A new equation to correlate liquid kinematic viscosities of multicomponent mixtures

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Fluid Phase Equilibria 329 (2012) 8–21

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Fluid Phase Equilibria

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A new equation to correlate liquid kinematic viscositiesof multicomponent mixtures

Griselda E. Nava-Ríosa, Gustavo A. Iglesias-Silvaa,∗, Alejandro Estrada-Baltazara,Kenneth R. Hallb,1, Mert Atilhanc

a Departamento de Ingeniería Química, Instituto Tecnológico de Celaya, 38010 Celaya, Guanajuato, Mexicob Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, TX 77843-3122, USAc Chemical Engineering Department, University of Qatar, 2713 Doha, Qatar

a r t i c l e i n f o

Article history:Received 24 December 2011Received in revised form 28 May 2012Accepted 29 May 2012Available online 6 June 2012

Keywords:Kinematic viscosityCorrelationBinary and multicomponent mixturesEyring theoryMcAllister equation

a b s t r a c t

The new equation in this paper has its basis in quadratic mixing rules for non-random mixtures. Testsfor the equation use the liquid kinematic viscosity of 232 binary mixtures and include comparison tothree, commonly used equations in the literature. The new equation correlates the data within an abso-lute average percentage deviation of 0.33% compared to 0.52, 0.48 and 0.38% for the McAllister (MC),Dizechi–Marschall (DM), and Moumouzias–Ritzoulis (MR) equations, respectively. The new equationdescribes several ternary mixtures.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Many engineering problems in mass and heat transfer and fluidflow applications require viscosity and its dependence upon com-position. This composition dependence is generally nonlinear andcan present a maximum, a minimum, both or neither. Andrade [1]presents one of the first models to correlate the liquid viscosity ofpure liquids. He represents the logarithm of the liquid viscosity forpure substances as a function of the inverse temperature. Later,Eyring [2] provides a theoretical interpretation of the Andradeequation using a theory based upon absolute rate of reactions. Hismodel has provided the base to create other correlative models.For example, McAllister [3] uses the Eyring model and considersthe intermolecular interactions among three adjacent molecules.He suggests that if the molecule sizes vary by more than a factor of1.5, it is necessary to consider more interacting molecules. Later,Dizechi and Marschall [4] try to improve the three-body McAl-lister equation using temperature dependence similar to that ofthe Antoine equation for vapor pressures. Goletz and Tassios [5]use this dependence in the Andrade equation. Moumouzias and

∗ Corresponding author.E-mail address: [email protected] (G.A. Iglesias-Silva).

1 Current address: Texas A&M University at Qatar, Qatar.

Ritzoulis [6] extend Dizechi and Marschall [4] work using a four-body McAllister model. They test their equation with the viscosityof the methanol + water system obtaining better results than theMcAllister model. Moumouzias and Ritzoulis [6] have not testedtheir expression with other binary systems. On the other hand, Kre-glewski et al. [7] have suggested that a random mixture can have aquadratic mixing rule for the Gibbs energy. They also claim that trueinteraction parameters depend upon composition. Their assump-tions provide expressions that predict excess Gibbs energy modelsand activity coefficients accurately.

This work combines the Kreglewski et al. [7] work with theEyring theory of absolute rates to develop a new viscosity equa-tion. The new equation can correlate the liquid viscosities for binarysystems. The performance of the new equation compares to theDizechi–Marschall [3] and Moumouzias–Ritzoulis [6] equations.The new equation extends to multicomponent mixtures and corre-lates the liquid viscosity of ternary mixtures.

2. Development

One of the most popular expressions to correlate the viscosityof fluids is the Andrade equation [1]

� = AeB/T (1)

0378-3812/$ – see front matter © 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.fluid.2012.05.022


G.E. Nava-Ríos et al. / Fluid Phase Equilibria 329 (2012) 8–21 9

in which � is the dynamic viscosity and A and B are characteris-tic constants. Eyring [2] provides the theoretical meaning of theequation

� = hN

Vexp[

�G∗

RT

](2)

or

� = hN

Mexp[

�G∗

RT

](3)

in which h is the Planck constant, N is the Avogadro number, V ismolar volume, M is molecular weight, R is the universal gas con-stant, T is temperature, �G* is the activation energy and v is thekinematic viscosity. According to Eyring, liquid molecules continu-ously rearrange themselves by moving from their lattice positionsonto adjoining positions of the same magnitude in the lattice cage.The energy necessary to break the barrier of the cage is the activa-tion energy.

McAllister [3] assumes that three-body interactions exist forbinary mixtures, and that all occur in a plane. Also, he assumes thatthe rate of each individual interaction is proportional to the energyof activation, and postulates that the logarithm of the kinematicviscosity is

ln � = x31 ln �1 + 3x2

1x2 ln �12 + 3x1x22 ln �21 + x3

2 ln �2

− ln[

x1M1 + x2M2

M1

]+ 3x2

1x2 ln

[2 + M2/M1

3

]+ 3x1x2

2 ln

[1 + 2M2/M1

3

]+ x3

2 ln(

M2

M1

)(4)

where �12 and �21 are temperature dependent parameters that canbe found from experimental kinematic viscosity measurements.This equation correlates the kinematic viscosity of binary mixturesat a given temperature well when the size difference between themolecules is small, and they have comparable polarity. Accordingto Kreglewski et al. [7], a random binary mixture follows bino-mial mixing rules. Extending this concept to the Gibbs energy ofactivation

�G∗m = x3

1�G∗1 + 3x2

1x2�G∗112 + 3x1x2

2�G∗122 + x3

2�G∗2 (5)

in which �G∗112 and �G∗

122 are the cross-interaction Gibbs energiesof activation. The McAllister equation assumes random mixtures. Ifclusters form, the number of cross-interactions decreases forminga non-random mixture. In this case, the Gibbs energy of activationis

�G∗m = x3

1�G∗1 + 3x2

1x2�G∗112 + 3x1x2

2�G∗122 + x3

2�G∗2 (6)

where �G∗112 and �G∗

122 are the apparent cross-interaction Gibbsenergies of activation. Following the analysis of Kreglewski et al.[7], the clusters disappear at infinite dilution, and �G∗

112 and �G∗122

become the true �G∗112 and �G∗

122. Therefore, the apparent inter-actions are functions of the mole fraction

�G∗112 = �G∗

112(1 − k1xn1) (7)

and

�G∗122 = �G∗

122(1 − k2xn2) (8)

in which n, k1 and k2 are constants. In a non-random mixture some111 and 222 clustering can occur. Hwang et al. [8] use n = 2 tocorrelate the excess Gibbs energy and activity coefficient. Then,substituting Eqs. (7) and (8) into Eq. (6)

�G∗m = x3

1�G∗1 + x3

2�G∗2 + x1x2[3x1�G∗

112(1 − k1x21)

+ 3x2�G∗122(1 − k2x2

2)] (9)

Applying x1 = 1 − x2 and x2 = 1 − x1 in the cubic terms

�G∗m = x2

1�G∗1 + x2

2�G∗2 + x1x2[3x1�G∗

112(1 − k1x21)

+3x2�G∗122(1 − k2x2

2) − x1�G∗1 − x2�G∗

2]

= x21�G∗

1 + x22�G∗

2 + x1x2[x1(3�G∗112 − �G∗

1)

+x2(3�G∗122 − �G∗

2) − 3k1x31�G∗

112 − 3k2x32�G∗

122]

(10)

Using x1 = 1 − x2 and x2 = 1 − x1 in the first two terms

�G∗m = x1�G∗

1 + x2�G∗2 + x1x2[x1(3�G∗

112 − �G∗1)

+ x2(3�G∗122 − �G∗

2) − �G∗1 − �G∗

2

−3k1x31�G∗

112 − 3k2x32�G∗

122] (11)

or

�G∗m = x1�G∗

1 + x2�G∗2 + x1x2[x1(3�G∗

112 − 2�G∗1 − �G∗

2)

+ x2(3�G∗122 − �G∗

1 − 2�G∗2) − 3k1x3

1�G∗112

− 3k2x32�G∗

122] (12)

Rearranging Eq. (3)

�m = hN

Mmixexp[

�G∗m

RT

]�G∗

m

RT= ln �m − ln

(hN

Mmix

) (13)

and

�ijk = hN

Mijkexp

[�G∗

ijk

RT

]�G∗

ijk

RT= ln �ijk − ln

(hN

Mijk

) (14)

with

Mijk = Mi + Mj + Mk

3(15)

and

Mmix =n∑

i=1

xiMi (16)

Substituting Eqs. (13)–(15) into Eq. (12)

ln �m = ln(

hN

Mmix

)+ x1 ln �1 − x1 ln

(hN

M1

)+ x2 ln �2

− x2 ln(

hN

M2

)+ x1x2

[x1

{3 ln �112 − 3 ln

(hN

M112

)−2 ln �1 + 2 ln

(hN

M1

)− ln �2 + ln

(hN

M2

)}+x2

{3 ln �122 − 3 ln

(hN

M122

)− ln �1 + ln

(hN

M1

)− 2 ln �2

+2 ln(

hN

M2

)}− 3k1x3

1

{ln �112 − ln

(hN

M112

)}−3k2x3

2

{ln �122 − ln

(hN

M122

)}](17)


10 G.E. Nava-Ríos et al. / Fluid Phase Equilibria 329 (2012) 8–21

Most of the hN terms cancel and regrouping the viscosity termsgives

ln �m = −ln(Mmix) + x1 ln �1 + x1 ln(M1) + x2 ln �2 + x2 ln(M2)

+ x1x2

[x1

{ln

(�3

112

�21�2

)+ ln

(M3

112

M21M2

)}+x2

{ln

(�3

122

�1�22

)+ ln

(M3

122

M1M22

)}+x3

1 ln(

�112M112

hN

)−3k1+ x3

2 ln(

�122M122

hN

)−3k2](18)

Assuming(�3

112

�21�2

)=(

�3122

�1�22

)= ı�12 (19)

simplifies Eq. (18). Eq. (19) is equivalent to assuming

3�G∗112 − 2�G∗

1 − �G∗2 = 3�G∗

122 − �G∗1 − 2�G∗

2 = ı�G∗(3,12)

(20)

This assumption indicates that removing the effect of two(1 + 1 + 1) interactions and one (2 + 2 + 2) interaction from three(1 + 1 + 2) results in two (1 + 1 + 2) and two (1 + 2 + 2) interactions.Removing the effect of two (2 + 2 + 2) interactions and one (1 + 1 + 1)interaction from three (1 + 2 + 2) results in identical remaining pos-sibilities. Substitution into Eq. (18) and rearranging terms resultsin an equation for the kinematic viscosity of binary mixtures

ln �m = −ln(Mmix) + x1 ln �1 + x1 ln(M1) + x2 ln �2 + x2 ln(M2)

+ x1x2

[ln(ı�12) + x3

1 ln ıg∗12 + x3

2 ln ıg∗21 + x1 ln

(M3

112

M21M2

)+x2 ln

(M3

122

M1M22

)](21)

with

ıg∗12 =

(�112M112

hN

)−3k1(22)

ıg∗21 =

(�122M122

hN

)−3k2(23)

using Eq. (15) for molecular weight. In the development of Eq. (21)we have not considered the temperature effect therefore Eq. (21)has three characteristic constants, ı�12, ıg∗

12, ıg∗21 that are temper-

ature dependent and available from experimental measurements.The delta indicates that the characteristic parameters are relatedto an activation energy difference.

3. Multicomponent mixtures

For a ternary mixture, the Gibbs energy of activation is

�G∗m = x3

1�G∗1 + x3

2�G∗2 + x3

3�G∗3 + 3x1x2(x1�G∗

112 + x2�G∗122)

+ 3x1x3(x1�G∗113 + x3�G∗

133) + 3x2x3(x2�G∗223 + x3�G∗

233)

+ 6x1x2x3�G∗112 (24)

Following the same procedure used above for a binary mixture,but applying it to a ternary mixture with x1 = 1 − x2 − x3

�G∗m = x1�G∗

1 + x2�G∗2 + x3�G∗

3 + x1x2[x1(3�G∗112 − 2�G∗

1

− �G∗2) + x2(3�G∗

122 − �G∗1 − 2�G∗

2) − 3k(12)1 x3

1�G∗112

−3k(12)2 x3

2�G∗122] + x1x3[x1(3�G∗

113 − 2�G∗1 − �G∗

3)

+x3(3�G∗133 − �G∗

1 − 2�G∗3) − 3k(13)

1 x31�G∗

113

−3k(13)3 x3

3�G∗133] + x2x3[x2(3�G∗

223 − 2�G∗2 − �G∗

3)

+x3(3�G∗233 − �G∗

2 − 2�G∗3) − 3k(23)

2 x32�G∗

223

−3k(23)3 x3

3�G∗233] + x1x2x3(6�G∗

123 − 2�G∗1

−2�G∗2 − 2�G∗

3) (25)

In terms of the kinematic viscosity,

ln �m = −ln(Mmix) + x1 ln �1 + x1 ln M1 + x2 ln �2 + x2 ln M2

+x3 ln �3 + x3 ln M3 + x1x2

[ln(ı�12) + x3

1 ln ıg∗12

+x32 ln ıg∗

21 + x1 ln

(M3

112

M21M2

)+ x2 ln

(M3

122

M1M22

)]+x1x3

[ln(ı�13) + x3

1 ln ıg∗13 + x3

3 ln ıg∗31 + x1 ln

(M3

113

M21M3

)+x3 ln

(M3

133

M1M23

)]+ x2x3

[ln(ı�23) + x3

2 ln ıg∗23

+x33 ln ıg∗

32 + x2 ln

(M3

223

M22M3

)+ x3 ln

(M3

233

M2M23

)]+x1x2x3[ln(ıg∗

123)] (26)

in which ıg∗123 = ((�123M123)6/(�1M1�2M2�3M3)2) and it can be

obtained from experimental measurements. If �G∗123 is a combi-

nation of the energy of activation of two species as suggested byHwang et al. [8], then ıg∗

123 becomes imbedded in the i–j terms, oth-erwise it appears explicitly. This combination suggests that it canbe obtained from binary data but using it as adjusting parametergives better results.

For multicomponent mixtures, Eq. (26) becomes

ln �m = −ln(Mmix) +C∑

i=1

xi ln �i +C∑

i=1

xi ln Mi

+C−1∑i=1

C∑j=i+1

xixj

[ln(ıvij) + x3

i ln ıg∗ij + x3

j ln ıgji

+xi ln

(M2

iij

M2i

Mj

)+ xj ln

(M2

ijj

MiM2j

)]

+C−2∑i=1

C−1∑j=i+1

C∑k=j+1

xixjxk ln ıg∗ijk (27)

It is useful to compare Eq. (21) to the equation developed byDizechi and Marschall [4]

ln vm = 1t + Cmix

[(t + C1)x31 ln(�1M1) + (t + C2)x3

2 ln(�2M2)]

− ln(Mmix) +(

3t + Cmix

)[(t + C12)x2

1x2 ln(�12M12)

+(t + C21)x1x22 ln(�21M21)] (28)

with

Ci = 239 + tb,iZi, (29)



Cmix =n∑

i=1

xiCi, (30)

and

Cij = 2Ci + Cj

3, (31)

Here �12, �21 and Zi are adjustable parameters and tb,i is theboiling temperature of specie i.

It is also useful to compare the new equation to the equationdeveloped by Moumouzias and Ritzoulis [6]. This is a four-bodyequation based upon the Dizechi and Marschall equation

ln vm = 1t + Cmix

[(t + C1)x41 ln(�1M1) + (t + C2)x4

2 ln(�2M2)]

− ln(Mmix) +(

4t + Cmix

)[(t + C1112)x3

1x2 ln(�1112M1112)

+(t + C2221)x1x32 ln(�2221M2221) +

(32

)(t + C21)x2

1x22

×ln(�1122M1122)] +(

ln(hN)t + Cmix

)[C1x4

1 + 4x31x2C1112

+4x1x32C2221 + 6x2

1x22C21 + C2x4

2 − Cmix] (32)

in which Eqs. (28) and (29) provide Ci and Cmix

Ciiij = 3Ci + Cj

4, Ciijj = Ci + Cj

2, Miiij = 3Mi + Mj

4

and Miijj = Mi + Mj

2

The adjusting parameters in Eq. (32) are �1112, �1122, �2221 andZi.

4. Results

In this work, Eq. (21) correlates the liquid viscosity of 232 binarymixtures taken from Wholfarth and Lechner [9]. Data have beentaken as reported by these authors. Comparisons of these results tothose given by the three-body McAllister [3] equation, the Dizechiand Marschall [4] (DM) equation and the Moumouzias and Ritzoulis[6] (MR) equation for alkanes systems appear in Table 1. The lat-ter two equations include the Z parameter, which is available forsome substances. Goletz and Tassios [5] develop an equation for theviscosities of pure components using the same function for Gibbsactivation energy as in the DM and RM equations. In this equation, ifthe value of Z is not available, the coefficient becomes an adjustableparameter increasing the number of parameters to three or four fora binary mixture in the DM equation and four or five in the RM equa-tion. The equation proposed herein contains only three adjustingparameters.

4.1. Methanol + benzene mixtures

Eq. (21) correlates the viscosity of methanol + benzene [10] from283.15 to 323.15 K. This system has a maximum and a minimum.Fig. 1 shows the comparison of Eq. (21) to data. The average per-centage deviation of the equation from the experimental data is0.08%. Comparison of DM, RM and the McAllister equation to thesame data produces average percentage deviations of 0.12, 0.10 and0.31%, respectively. The value of Z comes from Goletz and Tassios[5]. Simple functions of temperature correlate the parameters ofthe new equation

ı�12 = 10.0731 − 0.0536T + 7.8813 × 10−5T2, (33)

ıg∗12 = −7.9037 + 0.0598T − 0.9735 × 10−4T2, (34)

Fig. 1. Kinematic viscosity of methanol + benzene mixtures at different tempera-tures: �, 283.15 K; ©, 293.15 K; �, 303.15 K; �, 313.15 K; �, 323.15. Lines are Eq.(21).

ıg∗21 = −0.2827 + 0.00273T (35)

as shown in Fig. 2. The average absolute percentage error using Eqs.(33) and (35) is 0.1%.

4.2. Ethylene glycol + acetone and + methanol

These mixtures have a large range of viscosities. The new equa-tion correlates these data within an absolute average percentagedeviations of (0.57 and 0.95)% while the DM, RM, and MC equations

Fig. 2. Temperature dependence of the characteristic parameters of Eq. (21) for themixture methanol + benzene: �, ı�12; ©, ıg∗

12; �, ıg∗21. Lines are Eqs. (33)–(35).



Table 1Absolute average percent deviations from Eq. (21) for the kinematic viscosity of alkanes mixtures.

Component 1 Component 2 T/K This work MC [3] DM [4] MR [6]

2-Methylpentane Hexadecane 298.15 0.2068 0.4914 0.1202 0.09273-Methylpentane Decane 298.15 0.0265 0.0813 0.0983 0.0435

Hexadecane 298.15 0.0649 0.3671 0.4278 0.0515Hexane Methyl-cyclohexane 298.15 0.0817 0.1149 0.1239 0.0847

Heptane 293.15 0.0798 0.0983 0.0983 0.0821cis-1,2-Dimethyl cyclohexane 298.15 0.0652 0.0615 0.0829 0.0412Octane 293.15 0.0313 0.0383 0.0385 0.03072,2,4-Trimethylpentane 298.15 0.0858 0.0889 0.0901 0.0939Nonane 298.15 0.0827 0.1166 0.1195 0.0948Decane 293.15 0.0310 0.0461 0.0435 0.0399Dodecane 333.15 0.1055 0.1663 0.1764 1.6834

2,2-Dimethylhexane Hexadecane 298.15 0.0245 0.1570 0.0477 0.0646Decane 298.15 0.0044 0.0151 0.0163 0.0046

2,2-Dimethylpentane Hexadecane 298.15 0.0364 0.2586 0.4032 0.11612,3-Dimethylpentane Hexadecane 298.15 0.2714 1.3703 0.7362 0.29252,4-Dimethylpentane Hexadecane 298.15 1.2887 1.2179 1.1981 1.28613,3-Dimethylpentane Hexadecane 298.15 1.3810 1.3416 1.3729 1.3709Heptane Octane 293.15 0.0377 0.0410 0.0410 0.0377

2,2,4-Trimethylpentane 298.15 0.0383 0.0394 0.0385 0.0386Nonane 298.15 0.4639 0.5988 0.5971 0.4761Decane 298.15 0.0363 0.0405 0.0416 0.0337Dodecane 293.15 0.0428 0.0435 0.0504 0.0380Tetradecane 293.15 0.0871 0.1567 0.1774 0.0749Cyclohexane 310.95 0.0924 0.1589 0.1511 0.1028

2,2,4-Trimethylpentane Decane 298.15 0.0674 0.0659 0.0663 0.0669Tetradecane 298.15 0.0186 0.1011 0.1408 0.0129Hexadecane 298.15 0.1059 0.2957 0.1561 0.0994

Octane Undecane 293.15 0.0556 0.0554 0.0554 0.0556Tridecane 293.15 0.3208 0.3707 0.3703 0.5870Pentadecane 298.15 0.0283 0.0747 0.0938 0.0265Tetradecane 293.15 0.1566 0.1834 0.1899 0.1691Nonane 298.15 0.1343 0.1287 0.1287 0.1349Decane 293.15 0.0411 0.0416 0.0416 0.0415

Decane Tridecane 293.15 0.0240 0.0233 0.0233 0.0231Tetradecane 293.15 0.0685 0.0708 0.0703 0.0732

Undecane Tridecane 293.15 0.0241 0.0231 0.0229 0.0236Tridecane Pentadecane 298.15 0.0344 0.0335 0.0335 0.0348Tetradecane Pentadecane 293.15 0.0826 0.1168 0.1155 2.6604

Hexadecane 293.15 0.0298 0.0325 0.0325 0.0301Hexadecane Cyclohexane 298.15 0.0969 0.2102 0.2838 0.0138cis-1,2-Dimethyl-cyclohexane Hexadecane 298.15 0.0373 0.1261 0.1415 0.0679Methyl cyclohexane Decane 298.15 0.0893 0.0826 0.0842 0.0895

Hexadecane 298.15 0.1237 0.3653 0.2624 0.1819Heptane 293.15 0.5919 0.557 0.5715 0.5679

% error 0.1545 0.2288 0.2085 0.2553Std. Dev. 0.0034 0.0050 0.0070 0.0083

correlate them within (0.7, 0.71, 1.92)% for the acetone mixture and(1.04, 0.87, 1.37)% for the methanol mixture. The Z values for ace-tone and methanol are optimized values taken from Dizechi andMarschall [4]. The Z value for ethylene glycol comes from fitting thedata. Unfortunately, for both the DM and RM equations the param-eter Z for ethylene glycol is not statistically valid and convergenceoccurs at a negative and a positive value when they should be nearlyidentical.

4.3. 1-Alcohol + water mixtures

These systems present more difficulty for the correlations. Thiswork uses three systems with methanol to 1-propanol between283.15 and 323.15 K. Dizechi and Marschall [4] have already cor-related these systems using the McAllister equation, but they donot provide information about their weighting in the optimizationprocedure. Experimental data come from the same author and fromPang et al. [11]. The Zwater = −1.1 comes from Goletz and Tassios[5]. If Z = 0.4 for all alcohols, as proposed by these authors, the aver-age percentage deviations of the DM equation are 1.82, 5.46 and6.15% for the methanol, ethanol and 1-propanol aqueous systems,respectively. For the RM equation, the deviations are 0.4, 1.33 and

2.62%. The new equation has absolute average percentage devia-tions of 0.33, 1.47 and 2.76%. If the Z value is an optimized parameterin the DM equation, the average percentage deviations reduce to0.66, 2.32 and 0.59%. Changing the value of n to 3 in Eqs. (7) and(8) it changes the average percentage deviations to 0.9, 0.68 and1.32%. The average absolute percentage deviations for these sys-tems using the McAllister expression are 3.28, 8.13 and 8.49%. Forthe alcohol–water mixtures, the temperature dependence is dif-ferent. We obtain the temperature dependence of the parametersof the 1-propanol + water mixture and then we observed that theparameters of the methanol + water and ethanol + water mixtureshave the same functionality. The temperature functions are

ı�12 = k1 + k2T, (36)

ıg∗12 = k3 + k4T2, (37)

ıg∗21 = 1

(k5 + k6T)4.75(38)

where ki’s are the adjusting parameters and shown in Table 2. Theaverage absolute percentage deviations are 0.39, 1.55 and 2.78%for the methanol, ethanol and 1-propanol + water mixtures, respec-tively if the above functions are imposed in Eq. (21).



Table 2Paramaters of Eqs. (36)–(38) for water binary mixtures.

Parameter Methanol Ethanol 1-Propanol

k1 38.929 16.7761 11.2432k2 −0.0998 −0.0389 −0.0222k3 1.1198 9.2191 5.0124k4 −0.4433 × 10−5 −7.6974 × 10−5 −3.8517 × 10−5

k5 −1.5317 −1.151 −0.9847k6 0.00664 0.00438 0.00373

4.4. Acetone + water mixtures

These systems present a maximum in the kinematic viscos-ity with respect to composition. Experimental dynamic viscositiescome from Kalidas and Laddha [12], and experimental densitymeasurements from Estrada-Baltazar et al. [13] convert them intokinematic viscosities The Z value for acetone, 10.9, is an optimizedvalue from Dizechi and Marschall [4]. The average percentage devi-ations of the DM, RM and McAllister equations are 1.44, 1.07,and 8.82% in the temperature range from 283.15 to 323.15 K. The

new equation agrees with the experimental measurements within1.41%. Temperature dependence of the parameters can be repre-sented by

ı�12 = −0.5428 + 0.00464T, (39)

ıg∗12 = 0.0136T − 1.4621 × 10−4T2, (40)

ıg∗21 = 1980252 − 12600.7T + 20.0633T2 (41)

The average absolute percentage error using the above equa-tions is 1.48%.

For the 232 binary mixtures, we calculated the average absolutepercentage deviation and the average standard deviation using

AAPE =

∑NSj=1

{(100 ×

∑Ni=1|(�exp

i− �eqn

i)/�exp

i|)

/N}

j

NS(42)

Table 3Absolute average percent deviations from Eq. (21) for the kinematic viscosity of binary mixtures containing an alcohol.


Methanol Ethanol 283.15 0.1625 0.4489 0.4478 0.1588Ethylene glycol 303.15 0.9524 1.3678 1.4863 1.1813Acetone 303.15 0.3974 2.8514 0.1842 5.06451-Propanol 298.15 0.3829 1.5906 0.2525 0.36832-Propanol 298.15 0.0339 0.0650 0.0633 0.04091-Butanol 283.15 2.6819 6.8593 6.6961 3.16291,2-Dimethoxyethane 303.15 0.0969 0.4398 0.4224 0.1090Diethylene glycol 298.15 0.2787 0.9778 0.3425 0.3375Benzene 293.15 0.0998 0.3227 0.2257 0.1201Triethylene glycol 298.15 0.3645 1.4844 0.2401 0.1076Toluene 293.15 0.0859 0.1954 0.1663 0.1658

Ethanol Ethylene glycol 343.15 0.9641 1.4183 0.6505 0.70921-Propanol 303.15 0.1113 0.1131 0.1131 0.12372-Propanol 283.15 0.1235 0.2654 0.2655 0.13262-Butanone 298.15 0.4468 0.5932 0.4754 0.4280Ethyl acetate 298.15 0.4965 0.4515 0.4085 0.4054Diethylene glycol 298.15 0.3504 0.2728 0.3240 0.3317Triethylene glycol 298.15 0.2228 0.5240 0.7744 0.3010

1-Propanol 2-Propanol 283.15 0.1783 0.1931 0.1933 0.18032-Propanol 1-Chlorobutane 293.15 0.3124 1.2870 0.5608 0.3385

2-Butanol 298.15 1.1000 1.1079 1.1076 1.11121-Pentanol 293.15 0.1028 0.1067 0.1082 0.1154Cyclohexane 298.15 0.2286 0.5633 0.5308 0.2161Ethyl acetate 313.15 0.3392 0.6675 0.6126 0.3486Trichloromethane 293.15 1.2762 1.4377 3.1970 1.2648Acetonitrile 293.15 0.1753 1.3927 0.3581 0.1647

1-Butanol Pentane 298.15 0.6068 0.4046 0.3386 0.38861-Pentanol 313.15 0.1044 0.2244 0.2241 0.0919Hexane 298.15 0.4783 0.3249 0.3185 0.3465Heptane 298.15 0.1663 0.2221 0.2209 0.16541,4-Dimethylbenzene 298.15 0.0949 0.1355 0.1161 0.1019Ethylbenzene 298.15 0.1523 0.2679 0.1606 0.1589Octane 298.15 0.3505 0.3822 0.6028 0.20461-Nonanol 298.15 0.4206 0.4298 0.4297 0.43341-Decanol 298.15 0.0741 0.1041 0.1226 0.0841Ethyl acetate 298.15 0.2477 0.5573 0.2097 0.1793

2-Butanol Trichloromethane 293.15 2.1185 2.3669 3.0780 1.90241-Pentanol 1-Octanol 298.15 1.4049 1.4714 6.7603 1.37601-Hexanol Chlorobenzene 303.15 0.2510 0.2706 0.2397 0.2328

Hexane 303.15 1.4537 1.7288 1.4991 1.29981-Heptanol 1-Octanol 293.15 0.0706 0.0691 0.0691 0.07101-Octanol Toluene 303.15 1.6803 1.5829 1.5959 1.76101-Nonanol 1-Decanol 298.15 0.2110 0.2369 0.2368 0.22381-Decanol Heptane 298.15 0.0610 0.0974 0.1036 0.0659

Octane 298.15 0.0826 0.0867 4.4649 3.3850Nonane 298.15 2.1085 2.0161 2.0663 2.0901Decane 298.15 0.3557 0.7029 0.7684 0.3249

% error 0.5204 0.8655 0.9326 0.6782Std. Dev. 0.0220 0.0282 0.0316 0.0265



Table 4Absolute average percentage deviations from Eq. (21) for the kinematic viscosity of binary mixtures containing an aromatic component.


1,2-Dimethylbenzene Decane 298.15 0.4003 0.4934 0.4017 0.4045Tetradecane 298.15 0.0081 0.0426 0.1032 0.0233Ethyl acetate 313.15 0.0384 0.0420 0.0427 0.03961,3-Dimethylbenzene 273.15 0.2421 0.2540 0.2480 0.24301,4-Dimethylbenzene 303.15 0.0764 0.0756 0.0756 0.07651,2-Dimethyl cyclohexane 298.15 0.0560 0.0828 0.0813 0.06101,2-Dibromoethane 298.15 0.3235 0.4372 0.4553 0.3247Hexane 298.15 0.0130 0.0414 0.0594 0.0039Cyclohexane 298.15 0.0254 0.1087 0.0086 0.0029Acetone 298.15 0.5090 0.5050 0.3801 0.4623

1,2,4-Trichlorobenzene Decane 293.15 0.0918 0.2248 0.0806 0.0695Tetradecane 293.15 0.0377 0.0451 0.0474 0.0450

1,3-Dimethylbenzene 1,4-Dimethylbenzene 288.15 0.0720 0.1101 0.1101 0.07291,2-Dibromoethane 298.15 0.2516 0.4080 0.3126 0.2381

1,4-Dimethylbenzene 1,2-Dibromoethane 308.15 0.7248 0.7614 0.7645 0.73031-2-Dichloroethane 303.15 0.3330 0.3532 0.3470 0.3380Hexane 298.15 0.0097 0.0270 0.0393 0.0032Propyl propionate 298.15 0.0127 0.0907 0.0908 0.02222-Butanone 298.15 0.4795 0.6854 0.6468 0.4681Decane 298.15 0.0306 0.0571 0.0709 0.0527Tetradecane 298.15 0.0228 0.0336 0.0452 0.0256Ethyl acetate 313.15 0.0368 0.0495 0.0486 0.0368

Benzene Tetrachloromethane 298.15 0.0538 0.0774 0.0794 0.0503Trichloromethane 298.15 2.1178 2.6373 3.1612 1.88941,2-Dibromoethane 298.15 0.4789 0.5653 0.4914 0.4775Methyl formate 293.15 0.2965 0.2971 0.2988 0.2971Ethyl formate 293.15 0.0669 0.0737 0.0722 0.0729Acetone 298.15 0.0296 0.1946 0.2057 0.0020Ethyl acetate 298.15 0.0237 0.1507 0.1044 0.02261,4-Dioxane 293.15 2.1024 2.0522 2.2857 1.9393Propyl formate 313.15 0.1527 0.4634 0.4283 0.1485Cyclohexane 298.15 0.0144 0.1251 0.1251 0.0118Propyl propionate 298.15 0.0198 0.1773 0.1490 0.0111Hexane 298.15 0.0599 0.2270 0.2086 0.0725Toluene 298.15 0.0194 0.0195 0.0198 0.02081,4-Dimethylbenzene 298.15 0.0046 0.0224 0.0055 0.00442,2,4-Trimethylpentane 298.15 0.5924 1.5331 0.4563 0.2884Decane 298.15 0.0716 0.3252 0.3275 0.1167

Bromobenzene Toluene 283.15 1.4369 1.6907 0.8065 0.7684Chlorobenzene Hexane 303.15 0.2904 0.3095 0.2985 0.2903

Toluene 303.15 0.0655 0.0692 0.0690 0.0711Ethylbenzene Octane 313.15 0.0373 0.0639 0.0715 0.0414

Tetradecane 308.15 0.1451 0.1906 0.2419 0.11202-Etoxi-ethanol 298.15 1.5232 2.7940 0.2960 0.15392-Butanone 298.15 0.1183 0.1204 0.1200 0.1229Heptane 313.15 0.0067 0.0424 0.0640 0.0188Hexadecane 313.15 0.1643 0.2158 0.2691 0.1882

Toluene Methyl-cyclohexane 298.15 0.0214 0.0472 0.0467 0.01861,2-Dimethylbenzene 298.15 0.0019 0.0033 0.0034 0.00201,4-Dimethylbenzene 298.15 0.0012 0.0142 0.0143 0.0016Heptane 298.15 0.6520 0.6284 0.6312 0.6056Ethylbenzene 303.15 0.2340 0.2476 0.2475 0.2459Octane 308.15 0.0327 0.0941 3.8026 0.0507Decane 313.15 0.0612 0.1244 0.1486 0.0742Dodecane 313.15 0.0715 0.0640 0.0753 0.0612Tetradecane 308.15 0.1338 0.1795 0.2379 0.1272Hexadecane 308.15 0.0832 0.3407 0.4314 0.11601,4-Dimethylbenzene 298.15 0.1260 0.1579 0.1895 0.1599Trichloromethane 308.15 0.0598 0.0494 0.0439 0.04811,2-Dichloroethane 303.15 0.1141 0.1448 0.1273 0.1122Tetrahydrofuran 298.15 0.0159 0.0208 0.0221 0.0171Hexane 303.15 0.1364 0.1169 0.1187 0.13052,2,4-Trimethylpentane 293.15 0.2648 0.3264 0.3200 0.2588Tetrahydropyran 288.15 0.0205 0.0228 0.0240 0.0147Acetone 298.15 0.9775 0.9050 0.9284 0.9450Ethyl acetate 298.15 0.0116 0.0982 0.0971 0.0118Propyl propionate 298.15 0.0178 0.0940 0.0942 0.01742-Butanone 298.15 0.0337 0.0791 0.0802 0.0336

Trifluoromethyl-benzene 1,1,1-Trichloroethane 289.32 0.0203 0.0486 0.0421 0.0079

% error 0.2432 0.3359 0.3310 0.2028Std. Dev. 0.0035 0.0045 0.0040 0.0034



Table 5Absolute average percentage deviations of the kinematic viscosity for binary mixtures containing freons, esters, acids, etc.


1,2-Dimethoxyethane Dodecane 298.15 2.0800 1.9122 1.8158 1.88291,1,1-Trichloroethane Cyclohexanone 303.15 0.2381 0.2581 0.2562 0.2381

2-Methoxy-2-propanol 313.49 0.1111 0.2642 0.1020 0.11292-Methyl-tetrahydrofuran 289.32 0.0095 0.0167 0.0142 0.00852-Methyl-tetrahydrofuran 313.49 0.0025 0.0243 0.0219 0.0025

1,1,2-Trichloroethene Cyclohexanone 303.15 0.2525 0.5269 0.3159 0.21951,2-Dibromoethane Cyclohexane 298.15 0.9195 0.9330 0.9173 0.93011,2-Dichloroethane Cyclohexanone 303.15 0.1445 0.2364 0.1431 0.1332

Cyclohexane 303.15 0.0948 0.2380 0.1777 0.0971Quinoline 303.15 0.0451 0.0559 0.0490 0.0429

1-Chlorobutane Hexadecane 298.15 0.5046 1.0339 1.0882 0.70522-Butanone Ethyl acetate 298.15 0.0305 0.0333 0.0333 0.0329

Methyl-cyclohexane 298.15 0.0327 0.3227 0.3232 0.0225Heptane 298.15 0.0886 0.1787 0.1062 0.0916Tributylamine 313.15 0.2248 0.1813 0.1952 0.1937Propyl propionate 298.15 0.0527 0.0560 0.0557 0.0571Triethylene glycol 298.15 0.5532 1.9486 1.8931 1.0451

2-Methoxyethanol 1,2-Dimethoxyethane 298.15 0.1372 0.1955 0.1868 0.1675Acetic acid Acetone 323.2 0.6384 0.4245 0.4153 0.6155

Hexanoic acid 298.15 1.0875 1.8876 1.5087 1.0317Acetone 2-Butanone 298.15 0.1407 0.1565 0.1543 0.1960

Ethyl acetate 298.15 0.2215 0.2550 0.2383 0.2261Diethylene glycol 298.15 0.8332 0.7486 0.8323 0.6757Cyclohexane 298.15 0.3458 1.0899 0.8419 0.3489Propyl propionate 298.15 0.2247 0.2421 0.2483 0.1879Triethylene glycol 298.15 3.0963 9.2102 7.6248 3.2731Heptane 298.15 0.1347 0.3363 0.3334 0.0769cis-1,2-Dimethyl-cyclohexane 298.15 0.4130 1.8135 0.6057 0.2574

Acetonitrile Butyl benzoate 323.15 2.8148 3.0009 2.9549 2.45831-Chlorobutane 293.15 0.0701 0.1024 0.1215 0.0538Methyl benzoate 283.15 1.4312 1.5213 1.5216 1.5369Ethyl benzoate 353.15 1.5938 2.8791 1.9057 0.2619

Butyl acetate Heptane 298.15 0.0512 0.0796 0.0674 0.0509Butyl benzoate Heptane 298.15 0.0360 0.2424 0.4164 0.0336Cyclohexanone Cyclohexane 303.15 0.1613 0.1739 0.1800 0.1641

Dichloromethane 303.15 0.2710 0.4320 0.4960 0.3415Ethyl acetate 1,4-Dioxane 313.15 0.0726 0.0971 0.0977 0.0882

Cyclohexane 298.15 0.0531 0.5282 0.1217 0.0819Heptane 298.15 0.0400 0.1446 0.1460 0.0470

Ethyl benzoate Heptane 298.15 0.0793 0.1750 0.1254 0.0810Ethyl propionate Heptane 298.15 0.0320 0.1003 0.0933 0.0319

Decane 298.15 0.0746 0.1075 0.1023 0.0699Ethylene glycol Acetone 303.15 0.8496 0.6656 0.6588 1.0187

N,N-Dimethyl-formamide 263.15 1.3355 0.4755 1.7514 1.07311,2-Dimethoxyethane 268.15 5.1304 8.1974 1.1669 6.9100Diethylene glycol 298.15 0.0970 0.2326 0.2441 0.0948Triethylene glycol 298.15 0.0709 0.3663 0.0698 0.1714

Methyl acetate Heptane 298.15 0.0589 0.1507 0.1581 0.0678Octane 288.15 0.0955 0.1593 0.1247 0.0897Nonane 298.15 0.0937 0.0644 0.0766 0.2491Decane 298.15 0.1942 0.2783 0.2434 0.1774

Methyl benzoate Heptane 298.15 0.2911 0.3381 0.2822 0.3137Methyl butyrate Heptane 298.15 0.0528 0.1002 0.0933 0.0508

Octane 288.15 0.0135 0.0530 0.0537 0.0133Nonane 298.15 0.0796 0.0825 0.0840 1.0409

Methyl propionate Heptane 298.15 0.0903 0.1601 0.1636 0.0963Octane 298.15 0.0197 0.0552 0.0533 0.0342

Methyl cyclohexane Decane 298.15 0.0893 0.0826 0.0842 0.0895Hexadecane 298.15 0.1237 0.3653 0.2624 0.1819Heptane 293.15 0.5919 0.5570 0.5715 0.5679

Nitromethane Methyl acetate 293.15 0.0626 0.1509 0.1398 0.0708Ethyl acetate 293.15 0.1675 0.1683 0.1791 0.1657Propyl acetate 293.15 0.0386 0.0352 0.0265 0.0255Butyl acetate 293.15 0.0739 0.0877 0.0500 0.0500

Propyl acetate Heptane 298.15 0.0495 0.1305 0.1290 0.0412Decane 298.15 0.0668 0.1076 0.0972 0.0675

Propyl propionate Methyl-cyclohexane 298.15 0.0166 0.1298 0.0904 0.0157Heptane 288.15 0.0305 0.0636 0.0543 0.0302

Tetrahydrofuran Cyclohexanone 293.15 0.0291 0.0821 0.0331 0.0407Cyclohexane 298.15 0.0332 0.0616 0.0405 0.0270

Tetrahydropyran Cyclohexane 288.15 0.0307 0.0360 0.0324 0.0324Trichloromethane Nitromethane 293.15 0.0017 0.0020 0.0035 0.0016

Cyclohexanone 303.15 0.2321 0.3370 0.3685 0.2388

% error 0.4048 0.6567 0.4964 0.4277Std. Dev. 0.0330 0.0376 0.0369 0.0310



Table 6Characteristic parameters for Eq. (21) for binary mixture containing alkanes.

Component 1 Component 2 T/K ı�12 ıg∗12 ıg∗

21

2-Methylpentane Hexadecane 298.15 3.1414 2.2776 0.73903-Methylpentane Decane 298.15 1.3248 1.1398 0.9739

Hexadecane 298.15 3.2775 2.3879 0.7291Hexane Methyl-cyclohexane 298.15 0.8401 0.9923 0.9215

Heptane 293.15 1.0114 1.0183 1.0485cis-1,2-Dimethyl cyclohexane 298.15 0.7249 1.0819 0.8917Octane 293.15 1.0695 1.0184 1.01252,2,4-Trimethylpentane 298.15 0.9720 1.0758 0.9804Nonane 298.15 1.1547 1.1367 0.9887Decane 293.15 1.2780 1.0470 0.9348Dodecane 333.15 1.3258 1.2049 0.9650Tetradecane 298.15 2.3246 1.5525 0.7100Hexadecane 298.15 3.2062 2.0365 0.6521

2,2-Dimethylhexane Hexadecane 298.15 2.1588 1.4498 0.8546Decane 298.15 1.0733 1.0146 1.0054

2,2-Dimethylpentane Hexadecane 298.15 2.8510 1.8270 0.78362,3-Dimethylpentane Hexadecane 298.15 3.4827 0.5863 0.23442,4-Dimethylpentane Hexadecane 298.15 2.7881 0.7729 0.58643,3-Dimethylpentane Hexadecane 298.15 2.3150 2.2916 0.8683Heptane Octane 293.15 1.0217 0.9864 0.9941

2,2,4-Trimethylpentane 298.15 1.0238 0.9765 1.0036Nonane 298.15 1.1047 0.9259 0.7741Decane 298.15 1.1223 1.0362 0.9974Dodecane 293.15 1.3549 1.1108 0.9452Tetradecane 293.15 1.6985 1.3048 0.9013Hexadecane 298.15 2.1826 1.5790 0.8813Cyclohexane 310.95 0.6624 1.1891 0.7564

2,2,4-Trimethylpentane Decane 298.15 1.0891 0.9814 0.9787Tetradecane 298.15 1.6553 1.2524 0.9301

Octane Dodecane 298.15 1.1849 1.0574 1.0046Undecane 293.15 1.1048 1.0092 1.0078Tridecane 293.15 1.3784 0.9561 0.8484Pentadecane 298.15 1.6251 1.2147 0.9064Tetradecane 293.15 1.4415 1.2164 0.9495Hexadecane 298.15 1.8426 1.2937 0.8194Nonane 298.15 0.9989 1.0190 1.0328Decane 293.15 1.0587 1.0058 0.9883

Nonane Hexadecane 298.15 2.0085 7.4518 0.8989Dodecane 298.15 1.0886 1.0692 1.0194Tetradecane 298.15 1.2911 1.0537 0.8880

Decane Dodecane 298.15 1.0504 0.9946 0.9603Tridecane 293.15 1.0919 1.0097 0.9921Tetradecane 298.15 1.2105 0.9403 0.7767Tetradecane 293.15 1.2603 1.0138 0.9567Hexadecane 298.15 1.4152 1.0353 0.8478

Undecane Tridecane 293.15 1.0413 1.0063 0.9876Dodecane Tetradecane 298.15 1.0593 0.9232 0.8921

Hexadecane 298.15 1.0995 1.0275 0.9981Tridecane Pentadecane 298.15 1.0304 1.0142 1.0013Tetradecane Pentadecane 293.15 1.1851 0.9718 0.9299

Hexadecane 293.15 1.0206 1.0153 1.0000Hexadecane Cyclohexane 298.15 2.5254 0.7472 1.8142

and

ASD =

∑NSj=1

{√(∑Ni=1(�exp

i− �eqn

i)2/(N − NP)

)}j

NS(43)

where �expi

and �eqni

are the experimental and calculated from theequation kinematic viscosities; N is the number of data points; NSis the number of binary systems; and NP is the number of adjustingparameters in the equation. The AAPE of the new equation is 0.33%while the deviations for the McAllister (MC), Dizechi–Marschall(DM), and Moumouzias–Ritzoulis (MR) equations are 0.52, 0.48and 0.38%, respectively. Also, the ASD of the new equation is 0.016while for the MC, DM and MR is 0.019, 0.020, and 0.017. This is anindication that the new equation performs better for these systems.

Table 1 shows the average absolute percentage deviations foralkanes systems. The new equation performs slightly better thanthe other equations for the systems considered here. The average

absolute percentage deviation is 0.15% with an. The average abso-lute percentage deviations of the MC, DM and MR equations are0.23, 0.21 and 0.26%, respectively. The ASD for these systems is0.0034, 0050, 0.0070 and 0083 for the new, MC, DM and MR equa-tions. A logical conclusion is that the McAllister equation is a goodchoice for these systems because of its simplicity, and the fact that itcontains only two characteristic parameters. Table 3 compares thenew equation to other equations when correlating the kinematicviscosity of binary mixtures containing an alcohol. The averageabsolute percentage deviation from the experimental data is 0.52%with an average standard deviation of 0.022. The MC, DM and MRequations correlate the data within average absolute percentagedeviations of 0.87, 0.93, and 0.68%, respectively. The average stan-dard deviations for these equations are 0.028, 0.032 and 0.027,respectively.

The new equation contains fewer characteristic parame-ters than the Moumouzias–Ritzoulis equation. Perhaps, theDizechi–Marschall equation could correlate the kinematic viscosity



Table 7Characteristic parameters for Eq. (21) for binary mixture containing alcohols.


21

Methanol Ethanol 283.15 1.6359 1.5228 1.0040Ethylene glycol 303.15 1.3505 1.7809 2.0478Acetone 303.15 0.6726 0.6432 1.35652-Propanol 298.15 1.4049 1.1316 0.95541-Butanol 283.15 4.1171 220.5556 1.00001,2-Dimethoxyethane 303.15 0.7712 1.3759 1.3686Diethylene glycol 298.15 4.3224 2.5578 0.8572Benzene 293.15 1.1311 1.2835 0.5305Triethylene glycol 298.15 12.0824 5.3221 0.6384Toluene 293.15 1.2452 1.8222 0.5820

Ethanol Ethylene glycol 343.15 1.0028 0.4383 0.93961-Propanol 303.15 1.0407 0.9483 1.04672-Propanol 283.15 0.3287 0.8994 0.83662-Butanone 298.15 0.3956 0.6306 1.0805Ethyl acetate 298.15 0.3223 0.5023 1.6535Diethylene glycol 298.15 1.7337 1.1930 1.0148Triethylene glycol 298.15 4.6677 2.0221 0.6348

1-Propanol 2-Propanol 283.15 0.9832 1.0657 1.00702-Propanol 1-Chlorobutane 293.15 0.1990 0.3916 0.9552

2-Butanol 298.15 1.0220 0.9423 1.13841-Pentanol 293.15 1.1030 1.0261 0.9032Cyclohexane 298.15 0.3308 0.8498 0.6469Ethyl acetate 313.15 0.6560 0.6668 0.7602Trichloromethane 293.15 0.0920 0.3635 7.4014Acetonitrile 293.15 0.2487 0.3652 0.8155

1-Butanol Pentane 298.15 0.4878 1.8387 0.32471-Pentanol 313.15 1.0402 0.8908 0.8953Hexane 298.15 0.3889 1.4754 0.4621Heptane 298.15 0.3857 1.0665 1.09771,4-Dimethylbenzene 298.15 0.2848 1.1730 0.9823Ethylbenzene 298.15 0.2988 1.2255 0.9723Octane 298.15 0.3263 2.6259 0.44041-Nonanol 298.15 1.5988 1.2822 0.89081-Decanol 298.15 2.0170 1.0983 0.8189Ethyl acetate 298.15 0.2210 0.4559 1.4238

2-Butanol Trichloromethane 293.15 0.4174 2.7990 0.48231-Pentanol 1-Octanol 298.15 1.1187 1.1966 0.68001-Hexanol Chlorobenzene 303.15 0.5034 1.4030 0.5597

Hexane 303.15 0.3913 2.5101 0.55971-Heptanol 1-Octanol 293.15 1.0105 1.0175 0.98781-Octanol Toluene 303.15 0.6192 1.0923 0.50021-Nonanol 1-Decanol 298.15 1.0540 0.8440 0.98221-Decanol Heptane 298.15 0.9070 0.9675 0.9540

Octane 298.15 0.6893 1.0199 0.9409Nonane 298.15 0.5823 1.0880 0.7737Decane 298.15 0.4695 1.3382 1.1463

better by using the Z parameter as a different characteristic param-eter even for mixtures that contain the same alcohol. For mixtureswith an aromatic substance, the new equation and the MR correlatethe data equally well. The average absolute percentage deviation is0.24 and 0.20%, respectively. The MC and DM are not very differentfrom these two because they correlate the viscosity within abso-lute average percentage deviations of 0.34 and 0.33%, respectively.Table 4 shows the average absolute percentage error for thesesystems together with the average absolute percentage deviation.Finally for binary mixture containing esters, acids and ketones, thenew equation correlates the data within 0.40% while the MC, DMand MR equations correlate the data within 0.66, 0.50, and 0.43%,respectively. For these systems, the average standard deviation ofthe new equation is 0.033 comparing to 0.038, 0.037 and 0.031 forthe MC, DM and MR equations, as shown in Table 5.

The above results indicate that the assumptions in Eqs. (7) and(8), and the interactions assumed in Eqs. (19) and (20) could bevalid. Parameters for Eq. (21) are given in Tables 6–9. The tempera-ture function of the parameters varies according to the componentsinvolves in the mixture, however they are always a smooth func-tions that can be correlated to relative simple functions with aminimum deterioration to the fit.

4.5. Multicomponent mixtures

Correlations of four, ternary mixtures test the correlative abil-ity of the new equation as compared to those obtained by the DMequation. We did not include the other two expressions becauseof their performance (MC equation) and the number of adjustingparameters (MR equation). The binary parameters come from theviscosity of the binary mixtures, and only one ternary parame-ter requires adjustment to correlate the viscosity of the ternarymixture. The first ternary mixture is methanol + ethanol + waterfrom 283.15 to 323.15 K. Experimental measurements are thoseof Dizechi and Marschall [4]. The new equation correlates thekinematic viscosity within an average absolute percentage devi-ation of 2.51%, while the DM equation is within 4.11%. Thesecond system is acetone + methanol + ethylene glycol at 303.15 K.This system contains a component with relatively high kine-matic viscosity with respect to the other two components. Thenew equation and the DM equation agree with the experimen-tal measurements within an absolute percentage deviation of1.19 and 1.22%, respectively. The third ternary mixture is 1-heptanol + trichloroethylene + methylcyclohexane. IIoukhani andSamiey [14] measure the viscosities at 298.15 K. In this mixture the



Table 8Characteristic parameters for Eq. (21) for binary mixture containing an aromatic component.


21

1,2-Dimethylbenzene Decane 298.15 0.9777 0.7781 0.8992Tetradecane 298.15 1.5775 1.1859 0.8947Ethyl acetate 313.15 0.9504 0.8616 0.97021,3-Dimethylbenzene 273.15 0.9504 0.8616 0.97021,4-Dimethylbenzene 303.15 0.9712 1.0136 1.00171,2-Dimethyl cyclohexane 298.15 0.7084 0.9889 0.94161,2-Dibromoethane 298.15 0.7503 0.8690 0.8242Hexane 298.15 0.7179 0.9333 1.0148Cyclohexane 298.15 0.6017 1.1219 0.7729Acetone 298.15 1.0709 1.4969 1.0093

1,2,4-Trichlorobenzene Decane 293.15 0.6666 0.7075 1.1057Tetradecane 293.15 1.0717 1.0312 0.9793

1,3-Dimethylbenzene 1,4-Dimethylbenzene 288.15 1.0157 0.9481 0.95711,2-Dibromoethane 298.15 0.7208 0.9286 0.7011

1,4-Dimethylbenzene 1,2-Dibromoethane 308.15 0.5854 1.1289 1.23191-2-Dichloroethane 303.15 0.8922 1.0414 0.9407Hexane 298.15 0.7703 0.9664 1.0005Propyl propionate 298.15 0.9753 0.9522 0.93182-Butanone 298.15 1.1655 0.7171 0.7485Decane 298.15 0.8884 0.8842 1.0734Tetradecane 298.15 1.4101 1.0602 0.9679Ethyl acetate 313.15 1.0218 1.0128 1.0301

Benzene Tetrachloromethane 298.15 0.9119 0.9478 0.9779Trichloromethane 298.15 0.4145 2.7842 0.48421,2-Dibromoethane 298.15 0.5926 0.6927 0.8916Methyl formate 293.15 0.8576 0.8721 1.1610Ethyl formate 293.15 0.7422 0.9243 1.0747Acetone 298.15 0.8386 0.9391 0.8119Ethyl acetate 298.15 0.7987 0.8460 0.97341,4-Dioxane 293.15 0.8959 0.5466 1.2673Propyl formate 313.15 0.7453 1.3261 1.2353Cyclohexane 298.15 0.5297 0.9749 0.8690Propyl propionate 298.15 0.9518 0.8083 0.9258Hexane 298.15 0.5702 0.6957 1.1467Toluene 298.15 0.9543 0.9919 1.01411,4-Dimethylbenzene 298.15 0.9841 0.9594 1.00962,2,4-Trimethylpentane 298.15 0.7747 0.3069 0.7092Decane 298.15 0.9030 0.6564 1.0431

Bromobenzene Toluene 283.15 0.5971 4.0131 1.0731Chlorobenzene Hexane 303.15 0.8356 0.9976 0.9116

Toluene 303.15 1.0397 0.9745 1.0428cis-1,2-Dimethyl-cyclohexane Hexadecane 298.15 2.0302 1.5178 0.8329Ethylbenzene Octane 313.15 0.8229 0.9305 1.0206

Tetradecane 308.15 1.3721 1.2612 0.99982-Etoxi-ethanol 298.15 0.5798 1.9127 13.21612-Butanone 298.15 1.0449 1.0702 1.0274Heptane 313.15 0.7926 0.9165 1.0295Hexadecane 313.15 1.8468 1.4816 0.8149

Toluene Methyl-cyclohexane 298.15 0.7045 0.9843 0.95611,2-Dimethylbenzene 298.15 0.9629 1.0002 0.99631,4-Dimethylbenzene 298.15 1.0220 0.9917 0.9898Heptane 298.15 0.8741 1.0317 0.7430Ethylbenzene 303.15 0.8437 1.0653 0.9724Octane 308.15 0.8207 0.8662 1.0471Decane 313.15 1.0173 0.8477 1.0182Dodecane 313.15 1.2747 1.0331 0.9724Tetradecane 308.15 1.6768 1.3739 0.9210Hexadecane 308.15 2.2980 1.7803 0.85051,4-Dimethylbenzene 298.15 1.9348 1.1921 0.8520Trichloromethane 308.15 1.2100 0.9301 1.06191,2-Dichloroethane 303.15 0.8182 1.1142 0.9882Tetrahydrofuran 298.15 1.1794 0.9771 1.0126Hexane 303.15 0.8124 0.9656 1.11112,2,4-Trimethylpentane 293.15 0.7664 0.9448 0.8896Tetrahydropyran 288.15 0.9001 1.0433 0.9783Acetone 298.15 1.2805 0.4660 0.5122Ethyl acetate 298.15 0.9751 0.9560 0.9177Propyl propionate 298.15 1.0193 0.9520 0.93072-Butanone 298.15 0.9781 0.9405 0.9665

Trifluoromethyl-benzene 1,1,1-Trichloroethane 289.32 0.7728 1.0713 0.8769



Table 9Characteristic parameters for Eq. (21) for binary mixture containing esters, acids, etc.


21

1,2-Dimethoxyethane Dodecane 298.15 1.1947 1.4574 1.49141,1,1-Trichloroethane Cyclohexanone 303.15 1.0405 1.1411 1.0485

2-Methoxy-2-propanol 313.49 0.6519 0.6068 1.10472-Methyl-tetrahydrofuran 289.32 0.9863 0.9783 1.00682-Methyl-tetrahydrofuran 313.49 0.9996 0.9752 0.9935

1,1,2-Trichloroethene Cyclohexanone 303.15 1.1102 1.5817 1.04031,2-Dibromoethane Cyclohexane 298.15 0.4218 0.9005 0.79461,2-Dichloroethane Cyclohexanone 303.15 1.0539 1.2585 0.9959

Cyclohexane 303.15 0.5278 0.9639 0.7148Quinoline 303.15 1.0753 1.0618 0.9884

1-Chlorobutane Hexadecane 298.15 3.5524 2.6291 0.70702-Butanone Ethyl acetate 298.15 0.9706 1.0000 1.0427

Methyl-cyclohexane 298.15 0.8456 0.8848 0.8026Heptane 298.15 0.9213 0.9505 0.8683Tributylamine 313.15 1.5837 1.2492 0.8555Propyl propionate 298.15 1.1063 1.0871 0.9854Triethylene glycol 298.15 1.9351 0.1981 0.5362

2-Methoxyethanol 1,2-Dimethoxyethane 298.15 0.5993 0.8640 1.0353Acetic acid Acetone 323.2 1.2831 0.8662 0.6908

Hexanoic acid 298.15 1.0745 1.9508 3.1467Acetone 2-Butanone 298.15 1.0375 1.0983 0.9868

Ethyl acetate 298.15 1.0611 0.7994 0.9944Diethylene glycol 298.15 0.7335 0.2692 1.2461Cyclohexane 298.15 0.7442 0.5919 0.4696Propyl propionate 298.15 1.4554 0.8336 0.7980Triethylene glycol 298.15 4.7790 0.2468 0.1684Heptane 298.15 1.1283 0.8982 0.7569cis-1,2-Dimethyl-cyclohexane 298.15 1.1091 0.4899 0.2806

Acetonitrile Butyl benzoate 323.15 0.2112 6.3019 1.59881-Chlorobutane 293.15 1.0849 0.9843 0.9185Methyl benzoate 283.15 0.1976 1.9227 0.4872Ethyl benzoate 323.15 0.2572 6.2229 4.5198

Butyl acetate Heptane 298.15 0.7822 0.9848 0.9372Butyl benzoate Heptane 298.15 0.7382 0.7908 0.9767Cyclohexanone Cyclohexane 303.15 0.8378 1.0721 0.9656

Dichloromethane 303.15 1.3391 1.0212 1.2462Ethyl acetate 1,4-Dioxane 313.15 0.6865 1.1312 0.8558

Cyclohexane 298.15 0.4682 1.0225 0.5480Heptane 298.15 0.8080 0.9061 0.9172

Ethyl benzoate Heptane 298.15 0.5768 0.8257 0.9995Ethyl propionate Heptane 298.15 0.7870 0.9580 0.8811

Decane 298.15 1.1720 0.9268 0.7605Ethylene glycol Acetone 303.15 0.2149 1.4000 0.9807

N,N-Dimethyl-formamide 263.15 0.6346 1.5472 0.24771,2-Dimethoxyethane 268.15 0.0201 14.8895 156.8054Diethylene glycol 298.15 1.1961 0.9980 0.6941Triethylene glycol 298.15 2.4226 1.7047 0.6231

Methyl acetate Heptane 298.15 0.8612 0.9040 0.9565Octane 288.15 0.9908 0.9694 0.7921Nonane 298.15 1.1255 1.1536 0.8839Decane 298.15 1.4471 0.9936 0.5213

Methyl benzoate Heptane 298.15 0.4837 0.9258 1.4061Methyl butyrate Heptane 298.15 0.7527 0.9695 0.9237

Octane 288.15 0.8044 0.9317 0.8424Nonane 298.15 0.8374 0.9861 1.0118

Methyl propionate Heptane 298.15 0.8200 0.8877 0.9121Octane 298.15 0.8740 0.9367 0.8947

Methyl cyclohexane Decane 298.15 1.0738 0.9859 0.9691Hexadecane 298.15 2.4032 1.9086 0.8149Heptane 293.15 0.9059 1.2637 1.0644

Nitromethane Methyl acetate 293.15 1.0521 0.8781 0.9701Ethyl acetate 293.15 1.1191 1.1150 0.9116Propyl acetate 293.15 1.3164 1.0475 0.9327Butyl acetate 293.15 1.5669 1.2347 0.8822

Propyl acetate Heptane 298.15 0.7803 0.9358 0.9301Decane 298.15 0.9258 1.0382 1.1948

Propyl propionate Methyl-cyclohexane 298.15 0.7573 1.0054 0.8548Heptane 288.15 0.8034 0.9901 0.9173

Tetrahydrofuran Cyclohexanone 293.15 0.7464 1.0297 0.8387Cyclohexane 298.15 0.8104 1.0361 0.8778

Tetrahydropyran Cyclohexane 288.15 0.8456 1.0316 0.9501Trichloromethane Nitromethane 293.15 1.2684 0.8490 0.9111

Cyclohexanone 303.15 1.9498 1.6778 0.8351



Fig. 3. Determination of Z of the DM equation for trichloroethylene using a para-metric study of the sum of squares.

Z value used in the DM equation for trichloroethylene is unknown.Using the value found by correlating individually the kinematic vis-cosity of the binary pairs, values are different for different binarymixtures, and the asymptotic standard error is invalid. Using a Zvalue found from a parametric study to minimize the average abso-lute percentage deviation of the kinematic viscosity of the binarymixtures involved appears in Fig. 3. The absolute percentage devia-tion of the kinematic viscosity of the ternary mixture is 0.61% usingthe DM equation, while using the new equation predicts the vis-cosity within 1.03%. Finally we have tested both equations with thekinematic viscosity of methylbutanoate + n-heptane + n-octane. Inthis ternary mixture, the methylbutanoate and n-heptane act asdissolvents [15]. The performance of both equations is almost thesame having average absolute percentage deviations of 0.16 and0.09% for the new and the DM equations, respectively.

5. Conclusions

This paper presents a semi-theoretical equation to correlate thekinematic viscosity of liquids. The new equation improves the cor-relative ability of the McAllister (MC) equation and compares wellto the Dizechi–Marschall (DM), and Moumouzias–Ritzoulis (MR)equations. The new equation has three characteristic parametersfor binary mixtures compared to two and three for the DM andMR equations. Unfortunately, the latter two equations contain theparameter Zi, which could be derived from pure viscosity data,but generally is another adjustable parameter. When treated asan adjustable parameter in binaries mixtures, it does not have thesame value for a common component, and sometimes it is not sta-tistically valid, therefore it requires several binary mixtures witha common component in the correlating procedure. The correla-tive ability of the MR equation appears here. The new equationimproves the performance of the MC by considering that the molec-ular interactions are given in certain way by the expense of addingan extra adjustable parameter. The new equation correlates theviscosity data of 232 binary mixtures within 0.33%.

List of symbols

A constant in the Andrade equationB constant of the Andrade equationC temperature dependent parameter of the

Dizechi–Marschall and Moumouzias–Ritzoulis equationsG* Gibbs energy of activation (J/mol)G∗ apparent Gibbs energies of activation (J/mol)

h Planck constant (6.624 × 10−27 erg s/molecule,6.624 × 10−25 g mm2/(s molecule))

k1 constant of Eq. (7)k2 constant of Eq. (8)ki constants of Eqs. (36)–(38)M molecular weight (g/mol)n constant in Eqs. (7) and (8)N Avogadro’s number (6.023 × 1023) molecules/mol or

number of data points in Eqs. (42) and (43)NS number of binary systemsNP number of adjusting parameters in an equationR universal gas constantt temperature (◦C)T temperature (K)V molar volume (cm3/mol)x mole fractionZ characteristic parameter of Dizechi and Marschall equa-

tion

Greek lettersı�ij temperature dependent parameter of the new viscosity

equationıg∗

ijtemperature dependent parameter of the new viscosityequation

ıg∗ji

temperature dependent parameter of the new viscosityequation

ıg∗ijk

temperature dependent parameter of the new viscosityequation

� dynamic viscosity (mPa s)� kinematic viscosity (mm2/s)�12 temperature dependent parameter of the McAllister,

Dizechi and Marschall equations�21 temperature dependent parameter of the McAllister,

Dizechi and Marschall equations�1112 temperature dependent parameter of Moumouzias and

Ritzoulis equation�1122 temperature dependent parameter of Moumouzias and

Ritzoulis equation�2221 temperature dependent parameter of Moumouzias and

Ritzoulis equation

Subscriptsi, j, k component i, j, and km mixturemix mixtureb boiling

Acknowledgments

The authors wish to thank Texas A&M University, the Texas Engi-neering Experiment Station and Instituto Tecnólogico de Celaya forfinancial support during this work. The authors want to thank QatarNational Research Fund for financial support through Project No.NPRP-30-6-7-1.

References

[1] E.N.da.C. Andrade, Phil. Mag. 17 (1934) 497–511.[2] H. Eyring, J. Chem. Phys. 4 (1936) 283–291.[3] R.A. McAllister, AIChE J. 6 (1960) 427–431.[4] M. Dizechi, E. Marschall, J. Chem. Eng. Data 27 (1982) 358–383.[5] E. Goletz, D. Tassios, Process Des. Dev. 16 (1977) 75–79.[6] G. Moumouzias, G. Ritzoulis, Collect. Czech. Chem. Commun. 66 (2001)

1341–1346.[7] K. Kreglewski, K.N. Marsh, K.R. Hall, Fluid Phase Equilib. 21 (1985) 25–37.[8] C.-A. Hwang, J.C. Holste, K.R. Hall, G.A. Mansoori, Fluid Phase Equilib. 62 (1991)

173–189.



[9] C. Wholfarth, M.D. Lechner, Viscosity of Pure Organic Liquids and Binary LiquidMixtures, first ed., Springer-Verlag, Berlin, Germany, 2001.

[10] H.M.N.H. Irving, R.B. Simpson, J. Inorg. Nucl. Chem. 34 (1972) 2241–2247.[11] F.-M. Pang, C.-E. Seng, T.-T. Teng, M.H. Ibrahim, J. Mol. Liq. 136 (2007) 71–78.[12] R. Kalidas, G.S. Laddha, J. Chem. Eng. Data 9 (1964) 142–145.

[13] A. Estrada-Baltazar, A. de León-Rodríguez, K.R. Hall, M. Estrada-Ramos, G.A.Iglesias-Silva, J. Chem. Eng. Data 48 (2003) 1425–1431.

[14] H. IIoukhani, B. Samiey, J. Chem. Thermodyn. 39 (2007) 206–217.[15] J.S. Matos, J.L. Trenzado, E. Gonzalez, R. Alcalde, Fluid Phase Equilib. 186 (2001)

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A new equation to correlate liquid kinematic viscosities of multicomponent mixtures