chemical clock.

A safety map of peak reactor temperature over disturbance size and operating condition, with a runaway region — The runaway safety map: peak temperature against the size of a feed disturbance and the operating condition. The tolerable kick shrinks as the jacket warms — the parametric-sensitivity boundary that separates a reactor that recovers from one that runs away.

A PI controller holding the reactor at an open-loop-unstable setpoint that otherwise oscillates, settling cleanly — Control closes the loop: a PI controller manipulating coolant holds the reactor at the 350 K operating point that is open-loop unstable — where, left alone, it would oscillate. The most productive operating point is often the one you can only reach with feedback.

VALIDATION

Every number is one you can re-run.

The sign-off notebook re-derives each requirement from the same coupled balances the program builds.

	Result	Requirement
Steady states (Tc = 301 K)	3	multiplicity
Ignition/extinction hysteresis	5.1 K	303.2 / 298.1 K
Limit-cycle amplitude	~134 K	sustained (Hopf)
Runaway threshold	+20 K kick	+10 K absorbed
Requirements verified	7 / 7	PASS

HONEST SCOPE

One reaction, the whole zoo.

The reactor runs a single first-order exothermic reaction A→B in a perfectly-mixed lumped tank with constant physical properties — the Seborg-canonical model, not a detailed thermochemistry or a multi-reaction network. That minimal model is deliberately what shows the full nonlinear-dynamics zoo cleanly: multiplicity, hysteresis, the Hopf limit cycle, runaway, and feedback stabilization, all traceable by hand. It is exactly the fidelity reactor design and safety intuition need first — finding the operating points, mapping the runaway envelope, and designing the control — before a detailed kinetics or CFD model.

Model your own reactor.

Book a walkthrough and we'll set up your kinetics, heat of reaction and cooling, and run the multiplicity, runaway and control studies live.

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