, PURE THERMAL DIFFUSION ' f Thesis for theivDe'gre‘el‘bflPhtDy ‘; " > I 'CMICHIGANPSTATE UNIVERSITY: ‘ ' , ‘ TERRYGRANTANDERSON ' ' 1968 7 ' “i w. -- LIBRAI: W Michigan State _ Universit‘ “L! This is to certifg that the thesis entitled PURE THERMAL DI FFUSI ON presented bg TERRY GRANT ANDERSON has been accepted towards fulfillment of the requirements for PH.D degree in Chemistry Wflw Major professor Date September 9, 1968 0-169 u.” -. 1-:- "n «"4 :\. ":‘t ‘\ ABSTRACT PURE THERMAL DIFFUSION by Terry Grant Anderson The time—dependent theory of pure thermal diffusion in binary fluid mixtures is obtained, and experiments on the carbon tetrachloride—cyclohexane system are reported. The theory takes fUll account of the temperature and composition dependences of the ther— mal diffusion factor, thermal conductivity, mutual diffusion coef— ficient, and density. The second order partial differential equa— tions which describe simultaneous transport of heat and mass are solved approximately by means of series expansion methods in both time and space. Inclusion of the effects of time—dependent temper— ature and center of mass velocity gradients during the warming up period yields unambiguous identification of zero time. Inclusion of the variability of the coefficients makes it possible to evaluate the effects of the usual assumption of constant coefficients. A laser wavefront shearing interferometer is used for in_§itu_measure- ments of refractive index gradients. Improved cell design and care- fifl.temperature control have eliminated the effects of convection, previously the chief source of difficulty in pure thermal diffusion experiments. Measurements made during both the approach to the steady state (demixing) and the diffusional decay from the steady State following removal of the temperature gradient(remixing) are analyzed with the help of computerized curve fitting programs. Ex— n Terry Grant Anderson periments at four different mean temperatures and over the entire composition range yield, with a precision of about 1%, d1 = — 1.88 + 0.18xl + 0.0l(T — 25), 105 D = 1.29 + 0.19xl + 0.26(T — 25), where a1 is the thermal diffusion factor, D is the mutual diffusion coefficient in cm2 sec‘l, x1 is the mole fraction of C01”, and T is the temperature in degrees C. The thermal diffusion factors at 250 agree with the flow cell results of Turner, Butler, and Story (225231. Faraday Soc. 63, 1906 (1967)), and the mutual diffusion coefficients at 250 and 350 agree well with the results of Kulkarni, Allen, and Lyons (J. Phys. Chem. 69, 2491 (1965)). The temperature dependence of these parameters has not previously been available. New results are also reported for the temperature dependence of the refractive index of the pure components. It now appears that pure thermal dif— fusion can be a reliable experimental method when adequately de— scribed and carefully executed. PURE THERMAL DIFFUSION BY Terry Grant Anderson A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1968 [Iii-H git/ox; 3~/ 31H To Janie ii I am gri 599reciati°n to ie Completion ( Richard Menke of hohelped immea 5:1 parts 0f the heiigan State U who constructed kirk, who WrOte its extremely us I am gra for the faciliti' w the Departmen' equipment and COI “he form of a Science Foundati< 3334. purchased 5°! a Research As Iwish es :5 he Computer pr ht ahandling and ACKNOWLEDGMENTS I am grateful for this opportunity to express my appreciation to those who played such important roles in the completion of this work: Mr. Russell Geyer and Mr. Richard Menke of the Chemistry Department Machine Shop, who helped immeasurably with the construction of the cell and parts of the interferometer; Mr. Frank Galbavi of the Michigan State University Glass Fabrication Laboratory, who constructed the glass sample chamber; and Mr. Dwayne Knirk, who wrote a computer program for data plotting which was extremely useful. I am grateful also to: Michigan State University for the facilities provided and for a one—year fellowship; to the Department of Chemistry, which purchased permanent eqUipment and computer time and provided financial support in the form of a Teaching Assistantship; and t0 the National SCience Foundation, which, through its grant number NSF GP 05254, purchased specialized equipment and provided funds for a Research Assistantship. I wish especially to thank my colleague John Bartelt, WhOse computer programs MULTREG and MINIMIZE certainly eased data handling and yielded more information than would L otherwise have hours of enligl ' to both the the work. I can < Mr. F. H. Hc work so challer hours of labor, Finally helped in innum 55d whOSe ufider iViilab1e_ a =;.rr-.: _» -...' '. a -, 7—- otherwise have been obtained, and with whom I enjoyed many hours of enlightening discussions which contributed greatly to both the theoretical and experimental aspects of this work. I can only begin to convey my sincere appreciation to Dr. F. H. Horne for his participation, which made the work so challenging, and for his encouragement and countless hours of labor, which helped make it so rewarding. Finally, I am deeply grateful to my wife JaneAnn, who helped in innumerable ways, who endured my difficult times, and whose understanding and encouragement were always available. iv DEDICATION . . ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES LIST or APPEND] Chapter 1- INTRODUC 3- Motiv C.Plan H H EQUATION A. Intro - Equat C- Equat The ExPear - Bound Pur Simpl L’JU SOLUTION DEDICATION TABLE OF CONTENTS 0 o o o o o o a o a a o n o o 0 ACKNOWLEDGMENTS . . . . . . . . . . . . . . LIST OF TABLES . . . . . . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . . . . . LIST OF APPENDICES Ihapter I. II. o o c o a e I o 0 o o 9 INTRODUCTION . a . . . . o . . . . . A. The Phenomenon . . . . . . . . . . B. Motivation . . . . . . . . . . . . C. Plan of the Thesis . . . . . . . . EQUATIONS OF TRANSPORT . . . . . . . A. Introduction . . . . . . . . . . B. Equations of Hydrodynamics . . . . C. Equations of Nonequilibrium Thermodynamics . . . . . . . . . D. Experimental Transport Parameters E. Boundary Conditions for Pure Thermal Diffusion . . . . . F. Simplifying Assumptions . . . . . SOLUTIONS . . . . . . . . . . . . . . Previous Solutions . . . . . . . . New Solutions . . . . . . . . . . Center of Mass Velocity . . . . . Temperature Distribution . . . . . Composition Distribution . . . . . Working Equations . . . . . . . . Composition Distribution During Remixing . . . . . . . . Calculation of the Ordinar Diffusion Coefficient . . . . . Discussion . . . . . . . . . . . . Page ii UlLAJI-J I—‘ ONOWON 25 29 32 32 40 41 56 67 69 72 73 Chapter IV. EXPERIM A. Intr. B. The 1 C. Tempo am D. The I E. Work: V- EXPERIMI A. Weigh B. Step- C- DiSCI D- Publj CC] 3. Calik F- Methc “- EXPERIME A. Tabul r ErrOI - Resul - Discu . Tempe of menu: VH- CONCLUSI 1" Smuma 3- sugge BIBLIOGRAPHY . APPENDICES . . Ipter Page EV. EXPERIMENTAL APPARATUS . . . . . . . . . . 76 A. Introduction . . . . . . . . . . . . . . 76 B. The Cell 0 0 I O O I O I I O O I I O 80 C. Temperature Control and Measurement . . . . . . . . . . . 88 D. The Interferometer . . . . . . . . . . . 103 E. Working Equations . . . . . . . . . . . 114 V. EXPERIMENTS . . . . . . . . . . . . . . . . 117 A. Weighing Procedure . . . . . . . . . . . 117 B. Step-by—Step Procedure . . . . . . . . . 128 C. Discussion of Procedure . . . . . . . . 135 D. Published Data for CCl4 and C6H12 . . . . . . . . . . . . 139 E. Calibration of Interferometer . . . . . 149 F. Methods of Calculation . . . . . . . . . 151 E. EXPERIMENTAL RESULTS . . . . . . . . . . . 164 A. Tabulation of Data . . . . . . . . . . . 164 B. Error Analysis . . . . . . . . . . . . . 167 C. Results . . . . . . . . . . . . . . . . 180 D. Discussion . . . . . . . . . . . . . . 182 E. Temperature Dependence of Refractive Index . . . . . . . . . 196 I 0 CONCLUSION . o o I I I o I I o o o O o I o 2 01 A Summary . . . . . . . . . 201 B. Suggestions for Further Work . . . . . . 204 LIOGRAPHY . . . . . . . . . . . . . . . . . . . 209 ENDICES I o I a u o I o I o o I I I I I I I I I 214 vi Table N m Approxi: parame te 25° C , o 2b- Relative the flu) T = 25° “1 Steady s 2c' Levels c 5a. Lot anal bottle 1 ' Lot anal bottle 1 5“ SamoIe w. ' ”ensue, - Pore Com] ' t°mpositg lndex f0J at 200 C ‘ Thermal ,- results 1 BS! t = 7 mle 2b. LIST OF TABLES Approximate values of transport parameters for CCl4 — C6H12 at 25° C, w = 0.5 O O I O I D O l D 0 1 Relative contributions of forces to . _ o = the fluxes, CCl4 C6H12’ W1 1 0.5, Tm = 25° C, dT/dz = 5 deg cm' , steady state . . . . . . . . . . . Levels of assumptions . . . . . . . Lot analysis as given on bottle label for CCl4 . . . . . . . Lot analysis as given on bottle label for C6H12 . . . . . . Sample weighing form . . . . . . . Den51t1es of CCl4 - C6H12 m1xtures Pure component refractive indices . Composition dependence of refractive index for CCl - C H mixtures o6 12 at 20° C and 6563A. . . . . . . . Ordinary diffusion coefficient for CCl4 — C H 2 mixtures (Kulkarni et al., I965) . . . . . . . . . . Thermal diffusion factor 0‘1; previous results for mixtures of CCl4 ' C6H12 Measurements of fringe shape. Run B5, t = 70.00 min, photo #155 . . . Fringe position do(t) for Run F6 . Sample laboratory notebook record of a pure thermal diffusion experiment Summary Of experiments . . . . . . Experimental uncertainties . . . . . V711 Page 23 24 31 119 120 126 140 144 146 147 154 158 163 165 179 Figure 3.1 Compos z for w: (cc dT/dz: 3-2 Center as a fi C6HlZ’ differo SyntheI 4'1 Water < Overall 4'2 Assembl Positic Helium. 4‘4 Path 0 Shearin Interfe and P01 P1011 of (Cir Cle. 5.2 Plot of Z = 0 a (Circle CurVe. imprOpe 6.1 Experim, factOr functim at 250 I LIST OF FIGURES gure .1 Composition wi as a function of z for t/e = 0.33, 1.0, 2.0, 6.0; o _ o _ wl (CCl4) - w2 (C6H12) — 0.5, dT/dz = 5 deg cm'l . . . . . . . . . . . 2 Center of mass velocity at z = 0 as a function of time for CCl - C6H12’ AT/a = 5 deg cm‘l; finite difference solution (—-——) and synthetic function ( ) 1 Water deflecting channels in reservoir. Overall dimensions 8 in.><8 in. . . . 2 Assembled cell in tilted position for filling . . . . . . . . . . w Helium-neon laser and lenses L1 and L2 . 1 Path of light beam undergoing shearing in Q1 . . . . . . . . . . . . . Interferometer components and Polaroid camera . . . . . . . . Plot of measured fringe shape . (CirCleS)showing agreement w1th fifth- order polynomial (solid line) . . . . . Plot of fringe displacement at Z = 0 as a function of time (circles) and least squares curve. Dotted line shows improper extrapolation . . . . . . . . . EXperimental thermal diffusion factor for CCl4 — C6H12 as a function of mole fraction CCl4 at 25° C I o I o I a o o o I a n I o O 0 viii Page 39 42 82 86 105 108 111 155 159 182 Figure 6.2 as Experime factor f function at 25° C previous Experime factor a at three Experime; for CCl applied . xi = 0.5 BxPerimel coefficie fuIICtion Experimer Coefficie function CompariSc 7 Experimen coefficie function °°mpositi 3 .— Experimental thermal diffusion factor for CCl - C6H12 as a function of mole fraction CCl at 25° C; comparison with previous results . . . . . . . . 4 Experimental thermal diffusion factor as a function of temperature at three compositions for CCl4 — C6H12 . . Experimental thermal diffusion factor for CCl4 — C6H12 as a funct1on of applied temperature gradient; 0 — = O I I I X1 — 0.5, Tm 25 C o I I o o I 0 0 Experimental mutual diffusion coefficient for CCl4 - C6H12 as a function of mole fraction CCl4 at 25° C . Experimental mutual diffusion coefficient for CCl - C H 2 as a function of mole fraction at 25° C; comparison with previous results . . . . . Experimental mutual diffusion coefficient for CCl4 — C6H12 as a function of temperature at three compositions . . . . . . . . . . . . . . . ix Page 183 186 188 190 192 194 Appendix A. US ('3 a U1 ”:1 H521 L. A Relat: Coeffic: Simplifi Contribl the Ten; Temperat Conduct-,1 - Perturba Temperat - Perturba State Co Shearing SUBRooTI; SUBROUTII PROGRAM J LIST OF APPENDICES Idix A Relation Between Phenomenological Coefficients . . . . . . . . . . . . . Simplified Composition Distribution . . Contribution of Convection to the Temperature Distribution . . . . . Temperature Distribution Due to Heat Conduction During the Warming—Up Period Perturbation Solution for the Steady Temperature Distribution . . . . . . Perturbation Solution for the Steady State Composition Distribution . . . . Theory of the Wavefront Shearing Interferometer . . . . . . . . SUBROUTINE MULTREG . . . . . . . . . . SUBROUTINE MINIMIZE . . . . . - - . . PROGRAM ALPHA . . . . . . . . . . . - . Page 215 219 224 227 231 235 238 245 252 260 Thermal he to a temper. °f Studying the arrangement. I] Irate the compo; airY. or mutual, ThermOgI hikes use of the stable conVectiV thing a horizc Iethod (Turner E through a 1101.120 Radiant Causes menu“: and a he fluid into p CHAPTER I INTRODUCTION I Phenomenon ___________ Thermal diffusion is diffusion which takes place a temperature gradient. There are several methods dying the phenomenon, each with its own experimental ement. In all cases, thermal diffusion acts to sep— the components of a mixture and is opposed by ordi— >r mutual, diffusion. Thermogravitational thermal diffusion (Horne, 1962) lse of the earth's gravitational field to set up convective fluid flow in a vertical apparatus con— 'a horizontal temperature gradient. The flow cell (Turner et al., 1967) utilizes forced laminar flow a horizontal channel. A vertical temperature t causes a partial separation of the components of id, and a horizontal knife edge is used to divide id into portions for measurement of composition 1CeS. In a third method (Dicave and Emery, 1968), hers equipped with stirring devices contain the to be studied and are separated by a porous glass “a membrane. When the chambers are maintained at ferent temperatures, thermal diffusion takes place be— en them, and a composition difference develops. A fourth method, pure thermal diffusion, is con- ually the simplest. Here, the fluid mixture is con— ed between two horizontal flat metal plates attached eservoirs for individual temperature control. When a ical temperature difference is applied in such a way the densest portion of the fluid is closest to the er of the earth (i.e., when the fluid is heated from 2 except when its density increases with increasing Irature) thermal diffusion occurs. Thermal diffusive ing continues, opposed by the remixing tendency of ary diffusion, until a steady state is reached in the two effects balance each other, and a steady cal composition gradient is obtained. Given the appropriate mathematical description of Istem, one can calculate experimental values for the 11 diffusion transport parameters by measuring the 1P Of the composition gradient, its steady state value, 1 decay to zero following removal of the temperature ence in a remixing experiment (Gustafsson gE_§l;r Thermal diffusion has been studied for many years. 9W article by Grove (1959) lists over 900 references. heless, the phenomenological theory of pure thermal sion (Ludwig, 1856), or the Soret effect (Soret, 1879), zen inadequate in many cases. 1 \ Iivation Thermal diffusion has a wide range of applications. thermal transport in living systems is certainly of t to biologists. Some chemically indistinguishable s and isomers can be separated efficiently by means ned thermal diffusion techniques (Mulliken, 1922). fical mechanicians interested in fundamental knowledge 1liquid state require accurate experimental values of Irt parameters in order to judge the validity or range r theories (see, for example, Bearman and Horne, The most complete phenomenological theory of pure diffusion previously available (Bierlein, 1955), l adequate in many cases, is limited in the follow— . (1) Transport parameters are treated as constants, , for instance, that the temperature gradient is throughout the fluid. (2) The composition depen- density, or the "forgotten effect" (de Groot gp 3), is neglected. (3) "Warming up effects" (Agar, nsequences of the fact that the temperature builds up in the fluid not instantaneously but isurable period of time, are neglected. No allow~ .de for the possibility of convective transfer mass during the warming up period. Our primary purpose here is to obtain a phenomeno— 11 theory of pure thermal diffusion which is not sub— :0 the above restrictions and which, hopefully, will e the discrepancies which now exist in the literature ihomaes, 1951; Horne, 1962; Turner et al., 1967; ein, 1966) between the reported values of thermal 'on coefficients obtained from the different experi— methods. In addition, we hope to explain the dif— reported (Dicave and Emery, 1968) between ordinary on coefficients measured during a nonisothermal g experiment and those measured during isothermal 1g after a steady state has been reached. The second purpose of this work is to show that rect application of an adequate phenomenological of pure thermal diffusion to well designed experi- an lead (for the first time) to reliable results :e thermal diffusion. Hopefully, this technique 1 be used for any number of systems to obtain un— 5 results more easily than from the more complicated, 1 understood methods. )ur third purpose, implied above, is to present values of thermal diffusion parameters for the strachloride—cyclohexane system as a function of re and composition. By so doing, we shall demon— 3 significance of our theory and provide the first :ive data for the temperature dependence of thermal parameters. lan of the Thesis ( In the following treatment we make full use of the ions of hydrodynamics and nonequilibrium thermodynamics 1 L scribing the simultaneous transport of heat and mass in ,. rd system undergoing pure thermal diffusion. We solve a I gsulting set of partial differential equations for the l ) (f a two component fluid by means of a series expansion ji 1 which retains explicitly the temperature and composi— \ r I spendences of the thermal diffusion factor, ordinary fon coefficient, thermal conductivity, and density. 9 include time dependent boundary temperatures and the ‘ ility of convective transport. The results of the theoretical section are used in :ting values for the thermal diffusion factor and the y diffusion coefficient for mixtures of carbon tetra- e and cyclohexane over the entire range of initial tions. Both classical demixing and isothermal remix— ariments are described. A sensitive laser wavefront ' interferometer used to measure very small refrac- ex gradients in volatile liquids is also discussed. the results of our experiments with the CCl4 — stem are presented together with a discussion of ‘imental uncertainty and a comparison of our results ious results. CHAPTER II EQUATIONS OF TRANSPORT troduction In this chapter we present the differential equa— which describe macroscopic transport phenomena. lized equations for pure thermal diffusion and their iriate initial and boundary conditions are then pre- 1. We consider only continuous, isotropic, nonpolar— 2 fluids in which no chemical reactions occur and are subject to no external forces other than the ational field. For a more detailed discussion of nations which follow see, for example, works by (1966), Kirkwood and Crawford (1952), de Groot and 71962), and Fitts (1962). tions of Hydrodynamics For a fluid containing v components there are v lent equations of continuity of mass: (do/cit) + pV-g = 0 r (2.1) ll 0 p(dwa/dt) + V-ga a = l,...,v - l , (2,2) e p is the local fluid density, t is time, u is the er of mass, or barycentric, velocity, and Wu and jg respectively the mass fraction and diffusion flux of onent a. The barycentric velocity u is defined by (2.3) I "546 u: ~ a W u Q~a l = u is the velocity of component a with respect to a a ‘atory reference frame. The diffusion flux jg is de— [by id = pa(Bg ‘ E)' a = 1,...,v ' (2'4) pa = wap. The diffusion fluxes are not all indepen~ E , j = 0 . (2.5) g=1 ~a intial time derivatives d/dt are related to local time ltiVeS a/at by d/dt = (a/at) + u°V . (2.6) erator "del" is defined by lw +3: +1~< Z, (2.7) 3‘?!” SI” 0) V E i I j, and k, are the unit vectors of a three dimen— Cartesian coordinate system. The equation of motion of the fluid is p(du/dt) — V°g = 0g r (2.8) is the gravitational field, and where o is the s tensor, given approximately by the linear phenomeno— al relation g = — [p + (g n — ¢)(V-u” % + 2n sym Vu . (2.9) (2.9) p is the pressure, sym Vu is the symmetric If the tensor Vu, and n and y are the coefficients ar Viscosity and bulk viscosity, respectively. Com— on of Eqs. (2.8) and (2.9) yields the Navier—Stokes an: t) + V[(% n — ¢)(V.E)] — 2V-n sym Vu = pg — Vp (2.10) The general equation of continuity of total energy (BpEf/at) + V-gET = O , (2.11) E is the total flux of total energy, and where the T nergy E is the sum of the internal energy E and the T inetic energy of the center of mass: V1 E=E+W+Z§wu . (2.12) ‘2.12) we have further separated E into a thermal .nd an external potential part, where W1 is defined (2.13) 150 that 2 ua — Ba Ba . (2.14) The equation of energy transport can be expressed, 1e negligible terms or order j: are ignored, as p(dE/dt) + VojE = 0:Vu — pu-g , (2.15) E is the internal energy flux not due to bulk flow: QE 2 9E — pEE + E '9 . (2.16) tions of Nonequilibrium nodynamics We can recast the equations of the preceding section :e convenient form by making use of some of the re- ? nonequilibrium thermodynamics. In order to use such as temperature and entropy which are defined namically only for equilibrium states, i.e., for simultaneously in mechanical equilibrium, thermal ium, and chemical equilibrium (see Bartelt, 1968), cessary to postulate their existence in systems not ibrium. That postulate is (Fitts, 1962): Postulate 1 ‘or a system in which irreversible processes :aking place, all thermodynamic functions of s exist for each macroscopic volume element .e system. These thermodynamic quantities he nonequilibrium system are the same func- of the local state variables as the corres- ng equilibrium thermodynamic quantities. fac 10 Unfortunately, the historical and universal name of postulate is "the Postulate of Local Equilibrium." In , local equilibrium is ESE postulated. Instead, we are Alating that it is permissible to use the properties relationships defined in equilibrium thermodynamics :mostatics). An alternative approach is to construct EEEE a nonequilibrium thermodynamic theory in which )py, temperature, etc. are defined in context. This ach has been developed by the practitioners of con— m mechanics (Truesdell and Toupin, 1960; Coleman and 1963; and Muller, 1968), and its relationship to the tional postulatory extension of thermostatics is cur— ] under investigation (Bartelt). It appears that the >perational equations result from both approaches, .fferences between the approaches are therefore of (sequence for our present work. For simplicity of tion, we adopt the traditional postulatory approach. tulate l, we may use the Gibbs equation for dE v dE = TdS — pdv + Z padwa + dWl , (2.17) 05—1 2 Gibbs—Duhem equation, _ v _ _ + = 0 . SVT Wp + E waWa g , (2 18) a—l is the temperature, p is the pressure, E, 5, and respectively, the specific energy, specific entrOpy, 11 a specific volume, and “a is the chemical potential, 3 units, of component a. Each of the total specific iynamic functions is a weighted sum of the partial .c functions; for example, v E = Z waEa , (2.19) a=l E = (BE/8w )— — . (2.20) a a S’V’Wl’w8#a Application of the chain rule for differentiation p'wg’wl) yields ’dt) = (5% — pV )(dT/dt) — (TVB — pVB')(dp/dt) ~ + g (fig - E§)(dwa/dt) ~ u'g , (2.21) is the specific heat capacity at constant pressure tant external fields, 8 is thermal expansivity, __1 _ B E V (av/3T) , (2.22) p’wa'wl athermal compressibility, B' = — V‘l(avya ) (2 23) ‘ p T,wa,Wl ’ ' : partial Specific internal energy. Application .ain rule for differentiation to the equation of 0(T,p,wa) gives a similar relation, v-l dw dT dp 2 — _ — 98 a? + 08' 3E — p agl (Va vv) EE— . (2.24) 12 Lergy transport equation (2.15) can be restated by ,tuting Eqs. (2.1), (2.2), (2.20), and (2.24) into °V(Ha — H ) , (2.25) \) u—l dT _ dp _ _ . _ . 3? T6 dt ‘ ¢1 V 9 “El la $1 is the entropy source term for bulk flow, $1 E (o + pl):Vu , (2.26) Le heat flux, V — q 5 3E - Z gaHa , (2.27) “ ~ a=1 is partial specific enthalpy. One observes empirically that for nonisothermal the heat flux is proportional to the temperature : (Fourier's Law). Similarly, in an isothermal the diffusion flux is proportional to the composi- ldient (Fick's Law). The generalization of these :ions, as well as an extension to include cross a such as thermal diffusion, is expressed by the ostulate of nonequilibrium thermodynamics, that of henomenological equations (Fitts, 1962): Postulate 2 "The fluxes {a are linear, homogeneous functions of as Za' That is, V J = E L y " (2.28) 13 rces are "driving forces" for the fluxes; for example, the driving force for the heat flux in a single com— fluid. The phenomenological coefficients La 8 are ndent of the forces. The diagonal coefficients Lad conjugate fluxes and forces, while the off-diagonal ts La8(a # 8) give rise to cross phenomena. Although oice of fluxes and forces is to some extent arbitrary, guidelines are provided by the Second Law and by ial order (de Groot, 1962; Fitts, 1962; Bartelt, We shall use the set most convenient for our pur- We have already used Postulate 2 in writing Eq. Postulate 2 is demonstrably invalid for many ex- ntal situations, notably those in which chemical ans are occurring and those in which viscous dissi- is significant. It seems to be quite satisfactory, ?, for situations in which only heat and matter fluxes >ortant, such as thermal diffusion. The range of 1y of Postulate 2 is delineated in the continuum CS approach mentioned earlier. AS forces conjugate to the fluxes of heat and matter, se V Kn T and VT(uB — uv), B = 1,..., v - l, where = — = l ... v . (2.29) VTUB _ VuB + SBVT, B I I llate 2, v—l — (2.30) -g = (200V Zn T + 821 QOBVTWB 11v) 14 v—l = naov 2n T + 821 QGBVT(uB — “v’ a = 1,...,v —.1 ' (2.31) the 9's are the phenomenological coefficients. As a consequence of Eq. (2.5) we have ‘2’ 9 = 0 , B = 0,1,...,v . (2.32) a=1 dB due to the requirement of positive definite entropy :tion (see Appendix A) we have v Q = 0 , a = 0,1,...,v . (2.33) d=1 as An expression for the gradient of the chemical tial, which appears in Eqs. (2.30) and (2.31) can be led from thermostatics and the chain rule for rentiation: ml Vu = — VT + V v — , (2.34) B B BVP + “8 w 9 : (2.35) “80" - (BUB/awa)Trplgle§£a ation of Eqs. (2.29) - (2.34) gives the following sion for the fluxes: v—l v-l _ — v + 0 (u 8 VV) P Bil 3:1 a8 Ba (V VEl 9 V Kn T + 9 a0 8=l d8 _ “ )VWY , a = 0,1,...,v — l , where jo E q . (2.36) 15 The equations for the fluxes can be written in the 7ing compact notation: v+1 ' id = ygo DaYEY r 9 = O,1,...,v — l , (2.37) 0L0 OLO = — = ..., - 1 aY Bgl “as‘“ey uvY) , v 1, v v D — Z (2 )7 av 8:1 a8 8 v y.r\)+l = — 8;]. gas 0 E0 = V in T ~Y=VWY IY=lluco[v_l E. = Vp €v+1=‘VW1 We make the following associations with traditional antal transport parameters: D00 is related to thermal conductivity; DdO’ a = 1,...,v— l, are related to thermal diffusion coefficients; I a — 1,...,v— l, are related to mutual D 9Y diffusion coefficients; 16 l) D0y’ a = 1,...,v — 1, are related to Dufour coefficients; L) D d = 0,...,v — 1, av' are related to sedimentation; ) Dd,v+1 = 0, d = 0,1,...,v - 1. We now have a complete set of 4(v + 1) tranSport ons: (2.1), (2.2), (2.10), (2.25), and (2.37). The ans can be solved for the 4(v + 1) quantities: ature, pressure, v — l compositions, three components center of mass velocity, and the three components of 3 the v mass fluxes. At this point our description system is complete and valid for any number of com— . Before proceeding to the solutions, however, we restrict our consideration to binary fluid mixtures , whence Eqs. (2.1), (2.2), (2.10), (2.25), and become: (dp/dt) + pV-u = O p(dwl/dt) + V-ji p(du/dt) - 7'9 = pg — dT dp _ _ .- _ ' . — H pcp d? _ TB 3? _ qbl V 20 21 V(H1 2) 3 (2 39) _ 2d _ £0 DaYEY ' a = 0,1 r ° 17 DdO = Qdo a1 leull/WZ Dgz Qcalm-1 — V2) Dd3 _ 0 E0 = V in T Fl = le 52 = VP E3 = ‘ VW1 - (2.40) When the only external force is gravitational, a librium system containing v components may undergo ) types of transport processes in addition to vis— enomena. For a binary system, the six types are 2e contributions to the heat flux jo and the three ltions to the mass flux jl resulting from the gra— >f temperature, composition, and pressure. We make the following associations between the (Ological coefficients and the traditional experi- ransport parameters D, d1, Qi, Ki, and 51 which PeCtively, the mutual diffusion coefficient, the diffusion factor of component 1, the heat of trans— somPonent 1, the initial thermal conductivity of Jre (When awl/Bz = 0), and the sedimentation co- : of component 1: D00 = 900 = TKi D01 = Q01“11/w2 = pDQi D10 = 910 = ‘ pDalwlw2 D11 = Qllull/WZ = ”D 911 = stl . (2.41) Ionsequence of Eqs. (2.32) and (2.33), the six pheno— an be expressed in terms of four independent coef- ts. The experimental mutual diffusion coefficient D is d by Fick's law for isothermal, isobaric mutual dif- in the absence of external fields, - jg = DVcl (2.42) jg is the diffusion flux relative to the velocity of nter of volume, and cl is the concentration of com- 1 expressed in units of moles per cubic centimeter. Lvalent form of Fick's law in terms of mass fraction 1ter of mass velocity is = pDle . (2.43) ‘ 21 The thermal diffusion factor a1 is defined by con- 9 the steady state of a pure thermal diffusion ex— t in the absence of pressure gradients and external = O =Vw - a w w V Rn T . (2.44) 21 1 1 1 2 l9 ommon thermal diffusion parameters may be expressed the relations 1 = — TDT,l/D I (2.45) d1 = — Tol , (2.46) T,1 is the thermal diffusion coefficient of component 01 is the Soret coefficient of that component (Soret, It follows from Eq. (2.5) and the independence of ces that 02 = — al. The composition gradient of nt 1 has the same sign as the temperature if a1 is e. Alternative expressions for the diffusion flux can ten by using the relations (2.45) and (2.46): — — -1 = pDle + pDT 1leZVT + pDQ’l‘Wl — V2)w2ulle (2.47) I — — —1 = * - 2.48 pD[le + olwlwzw + Ql(Vl v2)w2ulle] ( ) l (2.47) emphasizes the existence of the two phenomena .sion and thermal diffusion, while Eq. (2.48) con- y allows removal of a common factor from the three For an isothermal binary system the heat of trans— is defined by at = on. - (“9’ rs from Eqs. (2.38) and (2.41) that 20 D01 = oDQ’i . (2.50) phenomenon which involves a heat flux due to a composi- gradient in an isothermal system in the absence of rnal fields is the Dufour effect (Dufour, 1873), which be considered the inverse of thermal diffusion. For ids the heat of transport Qi is very small, and reports easurements are still subject to question (Rastogi and n, 1965). We discuss the magnitude of Qi as well as rical values of the other transport parameters later in chapter. The thermal conductivity coefficient Ki, measured he beginning of a pure thermal diffusion experiment re the composition gradient develops is given by Fourier's of heat conduction, _. q : KiVT . (2.51) attempt to measure the thermal conductivity of a mixture Pplying a temperature difference necessarily results in 1evelopment of a composition gradient (unless a1 = 0), mere is consequently an additional contribution to the flux due to the heat of transport. Thus the effective a1 conductivity is the sum of two parts, one of which .ds on Ki and VT and the other on Q: and le. At the Y state of a pure thermal diffusion experiment in the Ce of external fields we have _ -l (2.44) le - le leZVT . 21 Llows that at the steady state ' 9 = KfVT , (2.52) Kf = Ki + pDQla1w1W2/T . (2.53) could measure the difference between the thermal :tivity of the mixture initially and that at the r state, we could calculate Q3 directly. That dif— :e, however, appears to be smaller than the experi— _ uncertainties which arise while attempting to re it with present equipment (see Table 2a). The sedimentation coefficient 51 is defined by lering the steady state of an isothermal experiment .ch the gravitational field is the only external _ )Vp , (2.54) = 2.55 sl wz/ull . ( ) The fluxes in Eq. (2.39) may now be rewritten en— in terms of experimental transport parameters: - q = KiVT + pDQinl + pDQilel - V2)Vp l = - prlwlsz—lVT + pDle + stl(Vl - V2)Vp (2.56) 22 Before proceeding to a solution to the equations of lsport, we examine the relative magnitudes of the pheno— l occurring simultaneously inside an experimental cell. Table 2a are presented estimates or typical values of aral important parameters for the system CCl4 — C6Hl at 2 I when wl = w2 = 0.5. In calculating a value for 51 we a used the relation for the specific chemical potential: RT Ul(T,p,X,) = ui(T,p) + MI in (flxl) , (2.57) re fl is the activity coefficient of component 1 and Tlp) is the chemical potential of component 1 in the 1dard state defined by pi = xii? “l . (2.58) )rder to obtain an approximate value for 51 we take 1. In the earth's gravitational field, the steady e composition gradient which would develop due to mentation is: 8w 1 = _ x 10.6 cm“1 . §E_ 0.8 The relative contributions of the gradients of Erature, pressure, and composition can be estimated by I Eqs. (2.57) and inserting reasonable values for all 1e quantities which appear. Consider the steady state Pure thermal diffusion experiment for CCl4 — C6H12’ T = 25°C and w? = 0,5, If we use the numerical m 23 .e 2a.——Approximate values of transport parameters for CCl4 - C6H12 at 25°C, wl = 0.5 ltity Reference Value a 1.4 X 10"5 cm2 sec—l b 4 x 10_8 cm2 sec—l deg--l b 6 x 10'3 deg"1 b —1.7 c 2.4 X 10—4 cal cm_l sec”l deg-l - Ki c 5 X 10—8 cal cm“l sec-l deg—1 C 6. cal g—l Eq. (2.55) -3.2 x 10'6 cm‘1 d 1.1 g cm—3 V2 d —0.66 cm3 g-1 e 0.2 cal deg—l 9—1 aKulkarni, et al., 1965. bTurner, et al., 1967; Beyerlein, (in press). c Horne, 1967. d Wood and Gray, 1952. eHodgman, 1962. 24 lues in Table 2a and specify a temperature gradient of 5 ; cm—l, then the six terms of interest have the values ren in Table 2b, where we have also used Vp = - pg , (2.59) ch follows from Eq. (2.8) at the steady state and tially. Since sedimentation effects are observable ther initially nor in the steady state, and since there no reason to expect observable departure from Eq. (2.59) any time, we henceforth neglect pressure effects. (See 1e 2b.) .e 2b.—-Re1ative contributions of forces to the fluxes; 0- _oil= cc14 — C6H12, wl — 0.5, Tm — 25 c, dz 5 deg cm— , steady state. Temperature Composition Pressure Units —3 —9 —11 cal l x 10 7 X 10 8 X 10 cm sec — - -10 z +1.2 x 10 7 -1.2 x 10 7 1 x 10 g cm sec >ther experimental situations, such as thermal diffusion L centrifuge or in a flow cell apparatus, the influence e Pressure gradient must be re—examined. Since we have no practical interest in the pressure, riginal 4(v + 1) independent equations have been 25 ad to eleven. Furthermore, by inserting the expres- (2.57) for the fluxes into the three continuity ons (2.1), (2.2), and (2.25) we effectively reduce mber of dependent variables to five: temperature, ition, and the three components of the center of mass ty. In order to solve the set of differential equa— we must specify an initial condition and two boundary ions for each of the five unknowns. zdary Conditions for Pure rmal Diffusion ______________ Pure thermal diffusion requires that a vertical .ture gradient be maintained across a layer of fluid S not undergoing any type of forced motion. More— he sign of the temperature gradient must be such e denser portion of the fluid is closer to the cen— the earth than the less dense portion. For ordinary this just means that the top must be warmer than the There are exceptions, however. For example, water 5 freezing point would be studied with the top cooler : bottom. If the temperature gradient is purely vertical and ‘nly external force is the gravitational field, there 0 non—vertical components of any of the forces, and ntly the fluxes have no horizontal components. Al— t iS plausible that the center of mass velocity also ) horizontal components, the existence of vertical gra- of temperature, composition and pressure is not suffi- to prove that uX = uy = 0. Instead, we have at best vertical density gradient gives (see Eq. (2.1)) Bu Bu 1 apuz +Tx+‘§§'= ‘5 32 , (2.60) uich the steady state relation follows: 0) 'O ‘O||-' °’l (1. _ a Zn 0 VE—‘UZT. (2.61) ; point we make the additional assumption that all is vertical and that ux = uy = 0. The possibility (zontal components of E has been considered by Bartelt but is beyond the scope of this work. We denote by "ideal" the boundary conditions which in a purely one dimensional system. The possibility zontal components of the fluxes or forces arises sofar as the actual experimental boundary conditions ideal. Since it is possible to eliminate effectively sence of spurious thermal gradients by proper cell and temperature control, we confine our interest to undergoing vertical motion only. The three quantities which remain as unknowns are )erature, the composition, and the vertical component .ocal center of mass velocity, each of which is a . of vertical position and time. We choose the three 27 itions describing the interrelations between the three :tions to be the equations of continuity of mass (2.1), s fraction (2.2), and energy (2.25). In one dimension, equations remaining are: (dp/dt) + p(3uz/3z) = 0 (2.62) p(dwl/dt) + (ajlz/az) = 0 (2.63) pcp 3% = $1 _ (3%?) = 312 %E (El - E2) ’ (2'64) _ qz = Ki 33—: + pDQ’i 8:: . (2.66) The domain of the independent variables t and z is emi—infinite strip defined by r (2.67) a is the cell height. The earth's radius vector points direction of increasing 2. The choice of the center call for z = 0 follows from the odd spatial symmetry hich the temperature and composition profiles develop. since we use Taylor series expansions in 2 about the of the cell, it is convenient to choose that point to origin. 28 Although it is possible to begin a pure thermal dif— Jn experiment with an arbitrary set of initial conditions, icilitate comparison with experiment, we choose the ini— state (t = 0) to be an equilibrium one in which the erature and composition of the fluid are uniform and the er of mass velocity is zero. Thus we have: T(z,0) = Tm o wl(z,0) — wl a a uz(z,0) — 0 , - 7 < z < 7 , (2.68) Tm is any chosen temperature, and w? is the chosen Lng composition. The temperatures maintained at the upper and lower plates constitute the two boundary conditions for T, the impermeability of the boundaries provides the ting conditions. The complete set of boundary condi- is expressed as follows for t > O: T(a/2,t) = ¢h(t) T(-a/2,t) = ¢c(t) (2.69) jlz(a/2,t) = O le(—a/2't) _ 0 (2.70) uz(a/2,t) = 0 l o uZ(—a/2,t) — (2.71) 29 In Eqs. (2.70) the functions ¢h(t) and ¢C(t) express fact that a certain period of time is required to change boundary temperatures. Both quantities are functions of reservoir volume, water flow rate, metal plate material thickness, and temperature difference. Consequently are best determined empirically. The pure thermal diffusion problem has now been fully ented in terms of three differential equations (2.62), 3), and (2.64); three initial conditions (2.68); and 2 sets of boundary conditions (2.69), (2.70), and (2.71), the fluxes given by Eqs. (2.65) and (2.66). The various 1 of approximations usually made in going from first :iples to complete solutions are discussed in the fol— g section. @plifying Assumptions 1 We distinguish between three levels of assumptions ly made in obtaining working solutions to the equa— of transport. First are those assumptions inherent equilibrium thermodynamics and hydrodynamics such as which allow us to use equilibrium properties, linear enological relations for the fluxes, and continuum mechanics. These assumptions are fundamental and as re necessary starting points which must be retained. Second are assumptions of a more technical nature restrict our attention to certain types of systems, iCh can be realized experimentally and do not, in 30 nciple, introduce error. In this group are the assump— is of a two component fluid, purely vertical motion, :ial ideality of boundary temperatures, absence of ex- )al fields other than gravity, and the insignificance of ssure gradients. Since it is possible experimentally to .eve the requirements imposed by these assumptions, it :0 our advantage to incorporate them into the phenomeno— cal theory, the net effect being a simplification of differential equations. Third are assumptions which have been made in all ious descriptions of pure thermal diffusion, but which demonstrably incorrect and can lead to significant rs in the description of the phenomenon. This group ldes the assumptions of time-independent boundary aratures; uniform temperature gradient; no convective sport; and constant diffusion coefficients, thermal lsion factors, and density. The three types of assump- ) ‘, viz., (l) necessary, (2) unnecessary but desirable, 3) unnecessary and undesirable are summarized in Table In the next chapter we discuss in more detail the tions of the third group and the solutions to the ort equations which one obtains both with and without assumptions. Our goal is to obtain a description of hermal diffusion subject only to a minimum number of tions. 31 Le 2c.-—Levels of Assumptions. Unnecessary but Unnecessary and :essary Desirable Undesirable ;inuous Binary system Linear temperature u1d Vertical motion distribution ulate l Spatially uniform Tegpiiaggriiigdepen- ulate 2 boundary tempera— tures No external fields except gravity Sedimentation negligible Constant p, D, a 1! Zero convective velocity $1 = 0, Eq. (2.26) O) 312 5? (H1 ‘ H2) ‘ K . l 0 CHAPTER III SOLUTIONS >revious Solutions __________________ Previous phenomenological theories of pure thermal Fusion (see, for example, deGroot, 1945; Bierlein, 1955) :been obtained only after the following simplifications made. 1. The temperature distribution does not vary with Although not experimentally realizable, this assump— has been made in the past with the explanation that initial period of time during which the temperature is Jing is so much smaller than the time required to comm 3 an experiment that it may be ignored. Since the im— 1 discontinuity in the temperature gradient cannot be éved experimentally, there has been an uncertainty in lefinition of "zero time." The "warming up period" 15 at the start of an experiment and lasts until no .er changes are observable in the temperature distribu— Its length, of course, depends on the apparatus used, YPiCally may be three to seven minutes. The relaxation 9 for pure thermal diffusion increases with the square 32 33 e cell height. For most mixtures of carbon tetrachloride yclohexane near room temperature, for example, 0 = 120 nutes when the cell height a is expressed in units of meters. Consequently, for a cell height of one or two neters, which is not uncommon, the warming up period may significant portion of the total time for an experiment. (gar (1960) has considered the warming up period. He ) by shifting the time axis in order to compensate for (me during which the temperature gradient does not have eady state value. His subsequent treatment was other— nmodified and required that dT/dz = AT/a (3.1) 1 values of time, where AT is the temperature difference. 2. The temperature distribution is a linear function vertical position in the cell. This assumption is ob— ! not true while the temperature profile is changing .me. Moreover, it is true for the steady temperature >ution only if the thermal conductivity Ki is a con- .ndependent of temperature and composition and if the )Ution of the heat of transport is not considered. 3. The center of mass velocity uz is zero. It fol— om Eq. (2.1) that only if the velocity is nonzero may n99 in the density occur. Since density certainly from point to point, a consistent theory requires that onzero. Actually, both uZ and its effect on the com- a distribution are usually very small. Nevertheless, 34 ; desirable to retain the velocity as a variable so that effects can be discussed quantitatively. 4. The density p is independent of the temperature. assumption (see deGroot, 1945) results in a great sim- .cation (3f the differential equation (2.63). Although :omposition dependence of density, "the forgotten effect" :oot, §t_al., 1942), usually has only a small influence 1e composition distribution in a pure thermal diffusion riment, it cannot be ignored if one wants to be consis— . because both optical and gravimetric techniques depend :nsity changes due to composition changes. It is also fable to retain the "forgotten effect" in order to be to discuss quantitatively its effect. 6. The mutual diffusion coefficient D is independent :mperature. This assumption also simplifies Eq. (2.63) S not generally valid. A change in D of about one per— per degree is not unusual (Longsworth, 1957), nor are iments with twenty degree temperature differences oot, 1945). Hence, a complete treatment must allow for variations. 7. The mutual diffusion coefficient D is independent mposition. The remarks of paragraph 6 apply here as Note, however, that the range of compositions en— Ered in a pure thermal diffusion experiment is much 3r (about 1000 times) than the temperature range. Con— ltly, we anticipate a much smaller effect due to the sition dependence of D. 35 8. The function dl/T is independent of temperature. marks of paragraph 6 again apply. For mixtures of tetrachloride and cyclohexane near room temperature, ample, the function dl/T decreases about five percent gree. 9. The thermal diffusion factor ml is independent position. The remarks of paragraph 7 apply, with V1)T = 0.18 at 25°C, while d1 is about —l.75 (see : VI). 10. The product w w2 in Eq. (2.65) can be replaced This assumption of deGroot (1945) limited his ant to very dilute solutions, and is obviously not :e. 11. The product w in Eq. (2.65) can be replaced 1W2 leading terms of the Taylor series expansion about .nt wi: —°° —0 —°. 3.2 wlw2 — wlw2 + (l 2wl)(wl wl) ( ) n (1955) used this "tangential approximation" to ze the term in Eq. (2.65) which contains the pro— Wz, a parabola. The function can be approximated Y Point w? by its Taylor series expansion at that 2 O —°.3.3 = wiwg + (l — ZWI)(W1 - wl) — 2(wl wl) ( ) 36 pure thermal diffusion experiments, the maximum value — w?) is on the order of 10_3, so that neglect of the d term in Eq. (3.3) is justified. The use of Eq. (3.2) ivalent to replacing a small segment of the curve in ighborhood of W? by the tangent to the curve at that 12. The entropy source term 01 in Eq. (2.64) does not cute significantly to the temperature distribution. ssumption is reasonable, since ¢1 is due to bulk flow, is very small in a pure thermal diffusion experiment. dimension we have, approximately, Bu 2 z _§E . (3.4) 4’1 = (5“ +50) v below that for systems of interest the maximum value 7/32) is about 10_6 sec—l, making ¢l very small indeed. . a — — '3123'2(H1 H2) 1. For mixtures of carbon tetrachloride and cyclo- 13. The term in Eq. (2.64) can be with dT/dz = 5 deg/cm, that term, which is zero .ly and zero at the steady state, has a maximum value It 5 X 10“8 g? (El - H2). This can be ignored when td with the term uzpcb(8T/Bz) which is itself very We now consider the most complete solutions pre— available for the functions T, uz, and W1 for a ermal diffusion experiment. The theoretical 37 :iption under consideration is that of Bierlein (1955) 1 follows from all of the assumptions listed except :rs 4 and 10. Assumptions 1, 2, 12, and 13 result in a tempera— distribution of the form AT T(Z) —Tm+—2 mIN / (3.4) no mention of time dependence. The center of mass ity is simply stated in the third assumption: u = 0 , (3.5) 2 11 values of z and t. The remaining assumptions (5 - 9) simplify the Lon of continuity of mass fraction: 2 3w ._ aw1 AT Blimp 01 _ 0 l 32 a w m 1 0‘1 AT 2 3 in p owo + (l _ 2Wo)(w _ wo) . _T— —a ——aT W12 1 l 1 m wl (3.6) ation of the separation of variables technique to ove equation and imposition of the above-mentioned l and boundary conditions results (see Appendix B) following solution, designated wi in order to dis— 3h it from a later expression for wl: o 0 W1 + alwlw HlD H 2 7 W:(Z,t) = + g? S) , (3- ) WIN O 2 m 38 re S: E k'3vw ( k2 k=1 k k eXp - t/e - pz/a - p/2) , (3.8) v = 1 - (1)k ex (P) k p , (3.9a) _ 1 a (in ' 0‘1 W1 m Wk = B sin 6 + kw cos C , (3.9c) z 1 c = M (a + j) , (3.10) e = aZ/(wzn), (3.11) p = — oclAT/Tm , (3.12) zs—ATab‘p +a—lAT(1—2°) (313) — T- T wl I 0 W m 1 re 3.1 shows the general shape of wI(z,t). The conver— 3 prOperties of the infinite series are of interest and be discussed below. Although the above expressions (3.4), (3.5), and for the temperature distribution, center of mass iitY: and composition distribution have been used to late thermal diffusion factors for a large number of ms reported in the literature (see, for example, fsson et al., 1965; Meyerhoff and Nachtigall, 1962) 39 .930 (758‘ 586~ 758- 330 l | 1 -0.5 —0.3 -0.1 0.1 0.3 0 5 z/a ' ' function of z for Figure 3.1——CompOSition wi as a 0. wo (CCl ) = 0.5, t/0 = 0.33, 1.0, Ecol 60 I l 4 dT/dz = 5 deg cm 40 of them is ever exact. Moreover, the degree of in— ness has not previously been examined quantitatively. There are several shortcomings of the above solu- . First, the steady temperature profile in the fluid t exactly linear. Variations in thermal conductivity iuce a slight curvature. Also, for several minutes the temperature difference is first applied to the and while the temperature profile is being built up, a variation exists in the local temperature gradients. gradients near the metal plates may accelerate the ng, or smaller gradients near the middle of the cell pede it. Semi-empirical corrections involving a shift in the time scale to take account of warming ects have been suggested (Agar, 1960), but no rigorous ent has been published. The exclusion of the possi— Eof convective motion and the restrictions to systems I J (constant transport parameters are additional short- 3 with which we concern ourselves in the following Ii). TSolutions ) In order to keep our treatment very general and to 1 ;the most complete description of the pure thermal (on phenomenon, we make only the last three of the gisted assumptions of the third type. The use of igential approximation (11) and the neglect of ¢1 1 41 . 8 — . . . . and 312 5; (H1 — H2)(l3) are certainly justified. We ct that any error introduced at this point will be much ler than the limits of experimental measurability. To avoid a cumbersome simultaneous solution of three ial differential equations, we adopt a scheme which de— . 3 on the fact that the calculation of an experimental 3 for the thermal diffusion factor ml is most strongly lenced by the accuracy with which the composition gra- :is known, next on that for the temperature gradient, inally, to a lesser extent, on that for the velocity. tice also that the composition gradient depends mostly e temperature gradient and only partly on the velocity. emperature gradient is a function mainly of the thermal ctivity of the fluid and some apparatus parameters. 1y, the velocity is quite small and can be determined ciently well from existing expressions for the tempera— ?nd composition. Hence, we can work backwards, first hg the velocity, and then using it to obtain an improved %on for the temperature. The final step involves the : both uz and T to determine the solution for the com- pon. The simultaneous solutions can be approached by (tion of the three—step cycle until self—consistency iained. i. Lter of Mass Velocity i In a uniform fluid mixture at equilibrium, such as 4 1 huid in a pure thermal diffusion cell at its initial ‘ 1 1 J? 42 (ignoring sedimentation), the center of mass of the is located at the geometric center of the cell. arly, the center of mass of each small volume element ies the geometric center of that volume element. How— at the steady state of a pure thermal diffusion ex- ant a vertical density gradient exists, and the center as of each volume element is displaced vertically ward) from the geometric center. This displacement : center of mass during some time interval gives rise a vertical component of the center of mass velocity liCh is nonzero as long as the density changes with It should be noted that the velocity with which we ncerned results from an uneven expansion and contrac— f the fluid as the temperature and composition change. ot due to any sort of forced flow. In order to obtain a mathematical expression for uZ the equation of continuity of mass: (dp/dt) + p(8uz/Bz) = 0 . (2.62) dimentation is ignored, the chain rule for differen- of the density gives (CLO) = $1) (9.1).. fl—Mffl) (3.14) ining Eqs. (2.62) and (3.14) with the balance equa- or energy (2.64) and mass fraction (2.63), we can 43 :e time as a variable and obtain a differential equa— 2 only: Buz _ _ B qu) + l 8 in p 3312 (3 15) 32 _ p— 52 p SW 82 ' ‘ c 1 p T dary condition is simply a — uz( 5 ,t) — 0 . (3.16) dependence of the velocity is still contained in assions for the fluxes. If these were known exactly 1 integrate Eq. (3.15) directly to find an expres— uz. However, because of the simultaneous nature 'oblem jlZ and qZ can be known completely only when )wn. Nevertheless, we can learn a good deal about :ity by using the approximate expressions for the (tained when w 2,t) is taken to be Bierlin's (1955) 1( Eq. (3.7), and T(2,t) is the temperature in a solid tant thermal conductivity (see T*(2,t) in Appendix C). s a test, the following hypothetical system was w0 = 0.5 , CCl - l 4 C6H12 ' Tm = 25°C , AT = 4°C, cell height = 0.741 cm , d = — 1.72 , l —1 -l 4 cal deg_ sec cm K. = 2.45 x 10' l 44 fation of the resulting expression for (Buz/az) is rhtforward, but extremely messy. In order to obtain :rical solution more easily, we used a finite differ— ntegrating technique (Ralston and Wilf, 1960) and a 1 Data Corporation 3600 digital computer. The dis— age of the numerical method is that no analytical on for uz is obtained. For purposes of illustration, r, the calculated velocity at the center of the cell vn as a function of time by the solid curve in Figure The velocity is very small in magnitude (less than '7 cm. sec-l) and is short lived. The whole effect disappears when the thermal steady state (ET/3t = 0) :hed. The spatial distribution of the velocity at 1e must be representable by a function which vanishes ( the top and bottom boundaries of the cell. In order to facilitate the use of the velocity as ion and to avoid having to perform a finite differ— tegration each time, we have used the boundary, , and steady state conditions on uZ as well as the e of Spatial and temporal extrema (see Figure 3.2) 'n a synthetic expression for uz(z,t). Figure 3.2 that the time part of uz is some sort of Morse—type . In fact, we found that the function 4u 2 _ t £n2 _ t KHZ ,t) = 00 [22 _ (g) ]e t0 1 - e t0 , a 45 oma .A v nonhuman onumnucsm 6cm Annuxv genusaom monwummmap whaqnm “76 amp m n 6\e< .mammo I vaoo How wasp mo dewuocsm o mm o N no wuflooaw> mmmE mo kucmollm.m mudmflm mpcoomm .u ONH om 1 u 0v NMI o 5 ADV moa mal 46 the finite difference solution very well. The dotted in Figure 3.2 is a plot of Eq. (3.17) for the example when u00 and t0 are obtained from the finite difference ion. The difference between the two methods for express— :(Z't) is very small, and no significant additional error 3e introduced if the more convenient formula (3.17) is , Equation (3.17) is only an approximation, and we use Ly to estimate the contribution of terms which are quite >rtant. It satisfies the conservation equations for 1nd energy, but it does not satisfy the equation of 1, (presumably because we have taken 01 = 0 and Ap = -pg). .on (3.17) may be regarded as the leading term of the solution. For experimental situations in which verti— >nvection is important, such as approximations of l diffusion in living systems, a more refined analysis be required. Equation (3.17) suffices to indicate that e present experiments the maximum value of uz is about m sec—l, and the maximum occurs at about 15 seconds the beginning of the experiment. Convection thus essentially no contribution to the measured value of note that this is a conclusion rather than an ion. erature Distribution All previous theories of pure thermal diffusion are n the assumption that the temperature gradient inside id can be expressed by the constant AT/a independent 47 Lme and position. However, in an experiment some mea— >1e time period is required before a steady temperature Lent is established in the fluid. Even if the plate aratures could be changed instantaneously, the heat 1ction process would still result in a time lag. The :ional contribution of time dependent plate temperatures .ts in a warming up period which may not be insignifi— as previously assumed, when compared with the relaxa— time 0 for diffusion. In our experiments, described I, the warming up period of six minutes was about ten :nt of the relaxation time 0 for a cell height of 0.741 Thermal diffusion studies are also being made with smaller cell heights (see, for example, Meyerhoff and Iigall, 1962), and since for CCl4 - C6H12 mixtures near 0 = 120 a2 minutes, a cell height of less than 0.25 cm 0 comparable to the length of the warming up period. ver, once a steady temperature distribution is estab— d, it is not perfectly linear because of variations in l conductivity. Rather than ignore both the time and space dependences temperature gradient, we obtain an explicit formula includes them, and which can be used in solving the ential equation for the composition distribution. 9 in chronological order, we confine our attention to the temperature distribution during the warming iod. 48 Time Dependent Temperature Distribution According to the assumptions which we are making, (11), nd (13), the equation of energy transport (2.64) can ten as 3T 2 ST + u __ _ _ qu 3t 2 3 — pcp §§_ (3.18) z is given by Eq. (2.66) and the auxiliary conditions (2.68) and (2.69). Since our main interest at this point is the time ice of temperature, we can tolerate the very small atroduced by assuming that Q1 and (awl/az) are known )— - the thermal conductivity Ki is constant. Except term containing the velocity, Eq. (3.18) is anal- ; the problem of one dimensional heat conduction in An additional complication is the presence of ependent boundary conditions. As usual an infinite series solution is expected. The following method Duhamel's integral formula (see Bartels and 1, 1942), is a convenient one for treating the in— ous equation with time dependent boundary conditions. ome of our early work with numerical solutions of 8) indicated that during the warming up period the re can be well represented by the sum of two func- e representing temperature changes due only to uction (as in a solid), and the other representing ibutions of the heat transfer by convection and 49 on. It is, of course, not unreasonable to neglect ve heat transfer in our apparatus. Accordingly, we write * T(z,t) = T*(2,t) + buz(z,t) g: (2,t) , (3.19) *(z,t) is the solution to — 3T* 32T* pc at = Ki ——7— , (3.20) p 82 )lem of heat conduction in a solid of uniform thermal Lvity with time dependent boundary temperatures, and :onstant whose value is to be determined. Equation (3.19) takes account of the following When the velocity is zero the temperature is just what it would be in a solid with the appropriate thermal conductivity, density, and heat capacity. When the temperature gradient (8T*/32) is zero the velocity causes no measurable change in the tempera— ture distribution. The velocity has a larger effect on the temperature istribution when the temperature gradient is large. he effect of the velocity on the temperature dis— ribution depends on the sign and magnitude of the elocity. n explicit expression for the constant b can be by combining Eqs. (3.18), (3.19), and (3.20)(see 50 C). Since b does not depend on 2 or t,it is con— to evaluate all quantities at t = t0 and 2 = 0. noring terms of order u: with respect to terms of , we obtain 2 _ a pc b: ——8—r- (3.21) 1 Hence, the temperature during the warming up period from Eqs. (3.19) and (3.21): 2 _. a QC _ 311* (2,t) — T*(2,t) — W72 uz(z,t) 82 (2,t) . (3.22) (3.22) satisfies our intuitive requirements for erature, it satisfies the differential equation 1d it satisfies the initial, boundary, and steady 1ditions. Therefore, to describe the temperature e warming up period we have only to find an ex— for T*, the conductive part of the temperature isfies Eqs. (2.68), (2.69), and (3.20). uhamel's integral formula (Bartels and Churchill, vides a means for solving the heat conduction ith time dependent boundary conditions. Param— n breaks the time domain into a number of small . Within each interval the boundary conditions nt and depend on the parameter 1. Let F(z,t,x) ution of the same problem except that the boundary ¢C(t) and ¢h(t) have been replaced by ¢C(A) and 51 their values at time A. Then the solution to Eqs. (2.68) , and (2.70) is t T*(2,t) = T +./' 3— F(z,A,t — ))d) . (3.23) m 0 3t w in Appendix D that T*(2,t) = rl(2,t) + r2(2,t) , (3.24) -W2Ki (2n+-1)2t 4Tm w -l . z 1 = _F_ 2 (2n-+1) Sln (2n-+1)w(a-+§J exp _ 2 , n=0 pcp a 2Kiw w 2 1' —Ki02n2t = _ 2 E n s1n[nn(a + 3’]I exp ————§——— , pc a n=1 a P t K.n2w21/a2p5 n =f e l p 4cm — (—1) ¢h())61. 0 YPiCal experiment we may have, for example, -t/t h ¢h(t) = Tm + A; (1 — e ) , (3.24a) —t/t AT c ¢C(t)=Tm-—21-e ), q and tC are some experimental relaxation times. In 5e we can write I = 1C —(—1)n 1h , (3.24b) 52 t t(K-l/t) AT Kt C C (Tm—7W6 '“HETTie "li' t(K — l/t ) h -1], t 1 AT) Kt _ h k'iTm+'—2‘ (e l)+K———-—th_l[e Kin2w3/(a2p5p) . The expression which we now have for the tempera— : the fluid in a pure thermal diffusion cell is :e during the warming up period, which for us is .ly six to eight minutes, when we are more interested time dependence of temperature than in its precise distribution and can tolerate the use of a constant conductivity. After this initial period and while all of the thermal diffusion occurs, the temperature t remains constant, within the limits of experimental oility, but it is in most cases not perfectly linear. itity Ki varies both with temperature and with :ion. Steady Temperature Distribution According to the discussion following Eq. (2.51), active thermal conductivity of the fluid initially from that when a composition gradient exists. The ion is that the temperature distribution continues e slightly until the steady state of thermal dif- 5 reached. In practice, however, the effective 53 11 conductivity is indistinguishable from Ki, which we : henceforth as K. After the warming up period Eq. becomes simply d dT EE[K(Z) 32] = 0 , (3.25) K varies from point to point in the cell. The boundary ions after the warming up period are T T(a/Z) h (3.26) T(—a/2) = Tc Equation (3.25) can be integrated by means of a nation technique. Since K varies only slightly with may write the expansion 2 n n K0(l + sklz + ezkzz +...+ s knz +...) , (3.27) = (Kon!)_l(dnK/dzn)0 , n = 1,2, .., can be found [8 of the chain rule. The zero subscript means the :y is evaluated at z = 0, and e is an ordering param— .ich allows us to keep track of the spatial dependence ature and composition dependence) of the thermal con— ty. Note that e is merely an index which does not The solution for the temperature has the form 2 T — TO + 5T1 + a T2 +... , (3.28) are the subscripts refer to the order of the per- Dn. The integration is straightforward (see Appendix to terms of order 62, yields 54 2 2 T(z) = Tm + A; z + ekl (g— — §—) +€Z(k — k2) ii—Z—ZL — E— + 0(53) (3 29) 2 1 12 3 ' ° terms are not necessary because they involve third atives of K, which are unmeasurable, and products of and second derivatives of K, which are extremely Discussion It is convenient both at the thermal steady state ring the approach to it to use a function f(z,t) which es the departure of the temperature gradient from the nt value AT/a and expresses its time dependence. The on f(z,t) is defined by BT : AT '3"; (2,t) — j; + f(Z,t) . (3.30) The portion of the temperature gradient which is in is just what has previously been ignored. Note that is not, in general, negligible when compared to AT/a. O, for example, f(z,t) = — AT/a. Inclusion of f(z,t) remainder of our theoretical treatment automatically lccount of warming up effects and deviations from a It steady gradient and leaves no uncertainty about :mixing begins or what to take as "zero time." Our tents begin precisely at the instant the temperatures metal plates begin to change, not when the tempera- adient is fully established. 55 As expected, the temperature gradient builds up slowly cally during four to six minutes) near the center of all; consequently, the diffusion flux in that region aehind what it would have been if there were no warming riod. Near the metal plates, however, transient large ants develop, causing an acceleration of the diffusion .n those regions for a short time. Agar (1960) has suggested a semiempirical correction lese effects. According to him we need not be concerned : acceleration of the flux is balanced by the decelera— :1sewhere. If the two effects do not balance, however, net effect can be negated by shifting the time axis in prOpriate direction in order to pretend that an unper— amount of diffusion has been going on for some slightly r or longer time. For example, when the boundary ature are given by _ AT _ -t/T T(a/2,t) — Tm + —E (1 e ) (3.31) AT —t T T(-a/2,t) =Tm-—a— l-e / . . _ o= = 8 experiment With CCl4 C6H12’ wl 0.5, Tm 25 C, ’C, a = 0.741 cm, I = 46 sec, the time shift t* is )Y D *= _— t (I 12K (3.32) I is the function K/pE . Choosing P 56 K = 2.4 X 10_4 cal cm_l sec_l deg-l p = 1.1 g cm"3 Eb = 0.2 cal deg—l g—l , find t* = 46 sec. No such manipulations are necessary with our method ccounting for warming up effects, which is automatic unambiguous. The function f(z,t) is also important after a steady erature distribution is attained. Nonlinearities due to ations in thermal conductivity which have previously been acted appear explicitly, and their effects on any measure- 5 are readily calculable. The temperature distribution has now been fully acterized. The external information on which it depends .sts of the thermal conductivity of the fluid and the dependence of the boundary temperatures. Our descriptions of the center of mass velocity and .emperature will next be used to obtain an expression he composition of the mixture as a function of position ime. mposition Distribution Steady State Because it is a great deal simpler and because its ion can serve as a test for the large—time limit of 57 umplete solution, the steady state case will be con— :d first. The steady state is defined by the vanishing . local time derivatives. It follows from Eq. (2.62) .e impermeability of the cell boundaries that u2 = 0 : steady state. Equation (2.63) then implies that = 0 , (3.33) ,nce jlz is zero at the boundaries, it must be zero 'here. Consequently, we have from Eq. (2.65) that at :eady state dw d 1 _ 1 dT (H) ’ T_ Eiwfl’z ° (3'34) Ipropriate boundary condition for Eq. (3.34) follows (qs. (2.70): a/2 i j, w (2)dz = w? . (3.35) a —a/2 l The steady state solutions of both Bierlein and t can be obtained by integrating Eq. (3.34) with a Tl wlw2 = constant , (3.36) d_T=AE , (3.37) d2 a ;ult in that case is simply a o l o _ 0 A3 3.38) Wl(Z) = W:L + ff; Wl(l W1) a l ( 1de m .0 (he 58 is readily calculated from the steady state composi- radient: (dw /d2) T a = ___l_£_m_, (3039) o 0 AT wl(l - wl) O‘1 eady state solution is obtained by integrating Eq. without making the simplifications given by Eqs. and (3.37). Since (dT/dz) and (alwlwz/T) are rarely nts, we make use of the following expansions: dlwlwz/T = 3(2) , (3.40) (dT/dz) = g3 + f(z) , (3.41) 00 n n 8(2) = 2 e 5nz . (3.42) n=0 BY Eq- (3.29)! 2 AT 2 — 2 a — 2 + 0 e3 (3.43) 3‘ [—8klz + 8 (k2 ki) 12 Z ( ) lows that n 4 = $— 9—£ , n = 0,1,..., (3.44) n n! dzn 0 at 2 n 3 f(2) = Z a f + 0(e ), (3.45) n n=1 _ _ (3.45a) fl - klz 2 a2 2 (3 45b) f2=(k2—kl) -]--2--Z). . 59 The method of integration is discussed in Appendix F. >lution to Eq. (3.34) through terms of order 82 is 2 2 E. _ i_ 2 24)] 2 522(k—k)az 235‘ —(k+ )23 +0(3 02 1T2'3—‘R 511523— ‘5) 0 AT 1 + 3— [602 + 5(41 - klbo) Equation (3.46) shows explicitly the influence on imposition distribution of variations in (dT/dz) and '2/T). Deviations from a linear temperature distri- ‘ are accounted for by the quantities kl and k2. The ties Al and A express the temperature and composition 2 (ences of (alwlwz/T). Comparison of Eqs. (3.38) and gives immediately the difference between our steady solution and the previous one: 2 2 2 3 3) z a 2 _ 2 a z_ §__ §_ 'ewl'kiéo) 7'24) +8 [50“‘2 k1)( 1‘2 3 48 LA) 2 3 (3.46a) (41kl + 42) §—] + O(€ ) . ation for calculating d1 from the measured steady composition gradient will be presented in Section F 5 chapter. Approach to Steady State In deriving the corrections which arise due to 9 up effects, variable coefficients, etc., we use the on G(Z,t), which is defined to be the difference be— the true composition wl(z,t) and the simplified Hhe 60 ;sion given by Eq. (3.7) which we shall denote hereafter (2,t): wl(z,t) = wi(z,t) + G(z,t) . (3.47) The equation of continuity of mass fraction remains 0(dWl/dt) = - (Bjiz/BZ) , (2.64) w* a DD £21 + E; - Tl g; w$(l - w?) + (1 — 2w$)(wi + G - w?) . (3.48) antion G(z,t) must obey a differential equation of the 2 8G 3 G 8G 5? = Pl(z,t);—§ + P2(z,t)§E + P3(z,t)G + P4 , (3.49) z It) = D I 3 Z T 0 3D 3 Kn p _ It)=—DOLl—a—:——-(l—2Wl)+a—Z-+D——3—z—— 1.12, 8 a fin T It) = " % (l — 2W?) a—Z— (pDocl T I 2 * 3 w* 3W 0 o l o l w; — [w (1 — w) 2 3w* 1 (1 _ 2 O)( * _ w°)] §_. 2.1212: 3 + W1 w1 1 32 1 32 z O O - a1 §_§§_E [wi(l — wi) + (1 - 2wl)(wi — wl)] (3.50) The func and and 61 The functions Pj(z,t) , (j = 1,...,4) are completely known functions of z and t since all of the quantities appearing in them can be determined without knowing G. The initial and boundary conditions are lim G(z,t) = o , — 9- < z < 3 , (3.51) 2 2 t+0 and lim jlz = 0 , t > o , (3.52) z+ia/2 or a/2 Jf G(z,t)dz = 0 , t > 0 . (3.53) —a/2 The coefficients Pj(z,t) , (j = 1,...,4), are ob— viously the sources of all the corrections to wi(z,t) with which we are concerned. Because of the factors Pj(z,t), EQ~ (3.49) cannot be simplified by separation of variables. Moreover, the factors Pj(z,t) are complicated functions (some parts are infinite Fourier series), and we have found no satisfactory integrating factors for simplifying Eq. (3-49). Integral transform methods are not usable because the spatial boundary conditions are two—point and finite. The only approach left is that of Frobenius. By Fuch's Theorem (Johnson and Johnson, 1965, p. 47) the z-dependence 0f G is given by (X) G = E gk(t)zk , (3.54) k=O if the functions P2/Pl and P3/Pl analytic (expandable in 62 Taylor's series) about z = 0. By inspection, these functions possess no singularities in any neighborhood of z = 0, and Eq. (3.54) is indeed the solution of Eq. (3.49). The time-dependent coefficients g(t) in Eq. (3.54) are completely determined by: (l) the form of the original differential equation (3.49); (2) the expansions for the co- efficients Pj(z,t), P.k(t)zk , j = 1,...,4 ; (3.55) P. z t = J( , ) 3 W "548 o and (3) the auxiliary conditions on G, Eqs. (3.51)-(3.53). Differentiation of Eq. (3.54), followed by substitution into Eq. (3.49) and use of Eq. (3.55), gives immediately the re— lationships between the desired coefficients gk(t) and the known coefficients pjk(t). As with any Frobenius—type method (see Irving and Mullineux, 1959), the preliminary result is a transformation 0f the problem from a single partial differential equation to a set of simultaneous ordinary differential equations in t for the coefficients gk(t). In this case EQ- (3°49) becomes so . k 00 n 0° k—2 z — p Z 2 k(k ‘ D9 2 kEO gk n20 1'“ k=0 k 00 n 0° _ z 2 kg Z 'nEO p2,n k=0 k 00 n °° k _ 2 g Z n20 p3’n kEO k I 3.56) _ 20 p4’nzn = O I ( n: 63 where ék E (dgk/dt) . (3.57) Since the functions {Zn} are linearly independent and form a complete set, and since the infinite series con— verge for —a/2 < z < a/2, the coefficient of each power of 2 must be independently equal to zero. The resulting set of ordinary differential equations can be expressed compactly by = AB + C , (3.58) ~ ~ (we where B is the column vector whose elements are gk(k = 0,1,...); C is a column vector whose elements are c (k = 0,1,...) ; (3.59) k = p4,k ' and A is a matrix whose elements aij(i,j = 1,2,...) are related to the coefficients pl n’ p2,n’ and p3 n(n = 0,1,...). I I A Portion of the matrix is given by ai1 = p3,1-1 ai2 = p2,1-1 + p3,1-2 ai3 = 2p1,1—1 + 2p2,1—2 + p3,1—3 a . = _ 14 6p1,1—2 + 3p2,1—3 + p3,1-4 aik = (k _ l)(k _ 2) + (k-l)P2,i—k+l + p3,i-k ' (3.60) pl,i-k+2 Where pij = 0, if i or j is less than zero. an... 0 any th fac the fie abc hi 0)); 64 By making use of the initial conditions gk(0) = O , k = 0,1,..., (3.61) one can solve the set of differential equations (3.58) for any number of coefficients gk(k = O,l,...,N) in terms of two constants of integration which must be evaluated from the boundary conditions (3.52) and (3.53). For the purpose of measuring a thermal diffusion factor d1 we do not need a complete solution for all of the gk's. In fact, because our experimental method measures the gradient of refractive index (which is directly related to the composition gradient) and because we have used series expansions about the center of the cell where z = 0, all that is required is the quantity (BG/Bz)z=0 = gl(t) . (3.62) In obtaining an expression for gl(t) we are justi— fied in bringing to bear all of the information we have about the function G, including the steady state solution, which must be approached asymptotically as t becomes infinite. One of the expressions which gl(t) must satisfy follows from Eqs. (3.59)-(3.60): ° 3.63 90 = p3,0‘3'0 + p2,0‘31 + 2p1,0‘5‘2 + p4,0 ( ) Since we expect no anomolous behavior at z = 0 due to unusual temperature gradients we write e't/e) , (3.64) go = Fo‘JL ' i (the: stat 5119s tiv tio COR ing 2e] H‘m C01 65 where F0 is an amplitude factor corresponding to the steady state value of go. It is not unreasonable to expect that gO is well behaved since our measurements (discussed below) suggest that for t 2 6/3, properties such as the refractive index gradient and the composition gradient near the center of the cell change smoothly and monotonically as a simple exponential function of time and do not exhibit the more unusual behavior observed near the metal plates. Since we are interested only in the first deriva- tive of G and not in G itself, only one constant of integra— tion, say g2(t), can be eliminated by means of the boundary condition (3.53), which gives 4 g2(t) = — g3 gO(t) . (3.65) The remaining constant F0 must be obtained from a condition on gl(t) for some extreme (0 or 00) value of time. The expression for gl(t) which one gets by rearrang— ing Eq. (3.63) is indeterminate in the limit as t approaches zero. (Note that this situation would not arise if the whole system of equations (3.58) were solved simultaneously.) Consequently, we use the alternative condition, lim gl(t) = gl(steady state) , (3.66) t—>oo Which is known from Eq. (3.46a). The second order perturba- tion solution (including terms of order 62) is 2 gl(st. st.) = 9% 60(k2 - kf)§7 + 0(83) , (3.67) 66 where 60, kl’ and k2 are given by Eqs. (3.44) and (3.45) respectively. Equation (3.46a) indicates that at the steady state G depends quite strongly on Al, the term expressing the temperature and composition dependences of the thermal diffusion factor. The first derivative of G at the center of the cell, as shown by Eq. (3.67), does not, to terms of order 83, depend on 61. Our choice of the center of the cell as a point about which to expand variable coefficients has resulted in this unexpected simplification. The simpli— fication is certainly a reasonable one, since we recall that in practice the temperature gradient at z = 0 is also unaf— fected by linear variations in the thermal conductivity. Note in what follows, however, that the time—dependent ex- pression for gl(t) does depend on Al and other factors that do not appear in Eq. (3.67). On combining Eqs. (3.63)—(3.66) we find F 48p (t) _ 0 —l —t/0 _ —t/e 1,0 _ gl(t) _ 53:2;FET[0 e + (l e ) -———jf———— p3,0(t)] P (t) - 4’0 , (3.68) 102,0“) where _ 2 3 Zn D 48F0 — a gl(st. st.) (T) _ z—O t=oo 2 0(1 0) E_ a §_£fl_1 (3 69) ‘awl ’w1 az1az _ ° z—O t=oo and 91(st. st.) is given by Eq. (3.67). 67 Our final expression for the composition gradient at the center of the cell is (awl/Bz)0 = (Bwi/Bz)0 + gl(t) , (3.70) where wi(z,t) is given by Eq. (3.7) and gl(t) is given by Eq. (3.68). Inspection of Eqs. (3.67—3.69) and (3.50) shows that gl is primarily the result of inclusion of variable coefficients and secondarily a result of inclusion of warm- ing up effects. The warming up part is of virtually no consequence after the steady temperature distribution is established, but it is, of course, all-important during the first few minutes. In order to calculate thermal diffusion factors by extrapolating to zero time, a popular practice, it is necessary in principle to use for the composition gradient Eq. (3.70) rather than the z—derivative of Eq. (3.7) alone. Since we have derived equations which fully characterize the experiment for all times, we may calculate thermal diffusion factors from measurements at any time. In particular, we may select those times for which the equations are the simplest. Working equations are presented in the next section. F. Working Equations The basic equation with which one can calculate d1 frOm measurements of the composition gradient is 8w 3w* (__1) =(__l) + gl(t) , t >, 0 , (3.70) 0 0 68 which is valid both during the approach to and at the steady state. The quantity on the left hand side if what is mea- sured (directly or indirectly). The first quantity on the right hand side is a known function of al, and the quantity gl(t) is a correction term which is also known. Upon rear— ranging Eq. (3.70) and transforming to refractive index gradients instead of directly measured composition gradients, we find Tm(N(t) — gl(t)) 0‘1=‘—T(’E)—_ , (3.71) where Tm is the mean temperature of the fluid, (— - (swig—50 N‘t) = (8n70wl) ' (3°72) T gl(t) is given by Eq. (3.68), and from Eq. (3.7), 3 dz a w V 2 dW 1 12.-k Jew—4-2m]. =1 k H(t) = ATwi(l - wi)[% + a? n k (3.73) When (an/3T) , (an/3w ) , and (ST/32) are known, W1 1 T 0 one can use Eq. (3.71) to calculate d1 from measured values Of the refractive index gradient for either steady state or non—steady state experiments. Clearly, similar equations hold for any other choice of measurement, such as electri— Cal conductivity or capacitance. All that is required is the gradient of the property being observed and information abOUt its temperature and composition dependences. L—_‘i . 69 G. Composition Distribution During Remixing By "remixing experiment" we mean one whose initial state is identical with the steady state of the correspond— ing demixing experiment. In practice one conducts the two experiments in succession with the same system. To initiate remixing one removes the temperature difference. The tem- perature gradient decays to zero during the next few minutes, and for a period of length 66 the composition gradient decays to zero by means of ordinary diffusion. Although no thermal diffusion takes place, the thermal diffusion factor d1 can still be measured since it determines the magnitude and di- rection of the original composition gradient. During remixing the velocity is the same as that given in Eq. (3.17) except for the sign of uOO' The temperature gradient is still given by Eq. (3.30) where now f is determined as in section D except that the final condition is lim T(z) = T , (3.74) m t+oo and the boundary conditions are T(-a/2) = 4 (t) , T(a/2) = ¢h(t) , (3.75) where now ¢c(t) and ¢h(t) are not the same as in Eq. (2.69). In particular, different flow rates and heat capacities be— tween the baths used in the two types of experiments lead 70 to different relaxation times th and tc. Also, both ¢C(t) and ¢h(t) must approach Tm as t increases. The solution for the temperature, however, still has the form of Eq. (C.l7) except that ¢h(t) and ¢C(t) are given by -t/t _ _ AT 0 ¢C(t) - Tm —E e , —t/t _ AT h ¢h(t) — Tm + —5 e , (3.76) and z l k(n+—l)w2t oo (2n+1)7r(-—+—) —_—_.2___ r = i T i-AE z Z 1 sin a 2 e a 1 N m a n-+ ' n=0 (3.77) The solution for the composition gradient at the center of the cell is obtained in the same way as for de— mixing experiments except for the final condition: . 0 lim wl(z,t) = wl . (3.78) t—>oo At z = 0 we have again 8w 3w*) 1 _ l (53—)0 _ (Bz 0 + gl(t) , (3.70) where now 3w* 01 0° V (1W 2 3.771 = T_1 W3” ”“3”... 2_3 g l; exp(-k t/B-p/Z) (A — 2 wk“, 0 m N k=l k dz a (3.79) lim gl(t) = 0 , 71 (3.80) t+oo lim g (t) = g (st. t+0 l 1 As in the case of Eq. st.) . (3.81) (3.68), we find , F 48p (t) _ 0 —l —t/0 -t/e l 0 _ 9'1“” ‘Wie (1 “e 1” (“:77— 93'0““ l p4,0(t) - p2 0(t) ’ I where _ 2 8 Zn pD 48FO — a gl(st. st )(— 2:0 t=0 2 o o 8 3 in T _ a Wl(l - Wl) [Eta]. T]] 2:0 . (3.82) t=0 Analogous expressions follow for the working equations: T (N(t) — g (t)) 0‘1 - T' 9-83) where N(t) and H(t) have the same form as in Eqs. (3.72) and (3.73), but (ST/82) (3.72), and 2 H = 0 _ 0 __ 0 is appropriately modified in Eq. w v k E ‘3‘ exp (—k2t/0 — p/2)(W; k=l k - p/a Wk) (3.84) 72 One determines ngin the same way as described above. The significant difference between the two methods is that no temperature gradient exists during most of the remixing. Consequently, there is no possibility of inducing convection by poor control of boundary temperatures. Moreover, after the first few minutes N(t) simplifies to (an/82)0 = (3375WIT_ . (3.85) T There is an extra advantage of the remixing method when an optical technique is used for in situ measurements of the refractive index gradient. Changes in (an/32)0 due to fluctuations in metal plate temperatures do not appear because there is no heat conduction through the fluid. All observed phenomena are due to composition changes only. As we shall show in Chapters V and VI, literature values of the composition dependence of refractive index are much more reliable than literature values of temperature dependence. H. Calculation of the Ordinary Diffusion Coefficient At least one measurement of (an/82)O must be avail— able to compute a from either of Eqs. (3.71) and (3.83). 1 When several values of (an/82)O are available at various times, they can be used in estimating the precision of the measurements. Also, when two or more values of (Sn/82)0 are obtained at different times, they can be used to 73 calculate not only d but also a second parameter, such as 1’ the relaxation time 6. Then, since the cell height a is known, the ordinary diffusion coefficient of the mixture can be calculated from 2 ' (3.86) Of course, only measurements during the approach to the steady state will exhibit the characteristic time dependence necessary to calculate 8. In principle, with sufficiently refined auxiliary equipment, we could also determine many other parameters such as thermal conductivity, heat of transport, tempera- ture and composition dependences of transport coefficients, etc. Even with our relatively simple equipment, we calcu- late from our equations the following properties: thermal diffusion factors plus their temperature and composition dependences, ordinary diffusion coefficients plus their temperature and composition dependences, and the tempera— ture dependence of refractive index. I. Discussion For the first time we have a phenomenological theory 0f pure thermal diffusion which takes complete account of transport parameters which vary with temperature and com- position, warming up effects, non—linear temperature dis— tribution, and transient convective transport. The results 74 of this chapter can be used to predict the effect on the composition distribution resulting from a temperature dis— tribution which varies slightly with position or with time (even an oscillating temperature difference). Alternatively, experiments can now be interpreted more accurately, and calculated values of d1 should be more reliable. The experimental time scale is clearly de— fined and leads to no ambiguous curve fitting or extrapola— tion to zero time. Because of our particular way of expressing the composition as = * w1 w1 + G , comparison of results calculated from our theory with those from the best previous theory is very easy: simply set G equal to zero in the latter case. In Eqs. (3.71) and (3.83) set gl(t) = 0, and (ST/32)O = AT/a. At the end of Chapter II we discussed three levels of approximations and stated our intention to derive a theory based on a minimum number of them. The "necessary" assumptions concerning the applicability of hydrodynamics and nonequilibrium thermodynamics have been retained as have the "unnecessary but desirable" assumptions which can be realized experimentally. Of the thirteen simplifying assumptions classed "unnecessary and undesirable" we have eliminated all but the last three. We feel justified in retaining the following approximations: (l) (2) (3) We have 75 _ o _ o _ o _ o . wlw2 — wl(l wl) + (l 2wl)(wl Wl)' This linearizes the differential equation and makes it tractable. The entropy source term $1 in the energy transport equation (2.64) is negligible. (H — H l 2) in Eq. (2.64) may be (3)0) N The term - jiz ignored. also justified two unclassified assumptions, viz., the absence of pressure effects (sedimentation) and the negligibility of the heat of transport Qi. For systems in which the gradients of temperature, pressure, composition, and velocity are as small as they are for our experiments, these simplifications certainly introduce no detectable error. CHAPTER IV EXPERIMENTAL APPARATUS A. Introduction There are three fundamental components of any pure thermal diffusion system: an appropriate sample container; a means of controlling the boundary temperatures; and a method for detecting small changes in composition. The first component, the cell, is much simpler for pure thermal diffusion than for any of the other methods. It requires no forced flow mechanism, no membrane or porous plate, and no stirring device. The cell dimensions are not as critical as they are in the case of a thermogravitational thermal diffusion column. In addition, a flat plate is generally easier to machine to a desired tolerance than is a narrow annulus. Moreover, expansion and contraction of the metal parts due to temperature changes cannot change critical dimensions since the plate separation depends only On the thickness of a piece of glass. For that reason also, the height of the cell is easily changed. A thermogravita- tional column lacks flexibility in that respect. The only criteria affecting the choice of cell di— mensions are convenience and sensitivity of the detection 76 77 system. The relaxation time for the thermal diffusion pro— cess, unlike that for the thermogravitational apparatus, does not depend directly on the composition or the temperature difference and is given (to within 0.01%) by e = a2/7T2D (4.1) where a is the cell height and D is the ordinary (mutual) diffusion coefficient of the mixture. For carbon tetra— chloride-cyclohexane mixtures at the temperatures and con- centrations of interest, say 25°C and W1 = 0.5, D is about 5 1.4 X 10- cm2sec_l. Consequently, 0 s 120 a2 min (4.2) Since the demixing is 99.75% complete when t = 60, and since, for experimental convenience, we wish to complete both demixing and remixing experiments in a 12 to 14 hour period, it follows that we should require a = 0.75 cm. (4.3) The length of the cell must be small enough so that a uniform temperature can be maintained, yet great enough so that the optical path through the liquid is suf- ficient for the desired sensitivity of the interferometer. After measuring the dimensions of the interferometer com— ponents and estimating the magnitude of the expected refractive index gradient, we concluded that a cell length 0f eight centimeters was suitable. Another pronounced difference between thermogravi— tational and pure thermal diffusion exists in the importance 78 of temperature control. In the former case the amount of composition separation does not depend on the magnitude of the temperature difference, but in the latter the tempera— ture distribution in the liquid is extremely important. In fact temperature control is the most troublesome part of pure thermal diffusion experiments. Large fluctuations at AT and drifting of Th and TC both result in changes in the diffusion flux. An addi— tional problem occurs when an optical method is used to detect composition changes. A very slight change in the temperature gradient can produce a change in the refractive index gradient nearly as great as that due to all of the thermal diffusion which has taken place. Consequently it is very important to maintain a constant temperature gra— dient as long as measurements of composition changes are being made. Our water circulation system was carefully designed to minimize temperature fluctuations. Although optical analysis of composition changes introduce the problem mentioned in the preceding paragraph, the advantages outweigh the disadvantages. Conductometric methods are restricted to electrolyte solutions and neces- sarily result in a great deal of spatial averaging of com— positions. Moreover, the imposed electric field constitutes another force which should be included in the phenomenologi- cal relations. Another method which has been used involves with— drawing an aliquot of the sample liquid at some predetermined 79 time from some known position and then determining the aver— age composition of the aliquot either chemically, conducto— metrically, or refractometrically. This method has the obvious disadvantage of disturbing the system and can provide at most a single reliable measurement for each experiment. Due to the time and work needed to carry out each experiment, we easily ruled out such sampling. Optical interferometric techniques have the decided advantage of providing an extremely large number of data while not disturbing the system in any significant way. Both electrolytes and nonelectrolytes can be studied, al- though dilute salt solutions require greater sensitivity. Interferometers suitable for diffusion studies utilize the composition dependence of the refractive index of the liquid. Our particular instrument was designed to measure the gra— dient of the refractive index, which completely determines the composition gradient if the temperature distribution is known. The wavefront shearing interferometer is at least as sensitive as any of the other types which have been used, and it has the advantage of being simpler to use. Our addi— tion of the laser as a light source resulted in increased intensity and improved accuracy. With this general idea of the apparatus in mind, we turn to a more comprehensive discussion of each of the Components. 80 B. The Cell Of primary importance in a pure thermal diffusion experiment is the cell. More than just a container, it must satisfy a long list of requirements. It must: 1 10. 11 12 13 have horizontal boundaries consisting of metal plates whose temperatures can be well controlled, have glass walls to permit in situ optical analysis, be fillable and sealable in some way which excludes a vapor phase, contain volatile liquids without permitting evapora- tion or leakage, be able to be accurately levelled, have reproducible geometry, be free of disturbing vibrations, have uniform temperature distribution over the metal plates, provide efficient heat transfer through the liquid, have a reproducible and measurable warming up time, not permit formation of impurities by means of chemical reactions between the sample liquid, the sealant, and the metal, be much larger in horizontal extent than in depth so that any anomalous behavior at the side walls or corners is negligible, provide proper control of boundary temperature so that convective remixing does not occur, 81 14. have reservoirs for circulating water with large heat capacity to minimize temperature fluctuations, 15. be easily dismantled, cleaned, and reassembled. Four preliminary designs were tested and found to be unsatisfactory with respect to the requirements of either temperature control, sealing, or inertness. We found that copper catalyzes the formation of oxides in aqueous solu— tions. Consequently all metal parts contacting the sample liquid were silver plated. The upper and lower plates were of copper, 6 in. x 6 in. X 1/4 in. Two filling tubes of 1/8 in. o.d. copper were soldered into holes in the upper plate 1/2 in. apart before the plates were machined flat. All of the copper pieces were then coated with 0.001 in. of silver deposited electrolytically (for $10 by Sarver Mfg. of Lansing, Michigan). Heating and cooling reservoirs were made from 8 in. X 8 in. X 1.5 in. magnesium blocks. The metal was chosen for its machinability, its availability, and because by rapidly exchanging heat with the circulating water it can help to damp temperature fluctuations. Channels were cut into the magnesium to form the reservoirs and to direct the flow of circulating water over the metal plates in such a way that spatial variations in the plate temperature were minimized. (See Fig. 4.1.) Each reservoir was supplied with one inlet and two outlet ports (3/8 in. dia.) in Order to maintain a symmetric flow pattern. 82 ——_.—.——.— Figure 4.1—-—Water deflecting channels in reservoir. Overall dimensions 8 in. X 8 in. 83 Each of the metal plates was secured to a reservoir with twelve brass machine screws (size 10—24) which passed through countersunk holes in the plates and into tapped holes in the magnesium block. The space between the plate and its reservoir was filled with a gasket (1/16 in. "Vellumoid") which was coated on both sides with "Lubriseal" stopcock grease. The resulting seal was completely effec— tive in preventing any leakage of the circulating water. Each reservoir had a capacity of about 300 ml. The vertical walls of the sample chamber were made of 3/8 in. thick Pyrex optical glass. Pyrex was chosen because its low thermal expansivity insures (1) that it will not crack when subjected to temperature gradients; (2) that there will be no change in cell volume when the temperature changes; and (3) that the thickness of the glass walls does not vary with the temperature. Four bars of width 8 mm, two 8.6 cm long and two 6.3 cm long, were cut from a single plate of Optical glass 3/8 in. thick. The four were positioned to form a rectangle with inside dimensions 6 X 8.3 cm. The alignment of opposite walls was made precisely parallel by means of coincidental back— reflection of a helium—neon laser beam. When properly aligned, the four pieces of glass were joined together with a two—part epoxy resin cement (Sears, "filled," gray in color). Earlier trials with a colorless epoxy always resulted in a breakdown of the 84 adhesive properties after several hours exposure to water or CCl4 - C6H12 mixtures. The colorless epoxy did not dissolve, but it became hard and brittle and would not adhere to the glass. The filled epoxy, however, was entirely satisfactory, remaining inert and securely bonded to the glass after 1000 hours of use. Construction of the sample chamber was completed by grinding the upper and lower surfaces of the glass assembly with carborundum until those two surfaces were uniformly flat and parallel to within 0.0005 cm (a sheet of paper 0.0005 cm in thickness could not be passed between the plate and a flat guage block held in contact with it. The height of the glass walls after grinding and polishing was 0.7410 cm i 0.0005 cm (by actual measurement with a micrometer). The material chosen for the sealant between the glass and the metal plates was a very viscous fluorosili— cone (Dow Corning "FS" stopcock sealant) which formed a leakproof seal and did not dissolve in or react with the liquids used. Assembly of the cell was accomplished in the fol— lowing way. The upper reservoir was inverted (metal plate up) and the glass wall assembly, to which a thin layer of sealant had been applied with a syringe, was placed on the metal plate in such a way that the long axis of the cell was parallel to the optical axis of the interferometer 85 system, and the two small filling holes just appeared inside one corner of the glass. Sealant was then applied to the top surface of the glass cell wall assembly, and the upper reservoir, with glass attached, was returned to position and lowered over four guide bolts until the glass contacted the lower metal plate. The entire unit was held together when four brass nuts (size 12—20) were applied to the four guide bolts which passed up through the upper reservoir housing. Four large holes in the corner of the bottom mag— nesium block fitted onto four upright 1/2 in. diameter threaded steel rods, each 18 in. in length. Steel nuts held the cell assembly to the threaded rods while allowing for height adjustment and levelling. The rods in turn were anchored to a steel I—beam 8 in. wide by 10 in. high and 15 ft long which was itself bolted to two 55—gallon drums filled with concrete. The entire structure, which weighed about 3000 lbs, was separated from the floor by 3/4 in. cushions of dense foam rubber and l in. thick plywood boards under the barrels of cement. To aid in filling the cell, the mounting was de- signed so that the cell assembly could be tilted about 25 degrees from the horizontal along a diagonal axis. (See Fig. 4.2.) Thus the two filling holes in the top plate occupied the highest corner of the sample chamber. While the cell was being filled by means of a syringe, all of 86 Figure 4.2.——Assembled cell in tilted position for filling- 88 the air was pushed to the top by the entering liquid and was easily expelled. Once filled, the cell was returned to its level posi— tion, and the glass walls were manually pushed sideways between the metal plates a distance of 1/8 in. Thus the filling holes were removed from the sample chamber, pre- cluding both evaporation and diffusion through the holes, and at the same time removing the slightest perturbation on the temperature distribution due to the tubes passing through the reservoir. This feature is an important inno- vation in our cell. Finally, strips of foam insulation were placed around the cell in the space between the reservoirs in order to prevent air currents across the metal plates and to avoid spurious heat transfer with the room air. Small flat glass plates were substituted for the foam along the optical path. 3 We have described the design, construction, and {assembly of the cell, but we shall postpone a discussion of fits operation until the next chapter. We consider next 1 (temperature control and measurement. ‘C. Temperature Control and Measurement We chose circulating water baths for temperature Icontrol devices rather than electric heating coils in com- ‘) )bination with cooling coils in order to avoid both spatial ) ) ( ) ) 89 variations in plate temperatures and the possibility of long term drifting. The main disadvantage of our water baths, namely fluctuations due to the off-on heaters, has since been eliminated by the substitution of proportional heating elements which are always on but supply slightly more heat if the temperature drops and less if it increases. Four baths were available for our experiments. The largest, a Lab—Line Tempmobile, was equipped with a com— pressor unit and served as a source of constant temperature cooling water for the other three baths. Tap water proved unsatisfactory for cooling even when its temperature was steady because only a trickle was needed, and fluctuations in pressure could change the effective cooling rate drastically. The Lab—Line bath had a capacity of 90 liters and was equipped with a built—in heating element. The point of balance between the heating and cooling actions was ad— justed by means of a Rota-Set mercury—contact thermoregula— tor connected to a relay switching mechanism which turned the compressor on and the heater off when the temperature fell below the preset level. Additional modes of operation were also available. The compressor or the heater or both could be shut off manually while the circulating pump continued to operate. For example, with the water temperature well below room 90 temperature, the heater could be disconnected, and then, the cooling of the compressor would have to be balanced by an absorption of heat from the room. Because of the in— sulation, this would be a slow process and would result in a very long cycling time for the compressor and an accom- panying slow drift of the water temperature. For a better balance and optimum temperature control, both the heater and the compressor were allowed to operate. The bath's circulating pump delivered water through a 3/8 in. i.d. fitting at an uninhibited rate of 1300 ml/ min. Near 25°C the temperature of the circulated water showed fluctuations of r0.l°C coinciding with the off—on Cycle of the compressor. A modification was made so that the used water was returned near the pump intake and the thermoregulator rather than to the opposite end of the bath, thus providing the needed increased mixing action. As a result, the fluctuations were reduced to r0.01°C. The two baths used to apply the temperature differ- ence to the cell were nearly identical. Both were Tamson model T—45, with 45 liter capacities, and were obtained from Neslab Instruments, Durham, New Hampshire. One oper— ‘ated with 110 V ac and the other with 220 V ac. Both had coils of 1/4 in. stainless steel tubing for external cool— ‘ing and both had quartz main heating elements. The quartz surrounded a piece of high resistance wire and served to dissipate the heat more slowly than the wire itself would, 91 hopefully providing a more constant source of heat than a conventional off—on heater. Both baths had booster heaters for rapid warm up when desired. That in the 110 V bath was quartz, and the other was stainless steel. The difference is of no conse- quence since the booster heaters were never used during an experiment. Control of the heating cycle was governed by Jumo mercury-contact thermoregulators (0—50°C) and mercury relay switches in each of the baths. The circulating pumps caused excellent stirring of the baths while delivering a flow of water through a 1/4 in. i.d. outlet pipe at a rate of 3500 ml/min with a 10 ft head. The manufacturer recommended that the cooling rate be adjusted so that the heater was on for about four seconds and off for about 16 seconds of each cycle. Such an ad— justment, however, resulted in fluctuations in the tempera— ture of the output water on the order of 0.01°C. We found that further reduction of the cooling rate, until a cycle of one second on and 30 seconds off was obtained, improved the fluctuations to about i0.005°C. This adjustment was quite delicate since it meant only a slight trickle of cooling water was flowing through the bath and any further decrease could shut off the cooling completely, resulting in a breakdown in the cycle. If that happened, the bath temperature would slowly increase due to the heat developed by mechanical stirring, and the heater would never be turned on. 92 The fourth bath was a Tamson model T-9 with a 10 liter capacity, and it was maintained at the mean tempera— ture. It was similar in most respects to the other Tamson baths. It had a quartz heater but no booster heater. The cooling coil was a length of 1/4 in. stainless steel tubing. This model had the same pump as the other two and conse— quently the same flow rate and head. Because of the smaller size, however, the temperature fluctuations of the unmodi— fied bath were on the order of i0.0l°C. We modified all four of the baths by making the outflow of each pass through a six foot length of 5/16 in. i.d. copper tubing formed in a 5 in. diameter coil. The coil was immersed in a 2000 ml beaker filled with water kept at the operating temperature of the bath. In the case of the T—9, the beaker was outside the bath (increasing the ieffective volume of the bath to 12 liters) and was stirred by means of a Sargent magnetic stirrer. The other three beakers and coils were positioned inside their respective baths and served as secondary semi-isolated thermostats. ‘ The purpose of the copper coils and their associated volumes of agitated water was to act as heat exchangers and absorb any pulse of excess heat as it passed through the COil or to give heat to the circulating water whose tempera- ture was slightly less than normal. In this way, fluctua— tions due to the off-on heating cycle were nearly damped out. The three Tamson baths operated routinely with 93 fluctuations of about r0.003°C, and, with very careful balancing of the cooling rates, could be made to operate with fluctuations of less than 0.001°C. All of the tem— peratures mentioned above were monitored continuously with 40 gauge copper—constantan thermocouple junctions attached to the metal plates, and two Sargent model SR strip chart recorders specially modified by our electronics technician to display temperature changes as small as 0.002°C. The water was suitable for use when it emerged from the copper coil. It was transported from there to the proper reservoir and back to the bath through 1/4 in. i.d. Tygon tubing. Joints between sections of the tubing were made with short lengths of 5/16 in. i.d. Tygon tubing. The material is soluble in methyl ethyl ketone, so when the ends of the tubing were dipped into the solvent for about one minute before slipping them together, a permanent bond was easily formed. Special Tygon Y—connectors, obtained from Scientific Glass Apparatus Co., were used in the same way. At the cell, the tubing was connected to 3/8 in. o.d. brass nipples screwed into tapped holes in the reservoirs. At no point along the line was the opening through which the water passed less than 1/4 in. in diameter. Between the baths and the cell all of the tubing, including return lines, passed through a two position clamping valve (see Fig. 4.2) specially designed to allow instantaneous switching of the cell from the isothermal to 94 the nonisothermal configuration and yige yersa. With both clamps closed, no water was admitted to the reservoirs. With only the left one open, a temperature difference was applied, and with only the right one open, the cell was isothermal at Tm. At no time could both clamps be open, or water would be transferred between the baths causing overflow. Bypasses had to be installed so that whenever a bath was isolated from the cell, its water could still circulate through the copper coil in order to maintain thermal equilibirium within the 2000 ml beaker. The starting time of all experiments was taken to be that instant when the clamp for the T—9 was closed and the clamp for the other two baths was opened so that the temperature difference was applied to the cell. Inside the reservoirs, the water flowed in the pattern shown in Fig. 4.1. There was a space of 3/32 in. between the baffles and the metal plates in order to elimi— nate the possibility of any dead Space. In order to promote a more uniform temperature dis- tribution, the reservoirs extended beyond the area covered by the sample chamber. The temperature distribution across the bottom plate was checked at 20°C by means of a 40 gauge Copper-constantan thermocouple junction held against the plate with a piece of styrofoam insulation and a 100 gram weight. The position of the thermocouple junction was measured, and it was allowed to remain undisturbed for two 95 minutes while the temperature at that point was measured by the strip chart recorder. Thirty seconds were usually re— quired for thermal equilibrium, but the additional time was used to insure that no further change in temperature would occur. The measurements were repeated at half—inch inter— vals across the whole plate. While the resulting 81 data points showed the presence of thermal gradients near the side walls of the reservoir, the temperature over the area occupied by the sample chamber remained constant to within 0.01°C with only randomly spaced variations. Since we did not wish to conduct thermal diffusion experiments with thermocouple wire inside the cell disturb— ing the temperature distribution and possibly the diffusion flux, it was necessary to establish whether any systematic difference existed between the temperature of the metal plate inside the sample chamber and the measured plate temperature somewhere outside the cell. For this check, a thermocouple wire was passed through one of the filling tubes and attached to the upper plate by means of a very small piece of tape. The cell was then assembled and filled with a mixture of CCl4 and C6Hl2' A second thermo- couple junction was mounted outside of but close to the sample chamber on the upper metal plate. The junction was first placed in contact with the plate and then covered with a one inch square of aluminum foil to insure that the 96 measured temperature represented that of the plate and not some average influenced by the air temperature. The wire leads from the junction were kept in contact with the plate for a distance of about three inches in order to eliminate thermal gradients in the wire. The junction, foil, and wire were then covered with a piece of black electrician's tape. When a temperature difference was applied to the liquid, and after a period of fifteen minutes passed, during which a thermal steady state was reached, the voltages of the two thermocouple junctions were recorded. Both refer— ence junctions were in the same ice-water bath. According to a Leeds and Northrup K-3 potentiometer and a previously prepared temperature-emf calibration chart, both junctions indicated the same temperatures to within 0.002°C. Measure— ments were repeated for thirty minutes, during which only small random differences between the two temperatures were observed. Consequently we felt safe in using the tempera— ture measured outside the cell as the plate temperature in the thermal diffusion experiments. The above—mentioned thermocouple junction and a similar one on the lower plate were next used to investigate the time dependence of the plate temperatures at the begin- ning of an experiment. During the change of configuration, the temperature of each plate was monitored with a separate strip chart recorder, and as expected, an exponential shape was observed. 97 Experimental curves were fitted to functions of the type ‘ -t/t AT h Th=Tm+——2(l—e )=¢h(t), -t/t _ AT c _ Tc_Tm__2 (l‘e )‘¢c(t" (4.4) where Th and TC are respectively the hot and cold plate temperatures, t is time measured from the instant of switching. The two constants tC and th are the two re- laxation times for heat conduction through the apparatus mentioned in Chapter II with the statement that they best determined experimentally. The results of the curve fitting were: t = 46 sec , c th = 46 sec . (4.5) Because the capacity of the T—9 bath is different from that of the others, the relaxation times were also measured for the initial part of the remixing experiment, which requires removal of an established temperature gra— dient. Here the functional form is: -t/t' a _ AT h T(§,t) — Tm + —2 e (4.6) -t/t' a _ _ AT C T(—§,t) _ Tm —E e (4.7) The relaxation times for the second case were found to be: 98 ‘ = th 54 sec té = 54 sec . (4.8) Measurement of the various temperatures required was accomplished with thermocouples made of 40 gauge matched copper and constantan wires. The wires, indivi- dually coated with Teflon, were wrapped together in an additional fabric insulation. A twelve inch length of the fine wire was soldered to about eight feet of more durable 20 gauge copper and constantan wires. The heavier wires were also of matched resistances, polymer coated and bound together by an outer clear plastic film. Both sets of wire were obtained from the Thermo—Electric Co., Inc., Saddle Brook, New Jersey. Sixteen thermocouples were prepared. A small arc welder, obtained from the Chemical Rubber Company, was used to fuse the two metals into spherical junctions with 0.4 mm diameters. Because of the thinness of the wires, the energy of the arc was sufficient to destroy about an inch of the metal before forming the junction. For more satisfactory performance, the welder was plugged into a 15 A variable transformer, and the voltage was cut from 115 V to about 25V. With the lower energy arc, the junction was more easily formed. The reference junctions were contained in an ice— water bath, with a four liter capacity, equipped to hold 1 “P ext( COM in: ( Fibl ter; th Dis ice 011$ anc' c S ta) 99 up to twenty such junctions. Copper and constantan wires extended into 1/4 in. diameter glass tubes 6 in. in length containing 1—1/2 in. of mercury. The tubes were immersed in an ice—water slush. The entire apparatus was surrounded by a l in. layer of styrofoam insulation and encased in Fiberglas. The top was covered with a wooden lid, and a terminal panel was provided for convenience in changing thermocouples. After filling, the bath retained a constant temperature for up to four hours before it needed attention. Distilled water was used in the bath along with machine—made ice cubes initially 3/4 in. X 3/4 in. X 1/4 in. in size. A sixteen junction Leeds and Northrup K—3 potenti— ometer was connected to an electronic null detector (Leeds and Northrup Model 9834) having nonlinear meter response and maximum sensitivity of i0.2 microvolt per division. EMF's could be read to the nearest 0.1 microvolt, permitting calculation of the temperature to the nearest 0.002°C. It should be noted that any shift in the temperature scale which might have developed due to a nonzero reference temperature or a decay of the standard cell in the potenti- ometer would be inconsequential,since only differences be— tween measured temperatures had to be very accurately known. Also available for measurements were two Sargent model SR potentiometric strip chart recorders. These showed full scale deflections of 200 microvolts, or 5 degrees Centigrade. Five different scales were available, 100 corresponding to the temperature ranges: 0—5°C, 15—20°C, 20—25°C, 25—30°C, and 30—35°C. Again, any inaccuracies introduced by the expansion of the recorder scale were not important since the recorders were used only as indicators. Any critical measurements of temperature were obtained with the potentiometer. The re— corders were entirely satisfactory; they showed fast response and a high degree of repeatibility, and registered tempera- ture changes on the order of 0.002°C. When the sixteen thermocouple junctions were compared against one another by placing them two at a time very close to each other in the same constant temperature bath at 25°C, they all registered the same voltage to within 0.1 microvolt or 0.002°C. Consequently it was not deemed necessary to per— form separate calibrations for each of them. This extra step would have been impractical anyway since most of the junctions were broken and replaced at some time during the experiments and since each calibration would have required about three days. The original calibration was carried out with a thermocouple which did not differ by more than 0.1 micro- volt at 25°C from any of the others. A platinum resistance thermometer was used along with a constant current source (2.0 mA) and a resistance box. By using the galvanometer Of the K—3 potentiometer,we could measure the resistances 0f the platinum wire at a series of temperatures. The known lOl temperature dependence of the resistivity of the platinum then allowed us to calculate the actual temperature. The platinum thermohm and the thermocouple junction in question were placed in contact with each other and into the small open port of one of the T-45 baths. The emf of the thermocouple could be monitored on one of the recorders as well as with the potentiometer. After the recorder in— dicated that a steady temperature had been reached in the bath, the following measurements were taken five times at one minute intervals: (1) resistance of the platinum wire at 2.0 milliamps, (2) emf of the thermocouple junction. If the measurements showed any large fluctuations or drifting, they were repeated until five consistent sets of values were obtained. Then the direction of the current was reversed, and the measurements were repeated. The same measurements were repeated at one degree intervals from 16°C to 34°C. For each value of the resis- tance the temperature was calculated by means of Eq. (4.9). R — T _ 3 _ T 0 T . --T _"-~ , T T ‘ _OLR0— +5 ”100 ' I) + B ‘1'0‘0 1’ (Too) , ‘4-9) where T = temperature in degrees Centigrade R = measured resistance, international ohms, at 2.0 mA R = 25.4884 int. ohms 102 a = 0.00392604 B = 0.1106 ; T < 0°C 8 = 0.0 ; T 2 0°C 6 = 1.4919. The data of interest consisted of a list of tempera- tures calculated from the measured resistances and a list of corresponding thermoelectric potentials. A FORTRAN IV program, EMFVST, was written for use with MULTREG, a multi— nomial regression analysis program. The Control Data Corporation 3600 digital computer calculated the best smooth curve through the experimental points to be: EMF = 2.33066 + 40.04151T + 1.300289 x 10'5T4 , (4.10) where EMF is in microvolts and T is in degrees Centigrade. The standard errors of the coefficients of T and T4 are respectively 5.52 x 10'2 and 8.13 x 10‘7. At 20°C and 30°C Eq. (4.10) gives EMF's of 0.8052 mV and 1.241 mV, respectively. The calibration table in the Handbook of Chemistry and Physics (44th edition) lists the correspond- ing numbers as 0.79 mV and 1.19 mV, respectively. In measurin differences in tem eratures, however, we used g the temperature coefficient of very nearly 0.4004 mV deg—l, which compares well with the handbook value of 0.40 mV deg—l. Equation (4.10) was used in preparing an extensive table with which a measured voltage could be rapidly 103 converted to a temperature. EMF‘s for all of the tempera— tures between l4.00°C and 35.99°C were printed out at 0.01°C intervals, and interpolation to the nearest 0.002°C was easily accomplished. The computer calculated the 2200 num— bers and printed them in tabular form. The table was hung on the laboratory wall for quick reference. D. The Interferometer Having decided to use optical rather than conduct— ometric or sampling methods for analysis of concentration changes, we next had to choose from among the various types of suitable interferometers available. Pure thermal dif- fusion experiments require that an instrument be able to detect differences in mass fraction as small as one part in 105 over distances of a few millimeters. The wavefront shearing interferometer described by Bryngdahl (1963), un- like Rayleigh or Gouy instruments, had not yet been applied to diffusion studies. Bryngdahl's interferometer promised to be at least as sensitive as any of the others in use and had the added advantage of not being difficult to use. Also it offered a chance to make the first application of a new design. While our work was in progress, however, Bierlein (Gustafsson, 1965) published an account of some experiments conducted with a similar instrument, both confirming its advantages and relating the results of some studies on de— Sign optimization. 104 Before building the interferometer, we modified the plans by substituting for the conventional light source a helium—neon gas laser (A = 6328A). This produced an intense beam of parallel, monochromatic, polarized light, all fea— tures required by the instrument, but not present in a sodium or mercury lamp. The laser chosen was a Siemens model LG-64 having output power in the TEM 0 uniphase mode 0 of six milliwatts. Since the light emerging from the laser was polarized in the vertical plane, the laser was rotated 45° about its long axis in order to provide the necessary orientation be— tween the polarization plane and the refractive index gra— dient in the cell. The laser was mounted between four vertical 1/2 in. steel threaded rods which were attached to the 15 ft horizontal steel I—beam mentioned above. The threaded rods provided flexibility in positioning the laser. See Fig. (4.2). The diameter of the beam emerging from the laser was 2.5 mm, much less than the cell height. A shutter was provided to keep light from the cell when not needed. A simple two lens system with focal lengths Ll,f = 17 mm and L2.f = 203 mm produced a parallel beam 35 mm in diameter. See Fig. 4.3. After traversing the cell, the initially flat wavefront was distorted if a refractive index gradient was present. A second simple lens system (L3,f = 371 mm, L4,f = 22 mm) reduced the height of the beam from 7.41 mm 105 Figure 4.3.--He1ium-neon laser and lenses L1 and L2. 107 to 0.5 mm in order that it be compatible with the dimensions of the beam splitters. The beam splitters (Bryngdahl, 1963), obtained from Valpey Optical Corporation, were modified Savart crystal quartz plates. These were cut at 45° from the axis and oriented so that the axes of the subplates were in the same plane but perpendicular to each other. An incident ray gave in the first plate an ordinary ray and an extraordi— nary ray. In order to get symmetrical light paths through the whole plate, the ordinary ray in the first subplate had to become the extraordinary ray in the second one and yige ygrsa. A half-wave plate inserted between the two subplates so that its principal plane bisected the 90° angle between those of the subplates interchanged the polarization planes of the two rays. Thus there was a compensation of path dif- ferences, i;e., no path difference was introduced by the beam splitter in parallel light. See Fig. 4.4. The net effect of the first beam splitter Q1 was the production of two identical beams of equal intensity having perpendicular polarization planes and separated vertically by a distance bl‘ The separation of the two beams is given by: (4.11) .HO nH mcHHmmnw mnflomHmpds Emma “coda mo spew v.v m . ll THU Hm 108 109 where e is the thickness of each part of the double plate, and I10 and ne are the principal refractive indices of the quartz. In convergent light, the beam splitter splits an entering wavefront into two wavefronts with polarization directions perpendicular to each other. In this case, the shear angle introduced results in an optical path differ— ence A between the two emergent wavefronts which depends on the x—coordinate via the corresponding incident angle 0 and on the thickness of the crystal plate. For the plate used, A = bl sin w cos Y (4.12) where w is the angle between the entering ray and the normal to surface and y is its azimuthal angle. The parallel light beam traverses the first beam splitter Q1 resulting in the formation of two identical beams displaced vertically from each other and having per- pendicular polarization planes. The second beam splitter was identical to the first but was turned through an angle of 90° in order to retain the proper orientation between polarization planes. Between the two beam splitters Q1 and Q2 a simple double convex lens L5 having focal length 22 mm produced the convergent light for Q2. After the second beam splitter, the interference fringes were made visible in image plane 2 110 by means of a polarizer (a Nicol prism) oriented at right angles to the polarization plane of the original laser radia— tion. A final lens L6, consisting of an ordinary microscope objective with a magnification factor of 5, made possible ad— justments in the beam size for convenient photographing. The interference fringes, representing the vertical refrac- tive index gradient in the cell, appeared within a sharp double image of the cell. The use of Q1 and Q2 rather than ordinary Savart plates caused the fringes to be presented in Cartesian rather than hyperbolic coordinates (Bryngdahl, 1963). A photograph of the interferometer is shown in Fig. 4.5. A theoretical discussion of the paths followed by the light beams inside the quartz plates is given in Appendix G. The working equation for the interferometer is x = A(An/Az) + B , (4.13) where A is a magnification—related apparatus constant which is best determined by means of a separate calibration (dis- cussed below), and B determines the family of fringes which is observed. B need not be known if we measure only the position of the same fringe at various times. The quantity (An/AZ) is a finite difference expression for the refractive index gradient. Through it we can relate measurements of fringe position and shape to expressions for the gradients of temperature and composition. 111 Figure 4.5.——Interferometer components and Polaroid camera. 113 The image of the cell and the fringe pattern was projected through a Polaroid MP-3 camera and onto a ground glass plate. At any time the plate could be moved aside and replaced by a Polaroid roll film back, and a photo— graph could be taken on Polaroid type 413 infra—red sensi— tive film. Even though visible light was used, this film was required because of its greater sensitivity in the red range of the spectrum. The photographs, developed in the camera in 15 seconds, were 3—1/4 in. X 4—1/4 in. black and white positive prints. A device to measure fringe positions on the photo— graphs was made by mounting on a 4 in. X 6 in.X l-1/2 in. block of aluminum a mechanical microscope stage with gradu— ations and Vernier scales which could be used in conjunction with a magnifying lens to determine the two dimensional shape of the fringes to 0.01 cm, or the equivalent of 0.13% of the cell height. Each photograph cost about $0.50 and required five to ten minutes to analyze. To permit more practical accum— ulation of large amounts of data, an alternate measuring device was also used. This consisted of the same microscope Stage mounted directly on the ground glass plate Of the camera. The arrangement allowed the fringe position at Z = 0 to be measured frequently and rapidlY~ A photograph could still be taken when more detailed information about the fringe shape was desired. 114 E. Working Equations Let the vertical refractive index distribution in the cell be given by the expansion: n(z,t) = z c (t)z , (4.14) where the coefficients ck are functions of time: c0(t) = n(0,t) cl(t) = (an/82)O 1 2 2 c2(t) = 5(8 n/Bz )O cn(t) = %T(8nn/an)0 . (4.15) The subscript zero means the derivative is evaluated at Z=O, The finite difference expression for the fringe shape requires the quantity: as An(Z,t) = n(z + %§,t) — n(z "‘Tf't) (4.16) It follows from Eqs. (4.13) and (4.16) that: 1 x = A{(cl + Z c3a s + 16 05a 5 + z + (2c2 + c4a s ) 2 2 2 3 _ + z + (4c + .)z + (3c3 + 2 c5a ) 4 + (5c +...)z4 + ...} + B . (4.17) The quantity 5 is the amount of shear (0.19). 115 The experimentally measured fringe shapes are well represented by the polynomial: 5 k x = X dk(t)n , (4.18) k=0 where n is the dimensionless vertical cell coordinate ob- served on a photograph, and the d the coefficients giving k the best least squares fit. Because of the double image which is due to the shear s, the vertical coordinate z in the cell is related to the vertical coordinate n in the photograph by: 22 1"] = W (4.19) where and s is the shear, or amount of overlap of the two images. The relationship between the coefficients dk and Ck is discovered by using Eq. (4.19) and equating coeffi— cients of like powers of z in Eqs. (4.14) and (4-l8)= 1 2 2 l 4 4 + + B d0 = A(cl + 4 c3a s + T6 c5a s ...) 2 d1 = g(a - as)(2c2 + c4a s +. ) 2 5 2 2 d2 = %(a _ as) (3c3 + 2 c5a s + ) _A _ 3 + d3 — g(a sa) (4C4 ) d = é— (a - as)4(5c + .) 4 16 5 __A __ 5 + .) (4.20) d5 — 3? (a as) (6c6 116 The coefficient dO represents a uniform lateral shift of the whole fringe pattern. Since dl is the coefficient of the first power of n, it accounts for a skewness in the photo- graphs which decays away as (82n/322)0, or c2, approaches zero, in exact agreement with Bierlein's observation. Inversion of Equations (4.20) gives: _ 16 —4 1 c5 — —X(a - as) (— d4 + ) _ 8 —3 1 c4 — g(a - as) (4 d3 +...) 4 2 1 1 2 2 a - as "2 C3 = g(a " as) [3 d2 — g d4a ( 2 ) +...] -2 _ 2 _ —l 1 1 2 2 a — as 1 c2 — g(a as) [2 dl § d3a S ( ) +- —2 _ 2 —1 l 2 2 a — as cl — g(a — as) [do — 5 dza s (7“) +. .] . (4.21) Thus all of the coefficients in the expansion for Sn/Bz which follows from Eq. (4.14) can be determined from measurements of d (k = 0...5), A, s, and a. kl The coefficients ck are related to the tranSport parameters d D, K , and Q3 through the expression for the l’ i temperature and composition distributions and through rela- tions such as: (33) (4.22) 0 Z C1 = %)T(O§Zl)0 + (%)W Where (Bwl/Bz)O is a function of 01, etc. In the next chapter calibration of the interferometer and the particular measurements involved are discussed, along with the other experimental details. CHAPTER V EXPERIMENTS A. Weighing Procedure Because we chose to work with carbon tetrachloride— cyclohexane mixtures in order to be able to compare our results with those of previous workers, we had to deal with the problem of evaporation. Such losses before and during an experiment can lead to miscalculations of the actual composition of the mixture. The following procedure was used in an attempt to avoid, or at least minimize, errors in the determination of the mass fractions of the components of the solution actually undergoing thermal diffusion in the cell. Four 25 m1 Pyrex pycnometers for volatile liquids were obtained from Scientific Glass Apparatus Company. The 80 gram capacity of our Mettler H—16 single pan analytical balance precluded the use of larger volumes of liquid. Each pycnometer consisted of a 25 m1 bottle, a capillary stopper, and a cover which prevented evaporation from the Open capil— lary tube. The three pieces fit together with ground glass joints, and each part was marked with the same number to prevent interchange between sets. 117 118 Only one of the pycnometers was used throughout the whole series of experiments. Before each use it was cleaned with a solution of potassium dichromate in 98% sulfuric acid, rinsed in distilled water, and dried in an oven at 105°C. When cool, it was placed in a water bath at 25.00°C for three minutes, removed, dried with Kimwipe tissues, and weighed on the previously zeroed H—l6 balance. The volume of the pyc— nometer was determined from a series of measurements during which it was weighed while filled with either distilled water, carbon tetrachloride, or cyclohexane at 25.00°C. The water was obtained from the distilled water tap in the laboratory. The other two liquids were obtained from the J. T. Baker Chemical Company. The labels of the bottles used are repro- duced in Tables 5a and 5b. All chemicals were used without further purification. An excess amount of the particular liquid below 25°C was poured into a clean, dry 250 ml Erlenemeyer flask provided with ground glass stopper. The liquid was removed from the flask by means of a 100 m1 capacity glass syringe fitted with a 12 in. length of Teflon tubing of 1/16 in. i.d. The Teflon tube was then replaced by a 1—1/2 in. size 18 stainless steel syringe needle, and the air in the syringe was removed. The liquid was then injected into the pycnom- eter bottle until the bottle was nearly full, at which time the capillary stopper was inserted, causing an overflow of the excess liquid and the exclusion of all air from the ... 1 1W. . . .4 . . I 4 ... want-4.411.111 .. .. .. “.11.. .n. . .... .. A. 119 Table 5a.—-"Baker Analyzed" reagent lot analysis as given on bottle label for CC14. 1 pt. (473.2 ml) 1513 ‘ CARBON TETRACHLORIDE CCl4 F.W. 153.82 "Baker Analyzed" REAGENT SPECTROPHOTOMETRIC LOT NO. 34532 Color (APHA) . . . . . . . . . . . . . . . . . . . . 5 Density (g/ml) at 25°C . . . . . . . . . . . . . 1.585 *Boiling Range 1-95 m1 . . . . . . . . . . . . . . . 0.1°C 95 ml—dryness . . . . . . . ... 0.2°C Residue after Evaporation . . . . . . . . . . . . . 0.0004% Acidity . . . . . . . . . . . . . . . . . Pass ACS Test Free Chlorine (Cl) Pass ACS Test Sulfur Compounds (as S) . . . . . . . . . . . . 0.003% Iodine Consuming Substances . . . . . . . Pass ACS Test Substances Darkened by H2804 . . . . . . . . . Pass ACS Test Solubility for use in Dithizone Test . . . . . Pass ACS Test rRecorded Boiling Point 76.7°C. 120 Table 5b.—-"Baker Analyzed" reagent lot analysis as given on bottle label for C H . 6 12 1 pt. (473.2 m1) 9206 CYCLOHEXANE CH2(CH2)4 CH2 .W. 84.16 'Baker Analyzed' Reagent ACTUAL ANALYSIS OF LOT NO. 34840 Color (APHA) . . . . . . . . . . . . . . . . . . 2 Density (g/ml) at 25°C . . . . . . . . . . . . 0.773 ’Boiling Range, 1—95 ml . . . . . . . . . . . . 0.1°C 95 ml—dryness . . . . . . . . . . . . 0.1°C Residue after Evaporation . . Substances Darkened by H2804 . . . . . . . Water (H20) . ’Recorded Boiling Point 80.7°C. . . 0.0008% Passes Test . 0.014% 121 container. The pycnometer was never touched directly. All handling was done with tissues or a wire holder consisting simply of a length of wire wrapped around the neck of the bottle. By means of the wire holder, the partially assembled pycnometer was transferred to a wire basket in the T—9 water bath which was maintained at 25.00°C. The heating and re— sulting overflow of the sample liquid continued for about four minutes. When the liquid level was just even with the top of the capillary tube, the pycnometer cover was put tightly into place. The assembly was then removed from the water, dried carefully with a Kimwipe as before, and weighed on the H-l6 balance. The air temperature, relative humidity, and barometric pressure were recorded at the time of the weighing for purposes of air buOyancy corrections. A Sargent hygrometer provided wet and dry bulb temperatures, and a Fortin—type mercurial barometer permitted determination of the atmospheric pressure. From the known densities of the liquids used, the 25° volume of the pycnometer could readily be calculated. In a substitution weighing, the beam is brought into . + eQuilibrium with a set of weights as the load and the scale reading are set to zero. Next an object is placed on the pan, and weights are removed to return the beam to equilibrium. +Following custom, we call the standard masses "weigh-ts . ll 122 The balance indicates two numbers: (1) the nominal value of the weights removed; and (2) the indicated difference be- tween the weight of the object and the weights removed. The effect of gravity and air buoyancy on the weights must be taken into account, while other forces must be avoided or eliminated. Various forces such as electro— static or magnetic forces, the "sail effect“ from moving air, and air buoyancy on the beam or other parts of the moving system may act to change the balance indication. As long as these forces remain constant, their effect will go out in the difference between the two readings. Conse— quently only gravity and air buOyancy need be considered. The balance equation is: (Mu — Vup)g = (MS — Vsp)g (5.1) where Mu = true mass of object Ms = true mass of weights used Vu = volume of object Vs = volume of weights p = air density g = acceleration of gravity Forces are eliminated by dividing by g, whence, upon re— arrangement: (5.2) 123 Equation (5.2) requires knowledge of the true mass of the weights used, which differs from that of the nominal values observed. In the United States, Normal Conditions are de— fined to consist of air density of 1.2 mg/cm3, temperature of 20°C, and standard weights having an ideal density of 8.4 g/cm3 at 0°C and coefficient of cubical expansion of 5.4 X 10_5 deg_1 C. From this the ideal density at 20°C is 8.3909 g/cm3. Usually the air density differes somewhat from 1.2 gm/cm3 as defined for normal conditions, and the density of the weights used differs from 8.3909 g/cm3 at 20°C. According to L. B. Macurdy, Staff Metrologist of the Mettler Corporation, the weights in the Mettler Model H-l6 balance are of one-piece stainless steel with a nominal density of 7.76 g/cm3 to be assumed at 20°C. In order to obtain the true mass of the weights Ms’ it is necessary to add the correction +ll.63 micrograms per gram to the indicated value to take account of the fact that the density of the weights is not 8.4 g/cm3. Also, since measured volumes are not usually available, Eq. (5.2) can be rearranged to give the true mass of the object in terms of densities: _ 2 2 3 3 Mu — Ms(l — p/DS)(1 + p/Du + p /Du + p /Du +...) , (5.3) wh = ' ' ere MS Mapparent X (1.00001163), DS 18 the denSity of the weights, and Du is the density of the object, both Calculated at the ambient temperature. Du is best 124 approximated by Du a Ms/Vu. Most of our weighings were ob— tained at an ambient temperature of 24°C rather than 20°C. If the coefficient of cubical thermal expansion of the stainless steel is assumed to be: 6 -l -8 = 51 x 10' deg , then at 24°C: 4 3 Du = 7.76(1 — 2.04 x 10‘ )g cm— Since Du was originally given with only three significant figures, the correction is certainly negligible. In order to calculate the air density, measurements of the barometric pressure, the relative humidity, and the ambient temperature were required. The standard temperature for the density of the mercury in the barometer is 0°C. Since the mercury and the brass scales have different co— efficients of thermal expansion, the pressure indications are affected by variations in the temperature. The manu— facturer of the barometer, Precision Thermometer and In- strument Company, Philadelphia, supplied Temperature Correction Tables which combined the corrections for length 0f the scales and density of the mercury. We used Gravity Correction Tables to take account of the latitudinal varia~ tion of the gravitational constant. The combination of these corrections usually contributed about —2.9 mm Hg. Tables supplied with the Sargent hygrometer were also used to calculate the relative humidity from the .1- 1-1.11-1111344.11-1-111.1 . . . . . . .... . ......c... 125 measured values of the ambient temperature and the depression (in degrees Farenheit) of the wet bulb thermometer in the Sargent electric hygrometer. For most of our weighings, the air density was about 1.16 mg/cm3. To speed the recording of data and the calculation of weighing corrections, a simple form was typed on a Ditto master, and spirit copies were used for all of the weighings. A completed sample form is shown in Table 5c. We used the data of Wood and Gray (1952) to obtain the densities of pure carbon tetrachloride and pure cyclo- hexane as well as the temperature and composition dependence of density. At 25°C: p1CC141= 1.58414 g/cm3 p(C6Hlj = 0.77383 g/cm3 . The volume of the pycnometer, based on the results of ten trials with water, CC14, and C6H12’ was taken to be: v = 25.7523 i 0.0025 cm3 Once the volume was known, the densities of mixtures of the two organic liquids could be determined by the same weighing technique. The density versus composition data of Wood and Gray at 25°C were expressed by the polynomial (from MULTREG, see Appendix H). l- wi = 1.99014 - 0.01505 — ——§%ll5 , (5.4) Where w? is the mass fraction of CCl4 in the mixture and D is the density of the liquid in g/cm3. The dimensions of 126 Table 5c.——Sample weighing form. Date 3—22—68 Time 10:00 a.m. Run No. -— Pressure, mm Hg 745.4 +742.5 Liquid CCl4 Tdry 77°F; Twet 54°F. Liquid Temp., °C 25.00 Room Temp., 23.9°C; 75.0°F Vol. Fraction 1.00 Rel. Humidity, 15% Pycnometer No. 375 Time in Bath, Minutes 5 Liquid Density (Approx), g/cm3 Du = 1.58414 Air Density, g/cm3 0 = 0.001158 Weight of Bottle and Liquid, g. 65.99890 65.99886 65.99888 Weight of Empty Bottle, g. 25.22384 25.22384 25.22384 Apparent Mass of Liquid, g. Ma = 40.77504 True Mass of Weights Used, g. MS = Ma (1.00001163) - 40.77551 True Mass of Liquid, g. _ 2 2 Mu - Ms(l — p/7.76)(l + p/Du + p /I3u + ...) = 40.77551 (.99985)(1.00073) = 40.79920 Pycnometer Volume, cm3. 25.7549 (calc.) Liquid Density, g/cm3. 1.58414 (lit.) Mass Fraction, Wl = 1.0000 127 he coefficients are appropriate to cancel those of p. The tandard errors of the two coefficients in Eq. (5.4) are 4 and 7.25 x 10'4. espectively 5.83 X 10— When the liquid being prepared was scheduled to ndergo thermal diffusion in the cell, the following modi- ications were made in the above procedure. The two com— )onents were mixed in the 250 ml flask in the approximate )roportions desired. For example, if the desired mole fraction was 0.6, then 40 m1 of cyclohexane were added to L0 ml of carbon tetrachloride. No precaution against :vaporation was taken at this point. The flask contained (1-1/4 in. Teflon coated magnetic rod, which permitted excellent mixing of the two liquids when the flask contain- .ng them was placed on a magnetic stirrer. When mixing was complete, the flask was chilled for a few seconds by placing it in contact with ice. This )as done in order to insure that the temperature of the iquid entering the pycnometer was below 25°C. After the hilling, the flask and its contents were returned to the tirrer for about another minute to insure a uniform emperature and composition. Then the stirrer was shut off, the stopper was emoved, and about 80 ml of the liquid was drawn through he Teflon tube into the large syringe. The tube extended e11 below the surface of the liquid. The tube was re- laCEd by a needle, and all air was removed from the syringe. 128 he first few milliliters of the liquid was discarded, and en the cell was filled as described in the following sec— ion. Immediately thereafter, a small quantity of the liquid as again discarded from the syringe (that portion which was 'n contact with air), and the pycnometer was carefully filled ithout disturbing the liquid surface or causing an unusual amount of evaporation. The pycnometer was overfilled so that the liquid close to the surface, whose composition may have changed by differential evaporation, was spilled out when the oottle was closed. The closing, thermal equilibration, and weighing of the filled pycnometer were the same as described above. The liquid in the pycnometer and the liquid in the cell were assumed to have the same composition. 3. Stepjbnytep Procedures All of the facets considered above, the cell, the temperature control system, the interferometer, and the eighing technique come together to fulfill their purposes 'n the actual execution of an experiment, which is most fficiently described by a series of steps. 1. Turn on all water baths to their desired preset temperatures at least 12 hours before the start of an experiment. 2. Switch on the potentiometer at least one hour before any measurements are to be made. 129 3. Dry the pycnometer parts in the oven at 105°C for at least two hours if not already dry. Handle only with tongs which have been cleaned in sulfuric acid, rinsed thoroughly in distilled water, and dried. 4. Check the water levels in all baths and in the beaker on the stirrer for T—9, and refill if necessary. 5. Refill the small (500 ml) beaker with fresh distilled water for rinsing the used pycnometer after cleaning. 6. Fill the thermocouple junction reference ice bath and allow to equilibrate. ' 7. Make certain the camera is loaded. 8. If pycnometer has been in oven for two hours remove, and let cool in air before assembling. Do not touch. 9. Clean the silver plated surfaces of the cell, remov— ing any oxide coating with silver polish. Rinse thoroughly with a CCl4 — C6H12 mixture and dry with— out leaving streak marks. 10. Position thermocouple junctions, each between its plate and a piece of foil. Hold in place with electrician's tape, making sure that two or three inches of the lead wire is in contact with the metal plates. ll. Clean the glass cell walls with a CCl4 — C6H12 mix— ture and Kimwipe tissue to remove all old sealant and any marks. 1 11111111141114.1315... .... . . . .. .. .4 . $13.14.: .. 4... . a p . . . . ..., . 12. 13. 14. 15 16. 17. 18. 19. 20. 21. 22. 130 Remove any air bubbles from bottom reservoir by holding it vertically so that the air exits through one of the ports at the highest point. Level the bottom plate by using a bubble indicator and the four adjusting nuts on the large threaded rods supporting the cell. Apply Dow FS Fluorosilicone Sealant to one surface of the glass cell wall assembly by means of a 10 ml glass syringe and needle. Position the glass cell wall assembly on the inverted top reservoir assembly, coated side down, and with proper alignment of the filling tubes in one corner. Apply silicone sealant to the upper side of the cell wall assembly with syringe. Assemble the cell, fasten retaining nuts with light pressure only. Turn on the laser and its timer. Align optics and focus interferometer On glass plate of camera. Tilt the cell for filling and provide a Kimwipe to absorb spilled liquid. Make sure the bath switch is in the isothermal con— figuration. Clean the glassware for solution preparation and rinse with CCl4 or C6H12' Of the solution in a 250 m1 Erlen- Prepare 100 m1 meyer flask with a ground glass stopper. 23. 24 25 26 27 28 29 30 31 32 o 131 Mix the solution with a magnetic stirrer, but not vigorously enough to aerate. Assemble the pycnometer, holding it with Kimwipe tissues. Prepare the balance by cleaning, levelling, and zeroing it. Record the room temperature, barometric pressure and wet and dry bulb temperatures on a weighing form. Using a wire holder, place the empty pycnometer in the 25.00°C water bath for l or 2 minutes. Re— move and dry it with two Kimwipes. Weigh the pycnometer immediately and record weight on form. Recheck the temperature of the 25°C bath with the potentiometer and temperature—emf chart. Reset if necessary. Chill the flask containing the sample liquid in the ice bucket for 30 seconds. Remove and dry with paper towel. Return flask to stirrer and mix gently for one minute. With a 100 ml syringe and Teflon tube, withdraw about 80 m1 of sample liquid from flask, keeping tube well below surface of liquid. 33 34 35 36 37 38 39 40. 41. 42 43. 132 Exchange tube on syringe for 18 gauge stainless steel needle and expel all air from syringe. Discard the 2 or 3 m1 of solution which has been in contact with air. Fill cell, using filling tube away from corner, until all air is removed from sample chamber and both filling tubes are full of liquid. Discard the next 2—3 ml from syringe and fill pyc— nometer rapidly without making bubbles or disturb— ing the surface. Insert the pycnometer's capillary top, but do not cover. Handle only with Kimwipes. Using wire holder, place pycnometer into rack in 25° water bath. While the pycnometer is in the bath, seal the cell by sliding the glass walls between the plates just enough to close Off the filling tubes. Tighten the nuts on the cell with a small wrench, being very careful to avoid breaking the glass. Turn on the recorders to monitor the plate tempera— tures. Observe the meniscus on top of the pycnometer, and place the cover on when the liquid is level with the capillary top. Remove the pycnometer from the bath and dry with two Kimwipes as before. 44 45 46 47 48 o 49. 50. 51. 52. 53 54 55 56 133 Immediately weigh the full pycnometer, and record the weight on form. Return the contents of the pycnometer and syringe to the flask for use as a rinsing solution for the next run. Do not touch the pycnometer. Place pycnometer parts in acid cleaner (K2Cr207 — H2804) for several hours, then into distilled water. Disassemble syringe and cover with tissues. Let the cell and contents equilibrate for about an hour. Insulate from room air. Use laser to realign interferometer if necessary. Set the experiment timer to 0.00 min. Take t = 0.00 photo for line spacing and shear measurements. Close the bypass valves in the two T—45 baths. Apply the temperature difference and start timer simultaneously. Change scale on recorder if necessary. Open the bypass valve on the T—9 bath. Realign the optics at t = 4 or 5 min when the temperature gradient has been established. When t = 8 or 10 min begin measuring the fringe 0 with the microscope stage appara- position at z = tus. Use intervals of l, 2, or 5 min until t = 0. .IIII...11.|..!J .14).». . . .. .n .. . . . . . . r. ..r. 57. 58. 59. 60. 61 62 63 64 65 66. 67 o 68 69. 134 Between measurements use the shutter to prevent light from passing through the cell. Check the plate temperatures regularly, and record the emf's on the charts. Refill the ice bath each hour. Take photos occasionally for fringe shape and spac— ing and note the time on the back of each. Continue taking measurements of fringe position at intervals of 10-15 min until t = 50 or 60. Obtain several steady state measurements for t > 60. Recheck temperatures. Reset timer to 0.00 min. Close the bypass on the T—9 bath. Remove temperature difference and restart timer simultaneously. Open bypass valves on T—45's. Check recorders; switch scales if necessary. Realign optics at t = 5 min. Repeat measurements of fringe position versus time. Take desired photos. Continue until t > 50. When finished, shut everything off unless another run is planned. Dismantle the cell. Press wire pins through the filling tube against the glass wall assembly in order to prevent leakage of the liquid from the bottom. 135 70. Empty the sample chamber with a syringe. Clean both metal plates and glass. 71. Record all data. 72. Coat and record photos. 73. Punch data cards. . Discussion of Procedures Some of the steps in the procedure enumerated above equire additional comment. It was found to be necessary 3 turn on the water baths well in advance of an experiment ven though they reached their nominal temperatures within n hour. The extra time was needed to allow the bath hous- 3g and insulation materials to reach steady temperatures. ecause of the common source of cooling water, the three aths were indirectly interconnected and had to be balanced gainst each other very delicately whenever a new tempera— dre range was set. During a series of experiments slight flanges in the mean temperature and/or the temperature dif- erence between runs arose primarily because of daily flanges in the room temperature. All of the carbon tetrachloride used was obtained tom the same lot, as was all of the cyclohexane. The 1emicals, in their one pint amber sealed bottles, were tored in their closed shipping cases, which were kePt in n exhaust hood. The liquids were exposed to light only ' ' ' an ex eriment. Jrlng the preparation for and execution of p 136 Each bottle was used an average of six times, and each bottle Of CCl4 was always used in conjunction with only one bottle of C6H12° The last 50 to 100 ml of the contents were never used. It was convenient to prepare a total volume of 100 m1 of solution, measured before mixing. The cell contained ap— proximately 35 ml, and the pycnometer required slightly more than 25 ml. The excess was used for rinsing the apparatus. The advantage of our weighing technique is that the composi— tion of the mixture is determined at nearly the same time that the cell is filled, rather than much earlier or much later. Materials with which the chemicals came into contact were: glass, stainless steel, Teflon, silver, and fluorosili— cone sealant. Horne (1962) showed that cyclohexane is oxidized in air to form small amounts of cyclohexylhydroperoxide: C H 612+0 2 + C6H1102H . (5.5) COpper and brass catalyze the reaction and consequently were avoided. The effects of the materials in contact with the sample mixture were tested by placing only one of the liquids at a time into the cell and applying a temperature difference to the pure component. In none of the cases was any thermal diffusion detectable interferometrically. We then concluded that no measurable amounts of thermally diffusing impurities Were present in the mixtures. 137 The effectiveness of the sealant was checked simply v watching for the appearance of air bubbles inside the imple chamber indicating evaporation. Specifically for 1is purpose the cell was left filled on occasion for as >ng as five days, and no loss of the volatile liquid was >served. Several times the application of the temperature .fference caused a net contraction of the liquid in the :11 resulting in a very small amount of air being drawn 1to the cell. The size of the bubble was less than that E a drop of water, say 0.05 ml, and its effect, if any, is neglected. In the later experiments however, at a mean emperature of 35°C, the increased vapor pressure of the .quid resulted in the formation of many pinpoint bubbles 1 the surface of the tOp metal plate during filling. lese were shaken loose before the run was started, but 1ey remained in one corner of the cell. It is doubtful lat they had any measurable effect because they comprised lch a small fraction of the total volume and because they .d not appear in the center of the cell where the measure— :nts were taken. By sliding the glass walls slightly between the metal .ates before tightening the retaining nuts, a better seal ls obtained, and there could no longer be any diffusion 1rough the filling tubes. In addition, the temperature .stribution across the upper plate was improved by removing 1e perburbation of the filling tubes from the area of interest. 138 The cell was filled while in the isothermal configura— lon so that the beginning of the experiment could be well afined. About one minute was required to fill the cell. Eter that the mixture was allowed to equilibrate for up to 1 hour both to insure a uniform temperature distribution ad to permit decay of the effects of any thermal diffusion liCh may have occurred when the liquid came in contact with 1e metal plates during filling. In all of our experiments, aro time was clearly taken to be that instant at which the alves controlling the temperature configuration of the cell are switched. When the 30° and 35°C runs were begun, we observed mat the steadiness of the interference fringes was very ensitive to the air currents passing the cell. This hap- ened because the temperature difference between the 35°C iquid and the 23° or 24°C air on the outside was sufficient 3 cause a horizontal heat flux through the glass walls. hus the temperature distribution of the liquid was upset, nd a slight tendency for convection was observed. This as eliminated in subsequent experiments by enclosing the pace outside the glass sample chamber but between the metal lates. Strips of styrofoam insulation were held in place Y black plastic tape. In the Optical path before and after he cell, standard 1 in. X 3 in. glass microscope slides were sed to keep out air currents. With the foam in place, the ir between the metal plates could attain the mean temperature 139 30°C or 35°C,resulting in no observable effects due to a izontal heat flux through the cell walls. The fringe tern remained very stable. Because of the cell height used and the value of the fusion coefficient of the CCl4 - C6H12 mixtures, the re- ation time 0 was very nearly one hour, but varied slightly to the temperature and composition dependence of the ual diffusion coefficient. Consequently at least six rs were required to reach a steady state. Remixing upied another six hours, so no more than one run could attempted in a day. Including preparation time, nearly teen hours of practically uninterrupted attention were uired to complete a full experiment. and C H Published Data for CCl4 6 12 The tables in this section summarize the essential a avilable in the literature for carbon tetrachloride, lohexane, and their mixtures. Authors and sources are ed where appropriate. From the results of Wood and Gray 52), shown in Table 5d, we obtain the following expres— ns related to the density of CCl4 - C6H12 mixtures. %.=a+bT+cT2+dT3, (5.6) re T is the temperature in degrees C, and a = 1.06913 - 0.59379 w + 0.08972 w2 4 1 1 + 0.00027 wl , 1 .1llll I .11 ...1.. 1| 1|. .11 Il...1...1 1.. . . I ... ....n ..N F . I 111111.111 111111 1111011111.... . . . . . . . .. . . u . 1111.“.113111.) a . . .. . 4. . ..I u. . u . ... I . . ... - 140 1e 5d.—-Densities of CCl4 - C6H12 mixtures (Wood and Gray, 1952). 1/p = a + bT + cT2 + dT3 ; T in Degrees C. 01 a 1031. 106C 109d St‘g' De)“ 4 10 A(l/p) 000 1.25479 1.4362 2.529 5.37 2.6 289 1.11832 1.3106 1.568 9.62 1.0 512 1.01133 1.1940 1.205 10.15 1.5 514 1.01141 1.1803 1.471 8.31 0.9 720 0.92189 1.0855 1.171 8.57 1.2 978 0.84224 0.9768 1.381 5.81 0.8 506 0.75911 0.9080 0.687 8.74 0.7 482 0.71306 0.8456 0.777 7.68 0.5 475 0.71332 0.8479 0.790 7.12 0.9 716 0.66057 0.7946 0.528 7.92 0.9 000 0.61233 0.7294 0.722 5.52 1.4 141 0 b = 1.25048 — 0.67403 w + 0.10391 w - 0.00034 w 1 0 c = 1.47942 — 1.11122 w + 0.12049 w + 0.00009 w , 2 1 2 l 1 4 l I 5 1 0 d = 7.71 . e have also, 3 Zn p/BT)W = ab + (2ac + b2)T + 3(ad + bc)T2 l + (4bd + 2c2)T3 + (3cd + 2bc)T4 + 3de5 , (5.7) da db 2 dc 3 dd = ___ ... ___ + T ——— . 5.8 8 Zn p/Bwl)T p dwl + T dwl + T dwl dwl ( ) 3e relationship which we used to determine the mass frac- ion of CCl4 in a mixture of known density at 25°C is: —1 (wl)25° = 1.99014 — 0.01505p — 1.53114p . (5.9) iis expression was obtained by fitting the data of Table i to a polynomial for wl in terms of p. The curve fitting Dutine MULTREG was used. See Appendix H. Table 5e contains reported values of the refractive ndices of each of the pure components at various tempera— ures (in degrees Centigrade) and wavelengths (in Angstroms). sing MULTREG, we obtained the following expressions. or CC14: —2 -4 = 1.44299 — 5.754 x 10‘4(T - 25) + 0-00499(A X 10 ) (5.10) —6 he standard errors of the two coeffiCients are 8.3 X 10 5 nd 4.3 X 10— , respectively. 142 1e 5e.——Pure component refractive indices. (Timmermans, 1950, 1959) . (International Critical Tables, 1933) CC14 C6H12 ° 0 °C MA n T,°C 1,2). n 3 6563 1.4599 10.85 6563 1.42910 0 6563 1.46005 13.5 6563 1.42777 0 6563 1.4576 15.0 6563 1.42670 0 6563 1.45461 16.1 6563 1.42626 3 5893 1.4656 20.0 6563 1.4242 0 5893 1.4631 20.0 6563 1.42405 0 5893 1.46305 25.0 6563 1.42134 0 5893 1.46325 44.6 6563 1.41056 0 5893 1.46005 10.85 5893 1.43119 0 5893 1.46044 14.8 5893 1.4292 0 5893 1.4602 15.0 5893 1.42886 0 5893 1.46023 20.0 5893 1.42623 0 5893 1.46026 20.0 5893 1.42615 0 5893 1.46036 20.0 5893 1.4262 0 5893 1.45704 20.0 5893 1.42637 0 5893 1.45732 20.0 5893 1.42630 0 5893 1.45759 20.0 5893 1.4263 0 5893 1.45732 25.0 5893 1.42358 0 5893 1.4576 25.0 5893 1.42354 0 5677 1.45833 25.0 5893 1.4233 5 5460 1.46086 30.0 5893 1.4210 3 4861 1.4726 25.0 5876 1.41825 0 4861 1.46970 25.0 5677 1.42440 0 4861 1.46400 23.5 5460 1.42643 0 4686 1.47405 10.85 4861 1.43668 3 4340 1.4835 13.5 4861 1.43531 0 4340 1.47530 15.0 4861 1.43430 0 4340 1.46954 16.1 4861 1.43381 20.0 4861 1.54157 25.0 4861 1.42878 44.6 4861 1.41785 15.0 4686 1.43762 10.85 4340 1.44116 13.5 4340 1.43972 15.0 4340 1.43870 16.1 4340 1.43820 20.0 4340 1.43592 25.0 4340 1.43310 44.6 4340 1.42214 . ‘I |.J - II lia1|ldw1ufl . lullflullllrflnl 1'9““. ... .....1. 1...... . .. .... r ”.32.... ... .1-.. _ ..- .. . . .... .. 143 12‘ 1.41215 - 5.337 x lo'4(T—25) + 0.00395(1 x 10'4)'2 - 5.3 x 10_6(T—25)(l x 10‘4)2 (5.11) standard errors are 6.6 X 10—6, 1.8 X 10—5, and X 10_6, respectively. Table 5f contains refractive index measurements at C and 6563K for mixtures of carbon tetrachloride and lohexane. From those reported values we obtain the fol- ing expression for the composition dependence of refrac— e index: + - 02146 w [1 + 2 68048w2 - 4 3514lw3 nlwl n2W2 ' w1 2 ' 1 ' 1 4 (5 12) - 2.01856wl] , _ . re I11 and n2 are respectively, the refractive indices of e CCl and pure C H at the temperature and wavelength 4 6 12 ired. The few measurements of thermal conductivity which available for the two compounds are shown below. The ues for water are also shown because water was used to ibrate the interferometer. We used the following ex- ssions for thermal conductivity: water: K = 1.429 x 10'3 + 3.5 x 10'7(T-2o) (5.13) 144 )1e 5f.--Composition dependence of refractive index in mixtures of CC14 and C6H12 at 20°C and 6563K. (Timmermans, 1959) ch14 wcol4 n 0.00 0.00 1.4242 0.10 0.17 1.4268 0.20 0.33 1.4297 0.30 0.45 1.4326 0.40 0.57 1.4359 0.50 0.66 1.4393 0.60 0.75 1.4425 0.70 0.82 1.4460 0.80 0.89 1.4497 0,90 0.95 1.4535 1.00 1.00 1.4576 145 or CCl4: _ -4 —7 K — 2.47 X 10 - 4.5 X 10 (T—20) (5.14) or C6H12: _ —4 K — 3.2 X 10 (5.15) nits of K are cal/(sec)(cm2)(°C/cm). (From the Handbook f Chemistry and Physics, 44th ed.) Table 5g contains measured values of the mutual dif— usion coefficient D for mixtures of CCl4 and C6H12' The esults of Kulkarni, Allen, and Lyons (1965) were chosen ver those of Hammond and Stokes (1955). Our values of 1 are not sensitive to the choice of D. We used the fol— awing expression for the "literature" diffusion coefficient: D x 105 = 1.481 - 0.201x + 0.0258(T—25) , (5.16) 1 1ere X is mole fraction CC14, and T is temperature in 2 sec -1. 1 agrees C. Units of D are cm Table 5h summarizes previously reported values of 18 thermal diffusion factor ml at various temperatures 1d compositions for mixtures of CCl4 and C6H12' The data of Horne are incorrect because no account is been taken for the temperature distribution in the eservoirs (Beyerlein, 1968). The results of Thomaes L951) were probably Vitiated by convection. The data of >rchinsky have been obtained without correction for the forgOtten effect," and the diffusion coefficients of immond and Stokes (1955) contributed a small error' 431434.44... ...I. ..........u.... NH . . ...... “I .. .. . .. 2:10. nu... . ....n» I.- . (lul.J.IIIA|l..1..IJ In. .4]. .N . l|.J1«l.|l.\I . . . h .... a u .... 4.|.. . 1444.14: . a .4 . . .H1. 4. » 146 able 5g.--Ordinary diffusion coefficient for CC14 - C6H12 mixtures (Kulkarni et al., 1965). Xl D X 105 cm2 sec_l xl D X 105 cmzsec_l 25°C .01655 1.481 0.6975 1.328 .02510 1.481 0.7958 1.311 .07134 1.476 0.9333 1.295 .1739 1.447 0.9744 1.285 .3002 1.417 0.9853 1.287 .3988 1.393 0.0237 1.768 .4868 1.374 0.4750 1.633 .6053 1.351 0.9764 1-515 147 1e 5h.——Therma1 diffusion factor x1; previous results for m1xtures of CC14 - C6H12' m Turner et a1. (1967), T = 25°C ‘1 "“1 X1 '“1 1 15 2.16 .724 1.71 g 9 1.83 .730 1.72 5. 9 1 78 .898 1.66 § 0 1.77 .904 1.70 6 1.78 .947 1.65 4 1.77 .9882 1.76 9 1.74 .530 (35.1oc) 1.60 m Beyerlein (1968), T = 25°C _ w _ 0.1 0L 1 1.84 .65 1.75 5 1.85 .80 1.71 0 1.88 .95 1.77 m Thomaes (1951), T = 26.13°C "“1 W1 _G1 2.10 .505 1.45 9 1.72 .605 1 38 0 1.80 .78 1.20 75 1.68 .87 1.17 3 1.62 .95 1.08 148 1e 5H Continued m Tichacek et a1. (1956), T = 40°C W1 '“1 4 .313 1.30 .646 1.27 .879 1.25 m Korchinsky (1965), T = 25°C W1 —0L1 .50 1.82 .80 1.74 m Horne (1968), T = 25°C ..a W — 1 G. 1 1 1.86 .495 (28°C) 1.98 0 1.94 .649 1.88 9 1.92 .796 1.81 0 1.88 .946 1.92 149 . Calibration of Interferometer The constant A in Eqs. (4.20) is the fringe displace— ent per unit refractive index gradient. It depends on the avelength of the light used, the path length through the 211, the amount of shear, the focal lengths of various anses and the distance between the last lens and the image lane where measurements are made. Because so many factors re involved, A is best determined experimentally. The agical method is to measure the fringe displacement caused { a known refractive index gradient. The value of A thus alculated is valid only for the particular optical con— Lguration used. If any of the lenses is moved, either A lSt be remeasured, or some relationship between the values E A for the two arrangements must be known. For our calibration we chose as a source of a known afractive index gradient a layer of water (in the cell) )ntaining a known temperature gradient. The calibration (periment consisted simply of filling the cell with dis— Llled water, letting it equilibrate for about 15 minutes, 1d then applying a vertical temperature difference with 1e warmer temperature on top. The motion of the fringe attern was observed on the ground glass plate of the imera. Measurements were made with the microscope stage lescribed below) and from photographs. Six trials were ide with water. Sample calculations are presented here. . . n... . . . ..JJMafiail..l.:.l.1lJJd1\. mum-.1“. 11.... ...1. m... . .a. ...1.....on. . .hfi..... 1.. 150 Temperature Difference: AT = 3.817 deg C Cell Height: a = 0.741 cm Final Temperature Gradient at z = 0: (dT/dz)0 = AT/a = 5.151 deg cm—l Temperature Coefficient of Refractive Index at 25°C: (dn/dT)25 = -9.80 x 10’5 deg'1 Final Refractive Index Gradient at z = 0: (dn/dz)0 = (dn/dT)25(dT/dz)0 = -5.05 x 10‘4 cm"1 Fringe Displacement at z = 0: 15.70 cm d0(w) = A(dn/dz)0 = 15.70 cm A = —3.11 X 104 cm2 The six experiments with water gave A = —(3.11 t 0.03) x 104 cm2. Displacement of a lens for focusing changed not y A but also the fringe spacing r. According to ngdahl (1963), the apparatus constant A is related to Y A = Lr/A , (5.17) re L is a function of cell length, lens focal lengths, ar, and various distances. Since the wavelength, cell gth, shear, and lens dimensions all remained the same all of our experiments, any change in A due to lens ement could be calculated from the Change in fringe cing. For our original calibration, the spacing was 4 cm. Thus, in more general form, we have (1.149111411uQI 4 .4“. . . 151 A = —(3.11 i 0.03) X 10 cm . (5.18) Methods of Calculation Raw data from each experiment consisted of tempera- 'e measurements and measurements of the interference fringe (pe and position. All of the information about the upper l lower plate temperatures was recorded on the strip (rts. The particular quantities available were the warm— ) up parameters th and tc, the steady temperature differ— te, the initial, mean, and final temperatures, and (perature fluctuations. The parameters t and tc were the same for all of h a pure thermal diffusion experiments. The initial tem— 'ature was inconsequential as long as it did not differ 'nificantly (more than 0.5 deg) from the mean temperature. : applied temperature difference AT and the mean tempera- e Tm varied for different experiments and were carefully ‘orded. For the purpose of error analysis, a record was 0 kept of temperature fluctuations. The continuously rating strip chart recorders showed automatically any perature drifting which would invalidate experimental ults. Information concerning the gradient of refractive ex inside the cell was obtained in two forms: photo- phs of interference fringes, and direct measurements of nge displacements. The photographs had the advantage 152 providing a permanent record of both the shape and the sition of the fringes at certain specified times. The device with which the photographs were analyzed lsisted of an aluminum block for a base, a clamp to hold :h photograph in place, and a standard, adjustable, aduated microscope stage which could be moved in two sections. A precision magnifier, with a reticle contain- ; several scales, was mounted on a 1 x 3 inch glass slide Ld in place by a lever on the mechanical stage. A parti- Lar point on the photograph could be sighted through the Jnifier, and its coordinates could be read from the :nier scales of the stage. The distance to a second Lnt was readily found by comparing its coordinates with >se of the first. In such a way, the shape of the fringe 3 characterized by a set of horizontal displacement x, :h corresponding to a certain height n in the cell. lally, 29 pairs of coordinates were recorded at intervals 0.1 cm vertically. The data were fit (by MULTREG) to the Lynomial 5 k x(z,t) = Z dk(t)n , (5.19) k=0 are the dimensionless vertical coordinate n is related to 3Y — .31— (5.20) “ ’ (1 — s)a ' 153 here 8 is the amount of shear (0.19), measured from the hotograph, and ‘1 g n S l - (5.21) The coefficients dk(t), k = 1,...,5, were obtained y curve fitting. See Appendile. The five terms of Eq. 5.19) fit the data to within measuring error. At this point the data consisted of five coeffi- ients d k = 1,...,5, for each value of time at which a k' hotograph was taken. Equations (4.15) and (4.21) provide he necessary relationships for calculating the various erivatives of the refractive index from the measured alues of the d's. In Table Si are representative measure— ents of fringe shape taken directly from photograph No. 155 or run B5. Figure 5.1 shows a plot of the same data to— ether with the smooth curve expressed by Eq. (5.19), where he five calculated coefficients and their respective stan- ard errors are k dk Std. Error 1 0.492 0.028 2 4.47 0.030 3 0.732 0.053 4 0.566 0.016 5 2.68 0.022 basurements of the fringe shape yield a great deal of in- ?ormation. With them one can calculate simultaneously he temperature and composition dependences of the thermal Uffusion factor and the thermal conductivity. Wlth 154 able 5i.——Measurements of fringe shape. Run B5, t = 70.00 min, photo No. 155. T] X,Cm n XICnl ‘ .686 1.825 -0.049 0.009 g: .637 1.576 —0 098 0.064 ‘E .588 1.350 —0.147 0.140 9 .539 1.150 -0.196 0.250 1 .490 0.932 -0.245 0.372 .441 0.732 -0.294 0.502 .392 0.565 —0.343 0.730 .343 0.432 —0.392 0.922 .294 0.311 —0.441 1.097 .245 0.210 —0.490 1.350 .196 0.112 -0.539 1.584 .147 0.052 —0.588 1.900 .098 0.014 —0.637 2.280 .049 0.002 -0.686 2.815 .000 0.000 .40 .24 .08 z/a -.08 -.24 -.40 155 5 5 5 C O 7 ’4 C '1 I '1 _’ a 1 C C ('6 \‘ 5‘ . O P U C 3 C 4 Q l__ J L L . . 0 0.68 1.36 2.04 2.72 3.40 x, cm Figure 5.1——Plot of measured fringe shape (circles) showing agreement with fifth-order polynomial (solid line). - .1 .161... 3.131141411111111 I111: . (I ll (1 . . .. . . «.n .. .. u. Iluwl u ...- .1. ufl‘ilifl11duu.llal1dflll].l- . . .k...:.. . stfiufinvu . I: 1......1 156 sufficient sensitivity, such measurements could also permit calculation of the heat of transport. One requires accurate refractive index data, however, in order to extract informa- tion from the fringe shape. As we show below, the refractive index data in the literature are too poor to be used with any reliability in measurements of this type. Since we were interested only in measuring 01 and D, me were able to use the second type of measurement, that of fringe displacement at z = 0. These measurements were ob- tained by means of the same microscope stage mounted directly on the ground glass plate of the camera. Operated vertically, the stage held a 1 x 3 inch glass slide which was marked in slack ink with a cross to be used as a reference point. Nearly an inch of clear plexiglass separated the glass slide from the viewing plate on which the fringe image appeared. Parallax errors were eliminated by requiring simultaneous alignment of the fringe image, the cross mark, and the re- flection of the cross mark in the glass plate. At prescribed or convenient intervals of time, vary— ing in length from 30 seconds to 20 minutes or more, as indicated by the timer which was started at t = 0, measure— ments of the fringe position at z = 0 were taken and re— corded along with the time. The fringes in this method were characterized by a set of numbers d0(t). With this method, many more data points could be measured efficiently, and once they Were recorded, no further treatment was 157 ’equired. The dO's were converted to measurements of the 'efractive index gradient by means of Eq. (4.20). Typical fringe displacement data are given in Table 5j. Figure 5.2 :hows a plot of the demixing data from Table 5j and a curve >f the form _ l _ I dO — Xl exp ( X2t) + X3 (5.22) )btained by means of a least squares treatment. The coefficient do, which depends mainly on [Bn/Bz)0, assumes a much wider range of values than does 1ny of the higher order coefficients d1, d2..., and is a >etter source of accurate measurements. Moreover, d0 is Less sensitive than the other coefficients to transient refractive index changes due to fluctuations in the metal >1ate temperatures. From Eq. (4.20) we have d =A[c +-];c (5.23) 2 2 1 4 4 0 l 4 1 + B O as +-6'C5.S +0.0 3 5 By virtue of the solutions T(z,t) and wl(z,t), discussed in Ihapter III, and the chain rule for differentiation, we 1ave expressions for c k = 1,2,..., in terms of the ex- kl perimental tranSport parameters 01, D, and Ki. In particu- Lar, we have C1 = (an/3T)wl(3T/Bz)0 + (Sn/3w1)T(3wl/Bz)0 158 Table 5j.-~Fringe position d0(t) for run F6. yuan. c0, cm t d0 t d0 )emixing: 130 9.41 20 7.49 10 4.76 140 9.46 25 7.10 15 4.95 150 9.74 30 6.84 20 5.38 160 9.58 35 6.50 25 5.71 170 9.80 40 6.22 30 5.97 180 9.82 45 5.85 35 6.22 190 9.85 50 5.59 40 6.54 200 9.90 55 5.32 45 6.75 212 9.93 60 5.15 50 7.13 222 9.95 72 4.72 55 7.30 250 9.96 85 4.48 60 7.63 252 9.97 98 4.12 65 7.90 266 9.96 112 3.88 71 8.16 280 9.99 128 3.66 75 8.39 300 10.07 155 3.31 80 8.45 314 10.07 170 3.23 90 8.66 364 10.07 192 3.14 100 8.78 Remixing: 212 3.06 110 9.11 10 8.18 230 3.04 120 9.31 15 7.87 256 3.00 273 2.96 —¥ fThe reference point is arbitrary. 159 .coflumaommuuxm Homoumfifl msonm mafia coupon .m>uso mmnmsvm ummma can Ammaouflov 08H» mo coauocdm 0 mm o oom mNN N um ucmEmomHmmflp mmcflum mo poamllm.m musmflm mmudsflfi .u omH me o ‘ n o o Eo.Auv U .I1.l1l I.|l.1|. ill-1|. . ... ..nw. .. I .....I1I ... .... 1141.41.41.41 . ..41 . aw... ... . .. ..r. .... . ...u.u.. .. .ml. m 160 = (8n/8T)wl(83T/8z3)0 + (an/8w1)T(B3wl/3z3)0 =3 2 + 3(8 n/8T2)W (BT/Bz)0(82T/822) 1 0 2 2 2 2 + 3(8 n/Bwl)T(3wl/8z)0(8 wl/az )0 + (83n/8T3) (8T/8z)3 W1 0 3 3 3 + (3 n/Bwl)T(8wl/BZ)O . (5.24) The N experimentally measured fringe displacements (ti), 1 = 1,..., N, coincide with the function in Eq. .23) when proper values are chosen for al, Ki, D, and B. practice we confined our measurements to values of time :ge enough so that the temperature distribution was not tnging. Then, by expressing all of our measurements as fferences, e.g., d0(t2) — d0(tl), both B and the terms rolving temperature disappear. Terms involving products derivatives are certainly negligible, since the maximum 3 -l Lue of [(Bwl/Bz)0| is about 5 x 10' cm When the Lution wl = wi + G from Chapter III is inserted where aded, we obtain the final working equation d0(t) — gl(t) = Xl exp (-X2t) + X3 , (5.25) are d0(t) represents the measured values of the fringe Splacement at z = 0 relative to some reference point (t )7 gl(t) is a correction term given by Eq. (3.68); r 1 the xj, j = 1,2,3, are given by (5.26) is the derivative of the first term of S (Eq. 3.8) with spect to z, evaluated at z = O, and multiplied by e-t/e. x2 = 0'1 = wZD/a2 , (5.27) l = — - — . .28 X3 d0(tr) gl(tr) Xl exp ( X2tr) (5 ) Our method of analysis consisted of fitting the Lnge position data to a function like Eq. (5.25) and :imizing the coefficients X1’ X2, and X3 according to Least squares criterion. (See PROGRAM ALPHA in Appendix ) The thermal diffusion factor 01 was calculated from a numerical value of X1 by means of Eq. (5.26), and the iinary diffusion coefficient D was calculated from X2 1 Eq. (5.27). Besides giving values for both 01 and D, our curve tting method has the advantage of smoothing the data and nbining all of the measurements from an experiment to tain a single set of results and an estimate of the andard deviation. Another approach could have been used if it were t desired to determine D, or if only a few data pairs vering a short period of time were available. If we had 3d a literature value for D, the time dependence of d0(t) 162 u1d have been fixed (since 0 = a2/Dn2). In that case, a ngle difference d0(t2) — d0(tl) would have sufficed to de— rmine al. The time dependence of d0(t) is shown in Figure 2. The tailing off for t < 0/4 reflects the fact that e infinite Fourier series solution for wi(z,t) does not nverge rapidly for t < 0/3. If only the leading term of e series is retained, one gets the function shown by the tted line. The figure shows clearly that if one wishes include in his measurements data for t < 0/3, enough rms of the Fourier series must be retained to insure nvergence to within some specified tolerance. The dis— vantage in this case is that the simple form of Eq. .25) is not obtained. If, on the other hand, only the first term of the urier series is retained, one must be certain to use 1y data corresponding to t < 0/3. Table 5k shows a sample laboratory notebook record a pure thermal diffusion experiment. 163 .ble 5k.——Sample laboratory notebook record of a pure thermal diffusion experiment. .te: 4-23-68 0 _ o = Ln: F6, CCl4 — C6H12’ xl — .250, wl .406 Th 37.008 1 0.004°C TC 33.032 1 0.004°C T 3.976 i 0.008°C T 35.020 i 0.004°C m Tf 34.950 1 0.004°C A —3.11 - 104 cm2 a 0.741 cm 6 64 min D 1.4 - 10'5 cm2 sec‘1 8 0.19 :arting Time: 12:57 pm ; 7:03 pm. (otos: 290 (arts: 105 :marks: After about two hours of continuous operation(,i the he frame to expan (ser generated enough heat to cause t _ ~ightly, causing the mirror alignment to change and result (g in a gradual diminishing in the light beam inten51fiy.t (e trouble was corrected by making an adgustment inlt e en— .on on the retaining rings hold1ng the mirrors 1n p ace. CHAPTER VI EXPERIMENTAL RESULTS LpTabulation of Data Table 6a summarizes the results of our experiments. iCh run is identified by a code consisting of a letter, a 1mber, and another letter. The first letter (A through H) anotes the bottles from which the chemicals were obtained. Lthough the same lot number in every case was used for 1ch of the liquids, a record was kept of the particular >tt1e used, and the CCl4 from any one bottle was mixed 11y with the C6H12 from the corresponding bottle. Some of 1e early sets of runs (A through D) either were trials : showed poor temperature control. The second figure in the code is a number (1 through 1 which denotes the number of the run in the series. sually, each chemical bottle was used about six times in reparing new mixtures. The last letter in the code is ither D or R, corresponding to demix1ng or rem1x1ng, lichever is appropriate for that tun. Not every number Spears with both a D and an R. The final two experiments, 3r example, which were conducted with temperature 164 165 )1e 6a.——Summary of experiments. 1 Date Num Tm wl xl AT DxlO -a 1968 5—13 29 293.42 .40731 .25867 4.147 1.304 1.856 5-14 24 293.04 .67369 .51178 4.160 1.228 1.784 5—15 28 293.00 .86107 .75886 4.152 1.230 1.741 5—15 20 293.06 .86107 .75886 4.152 1.234 1.714 3-21 33 298.22 .18045 .10055 4.066 71.463 1.777 3—7 44 298.14 .27577 .16201 4.040 1.451 1.840 3-7 40 298.22 .27577 .16201 4.040 1.511 1.812 3—8 46 298.14 .27577 .16201 4.064 1.503 1.785 3—8 43 298.22 .27577 .16201 4.064 1.420 1.777 3-6 30 298.20 .34326 .20972 4.178 1.492 1.810 3—6 37 298.27 .34326 .20972 4.178 1.415 1.778 3-5 42 298.26 .39899 .25209 4.194 1.481 1.778 3—5 40 298.26 .39899 .25209 4.194 1.494 1.763 5—10 31 298.16 .40731 .25867 4.077 1.438 1.841 2-5 36 298.16 .41454 .26444 4.058 - 1.754 3—1 31 298.08 .46666 .30760 3.970 - 1.793 3—1 42 298.24 .46666 .30760 3.970 1.437 1.752 2—26 44 298.16 .57738 .40957 4.100 1.402 1.761 1-30 50 298.21 .58439 .41655 4.170 1.316 1.739 2—23 37 298.19 .63477 .46878 4.108 1.422 1.757 2—23 36 298.24 .63477 .46878 4.108 1.385 1.732 5—8 36 298.22 .67269 .51065 4.114 1.424 1.759 5-8 30 298.17 .67269 .51065 4.114 1.375 1.721 2-9 64 298.22 .67465 .51287 4.160 1.331 1.734 3—11 42 298.14 .71725 .56293 4.088 1.388 1.720 3.11 41 298.20 .71725 .56293 4.088 1.374 1.706 2—13 66 298.20 .75781 .61371 4.146 1.326 1.702 2-13 22 298.19 .75781 .61371 4.146 1.396 1.698 3—12 39 298.14 .79224 .65942 4.076 1.377 1.705 3-12 46 298.20 .79224 .65942 4.076 1.301 1.702 298.15 .85922 .75603 4.063 1.340 1.699 42 298.18 .85922 .75603 4.063 1.371 1.722 5~9 38 298.22 .86074 .75835 4.098 1.361 1.683 5—9 28 298.16 .86074 .75835 4.098 1.368 1.653 5—16 32 298.16 .86107 .75886 7.922 1.309 1.691 5-17 29 298.20 .86107 .75886 12.092 1.322 1.728 3—18 44 298.16 .89337 .80967 4.082 1.332 1.682 3—18 31 298.19 .89337 .80967 4.082 1.316 1.678 3—19 29 298.15 .91520 .84568 4.047 1.307 1.643 3-19 30 298.16 .91520 .84568 4.047 — 1.698 5—6 30 303.15 .40669 .25818 3.812 1.536 1.757 5-6 33 303.17 .40669 .25818 3.812 - 1.753 4—30 29 303.10 .67307 .51108 4.015 1.440 1.669 WW I PIA 03w J; A tble 6a Continued l__ 166 1n Date Num Tm W? X? AT DxlO -al LR 4-30 23 303.15 .67307 .51108 4.015 1.563 1.668 !D 5—2 26 303.07 .86049 .75797 3.912 1.451 1.633 ZR 5-2 29 303.16 .86049 .75797 3.912 1.520 1.622 ifii 4-23 33 308.21’ .40775 .25902 4.036 1.673 1.713 3R 4-23 28 308.11 .40775 .25902 4.036 1.657 1.702 5R 4—22 37 308.11 .40818 .25936 3.950 1.715 1.682 2R 4-12 18 308.16 .67382 .51193 3.538 1.674 1.627 1D 4-18 26 308.29 .67403 .51216 3.700 1.671 1.623 1D 4—18 28 308.11 .67403 .51216 3.700 1.613 1.665 3D 4-15 33 308.52 .86135 .75928 3.514 1.609 1.589 3R 4-15 30 308.16 .86135 .75928 3.514 1.619 1.569 ,___ 167 lifferences of 8°C and 12°C, were not studied during demix- .ng because the large temperature gradients produced refrac— :ive index gradients great enough to deflect the light beam >ut of the Optical components of the interferometer. For :hose experiments, the demixing was allowed to continue un- >bserved until t = 60. Then the temperature difference was removed, and the isothermal remixing was monitored in the lsual way. The number of measurements of fringe position for each run is recorded in the column labelled Num. E;_Error Analysis Systematic Errors In this section several possible sources of systema— tic error are discussed. (1) Uncertainties in wi. Our technique for determining the initial composition w: is based 3n the assumption that the liquid in the filled cell has the same composition as that in the filled pyconmeter. We tried to achieve that condition by filling both vessels from the same syringe with the smallest time lapse possible. Of course, evaporation occurred during the filling, but we do not attribute a significant error to that for the following reasons. First, differential evaporation of the two compo— nents, acting to change wi, should occur in both containers. Second, such evaporation takes place only at the surface of the liquid, and by overfilling we discarded in both cases that portion of the liquid which was exposed to air during 1.... ialill . .-. “44...-.. . .. ...: 1. I 111-11.011.11 . ll- ill .... J1 I lilnlldll. . .. . .. ... .....f....u.... .. - . . . ..... . .....E. t . . .... . . ".168 lling.* Other systematic errors due to balance inaccuracies re effectively cancelled out since we always measured mass fferences. Probably the most significant source of sys- matic error in the determination of W? was the possible esence of impurities in the pure reagents. Both CCl4 and H12 were used without further purification. If, however, ch had nearly the same amount of impurities by weight, eir net effect very nearly cancelled in the calculation ‘ the mass fraction. We consider the possible effect of [purities on thermal diffusion below. An error of 0.01°C L the temperature of the bath in which the pycnometer was .aced would lead to a systematic error in W? of less than x 10—5. We estimate that the effect of any systematic rror here is less than 0.01% of w? and thus contributes ass than 0.01% to d1. (2) Uncertainty in T. The only significant systematic rrors in T arose when the plate temperatures drifted due > loss of bath control. Consequently, because those errors are obvious and large, we discarded the results of those :periments. (3) Uncertainties in cell geometry. Because of 1e nature of our cell, the Spacing between the plates did >t change with AT as it does in thermogravitational columns, >r example. The cell height depends only on the thickness 3 the glass wall assembly and on the amount of sealant 169 tween the glass and the metal. Throughout our experiments followed the same procedure for cleaning and replacing e glass, for applying the sealant, and for tightening e bolts holding the cell together. We found no change in e height of the cell during the course of our experiments. . is highly unlikely that any vibrations transferred to the .11 through its 3000 lb. support were significant. (4) Impurities. We discussed above the influence ‘ 1 tpurities may also thermally diffuse. We tested for such of impurities present in the reagents. Those initial L effect by placing one of the liquids, either carbon :trachloride or cyclohexane, into the cell and applying temperature difference. The steady temperature gradient 18 established within five minutes. During the next six >urs, no further change in the position of the interference :inges was observed except that attributable to temperature .uctuations. Since the interferometer was capable of de— :cting composition gradients of the order of l x 10..5 cm-l, : estimate that any impurities present (including cyclo— :xylhydroperoxide and phosgene) contributed less than 0.1% v the steady state composition gradient and hence less .an 0. . 1% to dl A second type, accumulated impurities, may be formed the mixture in the cell. Reactions between the components the solution, reactions with the metal plates, and re- tions with or dissolution of the sealant would cause an 170 lation of impurities, even after t = 60. We found, r, that all of our experiments reached a true steady (to within the sensitivity of the interferometer), .ting that if accumulated impurities were present they .ot detectable. (5) Convection. Previous pure thermal diffusion .ments have often been questionable on the grounds :onvective remixing caused incorrect results. We 7e that our isothermal remixing experiments were .tely free from convection, first, because no tempera— 1ifference existed, and there was no possibility for :tion to be induced by density inversions due to )ntal components of the temperature gradient. Second, :he uninsulated cell at Tm = 35.00°C the fringe pat— Jas highly unstable and moved erratically. Since the plate temperatures remained constant to within 0.005°C : 25°C), we attributed the phenomenon to convection >anying the horizontal heat transfer from the liquid a cooler room air. This claim was substantiated when ;ulated the cell with styrofoam. The same experimental Lions produced stable (non—fluctuating) fringes, be- the air surrounding the cell was allowed to reach the Lemperature distribution as the liquid in the cell, .ating the horizontal heat flux. Bartelt (1968) is investigating the degree of in- de required to produce convection due to heat loss 171 in through the vertical walls. In the meantime, our ferometric observations of convection (or the lack of nd the essential agreement of our results for demixing emixing techniques lead us to conclude that the demix— xperiments were also not affected by convection. (6) Uncertainty in the calibration of the inter- eter. We chose water for the calibration because the rature dependence of its refractive index is known r than that of either carbon tetrachloride or cyclo- e. Also, the thermal conductivity of water is better .cterized, and thus the temperature gradient in water own more accurately. Hence the refractive index gra- ., and consequently the apparatus constant, could be .fied with the greatest accuracy. The limiting factor e accuracy of the apparatus constant is the fringe ion, which is discussed under random errors. There was stematic change in the apparatus constant, since the ength, cell length, and lens focal lengths all remained nged. (7) Uncertainty in (Sn/8w1)T. The composition de- nce of refractive index is the only quantity from the ature which enters directly into the calculation of hermal diffusion factor. Equation (5.12) was obtained gh curve fitting, and the random error resulting from er of the data is discussed below. Any systematic or bias is small and disappears at wl = 0 and w1 = 1, 172 iere Eq. (5.12) reproduces the refractive indices of the ire components. For wl less than 0.4, n changes slowly .th increasing wl, and an uncertainty in wl of 0.1% con— :ibutes less than 0.005% to n. For wl greater than 0.8, :increases more rapidly with increasing wl, but any sys- ématic errors must remain small in order for n to approach ie correct limiting value. We estimate that any such un— zrtainties contribute less than 0.05% to d1. (8) Uncertainty in time. An electric timer which 1dicated digitally minutes and hundredths of minutes was ;ed. The combination of its inherent inaccuracy and the 1certainty in starting it and reading it was less than .01 min. Since we recorded fringe position as a function 5 time, we compare the ratio of those relative uncertain— Les and observe that the effect of the time uncertainty 1 d1 is about 1% of the effect of the uncertainty in the :inge measurement, which itself contributes less than 1% 5 d1. (9) Uncertainty in gl(t). The term gl(t) makes its argest contribution for very small (t < 0/3) values of ime and becomes less significant as the steady state is pproached. Systematic errors in gl(t) are due almost en— irely to uncertainties in the initial values chosen for he temperature and composition derivatives of the experi— ental transport parameters, of which we were confident Q better than 5% initially. Those values were improved 173 iterating the calculations so that they contributed at a 2% uncertainty in gl(t), which itself influences the ulated value of d by about 0.1%. 1 Consequently, we esti— that systematic uncertainties in gl(t) contribute less 0.01% to al. (10) Uncertainty in fringe position. Any systematic rs involved in the measurement of fringe position due arallax disappeared in taking differences. (ll) Neglect of $1, Bp/Bz, and jiz 8(fii — fi2)/32. ‘entropy source term ¢l due to bulk flow is, for a pure rmal diffusion experiment, zero except for about two- utes during the warming up period. Sedimentation due the pressure gradient contributes about 0.1% to the position gradient due to thermal diffusion and conse- ntly less than 0.1% to al. The term jiz 8(fil — fi2)/Bz approximately 3T — — 8T 91337“: _Cp2)8-E’ about 2 X 10—4 at most for very small values of time, decreases to zero at the steady state. All other sible sources of systematic error are related to the ic assumptions we made and justified in Chapter II. the values of time which we used to calculate d1 they tainly contribute less than 0.1%. We now show that the tematic errors discussed above are much smaller than dom errors which occur. 174 Random Errors The formula which allows us to calculate a thermal diffusion factor, Eq. (5.26), depends ultimately on such direct measurements as the fringe position, refractive in— dex, thermocouple emf, and mass of the liquid in a pycnom— eter. We investigate now the propagation of the uncer- tainties in each of those direct measurements resulting in some uncertainty in the calculated value of the thermal diffusion factor. Consider a general derived property U which is re— lated to the directly measured properties X1, X2,...,Xm by the functional relation U = U(Xl, X2,...Xm) , (6.1) which is continuous and differentiable over the region of interest. The uncertainty 8U in U is obtained from the formula (Parratt, 1961) 2 2 2 EU 2 EU 2 g = ___ 5 + ———- E +... U (3X1) Xl 3X2) X2 m 2 _ EU 2 ‘ .2 ax.) Ex. ' (6‘2) 1:1 1 1 where EX is the estimated uncertainty in Xi. i In principle, Eq. (6.2) can be applied only to statistical uncertainties of the same kind. That is, all 8's must be standard deviations, or all must be probable errors, or all must be 90% confidence limits, etc. 175 Furthermore, Eq. (6.2) is valid only if each s is independent. Our uncertainties are not all independent. For example, an uncertainty in w: is due in part to temperature uncertainties. We said in the preceding section, however, that such linkages are extremely weak, and we now assume that we can use Eq. (6.2). We express all of our uncertainties, or estimates thereof, as standard deviations. Because of the small size of the correction term gl(t) in our theory, uncertainty in gl(t) has virtually no effect on the uncertainty in al, which is given by the fol— lowing expression derived from Eqs. (5.26) and (6.2): 2 2 2 2 e s 6 2 e 061 X1 Tm EA Bn/Bwl "‘7 = ”x— + T + ‘X + —_3n/3w d l m 1 1 2 e o o 2 e I 2 s w w S + AT + 1 2 + J) . (63) AT Wowo 0 l 2 In order to obtain an estimate of the expected uncertainty in d1, we estimate the uncertainties in Eq. (6.3) in the following way. (1) Uncertainty in X1. The quantity X1 is essen- tially a measurement of fringe displacement. Neglecting experimental scatter, which we Consider later, we find that the uncertainty in X1 is due completely to the uncertainty in measuring the fringe position. The magnitude of X1 is abOUt 5 cm, and repeated measurements of the same stationary fringe Show a standard deviation of about 0.005 cm. Thus, 176 we have 8 i X 2 _ -6 — 1 X 10 - (6.4) H (2) Uncertainty in Tm. Random errors in Tm are due mainly to random fluctuations in the temperatures of the water baths, and only insignificantly to variations in the thermocouples, the reference ice bath, or the potentiometer. We estimate the standard deviation of measurements of Tm in a single experiment to be 0.0056°C, so that (6.5) (3) Uncertainty in the apparatus constant. The uncertainty in A is itself a function of two other uncer— tainties, that of the measured value for r = 1.74 cm, and that of the value of r for the experiment at hand. We estimate the standard deviation of A in the original deter- mination to be 0.01 X 104 cm2, and that of the fringe SPaCing r to be 0.01 cm. There results 2 A (4) Uncertainty in (an/awl)T. 2 = 5 x 10‘6 (6.6) This uncertainty depends on the value of wi. The standard deviation ob- tained in the fitting of the data of Table 5f to a poly— is 4.6 X 10—5. The values of (an/awl)T for nominal in w: 177 O T = 25°C and A = 6328A and the uncertainties at the three compositions are 2 8 O (8_n_ all/SW1 wl awl T Bn/Bwl 0.25 0.0186 2.80 x 10'6 (6.7) 0.50 0.0284 3.10 x 10‘6 (6.8) 0.75 0.0438 5.20 x 10’6 (6.8a) (5) Uncertainty in AT. The remarks of paragraph (2) apply here. We estimate the standard deviation of the plate temperatures to be 0.004°C. Hence the standard de— viation of AT is 0.0056°C, and 2 —6 = 2 x 10 . (6.9) (6) Uncertainty in wiwg. This term is also com- position dependent. There are three possible sources of . . 0 random error in the determination of wl: the polynomial in p, the calculated mass of liquid in the pycnometer, and the calculated volume of the pycnometer. Estimating the uncertainty in the mass at w? = 0.5 to be 0.00025 g, and calculating the uncertainty in the volume to be 0.02 cm3, we obtain as upper limits O O 2 E w1W2 O O 0.25 1.5 x 10’6 0 50 1.3 x 10‘6 (6.10) 1.5 x 10‘6 178 (7) Uncertainty in 85. The quantity 85 is an analyti— cal function —02/2a, and its uncertainty is related only to that of the cell height. From Chapter IV, 6a = .0005 cm, hence ES‘ 2 (—§$) = l x lo—8 . (6.11) 0 With the above estimates, Eq. (6.3) gives ‘ 0!. "a; = 15 x 10‘6 (6.12) or an estimated standard deviation of (6.13) —3 €81 _ 4 x 10 lull Thus, the a priori estimated standard deviation is 0.4% of I01 The expression from which the uncertainty in the ordinary diffusion coefficient is obtained is 2 2 e 8 D We estimate that our measurements sh E .3) , (6.14) ould give 0 to within 2% or better, so (as) = 4 x 10'4 , (6.15) and 179 2 E a _ [—5) = 7 X 10 7 ' (6.16) Thus, a 2 (—%i = 4 X lO—4 - (6.17) We can expect deviations in measured values of D of about 30.03 x 10'5 cm2 sec—l. The numbers listed in the preceding paragraphs are estimates. They merely suggest anticipated values for the uncertainties in the thermal diffusion factor and the ordi— nary diffusion coefficient. The actual experimental stan— dard deviations, which are measures of the scatter, must be calculated from the data. At several different composi— tions we had enough replicate experiments to calculate standard deviations for 01 and D. The results of those measurements are shown in Table 6b. Table 6b.-—Experimenta1 Uncertainties. 4_—______———————i e /D xl Sal/0cl D Observed Estimated Observed Estimated 0.162 0.0066 0.0039 0.014 0.010 0.511 0.0046 0.0041 0.012 0.010 0.010 0.010 0.759 0.0153 0.0045 180 C. Results In Figure (6.1)—(6.3)(Section D) are plotted the results of our calculations of the thermal diffusion factor for CCl4 - C6H12 at various temperatures and compositions. For the results at 25° we used MULTREG and allowed for the possibility of a fourth order polynomial in x1. The data from Table 6a at 25° produced the smallest standard error when a straight line was fit to them. The composition de— pendence of ml at 25° is given by — 0.181X (6.18) —d = 1.82 l l 7 with a calculated standard error in 01 of 0.022. We obtained the temperature dependence of ml by finding the least squares straight line through the data at four temperatures. The same calculation was made for three compositions: = _ = 7 — .0098 T - 25) (6.19) X]. 0.259 0L1 1. 98 0 ( r — . _ _— 7 — ,() ()0 T — 25) (6.20) X1 0.512 . 061 1. 33 0 ( r — ° — — 0.0102(T 5) u ( I ) X1 0.759 . 061 1.682 — " 2 6 21 The temperature and composition results can be expressed by the single function _ _ — — 0.0100(T — 25) -d1 — 1.74l — 0.181(Xl 0.5) + 0.0008(Xl - 0.5)(T — 25) I (6.22) Where the calculated standard error of 01 is 0.019, and the CouPling term (XlT) contributes only about 0.1% to dl. 181 Similar calculations for the ordinary diffusion co— efficient were carried out for the composition dependence and for the temperature dependence. The results in this case are, for 25°, (units of cm2 sec—l): 105D = 1.48 - 0.187x 2 1 , (6.23) with an uncertainty of 0.03 x 10_5 in D. Measurements of D at four temperatures for each of three compositions yielded the least squares lines: x1 = 0.259 : 1050 = 1.438 + 0.0250(T — 25) , (6.24) x1 = 0.512 : 1050 = 1.390 + 0.0256(T — 25) , (6.25) x1 = 0.759 : 1050 = 1.350 + 0.0261(T — 25) . (6.26) The combined temperature and composition formula is 1050 = 1.388 - 0.187(xl — 0.5) + 0.00256(T — 25) + 0.0024(xl — 0.5)(T — 25) , (6.27) With a calculated standard error in D of 0.03. The results and their significance are discussed in more detail in the next section. D. Discussion The results of our calculations of the thermal dif— fusion factor for the carbon tetrachloride—cyclohexane System at 25°C are presented in Figure 6.1. Figure 5‘2 15 ' is no a cOmparison with previously reported values. There 182 .. _ y . . ,. 45W.W...~w.13:u ".W.... ,3: , .3 pm foo coflomum 308 no .0 coauocsw 6 mm Nammo I 1.300 How Houomm GOHmSMMHU H3593 HmeamfifluomeIla .m munmflm Hon 0 H. N. m. w. m. m. h. w. m. of.“ . . _ _ . _ _ . . m . H maflxflfiom O I w . H mcflxflfimm D 0 0mm ..8 1:80 60.6.08an 39: mo ~ 0 .muanmmu m50fl>mum Sees GOmHmefioo SOADUGSM 6 mm mammo I vHOO How Moscow cosmSMMAU Hmsumnu HmquEHHmmmeIm.m mudmflm fix 0 a. N. m. w. m. m. S. m. m. o.H _ _ _ _ . . . . . O.H mmmfione Q Gawanmmmm 8 Q mmeHSOHOM I .an um M0539 O 3 .an #0 950m B measmms H50 q 4 Imim m < I44 «.qu 184 significant difference between our results and those of Turner, Butler, and Story, who used a flow cell method. The agreement between the two sets and the internal con— sistency within each set clearly indicate that the com— position dependence of the thermal diffusion factor for CCl4 - C6H12 at 25°C is a linear function of the mole fraction. The thermogravitational results of Beyerlein and Bearman show large scatter, but three of their points coin— cide with our results, and two differ from ours by less than their reported uncertainties. Their value for X1 = 0.32 is definitely incorrect, however. The compari— son demonstrates that although a rigorous phenomenological theory (Horne and Bearman, 1968) was used, the results are not reliable. There is obviously some large random error producing the observed scatter, which is not attributable to either uncertainty in w? or the effects of impurities. The source of the experimental difficulty must be discovered before the thermogravitational technique can be trusted. The only other results for the CCl4 — C6H12 system obtained by means of pure thermal diffusion are those of Thomaes, which have always been questionable, which for twenty years cast a shadow over pure thermal diffusion in general, and which are now discredited. The claim that Thomaes' results were invalidated by convection is probably true. 185 Horne and Bearman and Korchinsky independently re- ported values for the thermal diffusion factor of this system obtained from thermogravitational studies. In both cases the slope of the 01 vs xl line is close to ours, but their absolute values of 01 are higher. Such a systematic dif— ference could, as we pointed out earlier, result from the fact that Horne and Bearman did not account for the possible effects of temperature gradients in their reservoirs, and Korchinsky did not include the forgotten effect, which amounts to about 1% of al. Our experimental standard errors of calculated thermal diffusion factors were, for most experiments, less than 1% of 01. At 25°C our results can be expressed by the function —d1 = 1.827 - 0.181xl , (6.28) with a standard error of 0.0159. Ours has been the first systematic study of the temperature dependence of the thermal diffusion factor for CCl4 — C6H12' Figure 6.3 shows that the absolute value of 01 decreases with increasing temperature. The function which characterizes our data for the range 20-35°C is —d1 = 1.74 — 0.181(x 1 1 — 0.5) — 0.0100(T — 25) + 0.0008(Xl — 0.5)(T — 25) (6.29) which has a calculated standard error of 0.019. The single Point for X1 = .5 reported by Turner, Butler, and Story at 186 Gout» um mufipmnmmfiou mo GOHpossm 6 mm Houomm coflmSMMHp Hmsumnu HousoEHHmmmeIm.m onsmflm ov mm o 606 om .mammo I waoo How msofluflmomfioo ..e mm ON - B .am pm Moomsofla mmh.o Nam.o mmN.o OD<1 187 35.1°C agrees within less than 2% with our value there. The(stirred diaphragm method) results of Tichacek, Kmak, and Drickamer at 40°C, however, are 20% lower (in abso— lute value) than the numbers we obtain by extrapolating our lines to 40°. Beyerlein and Bearman (1968) have just shown that for thermogravitational experiments the thermal diffusion factor shows a significant dependence on the magnitude of the applied temperature gradients in the upper and lower reservoirs. A pure thermal diffusion cell has no such reservoirs containing the sample liquid, but it is con— ceivable that the apparatus material or construction might in some way effect the shape of the temperature distribu— tion in the liquid. In fact Longsworth (1957) observed that for his apparatus the temperature distribution depended on the type of seal present between the glass cell walls and the metal plates. In order to determine whether our calculated thermal diffusion factors would be influenced by the size of the temperature gradient, we conducted a set of experiments in which all conditions were identical except AT. Our results (Figure 6.4) are certainly not compre— hensive in this area, but they do indicate that there is very likely no significant dependence of the calculated value of d on AT. Of course, for extremely small gradients 1 (less than 1 deg cm-l) very little thermal diffusion takes I 1.33.4422“ 1.1:. .- I'v . I u 188 .0 6mm u EB .m.o u Mx “unmecmum ousumsmmfimu pmflammm mo coflpocdm m mm mammo I vaoo now Hoyomm GOHmSMMHc Hmsnocu HmucosfluomeIIv.m oudmflm HIEU mop .m\B< 3 3 m 189 place, and d1 is difficult to measure. For large gradients the terms involving the temperature and composition depen— dences of mi and D become more important. For very large gradients the linear phenomenological relations fail. Two quite different tests showed that convection was absent except for the unimportant (for our method) first minute or two of an experiment. In the demixing ex- periment poor temperature control or poor insulation of the cell can result in horizontal components of the tempera- ture gradient which cause density inversions and convection. Convection, when it occurs, causes remixing of the solution (in addition tothat due to ordinary diffusion. During a remixing experiment, however, no temperature gradient exists, and perturbing convection is much less likely to occur. Our results for both thermal diffusion factor (Figure 6.1) and ordinary diffusion coefficient (Figure 6.5) are identical for demixing and remixing: At 25°, Demixing: -01 = 1.844 — 0.212 xl , (6.30) Remixing: ~91 = 1.840 - 0.201 xl , (6.31) . . 5 2 —l DemiXing: 10 D, cm sec = 1.493 - 0.203 xl , (6,32) . . 5 2 -l RemiXing: 10 D, cm sec = 1.488 — 0.208 x1 , (6.33) Since there is no convection during remixing, and since the demixing results are the same as the remixing results, we conclude that there is no convection during demixing. 190 .0 6mm pm. waoo coauomnm mHoE mo coflnocafi 6 mm Nammo I v.80 How ncofloflmmooo coumwsmmunp accuse HmucmEHHmmeIImd unsung mqflxflswo D mcflxdeom O m .H own so Qmoa «In min 191 Our experience with cell insulation substantiates this conclusion. Tests with an uninsulated cell at 35°C showed heat loss from the warm liquid to the cooler room air. The horizontal component of the temperature gradient caused density inversions and resulted in convection which was visible interferometrically as very unsteady fringe patterns. When the cell was insulated so that the air immediately surrounding the glass sample chamber could reach thermal equilibrium with the glass and the liquid, the horizontal heat flux was eliminated and the fringe pattern was steady. Our calculated values of the ordinary diffusion coefficient show more scatter than do the results for the thermal diffusion factor. The apparent reason for this is that fluctuations in the metal plate temperatures can change the apparent time—dependence of the diffusion process (Change the calculated 0 and hence D) without changing the final value of al. The effect of the tempera— ture fluctuations is also reflected in standard error of 01. At 25° we obtain the following expression for the composition dependence of D: 1050 = 1.38 — 0.187(xl — 0.5) . (6.34) 8 where the standard error in D is 0.035, and where D has units of cm2 sec—l. Present diffusion coefficient results are plotted, along with results of others, in Figure 6.6. Our results for D compare quite well with those of 192 .muadmmu m50fl>mnm ayes GOmHMmmfioo no 6mm #6 coauomnw oHoE mo coapocdm H00 Mom ucmflOmewoo coamdmwflp Amanda HmucmsnummeIIo.m wndmflm 6 mm «ammo I a Hun o a. N. m. a. m. m. a. m. a. 04 u in) - x u u — ~ q < HoH D -NJ mmxoum pcm ccoEEdm .8 06 2803.31 0 mIIZHSWGH H50 193 Kulkarni, Allen, and Lyons and substantiate their conclusion that the stirred diaphragm results of Hammond and Stokes are invalid. (Figure 6.5) Previously, very little information about the tem- perature dependence of D has been available. Our result (see Figure 6.7), 105D = 1.388 — 0.187(xl - 0.5) + 0.0256(T — 25) + 0.0024(xl — 0.5)(T — 25) , (6.35) gives at x1 = 0.5, 1058D/8T = 0.0256 , (6.36) which is essentially the same as the 0.0258 of Kulkarni sai- Dicave and Emery have claimed that the ordinary diffusion coefficient measured when a temperature gradient is present.(when thermal diffusion is occurring) necessarily differs from that measured in an isothermal remixing experi— ment. According to the phenomenological theories of the various types of thermal diffusion the diffusion coefficient D should not change from one type of experiment to another. Our results indicate that no systematic difference exists between the ordinary diffusion coefficients measured in our two types of experiments. We suggest that perhaps Dicave and Emery's experi— ments were perturbed by one or more of the following effects: (1) The phenomenological theory of their stirred 194 .mcofluflmomEoo mmncu um onsumnomfimu mo scenocsw m mm mammo I vHoO How ucmflowmmmoo GOHmDMMMU andudfi HmucmfiflnwmeIIh.m mnsmflm camps mm om mm ON 195 diaphragm does not adequately account for the remixing due to the constant stirring; (2) The horizontal temperature gradient across the porous glass plate resultsin local fluid density inversions and unaccounted—for convection in the glass disc; (3) The stirring of the fluid near the disc causes a mixing flow through a portion of the disc near the surface, changing the effective thickness of the disc and, consequently, chang— ing the value of D calculated from a given measurement of the relaxation time 0; (4) The temperature gradients they used were large enough to make the (ignored) temperature derivatives of transport parameters significant. We conclude that there is no difference between the isothermal and non— isothermal ordinary diffusion coefficients (provided they are referred to the appropriate temperatures). It is interesting to note that the entire class of stirred diaphragm techniques is suspect since our results indicate that such methods lead to incorrect answers in three different cases: (1) The thermal diffusion factors of Tichacek, Kmak, and Drickamer at 40°C appear to be 20% too low in magnitude. (2) The ordinary diffusion coeffi— cients of Hammond and Stokes are only a few percent too high, but have a parabolic, rather than linear, composi— tion dependence. (3) We have shown that there is no difference between the isothermal ordinary diffusion coefficient and the nonisothermal one, and we therefore reject the contrary conjecture of Dicave and Emery, which 196 was based on stirred diaphragm thermal diffusion and diffu- sion experiments. E. Temperature Dependence of Refractive Index While testing for the posSible effects of thermal diffusion of impurities in the "pure" carbon tetrachloride and the "pure" cyclohexane, we discovered some new informa— tion about the temperature dependence of the refractive index in each case. The theory of the interferometer pre- dicts (Eq. (4.20)) that a uniform nonzero refractive index gradient should produce interference fringes indentical to those for a zero gradient but shifted horizontally by some fixed distance. What we in fact observed for both CCl4 and C6H12’ when a temperature difference was imposed vertically, and a steady temperature distribution developed, were curved interference fringes of a generally parabolic shape much like that in Figure 5.1. Analysis of the refractive index data from the literature for the two pure compounds showed that only a linear dependence on temperature was statistically signifi— cant. Use of those data and the working equations for the interferometer required that the temperature distribution inside the liquid be sigmoidal in shape in order to explain the shape of the interference fringes. Such a temperature distribution would, in turn, re- quire either anomalous variations in the thermal conductivity 197 of the liquid or some inexplicable apparatus effect. Be— lieving that the thermal conductivity is a well—behaved function of the temperature, and that our apparatus caused no strange effects (since the same temperature difference, applied to water, gave the expected straight fringes), we turned our attention to the validity of the reported values of the temperature dependence of the refractive index for CCl4 and C6H12' Since only second order thermal conductivity temperature dependence affects the value of (dT/dz)0, and since the effect is less than 0.1°, we have, at the center of the cell, (dT/dz)0 = AT/a. The coefficients cj defined by l j j j ET-(d n/dz )0 , (6.37) O I) then become cj = (4§)j(ajn/3Tj)0 . (6.38) Thus, to evaluate the temperature derivatives of refractive index, we need only the cj of Eq. (4.20). For j > 1, these are directly related to the dj—l which describes the fringe shape, and therefore second and higher temperature deriva— tives are obtainable from fringe shape analysis. c1 and therefore first derivatives are proportional to do, but do contains an arbitrary reference point and is therefore unobtainable from a single experiment. However, by per— forming experiments at two different values of AT and 198 determining the shift in do, we may calculate (Sn/8T)0 according to 1 cl(2) - 61(1) = X[40(2) — 60(1)] ll 1 AT AT g(Bn/BT)0 [P75 - {—5} ] . (6.39) 2 1 In the CC14 experiments, the temperature gradients were (AT/a)l = 4.534 deg cm_l, (AT/a)2 = 5.108 deg cm_l, the measured shift in d0 was [d0(a) — d0(l)] = 9.84 cm, the measured value of the fringe spacing was 1.61 cm, and the mean temperature was 25°C. By Eqs. (4.20) and (6.39), 25°, 6328A: 33 = —5.96 x 10'4 deg-1 . (6.40) CC1 8T 4’ -l . The fringe shape for (AT/a) = 4.534 deg cm was fit by MULTREG, with the result 5 3 + .04882‘ , (6.41) l 2 — 0.3412 l x= 1.454z' where z' is the vertical distance (in cm) on the photograph. By Eqs. (6.38) and (4.20): o 2 c014, 25°, 6328A: (BZn/BT ) = 0 (6.42) 3 3 -6 —3 (3 n/BT ) = —l.00 X 10 deg . (6.43) Higher order coefficients are also calculable from the numbers in Eq. (6.41). In the C6H12 experiments, the temperature gradients -1 -1 _ were (AT/a)l = 5.177 deg cm , (AT/a)2 — 4.563 deg cm , 199 the measured shift in d0 was [d0(2) — d0(l)] = 9.63 cm, the measured value of the fringe spacing was 1.62 cm, and the mean temperature was 25°. The corresponding results are, 0 _ _ 25°, 6328A: 39 = _ 5.44 x 10 4 deg l . (6.44) C6H12’ T The fringe shape for AT/a = 4.564 deg cm“1 was '2 '3 '4 x = 0.7632 + 0.05632 + 0.0307z . (6.45) As above, these lead to ° I 2 2 C6H12’ 25°, 6328A: (8 n/BT ) = 0 (6.46) (83n/8T3) = —0.516 x 10‘6 deg—3. (6.47) We conclude from these measurements that the sensi— tivity of the wavefront shearing interferometer has permitted us to measure the temperature dependence of refractive in— dex more precisely than it has previously been measured. Classical techniques have required measurements of the absolute refractive index at various temperatures. Analysis Of the rather scarce literature data yields for 25° and ° —4 —1 = - . 5 X 10 de and (an/8T) = 6328A, (En/8T)ccl4 5 7 g , C6H12 - 5.47 ><10_4 deg—l. These agree well with our results, and it is likely that our results are to be preferred, since we determine this coefficient directly. Derivatives of higher than first order have previously been undetected because they are much smaller than the experimental errors involved. With our method we sacrifice knowledge of the 200 absolute refractive index, but gain significant information about variations in the third and higher decimal places. Our results require more verification before we can ake a definite statement about the temperature dependence of refractive index. We are confident that our numbers for the first three derivatives are accurate to better than 1%, but higher derivatives are probably less accurate. We can conclude that the curved interferometric fringe shape ob- served for the pure components can be explained by the temperature dependence of refractive index, and that the laser wavefront shearing interferometer can be extremely ivaluable in studies of refractive index. Clearly, detailed temperature dependence will be most useful in testing microscopic theories of refractive index. CHAPTER VII CONCLUSION A . Summary In the preceding chapters we have set forth, for the first time, a phenomenological theory of pure thermal diffusion which is not restricted by traditional mathemati- cal simplifications. Our use of the series expansion tech— nique has allowed us to take full account of the temperature and composition dependences of the transport parameters involved. Our solution for the composition of the fluid in a pure thermal diffusion cell as a function of position and time contains explicitly the effects of transient vertical convection and a varying temperature gradient during the warming up period. By allowing for time-dependent tempera— ture gradients in our differential equations, we have been able to match the theoretical boundary conditions to the ones which are observed experimentally. Consequently, there is no doubt about when an experiment begins. We have consistently chosen the zero of time to be just that instant when the temperatures of the metal plates begin to change. 201 202 We have eliminated an additional source of error by allowing for a variable thermal conductivity. It has not been necessary to assume that the temperature gradient in the fluid has a constant, uniform value. That all of our measurements of fringe displacement were made at the center of the cell was not accidental. We chose to avoid the ambiguities introduced by previous workers (Gustafsson, 1965) in measuring differences between refrac— tive index gradients at two positions in the cell where the temperature gradients may not be the same. We are convinced of the absence of convection. Both ithe interferometric observation of induced convection and the agreement of demixing and remixing results support this claim. The laser wavefront shearing interferometer provides much more information than has heretofore been available from a single experiment. Its sensitivity and ease of Operation make it far superior to other types of inter— ferometers which have been used in the past. We have also introduced computer technology to the study of thermal diffusion. Automatic analysis of fringe displacements makes feasible the use of more data. An automated data gathering device, when coupled with a com— puter, can remove the necessity for tedious manual measure- ments and permit routine analysis of thermal diffusion experiments. V 203 Our theoretical and experimental investigations have combined to eliminate the doubts and questions prompted by the conflicting reports of previous thermal diffusion ex— periments. We have shown that when properly executed and when adequately described, pure thermal diffusion can be a reliable experimental technique. With respect to numerical results, we conclude the following: (1) The thermal diffusion factor of CC1 - C H 4 612 at 25°C is given by -dl = 1.83 - 0.18X1 , with standard error no greater than 1.2%. This result is in close agreement with the flow cell result of Turner et a1. (1967). It appears also to be in agreement with the thermogravitational results of Beyerlein, whose scatter is rather large. The "pure" results of Thomaes are now clearly incorrect. The thermogravitational results of Horne and Bearman and of Korchinsky and Emery appear to have the same composition dependence as ours, but are a few percent higher in absolute value. With the close agree- ment of our results and those of Turner, Butler and Story, the thermal diffusion factor and its composition dependence at 25°C are now firmly established. (2) The temperature dependence of ml is given by ad 1 _ —l §T_i — 0.011 deg . X1 204 (3) Diffusion coefficients for this system are given with standard error of less than 3%, by D x 105 = 1.482 + 0.0256(T — 25) — 0.187 xl , which agrees well with the results of Kulkarni, Allen, and Lyons. The stirred diaphragm results of Hammond and Stokes appear to be incorrect. Further, we have refuted the claim of Dicave and Emery that diffusion coefficients in thermal diffusion experiments are different from those in isothermal experiments. (4) The temperature dependence of the refractive O 0 index of CC14 at 6328A is given by (CCl4, 6328A): 25° n = n — 5.96 x 10'4 (T — 25) — 1.00 x 10’6(T — 25)3 , while that for C6H12 is given by (C6H12; 6328A): n = n250 - 5.44 x 10_4(T — 25) — 0.516 x 10'°(T - 25)3 . The standard error in each of the above coefficients is less than 2%. B. Suggestions for Further Work It has not been our purpose to collect thermal diffusion data for a large number of systems. Rather, we have shown that pure thermal diffusion can be a useful experimental tool, and that both our phenomenological theory and our analytical method can be used in routine studies. Further improvement can be made, however, especially in the area of temperature control, the element which most 205 limits the precision of the method. We have considered re— designing the cell, eliminating the circulating water, and substituting sealed chambers which make use of the constant temperatures of phase changes. The reservoirs then would be hollow metal blocks lined on the inside with a porous fiber material. The upper portion of the top reservoir would be heated electrically just enough to vaporize a portion of the liquid in the pores of the fiber. Heat would be trans- ferred by the gas downward to the bottom of the upper reservoir where the gas would condense at a constant tem— ‘perature determined by the particular substance chosen and ithe pressure inside the chamber. The newly formed liquid would move by capillary action up the side walls of the chamber and back to its original position. By means of such an arrangement, the required constant temperature would be maintained at the top metal plate in contact with the fluid in the thermal diffusion cell. The heat which was put into the system electrically would flow downward through the sample fluid and become available at the metal plate forming the upper boundary of the lower chamber. There the heat would be used to vaporize a different liquid at another constant temperature. The gas produced would carry the heat downward to the bottom of the lower chamber where the gas would condense on porous fibers cooled electrically by means of the Peltier effect. 206 Heat transfer by convecting gases has been studied (Eastman, 1968). The efficiency of such heat transfer is much greater than that of pure heat conduction. In tests described in the above—mentioned article, the effective heat transfer coefficients were one to three orders of magnitude larger for the "heat pipes" than for a copper bar of similar dimensions. If this method can be adapted to pure thermal diffusion experiments, it should provide highly stable and uniform plate temperatures. Alternatively, further refinements in the more conventional method may be attempted. A second place for improvement is the method of collecting data. The main advantage of pure thermal dif- fusion is that it permits one to make a very large number of measurements without disturbing the system. We usually used 20 to 40 measurements of fringe displacement in our calculations of fringe displacement in our calculations. We were limited mainly by the time and labor involved in making each measurement. If some automated measuring device were available, more information could be obtained from each experiment. The simplest arrangement would consist of photoelectric sensors which could continuously monitor the fringe position and eliminate the need for intermittent manual measurements. A more sophisticated improvement would make use of stop motion photography to obtain hundreds, or even ;; wimp; ' 207 thousands,of records of the interference fringes. A neces- sary adjunct would be an optical scanning device to determine the shape and position of each fringe and store that infor— mation in the memory of a computer, where it would be acces— sible for programmed analysis. The use of the entire fringe shape would have the advantage of providing second and higher derivatives of the refractive index. Consequently, a single experiment could be used to determine not only the thermal diffusion factor and the ordinary diffusion coefficient, but also the thermal conductivity and the temperature and composition derivatives (of all three quantities. We mentioned in Chapter VI that the apparent thick— ness of the interference fringes could be decreased by using a more intense light source. If the interferometer then proved to be sufficiently sensitive, one could, in principle at least, measure the heat of transport by determining the thermal conductivity of the initial, uniform mixture and that of the mixture at the steady state of demixing. This could be done by watching interferometrically the time dependence of temperature changes when a temperature dif— ference is applied to or removed from the test liquid. The relaxation time for heat conduction is a function of the thermal conductivity. The difference between the two ther— mal conductivities is DDQ1WIWZG1' which is numerically about 1% of K. (f 208 Finally, in further work with thermal diffusion, the possible applicability of new techniques such as radioiso— tope tracing and nuclear spin-echo methods should not be overlooked. BIBLIOGRAPHY Abrahamowitz, M., and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series, 55, U. S. Government Printing Office, Washington, D. C., 1964. Agar, J. N., Trans. Faraday Soc. 56, 776 (1960). , and J. C. R. Turner, J. Phys. Chem. 64, 1000 (1960). Bartelt, J. 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APPENDICES 214 APPENDIX A A RELATION BETWEEN PHENOMENOLOGICAL COEFFICIENTS+ Postulate: There exists a scalar, invariant, posi— tive, bilinear function (the entropy production) defined by \) a = ) ga-ga 2 o , (A.1) d=0 where the Xa form a linearly independent set of vectors. The vectors Ja are specified uniquely by the scalars QQB defined by v a = Z Qa8§8 , a = O,l,...,v . (A.2) ~ B=0 We seek a proof of the theorem \) "If E n = 0, then d=1 d8 v 2 Q = 0 , a = O,l,...,v ." (A.3) B=l d8 Lemma I: The quadratic form q(Y), T all a12 Y1 = Y AY = Y Y , (A.4) 3 ~ ~ 1 2 a12 a22 Y2 1LThis proof is due to Bartelt (1968). 215 216 is positive (i.e., q 2 0) for any Y if and only if both 2 0 (A.5) and — a 0 . (A.6) a11a22 12 2 Proof: 2 2 q = allYl + 2a12YlY2 + a22Y2 a 2 +3.2) + all 2 a - 553) Y: . (A.7) 11 q = all Yi a22 . 2 Thus, if for any Y1 and Y2, all 2 0 and all — a12 2 0, then q 2 0. On the other hand, if q 2 0 for any Y, and if Y2 = 0, then all 2 O . (A.8) a12 Furthermore, if Y + ———-Y = 0, then 1 all 2 a a — a2 > 0 (A.9) 11 22 12 ’ ' Definitions: E < 10) B I 0 , A. 6 d=1 d6 v C6 2 X “as ; 6 = O,l,...,v . (A.11) B=l Lemma II: \) 1 2 _ + 6 = 0 1 ... v . (A.12) 955 2 B5 2 4(BCS C0) ' I ' ’ 217 1299;: For eaCh 6 = O,l,...,V , Choose 50 = 9 , gd = §y ’ a = 1’2’ IV Y # 6 0!. 76 6 . (A.13) Then V V O '— 0 6:0 8:0 9GB§Q g8 ' V V = Q X o 621 3:1 “B~“ §B ' E E V c = Q x -x + 2 0 x -x 8:1 8:1 “B“Y ~Y 6:1 ”5“Y ~5 B#5 B#6 d#6 V + 821968¥6°§y + 955§5°65 , B#5 E 2 o = B — B - c + Q (X ) 8:1 5 6 6 66 y 2 + {B6 + c 0 , (A.15) 66 ’ 218 .nd v 1 . 2 Q B — — _ _ _ 55 5:1 8 B5 C6 + 966 2(B6 + C6) 966 3 0 - (A.16) in other words, V 1 2 966 )1 BE 2 T(B6 + c5) , 6 = O,l,...,v . (A.17) 6: Theorem: v If £1 has 0 , 6 = 0,1,. .,v , :hen E 0 = 0 , 6 = 0 1,...,v . 8:]- 68 I ?roof: According to our postulate, B6 = 0 , 6 = 0,1,. ,v . Dhus v 2 B8 = O I (A.18) 8:1 and by Lemma II or v c6 = E 968 = o , 6 = O,l,...,v . (A.20) APPENDIX B SIMPLIFIED COMPOSITION DISTRIBUTION The following derivation is essentially that of arlein (1954) and de Groot (1945). The introduction of ;s fractions instead of mole fractions, however, is ours. Consider the continuity equation (3.6) * * Bwi - D 8 WE _ dlA (1 _ 2W0) awl + 8 in p awl 8t _ 2 T a 1 82 32 Bz 82 m OLlA o o (1 2 o)( * _ wo)] 3 Zn p ' Tma [W1(l ' W1) + ' w1 W1 1 —az ' (B.1) are 0 > z > a , i t > O . define the quantity R by 3 £413 = R AI , (13.2) Bz a are R is assumed to be constant. The other symbols are scussed in Chapters II and III. For convenience we define the following dimenSion— as quantities: 219 E = z/a r = Dt/z2 A = RAT P = -alAT/Tm - (B.3) iation (B.1) becomes 15 32w: 0 3W1 0 02 F = £2 + [A + p(1 — 2wl)] 35 + pri(l — 2wl) + prl . Kiliary conditions are (B'4) lim w: = w? , 0 > E > 1 , (B.5) r+0 8w* 2 lim [3&1 + p(1 — 2w$)wi + pwi ] = 0 , r > 0 . (B.6) 5+0 5+1 The solution of these equations in a Fourier series facilitated by rewriting them as functions of a new riable: 2 wo 1 o 6 = w* + ————l———— exp —[A + p(1 — 2wl)]E . 1 o 2 (B.7) 3 equation to be solved becomes 32 1 2 o 39L=_1_ —Q —Ap(1—2wl)¢, (B.8) Br 862 4 are Q=A+p(i-2w‘i). 221 {iliary conditions are %Q€ 1im 6 = e , o < 6 < 1 , (3.9) r+0 lim 8¢ %¢[ Q - (1 — 2w°> 1 = o (B 10) a 3: 2 1 P ' ' €+l Equation (B.8) can be solved by the separation of 1es technique. When the initial and boundary condi— are imposed, the solution is seen to be _ e o _ o . wi — wl — 2prl(l wl)S , (8.11) we is the steady state term given either by 6_ 1 o pH-+N)em>b@ND N- 1 B 2 W1 — 3 W1 [—w—I_ exp (—pN) + N I ( .1 ) _ o N:l-2Wl7£0, 8 1 1 1 _ = _ - _ — N - 0 B.13) wl 2[1 + 2 p(2 6)], when ( nction S' is given by 2 2 2 oo — k " ] s' = Z kaWk exp [ (B +2 V :rz p5 . (3.14) = (32 + k2n2)(P + k W ) antities appearing in the summation are 222 B:iflbs-pN) 2 l P = E (A + PN) (B.15) Vk = l — (l— 1)k exp P Wk = B sin kHE + kn cos kfig Since the complete solution is so cumbersome, it seful to derive from it some convenient expressions h, due to the magnitudes of the parameters involved, still accurate. If we consider the usual ranges of parameters which determine B and P (yi§., — 0.002 < O, 3y> — ”I > 0, 20 > AT > 0, 1 > N > — 1), it is r that B and P will typically lie between the limits 1. Thus the squares of these constants are negligible ared with kznz. Also, since 0.2 > p > 0, we may, with . . e igible error, represent W1 by 8 o l w =wl[1+p(1—w‘f) (5 —a>1. (B.16) sual, we use the formula for the relaxation time 2 9.— (B.17) Our final expression for the solution is o 0‘1 O _ 0 E _ s w: = wl + T_ ATWl (1 W1) (a+ 113 1 ’ m 223 where _ °° -3 2 s- 2 k vkwkeXP (-1. t/z—pz/a—p/z), k=l Vk = l - (- l)k exp P, w = B sin kn[£ + l-] + kn cos kw[3 + l) k a 2 a 2 ’ and where we have transformed variables so that Eq. (B.18) is valid for - a/2 < z < a/2 t > 0 . APPENDIX C CONTRIBUTION OF CONVECTION TO THE TEMPERATURE DISTRIBUTION The temperature inside the fluid in a pure thermal .iffusion apparatus is described by Eq. (3.18): Sq — 8T _ z — 8T . 00p E "37‘ ' pcp uz az , ”I“ a a —§- < Z < -2- , t > 0 , 8w _ 8T * 1 2 Ihere -qz - Ki 5; + pDQl BZ , (C. ) 1nd the auxiliary conditions are T (2,0) = T a t) = ¢ (t) (C.3) T (2,t) = 9h (t) We identify by T* the contribution to the tempera- ;ure due to pure heat conduction, so that the equation 63. T* (c.4) 2 Z :3: Blhf 225 .5 equivalent to the equation describing heat conduction Ln a solid with constant thermal conductivity. To obtain an approximation to the complete local temperature, we add a contribution due to convective heat transport when a temperature gradient exists and when the center of mass velocity is nonzero. If the temperature gradient is zero, convective flow will produce no net transfer of heat through any fixed volume element. When a nonzero temperature gra— dient does exist, the convective contribution to the local temperature is proportional to the magnitude of the velocity. Accordingly, we write T (2,t) = T* (2,t) + b uz (2,t) %§*(z,t), (c.5) where b is a proportionality constant which must be chosen to make Eq. (c.5) satisfy Eq. (C.1). When Eq. (C.5) is inserted into Eq. (C.1) and Eq. (c.4) is subtracted, there remains 8T* 2 b u §__(8Tf]= Kl 82T* b 82 8t 2 t 82 pcp 822 2 8 u 8n 2 * 8 T* +b3_.£* Zz+bazzag+bu z 82 82 82 (C.6) 82T* 8uz 8T*_ u b 32* 8uz + 2 82 _ uz 82 82 32 82 82T* 226 the quantities in Eq. (C.6) are evaluated at z = 0 t = t0 (i.e., when uZ = uOO) we can make use of the tional relations ' 0 8u z —__ = .7 [32 ] 0 I (C ) [82uz] 24 u00 2 _ 2 ’ 82 Orto a stain -l 2 2 a [[6 av) 24 Ki (83T*/8z3) + u (a T:/8z )] ‘_ n_ ' —' 00 (3T7az) t 82 a2pcp (8T*/8z) 0 ' z: t=t 0 1 is very nearly 9) D 0| b = __R _ ((1.8) N .b 75 |-" APPENDIX D TEMPERATURE DISTRIBUTION DUE TO HEAT CONDUCTION DURING THE WARMING UP PERIOD The conductive part T* of the temperature of a d in a pure thermal diffusion cell during the warming eriod, when variations in the thermal conductivity unimportant, satisfies Eq. (3.20): 8T* K' 3211* (13.1) H auxiliary conditions T* = 6 (t) , when 2 = 0 c T* = 6h (t) , when 2 = a (D.2) T* = Tm , when t = 0 . (D.3) Le K = Ki/(pEb). +Note the limits on the variable 2. A transforma— to (— a/2 < z < a/2) is made at the end of this section. 227 ALOLRL" 228 L *= et T r1 + r2 ' (D-4) are 8r. 82r 1=K 1 at 2 r 0 0 , r1 = O , when 2 = O , r1 = 0 , when 2 = a , (D.6) r1 = Tm , when t = O , d 8r 82r 2 _ 2 51?. _ K 2 , O < 2 < a , (D.7) 82 t > 0 , r = ¢ (t) , when 2 = 0 , 2 c r2 = ¢h (t) , when 2 = a , (D.8) r2 = 0 , when t = 0 . The solution for rl is well known to be (see, for ample, Carslaw and Jaeger (1959), p. 96) 4T w _ m 1 . (2n+1)nz _§ 2 2 rl — -F_ n£0(2n+l Sln [———-?;———]exp[ a2(2n+1) n t]. (D.9) To obtain r2, we use Duhamel's integral formula .rtels and Churchill, 1942) which expresses the solution ' boundary tempeatures 6C (t) and ¢h (t) in terms of the ution to the same problem with constant boundary 229 :emperatures Tc and Th. The formula, as stated in Carslaw 1nd Jaeger, p. 30, (1959) is: "If v = F(x,y,z,l,t) repre— sents the temperature at (x,y,2) at the time t in a solid Ln which the initial temperature is zero, while its sur— face temperature is ¢(x,y,z,1), then the solution of the problem in which the initial temperature is zero, and the surface temperature is ¢(x,y,z,t) is given by t V =5 E F (XIYIZIAIt " A) d>"" (D'lo) Q) In this case the temperature at time t, when the temperature through the fluid at t = A is zero, and the plates are kept at 61 (A) and 62 (A) from t = A to t = t, is given by z 2 m 1 . nwz _ 2 2 t- 2 r2 = ¢1 la I-‘ l kflN x T + k1 O 233 sccording to Eqs. (E.2), (E.4), and (E.6), the boundary con— litions are Tl (a/2) = 0 , (E.ll) T (-a 2 = 0 . l /) Dhus, Cl=0 I (13.12) 2 . _ AT a cl — kl 3— § , (B.13) and the first order solution is 2 2 _ AT a _ z Tl—kla—[-§ §)' (E.l4) Dhe quantity k1 is given by dw 3£n K dT 8£n K 1 k=( JH+[ H—Jr 1 ST W dz 0 awl T dz 0 l rhere the zeroth-order solutions for (dT/dz)0 and (dwl/dz)0 must be used. Higher order solutions for the steady tempeature [istribution are obtained by successive iterations of .he above procedure. The complete solution through terms if order 82 is (E.lG) 3 + 82 (k2 - ki) [——— - g ] + 0 (e ) erms of order s3 and greater involve third and higher erivatives of thermal conductivity and products of erivatives which are extremely small. At the thermal teady state the correction f(z) to the temperature gra— ient is dT AT AT 2 2 a2 2 f(Z) = a; — r = a— ["€klZ + 8 (k2_kl) (TE " Z )1 3 (E.l7) + 0 (e ) t is obviously advantageous to make all measurements at L r 0, since deviations of the temperature distribution :here are of second and higher order. APPENDIX F PERTURBATION SOLUTION FOR THE STEADY STATE COMPOSITION DISTRIBUTION At the staedy state of a pure thermal diffusion experiment the diffusion flux jlz vanishes, and we have from Eq. (3.34) dw a w w 21 _;_£_§ 93 (F.l) T dz d Do take account of the variation with temperature and :omposition of the quantity (alwlwz/T) we introduce the expansion 0L W W 0° LE E S = 2 Sn 3 Zn I (F.2) where e is an ordering parameter, and where an = $7 (dnS/dzn)0 (F.3) By the chain rule for differentiation we have, for example, d S = [ii] £2 + [Bi ] [ail] , (E.4) 235 236 where the zeroth—order solutions for (dT/dz)0 and (dwl/dz)0 must be used. The temperature gradient is given by Eqo (E.l7): dT _ AT 1 2 2 a2 2 3 3E - g— [ - eklz + e (kz‘k1)(I§ — z )1 + 0 (e ). (E.l7) Due to the nature of the perturbation approach, the formal solution is + e w + ... , (F.5) where the second subscripts denote the order of the solu— tion. A sufficient boundary condition for Eq. (F.l) is f 2 w dz = a w0 , (F.6) which merely expresses the fact that the average composi— tion does not change from w? during an experiment. As in Appendix E we first neglect all terms con— taining e explicitly. We obtain A—T (F.7) wl,0 $0 a z + cO . The value of the integration constant follows from Eq. (F.6) and 3 wl,0 dz a wl . 237 The zeroth order solution is _ O . AT wl’0 — wl + 80 a— z , (F.9) where o O s = £3$19_Yl_ii:3li_ (F 10) 0 T l 0 Repeated application of the same procedure through successively higher orders of 6 gives 2 2 _ 0 AT _ z _ a_ wl-wl+a—{[soz+€(sl klso) 2 24 ] 2 3 3 3 2 2 a z z a _ z + E [50(1‘2'1‘1) [—12— ' § ' E] <31k1+32) 3 J} + 0 (e3) (F.ll) Higher order terms involve third and higher derivatives of K and S and products of first and second derivatives which are extremely small. APPENDIX G THEORY OF THE WAVEFRONT SHEARING INTERFEROMETER The wavefront shearing interferometer of Bryngdahl (1963) provides a comparison of the wavefront with a sheared Lmage of itself. The method utilizes birefrigence inter— ferences. We can describe the vertical component of the light wave entering the cell by a transversal electric field strength amplitude vector referred to the basic sys— tem of vectors E and n and to the object plane by u = (a + n) (A/fz‘) eiky , (G.1) where k = Zw/A , X is wavelength, and A is the scalar amplitude. If we denote the refractive index of the sub— stance in the cell by n(x,y,z) and the thickness of the cell by h, then the optical path through the cell will be W (x,z) = [h n (x,y,z) dy . (G.2) O The amplitude vector of the light leaving the cell, referred to the object plane, will therefore be ((5.3) U = (g + n)(A//2) exp {ik[W (x,z) + y0]} , 238 239 where y0 is an arbitrary reference plane. Next, the light passes through a lens system, the purpose of which is to effect a scale reduction in order to keep down the dimensions of the beam Splitters. Denoting the reduction factor by r, we have to introduce a new func— tion w(rx,rz) = W(x,z) , (G.4) and we can then write the amplitude vector of the wave entering the first beam splitter Ql in the following way, as referred to the first image plane: 9 = (x + z) Eééa exp {ik[w(x,z) + yl]} , (G.5) since the laser light is polarized in the E direction. New constants yl,y2,y3,... are introduced after each trans- formation. In passing Ql’ the component of U in the x direc— tion is displaced downward by an amount % b1 and becomes polarized in the z direction, while the component of g in the z direction is displaced upward by the same amount See Fig.(4.5). After and polarized in the x direction. Ql we have, therefore, the following amplitude vector, also referred to the first image plane, 9 = §(A/r/2) exp {ik[w(x,z + bl/2) + y2 + x/ZI} (G 6) + y(A/r/2) exp {ik[w(x,z - bl/Z) + Y2 ' X/21} 240 Referred to the second image plane, the light leaving Ql is described by the vector mb 9 = §(A/rm/2) exp {ik[V(x,z + —§l) + y3 + x/Zl} b ((5.7) m + y(A/rm/2) exp {ik[V(x,z — —;2L-) + y3 - x/21}- We have introduced the quantity x, the path difference be— ? tween the two sheared wavefronts, related to a possible tilting angle of the beam Splitter. When we refer to the second image plane, we must introduct the magnification factor m and the new function V(mx,mz) = W(x,z) . (G.8) On passing the second beam splitter Q2, there is introduced first a lateral displacement and second an optical path displacement A according to E G.9 l d cos w , ( ) where d is the distance between the focal plane of the lens L5 and the second image plane, and w is the angle between the crystal surface normal and the entering ray. The expression for the wave emerging from Q2, referred to the second image plane is 241 _ b mb 9 = x(A/rm/2) exp {ik[V(x — f; , z — —§£ ) + A/2 + Y4 - x'/21} + y(A/rm/2) - b mb - exp {ik[V(x + 2; , z + -§£) A/2 + Y4 + x'/2] (G.lO) After the beam passes the polarizer, the ampli— tude is U=g~ (x+z)//'2— U = (A/2rm) exp {ik[V(x - bl/Z, z — mbl/Z) + A/2 + y5 — X'/2] + (A/2rm) exp {ik[V(x + bl/2, z + mbl/Z) — A/2 + y5 + x'/2]} . (G.ll) Hence, the image intensity becomes I = [U12 = % (gm)2 (1 + cos ¢)- (G.12) where ¢ = k[V(x + bl/2, z + mbl/2) — V(x - bl/2, z — mbl/2) - A + x'] . Destructive interference is obtained for ¢ = 2nw (n — 0,1,...) with crossed polarizers. The expression 242 for A can be written A = bl g [1 — 0 (w2)] , (G.l3) and the equations of the curves of constant intensity (constant ¢) can be written x = dVé (x + 83 bl/2, z + 84 mbl/Z) I + dez (x + 63 bl/2, z + 64 mbl/Z) h 2 — 51 [E- x'] + x 0(1)) , (G.l4) where — l < 03 < 1 , and — 1 < 64 < 1 . If the substance in the cell has a one—dimensional refractive index gradient (n constant in the x and y direc— tions), then Eq. (G.l4) becomes 2 H X = de' (z) + 64 (m dbl/2) V (z + 9466 mbl/Z) —§ HE) - x'1+ 0 (V) , (G.lS) l where — l < 64 < 1 O < 66 < 1 Now, V(z) = w (E) = w (E?) , 243 whence V' (z) = £— W'(z—) (G 16) um mr ' ' and n l I! Z v (z) = <—2-=—2-) w (IE). m r Thus, the final equation is db ..d l2 l n z l X‘EW(In—r)+94—2W ‘5?"94962—5’ 2r d 2 —5 (g-x')+xo(w) , (G.l7) l where — l < 64 < 1 ; O < 66 < 1 According to Eq. (G.2), W(z) fh n (z) dy.= hn(z) , (G.l8) 0 for a one-dimensional variation in the refractive index. Thus, we obtain the equations A“ (c.19) X = A (A?) + B r where now Az'= bl/r , 244 and d B=-g [¢/k)-x'1- 1 The quantity 2' = g? is the height coordinate in the object plane, while 2 refers to the second image plane. It is seen that an increase in ¢ of 2n , which means passing from one interference fringe to the next, changes x by the same amount as a change in An/Az' amounting to A/h. As appears from the above derivation, precise adjust- ment of the crystal plates is not critical. This makes the method very easy to adjust and insensitive to mechanical vibrations. It is apparent that there are two sources of sys— tematic error. One is the term n . 2 64 dblh n (z + 6466 bl/2r)/(2r ) , in Eq. (G.l7), which is the difference between the differ— ence quotient An/Az' actually registered by the method and the corresponding differential quotient dn/dz' which one wishes to obtain. The magnitude of this error can be re— duced at will by making Az' = bl/r sufficiently small. In practice bl is fixed and r chosen to optimize sensitivity and accuracy. The second source of error, inherent in the term 2 0 ($2)” is very small. With suitable dimensioning, w < 10 , —4 and the relative error in x will be below 10 . APPENDIX H SUBROUTINE MULTREG MULTREG is a FORTRAN subroutine, written by J. L. Bartelt (1966), which can be used to obtain a polynomial expression for a number of experimental data points. One particular advantage is its multidimensionality. For example, when provided with a set of measured refractive indices of a substance obtained at various temperatures, concentrations, and wavelengths, the subroutine permits calculation of the coefficients of various powers of the temperature, concentration, wavelength, and cross terms appearing in a prescribed polynomial. The method consists of a multiple regression analysis, the theory of which is described by Ralston and Wilf (1960). The subroutine can be called from a FORTRAN program by means of the statement CALL MULTREG (X,W,N,M,NPLUS,A,SIGMA,B,SB,Y,DEV,IPRINT). The parameters which must be specified are: (l) N = the number of independent variables plus one dependent variable, (2) M = the number of data pairs, (3) NPLUS = N + 1, 245 246 (4) W(I) = 1.0, I = 1, ..., M, (5) X(N,M) = the variables arranged so that X(J,K), J = l, ..., N - l, are independent variables, and X(J,K), J = N, are dependent variables. (6) IPRINT = a parameter equal to zero if printing of intermediate results is not desired and equal to unity if it is. For example, to obtain a fourth order polynomial expression in terms of temperature for the density of a fluid from 20 data pairs, write X(l,K) = T(K) X(2,K) = T2(K) x(3,K) = T3(K) X(4,K) = T4(K) X(5,K) = p(K), K = l, ..., 20. In this example, N = 5 M = 20, NPLUS = 6. When MULTREG is used, the following dimension statement must appear in the calling program: DIMENSION X(N,M), W(M), A(N + 1, N + 1), SIGMA (N + 1), B(N + 1), SB (N + 1), Y(M), DEV(M), 247 where the correct numbers are inserted for the letters. When MULTREG is to be called more than once in the same main program, the two—dimensional arrays, X and A, must be the same size for each calling, i.e., N and M must not change under any one dimension statement. The output of the subroutine consists of a list of the coefficients of the variables X(J,K), J = l, ..., N — 1, plus any constant term which appears in the poly— nomial. Also available are calculated standard errors of each of the coefficients. Listing of MULTREG 249 SUBROUTINE MULTREG(X,W,N,M,NPLUS,A,SIGMA,B,SB,Y,DEV,IPRINT) DIMENSION X(N,M),W(M),A(NPLUS,NPLUS),SIGMA(NPLUS) DIMENSION B(NPLUS),SB(NPLUS),Y(M),DEV(M) FORMAT(lOX,*CONSTANT*,20X,*VARIABLE*,20X,*COEFFICIENT*, L20X,*STD ERROR OF COEFF*) FORMAT(lHO,5X,El4.6) FORMAT(36X,Il4.l6X,El4.6,l6X,El4.6) FORMAT(lH-,45X,*PREDICTED VS. ACTUAL RESULTS*,/,24X,*OBS. NO.*,18X l,*ACTUAL*,23X,*PREDICTED*,21X,*DEVIATION*,/) FORMAT(26X,l4,l6X,El4.6,l6X,El4.6,l6X,El4.6) FORMAT(* VARIABLE LEAVING =*,13,/,* F LEVEL =*,El4.6) FORMAT(* VARIABLE ENTERING =*,13,/,* F LEVEL =*,El4.6) FORMAT(* STANDARD ERROR OF I =X,El4.6) NLES=N—1 Fl=0.0 E2=0.0 AMIN=l.0E200 TOL=0.000l DO 1 I=1,NPLUS DO 1 J=l,NPLUS A(I,J)=0.0 DO 100 I=l,M A(N+l,N+l)=W(I)+A(N+l,N+l) DO 100 J=l,N A(N+1,J)=A(N+1,J)+/(I)*x(J,I) DO 100 K=J,N A(J,K)=A(J,K)+W(I)*X(J,I)*X(K,I) DO 101 I=l,N AN(+1,I)=A(N+1,I)/A(N+1,N+1) DO 301 I=l,N DO 301 J=I,N A(I,J)=A(I,J)=A(N+1,N+l)*A(N+lr1)*A(N+lrJ) DO 102 I=l,N SIGMA(I)=SQRT(A(I,I)) A(I,I)=l.0 DO 103 I=l,NLES ID=I+l DO 103 J=ID,N A(I,J)=A(I,J)/(SIGMA(I)*SIGMA(J)) A(J,I)=A(I,J) PHI=A(N+l,N+l)=l.O VMIN=l.0E200 VMAx=0.0 NMIN=O NMAx=0 A(N,N+l)=SIGMA(N)*SQRT(A(NIN)/PHI) 250 PRINT 211,A(N.N+1) IE(A(N,N+1).LE.AMIN)31.30 Fl=FX + TOL F2=FX +TOL GO TO 32 AMIN=A(N,N+l) DO 104 J=1,NLES B(J)=0.0 I=l IF(A(I,I).GT.TOL)7,l4 A(I,N+1)=A(I,N)*A'N,I)/A(I,I) IE(A(I,N+1))11,14,9 IF(A(I,N+l).GT.VMAX)lO,l4 VMAx=A(I,N+1) NMAx=I GO TO 14 B(I)=A(I,N)*SIGMA(N)/SIGMA(I) SB(I)=A(N,N+l)*SQRT(A(I,I))/SIGMA(I) IE(ABS(A(I,N+1)).LT.ABS(VMIN))13,14 VMIN=A(I,N+1) NMIN=I IF(I.EQ.NLES)l6,lS I=I+l GO TO 6 BO=A(N+l,N) DO 105 I=1,NLES BO=BO-B(I)*A(N+1,I) IF(A(N,N))500.l9 FX=ABS(VMIN*PHI/A(N,N) IF(FX.LE.F2)18,l9 K=NMIN PHI=PHI+l.0 PRINT 209,K,FX GO To 21 IF(A(N,N)-VMAX)400,401 Fx=VMAx*(PHI-1.0)/(A(N,N)—VMAX) IF(FX.GT.F1)20,22 - IF(PHI—l.0)20,22 5 3 3 9 K=NMAX PHI=PHI-l.0 PRINT 2lO,K,FX DO 113 I=1,N DO 113 J=1,N IF(I.EQ,K.OR,J.EQ,K)GO TO 113 A(I,J)=(A(K,K)*A(I,J)-A(I,K)*A(K,J))/A(K,K) CONTINUE DO 313 I=1,N DO 313 J-l,N IF(I.NE,K,AND,J,EQ,K)108,109 A(I,K)=—A(I,K)/A(K,K) IF(I.EQ,K,AND,J.NE,K)110,313 251 A(K,J)=A(K,J)/A(K,K) CONTINUE A(K,K)=l.0/A(K,K) GO TO 5 DO 115 J=1,M Y(J)=BO DO 114 I=1,NLES Y(J)=Y(J)+B(I)*X(I,J) DEV(J)=X(N,J)—Y(J) PRINT 201 PRINT 202,BO DO 117 I=1,NLES IF(A(I,N+l))ll6,ll7,ll7 PRINT 203,I,B(I),SB(I) CONTINUE IF(IPRINT.EQ,O)GO TO 1599 PRINT 206 PRINT 207,(I,X(N,I),Y(I),DEV(I),I=l,M) CONTINUE RETURN END APPENDIX I SUBROUTINE MINIMIZE+ The subroutine described herein was written in FORTRAN for use with a CDC3600 digital computer. It is designed to minimize a function of up to ten variables by choosing conjugate search directions. This assures that a quadratic function of n variables will be minimized in at most n steps. (If the number of variables exceeds ten, the program must be redimensioned.) For a theoretical description, see the article by Powell (1964), "An Effi— cient Method for Finding the Minimum of'a Function Without Calculating Derivatives." There are three considerations for use of the sub— routine which must be tailored to the individual purpose. 1. Calling statement. The subroutine may be called from a FORTRAN program by program by means of the following statement: CALL MINIMIZE (X,N ,EPS ,ENDNORM, ITMAX,IPRINT, SUCCESS) . . +Both the subroutine and this description were written by J. L. Bartelt (1967). 252 2. 3. Parameters. a. H) a LG X = a linear array dimensioned for the number of variables. The program should be called with a set of initial guesses for the variables stored in X. The solution will be returned in X. N = the number of variables (less than ten). EPS = a convergence criterion parameter. The change in each variable from the last step is compared with EPS times the current value, and convergence is assumed if the change is smaller. ENDNORM = a convergence criterion parameter. The function value at the current point must be less than ENDNORM to obtain convergence. ITMAX = the maximum desired number of iterations. IPRINT = an option. If IPRINT equals unity, the program will cause the results to be printed. If IPRINT is zero, no results will be printed. SUCCESS = a logical variable to indicate con- vergence. If SUCCESS is unity, the process has converged. If SUCCESS is zero, the method has failed to converge, and a statement will be written to indicate the reason for termination. Required subroutines. a. QUADMIN. This is a routine required by MINIMIZE and is furnished with the deck. 254 b. FNORM. This is a function-subroutine where the function to be minimized is placed. It must have the following form: FUNCTION FNORM (X,N) DIMENSION X(N) (any necessary calculations) FNORM = f(x(1),x(2), ..., x(N)) RETURN END Where f is the function to be minimized, and X and N have their previous meanings. For the purpose of analyzing data we choose FNORM to contain a function which is the sum of squares of the deviations of experimental points from some analytical function. The actual FNORM used is shown in the listing of ALPHA in Appendix J. Listing of MINIMIZE and QUADMIN 256 SUBROUTINE MINIMIZE(X,N,EPSl,EPSZ,ITMAX,IPRINT,SUCCESS) DIMENSION X(N),XO(lO),Y(lO),P(lO,lO) COMMON/MIN/LASTNORM,KOUNT TYPE REAL NORM, LASTNORM TYPE LOGICAL SUCCESS IF(N.GT.10)GO TO 5000 ITER=O KOUNT=O D01 I=1,N XO(I)=X(I) P(I,I)=0.l*XO(I) IF(XO(I).LT.(l.OE—7))P(I,I)=0.0l L=I+1 DO 1 J = L,N P(I,J)=P(J,I) =0.0 LASTNORM = FNORM(X,N) KOUNT=KOUNT+l NM=N—l IF(IPRINT)PRINT lOO,LASTNORM,X FORMAT(lHl,*THE INITIAL VALUES ARE*,//,5X,*NORM*,lOX,*XO (l).....XO(N)*,//, l(N)*,//,9E15.6,/,(15X,8E15.6) IF(IPRINT)PRINT 110 FORMAT(lH6,*ITER INC*,5X,*NORM*,lOX,*X(l).....X(N)*,//) ITER =ITER+1 IF(ITER.GT.ITMAX)GO TO 3000 DELTA=1.0E—100 M=O Fl=LASTNORM DO 2000 I=1,N DO 2 J=1,N Y(J)=P(J,I) CALL QUADMIN(X,Y,NORM,N) IF(IPRINT)PRINT lOl,ITER,I,NORM,X FORMAT(215,8ElS.6,/,(25X,7#15.6) IF((LASTNORM-NORM).GE.DELTA)3,4 M=I DELTA = LASTNORM—NORM LASTNORM = NORM CONTINUE F2=NORM IE(ITER.GT.N)15,16 IF(NORM.GT.EPSZ)l6,l7 DO 18 I=1 N IF(ABS(x(I)—XO(I)).GT.ABS(EPS1*X(I)))l6,l8 CONTINUE K.) 3 257 GO TO 4000 DO 5 I=1,N Y(I)=2.0*X(I)-XO(I) F3=FNORM(Y,N) KOUNT=KOUNT+1 IF(F3.GE.F1).OR.(((F1-2.0*F2+F3)*(Fl—F2-DELTA)** 12).GE.(DELTA*((F1—E2)**2)/2.0)))6,7 DO 8 I=1,N XO(I)=X(I) GO TO 1000 DO 9 I=1,N Y(I) =X(I)-XO(I) CALL QUADMIN(x,Y,NORM,N) DO 10 I=1,N XO(I)=X(I) DO 11 I=M,NM DO 11 J=1,N P(J,I)=P(J,I+1) DO 12 I=1,N P(I,N)=Y(I) LASTNORM = NORM GO TO 1000 PRINT 102 FORMAT(1H4,*THE MAXIMUM NUMBER OF ITERATIONS HAS BEEN EXCEEDED*) SUCCESS =0 PRINT 5004,KOUNT RETURN PRINT 103,ITER FORMAT(1H4,*THE PROCESS HAS CONVERGED IN*,16,3X*ITERATIONS*) SUCCESS =1 PRINT 5004,KOUNT FORMAT(lH-,*THE NUMBER OF FUNCTIONAL EVALUATIONS WAS*,I10) RETURN PRINT 5001 FORMAT(1H4,*MORE THAN 10 VARIABLES, PLEASE REDIMENSION,*) SUCCESS=0 RETURN END SUBROUTINE QUADMIN(X,P,NORM,N) DIMENSION PHI(3),VT(3),X(N),P(N) COMMON/MIN/LASTNORM,KOUNT TYPE INTEGER UPPER TYPE REAL NORM ,LASTT, LASTNORM DO 9 I=1,N X(I) =X(I) +P(I) LASTT = 1.0 T=0.0 ITER = ITER = ITER + 1 NORM = FNORM(X,N) 258 KOUNT=KOUNT+1 IF((ABS(T—LASTT).GT.(.01*ABS(T)).AND.ITER.LE.20).OR.(ITER.EQ.2)) 111,12 IF(ITER.EQ.1)13,14 VT(1)=0.0 VT(3) =1.0 PHI(l) =LASTNORM PHI(3) =NORM IF(PHI(1).GT.PHI(3))1,2 T=2.0 LOWER=1 MID=3 UPPER=2 K=2 GO TO 1000 T=1.0 LOWER=2 MID=1 UPPER=3 K=2 GO TO 1000 PHI(II)=NORM xw =VT(2)—VT(3) XX=VT(3)—VT(1) XY =VT(1)-VT(2) xw =-(PHI(l)*XW+PHI(2)*XX+PHI(3)*XY)/(XW*XX*XY) XX=(PHI(l)hPHI(2))/XY—XW*(VT(1)+VT(2)) LASTT =T IF(XW.GT.0.0)15,16 T=—XX/(2.0*XW) GO TO 19 IF(PHI(UPPER).GT.PHI(LOWER))17,18 T=3.0*VT(LOWER)-2.0*VT(MID) GO TO 19 T=3.0*VT(UPPER)-2.0*VT(MID) IF(T.GT.VT(UPPER))20,21 I=LOWER LOWER =MID MID = UPPER UPPER =I K=UPPER GO TO 1000 IF(T.LT.VT(LOWER))22,23 I=UPPER UPPER =MID MID =LOWER LOWER=I K=LOWER GO TO 1000 23 24 25 1000 1001 12 259 IF(T.GT.VT(MID))24,25 I=LOWER LOWER =MID MID=I K=MID GO To 1000 I = UPPER UPPER =MID MID=I K=MID II=K VT(K)=T DO 1001 J=1,N X(J)=X(J)+(T—LASTT)*P(J) GO TO 10 IF(NORM.LE.LASTNORM)RETURN NORM=LASTNORM DO 7 I=1,N X(I)=X(I)-LASTT*P(I) RETURN END APPENDIX J PROGRAM ALPHA This program is designed to permit computerized analysis of the raw data from a pure thermal diffusion experiment and calculation of final results. PROGRAM ALPHA consists of three main segments. In the first, measured values of plate temperatures, liquid densities, and interferometric fringe spacings are read and converted to statements of temperature differences, mean temperatures, initial compositions, and apparatus constants. The second section utilizes SUBROUTINE MINIMIZE to fit a smooth function of three variables to the measured values of fringe position versus time. In the third sec- tion, the results of the previous sections are used to calculate values for the thermal diffusion factor and the ordinary diffusion coefficient. A written record is made of all of the information from each experiment. A listing of the program follows. In addition to what is shown here, the deck must also contain subroutines MINIMIZE and QUADMIN and the fol— lowing data cards: (1) a card specifying the number of experiments to to analyzed; 260 261 (2) a card for each experiment stating the number of data points in that experiment, the code number of the experiment, the initial liquid density, the fringe spacing, and the final mean tempera- ture; (3 a card for each experiment containing a set of V initial guesses for the three variables which are to be determined by MINIMIZE; (4) a deck of cards for each experiment, each card containing a measured time and a measured fringe position. The format for each of the cards is shown in the listing below. Listin of ALPHA 263 PROGRAM ALPHA DIMENSION Xi3)oXO(3)wY(3>vp(303) DIMENSION Atté4iqDIZK64jyX11164} (OMMON/fl/D(10019T(IOO?¢F GO TO 53 DO 3 lecNuM EMTOT:E**iuT(J)/THETA) GRADWJSZERO*TAU/AH*Cla—4o/p1*EMTOTE GRAD2W2—4.SSZERO*TAU/AH/AH*BB/PI*EMTOT SONE:ASZEQO*\52*TAFvSSSNGRADW> 265 2*w10—4.351411*WIO**3+3.018563*WIO**43 READ 102c ALPHA]$—XO(1)*TM/DENOM CONTINUE IF(KKOUNT.EO-O) GO TO 60 K1=1./KAPPA*(»4.57E»7*TAU/AH> SZERO=ALPHA1P*WIO*W20/TM GlINFP2—TAU/AH*SZERO*K1*K1*AH*AH/12¢ 267 ALPHAI=‘AH*TM/WlO/W20/TAU*(PI*XO(II/é./A/MUBYW+GIINFP) ALPHA1==ALPHA1 6 O CONTINUE PRINT 3029NUM PRINT 3019R IT:TM TR=IT+016 ACORR=ALPHAIDoOO7*(TMATR) IF(KKOUNT.FO¢1)GO TO 54 ACOR2=ALPHAI ACOR3=ACORR 5 I} CONTINUE KKOUNT:KKOUNT+I IF(KKOUNT.EQ.1) GO TO 52 PUNCH 741~TM9X109WIO¢ACOR20ACOR3oALPHAIoACORRoTAU RRINT 201 1 PRINT 2020RUNgTMquOoXIO.TAU9AoMUBYWoDLITqDEXPqACORZqACORBo IALPHAlqACORR IOO FORMAT(I2) 101 FORMAT(I2o3XoA3.3F6030F5.3wF7.50F6.3) 102 FORMAT(F6¢2¢F4¢3.F5.ZI 103 FORMAT(F7.2~F5.2) 20 FORMATtIx.*RUN*.5X.*TM*.8X.%WIO*.7x.*x10*.ox.*TLu*. ,— 18X'*A*09X0*MUBYW*95X0*DLIT*09XQRDEXP*Q 17Xo*—ALPHA1*03Xo*"ALPHA2*o3Xo*‘ALPHA3*¢3X~**ALPHA4*I 202 FORMAT(lX0A3.3X9F7o303X0F70593X0F70502X0F69303X0510.393X0 iFéoav3XvE10.393X9E100304(3X0F7o4)) 301 268 FORMATIIXoElZoSI 302 FORMATIIXQIS) 74' 5‘ FORMAT(F7O3Q7(F7057) END FUNCTION FNOPM(XQN) DIMENSION X(3)9Y(3) COMMON/A/D(IOO>~T(lOO)aF(IOO)9L E=2.7182818459 SUMSO=Oo DO 1 I=I'L SUM=Oc FII)=X(1)*E**(—X(2)*T(I))+X(3) SUMSO=SUMSO+(F(I)SDII))**2 FNORM=SUMSO RETURN END I.‘..\.