Stages required

How a whole number falls out of two ratios, and why the rounding is delicate

Stages compose by multiplication. Two stages each doing a ratio of three deliver an overall ratio of nine, not six, and n stages at a per-stage ratio deliver that ratio to the power n. Inverting it is where the logarithm comes from: the count is the logarithm of the overall ratio divided by the logarithm of the per-stage limit, rounded upward because a machine can only have a whole number of stages.

The per-stage limit is an INPUT, and treating it as a fixed constant is the main way this calculation is misused. Figures around three to four are commonly cited, and the pack ships a default in that band while noting it is a rule of thumb, not a standard value. The real limit comes from the discharge temperature the stage would reach, from rod loading on a reciprocating frame, and from the vendor curve for the equipment in mind — so two engineers considering different machines will correctly use different numbers.

Rounding up is not rounding, and the difference is expensive. If the arithmetic lands a hair above a whole number, the ceiling adds an entire stage — a cylinder or wheel, an intercooler, a separator and its share of the foundation. That makes the result unusually sensitive to numerical noise near an exact boundary, and such boundaries are common, because duties are often specified as neat squares and cubes of a round ratio.

So the resolution is handled deliberately rather than left to the hardware. A limit of 1.2 applied twice reaches 1.44 exactly, and a limit of three applied twice reaches nine exactly, yet in binary floating point the logarithm quotient can land a fraction above two, and a naive ceiling would return three. The pack resolves the exponent to fourteen significant decimal digits before rounding, so a duty exactly reachable in two stages returns two. Both cases ship as declared test vectors, so the behaviour is checked rather than asserted.

When this calculation is used

• Screening a duty at concept stage to establish how many stages, and therefore roughly what kind of machine, the pressures imply.
• Testing how sensitive an arrangement is to the per-stage limit, by running the same overall ratio against the range of limits the candidate equipment would support.
• Checking whether a declining suction pressure will eventually push an installed machine past the stage count it was built with.
• Sanity-checking a proposed configuration where the quoted stage count looks low or high for the ratio being asked of it.

Worked example

Take this pack’s anchor case, the same machine the rest of this cluster describes: an overall ratio of 9.0 — the 100 psia to 900 psia duty — against a per-stage ratio limit of 3.5.

The count that comes back is the smallest whole number of stages reaching the duty without any stage exceeding the limit. Read it with what it implies for the per-stage ratio: dividing the duty evenly across that many stages lands comfortably below the limit supplied, and that figure is what the discharge temperature page should be given.

Now lower the limit from 3.5 to 3.0. The duty is exactly reachable there, because three squared is nine, so the count does not change — the boundary case the decimal resolution exists to protect, where a naive implementation would add a third stage for no physical reason.

What each input represents

Assumptions and limits

• Every stage is treated as taking an equal share of the overall ratio, the usual screening convention and not how every real machine is arranged.
• The per-stage limit supplied is a rule of thumb rather than a standard value, and the answer is only as defensible as that input.
• No pressure loss across interstage coolers, scrubbers or piping is included, so a real machine generally needs slightly more than the ideal split.
• The result is a minimum driven by ratio alone. Capacity, rod load, driver size and frame availability can all push the real machine higher, and nothing here checks that the per-stage ratio is thermally acceptable for the gas.

What the guards protect against

• The overall ratio must exceed one. At or below one there is no compression duty and no stage count describes it, so the input is refused as out of range.
• The per-stage limit must also exceed one. A limit of exactly one could never reach any duty however many stages were added, and the logarithm underneath has no meaning there.
• Both inputs are bounded above. A ratio or limit far outside ordinary service is refused rather than answered, because at that distance the value is far more likely a data-entry error than a duty.

Provenance

Standard staging relation, n = ceiling of ln r / ln r_max

The stage exponent is the ratio of natural logarithms, resolved to fourteen significant decimal digits before being rounded up to the next whole stage, with a floor of one stage.

Screening and reference material, to be checked against the governing standard and a qualified engineer; not a design determination. The signed pack carries its own citation, and the page reports the verification state of the release it mounted.