Correlation for Viscosity Data of Liquid Mixtures - American Chemical

Ind. Eng. Chem. Process Des. Dev. 1982, 27, 282-289

282

(6*CE,), = limiting value of 6*CE, for low porosity membrane surface In A* = quantity defined by eq 22 when polar, steric, and nonpolar effects are each set equal to zero e = dielectric constant O(structura1group) = structural contribution to (6*CE,)li, O,, = constant in eq 26 vt = kinematic viscosity of mixed solution, mz/s 7r = osmotic pressure, kPa p = density, kg/m3 pt = density of mixed solution, kg/m3 pw = density of water, kg/m3 w* = coefficient associated with nonpolar parameter, s* and

cs*

q = viscosity, kPa s qt = viscosity of mixed qw

solution, kPa s

= viscosity of water, kPa s

Subscripts 1 = bulk feed solution 2 = concentrated boundary solution on the high pressure side of membrane 3 = membrane permeated product solution on the low pressure side of membrane g = glucose m = D,L-malic acid i = ionized (dissociated) D,L-malic acid u = nonionized (undissociated) D,L-malic acid Literature Cited Agrawai, J. P.; Sourirajan, S. I d . Eng. Chem. Process Des. Dev. 1870, 9 , 12. Baxter, A. G.; Bednas, M. E.; Matsuura, T.; Sourirajan, S. Chem. Eng. Commun. 1980, 4 , 471. Clark, J. M., Jr., Ed. "Experimental Biochemistry", W. H. Freeman: San Francisco, 1964. Hsieh, F.; Matsuura, T.; Sourirajan, S.Ind. Eng. Chem. Process Des. Dev. 18788, 18, 414.

Hsieh, F.; Matsuura, T.; Sourirajan, S.J. Sep. Process Techno/. 1979b, 1 , 50. Matsuura, T.; Baxter, A. G.; Sourirajan, S. Acta Alimentaria 1873, 2 , 109. Matsuura, T.; Baxter, A. G.; Sourirajan, S. J. Food Sci. 1874a, 39, 704. Matsuura, T.; Baxter, A. G.; Sourirajan, S. J. Food Sci. 1975a, 40, 1039. Matsuura, T.; Bednas, M. E.; Sourirajan, S. J. Appl. folym. Sci. 1974b, 18, 567. Matsuura, T.; Dickson, J. M.: Sourirajan, S. Ind. Eng. Chem. Process Des. Dev. 1876a, 15, 149. Matsuura, T.; Dickson, J. M.; Sourirajan, S. Ind. Eng. Chem. Process Des. Dev. 1976b, 15, 350. Matsuura, T.; Pageau, L.; Sourirajan, S. J. Appl. Polym. Sci. 1875b, 19, 179. Matsuura, T.; Sourirajan, S. Ind. Eng. Chem. Process Des. Dev. 1971, 10, 102. Matsuura, T.; Sourirajan, S. J. Appl. f d y m . Sci. 1973, 17, 3661. Matsuura, T.; Sourirajan, S. AIChESymp. Ser. 1978, 74, 196. Monk, C. 6. "Electrolytic Dissociation"; Academic: New York, 1961; pp 272-273. Pageau, L.; Sourlrajan, S. J. Appl. folym. Sci. 1872, 16, 3185. Pereira, E. N.; Matsuura, T.; Sourirajan, S. J. Food Scl. 1878, 4 1 , 672. Rangarajan, R.; Matsuura, T.; -hue, E. C.; Sourirajan, S. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 46. Rangarajan, R.; Matsuura, T.; Goodhue, E. C.; Sourirajan, S. Ind. Eng. Chem. Process Des. Dev. 1878, 18, 278. ReM, R. C.; Sherwood, T. K. "The Properties of Gases and Liquids"; MceawHili: New York, 1958; p 284. SourireJan, S."Reverse Osmosis"; Academic: New York, 1970a: Chapter 3. SourIraJan, S. "Reverse Osmosls"; Academic: New York, 1970b; Chapter 6. Sourlrajan, S.; Matsuura, T. "A Fundamental Approach to Application of Reverse Osmosis for Water Pollution Control"; in Proceedings, EPA Symposium on Textile Industry Technology, Wllllamsburg, Dec 5-8, 1978, EPA600, 2-79-104, May 1979. Timmermans, J. "Physicochemical Constants of Binary Systems in Concentrated SOlvtjOnS", Vol. 4; Interscience: New York, 1960; pp 282-286 and 398-400. Weest, R. C., Ed. "Handbook of Chemistry and Physics"; CRC Press: Cievelend, 1977; pp D-218-D-267. Wiike, C. R.; Chang, P. AIChE J. 1855, 1 , 264.

Received for reuiew December 1, 1980 Accepted October 10, 1981

Issued as NRC No. 19974.

Correlation for Viscosity Data of Liquid Mixtures Manuchehr Dlzechl Teledyne Inef, Torrance, California 90509

Ekkehard Marschall' Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, California 93 106

McAllister's theory of viscosity for liquid mixtures was modified in order to account for temperature dependencies different than that predicted by the Eyring model. The resulting equation was tested with the experimentally obtained viscosity data of 41 binary and 16 ternary mixtures. The improved version of the McAllister equation correlates viscosities of liquid mixtures very well, whether they contain polar compounds or not.

Introduction A reliable and generally valid theory for the quantitative prediction of viscosities of liquid mixtures from the properties of the pure components has not been established. For this reason, information on viscosity data of liquid mixtures, especially of mixtures containing polar components, continues to be based on experimental investigations. Numerous semitheoretical equations have 0196-4305/82/112 1-0282$01.25/0

been proposed for the correlation of experimental viscosity data. There are several desirable properties which such equations should have. The equations should correlate experimental data as closely as possible; that is, they should allow for only a very small standard deviation. The number of necessary experimental data should be kept to a minimum in order to reduce.costly experiments. Finally, the equation should be generally applicable and should 0 1982 American Chemical Society

Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982 283 Table I. Values of 2 for Some Polar Liquids no.

component

1

water acetone methanol ethanol 1-propanol DMA ethylene glycol ethylene glycol-dimethyl ether

2 3 4 5 6 7 8

z -1.10

+10.90 +5.30 t 4.40 + 8.60 t 1.90

-1-1.60

+ 7.40

accommodate mixture viscosities whether the mixture components are nonpolar or polar. Unfortunately, none of the published equations meets all of these desired requirements. Reid and Sherwood (1977), who studied this problem extensively, regard the McAllister theory of viscosities (McAllister, 1960) as the best correlating technique available. While McAllister's theory was originally formulated for binary mixtures, it can be easily extended to correlate viscosity data of multicomponent mixtures. For ternary mixtures, this has been shown by Chandramouli and Laddha (1963), Kalidas and Laddha (1964), and others. Since McAllister based his approach on Eyring's theory of absolute reaction rates (Glasstone et al., 1941) which states that

satisfactory correlation of viscosity data is achieved whenever the viscosity of the pure components or the interaction viscosities can be related to temperature in the following form v = B&/T

(2)

Unfortunately, pure liquid viscosities and interaction viscosities of many liquid mixtures such as alcohol-water mixtures do not exhibit the behavior indicated in (1) and (2) at all. For these mixtures, which in general contain polar fluids, McAllister's correlation technique fails. Recently, Goletz and Tassios (1977) presented an Antoinetype equation which was used successfully to correlate viscosities of various pure hydrocarbons and polar liquids to temperature. This equation reads v = BeA/(t+C) (3)

+ Ztb

In eq 6, xi is the mole fraction of component i in the mixture, while average molecular weights for three interacting molecules are computed from

M12 = (2Ml + &)/3 M21 = (Mi + 2M2)/3 M13 = (2M1 + M3)/3 M31 = (MI+ 2M3)/3 M23 = (2M2 + M3) / 3 M32 = (M2 + 2M3)/3 MI23 = (MI + M2 Ma, = x,M,

+ M3)/3

+ x2M2 + x3M3

(7)

The same type of relationship is assumed to hold for the values of C, yielding

where

C = 239

3x,x,2c,, +3X12X3Cl3+ 3X1x32c31+ 3x22x3c23+ 3X2xZc32 + 6~1~2~3C123 - Cav) (6)

(4)

In the following, the approach taken by Goletz and Tassios to correlate viscosities of pure liquids is used to modify McAllister's model of liquid mixture viscosities in order to establish a more generally valid correlation technique for multicomponent mixtures.

Modified McAllister Model Following strictly McAllister's approach as presented by Kalidas and Laddha, the shear between three parallel layers of liquid of a ternary mixture is represented by interactions between molecules of the pure components. Restricting the interactions to three-bodied interactions in a single plane, as done by Kalidas and Laddha (1964), however replacing Eyring's eq 1 with the Antoine-type equation (5)

yields the following relationship for the kinematic viscosity of ternary mixtures

c 1 2

= (2c1

c2)/3

+ 2c2)/3 c 1 3 = (2c1 + c3)/3 c 3 1 = (ci + 2c3)/3 c21

=

c 2 3

= (2c2 + c3)/3

c3z

=

c123

=

(c1

(c2+ 2c3)/3

(c1

+ c2 + c3)/3

cav= X l C l + x2c2 + x3c3

(8)

According to eq 6, the ternary mixture viscosity can be determined if properties of the pure components as well as the six binary constants vij and the ternary constant vl= are known at a given temperature. Information on the pure liquid properties is abundant; some information on the binary constants vij is available; however, information

Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982

284

Table 11. Comparison of McAllister's Eq 9 and Eq 6 for Liquid Mixtures A. NonDolar Binarv Mixtures no.

mixture

temp, "C

ref

ea 9

ea 6

37.80

v I 2 = 0.6287 v z l = 0.5762 D = 0.130

v 1 2 = 0.6280 v z l = 0.5761 D = 0.130

McAllister (1960)

benzene-toluene

25.00

v = 0.6622 v 2 ] = 0.6490 D = 0.025

v 1 2 = 0.6623 v 2 ] = 0.6492

McAllister (1960)

benzene-n-hexane

25.00

v l 2 = 0.4528 v Z I = 0.4543 D = 0.300

v 1 2 = 0.4532

v I 2 = 0.4084 v,, -. = 0.4181 D = 0.292

Heric and Brewer (1969)

= 0.5925 vZ1 = 0.6181 D = 0.138

Heric and Brewer 1969)

v l 2 = 1.8076 v 2 ] = 1.2393 D = 0.261


v l z = 2.3364 v z I = 1.7758


1

cyclohexane-heptane

2

3

4

carbon tetrachloride(n-hexane)

25.00

v 1 2 = 0.4082 v 2 ] = 0.4180 D = 0.293

5

carbon tetrachloride(n-hexane)

25.00

v I 2 = 0.5925 u 2 ] T 0.6182 D = 0.1377

6

(n-tetradecane)-( n-hexane)

25.00

v I 2 = 1.7740 v z l = 1.1638

7

(n-hexadecane)-( n-hexane)

25.00

v , = ~ 2.3055 vz1 = 1.6115 D = 0.578

8

(n-hexadecane)(2-bromobutane)

25.00

v 1 2 = 2.2938 v Z I = 1.7656 D = 0.488

9

(n-hexadecane)carbon tetrachloride

25.00

v I z = 2.3434 v z l = 2.2504

D = 0.246

10

(n-hexadecane)-benzene

25.00

11

(4-methylcyc1ohexanone)(n-hexadecane)

25.00

12

(n-tetradecane)-(2-bromobutane)

25.00

13

(4-methylcyc1ohexanone)(n-tetradecane)

25.00

14

(2-bromobutane)-(n-hexane)

25.00

15

(4-methylcyc1ohexanone)(n-hexane)

25.00

16

(n-hexadene)-( n-tetradecane)

25.00

17

(n-hexadecane)-( 1-chlorobutane)

25.00

18

(n-hexadecane)-( 1-chloropentane)

25.00

19

(n-hexadecane)-(1-chlorohexane)

25.00

20

(n-hexadecane)-(1-chlorooctane)

25.00

21

(n-hexadecane)-(1-chlorodecane)

25.00

22

(n-hexadecane)-(1-chlorodecane)

25.00

D = 1.141 v I z = 2.9913 v z l = 1.8576 D = 1.008 v 1 2 = 2.6774 v z l = 3.1130 D = 0.404 v I 2 = 1.7827 v z l = 1.2603 D = 0.241 v I 2 = 2.0800 v z l = 2.3263 D = 0.255 v I z = 0.4178 v z l = 0.41i4 D = 0.143 v 1 2 = 0.9234 vZ1 = 0.5813 D = 0.429 v I Z = 3.4222 v z , = 3.1394 D = 0.313 v 1 2 = 2.4077 v z l = 1.9107 D = 0.909 v I 2 = 2.5421 v 2 , = 1.8182 D = 0.446 v l Z = 2.6176 u z l = 1.9025 D = 0.457 v I 2 = 2.9479 v,, = 2.1359 D = 0.057 v I z = 3.2216 v z l = 2.6526 D = 0.106 v , = ~ 3.5538 v z l = 3.3250 D = 0.156

D = 0.025 Heric and Brewer (1969)

vZ1 = 0.4542

D = 0.297

tilz

D = 0.07711 2.2768 1.9155 D = 0.650 v I 2 = 2.2746 v 2 1 = 2.4368 D = 1.391 v 1 2 = 3.0572 v.. = 2.0154 D = 1.012 v I z = 2.7341 vZ1 = 3.1511 D = 0.390 v I z = 1.7951 v z I = 1.3246 D = 0.321 v I 2 = 2.0975 v Z 1 = 2.3434 D = 0.250 v I z = 0.4184 v 2 ] = 0.4116 D = 0.145 v l Z = 0.9425 v 2 1 = 0.5866 D = 0.433 v I 2 = 3.4281 v 2 , = 3.1447 D = 0.309 v , 2 = 2.4395 v z l = 2.0924 D = 1.086 v I z = 2.5884 vZ 1 = 1.9373 D = 0.539 v I 2 = 2.6564 v z l = 1.9903 D = 0.493 v , = ~ 2.9817 vZ 1 = 2.1771 D = 0.069 v 1 2 = 3.2372 v 2 ] = 2.6695 D = 0.107 V 1 2 = 3.5574 v Z l = 3.3277 D = 0.156 v12= vZ1=










Coursey and Heric (1969)


Coursey and Heric (1969) Coursey and Heric (1969) Coursey and Heric (1969) Coursey and Heric (1969)

Ind. Eng. Chem. Process Des. Dev., Vol.

21, No. 2, 1982

Table I1 (Continued) no.

23 24 25 26

27

temp, "C

mixture

eq 9

25.00 u12 = 4.0038 u z l = 4.2014 D = 0.039 (n-hexadecane)-(1-chlorohexadecane) 25.00 u I z = 4.5453 u z l = 5.2988 D = 0.105 (n-hexadecane)-(1-chlorooctadecane) 25.00 u I z = 5.1645 u z 1 = 6.7046 D = 0.049 (methylcyc1ohexane)-(heptane) 25.00 u l 2 = 0.7705 u z l = 0.6648 D = 0.456 (toluene)-(heptane) 25.00 u l 2 = 0.6017 u z l = 0.5496 D = 0.688 (n-hexadecane)-(1-chlorotetradecane)

eq 6 u I z = 4.0039 u z l = 4.2019 D = 0.039 uI2 = uZ1

ref Coursey and Heric (1969)

4.5542 Coursey and Heric (1969)

= 5.3114

D = 0.105 u l 2 = 5.1988 u Z 1 = 6.7464 D = 0.050


0.7705 Mussche and Verhoeye (1975) 0.6649 D = 0.456 u l 2 = 0.6012 Mussche and Verhoeye (1975) u z l = 0.5494 D = 0.687 uI2 = uZ1 =

B. Binary Mixtures Containing Polar Components no.

ref

mixture

28

acetone-water

29

methanol-water

30

ethanol-water

31

1-propanol-water

32

water-DMA

33

ethylene glycol-water

34

ethylene glycol-dimethyl ether-water

35

methanol-ethanol

36

acetone-methanol

37

methanol-toluene

38

ethanol-( 1-propanol)

39

acetone-ethylene glycol

40

met hanol-et hylene glycol

41

methanol-( 1-propanol)

30.00 u l 2 = 0.1670 u 1 2 = 0.4784 u z l = 5.7191 u z l = 1.9395 D = 10.314 D = 0.168 30.00 u I 2 = 0.8338 u I 2 = 1.4565 u z I = 4.0369 uZI = 1.9367 D = 4.473 D = 0.457 30.00 u I z = 0.2895 u I 2 = 1.8217 ual = 22.0020 u21 = 3.9085 D = 16.854 D = 3.148 30.00 u I z = 0.0270 u I z = 2.1339 u Z 1 = 72.4616 u g l = 3.1290 D = 26.354 D = 0.572 24.05 u I 2 = 27.0674 u l 2 = 10.5842 u Z 1 = 1.4723 u Z I = 1.7329 D = 12.705 D = 2.376 30.00 u I z = 5.2571 u I 2 = 7.4933 u t l = 10.5816 u Z 1 = 2.6284 D = 4.849 D = 0.148 25.00 u l 2 = 0.0001 u I z = 0.3918 u Z 1 = 228.2938u z l = 5.4594 D = 29.988 D = 4.606 30.00 u l 2 = 0.8493 u I 2 = 0.8490 u Z I = 1.0369 u z 1 = 1.0366 D = 0.255 D = 0.256 30.00 u I 2 = 0.4210 u I 2 = 0.4134 u Z 1 = 0.4506 u z l = 0.4573 D = 0.391 D = 0.449 20.00 u I 2 = 0.9480 u I z = 1.0086 u Z 1 = 0.6265 u z l = 0.6839 D = .622 D = 0.653 30.00 u I z = 1.5316 u I 2 = 1.4404 u Z l = 1.8737 u Z 1 = 1.7964 D = 0.111 D = 0.099 30.00 u I z = 0.5968 u I 2 = 0.6833 u Z 1 = 3.0003 u Z 1 = 3.7229 D = 1.438 D = 0.975 30.00 u l 2 = 1.8557 u l 2 = 1.9019 u z l = 6.4101 u z l = 6.4961 D = 1.849 D = 1.813 30.00 u I 2 = 1.0828 u I z = 0.9211 u z I = 1.6053 u Z l = 1.4788 D = 0.489 D = 0.530

Dizechi (1980)

Dizechi (1980)

Dizechi (1980)

Mikhail and Kimel(l963)

Peterson (1960)

Dizechi (1980)

Wallace and Mathews (1963)

Dizechi (1980)

Wallace and Mathews (1969)

Hammond et al. (1958)

Dizechi (1980)

Kalidas and Laddha (1964)


Dizechi (1980)

C. Ternarv Mixtures no. 1

mixture (carbon tetrachloride)(n-hexane)-(benzene)

temp, "C

25.00

eq9 u I z 3 = 0.4161 u I 2 = 0.4082

eq 6 ulZ3= ul2 =

0.4164 0.4085

ref Heric and Brewer (1969)

285


286

Table 11 (Continued) no.

mixture

teomp, C 25.00

1

(carbon tetrachloride)-( n-hexane)(benzene)

2

(n-hexadecane)-( n-hexane)(2-bromobutane)

25.00

3

(n-hexadecane)-(benzene)(n-hexane)

25.00

eq 9 v z l = 0.4180 v I 3 = 0.5925 vJ1 = 0.6182 v z 3 = 0.4543 v 3 , = 0.4528 D = 0.339

1.6368 2.3055 1.6115 2.2938 1.1656 v 2 3 = 0.4114 vpz = 0.4118 D = 0.344 v123 = v12 = vzl = vI3 = vj, =

= 2.0012 v 1 2 = 2.9913 uzl = 1.8516

2.3055 1.6115 0.4528 0.4543 1.323 v 1 2 3 = 1.8546 U I Z = 2.3434 v z l = 2.2405 u I 3 = 2.9913 v 3 1 = 1.8576 v Z 3 = 0.5925 = 0.6182 D = 0.335 uI3 = vjl = uz3 = v3* = D=

4

(n-hexadecane)-(carbon tetrachloride)-( benzene)

25.00

5

(n-hexadecane)-(carbon tetrachloride)-( n-hexane)

25.00

v21 vlj v31 uZ3

vj2

D 6

(n-tetradecane)-(n-hexane)(2-bromobutane)

25.00

1.8982 2.3434 = 2.2405 = 2.3055 = 1.6115 = 0.4082 = 0.4180 = 0.416

vIz3= UIZ =

u 1 1 3=

1.1780

v 1 2 = 1.1140 v 2 ! = 1.1638 v 1 3 = 1.7827 v g 1 = 1.2603 u Z 3 = 0.4114 v J 2 = 0.4178 D = 0.202

7

(n-tetradecane)(4-methylcyc1ohexanone)(n-hexane)

25.00

v , 2 3= uIz = u21 =

8

(n-hexadecane)-( n-tetradecane)(4-methylcyclohexanone)

25.00

u l Z 3 = 2.9068 v l * = 3.4222

1.5236 2.3263 2.0800 u 1 3 = 1.7740 v 3 1 = 1.1638 u z 3 = 0.9234 va2 = 0.5813 D = 0.413

u Z 1 = 3.1394 v 1 3 = 3.1130 v p l = 2.6714 v z 3 = 2.3263 v3* = 2.0800

9

10

(n-hexadecane)-(n-tetradecane)(2-bromobutane)

25.00

(n-hexadecane)-( n-tetradecane)(n-hexane)

25.00

D = 0.398 2.1230 u l z = 3.4222 v 2 1 = 3.1394 U 1 3 = 2.2938 v3, = 1.7656 u z 3 = 1.7827 u3* = 1.2603 D = 0.573 u I z 3 = 1.9667 u 1 2 = 3.4222 u l Z 3=

eq 6 0.4181 0.5925 0.6181 0.4542 0.4532 0.338 u l Z 3 = 1.7901 v 1 2 = 2.3364 v 2 1 = 1.1158 v I 3 = 2.2168 v 3 , = 1.9155 v z 3 = 0.4116 u 3 2 = 0.4184 D = 0.312 u 1 2 3 = 2.2291 u 1 2 = 3.0572 v z l = 2.0154 v 1 3 = 2.3364 ujl = 1.7758 v Z 3 = 0.4532 v 3 2 = 0.4542 D = 1.416 v1*3 = 2.0029 v 1 2 = 2.2746 u z I = 2.4368 v I 3 = 3.0572 v P l = 2.0154 v Z 3 = 0.5925 v3* = 0.6181 D = 0.351 v 1 2 3 = 2.0992 v l Z = 2.2746 v I 1 = 2.4368 v 1 3 = 2.3364 v 3 ! = 1.7758 v 2 3 = 0.4085 v j 2 = 0.4181 D = 0.525 v l Z 3= 1.2480 v 1 2 = 1.8076 u 2 ! = 1.2393 v 1 3 = 1.7951 uJ1 = 1.3246 u 2 3 = 0.4116 v 3 2 = 0.4184 D = 0.206 u l z 3 = 1.5888 u 1 2 = 2.3434 u z l = 2.0975 v 1 3 = 1.8076 u 3 1 = 2.2393 u 2 3 = 0.9425 v 3 2 = 0.5866 D = 0.413 vzl = v,, ._= vJI = vz3 = v32 = D=

v123 = 2.9426 v 1 2 = 3.4281 v 2 1 = 3.1447

3.1511 2.7341 2.3434 2.0915 D = 0.413 v 1 2 3 = 2.1465 v 1 2 = 3.4281 vZ1 = 3.1447 v 1 3 = 2.2768 v J , = 1.9155 v 2 3 = 1.7951 v j z = 1.3246 D = 0.718 v l Z 3 = 2.0027 u l Z = 3.4281









v13 = uJ1 = uz3 = u3* =



Ind. Eng. Chem. Process Des.

Dev., Vol. 21, No. 2, 1982

287

Table I1 (Continued) no.

10

11

12

mixture

(n-hexadecane)-(n-tetradecane)(n-hexane)

(n-hexadecane)( 4-methylcyclohexanone)(n-hexane)

methanol-ethanol-water

temp, “C

25.00

eq 9 v z l = 3.1394 v , - = 2.3055 v31 = 1.6115 v z 3 = 1.7740 v g 2 = 1.1638 D = 0.580 v l z 3 = 2.0600 v I 2 = 3.1130 vzl = 2.6774 v I 3 = 2.3055 v g l = 1.6115 vz3 = 0.9234 v 3 z = 0.5813 D = 0.584 v l Z 3= 4.1884 v 1 2 = 0.8493 v z l = 1.0369 v13 = 0.8338 v g l = 4.0369 vz3 = 0.2805 V J Z = 22.0020 D = 23.644 vlZ3 = 7.2857 v l 2 = 1.0828 v z l = 1.6053 v 1 3 = 0.8338 v31 = 4.0369 vZ3 = 0.0270 v J Z = 72.4616 D = 27.234 v I z 3= 150.3454 v 1 2 = 1.5316 v z l = 1.8737 v 1 3 = 0.2805 Vjl = 22.0020 vz3 = 0.0270 v 3 2 = 72.4616 D = 617.397 v l Z 3= 0.9125 v l z = 0.4120 v z l = 0.4506 v 1 3 = 0.5968 v31 = 3.0003 v Z j = 1.8557 v 3 2 = 6.4101 D = 1.177 v l Z 3= 1.2697 v l z = 0.8493 v Z 1 = 1.0369 v 1 3 = 1.0828 v 3 , = 1.6053 v g 3 = 1.5316 v S z = 1.8737 D = 0.432 I”

25.00

30.00

13

methanol-( 1-propanol)-water

30.00

14

ethanol-( 1-propanol)-water

30.00

15

acetone-methanol-ethylene glycol

30.0

16

methanol-ethanol-( 1-propanol)

30.00

on ternary constants ~ 1 2 3is very limited. For C1 = Cz = C3 = 273.16 K eq 6 reduces to the equation proposed by Laddha et al. (1964).

eq 6 v Z 1 = 3.1447 v 1 3 = 2.3364 vgl = 1.7758 v B = 1.8076 vBz = 1.2393


D = 0.719 v l Z 3 = 2.2012 v 1 2 = 3.1511 v z l = 2.7341

v13 = v3, = vz3 = v32 = D=


vgl = vz3 = v32 = D=

2.3364 1.7758 0.9425 0.5866 0.604 1.7909 0.8490 1.0366 1.4564 1.9367 1.8217 3.9085 2.037

vlz3 = vl2 = vzl = v13 = vJ1 = vz3 = v32 = D=

2.0587 0.9211 1.4788 1.4564 1.9367 2.1339 3.1290 1.810

Dizechi 1980)

v l z 3 = 1.6353 v , ~= 1.4404 v Z 1 = 1.7964 v 1 3 = 1.8217 vjl = 3.9085 v z 3 = 2.1339 v g Z = 3.1290

Dizechi 1980)

v l Z 3= = .a vzl = v13 =

D = 1.992 vlz3 = 1.0559 v 1 2 = 0.4134 v z l = 0.4573 v 1 3 = 0.6833 v g l = 3.7229 vzj = 1.9019 v32 = 6.4961 D = 1.095 v l Z 3= vl2 = vZI = v13 = v3, = vz3 = v3z = D=

1.1585 0.8490 1.0366 0.9211 1.4788 1.4404 1.7964 0.374

Dizechi (1980)


Dizechi (1980)

Consequently, it can be expected that eq 6 and eq 9 correlate mixture viscosities equally well, whenever In v is linearly dependent on 1/T. For a binary mixture, that is for x,; x2; or x 3 = 0, eq 6 and 9 reduce to relationships containing just the two binary constant vij and vji. The constants vi, and vji are found from experimental data of viscosities of binary mixtures, for instance with the help of the least square method. Their values change with temperature, however, not with concentration. For nonpolar liquids, Goletz and Tassios (1977) found that the best value for 2 is given by 2 = -0.19. This value does not hold for most of the polar liquids. For water, Goletz and Tassios recommend 2 = -1.10, while Dizechi (1980), using an optimization technique, determined 2 values for various polar fluids, some of which are listed in Table 1. Even though


288

2 is actually dependent upon temperature, this depen-

dency is so weak that the selection of only one value of Z for each polar liquid is sufficient. With all properties of the pure liquids known and the binary constants vij determined, the ternary constant ulB can then be found from experimental data on ternary mixture viscosities.

Table 111. Comparison of Ternary Interaction Coefficients Obtained with Eq 6 and 12. The Interaction Coefficients Are Given in lo6 mZ/s(cSt)

no.

mixture carbon tetrachloride(n-hexane)-benzene acetone-methanolethylene glycol methanol-ethanol(n-propanol) methanol-ethanolwater methanol(n-propanol)-water

Results and Discussion The modified McAllister eq 6 was used to correlate experimental viscosity data of 41 binary and 16 ternary mixtures containing polar as well as nonpolar fluids. In comparison, the same data were also correlated with the original McAllister eq 9. Results of these correlations are compiled in Table 11. In Table 11, the first row provides a count, the second row identifies the mixture whereby the sequency of components corresponds to the subscripts 1,2, or 1 , 2 , 3 , and the third row lists the temperature at which the viscosity data were found. The ternary constant if applicable and the binary constants for eq 6 and 9 are listed in rows four and five accompanied by values for the mean deviations. The mean deviation on a percentage basis is calculated from the following definition mean deviation 1 m (viscosityi,e,p- viscosityi,dl D=-C x 100 (10) m i=l viscosityi,e,p where m is the number of data points. Row six tells from which reference the data were taken. Our data were all obtained with an automated viscosity measuring system using a Ubbelohde viscometer. Inspection of Table I1 shows that for mixtures of nonpolar liquids, eq 6 and 9 both correlate experimental data very well. However, eq 6 is by far superior to eq 9 when applied to mixtures of polar liquids, one of them being water. Thus,eq 6 provides for a much better correlation technique. In eq 6 , the last term

ethanol-# (n-propanol)-water

is so insignificantly small that it can be omitted without loss of accuracy. That is, the viscosity of ternary polar or nonpolar liquid mixtures can be safely expressed by In

u,

=

( + c1

-t) x 1 3 t + Cav

In ulMl +

(+

-t) x 2 .c2 t + Cav

from eq 11

25.00

0.4164

0.4164

30.00

1.0559

1.0559

30.00

1.1585

1.1585

30.00

1.7908

1.7908

30.00

2.0587

2.0587

30.00

1.6353 1.6352

With values for ul, u2, uij, and vi23 known at two temperatures, the respective values for AG*lR can be determined from eq 12. With viscosities and interaction coefficients known at a temperature t,, the unknown values at a temperature t, are then found from = uaeAG*/R(l/(t,+C)-l/(t.+C))

(13)

Finally, it should be pointed out that eq 11 can be extended to an n-component mixture. If one restricts the analysis to three-body interaction, viscosity of an n-component mixture may be expressed in the following form In

u,

l n = -C ( t

t

+ cavi=l + Ci) xi3 In uiMi - In Ma, +

In n

n

n

ifjzk

where n

Ma, =

xiMi i=l

Mij = (2Mi + M j ) / 3 Mijk = (Mi + Mj

+ Mk)/3

and

Ci = 239

+ Ztb,i

n

This is demonstrated in Table 111, in which ternary constants ~ 1 2 3for six ternary mixtures are listed. As can be seen, there is virtually no difference in the values obtained with eq 6 and eq 11. Since a similar result is obtained for the binary constants, the use of the simple eq 11 is rec-

VI23

from eq6

ommended over the use of eq 6 . Viscosity data for pure liquids and liquid mixtures obtained at a fixed temperature yield the interaction coefficients vij and ula at this temperature. In order to predict viscosities over a wide range of temperatures, at least one additional set of experimental data at a different but fixed temperature must be obtained. Rearranging and differentiation of eq 5 results in

YX

3x1x22c21 + 3xi2X3c13 + 3x1x32c31 + 3x22x3cz3 + 3 ~ 2 ~ 3 ~ C+36~1~2~3C123 2 - Cav)

'123

temp, "C

c,, = c xici i=l

Ind. Eng. Chem. Process Des. Dev. 1982, .?I, 289-296

The general validity of eq 14 remains to be checked at the present time. Nomenclature A = constant in eq 2 B = constant in eq 2 C = constant in eq 3 D = mean percentage deviation AG* = molar free energy of activation h = Plank’s constant m = number of experimental data points at a fixed temperature M = molecular weight N = Avogadro number R = gas constant T = absolute temperature, K t = temperature, “C tbi = boiling temperature of component i at atmospheric pressure, “C xi = mole fraction of component i Z = constant in eq 3 v = kinematic viscosity v, = kinematic viscosity of mixture, IO6 m2/s (cSt) Subscripts 1,2 = component 1, 2, respectively 1,2,3 = component 1, 2, 3, respectively

289

i j = refer to interaction of type i-i-j, etc 123 = refer to interaction between three molecules, one of component 1, one of component 2, and one component 3 av = average cal = calculated exp = experimental Literature Cited Chandramouii, V. V.; Laddha, G. S. Indlan J. Tech. 1963, I , 199. Coursey, B. M.; Herlc, E. L. J. Chem. Eng. Data 1969, 14, 426. Dizechi, M. Doctoral Dissertation, Unlversity of California at Santa Barbara, Santa Barbara, CA 93106, 1980. Glasstone, S.;Laldler, K. F., Eyring, H. “The Theory of Rate Processes”; McGraw-Hill: New York, 1941. Goietz, E.; Tassions, D. Ind. Eng. Chem. Process D e s . D e v . 1977, 16, 75. Hammond, L. W.; Howard, K. S.; McAllister, R. A. J. Phys. Chem. 1958, 62, 637. Heric, E. L.; Brewer, J. G. J. Chem. Eng. Data 1968, 74, 55. Herb, E. L.; Brewer, J. G. J. Chem. €ng. Data 1967, f 2 , 574. Kalldas, R.; Laddha, G. S. J. Chem. Eng. Data 1964, 9 , 142. McAlllster, R. A. AICHE J. 1960, 6 427. Mikhail, S. 2.; Kimel, W. R. J. Chem. Eng. Data 1963, 8 , 323. Mussche, M. J.; Verhoeye, L. A. J. Chem. Eng. Data 1975, 20, 46. Peterson, R. C. J. Phys. Chem. 1960, 64, 184. Reid, R. C.; Sherwood, T. K. “The Propertles of Gases and Liquids”; McGrawHiii: New York, 1977. Wallace, W. J.; Mathews, A. L. J. Chem. Eng. Data 1963, 8 , 496.

Received for review December 4, 1980 Revised manuscript received August 27, 1981 Accepted December 10,1981

Vapor-Liquid Equilibrium Calculations for Polymer Solutions Wllllam Schotte Englneering Technology Laboratory, E. I. du Pont de Nemours & Company, Wilmington, Delaware 19898

Bonner’s theory for the prediction of vapor-liquid equilibria of polymer solutions has been modified. A new equation of state and corresponding vapor-liquid equilibrium equation are proposed. Good results have been obtained for mixtures of nonpolar and moderately polar compounds, containing polymers and subcritical or supercriticai solvents. Predictions can be based on a single measurement of the infinite dilution activity coefficient or Henry’s law constant to determine a single interaction parameter for the mixture. Reliable calculations can then be made for the complete concentration range up to the highest vapor pressures. Vaporization of polymer can also be included in the calculations.

Introduction Vapor-liquid equilibrium theory for polymer solutions is still poorly developed. The Flory-Huggins theory (Flory, 1942; Huggins, 1942) has often been used to correlate experimental data. However, it contains an interaction parameter which is generally concentration dependent. It is Micult to use the theory for predictive calculations since the concentration dependence of the interaction parameter is not known. Abrams and Prausnitz (1975) have developed the UNIQUAC theory, which has been of much interest for mixtures of low-molecular-weight compounds. The authors have shown that the UNIQUAC equations can also give good results for polymer-containing mixtures. Two empirical parameters must be evaluated by fitting the UNIQUAC equations to experimental data. It is often difficult to make measurements over a significant concentration range as needed for the evaluation of two parameters. Measurements at infinite dilution of a solvent in a polymer can readily be made by gas chromatography. Such a single measurement cannot be used to evaluate two parameters. 0196-430518211121-0289$0 1.2510

A one-parameter theory would, therefore, be valuable. Bonner Theory Bonner and Prausnitz (1973) have extended Flory’s theoretical work for polymer solutions. The theory uses the follow,ing equation of state.

Pressure, volume, and temperature are expressed in terms of reduced properties p = P/p*; 0 = v/v*; = T/T* where p*, u*, and T* are characteristic parameters of the particular compound. They are determined empirically by fitting eq 1 to known P-U-T data. Equation 1 can also be used for mixtures. Bonner and Prausnitz have given mixture rules to obtain p* and T* as a function of concentration. They have also derived a vapor-liquid equilibrium equation to predict the activity of the solvent at low or moderate pressures. The theory is quite useful for 0 1982 American Chemical Society

Correlation for Viscosity Data of Liquid Mixtures - American Chemical

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