Damköhler Number (First Order)

The Damköhler number is a race between two clocks. One is the chemistry: a first-order reaction consumes its reactant with a characteristic time $1/k$. The other is the plumbing: the fluid is only in the vessel for a space time $\tau = V/v_0$. Divide the second by the first and you have $\mathrm{Da} = k\tau$, which asks how many reaction lifetimes the fluid gets before it is pushed out the far end.

Its power is that it is the ONLY input a first-order reactor calculation needs. Conversion in a stirred tank is $\mathrm{Da}/(1+\mathrm{Da})$ and in a tube $1 - e^{-\mathrm{Da}}$; neither expression contains volume, flow rate or rate constant separately. Two reactors differing by a factor of ten thousand in volume reach exactly the same conversion if their Damköhler numbers match. That is scale-up compressed into one number, and it is why the group is worth learning as a group.

Reading the value is straightforward once the two regimes are clear. Below about 0.1 the reactor is convection-limited: the fluid leaves before the chemistry has done much, conversion is roughly $\mathrm{Da}$ itself, and a tank and a tube perform almost identically because at low conversion the concentration barely falls anywhere in either. Above about 10 the reaction is essentially finished and further volume buys very little. Between the two is where the vessel geometry earns its keep, and where a designer actually works.

Two cautions about the name. This is the FIRST Damköhler number; there are others in the literature comparing reaction rate against mass transfer or against heat transfer, and they are unrelated quantities that happen to share a surname. And the group is defined for first order only in this simple form. For other orders it carries a concentration term as well — $\mathrm{Da} = k \tau C_{A0}^{\,n-1}$ — which means the feed concentration stops cancelling and the tidy scale-free property weakens. Above about $\mathrm{Da} = 10$ there is also a physical caution: a real vessel that fast is usually limited by mixing or by diffusion to a catalyst surface, not by the intrinsic kinetics this number is built from.