17: Vapor-Liquid Equilibrium via EOS
- Page ID
- 571
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Module Goal: To use cubic equations of state for the description of phase equilibrium.
Module Objective: To integrate the knowledge of qualitative phase behavior and the framework for quantification to actually build a phase equilibrium model.
- 17.1: Introduction
- This page highlights the importance of understanding gas and liquid co-existence in petroleum and natural gas applications, emphasizing vapor-liquid equilibrium and thermodynamic tools. It connects earlier equations of state to solving various applications, including phase diagrams, reinforcing the course's interconnected knowledge.
- 17.2: Equilibrium and Equilibrium Ratios (Ki)
- This page covers liquid-vapor equilibrium conditions, focusing on equal chemical potentials and fugacities. It introduces the equilibrium ratio \(K_i\) and methods for its calculation, emphasizing challenges in flash problems and advocating for an iterative composition determination. The page also compares the dual-fugacity coefficient method and dual-activity coefficient method for phase equilibria calculations, favoring the dual-fugacity method for its simplicity, despite some inaccuracies.
- 17.3: Solution Algorithms for VLE Problems
- This page explains how to simplify vapor-liquid equilibrium (VLE) problems by solving non-linear algebraic equations through the Rachford-Rice Objective Function and two other options. It emphasizes the difficulty in determining equilibrium ratios (Ki's) and presents iterative methods to find unknowns (Ki, yi, xi, αg). Two main solution techniques are described: computationally intense Newton-type methods and simpler but slower substitution-type methods.
- 17.4: Successive Substitution Method (SSM)
- In a substitution-type method, we start with initial guesses for all of the unknowns and loop around the equations to obtain “better” approximations for each of them. We test the goodness of the solution at every time step by comparing the new, better approximation to the previous guess. If the correction is small under certain convergence criteria, the procedure is stopped and we use the results from the last iteration as the final answer.
- 17.5: The Accelerated Successive Substitution Method (ASSM)
- This page addresses the convergence challenges in the Successive Substitution Method (SSM) near critical points due to composition-dependent fugacities. It presents the Accelerated and Stabilized Successive Substitution Method (ASSM) as a solution, detailing its implementation steps, efficiency criteria, and the verification of fugacity ratios. Additionally, it mentions other acceleration methods.
- 17.6: The Stability Criteria
- This page covers the complexities of predicting phase separation in mixtures at specified pressures and temperatures, emphasizing the necessity of single-phase detection and VLE calculations. It details the Michelsen stability test, which involves forming a second phase to evaluate stability through two scenarios—vapor-like and liquid-like phases. The stability is assessed based on convergence criteria and mole number sums, leading to a conclusion on whether the mixture is stable or unstable.