CSTR Design Equation (First Order)

A continuous stirred tank reactor is assumed perfectly mixed, and that assumption has a consequence people find genuinely uncomfortable: the entire vessel sits at the outlet concentration. Feed entering a reactor running at 90% conversion is instantly diluted to 10% of its inlet strength, so the reaction proceeds everywhere at the slowest rate in the whole process. Feed 5 L/s with $k = 0.01\ \ ext{s}^{-1}$ and want 80% conversion, and you need $0.005 \ imes 0.8/(0.01\ imes 0.2) = 2.0\ \ ext{m}^3$.

The $(1-X)$ in the denominator is the term that bites. Going from 80% to 90% conversion doubles the volume. Going from 90% to 99% multiplies it by ten. Chasing the last few percent in a single stirred tank is a losing game, and the standard answer is to put two or three smaller tanks in series, which approaches plug-flow behaviour and can cut the total volume dramatically for the same duty.

What the equation quietly assumes is worth listing, because a real vessel violates all of it to some degree. Perfect mixing, steady state, constant density, constant temperature, and no dead zones or short-circuiting. A tracer test on a real tank almost always shows some feed reaching the outlet faster than the nominal residence time, which means the achieved conversion falls short of the calculation. The compensating virtue is that a CSTR's isothermal, well-mixed nature makes it far easier to control for a strongly exothermic reaction than a tube, where a hot spot can run away.