Topic J — A Laminar Tubular Polymerization Reactor

axisymmetric flow with conversion-dependent viscosity · species and heat · exothermic kinetics · tier: advanced

Motivation

In a tubular polymerization reactor the viscosity rises by orders of magnitude with conversion. The fluid near the wall, which has stayed longest and polymerized most, slows down further, the velocity profile sharpens, the residence-time distribution broadens — and with an exothermic reaction a hot, viscous wall layer can form. The coupling of rheology, transport and kinetics decides conversion, molecular weight spread and safe operation; it is a classical problem of polymer reaction engineering.

Related fields: polymerization kinetics, polymer reactor modelling, process control and optimisation. Course weeks: 2 (axisymmetric weak forms), 11 (Navier–Stokes, mixed spaces, continuation), 12 (reactive flow, SUPG).

Model

Tube of radius R and length L, axisymmetric (r,z), steady state. Navier–Stokes (Reynolds number small) with a viscosity \eta(X,T)=\eta_0\exp\big(aX+\frac{E_\eta}{R_g}(\frac1T-\frac1{T_0})\big), monomer conversion X and temperature T: \mathbf u\cdot\nabla X-\nabla\cdot(D\nabla X)=k(T)(1-X),\qquad \rho c_p\,\mathbf u\cdot\nabla T-\nabla\cdot(k_T\nabla T)=(-\Delta H)\,c_{M0}\,k(T)(1-X), k(T)=k_0e^{-E/R_gT} (first-order pseudo-kinetics; free-radical kinetics with the gel effect as a stretch), parabolic inflow, X=0, T=T_0 at the inlet, wall at T_w (Dirichlet or Robin).

Design outputs: outlet conversion (mixing-cup average \bar X=\frac{\int uX\,r\,dr}{\int u\,r\,dr}), the velocity profile along the tube, the maximum temperature, against the flow rate and the viscosity parameter a.

Suggested parameters

Dimensionless: Damköhler number k(T_0)L/\bar u = 0.1–5, radial Péclet number \bar uR^2/(DL) = 10–10⁴ (polymer: small D), a = 0–10 (viscosity ratio e^a up to 2×10⁴), adiabatic temperature rise up to 50 K with Arrhenius number E/R_gT_0 = 20–30.

Required tasks

  1. Isothermal, constant viscosity: conversion against Da, compared with the limits below.
  2. Viscosity rising with conversion (a = 0 → 10 by continuation): velocity profiles, outlet conversion.
  3. Non-isothermal with wall cooling: maximum temperature and conversion against Da; Newton with continuation.
  4. Mesh independence of the outlet conversion and of the peak temperature.

Verification specific to this topic

  • Plug-flow limit (strong radial mixing, D\to large in r): \bar X=1-e^{-\mathrm{Da}}.
  • Segregated laminar-flow limit (D\to0, constant viscosity): each streamline is a batch reactor with residence time t(r); the RTD of laminar flow is E(t)=\frac{\tau^2}{2t^3} for t\ge\tau/2, so 1-\bar X=\int_{\tau/2}^\infty e^{-kt}E(t)\,dt — evaluate it with NumPy and compare.
  • Flow: constant viscosity gives Poiseuille exactly (P2–P1); with \eta(X), the flow rate must be conserved along the tube.
  • Energy balance: heat released = heat removed at the wall + enthalpy rise, to discretisation error.
  • MMS for the coupled axisymmetric system.

Stretch goals

  1. Free-radical kinetics with initiator decay and the gel (Trommsdorff) effect; moments of the molecular-weight distribution.
  2. Static mixer inserts (3D, or a 2D cross-section model) and their effect on the RTD.
  3. Thermal runaway map in (Da, wall temperature) with pseudo-arclength continuation.

Milestones

M1 Poiseuille + isothermal conversion with both limits · M2 viscosity coupling with continuation · M3 non-isothermal · M4 energy balance and mesh study · M5 parameter study.

Pitfalls

  • The axis: no condition on r=0 for the scalars, u_r=0 there for the flow, and the r-weight everywhere (Week 2).
  • Large Péclet numbers in z: SUPG for X and T (Week 12); check that the outlet conversion does not depend on the stabilisation parameter.
  • A viscosity contrast of 10^4 in one Newton step diverges: continuation in a with the previous solution as initial guess.

References

  • O. Levenspiel, Chemical Reaction Engineering, 3rd ed., Wiley 1999 — laminar-flow RTD, segregated flow model.
  • J. A. Biesenberger, D. H. Sebastian, Principles of Polymerization Engineering, Wiley 1983.