Predictive Corresponding-States Viscosity Model for the Entire Fluid

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A Predictive Corresponding States Viscosity Model for the Entire Fluid Region: n-Alkanes Lawrence T. Novak Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie400654p • Publication Date (Web): 26 Apr 2013 Downloaded from http://pubs.acs.org on May 8, 2013

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Industrial & Engineering Chemistry Research

A Predictive Corresponding States Viscosity Model for the Entire Fluid Region: n-Alkanes Lawrence T. Novak* Department of Chemical and Biomedical Engineering Cleveland State University 2121 Euclid Avenue, SH455 Cleveland, Ohio 44115-2214 Abstract Previous work by the author demonstrated a novel approach for correlating scaled self-diffusion coefficient, viscosity, and thermal conductivity with residual entropy, over the entire fluid region. It is shown here that a new entity-based scaled viscosity model correlates pure n-alkane components to a single semi-log line with entity residual entropy. With five fitting parameters, a 5.2% group average of the absolute relative deviations (AAD) is obtained over the entire fluid region for a group of seventeen n-alkanes, ranging from methane to 1280 Mw linear polyethylene. It was also found that these same five fitting parameters predict the fluid viscosity of some other alkanes, ethers, and olefins. Thus, the entity-based scaling approach introduced here provides a predictive corresponding states model for fluid viscosity of nonassociating molecules and is a practical approach to determining viscosity in process engineering, product engineering, oil and gas reservoir engineering, pipelines, and fracking applications. This entitybased scaled viscosity-entity residual entropy model is particularly useful in the critical region and at high pressures because the traditional saturated liquid and saturated vapor viscosity component correlations are not applicable. *

Tel: (216)687-2569, Fax: (216)687-9220, Email: [email protected]

Key words Viscosity, corresponding states, critical region, high pressure, alkanes, ethers, olefins, transport coefficients, PC-SAFT, process engineering, product engineering, oil and gas reservoir engineering, pipeline engineering, fracking

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Introduction Fluid viscosity is typically embedded in rate-based models used for detailed design and analysis of process equipment and in-situ processes. In multiphase systems, viscosity affects flow regimes, phase hydraulics, and interphase heat and mass transfer. Although fluid mixtures are typically encountered in practice, mixture transport coefficient models are available that utilize pure component transport coefficients.1-3 This paper extends previous work4,5 on a component-based approach to scaling self-diffusion coefficient, viscosity, and thermal conductivity over the entire fluid region (liquid, gas, and critical fluid) for pure components. A single model for the entire fluid region is of practical interest because common saturated liquid and saturated vapor viscosity correlations do not apply in the critical region, or at pressures significantly above saturation pressures. It was originally found4 that scaling pure component transport coefficients with Chapman-Enskog6,7 values resulted in semi-log relationships between the scaled-transport coefficients and residual entropy over the entire fluid region, and a wide range of temperature and pressure. This extended the applicability of Rosenfeld8 scaling from the dense fluid region to the entire fluid region. Also, Chapman-Enskog transport scaling is consistent with the use of residual entropy because, by definition, residual entropy is the difference in entropy between the real and ideal state for pure components. The correlation of component Chapman-Enskog scaled viscosity with residual entropy is based on the general idea that the magnitude of the pure component fluid viscosity is related to an appropriate measure of average molecular size, momentum, and molecular disorder. The Chapman-Enskog viscosity contains functionality for a measure of average molecular size ( σ )

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and momentum ( m k B T ) in the ideal gas state. From statistical thermodynamics, a pure component average molecular disorder in the ideal gas state is characterized by the Boltzmann relationship between component entropy (S) and component thermodynamic probability (W):

S = k B ln (W ) . Variables in the equations below are defined in the Nomenclature section of this paper. S res (ρ , T ) = S real − S ideal = k B ln (W real W ideal )

(1)

Use of component residual entropy ( S res ) provides a scaling variable for relative molecular disorder that is a function of both density (ρ ) and temperature (T ) . This leads to a more comprehensive approach to characterizing molecular disorder than just considering density. Calculation of component residual entropy from PC-SAFT has been reported previously.4,9,10 Although component Chapman-Enskog transport scaling provided simple two or three component parameter transport correlations with residual entropy over the entire fluid region, the approach is limited by the availability of Lennard-Jones potential parameters. To eliminate this limitation, a segment-based Chapman-Enskog transport scaling approach was proposed and evaluated.9 In the segment-based approach, Lennard-Jones collision diameters and energy were replaced with PC-SAFT10 segment parameters for pure components. Use of the PC-SAFT equation of state to calculate residual entropy10 also eliminated the need for critical properties used in highly accurate reference equations of state.4,11

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Evaluation of seventeen n-alkanes ranging from methane to 1280 Mw linear polyethylene illustrated that the log of segment Chapman-Enskog scaled viscosity (ηseg ) was linear with #

segment residual entropy (S res (N seg k B )) over the entire fluid region.9  η N seg # ln(η seg ) = ln  (η seg ) CE 

  = ln( Aseg ) + Bseg (S res (N seg k B ))  

(2)

Each n-alkane component had a different slope ( Bseg ) and intercept ( ln( Aseg ) ). For methane ( N seg = 1 ), it was found that − Bseg and Aseg were close to 1. This leads to the following ideal relationship for methane relating the magnitude of segment Chapman-Enskog scaled viscosity to a measure of relative segment thermodynamic probability. ideal real } (3) η seg # = exp{− (S res N segk B )}= {Wseg Wseg

The component parameters ( Aseg and Bseg ) were found to be related for n-alkanes.9 A two parameter relationship between Bseg and Aseg , and a four parameter relationship between Bseg and component molecular weight were found.9 These relationships provide the capability to predict component slope and intercept from molecular weight. Thus the prediction of n-alkane viscosity is possible from the segment-based approach when little or no component viscosity data are available. In the correlation mode, two parameters are used per component for viscosity. The segmentbased viscosity model was found to result in a group correlation squared (R2) of -0.998 and group average absolute relative deviation (AAD) of 3.9%,9 within the 3% to 5% range of typical

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viscosity measurement variation.12 In the predictive mode, six parameters were used to represent n-alkanes. A group average AAD of 6.7% was obtained for seventeen n-alkanes ranging from methane up to 1280 molecular weight linear polyethylene, over the entire fluid region and a wide range of temperature and pressure. By definition, at zero residual entropy the segment Chapman-Enskog scaled viscosity should be one. Because Aseg was found to decrease with increasing molecular weight, the use of PCSAFT segment values for Chapman-Enskog collision diameters is apparently too small for ideal gas transport in larger n-alkanes. If transport of momentum in the ideal gas state is essentially represented by some larger “entity” than the PC-SAFT segment size, the use of an appropriate entity size might result in an improved scaled viscosity-entropy model. To generalize scaled viscosity-entropy modeling to a corresponding states form, this work proposes and evaluates a new entity-based scaled viscosity model. The entity-based approach is based on the general idea that the magnitude of the entity fluid viscosity ( η N entity ) is related to an appropriate measure of average entity size ( σ entity ), entity momentum ( mentity k B T ), and entity molecular disorder. The appropriate measure of entity mass and average entity size is called the “entity mass” and “entity size”. The entity size ( σ entity ) and number of entities per molecule ( N entity ) for use in the entity Chapman-Enskog viscosity will be determined from viscosity data on the group of seventeen n-alkanes studied previously.9 PC-SAFT will be used to calculate residual entropy. And, entity residual entropy (defined below) will be used as a measure of average entity relative molecular disorder.

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Entity-Based Scaled Viscosity Model In the entity-based viscosity model, segments are replaced by entities.

(η )

=

entity CE

5 mentity k BT / π 16 σ entity 2Ω( 2,2 )*

(4)

Eq. 4 defines the entity Chapman-Enskog viscosity (ηentity )CE as a function of entity size ( σ entity ), momentum ( mentity k BT ), and collision integral Ω ( 2, 2 )* , based on ( k BT ε seg ). In this work, an empirical equation13 is used to calculate collision integral values. Differences, if any, between ε seg and ε entity are neglected. Entity mass ( mentity = m N entity ) is the molecular mass divided by the number of entities in the molecule ( N entity ). The entity Chapman-Enskog scaled viscosity is defined in a manner analogous to the segment Chapman-Enskog scaled viscosity introduced previously.9

ηentity # =

η N entity (ηentity )CE

(5)

For a viscosity corresponding states model, Aentity and − Bentity are taken as 1, as found in (eq. 3) for methane. Then, all components in the n-alkane class should follow the relationship.

(

)

ln ηentity = − (S res N entity k B ) #

(6)

We postulate that relationships exist between some molecular attributes (ξ ) and: a) entity and segment diameters (eq 7), and b) segment and entity numbers per molecule (eq 8). Various molecular attributes (ξ ) were considered to find the best attribute for the entity model: component molecular weight (Mw), Jernigan-Flory end to end distance,28 and PC-SAFT number of segments.10 6


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(σ

entity

σ seg ) 2 = f (ξ )

(7)

and

(N

seg

N entity ) = g (ξ )

(8)

Using eqs 7 and 8, the eq 6 can be written in terms of segment-based variables and molecular attribute (ξ ) .

(

)

ln η seg f (ξ ) g (ξ ) = − g (ξ ) (S res N seg k B ) #

(9)

With eq 9, published component slope and intercept values9 were used to study possible attributes (ξ ) and functional forms and for f (ξ ) and g (ξ ) . Based on data,9 it was found that a strong relationships exists (R2 = 0.993 and 0.995, respectively) for the following two empirical relationships in terms of ξ = N seg . Five empirical parameters (a1 – a5) are used for the entire range of n-alkanes studied here. The advantages of using Nseg for molecular attribute are a low n-alkane group AAD and a link to PC-SAFT component parameters.

  a4 g (N seg ) = a 3 exp    (N seg + a 5 ) f (N seg )

 g (N seg ) g (N seg ) =    a1 

7

(10)

(1 / a 2 )


(11)


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Results and Discussion Model Parameter Fitting (eqs. 9 − 11, a1 through a5 ) The components, and range of temperature and pressure, used to determine the entity model parameters (a1 through a 5 ) for n-alkanes are listed in Table 1. Footnotes provide a link to the original viscosity data sources, PC-SAFT equation, and component segment parameters used to calculate residual entropy, entity diameter, and number of entities per molecule. The seventeen n-alkanes cover the range from methane to 1280 molecular weight linear polyethylene. And, the entire fluid region is covered with temperatures and pressures ranging up to 650 °K and 4990 atm. A constrained Gauss-Newton method with multiple starting locations was used to adjust model parameters (eqs 10 − 11, a1 through a5 ) to provide a minimum sum of the relative square error of ηseg , for the data points9 calculated from eq 9. The resulting best parameter estimates #

for a1 through a 5 , and corresponding 2-σ confidence limits, are listed in Table 2. Using the entity model parameters in Table 2 and PC-SAFT segment parameters, the respective Nentityσentity and Nsegσseg chain length characteristics (A°) were calculated and plotted against one another in fig 1. Although the term “chain length characteristics” does not quantify an average molecular configuration size, it is used here for a simple comparison between segment-based and entity-based models. Fig 1 illustrates that the magnitude of the entity and segment chain length characteristics are related linearly and comparable in magnitude over the range of n-alkanes studied. In the entity-based approach, entity size (σentity) is larger than

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segment size (σseg), and the number of entities (Nentity) is smaller than the number of segments (Nseg) for n-alkanes. Fluid Viscosity and Entity Chapman-Enskog Scaled Fluid Viscosity Plots A plot of raw viscosity data9 used for methane, ethane, propane, n-butane, and n-pentane versus reciprocal temperature is given in fig 2 for contrast to the entity-based scaled fluid viscosity model plot in fig 3. The difficulty of modeling fluid viscosity over the entire fluid region is evident from fig 2. However, the saturated liquid and vapor viscosity vs. reciprocal absolute temperature data point curves evident in fig 2 suggests that a total of six parameters, per component, should fit the saturated viscosity data in the two phase region. However, liquid viscosity significantly above saturation pressure and fluid viscosity in the critical region would not be predicted by these empirical models. Traditional approaches in DIPPR and commercial process simulators use two viscosity-temperature correlations: one for saturated liquid and another for saturated vapor. For alkanes, three parameter viscosity-temperature correlations are typically used for each saturated phase. In some cases, more parameters are needed for a larger range of temperature. This practice is acceptable under saturation conditions, but may become problematic in other cases. Fortunately, the new entity-based fluid viscosity model introduced here provides an improvement by reducing all of the data plotted in fig 2 to the simple semi-log relationship in fig 3. In addition, all n-alkanes fitted from methane to 1280 Mw linear polyethylene also lie on the same semi semi-log line with intercept and negative slope of one. The semi-log dotted line with intercept and negative slope of one, in fig 3, is the entity Chapman-Enskog scaled viscosity model. The dotted line is included for comparison to the viscosity experimental data

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points calculated and plotted in accordance to the respective fig 3 ordinate and abscissa variables. These data9 cover the entire fluid region (liquid, gas, and critical) and are fit by the five n-alkane entity parameters in Table 2. This is a significant improvement in the traditional method of six parameters per component: for example, five parameters for n-alkanes in the entity-based approach for the entire fluid region, versus 6 x 17 = 104 traditional parameters for seventeen components under saturation conditions. With the PC-SAFT equation of state, PC-SAFT segment parameters, and five entity parameters from Table 2, n-alkane viscosity over the entire fluid region can be correlated and predicted, as illustrate in fig 4. Data plotted in fig 4 cover almost four orders of magnitude from dilute gas to dense fluid viscosity. It is reasonable to ask the following questions: 1) how does the entity-based viscosity model compare with other models in the literature? and 2) can the entity-based viscosity model with Table 2 entity parameters predict the viscosity of nonassociating components not used in the original parameter fitting? These questions are addressed, in order, below. Viscosity Model Comparisons Tables 3A and 3B provide a comparison of viscosity models in terms of characteristics and performance. Footnotes provide names to the various models and corresponding literature references. This work (EV-ES) and previous work (SV-SS)9 are the only models listed that cover the entire fluid region and have received a comprehensive evaluation with data over the entire fluid region. An advantage of the EV-ES and SV-SS models is that they do not require critical properties that may not be known, or may not exist, for certain molecules. When providing an over-all model performance by average of the absolute relative deviations (AAD), the components in the set and the data selected to represent the components in the set become

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uncontrolled variables between studies that affect comparison results. Nonetheless, Table 3A and Table 3B provide a limited comparison and suggest that correlations of 4 % and predictions of 5.2% are expected with the SV-SS9 and EV-ES models, respectively. The friction theory-free-volume viscosity model (FT-FVT-SAFT1) only considered the dense fluid regime and did not include methane because methane was not well represented by the model. The FT-FVT-SAFT1 could be considered as an evolution of the Batshinski model22 that related the intercept of reciprocal viscosity-reciprocal density plots to zero free volume in the Van der Waals equation of state. This led to the determination of the “b” parameter in Van der Waals’ equation. FT-FVT-SAFT1 predictions are based on fitting all components listed under FT-FVT-SAFT1 in Table 3B, except for n-hexadecane. Thus, the result for n-hexadecane is a true prediction of 8.3%. Entity-Based Chapman-Enskog Scaled Fluid Viscosity Model Predictions for Alkanes, Ethers, and Olefins Table 4a provides the scope of viscosity data predicted by the entity-based model (EV-ES) model introduced here. Components, data sources, and range of temperature and pressure are listed. Components are listed in four categories: n-alkanes, branched alkanes, ethers, and olefins. These components are defined as nonassociating components in the PC-SAFT model.10 The n-alkane predictions provide an indication of the EV-ES model capability for interpolating n-alkanes. The branched alkane, ether, and olefin predictions provide an indication of the EV-ES model capability for extrapolating to other classes of nonassociating molecules. Table 4B and fig 5 illustrate that the entity-based model proposed here has some capability to extrapolate to other compounds not included in the original determination of the five entity

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model parameters listed in Table 2. The semi-log dotted line, with intercept and negative slope of one, in fig 5, is the entity-based Chapman-Enskog scaled viscosity model and is included for comparison to the viscosity experimental data points calculated and plotted in accordance to the respective ordinate and abscissa variables in fig 5. It appears that C6 and larger mono-methyl branched alkanes can be predicted with a similar AAD as the n-alkanes. However, the presence of di-methyl and tri-methyl branching leads to extrapolations that are on the low side for these dense fluids. Alpha olefin extrapolations would appear reasonable when data are not available. Application of the Entity-Based Chapman-Enskog Scaled Fluid Viscosity Model to other Components and Component Classes Application of this model is restricted by the availability of PC-SAFT segment parameters and the applicability of the five entity-based parameters determined here for the class of nalkanes. Original tables of PC-SAFT segment parameters10 have been extended to other components by fitting vapor pressure and density data.23 And, group contribution methods24,25 are available for estimating PC-SAFT segment parameters for some molecules, and studies have reported approaches to improve PC-SAFT equation accuracy near the critical point26 and freezing point.27 Broader application of this entity-based fluid viscosity model will require studies, similar to this one, to determine entity parameters for other classes of components. Application to mixtures of similar molecules, such as lower alkanes, may provide useful results because of the corresponding states property found for this model.

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Conclusions The new results reported here demonstrate that a corresponding states model for fluid viscosity, over the entire region and a wide range of temperature and pressure, is obtained when a previous segment-based scaled viscosity model9 is transformed into an entity-based form. The entity size, based on viscosity data, appears to more realistically represent n-alkane viscosity in the ideal gas state compared to segment size. This entity-based corresponding states model has shown potential to also predict fluid viscosity of some nonassociating components that were not used in fitting the original five entity parameters. Average absolute relative deviations (AAD) for a group of seventeen n-alkanes, ranging from methane to 1280 Mw linear polyethylene, were found to be 5.2% in the predictive mode. Additional studies, similar to this one, with other classes of pure components and mixtures of similar components should provide useful practical results. Both the previous segment-based viscosity model9 and current corresponding states entitybased viscosity model reduce the number of viscosity parameters in databanks, and provide a more comprehensive and less complicated approach for calculating fluid viscosity over the entire region. Thus, the predictive corresponding states viscosity model introduced here provides a practical tool for predicting viscosity, over the entire region and a wide range of temperature and pressure, in process engineering, product engineering, oil and gas reservoir engineering, pipeline engineering, and fracking applications.

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Nomenclature AAD =

100 N D vis cos itycalc − vis cos ityexp , N D = number of datapoints ∑ ND 1 vis cos ityexp

Aseg and Bseg = PC-SAFT component viscosity segment parameters (eq 2) kB

= Boltzmann constant

m

= molecular mass

Mw

= molecular weight or molar mass, mass/mole

N

= number

P

= pressure (1 atm = 1.01325 ×105 Pa = 1.01325 bar = 760 torr = 14.6959 lbs/in2)

S

= entropy

T

= absolute temperature

W

= thermodynamic probability in S = kB ln (W)

Greek Symbols ξ

= a molecular attribute

ρ

= number density

σ

= Lennard-Jones collision diameter, or a characteristic size

ε

= Lennard-Jones characteristic energy of attraction between identical molecules

Ω ( 2, 2 )*

= Lennard-Jones collision integral for viscosity, a function of (k B T ε )

η

= Newtonian, or zero shear, dynamic viscosity for a pure component

η N entity = Entity viscosity

(η )

entity CE

=

5 mentity k BT / π = Entity Chapman-Enskog viscosity9 16 σ entity 2Ω ( 2, 2 )*

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η entity # =

η N entity (ηentity )CE

= Entity Chapman-Enskog scaled viscosity

Superscript ideal

= ideal gas state

real

= real fluid state

res

= residual, difference between a pure component real state value and ideal gas state

#

= Chapman-Enskog scaled viscosity for entire fluid region, S res k B ≤ 0

Subscripts CE

= Chapman-Enskog value

entity

= entity basis or units

seg

= segment basis or units

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(23) Tihic, A; Kontogeorgis. G.M.; von Solms, N.; Michelson, M.L. Applications of the simplified perturbed-chain SAFT equation of state using an extended parameter table. Fluid Phase Equil. 2006, 248, 29.

(24) Vijande, J.; Pineiro, M.M; Legido, J.L; Bessieres, D. Group-Contribution Method for the Molecular Parameters of the PC-SAFT Equation of State Taking into Account the Proximity Effect. Application to Nonassociated Compounds. Ind. Eng. Chem. Res. 2010, 49, 9394.

(25) Emami, F.S.; Vahid, A.; Elliott, J.R. Jr., Feyzi, F. Group Contribution Prediction of Vapor Pressure with Statistical Associating Fluid Theory, Perturbed-Chain Statistical Associating Fluid Theory, and Elliott-Suresh-Donohue Equations of State. Ind. Eng. Chem. Res. 2008, 47, 8401.

(26) Tang, X.; Gross, J. Renormalization-Group Corrections to the Perturbed-Chain Statistical Associating Fluid Theory for Binary Mixtures. Ind. Eng. Chem. Res. 2010, 49, 9436.

(27) Privata, R.; Gania, R.; Jaubert, J.N. Are safe results obtained when the PC-SAFT equation of state is applied to ordinary pure chemicals?. Fluid Phase Equil. 2010, 295, 76.

(28) Jernigan, R.L.; Flory, P.J. Configurational Correlations in Chain Molecules. J. Chem. Phys.

1969, 50(10), 4165.

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LIST OF CAPTIONS Figure 1: n-Alkane Chain Length Characterization Figure 2: n-Alkane Fluid Viscosity vs. Reciprocal Temperature: Methane through Pentane Figure 3: Entity Chapman-Enskog Scaled Fluid Viscosity vs Entity Residual Entropy: Methane through 1280 Mw LPE Figure 4: Predicted Fluid Viscosity vs Experimental Viscosity: Methane through 1280 Mw LPE Figure 5: Predicted* Entity Chapman-Enskog Scaled Fluid Viscosity vs. Entity Residual Entropy: Alkanes, Ethers, and Olefins *

Based on fitting 5 parameter entity model with viscosity data from 17 n-alkanes, excluding

n-pentane.

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Table 1: Scope of Viscosity Data Fitted and Sources* Table 2: Entity Chapman-Enskog Scaled Fluid Viscosity Model Parameters and Statistics** Table 3A: Comparison of Selected Viscosity Model Characteristics Table 3B: Comparison of Selected Viscosity Model Per Cent Average Absolute Relative Deviation (% AAD) Table 4A: Scope of Viscosity Data Predicted by Entity Chapman-Enskog Scaled Fluid Viscosity Model

Table 4B: Predictions Based on Table 2 Parameters, from Fitting Table 1 Components, excluding n-pentane

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Figure 1

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Figure 2

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Figure 3

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Figure 4

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Figure 5

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Table 1

Components

Range of Conditions* Temperature °K

Pressure atm. abs.

Methane

96-403

4-563

Ethane

105-650

10-4-494

Propane

112-478

0.4-1974

n-Butane

140-627

0.2-680

n-Pentane n-Hexane

139-511 175-393

1-2483 1-3886

n-Heptane n-Octane n-Nonane n-Decane n-Undecane

180-394 211-473 233-423 240-423 253-473

1-2456 1-4990 1-681 1-2510 1-616

n-Dodecane

262-473

1-4950

n-Hexadecane

293-453

1-2755

n-Octadecane

286-573

1-3553

n-Dicetyl

353-495

1

n-C64 LPE1280

373-573 395-494

1 1

* Data sources for fluid viscosity are listed in Table 19 Source of segment parameters is PC-SAFT10

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Table 2 Parameter Estimate and 2-Sigma Confidence Limits 2σ-Lower

Parameter

2σ-Upper

Limit

Estimate

Limit

a1

0.9461

0.9512

0.9563

a2

0.4188

0.4225

0.4263

a3

4.4847

4.5413

4.5979

a4

-4.2163

-4.1069

-3.9974

a5

1.6886

1.7512

1.8137

**Based on fitting viscosity data from all components listed in Table 1 of this work, excluding n-pentane.

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Page 34 of 37

Table 3A

Model Name Capability

Prediction,

Individual

this work EV-ES1

Correlations SV-SS2

SST

3

RHS

4

FT-FVTSAFT15

liq, gas, & crit liq, gas, & crit sat liq & gas dense fluid dense fluid*

Critical Region

yes

yes

no

no

yes*

Critical Data Needed

no

no

yes

no

no

EOS used

PC-SAFT

PC-SAFT

SRK

Tait

SAFT (SW)

Temperature Range (K°)

96-650

96-650

110-370

120-373

240-480

-6

Pressure Range (atm.) Fitting parameters Viscosity Model Correlation Group AAD Prediction Group AAD Total Data Points Used

-4

-4

10 -4990 5

10 -4990 2

10 -42 3

1-3000 6

40-600** 6

Nseg 5.2% 3122

Mw 3.9% 6.7% 3122

3.6% 53

3.3% 620

Mw 3.8% 917

AAD = average of the absolute relative deviations (see Nomenclature) 1

EV-ES = entity Chapman-Enskog scaled fluid viscosity-entity residual entropy model (eqs 4-11), this paper)

2

SV-SS = segment Chapman-Enskog scaled fluid viscosity-segment residual entropy model9

3

SST = significant structure theory14

4

RHS = rough hard sphere theory15

5

FT-FVT =friction theory-free volume theory12

*only dense fluid regime was used. **40-600 atm for ethane thru n-hexadecane (12 n-alkanes)

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Table 3B Prediction,

Individual

this work EV-ES1

Correlations SV-SS2

SST

Methane

3.9%

3.2%

-

-

-

Ethane

7.3%

4.7%

-

2.4%

5.0%

Propane

7.3%

6.4%

3.6%

3.4%

2.6%

n-Butane

8.4%

5.7%

-

4.0%

2.4%

n-Pentane

12.5%

7.4%

-

7.6%

2.3%

n-Hexane n-Heptane

4.3% 4.8%

3.3%

-

3.1% 3.4%

2.3% 2.3%

n-Octane n-Nonane n-Decane n-Undecane n-Dodecane

4.0% 4.9% 3.0% 3.8% 3.3%

4.8%

3.8%

-

4.5% 1.3% 4.1% 1.2% 4.0%

2.5% 2.1% 1.7% 4.5%

n-Hexadecane

4.9%

4.3%

-

3.1%

8.3%

n-Octadecane n-Dicetyl n-C64 LPE1280

5.4% 0.9% 7.7% 2.6%

4.5%

1.4%

-

-

-

AAD

5.2%

4.0%

3.6%

3.5%

3.3%

Model Name

4.9%

2.6% 3.6% 2.7%

0.5% 4.7%

3

RHS

4

FT-FVTSAFT15

AAD = average of the absolute relative deviations (see Nomenclature) 1 EV-ES = entity Chapman-Enskog scaled fluid viscosity-entity residual entropy model (eqs 4-11), this paper) 2

SV-SS = segment Chapman-Enskog scaled fluid viscosity-segment residual entropy model9

3

SST = significant structure theory14

4

RHS = rough hard sphere theory15

5

FT-FVT =friction theory-free volume theory12

35



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Page 36 of 37

Table 4A Components

Source of

Range of Conditions

Data for

Temperature

Pressure

Comparison

°K

atm. abs.

Interpolation of n-alkanes not in data set used for model fitting n-Tetradecane 16,17 278-424 1-987 n-Pentadecane 16,18 310-408 1-3158 n-Eicosane 2-Methylbutane 2-Methylpentane 3-Methylpentane 2-Methylhexane 2,2-Dimethylhexane 2,2,4-Trimethylpentane 2,2-Dimethyloctane Dimethylether Dibutylether Ethylene Propylene 1-Butene

19

310-615

Extrapolation to branched alkanes 19,21 200-353 19 220-335 19 220-335 20 200-360 20* 200-380 19 200-370 20* 250-430 Extrapolation to ethers 19 200-370 20* 200-415 Extrapolation to olefins 20 120-250 19 220-333 20 120-260

* PC-SAFT parameters from reference 23

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Saturation Saturation Saturation Saturation Saturation Saturation Saturation Saturation Saturation Saturation Saturation Saturation Saturation

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Table 4B Components

AAD Number of data points compared

n-Tetradecane n-Pentadecane n-Eicosane

Prediction

Correlation

This work

Previous work

EV-ES1

SV-SS2

Interpolation to other n-alkanes 45 6.7% 71 5.2% 62

5.7%

Extrapolation to branched alkanes 2-Methylbutane 37 10.7% 2-Methylpentane 46 3.1% 3-Methylpentane 46 3.6% 2-Methylhexane 66 4.9% 2,2-Dimethylhexane 74 11.9% 2,2,4-Trimethylpentane 39 11.1% 2,2-Dimethyloctane 74 13.3% Extrapolation to ethers Dimethylether 70 7.5% Dibutylether 44 12.2% Extrapolation to olefins Ethylene 54 3.8% Propylene 46 3.1% 1-Butene 58 10.8%

5.2% 2.6% 5.5% 2.8% 1.3% 1.7% 1.5% 2.8% 2.2% 1.5% 3.0% 4.4% 3.7% 1.3% 2.3%

AAD = average of the absolute relative deviations (see Nomenclature) 1 2

EV-ES = entity Chapman-Enskog scaled fluid viscosity-entity residual entropy model (eqs 4-11), this paper) SV-SS = segment Chapman-Enskog scaled fluid viscosity-segment residual entropy model9

+

37


Predictive Corresponding-States Viscosity Model for the Entire Fluid

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