Batch Polymerization Runaway

Directions

This simulation models bulk free-radical polymerization of methyl methacrylate (MMA) with AIBN initiator in a jacketed, well-mixed batch reactor. The reactor starts at the jacket temperature with no polymer.

Set the initiator loading, jacket temperature and heat-removal capacity with the sliders. The left plot shows reactor temperature (red) and monomer conversion (blue) for 300 minutes. The dashed orange line is the maximum allowable temperature of 100 °C, close to the normal boiling point of MMA. A run is labeled runaway if the reactor temperature exceeds it.

Uncheck gel effect to see how much of the runaway comes from the drop in the termination rate constant as the mixture becomes viscous. Check loss of cooling to cut the heat-removal capacity at a chosen time to a chosen fraction of its normal value; 0% means the reactor becomes adiabatic.

The right plot has three views:

Semenov diagram: heat generation Q_g versus temperature at the composition the reactor has at the selected time, with the heat-removal line Q_r. Drag on the left plot or use the time slider to move through the batch. The dotted green line (critical Q_r) is the critical removal line tangent to Q_g; when Q_r is less steep than it, there is no low-temperature intersection and the temperature must climb.

Semenov number: Ψ versus time, compared with the classical critical value 1/e.

Safe-limit scan: the peak temperature for a range of initiator loadings or jacket temperatures, with the other settings fixed. It reports the maximum safe value from the simulation (peak temperature = 100 °C) and from the Semenov criterion (largest Ψ = 1/e). Find the maximum safe initiator loading or jacket temperature and compare the two estimates.

Details

Kinetics

Initiator decomposition, propagation and termination:

$$\mathrm{I} \xrightarrow{k_d} 2\,\mathrm{R}^\bullet \ (\text{efficiency } f), \qquad \mathrm{P}_n^\bullet + \mathrm{M} \xrightarrow{k_p} \mathrm{P}_{n+1}^\bullet, \qquad \mathrm{P}_n^\bullet + \mathrm{P}_m^\bullet \xrightarrow{k_t} \text{dead polymer}$$

With the quasi-steady-state approximation for radicals, the total radical concentration is

$$\lambda_0 = \sqrt{\frac{2 f k_d [\mathrm{I}]}{k_t}}$$

Mole and energy balances

$$\frac{d[\mathrm{I}]}{dt} = -k_d [\mathrm{I}]$$ $$\frac{dX}{dt} = k_p (1-X)\,\lambda_0$$ $$\rho c_p \frac{dT}{dt} = \underbrace{(-\Delta H_p)[\mathrm{M}]_0 \frac{dX}{dt}}_{Q_g} \; - \; \underbrace{\frac{UA}{V}\,(T - T_j)}_{Q_r}$$

where X is monomer conversion, [M]₀ = 9.4 mol/L (bulk MMA), −ΔH_p = 57.8 kJ/mol and ρc_p = 1800 kJ/(m³·K). The adiabatic temperature rise for complete conversion is about 300 K. After cooling is lost at time t_f, UA/V is multiplied by the remaining fraction.

Gel (Trommsdorff) effect

As polymer accumulates, the viscosity rises and the large radicals can no longer diffuse together to terminate. The Schmidt–Ray correlation for MMA reduces k_t with conversion (T in K):

$$k_t = k_{t0}\, g_t, \qquad g_t = \exp\!\left[-2\left(A_1 X + A_2 X^2 + A_3 X^3\right)\right]$$ $$A_1 = 2.57 - 5.05\times10^{-3}\,T, \quad A_2 = 9.56 - 1.76\times10^{-2}\,T, \quad A_3 = -3.03 + 7.85\times10^{-3}\,T$$

Because the rate is proportional to k_t^−1/2, the rate accelerates as conversion increases, even at constant temperature. This autoacceleration and the Arrhenius rise of the rate with temperature reinforce each other.

Rate constants

constant	expression	value at 70 °C
k_d (AIBN), s⁻¹	1.58×10¹⁵ exp(−15 500/T)	3.7×10⁻⁵
k_p, L/(mol·s)	4.92×10⁵ exp(−2190/T)	830
k_t0, L/(mol·s)	9.8×10⁷ exp(−353/T)	3.5×10⁷
f	constant	0.58

Semenov analysis

At any instant, the heat generation rate depends on temperature through the rate constants, with composition (X, [I]) fixed:

$$Q_g(T) = (-\Delta H_p)[\mathrm{M}]_0\, k_p(T)\,(1-X)\sqrt{\frac{2 f k_d(T) [\mathrm{I}]}{k_{t0}(T)\, g_t(X,T)}}, \qquad Q_r(T) = \frac{UA}{V}(T - T_j)$$

A stable operating point needs an intersection where the removal line is steeper than the generation curve. The critical removal line passes through (T_j, 0) and is tangent to Q_g; its slope is the minimum over T > T_j of Q_g(T)/(T − T_j).

Approximating the rate with one apparent activation energy gives the Semenov number and the classical critical value:

$$\Psi = \frac{Q_g(T_j)}{UA/V}\,\frac{E_{app}}{R T_j^2} \le \frac{1}{e}, \qquad E_{app} = E_p + \frac{E_d}{2} - \frac{E_t}{2} = 81.2\ \text{kJ/mol}$$

Ψ is evaluated at the jacket temperature and the current composition. The gel effect raises Ψ during the batch. The Semenov criterion neglects reactant consumption, so it is conservative here: the batch can pass Ψ = 1/e and still finish below the allowable temperature if monomer or initiator is used up first.

Safe-limit scan

The scan runs the full model across the slider range of the chosen variable. The simulation limit is where the peak temperature reaches 100 °C. The Semenov limit is where the largest Ψ during normal operation (full cooling) reaches 1/e. The sharp rise of peak temperature with the scanned variable is the parametric sensitivity that marks runaway.

Assumptions

Well-mixed, constant volume (density change on polymerization is neglected), constant ρc_p and f, no glass effect on k_p, no depropagation (the ceiling temperature of PMMA is near 220 °C), and no boiling or pressure rise. Temperatures above about 100 °C would boil MMA in an open reactor, so the curves beyond that point show the thermal driving force, not a realistic trajectory.

References

A. D. Schmidt and W. H. Ray, “The dynamic behavior of continuous polymerization reactors—I,” Chem. Eng. Sci. 36 (1981) 1401–1410.
F. Stoessel, Thermal Safety of Chemical Processes, 2nd ed., Wiley-VCH (2020).
M. Morbidelli and A. Varma, “A generalized criterion for parametric sensitivity,” Chem. Eng. Sci. 43 (1988) 91–102.

Nomenclature

A₁, A₂, A₃	coefficients in the gel-effect correlation (dimensionless)
E_app	apparent activation energy of the polymerization rate (kJ/mol)
E_d, E_p, E_t	activation energies for initiator decomposition, propagation and termination (kJ/mol)
e	base of natural logarithms, 2.718
f	initiator efficiency, fraction of radicals that start chains
g_t	gel-effect factor, k_t/k_t0
ΔH_p	enthalpy of polymerization (kJ/mol monomer)
[I], [I]₀	initiator (AIBN) concentration and its initial value (mol/L)
k_d	initiator decomposition rate constant (s⁻¹)
k_p	propagation rate constant (L/(mol·s))
k_t, k_t0	termination rate constant with and without the gel effect (L/(mol·s))
[M]₀	initial monomer concentration (mol/L)
M, I, R^•, P_n^•	monomer, initiator, primary radical, growing polymer radical with n monomer units
Q_g	rate of heat generation per reactor volume (kW/m³)
Q_r	rate of heat removal to the jacket per reactor volume (kW/m³)
R	gas constant, 8.314 J/(mol·K)
T	reactor temperature (K in equations, °C in plots)
T_j	jacket (coolant) temperature
T_max	peak reactor temperature during the batch
t, t_f	time; time at which cooling fails (min)
UA/V	heat-transfer coefficient × area per reactor volume (kW/(m³·K))
X	monomer conversion (fraction of monomer polymerized)
λ₀	total concentration of growing radicals (mol/L)
ρc_p	volumetric heat capacity of the reacting mixture (kJ/(m³·K))
Ψ	Semenov number (dimensionless)

About

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.