Trust-Region phase envelopes and phase identification

Evidence class: verification. This report checks torch-flash against the computed Peng-Robinson phase regions in Figures 3 and 4 of M. Petitfrere and D. V. Nichita, “Robust and efficient Trust-Region based stability analysis and multiphase flash calculations,” Fluid Phase Equilibria 362 (2014) 51–68, doi:10.1016/j.fluid.2013.08.039. The reference figures contain model calculations, not experimental observations.

Systems, model, and protocol

The two fixed-temperature \(P\)-\(z_{\mathrm{CO2}}\) systems are:

Target	System	Temperature / K	Components	Grid
Figure 3	CO2 + Majlamar reservoir oil	305.35	11	81 by 81
Figure 4	CO2 + Oil B	307.60	16	81 by 81

The component properties, oil compositions, and nonzero binary interaction coefficients are the complete Tables 4 and 5 of Z. Li and A. Firoozabadi, “General strategy for stability testing and phase-split calculation in two and three phases,” SPE Journal 17 (2012) 1096–1107, doi:10.2118/129844-PA. The supporting paper spells the field “Maljamar”; the target Figure 3 caption uses “Majlamar.” Unlisted interactions are zero.

The model is Peng-Robinson with the Robinson-Peng 1978 high-acentric-factor extension for the heavy pseudo-components. flash_grid independently discovers one, two, or three phases at each state. polish_grid_equilibrium_with_trust_region then applies:

• the modified tangent-plane-distance Trust-Region objective to one-phase states; and
• the improved mole-number Gibbs objective of Petitfrere and Nichita, equations (8)–(15), to two- and three-phase states.

States of the same phase count are evaluated in PyTorch batches with independent exact Hessian blocks. The calculation uses float64 on CPU with four PyTorch threads, a Trust-Region residual tolerance of \(10^{-8}\), a material-balance tolerance of \(5\times10^{-11}\), at most 100 Trust-Region iterations, and a Trust-Region batch chunk of 128 states.

Phase-envelope reproduction

Colors are torch-flash equilibrium regions. Open markers and dashed/dotted lines are raster digitizations of the published calculations, not experimental data. The displayed identity map uses the Venkatarathnam-Oellrich phase-identification parameter; phase count and the three-phase boundaries do not depend on that labeling choice.

The Majlamar three-phase region is fully immersed between the V–L1 and L1–L2 regions and closes near two bicritical points. Oil B reproduces the single-phase L1 wedge, the V–L1 and L1–L2 regions, the narrow three-phase ribbon, and its CO2-rich bicritical termination.

Target	Boundary	Digitized points	MAE / bar	Maximum error / bar
Figure 3	lower	19	0.139	0.700
Figure 3	upper	19	0.343	0.530
Figure 4	lower	15	0.310	0.703
Figure 4	upper	15	0.221	0.401

The comparison includes the finite pressure spacing of the 81-point grid and the uncertainty of raster digitization. All four mean absolute errors remain below 0.35 bar and all maximum errors remain below 0.71 bar.

Solver diagnostics

Diagnostic	Figure 3	Figure 4
States	6,561	6,561
One / two / three phases	0 / 5,580 / 981	1,588 / 4,759 / 214
Phase-discovery failures	0	0
Trust-Region phase-count corrections	0	18
Trust-Region attempts / failures	6,561 / 0	6,561 / 0
Maximum Trust-Region iterations	0	1
Maximum log-fugacity residual	\(9.998\times10^{-9}\)	\(9.986\times10^{-9}\)
Maximum material-balance residual	\(1.110\times10^{-16}\)	\(1.110\times10^{-16}\)
Phase discovery / s	57.53	47.48
Trust-Region audit / s	11.41	19.27

The zero-iteration maximum for Figure 3 means the independently discovered states already satisfy the Trust-Region stationarity tolerance; their exact gradients and Hessians are still evaluated. For Figure 4, negative converged TPD minima correct 18 nominally one-phase seeds to physical two-phase states.

Comparison of phase-identification methods

Physical phase identification is a post-flash diagnostic. It changes the names assigned to converged compositions, not the equilibrium phase count, fugacity equalities, or material balance.

Petitfrere and Nichita do not define a separate scalar phase-identification criterion. Li and Firoozabadi define the figure labels only: V is vapor, L1 is the CO2-lean liquid, and L2 is the CO2-rich liquid. Their selection of the higher-molar-mass phase elsewhere in the supporting paper is a stability-test initialization rule, not a phase-identification method. No Nichita-named criterion is therefore inferred.

All six methods exposed by identify_grid_phases were attempted on the same Trust-Region-polished equilibria:

Method	Fig. 3 available / %	Fig. 3 agreement with PIP / %	Fig. 4 available / %	Fig. 4 agreement with PIP / %
Li pseudo-critical temperature	0.0	—	0.0	—
Pedersen \(V/b\)	100.0	51.4	100.0	66.7
Perschke negative flash	100.0	97.8	100.0	99.7
Pasad \(d\kappa/dT\)	100.0	100.0	100.0	100.0
Bennett \(d\alpha/dT\)	100.0	98.7	100.0	100.0
Venkatarathnam-Oellrich PIP	100.0	100.0	100.0	100.0

Li's criterion requires critical molar volumes that are absent from the published Tables 4 and 5. It remains explicitly unavailable rather than silently estimating missing inputs. Pedersen \(V/b=1.75\) classifies every two-phase state as vapor-liquid in these dense CO2-rich systems. The two response-derivative criteria and PIP recover the connected liquid-liquid region; Pasad and PIP agree on every grid cell in both systems.

The ambiguity bands contain 25 active phase slots for PIP in Figure 3 and none in Figure 4. Such ambiguity is retained as a diagnostic instead of changing a phase count or smoothing the categorical map.

Conclusion and limitations

torch-flash reproduces the two published phase-envelope topologies and all digitized three-phase boundaries within the declared one-bar mean and two-bar maximum gates. Every one of the 13,122 states passes phase discovery and the batched exact-Hessian Trust-Region audit.

This is verification of equations and numerical behavior for the published Peng-Robinson parameter sets. It is not experimental validation. The Trust-Region calculation is local: independent phase-count discovery and stability diagnostics remain necessary, particularly near coalescing bicritical phases. Phase-identification agreement is also not proof of a physical label; the criteria use different diagnostics and can disagree in dense supercritical regions.