This simulation models bulk free-radical polymerization of styrene in a 20 m jacketed tube. Monomer with dissolved initiator enters at the feed temperature, and the tube wall is held at the wall temperature. The temperature changes with radial and axial locations so the conversion and viscosity also depend on both radial and axial locations. The profiles also depend on whether plug flow, laminar flow, or laminar with viscosity rise is selected.
Select a flow model. Plug flow gives every fluid element the same residence time. Laminar, Newtonian keeps the parabolic velocity profile. Laminar, viscosity rise recalculates the velocity profile at each axial position from the local viscosity, so slow, viscous fluid near the wall pushes more of the flow into the center.
The plots show axial profiles of conversion, temperature and viscosity along the centerline, along the outer stream (the stream tube that enters at 90% of the radius), and for the mixing-cup average. The bottom row shows radial velocity profiles, the cumulative residence-time distribution, and the molecular-weight distribution of the polymer leaving the tube.
Increase the tube diameter or the wall temperature to see the hot spot grow into a thermal runaway. Increase the viscosity-rise parameter to see channeling in the core.
The gel effect lowers the termination rate constant as conversion \(X\) increases (Hui and Hamielec):
\[k_t = k_{t0}\exp\!\left[-2\left(A_1X+A_2X^2+A_3X^3\right)\right]\] \[A_1 = 2.57-5.05\times10^{-3}T,\quad A_2=9.56-1.76\times10^{-2}T,\quad A_3=-3.03+7.85\times10^{-3}T\]| \(k_d = 6.38\times10^{13}\,e^{-14842/T}\ \mathrm{s^{-1}}\) | initiator (benzoyl-peroxide-like), \(f=0.6\) |
| \(k_p = 1.051\times10^{7}\,e^{-3557/T}\ \mathrm{L/(mol\,s)}\) | propagation |
| \(k_{t0} = 1.255\times10^{9}\,e^{-844/T}\ \mathrm{L/(mol\,s)}\) | termination by combination |
| \(k_{th} = 2.19\times10^{5}\,e^{-13810/T}\ \mathrm{L^2/(mol^2 s)}\) | thermal initiation |
| \(C_M = 0.2198\,e^{-2820/T}\) | chain transfer to monomer |
Radial diffusion of polymer is negligible, so the tube is divided into 40 annular stream tubes. Each tube \(i\) carries a fixed volumetric flow \(q_i\) set by the inlet velocity profile. Along each tube, with \(dt = dz/\bar u_i\):
\[\frac{dX}{dt} = k_p[\mathrm{P}^\bullet](1-X) \qquad \frac{d[\mathrm{I}]}{dt} = -k_d[\mathrm{I}]\]Heat moves between tubes by radial conduction, and the wall is held at \(T_w\):
\[\rho c_p u\frac{\partial T}{\partial z} = \frac{k}{r}\frac{\partial}{\partial r}\!\left(r\frac{\partial T}{\partial r}\right) + (-\Delta H_p)R_p \qquad T(R,z)=T_w\]Conduction is solved implicitly at each axial step. The adiabatic temperature rise for full conversion is \(\Delta T_{ad} = (-\Delta H_p)[\mathrm{M}]_0/(\rho c_p) \approx 356\ \mathrm{K}\), which is why hot spots grow quickly in wide tubes.
For slowly varying viscosity (lubrication approximation), with \(G=-dp/dz\):
\[u(r) = \frac{G}{2}\int_r^R \frac{r'}{\mu(r')}\,dr' \qquad Q = \int_0^R 2\pi r\,u\,dr\]In the viscosity-rise model, \(G\) and the tube boundaries are found at each \(z\) so every stream tube still carries its own \(q_i\). Viscous wall tubes widen and slow down, and the core speeds up. The pressure drop is \(\Delta p = \int_0^L G\,dz\).
\(B\) lumps the effect of polymer concentration and chain length on viscosity. This is an illustrative correlation for teaching.
Polymer made at each instant follows a Schulz distribution with number-average degree of polymerization
\[\overline{DP}_{n,\mathrm{inst}} = \frac{R_p}{R_i/2 + C_M R_p}\]and shape parameter between 1 (all transfer) and 2 (all combination). The outlet distribution, plotted as weight fraction per decade \(W(\log M)\), is the sum over every axial step and stream tube, weighted by the polymer flow in each tube.
\(\rho = 900\ \mathrm{kg/m^3}\), \(c_p = 1.9\ \mathrm{kJ/(kg\,K)}\), \(k = 0.13\ \mathrm{W/(m\,K)}\), \(\Delta H_p = -69.9\ \mathrm{kJ/mol}\), \([\mathrm{M}]_0 = 8.7\ \mathrm{mol/L}\), \(M_{\mathrm{styrene}} = 104.15\ \mathrm{g/mol}\), \(L = 20\ \mathrm m\).
| \(A_1, A_2, A_3\) | gel-effect coefficients (dimensionless) |
| \(B\) | viscosity-rise parameter (dimensionless) |
| \(C_M\) | chain-transfer constant to monomer (dimensionless) |
| \(c_p\) | heat capacity (J/(kg K)) |
| \(D\) | tube diameter (cm) |
| \(\overline{DP}_{n,\mathrm{inst}}\) | instantaneous number-average degree of polymerization |
| \(f\) | initiator efficiency |
| \(F(t)\) | cumulative residence-time distribution |
| \(G\) | axial pressure gradient, \(-dp/dz\) (Pa/m) |
| \(\Delta H_p\) | heat of polymerization (kJ/mol) |
| \([\mathrm{I}],\ [\mathrm{I}]_0\) | initiator concentration, in feed (mol/L) |
| \(k\) | thermal conductivity (W/(m K)) |
| \(k_d\) | initiator decomposition rate constant (s−1) |
| \(k_p\) | propagation rate constant (L/(mol s)) |
| \(k_t,\ k_{t0}\) | termination rate constant, at zero conversion (L/(mol s)) |
| \(k_{th}\) | thermal initiation rate constant (L2/(mol2 s)) |
| \(L\) | tube length (m) |
| \(M\) | polymer molecular weight (g/mol) |
| \(M_n,\ M_w\) | number- and weight-average molecular weight (g/mol) |
| \([\mathrm{M}],\ [\mathrm{M}]_0\) | monomer concentration, in feed (mol/L) |
| \([\mathrm{P}^\bullet]\) | total radical concentration (mol/L) |
| \(\Delta p\) | pressure drop (Pa) |
| PDI | polydispersity index, \(M_w/M_n\) |
| \(q_i\) | volumetric flow rate in stream tube \(i\) (m3/s) |
| \(Q\) | total volumetric flow rate (m3/s) |
| \(r\) | radial position (m) |
| \(R\) | tube radius (m) |
| \(R_i\) | rate of radical initiation (mol/(L s)) |
| \(R_p\) | rate of polymerization (mol/(L s)) |
| \(t\) | time; residence time (s) |
| \(T\) | temperature (K in rate equations, °C on plots) |
| \(T_0,\ T_w\) | feed temperature, wall temperature (°C) |
| \(\Delta T_{ad}\) | adiabatic temperature rise at complete conversion (K) |
| \(u,\ \bar u\) | axial velocity, mean axial velocity (m/s) |
| \(W(\log M)\) | weight fraction of polymer per decade of molecular weight |
| \(X\) | monomer conversion |
| \(z\) | axial position (m) |
| \(\theta\) | dimensionless time, \(t/\tau\) |
| \(\mu,\ \mu_m\) | viscosity, viscosity of pure monomer at 100 °C (Pa s) |
| \(\rho\) | density (kg/m3) |
| \(\tau\) | mean residence time, \(L/\bar u\) (h) |
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.