Conversion in n Equal CSTRs in Series

Tanks in series is the bridge between the two ideal vessels, and it is one of the most satisfying results in the subject. Take a single stirred tank and divide it into $n$ equal compartments with the feed passing through them one after another. Nothing has been added but partition walls, and yet the conversion climbs — because each compartment now sits at ITS OWN outlet concentration rather than all of them sitting at the final, lowest one. The first tank works at high concentration, the last at low, and the staircase of concentrations is a crude approximation to the smooth profile down a tube.

The arithmetic makes the gain concrete. Three tanks each at $\mathrm{Da} = 1$ reach $1 - 1/2^3 = 0.875$. A single tank with the same TOTAL Damköhler number of 3 reaches only $3/4 = 0.750$. Twelve and a half points of conversion for two walls. Push $n$ upward with the total volume held fixed and the cascade converges on the plug-flow curve; the convergence is not fast, so a fair approximation to a tube typically wants ten to twenty compartments, but the direction is monotone and there is no catch.

Getting the basis right is where most errors in this equation live. The $\mathrm{Da}$ in the formula is per TANK — the rate constant times the space time of one compartment. If a fixed total volume is being divided into $n$ pieces, each piece's residence time is the total divided by $n$, so each tank's Damköhler number is the total divided by $n$ too. Substituting the whole cascade's $\mathrm{Da}$ into the per-tank slot overstates conversion enormously, and it is the classic homework mistake here.

The same equation runs backwards as a diagnostic, and that is arguably its more important use. Fit a measured residence-time distribution to this model and it returns an equivalent $n$ that need not be a whole number: $n \approx 1$ says the vessel behaves as one well-mixed tank, $n$ above about twenty says it is effectively plug flow, and something in between quantifies exactly how far from either ideal a real basin, kiln or fermenter sits. Used forward for design, remember that a fractional answer has to be rounded UP — you cannot buy nine-tenths of a vessel, and rounding down misses the target conversion.