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. 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 $\dot m \;=\; \rhos(Te, \mathrm{SH}, \mathrm{ref}) \cdot Vd \cdot N \cdot \etav(\Pi, \mathrm{ref})$ that depends on the operating frequency is the rotational speed $N$ itself. Both the suction density $\rhos$ and the volumetric efficiency $\etav$ are functions of state variables that the test pins. We therefore expect $\dot m(N) \;=\; k \cdot N, \qquad k \;=\; \rhos \cdot Vd \cdot \etav(\Pi)$ i.e. a single-parameter linear relation through the origin. Sample and method We choose Bitzer's 4HE-25Y 4-cylinder semi-hermetic compressor on R134a — a representative Octagon-family model with broad inverter coverage. The vendor publishes a separate EN 12900 polynomial for every supported inverter frequency under capacitycontroltype = "Inverter frequency (Hz)". Each polynomial is evaluated at $Te = -10\,^\circ\mathrm{C}$, $Tc = 40\,^\circ\mathrm{C}$. Shaft speed is computed from the kinematic relation $N = f \cdot 2/p$ with $p = 4$ motor poles. f [Hz] N [rev/s] ṁ [kg/h] ṁ / N [kg/h per rev/s] ------------ 25 12.5 267.4 21.39 35 17.5 374.1 21.38 50 25.0 533.6 21.34 60 30.0 638.7 21.29 70 35.0 732.9 20.94 The ratio $\dot m / N$ 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. Result A least-squares fit of $\dot m…