Prediction of Density and Viscosity of Bitumen Using the Peng

Ind. Eng. Chem. Res. 2009, 48, 10129–10135

10129

Prediction of Density and Viscosity of Bitumen Using the Peng-Robinson Equation of State Herbert Loria,* Pedro Pereira-Almao, and Marco Satyro Department of Chemical and Petroleum Engineering, Schulich School of Engineering, UniVersity of Calgary, Calgary AB T2N 1N4, Canada

Density and viscosity are important quantities required in engineering design for production, fluid transportation, and processing. However, there is no satisfactory theory for the calculation of these properties for bitumen. The principal objective of this paper is to obtain thermodynamic models to predict the density and viscosity of bitumen on the basis of the translated version of the Peng-Robinson equation of state. In the density calculation, a consistent correction to improve the liquid-phase volume estimation was applied. The density model evidenced a small percent average absolute error regarding experimental data (less than 1%). The model for viscosity was based on a modification of the Enskog’s equation. This modification allowed the prediction of bitumen viscosity using an equation of state along with a substance- and temperature-dependent parameter; this approach showed good accuracy with respect to experimental data. An important advantage of these models is the possibility of estimating viscosities at different pressures. 1. Introduction The development of convective-dispersive models to predict the concentration profile of ultradispersed particles transported through different viscous media inside cylindrical geometries is part of recent efforts1-3 to generate novel in situ heavy oil upgrading processes. One of the specific objectives of such efforts is to obtain results from simulations carried out at heavy oil upgrading conditions. These simulations will permit one to unveil the critical catalytic ultradispersed particle diameters and heavy oil conditions that permit particle suspension in such media and the conditions for which these particles will sediment in order to recover and reuse them in a recirculation system. In addition, the findings of these investigations can be extended to surface upgrading potential technologies that could make use of ultradispersed catalysts to avoid problems caused by the use of supported catalysts. To materialize the objectives stated in the previous paragraph, it is necessary to develop thermodynamic simulation models for the prediction of the density and viscosity of heavy crude oil and bitumen. These properties are necessary input parameters for the convective-dispersive models which will permit one to carry out simulations at heavy oil and bitumen upgrading conditions. The description of phase behavior and properties of hydrocarbon fluids is of fundamental importance in petroleum and chemical engineering for a wide variety of processes, including primary recovery, enhanced oil recovery, reservoir evaluation, and design of surface facilities, etc. The equations of state can be used accurately to predict properties, such as density, enthalpy, vapor pressure, fugacity and fugacity coefficient, and vapor-liquid equilibrium, among others. Furthermore, the use of equations of state is not only accurate but also convenient. One advantage of using an equation of state is that it can provide a unique and consistent model for all equilibrium properties. Extensive efforts have been made in the past 3 decades to improve the performance of the equations of state. Pe´neloux et al.4 first introduced volume translations or shifts to improve the accuracy of the Soave-Redlich-Kwong equation of state. One * To whom correspondence should be addressed. Tel.: +1 403 210 95 90. Fax: +1 403 210 39 73. E-mail: [email protected].

of the most popular cubic equations of state is that proposed by Peng and Robinson.5 The Peng-Robinson equation of state is widely used in chemical engineering process simulation, design, and optimization to estimate density values. Jhaveri and Youngren6 defined volume shifts for light hydrocarbons to be used with the Peng-Robinson equation of state and provided a correlation to define the volume shift for the hydrocarbon fractions. Another major improvement resulted from the introduction of parameters called binary interaction coefficients to better match the saturation pressure predictions. Katz and Firoozabadi7 proposed a fixed set of binary interaction coefficients for methane-hydrocarbon mixtures. On the other hand, transport properties such as viscosity, thermal conductivity and diffusion coefficients are important quantities required in engineering design for production, fluid transportation, and processing. However, there is no satisfactory theory of transport properties for real dense gases and liquids. The difficulties that are faced in the study of transport properties are 2-fold: one is the inherent difficulties involved in accurate measurements, and the other is the complexity involved in theoretical treatments.8 Therefore, the generally used correlations for the prediction of transport proprieties are either empirical or semiempirical. Enskog9,10 developed a popular theory for the transport properties of dense gas based on a distribution function. However, the Enskog’s theory was proposed for rigid spherical molecules. For real gases, some modification is needed. Following Enskog’s theory, several correlations have been proposed in the form of the reduced density and reduced temperature. Among those, Sengers11 examined Enskog’s theory with an emphasis on the behavior of transport properties in the critical region. The equations of state provide a good description of density, pressure, and temperature. They are attractive for the prediction of transport properties, such as viscosity, because of their application in phase equilibrium calculations and their use for high-pressure calculations. Little and Kennedy12 developed the first equation of state based viscosity model using the van der Waals equation of state. Unfortunately, the six coefficients involved in the model were not generalized, nor has it been adequately tested on fluid mixtures.13 Lawal14 proposed a

10.1021/ie901031n CCC: $40.75  2009 American Chemical Society Published on Web 09/02/2009

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Ind. Eng. Chem. Res., Vol. 48, No. 22, 2009

Table 1. Properties of the Different Pseudocomponents of the Athabasca Bitumen

(

Tc ) 341.7 + 811SG + (0.4244 + 0.1174SG)Tb +

i

pseudocomponent

vol. %

Tb, °C

SG

1 2 3 4 5 6

C5-320 320-400 400-650 650-700 700-975 975+

0.5 1.4 11.6 5 30.14 51.36

160 187.78 276.67 360 448.89 523.89

0.8031 0.8338 0.903 0.9309 0.9806 1.0639

viscosity model based on the four-parameter Lawal-LakeSilberberg equation of state, which is applicable to pure hydrocarbons and their mixtures. However, it is not a predictive model, and poor results were obtained when it was applied to reservoir oils.13 Guo et al.15 correlated viscosity data from hydrocarbons using a cubic equation of state type equation based on the geometric similarity of P-V-T and T-η-P diagrams. Sheng et al.8 combined Enskog’s theory with a simple cubic equation of state where they proposed a slight empirical modification of Enskog’s equation to improve the accuracy of prediction of viscosity coefficients for dense gases and liquids. The principal objective of this paper is to obtain thermodynamic models for the density and viscosity of bitumen. The predictions of fluid properties from an equation of state are subject to errors due to the inherent limitations on the accuracy of the equation of state and to the limitations in the characterization of the fluid. The inherent limitations of the equation of state can be minimized by a judicious choice of model and empirical enhancements. In this paper, the Peng-Robinson equation of state coupled with the Pe´neloux’s volume translation technique4 was selected for the bitumen density calculation. After the choice of the equation of state, the characterization of mixtures is perhaps the most important source of error when dealing with crude oils. The characterization of bitumen is one of the goals in this study. The characterization process traditionally requires the assignment of values of parameters, such as critical temperature (Tc), critical pressure (Pc), and accentric factor (ω), which define the thermodynamic behavior of the fluid as modeled by an equation of state. Enskog’s theory modified by Sheng et al.8 and the results of the density model developed with the Peng-Robinson equation of state were employed in order to obtain a thermodynamic model for the calculation of bitumen viscosity. 2. Thermodynamic Models 2.1. Development of the Density Model. The calculations for the density model were based on the data obtained from an assay of Athabasca bitumen with 8.8°API; this assay was acquired from the Alberta Chamber of Resources.16 The bitumen was divided into six different pseudocomponents named accordingly to their boiling points (°F): C5-320, 320-400, 400-650, 650-700, 700-975, and 975+. The specific gravity at 60 °F (SG) of each one of the pseudocomponents could be obtained directly from the bitumen assay, and the boiling point temperature (Tb) was volumetrically averaged. These properties are based on atmospheric pressure. The values of these properties are presented in Table 1. The properties in Table 1 were necessary to obtain the critical properties for each pseudocomponent. Kesler and Lee17 gave the following correlations for Tc and Pc, ω, and molecular weight (MW).

)

(0.4669 - 3.2623SG) × 105 /1.8 Tb

(

(

(1)

)

0.11857 0.0566 2.2898 - 0.24244 + + × SG SG SG2 0.47227 3.648 + 10-3Tb + 1.4685 + × 10-7Tb2 SG SG2 1.6977 101325 (2) 0.42019 + × 10-10Tb3 14.7 SG2

Pc ) exp 8.3634 -

(

(

(

)

)

)

6.09648 + 1.28862 ln Tbr Tbr 15.6875 / 15.2518 Tbr

ω ) ln Pbr - 5.92714 +

)(

0.169347Tbr6

13.4721 ln Tbr + 0.43577Tbr6

(

)

(3)

MW ) -12272.6 + 9486.4SG + (4.6523 - 3.3287SG)Tb

[

(

(1 - 0.77084SG - 0.02058SG2) 1.3437 -

] [

)

720.79 × Tb

107 /Tb + (1 - 0.80882SG + 0.02226SG2) ×

(

1.8828 -

)

)]

181.98 × 1012 /Tb3 Tb

(4)

In the equations above, the boiling temperature (Tb) is in Rankine, Tc is in kelvin, Pc is in pascals, and MW in grams per mole. Tbr is the reduced boiling point temperature, and Pbr is the reduced boiling point pressure. The next step is the calculation of the molar volume of each pseudocomponent at a specific temperature and pressure; to do this, the Peng-Robinson equation of state was employed. Peng and Robinson5 proposed the following two-parameter simple cubic equation: P)

RgT a(T) V-b V(V + b) + b(V - b)

(5)

where P is the pressure, T is the temperature, V is the molar volume, Rg is the gas constant, b is the van der Waals covolume and, a is the attraction parameter. The Peng-Robinson equation can be written in terms of the compressibility factor (Z): Z3 - (1 - B)Z2 + (A - 3B2 - 2B)Z - (AB - B2 - B3) ) 0 (6)

where A)

aP Rg2T2

(7)

B)

bP RgT

(8)

Z)

PV RgT

(9)

Equation 6 yields one or three real roots, depending upon the number of phases in the system. In the two-phase region, the largest root is for the compressibility factor of the vapor, while the smallest positive root corresponds to that of the liquid. For this work, the root of interest is the smallest one.

Ind. Eng. Chem. Res., Vol. 48, No. 22, 2009

The parameters a and b are calculated by applying eq 5 at the critical point: a(Tc) ) 0.45724 b(Tc) ) 0.0778

Rg2Tc2 Pc

RgTc Pc

(10)

(12)

b(T) ) b(Tc)

(13)

m ) 0.37464 + 1.54226ω - 0.26992ω2

(14)

where Tr is the reduced temperature. If the volumetric and phase behavior of a fluid mixture is calculated by means of an equation of state, certain translations along the volume axis, which leave the predicted phase equilibrium conditions unchanged, may be effected.4 This property may be exploited in the form of a consistent correction to improve volume estimations made by the Peng-Robinson equation of state. This volume consistent correction for each pseudocomponent (ci) is given by the difference between the volume calculated by the equation of state and the actual volume: ci ) Vi,PR - Vi,actual

(15)

In eq 15, Vi,PR is the molar volume of each pseudocomponent calculated by substituting the smallest root of eq 6 at 15.55 °C (60 °F) and 101325 Pa (1 atm) in eq 9. Vi,actual is the actual molar volume of the pseudocomponent that was calculated using the specific gravity given in the assay and the pseudocomponent’s molecular weight in the following way: Vi,actual )

MWi

(16)

60◦F SGi(Fwater )

Then, the mixture (in this case, the bitumen) liquid molar volume was calculated. The mixture parameters used in eq 6 are defined by the following mixing rules: b)

the Peng-Robinson equation of state, and Vcorrected is the corrected molar volume of the mixture. The density of the mixture was calculated using the volumetrically averaged molecular weight of the mixture with the following relationship: Fmix )

(11)

a(T) ) a(Tc)[1 + m(1 - Tr1/2)]2

∑xb

MWmix Vcorrected

(21)

In eq 21, Fmix is the density of the mixture and MWmix is the volumetrically averaged molecular weight of the bitumen (430.405 g/mol). 2.2. Development of the Viscosity Model. Many models for the calculation of transport properties at elevated densities are based on Enskog’s theory.10 According to Enskog’s theory, the viscosity (η) for a gas of rigid spheres is represented by the following expression: 1 + 0.8 + 0.7614(b′Fχ)] [ b′Fχ

η ) η0b′F

(22)

In eq 22, η0 is the dilute gas viscosity, F is the molar density, b′ is the covolume, and χ is the value of the equilibrium radial distribution function at a distance equal to the molecular diameter from the center of an individual molecule, for which a number of coefficients in its virial expansion are known:8 χ ) 1 + 0.625b′F + 0.2869(b′F)2 + 0.115(b′F)3 + 0.109(b′F)4 + ...

(23)

However, the interaction between the real molecules is not like the interactions of rigid spheres, and corrections to the theory are necessary. Enskog10 suggested that eq 22 could be used to represent the transport coefficients for a real gas over a large density range by attributing effective values to the parameters b′F and b′Fχ. In this way one can also partially account for the influence of attractive forces between the molecules. Enskog suggested that b′Fχ should be determined from compressibility experiments using the “thermal pressure” deduced from the compressibility isotherms of the substance which can be represented by8 b′Fχ )

(17)

i i

10131

V ∂P R ∂T

( )

V

-1

(24)

i

a)

∑ ∑xxa

i j ij

i

(18)

j

where xi and xj are the liquid molar fraction of the components i and j and aij is defined by aij ) (1 - δij)√aiaj

(19)

The parameter δij is an empirically determined binary coefficient characterizing the binary formed by component i and j. For this work it was considered that this parameter is equal to zero for all binary pairs. Once the molar volume was calculated by the Peng-Robinson equation of state, the following correction to the volume was applied according to Pe´neloux et al.:4 Vcorrected ) VPR +

∑cx

i i

(20)

Several correlations have been suggested in the literature for b′Fχ; however, Sheng et al.8 tried another approach with the Peng-Robinson equation of state incorporated into eq 22. Substituting the Peng-Robinson equation of state (eq 5) into eq 24: b′Fχ )

1 ∂a(T) V R R V-b V(V + b) + b(V - b) ∂T

[

(

(25) where

[

m + m2(1 - Tr0.5) ∂a(T) ) -a(Tc) ∂T (TTc)0.5

]

(26)

The calculation of the term b′F can be performed by considering the following limit for eq 23: lim χ ) 1

i

Ff0

In eq 20, xi is the pseudocomponent’s liquid molar fraction, VPR is the liquid molar volume of the mixture calculated with

)] - 1

Therefore

(27)

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Ind. Eng. Chem. Res., Vol. 48, No. 22, 2009

lim b′χ ) b′

(28)

Ff0

ωmix )

∑ ∑xxω

(35)

(ωi + ωj) 2

(36)

∑ ∑xxT

(37)

i j

i

Considering that F ) 1/V, an expression for b′χ can be obtained from eq 25: b′χ ) -

∂a(T) b 1 + 2 2 ∂T 1 Fb R(1 + 2bF - b F )

(

)

(29)

ωij ) Tcmix )

ij

j

i j cij

i

j

Tcij ) (TciTcj)0.5

From eq 28 lim b′χ ) Ff0

1 ∂a(T) + b ) b′ R ∂T

(

)

(30)

Pcmix )

∑ ∑xxP

i j cij

i

(38) (39)

j

Pcij ) (PciPcj)0.5

(40)

Therefore, the following equation can be obtained for b′F: 1 ∂a(T) 1 b′F ) b V R ∂T

[

(

)]

(31)

Equations 25 and 31 can be solved by employing the calculations previously performed in the density model. To improve the accuracy of eq 22 for real fluids, an empirical parameter (AP) was added to absorb the shortcomings of the theoretical model: η ) η0b′F

1 +A [ b′Fχ

P

]

+ 0.7614(b′Fχ)

(32)

AP is a substance- and temperature-dependent parameter and independent of pressure. To calculate the parameter η0 from eq 32, the model proposed by Sheng et al.8 uses the dilute gas viscosity of the pure components of the mixture instead of using their liquid viscosity to model the mixture viscosity in the liquid phase. In this paper, it is proposed that a better model for dense fluids and liquids can be created using the liquid viscosity of the mixture components for the calculation of η0. The value of η0i for each pseudocomponent was calculated using a relationship for the liquid-phase viscosity developed by Eyring et al.:18 η0i ) 0.1

( )

Tbi Nh exp 3.8 Vi T

(33)

In the equation above, N is the Avogadro number (6.023 × 1023 g/mol), h is the Planck constant (6.624 × 10-27 g cm2/s), Vi is the pseudocomponent’s liquid molar volume in cm3/mol, Tbi is the normal boiling point in kelvin of the pseudocomponent, T is the temperature in kelvin, and η0i is given in pascal seconds. The viscosity of the liquid mixture was calculated employing the following expression:19 ln η0mix )

∑ x ln η i

0i

(34)

i

For the calculation of b′F and b′Fχ it was necessary to know the values of the critical temperature of the mixture (Tcmix), critical pressure of the mixture (Pcmix), and the mixture accentric factor (ωmix). The following mixing rules given by Smith and Teja20 were applied. These simplified critical point calculations provided a convenient way to estimate critical properties; although, alternative estimates based on an equation of state are also possible.21

The parameter AP was calculated for each temperature by employing the viscosity experimental data at different temperatures obtained from Schlumberger Ltd.22 As it will be seen in Results and Discussion this parameter was not constant for each temperature; therefore, these values could be adjusted to obtain an expression that showed how AP varied with the temperature. 3. Results and Discussion 3.1. Results of the Density Model. The first part of the calculation consisted of the estimation of the critical properties, accentric factor, and molecular weight of each pseudocomponent. Next, using these properties, the liquid volume of each pseudocomponent was calculated at 15.55 °C and 101325 Pa, conditions at which the actual specific gravities of each pseudocomponent were provided in the bitumen assay. The liquid volume was calculated using the Peng-Robinson equation of state (eq 6), the tool SolVer from Microsoft EXCEL, which uses the generalized reduced gradient (GRG2) algorithm for optimizing nonlinear problems, was used to solve this equation. Once the liquid volumes of each pseudocomponent have been calculated, they were compared with the actual ones to obtain the difference between them. These differences were used in eq 20 to calculate the liquid molar volume of the whole mixture. Table 2 and Table 3 summarize the results. The next step was the calculation of the liquid molar volume of the whole mixture; this was done following a procedure similar to the previous one, but with the difference that the Peng-Robinson equation of state parameters are now calculated by applying the mixing rules shown in eqs 17 and 18. The programming was done in such a way that, for each temperature and pressure provided, it was only necessary to solve the Peng-Robinson equation of state (using the tool SolVer from Microsoft EXCEL) to obtain the liquid volume of the mixture. Once this has been done, eq 20 was applied to correct the calculated volume of the Peng-Robinson equation of state. The liquid density of the mixture could be found by means of eq 21. The bitumen assay provided the density of the whole mixture at 15.55 °C: 1008.6 kg/m3. The value that the simulation on this work provided was 1010.74 kg/m3. The percent absolute error (%AE), which is the absolute value of the difference between the experimental and estimated value divided by the experimental one and multiplied by 100, was 0.21%. The results of this model were also compared with five experimental densities at different temperatures for Athabasca bitumen given by Strauz and Lown.23 The percent average absolute error (%AAE) between the experimental data and the model was 0.83%.

Ind. Eng. Chem. Res., Vol. 48, No. 22, 2009

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Table 2. Critical Properties and Accentric Factor of the Different Pseudocomponents of the Athabasca Bitumen i pseudocomponent Tc, °C Pc × 10-6, Pa 1 2 3 4 5 6

C5-320 320-400 400-650 650-700 700-975 975+

349.85 381.85 475.85 548.85 630.85 717.85

2.79 2.67 2.18 1.66 1.34 1.31

Tbr

ω

MW, g/mol

0.695 0.703 0.734 0.770 0.798 0.804

0.407 0.445 0.593 0.785 0.969 1.034

125.32 143.07 213.02 310.21 437.16 498.04

Figure 1 shows the liquid density of the bitumen calculated at different temperatures as well as the corresponding experimental points given by Strauz and Lown;23 all the densities were related to atmospheric pressure (101325 Pa). Density values from methods proposed by Crane Co.,24 Go´mez,25 Valderrama and Abu-Sharkh,26 and Noor27 were calculated in order to compare them with the present model. In the case of the procedure proposed by Crane,24 the specific gravity of the bitumen is the only parameter which is needed for the calculation. The equation proposed by Go´mez25 uses the specific gravity and the average boiling point temperature of the bitumen as parameters. The methods given by Valderrama and Abu-Sharkh26 and Noor27 need a rigorous characterization scheme to evaluate pseudocritical properties of the bitumen, which is described by a series of correlations given in their works. Table 4 shows the experimental density values and the %AAE of each of the compared methods. 3.2. Results of the Viscosity Model. The first part of this calculation was the estimation of the value of η0 for each pseudocomponent that forms the mixture using eq 33. Then, eq 34 was used to obtain the viscosity of the liquid mixture. The next step was to calculate the critical properties and the accentric factor of the mixture by using the mixing rules provided in eq 35-40. The volume of the liquid mixture, which was necessary to calculate the parameters of eq 32, was obtained in a way similar to how was done in the calculation of the density. Using the experimental viscosities at different temperatures from Schlumberger Ltd.,22 the parameter AP for each temperature could be obtained. Specific details regarding the testing and state of the bitumen sample analyzed by Schlumberger Ltd.22 were not offered in their work. In this case, 27 experimental data were used; therefore, 27 different values of AP were found for each case. These parameters were correlated with the temperature and the following expression was obtained: AP ) 0.0642T - 47.087

(41)

where T is in kelvin. Thus, eq 32 can be rewritten as

[

ηmix ) η0mixbFmix

1 + (0.0642T - 47.087) + bFχmix 0.7614(bFχmix)

]

(42)

The experimental viscosities at different temperatures given by Schlumberger Ltd.22 were compared with the results from

Figure 1. Results of the density mathematical simulation model compared with the experimental data. Table 4. Comparison of Density Prediction Methods F, kg/m3 exptl (Strauz T, °C and Lown23) 0 15.55 50 100 150

this work

Crane Valderrama and Co.24 Go´mez25 Abu-Sharkh26 Noor27

1025 1013.51 1017.03 1018.65 1008.6 1010.74 1008.60 1008.60 995 1003.92 989.67 986.35 983 992.13 961.54 954.05 968 977.67 932.55 921.74 %AAE ) 0.83 1.43 1.84

902.76 883.19 841.35 784.13 730.79 16.91

1240.62 1228.51 1201.70 1162.78 1123.85 19.60

the eq 42. The %AAE between the experimental data and the model was 6.93%. Figure 2 shows values of the liquid viscosity of the bitumen calculated at different temperatures compared with the respective experimental values; all the viscosities were related to atmospheric pressure. This thermodynamic model can also calculate the viscosity of the mixture at pressures different than atmospheric. This calculation was performed by employing the same procedure that was explained before and since the parameter AP is independent of the pressure, eq 42 is valid to obtain viscosities at pressures different than the atmospheric. Figure 3 shows the viscosities predicted by the model at different temperatures and pressures. Viscosity values from methods proposed by Mehrotra,28 Singh et al.,29 Glasø,30 and Guo et al.15 were calculated in order to compare them with the present model. The first method depends on the density of the fluid at 25 °C and the temperature. In the case of the method proposed by Singh et al.,29 it is necessary to have a known viscosity value at a determined temperature. The equation proposed by Glasø30 can be used to approximate the viscosity of crude oils as a function of the temperature, the only parameter which is required is the crude oil’s API gravity. The Guo et al.15 correlation is based on the geometric similarity of the P-V-T and T-η-P diagrams, and no adjustable parameters are involved. Table 5 shows the experimental viscosity values and the %AAE of each of the compared methods.

Table 3. Peng-Robinson Equation of State Parameters, Compressibility Factor, and Calculated and Actual Molar Volumes (at 15.55 °C) of the Different Pseudocomponents of the Athabasca Bitumen i

pseudocomponent

A

B

Z

VPR × 104, m3/mol

Vactual × 104, m3/mol

ci × 106, m3/mol

1 2 3 4 5 6

C5-320 320-400 400-650 650-700 700-975 975+

0.132 0.160 0.301 0.563 0.979 1.304

0.0060 0.0066 0.0093 0.0135 0.0183 0.0205

0.0067 0.0073 0.0100 0.0142 0.0190 0.0212

1.61 1.75 2.37 3.37 4.52 5.04

1.56 1.72 2.36 3.33 4.46 4.68

4.59 3.25 1.13 3.59 6.13 3.55

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Ind. Eng. Chem. Res., Vol. 48, No. 22, 2009 Table 5. Comparison of Viscosity Prediction Methods η, Pa s × 103 T, °C

Figure 2. Results of the viscosity mathematical simulation model compared with the experimental data.

120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250

Schlumberger’s data22 70 58 50 40 35 30 25 22 20 18 17 16 15 14 12.5 11 10 9 8 7.5 7 6.5 6 5 4.5 4.2 4 %AAE )

Figure 3. Viscosities from the mathematical simulation model at different pressures and temperatures.

3.3. Discussion of Results. The results presented for the density of the liquid phase showed the logical tendency to decrease as the temperature increases. The %AAE between the experimental data and the model was calculated to be around 0.83%. This error resulted smaller as compared to the ones corresponding to other proposed methods, but in the same order of magnitude as the methods proposed by Crane Co.24 and Gomez.25 In the case of the viscosity model, it demonstrated good accuracy between the experimental and the calculated data. The %AAE between the calculated and experimental viscosities resulted to be smaller as compared with the other compared methods and with a magnitude order similar to the one corresponding to the method proposed by Singh et al.29 The model is also able to predict the viscosity behavior at different pressures. At very high values, pressure has a significant effect on viscosity for oils.31 For example, Fuller31 shows that viscosity may change by a factor of 2 when pressure changes from atmospheric to 3000 psi. This effect can be observed in Figure 3: the change in viscosity is small at low pressures; however, at high pressures (>100 atm) the increment of viscosity with respect to pressure starts to be more evident. 4. Conclusions The basic principle applied in the density model developed for this work was the volume translation developed by Pe´neloux et al.,4 which improved the results of the Peng-Robinson equation of state. The development of the viscosity model was based on Enskog’s theory as modified by Sheng et al.8

this work

Singh et al.29

Mehrotra28

Guo el al.15

Glasø30

62.38 54.43 47.61 41.73 36.65 32.27 28.48 25.19 22.34 19.86 17.70 15.82 14.17 12.74 11.48 10.38 9.41 8.57 7.82 7.17 6.59 6.08 5.63 5.23 4.87 4.56 4.28

77.40 63.88 53.29 44.90 38.19 32.76 28.34 24.70 21.68 19.16 17.04 15.24 13.71 12.40 11.26 10.28 9.43 8.68 8.02 7.44 6.92 6.46 6.05 5.69 5.36 5.06 4.79

150.38 119.05 95.50 77.55 63.69 52.87 44.32 37.49 31.98 27.50 23.82 20.78 18.24 16.11 14.30 12.77 11.46 10.32 9.35 8.49 7.75 7.09 6.52 6.01 5.55 5.15 4.78

9.73 9.11 8.53 7.99 7.49 7.03 6.60 6.20 5.83 5.48 5.16 4.86 4.58 4.31 4.07 3.84 3.63 3.43 3.24 3.07 2.91 2.75 2.61 2.48 2.35 2.23 2.12

73.01 65.36 58.73 52.96 47.92 43.50 39.59 36.14 33.08 30.35 27.91 25.72 23.75 21.98 20.38 18.93 17.61 16.41 15.32 14.32 13.41 12.57 11.80 11.09 10.43 9.83 9.00

6.92

8.35

42.00

66.44

70.97

The results presented for the density of the liquid phase resulted with a small %AAE between the experimental data and the model (less than 1%). In the case of the viscosity model, the presented results also showed good accuracy in terms of error estimation. Significant improvement in density and viscosity predictions for bitumen was observed from the developed models in this work as compared to previously published correlations and models. This work also demonstrates that the parameter AP from eq 32, at least for the case of Athabasca bitumen, is temperature-dependent. An advantage of the thermodynamic viscosity model is the possibility of estimating the bitumen viscosity at different pressures, since usually other models are pressure-independent. The model was not tested at conditions close to the critical point of the fluid, and it is expected that the viscosity predictions would be less accurate at those conditions. Overall, this study can have real-world implications in the near future due to the significance of the density and viscosity for reservoir characterization, primary and enhanced oil recovery processes, and design of upgrading facilities. Acknowledgment This work was supported in part by the National Council for Science and Technology of Mexico, The Alberta Ingenuity Centre for In Situ Energy funded by the Alberta Ingenuity Fund and the industrial sponsors: Shell International, ConocoPhillips, Nexen Inc, Total Canada and Repsol-YPF and The Schulich School of Engineering at the University of Calgary, Canada. Literature Cited (1) Lorı´a, H.; Pereira-Almao, P.; Scott, C. E. A Model To Predict the Concentration of Dispersed Solid Particles in an Aqueous Medium Confined inside Horizontal Cylindrical Channels. Ind. Eng. Chem. Res. 2009, 48, 4088–4093.

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ReceiVed for reView June 25, 2009 ReVised manuscript receiVed August 19, 2009 Accepted August 20, 2009 IE901031N