High-pressure dynamic viscosity and density of two synthetic hydrocarbon mixtures representative of some heavy petroleum distillation cuts

Fluid Phase Equilibria 212 (2003) 143–164

High-pressure dynamic viscosity and density of two synthetic hydrocarbon mixtures representative of some heavy petroleum distillation cuts C. Boned∗ , C.K. Zéberg-Mikkelsen, A. Baylaucq, P. Daugé Laboratoire des Fluides Complexes, UMR CNRS Faculté des Sciences et Techniques, Avenue de l’ Université, BP 1155, 64013 Pau Cedex, France

Abstract The dynamic viscosity and density of two hydrocarbon mixtures representative of some heavy petroleum distillation cuts at 515 K have been studied in the temperature range 293.15–353.15 K and up to 100 MPa. The pure hydrocarbons used are n-tridecane, 2,2,4,4,6,8,8-heptamethylnonane, heptylcyclohexane, heptylbenzene, and 1-methylnaphthalene. The normal boiling point of these five compounds are 508.0, 513.1, 518.1, 519.1 and 515.0 K, respectively. The studied mixtures contain respectively three hydrocarbons (n-tridecane, heptylcyclohexane, heptylbenzene) and five hydrocarbons. The viscosity was measured with a falling-body viscometer, except at atmospheric pressure, where a classical capillary viscometer was used. The experimental uncertainty for the dynamic viscosity is 2%, except at atmospheric pressure, where the uncertainty is 1%. For the density, the uncertainty is less than 1 kg m−3 . The viscosity data obtained for the two mixtures (84 experimental points) have been used to evaluate the performance of seven different representative models, applicable to hydrocarbon fluids, incorporating the effects of temperature, pressure and composition. The evaluated models are based on the classical mixing laws, the self-referencing model, the hard-sphere scheme, the free-volume viscosity model, the friction theory, and the Lohrenz–Bray–Clark (LBC) correlation. It follows from the discussion that some of the schemes are able to predict the viscosity of these two mixtures being simple representations of some petroleum distillation cuts at 515 K. This work shows the potential extension of these viscosity approaches to real petroleum fluids. In addition, using the experimental densities, the variation of the internal pressure versus temperature and pressure for the two synthetic hydrocarbon mixtures has been evaluated. © 2003 Elsevier B.V. All rights reserved. Keywords: Density; High-pressure; Measurement; Modeling; Petroleum cut; Viscosity

∗

Corresponding author. Tel.: +33-5-59-40-76-88; fax: +33-5-59-40-76-95. E-mail address: [email protected] (C. Boned). 0378-3812/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0378-3812(03)00279-6

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1. Introduction Despite the importance petroleum fluids have had for many years and the tremendous number of works already completed in relation with the oil industry, there is still a lack of studies concerning the viscosity versus temperature and pressure of some of the components in petroleum fluids, in particular of the heavy components. Since the viscosity is an important property required within many engineering disciplines ranging from the design of transport equipments to simulation of petroleum reservoirs, reliable and accurate viscosity models applicable to wide ranges of temperature, pressure, and composition are required. The compositional dependent viscosity models, which have been developed, have only been derived based on pure hydrocarbons and binary hydrocarbon mixtures. However, it should be stressed that petroleum and reservoir fluids are generally not suitable in order to derive these kind of models, due to the lack of basic compositional information. This is the reason why studies of the viscosity versus temperature and pressure are required for well-defined multicomponent mixtures. Although these systems are only simple representations of real reservoir fluids, they can be used to evaluate the performance of viscosity models for the potential extension to real reservoir fluids and the application within the oil industry. In this work, a study of the dynamic viscosity, η, and the density, ρ, of two synthetic hydrocarbon mixtures containing three and five components have been carried out at temperatures, T, from 293.15 to 353.15 K and for pressures, p, up to 100 MPa, in order to simulate some heavy petroleum distillation cuts at 515 K. The three chemical species which intervene most abundantly in the constitution of oils and petroleum fluids are the paraffins, the aromatics, and the naphthenes. Their simultaneous participation is essential in order to simulate real petroleum distillation cuts by simplified mixtures. Further, it is necessary that each of the chosen compounds has its normal boiling point as close as possible to the boiling point of the real cut in order to consider the synthetic system as a simple representation of a real distillation cut. Following a discussion with Total in Pau (France) it appeared reasonable for some heavy petroleum distillation cuts to admit a mass distribution of about 40% paraffins, 35% naphthenes, and 25% aromatics. This distribution results from an average of compositions evaluated on a broad crude oil interval. Based on this evaluation and in order to study some simple mixtures containing three and five components, the following pure hydrocarbons have been chosen: two alkanes (n-tridecane C13 H28 , boiling point Tb = 508 K and 2,2,4,4,6,8,8-heptamethylnonane C16 H34 , Tb = 513.1 K), one naphthene (heptylcyclohexane C13 H26 , Tb = 518.1 K) and two aromatics (heptylbenzene C13 H20 , Tb = 519.1 K and 1-methylnaphthalene which contains two aromatic rings C11 H10 , Tb = 515 K). The studied ternary mixture was composed of n-tridecane + heptylcyclohexane + heptylbenzene with the mole fractions 0.3940, 0.3485 and 0.2575, respectively (i.e. the weight fractions are 0.40, 0.35 and 0.25), whereas the quinary system was composed of n-tridecane + 2,2,4,4,6,8,8-heptamethylnonane + heptylcyclohexane + heptylbenzene + 1-methylnaphthalene with the mole fractions 0.1994, 0.1623, 0.3527, 0.1564 and 0.1292, respectively (i.e. the weight fractions are 0.20, 0.20, 0.35, 0.15 and 0.10). The study of the dynamic viscosity and the density of the involved pure compounds have previously been performed [1–3] in the same temperature and pressure ranges. The measured viscosity and density data for the two mixtures have been used in an evaluation of the performance of seven representative models, applicable to hydrocarbon fluids, incorporating the effects of temperature, pressure and composition. The evaluated models range from simple mixing laws through empirical correlations to models with a physical and theoretical background. The considered models are the mixing laws of Grunberg and Nissan [4] and Katti and Chaudhri [5], the self-referencing model [6], the hard-sphere scheme [7,8], the free-volume model [9,10], the friction theory [11,12] and the

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Lohrenz–Bray–Clark (LBC) correlation [13], which is widely used in petroleum engineering. These viscosity approaches are conceptually very different. In addition, it should be mentioned that the knowledge of the density makes it possible to calculate by numerical analysis, quantities such as the isothermal compressibility factor κT = 1/ρ(∂ρ/∂p)T , the isobaric thermal expansion coefficient αp = 1/ρ(∂ρ/∂T)p , or the internal pressure π = T(∂p/∂T)v − p = Tαp /κT − p, which is a quantity characteristic of the cohesion forces within the fluid. 2. Experimental techniques The dynamic viscosity was measured up to 100 MPa using a falling-body viscometer of the type designed by Ducoulombier et al. [14]. In this apparatus, a stainless steel cylinder falls through the fluid of unknown viscosity under selected conditions. The viscosity is a function of the falling time, the difference between the density of the cylinder and the one of the studied fluid, and an apparatus constant, which is determined by calibration for each considered temperature and pressure condition. The technical details for this viscometer are described by Et-Tahir et al. [15]. The calibration was performed using toluene [16], and n-decane [17,18] was used to verify the calibration. At 0.1 MPa, the dynamic viscosity was obtained by measuring the kinematic viscosity, ν = η/ρ, by a classical capillary viscometer (Ubbelohde). For this purpose, several tubes connected to an automatic Schott Geräte analyzer, were used. The density was measured for pressures between 0.1 and 65 MPa with an Anton-Paar DMA60 resonance densimeter combined with an additional 512P high-pressure cell. The calibration of the densimeter was performed using water and vacuum as described by Lagourette et al. [19]. The density measurements were extrapolated up to 100 MPa using the following Tait-type equation [15] for the variation of the density versus pressure:
 1 p − p0 1 + A ln 1 + = (1) ρ ρ0 B where A and B are two adjustable constants and ρ0 is the density at p0 , which in this work has been chosen to be 0.1 MPa. For the viscosity measurements performed with the falling-body viscometer and the density measurements, the uncertainty in the temperature was estimated to be ±0.5 and ±0.05 K, respectively. The uncertainty in the pressure was estimated to be ±0.1 MPa for the viscosity measurements and ±0.05 MPa for the density measurements (except at 0.1 MPa). The overall uncertainty in the reported density values are lower than 1 kg m−3 , while the relative uncertainty in the viscosity is of the order of 2% at high-pressure. As it has already been discussed in [15,20–22], this uncertainty is comparable with the uncertainties obtained by other authors for similar experimental devices. For the measurements of the kinematic viscosity performed with the classical capillary viscometer at atmospheric pressure, the uncertainty in the temperature was ±0.05 K. After multiplying the kinematic viscosity by the density, the dynamic viscosity is obtained with an uncertainty of less than 1%. The studied mixtures were prepared by weighing at atmospheric pressure and ambient temperature using a Mettler balance with an uncertainty of 0.001 g. The five pure compounds used to prepare the two mixtures are commercially available chemicals with the following purity levels: n-tridecane (C13 H28 : Tokyo Kasei, chemical purity >99% (GC), molecular weight M = 184.37 g mol−1 ), 2,2,4,4,6,8,8-heptamethylnonane (C16 H34 , Aldrich, chemical purity >98% (GC), M = 226.44 g mol−1 ), heptylcyclohexane (C13 H26 , Tokyo

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Kasei, chemical purity >99% (GC), M = 182.35 g mol−1 ), C13 H20 (Fluka, chemical purity >99% (GC), M = 176.30 g mol−1 ), and C11 H10 (Aldrich, chemical purity >98% (HPLC), M = 142.201 g mol−1 ). No further purification treatment of the pure compounds was performed.

3. Results The dynamic viscosity was measured at seven temperatures (293.15, 303.15, 313.15, 323.15, 333.15, 343.15 and 353.15 K) and at six pressures (0.1, 20, 40, 60, 80 and 100 MPa), resulting in 42 experimental points for each mixture. The density measurements of each mixture were performed at the same temperatures at pressures from 0.1 to 65 MPa. These values were extrapolated up to 100 MPa using Eq. (1) in order to obtain the densities at 80 and 100 MPa. Table 1 presents the measured viscosity and density values as a function of pressure and temperature for the two mixtures. Fig. 1 shows the variation of the viscosity with temperature for the ternary mixture at several pressures, whereas the variation of the viscosity with pressure is shown in Fig. 2 for the quinary mixture at several isotherms. In Fig. 3, the variation of the density with pressure is shown for the ternary mixture at several isotherms and Fig. 4 presents the variation of the density with temperature for the quinary mixture at several pressures. These figures reveal a general behavior consistent with previous observations made by other authors and by ourselves on different systems. The pressure coefficient of viscosity at constant temperature (∂η/∂p)T , is greater than zero for both mixtures and the shape of the curves for changes in the viscosity with pressure is sharply increasing. On the contrary, the temperature coefficient of viscosity at constant pressure (∂η/∂T)p , is always less than zero. For the density, at constant temperature, the curves are concave, which is associated with a negative second derivative. The shape of the isothermal curves of the density 6.5 0.1 MPa 20 MPa 40 MPa 60 MPa 80 MPa 100 MPa

Viscosity (mPa s)

5.5

4.5

3.5

2.5

1.5

0.5 293.15

303.15

313.15

323.15

333.15

343.15

353.15

Temperature (K)

Fig. 1. Dynamic viscosity vs. temperature for the ternary mixture at various pressures.

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Table 1 Dynamic viscosity η and density ρ of the ternary and the quinary mixture vs. temperature T and pressure p T (K)

293.15 293.15 293.15 293.15 293.15 293.15 303.15 303.15 303.15 303.15 303.15 303.15 313.15 313.15 313.15 313.15 313.15 313.15 323.15 323.15 323.15 323.15 323.15 323.15 333.15 333.15 333.15 333.15 333.15 333.15 343.15 343.15 343.15 343.15 343.15 343.15 353.15 353.15 353.15 353.15 353.15 353.15 a

p (MPa)

0.1 20 40 60 80 100 0.1 20 40 60 80 100 0.1 20 40 60 80 100 0.1 20 40 60 80 100 0.1 20 40 60 80 100 0.1 20 40 60 80 100 0.1 20 40 60 80 100

Extrapolated values using Eq. (1).

Ternary

Quinary

η (mPa s)

ρ (g cm−3 )

η (mPa s)

ρ (g cm−3 )

2.12 2.65 3.31 4.03 4.99 6.23 1.82 2.33 2.83 3.41 4.07 4.85 1.56 1.95 2.36 2.84 3.39 4.01 1.34 1.68 2.04 2.44 2.86 3.31 1.13 1.44 1.73 2.06 2.43 2.84 0.983 1.27 1.51 1.80 2.11 2.47 0.873 1.11 1.32 1.57 1.85 2.19

0.7985 0.8110 0.8219 0.8317 0.8406a 0.8488a 0.7912 0.8042 0.8155 0.8256 0.8347a 0.8431a 0.7839 0.7975 0.8093 0.8197 0.8290a 0.8376a 0.7767 0.7909 0.8031 0.8138 0.8235a 0.8322a 0.7695 0.7844 0.7971 0.8082 0.8181a 0.8271a 0.7623 0.7780 0.7912 0.8026 0.8128a 0.8220a 0.7551 0.7716 0.7853 0.7971 0.8076a 0.8170a

2.36 3.02 3.73 4.63 5.83 7.39 1.91 2.51 3.10 3.80 4.60 5.55 1.58 2.10 2.59 3.16 3.83 4.59 1.36 1.79 2.18 2.63 3.16 3.78 1.17 1.52 1.85 2.23 2.67 3.16 1.02 1.33 1.59 1.90 2.27 2.73 0.885 1.15 1.40 1.67 1.98 2.36

0.8170 0.8294 0.8404 0.8502 0.8591a 0.8672a 0.8099 0.8230 0.8344 0.8444 0.8535a 0.8617a 0.8028 0.8165 0.8283 0.8388 0.8482a 0.8568a 0.7957 0.8100 0.8224 0.8332 0.8429a 0.8517a 0.7885 0.8035 0.8164 0.8276 0.8376a 0.8466a 0.7814 0.7973 0.8106 0.8222 0.8324a 0.8416a 0.7743 0.7909 0.8048 0.8168 0.8274a 0.8369a

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293.15 K 303.15 K 313.15 K 323.15 K 333.15 K 343.15 K 353.15 K

Viscosity (mPa s)

6.5 5.5 4.5 3.5 2.5 1.5 0.5 0

20

40

60

80

100

Pressure (MPa)

Fig. 2. Dynamic viscosity vs. pressure for the quinary mixture at various temperatures.

versus pressure is compatible with the logarithmic relationship used in the Tait-type density relation used to model the influence of pressure on density. Furthermore, it should be noted that the variations of the density with temperature are practically linear due to the small temperature interval (293.15–353.15 K) considered in this investigation, because the aim was to observe the variations of density and viscosity with pressure. 0.86

0.82

-3

Density (g cm )

0.84

0.80

293.15 K 303.15 K 313.15 K 323.15 K 333.15 K 343.15 K 353.15 K

0.78

0.76

0.74 0

20

40

60

80

Pressure (MPa)

Fig. 3. Density vs. pressure for the ternary mixture at various temperatures.

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149

0.86

-3

Density (g cm )

0.84

0.82

0.80

0.1 MPa 20 MPar 40 MPa 60 MPa 80 MPa 100 MPa

0.78

0.76 293.15

303.15

313.15

323.15

333.15

343.15

353.15

Temperature (K)

Fig. 4. Density vs. temperature for the quinary mixture at various pressures.

4. Discussion Since the studied mixtures can be considered as being simple representations of some petroleum distillation cuts at 515 K, the obtained data have been used in order to evaluate the performance of seven different viscosity models for the potential extension to real petroleum fluids. The considered models are applicable to hydrocarbon mixtures over wide ranges of temperature and pressure. In order to validate and compare the performance of the considered models, the following quantities are defined: 
ηcalc,i devi (%) = 100 1 − ηexp,i 1  AAD = |devi | NP i=1 NP

1  devi bias = NP i=1 NP

MxD = max|devi | in which NP is the number of experimental points, ηexp the measured viscosity and ηcalc the value calculated using a given model. The quantity AAD (average absolute deviation) indicates how close the calculated values are to the experimental values and the quantity bias indicates how well the experimental points are distributed around the calculated points. If bias is equal to AAD, then all of the calculated values are higher than the experimental points. Finally, MxD characterizes the maximum absolute deviation obtained with a given model.

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4.1. Classical mixing laws Several mixing laws have been developed for calculating the viscosity of liquid mixtures. The objective of these mixing laws is to predict the viscosity based on the viscosity and the density of the pure compounds along with the composition. Two of the most well-known mixing laws are the one of Grunberg–Nissan [4] and the one of Katti–Chaudhri [5]. For a multicomponent mixture, the Grunberg–Nissan mixing law can be written as follows: ln ηmix

n  = xi ln ηi

(2)

i=1

and the Katti–Chaudhri mixing law can be expressed as ln(ηmix vmix ) =

n 

xi ln(ηi vi )

(3)

i=1

where vi = Mi /ρi is the molar volume, Mi the molecular weight, ρi the density, ηi the viscosity,  and xi is the mole fraction of component “i”. For mixtures the molecular weight is defined as Mmix = xi Mi . Both mixing laws are totally predictive in the sense that only properties of the pure compounds are required. In this work, the viscosity calculations with the Katti–Chaudhri mixing law were performed using the experimental densities of the mixtures in order to obtain vmix . With the Grunberg–Nissan mixing law an AAD of 1.35%, a MxD of 5.57%, and a bias of 0.23% are obtained for the ternary mixture, whereas for the quinary mixture the AAD, MxD, and bias are 11.0, 20.0 and −10.7%, respectively. Applying the Katti–Chaudhri mixing law to the ternary mixture, an AAD of 1.45%, a MxD of 5.32%, and a bias of 0.55% are obtained, whereas an AAD of 8.94%, a MxD of 17.8%, and a bias of −8.94% are obtained for the quinary mixture. For both mixing laws, the MxD is obtained at 293.15 K and 100 MPa with ηexp = 6.230 MPa s for the ternary mixture and ηexp = 7.390 MPa s for the quinary mixture. The results obtained with the two mixing laws are remarkable as they combine simplicity and good performances. Finally, by including the experimental densities in the Katti–Chaudhri mixing law, additional information about the mixture is incorporated into the viscosity predictions, resulting in a slightly better performance compared with the mixing law by Grunberg and Nissan, especially for the quinary mixture. 4.2. The self-referencing model A completely different model is the self-referencing model [6] developed in our laboratory in order to model the viscous behavior of petroleum cuts which complex composition is difficult to handle. For this kind of fluids, it is difficult to use equations based on physical properties, such as the molecular weight, the critical pressure and the critical temperature or the acentric factor, as for a mixture these properties have to be known for each pure component in the mixture. The formulation has the advantage that it only requires one experimental determination of the viscosity at atmospheric pressure and a selected reference temperature T0 . In this way, it is assumed that this measured viscosity value contains sufficient information about the studied fluid, and it is the reason why this method can be referred to as a self-referencing model. The method does involve neither the molecular weight, nor any other physical properties or critical parameters. It can be applied without restriction indifferently to pure substances, synthetic mixtures or

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chemically very rich systems such as petroleum cuts for which the method was originally developed. The method involves nine coefficients determined by numerical analysis on viscosity data of linear alkanes and alkylbenzenes. Based on the knowledge of this set of coefficients, the method can be used directly without further adjustment and can be considered general and predictive. The formulation of this method is

 
η(p, T) p − 0.1 1 1 2 2 ln = (ay + by + c) ln 1 + 2 + (gy0 + hy0 + i) − η(0.1 MPa, T0 ) dy + ey + f T T0 (4) where y = y0 + (gy20 + hy0 + i)(1/T − 1/T0 ) and y0 = ln[η(0.1 MPa, T0 )]. p is the pressure in MPa. Excluding the reference point, which in this work has been selected to T0 = 293.15 K, an AAD of 5.00%, a MxD of 11.6% and a bias of 4.30% are obtained for the ternary mixture, whereas for the quinary mixture the AAD, the MxD and the bias are 3.60, 10.6 and 2.80%, respectively. In the latter case, the results are better than for the classical mixing rules, with the advantage of only requiring the viscosity at atmospheric pressure for a given reference temperature and not the composition nor the density and the viscosity at a given T, p condition. Recently, a modification of the self-referencing method has been proposed [23] in which the nine coefficients required in the model are obtained from the composition and the respective nine coefficients of the pure compounds by using the following simple mixing rule αmix = xj αj with (α = a, b, . . . , i). In the pressure–temperature range considered in this work, the required nine coefficients a, b, . . . , i for each pure compound have been fitted separately using the pure compound viscosity data given in [1–3]. For the ternary mixture, the obtained AAD is 3.13% with a bias of 2.80% and a MxD of 7.35% found at 343.15 K and 20 MPa (ηexp = 1.270 mPa s), whereas an AAD = 1.93%, a bias = −1.05% and a MxD = 5.35% at 353.15 K and 80 MPa (ηexp = 1.980 mPa s) are obtained for the quinary mixture, which are very satisfactory results for a predictive method. This clearly shows, as suggested in the original article, that the use of the reference measurement point is important, because this measurement contains and provides useful information on the studied system. 4.3. Hard-sphere viscosity scheme A scheme has been developed for the simultaneous correlation of self-diffusion, viscosity, and thermal conductivity of dense fluids [7,8]. The transport coefficients of real dense fluids expressed in terms of vr = v/v0 , with v0 the close-packed volume and v the molar volume, are assumed to be directly proportional to values given by the exact hard-sphere theory. The proportionality factor, described as a roughness factor Rx (for the property x), accounts for the molecular roughness and departure from molecular sphericity. Universal curves for the viscosity have been developed and are expressed as
∗  
i 7 ηexp 1 ln (5) = aη,i Rη v r i=0 with η∗exp


= 6.035 × 10

8

1 MRT

1/2 ηexp v2/3

(6)

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where R is the gas constant, M the molecular weight and ηexp is the experimental viscosity of the considered fluid. The coefficients aη,i are universal, independent of the chemical nature of the compound. This has been verified by Baylaucq et al. [24,25] who used the hard-sphere scheme to model the viscosity of two ternary systems composed of n-heptane + methylcyclohexane + 1-methylnaphthalene and water + 2-propanol + 4-hydroxy-4-methyl-2-pentanone. In order to apply the hard-sphere scheme to mixtures, the following linear mixing rules have been suggested by Assael et al. [26]: v0,mix =

n 

xi v0,i ;

Rη,mix =

i=1

n  xi Rη,i ; i=1

Mmix =

n 

xi Mi

(7)

i=1

For various compounds it has been observed that v0 is temperature dependent, whereas Rη is temperature independent for pseudo-spherical molecules, such as n-alkanes, but shows a temperature dependency for molecules that either depart too much from sphericity or have hydrogen bonds. Correlations for v0 and Rη are not given in the literature for all of the considered compounds in this work. In a previous work, Daugé [27] carried out a direct modeling of v0 and Rη for each compound using experimental viscosities and densities given at the same temperature and pressure conditions considered in this work. The estimation was performed with the following assumptions that Rη is temperature independent and the aη,i parameters are universal. Using the estimated pure compound, v0 and Rη values and the mixing rules given in Eq. (7), an AAD of 2.84% is obtained for the ternary mixture with a bias of 2.64% and a MxD of 5.74% at 293.15 K and 100 MPa (ηexp = 6.230 mPa s). For the quinary mixture, the AAD, the MxD and the bias are 10.9, 14.8 and −10.9%, respectively. Here, the MxD is found at 303.15 K and 100 MPa (ηexp = 5.550 mPa s). Overall, these results can be considered satisfactory, since the method is predictive for mixtures. However, it should be mentioned that the representation of the viscosity of 2,2,4,4,6,8,8-heptamethylnonane with this model is less accurate (AAD = 3.59% and MxD = 23.8%) compared to the performance for n-tridecane (AAD = 2.81%, MxD = 8.30%) or heptylcyclohexane (AAD = 1.25%, MxD = 5.85%), see also [1,28]. This compound is present in the quinary mixture, but not in the ternary mixture. This could be an explanation for the relative higher discrepancies between the ternary and quinary results. In addition, the seven order polynomial development of the model in the reduced volume leads to a high sensitivity in the calculations of the viscosity. A direct fitting of the v0 coefficients versus the experimental data of the quinary mixture with Rη calculated by Eq. (7) reduces the AAD to less than 2% with an MxD of 8%. 4.4. Free-volume viscosity model Another approach, based on the free-volume concept, has very recently been proposed [9,10] in order to model the viscosity of Newtonian fluids in the gaseous and dense states. The total viscosity η can be separated into a dilute gas term η0 and an additional η term, in the following way: η = η0 + η

(8)

The term η characterizes the passage in a dense state. The model has been successfully applied to various hydrocarbons. This approach connects the term η to the molecular structure via a representation of the free-volume fraction. The viscosity, in this theory, appears as being the product of the fluid modulus ρRT/M and the mean relaxation time of the molecule defined by Na L2 ζ/(RT) leading to the following

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expression for the viscosity in excess of the dilute gas viscosity: η =

ρNa ζL2 M

(9)

where Na is Avogadro’s number, ζ the friction coefficient of a molecule, and L2 an average characteristic molecular quadratic length. The friction coefficient ζ is related to the mobility of the molecule and to the diffusion process (diffusion of the momentum for viscosity). Moreover, the free-volume fraction fv = vf /v (with vf = v − v0 , v the specific molecular volume and v0 the molecular volume of reference or hard-core volume) is for a given temperature T defined by
3/2 RT fv = (10) E assuming that the molecule is in a state in which the molecular potential energy of interaction with its neighbors is E/Na . It has been assumed [9] that E = E0 + pM/ρ, where the term pM/ρ = pv is connected to the energy necessary to form the vacant vacuums available for the diffusion of the molecules and where E0 is related to the energy barrier that the molecule has to exceed in order to diffuse. For dense fluids E0 can be considered constant [9]. Combining the empirical relation of Doolittle [29]
 B η = A exp (11) fv where B is the characteristic of the free-volume overlap, with Eq. (9) has lead to the writing of ζ in the form
 B (12) ζ = ζ0 exp fv and by combining Eqs. (9) and (12) gives η =

ρNa L2 ζ0 exp(B/fv ) M

Further, it has been demonstrated [10] that  E M ζ0 = Na bf 3RT

(13)

(14)

where bf is the dissipation length of the energy E. By inserting Eqs. (13) and (14) into Eq. (8) leads to 

   E0 + (pM/ρ) E0 + (pM/ρ) 3/2 η = η0 + ρl (15) exp B √ RT 3RTM or



 RT −2/3 B fv exp η = η0 + ρl 3M fv

(16)

where # = L2 /bf is homogeneous with a length. This equation involves three physical parameters #, E0 and B which are characteristic of the molecule. It should be mentioned that for dense fluids, such as those

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considered here, η0 could be neglected, giving 

  E0 + (pM/ρ) 3/2 E0 + (pM/ρ) exp B η = ρl √ RT 3RTM or alternatively   RT −2/3 B exp fv η = ρl 3M fv

(17)

(18)

The model has been adapted to mixtures [30] using the following mixing rules Mmix =

n  xi Mi ;

E0,mix =

i=1

(19)

i=1 j=1

n

 xi 1 = ; Bmix B i=1 i

n n  

xi xj E0,i E0,j

#mix =

n 

xi #i

(20)

i=1

For the viscosity predictions of the two studied mixtures, the required pure component parameters are those reported in [1,28]. For the 42 points of the ternary mixture, an AAD of 1.82%, a MxD of 4.03% at 303.15 K and 20 MPa (ηexp = 2.330 mPa s), and a bias of 0.82% are obtained, whereas for the quinary mixture the obtained AAD is 3.23% with a bias of −3.21% and a MxD of 7.54% found at 303.15 K and 100 MPa (ηexp = 5.550 mPa s). These results are very satisfactory for a predictive model concerning the dynamic viscosity, which only requires the knowledge of three characteristic parameters for each pure compound. Another possibility has been proposed for systems  containing gases (methane) [30] which uses a mixing rule on the free-volume fraction fv : 1/fv,mix = i xi /fv,i and the mixing rules for Bmix and #mix in Eq. (20). In this case, the obtained AAD for the ternary mixture is 1.78% with a bias of 0.72% and MxD of 3.88% at 303.15 K and 20 MPa for ηexp = 2.330 mPa s, whereas for the quinary mixture the obtained AAD is 5.04% with a bias of −5.04% and a MxD of 9.84% at 303.15 K and 100 MPa for ηexp = 5.550 mPa s. 4.5. The friction theory Recently, starting from basic principles of classical mechanics and thermodynamics, the friction theory (f-theory) for viscosity modeling has been developed [11]. In the f-theory, the total viscosity can be written as η = η 0 + ηf

(21)

where η0 is the dilute gas viscosity and ηf the residual friction contribution. The friction contribution is related to the van der Waals attractive and repulsive pressure terms, pa and pr , of a cubic equation of state (EOS), such as the Peng and Robinson (PR) EOS [31]. Based on this concept, a general f-theory model [12] has been proposed with 16 universal constants and one adjustable parameter—a “characteristic” critical viscosity. For hydrocarbons with a simple molecular structure it has been shown that the f-theory models [11,12] consisting of a linear correlation on pa and a quadratic correlation on pr suffices to accurately represent the viscosity over wide ranges of temperature and pressure. However, in some cases,

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such as 2,2,4,4,6,8,8-heptamethylnonane [28], the molecular structure of the compound may induce an interlinking effect that results in an important reduction of the fluid mobility (high viscosity) when brought under pressure. For many systems, such dragging effects can be taken into account by a simple extension of the f-theory models repulsive pressure dependency from quadratic to third-order. A third-order f-theory model can be written as ηf = κa pa + κr pr + κrr p2r + κrrr p3r

(22)

where the κ parameters are the temperature dependent friction coefficients. According to the foregoing, in order to apply the general f-theory model [12] to fluids with high dragging effects, it is necessary to introduce a third-order corrective term to the general model. Thus, Eq. (22) can be written as ηf = ηGM + ηIII

(23)

where ηGM is the friction viscosity contribution of the general f-theory model as defined in [12] and ηIII = κrrr p3r is the third-order correction to the general f-theory model. The following expression for the third-order friction coefficient has been proposed [28] κrrr = d2 (exp(2Γ) − 1)(Γ − 1)3

(24)

with Γ =

Tc T

(25)

and where d2 is a component related parameter. In this work, the general f-theory model [12] with a third-order correction term and in conjunction with the PR EOS has been used for the viscosity predictions of the two studied mixtures. The required critical properties for 1-methylnaphthalene and n-tridecane have been taken from Reid et al. [32], whereas the critical properties for heptylbenzene and heptylcyclohexane have been taken from Yaws [33]. For 2,2,4,4,6,8,8-heptamethylnonane the used critical properties are those reported in [2]. The characteristic critical viscosity and the d2 parameter for 1-methylnaphthalene, n-tridecane, and 2,2,4,4,6,8,8-heptamethylnonane are reported in [28], whereas these parameters are given in [1] for heptylbenzene and heptylcyclohexane. In the PR EOS, the regular van der Waals mixing rules have been used without any binary interaction parameters. For the two mixtures, the dilute gas viscosity is obtained by the proposed mixing rule given in [11,12] which is based on the dilute gas viscosity of the pure components, which have been estimated by the dilute gas viscosity model of Chung et al. [34]. The general PR f-theory model mixture contribution, ηGM , in Eq. (23) is treated according to the mixing rules proposed in [12] and the mixture third-order friction coefficient, κrrr,mix , is obtained with an exponential mixing rule [28] of the form ln(κrrr,mix ) =

n  xi ln(κrrr,i )

(26)

i=1

where subscript i refers to pure component i. Using this f-theory approach an AAD of 3.28%, a MxD of 7.88% and a bias of 2.91% are obtained for the ternary mixture, whereas for the quinary mixture the AAD, the MxD and the bias are 3.78, 16.9 and 1.34%, respectively. For both mixtures, the MxD

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is obtained at 100 MPa and 293.15 K corresponding to ηexp = 6.230 mPa s for the ternary mixture and ηexp = 7.390 mPa s for the quinary mixture. The obtained results are satisfactory taking into account that they have been obtained by a totally predictive method based on pure component properties and in conjunction with a simple cubic EOS. A comparison of the results obtained for the f-theory with those obtained for the free-volume model using experimental densities shows that the free-volume model gives better results for the maximum deviation. One reason could be, as already mentioned under the modeling of the ternary system 1-methylnaphthalene + n-tridecane + 2,2,4,4,6,8,8-heptamethylnonane [28], that the molecular structures of the components in the studied mixtures in this work may introduce a possible interlinking effect, resulting in a higher real viscosity. This interlinking effect may be better related to the free-volume than to a cubic EOS. Thus, an advantage of the f-theory is that it does not require the density of the considered fluid, and it is related to simple cubic EOS, which are commonly used in the petroleum industry as well as other industries. 4.6. The LBC model In addition and for comparison purposes, the viscosity of the two studied synthetic mixtures has also been predicted with the LBC model [13], which is widely used in the oil industry: [(η − η0 )ξ + 10−4 ]1/4 = d0 + d1 ρr + d2 ρr2 + d3 ρr3 + d4 ρr4

(27)

where ρr is the reduced density, η0 the dilute gas limit and ξ the viscosity-reducing parameter. Since the LBC model is a 16th degree polynomial in the reduced density, makes the calculated viscosities very sensitive and dependent on the accuracy of the used density values. In principle, the optimal performance with the LBC model should be obtained, when the experimental densities are used. Due to this, the estimation of the viscosity of the ternary and quinary mixtures has been performed using the experimental densities and according to the calculation procedure originally derived for the LBC model [13]. For the ternary mixture, the obtained AAD is 24.4%, whereas the bias is 24.4% and the MxD is 48.9% obtained at 293.15 K and 100 MPa (ηexp = 6.230 mPa s). For the quinary mixture the AAD, the bias and the MxD are 26.7, 26.7 and 56.1%, respectively. The MxD is obtained at 293.15 K and 100 MPa (ηexp = 7.390 mPa s). Although the experimental densities have been used, very large deviations are obtained with the LBC model for these two synthetic mixtures compared with the results obtained for the other models. Thus, an important property in the LBC model is the critical density. Due to this, a very common procedure within the oil industry in order to improve the viscosity calculations with the LBC model is to tune the model with respect to the critical density of the considered fluid. By adjusting the critical density for the optimal performance of the LBC model, an AAD of 10.5%, a MxD of 31.5% at 293.15 K and 100 MPa and a bias of 0.94% are obtained for the ternary mixture, whereas for the quinary mixture an AAD of 13.3%, a bias of 4.41% and a MxD of 41.2% at 293.15 K and 100 MPa are obtained. Despite the tuning of the LBC model, the obtained results are still higher than the results obtained with the other models, which, for mixtures, are totally predictive. Further, it should be stressed that the deviations obtained with the LBC model increase significantly as the fluids become more viscous (higher viscosity). The reason is that the LBC model and its parameters were originally derived from experimental viscosity and density data of light fluids and hydrocarbons, which have much lower viscosities than those of the considered mixtures in this work. Due to this, the LBC model is not applicable for calculations of the viscosity of heavy hydrocarbons.

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4.7. The internal pressure In addition, it is important to stress here that from the experimental density values it is possible to calculate by numerical analysis quantities such as the isothermal compressibility factor: κT =

1 , ρ(∂ρ/∂p)T

(28)

the isobaric thermal expansion coefficient αp =

1 , ρ(∂ρ/∂T)p

(29)

and the internal pressure
 κp ∂p π=T −p=T −p ∂T v κT

(30)

The internal pressure π is an important factor in order to study the behavior of liquids. It is a measure of cohesive forces acting in a liquid and corresponds to the attractive pressure term α(T)/v2 in van der Waal’s equation. Fig. 5 shows the internal pressure π of the ternary mixture as a function of temperature and for pressures up to 60 MPa. The restriction in the pressure range corresponds to the highest experimental pressure for which the density has been measured. The general trends of the variations with pressure and temperature are the same for both mixtures. The internal pressure decreases with temperature. Further, since the internal pressure is a function of some derivatives of the density with temperature and pressure, the uncertainty in the used density values is very important. In this study, although the uncertainty of the measured densities is less than 1 kg m−3 , the uncertainty for the obtained internal pressures is estimated to be of the order of 2%.

340 330 310 300

π (MPa)

320

290 280 .1 5 303 .1 5 313 Te 5 mp 23 .1 era 3 .1 5 tur 3 33 .1 5 e (K 3 43 ) .1 5 0 353

270

10

20

30

40

50

Pa) re (M u s s Pr e

Fig. 5. Variation of the internal pressure π vs. pressure and temperature for the ternary mixture.

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For mixtures it has been proposed [35,36] that the internal pressure can be defined as n  πmix = xi πi

(31)

i=1

where πi is the internal pressure of the pure component. This expression has been used in this work in order to predict the internal pressure of the ternary and quinary mixtures in the pressure range 0.1–60 MPa. The internal pressure of the pure compounds for a given pressure–temperature condition has been estimated using the density values given in [1–3]. For the ternary mixture, the obtained AAD is 2.87%, the MxD 6.88%, and the bias 2.41%, whereas for the quinary mixture the AAD is 2.01%, the MxD 6.90%, and the bias −2.01%, which are satisfactory results. 5. Conclusion The dynamic viscosity and the density of a ternary and a quinary hydrocarbon mixture, which are simple representations of some petroleum distillation cuts at 515 K, were measured in the temperature range 293.15–353.15 K and up to 100 MPa. The measured values were used to evaluate the performance of different viscosity models, which range from simple empirical equations to semi-empirical models having a physical and theoretical background. It follows from the discussion that some simple predictive models, except the LBC model, are able to predict the viscosity of such low associating systems within an acceptable uncertainty for most applications within the oil industry. For comparison reasons Fig. 6 shows for all T, p conditions of the ternary and quinary mixtures the obtained deviations versus the experimental values for the Grunberg–Nissan mixing law, the self-referencing method with parameter mixing rules, the hard-sphere scheme, the free-volume model using the mixing rules in Eqs. (19) and (20), the f-theory model, and a tuned optimal LBC model, whereas Figs. 7 and 8 show the obtained deviations versus pressure and temperature, respectively. For each of the studied mixtures these figures show no significant fluctuation in the variation of the obtained deviations by a given model with temperature and pressure, except for the LBC. However, the largest deviations for most of the models are obtained at the lowest temperature and highest pressure, except for the self-referencing model. Further, these figures also show that higher deviations are obtained for the quinary mixture than for the ternary. The quinary mixture contains 2,2,4,4,6,8,8-heptamethylnonane, which has a more pronounced viscosity increase when brought under pressure compared to the other involved compounds. The viscosity behavior of 2,2,4,4,6,8,8-heptamethylnonane is more complex to handle for the different models compared to the other involved compounds, see [1,28]. This may reflect more the higher deviations obtained by the considered models for the quinary mixture than the general performance of the models, except the LBC model. Thus, it should be stressed that for a given fluid, the performance of a specific model can be improved, if the parameters or the model itself, are readjusted or changed. In this case, the model will only be adequate for the considered system, and will no longer be considered as a general model. Although the self-referencing model gives the overall best performance for the two mixtures, it can only be applied to liquids and dense fluids, but not gases, which are common within the oil industry. This is also the case for the mixing laws and the hard-sphere viscosity model. The free-volume model and the friction theory are applicable to gases, liquids, and dense fluids, and due to this are more applicable within the oil industry. Moreover, the friction theory does not require the density which is an advantage for practical use compared to the other methods.

C. Boned et al. / Fluid Phase Equilibria 212 (2003) 143–164 50

50 Grunberg-Nissan Mixing Law

Self-Referencing Model

40

30

30

20

20

Deviation (%).

Deviation (%).

40

10 0

10 0

-10

-10

-20

-20

-30

-30 0

2

4

6

8

0

Experimental Viscosity (mPa s)

4

6

8

50 Hard-Sphere Scheme

40

Free-Volume Model

40

30

30

20

20

Deviation (%).

Deviation (%).

2

Experimental Viscosity (mPa s)

50

10 0

10 0

-10

-10

-20

-20

-30

-30 0

2

4

6

8

0

Experimental Viscosity (mPa s)

2

4

6

8

Experimental Viscosity (mPa s)

50

50 Friction Theory

40

Tuned LBC Model

40

30

30

20

20

Deviation (%).

Deviation (%).

159

10 0

10 0

-10

-10

-20

-20

-30

-30 0

2

4

6

Experimental Viscosity (mPa s)

8

0

2

4

6

8

Experimental Viscosity (mPa s)

Fig. 6. Deviations between experimental viscosities and calculated values by different models vs. the experimental viscosity for all points of the ternary ( ) and the quinary (×) mixtures.

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50 Grunberg-Nissan Mixing Law

40

30 Deviation (%).

30 Deviation (%).

Self-Referencing Model

40

20 10 0

20 10 0

-10

-10

-20

-20

-30

-30 0

20

40

60

80

100

0

20

Pressure (MPa) 50 Hard-Sphere Scheme

80

100

80

100

80

100

Free-Volume Model

40 30 Deviation (%).

30 . Deviation (%).

60

50

40

20 10 0

20 10 0

-10

-10

-20

-20

-30

-30 0

20

40

60

80

100

0

20

Pressure (MPa)

40

60

Pressure (MPa)

50

50 Friction Theory

40

Tuned LBC Model

40 30 Deviation (%).

30 Deviation (%).

40

Pressure (MPa)

20 10 0

20 10 0

-10

-10

-20

-20

-30

-30 0

20

40

60

Pressure (MPa)

80

100

0

20

40

60

Pressure (MPa)

Fig. 7. Deviations between experimental viscosities and calculated values by different models vs. pressure for all points of the ternary ( ) and the quinary (×) mixtures.

C. Boned et al. / Fluid Phase Equilibria 212 (2003) 143–164 50 40

50 Grunberg-Nissan Mixing Law

40

Self-Referencing Model

30 Deviation (%).

Deviation (%).

30 20 10 0

20 10 0

-10

-10

-20

-20

-30 293.15

313.15

333.15

-30 293.15

353.15

Temperature (K)

40

Deviation (%).

Deviation (%).

353.15

Free-Volume Model

30

20 10 0

20 10 0

-10

-10

-20

-20

-30 293.15

313.15

333.15

-30 293.15

353.15

Temperature (K)

313.15

333.15

353.15

Temperature (K)

50

50 Friction Theory

40

20 10 0

20 10 0

-10

-10

-20

-20

-30 293.15

Tuned LBC Model

30 Deviation (%).

30 Deviation (%).

333.15

50 Hard-Sphere Scheme

30

40

313.15

Temperature (K)

50 40

161

-30 313.15

333.15

Temperature (K)

353.15

293.15

313.15

333.15

353.15

Temperature (K)

Fig. 8. Deviations between experimental viscosities and calculated values by different models vs. temperature for all points of the ternary ( ) and the quinary (×) mixtures.

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The obtained experimental data could be included in databases and used to carry out further tests of other models more sophisticated, as for instance models based on molecular dynamic simulation. This study should be considered only as a fragmental part of a more general study concerning various systems (associative and non associative mixtures, various binaries with different compounds, ternary systems and systems with more than three constituents, and systems containing dissolved gases such as methane) that has been undertaken in the laboratory during the last years. Nevertheless, this work shows the potential extension of some viscosity approaches to petroleum fluids. List of symbols bf dissipative length B characteristic of the free-volume overlap E energy E0 barrier energy fv free-volume fraction # = L2 /bf characteristic molecular length L2 average characteristic molecular quadratic length M molecular weight Na Avogadro’s constant p pressure pa attractive pressure term pr repulsive pressure term R gas constant Rη roughness factor for viscosity T temperature Tc critical temperature v molar volume v0 hard-core volume vf free-volume vr reduced molar volume x mole fraction Greek letters αp isobaric thermal expansion coefficient Γ defined in Eq. (25) ζ free-volume friction coefficient η viscosity η0 dilute gas viscosity ηf residual friction term κa linear attractive friction coefficient κr linear repulsive friction coefficient κrr quadratic repulsive friction coefficient κrrr third-order repulsive friction coefficient κT isothermal compressibility factor ξ viscosity reducing parameter

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π ρ ρr

163

internal pressure density reduced density

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