Ind. Eng. Chem. Res. 2006, 45, 7329-7335
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Entity-Based Eyring-NRTL Viscosity Model for Mixtures Containing Oils and Bitumens Lawrence T. Novak* Department of Chemical and Biomedical Engineering, CleVeland State UniVersity, 2121 Euclid AVenue, SH455, CleVeland, Ohio 44115-2214
A segment-based Eyring-NRTL viscosity model was recently presented and evaluated using data from a variety of mixtures containing polymers. In this paper, the segment-based Eyring-NRTL viscosity model is modified to provide a practical theory-based model for the viscosity of liquid mixtures containing complex components that cannot be typically described by the segment-based approach. To reflect these changes, this new model is referred to as an entity-based Eyring-NRTL viscosity model, because entities are viewed as abstract segments that represent complex components but are not well-defined. The entity-based EyringNRTL viscosity model provides a theory-based equation for liquid mixture viscosity as a function of composition, temperature, pressure, and shear rate. The correlative and predictive capability of the entitybased Eyring-NRTL viscosity model is demonstrated using component and mixture viscosity data from the following systems: lubricating oil blends, crude oil blends, bitumen-diluent mixtures, heavy oil-diluent mixtures, and acetone-ethanol-2,2,4-trimethylpentane mixtures. Ternary lubricating oil blend viscosities and ternary acetone-ethanol-2,2,4-trimethylpentane mixture viscosities are predicted from component and binary viscosity parameters determined from only component and binary mixture viscosity data. The entitybased model should be useful for modeling viscosity in a number of applications containing complex components, such as petroleum, petrochemicals, polymers, lubricants, coatings, and personal care products. Introduction The viscosity of liquid mixtures is often an important physical property in process and product research, development, and engineering (RD&E). In the process area, viscosity typically appears in correlative and computational transport models that are used for the detailed design and analysis of process equipment and processes. In the product area, viscosity is often used as a performance specification target for product RD&E and manufacturing. Application of comprehensive viscosity models should lead to more effective RD&E and manufacturing. A comprehensive viscosity model for liquids encompasses the effect of temperature, pressure, composition, and shear rate. In this paper, a theory-based comprehensive viscosity model for liquid mixtures will be developed and then evaluated using component and mixture viscosity data from the following systems: lubricating oil blends, crude oil blends, bitumendiluent mixtures, heavy oil-diluent mixtures, and acetoneethanol-2,2,4-trimethylpentane mixtures. A comprehensive viscosity model for liquids is commonly written as
η ) η0(T,P, x)g(γ˘ )
and polymer-polymer systems.2 This new theory-based model, called the segment-based Eyring-NRTL viscosity model, defines liquid mixture dynamic viscosity (η0(T,P ) 1 atm,x)) at atmospheric pressure and zero-shear rate over the entire range of composition and a range of temperature. In addition to being more physically realistic for mixtures containing polymers, the segment-based approach was found to have improved correlative capability, over the component-based approach,3 for mixtures containing polymers. The segment-based approach was also found to have predictive capability in several cases, because viscosity binary parameters were essentially independent of polymer molecular weight (Mn) over the ranges studied.2 In the segment view, diffusion and flow of large molecules2,3 are viewed to occur by a sequence of individual segment jumps into liquid lattice vacancies, as opposed to the entire molecule jumping into a vacancy. For systems containing polymers, the segments are defined as the repeat units of polymers. Oligomers and relatively small solvent molecules are typically treated as a single segment, or a component. Model Development
(1)
where η is the viscosity of the liquid mixture at a given temperature (T), pressure (P), composition (x), and shear rate (γ˘ ). η0 is the zero-shear viscosity, and g(γ˘ ) models the effect of shear rate on viscosity. The form of g(γ˘ ) depends on rheology. For shear-thinning liquids, the Carreau-Yasuda model has sufficient flexibility to fit a wide variety of g(γ˘ ) experimental curves for polymer melts, polymer solutions, and soap solutions.1 A new viscosity model for mixtures containing polymers was presented and evaluated for solvent-solvent, polymer-solvent, * Tel.: (216) 687-2569. Fax: (216) 687-9220. E-mail: lt_novak@ yahoo.com.
Following the form of eq 1, the segment-based EyringNRTL viscosity model will first be extended to an equation that includes the effects of pressure and shear rate (eqs 2-10). Then, a modification will be made to convert the segment-based model to the entity-based model. The entity-based model is a more practical equation for mixtures containing complex components that cannot be typically modeled with the segmentbased approach.
ln(ηVm) )
g*e
∑i xi ln(ηiVi) + RT +
10.1021/ie060516c CCC: $33.50 © 2006 American Chemical Society Published on Web 09/09/2006
P V*e dP ∫P)1atm
RT
+ ln(g(γ˘ )) (2)
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xi )
∑I ri,I x˜ I ∑j ∑J rj,J x˜ J
ηi )
∑I ri,I ηI (4)
∑I ri,I
P VI* dP ∫P)1atm
ln{ηI} ) ln{ηI (T)} +
(3)
RT
≈
ln{ηI (T)} + Vm )
g*
∑i xiVi + V e ≈ ∑i xiVi
e
RT
VI*(P - 1 atm) (5) RT
)
∑i xi
( )
ln η0 )
∑j xjGjiτji
(7)
∑j xjGji
(8)
Gji ) exp(-Rijτji)
(9)
( ) ∂g*e ∂P
(10)
T, x
Variables in the above equations are defined in the Nomenclature section. Lower case indexes (i, j) refer to segments. Upper case indexes (I,J) refer to components, such as simple molecules, solvents, oligomers, synthetic polymers, and complex components. The first and second terms on the right-hand side of eq 2 are referred to as the ideal term and the excess term, respectively. These terms model liquid mixture viscosity at atmospheric pressure and zero-shear rate as a function of temperature and composition. The segment-based molar excess Gibbs free energy of activation for diffusion and flow (g*e) is defined by the NRTL equations4 (eqs 7-9). In this work, a theory-based temperature dependency4,5 of the form A/T + B is used for ln{η(T)} and τij. If needed, more general empirical equations6 could be used. Pressure and shear-rate effects are described by the third and fourth terms in eq 2. These modifications extend the applicability of the original segment-based Eyring-NRTL viscosity model.2 Because the NRTL equations do not contain pressure dependency, V*e in eq 2 cannot be calculated from eq 10 and the NRTL equations. However, when V*e can be treated as constant, eq 2 reduces to
ln(ηVm) )
∑i
xi ln(ηiVi) +
g*e RT
+
V*e(P - 1 atm) RT
Simplified Segment-Based Eyring-NRTL Viscosity Model (General Form)2
(6)
Gij ) exp(-Rijτij)
V*e )
the volume of activation for diffusion and flow (VI*) to be approximately one-sixth the molar volume of simple pure liquids. The pressure adjustment term for mixtures will be discussed further in the section on the viscosity of Kuwait crude oil blends. When diffusion and flow in mixtures are viewed to occur by a sequence of similar-size individual segment jumps into liquid lattice vacancies, segment molar volumes (Vi) are considered similar and the following simplified segment-based EyringNRTL viscosity model may be written for zero-shear viscosity at atmospheric pressure.2 The following simplified segmentbased Eyring-NRTL viscosity model (General Form) is now available in commercial software.6
+ ln(g (γ˘ )) (11)
The linearity in pressure found in eqs 5 and 11 is consistent with empirical observations and empirical models for the effect of pressure on viscosity. Eyring and co-workers5 found
∑i xi ln ηi + ∑i xi
( ) ∑j xjGjiτji ∑j xjGji
(12)
For applications involving pressure above ambient and nonNewtonian conditions, the third and fourth terms in eq 2 can be added to the right side of eq 12 to make appropriate adjustments for pressure and rheology in the calculation of mixture viscosity (η). The segment-based approach works well for synthetic polymer components, because synthetic polymers are usually relatively well-defined mixtures that can be described by well-defined repeat units, repeat unit average composition, and molecular weight distributions. Synthetic polymer repeat units, oligomers, solvents, and simple molecules are commonly chosen to be the well-defined segments in the segment-based approach.2 In contrast, complex components (such as nonsynthetic oils and bitumens) are mixtures that are typically not quantitatively defined in terms of detailed molecular structures. Therefore, information needed in eqs 3 and 4 is typically not available. So, complex components cannot be typically defined by the segment-based approach.2 Another approach, such as the entitybased approach described below, provides a practical theorybased approach for modeling the viscosity of mixtures containing complex components. Entity View In the entity view, diffusion and flow are viewed to occur by a sequence of individual entity jumps into liquid lattice vacancies. For complex components, an entity is considered to be a single unique abstract segment that represents the component. Since an abstract segment is not well-defined, it is called an entity to differentiate it from the well-defined segments used in the segment-based approach.2 For mixed systems containing noncomplex components (simple molecules, solvents, oligomers, and synthetic polymers) and complex components, there will be well-defined segments and abstract segments. For mixtures of noncomplex and complex components, eq 3 is written in an equivalent form in terms of component mass fractions (wI), segment mass fractions (fi, I), and segment molecular weights (Mw,i).
xi )
∑I
ri,I x˜ I )
∑j ∑J
rj,J x˜ J
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( ) ∑∑( ) ∑I
wI fi,I Mw,i
Mw (13)
wJ fj,J
j
J
Mw, j
Mw
When segment molecular weights are considered similar, eq 13 may be approximated and simplified by assuming equal molecular weights for well-defined and abstract segments.
xi )
∑I wI fi,I
(14)
Substitution of eq 14 for the segment-based mole fraction (x) modifies the segment-based Eyring-NRTL viscosity model (eqs 2-12) to a more general entity-based Eyring-NRTL viscosity model for mixtures containing noncomplex and complex components. For complex components, a single unique abstract segment, or entity, is used to represent the complex component (so, fi,I ) 1). In this case, eq 4 reduces to ηi ) ηI. For mixtures containing only components that are modeled as complex components, eq 14 further simplifies to
x i ) wI
(15)
The concept of similar segment molar volumes and segment molecular weights implies essentially identical component densities. The assumption of similar segment molar volumes has been used previously to arrive at the commercial simplified general form (eq 12) of the segment-based Eyring-NRTL viscosity model.2 Although these assumptions are not precisely true, the mathematical impact on the resultant viscosity model may be acceptable from a practical viewpoint because zeroshear mixture viscosity is typically more strongly dependent on component viscosities, composition, and temperature. Substituting eq 15 into eq 12 and including adjustments for pressure above ambient and non-Newtonian rheology results in the following entity-based Eyring-NRTL viscosity model for mixtures containing components that are complex or are modeled as complex. Entity-Based Eyring-NRTL Viscosity Model for Complex Components
ln η )
∑I wI ln ηI +
∑I wI
( ) ∑J wJ GJIτJI ∑J
wJ GJI
+
V*e(P - 1 atm) RT
+ ln(g(γ˘ )) (16)
GIJ ) exp(-RIJτIJ)
(17)
GJI ) exp(-RIJτJI)
(18)
τJI ) (gJI - gII)/RT
(19)
For mathematical simplicity, component notation has been associated with the abstract segments, or entities, in the excess terms of eqs 16 and 20. Although eq 16 appears to be the component-based Eyring-NRTL viscosity model,3 it is not, because the ideal term (first term in eq 16) is in component mass fraction units instead of component mole fraction units.
Also, the component viscosity binary parameters, in the NRTL equations (eqs 17-19), physically refer to interactions between the abstract segments, or entities, that represent the complex components in the entity-based Eyring-NRTL viscosity model. These entities are generally viewed to be significantly smaller than most molecules in the mixture modeled as a complex component. Entity sizes would be expected to be similar to the dimensions of the vacancies in a corresponding liquid lattice. As in the case of the segment-based Eyring-NRTL viscosity model, these interactions are viewed to include both classical attraction-repulsion and molecular entanglement.2 Since the above assumptions imply that component densities are considered to be identical, dynamic viscosities in eq 16 can be replaced by kinematic viscosities.
ln ν )
∑I wI ln νI +
∑I
( ) ∑J wJ GJIτJI
wI
∑J wJ GJI
+
V*e(P - 1 atm) RT
+ ln(g(γ˘ )) (20)
Similar segment molar volumes and segment molecular weights jumping into vacancies results in weight fraction weighting in the ideal term (i.e., ∑I wI ln ηI). This weight fraction weighting rule is commonly used in the ideal term of empirical viscosity models, such as the “Aspen viscosity model”.6,7 The above theory-based model (eqs 5 and 16-20) is proposed for mixtures containing complex components that cannot be typically described by the segment-based approach. Equation 20 may be particularly useful in applications in which only kinematic viscosity data is available or needed. Results and Discussion A variety of mixtures were used to evaluate the correlative and predictive capability of the entity-based Eyring-NRTL viscosity model at ambient pressure. These mixtures include binary and ternary Repsol lubricating oil mixtures, binary and ternary Kuwait crude oil mixtures, binary Canadian bitumendiluent mixtures, and binary Alberta heavy oil-diluent mixtures. Additionally, binary and ternary mixture viscosity data on the acetone-ethanol-isooctane system were used to provide a comparison between the entity-based and segment-based Eyring-NRTL viscosity model fitting results and a comparison between average absolute deviations (AADs) for complex components (see Tables 1-4) and noncomplex components (Table 5) consisting of simple molecules. Viscosity of Repsol Lubricating Base Oil Blends as a Function of Composition and Temperature. Table 1 summarizes the scope of experimental viscosity data on lubricating base oils and lubricating oil blends8 that were used to evaluate the entity-based Eyring-NRTL viscosity model. The viscositytemperature data set consisted of 3 lubricating base oil components, 12 binary mixtures of these components, and 5 ternary mixtures of these components. Viscosity-temperature parameters for the component base oils and corresponding average absolute deviations (AADs) are listed along with the corresponding binary parameters and AADs. Using only component and binary pair viscosity data and parameters, the entity-based Eyring-NRTL viscosity model was found to predict ternary lubricating oil blend viscosities with an AAD of 6.4%. This AAD is comparable to the AADs for component viscosity-temperature fits and binary pair fits over a range of
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Table 1. Viscosity Model Parameters for Repsol Lubricating Oils and Blends8
a
light-medium-heavy base oil ternary mixture viscosity prediction
AAD
entity-based Eyring-NRTL viscosity model weight fraction weighted ideal viscosity modelb
6.4% 10.9%
component 1
Ac
Bc
AAD
RIJ d
τIJ d @ 25 °C
τJI d @ 25 °C
1.472 58 1.965 49 2.357 25
-3.306 44 -4.270 11 -4.849 79
4.8% 7.5% 6.3% 6.4% 5.1% 5.8%
0.20 0.20 0.20
-0.052 -0.246 0.000
-0.085 -0.393 0.000
component 2
light base oil medium base oil heavy base oil light base oil light base oil medium base oil
medium base oil heavy base oil heavy base oil
a Data source is ref 8. Range of experimental conditions: 25-100 °C, 0-100 wt %, and 5-550 cSt. b ln ν ) ∑ w ln ν . c log {ν(cSt)} ) 1000A/T(K) 0 I I I 10 + B. d Entity-based binary parameters were fit using only pure-component and respective binary mixture viscosity data.
Table 2. Viscosity Model Parameters for Kuwait Crude Oils and Blends9
a
light-medium-heavy crude oil blend viscosity correlation
AAD
entity-based Eyring-NRTL viscosity model weight fraction weighted ideal viscosity modelb
7.1% 30.4%
component 1 light crude oil medium crude oil heavy crude oil light crude oil light crude oil medium crude oil
component 2
Ac
Bc
AAD
RIJ d
τIJ d @ 25 °C
τJI d @ 25 °C
1.046 36 1.841 4 2.802 01
-2.569 98 -4.332 35 -6.586 66
4.5% 6.8% 5.6% 7.1% 6.5% 7.5%
0.20 0.20 0.20
-0.628 -1.029 0.618
-0.297 -1.188 -1.339
medium crude oil heavy crude oil heavy crude oil
a Data source is ref 9. Range of experimental conditions: 10-50 °C, 0-100 wt %, and 5-2200 cSt. b ln ν ) ∑ w ln ν . c log {ν(cSt)} ) 1000A/T(K) 0 I I I 10 + B. d Entity-based binary parameters were fit using all pure-component and mixture viscosity data. Binary parameters fit using only pure-component and respective binary mixture viscosity data resulted in ternary blend predictions with an AAD ) 19.7%.
Table 3. Viscosity Model Parameters for Canadian Bitumen, Diluent, and Mixtures11 component 1 Syncrude coker feed Suncor coker feed Strachan condensate toluene naphtha Syncrude coker feed Syncrude coker feed Suncor coker feed
component 2
cSt @ 30 °C
a
AAD
RIJ
τIJ @ 30 °C
τJI @ 30 °C
3.6%b 10.1%c 3.6%d
0.22 0.22 0.22
-1.738 -1.617 -1.051
-8.075 -8.012 -6.697
157 279 26 988 0.53 0.61 0.70 Strachan condensate toluene naphtha
a Data source is ref 11. Range of experimental conditions: 30 °C, 0-100 wt %, and 0.5-157 300 cSt. b For comparison, the Miadonye et al. empirical model AAD was reported to be 12.6%.11 c For comparison, the Miadonye et al. empirical model AAD was reported to be 7.4%.11 d For comparison, the Miadonye et al. empirical model AAD was reported to be 5.2%.11
Table 4. Viscosity Model Parameters for Alberta Heavy Oil, Diluent, and Mixtures11 component 1 Lloydminster heavy oil naphtha Lloydminster heavy oil
component 2
cSt @ 30 °C
a
AAD
RIJ
τIJ @ 30 °C
τJI @ 30 °C
3.6%b
0.22
-0.596
-5.415
2 967 0.70 naphtha
Data source is ref 11. Range of experimental conditions: 30 °C, 0-100 wt %, and 0.7-2967 cSt. b For comparison, the Miadonye et al. empirical model AAD was reported to be 5.1%.11 a
temperature. Ternary predictions using only the ideal term resulted in a 10.9% AAD. Correlation of all component, binary, and ternary mixture viscosity data using the ideal term resulted in an average AAD of 10%.8 In contrast, the comparable average of AADs in Table 1 for the entity-based model is 6.0%. Therefore, the entity-based Eyring-NRTL viscosity model results in a lower average of AADs, in comparison to the ideal term, and provides true predictability of ternary blend viscosity from only component and binary pair viscosity data and parameters. Viscosity of Kuwait Crude Oil Blends as a Function of Composition and Temperature. Table 2 summarizes the scope of experimental viscosity data on Kuwait crude oils and crude oil blends9 that were used to evaluate the entity-based EyringNRTL viscosity model. The viscosity-temperature data at various pressures (14.7-8 000 psia) consisted of 3 Kuwait crude oil components, 9 binary mixtures of these components, and 7
ternary mixtures of these components. These crude oils were considered stabilized crudes (dead crudes) because dissolved gas concentrations were low (
