Why does a CSTR need a much larger volume than a plug flow reactor for the same conversion?

In an ideal CSTR the contents are perfectly mixed, so the whole volume sits at the outlet composition. At 90 percent conversion the concentration of A everywhere in the tank is one tenth of the feed concentration, and a first-order reaction therefore runs at one tenth of its initial rate throughout. In a plug flow reactor the fluid starts at the feed concentration and the rate falls gradually along the length, so most of the reactor works at a higher rate than the outlet.

The design equations show it. CSTR: τ = X / (k × (1 - X)) = 0.9 / (0.1 × 0.1) = 90 min, so V = 9000 L. PFR: τ = ln(1 / (1 - X)) / k = 2.303 / 0.1 = 23 min, so V = 2300 L.

The penalty grows quickly with conversion. At 99 percent, the CSTR needs τ = 0.99 / (0.1 × 0.01) = 990 min while the PFR needs ln(100) / 0.1 = 46 min, a ratio of more than 20. At 50 percent conversion the figures are 10 min and 6.9 min, and the difference hardly matters.

It also depends on the kinetics. The higher the reaction order, the larger the gap. For a zero-order reaction, whose rate does not depend on concentration, the two reactors need the same volume. For autocatalytic reactions, where the rate rises as product forms, a CSTR can be the smaller reactor over part of the conversion range.

Staging recovers most of the difference. For N equal CSTRs in series with a first-order reaction, 1 - X = 1 / (1 + k × τ_i)^N, where τ_i is the residence time of each tank. For 90 percent conversion:

• 2 tanks: k × τ_i = sqrt(10) - 1 = 2.16, so 21.6 min each and about 4300 L in total.
• 3 tanks: k × τ_i = 10^(1/3) - 1 = 1.15, so 11.5 min each and about 3500 L in total.

Only the last tank operates at the final low concentration. As N increases the total approaches the 2300 L of the plug flow reactor, which is why a train of two to four stirred tanks is such a common arrangement.

Stirred tanks are chosen for reasons other than volume efficiency. The contents are uniform in temperature and have a large thermal mass, so a strongly exothermic reaction is far easier to control with a jacket or coil than in a tube, where hot spots develop. Tanks cope with slurries, viscous liquids, gas sparging and solids that would block a tubular reactor. They are easy to clean and inspect, and they smooth out fluctuations in the feed.

Selectivity can favour them too. If an unwanted side reaction is of higher order in the reactant than the desired one, the low reactant concentration in a CSTR suppresses it, and the larger vessel is a fair price for a purer product.