Napier's Steam Leak Rate

Read the unit convention first, because the bare 70 means nothing without it: Napier's rule is ṁ in POUNDS PER SECOND, A in SQUARE INCHES, and P in POUNDS PER SQUARE INCH ABSOLUTE. Per hour the same statement is the ASME Section I relief-capacity form ṁ = 51.5 A P, since 3,600/70 is 51.43. The solver converts for you, so you can type square millimetres and kilopascals and get kilograms an hour back, but the constant belongs to those three units and quoting the 70 against pounds per hour understates a leak by a factor of 3,600. Now the number everybody actually wants. A 1/8 inch hole is 0.0123 in². At 100 psig, which is 114.7 psia, Napier gives 0.0123 × 114.7 / 70 = 0.0201 lb/s, or 72 lb/h. Left alone for a year that is 634,000 lb of steam, about 288 tonnes, and at a typical $12 per 1,000 lb it is roughly $7,600 walking out of one hole you could cover with a thumbnail.

Napier's rule is an approximation and it is worth knowing exactly which one. It is a straight-line fit to CHOKED flow, meaning the steam is moving at its own speed of sound in the throat, which happens whenever the downstream pressure is below about 58% of the upstream absolute. Above that ratio the orifice is not choked, the flow depends on the downstream pressure too, and the rule reads high. The fit was made for saturated steam and is honest to a few percent from roughly 25 psia up to about 1,000 psia, drifting low above that. As a check on the constant itself, an ideal isentropic choked-flow calculation for that same 1/8 inch hole at 114.7 psia gives 69.8 lb/h against Napier's 72.4, which is the few percent high the rule is known for. Two field cautions. The area is the hole, not the pipe, and holes are rarely round or sharp-edged, so on a corroded gasket face or a worn valve seat you are estimating an equivalent area and the answer carries that uncertainty with it. And a plume is not a hole: plumes look the same at very different flows, so estimate the leak size from the metal, not from the cloud.