literature Cited
Nomenclature
Abas-zade, A. K., Dokl. Akad. Nauk SSSR. 99,227 (1954). Abas-zade, A. K., Amiraslanov, A. M., Zh. Fiz. Khim. 31, 1459 (1957). Eucken, A., Physik. 2. 14,324 (1913). m ,n Groenier, W. S., Thodos, George, J . Chem. Eng. Data 6 , 240 M (1961). P, Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” p. 534, Wiley, New York, 1964. R Hofker, Hinrich, Programm-Gymnasium, Piattenscheid, 1892. S Hougen, 0. A., Watson, K. M., Ragatz, R. A., “Chemical ProT cess Principles,” Part I, p. 88, Wiley, New York, 1954. TC Kennedy, J. T., Thodos, George, A.I.Ch.E. J . 7,625 (1961). TR Keyes, F. G., Trans. Am. SOC.Mech. Engrs. 76,809 (1954). X Kobe, K. A,, Lynn, R. E., Jr., Chem. Revs. 52,117 (1953). Lambert, J. D., Cotten, K. J., Pailthorpe, M. W., Robinson, x 1 A. M., Scrivins, J., Vale, W. R. F., Young, R. M., Proc. Roy. Soc. (London)A231,280 (19%). AX1 Lambert, J. D., Staines, E. N., Woods, S. D., Proc. Rou. Soc. (London)A200,262 (1950). 6x1 Markwood, R. H., Benning, A. F., Refrig. Eng. 45, 95 (1943). Nasi&,A. P., Valle Bracero, A., Barrales Rienda, J. M., Ann. Real Vc SOC.Es aii. FZs. Quim. Ser. A , 60 (5/6), 89 (1964). zc Mathur, P., Thodos, George, A.I.Ch.E. J . 11,164 (1965). hlilverton, S. W., Proc. Roy. SOC.(London)A150,287 (1935). Moser, E., dissertation, Friedrich-Wilhelms Universitat zu BerGREEKLETTERS lin, 1913. Roy, Dipak, Thodos, George, Can. J . Chem. Eng. 46, 108 (1968a). CY = coefficient, Equation 1 Roy, Dipak, Thodos, George, IND.ENG.CHEM.FUNDAMENTALS C Y , p, y, 6 = polynomial coefficients, Equation 8 7,529 (1968b). = thermal conductivity parameter, 1W1~2Tc1/6/Pc2/aSenftleben. H.. 2.Anuew. Phus.. 5. 33 119531. x Senftleben; H.; Gladikh, H.,Z.’Pkysik. 125; 653 (1949). Sherratt, G. G., Griffiths, E., Phil. Mag. 27(7),68 (1939). SUBSCRIPTS Shushpanov, P. I., J. Erptl. Theor. Phys. USSR. 9, 875 (1939). Vines, R. G., Bennett, L. A., J . Chem. Phys. 22,360 (1954). r = rotational t = translational RECEIVED for review January 2, 1969 ACCEPTED September 25, 1969 V = vibrational = constant in Equation 5
C
thermal conductivity of gas at normal pressures, cal/sec cm OK = exponents, Equation 1 = molecular weight = critical pressure, a t m = gas constant = exponent, Equation 6 = temperature, O K = critical temperature, O K = reduced temperature, T / T c = rotational and vibrational contribution to k*A, [(k*A), (k*A),l = value of X a t T R = 1.00 = contribution to X 1 in replacement of a hydrogen atom by different functional groups = individual contributions for different groups in cyclic compounds = critical volume, cma/g-mole = critical compressibility factor, PcvC/RTc
k*
=
+
8.
Prediction of Viscosity of Liquid Hydrocarbons S. I. Kreps and M. 1. Druin’ Department of Chemical Engineering, Newark College of Engineering, 323 High St., Newark, N . J . 07109
The liquid viscosity of four series of homologs-n-paraffins, n- 1 -alkenes, n-alkylcyclohexanes, and n-alkylbenzenes-is estimated from density and molecular weight data. An empirical equation for each of the series, n-paraffins to n-alkylbenzenes, gives average errors of 1.78, 1.95,2.39,and 3.46%, respectively.
A
RECEKT review by Reid (1965) indicated the need for methods of computing the viscosity of a liquid in the absence of experimental data. Gambill (1959) and Reid and Sherwood (1958) concisely present the available methods and conclude that none are reliable. Of the methods available, those of Thomas (1946) and Souders (1938) are recommended only for rough estimates, which are usually good t o within 30% but sometimes are in error by greater amounts. The rheochor is suitable for rough approximations of viscosity a t the normal boiling point.
Viscosity of Monatomic liquids
From the kinetic theory of gases it has been shown (Bird
et al., 1960) that the viscosity of a monatomic gas is given by the relationship I* =
1/3
iipA
I n adapting this relationship for liquids it is here assumed that the liquid molecules are arranged in a cubic lattice, with center-to-center spacing equivalent to
A = (V/N)1’3 I t is further postulated that momentum is transferred from one lattice plane to a n adjacent plane at the sonic velocity, us, for the liquid. This is similar to a model for energy transport proposed by Bridgman (1923) which predicts the thermal conductivity of pure liquids, with a cubic lattice similar to that of the solid state. Energy is transferred from plane t o plane a t the sonic velocity by collisions arising out of molecular vibrations about the equilibrium lattice positions. Bridgman’s equation for the thermal conductivity is given (Bird et al., 1960) as
(1)
K Present address, Celariese Research Co., Box 1000, Summit, S . J . 07901 1
=
3(N/V)*/3kv,
(2)
I n the case of momentum transfer, molecules reaching a VOL. 9 NO. 1 FEBRUARY 1970
I&EC FUNDAMENTALS
79
Table
I. Accuracy of Viscosity Predictions by Equation 8
Series
nParaffins
n-lAlkenes
Monan-alkyl cyclohexanes
No. of carbon atoms 1-20 1-20 6-22 Av. % errora 33.6 39.9 51.8 Average % error calculated as [ ( p - p v ) / p r ] 100.
Monon- alkyl benzenes
6-22 44.3
v, = V,'(V/NV,)'/3
given plane have, on the average, suffered their last collision a t a distance A from that position, where A = 2/3 X = (V/N)''3
= 2
IaLul
(4)
I I
GL
=
p =
pv,(V/N)"3
(5)
Rao (1940) and Sakiadis and Coates (1955) show that the velocity of sound in a liquid may be calculated by = (PP/W3
(6)
where p is an empirical structural contribution factor (Reid and Sherwood, 1958). Using Equation 6 to replace v, in Equation 5 and replacing the molar volume, V, by M/p gives p =
p4P3_~~'/3/(-~P3N''3pl/3)
(7)
which may be simplified to =
p11/3p3/*Jf8/3N1/3
(8)
Ratio KIP
If Bridgman's equation for thermal conductivity is divided by Equation 5 , the ratio K / p is given as
K / p = 3kN/Vp Since k
=
(9)
R / N and V = X / p , there results
M(K/p)
=
3R
(10)
When the viscosity, p, is in poises and k is in calories per second per centimeter per degree Kelvin, the gas constant, R, is very nearly 2 cal per gram mole per degree Kelvin. Substitution of this value into Equation 10 results in
M(K/p)
=
6
(11)
This result is identical with an equation developed by Mohanty (1951), who combined the Andrade viscosity equation (Andrade, 1934) with the thermal conductivity equation of Osida (1939). The works of both Andrade and Osida are based on liquid models similar to that postulated here. Viscosity and Sonic Velocity
Equation 8 can be developed by a second method. The method used by Powell, Roseveare, and Eyring (1941) for estimating the thermal conductivity of pure liquids has been employed here for developing the viscosity-density correla80
I&EC FUNDAMENTALS
(13)
( 8 k T / m ~ ) '(V/NV,)1/3 /~
(14)
0.5Op(V/N)'/3 ( 8 k T / m ~ ) " ~(V/NV,)'/3
VOL. 9 NO. 1 FEBRUARY 1970
(15)
The free volume, V,, is eliminated by combining Equations 12 and 15 with
us'
1 1
Us
a(V/NV,)'/3
aL =
From this, the viscosity of a monatomic liquid is
where a, is the absolute value of the y component of the mean molecular velocity. Similar relationships apply to the velocity components in the 2 and z directions. Combination of Equations 1,3,and 4, and replacing aLY by the sonic velocity results in p =
(12)
Analogously for a liquid, Equation 1 must be modified by replacing the mean free path, A, by 3/2(V/N)1/3. I n accord with the speed of sound arguments, the mean molecular speed of the liquid molecules should be designated
(3)
The mean molecular speed is
a
tion. The expression presented below differs from Equation 8 only by the numerical coefficients. Powell et al. (1941) have shown that v, in liquids is from five to ten times greater than the mean speed of the molecules. Upon collision of two molecules, sound is transferred between molecules almost instantaneously. I n terms of the intermolecular distances,
= (y
kT/m)1/2
(16)
where y is the specific heat ratio. Thus p =
0.80 ~ , p ( V / / n ~ ) ~ ~ ~ y - ~ / ~
(17)
For liquids, y is nearly unity except near the critical point. Accordingly, the viscosity of a monatomic liquid is p =
0.80 ~ V , ( V / N ) ' / ~
(18)
Equations 5 and 18 differ only in the value of the constant coefficients, 1.0 and 0.8, respectively. Viscosity of Hydrocarbons
Equation 8 was used to compute the viscosities of normal paraffins, 1-alkenes, monoalkylcyclohexanes, and monoalkylbenzenes at each lO'K interval extending over a 1' to 300' range. Densities were taken from the API compilations (American Petroleum Institute, 1953) and the values of the additive constitutive constant from Reid and Sherwood (1958, Table 7-6). The results are summarized in Table I. The precision of the calculated viscosities is about the same order as is provided by the methods of Thomas (1946) and Souders (1938). However, when the individual point deviations, Z = p / p , , are plotted against the reduced temperature with respect to the normal boiling points, T r B = T / T B , on log-log coordinates, it is seen (Figures 1, 2, 3, and 4) that the predicted viscosities for each series of homologs differ from the reported values in a regular manner. For each series, a family of curves, parametric in the number of carbon atoms, may be drawn through these 2 points. The Z function approaches a limiting value a t about 20 carbon atoms. The parametric curves for each series of homologs may be approximated by a single line of best fit in the form Z = exp[a
+ 6 ln(TrB) + c l n * ( T r ~ ) ]
(19)
Coefficients for Equation 19 were obtained by nonlinear regression techniques. The 2 functions for the four series of homologs are plotted in Figure 5, and the coefficients for Equation 19 are listed in Table 11. Either the graph or the equation may be used to determine the appropriate 2 function to correct the estimate provided by Equation 8. The corrected viscosity estimate is thus = p1113 P 3 1 (-1/8/3*y1/32) (20)
*
'
O
;I
V
1.5
0.9 0.8 0.7 0.6
. 2
0.5
a I,
0.4
-
8 . 0
-
PI!
0.2
-
0. I 5
-
0.3 0.25
0
KEY -
A
d A
A I
0 I
0.1
I
I
I l l 1
t
.
0.5
.I
I
0.4
$1
N
0.3 0.2 5
0.15
G'2
1
4' T,n
Figure 3.
T/TR
Z as a function of TrB for n-alkylcyclohexanes
*'@ r-7 1. 5
S"O0
I. 0
0
KEY -
8
A
A 8
OA A 0.25
A
om
0.2
A
0. I 5
0. I 0 3
G d
0 6
''6
1 0
1
0. 2 5
00.2 '3
E
& 8
A
3
1 5
T r =~ T I T B
Figure 2.
Z as a function of TIB for n-1-alkenes
Table I1 also indicates the error experienced in using Equation 20 as a viscosity predictor. The improvement in quality of viscosity prediction provided by Equation 20 is evidenced by the ratio of average errors for Equations 8 and 20; the Z factor reduces average errors by a factor of 4.9 to 9.0 for the individual series. The maximum errors are generally produced by predictions a t or near to the freezing points of the hydrocarbons; Equation 20 rapidly gives much more accurate results even only 10" above the freezing point. Prediction of Viscosities
Equation 20 was further tested by prediction of the viscosities of octacosane (CBHB) and hexatriacontane (C38H74).
The predictions were compared with the experimental data of Doolittle (Doolittle and Peterson, 1951). Average errors were 8.96% for C28H58 and 8.0% for C36H74. Thus Equation 20 may be used for extrapolation to compounds of moderately high molecular weight with fair precision. Single Viscosity Equation for Normal Hydrocarbons
The 2 functions for the hydrocarbon series are sufficiently similar (Figure 5) t o be estimated t o be identical, and all the data available can be used to evaluate the coefficients for Equation 19. Alternatively, the data for two or three of the series can be used for the correlation, and these data can then be extrapolated to any other series of hydrocarbons to predict VOL. 9 NO. 1 FEBRUARY 1970
l&EC FUNDAMENTALS
81
Table 11.
Prediction of Viscosities b y Equation 20 nParaffins
Series
No. of carbon atoms 5-20
n-lAlkenes
Monon-alkylcyclohexanes
Monon-alkylbenzenes
5-20
8-22
7-22
Coefficients for Equation 19 0.422 0.323 0.552 0.337 0.239 0.405 0.842 -0.270 -4.60 -3.56 -3.88 -3.20
a
b C
Viscosity Calculated by Equation 20
No. of poiiit,s Av. yo error Max. yo error Fract’ion of errors >IO% Relative average error, Eq. 8/Eq. 20
37 8 6.8 49.6
177 4.9 47.5
0.143
177 7.9 29.0
0.074
265 4.9 27.8
0.282
0.121 0
4.9
8.1
6.6
Figure 5. Nonlinear least-squares fits as a function of for four series of homologs
9.0
the viscosities. Both propositions were tested (Table 111). The constants listed in this table for the case where all four series were correlated represent the equation of best fit to all the data used, based on a total of 997 points for 63 hydrocarbons in the four-homolog series. It is seen that Equation 20 can be used with fair accuracy for prediction of viscosities of hydrocarbons where no experimental data exist. Prior to this work, no reliable method was available for the estimation of the viscosity of a liquid in the absence of even a single viscosity measurement (Reid and Sherwood, 1958). If a few experimental points are available for even one compound, the 2 function line might thereby be located with better precision, and considerably better predictions of viscosity should result for the entire series of homologs. Parametric Predictor Equations
The 2 function was further refined by introducing a second variable, C, representing the number of carbon atoms in each compound-Le., 2 = Z(T,B,C).Thus, the constants of Equation 19 were correlated with C, using a least-squares nonlinear regression procedure, and the polynomial of best fit was developed. These are in the form a = a’
+ b’C + d’C2
(21)
and similarly for 6 and c constants (Table IV).
Equation 21 was used rvith Equations 19 and 20 to predict the viscosities of the 63 hydrocarbons considered here (Table V). The use of the carbon parameter decreases the average error for the series of homologs by factors ranging between 1.4 and 3.8. The alkylbenzenes and alkylcyclohexanes were correlated with slightly greater accuracy by using the number of carbon atoms in the alkyl side chain rather than the total number of carbon atoms. The parametric equations for the paraffins were tested by extrapolation of the equations t o the lower paraffin hydrocarbons, methane to butane, which were not used in the original correlations. Average errors in the predicted viscosities were methane 7.39, ethane 8.48, propane 8.86, and butane 6.64y0, respectively. Extrapolations to the higher hydrocarbons were not successful when the true parametric value for number of carbon atoms was used. The parameter approaches a limiting value a t about 20 carbon atoms. Significance of Z Function
The 2 function employed here to correct the viscosity predictions corresponds, in principle, to the compressibility factor for gases and liquids. The idealized liquid state model which is the basis of the present development, like the ideal gas law, does not adequately predict the viscous behavior of real liquids. However, the deviations from the model are
Table 111. Prediction of Viscosities for Series of Homologs Method
Av. % Error
Max. % Error
%-Paraffins Olefins Alkylbenzenes Alkylcyclohexanes
Correlated Correlated Predicted Predicted
7.59 8.53 10.1 18.4
54.2 52.8 35.2 37.9
0.159 0.310 0.426
0 341
n-Paraffins Olefins Alkylbensenes Alkylcyclohexanes n-Paraffins Olefins Alkylbenzenes Alkylcyclohexanes
Correlated Correlated Correlated Predicted Correlated Correlated Correlated Correlated
8.60 6.98 8.07 19.4
61.6 49.6 31.1 36.1
0.318 0,254 0.253 1.000 ( 0 . 2 3 > 25%)
0.338
-0.231
-4.36
7.51 10.2 12.1 15.2
77.6 57.8 31.1 30.4
0.180 0.305 0.590 1.000 (0.096 > 25%)
0.337
-0.161
-4.43
Series
82
I&EC FUNDAMENTALS
VOL. 9 NO. 1 FEBRUARY 1970
TIB
Fraction of Errors 10%
>
a
Constants for Eq. 1 9 b
0.0157
C
-3.97
1.000 ( 0 . 2 2 > 25y0)
relatively consistent, and may be taken into account b y a relatively simple empirical function, applicable to a wide variety of liquids. It is proposed that the 2 function varies with TTB in the reported manner because the development employed here was based on a reinterpretation of the rigidsphere gas theory to apply to a monatomic liquid. Thus, the maximum deviations from the idealized liquid state model generally occur near or a t the freezing point. The variation of 2 with carbon number, C, may be due to variations in molecular shape, size, and forces which are not considered in the idealized liquid state model (monatomic liquid). At low carbon numbers, small increases in C may significantly affect the magnitude of molecular forces. This effect diminishes as C increases, until it approaches a limiting value. The 2 function employed here approaches a limiting value at about 20 carbon atoms. Introduction of factor 2 into Equation 5 results in the relationship
M(K/g)
=
62
Acknowledgment
The authors thank the Sewark College of Engineering Reqearch Foundation for financial support of this project. The asqistance of Margaret Sichols in carrying out the computations is gratefully acknowledged.
k 111
na
s R T TB TI^ ii? c,, IhU
v ” ’ ;j
= = =
= = =
disposable constants disposable constants disposable constants disposable constants number of carbon atoms thermal conductivity coefficient cal/(sec) (cm)
(C)
Boltziiian constant, R / S niolecular weight = niass of molecule, g = .Ivogndro iiumber, 6.0238 X l o L 3niolecules/g mole = gas constant, 1.987 cal/o(g mole) (OK) = absolute temperature, K = &solute nornial boiling point, “ K = reduced temperature with respect t o boiling point, T / T B = mean molecular speed of gas, liquid molecules, crn/ sec = absolute value of mean molecular velocity, cm/see = molar volume, cni3/g mole = molecular free volume, cm3/molecule = speed of sound in liquid, gas, cm/sec = deviation, p / p 7 = =
GREEKLETTERS = empirical structural contribution factor 6
nParaffinsa
n-lAlkeneso
a = a’ a’
6’ d‘
0.496 -0.0147 0.0
g’
-0.279 0.0978 -0.00498
=
e‘
Monon-olkyl* cyclohexanes
Monon-alkyP benzenes
0.573 -0.141 0.00988
1.32 -0.240 0.0103
+ b‘C + d‘CZ
1.01 -0.0642 0.0 b
e’ f’
+ f’C + g’C2
1.94 -0.213 0.0
2.00 -0.705 0.0406
3.76 -1.08 0.0472
+
+
c = h’ j‘C k‘C2 -3.15 0.950 -2.26 -0.0534 j‘ -0.0573 -0.634 -0.810 -1.22 k’ 0.0 0,0144 0.0449 0.0564 C represents total number of carbon atoms in compound. C represents only number of carbon atoms in alkyl side chain.
h’
Table V. Accuracy of Viscosity Prediction for Parametric Equations
Series
S o . of points
No. of carboii
nParaffins”
Alkenes“
378 5-20
177 5-20
Monon-alkylh cyclohexanes
n-l-
177 8-22
Monon-alkylh benzenes
234 9-22
atoms Xv. yo error 1 78 1 95 2 39 3 46 Mas. yo error 14 8 6 29 19 5 17 1 Fraction of error< > 10% 0 0132 0 0 0 0057 0 047 Relative av. Yo error, Z(T,B): ~ ( T T BC), 3 82 2 51 3 31 1 42 a C represents total number of carbon atoms in compound. C represents number of carbon atoms in alkyl side chain.
A P
Nomenclature
a’,b‘, d’ e’, f’, g’ h’, J ’ , k’ C K
Series
(22)
l l o h a n t y tested his relationship (Equation 11) at the boiling point for a group of chemically unrelated compounds and shoited the mean value of the constant to be 10.8. Thus he, in effect, demonstrated deviations equivalent to a n average value of 2 of 1.8 a t T,, = 1. For the n-paraffins, the present work shows that 2 a t the boiling point is 1.4, and this varies from a high of 1.75 for the n-1-alkenes to a low of approsimately 1.35 for the alkylcyclohexaiies. The members of the series of homologs tested here all show essentially similar deviation factors, 2, when examined at corresponding conditions of reduced temperature. Thus Equation 22 is applicable over a wide temperature range.
a , b, c
Table IV. Constants for Equations 21
Y
x I.(
PT
= = = =
= =
mean intermolecular distance, cm density, g/cma specific heat ratio mean free path, crn viscosity, g/(cm)(sec) or poise reported viscosity, g/(cm) (sec) or poise
Literature Cited
American Petroleum Institute, “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,” Project 44, Carnegie Press, Pittsburgh, Pa,, 1953. Andrade. E. N. daC.. Phil. M a o . 17.497 11934). -~ Bird, B.‘ R., Stewart, W. E:, Lightfoot, E. K.,“Transport Phenomena,” Wiley, New York, 1960. Bridgman, P. W., Proc. Am. Acad. Arts Sci. 59, 141-69 (1923). Doolittle, A. K., Peterson, R. H., J . A m . Chem. SOC.73. 2145-51 (1951). Gambill, W. R., Chem. Eng. 66, 127-30 (1959). hlohanty, S.R., A-ature 168,42 (1951). Osida, I., Proc. Phys. Math. SOC.Japan 2 1,353 (1939). Powell, R. E., Roseveare, W. E., Eyring, H., Ind. Eng. Chem. 33, 430-5 (1941). Rao, R., Current Sci (India)9, 534 (1940). Reid, R. C., Chem. Eng. Progr. 61, 58-64 (1965). Reid, R. C., Sherwood, T. K., “Properties of Gases and Liquids,” 1IcGraw-Hill, New York, 1958. Sakiadis, B. C., Coates, J., A . I . Ch. E . J . 1,275 (1955). Souders, AI., J . A m . Chem. SOC.60, 154 (1938). Thomas, L. H., J . Chem. SOC.1946,573. I-
RECEIVED for review February 7, 1969 ACCEPTED November 25, 1969 VOL. 9 N O . 1 FEBRUARY 1970
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83
