January 1948
INDUSTRIAL AND ENGINEERING CHEMISTRY
(5) Cochran, W.,Nature, 157,231,872 (1946). (6) Dahlberg, H. W., Proc. Am. SOC.Sugar Beet TechnoE., 3, 323 (1942). (7) Grut, E.W.,Z . Zuckerind. eechoslovak. Rep., 61, 345,356,373 (1937): Lis& Cukrovar, 56, 37,53,62 (1937). (8) Holven, A. L., IND.ENG.CHEM.,34, 1234 (1942). Proc. Am. SOC.Sugar Beet Technol., 3, 499 (9) Hungerford, E. H., (1942). (10) Jackson, R.F.,and Silsbee, C. G., U. S. Bur. Standards, Tech. Paper 259 (1924). (11) Jenkins, G. H.,Dept. Agr., Brisbane, Queensland, Bur. Sugar Ezpt. Sta., Tech. Comm. 3 (1941). (12) Krasil’shchikov, B. E., and Tsap, M. M., Novoe v Nauke i Tekhnike Sakharnogo Proizvodstva, V S N I T O (Moscow), 1940, 24-8. (13) Kucharenko, I. A.,Planter Sugar Mfr., 75 (May-June 1928) (14) McCabe. W. L..IND.ENG.CHEM..38. 18 (1946).
89
(17) Nakhamovich and Zelikman, Nauch. Zapiski, 6,32, 109 (1928) (18) Nees, A. R.,and Hungerford, E. H., IND.ENG.CHEM.,28, 893 (1936). (19) Orth, P.,Bull. assoc. chim., 55, 105 (1938). (20) SmolBnski, K., and Zelazny, A., Gat. Cukrownicza, 74, 303-17 (1934). (21) Thieme, J. G.,“Studies in Sugar Boiling,” tr. by 0. W. Willoox. New York, Facts About Sugar, 1928. (22) Van Hook, A.,IND.ENG.CHEM.,36, 1042, 1048 (1944); 37, 782 (1945); 38,50 (1946): .Trans. A m . SOC.Sugar Beet Technol. 4,558 (1946);J . Am. Chem. Soc., 67,370 (1945). (23) Webro, A. L.,Mech. Eng., 58, 99 (1936); Mem. assoc. tecnicos UZUCUT. Cuba, 19,265-75 (1945). (24) Wohryzek, O., “Chemie der Zuckerindustrie,” Berlin, Juliup Springer, 1928. RECEIVEDNovember 25, 1946. The essential features of the paper were presented before the Division of Sugar Chemistry in the Symposium OD Current Progress in Carbohydrate Chemistry a t the 110th Meeting of the AMERICAN CHEMICAL SOCIETY,Chicago, Ill.
Thermal Conductivity of Liquids GERALD PALMER1 University of Rochester, Rochester, N . Y . Hydrogen bonding in associated liquids is used to explain apparent anomalies in the thermal conductivity of glycols and alcohols. A plausible mechanism of heat conduction in these liquids is suggested, and, on the basis of this theory, an improved equation has been devised and tested on forty-eight liquids.
I
N T H E past twenty years very little progress has been made in correlating thermal conductivity with other data. Although numerous equations relating thermal conductivity to other physical properties have been proposed, they have met with indifferent success. The theorctical ones have not chocked very well, and their use for engineering purposes is out of the question, since data on compressibility and velocity of sound are harder t o find than data on thermal conductivity. Among the empirical equations that of Smith (11) gives the best checks of any proposed to date, but in certain cases i t is subject to large errors. I n a summary of the field of thermal conductivity of materials in various states of aggregation, Eucken (7) pointed out that in the case of liquids it was impossible to predict when large deviations from the equations might occur. There seems to be a tendency, however, for liquids t o divide into two classes, those that can be fitted with empirical equations and those that deviate considerably. I n this latter class one finds the associated liquids such as water and the alcohols. This should be expected, for if there are additional forces constraining the molecules, the transmission of heat from one molecule to the next will be altered. The additional forces due to hydrogen bonds can be used t o explain certain anomalies in thermal conductivity which are considered in detail below. Several years ago the author and George W. Haeeard, in a n unsuccessful attempt to find some regularity t o use for empirical equations, tried using the law of corresponding statcs to bring out relationships that had not appeared before. The result is shown in Figures 1 and 2 as a logarithmic plot of the thermal conductivity at 0.56 of the critical temperature. The fraction 0.56 was chosen arbitrarily, as it gave the most temperatures for which experimental values could be found. Where it was impossible to find experimental values of the critical temperature, a value was estimated as being one and a half times the absolute Present address, Department of Chemistry, S t Lawrence Universit Canton, N. Y. 1
boiling point. At longer chain lengths the alcohols reach a constant value nearly the same as that of the hydrocarbons t o which they are related. Obviously, one of €he most important factors in thermal conductivity is molecular weight, conductivity generally decreasing with increasing molecular weight. However, there are glaring exceptions. S,ymmetry may be a factor, since carbon tetrachloride with a higher molecular weight also has a higher conductivity than chloroform. Similarly, symmetrical trans-dichloroethylene has a substantially higher conductivity than the cis form. Ethylene glycol and glycerol (3,4,6) are likewise abnormal, having conductivities at least twice as great as would be expected for the molecular weight, while propylene glycol is nearly normal Other isomeric substances also possess different conductivities, an indication t h a t more than molecular weight is involved. Ethyl ether and the butyl alcohols, for which the coriductivities a r r listed in Table I, are particularly interesting since they afford H clue t o the cause. THE HYDROGEN BOND
This leads obviously t o a consideration of the hydrogen bond. Water and the alcohols are known t o form hydrogen bonds They also have abnormally high conductivities, so that it is reasonable to expect n-butyl alcohol t o have a higher conductivit) than ethyl ether which does not form hydrogen bonds. Actuallj , the value for n-butyl alcohol is 18% higher than that for ether Furthermore, the n-butyl alcohol has lost most of the hydrogen bond effect of the alcohol series, as can be seen from Figure 1, where it is almost down t o the flat part of the curve. If t h r mechanism is correct, methyl ether and ethyl alcohol should shop a greater difference than the above example, but no data art available t o test this assumption. Methyl ethyl ether and npropyl alcohol would make a good pair t o investigate. On thr basis of the previous reasoning, a conductivity for methyl ethyl
TABLE I. CONDUCTIVITIES Substance Ethyl ether Isobutyl alcohol n-Butyl alcohol
0.56 Critical Temperature,
c.
k X 106
-1 1 28
34
41
40
38
'
INDUSTRIAL A N D E N G I N E E R I N G C H E M I S T R Y
90
30
! 09
50 MOLECULAR
WEIGHT-
Figure 1
ether about 20% less than that of n-propyl alcohol should be expected. As a result of hydrogen bonding all hydroxyl compounds probably have some tendency $0 form chains or aggregates. And since glycerol has three OH groups and ethylene glycol and propylene glycol have two each, similar behavior might be expected for them, resulting in high thermal conductivities. As indicated in Figure 1, the values for glycerol and ethylene glycol are high as expected, but the conductivity of propylene glyco! is nearly normal. Ordinarily, on addition of a CH2 group one would expect a few per cent reduction in conductivity; actually it is nearly ten times as large, about 20ojO0. This may be due to some internal hydrogen bonding or chelation. Internal hydrogen bonding frequently takes place in preference t o bonding between molecules, resulting in lower melting point and boiling point, and there is no reason t o believe that it would not show up in thermal conductivity. Usually, only six-membered rings are formed, though according to Pauling ( I O ) there is evidence that weak bonds are formed occasionally in the case of five-membered rings. Certainly, whatever the cause, there must be a greater percentage of monomers, as normally boiling point increases with molecular weight, but propylene glycol with a higher molecular weight has a lower boiling point than ethylene glycol. Perhaps a more clear-cut case is afforded by the work of Weishaupt (IS) on 1,3 and 1,4 butanediols. The 1,3 compound can form the six-membered ring
Vol. 40, No. 1
Since these examples involve a kind of polymerization, a study of the thermal conductivity of styrene during polymerization should be particularly interesting. As the molecular weight is known as a function of time, it should be possible to correlate it with the theory. It is proposed to investigate this in the near future. Using boiling points of isomeric substances as an indication of internal hydrogen bonding, one concludes that for alcohols the is0 forms have more chelation, or certainly a greater percentage of monomer. According t o Pauling ( I O ) the amount of monomer increases in the order primary, secondary, k r t i a r y for amyl alcohols. In any event, the correspondence between boiling point and conductivity seems to hold here also. It is difficult to assemble data to demonstrate this. as the spread of values for one compound by different investigators is frequently EO 0 larger than the differences between compounds. Xevertheless, after considering all the data available, there still seems to be a definite tendency for the is0 form t o have a lower conductivity. For alcohols where several isomers are possible there is always a question which one was measured, and further doubt about its purity. This may be of considerable importance among isomeric alcohols, though it does not seem to make much difference for some other materials. I n the older literature no distinction is given indicating which is0 alcohol was used. There is some likelihood that they may even be mixtures. This might result i n erroneous values for the conductivity, for according to Pauling it has been shown that there is no monomer in mixtures of tert-amyl alcohol w-ith methyl alcohol. These speculations about the hydrogen bond now lead to a consideration of water. The high conductivity and the positive temperature coefficient have always been a problem, but no speculations as to the mechanism have appeared in the literature. Certainly heat conduction in normal liquids is related t o vibration or collision of the particles, but for associated liquids the picture is altered. The formation of hydrogen bonds probably assists in conduction of heat in two Rays: (1) by causing orientation of the molecules in the direction of heat flow, and (2) by affording a n
TABLE TI. CONDUCTIVITIES Substance Glycerol Ethylene glycol Propylene glycol 1,3-Butanediol 1,4-Butanediol n-Propyl alcohol Isopropyl alcohol n-Amyl alcohol Isoamyl alcohol
B,.
g,
k X 10s
290 197 188 206 230 98 82 138 132
680 670
505
472 500 380
372 388 354
TABLE 111. CONDUCTION OF HEAT while the same type of arrangement for the 1,4compound results in a seven-membered ring. Accordingly, more chelation and more monomer would be expected for the 1,3 compound. The boiling point of the 1,3 compound is about 25" lower, as seen in Table 11,and as expected the thermal conductivity is appreciably lower. It appears, therefore, that the boiling points of isomeric substances can be used as an index of relative conductivities. And to some extent, the difference in boiling point gives a rough quantitative indication of the difference in conductivities.
Substance Water Methyl alcohol E t h y l alcohol n-Propyl alcohol n-Butyl alcohol n-Amyl alcohol n-Hexyl alcohol n-Heptyl alcohol n-Octyl alcohol n-Nonyl alcohol
% Carried by Hydrogen Bonds 80 32 23 17 14 11 9. 9 9
?
naive but useful picture of this can be formulated with the chains oriented in the direction of heat flow and the bonds breaking at one end being replaced at the other. The temperature coefficient must be related in some way t o this hydrogen bonding. Possibly it may be related
4
70
60
Y
9
I N D U S T R I A L A N D E N G I N E*ER I N G C H E M I S T R Y
92
gram/ O C., and M is the molecular weight, checked surprisingly well for such a simple’equation. By modifying the original constant proposed by Weber, he reduced the average error to 14.8q10 for 46 liquids. Of all the empirical equations, this is the simplest and therefore most satisfactory for the present purpose. Since values calculated from this equation averaged consistently low for normal liquids and consistently high for asspciated liquids, readjusting the constant to improve the fit for normal liquids would cause the values for the others to be about 25 to 3Oy0 high. However, bn dividing the normal Trouton’s constant bv the experimental value, a multiplying factor is obtained that will modify the values for associated liquids by about 25%, but will not greatly change the calculated values for normal liquids. By assuming an arbitrary value of 21 for Trouton’s constant, the equation of Weber is modified to
When the two constants are combined, the new general constant becomes approximately 0.09. However, for the actual evaluation of this constant in the present work, no value of Trouton’s constant was assumed, since the over-all constant was determined by the method of averages, using all the data (9). Using this method the best value of the constant turns out t o be 0.0947 and the equation becomes
lri order to find data to evaluate this constant and test the assumption, a literature search was made to find the most recent and most yeliable values of the thermal conductivity and the other factors entering the equation. Most of the data came from the International Critical Tables ( 8 ) , the Annual Tables of Constants ( 8 ) , the American Petroleum Institute Research Project ( I ) , and the work of Stull ( l a ) on vapor pressure of pure organic compounds. However, several values of the specific heat came from original sources in the literature. In about one quarter of the cases, values from different sources were not in agreement and some weighting was necessary. Eleven values of the heat of vaporization were determined from the vapor pressure data of Stull, using the Clausius-Clapeyron equation. The vaIues used to test the equation along with the calculated conductivities are listed in Table IV. I n all, enough data were found to check the equation for 48 liquids, for which the average error is 8.8%. Thus by applying a correction factor for hydrogen bonding the error is substantially reduced. Smith also modified Weber’s equation by splitting up the factors to produce the five-constant equation
k
= 0.000011
1J/TM + (cp -1550.45)3 +800
~
Vol. 40, No. 1
worth while. I t is difficult t~ tell whether the values for which large cirors are obtained are significant or simply poor data. Thc associated liquids that were formerly rather difficult to fit anc subject to largc, m o r s , in the present work fit much better thar the other liquids, the average error being 5.574 aq opposed to an over-all average of 8.8%. After thc present work had been completed, an aquation by Dcnbigh (6’1 was called t o the author’s attention. Indirectly, through the Prandtl number ( r p v / k ) it relates thermal conductivity to the latent heat ot vaporization bv the dimensionlev equation log ( P ? )= aA€I/RI/‘
-C
b
where a atid b are +0.183 and -2.2 for water, and Jr0.2 and -1.8 for organic liquids; AH is the molal enthalpy of vaporization a t 1 at,mosphcro; R is t,he gas constaut; and T is the act,ual absolute tjempcrature. An entirely different approach also led to the use of entropy of vaporization. KO calculation of the average percentage error was given by Denbigh, but, it could bc computed from information given in the paper. For all 28 liquids, at 30” C. only, t.he average error was 28.5y0. This is hardly in the same class as other equations mentioned in this paper. In common with many other equations, the liquids fall into two classes, the hydrocarbons on one side of the mean and the hydroxyl compounds on the other. The cquation was developed for estimat,ing film coefficients :or heat, transfer in which the Prandtl number is useful, but it is hardly accept,able for predicting thermal conductivity. However, the aut,hor did not claim high accuracy for the work and did not, suggest that it be used to predict the coefficient of thermal conduct,ivit,y. SUMRIAR’Y
A mrchanism for the conduction of heat in associated liquids explaining differences in thermal conductivity of isomeric substances has bren suggested. Using the entropy of vaporization as a measure of thc hydrogen bond effect, a multiplying correction factor has bpen applied to the equation of Weber rcducing the percentage error for 48 liquids to 8.877.. ACKNOYLEDGMENT
The author is indcbted to E. 0. Wiig and W. D. Waltcrs of the Department of Chemistry for assistance in clarifying some of the ideas concerning the hydrogen bond mechanism of heat transfer. H e also wishes to express his appreciation to H. E. Gunning for suggestions with respect to styrene.
ul/g
10,000
where u is the kinematic viscosity in centistokes. This equation checked with an average error of 6.7%. For purposes of comparison with this equation, the percentage error for the 36 liquids that are common to the present work and that of Smith was recalculated using the best values of the thermal conductivity from more recent work. For these 36 liquids the average error for Smith’s equation was 8.4% while that of the author’s equation was 8.5OjoO.Thus the errors are of the same order of magnitude. Inasmuch as Smith’s equation is a n adaptation of Weber’s, it would probably be possible to improve the fit by modifying a multiple constant equation like Smith’s, When an empirical equation is used for prediction of new values, other things being equal, the fewer constants the bet,ter. Certainly the equation given in the present work is still practically as simple as the original form of Weber’s equation. Until some of the values of the thermal conductivity a r e m determined, it is doubtful that a further modification would be
LITERATURE CITED
(1) Am. Petroleum Inst. Research Project 44, Natl. Bur. Standards,
1945.
(2) Annual Tables of Physical Constants and Numerical Data,
Frick Chemistry Laboratory, Princeton, N. J., 1941. (3) Bates and Hasaard, IND.ENG.CHGV.,37, 193-5(1945). (4) Bates, Hazzard, and Palmer, I h i d . , 33, 375-6 (1941). (5) Bates, Hazzard, and Palmer, IND. ENG.CHEM.,ANAL.ED., 10, 314 (1938). (6) Denbigh, K. G., J . SOC. Chem. Ind., 65, 61-3 (1946). (7) Eucken, Forsch. Gebiete Ingenieztrw., 11B,6-20 (1940). (8) Inteinational Critical Tables, New York, McGraw-Hill Book
cr,
1928.
(9) Lipka, “Graphical and Mechanical Computation,” New York, John Wiley & Sons, 1918. ( I O ) Pauling, “Natuie of the Chemical Bond.” Chap. IX, 2nd ed., Ithaca, Cornel1 Univ. Press, 1945. (11) Smith, Trans. Am. Soc. Mech. Engrs., 58, 719-25 (1936). (12) Stull, IND. E N G CHCM., . 39, 517-50 (1947). (13) Tcishaupt, Forsch. Gebicte Ingeniatrrw., I I B , 20-35 (1940).
RECEIVED J u n e 23, 1947.
