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The Helmholtz energy change due to association is calculated from Modeling α ⎡ ⎛ ⎞ ⎤ assoc α X ∑ ⎢∑⎜ i M ⎟ i A = RT xi ⎜ ln X i − ⎟ + ⎥ (III.8) i ⎢⎣ α ⎝ 2 ⎠ 2 ⎥⎦ where Mi is the number of association sites on each molecule of specie i, Σα represents a sum over all associating sites (on molecules of specie i) and Xi α is the fraction of nonbonded sites α of molecules i, defined as X α i = 1+ N ρ A β ∑xj∑XjΔ j 1 β α β i j (III.9) All the non-zero site-site interactions should be defined previously in order to solve αiβ j the equation. Δ involves an unweighted integral over all the orientations and an integration over all separations of molecules 1 and 2, defined as α β ∫ i j ij i j Δ = g ( 12) f ( 12) d( 12) (III.10) LJ α β with gLJ ij (12) the pair distribution function of the reference fluid, f αiβj (12) = exp (εAB HB /kBT) - 1 is the Mayer function of the association potential, and d(12) denotes an unweighted average over all orientations and an integration over all separations of molecules 1 and 2. The integration of Equation III.10 is not straightforward, since the pair distribution function is not readily available. An assumption is made that, for the purposes of the integration, the reference fluid pair correlation function is equivalent to that of the segment as part of a chain. This is a reasonable approximation if the bonding site is thought to be diametrically opposed to the backbone of the chain (Blas and Vega, 1998a). As mentioned before, further refinement of this approximation requires higher-order theories. The pair distribution function of the Lennard-Jones chain fluid was then replaced by the pair distribution function of the Lennard-Jones segment fluid, evaluated at the same temperature and segment density. In order to accurately calculate the integral, the expression from Muller and Gubbins (1995a) for a particular position of the association site inside the Lennard-Jones sphere has been used. Different association schemes can be assumed including cross-association. For more detail on the equations to use in each case see reference (Pàmies, 2003). - 101 -
Modeling The model is easily extended to mixtures. In fact, only the reference term needs to be extended to mixtures since the chain and association terms depend explicitly on composition and thus they are already applicable to mixtures. Mixtures consist in chains with different number of segments, which in turn can be of different size and/or dispersive energy and because of that, averaged parameters that simulate an “averaged” fluid that has the same thermodynamic properties as the mixture are needed. This is done by means of mixing rules. Several options exist (Economou and Tsonopoulos, 1997) but the van der Waals' one fluid theory is the most used. It is a well-established conformal solution theory that defines parameters of a hypothetical pure fluid having the same residual properties as the mixture of interest. Van der Waals' mixing rules are in good agreement with molecular simulation data for spheres of similar size (Rowlinson and Swinton, 1982). The corresponding expressions for the size and energy parameters of the conformal fluid are: ∑∑ 3 σ = i j 3 xi x jmi m jσ ij 2 (III.11) ⎛ ⎜ ⎝ ∑ i ∑∑ x m i ⎞ ⎟ ⎠ i 3 εσ = i j 3 xi x jmi m jε ijσ ij 2 (III.12) ⎛ ⎜ ⎝ ∑ i x m i ⎞ ⎟ ⎠ i The expressions III.11 and III.12 involve the mole fraction xi and the chain length mi of each of the components of the mixture of chains, denoted by the indices i and j, and the unlike (j ≠ i) interaction parameters σij and εij, which are determined by means of combination rules. The Lorentz-Berthelot combining rules are commonly used and were also employed in this work σ ii + σ ij σ ij = ηij (III.13) 2 ε = ξ ε ε (III.14) ij ij ii jj - 102 -
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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 ⎞
Page 69 and 70: 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: Chain Term Modeling Originally Wert
Page 123 and 124: ( ) ( ) ∑∑∑ 3 2 2 2 2 qq 32π
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
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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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