Sherwood Number

The Sherwood number is the answer a mass-transfer correlation gives you. It compares convective transport to the transport that pure diffusion would manage over the same distance, $\mathrm{Sh} = k_c L/D$, and it exists because there is no instrument that reads a mass-transfer coefficient. You cannot put a probe in a stream and measure $k_c$ the way a thermocouple reads a temperature. It has to be inferred, and the inference is always the same shape: measure Reynolds, look up Schmidt, apply a correlation, get Sherwood, divide out to recover the coefficient.

The name honours Thomas Sherwood, whose 1934 work with Edwin Gilliland on wetted-wall columns produced the first correlation of the modern form and set the template — a Reynolds number to a power near 0.8, a Schmidt number to a power near one third — that essentially every gas-phase correlation since has followed. The direct analogy to the Nusselt number of heat transfer is not a coincidence but the entire basis of the field's method: because momentum, heat and mass are carried by the same eddies, a heat-transfer correlation can usually be converted into a mass-transfer one by exchanging Nusselt for Sherwood and Prandtl for Schmidt.

There is a floor worth committing to memory. A sphere sitting in a perfectly stagnant fluid still transfers by diffusion alone, and solving that problem exactly gives $\mathrm{Sh} = 2$. Every correlation for particles is built around it, in the form $\mathrm{Sh} = 2 + C\,\mathrm{Re}^a\mathrm{Sc}^b$, with the 2 as the no-flow limit and the second term as whatever the flow adds. A calculated Sherwood number below 2 for a particle is a signal that something is wrong, and the usual culprit is the length.

Which brings us to the mistake that dominates this group. Sherwood has no meaning apart from the characteristic length it was defined on, and that length is the tube diameter for internal flow, the particle diameter for a packed bed, and the plate length for flow along a surface. Those differ by orders of magnitude in the same apparatus. Two papers reporting Sherwood numbers a factor of four apart are, more often than anyone admits, reporting the same physics on different lengths. Before a published correlation is used, find the sentence that says what $L$ is — and if the paper does not say, the correlation is not usable.