PFR Conversion (First Order)

A plug-flow reactor is the opposite idealisation to a stirred tank. Fluid moves down the tube as a series of independent slugs that do not mix with the slugs ahead or behind, so each slug is a little batch reactor whose reaction time is its position along the tube. Concentration therefore falls smoothly from the inlet value to the outlet value, and most of the tube is still working at concentrations well above the exit value — which is precisely the volume a stirred tank throws away.

Integrate a first-order rate law along that profile and the reactant decays exponentially with distance, giving $X = 1 - e^{-\mathrm{Da}}$. The comparison with the tank is the whole point of the chapter. At $\mathrm{Da} = 1$, the tank reaches exactly 50.0% and the tube 63.2%. At 90% conversion the tube needs $\mathrm{Da} = \ln 10 = 2.30$ where the tank needed 9.00, a vessel 3.9 times smaller. At 99% the tube needs 4.61 against the tank's 99 — a factor of twenty-one. The ratio is 1 at zero conversion and grows without bound, so the argument for tubular geometry gets stronger exactly where conversion targets get harder.

Neither curve literally reaches $X = 1$ in finite size; the difference is in the SHAPE of the approach. Unconverted reactant leaves a tube as $e^{-\mathrm{Da}}$, so every additional unit of Damköhler number divides what is left by 2.7 — the remainder falls geometrically, and a tube of ordinary length gets arbitrarily close to complete conversion. It leaves a tank as $1/(1+\mathrm{Da})$, which only ever chips away at the remainder linearly. That is the difference between a duty that is expensive and a duty that is impossible.

Two honest qualifications. This is an IDEAL tube, and a real one is not one: axial dispersion mixes neighbouring slugs, and the parabolic velocity profile in laminar flow means fluid at the wall spends far longer inside than fluid at the centre. Both drag performance back toward the stirred tank's, and both are measured with a tracer test rather than assumed away. And the tube's virtue in conversion is a vice in heat: an exothermic reaction in a tube concentrates its heat release near the inlet where concentration is highest, giving a hot spot with no thermal mass around it to absorb the excursion. Tube-side runaway is a real hazard, and it is one of the main reasons a plant sometimes accepts the tank's larger volume on purpose.