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Modeling where the factors ηij and ξij modify the arithmetic and geometric averages, respectively, between components i and j. Both parameters are usually set to one for mixtures of segments of similar size and energy. In this case, Equations III.13 and III.14 reduce to the simple Lorentz-Berthelot rules. Further information about the appropriateness of several combination rules can be found in reference (Diaz et al., 1982). III.2.1 The Quadrupole Moment To account for electrostatic contribution for the thermodynamic properties of a given system where at least one of the components has a quadrupole moment, a polar term, A polar , may be included into Equation III.3. The leading multipole term for fluids of linear symmetrical molecules, like carbon dioxide, nitrogen, benzene, etc., is the quadrupole- quadrupole potential (Gubbins and Two, 1978). An expansion of the Helmholtz free energy in terms of the perturbed quadrupole-quadrupole potential with the Padé approximation was proposed by Stell et al. (1974): A qq = A qq 2 ⎛ ⎜ ⎜ 1 ⎜ qq ⎜ A3 A + A 1− ⎜ qq ⎝ A2 qq 3B ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (III.15) Expressions for A2 and A3, the second and third-order perturbation terms, are calculated according to the segment approach presented by Jog et al. (2001) which is a modification of the molecular approach previously derived by Two and Gubbins (Two and Gubbins, 1975; Two, 1976) for an arbitrary intermolecular reference potential. The expressions for A2 and A3 were derived in the works of Gubbins and Two (1978) and Jog et al. (2001) and are presented here for completeness: 2 2 qq 14πN mρ Qi Q j A 2 = − ∑∑xi x jmi m j x p x i p I j 7 2, ij 5kBT i j σ ij ( ) ∑∑ 3 3 qq 144πN mρ Qi Q j A 3A = − xi x jmi m j x pix pj I 2 12 2, ij 245 kBT i j σ ij - 103 - (III.16) (III.17)
( ) ( ) ∑∑∑ 3 2 2 2 2 qq 32π 1/ 2 Nρ Qi Q j Qk A 3B = 2, 002π xix j xk mim jmk x pi x pj x pk I 2 3 3 3 3, ijk 2, 025 kBT i j k σ ijσ ikσ jk Modeling (III.18) The integrals I are calculated using molecular dynamic results for a pure Lennard- Jones fluid, and the resulting values were fitted to simple functions of reduced density and temperature as reported by Gubbins and Two (1978). As can be observed from Equations III.16 to III.18, when the quadrupole moment is explicitly taken into account, two more parameters have to be considered in the model, the quadrupolar moment Q (C.m 2 ) and xp, defined as the fraction of segments in the chain that contain the quadrupole as will be discussed latter in session III.4.3. III.2.2. The Crossover Approach As any other equation of state, soft-SAFT is unable to correctly describe the scaling of thermodynamic properties as the critical point is approached giving systematically higher predictions near this point. The mean-field equations of state provide a reasonable description of fluid equilibrium properties far away from the critical point. However, near the critical point, due to density and/or concentration fluctuations caused by long-range correlation between molecules, the thermodynamic properties of a fluid show singularities and therefore, classical analytical equations fail. Attempts to deal with this problem include the rescaling of molecular parameters so that the experimental critical point of the pure compound is matched (Blas and Vega, 1998b; McCabe and Jackson, 1999 and Pàmies and Vega, 2002). The simple rescaling of molecular parameters improves predictions of the critical behaviour of several mixtures, but a new set of different molecular parameters for each family of components is required for the near-critical region. This approach can be viewed as a practical one but it does not correctly solve the problem since the fluctuations inherent to the critical region are still ignored in this case. Renormalization-group methods (RG) have been very successful in describing the properties of systems near their critical point (Wilson, 1971; Wilson and Fischer, 1972). However, for engineering applications, the ideal situation would be a model that could correctly describe, using the same set of parameters and equation, the thermodynamics of the system far and close to the critical point. There are different approaches in which the long-wavelength density fluctuations can be taken into account in the near-critical region, searching for a global equation for real - 104 -
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Ana Maria Antunes Dias Universidade
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o júri presidente Prof. Dr. Joaqui
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palavras-chave resumo perfluoroalca
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Contents Notation List of Tables Li
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Notation Abbreviations AAD EoS LCST
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List of Tables Table I.1 Average Bo
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Table III.6 Adjusted Binary Paramet
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Figure II.9 Comparison between corr
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Figure III.8 Temperature-density di
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Figure III.25 Vapor-phase mole frac
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I.1. Fluorine Properties General In
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Table I.2. Physicochemical Properti
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General Introduction order to compa
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General Introduction the numerous a
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General Introduction carbon dioxide
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General Introduction animals. That
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General Introduction Table I.3. Lit
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References General Introduction Ban
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General Introduction Hildebrand, J.
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General Introduction Rowinsky EK. N
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II. Part EXPERIMENTAL METHODS, RESU
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Experimental Methods, Results and D
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Table II.3. (continued) T K ρexp g
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ρ / g.cm-3 1.750 1.730 1.710 1.690
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I. 3. Vapour pressure I.3.1. Biblio
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I.3.2. Apparatus and Procedure Expe
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I.3.3. Experimental Results and Dis
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Table II.8. (continued) T K Pexp kP
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ΔH vap = TΔS vap 2⎛ d ln P ⎞
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I.4. Solubility at atmospheric pres
Page 71 and 72: Experimental Methods, Results and D
Page 79 and 80: x2 (T,P2) 7.0E-03 6.0E-03 5.0E-03 4
Page 81 and 82: L 2,1 0.70 0.60 0.50 0.40 0.30 285
Page 93 and 94: P / MPa 6 5 4 3 2 1 0 P / MPa 14 12
Page 95 and 96: II.6. Liquid - Liquid Equilibrium I
Page 101 and 102: 0 τ β Δ1 2Δ1 [ 1+ B τ + B τ +
Page 103 and 104: References Experimental Methods, Re
Page 111 and 112: III.1. Introduction Modeling Most c
Page 113 and 114: Modeling The pioneering work of Wer
Page 115 and 116: III.2. Soft-SAFT Model Modeling A S
Page 117 and 118: Modeling The equation of state is w
Page 119 and 120: Chain Term Modeling Originally Wert
Page 121: Modeling The model is easily extend
Page 125 and 126: Modeling HRT is a promising theory,
Page 127 and 128: ( ρ) ρ , 0 ≤ ρ ( ρ) 2 Modelin
Page 129 and 130: III.3. Application to Pure Compound
Page 131 and 132: Modeling From the optimised paramet
Page 133 and 134: T / K 400 350 300 250 200 150 100 5
Page 135 and 136: ln Pvap 1.0E+01 1.0E+00 1.0E-01 1.0
Page 137 and 138: Modeling Table III.2. Absolute Aver
Page 139 and 140: Modeling The correlation coefficien
Page 141 and 142: Pvap (MPa) 4.00 3.50 3.00 2.50 2.00
Page 143 and 144: Modeling The mixture parameters a a
Page 145 and 146: Modeling the assumptions made by th
Page 147 and 148: Modeling between oxygen and perfluo
Page 149 and 150: xSolute 5.5E-03 5.0E-03 4.5E-03 4.0
Page 151 and 152: Modeling previous work, dealing wit
Page 153 and 154: Modeling These results confirm that
Page 155 and 156: Modeling CO2 binary mixtures using
Page 157 and 158: Modeling average deviation (AAD) be
Page 159 and 160: P / MPa 14 12 10 8 6 4 2 0 0 0.2 0.
Page 161 and 162: Modeling Figures III.16, III.17 and
Page 163 and 164: Modeling calculations from the orig
Page 165 and 166: Table III.9. References for VLE exp
Page 167 and 168: P / MPa 20 18 16 14 12 10 8 6 4 2 0
Page 169 and 170: Modeling Finally, Figure III.24 pre
Page 171 and 172: III.4.4. VLE and LLE of Alkane and
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Modeling number of perfluro-n-alkan
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y C6F14 1.0 0.8 0.6 0.4 0.2 0.0 0.0
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P / MPa 0.08 0.07 0.06 0.05 0.04 0.
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P / MPa 0.12 0.10 0.08 0.06 0.04 0.
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Modeling approach based on the meth
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Modeling Blas, F. J.; Vega, L. F.,
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DIPPR, Thermophysical Properties Da
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Modeling Hildebrand, J. H.; Fisher,
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Modeling McCabe, C.; Jackson, SAFT-
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Modeling Poling, B.; Prauznitz, J.;
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Modeling Wertheim, M. S., Fluids wi
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