A Liquid Viscosity-Temperature4 hemical Constitution Relation for Organic Compounds R. lopes Cardozo,'
D. van Velzen,*
and H. Langenkamp
C.C.R. Euratom, Ispra, Italy
A new empirical relation between liquid dynamic viscosity, temperature, and chemical constitution of organic compounds is proposed. The method i s based on the De Guzman-Andrade equation and the introduction of a new property, the equivalent chain length of a compound, defined as the chain length (in carbon atoms) of the hypothetical n-alkane having a viscosity equal to 1 CP at the same temperature as the compound in question. Cumulative constitutional correction factors for the equivalent chain length and the slope of the log viscosity-temperature curve are proposed. With the aid of these data, the viscosity between boiling and melting points of many compounds can be predicted. The method proves to be more accurate than the existing ones and it does not make use of any other physical property.
T h e literature dealing with liquid viscosity is very extensive and many attempts have been made to correlate the viscosity of liquids to temperature and chemical constitution, both theoretically and empirically. Truly theoretical efforts have met with little success; at present there is no theory which allows liquid viscosities to be calculated by simple equations. Therefore empirical estimation techniques must be used. Effect of Temperature
The best known relation to correlate liquid viscosity and temperature is 7~ =
AeB'T
(1)
where A and B are positive. Equation 1 was originally proposed by D e Guzman (1913) and has since become known as the Andrade equation because Andrade (1930) suggested this form as a result of a n analysis of the mechanism of liquid viscosity. The equation represents the viscosity- temperature relation fairly well within the temperature range which extends from the normal boiling point to the freezing point. For associated liquids and hydrocarbon mixtures of higher viscosity the relationship between log q L and l / T becomes slightly curved. Consequently, in these cases the D e GuzmanAndrade relation fails. T o overcome this difficulty, a large number of modifications of the original equation have been proposed, generally introducing, in one way or another, a third constant to correct the equation for the observed curvature. If many data points are available for a single compound, these can be more accurately fitted by such equations, but for general purposes the D e Guzman-Andrade relation is still considered the best simple function. Effect of Chemical Constitution
Various attempts have been made to predict viscosities as a function of chemical constitution: none is reliable; all are empirical. Hitherto all of them involve some other physical property and no relation has been developed by which the
viscosity can be estimated from the chemical constitution alone. Reid and Sherwood (1966) judge the methods of Souders (1938), Thomas (1946), and Stiel and Thodos (1961, 1964) as the three best estimations available. These three methods will be used as a basis of comparison for the relation proposed by the authors. Reid and Sherwood (1966) tested the three methods for 40 different compounds a t various temperatures. The error distribution is given in Table I and Figure 1. It is concluded that the best approximate method available is the Thomas relation. A serious drawback is the rather poor accuracy (15% of the values are predicted with a n error larger than 50%) and the fact that the critical temperature and the density of the compound must be known. Proposed Relation for the Homologous Series of the Alkanes
The aim of the development of another viscosity- temperature-chemical constitution relation was to establish a relation in which no other physical property occurs. The basis of the relation is the De Guzman-Andrade equation, written as
When q~ = 1 cP, log 7~ = 0 and it follows that T = To = - B / A a t q L = 1, from which a modification of the original equation can be derived r,
This equation contains two unknown factors, To, which is the intercept of the log viscosity-temperature line with the abscissa, and B , the slope of this line. For the homologous series of the n-alkanes B as well as T oappear to be functions of N , the number of carbon atoms. By regression analysis it is found that for N 6 20
To
=
and for N 1
A.K.Z.O., Divisie Zoutchemie, Boortorenweg 2 Hengelo,
Holland. 20 Ind.
Eng. Chem. Fundam., Vol. 11,
No. 1, 1 9 7 2
,i
28.86
+ 37.439N - 1.3547N2 + 0.02076N3
(3)
+ 238.59
(4)
> 20 To
=
8.164N
Table 1. Cumulative Error. Distribution of Various Methods for Calculation of liquid Viscosity. Examples of Reid and Sherwood Percentage of the predicted values Stiel and Proposed Souders Thomas Thodas method
Error
36.0 47.5 53.3 58.3 63.3 69.1 77.0 80.6 82.8 85.7 88.6 80% 90.8 90% 1 0 0 ~ o 92.2 calcd - exptl a Error = exptl
Less than Less than Less than Less than Less than Less than Less than Less than Less than Less than Less than Less than Less than
[
3
1
5
1
1
XI
0
30.2 46.5 54.3 65.9 70.6 76.0 81.4 85.3 90.7 93.0 96.9 99.9 99.9
5% 10% 15% 20y0 25% 3oy0 4oy0 5oy0 60% 707,
12.6 24.4 36.2 47.1 53.0 58.9 74.0 79.9 88.3 95.0 97.5 99.2 99.2
41.7 61.4 72.4 78.7 82.6 85.7 92.8 94.4 94.4 95.2 95.2 96.0 96.0
1
’
Y ) x I w o E m (%)
Figure 1 . Error distribution of various methods. Examples from Reid and Sherwood ( 1 966)
Compound
for N 6 20 B
for N
=
24.79
Table II. Mean Errors for Paraffinic Compounds
+ 66.885N - 1.3173N2 - 0.00377Na
(5)
+ 13.740N
(6)
> 20 B
=
530.59
Figures 2 and 3 show the calculated lines and experimental points. The viscosity data were obtained from “Selected Values of Hydrocarbons” (1953). With the aid of eq 3-6 the viscosities of all n-alkanes a t all temperatures between the melting and the boiling points can be predicted. With the exception of methane, the calculated and experimental results agree reasonably, as appears in Table 11. As a yardstick for the reliability of the method the “mean error” for each compound has been computed. The mean error is defined as
Methane Ethane Propane Butane Pentane Hexane Heptane Octaiie Nonaiie Decane Undecane Dodecaiie Tridecaiie Tetradecane Pentadecane Hesadecane Heptadecane Oct,adecaiie Nonadecane Eicosaiie Hexacosane Triacontane
No. of data points
Mean error,
4 18 29 18 39 49 54 52 46 47 48 45 55 49 53 49 62 58 59 61
111.8 9.0 12.3 9.3 4.0 3.5 2.7 1.8 2.4 1.8 1.9 1.6 2.1 1.8 2.2 2.1 2.3 2.2 2.3 1.9 6.4 8.6
1 1
%
n IEl = absolute value of (calculated value - experimental value)/(experimental value) and n is the number of data points available for each compound. The equations developed for the n-alkanes form the basis of the general relation for other compounds. General Relation
The principle of the method is the introduction of the equivalent chain length ( N E ) ,which is the chain length of a hypothetical n-alkane with viscosity equal to 1 CP a t the temperature a t which the viscosity of the compound in question is also 1 cP. N E is calculated as the sum of the total
number of carbon atoms of the compound ( N ) and one or more structural and/or configurational factors ( A N ) . I n the same way, B is calculated as
B
=
B.
+ AB
where B , is the value of B for the hypothetical alkane with the equivalent chain length N E (calculated by eq 5 or 6, or from Figure 3) and AB is a correction factor depending on the chemical constitution of the compound. Generally, &V‘ is not a constant, but a function of N , whereas in most cases AB is a function of N E . It appears that for compounds with more than one funcInd. Eng. Chem. Fundam., Vol. 1 1 , No. 1, 1972
21
To I O C 1
LOO
100
3 50 50 300
0 250 ,50 200
,100 150
-I 50 100 0
2
L
8
6
10
12
IL
16
18
20
22
24
N
Figure 2. Dependence of To on N for n-alkanes
800
700
600
500
400
300
200
100
0
2
4
6
8 IO 12 14 I6 18 Figure 3. Dependence of 6 on N for n-alkanes
tional group both AN and AB are cumulative factors. Thus, the equivalent chain length can be found from
NE
=
N
-f ANI
+ ANz
.
Once N E is known, To can be calculated from eq 3 or 4, inserting N E for AT. B, is found from eq 5 or 6, inserting B , 22
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
20
22
24
N
and N E for B and N . Subsequently the required viscosity is calculated by eq 2, as both factors B and Toare now known. Table I11 gives the various functions for AN and AB for a number of functional groups and structural configurations. There exists one important difference between the calculation of N E and B. If the compound contains two or more identical functional groups, the AN correction for this group should be
applied twice, thrice, etc. On the contrary, for the calculation of B , the functional correction has to be applied only once. A few examples follow. I. Pentanoic Acid (Valeric Acid). Calcula18ion of N E from N and Table 111 AN(acids): 6.795
+ 5 X 0.365
N
= =
NE
=
5.00 8 62 13.62
= 24.79
+
+
Thus: log
Calculation of B from N E , eq 5, and Table 111
+
66.885 X 13.62 - 1.3173 X 13.623 0.00377 X 13.623 = 681 86 AB (acids): -249.12 22.449 x 13.62 = 56 64 B = 738 50 Calculation of TOfrom N E (eq 3) To = 28.86 37.439 X 13.62 - 1.3547 X 13.62' 0.02076 X 13.633 = 339.93'K
B,
VL =
+
[k -
738.50 -
~
33:.93]
Table 111. Functions for A N and A 6 AB
AN
No. of compounds used
%-Alkanes Alkene Acid 3 6 A: 6 10 N > 10 Ester Primary alcohol Secondary alcohol Tertiary alcohol Diol Ketone Ether Primary amine Secondary amine Tertiary amine Fluoride Chloride Bromide Iodide Aromatic and 1-nitro 2-Nit ro 3-Nitro 4-, 5-Nitro
Functional Groups ... -0.152 - 0.042N 6.795 0.365N 10.71 4.337 - 0.230N 10.606 - 0.276N 11.200 - 0.605N 11.200 - 0.605N Alcoholic correction configurational factor 3.265 - 0.122N 0.209N 0.298 3.581 0.325N 1.390 0.461N 3.27 1.43 3.21 4.39 5.76 7.812 - 0.236N 5.84 5.56 5.36
+
+
+ + +
+
-44.94 5.410NE - 249.12 + 22.449NE -249 12 + 22.449NE - 149.13 18.695NE -589.44 70.519h'E 497.58 928.83 557.77
+ +
-117.21 -9.39 25.39 25.39 25.39 5.75 -17.03 - 101.97 -85.32 -213.14 -213.14 - 338.01 - 338.01
+ 15.781NE + 2.848NE + 8.744NE + 8.744NE + 8,744NE
22 19 10 4 22 6 1 2 2
+ 5.954NE + 18.3301VE + 18.330ArE + 25.086A'E + 25.086NE
9 5 6 5 3 2 4 9 4 4 6 9 6
13.869NE ++ 13.869NE + 9.558NE + 13.869NE
9,4,4,5,3 7 3 3
Configurational Factors Correction for Aromatic Nucleus Alkyl-, halogen-, nitrobenzenes, secondary and tertiary amines 8 6 N 6 15 0.60 N > 15 3.055 - 0.161N Acids 4.81 Esters -1.174 0.376N Alcohols: OH attached t o nucleus: take for all phenolic compounds N E = 16.17. Alcohols: OH in side chain -0.16 Ketones 2.70 Ethers: take for all aromatic ethers ATE = 11.50. Primary amines: NH2 attached to nucleus: take for all aniliiiic compounds N E = 15.04. Primary amines: NH2 in side chain -0.16 Various Polyphenyls -5.340 0.815N Ortho configuration: OH group present 0.51 without OH Meta configuration 0.11 Para configuration -0.04 Cyclopentane 7 6 N 6 15 0.205 0.069N N > 15 3.971 - 0.172N Cyclohexane 8 6 A' 6 16 1.48 N > 16 6.517 - 0.311N
+
- 140.04 -140.04 - 188.40 -140.04
213.68 213.68 - 760.65 -140.04
+
- 188.40
+
54.84 27.25 -17.57 -45.96 -339.67 -272.85 - 272,85
- 571.94
8 2 2
+ 50,478NE + 13.869NE
9
... ...
7 1
+
9.558NE
4 2 5
+ 2.224NE + 23.1351C'E + 25.041ArE + 25.0411VE
6 6 9 6 9 6 (Continued on nest p a g e )
Ind. Eng. Chem. Fundam., Vol. 11, No. 1, 1972
23
(Table 111.
Continued) No. of
AN
Aa
Is0 Configuration 1.389 - 0.238N 0.93 1.389 - 0.238N 0.24 -0.24 -0.24 -0.50
Alkanes Double is0 in alkanes (extra correction) Alkenes Alcohols Esters, alkylbenzenes, halogenides, ketones Acids Ethers, amines
compounds used
15.51
4 3 3 5 511,5,1 2 4,2
8.93 94.23 8.93
... 8.93
Various C(Cl), configuration 1 . 9 1 - 1.495X 4 -26.38 -CCl-cc10.96 2 -C (Br) z2 81.34 - 86.850X 0.50 -CBr-CBr1.60 -57.73 3 CF3- (in alcohols) 2 341.68 -3.93 (other compounds) -3.93 6 25.55 2 Diols -2.50 N See alcohols a Other substituents, such as C1, CH,, NO*, are neglected for the determination of NE. For the calculation of R, they have to be taken into account.
+
~~
~
Table IV. Viscosity Data Viscosity, CP Temp
Calcd
Exptl
Pentanoic Acid 2.38 2.22 2.015 1,296 0.954 0.726
2.41 2.30 2.050 1.315 0.986 0.753
O C
16.5 20 25 50 70 90
Chloroform 0 10 20 30 40 50 60
0.774 0.680 0,602 0.538 0.483 0.438 0.399
25 55 95
Benzophenone 12.4 5.11 1.96
0.700 0.625 0.563 0.502 0,465 0.425 0.389 13.6 4.67 1.74
~
Table V. Error Distribution No. of compounds
0-501, 5-1070 10-15% 15-2070 20-25% 25-30% 30-40% 40-50’% 50-6070 60-70ajO 70-8070 80-9070 90-100yo >100%
177 68 27 13 8 5 8 3 2
~~
24
~
Percentage
56 21 8 4 2 1 2 1 0
4 7 6 1 5 6 5 0 6
1
0 3
2
0 6
~~
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
11. Trichloromethane (Chloroform). Calculation of NE N = 1.00 ANl: 3 X (chlorine) = 3 X 3.21 = 9.63 AN2: (C(C1)a configuration) = 1.91 - 3 X 1.459 = -2.47 N E = 8.16 Calculation of B B.: 24.79 66.885 X 8.16 - 1.3173 X 8.162 - 0.00377 X 8.16’ = 480.81 AB1: (chlorine) = -17.03 ABz: (c(c1)3 configuration) = -26.38 B = 437.40
+
From eq 3 it follows that
To = 255.44’K and
111. Benzophenone (Diphenyl Ketone). Calculation of
NE N ANl: (ketones) 3.265 - 13 X 0.122 ANz: 2 X (aromatic (ketones)) = 2 X 2.70
= =
NE Calculation of B (eq 6) B,: 530.59 13.74 X 20.08 ABl: (ketones) - 117.21 15.781 X 20.08 A B z : (aromatic (ketones) - 760.65 f 50.418 X 20.08
+
=
+
= =
B From eq 4 it follows that log
7~
=
=
=
13.00 1.68 5.40 20.08
806.49 199.67 252.95 1259.11
TO= 402.50’K and
1259.11
[i -
-
40i.501
A comparison of the viscosity data and the calculated values for these examples is given in Table IV. The viscosity data for pentanoic acid and benzophenone were obtained from the “International Critical Tables” (1930) ; the chloroform data, from Landolt-Bornstein (1955). Discussion
The relationships given in Table I11 have been developed and tested by consideration of a large number of data, cover-
ing 314 different compounds with nearly 4500 data points. A survey of these compounds with all viscosity data and calculated values will be issued as a separate report (van Velzen and Langenkamp, 1971). It appears that the agreement between the calculated aiid experimental values is generally satisfactory. The error distribution is shown in Table V. The very large errors are usually fouiid iii the first numbers of a homologous series, e.g. methane ethene propene methaiioic acid ethaiioic acid
111 .8% 45.201, 25.4% 58.0% 26.370
methanol 172.0% ethanol 84.7% benzene 30.770 cyclohexane 33. 4y0 cyclopeiitaiie 33.1%
This may partially be due to the fact that, in a number of cases, the applied relation is not necessarily t,he right one for these compounds. For instance, benzene has been calculated as a n alkylbeiizene with N = 6, Le., as a n alkylbeiizene without a n alkyl group, no better relat’ioii being available. The above error distribution compares very favorable with the error distribution given in Table I for the Thomas, Souders, and Stiel aiid Thodos methods. However, a direct comparison would be misleading because the data from Table I are based on other compounds (less numerous) than the present work. For this reason the example of Reid and Sherwood for the three met’hods has been extended by the new met,hod. The results are fouiid in Table I, last column. It appears that in this case the errors are greater thaii for the total data, which is caused by the fact that the 40 compouiids chosen by Reid aiid Sherwood comprise eight of the ten “aiionialous” compounds mentioned above (met’haiie,ethene, ethaiioic acid, methanol, ethanol, benzene, cyclohexane, and cyclopeiitaiie). 111 comparison with the other methods the reliability of the new method is quite good: only 14y0 of the values showed aii error above 30% as opposed to 24% for the Thomas, 31y0 for the Souders, and 41y0 for the Stiel and Thodos method. Two final remarks have to be made. (1) A number of the relations given in Table I11 are based on a limited number
of data points. This implies that extrapolation beyond the range for which it has been developed may yield erroneous results. The relationships given in Table I11 must therefore not be considered as definitive; if more data become available, in certain cases other relationships will have t o be inserted to match all experimental data a s well as possible. (2) The proposed relation is based on the D e GuzmanAiidrade equation, although the modified forms of this equation generally yield a better fit of experimental and calculated points for single compounds. I n the course of the present study the authors have made many attempts to base the relation on the formula of Gutmaiin and Simmons (1952) and Girifalco (1955). These attempts were not successful, as the introduction of a third coiistaiit (C) severely influences the value of B. No simple and cumulative relations could be developed for B and C. Generally N E is not seriously affected by the use of modified forms of the D e Guzmaii-Andrade equation. The filial conclusion is that the aim to develop a relationship in which no other physical properties occur and which is reasonably accurate has adequately been reached. literature Cited
Aiidrade, E. N . da C., IYatuw 125, 309 (1930). De Guzman, J., An. SOC.Espan. Fzs. Quim. 1 1 , 153 (1913). Girifalco, L. A,, J . Chcnz. Phzs 23, 2446 (1955). Gutmann, F., Simmons, L. Rf!,‘J. i l p p l P h y s . 23 (9), 977 (1952). “Interiiational Critical Tables.” RIcGraw-Hill. New York. N. Y.. 1930.
“Laiidolt-Bornstein Tabellen,” 6. Aufl., Bd IV-1, SpringerVerlag, Berlin, 1955. Reid, It. C., Sherwood, T. K., “The Properties of Gases and Liquids,” RIcGraw-Hill, New York, N . Y., 1966. “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and lielated Compounds,” American Petroleum Institute, Project 44, Carnegie Press Pittsburgh, Pa., 193.1
SoLide&, X, Jr., J . Anlo.. Chma. SOC.60, 154 (1938). Stiel, L. I., Thodos G., A.I.Ch.E. J . 7, 611 (1961). Stiel, L. I., Thodos G., A.I.Ch.E. J . 10,275 (1964). Thomas, L. H., J . Chem. SOC.373 (1946). van Velzeii, l)., Langenkamp, H., Euratom Report, EUR 4735e (1971).
RECEIVED for review June 10, 1970 ACCEPTEDJune 1, 1971
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1, 1972
25
