Generalization of the Friction Theory for Viscosity ... - ACS Publications

12820

J. Phys. Chem. B 2006, 110, 12820-12834

Generalization of the Friction Theory for Viscosity Modeling Sergio E. Quin˜ ones-Cisneros* and Ulrich K. Deiters Institute of Physical Chemistry, UniVersity of Cologne, Luxemburger Str. 116, D-50939 Koeln, Germany ReceiVed: March 24, 2006; In Final Form: April 28, 2006

The friction theory (FT) approach relates the viscosity of a fluid to its equation of state (EoS), and it is known to give good results for a large number of compounds over wide ranges of temperature and pressure. Previous FT versions were restricted to use EoS of the van der Waals type, i.e., EoS explicitly consisting of a repulsive and an attractive term, which limited the number of usable EoS as well as the accuracy of the viscosity predictions. In this work, the restriction is removed by means of a pragmatic generalized definition of repulsive and attractive terms based on the internal pressure concept. As a result, the FT theory can be extended to practically all types of EoS, from theoretical ones (e.g., EoS based on thermostatistical or renormalization theories) to the highly accurate empirical reference EoS. In combination with the later, the FT is shown to represent experimental viscosity data for several fluids, including water, with an accuracy as high as that required for reference models. Additionally, some relevant phenomena, such as the critical anomaly, appear to follow naturally from the physics already built into the EoS.

1. Introduction Many technical applications of fluids require reliable and accurate viscosity data, along with other thermophysical properties such as caloric or volumetric data; an obvious example is refrigeration technology. Thus, multiproperty models capable of providing accurate descriptions not only of thermodynamic but also of transport properties of fluids are required. Such models should naturally cover all regions of interest: low temperatures, high pressures, and phase transitions. In particular, viscosity is an extremely important property for which many different models already exist; extensive critical reviews of such models are readily available.1,2 The viscosity models found in the literature range from highly theoretical to simple empirical correlations. Many viscosity models, however, are only applicable to either the liquid or the gas phase. In the case of the dilute gas limit (the zero pressure gas limit), the ChapmanEnskog kinetic theory of gases has been the basis for the development of semitheoretical models for the viscosity modeling of light gases such as the general Chung et al.3 model. For dense fluids, however, the complexity of the relevant intermolecular forces makes even a semitheoretical description based on thermostatistical concepts extremely difficult. According to Monnery et al.,1 the only methods that can be applied to both liquids and gases are semitheoretical methods based on either the corresponding-states principle, the hard-sphere theory, the modified Chapman-Enskog theory, or the empirical residual concept. Alternatively, the more recent friction theory (FT) development differs from all previous approaches as it has been based on the friction principles of classical mechanics, in particular on the extension of the Amontons-Coulomb friction law to the description of the shear stress. As applied to viscosity, the FT4 provides a way to link accurate viscosity models to the repulsive and attractive pressure terms of an EoS. The basic idea of the approach is to use the analogy between shear flow and two sliding surfaces under shear * Corresponding author. E-mail: [email protected].

stress, which is linked to normal stress through temperaturedependent friction coefficients. In the FT, the normal stresses are given by repulsive and attractive contributions to the thermodynamic pressure, and therefore, the shear stress naturally follows from a balance between repulsive and attractive terms, too. The FT has previously been applied to several widely used cubic EoS of the van der Waals type such as the SoaveRedlich-Kwong5 (SRK) EoS or the Peng-Robinson6 (PR) EoS as well as to the PC-SAFT7,8 EoS, a noncubic EoS consisting of separate repulsion and attraction terms. However, although some attempts in extending the approach to other type of EoSs have been previously made,9,10 such attempts were based on an arbitrary distinction between repulsive and attractive parts if the involved equation was not of the van der Waals type. With regard to performance, the FT (which is a phenomenological theory) has consistently been shown to represent the viscosities of a wide range of fluids rather well, including complex cases such as reservoir fluids.11 Reasonably good results could even be obtained with simple cubic EoS. It is, therefore, consequential as well as practically desirable to extend the FT toward other, more physically robust, EoS in order to achieve even better accuracies or to cover wider ranges of temperature and pressure. This, however, requires a rational method for splitting an arbitrary EoS into a repulsive and an attractive part. Another reason for studying the proper identification of repulsive and attractive contributions to the EoS is the fact that the FT concept, through the Amontons-Coulomb friction law, also formulates the shear stress as a balance between a repulsive and an attractive term. Evidently, this balance is fundamental for the performance of the model. In this work, we will review and reformulate the FT in a more general form than previously presented. We will introduce a rational method for splitting an EoS of a general type into attractive (pa) and repulsive (pr) pressure terms. The method is then applied to a group of representative polar and nonpolar fluids, including water, where the viscosity, similarly to density, shows relevant low-temperature anomalous behavior. Therefore,

10.1021/jp0618577 CCC: $33.50 © 2006 American Chemical Society Published on Web 06/06/2006

Friction Theory for Viscosity Modeling

J. Phys. Chem. B, Vol. 110, No. 25, 2006 12821

Figure 2. Shear flow between parallel plates. F, force acting on the upper plate that moves at speed uo, h, distance between the plates, u, local fluid velocity.

Figure 1. Basic forces acting in the case of a block moving under mechanical friction. N, normal force, F, tangential force responsible for movement, Fk, friction force opposing F, u, resulting velocity.

with the approach presented here, the FT essential features are readily extended to all type of EoSs, resulting in accurate viscosity models that can even be coupled to highly accurate reference EoS, as it would be the case of FT viscosity models coupled to the Span-Wagner12 CO2 or the Wagner-Pruss13 ordinary water reference EoSs. Additionally, the FT performance in the critical region is studied by applying the approach to a renormalized EoS: the generalized cubic crossover (GCC) EoS proposed by Kiselev and Ely.14 2. The Friction Theory

stress and σ the normal stress (pressure) acting on the contact surface. Clearly, the Amontons-Coulomb friction law follows from a simple first-order Taylor series expansion of the shear stress as a function of the normal stress. That is, if it is assumed that τ is an analytical function of σ, a first-order series expansion at σ0 ) 0 gives

τ ) τ0 +

(dσdτ) σ + O(σ ) 2

0

(3)

where the condition of no friction force under no normal force implies

lim [τ] ) 0 w τ0 ) 0 σf0

(4)

and eq 2 follows from setting

In the FT for viscosity modeling, the total shear viscosity η is separated into a dilute gas viscosity term η0 and a residual friction viscosity term ηf

η ) η0 + ηf

(1)

The dilute gas viscosity is well understood and defined as the viscosity given by the kinematics of particles of a finite mass but in the zero density limit, while the residual viscosity term, under the FT perspective, is related to friction concepts of classical mechanics. With the exception of the models discussed in Section 4 of this work, any in-depth discussion related the dilute gas viscosity term is outside the scope of this work. However, a revision of the original derivation of the friction viscosity term4 is relevant for the purpose of the generalized approach presented in this work. The fundamental mechanical concept of friction is wellknown and can be found in almost all elementary engineering textbooks on statics and dynamics.15 Mechanical friction deals with the macroscopic description of the forces that oppose the sliding of two bodies against each other. As pointed out earlier,4 the concept of mechanical friction was first formally formulated by Coulomb in 1781, although it appears to have been known to Leonardo da Vinci and experimentally studied by the French engineer Amontons in 1699.15 Coulomb also distinguished between the force necessary to initiate motion (static friction) and the force needed to maintain motion (kinetic friction), which is the focus of this work. According to the Amontons-Coulomb friction law, the kinetic friction force Fk parallel to the contact surface in Figure 1 and opposing the movement is given by

Fk ) µkN

(2)

where µk is known as the coefficient of kinetic friction. For rigid bodies, µk is supposed to depend only on the smoothness and the material of which the surfaces are made and not at all on the nominal area of contact (A). Thus, such forces can be represented as Fk ) Aτ and N ) Aσ, where τ denotes the shear

µk )

(dσdτ)

0

(5)

In the case of dense fluids, if a shear stress is applied to any portion of a confined fluid, the fluid will move (or deform as in the case of viscoelastic fluids). For instance, considering a fluid placed between two close but large parallel plates of area A (Figure 2), if an appropriate dragging force F is applied to the upper plate in order to achieve a constant velocity uo, while keeping the lower plate fixed, a shear stress τ ) F/A will be exerted on the fluid between the plates. Experiments show that the fluid in immediate contact with the solid boundaries has the same velocity as the boundaries (the no-slip boundary condition), and that F is, for many fluids, directly proportional to A as well as uo, and inversely proportional to the thickness h. That is, by considering the fluid made up of many layers of differential thickness parallel to the flat surface, the flow may be described in terms of the rate of shear (du/dh), i.e., the change of rate at which one layer moves relative to an adjacent layer. Thus, the flow is properly described by the Newton’s law of Viscosity:

du τ)η dh

(6)

where, in general, η is not a constant, and fluids are classified as Newtonian or non-Newtonian depending on whether η is, for practical purposes, found to be independent of du/dh. Furthermore, there are fluids that under some conditions behave like an elastic solid body instead of flowing spontaneously. All these concepts, definitions, and derivations can be found elsewhere, and for a better discussion, the reader should be referred to the extensive available rheology literature. However, for the purpose of this work, it will be assumed that the fluid is an ideal Newtonian one. In the case of a fluid at rest, for the sake of generality, let us assume now that the normal stress, i.e., the isotropic total pressure (p), is given by some contributions that may be of different physicochemical nature, i.e.

12822 J. Phys. Chem. B, Vol. 110, No. 25, 2006

Quinõnes-Cisneros and Deiters

p ) pI + pII + pIII + ...

(7)

These contributions can be such as a repulsive pressure term, pr, an attractive pressure term, pa, or even a renormalization scaling term, psc, according to the specific physicochemical concepts that may be built into the specific EoS. Next, as the fluid is subjected to an external dragging force, we assume that each one of the involved different physicochemical interactions may also give rise to a shear stress (friction force) contribution of a related nature, i.e.

τ ) τI + τII + τIII + ...

(8)

Then, by analogy with the general derivation of the AmontonsCoulomb law of friction, we assume that each one of the contributions to the shear stress follows from a series expansion of its corresponding normal stress. That is, n

τX )

µX,i pXi ∑ i)1

(9)

Here, the µX,i are the kinematic friction coefficients associated with one of the possible contributions contemplated on the righthand side of eq 8. Consequently, substituting eq 9 into eq 8 and this into the Newton’s law of viscosity, eq 6, yields nI

ηf ) (

nII

nIII

κI,i pI ) + (∑ κII,i pI ) + (∑ κIII,i pI ) + ... ∑ i)1 i)1 i)1 i

i

i

(10)

where the different κ are friction coefficients given by

κX,i )

µX,i (du/dh)

(11)

and which in the case of Newtonian fluids must be constants; it should be noticed that the relaxation of this condition leads to a general model for non-Newtonian fluids. Therefore, substitution of eq 10 into eq 1 represents a general FT viscosity model. To this point, the FT approach has been successfully applied to many different fluids, from simple ones even to complex heavy oils,16 based on simple EoSs of the van der Waals type. That is, following the simple van der Waals separation of the total pressure into a repulsive (pr) and an attractive contribution (pa),

p ) p r + pa

(13)

According to eq 9, the contributions to τ are given by na

τa )

τa,i pai ∑ i)1

∑ i)1

nr

κa,i pai +

κr,i pri ∑ i)1

(16)

where, assuming Newtonian behavior, the κa/r,i are temperaturedependent friction parameters related to friction coefficients between sliding surfaces subject to the pa and pr normal stresses. For applications involving cubic EoS, due to the strong highpressure dominance of the repulsive term given by the hardcore excluded volume, a linear model for the attractive contribution, together with a quadratic model for the repulsive (na ) 1, nr ) 2), has been found to be sufficient for many engineering applications.16-22 In fact, in most cases, this quadratic model can represent the pressure viscosity dependency up to around 100 MPa with good accuracy. However, if the application involves simple molecules and moderate pressures, a simple linear model (setting na ) 1, nr ) 1) may suffice. In contrast, for fluids involving molecules of complex structure capable of developing strong molecular interlinking at elevated pressures such as the 2,2,4,4,6,8,8-heptamethylnonane,23 a higher-order model may be required. Another important remark deals with the separation of the repulsive term into the ideal part and a residual term,

pr ) pid + ∆pr

(17)

In the related FT work by Tan et al.,8 the authors appear to have excluded the ideal gas term from their definition of the repulsive term; normally, this leads to good results. However, the ideal term is first order in density, while the residual and the attractive terms are of higher order. Therefore, if the ideal gas term is excluded, in some cases the model may not be able to properly correlate the transition area (where first-order effects are relevant) between the dilute gas limit (zero pressure) and the dense state. In fact, if the ideal gas term is excluded from the correlation, the zero pressure gas limit (dilute gas limit) is always approached with a zero slope with respect to pressure or density. The experimentally observed nonzero slope, which can be positive or negative, has been shown by theoretical studies24 to be a consequence of multiple-body contributions to the dilute gas limit. 3. The Thermodynamic Equation of State The thermodynamic relation that follows from the constant temperature derivative of the molar internal energy, uˆ ,

πT ) (∂uˆ /∂V)T ) T(∂p/∂T)V - p

(18)

(12)

and the shear stress is given by

τ ) τ r + τa

na

ηf )

is often called internal pressure, as it has dimensions of pressure, and it is assumed to be related to molecular attraction and repulsion. Although, as pointed out by van Uden et al.,25 the literature concerning the internal pressure is confusing, it is a useful thermodynamic property. Indeed, for the subject of interest of this work, it is of relevance to point out that from the original van der Waals EoS,26

(14)

and

p)

RT a V - b V2

(19)

a V2

(20)

the internal pressure nr

τr )

τr,i pri ∑ i)1

(15)

Thus, the friction contribution to the viscosity may be represented by the following series:

πT )

is derived. This leads to the interpretation of πT as a property related to intermolecular cohesion. Furthermore, the substitution



Figure 3. Separation of the pressure into attractive and repulsive parts based on internal pressure eqs 21-22 for the Span-Wagner CO2 reference EoS.12 Only stable branches are depicted at 220 K, 240 K, 260 K, 280 K, 300 K, Tc, 350 K, 400 K, 500 K, and 600 K.

of the classical van der Waals repulsive term by the hard-spheres term of Carnahan-Starling27 also naturally leads to similar results. In fact, eq 20 holds for all EoS that consist of a van der Waals attractive term and a repulsive term that is linear in T. Thus, for the purpose of this work, the following pragmatic definitions are proposed for the attractive and the repulsive part of the EoS, respectively:

pa ) - πT

(21)

pr ) p - pa ) T(∂p/∂T)V

(22)

and

Clearly, the definition of the repulsive term, eq 22, includes the ideal gas term along with a residual repulsive term. That is, the repulsive term can be further written as

pr ) pid + ∆pr

(23)

where ∆pr should be understood as a residual of the repulsive term. In general, this definition for the separation of an EoS into a balance between repulsive and attractive terms gives appropriate and consistent results qualitatively similar to those shown in Figure 3, which follows from the highly accurate Span-Wagner CO2 Helmholtz-energy-based reference EoS.12 From these results, it can be appreciated that the repulsive and attractive terms consistently remain positive and negative, respectively, and that their magnitude increases as the temperature approaches (from above) the triple point. Most fluids of practical interest behave in this “normal” manner, and in this case, it is straightforward to couple a simple but accurate FT model for the fluid viscosity covering the entire range of application (namely from the triple point to temperatures as high as 1000 K and pressures up to 100 MPa or higher). However, an exception to the “normal” behavior described above is the most important fluid in nature: water. The internal pressure of water is known to vanish and even change sign in the vicinity of the density anomaly (around 4 °C). This unusual behavior of the internal pressure, as calculated from the highly

Figure 4. Separation of the pressure into attractive and repulsive parts, based on internal pressure eqs 21-22, for the Wagner-Pruss water reference EoS.13 Only stable branches are depicted at 273 K, 300 K, 350 K, 400 K, 450 K, 500 K, 600 K, Tc, 800 K, and 1000 K.

accurate Helmholtz-energy-based International Association for the Properties of Water and Steam (IAPWS) reference EoS by Wagner and Pruss,13 is shown in Figure 4. Clearly, the assumption of a positive repulsive term and a negative attractive pressure term, as described above, breaks down in this case. However, despite this anomaly, the pressure can still be interpreted as a balance of the resulting pa and pr terms as defined by eqs 21 and 22 or 23. Consequently, as the FT models are actually built on such balance, a highly accurate FT water model based on the IAPWS reference EoS and recommended viscosity data28 can also be constructed. 4. The Modeling of the Dynamic Viscosity 4.1 The Dilute Gas Limit. The dilute gas limit, η0, is due to particle kinematics, a subject extensively studied and reported in numerous articles and textbooks. It is relevant for the gas phase only and may actually be neglected for engineering applications solely involving condensed phases. In previous FTrelated works, the semiempirical predictive model by Chung et al.3 has been extensively used for the calculation of η0. This model is applicable to the prediction of the dilute gas limit for both nonpolar and polar fluids over wide ranges of temperature with reasonably low absolute average deviations (AAD) of around 1.5% for nonpolar compounds. Alternatively, a more rigorous approach can be derived based on the kinetic theory of dilute monatomic gases by Chapman and Enskog.29 However, the extension of the kinetic theory to more complex polyatomic molecules such as water is, in general, not currently possible. Therefore, a semiempirical approach consisting of estimating an “effective” collision cross section through a fit of the theoretical relationships for the first-order approximation of the kinetic theory30 to experimental viscosity data is commonly used, as in the recent studies of the vapor-phase viscosity of the polar compounds water31 and methanol.32 However, as the dilute gas limit is neither part of the FT nor is it the objective



TABLE 1: Dilute Gas Limit Model Parameters (mPa‚s) do d1 d2 d3 AAD (%) ∆T (K)

methane

ethane

propane

butane

pentane

hexane

heptane

octane

HFC134a

CO2

H2O

0.00260536 -0.0185247 0.0234216 0 0.31

0.0159252 -0.0497734 0.0434368 0 0.55

0.0123057 -0.0425793 0.0403486 0 0.53

0.0183983 -0.0571255 0.0493197 0 0.62

0.0176805 -0.0556942 0.0487177 0 0.15

0.0169975 -0.0542985 0.0480065 0 0.35

0.0196036 -0.0597839 0.0507528 0 0.19

0.0167562 -0.0531705 0.0469105 0 0.38

0.0312515 -0.0896122 0.0730823 0 0.25

0.0691842 -0.215862 0.210944 -0.0490494 0.02

0.151138 -0.444318 0.398262 -0.0817008 0.06

95-600

100-500

100-600

150-500

200-550

200-600

200-600

250-600

200-475

200-1000

275-1100

of this work, we therefore use a simple, yet accurate empirical equation for η0:

η0 ) d0 + d1 Tr0.25 + d2 Tr0.5 + d3 Tr0.75

(24)

where Tr is the reduced temperature (Tr ) T/Tc). The specific substance parameters in eq 24 have been derived by regressing low-pressure viscosity data against a model that combines eq 24 with a simple linear FT model to account for the initial-density viscosity contribution. In all cases, the absolute average deviation (AAD) was around 0.5% or lower for the range of interest and therefore quite satisfactory. In most cases, the d3 parameter in eq 24 was set to zero except for CO2 and water, for which more accurate models of reference quality are derived in this work. The dilute gas model parameters used in this work, ranges of application, and AADs are reported in Table 1. 4.2 The Friction Theory General Model. As previously pointed out, the model proposed in eq 16, quadratic in the repulsive term (nr ) 2) and linear in the attractive term (na ) 1), can be satisfactorily applied to many systems of industrial relevance in wide ranges of temperature and pressure. For engineering applications, this model may suffice, and its quality has been extensively reported in several previous FT publications. However, in this work, we intend to establish the basis for a more general approach that may further lead to FT viscosity models of higher accuracy. Therefore, we start this discussion by analyzing the performance of the repulsive term in the initialdensity region, i.e., the transition region from the zero pressure to elevated pressures. In previous FT applications, compounds of rather low critical pressure (pc < 10 MPa) were studied. For this type of compound, the entire region, from low to elevated pressures, can be described in an acceptable manner by a basic quadratic model of the form

η ) η0 + κr pr + κa pa + κrr pr2 + κaa pa2

(25)

For this type of model, the transition between the zero-order dilute gas term and the higher-order terms is enclosed in the repulsive term, which includes the linear (in density) ideal gas term. In the case of compounds with low critical pressure, the plausible deviations that may follow from not properly describing the initial-density behavior might, in practice, be “hidden” by the close proximity of the vapor saturation curve. However, for more general cases, an accurate description of the lowdensity phase requires an explicit separation of the linear ideal gas term as, in the FT formulation, the ideal gas term provides the linear initial density viscosity dependence responsible for the behavior of the second viscosity virial coefficient and therefore its consistency with theory.24 Consequently, for general purposes, the following quadratic general FT model is proposed:

η ) η0 + κi pid + κr∆pr + κa pa + κii pid2 + κrr∆pr2 + κaa pa2 (26)

Figure 5. Viscosity of CO2 as a function of pressure in the vicinity of the critical point, at 304 K. (- - -), basic FT model (ideal gas term included in the repulsive term), (s), extended FT model (separate ideal gas term, eq 26), (b), recommended CO2 viscosity data by Fenghour et al.33

Figure 6. Viscosity of water as a function of pressure in the vicinity of the critical point, at 748 K. (- - -), basic FT model (ideal gas term included in the repulsive term), (s), extended FT model (separate ideal gas term, eq 26), (b), IAPWS recommended viscosity data.28

To illustrate the advantages of the general FT model, first consider the case depicted in Figure 5, where the CO2 recommended viscosity data by Fenghour et al.33 at 304 K is modeled using both FT models, a temperature that by being close to the critical one better illustrates the transition from the dilute gas limit to denser states. Noticeably, although the general FT model (eq 26) is superior to the basic FT model (eq 25), for practical purposes, the basic model may suffice. In contrast, Figure 6 shows the corresponding modeling results for the water nearcritical isotherm (T ) 648.15 K), which has a notoriously high critical pressure (pc ) 22.06 MPa). Clearly, in the case of water, while the extended model delivers a practically perfect performance, the basic model cannot possibly accurately describe the transitional region, at least for scientific purposes. This makes it clear that, in general terms, the accurate modeling of viscosity (including the discussed transitional region) requires treating the initial-density region independently. Therefore, in this work, the approach will be subsequently illustrated after the general



FT model given in eq 26 for a representative number of compounds, including the most important compound in nature: water. The temperature dependence of the friction coefficients proposed in this work, eqs 27-35, reflects the exponential behavior of viscosity, which has been observed (and to some extent theoretically derived) for most compounds:

κa ) (a0 + a1ψ1 + a2ψ2)Γ

(27)

κaa ) (A0 + A1ψ1 + A2ψ2)Γ3

(28)

κr ) (b0 + b1ψ1 + b2ψ2)Γ

(29)

κrr ) (B0 + B1ψ1 + B2ψ2)Γ3

(30)

κi ) (c0 + c1ψ1 + c2f(Γ))Γ

(31)

κii ) (C0 + C1ψ1 + C2ψ2)Γ3

(32)

ψ1 ) exp(Γ) - 1

(33)

where

1,1,1,2-tetrafluoroethane (HFC-134a) in combination with the short Span-Wagner EoSs.35-37 In the case of the light alkanes (methane trough octane), the model has been regressed against the recommended smoothed viscosity database by Ze´berg-Mikkelsen,38 which contains all experimental data, critically assessed, from literature until 2001. This database covers wide ranges of pressures and temperatures. The original sources for the experimental data are numerous and they have already been listed in detail elsewhere,4 along with pressure and temperature ranges. Table 2 contains the results of the FT regression based on the Ze´berg-Mikkelsen database, along with the temperature range, the maximum pressure considered in the regression, the AAD, the bias, and the maximum absolute deviation (max dev) for each one of the considered light hydrocarbons. Regarding the pressure range of application, although the available data did not reach 100 MPa in every case for the full temperature range, the models presented in Table 2 are considered reliable up to at least 100 MPa. The formulas for the statistical information reported in Tables 1 and 2 are the following:

AAD )

1

nT

∑ |Devi|

nT i)1

(36)

with

and

ψ2 ) exp(Γ2) - 1

(34)

Bias )

with

Γ)

Tc T

(35)

Therefore, eq 26, in combination with the foregoing parametrization of the friction coefficients, represents a highly flexible FT model with 18 adjustable parameters capable of delivering an accurate performance for applications of theoretical and industrial relevance. It should be pointed out, however, that depending on the compound and the required accuracy, a lower number of adjustable parameters may suffice. Also, it is relevant to remark that eq 31, in combination with the linear ideal gas term in eq 26, accounts for the viscosity initial density dependence and, in particular, for the second viscosity virial coefficient. The mathematical form of eq 31 is consistent with theory24 in a temperature range covering most practical applications, which involves the transition from negative values (at low temperatures) to a positive maximum (at a temperature that, for most compounds, is beyond the range of practical relevance). However, for the sake of generality, a second term has been included in eq 31, f(Γ), which should monotonically lower the value of the second viscosity virial coefficient at high reduced temperatures; this correction is relevant for cases such as helium only and not required for the examples presented in this work, not even for methane. Depending on the compound and the correlation desired accuracy, the mathematical form of f(Γ) can be as simple as a linear function (i.e., f(Γ) ) Γ) or as elaborate as the recommended empirical equation for the second viscosity virial coefficient of Vogel et al.34 The general FT model performance has been demonstrated with the precise modeling of the viscosity of CO2 and H2O, using as the core models the Span-Wagner12 reference CO2 EoS and the Wagner-Pruss13 reference water EoS. As a further test, the approach has been applied to the viscosity modeling of the normal alkanes family (methane trough octane) plus

Devi )

1

nT

∑ Devi

(37)

ηcal,i - ηref,i ηref,i

(38)

nT i)1

where nT is the total number of points, and for each point i, ηcal is the estimated viscosity, and ηref the reference experimental or recommended value. Propane. Figure 7 shows the FT model performance in relation to the data used for the model derivation, which corresponds to a survey of the available literature up to 2001. According to this, for propane, the highest pressure attained by experiments is only 60 MPa in a temperature range between 296 and 477 K, and much lower elsewhere. To demonstrate the reliability of the presented FT model for high-pressure extrapolation (up to 100 MPa), in Figure 8, the FT model predictions are compared to the recommended viscosity by Vogel et al.34 (extending up to 100 MPa), where all estimations outside the ranges shown in Figure 7 ought to also be considered extrapolations. For most of the range of practical relevance (150-500 K), the results in Figure 8 show a remarkable good agreement between both models, which only deteriorates outside the reported temperature range, particularly in the low-temperature region, where viscosity is most sensitive to temperature. Methane. The reliability of the reported FT models is further illustrated in Figure 9, where a deviation diagram with respect to the original experimental data, used in the derivation of the Ze´berg-Mikkelsen recommended database, is presented. As reported by Quinõnes-Cisneros et al.,4 the original methane data consist of some 750 measurements from 15 different sources having wide ranges of experimental uncertainty. Yet the model can reproduce all of the original experimental data with an AAD of 1.72% and bias of only 0.09%, even better than the value reported in Table 1 for the smoothed data. Furthermore, as indicated in Figure 9, 72.7% of the total original data are reproduced with an accuracy of (2%. Of the values, 12.6%

methane

hexane

-7.50686 × 10-6 -1.50327 × 10-6 0

-9.34268 × 10-6 -4.93309 × 10-5 0

-1.34111 × 10-5 -8.56588 × 10-5 0

1.08193 × 10-5 -4.70699 × 10-5 0

-6.63501 × 10-5 -2.14252 × 10-5 0

5.98858 × 10-5 -4.91143 × 10-5 0

6.72862 × 10-5 -4.36451 × 10-5 0

9.60710 × 10-5 -8.18031 × 10-5 0

1.49860 × 10-4 -1.71134 × 10-4 0

1.21502 × 10-4 -9.84766 × 10-5 0

1.64280 × 10-4 -1.34908 × 10-4 0

3.49668 × 10-5 -1.73176 × 10-5 0 -8.52992 × 10-10 -3.58009 × 10-10 0

3.88040 × 10-5 -1.38524 × 10-5 0 7.68043 × 10-10 -1.32048 × 10-10 0

7.68800 × 10-5 -4.18871 × 10-5 0 -8.49309 × 10-9 -4.91415 × 10-10 0

3.53018 × 10-7 -1.93040 × 10-5 0 -3.63389 × 10-9 -7.73717 × 10-10 0

-5.08307 × 10-5 -1.07000 × 10-5 0 -2.10025 × 10-10 -1.56583 × 10-9 0

7.25571 × 10-5 -3.12153 × 10-6 0 1.45984 × 10-9 -8.15150 × 10-10 0

1.60099 × 10-8 8.50221 ×10-10 0

9.15407 × 10-9 4.13028 × 10-10 0

2.08795 × 10-8 9.21785 × 10-10 0

3.70980 × 10-8 2.07659 × 10-9 0

1.98521 × 10-8 2.05972 × 10-9 0

2.59524 × 10-8 1.69362 × 10-9 0

C2

-3.55631 × 10-7 2.80326 × 10-7 0

-1.45842 × 10-7 2.39764 × 10-7 0

-4.05944 × 10-7 1.31731 × 10-7 0

-1.12496 × 10-7 7.66906 × 10-8 0

-1.18487 × 10-7 1.69571 × 10-7 0

∆T (K)

100-500

max p (MPa) AAD (%) bias (%) max dev (%) (T/Tc, p/pc)

138

100500 55

110500 55

145500 68

0.83 0.15 6.02

0.68 0.17 4.50

1.18 0.24 6.59

(0.80, 0.20)

(1.00, 0.40)

(0.90, 0.40)

a1 a2 bo b1 b2 co c1 c2 Ao A1 A2 Bo B1 B2 Co C1

a

ethane

propane

butane

octane

HFC134a

3.76297 × 10-5 0

8.68736 × 10-5 0

-4.40242 × 10-5 1.38068 × 10-4 0

-2.69591 × 10-5 1.46267 × 10-4 0

1.07271 × 10-4 -4.41655 × 10-5 0

-9.11096 × 10-5 9.93871 × 10-5 -6.36533 × 10-6 0 -3.76786 × 10-9 1.92500 × 10-9 0

-5.44584 × 10-5 1.28673 × 10-4 -1.76442 × 10-5 0 -2.40884 × 10-9 5.20715 × 10-11 0

0

0

-2.29226 × 10-6 1.18011 × 10-6 0

9.75463 × 10-9 2.71874 × 10-9 -1.24466 × 10-6 8.83261 × 10-7 0

6.62141 × 10-9 1.60012 × 10-9 -9.50545 × 10-7 1.03767 × 10-6 0

300550K 151 MPa

270550K 151 MPa

300550K 150 MPa

0.80 -0.14 5.00

0.65% -0.12% 4.13%

1.30% -0.41% 7.54%

(0.90, 0.40)

(0.90, 0.40)

(1.08, 1.79)

ai, bi, and ci values are in units of (mPa‚s)/bar. Ai, Bi, and Ci values are in units of (mPa‚s)/bar2.

heptane

H2O (a)

H2O (b)

1.19805 × 10-4 -1.25861 × 10-4 5.48871 × 10-5 3.15921 × 10-5 -2.60469 × 10-5 7.09199 × 10-6 1.80689 × 10-5 -7.41742 × 10-6 0 -2.31066 × 10-9 0

-1.17407 × 10-5 -3.78855 × 10-7 3.56743 × 10-8 1.62216 × 10-6 -8.36595 × 10-6 9.10863 × 10-8 1.92707 × 10-5 -1.28680 × 10-5 0 -3.30145 × 10-10 0

-8.42696 × 10-6 7.53418 × 10-8 4.70754 × 10-8 3.25301 × 10-6 -7.53004 × 10-6 6.08039 × 10-8 1.90150 × 10-5 -1.14044 × 10-5 0 -1.88986 × 10-10 0

5.42486 × 10-10 1.04558 × 10-8 -2.20758 × 10-9 0

1.02931 × 10-11 5.03140 × 10-10 1.82304 × 10-10 0

7.96901 × 10-12 3.00926 × 10-10 1.60797 × 10-10 0

280600K 149 MPa

4.81769 × 10-7 -1.17149 × 10-7 200425K 100

1.03255 × 10-6 -8.56207 × 10-7 3.84384 × 10-7 2001000K 100

8.01449 × 10-10 5.65614 × 10-9 1.10163 × 10-10 2731100K 100

2.41284 × 10-9 5.51425 × 10-9 1.03767 × 10-10 2731100K 200

0.72% 0.14% 4.90%

0.92% 0.12% 5.36%

1.81 -0.07 7.74

0.21 0.03 1.49

0.33 0.00 2.61

0.39 0.09 2.34

0.34 0.06 2.17

(1.00, 0.40)

(0.9, 0.40)

(0.80, 0.24)

(1.76, 9.49)

(1.35, 2.49)

(1.35, 2.49)

(1.12, 2.37)

1.66457 × 10-4 -4.80293 × 10-5 0 8.08333 × 10-5 -4.90360 × 10-5 0 -2.12476 × 10-8 2.81647 × 10-9 0 1.35594 × 10-8 0 3.17550 × 10-10 0

CO2

CO2 (PT-CO) -1.38364 × 10-4 1.78519 × 10-4 -7.77243 × 10-5 -6.96576 × 10-6 2.18138 × 10-5 6.78040 × 10-6 2.30314 × 10-5 -8.00196 × 10-6 0 0 0 0 0 0 0 -6.39590 × 10-7 1.06222 × 10-6 -5.27125 × 10-7 2001000K 20


pentane

-3.12118 × 10-5 1.99422 ×10-7 0

ao


TABLE 2: General Friction Theory Model Parametersa



Figure 9. Methane viscosity deviation diagram for the FT short SpanWagner model compared to the original experimental data. (+), Deviations above 2%, (O), deviations between (2%, (-), deviations under -2%.

Figure 7. Performance of the general FT viscosity model for propane based on the short Span-Wagner EoS.36 (s), FT model, (b), smoothed viscosity data by Ze´berg-Mikkelsen.38

Figure 10. Performance of the general FT viscosity model for HFC134a based on the short Span-Wagner EoS for polar fluids.37 (s), FT model, (b), experimental data as listed in Comunãs et al.10

Figure 8. Viscosity of propane at high pressures (beyond the pressure range used for the regression). (s), FT model predictions, (b), recommended viscosity of Vogel et al.34

are overestimated by more than 2%, while the remaining 14.7% are underestimated by less than 2%. These results clearly show a nonbias even performance of the model when tested against such a diverse range of measurements. Results of similar quality are also found for the remaining hydrocarbons reported in Table 2. HFC-134a. In the case of the 1,1,1,2-tetrafluoroethane (HFC134a), the data used in the regression is the same database as previously reported in the work of Comunãs et al.,10 i.e., the regression has been directly done against experimental data. Although the HFC-134a FT model reported in the original work of Comunãs et al.39 is already based on a noncubic EoS, the generalized FT model of this work delivers a better performance as a direct consequence of (1) the use of the more accurate SW EoS and (2) the improved performance of the general FT model in the transitional area from the dilute gas limit to the dense state. Figure 10 illustrates the model performance in comparison with the data used in the regression.

Carbon Dioxide. The application of the general FT approach to the reference models for CO2 and water constitutes the two main examples illustrating the extent of the FT capabilities. In the case of CO2, the model has been regressed against the recommended viscosity data by Fenghour et al.33 The regression has been done by considering data only in a temperature range from the CO2 triple point up to 1000 K and pressures up to 100 MPa; the results are depicted in Figure 11 and the performance statistics reported in Table 2. Clearly, because in this case smoothed recommended data were used along with one of the most accurate reference EoS for CO2, the FT model results are of equally consistent quality. Furthermore, the original recommended data by Fenghour et al. actually covers pressures up to 300 MPa and a temperature range up to 1500 K. Although such extended rages were not used in the model regression, the application of the model for the entire range delivers an overall AAD and bias of only 0.53 and 0.10%, with a max dev of 5.34% at 800 K and 300 MPa. Figure 12 illustrates the extrapolation capabilities of the model up to 300 MPa, showing an excellent agreement in comparison with the recommended data for pressures three times above the range used in the regression. Water. In the case of water, the model has been regressed against the water critically evaluated experimental data presented in the 2003 IAPWS release.28 The recommended water data


Figure 11. Performance of the general FT viscosity model for CO2 based on the Span-Wagner reference EoS.12 (s), FT model, (b), recommended data of Fenghour et al.33

Figure 12. Extrapolation (from 100 to 300 MPa) of the general FT viscosity model for CO2 based on the Span-Wagner reference EoS.12 (s), FT model, (b), recommended data of Fenghour et al.33

covers a temperature range from 0 to 800 °C and pressures up to 100 MPa. The water model results are shown in Figure 13, and the corresponding substance constants and performing statistics reported under the “H2O (a)” column of Table 2. In all cases, without exception, the model reproduces every single recommended point within the reported uncertainty. In fact, on average, the model reproduces all of the IAPWS-recommended experimental data with a deviation seven times lower than the reported uncertainty. Here, it is most interesting to remark the sharp qualitative difference in the low-temperature viscosity behavior of the CO2 (Figure 11) in comparison with H2O (Figure 13). CO2 can be considered to show a “normal” viscosity behavior at low temperatures and elevated pressures; i.e., a sharp viscosity increase with pressure and with decreasing temperature. In contrast, water does not follow this pattern; its response to decreasing temperature is not as sharp as expected, and its response to increasing pressure is not monotonically increasing but, at low temperature, the water viscosity passes with increasing pressure through a minimum before it starts increasing again. The described anomalous behavior is matched by the FT, not by allowing unusual values of the friction parameters (the parameters of the CO2 and water behave in a qualitatively similar way), but rather by the physics already built into the


Figure 13. Performance of the general FT viscosity model for water based on the Wagner-Pruss reference EoS.13 (s), FT model, (b), IAPWS recommended viscosity data.28

Pruss-Wagner EoS used in the FT model (see Section 3). The unusual behavior depicted in Figure 4, which causes the wellknown density anomaly of water at low-temperatures, abolishes the usual dominance of the repulsive forces at low temperatures. As a result, in terms of the FT formulation, anomalous viscosity behavior develops; this is a direct mathematical consequence of the sort of behavior shown by the competing pa and pr terms depicted in Figure 4. Additionally, we should remark that, in the 2003 IAPWS release, an interpolating equation for the calculation of the water viscosity is also presented. With respect to the recommended IAPWS equation, there is an excellent agreement between the IAPWS equation and the FT H2O (a) model for the entire range of temperature and pressures implied in the data used in the regression (between 0 and 800 °C and pressures up to 100 MPa). However, in the low-temperature region, as the FT model is extrapolated for pressures above 100 MPa, a discrepancy between the FT model and the IAPWS equation develops, which reaches 3.3% at 0 °C and 300 MPa. This discrepancy is also a direct consequence of the low-temperature water anomalous behavior as, although the physics is reproduced correctly, it is necessary to anchor the location of the minimum in the viscosity in order to achieve the high accuracy sought in the models. In spite of the fact that we remain uncertain where such a minimum is actually located, the discrepancy between models can be easily corrected by adding further data to the regression database in the range between 100 and 300 MPa for isotherms between 0 and 100 °C. Regressing the general FT model against this extended database results in the model parameters and statistics listed under the H2O (b) column of Table 2. For all practical purposes, the H2O (b) FT model is in full agreement with both the IAPWS-recommended experimental data and the IAPWSrecommended interpolating equation. The performance difference between the different models is shown in Figure 14 for the low-temperature region, also depicting in detail the water anomalous behavior. Finally, to reassure the initial-density FT model performance, a comparison with the recent low-density water vapor data of Teske et al.31 was carried out. Figure 15 shows all of the water vapor isochors reported by Teske et al. along with isochoric curves obtained from the FT H2O (a) model. Although their



Figure 14. General FT Wagner-Pruss reference EoS13 models performance: low-temperature extrapolation and prediction of the water anomaly. (s), FT H2O (a) model, (- - -), FT H2O (b) model, (b), IAPWS interpolating equation for water.28

Figure 16. Reduced second viscosity virial coefficient of water. NA, Avogadro’s number, kB, Boltzmann’s constant, σ ) 0.48873 nm, and /kB ) 459.85 K (from Teske et al.31). (- - -), IAPWS interpolating equation for water.26 (‚‚‚), Vogel et al.34 model (as implemented in Teske et al.31), (s), FT H2O (a) model.

such as water has, having the wrong initial slope will result in substantial discrepancies, particularly at temperatures close to the critical one. 5. Critical Region

Figure 15. Low-density water vapor performance of the general FT viscosity model. (s), FT H2O (a) model, (b), Teske et al.31 water vapor viscosity data.

data were not used for the fitting of the FT H2O (a) model parameters, they are reproduced with an AAD of only 0.18%. In their work, Teske et al. also reported semiempirical models for the water dilute gas limit and initial density dependence. Their dilute-gas limit model yields practically the same results as the model used in this work, and both dilute-gas limit models can be interchanged without affecting the results. The initialdensity water model reported by Teske et al. is based on the empirical equation for the second viscosity virial coefficient (Bη) proposed by Vogel et al.;34 Bη is defined as

Bη )

1 (∂η/∂F)F)0 η0

(39)

and in terms of the FT formulation given by:

Bη )

RT κ η0 i

(40)

The Teske et al. model, however, tends to over predict Bη as the temperature approaches the critical temperature, as shown in Figure 16, where the Bη data calculated for the Teske et al. model, the IAPWS-recommended viscosity interpolating equation,28 and the FT H2O (a) model of this work are compared; there is an excellent agreement between the IAPWS interpolating equation and the FT model, particularly close and above the critical temperature (kBTc/ ) 1.407), where the water dilute gas limit is approached with a small negative slope (also shown in Figure 6). Therefore, in a fluid with a high critical pressure

Of theoretical relevance is the study of the FT performance in the neighborhood of the critical point. Here, a distinction should be made right away between a classical model, even a highly accurate one as the reference CO2 and water models used in the previous sections, and those models that attempt to incorporate the power laws that follow from renormalization theory such as the renormalized GCC EoS proposed by Kieselev and Ely.14 Along this line, it is particularly interesting to study the reproduction or plausible prediction of any sort of critical viscosity anomaly, as implied, e.g., by the microgravity nearcritical measurements by Berg and Moldover.40 However, it must be remarked that it is not the purpose of this work to engage in a theoretical discussion regarding the nature of what appears to be a critical anomaly on the viscosity. The objective of this section is merely the application of the FT approach to a nonclassical EoS and to simply find out what sort of viscosity behavior the approach predicts. Of course, the nature of the results to be presented in this section clearly constitute the preamble for future works’ focus on the near-critical viscosity behavior. For the analysis presented in this work, it is relevant to make a formal distinction between a “critical anomaly” and a “critical singularity”. Let us then define a critical anomaly as a smooth phenomenon that may follow from competitive long-range interactions. For instance, if we consider the pressure versus density performance along the critical isotherm, it is well-known that, as the critical point is approached, the long-range terms become more and more relevant, inducing an anomalously enhanced flattening of the critical isotherm. Therefore, if the pressure-density critical isotherm is then modeled with a function that is intrinsically slow-convergent at the critical point, i.e., requiring a high-order series expansion to achieve convergence (such as the exponential functions), it is a mathematical fact that the isotherm can be flattened as much as necessary. That is, for practical purposes, the critical isotherm can be represented to any desired degree of accuracy by a sufficiently slow-convergent analytic function (although no analytic function will ever yield the correct critical exponent). In contrast to this, “critical singularities”, i.e., divergent nonclassical phenomena of some thermodynamic properties, cannot be modeled by


Quinõnes-Cisneros and Deiters Ely in the GCC EoS.14 From eq 41, it follows that, along the critical isochore, a weak divergence of ∂psg/∂T is found according to

∂psg f Cic∆T -R/2 ∂T

(42)

where the proportionality constant is given by

Cic )

5(2/3)1-R/2 Gi-1+R/2RTcR/2(V0c - Vc)(R - 2) 7(b - V0c)Vc

(43)

Figure 17. Critical enhancement predicted by the general FT viscosity model based on the short Span-Wagner EoS36 for propane.

Consistently, along the critical isotherm, the following singular behavior is found:

analytical models at all; an example is the λ-type divergence of the isochoric heat capacity. In the case of the viscosity-density near-critical behavior, it is well-known that some fluids develop a clear (as opposed to weak) near-critical smooth viscosity enhancement. In terms of the FT, this appears to be associated to the enhanced flattening of the pressure-density critical isotherm. Such enhanced flatness follows from an increased competitiveness in the balance between the pa and the pr terms. Consequently, the introduction of the friction coefficients in front of the pa and the pr terms will necessarily break such balance, and in some cases, this may result in a natural viscosity enhancement of the near-critical region. One such example is the FT SW propane model reported in Table 2, which naturally develops a clearly visible nearcritical viscosity enhancement, as depicted in Figure 17. In fact, we have even found and successfully modeled this type of viscosity enhancement after shear viscosity simulation results for the Lennard-Jones 12-6 fluid based on the FT and an analytical EoS.41,42 Consequently, the modeling of this type of behavior simply requires a more detailed correlation of the nearcritical region and, as far as we can tell, does not require using the renormalized critical phenomenon theory (the phenomenon is clearly linked to fluctuations). However, while the viscosity enhancement previously discussed does not reveal any major modeling problems, nearcritical measurements such as those by Berg and Moldover40 appear to indicate the existence of a weak “critical singularity”, a phenomenon that cannot be possibly modeled with any one of the classic EoS so far studied. To study this singularity, as outlined in the Appendix, we have combined the FT with the GCC EoS. For this, we have extracted the leading lowest-order terms of the GCC Helmholtz energy and used them to calculate the pressure contribution, psg, and its temperature derivative,

∂psg f Cit|∆V|-R/2β ∂T

∂psg q2-RRTc(V0c - Vc)(R - 2) × ) ∂T (b - V0c) (1359d1Gi2q4Tc(1 + V1)∆V + 250Gi2βm02q4βVc(Gi q2Tc + ∆T) (1359Gi2q4Tc(1 + V1)(2β - 1)∆V(Tc(1 + V1)∆V + 2d1Vc∆T) + 250Gi2βm02q4βVc2(3Gi2q4Tc2 + ∆T2) (41) where b is the van der Waals co-volume, ∆T ) T - Tc and ∆v ) V - Vc; q and V0c are the renormalized distance to the critical point and the classical critical molar volume; R ) 0.11, β ) 0.325, γ ) 2 - 2β - R ) 1.24, and ∆1 ) 0.51 are the universal nonclassical critical exponents, and all remaining parameters appear in the crossover sine model for q used by Kiselev and

(44)

with

Cit )

(250/453)R/4βm0R/2βR(V0c - Vc)(R - 2) 6βGi1-R/2(b - V0c)(1 + V1)R/2βVc1-R/2β

(45)

According to eqs 21 and 22, psg may be separated into the following attractive and repulsive contributions:

pa,sg ) -T(∂psg/∂T)V

(46)

pr,sg ) T(∂psg/∂T)V

(47)

and

where the psg term in eq 47 that may be expected from eq 22 has been neglected because, as the critical point is approached, psg vanishes while T(∂psg/∂T)V diverges. Next, by applying the FT approach, it is straightforward to deduce that the terms related to psg, eqs 46 and 47, predict a linear FT viscosity contribution of the form

ηsg ) (κr - κa)T(∂psg/∂T)V

(48)

Therefore, from eq 48, it follows that, as the critical point is approached along the critical isochore, the linear FT viscosity contribution will weakly diverge with the power of -R/2 (≈ -0.055). Clearly, given the singular nature of the pressure contribution implied in eqs 46 and 47, if high-order corrections beyond the FT linear term given in eq 48 are required, this cannot be in the form of the power series proposed in eq 9, as it will imply a divergent (rather than convergent) series expansion. However, a more general FT formulation for a renormalized type of EoS should be addressed in an independent work, and at this point, we will simply restrict the model to a linear formulation of the singular term, which appears to be consistent with the experimental evidence. For illustration purposes, we have combined the FT approach with the GCC EoS (in its full Patel-Teja form) for the viscosity modeling of CO2. However, it should be remarked that the current version of the GCC EoS14 develops thermodynamic inconsistencies at high pressure (above 50 MPa), which the authors of the model will address in a forthcoming publication.43 Therefore, at present, the GCC model cannot be applied at such conditions and consequently had to be restricted to a maximum pressure of 20 MPa (the model parameters were actually correlated for pressures below 10 MPa). Nevertheless, as the


Figure 18. Log-log plot of the predicted viscosity singular term, eq 48, along the critical isochore based on the CO2 FT GCC model. The log(ηsg) vs log(t) function asymptotically convergences to a line with slope -R/2 ) -0.055.

emphasis in this section is on the critical region, the highpressure problems are irrelevant. Additionally, to avoid potential problems with the high-order terms, we have carried out the FT GCC viscosity modeling of CO2 by simply eliminating the quadratic terms for pa and ∆pr from eq 26, which actually are not necessary for the considered reduced pressure range. Thus, all of the model parameters and deviation statistics that follow from the regression of the FT GCC model against the Fenghour et al.33 CO2 data are reported in Table 2. Clearly, at least for the considered range of application, the derived FT GCC CO2 viscosity model performs with a highly competitive accuracy. Within the considered range of application (up to 20 MPa), a plot of the FT GCC CO2 viscosity model will hardly reveal any difference in comparison to Figure 11. However, it is straightforward to deduce that the terms related to psg will necessarily predict a weak viscosity divergent contribution according to eq 48. The actual near-critical ηsg viscosity contribution, as predicted by the FT GCC CO2 model, is depicted in Figure 18 in units of µPa s (103 times smaller units than Figure 11). However, when our Figure 18 is compared to Figure 1 of the Berg and Moldover manuscript,40 the qualitative and order-of-magnitude agreement is striking; more so considering that no near-critical viscosity data has been used in the correlation of the FT GCC CO2 viscosity model. Even more remarkable is, given all involved uncertainties, the close agreement between Berg and Moldover’s estimated exponent of -0.0435 with the prediction of this work, -R/2 ) -0.055. 5. Conclusions This work presents an important step forward for the FT, as the approach is now extended to virtually all types of EoS. This extension is achieved by the introduction of a rational method for splitting a given EoS into separated predominantly repulsive and attractive terms. This method takes advantage of the internal pressure concept, a thermodynamic relation commonly linked to intermolecular cohesion. Although the internal pressure is a rather seldomly used thermodynamic function, it is still a commonly cited property, and the results obtained in this work strongly support the use of the internal pressure as a pragmatic tool for the separation of the total pressure into predominantly attractive and repulsive terms. In fact, this separation turns out to be not only a useful mathematical scheme, but also as a powerful tool for the revelation of important physical properties otherwise hidden in the total pressure. With the new rational splitting method, the FT can now be applied to highly accurate reference EoS, theoretically oriented EoS such as those based on the renormalization critical theory,

J. Phys. Chem. B, Vol. 110, No. 25, 2006 12831 or models for specific applications (e.g., polymer blends, for which EoS like the PC-SAFT44 are intended). Clearly, this is only the starting point for many potential future ramifications and improvements over previous work, as it may constitute a plausible improvement (particularly in prediction capabilities) of the previously developed one-parameter models45 that have been successfully used in applications related to reservoir fluids.16 Moreover, the combination of the generalized FT approach of this work with highly accurate reference EoS has been shown to yield results of an accuracy consistent with that expected from reference models, thereby readily extending the multiproperty capabilities of some of the reference EoS to the dynamic viscosity. One clear example is the FT model that has been presented for the viscosity water, which not only is highly accurate but also, because of the physics already built in the EoS, yields a consistent prediction of important low-temperature viscosity anomalies. To emphasize the relevance of these results, a direct comparison with a FT viscosity model based on another highly accurate reference EoS for a “normal” fluid (CO2) has also been presented. Previous experience with the FT modeling of the dynamic viscosity of numerous systems already hinted at the physical relevance of the Amontons-Coulomb friction law. Consistently, it has been observed that, even with fluids as complex as reservoir fluids, the better the EoS is, the more reliable the derived FT viscosity model turns out to be. Therefore, an interesting and necessary test for the approach is applying it to a renormalized EoS and finding out what type of viscosity behavior the approach may predict for the critical region. Although this is a complex topic that should be independently addressed, the initial results obtained here, based on a renormalized crossover type of EoS (the GCC EoS), have substantially surpassed the authors’ expectations. The remarkable agreement found between the predicted critical behavior of the dynamic viscosity with such relevant results as the Berg and Moldover microgravity measurements is striking. Particularly, the fact that, to the extent of the knowledge of the authors, this approach is the only one that has so far been able to predict the much-discussed weak viscosity divergence at the critical point, again a result that directly follows from the physics already built into the EoS. Pending future work specifically focused at the critical region, these preliminary results are perhaps the most important supporting arguments in favor of the high relevance of the internal pressure as a fundamental tool for the proper rational separation of the total pressure. Clearly, the combination of this phenomenological approach with more fundamental results from statistical physics in order to achieve a better understanding of the physics behind the FT is a research objective of further interest. Especially important might be the transitional region from the dilute gas limit to the dense state, as there already exist a substantial fundamental understanding of viscosity24,30 at this level. Therefore, the current theoretical knowledge can already provide the theoretical background for the further development of semiempirical predictive models for at least up to the first-order viscosity term (the initial-density viscosity term). This may lead to an accurate semitheoretical integral approach describing the transition from the dilute gas limit to the dense state where the FT can provide an appropriate description. Acknowledgment. To a large extent, this work addresses many commentaries and/or observations of several colleagues in recent years. As such, we would like to particularly acknowledge Prof. Josefa Fernańdez for all of her relevant



observations and Prof. Sergei B. Kiselev for all of his help in the implementation of the GCC EoS. The authors would also like to acknowledge funding from the Fonds of the German Chemical Industry.

pj0(T) )

p ) pR + p A

(A.6)

µ(V0c, T) RT

(A.7)

and

Appendix The generalized cubic crossover (GCC) EoS by Kiselev and Ely14 is the model on which our preliminary analysis of the critical region has been based. Consequently, it is of relevance to outline the steps in the implementation and simplification of the model leading to eq 39. However, for more detailed information, the interested reader should refer to the original publications of the crossover model.14,46 1. The Basic Cubic EoS. Let us start by considering a general cubic type of EoS of the form

p(V0c, T)V0c RT

µ j 0(T) )

the residual dimensionless Helmholtz energy is calculated according to

∆aj )

aˆ V j 0(T) + pj (T) - µ RT V0c 0

(A.8)

and from here, we substitute V f (∆V + 1)Vc and T f (∆T + 1)Tc to obtain

(A.1)

∆ajc(∆V, ∆T) ) ∆aj((∆V + 1)Vc, (∆T + 1)Tc) (A.9)

where the capital letters “R” and “A” have been used to differentiate the classical van der Waals (vdW) repulsive pressure term

Next, the classical values for ∆V and ∆T are replaced by the following renormalized values

∆V ) φΓ(γ-2β)/(4∆1) + (1 + φ)∆VcΓ(2-R)/(2∆1) (A.10)

RT pR ) V-b

(A.2)

and

∆T ) τΓ-R/(2∆1)

and the EoS-specific attractive term

pA )

a(T) 1 buV + V + b2(u2 - ∆2) 4

(A.3)

2

in order to obtain ∆ajc ) ∆ajc(φ, τ). In eq A.11, Γ ) Γ(φ, τ) denotes a simple crossover function given by

Γ) from the internal-pressure-based definitions for the attractive and repulsive pressure terms proposed in this work. From eq A.3, with u ) ∆ ) 0 w, the fundamental van der Waals EoS26 is recovered with u ) 1, ∆ ) 1, the Soave5 modification of the Redlich and Kwong47 (SRK) EoS follows, with u ) 2, ∆ ) 2 x2 the Peng and Robinson6 (PR) EoS is obtained, and with u ) 1 + c/b, ∆ ) x1+6c/b+(c/b) the Patel and (PT) EoS is also obtained. Then, according to this formulation, the general expression for the molar Helmholtz energy is given by 2

aˆ ) RT ln

[1V] - RT ln[1 - bV] -

a(T) ln

[

µ ) aˆ + pV

K)

(A.5)

2. The Generalized Cubic Crossover EoS. As we do not intend to repeat the work of Kiselev and Ely, here we will just outline the steps we followed in the implementation of the GCC EoS solely as a background for the simplifications leading to eq 41. First, to calculate the dimensionless critical part of the Helmholtz energy, ∆ajc, from

2∆1

(A.12)

τ2 (a (Γ-R/∆1 - 1) + a21(Γ-(R-∆1)/∆1 - 1)) 2 20

(A.13)

of φ andτ by

]

where the first two terms on the right-hand side of eq A.4 are related to pR, the third term to pA, and f+(T) is a residual function of T of relevance only for the caloric properties. Also, to recover the molar Helmholtz energy for the fundamental van der Waals EoS, it is pointed out that L’Hopital’s rule should be applied to the third term in eq A.4. From eqs A.3 and A.4, the molar Gibbs free energy readily follows:

(1 +q q)

where q is a renormalized distance to the critical point. Then, considering the back-substitution in ∆ajc(φ, τ) and in the kernel

Teja48

2V + b(u + ∆) 2V + b(u - ∆) + b∆ f + (T) (A.4)

(A.11)

φf

V -1 Vc

(A.14)

τf

T -1 Tc

(A.15)

and

the crossover molar Helmholtz energy is given by

(

aˆ ) RT ∆ajc(V, T) - K(V, T) -

V pj (T) + µ j 0(T) V0c 0

)

(A.16)

For the renormalized distance, q is calculated from a solution to the crossover sine model proposed by Kiselev and Ely14

( )[ q2 -

(

)]

τ p2 τ 1- 2 1- 2 Gi 4b q Gi

(

) b2

)

(1 + V1 exp[-10φ])φ + d1τ m0Giβ

2

Γ(1-2β)/∆1 (A.17)

where, in addition to p2 ) b2 ) 1.359, all other relevant



parameters are properly defined in the original work by Kiselev and Ely14 to which the interested reader is referred. From eq A.16 the pressure can be derived, and therefore, the general FT approach can be readily implemented. 3. The Neighborhood of the Critical Point. For the analysis of the near-critical GCC EoS-based FT model, only the leading (lowest-order term) contributions given by the crossover model need to be considered. Also, it is convenient to separate such contributions into the ones that follow from the van der Waals attractive term, eq A.3, and the van der Waals repulsive term, eq A.2. 3.1 The van der Waals Attractive Contribution. After applying the generalized crossover approach to the third term of eq A.4, we proceed by expanding all transcendental terms in the resulting expression, simplifying and deleting all but the lowest-order terms. This procedure gives as a result the following vdW attractive term for the neighborhood of the critical point:

Ra20q-2R∆T2 + 2Tc

acVc -ac 1 buV0c + V0c2 + b2(u2 - ∆2) 4 V0c 2V0c + b(u + ∆) 1 + ln b∆ 2V0c + b(u - ∆) 1 buV0c + V0c2 + b2(u2 - ∆2) 4 (A.18)

aˆ A ) -

(

])

[

where ac ) a(Tc). From this, the pressure contribution can be derived as well as its partial derivative in the vicinity of the critical point:

(∂pA/∂T)V ) -

5.436Ra20d1Gi2-2β(1 + V1)Rq4-2R-4β∆T2 m02Vc(3Gi2Tc2q4 + ∆T2)

(A.19)

This term vanishes as the critical point is approached and is, therefore, not of further interest. 3.2 The van der Waals Repulsive Contribution. A similar analysis shows that, in the neighborhood of the critical point, the contribution from the classical vdW repulsive term to the crossover model approaches

aˆ R )

RTc(V0c - Vc)(bVc + V0c∆V)q2-R

- RTc

(

V0c

+ (b - V0c)V0cVc (b + V0c) R(Tc + ∆T)(Vc + ∆V - (b - V0c) ln[V0c]) b + ln 1 V0c (b - V0c) (A.20)

[

])

where, essentially, as we carry out the required volume and temperature derivatives, all terms drop out except for the first one, which we denote by aˆ sg:

aˆ sg )

RTc(V0c - Vc)(bVc + V0c∆V)q2-R (b - V0c)V0cVc

(A.21)

From eq A.21, the leading term of the pressure in the vicinity of the critical point is obtained as

psg ) -

RTc(V0c - Vc)q2-R

and from this, eq 41 follows.

(b - V0c)Vc

(A.22)

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