(555b) Curvature Analysis on Parameterized Tangent Plane Distance Analysis for Multi-Component Multi-Phase Equilibrium System

Simulating multi-phase equilibrium in separation and distillation processes can be extremely challenging due to possible phase flipping between two liquid phases and the difficulty of distinguishing between two indistinguishable liquid phases. These issues can introduce numerical instability to process simulation, especially when starting from a single liquid phase region and altering the feed composition for critical compounds until the system enters the liquid-liquid equilibrium region. Without knowing the identity of the original single liquid phase region, numerical difficulties may arise, causing unreliable simulation results that can hinder the effectiveness of the process.

Although tangent plane distance analysis (TPDA) offers a fundamental analysis of a system based on thermodynamic rules and provides a theoretical guideline for phase identity and possible phase splitting areas, significant numerical difficulties can arise during TPDA analysis when dealing with complex systems that have complicated underlying thermodynamic models. In fact, even for a relatively simple binary system like H2S + Methane, TPDA analysis can be challenging, as it shows that liquid-liquid phase splitting can occur at low temperatures (190K), pressure at 40.53 bar, and a 50/50 mole fraction feed. The true TPDA curve for this system is represented by the solid curves in Figure 1 (vapor in grey and the liquid in blue, SRK model), and to draw conclusions from this curve, a comprehensive global minimization scheme must be employed to rigorously solve the two valleys on the blue curve. However, this solution scheme can be time-consuming, and Hua et al. (1999) noted that TPDA analysis of complex systems can be particularly difficult, even with advanced computational methods.

It's important to note that despite the challenges of solving TPDA analysis efficiently, it remains a crucial tool for analyzing phase equilibrium in complex systems. However, when dealing with multi-component mixtures, the computational effort required to solve the global minimization scheme can grow exponentially, making it challenging to apply in real-time dynamic simulations. Therefore, promising approaches include simplification while retaining numerical stability and improving computational efficiency.

This paper introduces a new approach, the parameterized TPDA analysis. The parameterized TPDA curve requires only the true TPDA values at the feed and two near-pure composition points, which are then parameterized using a scheme called α. This scheme reduces the composition domain into a single-dimensional α domain, allowing us to write the parameterized TPDA curve in polynomial form.

Curvature analysis can then be done with minimal computational effort, providing accurate predictions of possible phase splitting. We plotted the parameterized TPDA curve alongside the true TPDA curve (liquid phase) in Figure 1 for the binary system H2S + Methane. Our approach has been validated by real customer projects for complex systems with large numbers of components and complicated thermodynamic models. By using this method, we can simplify the TPDA analysis procedure and achieve computational efficiency without sacrificing accuracy.

This parameterized TPDA analysis is generally applicable to any thermodynamic model.

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