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Construction of Global Phase Equilibrium Diagrams Martín Cismondi Universidad Nacional de Córdoba - CONICET Construction of Global Phase Equilibrium Diagrams Martín Cismondi Universidad Nacional de Córdoba - CONICET

Introduction • A real binary system show one of 5 (or 6) different types Introduction • A real binary system show one of 5 (or 6) different types of phase behaviour. • EOS modelling leads to the same possible types (+ other). • Correspondence between real and predicted type depends on the model and parameters.

Type I: Unique Critical Line (LV) A Pv 1 Pv 2 Type I: Unique Critical Line (LV) A Pv 1 Pv 2

Type II: Also a LL critical line and LLV A B Ps 1 LLV Type II: Also a LL critical line and LLV A B Ps 1 LLV Ps 2

Type IV: Discontinuity in the LV critical line and second LLV region E UCEP Type IV: Discontinuity in the LV critical line and second LLV region E UCEP LLV D LCEP E B D LLV

Type IV: T-x projection E UCEP LCEP B UCEP D Type IV: T-x projection E UCEP LCEP B UCEP D

Type III: “rearrangement” of critical lines C D UCEP Type III: “rearrangement” of critical lines C D UCEP

Type III: T-x projection C UCEP D Type III: T-x projection C UCEP D

Type V: Just like type IV but without LL immiscibility at low T E Type V: Just like type IV but without LL immiscibility at low T E E D LLV LCEP D UCEP

Azeotropic lines and… Azeotropic End Points (AEP) • PAEP (Pure, meeting a vapour pressure Azeotropic lines and… Azeotropic End Points (AEP) • PAEP (Pure, meeting a vapour pressure line) • CAEP (Critical, meeting a critical line) • HAEP (Heterogeneous, meeting a LLV line)

One example of azeotropic line (P-T) CAEP HAEP One example of azeotropic line (P-T) CAEP HAEP

The same example in T-x CAEP HAEP The same example in T-x CAEP HAEP

Cases with two azeotropic lines! C H P H Cases with two azeotropic lines! C H P H

Objectives • Identification of predicted type • Automated calculation of global phase equilibrium diagrams Objectives • Identification of predicted type • Automated calculation of global phase equilibrium diagrams • Automated calculation of Pxy, Txy and isoplethic diagrams from limiting points

What do we need? • Strategy for construction of a GPED without knowing the What do we need? • Strategy for construction of a GPED without knowing the type in advance. • General method for CRIT lines calculation. • Location of isolated LL critical lines. • General methods for LLV and AZE lines. • Detection of CEP’s and AEP’s (critical and azeotropic end points). • Classifications of Pxy, Txy and isoplethic diagrams in terms of limiting points. • Methods for calculation of Pxy, Txy and isoplethic segments.

Algorithm: Basic Structure Algorithm: Basic Structure

Some remarks about the methods… • Formulation in T, v and x, y, w… Some remarks about the methods… • Formulation in T, v and x, y, w… • Solve using Newton J ΔX = -F ; Fn= XS - S • Michelsen’s procedure for tracing lines J (d. X/d. S) = (d. F/d. S) → Xnew= Xold + (d. X/d. S) ΔS • ΔSnew = min (4 ΔSold / Niter , ΔSmax) • The variable to be specified depends on d. X/d. S

Calculation of critical points: Criticality conditions tpd 2=0 b = smallest eigenvalue λ 1=0 Calculation of critical points: Criticality conditions tpd 2=0 b = smallest eigenvalue λ 1=0 tpd 3=0 c = n 1 = z 1 + s = 0 u 1; n 2 = z 2 + s u 2

How to locate an isolated LL critical line? Must be 0 and min at How to locate an isolated LL critical line? Must be 0 and min at (T, P)

LLV equilibrium and CEP’s LLV equilibrium and CEP’s

Use of stability analysis in the search for a Critical End Point (CEP) Use of stability analysis in the search for a Critical End Point (CEP)

Calculation of a Critical End Point Calculation of a Critical End Point

Calculation of LLV lines Calculation of LLV lines

Examples: type II minimum composition Examples: type II minimum composition

Split of LV critical line in type IV or V Split of LV critical line in type IV or V

Transition III IV II Transition III IV II

Our Classification for Adding Azeotropy Our Classification for Adding Azeotropy

Detection of AEP’s • PAEP: compute along each vapour pressure line • CAEP: Pseudocritical Detection of AEP’s • PAEP: compute along each vapour pressure line • CAEP: Pseudocritical point compute 1 st derivative along the LV critical line • HAEP: crossing between L and V composition compute y 1 – x 1 along LLV line

Calculation of azeotropic lines: variables and equations Calculation of azeotropic lines: variables and equations

Illustration: Negative Azeotropy Illustration: Negative Azeotropy

Double Azeotropy: Minimum T in the azeotropic line Double Azeotropy: Minimum T in the azeotropic line

PAEP Bancroft point (Pv 1 = Pv 2) PAEP Tmin PAEP Bancroft point (Pv 1 = Pv 2) PAEP Tmin

Automated construction of complete Pxy and Txy diagrams • Reading and storing the lines Automated construction of complete Pxy and Txy diagrams • Reading and storing the lines and points of the Global Phase Equilibrium Diagram. Identification of type. • Detection of local temperature and pressure minima or maxima in critical lines. • Determination of the pressures (or temperatures) at which the different lines intersect at the specified temperature (or pressure). • Deduction, from the points obtained, of how many and which zones there will be. • Calculation of each zone or two-phase region.

T specified (NVP=2, NC=2, NLLV=1) T specified (NVP=2, NC=2, NLLV=1)

Translating from limiting points to diagrams Translating from limiting points to diagrams

variables and equations… variables and equations…

Examples: Closed loops in Pxy diagrams Examples: Closed loops in Pxy diagrams

Generation of Complete Isopleths • • Detection of composition local minima or maxima in Generation of Complete Isopleths • • Detection of composition local minima or maxima in critical lines, as well as in vapour or liquid branches of LLV lines. Location of intersection points at specified composition. Deduction of the number and nature of the segments the isopleth will be constituted of. Calculation of each segment of the isopleth.

Location of intersection points Location of intersection points

Number and nature of segments Number and nature of segments

Calculation of each segment Numerical continuation method • Sensitivities are used to – Choose Calculation of each segment Numerical continuation method • Sensitivities are used to – Choose which variable to specify for next point – Estimate values for all variables

Global Diagram: P-T projection Global Diagram: P-T projection

Global Diagram: T-x projection Global Diagram: T-x projection

Ethane-Methanol. RK-PR EOS. Ethane-Methanol. RK-PR EOS.

Ethane-Methanol. RK-PR EOS. Ethane-Methanol. RK-PR EOS.

Ethane-Methanol. RK-PR EOS. Ethane-Methanol. RK-PR EOS.

Ethane-Methanol. RK-PR EOS. Ethane-Methanol. RK-PR EOS.

Modular Approach • One general subroutine for calculation of P, and derivatives wrt T, Modular Approach • One general subroutine for calculation of P, and derivatives wrt T, V and n (given T, V and n) • Model specific subroutines for calculation of Ar and derivatives wrt T, V and n

Conclusions • We have provided strategies for constructing GPED’s from scratch. • Types I Conclusions • We have provided strategies for constructing GPED’s from scratch. • Types I to V, with or without azeotropy. • Pxy, Txy and Isopleths can be derived. • Strength: based on the GPED • Weakness: everything is based on the GPED www. gpec. plapiqui. edu. ar www. gpec. efn. uncor. edu

References • Global phase equilibrium calculations – – – • Cismondi, M. , Michelsen, References • Global phase equilibrium calculations – – – • Cismondi, M. , Michelsen, M. “Global Phase Equilibrium Calculations: Critical Lines, Critical End Points and Liquid-Vapour Equilibrium in Binary Mixtures”. The Journal of Supercritical Fluids, Vol. 39, 287 -295. 2007. Cismondi, M. , Michelsen, M. “Automated Calculation of Complete Pxy and Txy Diagrams for Binary Systems”. Fluid Phase Equilibria, Vol. 259, 228 -234. 2007. Cismondi, M. , Michelsen, M. L. , Zabaloy, M. S. “Automated generation of phase diagrams for binary systems with azeotropic behavior”. Industrial and Engineering Chemistry Research, Vol. 47 Issue 23, 9728– 9743. 2008. GPEC (the program) – • Cismondi, M. , Nuñez, D. N. , Zabaloy, M. S. , Brignole, E. A. , Michelsen, M. L. , Mollerup, J. M. “GPEC: A Program for Global Phase Equilibrium Calculations in Binary Systems” (Oral Presentation). EQUIFASE 2006. Morelia, Michoacán, México. October 21 -25, 2006. Models and their pure compound parameters – – Cismondi, M. , Mollerup, J. “Development and Application of a Three-Parameter RK-PR Equation of State”. Fluid Phase Equilibria, Vol. 232, 74 -89. 2005. Cismondi, M. , Brignole, E. A. , Mollerup, J. “Rescaling of Three-Parameter Equations of State: PC-SAFT and SPHCT”. Fluid Phase Equilibria, Vol. 234, 108 -121. 2005.