Cylinder-count symmetry: η_v(Π) is invariant to N_cyl in semi-hermetic reciprocating compressors
By DigiEntropy Engineering · 2025-12-21 · 7 min read
The kinematic mass-flow law treats cylinder count as a simple multiplier on swept volume. We verify the prediction by computing the empirical volumetric efficiency η_v(Π) for several Bitzer HE-family models on R404A and observe that the curves overlap within ~2 % across cylinder counts.
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 Cylinder count $N{\mathrm{cyl}}$ enters the kinematic mass-flow equation only as a multiplier on the total swept volume. If the per-cylinder bore and stroke are held fixed, the volumetric efficiency $\etav(\Pi)$ should be independent of $N{\mathrm{cyl}}$. We test this prediction across the Bitzer HE-family of semi-hermetic reciprocating compressors on R404A at 50 Hz. The $\etav(\Pi)$ curves of 4-cylinder and 6-cylinder models overlap within 2 % coefficient of variation. A 2-cylinder representative of the closely related HES geometry is included for visual contrast. Hypothesis Starting from the unified kinematic form $\dot m \;=\; \rhos \cdot \underbrace{N{\mathrm{cyl}} \cdot V{d,\mathrm{cyl}}}{V{d,\mathrm{total}}} \cdot N \cdot \etav(\Pi, \mathrm{ref})$ we extract the empirical volumetric efficiency from the vendor polynomial as $\etav^{\mathrm{emp}}(\Pi) \;=\; \frac{\dot m^{\mathrm{poly}}(Te, Tc)}{\rhos(Te, \mathrm{SH}) \cdot V{d,\mathrm{total}} \cdot N}$ with $\Pi = pd(Tc) / ps(Te)$. If the cylinder-count multiplier is the only role $N{\mathrm{cyl}}$ plays, two models with identical $(D, L)$ and different $N{\mathrm{cyl}}$ must yield the same $\etav^{\mathrm{emp}}(\Pi)$ curve. Sample and method The Bitzer HE family uses $D = 70\,\mathrm{mm}$, $L = 55\,\mathrm{mm}$ across cylinder-count variants. The HES family uses a smaller bore for the 2-cylinder version. We pick one representative from each family on R404A at 50 Hz with capacitycontroltype = "Inverter frequency (Hz)", then evaluate the vendor's $\dot m$ polynomial on a 25-point $Te$ grid at fixed $Tc = 40\,^\circ\mathrm{C}$. The Bitzer convention pins suction-gas temperature at $T{\mathrm{suct}} = 20\,^\circ\mathrm{C}$, so $\mathrm{SH} = 20 - Te$. Suction density is taken from CoolProp. Model Ncyl D × L [mm] Vd,total [cm³] ------------ 2HES-1Y 2 30 × 35 49.5 4HE-18Y 4 70 × 55 846.7 6HE-28Y 6 70 × 55 1270.1 Result Figure 1 overlays the empirical $\etav(\Pi)$ curves for the three models. The 4HE and 6HE curves coincide to within 2 % across the full pressure-ratio range — the residual gap is consistent with the small-bore correction examined in our…