Multiple States in CSTR Polymerization

coolant inlet T_c

coolant flow ṁ_c

residence time τ

monomer

reactor start

Heat generation and heat removal

Steady-state temperature vs.

Free-radical solution polymerization of styrene or methyl methacrylate (MMA), initiated by AIBN, runs in a jacketed CSTR. The left plot shows the heat generated by polymerization, G(T) (red), and the heat removed by the outflow and the coolant, R(T) (blue), per unit reactor volume. Steady states are the intersections, numbered to match the table below the plots: filled circles are stable and open circles are unstable; a red-outlined open circle is unstable to oscillations.

Change the coolant inlet temperature or coolant flow rate to shift the removal line. Its slope depends on coolant flow and residence time; its intercept depends on coolant temperature. Where the line cuts the S-shaped curve three times, the reactor has two stable steady states.

The right plot is the steady-state reactor temperature versus coolant temperature (or coolant flow). Its coolant-temperature axis normally runs from 280 to 420 K; when the ignition or extinction point lies outside that range, the axis widens to show it. Stable branches are solid green and unstable branches are dashed red. On the dotted red part of the upper branch, the steady state is unstable to oscillations: the reactor temperature cycles instead of settling, so the reactor can fall off the hot branch before reaching the extinction fold. The yellow diamond (on both plots) is the reactor's current state, which follows the dynamic mass and energy balances as you move sliders. Press sweep up to raise the coolant setting steadily to its maximum, and sweep down to lower it to its minimum; pause sweep stops a sweep where it is, and pressing either sweep button continues from there. Starting cold and sweeping up, the reactor ignites at the ignition point; sweeping back down, it stays hot until the extinction point. An orange path records the reactor's temperature during the sweeps, so an up sweep followed by a down sweep traces the hysteresis loop. The path stays on the plot until you move a slider by hand or press another button. When there are multiple steady states, use cold or hot to restart the reactor on the lowest or highest one.

Kinetics

With the quasi-steady-state assumption for radicals, the rate of polymerization is first order in monomer:

	(L/mol/s)	(kJ/mol)	(L/mol/s)	(kJ/mol)	(kJ/mol)
styrene	4.27 × 10⁷	32.5	1.8 × 10⁹	8.0	69.9
MMA	2.67 × 10⁶	22.4	7.1 × 10⁷	2.9	57.8
AIBN	, kJ/mol, efficiency

Dynamic balances

Heat generation and removal curves

At steady state, both concentrations follow from the temperature:

is S-shaped because conversion saturates at high temperature; initiator burnout () flattens its upper part. is a straight line. The coolant is modeled as a jacket with inlet temperature , flow , and heat capacity , so the effective heat-transfer coefficient is

which approaches at high coolant flow and at low flow. Raising shifts the line to the right; raising or lowering makes it steeper.

Stability

A steady state where the removal line is less steep than the generation curve () is always unstable (a saddle). Passing the slope test is necessary but not sufficient, so the simulation also evaluates the Jacobian of the three balances at each steady state. With characteristic polynomial , the Routh–Hurwitz conditions for stability are

where , is the sum of the principal 2 × 2 minors, and . A violated condition corresponds to the slope condition; a violated condition gives an oscillatory instability (a Hopf bifurcation). This occurs on the upper branch just above the extinction point: the slope test is satisfied there, but the interaction between monomer depletion and temperature produces growing oscillations that settle into a limit cycle.

Parameters and assumptions

mol/L (monomer in toluene), mol/L, K, J/(L·K), L, W/K, J/(kg·K) (water). The kinetic constants are representative literature values. Constant volume and physical properties, perfect mixing, a quasi-steady jacket, and no gel (Trommsdorff) effect are assumed. Including the gel effect makes multiplicity more pronounced, especially for MMA. At high temperatures on the upper branch, a real reactor would be limited by solvent and monomer boiling, and the polymer's ceiling temperature.

Definitions

symbol	definition	units
	pre-exponential factors for initiator decomposition, propagation, and termination	s⁻¹; L/(mol·s)
	coefficients of the characteristic polynomial of	s⁻¹, s⁻², s⁻³
	heat capacity of the coolant (water)	J/(kg·K)
	activation energies for decomposition, propagation, and termination	kJ/mol
	initiator efficiency	—
	rate of heat generation by polymerization per reactor volume	kW/m³
	heat of polymerization	kJ/mol
	initiator concentration in the reactor and in the feed	mol/L
	Jacobian matrix of the three dynamic balances	—
	overall first-order polymerization rate constant	s⁻¹
	initiator decomposition rate constant	s⁻¹
	propagation and termination rate constants	L/(mol·s)
	monomer concentration in the reactor and in the feed	mol/L
	coolant mass flow rate	kg/s
	rate of heat removal (outflow plus coolant) per reactor volume	kW/m³
	ideal gas constant, 8.314	J/(mol·K)
	rate of polymerization	mol/(L·s)
	time	s
	reactor temperature	K
	coolant inlet temperature	K
	feed temperature	K
	jacket heat-transfer coefficient times area	W/K
	effective heat-transfer coefficient, including coolant warming	W/K
	reactor volume	L
	monomer conversion	—
	eigenvalue of	s⁻¹
	volumetric heat capacity of the reacting solution	J/(L·K)
	residence time, divided by volumetric flow rate	s (slider in h)