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Thermal Conductivity Of Binary Liquid Mixtures

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Thermal Conductivity Of Binary Liquid Mixtures

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Content INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. 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Xerox University Microfilms 300 North Z eeb Road Ann Arbor, Michigan 48106 74-21,495 PARKINSON, William Jerry, 1939- THERMAL CONDUCTIVITY OF BINARY LIQUID MIXTURES. University of Southern California, Ph.D., 1974 Engineering, chemical University Microfilms, A X ERO X Com pany, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. THERMAL CONDUCTIVITY OF BINARY LIQUID MIXTURES by William Jerry Parkinson A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Chemical Engineering) June 1974 UNIVERSITY O F SOUTHERN CALIFORNIA TH E GRADUATE SC H O O L U NIVERSITY PARK LOS A N G ELES. C A LIFO R N IA 9 0 0 0 7 This dissertation, written by William Jerry Parkinson under the direction of h%3.... Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Gradu ate School, in partial fulfillment of require ments for the degree of D O C T O R OF P H IL O S O P H Y Dean DISSERTATION COMMITTEE / ^ / / Chairma ACKNOWLEDGMENTS I would like to express my appreciation to the following individuals for their assistance and encourage ment in this work: To Dr. C. J. Rebert, my advisor, for his interest and guidance during the course of the research; The members of my research committee: Dr. F. J. Lockhart and Professor J. B. Vernon, for their aid; Mr. J. M. Scott, the laboratory mechanic, and Dr. J. M. Lenoir for their help and advice; The staff of H. T.R.I. for their aid in the initial phases of the work; My mother, Mrs. V. F. Parkinson for her prepara tion of the rough draft; The Fluor Corporation for tuition reimbursement for class credits; Fractionation Research, Inc. for giving me part- time employment during the final stages of this work; Finally, to Mrs. Ruth Toyama for her patient preparation of the final manuscript and also for keeping me in line during most of the research work. ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS ..................................... ii LIST OF TABLES ...................................... iv LIST OF FIGURES ..................................... V Chapter I. INTRODUCTION................................ 1 II. LITERATURE SURVEY........................... 9 III. THEORY...................................... 25 IV. APPARATUS AND PROCEDURE.................... 47 V. EXPERIMENTAL DATA........................... 62 VI. DISCUSSION AND CORRELATION................. 113 NOMENCLATURE ........................................ 152 REFERENCES ........................................ . 154 APPENDICES.......................................... 162 A. Details of the Solution of Fourier's Equation..................................... 163 B. Method for Calculating Average Cross Sectional Areas ............................. 168 C. Calculation of the Standard Energy of Vaporization ................................ 176 iii LIST OF TABLES Table Page 1-1 Binary Mixtures Measured in This Work....... 5 1-2 Binary Mixtures Obtained from Literature .... 6 II-l Values of b for Equation (11-10) ............ 17 III-l Thermal Conductivity and Normal Boiling Point for Some Isomers ...................... 39 V-l Pure Components Used - Suppliers and Grade .. 63 V-2 Pure Component Thermal Conductivities at 0°C...................................... 64 V-3 Precision and Comparison with Jamieson (34) . 66 V-4 Pure Component Thermal Conductivity Comparisons 0°C.......................... 69 V-5 Experimental Binary Data .................... 74 VI-1 Abbreviations for Experimental Compounds .... 114 VI-2 Pure Compounds Forming the Mixture® Studied in This Work.............. 116 VI-3 Data Sets Listed by Subdivision ........ 128 VI-4 Equations for B .......................... 142 VI-5 Popovics' Statistics ...............».......... 146 VI-6 Average Statistics for Various Correlations » 147 (In Appendices) B-l Cross Sectional Areas of Molecules Studied .. 174 B-2 Atomic Properties ............................ 175 C-l Standard Energy of Vaporization for Compounds Used in This Study....................... 179 iv LIST OP FIGURES Figure Page III-l Thermal Conductivity at 0°C and 1/ - / m"...... 36 IV-1 Relative Transient Hot-Wire Thermal Conductivity Apparatus ..................... 52 IV-2 Thermal Conductivity Cell .................. 55 IV-3 Actual x-y Recorder Plots .................. 58 IV-4 Measured Thermal Conductivity of Carbon- tetrachloride and Toluene Compared with Jamieson's Data (34) ......... 60 V-l Thermal Conductivity at 20°C of Liquid Toluene from (50) .......................... 73 V-2 Thermal Conductivity of Acetone and n- Heptane at 0°C............................. 85 V-3 Thermal Conductivity of Acetone and Methyl- ethylketone at 0°C......................... 86 V-4 Thermal Conductivity of Acetone and 2,2,4- Trimethylpentane at 0°C.................... 87 V-5 Thermal Conductivity of Acetone and 2,4,4- Trimethylpentene-1 at 0°C.................. 88 V-6 Thermal Conductivity of 1-Butanol and 2.2.4-Trimethylpentane at 0°C.............. 89 V-7 Thermal Conductivity of 1-Butanol and 2.4.4-Trimethylpentene-l at 0°C ............ 90 V-8 Thermal Conductivity of Carbontetrachloride and n-Heptane at 0°C....................... 91 V-9 Thermal Conductivity of Carbontetrachloride and Toluene at 0°C......................... 92 v Figure Page V-10 Thermal Conductivity of Carbontetrachloride and 2,2,4-Trimethylpentane at 0°C......... 93 V-ll Thermal Conductivity of Cyclopentane and n-Heptane at 0°C........................... 94 V-12 Thermal Conductivity of Cyclopentane and Methylcyclohexane at 0°C................... 95 V-13 Thermal Conductivity of Cyclopentane and Methylethylketone at 0°C................... 96 V-14 Thermal Conductivity of 2,3-Dimethylbutane and 2,2,4-Trimethylpentane at 0°C......... 97 V-15 Thermal Conductivity of 2,3-Dimethylbutane and 2,4,4-Trimethylpentene-l at 0°C....... 98 V-16 Thermal Conductivity of n-Heptane and n-Hexane at 0°C............................ 99 V-17 Thermal Conductivity of n-Heptane and n-Octane at 0°C............................. 100 V-18 Thermal Conductivity of n-Heptane and Toluene at 0°C............................... 101 V-19 Thermal Conductivity of n-Heptane and 2.2.4-Trimethylpentane at 0°C............... 102 V-20 Thermal Conductivity of n-Heptane and 2.4.4-Trimethylpentene-l at 0°C .............. 103 V-21 Thermal Conductivity of n-Hexane and n-Octane at 0°C............................. 104 V-22 Thermal Conductivity of Methylcyclohexane and 2,2,4-Trimethylpentane at 0°C............ 105 V-23 Thermal Conductivity of Toluene and 2,2,4- Tr imethylpentane at 0°C..................... 106 V-24 Thermal Conductivity of Toluene and 2,4,4- Trimethylpentene-l at 0°C................... 107 V-25 Thermal Conductivity of Toluene and 0-Xylene at 0°C.............................. 108 vi Figure Page V—26 Thermal Conductivity of 2,2,5-Trimethy1- hexane and 2,2,4-Trimethylpentane at 0°C ... 109 V-27 Thermal Conductivity of 2,2,4-Trimethyl pentane and 2,4,4-Trimethylpentene-l at 0°C 110 V-28 Thermal Conductivity of 2,2,4-Trimethyl pentane and 0-Xylene at 0°C................ Ill V-29 Thermal Conductivity of 2,4,4-Trimethyl- pentene-1 and 0-Xylene at 0°C.............. 112 VI-1 B for Subdivision I-A....................... 131 VI-2 B for Subdivision I-B....................... 132 VI-3 B for Group I ............................... 133 VI-4 B for Subdivision II-A or I-C.............. 134 VI-5 B for Subdivision II-B..................... 135 VI-6 B for Subdivision II-C...................... 136 VI-7 B for Subdivision II-D...................... 137 VI-8 B for Group II .............................. 138 VI-9 B for Group III ............................. 139 VI-10 B for Group I V .............................. 140 VI-11 B for Group V ............................... 141 (In Appendices) A—1 The Function y » Ei(-x) .................... 167 C-l Carbon and Hydrogen......................... 177 vii CHAPTER I INTRODUCTION The thermal conductivity of a substance, in this case a liquid, is defined by Fourier's law as a propor tionality constant between heat flux and the temperature gradient applied to the substance. More simply, the ther mal conductivity is a heat transfer rate constant which is a physical property of the material, or liquid, in ques tion. In industry, liquid mixtures are encountered more often than pure liquids. In all heat transfer calcula tions dealing with liquid mixtures, the thermal conductiv ity of the mixture must be known or estimated. For esti mation purposes, ordinarily a weight or mole average of pure component thermal conductivities is used. The experi mental evidence of this project and many others (31) has shown that this "linear rule" is usually unsatisfactory. Tsederberg (81) has shown that in the case of highly non ideal liquid mixtures, deviations from the linear rule can be greater than 25%. However, the average deviation from linearity measured in this work was never greater than seven percent. This is almost within the known accuracy of many pure component thermal conductivities. The purpose of this work is not to provide a 1 2 "specie dependent" mixing rule to industry to increase their precision a few percent. Jamieson (33) has done this. The purpose is not to provide an entirely theoret ical approach such as those described by McLaughlin (49) for both pure components and binary mixtures. Such an approach is for theoreticians who are not really interest ed in numerical results. These two approaches will be discussed more thoroughly in Chapters II and III. The purpose of this work is three-fold. 1. The data given should indicate to an engineer in industry when he can use the "linear rule" and the magnitude of the errors he can expect. 2. The correlation provides a simple method for estimating the thermal conductivity of binary liquid mixtures when only limited data are available and a better estimate than the "linear rule" is needed. 3. The data and correlation presented here should provide more insight into the mechanism of the thermal conductivity of binary liquid mixtures and, hopefully, provide a useful first step for predicting simply the thermal conductivity of of multicomponent mixtures as used in industry. Two general methods have been used to estimate the thermal conductivity of binary liquid mixtures. The 3 first approach, tried by Tsederberg (81) and Rodriguez (70), is to correlate mixture conductivity entirely with other average mixture physical properties. Often these properties are harder to obtain than the mixture thermal conductivity. In the case of Tsederberg, errors greaJtsx: than 30% were reported. This is unsatisfactory for engi neering work. The second method is to estimate the mixture thermal conductivity as a function of composition and pure compo nent thermal conductivities. This approach has been tried by several investigators (8,21,33,46,49,76), with varying degrees of success. The above methods will be discussed further in Chapter III. Most earlier investigators (3,21,33,42,70) have measured thermal conductivities of mixtures which are highly non-ideal solutions. These mixtures have larger deviations from linearity than more ideal solutions. One possible reason for measuring such mixtures is that the older, steady-state devices generally used are not capable of measuring thermal conductivities with the precision required to properly describe a solution with a smaller deviation. The advent of the fast and accurate transient hot wire apparatus which was introduced by Grassmann (27) in 1960 and constructed at the University of Southern Cali- 4 fornia by Malian (47), has made it possible to measure all mixtures more precisely. A modification of Malian's apparatus was used in this work. Some of the solutions measured were nearly ideal. These solutions were measured in order to better assess the contribution to the mixture conductivity by factors such as molecular weight differences, density dif ferences and molecular structure without the apparently overwhelming complications of high dipole-moraents and hydrogen bonding. Highly non-ideal solutions were mea sured also, to assess these latter factors. The work of other selected investigators was in cluded in the correlation for the sake of completeness. The correlation is based on mixtures including paraffins, olefins, naphthenes, aromatics, ketones, alcohols and carbontetrachloride and a few other miscellaneous com pounds. There are good data available (32) on binary mix tures containing such compounds as ethers, glycols, car- boxylic acids, and water, etc. Because of the effort in volved, the inclusion of these mixtures into the correla tion will be left to a future investigator. Tables 1-1 and 1-2 list the binary mixtures whose thermal conductivity data were used in obtaining the cor relation. 5 TABLE 1-1 BINARY MIXTURES MEASURED IN THIS WORK Mixture No.____________________Compounds______ Temp.*C 1. Acetone - n-Heptane 0 2. Acetone - Methylethylketone 0 3. Acetone - 2,2,4-Trimethylpentane 0 4. Acetone - 2,4,4-Trimethylpentene-l 0 5. 1-Butanol - 2,2,4-Trimethylpentane 0 6. 1-Butanol - 2,4,4-Trimethylpentene-l 0 7. Carbontetrachloride - n-Heptane 0 8. Carbontetrachloride - Toluene 0 9. Carbontetrachloride - 2,2,4-Trimethylpentane 0 10. Cyclopentane - n-Heptane 0 11. Cyclopentane - Methylcyclohexane 0 12. Cyclopentane - Methylethylketone 0 13. 2,3-Dimethylbutane - 2,2,4-Trimethylpentane 0 14. 2,3-Dimethylbutane - 2,4,4-Trimethylpentene-l 0 15. n-Heptane - n-Hexane 0 16. n-Heptane - n-Octane 0 17. n-Heptane - Toluene 0 18. n-Heptane - 2,2,4-Trimethylpentane 0 19. n-Heptane - 2,4,4-Trimethylpentene-l 0 20. n-Hexane - n-Octane 0 21. Methylcyclohexane - 2,2,4-Trimethylpentane 0 22. Toluene - 2,2,4-Trimethylpentane 0 23. Toluene - 2,4,4-Trimethylpentene-l 0 24. Toluene - 0-Xylene 0 25. 2,2,5-Trimethylhexane - 2,2,4-Trimethylpentane 0 26. 2,2,4-Trimethylpentane - 2,4,4-Trimethylpentene-l 0 27. 2,2,4-Trimethylpentane - 0-Xylene 0 28. 2,4,4-Trimethylpentene - 1-0-Xylene 0 T A B L E 1-2 BINARY MIXTURES OBTAINED FROM LITERATURE Mixture No. Compound Temp.*C Reference 29. Acetaldehyde - Toluene 0 Jamieson (32,33) 30. Acetone - Benzene 15 Filippov (20,32) 31. Acetone - Benzene 20 Riedel (32,67) 32. Acetone - Benzene 15 Bashirov (4,32) 33. Acetone - 1-Butanol 0 Jamieson (32,33) 34. Acetone - Isobutanol 0 Jamieson (32,33) 35. Acetone - Isobutanol 15 Filippov (20,32) 36. Acetone - 2-Butanol 0 Jamieson (32,33) 37. Acetone - Tertbutanol 0 Jamieson (32,33) 38. Acetone - Carbontetrachloride 0 Jamieson (32,33) 39. Acetone - Carbontetrachlorlde 15 Filippov (20,32) 40. Acetone - Chloroform 25 Rodriguez (32,70) 41. Acetone - Ethanol 25 Rodriguez (32,70) 42. Acetone - 1-Heptanol 0 Jamieson (32,33) 43. Acetone - 1-Hexanol 0 Jamieson (32,33) 44. Acetone - Methanol 0 Jamieson (32,33) 45. Acetone - 1-Pentanol 0 Jamieson (32,33) 46. Acetone - 2-Propanol 0 Jamieson (32,33) 47. Acetone - Toluene 0 Jamieson (32,33) 48. Acetone - Toluene 15 Bashirov (4,32) 49. Acetone - Toluene 40 Geller (25,32) 50. Benzene - Bromoform 20 Riedel (32,67) 51. Benzene - 1-Butanol 25 Schroff (76) 52. Benzene - Carbontetrachlorlde 20 Riedel (32,67) 53. Benzene - Carbontetrachlorlde 30 Filippov (19,32) 54. Benzene - Carbontetrachlorlde 40 Geller (25,32) 55. Benzene - Carbontetrachlorlde 15 Venart (85) 56. Benzene - Carbontetrachlorlde 31 Venart (85) 47. Benzene - Carbontetrachlorlde 46 Venart (85) 58. Benzene - Carbontetrachloride 61 Venart (85) 59. Benzene - Chloroform 15 Filippov (19,32) 60. Benzene - Chloroform 20 Riedel (32,67) 61. Benzene - Cydohexane 20 Riedel (32,67) 62. Benzene - Dlchloromethane 0 Jamieson (32,33) 63. Benzene - Ethanol 37 Barnette (3,32) 64. Benzene - Ethanol 6 Tsederberg (32,82) 65. Benzene - Ethanol 20 Tsederberg (32,82) 66. Benzene - Ethanol 40 Tsederberg (32,82) 67. Benzene - Ethanol 60 Tsederberg (32,82) 68. Benzene - Ethanol 75 Tsederberg (32,82) 69. Benzene - Methanol 0 Jamieson (32,33) 70. Benzene - Methanol 15 Filippov (20,32) 7 T A B L E 1-2 (c o n 't.) Mixture No._____________Compound________ Temp.*C_______Reference 71. Benzene - Methanol 20 Riedel (32,67) 72. Benzene - Methanol 37 Barnette (3,32) 73. Benzene - Methanol 40 Geller (25,32) 74. Benzene - Methanol 25 Schroff (76) 75. Benzene - 1-Propanol 36.7 Barnette (3,32) 76. Benzene - 2-Propanol 36.7 Barnette (3,32) 77. Benzene - Toluene 0 Jamieson (32,33) 78. Benzene - Toluene 15 Filippov (19,32) 79. Benzene - Toluene 15 Bashirov (4,32) 80. Benzene - Toluene 15 Venart (32,85) 81. Benzene - Toluene 31 Venart (32,85) 82. Benzene - Toluene 46 Venart (32,85) 83. Benzene - Toluene 61 Venart (32,85) 84. 1-Butanol - Carbontetrachlorlde 0 Jamieson (32,33) 85. 1-Butanol - Toluene 0 Jamieson (32,33) 86. Isobutanol - Carbontetrachlorlde 0 Jamieson (32,33) 87. Isobutanol - Carbontetrachlorlde 15 Filippov (20,32) 88. 2-Butanol - Carbontetrachlorlde 0 Jamieson (32,33) 89. 2-Butanol - Toluene 0 JamieBon (32,33) 90. Tertbutanol - Carbontetrachlorlde 0 Jamieson (32,33) 91. Tertbutanol - Toluene 0 Jamieson (32,33) 92. Carbontetrachlorlde - Chlorobenzene 15 Filippov (19,32) 93. Carbontetrachlorlde - Chloroform 15 Filippov (19,32) 94. Carbontetrachlorlde - Cydohexane 20 Venart (32,85) 95. Carbontetrachlorlde - Cydohexane 40 Venart (32,85) 96. Carbontetrachlorlde - Cydohexane 60 Venart (32,85) 97. Carbontetrachlorlde - Ethanol 36.5 Barnette (3,32) 98. Carbontetrachlorlde - 1-Heptand 0 Jamieson (32,33) 99. Carbontetrachlorlde - 1-Hexanol 0 Jamieson (32,33) 100. Carbontetrachlorlde - Methanol 0 Jamieson (32,33) 101. Carbontetrachlorlde - Methanol 15 Filippov (20,32) 102. Carbontetrachlorlde - Methanol 36.5 Barnette (3,32) 103. Carbontetrachlorlde - 1-Pentanol 0 Jamieson (32,33) 104. Carbontetrachlorlde - 1-Propanol 36.5 Barnette (3,32) 105. Carbontetrachlorlde - 2-Propanol 0 Jamieson (32,33) 106. Carbontetrachlorlde - 2-Propanol 36.3 Barnette (3,32) 107. Carbontetrachlorlde - Toluene 0 Jamieson (32,33) 108. Carbontetrachlorlde - Toluene 14.5 Venart (85) 109. Carbontetrachlorlde - Toluene 31 Venart (85) 110. Carbontetrachlorlde - Toluene 46 Venart (85) 111. Carbontetrachlorlde - Toluene 61 Venart (85) 112. Chlorobenzene - Methanol 15 Filippov (20,32) 113. Chloroform - Diethylketone 30.6 Kerr (32,42) 114. Chloroform - Isobutylmethylketone 30.6 Kerr (32,42) 8 T A B L E 1-2 (c o n 't.) Mixture No._____________Compound________________Temp.*C Reference 115. Chloroform - Methylethylketone 30.6 Kerr (32,42) 116. Chloroform - Toluene 30.6 Kerr (32,42) 117. Cydohexane - Ethanol 36.7 Barnette (3,32) 118. Cydohexane - 1-Propanol 36.7 Barnette (3,32) 119. Cydohexane - 2-Propanol 36.5 Barnette (3,32) 120. Methylethylketone - Toluene 40 Geller (25,32) 121. 1-Heptanol - Toluene 0 Jamieson (32,33) 122. 1-Hexanol - Toluene 0 Jamieson (32,33) 123. Methanol - 2-Propanol 40 Rastorguev (32,65) 124. Methanol - Toluene 0 Jamieson (32,33) 125. Methanol - Toluene 25 Schroff (76) 126. n-Octane - 2,2,4-Trimethylpentane 40 Rastorguev (32,65) 127. 1-Pentanol - Toluene 0 Jamieson (32,33) 128. 1-Propanol - Toluene 25 Schroff (76) 129. 2-Propanol - Toluene 0 Jamieson (32,33) CHAPTER II LITERATURE SURVEY The literature survey included the study of pure component liquid thermal conductivity correlations as well as correlations for binary liquid mixtures. Also included was the search for existing binary liquid mixture thermal conductivity data. The chemical abstracts indicate that the Russian literature has carried most of the publica tions on liquid thermal conductivity work in recent years. Most of this work, however, is for pure components. Much of the Russian literature is still almost inaccessible and, therefore, only the most pertinent Russian articles were sought out. Pure Liquid Thermal Conductivity Correlations Thermal conductivity correlations for pure liquids range from totally empirical to completely theoretical statistical mechanical modesl. Kerr (42) points out that some of the more complex theoretical models predict thermal conductivities which differ from experimental values often as much as 100%. In a study of the various correlation methods, Reid and Sherwood (66) state: "Most are empiri cal, though a few rest upon rather tenuous theoretical models." Possibly because the highly theoretical models 9 10 predict so poorly, they were not included in the study of "estimating techniques.* Of all the techniques studied, Reid and Sherwood recommend only one method for "more ac curate" estimates, which was the method of Robbins and Kingrea (69). Unfortunately, many of the empirical tech niques are based on older data which tend to be high be cause of undetected natural convection which occurs in some of the older steady state devices used for making thermal conductivity measurements; hence, many of these techniques give poor results. A review of most of the very theoretical models is given by McLaughlin (49). Reviews of some of the other correlations are given by Tsederberg (81) and Gambill (23,24). The correlations of historical interest and of most interest in this work are listed below. Some of the other more interesting methods are given in the following references: (39,40,47,54,55,58,63,72,77,86,87). Apparently the first attempt to correlate thermal conductivity of liquids empirically was made by Weber (89). Weber's first equation was of this form: k = A f > Cp (II-l) where k = thermal conductivity, f> - density, Cp is the heat capacity, A is a constant. He later modified the equation to this form: 11 1/3 k = Bp Cp ( f*/M) (II-2) M is the molecular weight and B is another constant. Equa tion (II-2) has been modified a number of times (55,68,77). The best modification to date is that of Robbins and Kingrea (69): k = £[(88.0 - 4.94H) (10”3)]/AS*| Cp ( f > (II-3) k = thermal conductivity - cal/(cm)(sec)(°K) Tr = reduced temperature - T/Tc Cp = heat capacity - cal/(g-mole)(°K) f* - density - g-moles/cm AS* = modified Everett entropy of vaporization, *Hvb/Tb + R In (273/Tb) AHV, = heat of vaporization at the normal boiling point cal/g-mole Tjj = normal boiling point °K N depends upon the liquid density at 20°C N = 0 for p > 1.0 g/cm^ N - 1.0 for p < 1.0 g/cm^ H is called the hindrance factor and depends upon the shape of the molecule or the functional groups. It is an integer ranging in value from -1 to 6. See references (66) or (69) for H values. Bridgman (11) is a pioneer in the theoretical ap proach to the thermal conductivity of liquids. His 12 equation is: k = 3K Us/6 2 S = (m//> )1/3 (H-4) < T = mean distance between centers of molecules K = Boltzmanns constant Us = the velocity of sound in the liquid m = the mass of one molecule Kardos (41) modified Bridgman's theory, Sakiadis and Coates (73,74), in turn, modified the Kardos equation slightly and used it in this form: where L is the distance between the surfaces of the mole cules. The distance L is determined by X-ray diffraction techniques. By assuming the liquid molecules form a face centered cubic lattice and that they are spherically sym metric with a Lennard-Jones (6-12) intermolecular poten tial energy function, Horrocks and McLaughlin (30,83) predict liquid thermal conductivities with the following equation: k = Cp Us p L (II-5) 2 77 (H-6) v* = cr3/(V/N) * O ' " = molecular diameter 13 V = molar volume N * Avogadro'a number Cv = constant volume heat capacity m * molecular mass a = nearest neighbor distance for a face centered cubic crystal = • ( /Sv/N)1^3 Z = coordination number for a face centered cubic crystal * 12 a lattice summation constant ** 22.11 * lattice summation constant = 10.56 6 ** minimum potential energy in the Lennard-Jones (6-12) potential function. The theoretical details of these correlations will be described more fully in Chapter III. Binary Liquid Mixture Correlations To date thermal conductivity correlations for bi nary liquid mixtures range from extremely simple empirical mixing rules to very complex theoretical formulas. Two of the theoretical formulas are discussed by McLaughlin (49). In general, this type of formula is for very simple mole cules and utilizes very crude approximations and/or proper ties of the system which are generally more difficult to obtain than the thermal conductivity itself. From an engineering standpoint, these models predict thermal con ductivities too poorly to be of practical use. Barnette (3) obtained an equation for systems of 14 alcohol and non-polar liquid mixtures which he treated as ternary mixtures of monomer alcohol, hydrogen-bonded polymer alcohol and the non-polar liquid. While this ap proach gives some insight into the mechanism of thermal transport, the equation requires such properties as hydro gen bond energies, chemical equilibrium constant, diffu sion coefficients, and activity coefficients. These prop erties are often more difficult to obtain than thermal conductivities and, therefore, this approach is not parti cularly useful. Kerr (42) also presented an equation for predicting thermal conductivities of hydrogen bonded binary solutions. Again, insight into the mechanism is provided but the equa tion is not suitable for engineering application. Rodriguez (70) attempted to use the Kardos equation (II-5) to predict binary mixtures. He used mixture values for the properties Cp, Us, p and L to predict the mixture k. The mixture value of L is estimated from excess Gibbs free energy data. The approach is interesting but not suited to engineering calculations. Tsederberg (81) used weighted average pure compo nent physical properties in the Predvoditelev-Vargaftik equation with limited success. This equation is a modifi cation of the Weber equation with a term to account for molecular association. 15 The difficulties encountered in using these methods indicate that a less sophisticated approach be taken, such as using a mixing rule based on pure component thermal conductivities. Recent improvements in measurement tech niques have made available enough good pure component thermal conductivity data to make a pure component mixing rule justifiable. Good data reduces the need for methods such as those of Rodriguez and Tsederberg, which predict mixture thermal conductivity entirely from mixture proper ties. There are several of these correlation methods available. They are for the most part empirical or margin ally theoretical. Most of the known methods are listed below; Jordan and Coates (38) derived an equation for the thermal conductivity of binary liquid mixtures based on an analogy for viscosity of binary liquid mixtures (28). Ink = x^lnk^ + x2lnk2 + XjX2ln (exp| k^-k2| - (II-7) x = weight fraction The units on k are BTU/hr ft °P. Filippov (19,20) presents the following empirical equation: k = X]k^ + X2k2 - Cx^X2|k^ - k2| (II-8) 16 C « 0.72 for non-associated solutions C * 0.76 fox associated solutions k is in cal/cm-sec °K Blaha (7) went one step further than Filippov and suggested the following equation: k = (xiki + x2k2 ^ 1 ” xlx2 [ b + c(xi“x2^ + d(xi”x2)2 + •••]) (II-9) He states that constants c and d can be neglected and the equation reduces to: k * (xiki + x2k2^ ^ ~ xix2b) (11-10) k is in the units Watts/m °K b is a constant depending on the class of liquids used in the mixture. The liquid classes used by Blaha are the classes defined by Ewell (18). The liquids are grouped based on their potential to form hydrogen bonds. A brief descrip tion of Ewell's classification follows: Class 1: Liquids which form three-dimensional net works of strong hydrogen bonds; e.g., water, glycols, glycerol, etc. Class 2: Other liquids with molecules which have both active hydrogen atoms and donor atoms (oxygen, nitrogen and fluorine); e.g., alcohols, acids, phenols, etc. 17 Class 3: Liquids with molecules of donor atoms but no active hydrogen atoms; e.g., ethers, ketones, aldehydes, etc. Class 4: Liquids with molecules containing active hydrogen atoms but no donor atoms; e.g., chloroform, dichloromethane, etc. Class 5: Liquids having no hydrogen bond forming capabilities; e.g., hydrocarbons, carbon- tetrachloride, etc. Blaha puts combinations of these classes into three groups and assigns a value to b for each group. Table II-l lists the values of b for each group. TABLE II-l VALUES OF b FOR EQUATION (11-10) GROUP I II III Class 2-2 2-3 1-1 1-2 Combinations 3-3 3-5 2-4 1-3 4-4 4-5 2-5 1-4 5-5 3-4 1-5 b 0.12 0.40 0.84 18 Jamieson (32,33,34) measured 59 binary liquid mix tures and fit his data to the following equation: k = k^x^ + k2x2 “ ^2 “ kl^ ^ " ^~*2^x2 (H-li) where is an empirical constant obtained for each binary measured except for a few which could not be fit to Equa tion (11-11). These constants may be found in (33). Jamieson states that Equation (11-11) with c< =1.0 will predict thermal conductivities with an accuracy of ±6%, if no other values for are available. Dul'nev (16) presents a liquid mixture structure oriented equation: k = kx Z2 + v(l-Z)2 + 2vZ(1-Z)/ vZ + (1-Z) (11-12) where v = k2/ki# Z is a lattice parameter defined by Equation (11-13): P = 2Z3 - 3Z2 + 1 (11-13) P is the volume concentration of component 2 = V2/V where V 2 is the volume of component 2 and V is the volume of the system. Article (16) does not describe the quantity P very well. Another article by Dul'nev (17) describes P for a system such as a gas and a solid. Based on the informa tion in article (17), it is assumed that when dealing with mixtures of liquids, Dul'nev uses pure component densities 19 to determine P. This is not clear. Rastorguev (65) states that the thermal conductiv ity of a liquid mixture is a function of intermolecular forces and the size of the component molecules. He then uses three different equations based on the relationship of to V2 to predict mixture thermal conductivity. and V2 are the molecular volumes of components 1 and 2, respectively. If Vx = V2, k = n ^ + n2k2 (11-14) where n is mole fraction. If > V2 k2n2 k1(2V1A 2“1)nl * = ri2 + 'C2VJ[yV2-l )nx + n ' 2'" + ( 2V1/vfi)n1 t11*15) If vx < V2 k2(2V2A 1-l)n2 kini nx + (2V2/V1-l)n2 nx + (2V2A 1-l)n2 (II“16) Preston (61) uses the following mixing rule for binary mixtures of spherical molecules: k = 0i ki = 20102k12 + 02^2 (11-17) nl Vci n2 vc2 01 " nlVci + n2VC2 02 ~ nxVCi + n2VCz (II-18) Vc is the critical volume. k^2 t* ie mixture thermal conductivity. The latter may be calculated in the follow ing manner (the pure component thermal conductivities may 20 also, if they are unknown: log1()k* = -0.4404 + 1.4665/T* - 0.3991/(T*)2 (11-19) k* is the reduced thermal conductivity T* is the reduced temperature (T* = KT/€) T is the absolute temperature K is Boltzman's constant € is the characteristic energy parameter which appears in potential functions such as the Lennard-Jones (6-12). r- - * 4 1 (” -20) k K & rQ is the characteristic size parameter appearing in the same equation as € ■ . m is the molecular mass. The following mixing rules apply for e , r0 and ra for estimating k-j.2s * 12 - “ i V * (II-21) *012 = *<*0x + r02> (11-22) 2mim2 m “ ml m2 (11-23) Bondi (8) points out that € and rQ can be uniquely deter* mined only for monoatomic molecules and suggests the use of other parameters that represent these quantities in a pure component model. The term representing rg is dw 21 which is defined i=n dw = ~ y~. dw(i) (11-24) where dw(i) are the Van der Waals diameters of all outer atoms of a molecule. The energy term which also appears in his mixing rule is the standard energy of vaporization E°. E° = 4HV - RT (11-25) at the temperature where V/Vw is 1.7. 4 hv is the heat of vaporization T is absolute temperature R is the gas law constant V is molal volume Vw is Van der Waals volume o Bondi gives tables for estimating E based on the func tional groups and structure of the molecules (see Appendix C). His mixing rule for non-associating mixtures is: k = n-^k^ + n2k2 - f.b.n^r^ (11-26) f 4 Ei<V - ( ^ V b is a constant equal to 4.5 x 10”5 when E° is expressed in calories per mole and k is in cal/(cm)(sec)(°K). M is mole weight and n is mole fraction, if neither specie 22 has a skeletal chain of seven carbons or greater. For mixtures containing the longer chains, n is volume frac tion. Losenicky (46) presents the following interesting mixing rule: k =* (Pn + P]^) *1 * 1 + (?22 + P21^ x2^2 (II-27) where P1X = C P12 * C D^DjNj, (11-28) p2l = c P22 - C D22H2. The constant C can be determined by including the equa tion: <PU + P12) + <P22 + P21) x2 = 1 (H-29) N is the number of molecules present, D is the average cross sectional area of the mole cule. The average cross section D can be calculated from the actual size and shape of the molecule. With bond angles, covalent radii and packing radii as given by Pauling (56), one can determine the orthogonally projected areas of the molecule. D is the average of the three projected areas. The method and a sample calculation are outlined in Ap pendix B. A table of cross sectional areas is provided also. Some of the theoretical aspects of the above 23 methods will be discussed in Chapter III. Their predic tive abilities will be discussed in Chapter VI. Available Experimental Data A search of the literature yielded thermal con ductivity data on approximately 300 binary pairs. Much of the data before 1968 has been collated by Jamieson (32). With the exception of Mukhamadzyanov (32,53) and Rastorguev (32,65), no mixtures containing paraffins were measured. No mixtures containing olefins are reported. The only naphthene used in a mixture was cyclohexane. Many of the literature data are for highly non ideal solutions such as hydrogen bonding alcohols and glycols and other polar molecules such as ketones and ethers. The apparent reason is that these non-ideal mix tures produce large deviations from the additive rule and the larger deviations are easier to measure. Some of these data were eliminated from the accumu lated data bank. Much of the data were not of interest to this study because they did not fit into the categories described in Chapters I and VI. Those binary systems for which only a single mixture point was reported were re moved on the contention that one mixture point was not sufficient to describe the functional relationship between conductivity and composition. Others were eliminated because the conductivities were obtained with a different temperature at each composition. This made it difficult to determine the relationship between conductivity and composition. Finally, one set, ethanol-methanol, was discarded because the results appeared erratic. CHAPTER III THEORY The thermal conductivity of a liquid is defined by Fourier's equation in one dimension: ^3 = -kA (III-l) d© dx ||3 is the heat flux across the temperature gradi- dT ent, normal to the area A; k is a proportionality con stant, dependent upon the heat transfer medium. The thermal conductivity k can be defined by the more general form of Fourier's equation: -f§ = -75 V 2T + q(x,y,z,0) (HI-2) q(x,y,z,G) is a heat generation term V is the differential operator defined Although the subject of this work is the thermal conductivity of binary liquid mixtures, the theory of the thermal conductivity of pure liquids must be discussed before one can speculate about mixture behavior. Pure liquid thermal conductivities have been stud ied for almost 100 years. Yet, a purely theoretical model which will predict pure liquid thermal 25 26 conductivities with the accuracy required for engineering calculations has not been developed. The main reason is that the liquid state is still not well understood; con sequently, the thermal conductivity model is highly depend ent upon the assumed liquid model. Some information about liquid structure may be ob tained by X-ray diffraction techniques. X-ray diffrac tion yields a description of the equilibrium distribution of the molecules. This information is used to derive a radial distribution function, or the distribution of neighbor molecules about a particular molecule. The thermal conductivity of a liquid may be calculated from a complicated integro-differential equation giving the thermal conductivity as a function of the disturbance of the radial distribution function. Bondi(8) points out that extremely crude approximations must be made to solve these equations. The assumptions that must be made to solve the equations essentially nullify the effectiveness of the model. McLaughlin (49), however, reports that "Integration of the experimental radial distribution func tion for liquid argon at its melting point shows that the number of nearest neighbors has decreased from 12 in the solid to about 10.8 in the liquid ..." This indicates that "pseudo" lattice theories as a simplified model of the liquid state may have some validity. 27 These models, to date, have produced the best re sults for theoretical models for thermal conductivities of pure liquids and three of these models are discussed in some detail. Pure Liquid Thermal Conductivity Theories Bridgman (11) considered liquid molecules to be in a cubic lattice with centers a distance <$ apart. Where S 1/3 is (m/f) , /> is the liquid density and m is the molecu- dT lar mass. The temperature gradient is If a molecule possesses ~ KT kinetic energy and KT potential energy where K is Boltzman's constant, the difference in energy dT between neighboring molecules is -3K<£ — . If the energy is propagated through the liquid with the velocity sound Us, in one unit of time, it travels through Us/ 6 rows of molecules and the energy change through these rows is: % = -3K 6 ££ = -3K US^ (III-3) d© dx 6 dx 2 A unit cross-sectional area A is equal to & , and the energy transport per unit time per unit area is: dg_ = ~3, K dT (m-4) dOA § 2 dx # Equation (III-4) is compared with Fourier's equa tion (III-l), resulting in Bridgman's equation: 3K Us k = “ T2“ (II-4) d 28 Tyrell (84) points out that for such a simple theory, Bridgman's equation is surprisingly good. Other approaches (6,53,84) based on the kinetic theory of rigid sphere gases yield equations for the ther mal conductivity of liquids similar to Equation (II-4). An interesting variation of the Bridgman equation was derived by Kardos (41). The basic difference is that the mean free path in the Kardos equation is L, the dis tances between the surfaces of the molecules rather than $ , the difference between the centers. Sakiadis and Coates (73) used a slightly modified version of the Kar dos equation in their work. The derivation of this equa tion is as follows, as given by Rodriguez (70). Again, the molecules are assumed to be in a cubical lattice. Energy is transferred down a column of molecules as the molecules oscillate about their equilibrium posi tions and collide. There is an energy drop, Au» per mole cule across the distance L of: (III-5) Q “ heat content per mole N ■ Avogadro's number (III-6) 29 If energy is transmitted through the liquid with the velocity of sound, the rate of energy transfer down the column of molecules is: A comparison of Equation (III-8) with Equation (III-l) shows that: Substitution into (III-9) yields: k * Cp US^L (II-5) The use of L rather than & as a mean free path is tanta mount to assuming that the controlling resistance to heat transfer is in the "hole" between the molecules and that the energy change within the molecule itself is essential ly instantaneous. The difficulty of obtaining L makes it difficult 2 Since S is the unit area per molecule normal to the direction of heat transfer, (III-8) (III-9) Since: UII-10) and: (III-ll) 30 to determine if Equation (II-5) is better than Equation (II-4). Horrocks and McLaughlin (30,49,83) derive a slight ly more sophisticated equation for the thermal conductiv ity of liquids. The model they use assumes a quasi crystalline liquid in a face-centered cubic lattice for spherical molecules, interacting with a Lennard-Jones potential. They show that thermal conductivity is the sum of vibrational and convective (molecules jumping from lat tice sites) contributions, but that the convective term may be very small (about 1%). For the vibrational contri bution only, the energy difference between successive planes, a distance i apart, due to a temperature differ ence, is $ If the energy is transferred down the temperature gradient with a frequency Pr, the rate of heat flow, is defined by the following equations fay . -2n Pr i (111-12) P is the probability that energy is transferred when two vibrating molecules collide, v is the mean vibrational frequency, n is the number of molecules per unit area of the liquid lattice normal to the heat flow. The factor 2 accounts for the fact that the molecule crosses a plane at right angles to its motion twice every vibration. Noting that ^ *» ~ and again substituting 31 into Equation (III-l), the thexmal conductivity becomes* k « 2PnvJl% (111-13) dT Horrocks and McLaughlin show that n “ where a is the nearest neighbor distance, which for a face-cen- r- 1/3 tered cubic is ( V2V /N) . P is not known and set equal to unity. If is equal to Cv and / » a then Equa tion (111-13) becomes* k = T2 v Cv/a (111-14). If the potential energy function, 0(r), between a pair of molecules can be described by the Lennard-Jones (6-12) potentials 0(r) « 46 [(f)12 - (f) 6j (HI-15) r is the distance between the centers of the molecules; o ' is defined as the distance r at which the potential energy function 0(r) is equal to zero. The minimum value of 0(r) is -£, and rQ is the value of r at which 0(r) is a minimum (0(ro) ■ - € ). r03 * Ti"^3 (111-16) Corner (15) has shown that the vibrational frequency for a harmonic oscillator is* 32 V “ [ U C 1 4 ( ? ) 14 ' 5 C8( ? ) " i f (111-17) where m is the mass of the molecule and and Cg are lattice summation constants derived by Lennard-Jones (43), for crystals, to take into account the potential energy effect on one molecule due to the rest of the molecules in the crystal. (The values of Cg given by Lennard-Jones and that given by Horrocks and McLaughlin are slightly differ ent. ) The thermal conductivity now becomes: k c i 4 ( ? ) 1 4 -5 c 8 ( ? ) 8 if ^ (111-18) Horrocks and McLaughlin have solved this equation for some simple molecules and found that the calculated values agree with measured values, on an average, within about 20%, as compared to about 35% for Equation (11-4). It can be shown that Equation (IIIr?18) is equal to (II-6). \/"2 C m r r *4 *2 i i^5 . * " [28e{‘ iv - M iv } ] (II-6>- If Cy « 3K and for liquids Cp » Cv, then some comparisons may be made. Equation (II-4) becomes: Equation (111-14) becomest 3,— /P N\ ^ 3 k « °JT Cpv(V") (III—2 0 ) These two equations can be compared with the Kardos equa tion: k - CpUs^L (II-5) and the empirical Weber's equation: k = B Cp (/>/M)1/3 (II-2) which is the basis for most modern empirical thermal con ductivity equations. Comparing Equations (111-19) and (111-20) shows that V , the frequency of a harmonic oscil lator, apparently may be compared to Ug, the velocity of sound. Rao (64) presents an empirical equation for the velocity of sound: Us - (/°R/M)3 (111-20) where R is a constant independent of temperature, but dependent upon molecular weight and the chemical species. r = + b (111-21) k is a constant for all homologous series and B is a 34 constant which is series dependent. The point here is that thermal conductivity is, theoretically, dependent upon density, but the exact de pendence is not really known. Since density is a property which does change upon mixing, it appears that mixture density should be consider ed in a correlation for the thermal conductivity of binary liquid mixtures. An intensive study of the binary density data of Timmermans (78,79) for over half of the 129 binary data sets studied here, however, failed to reveal a cor relation between the two mixture properties. The potential energy term of Equation (111-15) which appears in a slightly altered form in Equation (HI- 18) , indicates that the thermal conductivity of a molecule is dependent upon the forces which affect the molecules. Hirschfelder et al. (29) state that the force of inter action between two spherical non-polar molecules is a func tion of inter-molecular separation. For most purposes it is more convenient to use a potential energy of interac tion 0(r) rather than the force of interaction F(r). The two functions are simply relatedt A commonly-used potential function for non-polar molecules (111-22) 35 is the Lennard-Jones (6-12) potential (111-15). At this point it is interesting to compare the form of Horrocks and McLaughlin's theoretical equation to actual pure component data. Equation (111-18) contains the term V /In; m is the molecular mass which is related to mole weight M by Avogadro's number N. Figure III-l is a plot of thermal conductivity, k vs. mole weight, M, and *•/ /M vs. M. This is like plotting k which is proportional to *■/ /TT vs. M, at 0°C for real liquids. Many of the data points are from this work; the other sources are consistent with this data. The chemical families plotted are: normal alcohols, normal paraffins, normal ketones and aromatics with one straight chain branch. Two distinct trends can be observed in Figure (III-l)j (1) The spherical type molecules, for which Equa tion (111-18) was derived, do, indeed, follow the shape of the curve V /m or •*•/ /m vs. M. (2) As the same families become more chainlike, they tend to follow a curve characteristic of the normal paraffins. This is an indication of the strong dependence on structure of liquid thermal conductivity. Bondi (9) gives an explanation for the behavior Figure III-l Thermal Conductivity at 0°C and 1//~M 9 ■ normal alcohols O * aromatics with one chain branch ■ * normal paraffins + « normal ketones t ■ extrapolated value 160 0.12 0.11 150 0.10 140 0.09 130 t 100 120 110 80 Mole Weight (M) u> 37 exhibited in Figure III-l. He used Eyring's equation (58) to predict the thermal conductivity of the spherical type molecules, benzene and glycerol. Eyring's equation ist The equation predicted these thermal conductivities very well. Then he used the equation to predict the thermal conductivity of long chain molecules like n- hexane, n-decane, olive oil and rubber. As shown in Fig ure III-l, he found that for these molecules, as the chain length increases, so does the actual thermal conductivity, but the predicted conductivity decreases. Bondi then used Equation (111-23) to predict the molal volume V. The volume predicted by Bondi, the effec tive volume, was relatively constant for all molecules. For the spherical type, e.g., benzene and glycerol, the effective volume was very close to the actual molal vol ume, but for the long chain molecules, the ratio of effec tive volume to actual volume decreased as chain length in creased. Bondi explains that this indicates that mole cules of long flexible chains move in segments rather than the whole molecule moving in one direction at one time. This means that the collisions involved in the thermal conductivity of chain-like molecules are segmental (111-23) 38 collisions rather than molecular collisions. The model for these thermal conductivities should be more compli cated than those presented here. Bird (6) states that when £ from Equation (111-18) is unknown, it may be estimated by the following equation: € * 1.15 TbK (111-24) where Tb is the normal boiling point in °K, and K is Boltz- man's constant. Substituting Equation (111-24) into (III- 18) shows k proportional to >/t^. Table III-l lists boiling points and thermal con ductivities for some isomers. It indeed appears that there is some relationship between thermal conductivity and normal boiling point. Morrison and Boyd (52) explain that branched chain isomers have lower boiling points than straight-chain isomers because, with branching, the shape of the molecule approaches that of a sphere; as this hap pens, the surface area decreases. This decreases the intermolecular forces which are overcome at a lower temper ature. Evidently similar forces come into play when deal ing with thermal conductivities. This argument reinforces the idea that thermal conductivities are indeed a function of the forces applied to molecules. These forces are, in tuin, functions of density, molecular mass and molecular structure. 39 THERMAL CONDUCTIVITY TABLE III-l AND NORMAL BOILING POINT FOR SOME ISOMERS Compound Tb #C Reference 1-Butanol 153.6 117.1 Parkinson 2-Butanol 139.7 99.5 Jamieson (34) Tertbutanol 114.3 82.9 Jamieson (34) Isobutanol 137.7 107.8 Jamieson (34) n-Hexane 130.1 69.0 Parkinson 2,3-Dimethylbutane 110.7 58.0 Parkinson n-Octane 134.5 125.7 Parkinson 2,2,4-Trimethylpentane 103.5 99.2 Parkinson 2,4,4-Trimethylpentene-l 106.0* 101.4 Parkinson n-Nonone 140.5 150.5 Missenard (51] 2,2,5-Trimethylhexane 107.8 124.1 Parkinson JL This is not an isomer of n-Octane, but an olefin with a shape similar to that of the branched paraffin isomer. 40 The forces discussed so far are simple ones. For spherical molecules, to this point, the only forces dealt with have been the London dispersion forces (44,45). Roughly, London's dispersion forces may be related to the attractive term in the Lennard-Jones (6-12) potential func tion. The potential of these dispersion forces ist Discussions on forces between molecules can be found in several texts (29,60,62,71). Other physical forces which should be considered are electrostatic forces between two permanent dipoles and induction forces between a permanent dipole and an induced dipole. The only chemical force considered in this study is that of hydrogen bonding. The experiment was designed in an effort to isolate the effect of these forces on the mixture thermal conduc tivity. The experimental design is discussed in Chapter VI. In more simple terms, thermal conductivity may be thought of as a rate constant. Rate constants are depend ent upon collision frequency, which depends upon these three thingss (dispersion) 0(r) « (111-25) 41 (1) How closely molecules are crowded together, or density; (2) How large the molecules are, or structure; (3) How fast the molecules are moving. This — depends upon the forces upon the molecule and also how heavy it is. A heavy molecule would move more slowly, thus re ducing the collision frequency. Heavy molecules, how ever, are often larger, thus increasing the collision probability and increasing the collision frequency. These two effects often cancel out. An exception to this rule are the halogenated methanes like carbontetrachlor- ide. Carbontetrachloride is very heavy, yet rather small as seen by its small cross section area (see Appendix B). Carbontetrachloride has a lower thermal conductivity than, say, large cross sectional area hydrocarbons of similar molecular weight. Binary Theory Bearman (5) has developed an equation for binary liquid mixtures from integration of the radial distribu tion function; 42 (IXX-26) Dl*' d2* are diffusion coefficients of each component in the mixture; (D^*)°, (D2*)° are self-diffusion coeffi cients of the pure components, v is the mean molar vol ume (njV^ + n2V2). I is an integral which vanishes if the molecules behave as hard spheres. Tyrrell (84) states that Equation (111-26) was used (with assumptions about self-diffusivities) on the binary system carbontetra- choride-benzene, and, although the predictions were not of the correct magnitude, the general shape of the thermal conductivity-composition curve was predicted. Clearly, Equation (IXX-26) is not suitable for engineering calculations. An interesting equation was presented by Losenicky (46). As described in Chapter II, Losenicky*s equation is as followst k ■ * pi2^xl^l ^ (^22 ^21^x2^2 (11—27) where t The C in Equation (11-28) can be calculated from the mix ing rule used by Losenicky: <PU + PX2> Xx + <P22 + P21) x2 - 1 (IX-29) D is the average cross-sectional area of the molecule which can be calculated from actual molecular dimensions. Losenicky refers to London's work (44) and apparent- 2 ly uses D as an approximation to O’ as shown in Equation (111-25). If the molecule were a sphere, D would then be the projected surface area of the sphere equal to (7r/4)d where d is the diameter of the sphere or an approximation to f f * . Losenicky's equation predicts the thermal conduc tivity of mixtures with widely differing cross sections fairly well, indicating that differences in structure between molecular pairs is one item to consider when deal ing with mixture thermal conductivities. In simple terms, the thermal conductivity of a binary mixture may be thought of as the sum of three types of thermal conductivities. (1) The thermal conductivity represented by colli sions between molecules of component one only. 44 (2) The thermal conductivity represented by colli- sions between molecules of component two only. (3) The thermal conductivity represented by colli sions between molecules of component one with component two. The probability of one of these types occurring is a function of the number of molecules present as well as the structure or size of the molecule present. It is observed that, generally, binary mixtures ex hibit negative deviations from linearity or the "additive rule." It would be convenient to describe this deviation in terms of the three types of collisions described above. These deviations appear to be, rather, some complex com bination of the differences between the molecules in the mixtures; i.e., force differences, size differences, weight differences, etc. Some of the ideas presented in this chapter were used as starting points for several theoretically based mixture correlations. Each correlation predicted some mixtures well and other mixtures very poorly. This is generally the case of the equations described in Chapter II. The more theoretical the equation, the better is the prediction in some cases, and the worse is the failure in the other cases. A more complete discussion of the quantitative predictive abilities of these equations is 45 given in Chapter VI. Apparently, the mechanism of thermal conduction is often more complex than the simple models derived in this chapter. One example of a more complex situation is the segmental collisions described by Bondi. Rodriguez (70) points out further complications for binary mixtures. He states that molecules with simi lar diameters are only involved in two-body collisions. The thermal conductivity produced by these simple colli sions may be described by the equations in this chapter and in Chapter II. As the ratio of molecular diameters increases from one to about 1.5 and greater, more than two molecules may be involved in one collision. When three- body (or more) collisions occur, more complicated expres sions are needed to describe the thermal conductivity of these mixtures. Hydrogen bonding can present "mechanism" problems as well as "force" problems. One question which arises is: Do the bonded mole cules collide as individuals or as segments of chains? Another question: Do the bonded molecules present individual targets or combined targets which would be involved in multibody collisions? The model problems caused by these situations and the repeated failure of the theoretical models suggest a more empirical approach to the correlation problem. CHAPTER IV APPARATUS AND PROCEDURE Many of the items discussed in this chapter are discussed in greater detail by Malian (47). Background Over the years the most popular method of measuring the thermal conductivity of liquids has been the steady state method. This method consists of using a device which can supply the boundary conditions for Fourier's steady state equation in one dimension, <*«-» in any coordinate system. In a steady state apparatus dT the temperature gradient, and the rate of heat trans fer, are observed. The heat transfer area. A, is con- w stant leaving only k, the thermal conductivity to be calcu* lated. The apparatus can have one of several different geometries. This changes the form of Equation (III^l) slightly. Some configurations arei parallel plate (Car tesian coordinates), concentric cylinders (cylindrical coordinates), concentric spheres (spherical coordinates), and a hot wire, which is a variation of the concentric 47 48 cylinder type. The hot wire configuration consists of a cylinder with a wire suspended axially in it. The wire is the source of the heat flux d© Steady state devices have two major problems which tend to aggravate one another. The first is that the apparatus must be operated at steady state, which requires several hours to attain and a number of measurements to as sure. Steady state may be reached more quickly if the temperature gradient is increased; however, the increase in temperature gradient encourages natural convection. Natural convection is the second major problem en countered. It is the largest single cause of error in thermal conductivity measurements. Natural convection generally cannot be detected with a steady state device. These two problems can be essentially eliminated if a transient apparatus, which can detect natural convec tion, is used. The transient hot wire apparatus used in this ex periment is such a device. This device consists of a fine wire suspended vertically in a cylindrical tube filled with the liquid of interest. The temperature rise in this wire as a function of time can be predicted by solving Fourier's general equation. ~-f - ^ V2T + q(x,y,z,0) (III-2) 49 As shown by Carslaw and Jaeger (13), this equation reduces to q(x,y,z,9), is a step input and introduced into the equa tion as an initial conditions. More details on the solu tion by Carslaw and Jaeger are given in Appendix A. The solution for Equation (IV-1), for this device is T « the temperature rise in the wire T9 - Tc » heat generated per unit time and per unit length r a radial distance from the center of the wire 9 a time a thermal diffusivity of the liquid a k//> c k a thermal conductivity of the liquid p a density of the liquid C a specific heat of the liquid Ei(-x) s the exponential integral. At the boundary of the wire (raR), the function Ei ( ^ ) can be expanded in the following series1 < 5 9 1 or (IV-1) 2 2 where r = x + y. In this case the heat generation term where: 50 T(R»9) + 4 ^ [in 0 + In ^ 7 + + ... - V ] (IV-2) where y * 0.5772 is Euler's constant. It has been demonstrated (31,47) that for small values of R^/4*0, (R^/4®*0, < 0.12 at 0^ > 0.1), and that Equation (IV-2) can be reduced to with only a +0.1% error. A negative departure by the ap paratus from this equation is the means of detecting natural convection. The calculations involved in Equation (IV-3) can be simplified still further by using a relative apparatus such as the ones used by Grassman (27), Malian (47), and the one used in this experiment. This is done by compar ing the temperature rise of a hot wire immersed in a fluid of unknown thermal conductivity to the temperature rise of a hot wire immersed in a liquid of known thermal con ductivity. This simplifies the Equation (IV-3) as follows: By writing Equation (IV-3) for both cells / x \ / 1\ (4'?FTc" ) *n\3 / = C® H with known fluid) X' 2 f *^v \ /®1\ 14TFk"/ ^n(©2/ ^ with unknown fluid) 51 and dividing A Tx fqix\ ^7 ‘ l 5 I 7 ^ Equation (IV-4) is obtained. A Tx ky * kx(constant) ^ ‘ ip (IV-4) qlx In this system the ratio is a constant since the power output, is dependent upon a constant energy sup ply and the cell resistance. The transient-relative hot-wire apparatus used in this work is an excellent analog of the mathematical model described above. Details of the Apparatus Two hot-wire cells whose temperature rise as a function of time are described by Equation (IV-3) and con nected into two similar Wheatstone bridges (one for the known liquid, one for the unknown liquid), as shown in Figure IV-1. In the Wheatstone bridges shown in Figure IV-1, the usual galvanometer which detects bridge unbal ance is replaced by one axis of an x-y recorder. (Record ing the output unbalance of a single bridge as a function of time would result in a logarithmic function as des cribed by Equation (IV-3). The bridge is unbalanced by applying a sudden step input of constant voltage. Figure IV-1 Relative Transient Hot-Wire Thermal Conductivity Apparatus x axis y axis x-Bridge (reference liquid) y-Bridge (unknown liquid) cel; *-y recorder (Power balance) 53 The thin wire in the cell begins to heat up at a rate inversely proportional to rate at which heat is being conducted away or inversely proportional to the thermal conductivity of the liquid surrounding it. The resistance of the wire increases at a rate proportional to the tem perature rise in the wire. The potential difference measured by the x-y recorder is proportional to resistance change in the hot wire. This potential difference appears on the recorder graph paper as Ay or Ax. If two bridges are connected in parallel as shown in Figure IV-1 and power supplied simultaneously, the slope on the recorder A®! graph paper (Ay/Ax) is a measure of 25ri which ATv x equals A tx * value substituted into Equation (IV-4) results in: ky =» kx(constant) (IV-5) If a calibration liquid of known thermal conduc tivity is placed in both cells, the constant in Equation (IV-5) is (Ay/Ax) q. With the calibration liquid always in cell x, Equation (IV-5) can be written ky ** hx (A y/A x) c/ (A y/A x) (IV-6 ) The quantity kx(Ay/Ax)c is defined as a cell constant, C, so that the expression for an unknown thermal conduc tivity becomes 54 ky - C/( Ay/ a x ) (IV-7) The calibration liquid, toluene, has a thermal conductiv ity according to Ziebland (91) of k * 140.7 - Malian (47) studied possible choices for a reference fluid and chose Ziebland's values for toluene over the National Bureau of Standards values (59) because he felt that the N.B.S. incorporated too many questionable sources into the fit of their data. Since this experimental work was performed at 0°C, mW the k-value used for toluene was 140.7 in metric units. (Multiplying by a factor of 5.778 x 10- 4 gives k in units of BTU/hr. ft. °F). The working equipment consisted of one x and three y cells. This made it possible to make measurements on three different samples over a short period of time, one y at a time. All four cells were very similar. They consisted of a six-inch one mil diameter platinum wire fixed in a pyrex tube 3/8 inch o.d. and six-and-one-half inches long. The wire was connected to a Tungsten ele ment at each end of the tube. The cell construction, simi lar to one used by Jamieson (34) is shown, complete with filling arms, in Figure IV-2. All the cells are immersed in a constant tempera ture bath maintained at a constant temperature of 0°C. (Units are milliwatts per meter per °C). 55 Figure IV-2 Thermal Conductivity Cell V 56 The bath, a Lauda Ultra Kryostat model UK-50 SKW Is ca pable of maintaining a constant temperature to +0.03°C. The power supply shown in Figure IV-1 as Ex and Ey is a Hewlett Packard 6227B Dual DC Power Supply, with a range of 0-25 volts and 0-2 amps. The x-y recorder is a Hewlett Packard 7000A model. It was used at a recording range of 0.2 mV/inch. It is capable of ranges from 0.1 mV/inch to 20 V/inch. The rest of Figure IV-1 is physically condensed on to a circuit board along with a microammeter, two preci sion potentiometers and several switches. Operating Procedure All liquids were degassed. This was accomplished by first placing the pure liquids into an open container which was then placed in an ultra-sonic bath for about 2 0 minutes. The container was then closed and brought to a boil by using a vacuum pump. The samples were then weighed on a type P-120 Mettler balance with scale divisions of 10 mg. For each binary system a total of eight samples were prepared con sisting of two pure components and six mixtures of about 15, 30, 45, 55, 70 and 85 weight percent. After the samples were prepared and placed in the conductivity cells, the experimental equipment was 57 readied. The constant temperature bath was turned on and checked to make sure that the temperature was at 0°C. Next, the power supplies were turned on. They were set at 2.8 volts. The 2.8 volt supply produced a good pen movement on the x-y recorder. The 2.8 volt setting was checked externally with a voltmeter. The power supply circuit was isolated and the current checked with a micro ammeter. When the meter read zero, the power supplies had equal voltages. The Wheatstone bridges were balanced by passing a weak current through the system and adjusting the variable resistors (Precision Potentiometers), Ry4 and Rx4* The experiment is carried out in the following steps s (1) The pen on the x-y recorder is placed in the proper position on the graph paper. (2) The switches (Sy and Sx - Figure IV-1) are closed for a period of about eight seconds. The recorder plots a straight line similar to those shown in Figure XV-3. (3) The slopes of these straight lines were measured and used in Equation (XV-7) to determine the unknown thermal conductivity. After waiting three minutes, the switches are 58 Figure IV-3 Actual x-y Recorder Plots 59 closed again and anothex line is recorded. This procedure is repeated at least ten times for each sample in order to establish a 95% confidence interval on each datum point. The procedure is repeated for each of the eight samples which make up the binary system. Figure IV-3 shows that the pen on the x-y recorder actually travels backwards before making the straight line. This is done purposely by actually unbalancing the Wheat stone bridges slightly in the opposite direction. It is done so that the mechanical transients in the recorder have time to dissipate before the pen gets off the page. Before and after each binary system was run, each of the y (unknown) cells were calibrated with toluene. The binary system toluene-carbontetrachloride was measured at 0°C and compared to the same system measured by Jamieson (34), using another method, as a test for the consistency of the apparatus and procedure. The compari son is shown in Figure XV-4. The pure component data ob tained compared closely with that of Jamieson, but with a much smaller interval at the 95% confidence level. This comparison and others are discussed in Chapter V. Based on these comparisons, it is felt that the data obtained were excellent. 60 Figure IV-4 Measured Thermal Conductivity of Carbontetrachloride and Toluene Compared with Jamieson's Data (34) 140 O » Parkinson £ ■ Jamieson (34) 135 130 125 120 115 110 105 : 0 0.2 0.4 0.6 0.8 1.0 x(Weight Fraction Toluene) 61 Exper imental Difficulties The two chief experimental problems encountered weres ( 1 ) variability of the cell resistances, and (2 ) x- y recorder failure. (1) The cell resistances changed slightly over the period of the experiment. This was probably due to the continuous hard use and the method of wrapping the plati num wire onto the Tungsten element (see Figure IV-2). The problem was overcome by continually recalibrating as is demonstrated by the extremely good precision of the ex periment. It should be noted that these cells are ex tremely difficult to fabricate. (2) The x-y recorder failed three times during the experimental period. This was probably because the instrument was being used very near to its sensitivity limit and due to the rapidity of the transient recorded (e.g. 1 second), which caused the continuous jerking of the pen as shown by the loops in Figure IV-3. CHAPTER V EXPERIMENTAL DATA The original experimental measurements of thermal conductivity in this work are for the 28 binary solutions listed in Table 1-1. All measurements were made at 0°C. The values of thermal conductivity reported are relative to Ziebland's (91) value for toluene at 0°C, 140.7 j-jwr. uk The thermal conductivity units used in this work are milliwatts per meter per degree Kelvin. To convert this to BTU/hr ft °F, multiply by 5.778 x 10~4. Fifteen pure components were used in making the 28 binary sets. Table V-l is a list of these pure compo nents, their suppliers and purity (grade). The pure component thermal conductivities of these liquids and their 95% confidence intervals are listed in Table V-2. Pure component thermal conductivities were usually measured at least twice with each binary run. Based upon the following arguments, it is claimed that the average accuracy of all of the data recorded is better than +1.5% and the average precision can be repre sented by a 95% confidence level of +0.20% based on the pure component precision listed in Table V-2 and binary precision listed in Table V-5. 62 T A B L E V -l PURE COMPONENTS USED - SUPPLIERS AND GRADE Compound Supplier Grade 1. Acetone Mallincrodt Chemical Analytical Reagent 2. 1-Butanol Mallincrodt Chemical Analytical Reagent 3. Carbontetrachloride Mallincrodt Chemical Analytical Reagent 4. Cyclopentane Phillips Petroleum Pure (99 mole Z-min.) 5. 2,3-Dlmethylbutane Phillips Petroleum Pure (99 mole Z-min.) 6. n-Heptane Phillips Petroleum Pure (99 mole %-min.) 7. n-Hexane Phillips Petroleum Pure (99 mole %-min.) 8. Methylcyclohexane Phillips Petroleum Pure (99 mole Z-min.) 9. Methylethylketone Mallincrodt Chemical Analytical Reagent 10. n-Octane Phillips Petroleum Pure (99 mole %-min.) 11. Toluene Mallincrodt Chemical Analytical Reagent 12. 2,2,5-Trimethylhexane Phillips Petroleum Pure (99 mole Z-min.) 13. 2,2,4-Trimethylpentane Phillips Petroleum Pure (99 mole Z-min.) 14. 2,4,4-Trimethylpentene-l Phillips Petroleum Pure (99 mole Z-min.) 15. 0-Xylene Matheson, Coleman & Bell (MCB) Fine (B.P. 143.5-144. 64 T A B L E V-2 PURE COMPONENT THERMAL CONDUCTIVITIES AT 0*C Average ,/mW \ 95% Confidence No. of Component Interval Observations 1. Acetone 170.0 (169.6-170.4) 13 2. 1-Butanol 153.6 (153.4-153.8 6 3. Carbontetrachloride 107.1 (106.9-107.3) 6 4. Cyclopentane 140.7 (140.5-140.9) 8 5. 2,3-Dimethylbutane 110.7 (110.4-111.0) 11 6. n-Heptane 132.2 (132.1-132.3) 14 7. n-Hexane 130.1 (129.6-130.6) 9 8. Methylcyclohexane 116.2 (116.0-116.4) 7 9. Me thyle thyIketone 157.1 (156.7-157.5) 7 10. n-Octane 134.5 (134.3-134.7) 12 11. Toluene 140.7 - Ziebland 12. 2,2,5-Trimethylhexane 107.8 (107.6-108.0) 5 13. 2,2,4-Trimethylpentane 103.5 (103.4-103.6) 18 14. 2,4,4-Trimethylpentene-l 106.0 (105.9-106.1) 18 15. 0-Xylene 136.8 (136.6-137.0) 9 65 The precision of the measurements is one of the most important points in evaluating or creating a mixing rule (the relationship between pure component thermal con ductivities and mixture thermal conductivities). If, for example, because of the experimental setup, both pure com ponent thermal conductivity values are high by the same proportion, then presumably, all mixture points are high by the same proportion. If the experimental precision is good, then the data are still valuable. The precision of the pure component thermal conduc tivities of this work and some of those of Jamieson (34) are given in Table V-3. The work of Jamieson was chosen because his work is considered reliable and he supplied data with which to determine the precision and confidence levels. The average 95% confidence interval on the pure component data of this work is +0.19% for 14 different pure components. Jamieson measured several other pure components besides those listed in Table V-3. His average 95% confidence interval was +1.86% for 20 different pure components. This is not a perfectly fair comparison be cause in some cases Jamieson made only two or three mea surements in which cases the confidence intervals are large. (In this comparison one of Jamieson's sets of only two widely varying points was not included because of the extremely large confidence interval.) T A B L E V-3 PRECISION AND COMPARISON WITH JAMIESON (34) Average Thermal Conductivity /mW No. of 952 Standard Confidence Component Investigator Vm*?/ Observations Deviation Interval Range 1. Acetone Parkinson 170.0 13 0.688 40.242 1.29 Acetone Jamieson 171.1 16 1.94 ±0.602 5.03 2. 1-Butanol Parkinson 153.6 6 0.206 ±0.142 0.39 1-Butanol Jamieson 154.5 3 1.92 +3.082 2.27 3. Carbontetrachloride Parkinson 107.1 6 0.195 +0.192 0.47 Carbontetrachloride Jamieson 108.2 11 1.28 ±0.7922 4.16 4. Cyclopentane Parkinson 140.7 8 0.276 ±0.162 0.64 5. 2,3-Dimethylbutane Parkinson 110.7 11 0.396 ±0.242 1.08 6. n-Heptane Parkinson 132.2 14 0.232 ±0.102 0.53 7. n-Hexane Parkinson 130.1 9 0.668 ±0.392 1.38 8. Methylcydohexane Parkinson 116.2 7 0.221 ±0.182 0.52 9. Me thylethyIketone Parkinson 157.1 7 0.484 ±0.282 0.83 a* < r » T A B L E V-3 (con’t . ) Component Investigator Average Thermal Conductivity ( mW \ ' m*K/ No. of Observations Standard Deviation 95% Confidence Interval % Range 10. n-Octane Parkinson 134.5 12 0.296 +0.14% 0.74 11. Toluene Ziebland (91) 140.7 Toluene Jamieson 140.0 * 22 1.37 +0.43% 2.70 12. 2,2,5-Trimethylhexane Parkinson 107.8 5 0.164 +0.19% 0.37 13. 2,2,4-Trimethylpentane Parkinson 103.5 18 0.291 +0.14% 1.06 14. 2,4,4-Trimethylpentene--1 Parkinson 106.0 18 0.279 +0.13% 0.35 15. 0-Xylene Parkinson 136.8 9 0.260 +0.14% 0.66 t The calculated average here was 141.0, but Jamieson used the value of 140.0. <n 68 The 95% confidence interval is calculated by the formula (The term used here for confidence is + *0.975 ^x)» where x is the average thermal conductivity, * £x/n; n is the number of data points. *0.975 is the 95% t value (two-tailed test) obtained from a t table found in standard textbooks on statistics ( 8 8 ). Table V-4 was prepared in an effort to estimate the accuracy of data in this work. It shows a comparison between the pure component thermal conductivities taken here and thermal conductivities measured or tabulated by several other investigators. Comparisons are also made with thermal conductivities predicted by the method of Robbins and Kingrea (69). Reid and Sherwood ( 6 6 ) consider the Robbins and Kingrea method to be the best available today for predicting pure component thermal conductivities. Malian (47), however, has pointed out that the method of Robbins and Kingrea is based heavily upon thermal conduc tivities measured by Sakiadis and Coates (73,74) some of which now appear to be high. The predicted values and some of the apparently i = > x + *0.975 ^x (V-l) (V-2) n(n-l) T A B L E V-4 PURE COMPONENT THERMAL CONDUCTIVITY COMPARISONS 0#C Component 1. Acetone 2. 1-Butanol 3. Carbontetra- chloride 4. Cyclopentane 5. 2,3-Dimethyl- bntane Thermal Conductivity Thermal Conductivity Calculated . mW _ . mW Robbins & This Work, j g Others, Klnijree (69) 170.0 171.1(34), 170.0B(35,36) 180.0 169.9A(35,68)(b) 171.5A(35,48)(b) 169.0(51) [177.0C(26,35), 184.0C(90,95)](a) 153.6 154.5(34), 157.0A(35,68)(b) 156.5 155.0(51), 151.0(37)(d) [181.0C(2,35)](a) 107.1 108.2(34), 108.3A(35,68)(b) 107.1 109.7A(14,35)(b) 106.0(47)(c) 106.5(51), 104.4(80)(d> [111.4C(26,35)](a) 140.7 140.IB(c)(35,74) 135.5 110.7 112.0B(c)(35,74) 116.2 Estimated Relative Accuracy of Data Point +0.44% +1.35% +1.45% +0.42% +1.18% os SO Component 6. n-Heptane Thermal Conductivity This Work, m K 132.2 7. n-Hexane 130.1 8. Methylcyclo- hexane 116.2 9. Methylethyl- ketone 10. n-Octane 11. Toluene 157.1 134.5 140.7 Ziebland (91) T A B L E V-4 (c o n 't.) Thermal Conductivity mW m°K Others, —g- 132.4B(22,35)(c) 129.2(47)(c) 134.0(51), 132.0(37)(d) [140.3B(35,73)](a)(c) 130.5B(22,35)(c) 132.0(47)(c) 128.5(51), 127.0(37)(d) [140.0B(35,73)](a)(c) 102.2B(22,35)(c) 110.4B(12,35)(c) 117.0(47)(c) 157.0(51) [166. 8B (35,74) ] (a) (c) 137.5(51), 138.0(37)(d) [143.4B(35,73)](a)(c) 140.0(34), 140.0A(35,68) (b) 141.1A(35,68Hb) 144.3B(14,35)(b) 145.8B(35,75)(b), 140.5(51) 136.2(80)(d) [146.1C(26,35)](a) Calculated Robbins & Kingrea (69) 140.0 133.0 139.0 193.0 t 147.0 120.0 f Estimated Relative Accuracy of Data Point +0.98% +1.35% +4.23% +0.07% +2.42% +1.51% o T A B L E V-4 (c o n 't.) Thermal Conductivity Thermal Conductivity Component 12. 2,2,5-Trimethyl- hexane 13. 2,2,4-Trimethyl- pentane 14. 2,4,4-Trimethyl- pentene-1 15. 0-Xylene This Work, —^ mW m°K 107.8 103.5 106.0 136.8 Others, mW * m°K [124.4B(35,73)](a)(c) 104.2B(22,35)(c) [109.1B(35,73)](a)(c) 141.0A(35,68)(b), 138.0(51) [144.0C(26,35)](a) Estimated Calculated Relative Robbins & Accuracy of Kingrea (69) Data Point 122. 01 107.9 126.0 * 158.0 * No compari son made. +0. 68% No compari son made. +1.99% Average +1.38% Grading by Jamieson (35) where available. A * accurate to within +2% B » accurate to within +5% C ■ less accurate than +5% + Much of the error in these predicted values is due to uncertainty in the physical properties used for the prediction. (a) ■ values not used in the estimate (b) v values calculated by slope (<*) given in (35), where kg0Q ■ kf c + <*t, where t is the temperature of kf c (35), in °C. (c) ■ extrapolated beyond correlation limits (d) ■ values taken from a graph. H* 72 high values listed in Table V-4, as indicated, were not used in the actual accuracy estimate. The accuracy estimate for each pure component is listed in the right hand column of the table. It was simply calculated by the formula: % accuracy = ±100.0 ( £|k* - ki|)/k* (V-3) * where k is the thermal conductivity value measured in this work and the kj,'s are the thermal conductivities chosen for comparison. The average accuracy for the pure components is ±1.38%. It is interesting to note that in most cases the thermal conductivities measured here were slightly lower than most of those used for comparison. Generally, more recent thermal conductivity values are lower than the older ones, This is probably because the recognition of natural convection errors and the correction for them has increased over the years, causing the value of thermal conductivities to drop. Figure V-l is a plot of the thermal conductivity of toluene at 20°C versus year of measurement taken from Michaelian (50). This figure bears out the above statement and strengthens the case for the good accuracy claim for the data of this work. Table V-5 lists the experimental binary data measured in this work and their 95% confidence intervals. BTU/HR-FT- 73 Figure V-l Thermal Conductivity at 20°C of Liquid Toluene from (50) 0.08R 0.086 0.084. 0.082 O 0.080- i 0.078 0.074 1940 1950 Year of Measurement 1920 1930 1960 1970 74 TABLE V-5 EXPERIMENTAL BINARY DATA Data Set #1 Acetone - n-Heptane 0°C Wt.% 95% Confidence Interval (minimum of 10 measurements for each mixture point) 0 132.2 ±0.1, ±0.101% 14.9 133.0 ±0.2, ±0.237% 29.9 135.8 ±0.2, ±0.176% 44.9 141.6 +0.2, ±0.164% 54.8 144.1 ±0.3, ±0.177% 70.1 151.5 ±0.3, ±0.167% 85.0 158.6 ±0.5, ±0.339% 100.0 (Acetone) 170.0 ±0.4, ±0.244% ita Set #2 Acetone - Methylethylketone 09C 0 157.1 ±0.4, ±0.284% 15.0 159.3 ±0.4, ±0.264% 30.0 160.9 ±0.4, ±0.230% 45.1 162.2 ±0.5, ±0.319% 55.1 163.9 ±0.4, ±0.261% 69.9 165.7 ±0.4, ±0.241% 84.9 167.9 ±0.4, ±0.266% 100.0 (Acetone) 170.0 ±0.4, ±0.244% ita Set #3 Acetone - 2,2,4-Trimethylpentane 0“C 0 103.5 ±0.1, ±0.139% 15,1 107.9 ±0.2, ±0.151% 29.9 114.2 ±0.2, ±0.178% 45.0 122.7 ±0.2, ±0.138% 55.3 130.1 ±0.3, ±0.237% 70.1 140.9 ±0.4, ±0.260% 85.1 153.7 ±0.5, ±0.329% 100.0 (Acetone) 170.0 ±0.4, ±0.244% 75 TABLE V-5 (con't.) Data Set #4 Acetone - 2,4,4-Trimethylpentene-l 0°C 95% Confidence Interval ./mW \ (minimum of 10 measurements Wt.% ” m°K/ for each mixture point) 0 106.0 ±0.1, ±0.130% 14.9 110.3 ±0.3, ±0.255% 30.0 117.6 ±0.3, ±0.223% 44.8 124.7 ±0.4, ±0.321% 55.0 132.0 ±0.4, ±0.280% 69.8 142.0 ±0.3, ±0.233% 84.8 154.9 ±0.3, ±0.188% 100.0 (Acetone) 170.0 ±0.4, ±0.244% ita Set #5 1-Butanol - 2,2,4-Trimethylpentane 0°C 0 103.5 ±0.1, ±0.139% 15.0 108.5 ±0.3, ±0.241% 29.7 113.4 ±0.3, ±0.242% 44.8 119.2 ±0.3, ±0.243% 55.1 124.2 ±0.2, ±0.176% 69.9 132.2 ±0.2, ±0.158% 84.9 142.9 ±0.3, ±0.236% 100.0 (1-Butanol) 153.6 ±0.2, ±0.140% ita Set #6 1-Butanol - 2,4,4-Trimethylpentene-l 0°iC 0 106.0 ±0.1, ±0.130% 14.8 110.2 ±0.2, ±0.224% 30.0 115.3 ±0.3, ±0.254% 44.8 120.8 ±0*3, +0.277% 54.7 125.5 +0.3, +0.254% 69.9 134.6 ±0*3, ±0.220% 85.1 143.6 ±0.4, ±0.277% 100.0 (1-Butanol) 153.6 ±0.2, +0.140% TABLE V-5 (con’t.) Data Set #7 Carbontetrachloride - n-Heptane 0*C 95% Confidence Interval y/mW \ (minimum of 10 measurements Wt.% lm*K/ for each mixture point) 0 107.1 ±0.2, +0.190% 15.0 102.7 ±0.3, ±0.259% 30.2 105.3 ±0.1, ±0.138% 45.0 109.8 ±0.3, ±0.254% 55.0 113.8 ±0.2, ±0.158% 70.0 119.6 ±0.3, ±0.234% 85.3 126.5 ±0.2, ±0.174% 100.0 (n-Hep tane) 132.2 ±0.1, ±0.101% Data Set #8 Carbontetrachloride - Toluene 0“C 0 107.1 ±0.2, ±0.190% 15.2 107.9 ±0.2, ±0.162% 30.3 111.7 ±0.2, ±0.144% 45.0 116.3 ±0.2, ±0.192% 54.6 120.6 ±0.1, ±0.109% 70.4 127.3 ±0.3, ±0.244% 84.5 133.7 ±0.2, ±0.129% 100.0 (Toluene) 140.7 - Data Set #9 Carbontetrachloride - 2,2,4-Trimethylpentane 0* 0 103.5 ±0.1, ±0.139% 15.0 101.4 ±0.2, ±0.189% 29.8 99.7 ±0.2, ±0.170% 44.9 97.9 ±0.2, ±0.153% 55.1 97.7 ±0.2, ±0.153% 70.0 98.0 ±0.2, ±0.156% 85.0 100.2 ±0.2, ±0.161% 100.0 (Carbon tetrachloride) 107.1 ±0.2, ±0.149% 77 TABLE V-5 (con't.) Data Set #10 Cyclopentane - n-Heptane 0aC 95% Confidence Interval 1/mW \ (minimum of 10 measurements Wt.% \m°k/ for each mixture point) 0 132.2 ±0.1, ±0.101% 15.1 133.0 ±0.3, ±0.247% 30.0 133.5 ±0.3, ±0.247% 44.9 134.2 ±0.2, ±0.156% 55.1 135.3 ±0.3, ±0.188% 70.0 136.4 ±0.3, ±0.203% 84.9 137.9 ±0.2, ±0.162% 100.0 (Cyclopentane) 140.7 ±0.2, ±0.164% Data Set #11 Cyclopentane - Methylcyclohexane 0°C 0 116.2 ±0.2, ±0.175% 14.9 119.1 ±0.3, +0.230% 30.0 122.4 ±0.2, ±0.191% 44.9 125.8 ±0.2, ±0.196% 54.9 128.1 ±0.2, ±0.170% 69.8 132.5 ±0.2, ±0.125% 84.8 136.7 ±0.2, ±0.181% 100.0 (Cyclopentane) 140.7 ±0.2, ±0.164% Data Set #12 Cyclopentane - Methylethylketone 0°C 0 140.7 ±0.2, ±0.164% 15.0 140.6 ±0.3, ±0.193% 30.0 141.3 ±0.3, ±0.234% 45.0 143.6 ±0.3, ±0.236% 55.0 145.2 ±0.3, ±0.237% 70.0 148.8 ±0.4, +0.289% 85.0 152.8 ±0.4, ±0.290% 100.0 (Methylethyl- ketone) 157.1 ±0.4, ±0.284% 78 TABLE V-5 (con't.) Data Set #13 2,3-Dimethylbutane - 2,2,4-Trimethylpentane 0"C 95% Confidence Interval Wt.% (minimum of 10 measurements for each mixture point) 0 103.5 +0.1, +0.139% 15.0 104.1 +0.3, ±0.250% 30.1 105.7 ±0.2, ±0.166% 45.2 106.0 +0.3, +0.278% 55.2 107.0 ±0.2, ±0.169% 70.1 107.9 ±0.3, ±0.261% 84.8 109.7 ±0.2, ±0.155% 100.0 (2,3-Dimethyl- butane) 110.7 ±0.3, ±0.240% Data Set #14 2,3-Dimethylbutane - 2,4,4-Trimethylpentene-l 0*C 0 106.0 ±0.1, ±0.130% 15.1 106.6 ±0.2, ±0.154% 30.1 106.8 ±0.2, ±0.195% 45.0 107.9 ±0.2, +0.141% 55.0 108.6 ±0.2, ±0.206% 69.9 109.2 ±0.3, ±0.232% 84.8 111.2 ±0.2, ±0.182% 100.0 (2,3-Dimethy1- butane) 110.7 ±0.3, ±0.240% ita Set #15 n-Heptane - n-Hexane 0°C 0 130.1 ±0.5, +0.394% 15.2 130.4 +0.3, +0.192% 29.9 130.7 +0.3, +0.234% 44.9 131.2 ±0-3, ±0.243% 55.0 131.5 +0.3, +0.248% 70.0 131.5 40.3, ±0.206 85.1 132.4 ±0.4, ±0.274% 100.0 (n-Heptane) 132.2 +0.1, ±0.101% TABLE V-5 (con't.) Data Set if 16 n-Heptane - n-Octane 0#C Wt.% 95% Confidence Interval (minimum of 10 measurements for each mixture point) 0 132.2 +0.1, +0.101% 14.7 133.4 +0.3, ±0.238% 29.7 133.1 +0.3, ±0.248% 44.9 133.5 +0.3, ±0.191% 54.6 133.6 ±0.3, +0.221% 69.6 134.1 ±0.3, ±0.215% 84.9 134.2 ±0.3, ±0.222% 100.0 (n-Octane) 134.5 ±0.2, ±0.139% Data Set #17 n-Heptane - Toluene 0°C 0 132.2 ±0.1, ±0.101% 15.0 131.9 ±0.3, ±0.232% 29.8 132.7 ±0.2, ±0.175% 44.9 133.6 ±0.3, ±0.214% 55.2 134.8 ±0.2, ±0.178% 70.0 135.6 ±0.4, ±0.265% 85.0 138.1 ±0.2, ±0.146% 100.0 (Toluene) 140.7 Data Set #18 n-Heptane - 2,2,4-Trimethylpentane 0°C 0 103.5 ±0.1, ±0.139% 15.0 106.7 ±0.2, ±0.175% 30.0 110.5 ±0.3, +0.241% 45.1 114.8 ±0.2, ±0.147% 55.0 117.2 ±0.2, ±0.147% 70.0 122.5 ±0.2, ±0.173% 85.1 126.4 ±0.3, ±0.242% 100.0 (n-Heptane) 132.2 ±0.1, ±0.101% 80 TABLE V-5 (con't.) Data Set #19 n-Heptane - 2,4,4-Trimethylpentene-l 0*C 95% Confidence Interval wt.% * ( & ) (minimum of 10 measui for each mixture pc 0 106.0 ±0.1, ±0.130% 15.0 109.3 ±0.2, +0.185% 30.0 113.0 ±0.3, ±0.230% 45.0 116.5 ±0.3, +0.227% 55.2 118.8 ±0.3, ±0.233% 70.0 123.1 ±0.3, ±0.247% 85.0 127.5 ±0.3, ±0.250% 100.0 (n-Heptane) 132.2 ±0.1, ±0.101% Data Set #20 n-Hexane - n-Octane 0*C 0 130.1 ±0.5, ±0.394% 15.0 130.6 ±0.3, ±0.259% 30.2 131.3 ±0.2, ±0.178% 45.0 131.7 ±0.3, +0.234% 54.9 132.3 ±0.2, ±0.178% 70.0 133.3 ±0.3, +0.255% 84.8 133.4 ±0.2, +0.185% 100.0 (n-Octane) 134.5 ±0.2, ±0.139% Data Set #21 Methylcyclohexane - 2,2,4-Trimethylpentane 0*C 0 103.5 ±0.1, ±0.139% 15.1 104.9 ±0.2, ±0.151% 30.0 106.4 ±0.2, ±0.203% 44.9 108.1 ±0.2, ±0.203% 54.9 109.4 ±0.2, ±0.227% 70.0 111.5 ±0.2, ±0.136% 84.9 113.7 ±0.2, ±0.151% 100.0 (Methylcydo- hexane) 116.2 ±0.2, ±0.175% 81 TABLE V-5 (con’t.) Data Set #22 Toluene - 2,2,4-Trimethylpentane 0°C 95% Confidence Interval k/mW \ (minimum of 10 measurements Wt.% \m*K/ for each mixture point 0 103.5 +0.1, +0.139% 15.0 107.1 ±0.2, ±0.174% 30.0 111.6 ±0.2, ±0.188% 44.8 115.9 ±0.2, ±0.172% 54.9 119.2 ±0.3, ±0.228% 70.1 125.7 ±0.3, ±0.215% 85.2 132.5 ±0.2, ±0.173% 100.0 (Toluene) 140.7 - Data Set #23 Toluene - 2,2,4-Trimethylpentene-1 0 0 C 0 106.0 ±0.1, ±0.130% 15.1 111.1 ±0.3, ±0.225% 29.8 113.8 ±0.3, ±0.297% 44.9 117.8 ±0.2, ±0.145% 55.2 121.7 ±0.3, ±0.233% 70.0 126.9 ±0.3, ±0.207% 85.0 132.8 ±0.3, ±0.231% 100.0 (Toluene) 140.7 Data Set #24 Toluene - 0-Xylene 0"C 0 136.8 ±0.2, ±0.145% 14.8 136.9 ±0.3, ±0.246% 30.1 137.1 ±0.2, ±0.173% 44.9 137.2 ±0.4, ±0.281% 55.1 137.9 +0.3, ±0.193% 70.1 139.0 ±0.3, ±0.190% 84.7 139.8 ±0.3, ±0.187% 100.0 (Toluene) 140.7 - 82 TABLE V-5 (con’t.) Data Set #25 2,2,5-Trimethylhexane - 2,2,4-Trimethylpentane 0°C 95% Confidence Interval j/mW \ (minimum of 10 measurements Wt.% \m°K/ for each mixture point) 0 103.5 ±0.1, ±0.139% 15.0 104.5 ±0.2, ±0.154% 29.8 104.8 ±0.1, ±0.138% 44.9 105.6 ±0.2, ±0.159% 55.0 106.1 ±0.1, ±0.139% 70.0 106.7 ±0.2, ±0.161% 85.0 107.3 ±0.2, ±0.155% 100.0 (2,2,5-Trimethyl hexane) 107.8 ±0.2, ±0.189% Data Set #26 2,2,4-Trimethylpentane - 2,4,4-Trimethylpentene-l 0°C 0 103.5 ±0.1, ±0.139% 15.1 104.2 ±0.1, ±0.122% 30.3 104.1 ±0.1, ±0.139% 45.0 104.3 ±0.2, ±0.156% 55.1 104.8 ±0.2, ±0.154% 69.9 105.5 ±0.2, ±0.169% 84.9 105.5 ±0.2, ±0.159% 100.0 (2,4,4-Trimethy1- pentene-1) 106.0 ±0.1, ±0.130% Data Set #27 2,2,4-Trimethylpentane - 0-Xylene 0°C 0 103.5 ±0.1, ±0.139% 14.8 107.0 ±0.4, +0.338% 29.9 111.7 +0.2, +0.205% 44.8 115.4 +0.3, +0.278% 55.2 119.2 +0.3, +0.247% 70.1 124.8 +0.2, +0.186% 84.9 130.3 +0.4, +0.276% 100.0 (0-Xylene) 136.8 +0.2, +0.145% 83 TABLE V-5 (con't.) Data Set #28 2,4,4-Trimethylpentene-l - 0-Xylene 0°C wt.% K 5 ) 95% Confidence Interval (minimum of 10 measurements for each mixture point) 0 106.0 ±0.1, ±0.130% 14.9 109.2 +0.2, ±0.222% 29.8 114.0 ±0.3, ±0.257% 45.1 117.3 ±0.3, ±0.262% 54.8 120.5 +0.4, ±0.311% 69.9 125.4 ±0.4, ±0.311% 84,8 131.6 ±0.3, ±0.241% 100.0 (0-Xylene) 136.8 +0.2, +0.145% (Percent confidence values taken before round-off.) 84 Figures V-2 to V-29 are graphical representations of this data showing comparison with the correlation of Chapter VI. Note the scale change in Figures V-2, V-4, V-5, V-6 , V-7 and V-23. If at any time in the future a better value for the thermal conductivity of toluene at 0°C is obtained, the thermal conductivities listed here may be adjusted by the following formulat (mW/m°K) (V-4). X4U • 7 85 Figure V-2 Thermal Conductivity of Acetone and n-Heptane at 0“C 175 O “ data - ” correlation 165 155 k 145 Note scale 135 125 j 0.2 0.4 0 0.6 0.8 1.0 x (Weight Fraction Acetone) 86 Figure V-3 Thermal Conductivity of Acetone and Methylethylketone at 0°C 175i 170h 165 i W 160 155 150 data correlation -li 0.2 0.4 0.6 0.8 1.0 x (Weight Fraction Acetone) 87 Figure V-4 Thermal Conductivity of Acetone and 2,2,4-Trimethylpentane at 0°C 9 0 data 170 correlation 160 150 140 130 Note scale 120 110 100' 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x (Weight Fraction Acetone) 170 165- 160 155 150 145 e) 140 135 130 125 120 115 110 10^ 88 Figure V-5 iductivity of Acetone and 2,4,4-Trimethylpentene-l at 0°C • ■ data - ■ correlation Note scale 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1. x (Weight Fraction Acetone) 89 Figure V-6 Thermal Conductivity of 1-Butanol and 2,2,4-Trimethylpentane at 0°C 155 O * * data 150- correlation 145. 140 135 130 125 120 Note scale 115 105 i 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x(Weight Fraction 1-Butanol) 90 Figure V-7 Thermal Conductivity of 1-Butanol and 2,4,4-Trimethylpentene-l at 0°C 155 O “ data - ■ correlation 150 145 140 135 130 125 Mote scale 120 115 110 105 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x(Weight Fraction 1-Butanol) 91 Figure V-8 Thermal Conductivity of Carbontetrachloride and n-Heptane at 0aC 135 data correlation 130 125 120 t 115 110 105 100 a 0 0.1 0.2 0.3 0,4 0.5 0.6 0.7 0.8 0.9 1.0 x (Weight Fraction n-Heptane) 92 Figure V-9 Thermal Conductivity of Carbontetrachloride and Toluene at 0°C 0 0,1 0.2 0,3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x (Weight Fraction Toluene) 93 Figure V-10 Thermal Conductivity of Carbontetrachloride and 2,2,4-Trimethylpentane at 0°C 115. O ” data - ■ correlation iia; l°5-g:: M 100 90 0.2 0 0.4 0.6 0.8 1.0 x (Weight Fraction Carbontetrachloride) Figure V-ll Thermal Conductivity of Cyclopentane and n-Heptane at 0°C 150 O ” data correlation 145 140 135 130 125 0 0.2 0.4 0.6 0.8 1.0 x(Weight Fraction Cyclopentane) Figure V-12 Thermal Conductivity of Cyclopentane and Methylcyclohexane at 0°C 145 O “ data - ■ « correlation 140 135 : : : 130 125 120 115 0 0.2 0.6 0.8 1.0 x(Weight Fraction Cyclopentane) Figure V-13 Thermal Conductivity of Cyclopentane and Methylethylketone at 0#C 160 O ■ data correlation 155 150 145 140 L 135 t 0 0.2 0.4 0.6 0.8 1.0 :X (Weight Fraction Methylethylketone) 97 Figure V-14 Thermal Conductivity of 2,3-Dimethylbutane and 2,2,4-Trimethylpentane at 0°C 120 115 110 105 100 95 O » data - ■ correlation 0.2 0.4 0.6 0.8 x(Weight Fraction 2,3-Dimethylbutane) 1.0 98 Figure V-15 Thermal Conductivity of 2,3-Dimethylbutane amd 2,4,4-Trimethylpentene-l at 0°C 120 O « ■ data correlation 115 110 105 100 0.6 0.4 0.8 0.2 1.0 0 x (Weight Fraction 2,3-Dimethylbutane) 99 Figure V-16 Thermal Conductivity of n-Heptane and n-Hexane at 0°C 145 O “ data 140 135 130' 125 120 0 0.2 0.4 0.6 0.8 1.0 x (Weight Fraction n-Heptane) 100 Figure V-17 Thermal Conductivity of n-Heptane and n-Octane at 0°C 145 data correlation 140 135 130 125 120 0.2 0.4 0.6 0 0.8 1.0 x (Weight Fraction n-Octane) 101 Figure V-18 Thermal Conductivity of n-Heptane and Toluene at 0°C 150 145 140 k( S ) 135 130 a O *■ data - « correlation 125 0.2 0.4 0.6 0.8 1.0 x(Weight Fraction Toluene) 102 Figure v-19 Thermal Conductivity of n-Heptane and 2,2,4-Trimethylpentane at 0°C 135 O *• data correlation 125 120 115 110 105 100 0.4 0 0.6 0.2 1.0 0.8 x (Weight Fraction n-Heptane) 103 Figure V-2Q Thermal Conductivity of n-Heptane and 2,4,4-Trimethylpentene-l at 0°C 135 O ■ data - “ correlation 130 125 120 ^ 115 110 105 E 100 0 0,2 0.4 0.6 0.8 1.0 x (Weight Fraction n-Heptane) 104 Figure V-21 Thermal Conductivity of n-Hexane and n-Octane at 0°C 145 O * data correlation 140 135 125 120 ® 0 0.2 0.4 0.6 0.8 1.0 x (Weight Fraction n-Octane) 105 Figure V-22 Thermal Conductivity of Methylcyclohexane and 2,2,4-Trimethylpentane at 0°C 120 O “ data - m correlation 115 110 105 100 f 0 0.2 0.4 0.6 0.8 1.0 x(Weight Fraction Methylcyclohexane) 106 Figure V-23 Thermal Conductivity of Toluene and 2,2,4-Trimethylpentane at 0°C 150 O » data correlation 140 130 k 120 Note scale 110 100 0 0.2 0.4 0.6 0.8 1.0 x(Weight Fraction Toluene) 107 Figure V-24 Thermal Conductivity of Toluene and 2,4,4-Trimethylpentnne-l at 0®C O =* data - ■ correlation 140 135 130 125 mW m°] 120 115 110 1051 0 0.6 0.4 0.8 1.0 0.2 x (Weight Fraction Toluene) 108 Figure V-25 Thermal Conductivity of Toluene and 0-Xylene at 0®C 150 145 140 135 P ■ data - ■ correlation 130 0 0.2 0.4 0.6 0.8 1.0 x(Weight Fraction Toluene) 109 Figure V-26 Thermal Conductivity of 2,2,5-Trimethylhexane and 2,2,4-Trimethylpentane at 0°C 120 O « data correlation 115 110 k 105 100 0 0.2 0.4 0.6 0.8 1.0 x (Weight Fraction 2,2,5-Trimethylhexane) 110 Figure V-27 Thermal Conductivity of 2,2,4-Trimethylpentane and 2,4,4-Trimethylpentene-l at 0°C 120 data correlation 115 110 mw 105 100 0 0.2 0.4 0.6 0.8 1.0 x (Weight Fraction 2,4,4-Trimethylpentene-l) Ill Figure V-28 Thermal Conductivity of 2,2,4~Trimethylpentane and 0-Xylene at 0°C O B data 135 correlation 130 125 120 115 110 105 100 0 0,2 0.4 0.6 0.8 1.0 x(Weight Fraction 0-Xylene) 112 Figure V-29 Thermal Conductivity of 2,4,4-Trimethylpentene-l and 0-Xylene at 0°C O m data correlation 135 130 125 120 115 110 105 1.0 0.8 0.6 0.4 0.2 0 x (Height Fraction 0-Xylene) CHAPTER VI DISCUSSION AND CORRELATION The experiment was designed with the following ideas in minds Mixtures of paraffins, olefins, naphthenes and aromatics were measured for three reasonst ( 1 ) thermal conductivities of mixtures of these families are very scarce in the literature; ( 2 ) these mixtures appear more often in industry than mixtures of, say, benzene and methanol; (3) it was hoped that insight could be obtained into deviations from the additive mixing rule caused by only simple non-idealities such as differences in weight, size, shape, etc. Next, mixtures containing ketones were used to determine the effect of non-hydrogen bonding polar com pounds. Then mixtures containing alcohols were studied to determine the effect of strong hydrogen bonding. Fi nally, mixtures containing carbontetrachloride were used because of its high molecular weight and apparent strange behavior in binary mixtures. Table VI-1 lists formulas and abbreviations for the compounds used in this study. This nomenclature is used in order to make the following discussion more compact. 113 TABLE VI-1 ABBREVIATIONS FOR EXPERIMENTAL COMPOUNDS No. Compound Abbreviat: 1. Acetaldehyde ACETAL 2. Acetone ACET 3. Benzene C6H6 4. Bromoform CHBR3 5. 1-Butanol 1-B 6. 2-Butanol 2-B 7. Isobutanol I-B 8. Tertbutanol 3-B 9. IsobutylmethyIketone IBMK 10. Carbontetrachloride CCL4 11. Chlorobencene C6H5CL 12. Chloroform CHCL3 13. Cyclohexane CYC6 14. Cyclopentane CYC5 15. Dichloromethane CH2CL2 16. Die thyIke tone DEK 17. 2,3-Dime thy lbutane IC6 18. Ethanol C2H50H 19. n-Heptane NC7 20. 1-Heptanol 1-HEPT 21, n-Hexane NC6 22. 1-Hexanol 1-HEX 23. Methanol CH30H 24. Methylcyclohexane CTC7 25. Me thyle thyIke tone MEK 26. 2,2,4-Trimethylpentane IC8 27. n-Octane NC8 28. 1-Pentanol 1-PENT 29. 1-Propanol 1-PROP 30. 2-Propanol 2-PROP 31. Toluene TOL 32. 2,2,5-Trimethylhexane IC9 33. 2,4,4-Trimethylpentene-l IC8- 34. 0-Xylene OX 115 Table VI-2 lists the pure components which were combined to form the binary mixtures whose thermal conductivities were studied in this work. Table VI-2 is divided into blocks representing chemical family and size (molecular weight). Binaries to be studied were combinations of blocks in the table. The binary mixtures measured were various combinations of the compounds shown in parentheses. Mixtures measured by other investigators, which were various combinations of these blocks, were the ones chosen for additional study. A group of 59 different binary mixtures, many of which fall into the pattern of Table VI-2 were measured by Jamieson (32,33,34) at 0°C and atmospheric pressure. For this reason 0°C and atmospheric pressure were the con ditions chosen for this experiment. The sum total of 129 binary mixtures listed in Tables 1-1 and 1-2 were the mixtures studied and cor related in this work. Several of the mixtures were mea sured by more than one investigator and several were mea sured by the same investigator at different temperatures. The 28 mixtures in Table 1-1 were selected specif ically to fit within the scheme of Table VI-2. The rest (Table 1-2) are literature data. Much of the data are at 0°C. The great majority are below 30°C but some are at temperatures as high as 75°C. TABLE VI-2 PURE COMPOUNDS FORMING THE MIXTURES STUDIED IN THIS WORK Molecular Weight Range CHEMICAL FAMILY Normal Paraffins Branched Paraffins Olefins Naphthenes Aromatics Aldehydes & Ketones Alcohols MW <60 ACETAL (ACET) CH30H C2H50H (C5) 60<MW <74 (CYC5) (MEK) 1-PROP 2-PROP (C6) 74<MW <88 (NC6) (IC6) CYC 6 C6H6 DEK (1-B) 1-B 2-B 3-B (C7) 88 < MW < 102 (NC7) (CYC7) (TOL) IBMK 1-PENT (C8) 102 < MW <116 (NC8) (IC8) (IC8-) (OX) 1-HEX (C9) 116 < MW <130 (IC9) 1-HEPT Miscellaneous CH2CL2 C6H5CL CHCL3 MW >130 (CCL4) CHBR3 117 Correlation Several theoretical models were tried based on the theories of Chapter III. The models were tested with the 129 data sets. In each case a particular model described some data very well, but the remainder rather poorly. Several of the methods of Chapter II were also used to predict this data. The more theoretical ones did the same thing, a very good job on some of the data and a very poor job on the rest. (These calculations will be discussed in the next section.) It was finally decided that the mechanism of therm al conductivity is too complex to attempt to describe all types of molecular collisions with one theoretical equa tion; therefore, a more empirical approach was taken. In the calculations mentioned above, several obser vations were made. (A) Filippov's (19,20) equation: k =* + x2 ^ 2 " ^l^j^l ~ (II-8 ) did a reasonably good job of predicting the data except for solutions where k^ * k2; e.g., CCL4, k^ =» 107.1 and IC8 , k 2 “ 103.5. (B) Bondi's (9) equation: k = njk^ + n2k 2 “ f-b-n^nj (11-26) 118 did an excellent job on the non-associating mixtures, but a very poor job on mixtures which form hydrogen bonds. Bondi's correlating parameter, f * B f M 2 E jO Mi is rather difficult to obtain for compounds which E° has not tabulated. (See Appendix C.) Therefore, a more easily obtainable parameter was used, which proved to be a good one. The parameter F * ~ ~ is similar to Bondi's f because Bondi correlates pure component thermal conductivities in terms of E° also. Since Equa tion (11-26) predicted experimental data well in the cases of normal liquids and poorly in the cases of associating liquids, it was decided to study the data in terms of Ewell's (18) classification of hydrogen bonding compounds as Blaha (7) did. Ewell's classification as shown in Chapter II is again Class Is Liquids which form three dimensional net works of strong hydrogen bonds} e.g., water, glycols, glycerol, etc. Class 2s Other liquids with molecules which have both active hydrogen atoms and donor atoms (oxygen, nitrogen and fluorine); e.g., alcohols, acids, phenols, etc. Class 3s Liquids with molecules of donor atoms but no active hydrogen atoms; e.g.. 119 ethers, ketones, aldehydes, etc. Class 4s Liquids with molecules containing active hydrogen atoms but no donor atoms; e.g., chloroform, dichloromethane, etc. Class 5s Liquids having no hydrogen bond forming capabilities; e.g., hydrocarbons, carbon- tetrachloride, etc. The 129 data sets used in this study were divided into the following eight groupss Group Is Consists of mixtures of Class 5 molecules with other Class 5 molecules. No hydro gen bonds are formed in this group and the mixture thermal conductivities should depend upon simple structural type dif ferences alone. There are 47 data sets in this group. Group IIs Consists of mixtures of Class 2 molecules with Class 5 molecules. Strong hydrogen bonds are formed by one set of molecules. There are 46 data sets in this group. Group Ills Consists of mixtures of Class 3 molecules with Class 5 molecules. No hydrogen bonds are formed, but one set of molecules is strongly polar. There are 14 sets in this group. 1 2 0 Group IVi Group Vt Group VIt Group VIIi Group VIIIt To study rewritten ast Consists of mixtures of Class 3 molecules with Class 2 molecules. Two types of hydrogen bonds are formed. Also, both sets of molecules are highly polar. There are 11 data sets in this group. Consists of mixtures of Class 3 molecules with Class 4 molecules. Hydrogen bonds are formed between the different mole cules. There are four data sets in this group. Consists of mixtures of Class 3 molecules with other Class 3 molecules. No hydro gen bonds are formed, but both sets of molecules are polar. There is only one data set in this group. Consists of mixtures of Class 5 molecules with Class 4 molecules. No hydrogen bonds are formed. There are five data sets in this group. Consists of mixtures of Class 2 molecules with other Class 2 molecules. Strong hydrogen bonds are formed. There is only one data set in this group. the data, Bondi's equation (11-26) was 121 k « ni*i + n2^2 “ ® nln2 (VI-1) The "least squares" values for B were obtained for each data set. Their values were then plotted as a func- sis, the data of the major groups fell along families of straight lines (see Figures VI-1 to VI-11). B was ob tained for each line in the form: The values of a and b are dependent upon the character istics of the molecules involved, such as structural dif ferences, like size, shape and weight, polarity differ ences and the hydrogen bonding character of the mixture. The results of the correlation and the study are presented below. Group I divided distinctly into three categories (see Figure VI-3). This group was measured primarily to determine the effects of the simple non-idealities caused by structural differences in the molecules. Subdivision I-A is the first division, consisting of chain type molecules mixed with other chain type mole cules. The chain type molecules are the normal paraffins and their branched isomers, normal olefins and their branched isomers and the naphthenes or cycloparaffins. tion of the parameter F 30 rr - —=■ . Under this analy Mi M2 (VI-2) 122 Naphthenes behave like branched paraffins in these mix tures. These mixtures tend to deviate only slightly from the additive or linear mixing rule. These mixtures are probably different from the rest because, as discussed in Chapter III, all of the molecular collisions are segmental rather than one on one-two body collisions. Subdivision I-B is the second division. This divi sion consists of mixtures containing carbontetrachloride, with the exception of carbontetrachloride mixed with aro- matics; e.g., toluene and benzene. Carbontetrachloride- alcohol mixtures from Group II also fit into this sub division, with the exception of mixtures with alcohols lighter than butanol. Carbontetrachloride mixtures from Groups III and V fit well into this subdivision also. Finally, the binary mixture of benzene and bromoform also fits into this subdivision. These mixtures have large deviations from the additive rule and often have minimums. This subdivision is perhaps the most "non-ideal" of all. The only explanation here is that carbontetrachloride is an extremely heavy compact molecule with a rather small average cross sectional area or target area (see Appendix B). The dependence of thermal conductivity on molecular weight is discussed in Chapter III. The carbontetrachlo ride molecule is structurally very different from almost any compound it was mixed with, and hence, these mixtures 123 produce large deviations from linearity. This is in con trast to the mixtures of the structurally similar mole cules of Subdivision I-Af which have small deviations. The average cross sectional area of carbontetrachloride is similar to that of toluene and benzene. This is possibly why these mixtures do not fall into Subdivision I-B. These particular mixtures have a smaller deviation from linearity. Benzene and bromoform also have similar cross sec tional areas but in this case the molecular weights are extremely different. The reason for leaving mixtures containing propanols, ethanol and methanol out of Sub division I-B will be discussed later. This demarcation is not distinct, however. Subdivision I-C is the third division of Group I. This subdivision could be termed the average subdivision. All other Group I mixtures fall into this subdivision, including carbontetrachloride-aromatic mixtures. Also, the mixtures of Group VII fit best here, with the excep tion of the binary chloroform and carbontetrachloride, which fits best into Subdivision I-B. Many of the al cohol mixtures from Group II also fit into Subdivision I-C. So, for convenience. Subdivision I-C will be grouped with these alcohols and called Subdivision II-A. Group II divides distinctly into four families 124 (see Figure VI-8 ). The constant B can be defined by one of four parallel lines. Subdivision II-A consists of all mixtures from Sub division I-C plus all mixtures from Group II containing the alcohols, butanol and heavier, with the exception of the carbontetrachloride mixtures already discussed, which are included in Subdivision I-B. This is by far the largest subdivision and should probably be referred to as the average rule. Apparently the structural differ ences between the molecules in this subdivision which cause the deviations from the additive rule are about the "average" for most mixtures. Surprisingly, the fact that one of the pair of molecules involved in this subdivision may form hydrogen bonds makes little difference to binary mixture thermal conductivities. Apparently, for alcohols of C4 and greater (butanol and heavier), structural dif ferences are more important to mixture thermal conduc tivities than force differences caused by hydrogen bond ing. For C3 alcohols and smaller (propanol and lighter), the story is apparently different. The reason for this may be that, as the alcohol molecular chain gets longer it becomes more like a paraffin and the hydrogen bonding capability per molecule becomes less distinct. On the other hand, as the molecule becomes smaller, the hydrogen bonding capability per molecule becomes greater. As far 125 as the thermal conductivity of binary mixtures is con cerned, hydrogen bonding becomes significant at C3 and lighter. Subdivision IX-B consists of all mixtures from Group II containing propanol. This includes carbontetra chloride mixtures. This apparently is because the hydro gen bonding characteristics of the propanol molecule are more important than the structural differences between the molecules. Subdivision II-C consists of all mixtures from Group II containing ethanol. Subdivision II-D consists of all mixtures from Group II containing methanol. The one mixtures from Group VIII, methanol and 2-propanol, fits best in this group. The four subdivisions of Group II may be character ised by the hydrogen bonding capability per molecule as the alcohol molecule goes from C4 and heavier, to Cl. It is also apparent that as this happens, in general, the difference in molecular weight between the binary pairs becomes larger; therefore, the differences in this group may be attributed to both hydrogen bonding, or force dif ferences, and structural differences. Group III divides into two families. These sub divisions again form two parallel lines. The mixtures of this group consist of polar molecules and non-polar mole- 126 \ cules. No hydrogen bonds are formed. The molecules of this group attract and repel one another in a different manner than the molecules of the other groups; therefore, the mixture thermal conductivity is different. Subdivision III-A consists of mixtures of chain type molecules, as discussed above, with spherical type or other molecules from Group III. Evidently there are some segmental collisions and some of the two-body type. Subdivision III-B consists of all other mixtures in Group III except those containing carbontetrachloride, which are in Subdivision I-B. The one set from Group VI, acetone-methylethylketone, fits best into this subdivi sion. Group IV also divides into two families. These mixtures can form two types of hydrogen bonds and both sets of molecules are polar. Obviously, the thermal con ductivities of these mixtures will be different than other groups. Subdivision IV-A includes all mixtures in Group IV except those containing methanol. Subdivision IV-B includes mixtures in Group IV con taining methanol. There is only one data set to support this subdivision, but based on patterns formed in Groups II and III, a line parallel to Equation (VI-7) is drawn through this point (see Figure VI-10)• 127 Group VII has no subdivisions, possibly because there are only four data sets in this group. This group forms hydrogen bonds which are not present in either pure component liquid. It is reasonable then, to expect these mixtures to behave differently than others when analyzed in terms of pure component thermal conductivities; i.e., *1 ^2 I F =* —=■ - —s- . Mx M 2 I Table VI-3 lists the data sets used in each sub division. Figures VI-1 to VI-11 are plots of B vs. F * ^1 ^2 I ~ - — I for the various groups and subdivisions in 1 2 Table VI-3. Use of the Correlation Table VI-4 lists the equations to use for the pre viously described groups and subdivisions in order to find B for Equation (VI-1) to predict the thermal conduc tivity of the particular binary mixture. k * + n2lc2 ” B nln2 (VI-1) The n*s in Equation (VI-1) are ordinarily mole fractions. As mentioned in Chapter II, Bondi suggested using volume fractions in his equation when one molecule had a skeletal chain length of seven carbons or greater. For Equation (VI-1), a good rule is to use volume TABLE VI-3 DATA SETS LISTED BY SUBDIVISION GROUP I Subdivision I-A Subdivision I-B 10. CYC5 & NC7 7. CCL4 & NC7 11. CYC5 & CYC7 9. CCL4 & IC8 13. IC6 & IC8 50. C6H6 & CHBR3 14. IC6 & IC8- 94. CCL4 & CYC6 15. NC7 & NC6 95. CCL4 & CYC6 16. NC7 & NC8 96. CCL4 & CYC6 18. NC7 & IC8 84. 1-B & CCL4 (II)* 19. NC7 & IC8- 86. I-B & CCL4 (II) 20. NC6 & NC8 87. I-B & CCL4 (II) 21. CYC7 & IC8 88. 2-B & CCL4 (II) 25. IC9 & IC8 90. 3-B & CCL4 (II) 26. IC8 & IC8- 98. CCL4 & 1-HEPT (II) 126. NC8 & IC8 99. CCL4 & 1-HEX (II) 103. CCL4 & 1-PENT (II) 38. ACET & CCL4 (III) 39. ACET & CCL4 (III) 93. CCL4 & CHCL3 (VII) Numeralin parenthesis represents the group in which the mixture would normally belong. 129 TABLE VI-3 (con't.) GROUP II Subdivision II-A Subdivision II-B 5. 1-B f i . 1C8 6. 1-B f i . IC8- 51. 1-B f i > C6H6 85. 1-B f i . TOL 89. 2-B f i i TOL 91. 3-B f i . TOL 121. 1-HEPT f i . TOL 122. 1-HEX i S t TOL 127. 1-PENT & TOL 8. CCL4 & TOL (I) 17. NC7 f i < TOL (I) 22. TOL f i , IC8 (I) 23. TOL f i . IC8- (I) 24. TOL f i . OX (I) 27. IC8 & OX(I) 28. IC8- & OX (I) 52. C6H6 f i eCCL4 (I) 53. C6H6 & CCL4 (I) 54. C6H6 & CCL4 (I) 55. C6H6 & CCL4 (I) 56. C6H6 & CCL4 (I) 57. C6H6 & CCL4 (I) 58. C6H6 & CCL4 (I) 61. C6H6 & CTC6 (I) 77. C6H6 & TOL (I) 78. C6H6 & TOL (I) 79. C6H6 & TOL (I) 80. C6H6 & TOL (I) 81. C6H6 f i tTOL (I) 82. C6H6 & TOL (I) 83. C6H6 & TOL (I) 92. CCL4 & C6H5CL (I) 107. CCL4 & TOL (I) 108. CCL4 f i tTOL (I) 109. CCL4 & TOL (I) 110. CCL4 & TOL (I) 111. CCL4 & TOL (I) 59. C6H6 & CHCL3 (VII) 60. C6H6 f i tCHCL3 (VII) 62. C6H6 f i tCH2CL2 (VII) 116. CHCL3 ( S t TOL (VII) 75. C6H6 & 1-PROP 76. C6H6 & 2-PROP 104. CCL4 f i t1-PROP 105. CCL4 & 2-PROP 106. CCL4 f i t2-PROP 118. CYC6 f i t1-PROP 119. CYC 6 f i t2-PROP 128. 1-PROP f i t TOL 129. 2-PROP f i t TOL Subdivision II-C 63. C6H6 & C2H50H 64. C6H6 f i tC2H50H 65. C6H6 f i tC2H50H 66. C6H6 & C2H50H 67. C6H6 f i tC2H50H 68. C6H6 & C2H50H 97. CCL4 f i tC2H50H 117. CYC6 & C2H50H Subdivision II-D 69. C6H6 & CH30H 70. C6H6 & CH30H 71. C6H6 & CH30H 72. C6H6 f i t CH30H 73. C6H6 & CH30H 74. C6H6 f i t CH30H 100. CCL4 f i t CH30H 101. CCL4 f i . CH30H 102. CCL4 f i . CH30H 112. C6H5CL f i . CH30H 124. CH30H f i t TOL 125. CH30H f i t TOL 123. CH30H & 2-PROP (VIII) 130 TABLE VI-3 (con't.) GROUP III Subdivision III-A Subdivision 1II-B 1. ACET & NC7 24. ACETAL & TOL 3. ACET & IC8 30. ACET & C6H6 4. ACET & IC8- 31. ACET & C6H6 12. MEK & CYC5 32. ACET & C6H6 47. ACET & TOL 48. ACET & TOL 49. ACET & TOL 120. MEK & TOL 2. ACET & MEK (VI) GROUP IV Subdivision IV-A Subdivision IV-B 44. ACET & CH30H 33. ACET & 1-B 34. ACET & I-B 35. ACET & I-B 36. ACET & 2-B 37. ACET & 3-B 41. ACET & C2H50H 42. ACET & 1-HEPT 43. ACET & 1-HEX 45. ACET & 1-PENT 46. ACET & 2-PROP GROUP V 40. ACET & CHCL3 113. DEK & CHCL3 114. IBMK & CHCL3 115. MEK & CHCL3 131 Figure VI-1 B for Subdivision I-A 20 T 18 I8 9 (B) 1260 190 O 1 0 2 1 0 013 O 20 015 -2 0.25 0.50 0.75 0 1.0 132 Figure VI-2 B for Subdivision I-B Group I Group II Group III Group VII 100 38# •39 50# 1031 (B) 70 O 90 ■ 93 2.5 2.0 1.5 0.5 1.0 0 133 Figure VI-3 -B-for Group -I 100 50 (B) 0 0,5 0.25 0.75 1.0 1.25 F - ^ 2 MX " M2 134 Figure VI-4 B for Subdivision II-A or I-C 6 * Group II 100 Group I Group VII 55 50 (B) 116 jp .07 111 O 92 O 22 '^§ 2'27 *>5 & 024 ®77 _______ 0.25 0 0.50 0.75 1.0 1.25 135 Figure VI-5 B for Subdivision II-B 100 90 105 0106 (B) 1180 40 119 76 0 128 0129 1.5 2.0 1.0 2.5 0.5 0 Figure VI-6 B for Subdivision II-C 136 100 90 50 (B) 117© 1.0 1.5 2.0 2.5 3.0 ^1 *2 M i ^ 137 Figure VI-7 B for Subdivision II-D O » Group II Group VIII 200 180 O 101 168 140 120 oioo 100 (B) 102< 1250 O 112 0124 0 73 • 69 4.0 4.5 5.0 5.5 6.0 138 Figure VI-8 B for Group II 100 (B) ■s o 3.0 1.0 4.0 2.0 5.0 Figure VI-9 B for Group III 139 | 0 ■ Group III O ■ Group VI 140 Figure VI-10 B for Group IV 1 0 0r (B) 3.0 4.0 1.0 2.0 5.0 k- Figure VI-11 B for Group V 141 100 0 40 115 (B) 40 O 113 11 0 0,5 1,0 1.5 2.0 2.5 kl _ £ 2 Mi ft. TABLE VI-4 EQUATIONS FOR B 142 GROUP I Subdivision I-A Chain type molecules with other chain type molecules B - -1.961 + 17.615|^ - ^-1 (VI-3) 1 21 Subdivision I-B Carbontetrachloride mixtures, except with aromatics and alcohols lighter than butanol B - 19.329 + 34.51l|^ - H?. (VI-4) lM! 2 GROUP II I kl k2 I B - a + 47.324 TT-- 77- (VI-5) 1 2 Subdivision II-A All Group I mixtures not in I-A or I-B, all alcohol mixtures containing butanol and heavier except those containing carbontetrachloride a - -3.957 Subdivision II-B Propanol mixtures a - -15.978 Subdivision II-C Ethanol mixtures a - -67.375 Subdivision II-D Methanol mixtures a - -177.221 143 TABLE VI-4 (con’t.) GROUP III |kl k2 I B - a + 9.444 tt-- tt- (VI-6) 1 1 2 ' Subdivision III-A Group III mixtures containing chain type molecules a ■ 14.814 Subdivision III-B All other Group III mixtures a ■ 3.547 GROUP IV kl k2 B m a + 19.888 Mi M2 (VI-7) Subdivision IV-A Group IV mixtures not containing methanol a - -2.195 Subdivision IV-B Group IV mixtures containing methanol a - -42.697 GROUP V I kl k2f B - 17.054 + 25.439|^ - (VI-8) 144 fractions instead of mole fractions when the cross sec tional area ratio becomes greater than 1.5. This is apparently the approximate point where the size or volume of the molecule becomes more important than just the number of molecules. This is possibly the point where multi-body collisions occur or segmental collisions occur. (See Appendix B for Losenicky's cross sectional areas.) If cross sectional area data are not available, a fair rule of thumb is to use molar volume fractions when molar volume ratios are greater than about 1 .8 . Comparison of Methods The correlation of this work was compared with the correlations of the following investigators based on the 129 data sets listed in Chapter I. (A (B (C (D (E (F (G (H (I Jordan-Coates (38), Equation (II-7) Filippov (19,20), Equation (II-8 ) Blaha (7), Equation (11-10) Jamieson (33,34), Equation (11-11) Dul'nev (16), Equation (11-12) Rastorguev (65), Equations (I1- t14), (11-15), and (11-16) Bondi (9), Equation (11-26) Losenicky (46), Equation (11-27) The "linear rule" or straight line method. 145 The method of Pceston (61) was not compared because the properties needed were not readily available. Three statistics were chosen as a means of com paring the various correlations when they were used to predict the existing data. First, the average percent relative deviations km is the measured mixture thermal conductivity at a given composition; kp is the predicted thermal conductivity; N is the number of data points in the particular data set. This value was calculated for each of the 129 data sets. The average value for all 129 sets was then computed, represented by % I . These values were compared for each method and listed in Table VI-6 . The other two statistics used in the comparison were from Popovics' (57) statistical method for determin ing the goodness of fit. They are the fit coefficient, F, and the relative standard error of fit, RSEF. These statistics are defined in Table VI-5. The average fit coefficient, F, and average relative standard error of fit, RSEF, for the entire 129 data sets are listed in Table VI-6 . Popovics' method compares the points (x,y) generated by the measured and predicted values (measured. (IV-9) 146 TABLE VI-5 POPOVICS' STATISTICS k j j j « * measured mixture thermal conductivity kp ■ predicted mixture thermal conductivity N » number of points (k^.kp) per set k ■ value of the thermal conductivity on the line of equality kp - k,„(y - x) fcl ■ least squares estimated value of the thermal conductivity from the line k*- - a + bkm =» average value of ■ ^k^N kp ■ average value of kp * 2kp/N SL^ ■ variance of the kl points of the regression line about the line of equality ■ 1/N 2(k^ - k)2 Sy2 - variance of the points kp about the mean - 1/N S(kp - kp)2 S . 2 = variance of the calculated points about the line of equality - 1/N £ (k p- k^)2 SEF « standard error of fit « Jsp/ X2 RSEF ■ relative standard error of fit ■ SEF/kp F » fit coefficient ■ Jl - Sp/X2/(SL2 + Sy2) b » regression coefficient ■ ^kmEkp - (£km ) 2 a « regression coefficient « = » - bkjj 147 TABLE VI-6 AVERAGE STATISTICS FOR THE VARIOUS CORRELATIONS Author RSEF F %s 1. Parkinson 0.01321 (1) 0.91963 (1) 1.139 (1) 2. Jamieson 0.01800 (2) 0.87976 (2) 1.598 (2) 3. Filippov 0.01999 (3) 0.86758 (3) 1.754 (3) 4. Jordan-Coates 0.02109 (4) 0.84065 (4) 1.954 (4) 5. Losenicky 0.02565 (5) 0.81944 (5) 2.433 (5) 6. Linear mixing rule 0.02821 (6) 0.79568 (6) 3.036 (7) 7. Blaha 0.03670 (7) 0.74879 (10) 2.893 (6) 8. Rastorguev 0.03651 (8) 0.75914 (9) 3.755 (8) 9. Dul'nev 0.03670 (9) 0.76122 (8) 3.757 (9) 10. Bondi 0.04478 (10) 0.79291 (7) 3.943 (10) Note: The number in parentheses is the relative ranking according to the statistic. 148 km 9 x* predicted, kp * y ) with the line y * x. The correlation of this work was designed to fit all of the data almost equally. The other correlations were not; therefore, a few comments about their strengths and weaknesses relative to this data are in order. Jamieson's equation (11-11) does exceptionally well for a simple equation except in the cases of positive deviations (some of the hydrocarbon mixtures) and large minimums (usually carbontetrachloride mixtures, some chloroform mixtures and some propanol mixtures). Filippov' 8 equation (II-8 ) is acceptable for a simple equation. Again, it does not predict minimums or positive deviations. The Jordan-Coates equation (II-7) is a little more complicated but does a fair job, with the following ex ceptions: again, the thermal conductivity of mixtures with large minimums is not predicted well. Mixtures with pure component thermal conductivities which are fairly close to one another are generally predicted to deviate negatively from linearity by far too much. Several ex amples of this type appear in mixtures of Subdivision I-A. Losenicky's equation (11-27) is more difficult to use. The molecular cross sections which are needed for this equation can be obtained with some difficulty if 149 they have not previously been computed (see Appendix B). The equation generally "under-predicts" 7 that is, it pre dicts smaller than measured deviations from linearity. It does a fine job on mixtures with large differences in pure component thermal conductivities and large differ ences in cross sectional areas. If the cross sectional area of the component with the highest thermal conductivity is the largest, this equation will predict a positive deviation from linearity. This is the case with many of the Subdivision I-A mixtures which do, indeed, have small positive deviations. Un fortunately, several mixtures which show negative devia tions also have this property and, therefore, the mixture thermal conductivity is predicted in the wrong direction. With a meager knowledge of hydrogen bonding, one finds Blaha's equation (11-10) easy to use. This equa tion predicts the thermal conductivity of many solutions quite well. It is capable of predicting minimums and does it correctly in several cases. Unfortunately, it very often predicts minimums where they don't exist. If the pure component thermal conductivities are relatively close together, this equation predicts a large negative deviation, usually much larger than exists. Rastorguev's equations (11-14,11*15,11-16) are easy to use but generally predict larger positive deviations 150 from linearity for solutions containing carbontetrachloride or chloroform when in reality large negative deviations and even minimums are common for these mixtures. Dul'nev's equation (II-12) is difficult to use be cause one must "iterate" in order to evaluate Z from Equa tion (11-13) to use in Equation (11-12). This equation does a fine job on nearly linear mixtures. As with Rastorguev's equation, this equation also predicts large positive deviations for many of the solutions containing carbontetrachloride or chloroform. It does rather poorly on other mixtures which exhibit minimums. Bondi's equation (11-26) is difficult to use if the parameter E° is unknown. See Appendix C for more in formation on this parameter. This equation was intended for normal solutions only. (It predicts the conductivity of these solutions very well.) It usually predicts mini mums where they exist. Unfortunately, it does a very poor job on associating solution. It predicts large deviations and minimums for solutions containing alcohols. These deviations are generally not that large. It should be noted that each correlation does a good job on at least some of the mixtures, possibly the ones used to create the particular correlation. The correlation of this work takes the best fea tures of some of these correlations to do a better overall job on the group of mixtures considered. NOMENCLATURE A Area, cross sectional area a Nearest neighbor distance or a constant B; Correlation parameter or a constant, cross section al area b A constant C Specific heat or a constant, cross sectional area Cp Constant pressure heat capacity Cv Constant volume heat capacity D Diffusivity 0 Average cross sectional area E° Standard energy of vaporization F Interaction force, or correlation parameter f Correlation parameter H Hindrance factor K Boltzman's constant k Thermal conductivity L Length or distance between molecule surfaces 1 Distance term or confidence interval M Mole weight m Molecular mass N Avogadro's number or a constant n Mole fraction, sometimes volume fraction, number of molecules P Probability or a constant 152 153 Q Heat content per mole q Thermal energy R Electrical resistance, distance term, or a constant r Distance term T Temperature U8 Velocity of sound u Internal energy V Volume per mole x Weight fraction or distance y Distance Z Coordination number z Distance <x Thermal diffusivity or a constant S Distance between molecular centers e Energy term in Lennard-Jones function 0 Potential energy function 0 Time p Density cr Point where 0 ** 0 v Vibrational frequency REFERENCES 1. 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H., "The Thermal Conductivity of Binary Liquid Mixtures Experimental Data at 0°C," Nat. Enqr. Lab. Reot. No. 370. Dept, of Sci. and Ind. Research, Great Britain (1968). 157 35. Jamieson, D. T. and Tudhope, J. S., "The Thermal Con ductivity of Liquidsi A Survey to 1963," Nat. Enqr. Lab. Rent. No. 137. Dept, of Sci. and Ind. Research, Great Britain (1964). 36. Jamieson, D. T. and Tduhope, J. S., "A Simple Device for Measuring the Thermal Conductivity of Liquids with Moderate Accuracy," Nat. Enqr. Lab. Reot. No. 81. Dept, of Sci. and 2nd. Research, Great Britain (1963). (Original not seen.) 37. Jobst, W., "Measurement of Thermal Conductivities or Organic Aliphatic Liquids by An Absolute Unsteady- state Method," Int. J. Heat Mass Trans.. 7, 725 (1964). 38. Jordan, H. B., Prediction of Thermal Conductivity of Miscible Binary Liquid Mixtures from Pure Component Values. M.S. Thesis, Louisiana State University (1961). (Original not seen.) 39. 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Phvs. Chem.. 72. 4308 (1968). 47. Malian, G. M., Thermal Conductivity of Liquids. Ph.D. Dissertation, University of Southern California (1968). (Malian, G. M., Michaelian, M. S. and Lock hart, F. J., "Liquid Thermal Conductivities of Organic Compounds and Petroleum Fractions,” J. Chem. Eng. Data. 17, 412 (1972)). 48. Mason, H. L., "Thermal Conductivity of Some Industri al Liquids from 0 to 100 Degrees C," Trans. A.S.M.E.. 74, 817 (1954). 49. McLaughlin, E., "The Thermal Conductivity of Liquids and Dense Gases," Chem. Rev.. 64. 389 (1964). 50. Michaelian, M. S., Liquid Thermal Conductivities. M.S. Thesis, University of Southern California (1968). (Malian, G. M., Michaelian, M. S. and Lock hart, F. J., "Liquid Thermal Conductivities of Organic Compounds and Petroleum Fractions," J. Chem. Eng. Data. 17, 412 (1972)). 51. Missenard, A., "Recherches sur la Conductivite des Liquides et des Gaz Organiques," Cahiers de la Ther- mioue. Ser. C. (March 1971). 52. Morrison, R. T. and Boyd, R. N., Organic Chemistry. Allyn and Bacon, Inc., Boston (1960). 53. Mukhamadzyanov, G. Kl., Usmanov, A. G. and Tarzimanov, A. A., "Thermal Conductivity Measure ments of Organic Liquids and Their Mixtures," Izv. vvssch. ucheb. Zaved. Neft i Gaz.. 10. 70 (1964). (Or iginal not seen.) 54. Pachaiyappan, V., Ibrahim, S. H. and Kuloor, N. R., "A New Correlation for Thermal Conductivity," Chem. Eng.. 74, 140 (1967). 55. Palmer, G., "Thermal Conductivity of Liquids," Ind. Eng. Chem.. 40. 89 (1948). 56. Pauling, L., The Chemical Bond. Cornell University Press, Ithaca, N. Y. (1967). 159 57. Popovics, S., "A Method for Evaluating How Well Ob served Data Fit the Line y-x," Matl. Res, and Stand. ASTM. 2* 1 9 5 (1967). 58. Powell, R. S., Roseveare, W. E. and Eyring, H., “Diffusion, Thermal Conductivity and Viscous Flow of Liquids," Ind. Eng. Chem.. 33, 430 (1941). 59. Powell, R. W., Ho, C. Y. and Tiley, P. E., Thermal Conductivity of Selected Liquids. NBSDS-NBS-8 , Nov. 25, 1966, U. S. Govt. 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K., The Properties of Gases and Liquids. 2nd ed., McGraw-Hill Book Co., New York (1966). 67. Riedel, L., "Thermal Conductivity of Liquids," Mitt. Kaltetech. Inst. Karlsruhe. 2 (1948). (Original not seen.) 6 8 . Riedel, L., "New Measurements of Thermal Conductivity of Organic Liquids," Chem. Eng. Tech.. 21. 340 (1949) (Original not seen.) 160 69. Robbins, L. A. and Kingrea, C. L., "Estimation of Thermal Conductivity of Organic Liquids Over Useful Temperature Ranges," Proc. A.P.I.. 42. 52 (1962). 70. Rodriguez, H. V., Molecular Field Relationships to Liquid Viscosity. Compressibility and Prediction of Binary Liquid Mixtures. Ph.D. Dissertation, Louisiana State University (1962). 71. Rowlinson, J. S., Liquids and Liquid Mixtures. 2nd ed., Butterworths, London (1969). 72. Ruel, M. J. M., "A Simple Correlation for the Conduc tivity of n-Alkanes," Chem. Eng. (London) 239. 194 (1970). 73. Sakiadis, B. C. and Coates, J., "Studies of Thermal Conductivity of Liquids, Parts 1 and II," A.I.Ch.E. J. ±, 275 (1955). 74. Sakiadis, B. C. and Coates, J., "Studies of Thermal Conductivity of Liquids, Part III," A.I.Ch.E. J.. 3. 121 (1957). 75. Shrock, V. E. and Starkman, E. S., "Spherical Appara tus for Measuring the Thermal Conductivity of Liquids," Rev. Sci. Instrum.. 29. 625 (1956). 76. Shroff, G. H., "Measurement of Thermal Energy Trans port in Non-Ideal Liquid Systems," Proc. 8 th Confer ence Thermal Conductivity. 643 (1969). 77. Smith, J. F. D., "The Thermal Conductivity of Liquids," Trans. A.S.M.E.. 58, 719 (1936). 78. Timmermans, J., The Physico-Chemical Constants of Binary Systems in Concentrated Solutions. Vol. 1. Interscience Publishers, Inc., New York (1959). 79. Timmermans, J., The Physico-Chemical Constants of Binary Systems in Concentrated Solutions. Vol. 2. Interscience Publishers, Inc., New York (1959). 80. Tree, D. R. and Leidenfrost, W., "Thermal Conductiv ity Measurements of Liquid Toluene and Carbontetra chloride, " Proc. 8 th Conference Thermal Conductivity. 611 (1969). 161 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. Tsederberg, N. V., Thermal Conductivities of Gaaes and Liauida. M.I.T. Press, Cambridge, Mass. (1965). Tsederberg, N. V., "Thermal Conductivity of Binary Solutions of Benzene and Ethyl Alcohol," Nauch Dokladv Vvssh. Shkol Eneraetika. 4_, 189 (1958). (Original not seen.) Tye, R. P., Ed., Thermal Conductivity, Vol. 2. ■ Academic Press, New York (1969). Tyrrell, H. J. V., Diffusion and Heat Flow in Liquids. Butterworths, London (1961). Venart, J. E. S., "The Thermal Conductivity of Binary Organic Liquid Mixtures," 4th Svmo. Thermoohvsical Properties. Univ. of Delaware (1968). Viswanath, D. W., "On Thermal Conductivity of Liquids," A.I.Ch.E. J.. 13, 850 (1967). Viswanath, D. S. and Rao, M. B., "Thermal Conductiv ity of Liquids and Its Temperature Dependence," J. Phvs.. D 3., 1444 (1970). Volk, W., Applied Statistics for Engineers. 2nd ed., McGraw-Hill Book Co., New York (1969). Weber, H. P., Annalen Phvs. and Chem.. 10. 103 (1880). (Original not seen.) Weber, H. F., "Investigations of the Thermal Con ductivity of Fluids," Wied Ann.. 10. 103 (1880). (Original not seen.) Ziebland, H., "The Thermal Conductivity of Toluene, New Determinations and an Appraisal of Recent Ex perimental Work," Int. J. Heat Mass Trans.. 2, 273 (1961). APPENDICES 162 APPENDIX A DETAILS OF THE SOLUTION OF FOURIER'S EQUATION Much of the following derivation is given by Car- slaw and Jaeger (13) but with considerably less detail. The problem of interest is the solution of Fourier's unsteady state equation given in Chapter IV. It can be demonstrated that is a solution to (IV-1) for an instantaneous line heat source at r = 0 along the z axis of infinite length. The mathematicl approach is to try solution (A-l) in Equation (IV-1). If it works and satisfies the ap propriate initial and boundary conditions, it is a tractable solution. The conditions for this problem are: Initial condition: r > 0, 0*0, T » 0 boundary condition 1 : 9>0, r -*■<», T * 0 boundary condition 2 : (A-l) 0 > 0 , r » 0 , * 0 a r 163 164 The solution procedure is straightforward as follows: JT . qiexp(“ 4 ^ 9 ) / r2 _ A a 9 4F«< 02 V4<x 0 J a T _ _ qlr r2 \ T7= ~-T- T p ( KT§), a 2t qiexP f 4^ 9 ) / r2 _ ,\ 9 r2 8 F*C202 2<< 9 ' a t _ / a2T 1. a t\ qiexp(- 4 «q) , r 2 _ > . <90° l^r2 r ar/“ 477'o<©2 140(6 " 7 = qlexp 4^ 0 ) ( r2 , A 8 F<* ©2 ^2<*0 V = qlexp (~ T y ) ( r2 . A 4 7ToC <»2 \ 4 o C 6 / This shows that (A-l) is a solution to (IV-1). To test the initial condition, take the limit of T as 0—* ■ 0. This results in the indeterminate form T * q^/(4F< = * 0* ®° ). Therefore, L'Hospital’s rule must be used. Let q^0~^ be the top portion of T and 4 Fo< eXp ^ be the bottom. Taking the derivative of the top and the bottom, the following form is obtained: qi®“2 ?l ~— * -— , which in the 2 \ ~-2___/ r2 \ ' 165 limit as 6 approaches zero also approaches zero. The first boundary condition is easily satisfied* lim T . lim (— -----Jt) - 0 r-+oo x-+<x> 42T<*0 exp(~-gj To test the second boundary condition, take the derivative of T with respect to r. “ <11 & ---- * Avn l _ _ , 9) /© T\ m ( - <U \ r ^ r /q ^ 8 exp/ . r2 S~ '4o< 0 / ^ T -j-j’ = 0 when r = 0 and the second boundary condi tion is satisfied. Since the input in the problem is a step input rather than a pulse input, the solution (A>1) must be integrated with respect to time. Using the dummy vari able © - 0* to replace 0, (A-l) is integrated from 0 = 0 to © = © resulting in Equation (A-2). 0 T " ^ ^ 4^ ^0-01)) 0?0l (A"2) Finally, substituting u = r^/4c*(0 “ 6 ^)» Equation (A-3) is obtained. oo T - 4TFST f exp(-u) ~ (A-3) I r2 4 0 166 or I_2k. Eir^i) 4 7T<< El\4o<e' CO C exp(-u) -Ei(-x) « \ — " 1 1 1 ■ du is the exponential x integral which is a commonly occurring function in mathe matics and is tabulated in several Handbooks (1). The graph of the function Ei(-x) is plotted in Figure (A-l). The series expansion for the function is: oo Ei(-x) a» y + lnx + ^ l— i nn: n*l where # is Euler's constant equal to 0.57721 ... 167 Figure A-l The Function y - EiC-x) x 0 0.2 0.4 0.6 0.8 1.0 Ei(-x) - 1.0 - 2.0 APPENDIX B METHOD FOR CALCULATING AVERAGE CROSS-SECTIONAL AREAS Example: Acetone 0 Bond lengths Packing radii II c - 0, 1.24} c, 1.651 u \ /_Hh C - C, 1.541 0, 1.40A /C C\ C - H, 1.071 H, 1.151 H H Begin the calculation by placing the molecule with the oxygen directly above the carbon. ojf Ox / I I % . ^ 1 2 0°-^ The problem now is to determine the orientation of the methyl groups. Assume that the methyl groups are oriented in the plane like this: r'lQ? • 5 * * 40.5° Note: The other hydrogen atom is directly behind this one. 168 169 Now, consider a methane molecule arranged in the form of a tetrahedron. Note: The carbon atoms is in the center of the tetra hedron. The tetahedral angle is 109°, 28’ ~ (109.5°). Now, draw the tetrahedron in a cube. diagonal (X) 1 ^ Y l- *<109.5 ^109y5* Note: The carbon atom is the center or the cube. Calculate the length of the diagonal (d) and the length of a aide (a). d * 2(1.07) sin(—°y~-) * 1.75& ■ d a2 + a2 - (1.75)2 — ► 1.232^ » a 170 Now, put the cube on the methyl group. 0 diagonal (X) (viewed from the side) (PSUEDO) (H atom) C Notes The PSUBDO H atom and the angle of tilt ' 0 may be calculated as followss 40.5 tan 0 ■ -=2- y 2x y * x tan 40.5° - ■ 2x tan 0 x tan 30° tan 0 * *s(tan 40.5° - tan 30°) » 7.8° 171 Now, calculate the relative positions of the atoms in the box. (PSUEDO) (H atom) C 1 1 1 - 1.54 cos 30° - 1.333X - 1.07 cos 30° - 0.93A » 1.333 - .93 » 0.403A - 0.403 + 1.75 cos (7.8*) - 2.143A • 1.07 cos (54.75°) - 0.616A (from the tetra- . hedral angle) or H(1.232) - 0.616A (half the cube side) distance 1 to 4 - 1.333X + (.616) sin (7.8°) » 1.413A distance 1 to 3 distance 2 to 3 distance 1 to 2 distance 1 to 5 (Bond) distance 3 to 4 172 Draw View B. Note: Symmetry of molecule. vO 1.65X 1.15: 1/65& Draw View C. (1.54) cos 60 ^109,5°-^^ 1.06A - (1.07) cos (7.8°) 173 Draw View A. Notes Symmetry of molecule. Finally, integrate Views A, B, and C. with a planimeter. The one used in this work was a polar planimeter type F 4236, K & B Co. D “ (area A + area B + area C)/3. 174 TABLE B-l CROSS SECTIONAL AREAS OF MOLECULES STUDIED Area A Molecule A2 Acetaldehyde 15.8 Acetone 20.8 Benxene 36.0 Br onto form 26.8 1-Butanol 26.3 2-Butanol 26.1 Isobutanol 25.5 Tertbutanol 25.5 IsobutylmethyIketone 35.0 Carbontetrachloride 27.0 Chlorobenxene 41.4 Chloroform 27.0 Cyclohexane 32.0 Cyclopentane 25.6 Dichloromethane 22.1 DiethyIketone 29.9 2,3-Dimethylbutane 32.0 Ethanol 19.3 n-Heptane 36.3 1-Heptanol 39.2 n-Hexane 34.8 1-Hexanol 34.3 Methanol 14.9 Methylcyclohexane 37.5 MethylethyIketone 24.9 n-Octane 42.2 1-Pentanol 28.9 1-Propanol 21.1 2-Propanol 20.7 (Isopropanol) Toluene 39.3 2,2,5-Trimethylhexane 44.0 2,2,4-Trimethylpentane 39.7 (Isooctane) 2,4,4-Trimethylpentene-l 35.3 0-Xylene 46.4 Area B Area C Area D A2 A2 A2 Investlgat< 18.1 16.8 16.9 Parkinson 23.6 14.1 19.5 Parkinson 18.5 20.8 25.1 Losenicky 24.1 25.5 25.5 Parkinson 29.1 20.8 25.4 Parkinson 28.8 22.6 25.8 Parkinson 28.1 23.6 25.7 Parkinson 25.3 25.1 25.3 Parkinson 37.9 22.5 31.8 Parkinson 26.0 27.9 27.0 Losenicky 21.1 23.3 28.6 Losenicky 20.5 22.5 23.3 Losenicky 24.9 24.8 27.2 Parkinson 18.5 19.8 21.3 Parkinson 15.5 22.1 19.9 Parkinson 34.4 17.3 27.2 Parkinson 34.7 29.9 32.2 Parkinson 17.1 14.4 16.9 Losenicky 41.7 31.3 36.4 Parkinson 46.2 29.8 38.4 Parkinson 36.7 26.5 32.7 Parkinson 42.1 28.1 34.8 Parkinson 14.5 11.4 13.6 Losenicky 29.9 25.0 30.8 Parkinson 28.0 17.3 23.3 Parkinson 48.0 30.4 40.2 ParkinBon 34.3 25.5 29.6 Parkinson 24.1 19.4 21.5 Parkinson 23.1 21.2 21.7 Parkinson 25.1 18.9 27.8 Parkinson 45.5 41.4 43.7 Parkinson 39.0 35.5 38.1 Parkinson 40.0 31.0 35.4 Parkinson 26.8 25.3 32.8 Parkinson 175 Single Bond Double Bond Aromatic Bond TABLE B-2 ATOMIC PROPERTIES Covalent Radii A H C N 0 0.30 0.77 0.74 0.72 0.62 0.62 ~ 0.70* Cl 0.99 Br 1.14 Packing Radii X H C N 0 Cl Br 1.15 1.65 1.50 1.4 1.8 1.95 The Sp3 hybrid tetrahedral bond was taken as 109.5°. (e.g., the H-C-H bond In the methyl group, etc.) The Sp^ hybrid bond was taken as 120°. (e.g., H-C ■ C or C-C » 0.) The other Sp hybrids, such as the C-C-H bond in alcohol were taken as 108°. The bond length is calculated by adding the covalent radii. (e.g., C-C - 0.77 + 0.77 - 1.54A.) *The C - C aromatic bond was taken as 1.39A. APPENDIX C CALCULATION OF THE STANDARD ENERGY OF VAPORIZATION Bondi ( 8 ) uses E°, the standard energy of vaporiza tion, in his correlation (11-26) for the thermal conduc tivity of binary liquid mixtures. E° is defined by the equation, E° = 4HV - RT at T when V/Vw = 1.17. This means that if E° is not available from Tables ( 8 ), Vw must be obtained, then the temperature T and finally, the latent heat of vaporization, «AHV at that temperature. For most of the compounds in this study, E° could be found from the Tables of Article ( 8 ). E°, however, was not available for alcohols or halogenated hydrocarbons and had to be calculated for these compounds. A second article by Bondi (10) describes this calculation procedure fairly well. A simple example for the calculation of Van der O Waal's volume is given here for methane. Bondi uses 1.7A for carbon and 1.2& for hydrogen as Van der Waal's radii or packing radii, as opposed to 1.65& and 1.15& used by Losenicky in Table B-2. The Van der Waal's volume for carbon is then 4/3" (1.17) = 20.58A . The appropriate hydrogen volume segments must be added to this to obtain Van der Waal's volume for methane. This calculation followss 176 177 Figure C-l Carbon and Hydrogen 1.20A r2 » 1.70A I - 1.07A (See Table B-2.) is the volume of the hydrogen atom excluded by the chord C-C in Figure C-l. V2 is the volume of the car bon atom excluded by the chord C-C. The volume to be added to the carbon atom is - V2 for each hydrogen atom. From the geometry of Figure C-l it can be seen T l and h^ * r^ - m h 2 * r2 - ni - i . o 0 For this calculation m ■ 0.14A, h^ = » 1.06A, h2 * 0.49A. is given by ** 77"h^2 (r^ - and V2 by r22 that m = — — v2 0 17 h22(r2 “ 3^) * 178 For this calculation V-^ « 2.98A3 and V2 " 1.16A3. 0 Therefore, the additional volume is - V2 or 1.82AJ for each hydrogen atom. The Van der Waal's volume for methane is then °3 3 27.86A per molecule or 16.77 cm/g-mole. This compares closely with the value of 17.12 cm3/g-mole given by Bondi. This is Vw. In order to calculate E°, Vw is multiplied by 1.7; then density data is used to find the temperature at which the compound in question is at this volume. When the temperature is obtained, thermodynamic data is used to obtain the A Hy at that temperature. E° is then cal culated by the formula E° » AHV - RT. Table C-2 lists values of E° for the compounds used in this study. 179 TABLE C-l STANDARD ENERGY OF VAPORIZATION FOR COMPOUNDS USED IN THIS STUDY Compound E*(k cal/g-mole) Investigator 1. Acetaldehyde 7.17 Bondi 2. Acetone 7.60 Bondi 3. Benzene 9.31 Bondi 4. Bromoform 8.00 Parkinson 5. 1-Butanol 11.08 Parkinson 6. 2-Butanol 12.26 Parkinson 7. Isobutanol 11.30 Parkinson 8. Tertbutanol 11.81 Parkinson 9. Isobuty line thy lketone 8.99 Bondi 10. Carbontetrachloride 7.95 Parkinson 11. Chlorobenzene 10.09 Parkinson 12. Chloroform 7.28 Parkinson 13. Cyclohexane 7.83 Bondi 14. Cyclopentane 7.07 Bondi 15. Dichloromethane 6.82 Parkinson 16. Diethylketone 9.42 Bondi 17. 2,3-Dimethylbutane 7.86 Bondi 18. Ethanol 11.17 Parkinson 19. n-Heptane 9.55 Bondi 20. 1-Heptanol 15.31 Parkinson 21. n-Hexane 8.40 Bondi 22. 1-Hexanol 14.79 Parkinson 23. Methanol 10.12 Parkinson 24. Methylcyclohexane 7.87 Bondi 25. Me thylethyIke tone 8.45 Bondi 26. n-Octane 10.70 Bondi 27. 1-Pentanol 13.58 Parkinson 28. 1-Propanol 12.06 Parkinson 29. Isopropanol 14.55 Parkinson 30. Toluene 9.13 Bondi 31. 2,2,5-Trimethylhexane 10.68 Bondi 32. 2,2,4-Trimethylpentane 9.61 Bondi 33. 2,4,4-Trimethylpentene-l 9.95 Bondi 34. 0-Xylene 9.92 Bondi

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Language English

Contributor Digitized by ProQuest (provenance)

Advisor Rebert, Charles J. (committee chair), Lockhart, Frank J. (committee member), Vernon, James B. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-842784

Unique identifier UC11356458

Identifier 7421495.pdf (filename),usctheses-c18-842784 (legacy record id)

Legacy Identifier 7421495.pdf

Dmrecord 842784

Document Type Dissertation

Rights Parkinson, William Jerry

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA