Three‑dimensional graphical golden ratio optimization of a variable working fluid concentration auto‑cascade refrigeration cycle

The auto-cascade refrigeration cycle has broad application prospects in low-temperature freezers, biomedical equipment, environmental test equipment, and small-scale liquefaction devices. Compared with conventional multi-stage cascade cycles, improved low-temperature refrigeration performance can be achieved through composition and mass-flow redistribution by vapor–liquid separation. Three-dimensional thermodynamic diagrams can intuitively visualize these variations and provide a basis for graphical pre-optimization. In this study, a coupled vapor–liquid equilibrium separator and ejector model was established for an R600a/R134a concentration adjustment auto-cascade refrigeration cycle (CA-ACRC), and three-dimensional P-h-w and T-s-w diagrams were constructed. A golden-ratio-based composition-scaling criterion was proposed, with Δ w GR = (1/ φ )(1− w Ⅰ ) ≈ 0.618(1− w Ⅰ ) and G w = Δ w /(1− w Ⅰ ) ≈ 0.618, where w Ⅰ is the initial R134a mass fraction. This criterion can be used to identify high-performance candidate separation states before final thermodynamic optimization. Parameter scanning showed that the maximum-COP states satisfied D GR < 3% within w Ⅰ = 0.35–0.51, while the applicable range extended to approximately w Ⅰ = 0.54 when COP GR /COP opt ≥ 95% was adopted. Further cross-refrigerant VLE analysis indicated that the golden ratio represents a target composition-separation scale rather than a universal thermophysical law. Its feasibility requires the VLE separation envelope to intersect Δ w GR , establishing a physical connection between graphical pre-optimization and refrigerant phase-equilibrium properties.