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Thermodynamic models for wax precipitation

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Understanding the mechanisms and potential for wax precipitation are key factors in preventing production problems as a result of wax. Wax precipitation has a strong dependence on temperature and weak dependence on pressure. This page presents a general form of the thermodynamic relation used to define the K values for solid and liquid phases in equilibrium, and the effect of different simplifying assumptions and thermodynamic descriptions of the phases involved on the model results are examined.

Thermodynamics of solid/liquid equilibrium

The thermodynamic basis of solid/liquid equilibrium of components in a melt or dissolved in a solution is well established and is described in many standard texts (e.g., Prausnitz et al.). The basic principles continue to be applied to more complex systems as researchers attempt to develop more accurate models of solid wax precipitation. Lira-Galeana and Hammami reviewed experimental techniques and thermodynamic models for studying wax precipitation in petroleum fluids.

The predictive capability of the thermodynamic models is affected both by the form and assumptions of the models themselves and the characterization procedures used to quantify the number and properties of wax forming components present in a fluid.

Thermodynamic equilibrium

Thermodynamic models for predicting wax precipitation may be derived assuming single-component or multicomponent, single-phase or multiphase solid deposits. Regardless of which set of assumptions is chosen, the condition of thermodynamic equilibrium between phases is expressed as the equality of chemical potential for each component in all phases. For one solid phase in equilibrium with an oil, this condition is given by ....................(1)

where:

• μio and μis are the chemical potentials of component i in the oil and solid phases, respectively
• nc is the number of components. With the fundamental relation between chemical potential and fugacity of component i (nc) ....................(2)

the equilibrium relation also may be expressed in terms of fugacities: ....................(3)

where fio and fis are the fugacities of component i in the oil and solid phases, respectively.

Calculation of pure solid component fugacity

Calculation equations-of-state (EOSs) are not available to describe the volumetric behavior of the solid phase as a general function of temperature and pressure; therefore, thermodynamic solid-precipitation models are derived by relating the chemical potential of a pure solid to the chemical potential of the pure liquid at the same pressure and temperature in terms of experimentally known melting properties. Derivation of this expression is discussed in standard thermodynamics texts such as Prausnitz et al. The most general form of this relationship, including multiple solid-phase transitions, is ....................(4)

where:

• μpi,k = chemical potential of pure component
• ΔHif = enthalpy of fusion of component i,
• Tif = temperature of fusion (melting temperature) of component i
• ntr = number of solid state transitions,
• ΔHij,tr = enthalpy of the jth solid state transition of component i
• Tij,tr = jth solid state transition temperature of component i
• ΔCpi = (CPo,i - CPs,i), heat capacity of fusion of component i
• Pif = pressure of fusion (corresponding to Tif) of component i
• ΔCpij,tr = heat capacity of jth solid state transition of component i
• Δvi = (vo,i-vs,i), change of molar volume caused by fusion of component i

In the majority of wax precipitation models, multiple solid-state transitions are not considered, or the effects are lumped into the enthalpy of fusion and heat capacity of fusion terms. Removing these terms and applying the relation between chemical potential and fugacity given in Eq. 2, Eq. 4 can be written in terms of fugacities as ....................(5)

where is the fugacity of pure component i in phase state k (k = o, s).

K-value equations

Eq. 5 may be used directly to determine pure-solid-component fugacities, or it may be combined with activity- or fugacity-coefficient models to derive expressions for solid/liquid K values. This section gives the fundamental forms of these K-value equations. These equations then are used with various assumptions to perform solid/liquid or solid/liquid/vapor equilibrium calculations.

Activity-coefficient models

Activity coefficients can be defined in terms of fugacities as ....................(6)

where γi = activity coefficient of component i in a mixture, fi = fugacity of component i in the mixture, xi = mole fraction of component i in the mixture, and = standard state fugacity of component i. The standard state fugacity is the fugacity of component i in the same state and at the same temperature as the mixture and at an arbitrarily chosen pressure and composition. If the activity coefficients are defined with reference to an ideal solution in the sense of Raoult’s law, then the pressure is chosen as the system pressure and the composition is chosen as pure component i. The development of the equations presented here uses this definition.

An expression for solid/liquid K values in terms of activity coefficients can be derived with the use of the definition of Eq. 6 as ....................(7)

where:

For use with activity-coefficient models, the condition of equilibrium between the solid and liquid phases given in Eq. 3 can be substituted into Eq. 5 to yield the following relation in terms of pure component fugacities. ....................(8)

Substituting Eq. 8 into Eq. 7 then gives the general relationship for solid/liquid K values in terms of activity coefficients and melting properties: ....................(9)

Fugacity-coefficient models

For use with EOSs, it is convenient to write the solid/liquid K-value equation in terms of the wax melting properties and fugacity coefficients, as opposed to activity coefficients. Fugacity coefficients are defined as ....................(10) ....................(11)

where:

Substituting the fugacity-coefficient definitions, Eq. 5 can be rearranged to give the solid/liquid K-value expression ....................(12)

Mixed-activity and fugacity-coefficient models

The use of the fugacity coefficient as defined in Eq. 11 for the liquid phase and the activity coefficient as defined in Eq. 6 for the solid phase leads to the following equation for the solid/liquid K values when the equality of fugacity condition is applied. ....................(13)

This formula is convenient when the fluid-phase fugacities are determined with an EOS and the solid-phase activity coefficient is determined with another model.

Pure ideal solid model

In Eq. 9, the last term in the exponential accounting for the difference in molar volume between the solid and liquid as a function of pressure is usually the smallest and is most often neglected. The heat-capacity term is of larger magnitude but also is assumed negligible in many applications. If the nonidealities of the oil and solid phases also are considered to be small (i.e., γio/γis = 1) and the solid phase is assumed to be a pure component, the equation of ideal solubility results in ....................(14)

This equation may be regarded as being based on the Clausius-Clapeyron or van’t Hoff equations.

Reddy reported one application of the ideal solubility equation. Eq. 14 was used to determine the cloud points and amounts of precipitated wax for synthetic fuels and diesels. In this case, only n-paraffins were assumed to precipitate. For the synthetic fuels, measured quantities of n-paraffins were combined with a solvent. For the diesel fuels, the amounts of n-paraffins up to C27 were determined experimentally. The ideal solubility equation was used to convert the amounts of all n-paraffins in a system to an equivalent amount of reference paraffin. The solubility behavior of the reference component, n-eicosane, was determined experimentally. The mixtures then were treated as binary solute/solvent systems for computation. The predicted amount of wax precipitated at one temperature below the cloud point is compared with the experimental values in Fig. 1. These results illustrate the ability of the ideal solubility equation to correlate correctly experimentally observed trends, provided the distribution of wax-forming components is well defined.

The ideal solubility equation also was used by Weingarten and Euchner for predicting wax precipitation from live reservoir fluids. Experimental determination of wax crystallization temperatures (cloud points) for two reservoir fluids was performed at 10 different bubblepoint pressures during differential-liberation experiments. Constants relating the enthalpy of fusion and temperature of fusion were determined by linear regression to the experimental data. Fig. 2 shows a comparison of the crystallization temperature predicted by the model to the experimental values. In this implementation, no characterization of the feed is necessary. The precipitated wax is treated as a single component. The model is able to only approximately reproduce an important trend in the data as a function of pressure: at high pressures, at which only small amounts of the lightest gases are being liberated, the crystallization temperature increases slowly with decreasing pressure; at lower pressures, at which more gas and heavier gas components are liberated, the crystallization temperature increases more rapidly with decreasing pressure.

Solid-solution models

Wax models describing the precipitated solid as a single-phase multicomponent solution have been used in a large number of studies. The solid phase has most often been modeled as an ideal or regular solution. The fluid phases are modeled with the regular solution theory, Flory-Huggins theory, or EOSs.

Regular solid-solution models

Regular solution theory, as developed by Scatchard and Hildebrand, refers to mixtures with zero-excess entropy provided that there is no volume change of mixing. The Scatchard-Hildebrand equation for activity coefficients is ....................(15)

where:

The volume fraction average solubility parameter for a phase is given by: ....................(16)

At conditions far removed from the critical point, the solubility parameter for a component in the oil phase may be expressed in terms of the enthalpy of vaporization and the molar volume of the component. ....................(17)

Won proposed a modified regular solution theory in which the solubility parameter for a component in the solid phase is given by ....................(18)

With Eq. 15 and assuming that vis = vio, the activity-coefficient ratio can be described by ....................(19)

Substituting Eq. 19 into Eq. 9 and assuming the pressure and heat-capacity terms are negligible gives the final equation used by Won for the solid/liquid K values as ....................(20)

Won also presented correlations for the heat of fusion, temperature of fusion, and molar volume as functions of molecular weight and tabulates values of the solubility parameters for the liquid and solid phases. The correlations are applicable to normal paraffins. The heat of fusion is given by ....................(21)

where Mi is the molecular weight of component i. The heat of fusion from Eq. 21 is approximately equal to the sum of the heat of fusion and one-half the heat of transition for molecules heavier than C22 and approximately equal to the heat of fusion for odd carbon number molecules lighter than C22. The temperature of fusion is given by ....................(22)

and the molar volume is given by ....................(23)

In Won’s model, solid/liquid/vapor equilibrium is determined. Liquid/vapor K values are calculated with the Soave-Redlich-Kwong EOS. These K values are used with the solid/liquid K values in a three-phase flash algorithm to determine the solid/liquid/vapor-phase split as a function of temperature and pressure. There is an inconsistency in this technique in that the liquid properties are calculated from an activity-coefficient model for the solid/liquid K values and from an EOS for the vapor/liquid K values. Despite this inconsistency, Won’s technique has some important advantages over the ideal solubility models presented previously. These advantages include accounting for nonidealities in the solid and liquid phases and accounting for the simultaneous effects of pressure, temperature, and vaporization or solution of gas in the liquid on solid precipitation.

Won applied this method to a hydrocarbon gas defined as a mixture of single carbon number (SCN) fractions from C1 to C40. These SCN fractions are assumed to have paraffinic properties as given by Eqs. 21 through 23. The feed composition is determined by extrapolating the measured mole fractions of C15 through C19. Fig. 3 shows the effect of temperature on the molar-phase splits for this feed gas. The cloud-point temperature can be seen as the highest temperature at which the solid phase exists, just below 310°K. The amount of solid increases rapidly as the temperature is decreased below this point. Fig. 4 shows the effect of pressure on the phase equilibrium.

Regular solution theory model for liquid phase

Pedersen et al. use the general form of the solid/liquid K-value relation as given in Eq. 9, including the heat-capacity term but neglecting the pressure term. This results in the following equation for the K values: ....................(24)

The activity-coefficient ratio is calculated with the regular solution theory (Eq. 19), as in Won’s model. Correlations are given for the solubility parameters of paraffins in the oil and solid phases as ....................(25)

and ....................(26)

where Ci is the carbon number of component i. Won’s correlation for the enthalpy of formation (Eq. 22) is modified as ....................(27)

and the model is completed by defining a relation for the heat-capacity difference as ....................(28)

Constants a1 through a5 were determined by a least-squares fit to the data of Pedersen et al. as:

• a1 = 0.5914 (cal/cm3)0.5
• a2 = 5.763 (cal/cm3)0.5
• a3 = 0.5148
• a4 = 0.3033 cal/(g•K)
• a5 = 0.635×10-4 cal/(g•K2)

The oils were characterized on the basis of experimentally determined SCN fraction distributions. The fractions are subdivided into a paraffinic part and a naphthenic plus aromatic (NA) part. The NA fractions are given solubility parameters 20% higher than those obtained from Eqs. 25 and 26, while the enthalpy of formation for the NA fractions is set to 50% of the value calculated from Eq. 27.

Pedersen et al. compared experimental wax precipitation as a function of temperature with model predictions for 16 crude oils. Only liquid/solid equilibrium was calculated. See graphics in Pederson et al. for comparison of experimental results and full model predictions.

Internally consistent model with EOS for fluid phases

Mei et al. applied the mixed activity/fugacity coefficient model given in Eq. 13 with a three-phase flash algorithm, in conjunction with liquid/vapor K values obtained from the Peng-Robinson EOS. As opposed to Won’s model, this form maintains internal consistency with the use of the EOS for all fluid phase calculations and uses regular-solution theory only for the solid solution. The fugacity of the pure solid is calculated with Eq. 8, neglecting the pressure effect. Solid-solubility parameters required for regular-solution theory are calculated with a correlation given by Thomas et al. Won’s correlations for enthalpy of fusion, temperature of fusion, and molar volume are used with additional adjustable coefficients. A heat capacity of fusion correlation of the form given by Pedersen completes the model.

Fluids used in the study were characterized on the basis of experimental SCN analysis to C40. No further subdivision of the components into P, N, and A subfractions was performed. A good match to experimental cloud points and wax precipitation amounts as a function of temperature was attained through the adjustment of five correlation coefficients.

Ideal solid-solution models

Applying the assumptions that the solid phase may be considered an ideal solution, the heat capacity terms are negligible, and the pressure terms are negligible, the K-value expression from Eq. 9 can be written as ....................(29)

Flory-Huggins model for the liquid phase

Flory and Huggins derived expressions for the thermodynamic properties of polymer solutions. A key parameter in determining the properties of these mixtures was found to be the large difference in molecular size between the polymer and the solvent species. The same situation is found in petroleum fluids, in which the large molecules of the heavy end are in solution with much smaller hydrocarbons. Flory-Huggins theory has been applied to asphaltene precipitation modeling as discussed in Thermodynamic models for asphaltene precipitation.

Hansen et al. used the generalized polymer-solution theory given by Flory to derive an expression for the activity coefficient of a component in the liquid phase. Eq. 29 then was applied to liquid/solid equilibrium calculations. Characterization of the oils is done on the basis of experimental determination of the SCN fraction distribution to at least C20+. Each of the SCN fractions then is divided into two subfractions: the aromatic part and the combined paraffinic and naphthenic part. Flory interaction parameters are calculated between the subfractions with a group-contribution method. Although good results were obtained, the resulting expression is complicated and the model has not been used by other researchers.

Ideal solution model for the liquid phase

Erickson et al. used Eq. 29 with the additional assumption that the liquid phase is also an ideal solution. These authors note that the heat of fusion and melting-temperature terms are of much greater importance than the activity-coefficient terms for prediction of liquid/solid equilibria of stabilized liquids, justifying the use of the ideal solubility equation. Won’s correlation for melting temperature as given in Eq. 22 is used for n-alkanes. A modification of this expression is used for all other species in the fluid. A single constant multiplying Won’s enthalpy of fusion correlation is used as an adjustable parameter to enable a better fit of the experimental data.

Erickson et al. applied the model to stabilized oils with detailed experimental compositional analysis, which allows a direct determination of the amount of n-alkanes in each SCN fraction up to carbon numbers of 35 or 40. Extrapolation to C50 or higher is then performed. They also apply a "staged" equilibrium flash, which assumes that once a solid forms, it does not remix with additional solid that precipitates at lower temperatures. Fig. 5 compares model results with experimental data.

EOS models for liquid and vapor phases

Brown et al. used a simplification of the fugacity coefficient form of the solid/liquid K-value expression (Eq. 12) to study the effects of pressure and light components on wax formation. The assumptions used are that the heat capacity difference is negligible, the solid phase can be considered an ideal solution, Δvi is constant, and pif is small compared with p. Applying these conditions leads to the final K-value expression: ....................(30)

The melting temperature and heat of fusion terms are calculated with the correlations given by Erickson et al., and the molar-volume difference is correlated as function of molecular weight. Brown et al. used the simplified perturbed-hard-chain theory EOS to calculate the fugacity coefficients. A correlation was developed for binary-interaction parameters of the paraffin components. The fluid-characterization method is the same as that described for the model of Erickson et al.

Model predictions are compared with experimental data in Fig. 6 for a live fluid with a bubblepoint of 285 bar. The model predictions show that increasing the pressure from atmospheric (dead) oil causes a decrease in the cloud-point temperature as light ends dissolve in the oil phase. The light ends increase the solubility of heavy-wax components in the oil. A minimum in the cloud point is achieved at the bubblepoint of the oil. Further pressure increase in the single-phase region causes an increase in the cloud-point temperature.

Pedersen used the fugacity-coefficient model of Eq. 30 with the additional simplification that the pressure effects were neglected, resulting in the following expression for the solid/liquid K values: ....................(31)

The Soave-Redlich-Kwong EOS is used to determine the fugacity coefficients for liquid and vapor phases. The characterization of the fluid is performed on the basis of a standard extended compositional analysis in which the mole fraction, molecular weight, and density of each SCN fraction is given. Pedersen developed an empirical expression to calculate the mole fraction of the potentially wax-forming part of any SCN fraction. This approach is in contrast to the characterization procedure of Erickson et al., in which all the subfractions of a SCN fraction may potentially enter the wax phase, but the parameters of the nonnormal alkane fractions are defined such that they will enter the solid phase in lesser amounts.

Three adjustable parameters are used in the expression to determine the mole fraction of the wax-forming part of each SCN fraction. Fig. 7 shows example results for the model comparing the predicted and experimental amount of wax precipitated as a function of temperature.

Multiple-pure-solid-phase (multisolid) models

All the models discussed up to this point treat the deposited wax as a single phase, consisting of either a pure component or a mixture of components as a solid solution. As discussed in Asphaltenes and waxes, experimental work on binary-alkane mixtures shows that the components can separate into two immiscible solid phases, where each phase is essentially a pure component. Solid-phase transitions also are observed in crude oils.

Lira-Galeana et al. developed a thermodynamic model for wax precipitation based on the concept that the precipitated wax is made up of several solid phases, at which each phase consists of a single component or pseudocomponent. From stability considerations, a component may exist as a pure solid if the following inequality is satisfied: ....................(32)

The number of solid-forming components and the number of solid phases, n s , is determined from Eq. 32. Once the number of solid phases is known, the phase-equilibrium relationships for vapor, liquid, and solid are given by ....................(33)

and ....................(34)

Eq. 8 is used, neglecting pressure effects, to obtain the pure-solid fugacity. The pure-liquid fugacity is obtained from the Peng-Robinson EOS, as are the component fugacities in the liquid and vapor phases.

In the original multisolid-wax model presented by Lira-Galeana et al., the fluids are characterized by splitting the C7+ fraction into 7 to 12 pseudocomponents. No further subdivision of the pseudocomponents into P, N, or A fractions is performed. Instead, melting temperature and enthalpy of fusion correlations are developed to define properties that represent an average of the three subgroups. These correlations weight the aromatic properties more heavily for heavier molecular-weight pseudocomponents. The heat capacity of fusion is given by the correlation of Pedersen et al., as shown in Eq. 28. Fig. 8 shows experimental data and predicted results of the model.

Pan et al. also have used the multisolid-wax model but with a different fluid-characterization procedure. The characterization is based on experimental SCN analysis. Every five consecutive carbon number fractions are lumped together. The relative amounts of the P, N, and A subfractions are determined experimentally or with correlations. Melting temperature and enthalpy of fusion properties are assigned to the paraffinic subfractions with Won’s correlations. For naphthenes and aromatics, the correlations of Lira-Galeana et al. were used, with the exception of the enthalpy of fusion for aromatics, which was fit with a new correlation independent of the molecular weight. For the heat capacity of fusion, the correlation of Pedersen et al. was used for all components. Results of the model for a synthetic oil at 110 bar are shown in Fig. 9, illustrating the reduction in cloud point and also the reduction in amount of wax precipitated with the addition of methane to the system.

Multisolid-wax model including enthalpies of transition

Nichita et al. used Eq. 4 to derive an expression for the ratio between the pure-solid and pure-liquid fugacities including the effect of multiple solid-state transitions. Assumptions used in the derivation are that the heat capacity of fusion is constant, the heat capacities of transition are negligible, the solid/liquid molar-volume difference is constant, and the terms for the enthalpies of transition are all evaluated at the temperature of the first transition. The authors state that this treatment of the enthalpies of transition may lead to relative differences in results of up to 10% compared with lumping the enthalpies of transition in with the enthalpy of fusion. Applying these considerations results in

Ungerer et al. derived a similar expression with multiple enthalpies of transition; how-ever, the enthalpy terms are evaluated at the fusion temperature rather than the first transition temperature. The model is applied with a single pure-component-solid phase. Nichita et al. used Eq. 35 with the modified multisolid-wax model presented in Pan et al., including the correlations for all component properties except enthalpies of fusion and enthalpies of transition; new correlations are presented for these properties. The model of Nichita et al. is used with the Peng-Robinson EOS to calculate a pressure-temperature phase diagram, shown in Fig. 10, for a synthetic fluid with phenanthrene as the precipitating component. The binary-interaction parameter between methane and phenanthrene was adjusted to match the vapor/liquid dewpoint.

Excess Gibbs energy models

Activity coefficients are related to the partial molar excess Gibbs energy for a component i, , and the total excess Gibbs energy for a phase, GE, by ....................(36)

and ....................(37)

Excess free-energy models thus can be used with the solid/liquid K-value equation expressed in terms of activity coefficients for wax-precipitation modeling.

Coutinho and coworkers published a number of studies on modeling paraffin-wax formation from synthetic and real petroleum fluids. In these works, an equation is used for the pure-component solid to liquid-fugacity ratio similar to that given in Eq. 35, with the additional assumption that only a single enthalpy of transition term is used and the pressure effect is neglected. This results in the following solid/liquid K- value expression in terms of activity coefficients: ....................(38)

The liquid-phase activity coefficient is given by ....................(39)

where the combinatorial free-volume contribution, , is obtained from a Flory free-volume model, and the residual contribution, lnγior , is obtained from the UNIFAC model, which is based on the universal quasichemical (UNIQUAC) equation. Coutinho et al. contains more detail and references on these models.

Excess Gibbs energy models are used for the solid phase. A modified Wilson’s equation with one adjustable parameter was used initially. Then, a predictive version of the UNIQUAC equation was developed, which incorporates multiple-mixed-solid phases and is used to predict wax formation in jet and diesel fuels. An analysis of the amounts of the individual n-alkanes is required for the fluid characterization. The N and A subfractions of a SCN fraction can be treated separately or lumped as a single pseudocomponent. The accuracy of the model is very good (see images in Coutinho et al. showing model results versus experimental data).

Pauly et al. presented further development of the excess Gibbs energy model. In this model, the modified Wilson equation, as given by Coutinho and Stenby, is used for the activity coefficients in the solid phase at atmospheric pressure. The Poynting factor is used to determine the high-pressure solid fugacity from the fugacity determined at atmospheric pressure. The liquid phase is modeled with an EOS/GE model. This combination of fluid and solid treatments yields good results for prediction of solid/liquid and solid/liquid/vapor phase boundaries up to 200 Mpa for binary and multicomponent systems of n-alkanes. The quality of the predictions is a result of the treatment of the pressure effect on the solid phase and the EOS/GE model, which guarantees continuity between fugacities of the fluid and solid phases.

Comparison of models

Pauly et al. compared the models of Won, Pedersen et al., Hansen et al., Coutinho and Stenby, Ungerer et al., and the ideal solution model. The models are tested on systems composed of n-decane and a heavy fraction of normal alkanes from C18 to C30. Fig. 11 compares the total amount of solid precipitate as a function of temperature for the models with experimental data. The solid-solution models overpredict the cloud-point temperature and the amount of solid precipitated, while the multisolid model gives better results for the cloud point but underpredicts the amount of wax precipitated, at least for the higher temperature region. Coutinho and Stenby’s model gives a very good match of the data.

Nichita et al. also compared their modification of the multisolid-wax model with a solid-solution model. The solid phase is assumed to be ideal, and the liquid phase is described by the EOS. Results for the two models are compared in Fig. 12 for three synthetic mixtures of n-decane with n-alkanes from C18 to C30. As in the comparison performed by Pauly et al., the solid-solution model overpredicts the cloud-point temperature and the amount of wax precipitated, while the multisolid-wax model gives good estimates of the cloud-point temperature yet underpredicts the amount of wax precipitated.

Nomenclature

 a = constant a1-5 = constants A = deposition area, L2 b = constant Ci = carbon number of component i Cw = concentration of precipitated wax at the wall, m/m dij = interaction coefficient between component i and j D = shell thickness of the micelle, L Di = effective diffusion coefficient for component i, L2/t fao = fugacity of asphaltene component in oil phase, m/Lt2 fi = fugacity of component i, m/Lt2 = standard state fugacity of component i, m/Lt2 fig = fugacity of component i in the oil phase, m/Lt2 fik = fugacity of component i in phase k (k = o, s), m/Lt2 = fugacity of pure component i in phase state k (k = o, s), m/Lt2 = standard state fugacity of component i in phase k (k = o, s), m/Lt2 fio = fugacity of component i in the oil phase, m/Lt2 = fugacity of pure component i in oil phase, m/Lt2 = standard state fugacity of component i in oil phase, m/Lt2 fis = fugacity of component i in the solid phase, m/Lt2 = fugacity of pure component i in solid phase, m/Lt2 = standard state fugacity of component i in solid phase, m/Lt2 fℓ = fugacity of the asphaltene component in the pure liquid state, m/Lt2 fs = solid fugacity, m/Lt2 = reference solid fugacity, m/Lt2 G = total Gibbs free energy of the system, m/L2t2 GE = total excess Gibbs energy for a phase, m/L2t2 = partial molar excess Gibbs energy for a component i, m/L2t2 k* = empirical constant for mass transport of wax caused by shear dispersion Ka = ratio of rate constants of the adsorption/desorption reactions Kis = solid/liquid K value for component i mi = mass of component i, m mw = mass of wax, m Mi = molecular weight of component i, m nc = number of components ns = number of solid phases ntr = number of solid state transitions n1 = number of asphaltene molecules in the micellar core n2 = number of resin molecules in the micellar cell = number of asphaltene monomers in phase L1 = number of asphaltene monomers in phase L2 = number of micelles in phase L1 = number of resin monomers in phase L1 = number of resin monomers in phase L2 p = pressure, m/Lt2 p* = reference pressure, m/Lt2 pAℓ = lower pressure on the APE, m/Lt2 pAu = upper pressure on the APE, m/Lt2 pb = bubblepoint pressure, m/Lt2 pr = reservoir pressure, m/Lt2 ps = oil saturation pressure, m/Lt2 Pif = pressure of fusion (corresponding to Tif) of component i, m/Lt2 R = gas constant R = solvent to crude oil ratio t = time, t T = temperature, T T* = reference temperature, T Tf = melting point temperature, T Tif = temperature of fusion (melting temperature) of component i, T Tij,tr = jth solid state transition temperature of component i, T Til,tr = lth solid state transition temperature of component i, T uc = critical speed required to mobilize surface deposit asphaltene, L/t uo = oil velocity, L/t va = molar volume of pure asphaltene, L3/n vci = critical volume of component i, L3/n vcj = critical volume of component j, L3/n vik = partial molar volume of component i in phase k (k = o, s), L3/n vio = partial molar volume of component i in oil phase, L3/n vis = partial molar volume of component i in solid phase, L3/n vjk = partial molar volume of component j in phase k (k = o, s), L3/n vjo = partial molar volume of component j in oil phase, L3/n vjs = partial molar volume of component j in solid phase, L3/n vℓ = molar volume of liquid, L3/n vm = molar volume of mixture, L3/n vr = molar volume of resins, L3/n vs = solid molar volume, L3/n wi = weight fraction of component i, m/m x = mole fraction, n/n xi = mole fraction of component i, n/n xik = mole fraction of component i in phase k (k = o, s), n/n xio = mole fraction of component i in oil phase, n/n xis = mole fraction of component i in solid phase, n/n xjk = mole fraction of component j in phase k (k = o, s), n/n yi = mole fraction of component i, n/n yik = mole fraction of component i in phase k (k = o, s), n/n yio = mole fraction of component i in oil phase, n/n yis = mole fraction of component i in solid phase, n/n yjo = mole fraction of component j in oil phase, n/n yjs = mole fraction of component j in solid phase, n/n γ = shear rate, L/t γi = activity coefficient of component i in a mixture γik = activity coefficient of component i in phase k (k = o, s) γio = activity coefficient of component i in oil phase = combinatorial free volume contribution = residual contribution γis = activity coefficient of component i in solid phase δa = solubility parameter of asphaltene δi = solubility parameter for component i δik = solubility parameter for pure component i in phase k (k = o, s) = volume fraction average solubility parameter for phase k δm = solubility parameter of mixture δo = solubility parameter of oil phase = volume fraction average solubility parameter of oil phase δs = solubility parameter of solid phase = volume fraction average solubility parameter of solid phase ΔCp = heat capacity of fusion, mL2/nt2T ΔCpi = heat capacity of fusion of component i, mL2/nt2T ΔCpij,tr = heat capacity of jth solid state transition of component i, mL2/nt2T = expression for Gibbs free energy of formation of the micelle, mL2/nt2 ΔHf = enthalpy of fusion, mL2/nt2 ΔHif = enthalpy of fusion of component i, mL2/nt2 ΔHij,tr = enthalpy of the jth solid state transition of component i, mL2/nt2 ΔHil,tr = enthalpy of the lth solid state transition of component i, mL2/nt2 ΔHiv = enthalpy of vaporization of component i, mL2/nt2 ΔUv = molar internal energy of vaporization at the system temperature, mL2/nt2 Δvi = change of molar volume caused by fusion of component i, L3 = reference chemical potential of asphaltene component μam = chemical potential of asphaltene in the mixture μc = critical speed required to mobilize surface deposit asphaltene μi = chemical potential of component i, m/L2t2 μik = chemical potential of component i in phase k, m/L2t2 μio = chemical potential of component i in the oil phase, m/L2t2 μis = chemical potential of component i in the solid phase, m/L2t2 μpi,k = chemical potential of pure component i in phase k (k = o, s), m/L2t2 μpi,o = chemical potential of pure component i in oil phase, m/L2t2 μpi,s = chemical potential of pure component i in solid phase, m/L2t2 μr,m = chemical potential of resins on the surface of the asphaltene micelle, m/L2t2 μr,o = chemical potential of resins in the oil phase, m/L2t2 μs = chemical potential of the solid, m/L2t2 ρo = mass density of oil, m/L3 Φ = porosity Φ0 = initial porosity Φik = fugacity coefficient of component i in phase k = fugacity coefficient of pure component i in phase state k (k = o, s) Φio = fugacity coefficient of component i in oil phase Φa = volume fraction of asphaltene in the mixture Φcr = critical volume fraction of resins in the mixture Φik = volume fraction of component i in phase state k (k = o, s) Φr = volume fraction of resins in the mixture