Adiabatic head

Why a compressor makes head rather than pressure, and why the gas decides how much

Energy divided by weight leaves a length, which is why the answer arrives in feet — feet OF THE GAS BEING COMPRESSED, not feet of water, and nothing physically rises. The unit is bookkeeping: it lets head be multiplied by a mass flow and divided by an efficiency to reach shaft power.

A compressor develops HEAD, not pressure, and that is the practical reason this page exists. A wheel at a given speed puts roughly the same work into every pound of gas it handles, whatever the gas; the pressure that work produces depends entirely on the gas. It is why a machine moved from a rich stream to a light one fails to make its discharge pressure while doing nothing wrong.

The gas enters through the specific gas constant, 1545.349 divided by the apparent molecular weight. Head is directly proportional to it, so halving the molecular weight doubles the head needed for the same ratio: light, hydrogen-rich service is punishing where a heavy gas is comfortable, at identical pressures. Compressibility scales it the same direct way and is supplied rather than derived, because a relation that assumed one would be answering for a different gas than the one described.

The exponent is the same (k−1)/k that governs discharge temperature — head and temperature rise are two readings of one adiabatic path. The idealisation cuts the other way here, though. The temperature figure is optimistic because real losses arrive as heat; this is IDEAL work, so a real stage demands MORE, larger by the reciprocal of its efficiency. Vendors quote polytropic head, another idealisation of the same path, and the two must be converted before comparison.

When this calculation is used

• Turning a settled per-stage ratio into a work quantity, so shaft power follows once a mass flow and a stage efficiency are known.
• Screening centrifugal machines, where total head divided by the head one wheel develops sets the number of wheels on the shaft.
• Comparing gas compositions at the same pressures, to show why a lighter stream is a harder duty than the ratio alone suggests.
• Checking a head quoted on a vendor datasheet against an independent ideal figure, before efficiency or a polytropic correction.

Worked example

Close the cluster on the machine the other three pages describe: one of the two stages the staging page produced from the 9.0 duty, so a per-stage ratio of 3.0, with gas entering at 530 R, a specific heat ratio of 1.28, a compressibility of 0.95 and an apparent molecular weight of 18.825625 — the 0.65-gravity gas this pack anchors on.

Two figures come back: the specific gas constant the composition implies, and the head itself in feet of that gas. Read the head as the IDEAL work one stage puts into each pound. For a stage power, multiply by mass flow and divide by stage efficiency; two stages at this ratio need roughly twice this head between them.

Now change nothing but the molecular weight, dropping it to about half for a light, hydrogen-rich stream. The head required roughly doubles at identical pressures, temperature and ratio. That is why compression is sized on head rather than on pressure rise.

What each input represents

Assumptions and limits

• The path is adiabatic and reversible — isentropic — so this is IDEAL work per unit mass. A real stage needs more shaft work, by the reciprocal of its efficiency.
• The specific heat ratio and the compressibility factor are each one constant value across the stage, although both genuinely vary along the compression path.
• The gas is otherwise ideal: the specific gas constant is 1545.349 over the apparent molecular weight, and all real-gas behaviour rides on the supplied compressibility alone.
• The ratio is for one stage. Nothing here detects an overall machine ratio entered by mistake, and nothing here knows how many stages exist.
• This is adiabatic head, not the polytropic head vendors normally quote — same duty, different idealised path.

What the guards protect against

• The compression ratio must exceed one. At exactly one no work is done and below it the stage is expanding, so the input is refused as out of range rather than returning a head at or below zero — the pack ships that refusal as a declared test vector.
• Specific heat ratio, molecular weight and compressibility are each bounded to real gas mixtures. A specific gravity entered where a molecular weight belongs falls under the floor and is refused, instead of quietly producing a head roughly twenty times too large.
• Suction temperature is bounded to realistic absolute conditions, which catches this page’s commonest unit error: a Fahrenheit reading entered where Rankine is asked for lands below the floor.

Provenance

Standard adiabatic head relation, H = Z R Ts (k/(k-1)) (r^((k-1)/k) - 1)

The specific gas constant is 1545.349 divided by the apparent molecular weight and the adiabatic exponent is (k-1)/k. Head is compressibility times that constant times the absolute suction temperature, divided by the exponent, times the ratio raised to the exponent less one.

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.