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.
2 % - gap between the 4HE and 6HE curves - coefficient of variation
The 4HE and 6HE curves coincide to within 2 % across the full pressure-ratio range.
5–7 pp - the 2HES curve sits lower - smaller bore, leakage penalty
The 2HES outlier sits 5–7 percentage points lower because its smaller bore introduces a measurable surface-to-volume leakage penalty.
70 × 55 - bore × stroke in mm - 4HE-18Y and 6HE-28Y alike
In the sample table, the 4HE-18Y and the 6HE-28Y both have D × L = 70 × 55 mm; only the cylinder count differs.
0.9984 - R-squared, mass flow vs shaft speed - 4HE-25Y in the God Theory lab
God Theory lab, Collapse study, 4HE-25Y on R404A: mass flow is linear in shaft speed (R-squared 0.9984).
2.4% - forward model vs vendor capacity - mean absolute error
The forward physics model tracks the vendor capacity to 2.4 percent mean absolute error.
0.67× - 4HE-25Y to 6HE-35Y capacity - 4:6, the COP is invariant
Benchmark — overlay 4HE-25Y vs 6HE-35Y on R404A. The capacity scales near-perfectly 4:6 = 0.67×; the COP is invariant.
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
$\etav^{\mathrm{emp}}(\Pi) \;=\; \frac{\dot m^{\mathrm{poly}}(Te, Tc)}{\rhos(Te, \mathrm{SH}) \cdot V{d,\mathrm{total}} \cdot N}$
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