Polymerization Reactors — CSTR Cascade vs. Tubular (PFR)

A free-radical CSTR or PFR reactor with method-of-moments Mn/Mw/PDI, plus cross-stage MWD blending so a CSTR cascade reports a true cumulative molecular-weight distribution instead of just its last stage's local production — the real answer to whether a tubular reactor gives a tighter MWD than a CSTR train (it doesn't, and here's why).

polymerizationcstrpfrmolecular weight distribution

Concept

A free-radical polymerization reactor's job is to report not just how much monomer converted, but the molecular weight distribution (Mn, Mw, PDI) that determines the polymer's mechanical properties. Two reactor topologies are common in practice — a CSTR cascade (3-5 tanks in series) and a tubular (PFR) reactor — and it's a common assumption that the PFR's plug flow gives a "tighter" molecular weight distribution. That's true for the residence-time distribution (no long-tail under/over-reacted material, unlike a single CSTR), but it is not true for the instantaneous-chemistry MWD: a PFR's monomer concentration — and therefore its instantaneous chain length — drifts continuously from the inlet condition down to the outlet condition, while each CSTR stage makes its own chains under one fixed, local monomer concentration. Blending chains made at different instantaneous chain lengths always broadens dispersity above the local value.

The math

Show the governing equations

[R^\bullet] = \sqrt{\dfrac{2 f k_d [I]}{k_t}} \quad \text{(QSSA radical concentration)}

DP_n(z) = \dfrac{k_p\,[M](z)}{2\sqrt{f\,k_d\,k_t\,[I]}} \quad \text{(instantaneous chain length, drifts with }[M](z)\text{ along a PFR)}

\dfrac{1}{\overline{M_n}} = \sum_i \dfrac{w_i}{M_{n,i}}, \quad \overline{M_w} = \sum_i w_i M_{w,i} \quad \text{(mass-weighted blend of two populations, Odian)}

A CSTR evaluates DPn once, at its single outlet [M]. A PFR's [M] — and so its DPn — varies continuously along the tube; the cumulative Mn/Mw is the mass-weighted blend of every instantaneous slice, which is always ≥ the local pointwise PDI (2, under disproportionation termination).

Execution

A real, solved comparison

The same monomer feed, split 50/50 into two trains at equal total reactor volume (0.01 m³): a 3-stage CSTR cascade (each stage blending its own new polymer with the MWD already carried forward from the stage before it) versus a single PFR. Solving both trains gives:

Train	Conversion	M_n	M_w	PDI
CSTR cascade (3 stages)	81.3%	5,061	12,069	2.39
Tubular (PFR)	89.4%	5,566	15,444	2.77

Molar masses in kg/mol (this run's monomer/rate-constant parameters are illustrative, not a specific commercial styrene process — the comparison is what to trust, not the absolute magnitudes). Reproduce it yourself in the workspace below.

Launch in MaximaLabs Workspace

Bounded scope — read before assuming more

What this does not model

Single monomer only — no copolymer composition drift.
Disproportionation-only termination (PDI baseline 2, not combination-termination's ~1.5).
No initiator depletion along the PFR's length — the same constant-radical- concentration assumption the CSTR path makes, extended along the tube. This is exactly what makes the PFR case solvable in closed form; adding depletion would need a real ODE integration.
Method-of-moments only — no full chain-length distribution (no Flory/Schulz-Zimm shape), and no true residence-time-distribution solver anywhere in the platform today.