The pressure correction Δλv,P is calculated from the method of Stiel and Thodos, see Ref. 32, which is applicable for ρr<3, but is less accurate for H2, strongly polar gases, and gases with a high degree of hydrogen bonding, such as H2O and NH3

The mixing rules are as suggested by Yorizane, see Ref. 33:

However, in order to ensure 0th-order continuity at ρr = 0.5 and ρr = 2.0, the following coefficients for 0.5 ≤ ρr < 2.0 are recalculated from

The vapor thermal conductivity correlation must be available for all species. Also critical volumes, Vc,i, critical temperatures, Tc,i, molecular weights Mi, and acentric factors ωi must be specified for all species.

Lindsay and Bromley (see Ref. 34) provided an equation for the interaction parameters of the method of Wassiljewa (see Ref. 35) based on the kinetic theory, to provide mixture thermal conductivity from pure species values

where the pressure correction Δλv,P is calculated from Equation 2-133. Both vapor thermal conductivity correlation λi,v and the vapor viscosity correlation ηi,v must be available for all species. In addition, all normal boiling points Ti,b, molecular weights Mi, critical volumes Vc,i, critical temperatures Tc,i, and acentric factors ωi must be specified.

The International Association of the Properties of Water and Steam recommend an equation Ref. 36 which is valid in the following range:

To calculate the mixture liquid thermal conductivity, λl,m, the values of pure liquid thermal conductivity correlations are mixed ideally

The pressure dependence is based on the work of Missenard (Ref. 52) where

where Q is correlated as

All liquid thermal conductivity correlations must be specified. All values for critical temperatures, Tc,i, critical volumes, Pc,i and critical compressibility factors, Zc,i must be specified for all species i.

All liquid thermal conductivity correlations must be specified. The model is valid for pure compound thermal conductivity values that are no more apart than a factor of 2 (Ref. 53 and Ref. 54). The pressure dependence is introduced using Equation 2-152 through Equation 2-159. All values for critical temperatures, Tc,i, critical volumes, Pc,i and critical compressibility factors, Zc,i must be specified for all spices i.

The local composition model by Rowley (Ref. 54) uses an ideal and excess contribution

where Gj,i follows from Equation 2-57. The binary interaction terms follow from

with ϖi is the composition in the binary mixture of species i and j and the local composition is equi-molar

All liquid thermal conductivity correlations must be specified. The pressure dependence is introduced using Equation 2-152 – Equation 2-159. All values for critical temperatures Tc,i, critical volumes Pc,i, critical compressibility factors Zc,i, and molecular weights Mi must be specified for all compounds i. In addition, all NRTL binary interaction parameters Ai,j must be specified. Unspecified values for NRTL interaction parameters Bi,j are set to zero. The randomness parameters αi,j have values of zero on the diagonal and the matrix is symmetric. All off-diagonal values must be specified. NRTL model is presented in Equation 2-53 to Equation 2-64.

Rowley (Ref. 53) adapted the local composition model by replacing the mixing rule in Equation 2-165 by the following

which he found to produce better model predictions in most cases where both the Local Composition model and Power Law model have trouble. However, the model is not as generally applicable; for instance, systems containing H2O are not well described by this model due to the low molecular weight of H2O.

The International Association of the Properties of Water and Steam recommend an equation Ref. 36 which is valid in the following range:

Wilke, see Ref. 37, based his method for mixture viscosity of the vapor phase on kinetic theory:

The vapor viscosity correlation ηi,v must be available for all species. In addition, all molecular weights Mi must be specified.

Brokaw (see Ref. 38) uses the same basic equation as Wilke (Equation 2-173). However, Equation 2-174 is replaced by

The vapor viscosity correlation, ηi,v must be available for all species i. In addition, all molecular weights Mi must be specified. If Lennard-Jones energy εi (see Ref. 39) Stockmayer’s polar parameter δs,i(Ref. 40 and Ref. 41) are specified for both species i and j then

where ξ is calculated from the correlation of Jossi (Ref. 42), which is applicable for ρr < 3.0. It is less accurate for H2, strongly polar gases and gases with a high degree of hydrogen bonding such as H2O and NH3.

The values for critical volumes, Vc,i, critical temperatures, Tc,i, critical compressibility factors, Zc,i and molecular weights, Mi must be specified for all species i.

where ηv,Wilke is calculated from Equation 2-173.

The corresponding states viscosity model of Pedersen (Ref. 43 and Ref. 44) applies to both vapor and liquid phases of hydrocarbon mixtures. The selected reference species is CH4.

The CH4 viscosity is calculated from Ref. 45, modified by Pedersen and Fredunsland (Ref. 46) to avoid issues below 91 K where CH4 becomes solid

Here, ρCH4 is used in g/cm3; for the mass-mole conversion of ρCH4, a molecular weight of MCH4 = 16.042568 g/mol is used.

with the values of the parameters L1 through L25 are listed in Table 2-5 below:

Here, ρCH4 is used in g/cm3; the critical density is given by ρc,CH4 = 0.16284 g/cm3. The following equation by McCarty (Ref. 47) is solved for the density of CH4

where ρCH4 is used in mol/l.

With the viscosity and density of CH4 defined, the viscosity of any mixture, ηm, can be calculated from the corresponding states principle

where CH4 viscosity ρCH4,P0,T0 is calculated at temperature, T0 and pressure P0

The following mixing rules are used for the critical properties, see Ref. 48:

The parameter α is

with ρc,CH4 = 0.16284 g/cm3. For CH4

where Equation 2-204 is used. The mixture molecular weight is a function of the weight-averaged molecular weight and the number-averaged molecular weight

where the power in Equation 2-208 is determined by fitting to experimental viscosity data.

Note that pure species vapor viscosity correlations ηi,v are not required. However, for each species i, molecular weight Mi, critical temperature, Tc,i and critical pressure, Pc,i must be specified.

The International Association of the Properties of Water and Steam recommend an equation Ref. 49- Ref. 50 for industrial application which is valid in the following range:

The values of pure species log liquid viscosity, ln ηi,l are mixed ideally using mole fractions xi

where ηm,l is the mixture viscosity of liquids.

The Pedersen Corresponding States Model described above for the gas phase viscosity also applies to the liquid phase. Pure species liquid viscosity correlations are not required. However, for each species i, molecular weight Mi, critical temperature Tc,i, and critical pressure Pc,i, must be specified.

The model is noted in Ref. 51 to provide reasonable results for hydrocarbon mixtures of similar components.

The model requires that the log liquid viscosity correlation is available for all species i.

The model requires that the log liquid viscosity correlation is available for all species i.

The International Association of the Properties of Water and Steam recommend an equation Ref. 49- Ref. 50 for industrial application which is valid in the following range:

For dilute systems, the binary diffusion coefficient D0i,j represent the diffusivity of species i in a medium consisting of pure species j. This corresponds to the Fickian diffusion coefficient.

For any mixture, the binary Maxwell-Stefan diffusion coefficient , represents the inverse drag coefficient of species i moving past species j (Ref. 55 - Ref. 58). This property is referred to as the Maxwell-Stefan diffusivity. The Maxwell-Stefan diffusivity is symmetric, , and the diagonal elements are not used. T

When the Gas diffusivity property model is set to Automatic, the Fuller Schettler Giddings model is used, provided that the Fuller diffusion volume is known for both species (i and j), otherwise the Wilke-Lee model is used.

Fuller et al. (Ref. 59) modified the Chapman-Enskog relation to correlate binary diffusion coefficient for species i and j in the vapor phase according to the Fuller Schettler Giddings (FGS) model:

where T denotes the temperature (K), Mi the molecular weight of species i (g/mol) and P is the pressure (Pa). vi are the atomic diffusion volumes (Fuller diffusion volume), (cm3), which are estimated using group contribution for each species (Ref. 60):

Wilke and Lee (Ref. 61) also modified the Chapman-Enskog relation to correlate binary diffusion coefficient in vapor phase according to:

The length scale for the interaction is taken from the Lennard-Jones diameter parameters of species i and j:

where Vi,l,b is the molar volume of species at normal boiling point.

The collision integral is evaluated from (Ref. 62):

The energy scale for the interaction is taken from the Lennard-Jones energy parameters of species i and j:

where k is the Boltzmann constant, and is normal boiling point temperature.

All models for gas phase diffusion at infinite dilution are symmetric, the diffusivity of species i in species j equals that of species j in species i. In addition, under the ideal gas assumption, the Maxwell-Stefan diffusion coefficient matches the Fick diffusion coefficient. This implies that the Maxwell-Stefan gas diffusivities provided are independent of composition.

When the Liquid diffusivity at infinite dilution property model is set to Automatic, a selection for each solute i in solvent j will be made from the models according to the following rules. Except in the case of a temperature correlation and the Erkey-Rodden-Akgerman model, all of these rules require that the log liquid viscosity correlation, ln ηj is available in the database for species j and that the liquid volume at normal boiling point, Vi,l,b is available for the species i.

For the Siddiqi-Lucas method, a molecule is considered organic if it has at least one C atom bound to anything else than O or C atoms. This is determined by the SMILES formula, if available. If, for any pair of species, the required input data for none of the above models is available, the entire property liquid diffusion coefficient at infinite dilution is not available.

The correlation by Wilke and Chang (Ref. 63) for liquid phase diffusion coefficients at infinite dilution is:

Molecular weight, Mj and log liquid viscosity correlation, , for species j and liquid molar volume at normal boiling point for species i, , is required. The Wilke-Chang association parameter , if unavailable, is set to:

Species are identified by their CAS number or SMILES formula. The Wilke-Chang correlation is not suitable for diffusion of water. If water is the solute, the correction suggested by Kooijman (Ref. 65) is applied where liquid molar volume of water at normal boiling point, Vwater,l,b is multiplied by 4.5.

The Correlation by Tyn and Calus (Ref. 66) for liquid diffusion coefficients at infinite dilution reads:

The log liquid viscosity correlation should be available for species j, The liquid volume at normal boiling point, Vi,l,b and parachor, Pi should be available for both species i and j.

If the solvent is nonpolar (dipole moment is zero), and the solvent is methanol, ethanol or1-butanol or if the solvent is a mono-hydroxy alcohol, both the liquid volume at normal boiling point, Vj,l,b and the parachor, Pj are corrected by a factor of .

If the solute is water, both the liquid volume at normal boiling point, Vi,l,b and parachor, Pi for the solute are corrected by a factor 2. This factor also applies if it is detected from the SMILE formula that the species is an organic acid (a carboxyl group is found), except in the cases where the solvent is water, methanol or n-butanol.

Hayduk and Minhas (Ref. 67) suggested three different correlations for liquid diffusion coefficients at infinite dilution.

 Aqueous Solutions: in case the solvent is water (derived from CAS number or SMILES formula), the correlation reads:

 Normal paraffin solutions: In case both the solute and solvent are normal paraffins (derived from SMILES formula), the correlation reads:

If the solvent is nonpolar (dipole moment is zero), and the solvent is methanol, ethanol or1-butanol or if the solvent is a mono-hydroxy alcohol, both the liquid volume at normal boiling point, Vj,l,b and the parachor, Pj are corrected by a factor of .

If the solute is water, both the liquid volume at normal boiling point, Vi,l,b and parachor, Pi for solute are corrected by a factor 2. This factor also applies if it is detected from the SMILES formula that the species is an organic acid (a carboxyl group is found), except in the cases where the solvent is water, methanol or n-butanol.

Siddiqi and Lucas (Ref. 68) suggested correlations for liquid diffusion coefficients at infinite dilution for aqueous system (including gases) and for organic solutions. For normal paraffins systems, they recommended the Hayduk-Minhas model.

Where T is temperature, Vi,l,b is liquid volume at normal boiling point and is the viscosity of the solvent.

The correlation by Erkey et al. (Ref. 69) for liquid diffusion coefficients at infinite dilution in normal paraffins is:

NAv is Avogadro’s number, and the deviation from closest packing volume is given by

The correlation is fitted to normal paraffins, hydrogen, carbon monoxide and carbon dioxide diffusing in normal paraffins. The molecular weight Mi should be available for both solute and solvent. The Lennard-Jones diameter, σi is estimated from Bondi group contribution method (Ref. 70), and for some species are taken from Ref. 69 and Ref. 71 as:

To prevent the diffusion coefficients from becoming negative, the minimum difference of (V − Vjref) is considered to be 10−12 mol/m3.

The Maxwell-Stefan liquid diffusion coefficients are calculated from the liquid diffusion coefficients at infinite dilution. The diagonal values should be ignored and are set to zero. The Vignes (Ref. 72) model for diffusion in binary solutions can be extended to multi-component systems (Ref. 73):

where denotes the Maxwell-Stefan liquid diffusion coefficient for species i and j in the limited of pure species k. If k=j, it represents a binary system of species i and j where i is infinitely diluted. In addition, at infinite dilution the thermodynamics factor (activity) (Ref. 58) becomes unity and the Maxwell-Stefan diffusivity equals to Fick diffusivity:

and to ensure continuity if both species i and j vanish:

For binary systems, Equation 2-245 reduces to the Vignes interpolation formula.

Wesseling and Krishna (Ref. 73) proposed:

Kooijman and Taylor (Ref. 74) found on a limited number of systems that this provides better results:

Wesselingh and Bollen (Ref. 75) asserted that this is a reasonable estimate.

Krishna and van Baten (Ref. 76), on the basis of data obtained from molecular dynamic simulations, proposed the following Vignes-based (Ref. 72) interpolation:

The model reduces to the Kooijman-Taylor model for xi = xj. The value of is undefined in the limit of both and . From a physical point of view this is inconsequential as the value cancels out in the expressions obtained for the fluxes using the Maxwell-Stefan equations. Nevertheless, in order to obtain well-defined values and composition derivatives of itself, the equation is modified to:

where is taken as a small composition. The limiting case for both and also reduces to the Kooijman-Taylor model. This equation leads to: