19.2 Interstage Pressure Selection (Geometric Mean)

19.2 Interstage Pressure Selection (Geometric Mean)

Quick Answer: Convert suction and discharge to psia, then set P_int ≈ √(P_suc × P_dis). That geometric mean makes the two stage compression ratios approximately equal. For a −40°F freezer (10.4 psia) against an 86°F condenser (169.2 psia), P_int is about 42 psia (27 psig). Gauge pressures and vacuum inches do not go under the radical.

Once you accept that two-stage is required, the next CIRO skill is where to put the intercooler. Too low, and the high-stage does almost all the work. Too high, and the booster’s ratio and discharge temperature climb back toward the single-stage problem. The clean theoretical answer is the geometric mean of the two absolute pressures the plant is actually running.

Why geometric mean, not the average

Write the two stage ratios:

• CR_low = P_int / P_suc
• CR_high = P_dis / P_int

Set them equal: P_int / P_suc = P_dis / P_int. Cross-multiply: P_int² = P_suc × P_dis, so P_int = √(P_suc × P_dis).

That is the geometric mean. The arithmetic mean (P_suc + P_dis) / 2 sits far too high for a freezer. With 10.4 psia and 169 psia, the arithmetic mean is about 90 psia — a booster CR near 8.6:1 and a high-stage CR near 1.9:1. You just rebuilt an almost-single-stage booster and a lazy high-stage. Geometric mean is the formula that balances ratios, which is what keeps both discharge temperatures and both volumetric efficiencies in a sane band.

Equal ratios are the starting target when both stages handle about the same mass flow of the low-temperature load. Real plants add medium-temperature loads on the high-stage, different compressor displacements, economizer ports, and floating head pressure. The formula is still how you grade an interstage setpoint. It is not a law that the float valve must hit to the tenth of a pound.

Worked example: −40°F evaporator, 86°F condensing (psia)

Use class-standard ammonia saturation numbers and the vacuum conversion from the previous section.

Given

• Evaporator saturation: −40°F, suction 8.7 in Hg vacuum
• Condensing: 86°F, discharge pressure 169.2 psia (about 154.5 psig on the gauge)

Step 1 — Suction to psia

P_suc = 14.7 − (8.7 × 0.491)

8.7 × 0.491 = 4.27 psi below atmosphere

P_suc = 14.7 − 4.27 = 10.4 psia

(ASHRAE-style tables list anhydrous ammonia at −40°F as about 10.4 psia, so the conversion and the P/T chart agree.)

Step 2 — Discharge is already psia if you used the table; if you had a gauge, add 14.7

P_dis = 169.2 psia

If a problem gives 154.5 psig at 86°F: 154.5 + 14.7 = 169.2 psia. Same number.

Step 3 — Geometric mean

P_int = √(10.4 × 169.2) = √1759.7

Check nearby squares: 42 × 42 = 1764. 41.9 × 41.9 = 1755.6. 41.95 × 41.95 ≈ 1759.8.

P_int ≈ 41.95 psia ≈ 42 psia

Step 4 — Report it as a gauge reading the operator would set or recognize

Interstage gauge ≈ 42 − 14.7 = 27.3 psig (call it 27 psig on an exam)

Step 5 — Saturation temperature at that pressure

Ammonia at 15°F is about 43.1 psia (28.4 psig). At 10°F it is about 38.5 psia. 42 psia is about 14°F saturation. The intercooler liquid that will feed the −40°F loads is therefore in the mid-teens °F, not at 86°F condenser temperature.

Step 6 — Prove the ratios are equal

• CR_low = 42 / 10.4 ≈ 4.04
• CR_high = 169.2 / 42 ≈ 4.03

That is what “geometric mean” means on this exam: two stage ratios of about 4:1, not a magic pressure everyone memorizes as 33.5 psig.

Same freezer, 95°F / 181 psig condenser (the plant-diagram numbers)

Now apply the identical method to the representative two-stage diagram:

• P_suc = 10.4 psia (still 8.7 in Hg vac at −40°F)
• P_dis = 181 + 14.7 = 195.7 psia

P_int = √(10.4 × 195.7) = √2035

45.1 × 45.1 = 2034, so P_int ≈ 45.1 psia = 30.4 psig.

The diagram’s intercooler is 33.5 psig = 48.2 psia (20°F saturation on a standard ammonia chart — 20°F is 33.5 psig). That is a few pounds above the geometric-mean target of ~30 psig, not a contradiction.

Stage ratios at 33.5 psig: booster 4.6, high-stage 4.1. Still balanced enough to run. Why might operations hold 33.5 psig instead of 30 psig?

• Medium-temperature loads (docks, rooms, glycol) hanging on the intercooler / high-stage suction need a pressure that matches their evaporator, often near 20°F / 33.5 psig.
• Compressor displacement mix: a large high-stage and a smaller booster, or vice versa, shift the pressure that actually develops when both machines are loaded.
• Floating head: as condenser pressure falls at night, the geometric mean falls. A fixed 33.5 psig setpoint that was perfect in July is a different CR split in January.
• Level and vessel design: the open intercooler is also a liquid source. Operators may hold a pressure/temperature that keeps the HEV and recirculator happy rather than chasing a tenth of a ratio.

Exam move: compute the geometric mean, compute the actual interstage from the diagram, and be ready to say whether the plant is near the target. Do not mark the diagram “wrong” because it is not 27.3 psig.

Procedure you should be able to recite

1. Convert all pressures to psia. Vacuum: 14.7 − (in Hg × 0.491). Gauges: add 14.7.
2. Multiply P_suc × P_dis.
3. Take the square root. If you have no square-root key, use nearby integer squares (42² = 1764, 45² = 2025, 48² = 2304).
4. Subtract 14.7 if the question wants psig.
5. Look up saturation temperature on the ammonia chart — that is the intercooler temperature you expect if the vessel is a saturated flash drum.
6. Compute CR_low and CR_high. If they differ by a lot, either the plant is serving mixed loads or something is wrong (high-stage off-line, booster short-cycling, condenser in the weeds).

Traps that fail geometric-mean items

• √(psig × psig). At −40°F the suction gauge is a vacuum. There is no honest positive psig to multiply. Even at +20°F, √(33.5 × 181) is not the interstage in any unit system.
• √(8.7 × 181) mixing inches of mercury with psig.
• Arithmetic mean (10.4 + 169) / 2 ≈ 90 psia — a popular wrong answer because it looks “in the middle.”
• Forgetting to convert 181 psig and using 181 under the radical with 10.4, which understates P_int.
• Using 14.7 as suction because “vacuum means zero gauge, so 14.7 psia.” That is −28°F ammonia (atmospheric boiling), not −40°F.

Floating head pressure is a feature, not a bug: when condensing drops from 181 psig to 150 psig, recompute P_int. The geometric mean moves. Operators who nail the intercooler pressure to a winter value all summer are choosing a CR split, not “holding the book value.”

If a problem gives only temperatures, you still do not skip the chart. −40°F and 86°F are not pressures. Convert T → P_sat → psia, then take the square root. Temperature does not go under the radical.