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Ind. Eng. Chem. Res. 2003, 42, 3824-3830
A New Model for Calculating the Viscosity of Pure Liquids at High Pressures Rosana J. Martins,†,‡ Ma´ rcio J. E. de M. Cardoso,*,† and Oswaldo E. Barcia† Laborato´ rio de Fı´sico-Quı´mica de Lı´quidos e Eletroquı´mica, Departamento de Fı´sico-Quı´mica, Instituto de Quı´mica, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Bloco A, sala 408, Cidade Universita´ ria, cep. 21949-900, Rio de Janeiro, RJ, Brazil, and Departamento de Fı´sico-Quı´mica, Instituto de Quı´mica, Universidade Federal Fluminense, Outeiro de Sa˜ o Joa˜ o Batista s/n, cep. 24020-150, Nitero´ i, RJ, Brazil
In this paper we present a new model for correlating the dynamic viscosity of Newtonian liquids at high pressures. The proposed model is based on Eyring’s absolute rate theory for liquid viscosity and considers the activation energy for viscous flow as being a thermodynamic free energy. The viscosity of the system is described by a combination of a reference term, given by the Chapman-Enskog theory, and a deviation contribution. A virial-type expansion in pressure and a term comprising an expression for the residual Helmholtz free energy of the system account for the deviation from the nonattracting rigid sphere model viscosity behavior. Three cubic equations of state, Peng-Robinson, Soave-Redlich-Kwong, and Peng-Robinson-Stryjek-Vera, have been tested for evaluating the residual Helmholtz free energy. The model requires only two adjustable parameters for each pure liquid, at each temperature. The parameters have been determined by fitting literature viscosity data of 49 different pure liquid compounds from pressures of 0.1-250 MPa within the reduced temperature interval of 0.4-0.7. The performance of the model has been found to be insensitive to the choice of the equation of state, except at pressures above 100 MPa for which only the Soave-Redlich-Kwong equation of state has been able to describe the volumetric behavior of the liquids. The studied liquids are n-alkanes, substituted alkanes, n-alkenes, cyclic alkanes, aromatics, alcohols, esters, 1-butylamine, argon, nitrogen, oxygen, ammonia, and water. The calculated values are in good agreement with the experimental ones. The value of the overall average absolute deviation, for the 4380 data points correlated with the present model, is 1.22%. 1. Introduction The transport properties of fluids are of fundamental importance for the development and optimization of industrial processes. A great amount of models for estimating pure fluids and mixture viscosities have appeared in the literature. Thorough discussions about viscosity calculation may be found in the works of Poling et al.1 and Monnery et al.2 Most of the methods available for estimation of the viscosity of pure liquids and liquid mixtures are based on the principle of corresponding states,3-5 on the absolute rate theory of Eyring and co-workers,6 and, more recently, on molecular dynamic calculations.7 Nevertheless, few of them are appropriate to describe the viscosity of fluids under high-pressure conditions and, frequently, those that are suitable for pure fluid calculations cannot be readily extended to mixtures. The most reliable mixture viscosity models require the viscosity of the pure components as an input parameter.2 Therefore, there is an increasing demand of accurate models for pure liquid viscosities, particularly under high-pressure conditions. On the basis of the similarity between the P-V-T and T-η-P diagrams, Guo et al.8 developed models for * To whom correspondence should be addressed. Tel.: (55) (21) 25627172. Fax: (55) (21) 25627265. E-mail: marcio@ iq.ufrj.br. † Universidade Federal do Rio de Janeiro. ‡ Universidade Federal Fluminense. Tel.: (55) (21) 26207769.
the pressure dependence of the viscosity of pure fluids and mixtures. The viscosity equations are obtained by means of Patel-Teja9 and Peng-Robinson10 cubic equations of state, in such a way that P and T are interchanged in the expressions, the viscosity replaces the molar volume, and the gas constant is substituted by a model parameter. The model derived from the PengRobinson equation does not require any adjustable parameter and is applicable to hydrocarbons and respective mixtures. It is also suitable for high- as well as low-pressure viscosity calculations. For pure nalkanes, an overall average absolute deviation (AAD) of 6.18% was obtained with the PR viscosity equation. Using classical and fluid mechanic concepts, Quin˜ones-Cisneros et al.11 presented a new method for viscosity modeling, called f theory. In f theory, viscosity is treated as a mechanical property composed of a dilute gas contribution and a friction term. Repulsive and attractive pressure terms from cubic equations of state of the van der Waals family, like Soave-RedlichKwong12 and Peng-Robinson-Stryjek-Vera,13 are used in the description of the friction contribution. Five adjustable parameters are necessary for each pure fluid. This model was tested with hydrocarbons and respective mixtures over wide temperature and pressure ranges. In the case of pure fluids, the overall AAD obtained was 2.33%. Subsequently, this model was generalized14 by means of corresponding states concepts. Besides one adjustable parameter for each pure fluid, the generalized version proposed by the authors14 contains 16
10.1021/ie021017o CCC: $25.00 © 2003 American Chemical Society Published on Web 07/02/2003
Ind. Eng. Chem. Res., Vol. 42, No. 16, 2003 3825
universal constants. When the models were tested for n-alkanes, deviations of 2.56%, 2.83%, and 2.44% were obtained with the Peng-Robinson-, Soave-RedlichKwong-, and Peng-Robinson-Stryjek-Vera-based models, respectively. Assael et al.15,16 presented a method for simultaneous correlation and prediction of self-diffusion, viscosity, and thermal conductivity of dense fluids. This method is based on the exact hard-sphere theory of transport properties and assumes that the n-alkane transport coefficients are directly proportional to the smooth hardsphere coefficients. For pure n-alkanes,16 the overall root-mean-square percentage deviation is 2.8% for correlation and 6% for prediction. Calculated viscosity and thermal coefficients for binary, ternary, and quaternary n-alkane mixtures17 are in close agreement with experimental values. All experimental points are fitted to within 5%. Monnery et al.18 developed a semitheoretical threeparameter model based on square-well theory for simultaneous correlation of liquid- and gas-phase viscosities. The model parameters are obtained from viscosity data correlation and can be generalized with corresponding states expressions and group contribution concepts.19,20 The model correlates gas and liquid viscosities of a wide variety of compounds, including nonpolar, polar, and hydrogen bonding, with average deviations of 0.5% and 1.8%, respectively.20 The aim of this work is to present a new model for viscosity calculation of pure liquids, from atmospheric up to high pressures. The presentation of the rest of this paper is organized as follows. In section 2, we develop the main equations and discuss the proposed model for calculating pure liquid viscosity at high pressures. In section 3, the results obtained with the model for 49 different liquids are presented and discussed. Finally, in section 4, we present our conclusions. 2. Viscosity Model for Liquids at High Pressures According to the absolute rate theory approach of Eyring and co-workers,6 the viscosity of a Newtonian liquid is given by
η)
( )
∆F hq pN exp RT V h
(1)
where η is the dynamic viscosity of the liquid, p is Planck’s constant, N is Avogadro’s number, V h is the molar volume of the liquid, ∆F h q is the molar energy of activation for flow, R is the gas constant, and T is the absolute temperature. For the particular case of a fluid in the reference state, eq 1 can be written in the form
η0 )
( )
∆F h 0q pN exp RT V h0
(2)
where η0 and V h 0 are respectively the dynamic viscosity and molar volume of the fluid in the reference state. Taking the ratio η/η0 from eqs 1 and 2, one obtains
(
)
V h0 ∆F h q - ∆F h 0q η ) η0 exp RT V h
(3)
The basic hypothesis of the model is that the differh 0q, is ence of the molar activation energies, ∆F h q - ∆F equal to the difference of the molar thermodynamic free energies:21
∆F h q - ∆F h 0q ) ∆F h )F h -F h0
(4)
Substituting eq 4 into eq 3 leads to
V h0 ∆F h η ) η0 exp RT V h
( )
(5)
We have chosen the Helmholtz free energy, A, as the thermodynamic potential to be used in eq 5; thus
V h0 ∆A h η ) η0 exp RT V h
( )
(6)
where ∆A h is the difference between the molar Helmholtz free energy of the fluid at given T and P and that of the fluid in the reference state. The ratio between the molar volumes in eq 6 can be calculated by means of a virial-type expansion:
V0/V h ) 1 + B1(T) P + B2(T) P2
(7)
where B1(T) and B2(T) are temperature-dependent adjustable parameters, which are characteristic of each pure liquid. We have considered the reference state viscosity, η0, as the viscosity of a nonattracting rigid sphere gas (dilute gas) in the same temperature of the pure fluid. According to the Chapman-Enskog equation,1 η0 is given by
η0 ) 26.69 × 10-7(MT)1/2/σ2
(8)
where M is the molar mass of the fluid and σ is the hardsphere diameter of the molecule:1
σ)
3b (2πN )
1/3
(9)
Substitution of eqs 7 and 8 into eq 6 gives
η ) 26.69 × 10-7
(MT)1/2 [1 + B1(T) P + B2(T) P2] σ2 A hR (10) exp RT
( )
where A h R is the residual molar Helmholtz free energy, which is the difference between the molar Helmholtz free energy of the fluid and that of an ideal gas at the same conditions of temperature and pressure. To evaluate the residual molar Helmholtz free energy, we have used different cubic equations of state. Cubic equations of state represent a compromise between simplicity and an accurate representation of both liquidand vapor-phase behaviors.22 Cubic equations of state of the van der Waals family can be represented by means of the general expression1
P)
a RT - 2 V h -b V h + ubV h + wb2
(11)
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Substituting eq 16 into eq 10 leads to the following expression for calculating the dynamic viscosity of pure liquids:
Table 1. Characteristic Parameters for the Cubic Equations of State Employed in This Worka parameter EOS
u
w
b
a
2 -1 0.0778RTc
PR10
Pc
2
2
0.45724R Tc [1 + fω(1 - Tr1/2)]2 Pc where fω ) 0.37464 + 1.5422ω 0.26992ω2
SRK12
0 0.08664RTc 0.42724R2T 2 c
1
Pc
[1 + fω(1 - Tr1/2)]2
Pc
where fω ) 0.48 + 1.574ω 0.176ω2 PRSV13 2 -1 0.0778RTc
Pc
0.45724R2Tc2 R Pc
a PR ) Peng-Robinson; SRK ) Soave-Redlich-Kwong; PRSV ) Peng-Robinson-Stryjek-Vera; EOS ) equation of state; Pc ) critical pressure; ω ) acentric factor; Tr ) reduced temperature; κ1 ) constant characteristic of the substance.13
The parameters a and b may be determined by using the mathematical conditions which define the critical point,1 while the parameters u and w are characteristic of each cubic equation of state. Table 1 summarizes the u and w values, as well as the expressions for the calculation of a and b, according to the cubic equation of state employed: Peng-Robinson10 (PR), Soave-Redlich-Kwong12 (SRK), and PengRobinson-Stryjek-Vera13 (PRSV). One can rewrite eq 11 in terms of the compressibility factor, Z ) PV h /RT, as follows: 2
Z - (1 + B* - uB*)Z + (A* + wB*2 - uB* - uB*2)Z A*B* - wB*2 - wB*3 ) 0 (12) where
A* ) aP/R2T 2
(13)
B* ) bP/RT
(14)
and
We have found the roots of the cubic equation of state, eq 12, at the desired conditions of temperature and pressure and selected the lowest real root as the compressibility factor of the liquid. The expression for calculating the residual molar Helmholtz free energy can be written as follows:1
A hR )
a bxu2 - 4w
2Z + B*(u - xu - 4w) 2Z + B*(u + xu2 - 4w)
-
RT ln(Z - B*) (15) Rearranging eq 15 and introducing the notation k )
xu2-4w, one obtains
{[
]
2ZRT + bP(u - k) A hR ) ln RT 2ZRT + bP(u + k)
a/bkRT
×
[
] }
(17)
The thermodynamic properties of the pure fluids, such as the critical temperature, critical pressure, and acentric factor, are obtained from the literature.1,13
The proposed model has been used to correlate the experimental viscosity, 4380 data points, of 49 different liquids under the reduced temperature interval of 0.40.7 and pressures ranging from 0.1 to 250 MPa. The studied liquids were n-alkanes (C1-C16 and C18), ialkanes (C4, C5, and C8), n-alkenes (C6-C8), cyclic alkanes (cyclopentane, cyclohexane, methylcyclohexane, and ethylcyclohexane), aromatics (benzene, toluene, methylbenzene, n-butylbenzene, o-, m-, and p-xylene, 1,3,5-trimethylbenzene, and 1-methylnaphthalene), nalcohols (C1-C4), esters (n-propyl acetate and n-butyl acetate), 2-propanol, 1-butylamine, argon, nitrogen, oxygen, ammonia, and water. For each liquid, the model parameters B1(T) and B2(T) have been determined by least-squares fitting of the experimental viscosity data at a given temperature. The objective function, Fobj, used in the determination of the parameters was ND
Fobj )
(ηiexp - ηical)2 ∑ i)1
}
RT ZRT - bP (16)
(18)
where ND is the number of experimental data at a fixed temperature, ηi is the dynamic viscosity of the liquid under a given pressure, the superscript exp represents experimental data, and cal represents the viscosity values calculated by means of eq 17. A detailed compilation, containing the correlation results for all of the studied liquids, is available as the Supporting Information. This material consists of a table, which presents, for each liquid, the total number of correlated points, the pressure range, the values of the adjustable parameters for a given temperature, and the AAD between experimental and calculated viscosities. The values of the AADs were calculated by the expression
AAD (%) ) 100 ×
2
ln
{
3. Results and Discussion
where R ) [1 + κ(1 - Tr1/2)]2 κ ) fω + k1[(1 + Tr1/2)(0.7 - Tr)] fω ) 0.3789 + 1.4897ω 0.1713ω2 - 0.0197ω3
3
(MT)1/2 (1 + B1P + B2P2) × σ2 2ZRT + bP(u - k) a/bkRT RT ZRT - bP 2ZRT + bP(u + k)
η ) 26.69 × 10-7
1
ND
∑
ND i)1
|ηiexp - ηical| ηiexp
(19)
where ND is the total number of data points for a pure liquid at a fixed temperature, ηiexp represents the experimental value of the dynamic viscosity at a given pressure and temperature, and ηical is the value calculated by means of eq 17 at the same temperature and pressure conditions. It can be observed from the table presented as the Supporting Information that the deviations are nearly the same for the three different cubic equations of state
Ind. Eng. Chem. Res., Vol. 42, No. 16, 2003 3827 Table 2. Results of the Correlation of High-Pressure Viscosity of Liquids with the Present Model, with the SRK12 Equation of State family
NC
AAD (%)
alkanes alkenes aromatics condensed aromatics alcohols esters amines water nitrogen oxygen argon ammonia
24 3 8 1 5 2 1
1.30 0.91 1.30 1.41 0.78 0.67 2.18 1.11 0.75 2.05 1.56 1.30
a AAD ) average absolute deviation; N ) number of compoC nents of each family studied in the present work.
Figure 2. Pressure dependence of the dynamic viscosity of n-alkenes at 320 K: (9) n-hexene, (b) n-heptene, and (2) n-octene are the viscosity values recommended by Stephan and Lucas,24 and the solid line represents the correlation results obtained by means of eq 17 combined with the SRK equation of state.
Figure 1. Pressure dependence of the dynamic viscosity of n-alkanes at 323.15 K: (9) n-hexane, (b) n-octane, and (2) n-decane are the experimental viscosity values,23 and the solid line represents the correlation results obtained by means of eq 17 combined with the SRK equation of state.
used in the calculation. Nevertheless, for pressures higher than 100 MPa, the SRK equation was the only one that described the liquid state satisfactorily. It is important to remark that all of the deviations listed in the Supporting Information are inferior or close to the experimental errors reported in the literature for nonpolar as well as for polar liquids. Table 2 summarizes the results obtained with the present model, with the SRK equation of state, for all of the studied families. For n-alkanes, we have obtained an AAD of 1.30%, which is comparable with the value of 1.9% obtained by experimental data correlation by Monnery et al.20 The present model results are better than the ones obtained by Guo et al.8 (5.63%) and Quin˜ones-Cisneros et al.11 (2.83%). Figures 1-7 illustrate the applicability of the model proposed in this work, when the SRK equation of state is used to determine the residual molar Helmholtz free energy. Figure 1 shows the results obtained for some compounds of the n-alkanes family at 323.15 K and pressures ranging from 0.1 to 250 MPa. The agreement
Figure 3. Pressure dependence of the dynamic viscosity of benzene and some n-alkyl aromatics at 300 K: (9) benzene, (b) toluene, and (2) ethylbenzene are the viscosity values recommended by Stephan and Lucas,23 and the solid line represents the correlation results obtained with eq 17 combined with the SRK equation of state.
between experimental23 and calculated viscosities is rather satisfactory. The AADs obtained are 4.6% for n-hexane, 1.1% for n-octane, and 2.2% for n-decane. For pressures up to 100 MPa, the AADs are lower than 3% for most of the linear hydrocarbons listed in the Supporting Information. The experimental behavior of the dynamic viscosity of n-alkenes at the temperature of 320 K and in the pressure interval from 0.1 to 50 MPa is illustrated in Figure 2. The results obtained with the proposed model are also represented. A good agreement between the values recommended by Stephan and Lucas24 and those
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Figure 4. Pressure dependence of the dynamic viscosity of n-alcohols at 298.15: (9) methanol, (b) ethanol, (2) 1-propanol, and (1) 1-butanol are the experimental viscosity values,26 and the solid line represents the correlation results obtained by means of eq 17 combined with the SRK equation of state.
Figure 6. Pressure dependence of the dynamic viscosity of 1-methylnaphthalene 303.15 K: b are the experimental viscosity values,27 and the solid line represents the correlation results obtained by means of eq 17 combined with the SRK equation of state.
Figure 5. Pressure dependence of the dynamic viscosity of 1-butylamine at 298.15: b are the experimental viscosity values,26 and the solid line represents correlation results obtained by means of eq 17 combined with the SRK equation of state.
Figure 7. Pressure dependence of the dynamic viscosity of esters at 320 K: (b) n-propyl acetate and (9) n-butyl acetate are the viscosity values recommended by Stephan and Lucas,24 and the solid line are the correlation results obtained by means of eq 17 combined with the SRK equation of state.
calculated with the present model is observed. The AADs obtained are 0.83% for n-hexene, 0.61% for n-heptene, and 1.07% for n-octene. Figure 3 compares the recommended values24 for the viscosities of aromatic hydrocarbons at 300 K and pressures from 0.1 to 40 MPa with those obtained with the SRK viscosity based model proposed in this work. The AADs are lower than 0.5% for all of the compounds represented in Figure 3. The deviations obtained for o-, m-, and p-xylene are 1.55%, 0.66%, and 0.98%, respectively. The poorest agreement for aromatics has been observed for 1,3,5-trimethylbenzene,25 for which the AAD is 2.31% at 313.15 K and pressures from 0.1 to 50 MPa.
Besides the satisfactory results obtained for nonpolar compounds, the model showed to be rather adequate for viscosity modeling of polar liquids, as presented in Figures 4 and 5. Figure 4 presents the experimental26 and calculated values for the viscosities of n-alcohols at 298.15 K and under pressures ranging from 0.1 to 80 MPa. The calculated viscosity values deviated less than 0.5% from the experimental data. As showed in Figure 5, the model also correlates quite well the experimental viscosity data of 1-butylamine.26 In this case, the AAD is lower than 1%. The model also provides good results for 1-methylnaphthalene, an aromatic hydrocarbon with condensed rings. Figure 6 shows the experimental27 and calculated
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Figure 8. Relationship between the SRK model parameter B1 and the size of the linear chain: (1) linear hydrocarbons at 313.15 K, (9) linear hydrocarbons at 323.15 K, (b) linear hydrocarbons at 340 K; ([) benzene and n-alkyl aromatics at 320 K; (2) n-alcohols at 298.15 K. The solid line represents the linear fitting.
Figure 9. Relationship between the SRK model parameter B2 and the size of the linear chain: (1) linear hydrocarbons at 313.15 K, (9) linear hydrocarbons at 323.15 K, (b) linear hydrocarbons at 340 K; ([) benzene and n-alkyl aromatics at 320 K; (2) n-alcohols at 298.15 K. The solid line represents the linear fitting.
values for the viscosity of 1-methylnaphthalene at 303.15 K and along the pressure interval of 0.1-100 MPa. The AAD is 1.41%. The correlation results for n-propyl acetate and nbutyl acetate are presented in Figure 7. The proposed model agrees quite well with the recommended values.24 For n-propyl acetate, the AAD obtained is 0.63%, and for n-butyl acetate, the AAD is 0.54%. Besides the good agreement between experimental and calculated viscosity values, for the different families studied in this work, an interesting behavior is also observed for parameters B1(T) and B2(T). As shown in Figures 8 and 9, for the linear alkanes (from butane to octadecane) at 313.15 and 323.15 K and (from butane to dodecane) at 340 K, for benzene, toluene, ethylbenzene, and n-butylbenzene at 320 K, and for the n-alcohols (from methanol to butanol) at 298.15 K, it is possible to establish a linear relationship between the model parameters and the number of the carbon atoms in the linear chain (for the n-alkyl aromatics, we have used the number of carbon atoms in the substituent chain).
the structure of the molecule, suggesting that the model proposed in this work could be generalized by means of a group contribution approach. Finally, we remind everyone that other expressions for dynamic viscosity calculation could be derived, following the approach outlined in this work, if one adopts other types of cubic equations of state. We are now extending the model for viscosity calculation of high-pressure liquid mixtures.
4. Conclusions A new model is proposed for pure liquid dynamic viscosity at high pressures. The proposed model is applicable for both nonpolar and polar compounds. The results obtained by means of PR,10 SRK,12 and PRSV13 equations of state are comparable. Nevertheless, above 100 MPa only the SRK-based equation gives a reasonable description of the viscosity behavior of the liquid. The model has been tested with 49 different substances in the reduced temperature interval of 0.4-0.7 and pressures from 0.1 to 250 MPa. The agreement between experimental viscosity data and the values calculated with the proposed model are reasonably good. According to the results obtained with n-alkanes, n-alcohols, and substituted n-alkyl aromatics, the regressed values of the model parameters are related to
Acknowledgment The authors are grateful to the Brazilian agencies CAPES, CNPq, FAPERJ, FINEP, FJPF, and FUJB for financial support. Supporting Information Available: This material is a table that contains the correlation results obtained for the liquids used to test the model developed in this work. It presents, for each liquid, the total number of correlated points, the pressure range, the values of the adjustable parameters for different temperatures, and the AAD between experimental and calculated viscosities. The table also reports to the original literature source of the experimental viscosity data. This material is available free of charge via the Internet at http:// pubs.acs.org. Nomenclature a ) cubic equation of state parameter, eq 11 A* ) modified cubic equation of state parameter, eq 13 AAD ) average absolute deviation (%) b ) cubic equation of state parameter, eq 11 B* ) modified cubic equation of state parameter, eq 14 B1(T) ) temperature-dependent model parameter B2(T) ) temperature-dependent model parameter dA h ) infinitesimal variation of the molar Helmholtz free energy (J‚mol-1) ∆A h ) finite variation of the molar Helmholtz free energy (J‚mol-1)
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Fobj ) objective function ∆Fq ) molar energy of activation for flow (J‚mol-1) p ) Planck’s constant (6.626 176 × 10-34 J‚s) M ) molar mass (kg‚mol-1) N ) Avogadro’s number (6.022 045 × 1023 mol-1) ND ) number of data points P ) pressure (Pa) PR ) Peng-Robinson equation of state PRSV ) Peng-Robinson-Stryjek-Vera equation of state R ) gas constant (8.314 J‚mol-1‚K-1) SRK ) Soave-Redlich-Kwong equation of state T ) absolute temperature (K) u ) cubic equation of state parameter, eq 11 V h ) molar volume of the pure liquid (m3‚mol-1) V h 0 ) molar volume of the pure liquid in the reference state (m3‚mol-1) w ) cubic equation of state parameter, eq 11 Z ) compressibility factor Greek Letters η ) dynamic viscosity of the fluid (Pa‚s) η0 ) low-density limit dynamic viscosity (Pa‚s) κ and κ1 ) parameters of the Peng-Robinson-StryjekVera equation of state σ ) hard-sphere diameter (m) ω ) acentric factor Superscripts cal ) calculated property exp ) experimental data 0 ) reference state q ) activated state R ) residual
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Received for review December 12, 2002 Revised manuscript received May 20, 2003 Accepted May 21, 2003 IE021017O
