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Construction of Global Phase Equilibrium Diagrams

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 of phase behaviour. EOS modelling leads to the same possible types (+ other) .

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Construction of Global Phase Equilibrium Diagrams

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

  2. 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.

  3. Type I: Unique Critical Line (LV) A Pv1 Pv2

  4. Type II: Also a LL critical line and LLV A B Ps1 Ps2 LLV

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

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

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

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

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

  10. 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)

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

  12. The same example in T-x CAEP HAEP

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

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

  15. 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.

  16. Algorithm:Basic Structure

  17. 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 (dX/dS) = (dF/dS) → Xnew= Xold + (dX/dS)ΔS • ΔSnew = min (4 ΔSold / Niter , ΔSmax) • The variable to be specified depends on dX/dS

  18. n1 = z1 + s u1; n2 = z2 + s u2 Calculation of critical points:Criticality conditions tpd2=0 b = smallest eigenvalue λ1=0 tpd3=0 c = = 0

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

  20. LLV equilibrium and CEP’s

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

  22. Calculation of a Critical End Point

  23. Calculation of LLV lines

  24. Examples: type II minimum composition

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

  26. Transition IIIIV II

  27. Our Classification for Adding Azeotropy

  28. Detection of AEP’s • PAEP: compute along each vapour pressure line • CAEP: Pseudocritical point compute 1st derivative along the LV critical line • HAEP: crossing between L and V composition compute y1 – x1 along LLV line

  29. Calculation of azeotropic lines: variables and equations

  30. Illustration: Negative Azeotropy

  31. Double Azeotropy: Minimum T in the azeotropic line

  32. PAEP Bancroft point (Pv1 = Pv2) PAEP Tmin

  33. 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.

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

  35. Translating from limiting points to diagrams

  36. variables and equations…

  37. Examples: Closed loops in Pxy diagrams

  38. 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.

  39. Location of intersection points

  40. Number and nature of segments

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

  42. Global Diagram: P-T projection

  43. Global Diagram: T-x projection

  44. Ethane-Methanol. RK-PR EOS.

  45. Ethane-Methanol. RK-PR EOS.

  46. Ethane-Methanol. RK-PR EOS.

  47. Ethane-Methanol. RK-PR EOS.

  48. 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

  49. 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.arwww.gpec.efn.uncor.edu

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