A chemist's number becomes a tank

The equation, in words first

Volume equals what you feed in per second, times the fraction you want converted, divided by the rate. Feed, conversion, rate. Written down it is V = F_A0 X / (−r_A), and the only term in it that came from a laboratory is the one on the bottom.

F_A0 is how much reactant arrives every second, and it is a business decision: it is the plant's nameplate. X is how much of it you insist on converting before the stream leaves, and that is a separation decision, because whatever you do not convert you have to chase down and recycle. Both of those belong to the engineer. The rate is the chemist's, and it is the only one of the three you cannot argue with.

Where you evaluate the rate decides what you build

A stirred tank is mixed. That is not a detail, it is the definition: an impeller throws liquid at the wall, the wall splits the stream into two loops that turn the whole contents over, and within a few seconds there is no such thing as a fresh corner. Every millilitre in the vessel is at the composition of the stream leaving it.

That is a wonderful property for temperature control and a punishing one for size. The reaction runs at the concentration of the outlet, which is the lowest concentration in the process, so it runs at the slowest rate in the process, everywhere, all the time. You are dividing by the smallest number available. At 90% conversion the outlet rate is 0.01 mol/(L·s), a tenth of the rate the feed arrives capable of, and the volume that falls out is 900 L.

A tube never does that. Fluid enters at feed composition and leaves converted, and in between it passes through every concentration on the way down. Near the inlet it is reacting at 0.1 mol/(L·s), 10 times faster than anything ever happens in the tank. You add that up along the length instead of assuming the worst value everywhere, and the same duty closes in 230 L.

For any reaction whose rate falls as it proceeds, which is almost all of them, the tank is the bigger vessel. That is not a preference or a rule of thumb, it is what the two integrals do. The tank is still often the right answer, because it is easier to cool, easier to clean, and tolerant of solids, and those can be worth more than the steel. But you pay the 3.9 times in volume knowingly, or you find out later.

The rule of thumb that is not a law

Reaction rate doubles every ten degrees. Everyone is taught it and it is not a law, it is a single point on a surface. The real relationship is Arrhenius, and the gain you get for a ten kelvin step depends on both the activation energy and the temperature you start from.

Solve it backwards and the rule of thumb tells you what it was really assuming. A gain of exactly two per ten degrees starting from 25 °C corresponds to an activation energy of 52.9 kJ/mol. That is a perfectly ordinary value, which is why the rule survives, and it is the only place the rule is exact.

Take that same 50 kJ/mol reaction and run it at 100 °C, and a ten degree step now buys you 1.5 times. Take a reaction with a steeper temperature dependence, around 106 kJ/mol, and at room temperature the same ten degrees buys you four. Those are not edge cases. They are the ordinary spread of real chemistry, and the rule of thumb is silent about all of it.

So the chemist's second number is how steep that curve is, and it is the one nobody asks for until the tank is already built. It decides the size of the cooling system, how much a temperature excursion costs you, and whether a runaway is even reachable from your normal operating point. A reaction at 106 kJ/mol that loses its cooling is a fundamentally different problem from one at 50, and the design equation on its own will never tell you which you have.

Four numbers, one tank

1. 01
   Feed. How much reactant arrives every second. Yours, and it is the plant's nameplate.
2. 02
   Conversion. How much of it has to be gone before the stream leaves. Yours, and it is really a separation decision in disguise.
3. 03
   Rate. How fast the reaction goes at a stated concentration and temperature. The chemist's, and it sets the size.
4. 04
   Slope. How hard that rate moves when the temperature does. The chemist's, almost never handed over, and it sets the safety case.

Sources: the design equations and the standard result that a stirred tank needs more volume than a tube at the same conversion for positive order kinetics are from Fogler, Elements of Chemical Reaction Engineering. The Arrhenius treatment, and the note that a doubling per ten degrees near room temperature corresponds to an activation energy near 53 kJ/mol, follow Chem1, Arrhenius. The area and volume scaling behind the cooling argument is University of Illinois, scale up reaction safety. Every figure on this page is computed, not drawn to look right: the arithmetic is exp(E_a/R × (1/T₁ − 1/T₂)) and the two design integrals, evaluated for the case in the table above.