Cubic-Plus-Chain (CPC). I: A Statistical Associating Fluid Theory

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Thermodynamics, Transport, and Fluid Mechanics

Cubic-Plus-Chain (CPC). I: A SAFT-based Chain Modification to the Cubic Equation of State for Large Nonpolar Molecules Caleb J. Sisco, Mohammed I. L. Abutaqiya, Francisco M. Vargas, and Walter G Chapman Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.9b00435 • Publication Date (Web): 18 Mar 2019 Downloaded from http://pubs.acs.org on March 22, 2019

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Industrial & Engineering Chemistry Research

Cubic-Plus-Chain (CPC). I: A SAFT-based Chain Modification to the Cubic Equation of State for Large Nonpolar Molecules Caleb J. Sisco1,2, ‡, Mohammed I.L. Abutaqiya1, ‡, Francisco M. Vargas1,2, Walter G. Chapman1* 1. Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas-77005, USA 2. ENNOVA LLC, Stafford, Texas-77477, USA KEYWORDS. cubic-plus-chain, CPC, cubic equation of state, thermodynamic modeling, polymer phase behavior, chain molecules

ABSTRACT. An equation of state framework for nonpolar chain molecules is proposed that hybridizes the classical cubic equation of state with the chain equation of state from SAFT (Chapman et al., Fluid Phase Equilibria, 1989). The cubic equation of state serves as the physical description of the monomer beads, providing repulsive and attractive character to the segments, and the chain term bonds these monomer segments to form chains of homogenous beads. Whereas the model molecule of the standard cubic equation of state is a sphere with variable attraction energy and volume but whose fundamental shape remains roughly spherical, the model molecule of the proposed cubic-plus-chain (CPC) equation of state consists of beads bonded covalently to

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form linear chains. The CPC model molecule is a better representation of chain-like molecules, such as n-alkanes and polymers, and phase behavior modeling with the proposed CPC equation of state shows considerable improvement over the reference cubic equation of state, mainly due to the improved physical description of the molecule. Additionally, CPC compares well to PC-SAFT in its description of both hydrocarbon VLE and polymer/solvent LLE despite being substantially less computationally expensive. The polymer/solvent example illustrates that a relatively simple modification can be made to the classical cubic equation of state that significantly improves density predictions while retaining many of the properties of the classical cubic equation of state that make it a popular model.

1.

Introduction The efficient design of chemical processes often depends on an accurate description of the

thermodynamic properties of fluids across a wide range of temperatures and pressures, and equations of state are one such method for predicting fluid properties at conditions for which little to no experimental data exist. Early equation of state models were constrained to describing only simple molecules interacting by weak dispersion forces, but more recent work has focused on improving the description of complex molecules like polymers, alcohols, amines, etc. Many of the most popular of these newer equation of state models for describing complex fluid phase behavior stem from the Statistical Associating Fluid Theory (SAFT)1,2 – including PC-SAFT,3 SAFT-VR,4 and CPA5 – and Perturbed Hard-Chain Theory (PHCT)6 – including the Elliott-Suresh-Donohue (ESD)7,8 model. The new model proposed in this work, called the cubic-plus-chain (CPC) equation of state, is a framework for modifying the classical cubic equation of state by adding a term to account for


2

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energy change due to chain formation. For the CPC equation of state model, the classical cubic equation of state offers a description of the weak dispersion forces of a monomer segment, and the SAFT chain term acts to bond these monomers to form chains. This idea is conceptually similar to the cubic-plus-association (CPA) equation of state, where the cubic equation of state describes weak dispersion forces and the SAFT association term describes hydrogen-bonding forces like those that dominate the phase behavior of water and alcohols. The description of the CPC model molecule makes it attractive for describing the physics of nonpolar chain-like molecules, such as high molecular weight alkanes and polymers, which are known to be poorly described by the standard cubic equation of state models alone. Despite their many known shortcomings, the classical cubic equations of state remain among the most popular thermodynamic models for predicting the phase behavior of nonpolar hydrocarbon systems due to their unique combination of simplicity, sufficient accuracy for many applications, and computational speed relative to more complex models like PC-SAFT and CPA. The Redlich-Kwong (RK),9 Soave-Redlich-Kwong (SRK),10 and Peng-Robinson (PR)11 models are some of the more commonly used cubic equations of state. The pressure form of the RK and SRK equations of state can be written as:

𝑃=

𝑛𝑅𝑇 𝐵 𝐴 (1 + )− 𝑉 𝑉−𝐵 𝑉(𝑉 + 𝐵)

(1)

where the ideal gas, repulsive, and attractive contributions are written separately. The variables 𝐴 and 𝐵 represent the molecular interaction energy and excluded volume, respectively. 𝐴 and 𝐵 are typically calculated by the following expressions written in terms of 𝑇𝑐 and 𝑃𝑐 :


3


𝑅2 𝑇𝑐2 𝐴 = 𝑛 a 𝑖 = 𝑛 Ωa 𝛼 𝑃𝑐 2

(2)

2

(3)

1

𝛼 = (𝑇⁄𝑇𝑐 )−2 𝐵 = 𝑛 𝑏𝑖 = 𝑛 Ω𝑏

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𝑅𝑇𝑐 𝑃𝑐

(4)

where Ωa and Ω𝑏 are constants specific to the equation of state that force a match to the critical point, and the alpha-function in Eq. (3) is specific to the RK equation of state. The critical temperature (𝑇𝑐 ) and critical pressure (𝑃𝑐 ) are specific to the component and typically obtained experimentally. For RK and SRK, Ωa = 0.42748 and Ω𝑏 = 0.08664. In the CPA equation of state proposed by Kontogeorgis et al.5, the pressure of the fluid is calculated by combining Eq. (1) with the association term from SAFT, yielding:

𝑃=

𝑛𝑅𝑇 𝐵 𝐴 𝑛𝑅𝑇 1 1 𝜕𝑋 𝑆 + ∑[ 𝑆 − ]𝜌 (1 + )− 𝑉 𝑉−𝐵 𝑉(𝑉 + 𝐵) 𝑉 𝑋 2 𝜕𝜌

(5)

𝑆

where 𝜌 is the fluid density and 𝑋 𝑆 is the unbonded monomer fraction of associating site 𝑆. CPA requires an optimization or root-finding routine inside the volume solver to converge 𝑋 𝑆 . This makes CPA much more computationally expensive than the standard cubic equation of state models and more on par with models like SAFT. Also, CPA does not use the expressions in Eqs. (2)-(4) to calculate 𝐴 and 𝐵, so, unlike the standard cubic equation of state models, CPA is not forced to match the critical point. For the CPC equation of state proposed in this work, the pressure of the fluid is calculated by combining Eq. (1) with the SAFT chain term, yielding:


4

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𝑁𝐶

𝑛𝑅𝑇 𝑚 ̅ 2𝐵 𝑚 ̅ 2𝐴 𝑚 ̅𝐵 g ′ (𝛽) 𝑃= (1 + )− − 𝑛𝑅𝑇 2 [∑ 𝑛𝑖 (𝑚𝑖 − 1) ] 𝑉 𝑉−𝑚 ̅𝐵 𝑉(𝑉 + 𝑚 ̅𝐵) 𝑉 g(𝛽)

(6)

𝑖

where 𝑚 ̅ is the mole-average chain length or segment number and g(𝛽) is the radial distribution function (RDF) evaluated at contact. The reduced volume (𝛽) is defined as a ratio of the excluded volume of the chain to the total volume of the system: 𝛽 = 𝑚 ̅𝐵/𝑉. The repulsive and attractive contributions from the cubic equation of state are modified in CPC because the standard cubic equation of state models are describing the pressure of a monomer fluid whereas CPC is describing the pressure of a chain fluid. CPA does not have to be modified in this way because CPA is merely adding another physical interaction but using the same fundamental molecular shape used by the standard cubic equation of state models.

2.

Theory CPC uses a framework similar to the SAFT models where the molecular interactions are

treated as separate contributions of repulsive, attractive, and chain-bonding energies. For PCSAFT, which is currently the most widely used of the SAFT family of models in the hydrocarbon and polymer industries, the repulsive contribution is given by the Carnahan-Starling equation of state for hard spheres,12,13 the attractive contribution is given by the Barker and Henderson dispersion term,14 and the chain contribution is that proposed in Chapman’s original SAFT formulation.1 This yields a residual Helmholtz energy function of the following general form:

𝐹 𝑃𝐶−𝑆𝐴𝐹𝑇 =

𝐴𝑅 (𝑇, 𝑉, 𝒏) = 𝑚𝑎hs + 𝑎disp + 𝑎chain 𝑅𝑇


(7)

5


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CPC replaces the hard sphere and dispersion terms of PC-SAFT with the repulsive and attractive terms of the standard cubic equation of state models, yielding a residual Helmholtz energy function of the following general form:

𝐹 𝐶𝑃𝐶 =

𝐴𝑅 (𝑇, 𝑉, 𝒏) = 𝑚𝑎rep + 𝑚𝑎att + 𝑎chain 𝑅𝑇

(8)

Because of the addition of the chain term – and the new molecular shape associated with it – even the repulsive and attractive contributions of the standard cubic equation of state models need to be modified to account for chain-bonding. 2.1. Molecular Model The model molecule of the standard cubic equation of state is a sphere with attractive potential. The magnitude of the attraction force is described by 𝐴 and the excluded volume of the sphere is described by 𝐵, which are usually calculated from values of 𝑇𝑐 and 𝑃𝑐 as shown in Eqs. (2)-(4). The assumption of a spherical molecule works well for low molecular weight n-alkanes that could be reasonably represented by this model, but long-chain molecules like high molecular weight n-alkanes and polymers violate the spherical molecule assumption by any reasonable standard. Most versions of the cubic equation of state offer a modification to the attraction force to account for temperature-dependence, as in Eq. (3). This is often interpreted as an empirical shape factor that relaxes the hard-sphere assumption and gives better predictions for the phase behavior of long-chain alkanes. Even with this modification, the cubic equation of state model molecule is, at best, a sphere of variable attraction energy, volume, and rigidity. To address this shortcoming, CPC employs the cubic equation of state to describe the physics of the monomer and then uses the SAFT chain term to bond the monomer beads to form


6

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linear chains. Relative to the standard cubic equation of state, the CPC model molecule is a smaller, homogeneous monomer unit bonded tangentially to form chains. This molecular model is more consistent with actual polymer physics than the representation given by the standard cubic equation of state. A pictorial representation comparing the model molecule in the cubic equation of state to that in CPC is shown in Figure 1. (A) cubic EOS

(B) CPC

𝐴

𝐵

𝐵

𝐴

𝑚

Figure 1. (A) model molecule for the cubic equation of state. The interaction energy (𝐴) and excluded volume (𝐵) of the molecule can be modified, but the fundamental shape is spherical. (B) model molecule for CPC. The interaction energy and excluded volume of the monomer segments are extended into a linear chain of spherical segments by the SAFT chain term. 2.2. Equation of State In terms of the reduced residual Helmholtz, the cubic-plus-chain (CPC) equation of state is written:

𝐹

𝐶𝑃𝐶

𝐴𝑅 (𝑇, 𝑉, 𝒏) = = 𝑚𝑎mon + 𝑎chain 𝑅𝑇


(9)

7


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where the monomer term (mon) accounts for the repulsive (rep) and attractive (att) contributions of the cubic equation of state and the chain term accounts for the monomer bonding contribution from SAFT. Using RK for the monomer term, CPC can be written as: (10)

𝑎mon = 𝑎rep + 𝑎att 𝑎rep = −𝑛 ln(1 − 𝛽) 𝑎att = −

𝐴 ln(1 + 𝛽) 𝐵𝑅𝑇

𝑎chain = − ∑ 𝑛𝑖 (𝑚𝑖 − 1) ln g(𝛽) 𝑖

where the reduced volume (𝛽) is defined as a ratio of the excluded volume of the chain to the total volume of the system: 𝛽 = 𝑚 ̅𝐵/𝑉. In terms of the compressibility factor, CPC can be written:

𝑍 =1+

𝑚 ̅𝛽 𝑛 𝐴 1 g′(𝛽) [ − ] − ∑ 𝑛𝑖 (𝑚𝑖 − 1)𝛽 𝑛 1 − 𝛽 𝐵𝑅𝑇(1 + 𝛽) 𝑛 g(𝛽)

(11)

𝑖

Writing the equation of state in terms of 𝛽 instead of 𝑉 is useful for applying numerical methods to the volume root-finding procedure as 𝛽 is bounded between 0 and 1, whereas 𝑉 is bounded ̅ 𝐵) and ideal gas volume (𝑉ig ). between the excluded volume of the chain (𝑚

To extend CPC to multi-component mixtures, averages of the individual interaction energies, excluded volumes, and chain lengths are applied using the following mole-weighted mixing rules for 𝐴 and 𝐵:


8

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𝑚 ̅ = ∑ 𝑥𝑖 𝑚𝑖

(12)

𝑖

𝐴=

1 ∑ ∑ 𝑛𝑖 𝑛𝑗 𝑚𝑖 𝑚𝑗 a𝑖𝑗 𝑚 ̅2

(13)

1 ∑ 𝑛𝑖 𝑚𝑖 𝑏𝑖𝑖 𝑚 ̅

(14)

𝑖

𝐵=

𝑗

𝑖

The CPC mixing rules for 𝐴 and 𝐵 reduce to the standard cubic equation of state mixing rules when 𝑚𝑖 = 1 for all components. Also, the chain contribution to the compressibility factor is zero when all 𝑚𝑖 = 1. The combining rules for a𝑖𝑗 and 𝑏𝑖𝑗 are given by:

a𝑖𝑗 = √a𝑖 a𝑗 (1 − 𝑘𝑖𝑗 )

(15)

1 𝑏𝑖𝑗 = (𝑏𝑖 + 𝑏𝑗 ) 2

(16)

CPC is fully parameterized when the mole numbers (𝑛𝑖 ), monomer interaction energy (a𝑖 ), monomer excluded volume (𝑏𝑖 ), and chain length (𝑚𝑖 ) are defined for each component in the mixture. Also, the radial distribution function (RDF) must be defined. In this work, we use the RDF proposed by Elliott et al.7, which is given by:

g(𝛽) =

1 1 − 0.475𝛽

(17)

In the remainder of this work, this version of CPC is referred to as CPC-RKE (RK monomer term + Elliott RDF).


9


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2.3. Critical Point Matching with CPC For the standard cubic equations of state, the model is fully parameterized when the mole numbers (𝑛𝑖 ), molecule interaction energy (a𝑖 ), and excluded volume (𝑏𝑖 ) are defined for each component in the mixture. The a𝑖 and 𝑏𝑖 parameters are typically calculated from Eqs. (2)-(4) which are written in terms of the critical temperature (𝑇𝑐 ) and critical pressure (𝑃𝑐 ). Additionally, the parameters Ωa and Ω𝑏 (found in Eqs. (2) and (4), respectively) are set to force the model to produce a critical point at 𝑇 = 𝑇𝑐 and 𝑃 = 𝑃𝑐 . For the standard cubic equation of state models, Ωa and Ω𝑏 are constants. Because of the addition of the chain term, CPC will not match the critical point if Eqs. (2)-(4) are used to calculate a𝑖 and 𝑏𝑖 . To force CPC to match the critical point, Ωa and Ω𝑏 are re-derived as functions of the chain length (𝑚) and critical compressibility factor (𝑍𝑐 ), and a𝑖 and 𝑏𝑖 are rewritten as:

a𝑖 = Ωa (𝑚)

𝑅2 𝑇𝑐2 𝛼 𝑃𝑐

(18)

𝑅𝑇𝑐 𝑃𝑐

(19)

𝑏𝑖 = Ω𝑏 (𝑚) with Ωa (𝑚) and Ω𝑏 (𝑚) calculated by:

Ωa (𝑚) =

1 𝛽𝑐 𝑍𝑐2 𝜆chain (𝑚 + − 1)𝛽 𝑍 [ 𝑐 𝑐 mon ] 𝑚2 𝜆mon 𝜆

(20)

Ω𝑏 (𝑚) =

1 𝛽𝑍 𝑚 𝑐 𝑐

(21)

𝛽𝑐 is calculated by solving the 6th order polynomial shown in Table 1, and 𝑍𝑐 is calculated from the following expressions:


10

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(22)

𝑍𝑐 = 𝑚𝑍𝑐mon − (𝑚 − 1)𝑍𝑐chain 𝑍𝑐mon =

𝑍𝑐chain

mon

𝜆

−1 1 𝛽𝑐 [1 + mon ] (1 + 𝛽𝑐 ) 1 − 𝛽𝑐 𝜆

−1 𝛽𝑐 𝜆chain 40 𝛽𝑐 = + [1 + mon ] ] [ (1 + 𝛽𝑐 ) 1 + 𝛽𝑐 𝜆mon (40 − 19𝛽𝑐 ) 𝜆

𝛽𝑐2 + 2𝛽𝑐 − 1 = −𝛽𝑐 (1 + 𝛽𝑐 )2

𝜆chain =

840𝛽𝑐 (40 − 19𝛽𝑐 )2

Table 1. Coefficients for 6th order polynomial for solving 𝛽𝑐 for CPC-RKE. Coefficient

Expression

𝑐0 𝑐1 𝑐2 𝑐3 𝑐4 𝑐5 𝑐6

−128,000 − 128,000 𝑚0 566,400 + 566,400 𝑚0 −249,840 − 748,800 𝑚0 −145,562 + 188,800 𝑚0 36,366 + 182,400 𝑚0 45,486 −13,718 − 60,800 𝑚0 ▪ Polynomial of the form: 0 = 𝑐0 + ⋯ + 𝑐6 𝛽𝑐6 ▪ 𝑚0 = (1 − 𝑚)/𝑚

Note that the coefficients presented here for the calculation of 𝛽𝑐 and 𝑍𝑐 correspond only to CPCRKE. Using a different monomer term or RDF will yield a slightly different form for these equations. It should also be noted that each component in a mixture has its own Ωa (𝑚𝑖 ) and Ω𝑏 (𝑚𝑖 ) corresponding to their individual segment length. Thus, Eqs. (20)-(22) and the polynomial for 𝛽𝑐 must be solved for each component in the mixture. For the standard cubic equation of state, Ωa and Ω𝑏 and the critical compressibility factor (𝑍𝑐 ) are constants specific to each model and independent of the component type. For example,


11


the values Ωa , Ω𝑏 , and 𝑍𝑐 for the RK and SRK models are 0.42748, 0.08664, and 1/3, respectively. For CPC-RKE, these variables are functions of the chain length (𝑚), as shown in Figure 2 and Figure 3. Note that for 𝑚 = 1, Ωa (𝑚) and Ω𝑏 (𝑚) reduce to 0.42748 and 0.08664, respectively. The trends shown in Figure 2 indicate that as the chain length increases, Ωa (𝑚) and Ω𝑏 (𝑚) decrease to satisfy the criticality condition. Although 𝑍𝑐 is also a function of 𝑚 for CPCRKE, the dependence is weak and the predicted 𝑍𝑐 is approximately 1/3 regardless of chain length. The choice of the radial distribution function has some influence on the magnitude of the effect that 𝑚 has on 𝑍𝑐 , though this effect seems to be quite negligible regardless of RDF. 0.50

0.10

A

B

0.40

0.08

0.30

0.06

Wb

Wa

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Page 12 of 35

0.20

0.04

0.10

0.02

0.00

0.00 0

2

4

6

8

10

0

m (-)

2

4

6

8

10

m (-)

Figure 2. Dependence of the coefficients (A) Ωa (𝑚) and (B) Ω𝑏 (𝑚) on the chain length 𝑚 in CPC-RKE.


12

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0.40 0.37 0.33

Zc

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.30

0.27 0.23 0.20 0

2

4

6 8 10 m (-) Figure 3. Effect of chain length 𝑚 on the critical compressibility predicted by CPC-RKE.

Because of short-comings in the RK alpha-function (Eq. (3)) and Soave alpha-function,15 alternative formulations have been proposed by various researchers – including Mathias and Copeman,16 Melhem,17 and Twu18 – to correct physically inconsistent results in the standard cubic equation of state models and to improve vapor pressure predictions. Any of these or similar alphafunctions can be substituted into CPC for the standard alpha-function in Eq. (3). Modifying the alpha-function does not alter the expressions for Ωa (𝑚), Ω𝑏 (𝑚) and their auxiliary variables in Eq. (22) because Ωa (𝑚) and Ω𝑏 (𝑚) are fit strictly to match the critical point while the alpha-function describes deviations from the vapor pressure curve away from the critical point. The RK alpha-function (Eq. (3)) was used in this work to provide temperature dependence to the interaction energy instead of the more commonly used Soave alpha-function. Soave’s modification uses another tuning parameter (acentric factor, 𝜔) that is often interpreted as an empirical shape factor. In this work, the SAFT chain term is meant to provide shape to the molecule, not the alphafunction, so the simple RK alpha-function is preferred.


13


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2.4. Analogies to PC-SAFT In Sections 3 and 4, phase behavior modeling for nonpolar hydrocarbons are performed with RK, CPC-RKE and PC-SAFT. Before investigating the results, it is useful to understand where CPC and PC-SAFT are similar and the key differences that give PC-SAFT superior predictive capability relative to CPC. The molecular model and the parameters that determine the size, shape, and interaction energies of the molecules are nearly identical for CPC and PC-SAFT. Whereas CPC uses a𝑖 , 𝑏𝑖 , and 𝑚𝑖 to describe monomer interaction energy, monomer excluded volume, and chain length, respectively, PC-SAFT uses 𝜖𝑖 , 𝜎𝑖 , and 𝑚𝑖 to describe monomer interaction energy, monomer diameter, and chain length, respectively. The PC-SAFT model molecule is given in Figure 4. Instead of the van der Waals repulsion term used in CPC, PC-SAFT uses the hard-sphere term of Carnahan-Starling to describe molecular repulsion. Instead of the cubic equation of state attraction term used in CPC, PC-SAFT uses the dispersion term of Barker and Henderson to describe molecular attraction. Both the repulsion and attraction terms in PCSAFT are much more intricate than those in CPC, which contributes to both the increased predictive capability and computational time of PC-SAFT.

𝜖

𝜎

𝑚

Figure 4. Model molecule for PC-SAFT. The interaction energy (𝜖) and diameter (𝜎) of the monomer segments can be extended into a linear molecule of segments by the chain length (𝑚).


14

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Another consequential difference between the models appears in the monomer size parameter. In CPC, the excluded volume of the monomer (𝐵) is temperature-independent, whereas in PC-SAFT, the diameter of the monomer segment (𝜎) is corrected by a temperature-dependent term. This temperature-dependence on monomer diameter in PC-SAFT propagates throughout each contribution to the Helmholtz energy function, whereas only the attraction term in CPC carries temperature-dependence. The repulsion and chain terms in CPC, which are driven in large part by the monomer volume (𝐵), are temperature-independent. For mixtures, another important distinction between PC-SAFT and CPC appears: the radial distribution function (RDF) in PCSAFT is a 𝑁𝐶 x 𝑁𝐶 matrix where the size distinction between molecules 𝑖 and 𝑗 are considered explicitly, whereas the RDF in CPC takes the mixture value of 𝐵 to calculate a single value for the RDF.

3.

CPC Parameter Tuning CPC is fully parameterized when the mole numbers (𝑛𝑖 ), monomer interaction energy (a𝑖 ),

monomer excluded volume (𝑏𝑖 ), and chain length (𝑚𝑖 ) are defined for each component in a mixture. To determine these parameters, a𝑖 , 𝑏𝑖 , and 𝑚𝑖 for each component are tuned to some set of its pure-component data. For PC-SAFT and CPA, parameter tuning for pure components is performed over saturation pressure and liquid density data. For CPC-RKE, we use a mixed approach where the expressions for Ωa (𝑚) and Ω𝑏 (𝑚) in Eqs. (20) and (21) are used to write a𝑖 and 𝑏𝑖 in terms of 𝑇𝑐 and of 𝑃𝑐 and only 𝑚𝑖 is tuned to saturation pressure and liquid density data. The input parameters for standard molecules in CPC-RKE are 𝑇𝑐 , 𝑃𝑐 , and 𝑚. The critical temperature (𝑇𝑐 ) and critical pressure (𝑃𝑐 ) are often obtained experimentally and available in the


15


Page 16 of 35

literature for most pure compounds,19 and the chain length (𝑚) is tuned to match reference data for vapor pressure and density in the reduced temperature range 𝑇𝑟 = 0.5 − 0.9. The objective function to be minimized for parameter tuning is given by: 𝑁𝑝𝑡

𝑁𝑝𝑡

𝑠𝑎𝑡 𝑠𝑎𝑡 𝑃𝐶𝑃𝐶 − 𝑃 𝑠𝑎𝑡 𝜌𝐶𝑃𝐶 − 𝜌 𝑠𝑎𝑡 𝑂𝐵𝐽 = 𝛼𝑃 ∑ | | + 𝛼 ∑ | | 𝜌 𝑃 𝑠𝑎𝑡 𝜌 𝑠𝑎𝑡 𝑘=1

(23)

𝑘=1

where 𝛼𝑃 and 𝛼𝜌 are the weighting factors of saturation pressure and density, respectively. The sampling points are equally spaced between 𝑇𝑟 = 0.5 − 0.9 and reference data for saturation pressure and density are generated from correlations provided by DIPPR.20 The 𝑚 parameter for each component was tuned independently from other components to ensure that the expected trends – such as 𝑚 increasing as a function of n-alkane molecular weight – were observed without being forced. For each pure component, 𝑚 is allowed any positive value and the tuning procedure searches the solution space to find 𝑚 that minimizes the objective. The optimized values of chain length (𝑚) for select components, along with their critical properties, are provided in Table 2. The chain length in CPC-RKE increases with increasing carbon number for the n-alkanes, as expected. This suggests that 𝑚 in CPC-RKE can be correlated to 𝑀𝑊 for nalkanes in a similar manner as 𝑚 is correlated to 𝑀𝑊 in PC-SAFT.3 Figure 5 shows that the relation between 𝑚 and 𝑀𝑊 is virtually linear for n-alkanes. The regressed line in Figure 5 is forced to an intercept of zero. The resulting straight line indicates that 𝑚/𝑀𝑊 has, to a good approximation, a constant value of 0.04155 for n-alkanes.


16

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Table 2. Critical properties and CPC-RKE chain length (𝑚) for select n-alkanes, benzene derivatives, and nonhydrocarbon gases. 𝑚 for each component is tuned to DIPPR20 correlations for saturation pressure and density.

Component

𝑀𝑊

𝑇𝑐

𝑃𝑐

𝑚

[bar]

[-]

methane ethane propane butane pentane hexane heptane octane nonane decane undecane dodecane tridecane tetradecane pentadecane hexadecane heptadecane octadecane nonadecane eicosane

[g/mol] [K] n-alkanes 16.043 190.6 30.070 305.3 44.096 369.8 58.123 425.1 72.146 469.7 86.177 507.6 100.203 540.2 114.231 568.7 128.26 594.6 142.285 617.7 156.312 639.0 170.338 658.0 184.365 675.0 198.392 693.0 212.419 708.0 226.446 723.0 240.473 736.0 254.49 747.0 268.527 758.0 282.553 768.0

45.99 48.72 42.48 37.96 33.70 30.25 27.40 24.90 22.90 21.10 19.50 18.20 16.80 15.70 14.80 14.00 13.40 12.70 12.10 11.60

0.8733 1.3301 1.7047 2.0854 2.6278 3.1708 3.7948 4.4125 4.9937 5.6575 6.2173 6.8653 7.5387 8.0164 8.7008 9.1951 10.0490 10.6229 11.1571 11.7654

benzene toluene ethylbenzene p-xylene naphthalene 1-methylnaphthalene

benzene derivatives 78.114 562.05 92.141 591.75 106.165 617.2 106.165 616.2 128.17 748.4 142.20 772

48.95 41.08 36.06 35.11 40.5 36

2.2503 3.0329 3.4995 3.7226 3.5002 4.0738

nitrogen hydrogen sulfide carbon dioxide

28.014 34.081 44.010

gases 126.19 373.1 304.13

33.96 90 73.77

0.8100 1.1574 2.3921


17


16

m (-)

12

8 y = 0.04155x R² = 0.997

4

0 0

100

200 MW (g/mol)

300

400

Figure 5. Chain length (𝑚) in CPC-RKE as a function of molecular weight (𝑀𝑊 ) for n-alkanes. Saturation pressure and density plots comparing RK, CPC-RKE, and PC-SAFT to reference data20 are given in Figure 6 for propane, Figure 7 for n-decane, and Figure 8 for n-pentadecane. RK is chosen for comparison over the more accurate SRK and PR cubic equation of state models because the CPC-RKE model uses RK as the reference equation of state for the monomer term. The only difference between RK and CPC-RKE is the addition of the chain term. 10

1.0

A

6 2 -2

RK CPC PC-SAFT Exp

-6

-10 0.50

B

0.8

0.60

0.70

0.80

0.90

1.00

Density / g.cm-3

ln Psat / bar

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 35

0.6 0.4 0.2 0.0 0.50

Tr

0.60

0.70

0.80

0.90

1.00

Tr

Figure 6. Saturation pressure (A) and liquid density (B) for propane from 𝑇𝑟 = 0.5 − 0.9.


18

Page 19 of 35

10

1.0

A

B

0.8

2 -2

RK CPC PC-SAFT Exp

-6

-10 0.50

0.60

0.70

0.80

0.90

Density / g.cm-3

ln Psat / bar

6

0.6 0.4 0.2 0.0 0.50

1.00

0.60

0.70

Tr

0.80

0.90

1.00

Tr

Figure 7. Saturation pressure (A) and liquid density (B) for n-decane from 𝑇𝑟 = 0.5 − 0.9. 10

1.0

A

6

B

0.8

2 -2

RK CPC PC-SAFT Exp

-6

-10 0.50

0.60

0.70

0.80

0.90

1.00

Density / g.cm-3

ln Psat / bar

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6 0.4 0.2 0.0 0.50

Tr

0.60

0.70

0.80

0.90

1.00

Tr

Figure 8. Saturation pressure (A) and liquid density (B) for n-pentadecane from 𝑇𝑟 = 0.5 − 0.9. The chain term in CPC-RKE markedly improves the saturation pressure predictions relative to RK, particularly for longer chain n-alkanes where the predictions from RK become progressively poorer as molecular weight increases. The saturation density predictions for CPCRKE also show improvements relative to RK, but there are still noticeable deviations from the experimental data. Because the chain term of CPC-RKE has a negligible effect on critical compressibility (as shown in Figure 3), the critical compressibility predicted by CPC-RKE is nearly equivalent to that of the reference EOS. For that reason, the saturation density curves for CPC-RKE and RK (shown in Figures 5-7) converge to a similar value as the temperature approaches the critical point. Because the standard cubic equation of state models predict critical density poorly, CPC-RKE also predicts critical density poorly. If the chain term had a larger effect


19


on the critical compressibility factor, then the poor predictions given by the reference EOS could be corrected by the chain term, but, as currently formulated (with the Elliott RDF in the chain term and Ωa and Ω𝑏 tuned to force a match to 𝑇𝑐 and 𝑃𝑐 ) CPC-RKE can only negligibly shift the critical compressibility factor given by the reference EOS. Subcooled liquid density plots comparing RK, CPC-RKE, and PC-SAFT to reference data are shown in Figure 9 and Figure 10. Again, the density predictions for CPC-RKE are improved relative to RK, but there is still a noticeable deviation also due to the critical compressibility behavior of CPC-RKE.

RK CPC PC-SAFT Exp

n-octane

Density / g.cm-3

0.9 0.8

1.0

A

0.7 0.6

0.8 0.7 0.6 0.5

0.5 280 1.0

B

n-decane

0.9 Density / g.cm-3

1.0

290

300

310 320 330 Temperature / K

340

280

350

1.0

C

n-dodecane

290

300


340

350

D

n-pentadecane



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 35

0.8 0.7 0.6

0.8 0.7 0.6 0.5

0.5 280

290

300


340

350

280

290

300


340

350

Figure 9. Density as a function of temperature at 𝑃 = 1 bar for (A) n-octane (B) n-decane (C) ndodecane (D) n-pentadecane.


20

1.0

Density / g.cm-3

0.7 0.6 RK CPC PC-SAFT Exp

0.5 0.4 0.3

1.0

500 1000 Pressure / bar

0.7 0.6 0.5

0.3

0

1500 1.0

0.8 0.7 0.6 0.5


1500 D

Tr = 0.711


0.9

0.8

0.4

C

Tr = 0.628

B

Tr = 0.506

0.9

0.8

0

1.0

A

Tr = 0.445

0.9

Density / g.cm-3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Density / g.cm-3

Page 21 of 35

0.8 0.7 0.6 0.5 0.4

0.4

0.3

0.3

0


1500

0


1500

Figure 10. Density as a function of pressure for n-dodecane at (A) 𝑇𝑟 = 0.445, (B) 𝑇𝑟 = 0.506, (C) 𝑇𝑟 = 0.628, and (D) 𝑇𝑟 = 0.711. CPC-RKE outperforms RK for prediction of saturation pressure and density across the family of n-alkanes studied. For low molecular weight n-alkanes which are represented reasonably well by the spherical molecule assumption implicit in the cubic equation of state, the chain term is relatively unimportant and the predictions with CPC-RKE do not show significant improvement. However, for high molecular weight n-alkanes, the spherical molecule assumption in the cubic equation of state is especially poor and predictions suffer severely. The physics implicit in the CPC model – primarily in the representation of the model molecule as spherical beads covalently bonded to form chains – matches much more closely to how long-chain n-alkanes exist in reality, so CPC-RKE shows dramatically better predictions for these types of systems than RK shows. PCSAFT predictions for pure-component saturation pressure and density are consistently better than


21


Page 22 of 35

RK and CPC-RKE, but that is to be expected as the PC-SAFT parameters are tuned to match these properties only and PC-SAFT is a much more complex equation of state than RK and CPC-RKE.

4.

Phase Behavior of Mixtures with CPC Once the simulation parameters for pure components (a𝑖 , 𝑏𝑖 , 𝑚𝑖 ) are defined, the average

̅ ), interaction energy (𝐴), and excluded volume of the monomer (𝐵) can be chain length (𝑚 ̅ calculated for mixtures from the mixing and combining rules in Eqs. (12)-(16). With 𝐴, 𝐵, and 𝑚

defined, the CPC pressure equation, as well as any other thermodynamic property, can be calculated from the residual Helmholtz function (Eq. (9)) and its derivatives. In the next sections, phase behavior modeling is first performed for standard molecules that have experimentally measurable critical points and whose simulation parameters are determined from the tuning approach presented in Section 3. Then, a polymer/solvent cloud point calculation is shown. The systems modeled in this section were first modeled with PC-SAFT by Gross and Sadowski3,21 to show the superior performance of PC-SAFT over one of the earliest versions of SAFT proposed by Huang and Radosz.22,23 Here, we compare CPC-RKE to PC-SAFT to illustrate how well the proposed CPC framework does in predicting phase behavior relative to arguably the most widely used non-cubic equation of state for nonpolar hydrocarbons. It is not expected that CPC-RKE should outperform PC-SAFT in predicting the phase behavior of these systems for two key reasons. First, the PC-SAFT parameters for pure components are tuned to match saturation pressure while CPC-RKE is forced to match the critical point and only one parameter (𝑚) is tuned to saturation pressure. Second, PC-SAFT is a much more computationally expensive algorithm which accounts explicitly for effects that are not considered in CPC-RKE. For example, the RDF in CPC-RKE produces a single value that is independent of temperature while the RDF in PC-


22

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SAFT produces a 𝑁𝐶 x 𝑁𝐶 matrix with temperature-dependence on the diameter of the monomer segment. The primary utility of CPC is that it is a computationally inexpensive equation of state that accounts for the major contributions to the phase behavior of long-chain nonpolar components that are missing from the classical cubic equation of state. 4.1. Standard Molecules P-xy diagrams for three hydrocarbon binary mixtures are provided in Figure 11, Figure 12, and Figure 13 showing the predictive capability of CPC-RKE relative to PC-SAFT. These three systems pair a light gas with different types of hydrocarbon solvents. In the first system, methane is paired with a light n-alkane. In the second system, ethane is paired with a heavy nalkane. In the third system, propane is paired with an aromatic compound. The CPC-RKE simulation parameters for the pure components can be found in Table 2, and the PC-SAFT simulation parameters can be found in the work of Gross and Sadowski.3 With the pure component parameters defined, the only tunable parameter for these systems is the 𝑘𝑖𝑗 that modifies the cross-interaction energy between unlike components. For all cases shown in this section, 𝑘𝑖𝑗 = 0, so the predictions are performed without tuning to mixture experimental data. CPC-RKE shows good agreement to the experimental data and outperforms PC-SAFT in some cases.


23


160

PC-SAFT Exp (21 ℃)

120 Psat / bar

methane/n-butane

CPC

140

Exp (121 ℃)

100 80

60 40 20 0 0

20

40 60 xC₁ / mol%

80

100

Figure 11. Vapor-liquid equilibrium (VLE) of methane/n-butane mixture. Comparison of CPCRKE and PC-SAFT to experimental data. 𝑘𝑖𝑗 = 0 for both cases. Experimental data from Sage et al.24

140

ethane/n-decane

CPC PC-SAFT

120

Exp (171 ℃)

100 Psat / bar

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 35

Exp (238 ℃)

80 60

40 20 0 0

20

40

60

80

100

xC₂ / mol%

Figure 12. Vapor-liquid equilibrium (VLE) of ethane/n-decane mixture. Comparison of CPCRKE and PC-SAFT to experimental data. 𝑘𝑖𝑗 = 0 for both cases. Experimental data from Reamer and Sage.25


24

Page 25 of 35

80

CPC PC-SAFT Exp (71 ℃) Exp (138 ℃) Exp (204 ℃)

70

60 Psat / bar

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


propane/benzene

50 40 30

20 10 0 0

20

40

60

80

100

xC₃ / mol%

Figure 13. Vapor-liquid equilibrium (VLE) of propane/benzene mixture. Comparison of CPCRKE and PC-SAFT to experimental data. 𝑘𝑖𝑗 = 0 for both cases. Experimental data from Glanville et al.26 4.2. Polymers The classical cubic equation of state models, with a𝑖 and 𝑏𝑖 tuned to match the critical point, are incapable of describing polymer physics to any acceptable degree of accuracy. Other researchers that have attempted to perform polymer modeling with standard cubic models have focused on changing the expressions for calculating a𝑖 and 𝑏𝑖 of the polymer component and/or using mixing rules in terms of excess Gibbs.27–30 These types of models have not been shown to accurately predict solid-liquid or liquid-liquid equilibria at high pressures. CPC-RKE, which applies a simple chain modification to the cubic equation of state, is capable of modeling high-pressure liquidliquid equilibria for polymer/solvent systems.


25


To illustrate, a cloud point calculation with CPC-RKE is performed for a polymer/solvent system and compared to PC-SAFT.21 The results are shown in Figure 14.

100 Exp (10 wt%)

PP/n-pentane

Exp (15 wt%)

80

CPC (kij(T)) PC-SAFT (kij=0.0137)

Psat / bar

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 35

60 40

20 0 170

175

180 185 190 Temperature / C

195

200

Figure 14. Liquid-liquid equilibrium (LLE) of polypropylene (PP)/n-pentane system. Comparison of CPC-RKE and PC-SAFT to experimental data. Experimental data from Martin et al.31 𝑘𝑖𝑗 for PP/n-pentane with CPC is given by: 𝑘𝑖𝑗 (𝑇) = 0.0011 𝑇(℃) − 0.177. The performance of CPC-RKE in predicting saturation pressures for polymers is driven primarily by the chain term, which is largely dependent on the RDF. As mentioned before, the RDF used in this work for CPC-RKE is single-valued and temperature-independent, whereas the RDF in PCSAFT is a 𝑁𝐶 x 𝑁𝐶 matrix where each value in the matrix is temperature-dependent. This lack of temperature-dependence on the molecule volume in CPC-RKE is likely the primary reason that a temperature-dependent binary interaction parameter is required to capture the cloud point trends correctly. The PC-SAFT and CPC simulation parameters for polypropylene are given in Table 3 and Table 4, respectively.


26

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Table 3. PC-SAFT simulation parameters for polypropylene.

Component polypropylene (PP)

𝑀𝑊

𝑚/𝑀𝑊

𝜎

𝜖 ⁄𝑘

[kg/mol] 50.4

[-] 0.0235

[Å] 4.1

[K] 217

Table 4. CPC-RKE simulation parameters for polypropylene.

Component polypropylene (PP)

𝑀𝑊

𝑚/𝑀𝑊

Ωa

Ω𝑏

[kg/mol] 50.4

[-] 0.0120

[-] 0.177

[-] 0.046

Ωa , Ω𝑏 are used in Eqs. (18)-(19) to define a𝑖 and 𝑏𝑖

5.

Computational Time Comparisons Time comparisons for the three equations of state studied in the work (RK, CPC, PC-

SAFT) are shown in Figure 15. The calculations were performed in VBA for MS Excel (32-bit) with an Intel i7 processor operating at 1.70 GHz for the family of n-alkanes lighter than n-decane in the reduced temperature range 𝑇𝑟 = 0.5 − 0.9 and reduced pressure range 𝑃𝑟 = 0.5 − 0.9.


27


7 PC-SAFT

6

time ratio (EOS/RK)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 35

5

4 3 2 1

CPC-RKE

RK

0

EOS Figure 15. Time comparison for RK, CPC-RKE and PC-SAFT to solve volume loop and fugacity coefficient equations. Simulations were executed for methane to n-decane between 𝑇𝑟 = 0.5 − 0.9 and 𝑃𝑟 = 0.5 − 0.9. Comparison presented as ratio per time elapsed for RK. To solve the volume loop and fugacity coefficient equations, the computational time of CPC-RKE is approximately 1.5x that of RK whereas PC-SAFT is greater than 6x that of RK. Though PC-SAFT often generates good predictions for phase behavior, this comes at the expense of computational time, whereas the CPC-RKE time penalty with respect to the classical cubic equations of state is marginal. Much of the time penalty incurred for CPC-RKE relative to RK is in the calculation of 𝛽𝑐 and 𝑍𝑐 , which are used to calculate Ωa (𝑚) and Ω𝑏 (𝑚). This portion of the time penalty can be avoided altogether if CPC is not forced to match the critical point or, even if fitting to the critical point, the time penalty can be significantly reduced if the calculation of Ωa (𝑚) and Ω𝑏 (𝑚) are calculated outside the EOS function. For example, in a compositional simulation performed with CPC – regardless of the number of phase equilibrium computations performed – Ωa (𝑚) and Ω𝑏 (𝑚)


28

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only need to be calculated once for each component present as they are independent of any of the process variables that change during a compositional simulation. Thus, the only time penalty that is unavoidable with CPC is in the calculation of the radial distribution function and chain contribution to the compressibility factor, Helmholtz energy, and fugacity coefficient. These calculations yield an insignificant time penalty relative to RK.

6.

Conclusions The new equation of state proposed in this work is intended primarily as a proof-of-concept

of the CPC framework, showing that the addition of the SAFT chain term to the cubic equation of state can offer substantial improvements to the model physics without adding significant computational time or tuning parameters. The SAFT-modified cubic equation of state proposed in this work (CPC-RKE) proved to work well for describing vapor-liquid equilibrium of various binary mixtures and, most impressively, the liquid-liquid phase-splitting of a polymer/solvent system. The CPC framework is not restricted to a specific equation of state for the monomer or a specific radial distribution function for the chain term. For some systems, RK, SRK or PR paired with another radial distribution function – such as those proposed by Carnahan and Starling12 or Gow32 – could show better results than the CPC-RKE equation of state proposed in this work. There are numerous variations of CPC that can be constructed from hybridizing any cubic equation of state with a suitable radial distribution function, and future work should focus on identifying which combinations are best for which systems. The CPC framework also makes it an attractive model for heavy hydrocarbon systems like crude oils. Despite their known drawbacks, the classical cubic equations of state of SRK and PR


29


Page 30 of 35

are still commonly used in the oil and gas industry because they are computationally faster than the more advanced association equations of state and are also well understood across the industry. CPC is only marginally slower than RK and yet provides significant improvement in describing the phase behavior of large nonpolar molecules like polymers, suggesting that it will also be capable of describing heavy crude oil components like asphaltenes and waxes. CPC offers a reasonable balance between computational time and accuracy that make it an especially attractive model for engineering applications, and its foundation being the widely familiar classical cubic equation of state should appeal to practitioners in the oil and gas industries that still rely on the classical cubic models.


30

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Supporting Information Appendix A: Compressibility Factor Derivation for CPC-RKE (PDF) Appendix B: Fugacity Coefficient Derivation for CPC-RKE (PDF) Author Information Corresponding Author * Walter G. Chapman, email: [email protected] Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. ‡These authors contributed equally.


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Nomenclature Roman Symbols residual Helmholtz energy 𝐴𝑅 monomer interaction energy 𝐴 monomer excluded volume 𝐵 reduced residual Helmholtz energy 𝐹 radial distribution function (RDF) evaluated at contact g(𝛽) 𝑘𝑖𝑗 binary interaction parameter (BIP) ̅ 𝑚 average chain length 𝑚𝑖 chain length of 𝑖 𝑀𝑊 molecular weight total moles 𝑛 𝑛𝑖 mole number of 𝑖 pressure 𝑃 gas constant 𝑅 temperature 𝑇 volume 𝑉 𝑥𝑖 molar composition of 𝑖 compressibility factor 𝑍 𝑍𝑐 critical compressibility factor Greek Symbols reduced volume 𝛽 𝛽𝑐 critical reduced volume ̂ fugacity coefficient of 𝑖 in a mixture 𝜙𝑖 Ωa tuned to force standard cubic EOS to match the critical point Ω𝑏 tuned to force standard cubic EOS to match the critical point Ω a (𝑚 ) defined in Eq. (20) to force CPC to match the critical point Ω 𝑏 (𝑚 ) defined in Eq. (21) to force CPC to match the critical point Abbreviations att disp hs mon rep RDF

attractive dispersion hard sphere monomer repulsive radial distribution function


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References (1) (2) (3) (4)

(5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

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For Table of Contents Only

𝑃 𝐶𝑃𝐶 = 𝑃 c

ic

+ 𝑃 chain

monomer: cubic EOS

𝐴 𝐵

𝑚

chain: SAFT EOS


35

Cubic-Plus-Chain (CPC). I: A Statistical Associating Fluid Theory

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