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.
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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.
2.9 % - mean absolute error of the fit - over the whole Octagon corpus, R134a
The fit's mean absolute relative error is 2.9 % over the whole corpus.
4.4 % - error with no bore correction - βb forced to zero
If βb is forced to zero (no bore correction), the error rises to 4.4 % and a clear sorting of residuals by bore reappears.
3 - universal parameters per refrigerant - ε, n and βb
Three universal parameters (ε, n, βb) per refrigerant fit the entire Bitzer Octagon reciprocating family.
30-82 mm - bore range of the Octagon family - strokes from 27 mm to 60 mm
The Octagon family spans bores from 30 mm to 82 mm and strokes from 27 mm to 60 mm.
0.28 - fitted effective clearance ε - a real recip is typically 4–8 %
ε = 0.28 is well above the physical clearance fraction of a real reciprocating compressor (typically 4–8 %).
60 % - of the residual variance explained by bore - one linear factor in D
The single linear factor in D recovers roughly R² ≈ 0.6 of the residual variance.
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.
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