Geometry collapse: one volumetric efficiency curve fits the entire Octagon family
By DigiEntropy Engineering · 2025-12-21 · 8 min read
We extend the η_v(Π) analysis to all reciprocating geometries in the Bitzer Octagon family. A single universal clearance-volume law with three parameters (ε, n, β_bore) fits the empirical volumetric efficiency to a mean absolute error of ~3 % on R134a. The dominant residual is explained by cylinder bore.
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 A single closed-form expression for the volumetric efficiency, $\etav(\Pi) \;=\; \left(1 + \varepsilon - \varepsilon \, \Pi^{1/n}\right) \cdot \left(1 - \betab \, \frac{D{\mathrm{ref}} - D}{D{\mathrm{ref}}}\right)$ with three universal parameters $(\varepsilon, n, \betab)$ per refrigerant, fits the entire Bitzer Octagon reciprocating family to a mean absolute error of $\Delta\etav/\etav \approx 3\%$ on R134a. Cylinder bore is the single significant second-order predictor of the residual; once $\betab$ absorbs it, no further geometric variables improve the fit. Hypothesis The standard textbook derivation of the volumetric efficiency from clearance-volume re-expansion gives $\etav \;=\; 1 + \varepsilonc - \varepsilonc \, \Pi^{1/n{re}}$ where $\varepsilonc$ is the relative clearance volume and $n{re}$ the polytropic re-expansion index. The textbook form assumes that $\varepsilonc$ and $n{re}$ are geometry-independent. We test this assumption across all available bore $\times$ stroke combinations in the Bitzer Octagon family. A first-pass fit reveals a systematic residual that correlates with cylinder bore $D$: small bores run lower $\etav$ at any given $\Pi$. The physical interpretation is the surface-to-volume ratio of the cylinder — leakage past piston rings and through the discharge valve is proportional to the perimeter, while the swept volume scales with the cross-section. We absorb this into a multiplicative correction parameterised by $\betab$, leaving three parameters $(\varepsilon, n, \betab)$ per (technology $\times$ refrigerant) pair. Sample and method The Octagon family spans bores from 30 mm to 82 mm and strokes from 27 mm to 60 mm, distributed across several letter classes (HE, FE, GE, JE, EES, CES, …). For each model with both a polynomial record on R134a at 50 Hz and a known cylinder geometry, we sweep $Te$ across the model's published envelope at fixed $Tc = 40\,^\circ\mathrm{C}$ and accumulate $(\Pi, \etav^{\mathrm{emp}}, D)$ triples by the same construction used in the cylinder-count analysis. The Bitzer suction convention $T{\mathrm{suct}} = 20\,^\circ\mathrm{C}$ is preserved. A…