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Thermal conductivity and sonic velocity of liquids

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Thermal conductivity and sonic velocity of liquids

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Content THERMAL CONDUCTIVITY AND SONIC VELOCITY OF LIQUIDS \\ A Thesis Presented to The Faculty of the School of Engineering The University of Southern California In Partial Fulfillment of the Requirements for the Degree Master of Science in Chemical Engineering By Kenneth Craig Sadoian n ‘ January 1969 UMI Number: EP41795 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI EP41795 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 -1 3 4 6 ■5 / b This thesis, w ritte n by Kenneth Craig Sadoian under the guidance G^tiis F a cu lty Com m ittee and approved by a ll its members, has been presented to and accepted by the School of E ngineering in p a rtia l fu lfillm e n t o f the re quirements fo r the degree of Master of^...Science in Chemical Engineering D a te .. . . . January.. 1 9 . 6 . 9 . ... F a c u lty C om m ittee .. l Chairman <r/i ACKNOWLEDGMENTS The author would like to express his sincere appre ciation to the following peoples Dr. Prank J. Loekhart - for his guidance and steadying influence through the research period? Dr. Charles J. Rebert and Dr. E. G. Partridge - honorable members of the Research Committee; Mrs. Ruth Toyama - for preparing the final draft and her assistance when needed; The Fluor Corporation - for financial aid during the research period. ii TABLE OP CONTENTS Page ACKNOWLEDGMENTS ....................... ii LIST OF TABLES........................................ iv LIST OF FIGURES............................... V CHAPTER I INTRODUCTION ........... 1 CHAPTER II PREVIOUS RELATIONS INVOLVING k AND Us . . ............................ 5 CHAPTER III PRESENTATION OF CORRELATIONS ...... 9 CHAPTER IV CONCLUSIONS............................ 16 APPENDICES .......................................... 18 APPENDIX A SOME THEORETICAL SUPPORT FOR A k-Us RELATIONSHIP ..................... 19 APPENDIX B ANALYSIS OF DATA............. 24 APPENDIX C CORRELATION GRAPHS AND TABLES ..... 28 APPENDIX D REVIEW OF WORK DONE IN RELATING X to Us ......................... 36 BIBLIOGRAPHY .......................................... 50 iii LIST OF TABLES Table Page 1-1 Table of Liquid Data Sources............... 3 III-l Experimental and Calculated Data ...... 10 B-l comparison of Experimental and Calculated Values of Us for Some Liquids ........ 26 C-l Predicted Values of k from Eqs. III-l and III-2 ....................... 33 C-2 Data for High Molecular Weight Hydrocarbons 35 D-l Critical Values of k for Some Liquids . . . 43 iv i LIST OP FIGURES Figure Page C-l Plot of Eq. III-l Through Data for Hydrocarbons............................ 29 C-2 Plot of Eq. III-2 Through Data for Hydrocarbons..................... 30 C-3 Plot of Eq. III-l Through Data for Halogenated Hydrocarbons .................. 31 C—4 Plot of Calculated Thermal Conductivity Values for Heptane Using Nozdrev's Data . 32 D—1 Thermal Conductivity vs. Sonic Velocity at 77°F.........................................39 D-2 Thermal Conductivity vs. ^CpUs at 77°F for Hydrocarbons ................... 40 D-3 Thermal Conductivity in Critical Region for Pentane . . . ........................45 D—4 Thermal Conductivity in Critical Region for Hexane.............................. . 46 D-5 Thermal Conductivity in Critical Region for Heptane . . . ...................... 47 D- 6 Thermal Conductivity in Critical Region for Benzene .............................. 48 v CHAPTER I INTRODUCTION Existing experimental thermal conductivity data for organic liquids are relatively sparse at temperatures above 212°F and pressures much above atmospheric. This fact is due to the difficulty in measuring accurate values of ther mal conductivity (k) at elevated temperatures primarily be cause of the onset of natural convection at these tempera tures. A secondary reason is the temperature limitations of the various experimental apparatus. However, these lira- \ itations usually occur at much higher temperatures than those corresponding to the beginning of natural convection. Some empirical and semi-empirical equations for k; those of Weber (15)*, Scheffy and Johnson (20), and Robbins and Kingrea (15), have given reasonably good re sults in low temperature regions for many .organic com pounds, but are for the most part unreliable at elevated temperatures and pressures. Part of this unreliability is *Numbers in parenthesis refer to numbered items in the bibliography. 2 undoubtedly due to the lack of good k data in these temper ature and pressure spans. Furthermore, these equations for k also have not predicted values in as close agreement with the most recent experimental data (8,11). In addition, these equations involve the evaluation of a number of physical properties at the temperature at which k is de sired, which is another drawback to their use. The ultimate purpose of this study is to develop an equation for predicting k that is dependent on as few phys ical properties as possible and ones that can be easily evaluated or obtained at any temperature and pressure. Also imperative in any k relation is that it gives results in close agreement with experimental values over the entire temperature range for which experimental data exist. The property primarily under consideration to cor relate with k is the sonic velocity (Us) in liquids. As a result of this study a correlation has been developed be tween k and Us for liquid hydrocarbons which gives values agreeing well with experimental ones over the temperature range 77-250°F. A list of the liquids studied and the sources of the experimental k and Us data are given in Table 1-1. 3 TABLE 1-1 Table of Liquid Data Sources # Liquid k Reference Us Reference 1 Pentane Michaelian (li) Ref. (18) at 77°F. Calc, at other temps. 2 Hexane Malian ( 8) Ref. (18) at 77°F. Calc, at other temps. 3 Heptane Malian ( 8) Ref. (18) k Octane Michaelian (11) Ref. (18) 5 Nonane Michaelian (11) All values calculated. 6 Decane Michaelian (ll) Calculated 7 2-Methylbutane Vilim (2k) Ref. (18) 8 2-Methylpentane Sakiadis (l 6) All values calculated 9 2,3-Dimethylbutane Sakiadis (16) All values calculated 10 2,2-Dimethylbutane Sakiadis (16) Calculated 11 Cyclohexane Horrocks (5) Ref. (18) at 77°F. Calc, at other temps. 12 Methylcyclohexane Malian ( 8) All values calculated 13 Benzene Michaelian (ll) Ref. (18) Ik Toluene Ziebland (27) Ref. (18) 15 Ethylbenzene Riedel (lU) Ref. ( 1 8) 16 Isopropylbenzene Mason (9) Calculated 17 p-xylene Michaelian (11) Ref. (18) 4 It is hoped that the relation presented in this the sis can be extended to higher temperatures when more ex perimental 1c and Us data are available. CHAPTER II PREVIOUS RELATIONS INVOLVING k AND Ug First to propose a relation between k and Us was P. W. Bridgeraan (3,16) in 1923. By assuming that the liq uid molecules were arranged in a cubical lattice with a distance d between centers of adjacent molecules and that the heat energy was transported along a row of molecules with the velocity of sound, Bridgeraan derived the equation k = 2RUS/Nd2 where k = BTU/hr.-ft.-°F; R = gas constant, 1.987 BTU/lb. mole-°F; Us = ft./hr.; N = Avagadro'S No., 2.73 x 102 6 molecules/lb.mole; and d = mean distance of separation be tween centers of adjacent molecules, d is calculated by the formula d = (M/N p )1//3, where M = molecular weight, N = Avagadro's No., and p = liquid density, lb./ft.3. Bridgeraan himself acknowledged that the relation gave values only within an order of magnitude for some liquids compared to his experimental data. But, what im pressed him most was the simplicity of the equation and the fact that it /predicted an approximately correct value for water, gave the right sign for the temperature coeffi cient of k at atmospheric pressure for both ordinary or ganic liquids and for water, and gave the right sign for the pressure effect on k. These facts led Bridgeman to be lieve that he had developed an equation of significant value. In 1934, A. Kardos (7,16) revised Bridgeraan's equa tion. He assumed that the temperature drop in the liquid changes exponentially and that the energy drop in the liq uid occurs in the intermolecular spaces. This necessi tated the use of a factor L, the distance between adjacent molecular surfaces, and the elimination of d, the center to center distance between adjacent molecules. Similar to Bridgeraan, he visualized the heat energy being transferred along a row of molecules at the velocity of sound and pre sented the following equation for kt k = LUs^Cp where k = BTU/hr.-ft.-°F? L = distance between adjacent molecular surfaces, ft.; Us = ft./hr.; = liquid density, lb./ft.3; and Cp = heat capacity, BTU/lb.-°F. As a first 1 approximation Kardos assumed that L was a constant for all liquids and equal to 3.12 x 10”- 1 -® ft. Both Bridgeman's and Kardos' correlations were tested by B. C. Sakiadis and J. Coates (16) in 1955 using experimental k data for 28 liquids that they had measured. Large deviations from the observed values resulted. It was already known that Bridgeman's equation was only approxi mate for many liquids. Thus these large deviations were not surprising. Kardos' equation gave poor results pri marily because of the assumption of a constant L for all liquids. Sakiadis and Coates (16) took the Kardos equation and showed that when all the properties (including L) are properly evaluated, predicted values of k agreed well with their experimental data which included a wide variety of organic liquids. But, it is the evaluation of these pro perties, p , Cp/ US/ and particularly L, which limits the practical use of the equation. For example, to evaluate L the following variables must be known accurately: x, equal to d+L (d is the mole cular diameter) which is obtained from x-ray diffraction; vc, the critical volume; and vc/v0, the ratio of critical j 3 to minimum molecular volume. Then, a rather long procedure is needed to calculate L. In addition density and Cp data at elevated temperatures are usually not readily available, thus forcing the estimation of these properties which can severely limit the accuracy of the relation. The avail ability of Us data is discussed later, but is currently subject to the same limitations as density and Cp data. CHAPTER III PRESENTATION OF CORRELATIONS A correlation has been developed between X and Us for liquid hydrocarbons based on the assumption that a spe cific value of Us corresponds to a specific value of 3c. This was accomplished by plotting values of X vs. Us over the temperature range 77-250°F and noting that the plot ap proximated a straight line. A linear regression analysis was then performed on the data by use of a computer program to yield the constants of the linear relation. Table III-l gives the liquids used in the correla tion, their corresponding values of k and us at various temperatures and shows whether Us was calculated or ob tained from experimental data. The method used to calcu late Us where needed is discussed in Appendix B. All the data of Table III-l, except for 2,3-di- methylbutane, 2 ,2-dimethylbutane, methylcyclohexane, and isopropylbenzene, are well represented by the equation k = 0.02783 + 0.00322 (Us x 10“6) III-l TABLE III-l Experimental and Calculated Data # Liquid T °F k expt'l. BTU/hr-ft-°F Us expt’l (ft/hr) x 1 1 Pentane TT 0.06688 11.77 100 0. 061+ 1+ 1 * 1 0. 66* 150 0. 0586U 9.3U* 200 0.05381+ 7. 86* 250 O.0I + 80I + 6.35* 2 Hexane 77 0.06823 12.59 100 0.06593 11.75* 150 0.06083 10.39* 200 0.05577 8.78* 3 Heptane 77 0.07116 13.38 100 0.06922 12.75 150 0.06501 11.37 200 0.06081 9.98 250 0.05661 8.60 k Octane 77 0.07508 13.81 100 0.07250 13 • 20 150 0.06689 11.86 200 0.06128 10.52 250 0.05567 9.18 5 Nonane 77 0.071+99 13.97* 100 0. 0726U 13.1+2* 150 0.06753 12. 26* 200 O.062U2 1 1. 11* 250 0.05731 9.93* 6 Decane 77 0.07838 ll+.l 8* 7 2-Metbylbutane 77 0.06370 1 1. hk 8 2-Methylpentane 77 0.061+50 11.95* 9 2,3-Dimethylbutane 77 0.06112 12.52* 100 0.05930 11.78* 150 0.05530 10.52* 200 0.05196 9.05* 10 2,2-Dimethylbutane 77 0. 0582U 11.63* 11 Cyclohexane 77 0.06890 lit. 76 100 O.067IO 13.51* 150 0.06310 1 2. 02* 200 0.05915 10.79* 11 TABLE III-l, (con't.) # Liquid T °F k expt'l. BTU/hr-ft-°F 0S expt'l. (ft/hr) x 10-6 12 Methylcyclohexane 77 0.061*21 13.51* 100 0.0621* 3 12.93* 150 0.05860 11.7^* 200 0.05^76 10.50* 13 Benzene 77 0.0781*1* 15.35 100 0.07662 lk.67 150 0.07265 13.19 200 0.06869 11.72 250 0.061*72 10.2k Ik Toluene 77 0.07679 15.33 100 0.071*75 li*.70 150 0.07030 13.33 200 0.06587 12.00 250 0. 06ll *2 10.66 15 Ethylbenzene 77 0.07575 15.03 16 Isopropylbenzene 77 0.0711*7 15.1*8* IT p-xylene 77 0.07623 15.U8 % Indicates calculated values of Us. 12 where k = BTU/hr.-ft.-°F and Us = ft./hr. This same result is reached when a regression is performed only on the liq uids where both k and Us are experimental values. Eq. III-l is derived with data from 13 liquids and a total of 44 data points, 17 of which include calculated values of us* Eq. III-l gives an average deviation of ±2.3% from experimental data, and the deviations man from +6 .1% (ben zene at 250°P) to -9.4% (cyclohexane at 77°F). Deviations are within +3.0% for most data points, however. Table G-2 shows the calculated values and deviations for each liquid. Pig. C-l is a plot of Eq. III-l through the data. When the liquids that were excluded in the regres sion to obtain Eq. III-l are included the equation is al tered to k = 0.02676 + 0.00320 (Us x 10~6) III-2 This equation encompasses 17 liquids and 53 data points of which 27 have calculated values of Us. Eq. III-2 gives deviations ranging from +8.0% (ben zene at 250°P) to -10.2% (methylcyclohexane at 200°F) and has an average deviation from experimental data of +4.2%. 13 Refer to Table C-2 for calculated values of k and individu al deviations, and Fig. C-2 for a plot of Eq. III-2 through the data. The data points added to obtain Eq. III-2 are not well represented by this equation. Table C-l shows that deviations vary from about 6-10% for these points. It is important to note that for these liquids the predicted k values are all higher than the experimental values, indi cating that these calculated values of Us are high. The method used to calculate Us is discussed in Appendix B where an explanation for these seemingly high values of Us is given. For these reasons it is recommended that Eq. III-l be taken as the best fit to the data. As stated Eq. III-l is developed solely from liquid hydrocarbon data and, therefore, should be used only for liquid hydrocarbons. However, the equation was tested for some halogenated hydrocarbons where experimental data are available, and the results are shown graphically in Fig. C-3. In every case, except for chloroform, Eq. III-l gives results that are high. No attempt was made to explain this fact. 1 i Values of k were also calculated for two high mole- -14. cular weight hydrocarbons, P.S.U. #25 and #87, for which k data are presented in API Research Project 42 (1). Actual ly, k data are given for five high molecular weight hydro carbons, but Us could be calculated only for the two men tioned. The k data were measured by R. R. Dreisbach of the Dow Chemical Co. in 1957. Although the calculated results agree rather well with experimental values for three of the four points (see Table C-2 in Appendix C), no positive con clusions can be made since all values of Us are calculated and the reliability of the k data is unknown. Temperature and pressure limitations of the equation presented are not clearly known. The data used to Obtain Eq. III-l range in temperature from 77-250°F. The only Us i data avilable for liquid hydrocarbons above 250°P are those presented by Nozdrev (12) for isopentane up to 320°F, hexane up to 464°F, heptane up to 500°F, and toluene up to 356°F. The data for hexane, heptane, and toluene are a little higher by a constant amount than those used in the regression to obtain Eq. III-l, and since the data for each liquid follow a straight line almost to the critical point, they are probably also higher in the higher tempera ture regions. This explains why Nozdrev’s data for heptane, IS when used in conjunction with Eq. III-l predict values that lie above a straight line drawn through the calculated k values obtained from the Us values used to obtain Eq. III-l. This is shown in Pig. D-2. The important thing to note here, though, is that the results are reasonable con sidering the above mentioned point, except near the criti cal point where the line should begin to curve down. Sim ilar results are also obtained when Nozdrev's hexane and toluene data are used in conjunction with Eq. III-l. Thus it is possible to say that Eq. III-l is prob ably reliable at temperatures well above 250°F for most liquid hydrocarbons. It is recommended, however, that as more k and Us data become available at higher temperatures, a new regression analysis be performed to get a better fit to the data. CHAPTER IV CONCLUSIONS The results presented in this thesis show that a linear relationship exists between k and Us for liquid hy drocarbons over the temperature range 77-250°P. Where ex perimental data are available for k and Us predicted values of k using Eq. III-l are in excellent agreement with ex perimental data. For those liquids where calculated values of Us are used, results are very good except for 2,3- dimethylbutane, 2 ,2-dimethylbutane, methylcyclohexane, and isopropylbenzene. For these above four components it is believed that the sonic velocities calculated are high possibly due to the fact they are cyclic, branched and substituted compounds. This would account for the high predicted values of k. A linear relation does not exist between k and Us for polar liquids as k and Us are affected differently by polarity and association. This does not eliminate the possibility of a relation between k and us for polar liq- quids, however, as it is possible that other properties 17 become involved besides Us* Hopefully, the positive results presented in this thesis will stimulate further work in the field of relating Us to k. 1 APPENDICES APPENDIX A SOME THEORETICAL SUPPORT FOR A k-Us RELATIONSHIP No theories have yet been proposed that adequately predict k values for even the most simple liquids. This is primarily due to the fact that a good equation of state for liquids has not been developed. In a recent article on thermal conductivities by McLaughlin (10), a number of the ories on k are reviewed. They are all based on one of three liquid energy transfer mechanisms; hard sphere mole cules, molecules interacting with a square-well potential, and molecules interacting with spherically symmetric inter- molecular potentials. The theories proposed by Bridgeman, Kardos, and Sakiadis and Coates are not discussed nor is mention made of a relation between k and Us. One theory proposed by Horroeks and McLaughlin (10) based on the third energy transfer mechanism mentioned above does seem to have some similarities to the theories of Bridgeman, Kardos, and Sakiadis and Coates. In their theory Horroeks and McLaughlin assume a f.c.c. lattice- type structure for the liquid, and the excess energy due 20 to the temperature gradient is assumed to be transferred down the gradient with a frequency V which is determined by the molecular mass and the intermolecular forces, k is expressed as the sum of two terms. The first kconv is due to the motion of molecules between cells, and the second k j j f is due to an intracellular term arising from the inter molecular forces with the transfer taking place by the vi brational mechanism. The two terms are represented by ^conv “ 2k0Cv/a *0 = r?Vcv/a where a - nearest neighbor distance, V is the lattice fre- quency, and k0 is the frequency of diffusion displace ments. For the liquids considered (A, N2 CO, CH4 , CgHg, and CCI4 ) the convective term is negligible. Calculated values of k for the six mentioned liquids are in general agreement with experimental data, and in all cases the correct temperature dependence is obtained. This fact does seem to support the theory, at least for relatively simple compounds. It is possible that the velocity of sound could be related to the frequency V mentioned above. If so, this would mean that for the more simple compounds (namely non polar hydrocarbons) a linear relation could exist between k and Us over a certain temperature range. For polar liquids and particularly associated ones it is reasonable to assume that either kconv or some other factor that comes into play contributes more heavily to the total value of k while these factors have little ef fect on Us. It is known that in associated liquids ultra sonic absorption is much higher than in normal liquids (2). So, instead of the sonic velocity being affected greatly by polarity or association, as is k, it is the ab sorption of sound that is. It can, therefore, be conclud ed that a simple relation would not exist between k and Us for polar liquids for these reasons. It is realized that this is only one of many theo ries that has been proposed for k. But, many of these theories are closely related to each other and differ only slightly in molecular model or energy transfer mechanism chosen. Other support for a theoretical relationship be tween k and Us for liquids comes from the comparison of ex perimental data. The temperature coefficient of both k and us have the same sign for nearly all liquids including water, which has a positive temperature coefficient of k up to about 250°F. Exceptions to this observation are gly cerol, ethylene glycol and other highly viscous, related liquids which have positive temperature coefficients of k but negative temperature coefficients of Us. The tempera ture coefficient of k for water changes sign at about 250<>F while the temperature coefficient of Us changes sign at 165°F. The reasons for this difference are not clear, but this does offer a possible explanation as to why glycerol and ethylene glycol have temperature coefficients for k and Us of opposite signs at room temperature. Apparently, the phenomenon which causes water to have both coefficients positive has already been surpassed at room temperature for glycerol and ethylene glycol. Sufficient data are not available at lower temperatures for these liquids to prove this point. Experimental data also confirm that the change in k/ and Us with changing pressure has the same sign for all liquids where data are available. Both increase with in creasing pressure and decrease with decreasing pressure. From these observations it seems quite possible 23 that k and Us have a theoretical basis for being related, especially for liquid hydrocarbons. APPENDIX B ANALYSIS OF DATA The k data used for this study are believed to be reliable, many of them being recently measured at the Uni versity of Southern California (8,11). Data that are not available from these sources are taken from a survey on ex- permental k measurements by Jamieson and Tudhope (6 ). Nearly all experimental values of Us are taken from a survey done by Sakiadis and Coates (18). For many of the liquids experimental data are presented by more than one researcher, thus increasing the reliability of the data. In these cases average values of the data presented are used. Where temperature dependence is given most of the data have been measured only up to about 150°F, but are extrapolated up to 250°F since this is still in the straight line portion of the Us vs. T curve for liquid hydrocarbons. For liquids where good k data are available and no Us data exist, values of Us are calculated by the empirical equation 25 Us = (R^/M) 3 where Us = ft./sec.; R = structural contribution to velo city of sound; <5 = liquid density, lb./ft.3; M = molecular weight. This equation was fitted to existing data by Sakiadis and Coates by adjusting values of R for individual components of the structural equation of the liquid (16). Sakiadis and Coates calculated Us for 135 pure or ganic liquids of all chemical types using this equation and reported an average deviation from experimental values of ±2.6% and a maximum deviation of +8.0%. Values of R are contained in the same article. Table B—1 gives a comparison of calculated to ex perimental values of us for a number of liquid hydrocar bons showing the reliability of the method. Previously given as a reason for the discrepancy in the predicted values of 2 ,3-dimethylbutane, 2,2-dimethyl- butane, methylcyclohexane, and isopropylbenzene is the fact that their calculated values of Us are too high. It is possible that for these branched and substituted liquids the equation gives results in error up to 5 or 6%. If this is so this would bring the predicted k values for 26 TABLE B—1 Comparison of Experimental and Calculated Values of Us for Some Liquids Liquid T °F Us expt'l. Us calc. % diff* Heptane 77 13.38 13.M 2.50 100 12.75 12.kk 2.1*0 150 11.37 11.17 1.70 200 9.98 9 ' . 93 0.50 250 8.60 8.66 -0.60 Octane 77 13.81 13.1*9 2.30 100 13.20 12.92 2.10 150 11.86 11.67 1.60 200 10.52 10.1*9 0.20 250 9.18 9-33 -1.60 Benzene 77 15.35 15.**1 -0.30 100 ll+. 67 lU.73 -0.1*0 150 13.19 13.21* -0.30 200 11.72 11.81 -0.70 250 10.2l* 10.1*3 -1.80 Toluene 77 15-33 15.30 0.10 100 ll*.70 11*. 72 -0.10 150 13.33 13.1 * ! * -0.80 200 12.00 12.11* -1.10 250 10.66 10.81 -1.1*0 p-xylene 77 15.1*8 l6.ll* -1+.20 All Us values in (ft/hr) x 10 *% diff s (Ug expt'l - Us calc) x 100/Us expt'l. these liquids to well within 5% of the experimental values. The only other property affecting the calculated value of Us is the density. All density data at 77°F are accurate and are obtained from G. M. Malian's disserta tion (8), or from Weisberger, et al. (26). Density data at elevated temperatures are obtained from Timmermans' (22) which is a widely used and respected reference. APPENDIX C CORRELATION GRAPHS AND TABLES 29 FIGURE C-l Plot of Eq.. III-l Through Data for Hydrocarbons + 1 / r-+ 0.07-0 i®! 0.060. I e: :',. 4 - - + + - 050 ExptZlJ .Value s^Jofj-U Calc . Values of ui i 0.01*0 810 6.0 1 0 . 0 30 FIGURE C-2 Eq. III-2 Through Data for Hydrocarbons □ 0.070 I I + I I 4 9 0.060 + t ■ * I t - f - f t - * - + | + t t - + | t 4 + t 4 4 . 4 , * ; ■0.050 !1. Values, of lU-I Calc.-[Values Data"Roihts from Eq. -of—Us Excluded* I i r -1 o.oUo 1 ; 1210 . - t- • ft/hr: FIGURE C-3 Plot of Eq. III-l Through Data for Halogenated Hydrocarbons + t r i 0.070 07060 Legend 0.050 Car‘ bontbtrachi6ride! , Chloroform 1 ; . . .4 "Ethyl -Iodide ^Ethyl Bromide, 4 'Bromobenz e ne lodobenzene 0 ; 0U0- TCofrvn; ;; -1 * t ^x-lO-^ 32 FIGURE C-H Plot of Calculated Thermal Conductivity Values for Heptane Using Nozdrev’s Data Calc. xa + Regress ion from Used from Efbidrev,' s Us! Data + 4 - t+-f-+ + -t'i-i--+i- -***!• 4 - ^ f + / - i - i + I + t + ine Represents Experimental Data/-i- i - t + fM + f' (Dotted! ; ,.500;:: i t ;: ; 6joo: ^: 11; r Irob r;; t r T ' j T 't + + "f " + + + t i f f t 1 - 4 - + - + + - r 1 - + 4-r + +f', 'ji—t 1 " + ' ;;8ooi r : i + 4- + 1000 3 3 TABLE C - l Predicted Values of k from Bqs. III-l and III-2 # Liquid T°F K , expt'l. k pred. (lq.III-1) % diff* k pred. (Eq.III-2) % diff 1 Pentane T7 0.06688 0.06572 1.73 0.061*1*2 3.67 100 0.061*1*1* 0.06215 3.55 0.06087 5.51+ 150 0.0586k 0.05790 1.26 0.05665 3.1*0 200 0.0538U 0.05313 1.32 0.05191 3.59 250 O.OU80I * 0.0U827 -0.1+8 0.01*708 2.00 2 Hexane 77 0.06823 0.06836 -0.19 0.06705 1.73 100 0.06593 0.06566 0.1*1 0. 061*36 2.38 150 0.06083 0.06128 -0.71* 0.06001 1.35 200 0.05577 0.05610 -0.59 0.051*86 1.61* 3 Heptane 77 0.07116 0.07091 0.35 0.06598 2.23 100 0.06922 0.06888 0.1*9 0.06756 2.1*0 150 0.06501 0.061*1*1* 0.87 0.06311+ 2.87 200 O.O6081 0.05996 1. 1*0 0.05870 3.1*8 250 0.05661 0.05552 1.93 0.052*28 1*.12 1 * Octane 77 0.07508 0.07229 3.72 0.07095 5.50 100 0.07250 0.07033 2.99 0.06900 1 * . 83 150 0.06689 0.06601 1.32 0. 061*71 3.26 200 0.06128 0.06170 -0.69 0. 0601*2 1. 1 *0 250 0.05567 0.05738 -3.07 0. 05611* -0.81* 5 Nonane 77 0.07U99 0.07281 2.91 0. 0711*6 1+.70 100 0.07261* 0.07101* 2.20 0.06970 l*.0l* 150 0.06753 0.06730 0.31+ 0.06599 2.28 200 0. 0621*2 0.06360 -1.89 0.06231 0.17 250 0.05731 0.05980 -l*.3l* 0.05821* —2 .ll* 6 Decane 77 0.07838 0. 0731*8 6.25 0. 07211* 7.97 7 2-Methyl- 77 0.06370 0. 061*66 -1.51 0.06337 0.52 butane 8 2-Methyl- 77 0.061*50 0.06630 —2.80 0.06500 -0.78 pentane 9 2,3-Dimethyl- 77 0.06112 — — 0.06682 -9.33 butane 100 0.05930 - - 0.061*1*6 -8.69 150 0.05530 * ■ - 0. 0601*2 -9.27 200 0.05196 - - 0.05572 -7.2l* 10 2,2-Dimethyl- 77 O.0582U - - 0.06398 -9.81* butane 34 TABLE C-l (con't.) # Liquid T°F k expt’l. k pred. (Eq.III-l) % diff* k pred. (Eq.III-2) % diff 11 Cyclo- 77 0.06890 0.07535 -9.36 0.07399 -7.39 hexane 100 0.06710 0.07133 -6.30 0.06999 -1*.31 150 0.06310 0.06653 -5.1*1* 0.06522 -3.37 200 0.05915 0.06257 -5.78 0.06129 -3.61 12 Methylcyclo- 77 0. 061*21 ■- — 0.06999 -9.00 hexane 100 O.O62U3 — 0. 06811* -9.il* 150 0.05860 — — 0.061*33 -9.77 200 0.051*76 - - 0.06036 -10.23 13 Benzene 77 0. 0781* 1 * 0.07725 1.52 0.07588 3.26 100 0.07662 0.07506 2. 01* 0.07370 3.81 150 0.07265 0.07030 3.23 0.06897 5.07 200 0.06869 O.O6566 i*. 56 0. 061+26 6. 1* 1 * 250 0.061*72 0.06080 6.06 0.05953 8.02 lh Toluene 77 0.07679 0.07719 -0.52 0.07582 1.27 100 0.071*75 0.07516 -0.55 0.07380 1.27 150 0.07030 0.07075 -0.61* 0. 0691*2 1.26 200 O.O6587 0.0661+7 -0.91 0.06516 1.08 250 0. 0611+2 0.06215 -1.89 0.06087 0.89 15 Ethyl- 77 0.07575 0.07622 -0.62 0.071*86 1. xd benzene 16 Isopropyl 77 0.0711*7 — — 0.07630 -6.75 benzene 17 p-xylene 77 0.07623 0.07767 -1.89 0.07630 -0.08 k is in BTU/hr-ft-°F *% diff = (k expt'l - k calc) x 100/k expt'l. 35 TABLE C-2 . Data for High Molecular Weight Hydrocarbons Liquid M.W. T °F Us(ft/hr) k expt'l. k pred. (Eq. III-l) P.S.U. #25 352.7 106. ^ 15.56 x IQ-6 0.0762 0.0781+ 196.h 13.70 x 10“6 0.0760 0.0721 P.S.U. #87 31+1+. 6 103.6 16.16 x 10"6 0.0670 0.0801+5 198.8 1^.27 x 10-6 0.0726 0.071+05 k is in BTU/hr-ft-°F. APPENDIX D REVIEW OP WORK DONE IN RELATING k TO Us This section of the Appendix is a brief review of the work that was done in this study to develop a correla tion for k. Although most of the work is closely related, much of it is worth mentioning. The most valuable informa tion contained in this section is that pertaining to the prediction of k in the critical region. Attention was first focused on correlating k at 77°P for organic liquids with the hope that any correlation developed could be extended to higher temperatures. Upper most in mind was the Sakiadis and Coates equation, k = ^CpUsL which, although it gives reasonable results, is not used because of the difficulty in evaluating the properties involved, particularly L, the available intermolecular dis tance. And, since the sonic velocity seemed to vary simi larly with k for pressure and temperature for nearly all organic liquids, special attention was given to this pro perty. Other physical properties that were investigated besides Us were: density (^>), heat capacity (Cp), mole cular weight (M), heat of melting (£Hm), melting point temperature (Tm), heat of vaporization at 77°F (A 1^,77), heat of vaporization at normal boiling point (A % BP) , boiling point temperature (Tp), critical temperature (Tc), refractive index (nD), viscosity (/£), and surface tension (Y ) . Units employed were English and AH values were evaluated per lb. and lb.mole. Liquids under study were those in G. M. Malian's dissertation (8 ) which covers a wide range of polar and non-polar liquids. The procedure used to analyze the data was to plot k, k/Us, k/^> CpUg, and in some instances the Prandtl Num ber Cp/c/k vs. various physical properties and their com binations. Plotting of k/£? CpUg was an attempt to repre sent the factor L in the Sakiadis and Coates equation by some other physical property or properties. In all cases for the wide variety of liquids studied none of the graphs plotted approached straight lines. However, it was ob served that for hydrocarbons and particularly for the nor mal paraffins some of the plots did approximate straight lines. This was especially true for graphs involving Us, , Cp, and their combinations. 38 With these results in mind it was decided to con centrate only on hydrocarbons as it appeared a simple cor relation did not exist for polar liquids, or at least the same correlation could not be applied to both. More graphs were drawn with the result that the best correlating pro perties seemed to be Us, £? , Cp, and ,01^. In order to select»the best properties and a corresponding correlation the data were submitted to a computer program. A linear stepwise regression program was available for use from the Computer Sciences Laboratory at USC and was used to analyze the data. The computer program picked us and ln^lHm as the bestscorrelating parameters for k at 77°F. Since values of & H j j j are not available for many compounds and cannot be predicted accurately by any known method it was not given further consideration in a correlation for k. However, this result does offer the possibility of calculating <&. Hra from k values at 77°F. The relation between k and Ug was subsequently modified to fit a wide temperature range as described in Chapter III. Examples of graphs that were plotted are given in Figs. D—1 and D-2. A considerable amount of time was also spent on 39 lii. tt t * - + + ++• • ■ I — I - + 4 - t I ; I lit: 0.12 -o; li f e ' ?.! -p I b -B- i : X i. 0.10 0,09 I I =L | ; I ; :o.08‘ o:ot h+- -I- 0: 06- I ; ! I I i -t-1 I I I ■ :t: 0.05 I 1 I ' I + 4 + 4 f t: i;! i ' ■ ■ ' t * 4 I t : ♦ X - 4 - + + 1 i i ;: i I I ' i : * I i T T T I I J Z l l : I 1 1 FIGURE D-l Thermal Conductivity vs. Sonic Velocity at 77°F ■ ' " • y ; i . i I • • 5 #5 M i l ■ i,. - t - * 4 +■ I . I - I —I- -t t y t 4 t » t + It rrt t f ♦ + t + + + ► !■ !■ + i . ■ , • 6m: © _ - 1 +3 3 k •+ +■ P+- i I i -H-t 1® ik # 5::: • 1h f- -r A I * I ©5 i t ■ 4 — *- + -i -4 - t r • t t t + 4 + 4 + t + I - t I -*1 ’ I 1 » " " i -4 - * y t + f - + 4 + 4 4 + + io.o": t:! i2 .o; • u - + t + 4 - 4 * + 4 t r 4 1 111; 4 - - t - + f + 4 - + 4 t + +• 4 - t + < ■ 4 t f t 1 -r f t * + t + + f-+ 1 * f + 't + + + t t 4 f . t + i 1 -4-j- 4 - 4 + 1 - 4 t i 4 ; 1 4 - 4- + .+. + + - r + - - * —* - + + - + 4- + - + - J ~ f 4 - • - + + f -f- + t 1 f 4- i -+ t t t - + • f 4- | t + .+ ‘ + + - + 4 t- +- + t . f ► * 4 4 * - + + + T T T T 4 - 4 + V i i ■ 9 O * . j t* re:: T 4 - + f 4 . f ; i ; 4 Hydrocarbons> lU -2, -+ 'Alcohols + 4*- + -- -t+ + + j - * t 4 + -t- f ♦ + i . --g t-t-iC eftio iM iS I ■ ; t ; + r-1 . t t - * "! ^ * - 4 - 1 • t . • Jk+-> Nitro • Compounds -4* -J~ + “4- +-■*+■ 4-4- ------4 4 V ^ ' t t- 4--+ t t t • h T • + t H . f-+-i + I- - r - 4- -UXt t - - * f-4 ■ f - f ■ 4 . * -t - + + + - - * - • t * ' —p-1-*- • + - - -t--+ +- + • + t" t t-4 + t ; r " 1 " " j " 4 + t + 4 - + + + 4 t t + I 4 * + + + t t f j . 41; T r T 1 r -+ + 4- - ’ i + t + + - - t - - + - 4 r * ■ t + + + + + •+■ + f -4- I ■ + - * f . 1 - + 4 4 4-* 4 + 1 - 4 ' t + t 4 4 + + 4 ' * 4 . f •+*■77 " j ” • I - * 4-f- ■ 6--♦•n-Ac ids i ii .o; 4 r ; f16. b * ; t + +‘ 4 ■ ■ • | 1 1 4 t J ■ '] * ' ' ’ U”* TfvtVhr .0 ‘ 14 { r 4 i- * + - + - t - t * • + ■U 4 - | f’ + 1 -+ ■ 1 4 | l l n +j t - i -t-l + i * 1 +-I + - I H ; t 20 J o ; r t + - 4 - t t f + - 40 t + + + 4 + + - + ' f t + M l i : 11- +1 4 + + + I - i * 0. *08 f "ttj' M j + + + 4 -f • + + ■ + f 4 + 0.OT S o .41 . ■ ■ ,H ; i I I I I 4 + + + ! 0.Q6 H- 4 • + -4 0 .0 5 f - + + + ■ • i — ( - 4 t . 4 . .1 4 - . A -A 1 A ! i M-M n i i . i FIGUHE D-2 Thermal Conductivity vs. Q CpUs at 77°F for Hydrocarbons + + • 4 - • X -+ M- 1 4 - + - ( ■ 4 - + r+ f 1 + - + -U +-+ t X - 4 - f- + + —i — t - + + Xt + 4- T T T T -I .! t T t - t + 1 - + - + t 4 4 4 * + - 4- -t r i I + - I - + 4 - - 4 — j- I . + t It t * - 4 4 4 + + t 4 - + + + I 4 4 4 -4-4-1- | - t 9 I .1 i + - + + +- + 4- -+ + + + +• • + + 4 - + + + + + #8~ * 7 -+ + t • i * ■ t 4 U I i M l ' i 1 ++ + + 4 4 + 4 ! ■ i ‘ f + 4 + -+- +— i---4- I t -+ + 4 - + + 4+ t + 4- -+ + ■ + + -+ + + + + + + + + + ■+-14 + r + + + + + + * ■ M T t + f 4 - + 4 4 + + • 4 +- + - + 4 + + + 4 4 + 4 + + i ' ■ 1 9 & ♦ t + r 1 *1l2. ! - 9 + i * £.* ,10- + + -+ + 4- 4- - ‘ f • 4 4 - * + - 4 + f 4 + 4 ' - 4 -I -i -H Numbers +■ + r + i t r f + 4-1-4 4 + t H + 4 - t I ■p-H" 4 t + 4 ; ; m t r t | f t 4- + , + 4 4 j - | M '+ 4-4 4 + + + + + t + -t t-t t r t + + I I - t t 4-4 4 + + 4 + r 4 - + t +- f 4 + 4 4 + +-44 4 4 + 4 + 4 •+ 4 I ♦ t I n ) ■ 4 4-4 I 4 4 I M M * 4 4 4 I ! ' ! A t 4 -4- 4 t T I I 4 - 4 4 + i ! ! ■ ! ' 4 4 4 4 4 + i * t ' * 1 t + r f . +++444 + + 4 4 + t + t 11 +1 M M r r r • t - + - 4 M + - t + - 4 4 + + 4 l+ ! i i ? t " '4 + 4 < - -4 4 T T 4 4 •+ + 7 + T t -f +• 4 + -M-M- + +-4 I 4 I + 4 4 i i I t f -++• + + 4 4 4 f 1 ' I ! : * t * M + 4 t r . . . f + + | - -4 4 - 4 4- -+44+ 4 + + + + + ; i T + 1 : 4 * ■ + f * 4 4 -4- r 4 4 4™-^-' + f * -+ 4 f +•-* ■ I ; I +•+-+■ + + t t + + 4- +- 4 t ( | | * t 1-4 I > t 4 4 + T4T + + 4 P 4 t 7 ^ + + ; ■ + + + 4 . .~ + ... 4 + f c - h 4 ‘ To f + 4 V 4 t t ^ + i - + t 4 4 4 t + j I } 4 ; 4-- i + r + + + 4-44 + X ; + 4 + -f + Refer }to‘ Comtouads^ln^Tahl'e^I^l TT + +-+ + 4 +- 4 i + + -1—f M I ! 4 - + + -4- t - t t + I 4 -t - -+ 4 + + + r + +- tr t + t I + +~ 4 + 4 •+- M l } - -4-4 4 • + M + - -+-- f- + - f t + "I ■2T r 4 + 1291 t ^ ^pUs+ x + 44 + t ' X0“T: MtU / hr - ft M ° F 3M 35. i-tit predicting k values in the critical region for gases and liquids. Although no good method was found to do this, I believe that much of what was done in this area is valuable informat ion. Not much is known about absolute values of k in the critical region, but it is known that as the critical point is approached k decreases rapidly for liquids and increases rapidly for gases. At the critical point the k vs. T curve for the gas and liquid meet with an infinite slope. Very few experimental data exist in the critical region except for the more simple gases (inert gases, N2, 02, CO, C02, NH3, H2, ete.)J and the hydrocarbon gases methane, ethane, propane, and butane. Abas-zade (19) has measured liquid k values up to the critical point for ben zene, acetone, methanol, and ethanol as have Scheffy and Johnson (19) also for acetone. In order to estimate critical values of k for other compounds two methods were used. One was developed by Gamson (4) and is applicable to any type of organic com pound. It gives values that appear to be substantially high in most cases. The second method makes use of a graph developed by Stiel and Thodos (21) based on experi 42 mental data for gases and liquids. The graph is a plot of a function of k-k° vs. reduced density where k° is the low pressure thermal conductivity. The graph is not good for polar substances or for H2 and He and has an estimated ac curacy of +10 to 20% near the critical point. The graph ranges from a reduced density of 0.04 to about 3 which in cludes the liquid region. Table D-l gives both experiment al and calculated critical k values for a number of com pounds . To analyze these critical k values, graphs of k vs. T were plotted for pentane, hexane, heptane, and benzene for both the gas and liquid phase and along the saturation curves. Experimental gas data for hexane and benzene are those of Vines and Bennett (25). Gas values for pentane were obtained from data given in Reid and Sherwood (13). All other k values for gases were calculated using the graph of St iel and Thodos. Prom these plots of k vs. T it is observed that the k values calcaulted by the Stiel and Thodos graph fit the experimental data the best. This fact islobserved by drawing smooth curves through the gas and liquid data and extrapolating through the critical region. In many cases 43 TABLE D-l Critical Values of k for Some Liquids kc calc. k0 calc. kc expt'l. kc expt’l. Liquid Gams on Stiel & Thodos Abas-Zade Seheffy & Johnson Pentane 0.0425 0.0289 - - Hexane 0.01+45 0.0294 - Heptane 0.0449 0.0296 - - Octane 0.0474 0.0295 - - Nonane - 0.0299 - - Decane - 0.0299 _ - Benzene 0.0427 0.0294 0.02784 - Toluene - 0.0282 - - Acetone o.o4oo - 0.04212 0•04212 Methanol - - - - Ethanol 0.0486 0.03389 _ k is in BTU/hr-ft-°F. 44 the gas and liquid curves can be made to join with infinite slope very close to the values calculated using the Stiel and Thodos graph. These graphs are presented as Pigs. D-3 to D-6 . Some effort was also spent attempting to predict k at elevated temperatures by another method. Tsederberg (23) states that the ratio of X of a liquid at various temperatures is given by * 1 ^ 2 * ( ?i/?2)4/3 (D“I) This equation was tested up to 250°F for a number of polar and non-polar liquids. Results varied considerably for both types of liquids. Believing that an equation of the form of Eq. D-I could have some merit a number of ratios involving P and Us were tested. Two equations which give results usually within ±5% up to 250°P are * iA 2 - ie2/ e 1)(uSl/v82) (d-ii) and kl/ k 2 “ (^>2/ ^l)4/3 (Usi/us2) (D—III) The final correlation presented in this thesis 45 FIGURE D-3 Thermal Conductivity in Critical Region for Pentane + -t + 4 * ■ * 4 • 0“ ‘ 4 ‘ —1 | ‘ — j - 4 - > . ; P 1 - * • -Y-\o.oU 0.03 0.02 Expt'l^Liquid Data • Expt’l. Gas Data [‘ * + .*» 4 4 + - 4 + . . . f - - t - t 4 Calc. j Gas; Data f ' ; ; Calb+ . ;:dS%15ar")?5iirnt: a. o r t V Tim 960 • t . - i ~ + 46 FIGURE D-U Thermal Conductivity in Critical Region for Hexane HI E i ^ i : l ' p , + .jrLi, quidII)ata+ Expt'1.'Gas Calc .f Gas Data 0.01 Data 4 - * C al c . Critic al lRo iht 0.00 600' ' t 4 ' :800' ' { - • + - t - * + + + • • » + • r H . + • ; '9oo: t + • - • + • - » » - < - t + * t + i - + • • + 1000 i-, n + + 1 h + 47 FIGURE D-5 Thermal Conductivity in Critical Region for Heptane 0 .AT 0.06 :o.o5. o.oU 0.03 O'.Ol E ^ p - t j . l , , : _jLiq.uid_Data. Calct : G a ' s I Cat a , 1 " Calc I t Critical: Poin 0.00 600 : 800 560 900 1000 700 48 FIGURE D-6 Thermal Conductivity in Critical Region for Benzene o.'oT. ^ — I —4 - + 0.06 ; --+ — + + - 0^05 j •ir t - 3*^-+_Expt ' ' I . . Liqui’ dtData ^K-!-Expt'll > Gas bata I ■ + < - 4-1 . . . | T . j + ~P^~Calcv-Gao--l)ata • * + ■-+- - ■ f — 1— ‘ — I — I — +--i— | * f + * • | ' ‘ > + - + • + Calc.+ Critical ’ Point ; 0 • 00 ■ 6oo 500 900 TOO 1000 49 between k and Us for hydrocarbons gives better results than Eqs. D-II and D-III and is easier to evaluate, But, it is possible that for polar compounds some form of these relations may be useful in predicting k at elevated temp eratures. BIBLIOGRAPHY 1. American Petroleum Institute, “Properties of Hydrocar bons of High Molecular Weight,” Project 44, Penn sylvania State University, 1940-66. 2. Bhatia, A. B., "Ultrasonic Absorption," Clarendon Press, Oxford, 1967. 3. Bridgeman, P. W., Proc. Am. Acad. Arts Sci., j>9, 154 (1923). 4. Gamson, B. W. , Chem. Eng. Progr. , 415, 154 (1949). 5. Horrocks, J. K. , and McLaughlin, E., Proc. Roy. (Lon.), A273Y 259 (1960). 6 . Jamieson, P. T., and Tudhope, J. S., “The Thermal Conductivity of Liquids— A Survey to 1963," Nat. Engr. Lab. Report No. 137, Dept, of Sci. and Ind. Research (Great Britain), 1964. 7. Kardos, A., Forsch. Gebiete Ingenieurw, 5, 14 (1934), (Original not seen.) 8 . Malian, G. M., "Thermal Conductivity of Liquids," Ph.D. Dissertation, University of Southern California, 1967. 9. Mason, H. L., Trans. A.S.M.E., 76, 817 (1954). 10. McLaughlin, E., Chem. Revs., 64, 389 (1964). 11. Michaelian, M. S., "Liquid Thermal Conductivities," M.S. Thesis, University of Southern California, 1968. 12. Nozdrev, V. F., "Application of Ultrasonics in Mole cular Physics," Gordon and Breach Co., New York, 1963. 50 i 51 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. Reid, R. C., and Sherwood, T. K., "Properties of Gases and Liquids," 2nd ed., McGraw-Hill Book Co., New York, 1966. Riedel, L., Chem. Ingr. Tech., 23., 465 (1951), (Original not seen.) Robbins, L. A., and Kingrea, C. L., Hydrocarbon Pro cessing and Petrol. Refiner, 41 (3), 133 (1962). Sakiadis, B. C., and Coates, J., A.I.Ch.E. J., JL, 275 (1955). Sakiadis, B. C., and Coates, J., A.I.Ch.E. J., _3, 121 (1957). Sakiadis, B. C., and Coates, J., Studies of Thermal Conductivities of Liquids, Part IV,/ Louisiana State University, Eng. Expt. Sta. Bull 46, 1954. Scheffy, W. J., and Johnson, E. P., J. Chem. Eng. Data, 6., 245 (1961). Scheffy, W. J., and Johnson, E. P., Project SQUID, Tech. Report PR-85-R, Princeton Univ., Princeton, N. J. (1958). Stiel, L. I., and Thodos, G., A.I.Ch.E. J., 10, 26 (1964). Timmermans, J., "Physico-Chemical Constants of Pure Organic Compounds," Elsevier Book Co., New York, 1950. Tsederberg, N. V., "Thermal Conductivity of Gases and Liquids," The M.I.T. Press, Cambridge, Mass., 1965. Vllim, O. , Collection Czech. Chem. Commun. , 25., 993 (1960), (Original not seen.) Vines, R. G., and Bennett, L. A., J. Chem. Phys., 22. 360 (1954). 52 26. Weisberger, A., "Techniques of Organic Chemistry,” Vol. VII, Interscience Pub. Co., New York, 1955. 27. Ziebland, H., Int. J. Heat Mass Trans., 2, 273 (1961). J

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Creator Sadoian, Kenneth Craig (author)

Core Title Thermal conductivity and sonic velocity of liquids

Degree Master of Science

Degree Program Chemical Engineering

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Advisor Lockhart, Frank J. (committee chair), Partridge, Edward G. (committee member), Rebert, Charles J. (committee member)

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