olecular Transport Properties of Fluids ERNEST F. JOHNSON Princeton University, Princeton, N. .I.
T
HIS article briefly revicws some of the importmt publications dealing with the molecular transpoi t propcrties of singlephase fluids. The period of covei:tge is from May 1954 through October 1955, but articles appearing during this period which 71 ere d e s c r i l d in last year's Fundamentals Review havc bcen omitted. Refeiences to work on non-Ne-tonian fluids and extremely dilute solutions have also been omitted, as well as works in which various transport properties were measured incidentally or without interpretation.
General Various voluines of the monumental series "High Speed Aerodynamics and J e t Propulsion" havc bcgun t o appear. This series of 12 volumes is being published by the Princeton University Press under an editorial board headed by Theodore von K&rm&nand under support by various military agencics. The purpose is to consolidate the voluminous literature in the field that has appeared since 1940 and to stimulate further research by presenting advanced theoretical problems and solutions to these problems. Volume I, "Thermodynamics and Physics of hlatter," edited by F. 13. Kossini (43), is an escellent and scholarly compendium comprising 10 sections by a variety of authors. Five sections deal to at least some extent with transport properties and processes, and of these one section is devoted to the transport properties of gases and gaseous mixtures and another to the thermodynamics of irreversible processcs. As is typical of large cooperative ventures, the publication date is much later than the completion date of the manuscripts, with the result that the most recent, references are not later than 1951 and 1952. Much of thc material is condensed froin recent books of the authors. For cxainple, the two sections mentioiled above arc derived from "hlolecular Theory of Gams and Liquids" (23). The emphasis throughout is highly theoretical, and the treatment of transport properties is of more interest to the research theorist than to the design engineer.
Vis cos ity Three advances in mpasuring techniques are described by authors froin as many different countries. Thirion ( 5 3 ) presents details on thc use of a magnetostrictive steel blade inserted halfway into the coil carrying the exciting current, and the whole immersed in the fluid under study. With only 2 ml. of sample, viscosities can be nieasured from -120" to +350" C. a t pressures t o 800 pounds per square inch over ranges from 0 t o 50,000 centipoises for one instrument and up t o 500,000 centipoises in another. Below 2000 centipoises the precision is f 2 % , and above it is &5%, A detailed theoretical treatment of the oscillating cylinder viscometer wherein the test fluid is ronfinrd between concentric cylinders, the outer one oscillated about its axis, and the inner one supported by a torsion Wirc, is given by Ibrahim and Kabiel (2b). Experimental results agree well with the theory.
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The interesting phenomenon of resonance betrc ecn the liquid and the inner c j linder is discussrd at length. Junhins (251 describes an automntic reading viscometer for gasps, v\hlc.h measures the time for a fived drop in pressure t o occur in a chamber in front of a capillary through which the gas disrhargcs into a ronstdnt pressure. The precision of this instrument is 1 0 O?%. Measurement of the viscosity of argon froin 0 to 75' C. at pressurcs up to 2000 atni. by hlichels, Rotzen, and Schuurman ( 3 1 ) using the transpiration method reveals large discrepancies nith Enskog's dense gas theory. Lambert and others (28) determined the viscosity and thermal conductivity of 23 hydrocarbons, including unsaturated and aromatic compounds plus carbon tetrafluoride; the former property v-as measured by a capillary method nnd a swinging pendulum method, the latter by the hot lvire method. They found that the ratio of thernial conductivity to viscosity i s lower for more flexible moleculcs. In a broad yet succinct summary Andrade (2) survels our prescnt 1ino.rrledge of the viscosity of pure S e n toriian liquids. The limitations of this knowledge are the limitations of our h o d edge of the liquid state. Why is the simple two-constant Arrhenius or Boltzninnn-type evponential equation BO widely representative of the temperature dependency of viscositj ? Andrade suqgests that some very general principle is involved, which is independent of molecular specics, but wliich is not likely t o be identified from the many elaborate theoretical investigations of recent yeais. Grifing and others ( 2 0 ) studied the effect of temprrature on the viscosity of 11 halogenated aliphatic arid aromatic compounds over the range 0" t o 100" C and computed energies of activation for viscous flow. The availahlP data from the literature havr heen enamined, and comprehensive tabulations of the best available data on viscosity, thermal Conductivity, heat capacity, density, and vapor pressure of aqueous solutions of nitric acid and sulfuric ac'id from -40" to 300" F. are presented by Bump and Sihbitt (8). E:stimates of reliability are given for all the tahlcs. Using an Ostwald visrometei s i t h an accuracy of f O 5 % , Sclmah and Kolb ( 4 9 ) deterinincd the viscosity of 80 to 115% sulfuric acid froin 20" to 132" C. Their results showed that the temperature dependency does not follow the simple Boltzinnnn equation Discontinuities occur lmause of compound formation, ion dissociation increases visrosity, and molecular assoriation decreasrs viscosity. On the basis of the behavior of a numhcr of 1)iiiai-y mixtures of polar orgariic rompounds, Thaclrer and RoT\lir?son (56) conclude that the viscosity of the vapor is only qualit,atively related to the thermodynamic properties of the liquid mixture VIhen the r e l e tion is given hy an equation valid for spherical molecules The efiect of polar and other noncentral forccs on the thermodynamic properties of the liquid solutions i s greater than the eflect on vapor visrosity. When an electric field is applied t o a polar liquid, there is a sinal1 increase in viscosity proportional to the square of the field, and the effect increases parabolically with frequency. Andradc and Hart ( 3 ) present data for chlorobenzene, amyl acetate, and
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TRANSPORT PROPERTIES nitrobenzene, which show a connection between dipole rotation and viscosity. From modifications of Licht-Stechert's formula, Uchiyaina ( 5 4 ) presents three equations for computing the viscosity of g a m a t the critical temperature and atmospheric pressure. The choice of equation depends on the classification of the gas, which is determincd by the critical temperature, the critical compressihility factor, and the observed value of a constant in Uchiyama's equations. Hrancker (7) computes the fluidities (reciprocal viscositics) for gaseous helium, nitrogen, oxygen, hydrogen, carbon dioxide, and methane by means of an exponential equation in temperature with three constants. In an extension of his previous theoretical studies of the interrelation of viscosity, self-diffusivity, and isotopic thermal diffusivity, Holleran ( 2 4 ) modifies his previously developed Grst approximation to conform with the Rinipler and more accurate approxiination for the thermal diffusion ratio of Kihara ($7). His results agree well with experimental data for argon. Eisenschitz ( 1 6 ) also uses argon to test his molecular theory of nonequilibrium conditions in fluids. From his expressions giving the cffcct of temperature on viscosity and therinal conductivity, he is able to compute the internal friction of argon with sufficient agreement n-ith experiment to warrant continued development of the theory. From studies of the viscosit,y of three regular solutions (benzene and cycloheuane, benzene and carbon tetrachloride, and carbon tetrachloride and cyclohexane) Grunberg (If)calculated interchange energips which he helieves arc a closer approximation to the true heat of mixing than are interchange energies computed from vapor pressures. Pospelchov (S6) shon-s that the theoretical significance ascribed to the Graetz formula for liquid viscosity by many authors is unwarranted, since recent data indicate that t o and tl in the formula do not correspond to the critical temperature and melting point, respectively. A simple, new formula for the temperature dependency of liquid viscosity is proposed by Cornelissen and Waterman (IS). Although it resembles Souder's equation, it is more generally valid and is of the form log Y = B A/T" where Y is the kinematic viscosity. For materials as widely variant as aqueous solutions, hydrocarbons, and mercury, it rarely deviated from experimental valucs by more than 1.5%. Mitra and Chakravarty (SI) present a simple, tvio-constant equation relating viscosity and vapor pressure of liquids, and Reik (SQ) in a critical invrstigation derives foimulas for determining thc isothwmal vapor pressure diagram from the isothermal viscosity diagram for blnary mixturm. His equations are tested on eight binary systems over the range 1 4 " to 50' C. with good agrecmcnt with experimental data except for one system (ethyl iodide and carbon tetrachloride) (40). I n an exposition of some general relations on the viscosity of homologous liquids, Palit (36)shows that a t constant temperature the molar viscosity is a linear function of the molar surface energy, and plots of log molar viscosity us. 1/7' give a family of straight lines intersecting a t the same point on the ordinate axis. Gimons and TT-ilson ( 5 0 ) relate energy of evaporation, energy of artivation of viscosity,
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ERNEST F' 'OHNSoN is assistant professor of chemical engineering at Princeton University, where he directs research on the molecular transport properties of rocket fluids, H~ received his B.S. from Lehigh and ph,D. from the University of Pennsylvania. Johnson is an industrial consultant and a member of the American Institute of Chemical Engineers, ACS, American Society for Engineering Education, Tau Beta Pi, and Sigma Xi.
March 1956
and surface tension a t the boiling temperature for inert gases, normal fluorcarbons, and normal hydrocarbons. Although chemical engineers are most likely t o be concerned only with the shear coefficient (Erst coefficient) of viscosity, the rapidly expanding scope of chemical engineering will increasicgly make use of the dilational or volume coeficieiit (second coefficient). The volume coefficient of viscosity is the measure of a fluid's resistance to expansion in the absence of shear effects. It is important in sound absorptiou, and for most liquids it is largcr than the first coefficient. Our prcsent knodedge is outlined in a syniposium of 13 articles ( 4 2 ) by international authorities in the field. Roscnhead introduces the subject and hndrade gives the concluding summary.
Thermal Conductivity Using a Keyes-Sandell cell, Rothman and Bromley ( 4 4 ) measured the thermal conductivity of nitrogen, carbon dioxide, argon, and mixtures of carbon dioxide and nitrogen a t temperatures to 775' C. Recommended values for pure carbon dioxide and pure nitrogen are tabulated a t 100" intervals from 0" to 800" C. The Lindsay-Bromley correlation for mixtures is found to be low by about 3% above 370" C. This correlation is also found by Bennett and Vines (4)to be somewhat low in predicting the thermal conductivity of vapor phase mixtures of argon and benzene and to be entirely unsuitable for strongly polar mixtures. These authors studied a variety of binary organic vapors from 60" t o 125' C., observing that for nonpolar compounds of similar viscosity and molecular weight the viscosity is linear in composition, but for polar-nonpolar vapors there are pronounced maxima resulting from interactions betxeen the polar molecules on collision. I n a further extension of his application of the tlLeoremof corresponding states Riedcl (41) correlates compressibility, surface tension, and thermal conductivity in the liquid phase. Available data are relatrd by a "critical parameter" and reduced temprrature. Graphs and tables detail this prcscntation. Powers, Mattox, and Johnston ( S T ) describe a parallel plate cell which they used to determine the thermal conductivity of nitrogen, hydrogen, and deuterium in the liquid phase nith a reliability of better than 2%. The thermal conductivity of liquid deuterium a t 19" to 26" K. exceeds that of hydrogen by ahout 6%. On the basis of data for five different binary mixtures of normal liquids, Filippov and Novoselova ( 1 8 ) present a second degree equation for the thermal conductivity of the solution in terms of the constituent thermal conductivitirs and composition by weight. Ening, Seebold, Grand, and Miller (I?') describe their apparatus and procedure for measuring the thermal conductivity of molten metals. n a t a for mercury and tn o sodiumpotassium alloys arc reported for temperaturrs to 680" C. with a precision of 0 3%. The temperature coefficient of these ideal liquids is accurately predictable by elect1ical analogy. -4supplement to their comprehensive literature survey of liquid thermal conductivity has been iPsucd hy Sakiadis and Coates ( 4 6 ) . This list adds 110 liquids to the original list. Siitre both lists contain errors, they should be regarded more as comprehensive hibliographies than as definitive sources of data. These aut2~arshave also reported the results of their own studies of liquid thermal conductivity (45). Using a horizontal disk-type apparatus they measured the thermal conductivity and its temperature coefficient for 53 pure organic liquids over various temperature ranges between 86' and 172" F. with an estimated maximum error of f1.50%. They conclude from their work that most available thermal conductivity data are 4 to 5% too low because of failure to allow for irregularities in the metal surface at the liquid-metal interface. A correlation based on Kardos is presented for estimating liquid thermal conductivity with a maximum deviation of *6%. The relation is simply k = C,U,L, where U. is the
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FUNDAMENTALS REVIEW
Simplified relation for thermal diffusion ratio in liquids involves only pure component heats of vaporization and excess thermodynamic properties of the solution
...
velocity of sound and L is the intermolecular length. Although this correlation is reliable and simple, it is of limited engineering utility because of the particular properties involved in it.
and other data in the literature (a total of 251 solute-solvent systems) the authors (66) propose for dilute solutions the equation
Ordinary Diffusivity Nore general equations than have been used previously for detcrinining coefficients of self-diffusion by the method of heterogeneous isotope exchange are presented by Berthier (6). These equations are shown t o represent experimental results more completely. Carmichael, Reamer, Sage, and Tmey (9, 10) measured the Maxn-ell diffusion coefficients of n-heptane and n-hexane in the gas phase of binary systems with nicthane, ethane, and propane from 70" t o 220" F. and a t pressures to 70 pounds per square inch. Their data suggest that the Maxwell hypothesis ( d h either fugacity or partial pressure as potential) is an adequate description of the molecular transport of light hydrorarbons in the gas phase of other light hydrocarbons only a t low pressures. Diffusion coefficientsfor carbon dioxide weredeterniined (Figure 1) by O'Rein and Martin (33)from 0" to 100" C. and a t pressures to 200 atm. using carbon-14 as the tag. I n thesc ranges of coiiditions the diffusion coefficient is approximately proportional to the reciprocal of the density. Raw ( 3 8 ) measured gas diffusivities for the binary systems of boron trifiuoride and boron trichloride each with carbon tetrachloride a t 30' C. and 1 atm., obtaining results in excellent agreement with those calculated assuming the Lennard-Jones 12-6 model. A capillary cell
where z = association parameter (2.6 for water, 1.9 for methanol, 1.0 for benzene), M = molecular weight of solvent, and V = molar volunie of solute.
Using tritium as tag and a capillary type apparatus, Fishman (19) measured the sclf-diff usion coefficients for liquid n-pentane and n-heptane over most of the liquid temperature range. Molecular dimensions calculated from these data and viscositics ayr ee qualitatively with known dimensions. Assuming the Lennard-Jones 6-12 potential field Cohen, Offerhaus, and de Boer ( 1 2 ) coniputed quantum mechanically the self-diffusion coefficients and mutual diffusion coefficients for helium-3 and helium4 in first and second approximations. They also calculated second virial coefficient, thermal conductivity, thermal diffusivity, and viscosity. Results for the last quantity agree well with recent experimental data. Madan (SO) also used the Lennard-Jones 12-6 function and force constants calculated from thermal diffusion data to calculate the coefficients of self-diffusion of argon and methane. From a modified kinetic theory, Ottar (34) develops a simple equation relating selfdiffusivity and viscosity of liquids. Diffusion coefficients coniputed from viscosity data for a number of materials, including water and niercury, are in good agreement with experimental values. Wilke and Lce (57) compare the availahle POSITIVE ELECTROOET, ,,-POROUS M E T A L methods for predicting the diffusion coefficients of gases and vapors and suggest an empirical modifiration of the method of Hirschfelder, Bird, and Spotz. Average deviation of the modified expression is 4%, while for the original it is 7%, and for Gilliland's equation it is 20%. In an examination of the relation between self-diffusion coefficient and viscosity of liquids, SOLOEREDNNECTION F O R Li and Chang (29) show that the difference L O E D CABLE between early hydrodynamic theory and Eyring's CONNECTION FOR TUBING kinetic theory lies in the latter's derivation of viscosity. Agreement on the basis of these Figure 1. Diffusion cell used by O'Hern and Martin (33) theories is possible only for liquids having an approximate cubic packing structure.
niethod was used by Katts, Alder, and Hildebrand ( 5 5 )in determining self-diffusion isobars and isochorw for carbon tetrachloride, 25" to 50" C. and 1 to 200 atm. Activation energies are the same as for iodine in carbon tetrrwhloridt? and over the range studied & / T is constant within 7%. In a mathematical treatment of diffusion and mass transport in tubes, Taylor (51)suggests measuring diffusivity by introducing solute into solvent flowing slowly through a long capillary tube and measuring longitudinal dispersion. Chang and Wilke (If)used a diaphragm cell to measure diffusion coefficients of a variety of organic compounds and iodine in a vareity of organic solvents, 6' to 50" C. For some of the systems Dq/T is constant within 5% over wide temperature ranges. On the basis of these
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Thermal DiffuJion Coefficient The thermal diffusion factor LY of the nitrogen-hydrogen system was measured by the two-bulb method between 0" and 100" C. by Hirota and Sasaki (%a),who conclude that the LennardJones potential gives good agreement vvith expcriniental results for hydrogen concentrations below 80 mole %. Van Schooten and van Xes ( 4 8 ) describe a microthermal diffusion colunm, 28 cm. long with an annulus of 0.2 mm. and total liquid volume of 1 to 3 ml., with which they mcasured diffusivities in the systems octane and o-xylene and octanc and 3-methyl-3-ethylpentane. A theory previously derived from the thermodynamics of ir-
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TRANSPORT PROPERTIES reversible processes is modified by Dougherty and Drickamer (14)t o give a general expression for the thermal diffusion ratio in liquids involving only heats of vaporization of the pure components and the excess thermodynamic properties of the solutions. Their theory is tested on new data for carbon disulfide and a series of binary mixtures of isomers (16)with excellent qualitative agreemcnt, Bicrlein (6) presents an analytical Rolution of the classical Soret diffusion problem for the rase of unrcstricted conipositions in binary liquid systems. He discusscs with examples how the Soret coefficient and the ordinary diffusion coefficient can be obtained from data taken during the thermodiffusion unmixing period. On the basis that the rate-determining elementary step in self-thermodiffusion, self-diffusion, and dipole orientation is an activated rotation of the nonassociated molecules, Alexander ( 1 ) develops a model concept for self-diffusional processes in water. Confirmation of this concept is found in Raman frequencies for inhibited rotation and in the temperature coefficient of the heat of transport. I n computing intermolecul‘w force constants for simple gases from data on the thermal diffusion factor, for examplc, there is always some doubt regarding the restriction that only small gradients of physical quantities are involved and that therefore only the first approximation perturbation term need be considered. Saxcna and Srivastava ( 4 7 ) relieve this doubt by showing that the second approximation on the Lennard-Jones 12-6 model is unnecessary, since its effect is less than experimental error.
Bibliography Alexander, K. F., 2. physik. Chem. (Leiprig) 203, 357-82 (1954). Andrade, E:. N. da C., Endeavour 13, 117-27 (1954). ilndrade, E N. da C., Hart, J., Proc. Roy. SOC.London A225, 463-72 (1954).
Bennett, I,. A., Vines, R. G., J . Chem. Phys. 23, 1587-91 (1955). Berthier, G., J . chim. phys. 52, 41-7 (1955). Bierlein, J. A., J . Chem. Phys. 23, 10-14 (1955). Brancker, A. V., Znd. Chemist 30, 307-12 (1954). Bump, T. R., Sibbitt, W. L., IND.ENG.CHEM.47, 1665-70 (1955).
Carmichael, I,. T., Reamer, H. H , Sage, B. H., Lacey, W. N., Ibid.. 47. 2205 (1955).
Carrnichael, L. T.; Sage, B. H., Lacey, W. K., Am. Inst. Chem. Engis. J . 1 , 385-90 (1955).
Chang, P., Wilke, C. R., J . Phys. Chem. 59, 592-6 (1955). Cohen, E. G. D., Offerhaus. M. J., Boer, J. de. Physica20. 50115 (1954) (in English). Cornelissen, J., Waterman, H. I., Chem. Eng. Sci. 4, 239-46 (1955).
Dougherty, E. L., Drickamcr, H. G., J . Chem. Phys. 23, 295309 (1955).
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Eisenschitz, R., KoZZoid-2. 139, 38-43 (1954).
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(17) Ewing, C. T., Seebold, R. E., Grand, J. A., Miller, R. R., J . Phys. Chem. 59, 524-8 (1955). (18) Filippov, L. P., Novoselova, N. S., Vestnik Moskov, Univ. 10 (No. 3), Ser. Piz. Mat. i Estestven Y a u k , No. 2, 37-40 (1955) (19) Fishman, E., J . Phys. Chem. 59, 469-72 (1955) (20) Grifing, V., others, Ibid.,58, 1054-6 (1954). (21) Grunberg, L., Trans. Faraday SOC.50, 1293--303 (1954). (22) Hirota, K., Sasaki, K., Rzcll. Chem. SOC.Japan 27, 27-9 (1954). (23) Hirschfelder, C., Bird, W.,“11olecular Theory of Gases and Liquids,” Wiley, New York, 1954. (24) Holleran, E. M., J . Chem. Phys. 23, 847-53 (1955). (25) Ibrahim, A. A. K., Kabiel, 4.11. I., Z . angew. Math. u. Phys 5, 398-408 (1954). (26) Junkins, J. H., Rev. Sci Instr. 26, 467-70 (1955). (27) Kihara, T.,“Imperfect Gascs,” Asakusa Bookstore, Tokyo, 1949 (English translation by USOAR, Wright-Patterson Air Force
Base, Dayton, Ohio). (28) Lambert, J. D., others, Proc. Roy. SOC.London A231, 280-90 (1955). (29) Li, J. C. M., Chang, P., J Chem. Phya. 23, 518-20 (1955). (30) AIadan. M. P.. Ibid.. 23. 763-4 (1955). i31j Rfichels, A., Botzen, A:, Schuurman, W., Physica 20, 1141-8 (1954) (in English). (32) Mitra, S. S., Chakravarty D. N., J . Chem. Phys. 22, 1775-6 (1954). (33) O’Hern, H. A., Martin, J. J., INn. E m . CHEix. 47, 2081 (1955). (34) Ottar, B., Acta. Chem. Scand. 9, 344-5 (1955) (in English). (35) I’alit, S.R., Indian J . Phys. 26, 627-36 (1952). (36) Pospekhov, D. A., Zhur. Priklad. Khim. 27, 789-91 (1954). (37) Powers, R. W.,Mattox, R. W., Johnston, H. L., J . Am. Chein SOC.76, 5968-73 (1954). (38) Raw, C. J., J . Chem. Phys. 23, 973 (1955). (39) Reik, €I. G., 2. Elektrochm. 59, 35-45 (1955). (40) Ibid., pp. 126-36. (41) Riedel, L., Chem.-Zng.-Tech. 27, 209-13 (1955). (42) Rosenhead, I,., others, Proc. Roy. SOC. London A226, 1-65 (1954). (43) Rossini, F. D., ed. “Thermodynamics and Physics of Matter,” Princeton University Press, Princeton, N. J., 1955. (44) Rothman, A. J., Bromley, 1,. A., IND. ENG.CHEM.47, 899-906 (1955). (45) Sakiadis, B. C., Coates, J., Am. Inst. Chem. Engrs. J . 1,275-288 (1955). (46) Sakiadis, B. C., Coates, J., Louisiana State Univ. Eng. Expt. Sta. BUZZ.48 (1954). (47) Saxena, S. C., Srivastava, E. N., J . Chem. Phys. 23, 1571-4 (1955). (48) Schooten, J. van, Nes, K. van, Rec. trau. chim. 73, 980-90 (1954)
(in English). (49) Schwab, G. hf., Kolb, E., 2. physik. Chem. [N.F.] (Frankfurt) 3, 52-64 (1955). (50) Simons, J TI., Wilson, W. H., J . Chem. Phys. 23, 613-17 (1955). (51) Taylor, G. I., Proc. Roy. Sac. London 67B, 857-69 (1954). (52) Thacker, R., Rowlinson, J. S., Trans. Paraday SOC.50, 1158-63 (1954). (53) Thirion, B., CBnie chim. 73, 37-41 (1955). (54) Uchiyama, 11.. Chem. Eng. Japan 19, 310-15 (1955). (55) Watts, H., Alder, €3. J., Hildebrand, J. H., J. Chem. Phys. 23, 659-61 (1955). (56) Wilke, C. R., Chang, P., A m Inst. C‘hem. Engrs. J . 1, 264-70 (1955). (57) Wilke, C. R., Lee, C. V., IND.E m . CHEX. 47, 1253-7 (1955).
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