I ,
CHARLES A. SCHAEFER and GEORGE THODOS The Technological Institute, Northwestern University; Evanston, 111.
Reduced Thermal Conductivity Correlation Gaseous and Liquid Hydrogen Authoritative information is presented for the thermal conductivity of gaseous and liquid hydrogen at high pressures and extensive ranges -of temperature
CURRENT
ADVANCES in chemical technology to higher temperatures and pressures require a better comprehension of heat transfer operations. As most equations for heat exchange involve the coefficient of thermal conductivity, simpler and more reliable means for predicting this property over wide ranges of temperature and pressure are desirable. Various equations derived from theoretical and empirical considerations have been proposed for calculating thermal conductivity. The kinetic theory of gases is based either on the hypothesis that molecules are rigid elastic spheres or that they are point centers of forces, which vary inversely as the fifth power of the distance between them. These hypotheses have been worked out by Maxwell (32),Boltzmann ( 5 ) ,and Chapman (8). Utilization of the derived results for simple molecules was made by Gregory (77), Gregory and Dock (79), and Kannuluik and Martin (24). For polyatomic gases Eucken ( 7 4 , Sutherland (44), Enskog (72), Keyes (25), and Woolley, Scott, and Brickwedde (57)proposed theoretical and empirical relationships. Hirschfelder, Curtiss, and Bird (27) presented a first approximation to the thermal conductivity of monatomic gases by this expression:
where
k = thermal conductivity, cal./ second cm. O K. T = temperature, K. TN = normalized temperature,
TK/E
M = molecular weight' parameters in potential function, A. and K., respectively w ~ , ~ )[ TAr] * = integral for calculating transport coefficients U, E / K =
For the inert gases, Owens and Thodos (35),found good agreement between experimental values and the results obtained with Equation 1. For polyatomic molecules, a correction factor (70) to account for the transfer of energy between
translational and internal degrees of freedoni must be applied to Equation 1. The equations for thermal conductivity have several serious drawbacks to their application. In the first place, accurate usage is limited to the temperature range for which they have been developed. Secondly, only the equation of Enskog (72) is applicable to pressures above 1 atm. Because of the nature of these expressions, they are not convenient for rapid use. Hence, a graphical correlation based on experimental hydrogen data over wide ranges of temperature and pressure would be preferable. Comings and Nathan (9) attempted the prediction of thermal conductivity at high pressures by utilizing high pressure gas viscosities and pressure-volumetemperature-data for nitrogen, methane, and ethylene. Lenoir, Junk, and Comings (30) modified this correlation by
means of high pressure thermal conductivity values for nitrogen, methane, argon, and ethane. Gamson (75) developed a generalized state correlation based on thermal conductivity data for 27 gases. His contribution is limited to the gaseous state only and is reported to be reliable to 15%. Furthermore, it is too general and n.ot sufficiently accurate for hydrogen. The use of generalized correlations provides a n expedient means for predicting various properties of substances. This approach is valid only as long as the substances conform to a generalized corresponding states behavior. However, the structure and properties of hydrogen do not permit it to follow a generalized behavior well enough to be included in this type of correlation. This was found to be the case with density (39) and viscosity (6). T o account for such irregu-
45x Io-'
40
t HYDROGEN T,= 33.3'K.
30 \
Andrussow (2) Dickins (11) Eucken (13) Eucken (14) Godridge (IS) Gregory (17) Gregory and Archer(l8) Gregory ond Dock (19) lbbs and Hirst (22) Johnston and Grilly(23) Kornfrld and Hilferding ( 2 8 ) Stolyorov ( 4 2 ) Stolyorov, others (43) Ubbink ( 4 5 ) Wassiljewo ( 4 8 ) Weber (49,501
25 20
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Reduced Temperature, T =-
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T
Tc
Figure 1. The thermal conductivity data a t atmospheric pressure are correlated with temperature for a number of investigators VOL. 50, NO. 10
OCTOBER 1958
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HYDROGEN x Andrussow (2) Dlcklns (11) A Eucken (13) Eucken (14) A Godridge (16) f Gregory (17) 4 Grrgory and Archer (18) r Gregory and Dock (19) i t Ibbs and Hirrt(22) 0 Johnston and Grilly (23) Konnulyik and Martin ( 2 4 ) Kryes ( 2 5 26) Kornfrld dnd Hilfrrdinq (28) + Lenoir and Comings ( 2 9 ) 7 Stolyorov ( 4 2 ) L Stolyarov others (43) v Ubbink ( 4 5 ) F Vargoftik and Porfrnov ( 4 7 ) M Wosriljrwa (48) x Weber (49,50)
+
+
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1 1 1 1
4 5 6
8 IO
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20
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30 40
1
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60 80 IO?
Reduced Temperature , T
"=I Tc
Figure 2. Hydrogen thermal conductivity data a t atmospheric pressure referred to the corresponding value at the critical temperature are correlated with reduced temperature
2
50X10-5 Keyes ( 2 5 , 2 6 ) t Lenoir and Comings ( 2 9 ) A Powers, Mattox and Johnston ( 3 6 )
8
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.06.08.10
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Density, g./cm?
Figure 3. Residual thermal conductivities resulting from data obtained at high pressures produce a unique function with density
larities. some investigators considered t, u as a correlating parameter (37). Nelson and Obert (33) suggested the existence of a quantum parameter for hydrogen and helium. Studies by de Boer and Bird ( 4 )are concerned with the mechanical quantum parameter, A*, as a possible correlating variable. As a n alternative, it is possible to consider the properties of hydrogen separately from other substances. Since no thermal conductivity correlation is available exclusively for hydrogen, this study is directed toward its development on a reduced state basis.
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DeveloDment of Correlation Available literature data compiled for the thermal conductivity of hydrogen include values not only at atmospheric pressure and higher, but also for the liquid state. Because inherent difficulties are encountered in obtaining data at high pressures most investigations are conducted near atmospheric pressure and the results are presenied in Figure 1. Rectilinear coordinates are used to permit extrapolation to a temperature of absolute zero, Although data are available at elevated temperatures, Figure 1 is limited
INDUSTRIAL AND ENGINEERING CHEMISTRY
to those below 300' K. in order to establish more clearly the value a t T R = 1.0. At the critical temperature, T R = 1.0, k*T, = 5.40 X 10-bcal./second cm.' K. for hydrogen. Dividing k*, the thermal conductivity values at atmospheric pressure, by k*To, produced ratios which are plotted against the reduced temperature in Figure 2. Logarithmic coordinates are used to indicate the extent of available thermal conductivity values. While the relationship of this figure shows deviation from Ubbink's data (45)below TR= 1.O, a linear relationship of unit slope was constructed to maintain consistency Ivith similar reduced state correlations from thermal conductivity (35, 39) and viscosity (6, 40) studies for the inert and diatomic gases. In Figure 2, k * / k * T , = 1.O at T R = 1.0. To consider the thermal conductivity of compressed gases and liquids, and based on the theory of Predvoditelev (37), Abas-Zade (7) presented the residual thermal conductivity, k - k*, as a function of p , the density. This relationship was used successfully for correlating high pressure thermal conductivity (35, 39) and viscosity (6, 40) data. By means of Figure 2, the ratio k*/k*T,, was obtained at the experimental temperatures for the high pressure thermal conductivity data reported by Keyes (26):Lenoir and Cornings (29), and Powers, Mattox, and Johnston (36). Multiplying this ratio by k*Te = 5.40 X 10-5cal./second cm.' K., obtained from Figure 1, atmospheric thermal conductivity values: k * , are calculated, These in turn are subtracted from the experimentally determined high pressure values to produce the residual thermal conductivities, k - k*. Using a reduced state density correlation developed specifically for hydrogen (39), density values at the experimental conditions corresponding to the thermal conductivity data are plotted against k -k* (Figure 3 ) . Accurate experimental data in the high pressure region are difficult to obtain. This is evidenced by the only available data of Keyes ( 2 6 ) :Lenoir and Comings (29), Stolyarov (42), and Stolyarov, Ipatiev, and Teodorovich (43). While the high pressure data of Keyes and Lenoir and Comings are not in the best of agreement, those of Stolyarov and others were inconsistent within themselves and were also in complete disagreement with values reported by Keyes and Lenoir and Comings. The data of the latter investigators were considered more reliable than those of Stolyarov and others, already eliminated from this study. For the liquid state, no data exist to compare the work of Powers, Mattox, and Johnston (36). Utilizing these data in the best manner possible, the curve of Figure 3 has been adopted. This relationship enables the evaluation of k,, the thermal conductivity at the critical
THERMAL CONDUCTIVITY temperature and pressure. As given by Kobe and Lynn (27), the density of hydrogen a t the critical point was p% = 0.0310 gram/cu.cm. The residual thermal conductivity value obtained from Figure 3 for this density is k - k* = k, k*T, = 10.5 X lO-6cal./second cm. 'K. FromFigure 1, k*T, = 5.40 X 10cal./second cm. O K. Therefore, the thermal conductivity a t the critical point is k, = 15.9 X lO-5cal./second cm.' K. Using these values, the ratio, k,/k*T, = 15.9 X 10-6/5.4 X 10-6 = 2.944 was calculated. This is in contrast to the value of 2.60 reported by Gamson (75) and 3.047 reported by Owens and Thodos (35) for argon. Having established the thermal conductivity at the critical point, k, = 15.9 X 10-6cal./second cm.' K., a correlation of reduced thermal conductivities, kR = k/k,, was developed' by first obtaining k*/k*T,vahes at uniform temperature intervals from Figure 2 and using k*To = 5.40 X 10-5 cal./second cm.' K. to obtain k*, the thermal conductivity at atmospheric pressure. Dividing these values by k, enabled the construction of the base isobar, PR = 0.078 (1 atm.). Secondly, to include the effect of higher pressures, isobaric densities of hydrogen at uniform temperature intervals were obtained from a reduced state density correlation (39). These values and the relationship of Figure 3 produce residual thermal conductivities, k - k*, a t uniform temperature intervals. The values of k* used to develop the base isobar are added to the residual thermal conductivity, k - k*, to produce values of k. These values when divided- by k, give the reduced thermal conductivity, k R = .k/k,, which is plotted directly against reduced temperature for reduced pressure parameters. This final correlation (Figure 4) extends from the high temperature gaseous region, through the critical point, and into the liquid state. This correlation developed from only experimental hydrogen data enables the determination of thermal conductivity from atmospheric pressure to 1280 atm. and from 20' to 1500' I(.
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Discussion of Results The reduced state correlation (Figure 4) is developed from about 170 experimental hydrogen points representing the data available. T h e reliability of this type of correlation depends upon two factors. First, the experimental data used in developing this correlation must be dependable. Perfect agreement of the data for various investigators was not anticipated on account of difficulties inherent in thermal conductivity measurements. At higher temperatures and pressures, convection, radiation, and end effects become increasingly important. Even a t l
UJY 1 '
Y.
L
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5
U
6
B
U
L
Figure 4. Thermal conductivity values for gaseous and liquid hydrogen can be obtained from this correlation over wide ranges of femperature and pressure
atm. and a temperature of 300' K., 7.1% deviation was found between the experimental data of Eucken (73) and Weber (50). For the 11 investigators (2, 3, 73, 74, 76, 22, 24, 34, 42,48, 49) who reported data at these conditions, a n average conductivity value of 40.4 X 10-5 cal./second cm.' K. was obtained. This value agrees exactly with that obtained from Figure 4. The average deviation of the l l investigators from the value of 40.4 X 10-6 is 2.54%. I n developing this correlation, only those data (42,43)which were obviously out of line from the general trend were eliminated. The second consideration is the ability of the correlation to reproduce the experimental data. I n this regard, 100 experimental points have been checked against the developed correlation shown in Figure 4. Agreement was found to within 2.26% over the entire range-an improvement on the average deviation of 2.54% resulting from the combined data of the 11 investigators. In data of questionable accuracy, the reliability of this correlation is good. Elimination of the data of Stolyarov and others (42, 4.3) is justified because their data disagree by 7,1%. T o compare experimental results with modern theoretical developments, values of thermal conductivity were calculated with Equation 1 corrected with the Eucken factor (70). For these calculations, the collision integral values, W Z ) * [ T M ]as , well as the Lennard-Jones
force constants reported by Hirschfelder, Curtiss, and Bird (27) were used. For hydrogen, these constants are, cr = 2.968 A. and e / u = 33.3' K. A comparison of the calculated values and the experimental data is shown by the relationships of Figure 5. While these curves exhibit similarity, the calculated values at higher temperatures are notably lower than the experimental results. O n the other hand, at lower temperatures, the calculated values are higher. Experimental data yield a value at the criti-cal temperature of k*T, = 5.40 X 10-5 cal./second cm.' K. whereas the corresponding calculated value is 5.92 X 10-5 cal./second cm. 'K. As in previous studies with the inert gases (35) good agreement was found between experimental results and the values obtained with Equation 1 ; it appears that this form of the equation is valid. This equation was applied to monatomic gases and hence no Eucken correction factor was involved. The observed discrepancy of the hydrogen experimental and calculated values could be due to a number of causes. One of these is the heat capacity values used in the Eucken correction factor. As three different sources (20, 38, 47) of heat capacity data were consulted, this cause is not likely to offer an explanation. Another possibility is the Eucken correction factor, while still another cause could be the values given for cr and E / K . I n conclusion, a reduced state correlaVOL. 50, NO. 10
OCTOBER 1958
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Figure 5. Thermal conductivity values calculated with the equation of Hirschfelder, Curtiss, and Bird ugree reasonably well with experimental data at atmospheric pressure
tion specific to hydrogen has been developed from available experimental data. This enabled the accurate and rapid determination of thermal conductivity values over wide ranges of temperature and pressure. Its use is not subject to the drawbacks characteristic of theoretical and empirical equations. Owing to the comprehensive nature of its development, this correlation is offered as the best means a t present for predicting thermal conductivities. Acknowledgment
Grateful acknowledgment is extended to the Dow Chemical Co. for the fellowship grant that made possible this study. ’ Nomenclature
k
= thermal conductivity, cal./second
cm.O K. k* = thermal conductivity a t moderate pressures (1 atm.) cal./’second cm.’ K. kc = thermal conductivity a t critical point, cal./second cm. K. kb = reduced thermal conductivity, k/ kc
k*Tc = thermal conductivity a t moderate pressures (1 atm.) and the critical temperature, cal./ second cm. O K. M = molecular weight P = pressure, atm. P, = critical pressure, atm. Pa = reduced pressure, P/P, T = temperature,’ K. T, = normalized temperature, TK/E T, = critical temperature, K. TR = reduced temperature, TIT, z, = critical compressibility factor
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Greek = maximum energy of attraction of E two molecules for the LennardJones potential function, ergs K = Boltzmann constant, 1.3805 10-’6 ergs;’ K. A* = mechanical quantum parameter p = density, gram,’cu. cm. p, = critical density, gram/cu. cm. u = collision diameter for LennardJones function, A. a(2,2)*[TNJ = collision integral for Lennard-Jones potential function literature Cited (1) Abas-Zade, A. K., Zhur. Eksptl. 1 Teoret. Fzt. 23, 60 (1952). (2) Andrussow, Leonide, J. chzrn. phys. 52, 295 (1955). (3) Archer, C. T., Proc. Roy. SOC.(London) A165, 474 (1938). (4) Boer, J. de, Bird, R. B., Phys. Rev. 83. 1259 (1951). (5) Boltzmann, L., “Vorlesungen hber Gastheorie,” vol. I., J. A. Barth, Leipzig (1896). (6) Brebach, W. J., M.S. thesis, Northwestern University, Evanston, Ill., 1957. (7) Bromley, L. A., U. S. Atomic Energy Comm. Tech. Inform. Service, UCRL 1852 (1952). (8) Chapman, S., Phil. Trans. Roy. SOC. London A211, 433 (1912). (9) Comings, E. LV., Nathan, M. F., IND.ENG.CHEM.39, 964 (1947). (10) Curtiss, C. F., “Theories of Gas
Transport Properties,” Biennial Gas Dynamics Symposium, Northwestern University, Evanston, Ill., August 26-8,
1957. (11) Dickins, B. G., Proc. Roy. Soc. (London) A143, 517 (1934). (12) Enskog, David, Kgl. Svenska Vetenskapsakad. Handl. 63. No. 4 (1921). (13) Eucken, A:, Physdk. Z. 12, 1101 (1911). (14) Eucken, A., Zbid., 14, 324 (1913).
INDUSTRIAL AND ENGINEERING CHEMISTRY
(15) Gamson, B. W.,Chern. Eng. Progr. 45, 154 (1949). (16) Godridge, A. M., Bull. Brit. Coal Utzlzsation Research Assoc. 18, 1 (1954). (17) Gregory, H. S., Proc. Roy. Soc. (London) A149, 35 (1935). (18) Gregory, H. S., .4rcher, C. T., Zbid., A110, 91 (1926). (19) Gregory, H. S., Dock, E. H., Phil. M a g . 25, No. 7, 129 (1938). (20) Hilsenrath, Joseph, Beckett, C. W., Benedict, W. S., Fano, Lilla, Hoge, H. J., Masi, J. F., Kuttall, R. L., Touloukian, Y. S., Woolley, H. W., Natl. Bur. Standards, Circ. 564, p. 217, U. S. Department of Commerce, 1955. (21) Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” p. 534, Wiley, New York, 1954. (22) Ibbs, T. L., Hirst, A. A . , Proc. Roy. SOC.(London) A123, 134 (1929). (23) Johnston, H. L., Grilly, E. R., J . Chem. Phys. 14, 233 (1946). (2g) Kannuluik, W. G., Martin, L. H., Proc. Roy. Soc. (London) A144, 496 (1934). (25) Keyes, F. G., Trans. Am. Sac. ilfech. Engrs. 73, 589 (1951). (26) Ibid., 76,809 (1’954). (27) Kobe, K. A., Lynn, R. E., Jr., Chern. Revs. 52, 117 (1953). (28) Kornfeld, G., Hilferding, K., 2. phystk. Chem. Bodenstein-Festband, 792 (19’31). --, \ -
(29) Lenoir, J. M., Comings, E. W., Chern. Eng.Progr. 47, 223 (1951). (30) Lenoir, J. M., Junk, W. A , , Comings. E. W., Ibid., 49. 539 (1953). (31)-Lydersen, A. ‘L., ‘Greenkorn, R. A,, Hougen, 0.A , , Wisconsin Engineering Experimental Station, Rept. 4, October 1955. (32) Maxwell, J. C., Collected Works 2, 23 (1866). (33) Nelson, L. C., Obert, E. F., A.Z.Ch.E. Journal 1 , 7 4 (1955). (34) Nothdurft, Walter, Ann. Physik. 28, No. 5, 137 (1937). (35) Owens, E. J., Thodos, George, A.I.Ch.E. Journal 3,454-61 (1957). (36) Powers, R. W.,Mattox, R. W., Johnston, H. L., J. .4m. Chem. SOC. 76, 5968 (1954). (37) Predvoditelev, A. S., Zhur. Fiz. Khirn. 22, 339 (1948). (38) Rossini, F. D., Pitzer, K. S., Arnett, R. L., Braun, R. M., Pimental, G. C., Am. Petroleum Inst. Proj. 44, p. 631, Carnegie Press, Pittsburgh, Pa., 1953. (39) Schaefer, C. A., M.S. thesis, Northwestern University, Evanston, Ill., 1957. (40) Shimotake, Hiroshi, Ibid., Northwestern University (1957). (41) Souders, Matt, Matthews, C. S., Hurd, c. o.,IND.ENG.CxiEht. 41, 1037 (1949). (42) Stolyarov, E. A,, Zhur. Fit. Khirn.24, 279 (1950). (43) Stolyarov, E. A., Ipatiev, V. V., Teodorovich, V. P., Ibid., 24, 166 (1950). (44) Sutherland, W., Phil. Mug. 36, 507 (1893). (45) Ubbink, J. B., Physica 14, 165 (1948). (46) Ubbink, J. B., de Haas, W. J., Zbid., 10, 451 (1943). (47) Vargaftik, N. B., Parfenov, I. D., Zhur. Eksptl. i Teoret. Fiz., 8, 189 (1938). (48) Wassiljewa, Alexandra, Physik. Z. 5 , 737 (1904). (49) Weber, Sophus, Ann. Physik. 54, No. 4, 437 (1917). (50) Zbid., 82, No. 4, 479 (1927). (51) Woolley, H. W., Scott, R. B., Brickwedde, F. G., J. Research N a t l . Bur. Standards 41, 379 (1948).
RECEIVED for review November 7, 1957 ACCEPTED May 1, 1958
