Frequency collapse: mass flow is linear in shaft speed for reciprocating compressors
By DigiEntropy Engineering · 2025-12-21 · 6 min read
An empirical re-derivation of the kinematic mass-flow law from vendor polynomial coefficients. For a reciprocating compressor, ṁ scales linearly with rotational speed N to within ~0.2 % across 30–70 Hz inverter operation. We show the result for one Bitzer Octagon model and note that BOCK and Hanbell inverter ranges share the same form.
About this post — Authored by an AI assistant using DigiEntropy's polynomial corpus, the universal compressor predictor, and the project's chart-generation tooling. Charts are produced by Python scripts that read the same database the live site queries; tables and formulas are pulled from the same engineering modules. Findings reflect the project's current dataset and methodology — send corrections or deeper questions to admin@digientropy.com.
Abstract
The kinematic equation $\dot m = \rhos \cdot Vd \cdot N \cdot \etav$ predicts that the mass flow rate $\dot m$ of a positive-displacement compressor is linear in shaft speed $N$ whenever the suction state $(ps, Ts)$ and the pressure ratio $\Pi$ are held constant. We test this prediction against vendor EN 12900 polynomials for a single Bitzer Octagon reciprocating compressor across its full inverter range. The fit is straight-line to four significant figures ($R^2 = 0.9981$). The same form holds for the inverter-driven semi-hermetic ranges of BOCK and Hanbell.
0.9981 - R^2 of the straight-line fit - ṁ = k · N, through the origin
The fit is straight-line to four significant figures (R^2 = 0.9981).
21.4 - k, in kg/h per rev/s - the single fitted slope
A least-squares fit of ṁ = k · N to the full polynomial-evaluated set returns k = 21.4 kg/h per rev/s with R^2 = 0.9981.
2.1 % - spread of ṁ / N - across the full 25–70 Hz range
The ratio ṁ / N is constant to within 2.1 % across the full 25–70 Hz range. The slight high-frequency droop is consistent with a small valve-throttling loss at high N, not a violation of the linear law.
0.84 - effective volumetric efficiency - conventional band: 0.80–0.85
Inverting the fit gives a single effective volumetric efficiency of about 0.84 at this operating point, consistent with the conventional 0.80–0.85 band for semi-hermetic reciprocating compressors at Π ≈ 3.6.
46 - inverter-frequency curves - 4HE-25Y on R404A, 25 to 70 Hz
God Theory lab, Frequency study: 4HE-25Y on R404A carries 46 inverter-frequency curves from 25 to 70 Hz. Dividing by shaft speed collapses every curve onto one line, with a coefficient of variation of 0.90 percent at TE=-10, TC=40 C.
3 % - largest miss of the linear law - over the rated frequency window
We have not found a positive-displacement reciprocating range in which the linear relation fails by more than 3 % over the rated frequency window.
Hypothesis
For a positive-displacement compressor evaluated at a fixed evaporating temperature $Te$, fixed condensing temperature $Tc$, and fixed refrigerant, the only quantity in the kinematic mass-flow equation
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