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Prediction of viscosity of pure liquids, mixture of gases and mixture of liquids

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Prediction of viscosity of pure liquids, mixture of gases and mixture of liquids

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Content PREDICTION OP VISCOSITY OF PURE LIQUIDS, MIXTURE OF GASES AND MIXTURE OF LIQUIDS A Thesis Presented to The Faculty of the School of Engineering The University of Southern California In Partial Fulfillment of the Requirements for the Degree Master of Science in Chemical Engineering by Shami Gandhi January 1963 UM I Number: EP41775 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a com plete m anuscript and there are missing pages, th ese will be noted. Also, if material had to be rem oved, a note will indicate the deletion. D issipation Publishing UMI EP41775 Published by ProQ uest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQ uest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United S ta te s C ode ProQ uest LLC. 789 E ast Eisenhow er Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 -1 3 4 6 c.h 'ai G W S This thesis, written by S. K. Gandhi under the guidance ojbdUs Faculty Committee and approved by all its members, has been presented to and accepted by the School of Engineering in partial fulfillment of the re quirements for the degree of ........... .Chemical Engineering Date £asagxx.2$&3... Faculty Committee I Chairman ACKNOWLEDGMENTS The author is deeply indebted to Dr. Charles J. Rebert for his suggestions and expert guidance on this project. Further, he wishes to thank Dr. E. J. Lockhart and Dr. George V. Chilingar for their helpful sugges tions during the progress of the research. In addition, the author would like to express his gratitude for the help obtained from his friend, Ita Gordon, in the preparation of this thesis. He also wishes to acknowledge the assistance given in the preparation of graphs by Mr. Mustan Officewala. i i TABLE Of CONTENTS CHAPTER PAGE I. INTRODUCTION ....... 1 Statement of Problem ........... 1 Importance of Study .......... 1 History ........... 1 Recent Developments ........... 3 References ................ 4 II. MODERN THEORY OP VISCOSITY .... ....... 5 Andrade Theory .......... ....... 5 Eyring Theory . .... ........ 6 Mechanism Viscosity ... ............ 7 References ........ 11 III. VISCOSITY OP MIXTURE OP GASES........... 12 Wilke Equation ................ 1 £ 5 Budenburg and Wilke Equation ....... 16 Extension of Modern Theory ............ 17 Hirschfelder Equation ... ........... 18 Johnson Equation .. ....... ...... 37 Heraing and Zipperer Equation ...... 41 General Equation ..... ....... 42 Hirshfelder, Curtiss and Bird Equation . . 42 Buddenberg and Wilke .............. 43 i i i CHAPTER PAGE Wilke Equation........ 44 References ••«•••••••••••• 43 IV. VISCOSITY OP MIXTURE OP LIQUIDS....... 50 Arrhenius Early Equation .••••.•• 50 Kendal and Monroe Equation • ••••.. 51 Dolezalek Equation........ . • . • • 51 Taylor Equation • ••••••••••• 52 Vander Wijk Equation • ••••••••• 54 Kern Equation .•••••••••••. 55 Tamura and Kurata Equation ••••••• 56 Lima Equation............•••••• 60 Zdanovskii Equation 63 Shutela and Bhatnagar Equation ....... 64 Ishikawa Equation • •••..••.•• 65 Kendal Equation • ••;..•••.•• 67 Prehfcel Equation ....... • • 63 Eyring Equation •••••••••.»• 69 Cubic Equation ••••• 70 References ..••••••••••••• 93 V. PART I. LIQUID VISCOSITY . . ......... 94 Souder Equation 94 Trouton Equation •••••••••••• 97 Spring Modified Rate Theory ..•••• 98 Priend and Hargreaves Equation ..... 99 iv CHAPTER PAGE Thomas Equation • ••••••*••«•• 101 Grunberg and Nissan Equation........ • 103 Andrade Equation •••••••«••••• 104 Arrhenius Equation • •••••••••• 105 Gambill Equation • ••••*.••.••• 106 PART II* EPFEGT OP TEMPERATURE ON VISCOSITY 107 Andrade Equation *.•.••.*...•• 107 Bingham and Stookey Equation ••••••• 108 Modified Andrade Equation ......... 110 Thomas Equation • *••.•••••••• 112 When One Value is Known........ 113 Proposed Chart ....... ......... 115 References ••••.*.••••«•••• 118 v TABLE 2-1 3-1 3-2 4-1 4—2 4-3 4-4 4-5 4-6 4-7 5-1 5-2 5-3 LIST.01 TABLES PAGE Transport Processes • ••••••••• 3 Constants of Eq. 3-5 40 Maximum Error by Eq. 3-12 • ....... 47 Interaction Viscosity Constant • • • • • 59 Calculated Values of N-Octyl Alcohol + CCI4 Mixture • •••...•••••••• 61 Maximum Error in Calculated. Viscosities from Lima Equation ..•••••.•• 62 Observed and Calculated Viscosities • • 63 Observed and Calculated Values ..... Association Degrees of Liquids • • • • • 67 Enthalpies and Entropies of Activation for Viscosity for the Methanol-Toluene System 71 Viscosity Constitutional Constant ... 97 Eheochor Contributton ..••••• •• 101 Structural Contribution •••••••• 103 vi LIST OP FIGURES figure page 3-1 Viscosity of Mixtures with H-Butanol . . • 14 3-2 Viscosity of Ethanol— H-Propanol • • • • • 15 3-3 Viseosity of A-Xe Mixture............ . 20 3-4 Viseosity of Ee-Xe Mixture...... .. 21 3-5 Viscosity of Ee-Kr Mixture .••••.•• 22 3-6 Viscosity of le+A Mixture • 23 3-7 Viscosity of Hg-CO Mixture • 24 3-8 Viscosity of Hg+Eg Mixture • ••••••• 25 3-9 Viscosity of Hg+A Mixture 26 3-10 Viseosity of He+A Mixture • ••••••• 27 3-11 Viscosity of Hg+O Mixture • 28 3-12 Viseosity of CH^+Eg Mixture ....... 29 3-13 Viscosity of Hg+ 2-Butene Mixture ...» 30 3-14 Viseosity of He+02 Mixture • •...•• 31 3-1S Viscosity of CO+Hg............. . . . . 32 3-16 Viscosity of He+Steam Mixture ...••• 33 3-17 Viscosity of A+HS Mixture ........ 34 3-18 Viscosity of C0£*E© Mixture . ...... 35 3-19 Viscosity of Steam + Og Mixture . • • • • 36 3-20 Johnson1s Equation Constant ....... 39 4-1 Mixtures of Ethanol with Benzene and Water 53 4-2 Observed and Calculated Values, Eq. 4-7 . 57 4-3 Observed and Calculated values Eq. 4-7 . 58 vii PIGURE PAGE 4-4 Cyclohexane— Heptane ancl Benzene-Toluene Mixture ....... ........... ••••71 4-5 Acetone— Hater Mixture............ • 73 4-6 Methanol— Toluene Mixture ...... * . 74 4-7 Kinematic Viscosity as a Punction of Temperature 75 4-8 Mixture of OgHg+ H-C^Cl............ 777 4-9 Viscosity of CgHg+ U-C4H^Br Mixture ... 78 4-10 Mixture of CgHg+ Og^Cl.............. 79 4-11 Mixture of CgHg+ CgH5Br.............. 80 4-12 Mixture of CgHg+ CgH5I .............. 81 4-13 Mixture of OgHg+ CgHc^Hcj ........ 82 4-14 Mixture of OgHg+ CgH^OOH^ ........ 83 4-15 Mixture of CgHg+ Q-CgH4(CH} OH .... . 84 4-16 Mixture of CgHgt CgHgOHgOH ....... 85 4-17 Mixture of CgHg+ CgH5C H 0 ........... 86 4-18 Mixture of CgHg+ (02H5)2H H .......... 87 4-19 Mixture of CgHg+ (CjjHc^O ........ 88 4-20 Mixture of OgH5+ C2H5C00H ............ 89 4-21 Mixture of CgHg+ M-C4HgOH ........ 90 4-22 Mixture of CgHg+ HC00C2H5 . ........ 91 4-23 Mixture of CgHg+ CHjCOC-?^.......... 92 5-1 Lagemannfs Homograph............. . 96 5-2 Viscosity y u t as a function of Temperature 109 5-3 Plot of Viscosity vs Temperature .... 111 v i i i FIGURE 5-4 Viscosity changes -with temperature when one value is known • 5-5 Proposed alignment chart PAGE 114 116 CHAPTER I IHTROBTJCTIOI Statement of Problem The purpose of this study Is to make a recent survey of methods for the prediction of viscosity of pure liquid substances, vapor-mlxtures and llquld-mixtures• Chemical, petroleum and petro-chemical industries continue to grow day by day. Knowledge of viscosity values is indispensable in most unit operations such as heat transfer, mass transfer, and flow of fluids. A study such as this is greatly desired by all chem ical engineers, petroleum engineers and chemists who must deal with the viscosity of pure liquid substances, vapor and liquid-mixtures. History It is interesting to reflect that from earliest times man has been concerned with problems of fluid flow. About 3000 B.C., we are told, the Ancient Sumerians took as their unit of weight the amount of water which flowed out of a standard vessel in a given time. In the sixteenth century B.C. the Egyptian Amenemher, knight and keeper of 2 the King's seal, describes a water clock consisting of a conical vessel with a hole in the bottom, which, it claimed, told the time accurately during the night,(1^ It seems almost certain, in view of experiments by Hoppler^2^ that Amenemhefc even allowed for the drop In temperature which must have taken place during the night, although such a correction was most probably empirical. In his article Hoppler compares this very early form of clock with a commercial viscometer Invented by Engler in 1885, The similarity is most striking, The first hypothesis postulated by lewton in his principle is that the force required to maintain a rela tive motion between two parallel planes of fluid is dir ectly proportional both to the area of the planes and the relative velocity and inversely proportional to the dis tance between the planes, Mathematically 1*<*A As Area, Gm2 Vs Velocity cm/sec X: distance cm The coefficient required to make this proportionality an equality is called the coefficient of viscosity, ^ • Held and Eelenyessy^ modified the basic foiseuille equation as follows: M-Bis = 1q6. f t . r^M -rc2) - JUL. _____ '-1 16 M R TZ 87YL % Nomenclature f t , =s Viscosity of mixture, mlcropoises. r s Integrated fourth-power average radius of the capillary bore M s Molecular Weight -Pg s s Pressure drop in the capillary tube I t =s length of the capillary tube W s Mass flow rate E = Gas constant T = s Absolute temp*, °K Z = Compressibility factor Badius of lube Bg<mn§ 5f VSS!g" , ' g, dHtMbTg, ”5VT: aX5T m s Subscript mix a gas mixture IT = constant is equal to 3.14 This equation represents flow in a round capillary tube with a correction for end effect ~5L. 8 it I The correction factor may be estimated or determined from calibration runs using a vapor whose viseosity is known* Experimental results are compared with Wilke Eq* 3-1 in Pig* 3-1 and 3-2* Recent Developments If a shearing stress is applied to any portion of a confined fluid, the fluid will move and a velocity grad ient will be set up within the fluid, with a maximum 4 Telocity at the point where stress is applied* If the shear stress per unit area at any point is divided by the Telocity gradient, the ratio obtained is defined as the viscosity of the medium* It can be seen, therefore, that viscosity is a measure of fluid friction which tends to oppose any dynamic change in fluid motion, l*e* if the friction between layers of fluid is small (low viseosity), an applied shearing force will result in a large veloeity- gradlent* As the viscosity increases, each fluid layer exerts a larger frictional drag on adjacent layers and the velocity gradient decreases*^ Inasmuch as the viscosity is defined as a shearing stress per unit area divided by the velocity gradient, it should have the dimensions of force-time/( length)2 or mass/(length)(time)* Both dimensional groups are used, although the viscosities are expressed most frequently in texms of micropoises, centipolses, poises which are the metric unit equivalent of the mass-length-time dimensions* REgEREHORS 1* A* 0* Harringtons "Viscosimetry"; London, Edward Arnold and Co* 1949* 2* SoppierS "Koll Zeits' 1 2&» I* 1942* * » 32 B* 02 Reid and I* K2 Sherwoods "She Properties of Gases and Liquids"; McGraw Hill Co* 1958* 4* R. 0. Reid and L* I* Beleuyessy, M*I*T*, Cambridge, J* Chem* and Sugg* data 1960; CHAPTER II M0B1BI THEORY OP VISCOSITY Andrade Theory Inasmuch as the influence of temperature on the vis cosities of liquids is quite different from the behavior with gases, it has been proved that different molecular meehanisms are involved in the flow of liquids and gases* The realization that liquids possess a hind of structure has permitted the development of theories of viscosity which do not involve the transfer of translational energy from one layer to another by Interchange of molecules as postulated for gases* 1* N* da C* Andrade^ suggested that the temporary union between groups of molecules which appears to be equivalent to the formation of eybotactio groups^^ (existing because of the great length of mole cules) leads to the transfer of vibrational energy* According to the Maxwell distribution law, the number of molecules possessing sufficient energy is related to 0 -E/RT where e is the activation energy per mole, and hence it is possible to show that the viseosity of liquid will vary with temperature according to the exponential equation =AeB/^ where B may be replaced by E, the energy of activation* According to Guzman this is related 6 to the latent heat of fusion in many instances. Byring gheory of Viscosity A different point of view, involving the energy of activation for viseosity, and the latent heat of vaporiza tion, has been discussed by H, Byring, who has developed a general theory of the properties of liquids, The energy of aetlvation is regarded as consisting mainly of that required for the provision of a MHole”^ (Holes may be regarded as playing the same part in the liquid phase as molecules do in the gas phase, the molecules in the former state being equivalent to empty space in the latter) into which a molecule, or in some eases two molecules together, can be transferred by rotation. Suppose the n molecules forming a liquid are each bound to others by "bonds” add ing up to a total energy nu; to vaporize a single molecule would require energy i U, Each "bond1 1 is shared between two molecules, provided the rest of the molecules join up, so as to leave no hole in the liquid. If a hole is to be left, however, the vaporization of a single molecule requires energy TJ, ghe return of a molecule from the gas into a hole prepared for it in the liquid, would result in the liberation of energy § U and so ¥-•§ T J = § U is the energy necessary to make a hole in the liquid large enough for a molecule, as to evaporate a molecule without leaving a hole, ghe latter quantity clearly determines the latent 7 heat of vaporization, and so the activation energy of the viscosity process, which is required for the provision of a suitable hole, should be related to the heat of vapor ization. luring's theory was supported by experimental results. Mechanism of Viscosity Transport phenomena are the combined study of the basic fundamentals of momentum transport, energy transport, and mass transport, which represent fluid flow, heat transfer and mass transfer respectively. Viscosity is coefficient of one of the three transport processes which involve the transport of some property across a fluid. These are summarized^2*3*4-) Table 2-1. 8 TABLE 2-1 TRANSPORT PROCESSES Property Governing Law Flux Property transferred/ time x area Diffusivity Force,gradient (lengtn/t ime}’ of concentration of property Heat Btu. Fourier *s Heat Flux Law (conduction) q. q--Kd(PCpT) PCpdx BTU/hr.sq.ft. Thermal Diffusivity K/pcP sq.ft/hr. Temperature gradient d(PCpT) dx d(Btu/cu.ft)/dx Mass lb Fick1s First Law CMolecular Diffusion) Ja^MCa dx M : ■lass Flux Concentration gradient Molecular Diffusivity Ja, H D d(CA)/dx lbs-:/(Hr)(sq.ft)sq*ft/Hr d(lbs/cu.ft)/dx Momentum Newton's Law Momentum Viscous Velocity (Mass x (Viscous Flow)Flux diffusivity gradient Velocity) T gg ^ d ( V P ) / d x / x/ w g=-MlSl J / r . , d(lb.)(ft.)/ (lb.)(ft.)/Hr ' * dx lb.ft/for.sq.ft) sq.ft/Hr (Hr)(cu.ft)/dx Hr. / It is convenient to picture viscosity by visualizing a bulk of fluid flowing in viscous flow in a conduit* There is a velocity gradient with the maximum velocity occurring at the center and the minimum at the walls of the conduit*' Momentum is transferred across the fluid and is dissipated at the walls as heat* Quantitatively, the viscosity of a fluid Is a measure of the rate at which the momentum can be transferred across the fluid* The point 9 of interest here is the actual mechanism of transfer within the fluid itself* Consider first a diluted gas flowing in the conduit* In addition to the net forward velocity, the molecules of the gas diffuse in all direc tions, both parallel and perpendicular to the fluid stream* A molecule diffusing from the higher velocity central por tion of the stream into a slower portion of the stream nearer the conduit wall soon loses its excess forward velocity and assumes that of the slower stream* Similarly a molecule leaving the slower stream in favor of the faster will he accelerated to the higher velocity. Each such occurteuje results in a net transfer of momentum from the faster to slower stream* Ihe mechanism is known as the “Molecular” or "Dlffuslonal" transfer of momentum* Obviously, it is very dependent upon the diffusivity. Dif fusion in a gas increases with about the 1*5 or 2 power of temperature, whereas the number of molecules available for diffusion in a given volume is inversely proportional to temperature; The net result is an increase in diffusivity with temperature and accordingly an increase in viscosity with temperature. Similarly the diffusivity varies with reciprocal pressure while the number of molecules available increase linearly with pressure; Viscosity of a diluted gas is, therefore, found to be Independent of pressure; When gases become more concentrated as pressure is increased, the mean free path is lowered and diffusion_____ 10* decreases. The molecule traveling in a faster stream is then more likely to collide with a molecule of the slower stream at the ‘ 'interface" and rebound back into the faster stream; At eaoh such collision a small amount of momentum is transferred as the faster molecule loses some of its i ! velocity to the slower one* This is known as “collisional" transfer of momentum and is predominant in liquids and dense gases* It results in an increase in viscosity of a dense gas with increasing pressure solely because of the increase in the number density of the molecules. Simi larly, an increase in temperature decreases the number of moleeules in a unit volume and the viscosity thereby decreases* Gases therefore show an inversion point where, with Increasing pressure, the temperature coefficient changes from positive to negative* Nomenclature 0 s s Concentration, Ib./cu.ft. Opar Heat Capacity, Btu/(lb)(®P) £ » Heat flux, Btu/hr* sq.ft. k = Thermal conductivity, Btu/hr.sq.ft.(°P/ft.) d = Pipe diameter, in ft. P = density, lb/ft.^ T = Absolute temp. °R x,y,z s s Cartesian coefficient J a s nnr-.-of jHtess *, 1 K/bkeq#f$. 11 D = Diffusivity sq.ft./Hr. Ty = Shear Stress perpendicular to flow Ibf./sq.ft. V = Vector Velocity, ft9/hr. = Viscous diffusivity (Kinematic viscosity), sq.ft./hr. E = Activation Energy E = Constant REFERENCES 1. G-lasstone, Samuel, Text Book of Physical Chemistry, 2nd Edition, pp. 493-94. 1958. 2. Bird, R.B., ¥. E. Stewart and E. N. Dightfoot, "Transport Phenomena,” Wiley, New York, 1960. 3. Foust, A. S., L. A. Wenzel, C. W. Clump, 1. Maus and L. B. Anderson, "Principles of Unit Operations", Wiley, New York, 1960. 4. Smoot, L. Douglas, Chemical Engineering, August 26, 126-27, 1961. CHAPTER III VISCOSITY OP MIXTURE OP CASES TMs chapter contains the selected relationships for the prediction of viscosity of mixture of gases. Each formula Is presented with the author*s name, nomenclature, subscript, figures, comment and accuracy, ¥ilke Equation (30) /^Wf' Por Ideal /u. + 3-1 '+Ci)^ 2 3-lb Por non-ideal ?S -W * OS O'W 2 . \ 13 Nomenclature A^m m Gas viscosity, micropoises y s Gas phase mole fraction M = Molecular weight P s Density gms/cc at temperature and pressure ^2 = Function of pure oomponent properties defined by Eq.. 3-la* = Function of pure component properties defined by Eq. 3-1 fe. Subscript m = Gas mixture 1,2 = For a mixture of component 1 and 2, Figures Calculated values of Eq.* 3-1 are plotted in Fig* 3-1 and 3-2# Comment According to Wilhe^, this equation can be used for Ideal and non-ideal gas mixtures. a Accuracy An average error in representing A1 m for seventeen binary gas mixtures was only 0.97 percent, excluding Eg-A for which agreement was least satisfactory. VISCOSITIES OP VAPOR MIXTURES WITHJN-BUTANOL AT 1 ATM. ^ VISCOSITY OP ETHiNOL— N-PIOPANOL VAPOR MIXTURE AT 150®C AND 1 ATM. Budenburg and Wilke Equation^ (1951) 16 3-2 3-2a Bomenclature s r TFlseoslty of Gas, eentipoise y = Gas phase mole fraction ? = Density of Gas, Gms/ce* D12 = Biffusivity of component 1 In a mixture of 1 and 2, cm2/see. T s' Absolute Temp* °k P s Total pressure, atm* H a Molecular weight. V = s Specific volume, cc/gm. C*= Constant Subscript 1,2 = Component one and two m = Por Gas Mixture Comment This equation is good for low pressure* It can be used at high, pressure with reasonable accuracy* 0* is a |^vn = 14 . ( h±\ f a 17¥ a m D = — C - I ______ 5 P O,"9 t V?) + -L M, 17 constant for all compositions for gas mixtures* Johnson suggested the value of 0* is 0,0043* Accuracy An average error was 0.6$ for sixteen compositions of five gas pair* Br tension of Modem theory (6,8,9) (1954) _ _ m“ . os os r-* s=i+ Nomenclature jU. = Gas viseosity, Micropoises y s s Gas phase mole fraction M = Molecular weight Subscript 1,2 s s For a gas mixture of component 1 and 2* m = For a gas mixture* Comment Most accurate equation is for non-polar binary gas mixtures at low pressure* One simple case is that of binary mixtures of heavy Isotopes for which Mg/Mj approaches unity* This equation has been extended to multicomponent mixtures by Curtiss and Hirsehfelder (6,7)* Accuracy An average error was 0*5 to 1*0 percent error.______ 18 Hirschfelder Equation^10) (1954) I — t Yn Xn = = I + f t + ^*|X* - “ n.ft Xa ^a i A 5 X, Ma 3-4 3-4j a M, ^ . a \i- 4 M.ma 3 A* 5 A,a *) = '•a •Ht * * * \|_ 4Ml MAJ|jn, TQjJ I «5h,rv S3V ~^i.+ I J+xJ I 5-4, MaT Ma a ^,ft. T * tx ha 3-4/ -r* kt ^ = — °i, =i:(f:+05). Nomenclature i " ) = Viscosity of gas, micropoises. M = Molecular weight. x s Mole fraction. o- s s Collision diameter, Angstrom ia -Q^=: Function of collision integrals. * Ata= Function of collision integrals. 3-4e 3-4f 19 K s Boltzman Constant, ergs/°K 6 = Maximum energy of molecular attraction, ergs« T = Absolute temperature, °K* t^12 = Define by £q 3-4a T*2s Inaction of collision integrals* Subscript 1,2 s For Mixture component one and two. *0 a For gas mixture Figures - Fig* 3-3 to 3-19 represent the calculated values of Eg.* 3-4* Onmmftnt (12) K* J* Kenney and M* W. Thring' 'reported that it is possible to calculate the viscosity of mixtures up to 1000°C* with an accuracy of at least 2 percent* The vis cosity of mixture of carbon-dioxide and nitrogen for various composition were determined up to 900°G, and results obtained were agreed with Southerland's Law^1^. By ELrschfelder's equation it is possible to calculate the viscosity of gas mixture accurately for large temperature range with no experimental values for the gas mixture*' The values of and £ for various Substances, are reported in (6). While the values of , A12 &ud ^12 are tiie function of collision Integrals reported in (10)* This equation actually is a multicomponent mixture which has VISCOSITY OF A-XS MIXTURE AT 1 ATM. Fig. 3-3 -OS VISCOSITY OP Ne + A MIXTURE AT 1 ATM is Pig. 3-6 I L-<L •9y[ HIT I ST aanSXIH 00 + °H 40 XSISOOSIA VISCOSITY OF Hg + A MIXTURE AT 1 ATM H e Si: r w O > - VISCOSITY OF CH4 + Ng VISCOSITY OP H0 + 2 - BUTENE MIXTURE AT 1 ATM I Pig. 3-13 VISCOSITY OF He + 02 MIXTURE AT 300WC S s p e z ' i a i f e E ' l a Fig. 3-14 VISCOSITY OF CO + H2 MIXTURE AT 1 ATM Pig. 3-15 VISCOSITY OP He + STEAM MIXTURE AT 2000 PSIA Pig. 3-16 VISCOSITY OF A + m MIXTURE AT 1 ATM ■ ■Ml Fig. 3-17 VISCOSITY OF C0o + NO MIXTURE AT 1 ATM I ; I j I VISCOSITY OP STEAM + 0o MIXTURE AT 2000 PSIA 37 been specialized for the case of a binary equation. Johnson* s Equation (11) (1956) At />*• . , S** 3-5 (^•COo3l(% V “ ’ CV> ' 3- fCM = ................. .. .. ....... Crt.+ HzV3- >5a exf 5 k U ’ i;J 3-5* /^ = 0-75 - f c OR ^/n. “ 3~5e Nomenclature M = Viscosity of gas, Mlcropoises* y = Gas phase mole fraction* Z s Compressibility factor* M = Molecular weight I P = Absolute lempoerature, °K* £ ■ = Maximum Energy of Molecular Attraction, ergs. K 3 Boltzmam’s constant, ergs/degK. V 3 Molecular volume B s Johnson*s constant Subserlot 1,2 s Component of Mixture 1 and 2* C 3 At critical point* m 3 for gas mixture* 38 Figures Figures 3-12 to 3-19 represent the calculated values of Ecu 3-5* Constant B of Eq. 3-5 may he found by Fig* 3-20# It Is noted, according to Johnson, that this corre lation gives good agreement for binary gas mixtures at low pressure. It can be used at high pressure, til and B can be found from Table (3-1), for various gas mixtures. Accuracy Johnson claimed his equation to be accurate. An average error between the observed values and the calcu lated values was found to be 5*1 percent and the maximum error 33 percent. JOHNSON1S EQUATION CONSTANT 40 TABLE 3-1 CONSTANTS OF Eq. 3-5 Mixture ^12/K I Hg-Ne 34.94 .558 Hg—A 64.25 .758 He-Propylene 47.96 .561 E20-C3H8 245.3 .912 co-c2h4 150.2 .802 N2“°2H4 137.0 .818 W b 92.0 .700 Hg—Ogo 61.4 .709 Hg-H20 88.9 .691 Hg-Og 101.7 .734 OQt*C^H0 232.6 .871 Hg-COg 84.2 .765 2-Butene-He 60.6 .690 o2h4- o2 152.2 .789 He-Ne 19.10 .628 oo-o2 111.5 .737 Hg-Propylene 86.6 .673 Hg-He 18.4 .644 Xe-He 48.4 .648 He-A 35.6 .556 41 Heralng and Zipper Equation (5) (1936) (1958) Mi (M i) 3 - 6 Z yc (Mi) ' 5 Nomenclature Ja = Viscosity of gas, Mlcropolses y = Gas phase mole fraction M = Molecular weight Subscript m s For a gas mixture 1 = For 1 ^component Comment This well Imown correlation, according to Herning and Zipperer,^ id sometimes called the "Square Root Rule,*' Friend and Adler^have pointed out that for a gas mixtures with content in excess of about 25 percent this equation does not give accurate results* This equation is also good for Hydrocarbon Mixtures, Accuracy An approximate error was 1.9 percent for 25 indus trial multicomponent mixtures. Maximum error for hydro carbon mixture was 1.5 percent. 42 General Equation (3,16,18,19,31) 3-7 Nomenclature Tj = Viscosity of Gas, poise* X = Mole fraction of components 4>. = Ponetion of the properties of pure component i defined by 1q. (3-8), (3-9), (3-10), (3-11), Subscript m = Gas mixture 1) = Component i and j Comment This general type equation has been proposed by SutherlandP ®) Thlesen,^) SehudelP1^ ludenberg and Wilke, ^and Wilke It may be shown that all the functions adopted by various authors may be derived as approximations using the modem Kinetic theory of gases, and that there are disadvantages or inaccuracies involved in using the equations in many cases* Hlrschfelder. Curtiss and Bird (6) Nomenclature T| s Gas viscosity, poise = Define by equation, 3-8a M ss Molecular weight * \ i = Ratio of collision integrals T = Absolute temp*, °K ojj s Va.Cn Where o - < and are the collision diame ters, angstrom = Function of collision integrals R = Gas constant Subscript l,j = Component 1 and 3 Comment Hirschfelder, Curtiss and Bird^ used'; the modern kinetic theory of gases, which involve the extensive numerical work for mixture containing three or more com ponents* When properties of components 1 and 3 are identical, it gives a value of < $ > of 1*2.6 instead of unity* However this simple equation may represent the greatest accuracy* Buddenberg and Wilke (6) 3-9 3-9a 44 Nomenclature = Gas viscosity, poise P s density gms/co. t>ii = Defusivity of component i in a mixture of i and 3 Gm/sec. ] s M a Molecular weight J Aij= Function of collision integrals T s Absolute temp. °K P = Total pressure, atm. <nj = ‘ /a.c«v + where ° - * i and are the collision diameters of 1 and j reap., angstrom a a . Function of collision Integrals Subscript i,3 = Component i and 3 in mixture ¥113ce Equation (30) Nomenclature * 1 = 'Viscosity of gas, poise M 3 Molecular weight Subscript 1,3 = Component 1 and 3 mixture Comment Wilke replaced the diffusion coefficients in the 45 Buddenberg and Wilke(3-9) by an expression involving coefficients of self-diffusion and assuming that the Schmidt number, is a constant for all gases* Accuracy This equation shows good agreement with experiment for several binary mixtures with very irregular viscosity against concentration curve* Wilke’s equation fits the experimental data quite reasonably in many cases but gives large errors in binary mixtures where M^> and eg# H2+ H2+ A* °2* ®2+ °0* ete* Recent Development (6) 9 ^ 22 J Sii xi a ^ Mi+Mj Nomenclature K = Molecular weight <f = collision diameter 22 XI s Function of collision integrals* Subscript i,3 = Component i and 3 of mixture Comment This equation for , which may be expected to predict viscosities of multicomponent mixtures of non polar gases with an accuracy of about 2%, It has been 46 tested on a large number of binary mixtures of gases of widely differing molecular weights and viscosities. Such mixtures present the most severe test of an equation of this type. The results are summarized in Table 3-2, 47 TABLE 3-2 MAXIMUM ERROR BY M. 3-12 G-as Bair Tenn>.°K i ; Max-Error Ref. Hg + GO 195-523 + 1.9 20 Hg + Hg 195-523 - 1.0 20 Hg + Mg8 300-550 + 1.5 21 Hg hP gOg 300-550 —» 1.3 21 H2 + °3H8 300-550 + 2.7 21 Hg + CH^ 293-523 - 0.8 22 Hg + CgHg 293-523 + 1.0 22 gh4+ C3Hq 293-523 + 0.7 22 Hg + G3H6 293-523 + 5.0 23 He + C3:H6 293-523 + 0.8 23 He + A 298-474 + 0.6 26 He + A 298-474 - 0.8 26 He + He 90-474 - 0.8 26,27 Hg + He 90-474 + 1.7 26,27 Hg + Xe 293-550 -6.0 28 He + Xe 293-550 + 0.6 28 H2 + He 90-523 *» 3.7 29 Hg + A 293-550 - 2.0 29 Hg + Og 300-550 + 1.2 24 % + c2h4 195-523 2.0 25 48 REFERENCES 1* Bromley and C.R.Wilke: Ind. Engg. Chem., 43 1641 (1951) 2. Budenburg, J.W. and C.R. Wilkes J. Physics Colloid ehem., 1491-98 (1951) | 3* Buddenberg and Wilke: Ind. Eng* Chem. 4l_, 1345, 1949 i | 4. Friend, 1. and S.B. Adler,: "Transport Properties in Gases," edited by Gambel and Penn, northwestern Univ. Press, Evanston (111*) 1958 5* Heraing, P* and Zlpperers Gas-U, Wasserfach*, 79 69. (1936) “ ■ 6* Hirschfelder, J.O., C.F. Curtiss and R.B. Bird; "Molecular Theory of Gases and Moulds," Wiley John & Sons, Wew York (1954) 7* Hirschfelder, J.O* and C.F. Curtiss: J* Chem. Physics 12, 1345-47 (1949) 8* Hirsehfelder, J.O*, R.B. Bird and E.L. Spotz, Trans. ASME, 11 921-37 Hov. (1949) 9* Hirschfelder, J.O., B*B* Bird and E*L* Spotz, Ch'em. Revs., 44, 205-31. (1949) 1 A » ' w & 10. Hirschfelder, J.O., R.B. Bird and C.E. Curtiss: Trans. ASME 16, 1031 (1954) 11. Johnson, C.A.: "Viscosity of Gas mixtures" Syracuse University Res. Inst. Report Ho. Ch.E, 273-566 F.3, July 1, (1960) 12. Kenny, M.J. and M.W. Thring: British J. Applied Physics 1: 324-29 (1956) 13. Reid, R.C. and T.K. Sherwood: "The Properties of Gases and Liquids," McGraw Hill, Hew York, (1958) 14. Reid, R^C. and L.I, Belenyessy, M.I.T., Cambridge, J. Chem. and Engg. data 5 (1960) 15* Rowlllnson, J.S. and Townley: Trans. Paraday Soc., 4£ 20(1953) 49 16* Schudel, Schweiz, Ter* Gas-U* Wasserfach Monats- Ball* 22 112-131, 1942 17. Sutherland, ¥.: Phil Mag,, 36 507-31 (1893) 18. Sutherland, ¥,: Phil Blag,, 40 421 (1895) 19* Thlesen, Verhandl. deut. Physik. Gesi, 4, 348, 1902 20. Trautz and Baumann Ann, Physik, 2, 733, 1929 21. Trautz and Kurz Ann, Physik, 2, 981, 1931. 22. Trautz and Scrg, Ann. Physik 1,0, 81, 1931. 23. Trautz and Husseini, Ann. Physik 20, 121, 1934. 24. Trautz and Eipphan, Ann. Physik 2, 743, 1929. 25* Trautz and Zimmermann, Ann. Physik, 2£, 189, 1935. 26. Trautz and Heberling, Ann. Physik 20, 118, 1934. 27* Trautz and Binkele, Ann Physik, 3 561, 1930. 28. Trautz and Melster, Ann Physik, £, 409, 1930. 29. Trautz and Stauf, Ann. Physik, 2, 737, 1929. 30. Wilke, G.B.: J. Chem. Physics 18 517-19 (1950). CHAPTER IV VISCOSITY OP MIXTURE OP LIQUIDS This chapter contains the selected relationships for the prediction of viscosity of liquid mixtures. Each for mula is presented with the author’s name, nomenclature, subscript, figures, comment and accuracy. Arrhenius Early Equation^1^ Lv*\ - + JM.^ Nomenclature h = Absolute viscosity, centipoise. * = Mole fraction of liquid component Subscript Lm= Liquid Mixture 1,2 = Component one and two Comment According to Arrhenius^ that deviation from such simple additivity, is proportional with real-non-ideal mixtures to the energy and entropy of mixing* This equa tion is most accurate when component molecular weight and viscosity differences are small j U £15 Op. 51 Accuracy No calculated data Is available. Kendal and Monroe Equation (7) = 3t,|M.,/3+ y.^1* 4-2 i Nomenclature M. 5 5 Absolute viscosity of liquid, poise or eenti- poise = Mole fraction of component Subserlnt 1,2 s Component one and two L = s Liquid Mixture Comment It is noted according to Kendal and Monroe(7) that this equation is best applied to non-electrolytic, non- assoclated similar liquid pairs. Accuracy It has been shown to be accurate within 2-3$ for oil blends. It is most accurate when component molecular weight and viscosity differences are small, (j'-t,-fu) £ 15 Op Dolezalek Equation A I 2 A 52 Nomenclature ( M . a Viscosity, centipoise 01 s Mole fraction of component jtLi a , s Interaction viscosity Subscript 1,2 = Component 1 and 2 L s Liquid mixture Comment This equation is similar to Tamara and Kurata, It can be used for polar-polar, non polar-non polar, or non-polar-polar liquids, fu.,x can be obtained from Table 4-1, The temperature dependency of j^ixmay also satisfy the Eq, 5-10, Accuracy The comparison of this equation with others shown in figure 4-1, An average error between calculated and experimental values were about \0%. Taylor Equation (15) n C V /^d + /^c ) Nomenclature jK. = Absolute viscosity, poise or centipoise = Volume fraction of dispersed phase OBSERVED AND CALCULATED VALUES Fig. 4-1 I J 54 Subscript eff = Effective c = Continuous phase d s Dispersed phase j Onmmant This equation Is most accurate known for Immiscible liquid-liquid mixture* Best result Is obtained for volume percent of dispersed phase % 3%* Accuracy No comparison between calculated and experimental values are available. Taylor^^^claimed to be most accurate* Tander ffi.1k Equation (16) z x _ L09 = . Lo^ + “ X a , i - < > 9 f " ' * * * 3** Log 4-5 Nomenclature | u t = Viscosity, centipoise y . s Hole fraction of component fSa = Interaction viscosity Subscript 1 = liquid mixture 1,2 = Component 1 and 2 Comment This equation is similar to Tamura and Kurata Eq.4-7* 55 t u L can be obtained from Table 4-1* ' 12 Accuracy The comparison of this equation with others shown in Hg* 4-1 * An average error between calculated and experimental values was maximum t 10$* Kern1s Equation (6) 1 _ u ) | ( j J x r kJ - < 3 4—6a rr “ -7- — + --- (4-4) p-L-m - P'uJaior pLtn \ r t / 1 Nomenclature s Absolute viscosity, poise or centipoise* u) a Weight fraction of liquid component Subscript 1 = Liquid mixture 1,2= Component one and two Comment The Eq*4-6 is a rough estimation for organic liquid- liquid and organic liquid-HgO mixture for salts in water where concentration does not exceed 30 wt. $ and where it is known that a sirupy-type of solution does not result then use Eq.4-6 St Accuracy No calculated values are available but claimed^ to be most accurate• 56 Tamura and Kurata (14) os ^im - i j ' - * - * * . ^ 3. + C**i^i^0 4-7 Nomenclature ft " =ifViscosity of liquid, centipoise. ■* b Mole fraction of components ^ a Volume fraction of component f’ Sa = Interaction viscosity weentipoisew Table 4-1 Subscript Lm = Liquid mixture 1,2 = Component 1 and 2 Figures: Fig. 4-1 to 4-3 represent calculated and observed values of some binary mixtures. Comment Tamura and Kurata^^claimed this equation is better than Eq. 4-3 and 4-5* For ideal solution volume change on mixing can be neglected generally. This equation can be used for polar-polar liquids, non polar-polar liquid^ and non-polar-non-polar liquids. The temperature dependency ofmay also satisfy the Eq. 5-10. The values of f + , 1 for various mixtures may be obtained from table 4-1 Accuracy The comparison of this equation with the experimental OBSERVED AND CALCULATED VALUES IWjTPKMMHqEE 58 1 OBSERVED AND CALCULATED VALUES 59 values of viscosities, (observed), has been performed in regard to about 30 systems, which are chosen from various types of mixtures as shown in Jigs. 4-1, 4-2, An average error between calculated values and experimental values was about 10$. This equation, as well as others (Eq.4-3, 4-5)» seems not to be applicable to the mixture in which the difference between j U , and ^«,is very remarkable (for example the mixture of water and glycerine). System TABLE 4-1 INTERACTION AND VISCOSJTX CONSTANT OgH g + CHxCOOH CgHg + C2% 0H C6 H6 + C6 H5 C H 2C 0 2 C6 H5 H2 0 + CH3 OH H 2 0 + C2 H5 OH H2 0 + C3 H7 OH C H 6 1 3 + G2 Hcr0C 2 H c C H d p + CH3 CO 0 % CH3OH + C2H5OH i - C c H i P + H CONH2 i-CicH110H +C6H5NO2 i-C cH iiO H + CgHcNCCpHcJo C*hM , + C^HkCOoOoHk ^ CgHg + 0HC1 C6H5 + CH3O3 OCI4 + N-CsHwQH CS2 + CH3COOCH3 CS2 + C2H5OH 0&2 + C6H5CH3 C2H2OI4 + O6H5NO2 CO14 + CH3CQOH CCI4 + CH3COOCH3 CCI4 + Gpd4 OCI4 + C2H2CI4 76.5 25 25 25 25 40 40 50 25 25 25 30 25 25 25 0 25 35 25 25 0.752 0.973 0.688 0.474 0.589 0.657 0.538 1.112 O.O326 0.510 0.442 0.958 0.572 0.636 0,464 0.524 *905 1.510 1.721 4.50 0 25 76.5 80 76.5 25 0.730 1.560 O.503 0.559 0.782 60 Lima (8) L°S L o s K m = < * (; X.X, 4 xala Xk M, 4 X^Ma 4-8 1 = M Cuo3 Lo9 H- + K) /j 4“®* Nomenclature & = Density, Sm/cc |H a Viscosity, millipoise M = Molecular weight x = Mole fraction of component K = Constant 1 a Define by Iq* 4-8a Subscript 1,2= Component 1 and 2 Lm = liquid mixture Comment This equation is claimed best for ideal and non ideal solution and suggested^1 lvalue of K was 2*9* lima com pared his equation with other authors, summarized in Table 4-2 Accuracy It is concluded according to lima,^®^ his equation is most accurate* An average error between observed values and calculated values is 2*4$ for all concentration, Table 4-4* Maximum error in calculated viscosities from 61 equation is summerized in Table 4-3. TABLE 4-2 H-OCTYL ALCOHOL + CCL^ MIXTURE Wt % Sachanov ' Alcohol Eq.4-2 Rjachow- lq.4-11 Eq.4-1 Eq.4-8 Observed ____________ sfcy (9) Values 0 9.02 9.02 9.02 9.02 9.02 9.02 10 12.58 11.27 12.33 13.05 11.61 11.39 20 16.80 14.74 16.03 17.83 15.07 14.70 40 27.23 24.83 24.83 29.28 24.49 24.83 50 33.41 31.23 30.14 35.84 30.48 31.24 60 40.23 38.41 36.23 42.72 35.31 38.47 80 55.70 54.80 51.60 57.6 51.05 55.80 100 73.30 73.30 73.30 73.30 73.30 73.30 mximm error is 62 TABLE 4-3 CALCULATED VISCOSITIES FROM LIMA. EQUATION Comoonent 1 Component 2 Max.?£ error Methyl alcohol Carbon tetrachloride 6.4 Ethyl alcohol Carbon tetrachloride 5.3 Propyl alcohol Carbon tetrachloride 7.8 Betyl alcohol Carbon tetrachloride 10.8 Hexyl alcohol Carbon tetrachloride 11.4 Heptyl alcohol Carbon tetrachloride 7.3 Octyl alcohol Carbon tetrachloride 3.7 Decyl alcohol Carbon tetrachloride 5.4 Acetic acid Carbon tetrachloride 4.8 Butyllc acid Carbon tetrachloride 5.0 Oaproic acid Carbon tetrachloride 9.0 Heptylic acid Carbon tetrachloride 15.0 Gapryllc aeid Carbon tetrachloride 3.6 Butylie acid Benzene 5.5 Valeric acid Benzene 4.8 Oaproie aeid Benzene 3.4 Heptylie acid Benzene 4.2 Caprylic aeid Benzene 3.3 63 TABLE 4-4 OBSERVED AID CALCULATED VISCOSITIES Toluene + Ritro Bromobenzene + Toluene + Bromo benzene at nitrobenzene at benzene.at g5°0_______________2S£0______________§2!0________ Mole Pr. Hole Pr. Ritro Ritro Bromo benzene Obs. Calc, benzene Obs. Calc, benzene Obs. Calc. OOO 5.52 5.52 000 10.68 10.68 000 5.52 5.52 • 115 6.22 6.20 .117 11.06 11.40 .129 6.13 5.92 .227 7.05 6.95 .225 11.64 12.0 .267 6.72 6.51 .344 8.03 7.96 .345 12.27 12.75 .379 7.20 7.00 .457 8.94 9.09 .464 12.96 13.59 .505 7.86 7.65 .577 10.30 10.55 .554 13.52 14,17 .629 8.51 8.29 .675 11 .69 11.83 .678 14,65 15.15 .736 9.10 9.01 • 00 —4 13.74 13.80 .790 15.56 16.20 .853 9.81 9.72 • 00 - " 4 O 15.42 15.68 .887 16.73 17.20 .882 9.96 9.84 1.00 18.20 18.20 1.00 18.20 18,20 1 .00 18.20 18.20 Zdanovskii Equation (17) I W - f i Q », ] nomenclature " f \ s t Viscosity poise or centipoise x = Volume fraction of component P = Density, ®a/ee r - : Ouiijstaat foa? 64 t = Constant for components* Subscript 1,2 = Component one and two m = Id quid mixture Note This equation is good for ideal mixture, with no volume and thermochemical mixing effect. Accuracy No calculated values are available in literature' r/ but claimed to be accurate. Shufcla and Bhatnager Equation (10), (11) Nomenclature i \ = Viscosity of liquid, centipoise d = Density of mixture, &®s/ce x = Mole fraction & = Rheochor of component, defined by Iq. 4-1 Oa M = Molecular weight J* d R( + Rjj) 4-10 4-1 Oa * Values of ?/n for NaCl, IH4CI, BaCl2, Ia2C®3, P20c, 06H4(M2)p, CgHcCOOH, SbCl^, CsHcCHpCgHc, published in Zhur Fiz Khlm, 2£, 209-18 (1955). 65 Subscript * = Components 1,2 m = Liquid mixture P s Pseduo molecular weight Comment Shukla and Bhatnagar ^ ^modified the original Smith's equation. This equation is best known for multi component mixture. Calculated and observed values are summer!zed in Table 4-5. Accuracy Shukla and Bhatnagar claimed ^11'average error is only \% and maximum error is 4.4$. Ishikawa (5) - t 1iCl"Zrn) ^ + K Z n , (\-Zm) + K Z « , 4-11 hi ' na, 4-11a nomenclature = Liquid mixture viscosity, centipoise zm= Mole fraction of component 2 4> . nuiaitr = i M = Molecular weight a = Association degree of component K = Defined by Eq. 4-11a 66 Subscript 1,2 = Component 1 and 2 Comment This equation is good for ideal mixture* Ishikawa1s characteristic constant K can he calculated through Eq*4-11 , knowing the value of association degree of com ponents which are given in table 4-6* Accuracy Ishikawa claimed this equation is most accurate for organic mixture and error reported is TABLE 4-5 _____ _ ^ _ __ _ gmy c- c - - _______ 8c ** 3 d M p Calc. Ohs. CTHg-CH^OH-C^Hg 35 .1467 .1506 .7025 .8618 73.10 5*187 5*100 c7H8-C2Hc0H-c6H6 35 .0905 *2524 .6572 .8549 71*20 5*556 5*600 C7H8-G^g-CCl^ 55 .7929 .1385 *0688 .8946 94.34 5.171 5*171 CEUOH-gH^OH-C7H3 35 .1840 .1933 *6227 *8507 63.55 5*385 5*600 CH^OH-CgHgOH-CHoCOCHo 45 .5048 .2541 .2413 .7874 41.83 4.948 4 .8 2 CHoCOCEs-CgHcOH-CH^OH 35 *0771 *0834 .8395 .7916 3 5. 16 5*346 5*25 CXtt^-CgHg-C'pHs 35 .0688 .1385 .9046 .9046 94.34 5-515 5.64 Rheochor of Substances (10) Toluene £ 133*0 Methyl Alcohol ^ 49.9 Benzene = 109*8 Ethyl Alcohol = 79*59 Carbon Tetrachloride = 122.00 Acetone = 85.00 67 TABLE 4-6 ASSOCIATION DEGREES OP LIQUIDS Substance Ass. Degree Substance Ass.Degree N-C6H14 1.0 c9h7n 2.32 CHC13 1.0 c2h5002h5 1.27 CC14 1.0 CH3COOCH3 1.99 0H3I 1.0 CH3C00C2H5 1.85 c2h5i 1.24 C^CdOH 3.01 cs2 1.10 h2o 6.79 C2H5S1 1.29 CH3OH 2.22 CH3COCH3 1.74 C2£^OH 2.44 06^6 1.48 N-C3H7OH 2.87 O6H5OH3 urn isb-C3H70H 2.77 o6H5ei 1.84 N-C^OH 2.57 HCONHg 4.93 iso-C4H90H 2.25 OgH^NOg 1.38 CgHgOH 3.33 o6h5nh2 2.93 Kendall's Equation (7) Nomenclature a t Absolute viscosity, centipoise * = s Hole fraction of component in the liquid 68 Subscript 1.2 = Component 1 and 2 m s Liquid mixture Comment McAllister^claimed that this equation is accu rate* The drawback of this equation is to require the molar volumes of the mixture, component one and two, all to be equal* Accuracy Ho calculated data is available on mixture* Frenkel*s Equation (3) i a-. ^ a. 4-1 * Xt InN, + XaU/^a nomenclature Interaction viscosity, Table 4-1 N = Absolute viscosity, poise or centlpoise x = Mole fraction of component in the liquid Subscript m = Liquid Mixture 1.2 a Component one and two Comment McAllister^claimed that this equation is accu rate* The drawback of this equation is to require the molar volumes of the mixture, component one and two, all 69 to be equal* Accuracy No calculated data is available on mixture* Byring Equation * e.ACVfvr 4-14 Aof = AH*- T as* 4-14a 4-14* M«va = x,n, 4 Nomenclature S> s Kinematic Viscosity, Oentistoke h = Planck Constant, 6.6240x 10"^ Erg*-Sec*/ Molecule N = Avogadro Number, 6.023 x 102- > Molecule/g. mole* M = Moleeular Weight, ®/® mole* AG* = Molal Pree Energy of Activation for Viscosity, Cal*/g. mole* 1= Gas Constant, 1 **9.87 Cal./g. mole.) (°K) T = Absolute temperature AH*= Enthalpy of Activation of Viscosity K Cal./g. Mole. As*= Entropy of Activation of Viscosity K.Cal./g. mole Subscript avg = Average 1,2= Component 1 and 2 70 The Cubic Equation (18) InS) - at?M, + - + l ' n - \ ) ai + otf -!„[*, + X* ■ t£] + 3*f x, u[u * m*)/^7 + 1!^)/31 + y-l 4-15 * . -AS, AW, ^ = m 7 e R 'e RT 4-15e . . m -*4. ^ 5 V3 - 7^; ■ e R- e RT 4-15b ^ . • > 5 . p‘^'- ^ ** * M R G ftT 1 i a, 4-150 ★ k Kn ~ASia • AH u vR = — e r • e -rt 1 liz 4-15d M.n - <3,V1*+ * * » * . 3 4-15e ( V I M, * AM4 ' ■ a . i - —■ ■ ■ . 3 4-15f Nomenclature x = Mole Fraction Mgi = Define by Eq. 4-15f M^2 = 5 Define by Eq. 4-15e ■ > 0 12 = * Define by Eq. 4-15<j "O2I = Define by Eq. 4-15C Other Symbol follows Erring Equation 4-14 71 Figures Figs* 4-4, 4-5, 4-6, represent the calculated and observed values* Fig* 4-7 represents Kinematic viscosity as a function of Temperature for metbanol-Toluene mixture. Comment This is an equation relating the viscosity of a liquid mixture to both composition and temperature* In this equation the only constants not known a priori or not # # f t # involving the pure components are Sj2, H^2, S2j, H2i • If these constants are assumed Independent of temperature, a method is provided for extrapolating viscosity-composi- tion data at two temperature to other temperatures. Accuracy Accuracy of this equation claimed to be *5% TABLE 4-7 ENTHALPIES AND ENTROPIES OF ACTIVATION FOR VISCOSITY FOR THE METHANOL— TOLUENE SYSTEM H* 2.27 KCal/g.mole S»* -0.000369 KCal/g.mole °K Hf2 2.64 KCal/g.mole Sf2= -O.OOO5II KCal/g.mole °K H21 2.04 KCal/g.mole S21= -0.00246 KCal/g.mole Ok hJ 1.88 KCal/g.mole S*= -0.00364 KCal/g.mole Ok These constants were determined for the temperature range from 20° to 60°C. ^ 12and^2|Were determined by the method of least square. OYCLOHEXANE - HEPTANE AND BENZENE - TOULUENE MIXTURE — ’i Fig. 4-4 l i ACETONE - WATER MIXTURE 37.8°C Pig. 4-5 METH4N0L - TOLUENE MIXTURE iiiaait.i&u&aiKiiij Fig. 4-6 KINEMATIC VISCOSITY AS A FUNCTION OF TEMPERATURE [■■I Pig. 4-7 76 Metbanol— Toluene System Accuracy * .5$ (Max,) avg. t .2$ at 37.8°C, Cyclohexane--Heptane System Accuracy t .5$ (Max.) avg. t 2.$ at 37*8°C. " \ ) , a and "^are 0.6272 and .5782 Centistokes respectively. Benzene— Toluene System Maximum Difference .06$ and avg. difference .02$ at 250C. ' S ) la and ' \ ) !M are .6616 and .6493 Centistokes respectively. Acetone— Water System Avg. deviation 6.4$ and Maximum 15.8$ (|*7 ) Pigs. 4-8 to 4-23 are selected observed values of organic mixture. MIXTURE 03? C6H6 + N-C^Gl 77 78 VISCOSITY OP CgHg + N-C^HgBnr MIXTURE MIXTURE OE C6H6 + C6H5C1 80 MIXTURE OF + CzrHcBr Fig. 4-11 ■g-— | MIXTURE OP CgHg + OgHgl i i 109 i i MIXTURE 0%CgH6 + CgH^Up^ MIXTURE OP OgHg + O6H5OCH3 Pig. 4-14 MIXTURE OP CgHg + 0-C6H4(CH3)0H 84 "'1 i Pig. 4-15 MIXTURE 01 CgHg + C^HgOHgOH 86 MIXTURE OF C6H6 + OgHgOHO i i i i I MIXTURE OE O5H6 +(C2H5)2M i ! I 88 MIXTURE OP C^IL- +(CoH,-)o0 m Pig. 4-19 MIXTURE OP CgH6 + C2H5COOH | Pig. 4-20 MIXTURE OP CgH6+ N-G^OH iiairiiiiimiM'iimm Pig. 4-21 91 i MIXTURE Of C6H6 + HC©6C2H5 I fig. 4-22 MIXTURE OP CgHg + OH3COO2H5 Pig. 4-23 9-3 REFERBHQES 1. Arrhenius, S. A.: Z. Physlk Chem. J. ,25. 285 (1887) 2. Dolezalek, F.: Z. Physlk Ohem. 8^ 73 (1913) 3. Frankel, Y.I.: “Kinetic Theory of Liquids” Oxford University Press, London (1946) 4. Gruhberg, L. and A.H. Hissan: nature, 164 79 (1954) 5. Ishikawa, T.s Bull. Ohem. Soc., Japan, 31 524-29, (1958) 6. Kern, D.Q.: “Process Heat Transfer” 161. McGraw-Hill (1950) 7. Kendall, J. and K.P. Monroe's Jacs, 39 1787-1802 (1917) 8. Lima: J. Physical Chemestry 56 1052 (1952) 9. Saehanov, Rjaehowsky: Zhur Fiz Khim 50. 680-83 (1955) 10. Shukla and Bhatnagars J.Phys.Ohem. 22 988, (1955) 11. Shukla and Bhatnagars J.Phys.Ohem. 60 809, (1956) 12. Smith Et Al: J. Am. Ohem. Soc. £i 1736 (1929) 13. Souders, M.Jr.: J.Am.Chem.Soc., 60 154 (1938) 14. Tamara, M. and M. Kuratas Bull. Chem. Soc. Japan, 2£ 32 (1952) 15. Taylor, G.I.: Proc. Hoy. Soc.,London, 138A. 41, (1932) 16. Wijk, Yander A.J.A.: Mature, 128 845 (1936) 17. Zadamovskii: Zhur Fiz Khim, 22., 209-18, (1955) 18. McAllister, E.A.: The Viseosii^y of Liquid Mixture, A.I.C.H.E. J. 6, Sept., (i960) 19* Samuel Glasstone, Text Book of Physical Chemistry, 2nd ed. 493-94, (1958} CHAPTER V I. LIQUID VISCOSITY This chapter contains the selected relationships for the prediction of viscosity of liquids, and some selected relationships for the ehange of viscosity with temperature. Each formula is presented with the author’s name, nomen clature, subscript, figures, comment, and accuracy* Souder’s Equation (23) i / \ a-A-qo logtio/uc) = 10 a = Ift/M Nomenclature fc a Viscosity, Centipoise a = Define by Eq. 5-la ? = Density, grams/ee* M = Molecular weight I = A viscosity constant. Table (5-1) Subscript L = Liquid Accuracy It is noted from recent study that 39 liquids at 14400 were tested and an average and maximum errors of 5-1 5-1, 95 calculated and experimental values were 45$ and 770$ (21)* Acids, alcohols and some halogenated compounds give large errors* The error appears(23)to increase with increasing halogen substitution and with an accumulation of negative groups, especially when they are attached to adjacent carbon atoms* In these two cases, the calculated ftc is greater than that of observed. Comment The obvious drawback of equation is its requirements of very reliable values of and I for accuracy* Because of the double-log function, a small variation in density or in the viscosity constant will cause a considerable change in the value of ftu . Viscosity constant I, may be conveniently calculated from Lageman* s nomograph (Hg.5-1) relating I and molar (9) refraction for seven chemical classes* Banerji proposed that I approximate 9*5 times the molar refraction.^) 4 0 § I * 0 0 0 2 o ° o l i o ® o o o o X ! ft* v» i n ♦ k> H “ ! v L i t i t i n . I . i . i . i i i i l i 1 1 1 1 1 ■ 1 1 1 1 1 . 1 . 1 1 u 1 1 1 . 1 1 1 i i i l ml * \ $ ' y \ T l > s --a'5 • < 0 - X . A 8 < x i 'I 4 U A v<> « 6 » j i U ]c \ O ^ * 7. r v “ i l I J u « J 7 9 X l / l ^ A p v < y < u O * 8 ' J P i o . «*’ 5 < >iu S J - 0 IS y , - i I c r V . c r > jj | o*k»'x £ £ a t« ui o. c ^ £ o i ^ • + r ~ | | " 5 \ l - 1 - i* I t f a » S \ j| 3 s prm 111111 rn 1111111 1111111 ■ 1 1 nrir»i’fi mu'hVuihmii < r t at ao O o o o o o C»x O f - N K» ♦ i n I * —I -M 97 TABLE 5-1 VISCOSITY CONSTITUTIONAL CONSTANT Atomic and Group Values 55.6 2.7 50.2 29.7 57.1 90.0 104,4 80 37 60 79 110 13 10 5 6 8 4 -15.5 -24, -21 Side groups on 6-C ring when Mol. wt, of group less than 17-9 Mol. wt. of group less than 16 -17 Ortho and Para 3 Meta 1 Trouto^s Equation^2 5-2 _ r 3-83Tb \ |U u = c .x p '* t ) Nomenclature j u , = Viscosity, centipoise P = Density, B^/ee M = Molecular weight T = Absolute temperature, °K X Negative group R-Hydrocarbon CHo Hydrogen Carbon Oxygen Hydroxyl, OH COO COOH f a Chlorine Bromine Iodine CRa ECHO RCQOCH-z R2CHx RoCCHjoRo -CH=CHCH2x Double bond Five Carbon ring six carbon ring 98 Subserlr>t b = At normal boiling point L = Liquid phase Comment This equation is restricted to ’ 'normal1 1 liquids only* Accuracy Six liquids were tested and showed average and max imum errors of 67$ and 283%• Erring*s Modified Rate Theory (19) ^ o-4pu (d£gy») ^«-s “pp p 8 76 T For Ideal Vapor AEv = AH* - RT Fo* Q• n ' i Vapor AEv - AHm " pAv nomenclature f t , a Viscosity, Gentipoises P = Density, Srams/ee. M = Molecular weight Aev= Internal energy of vaporisation, Btu/lb. T x Absolute temperature, °K AHv = Latent heat of vaporization, Btu/lb* R = Gas constant, 10*73 (Psia)(ft^/l^*®ol®) (°*0 P = Pressure, Psla* 5-3a 5-3b 99 V = Specific volume, lbs/ft^. Subscript b = At normal boiling point v = Vapor phase Qftiwrnent, This empirical modification of luring*s absolute rate th.eory(19)f in which activation energy for viscous flow is correlated by internal energy of vaporization. This equa tion is good for simple molecules and applied for the range -200°C 300°0. Equation is not generally appli cable to either very long chain molecules (Polymers) or to alcohols and other highly associated liquids for which is greater than 14,000 Btu/lb,mole. Accuracy Wallace R. Gambill^ tested this equation with experimental data for 15 diverse liquids and found average and maximum deviations between calculated and experimental values were 43$ and 138$, Friend and Hargreaves (7) Q- H Q o j U Q 0125 g.4 fu-t- 2?v Nomenclature Jj, = Viscosity, centipoise R = Rheoehor 100 M = Molecular weight ^ = Density, gms/cc V = Volume cc/gm mole Subscript L = Liquid c 3 5 Critical b * At boiling point Comment This equation is used only at normal boiling point. The rheochor is additive and constitutive and relatively independent of temperature over the range in which the liquid structure remains the same. It is also independent of pressure up to about 6000 atm. R is evaluated at the normal boiling point, and 6v may be neglected for that temperature. Rheochor may be calculated by summation of the atomic and group contribution given in Table 5-2. Accuracy This equation was tested with experimental data for 35 organic liquids, gives average and maximum errors of the calculated values of |M.ubwith respect to experimental values of 19% and 97%. For many monomeric liquids 101 TABLE 5-2 BHEOCHOE CONTRIBUTIOU c 12.8 H in 0-H 5.5 H in C-OH 10.0 H in H-02 9.7 H in H-Br 12.6 H in H-I 15.0 CHg 23.8 0 (ethers) 10.0 0 (alcohols) 13.0 0 (Ketone, aldehyde) 13.2 GOO (acids, Esters) 36.0 (attached to aUcyl) 100.6 Cl 27.3 Br 35.8 N 6.6 HH 13.6 eh2 20.6 OHO 39.3 & o ro 38.9 OH 33.0 Thomas Equation (24) (a (l&l ? t .CS)loc * o c - 3 (t- Tr)/f r 5-5 5-5a 102 Nomenclature JsA = Viscosity, centipoises P = Density, grams/cc °C = Defined by equation S-5o- B = Viscosity constant, calculated from Table 5-3 T s Absolute temperature, degree E Subscript T = At reduced condition c = At critical point 1 = liquid Comment The best generalized expression for liquid viscosity when no experimental data is available. This equation applies only when Tr is less than or approximately equal to 0.7. Accuracy Thomas found for seven Mbad actors1 1 (benzene and alkylene halides) a mean error of 39$* For 108 other liquids, the mean error was 5$. In another^21) study of 36 liquids at 128°0, the average and maximum errors of calculated values with respect to experimental values were 21$ and 90$ respefetively. 103 TABLE 5-3 STRUCTURAL CONTRIBUTION Carbon -0*462 Hydrogen 0*249 Oxygen 0*054 Chlorine 0.340 Bromine 0.326 Iodine 0.335 Sulfur 0.043 c6h5 0.385 Double bond 0.478 CO (Ketones, esters) 0.105 ON (cyaaiide) 0.381 G-runbera and Nissan (11) Nomenclature 5-6 /a = Viscosity, Centipoise f = Density, ©Vee T = Temperature, °K M = Molecular weight k = constant for component Subscript c s s At critical point 104 This equation is Intended for the computation of viscosity at critical point for hydrocarbon* Values of K for CH4, C2Hg, C3H8 were 0.00575, 0.00552, 0.0058 respectively. K avg. was 0.00569. By means of K avg., the critical viscosities for straight-chain paraffins from O4 through Cg were calculated. As Grunberg and Mssan^1^ predicted, the computed critical values for I-C4 through I-Cg were constant at 0.023 cp. Accuracy Grumberg-Ni ssan equation for estimating critical viscosities was tested with the experimentally determined values of critical viscosity for methane, ethane and propane, and the error was 0.5?£. The results may be used to estimate viscosities in the reduced temperature range from 0.95-1.0^ Andrade Eqnation (i) nomenclature yU =; Viscosity, Oentipoises A = Atomic weight, Srams/grams-atom T = Temperature, °K V = Atomic volume, ^/gram-atom 0.5 5-7 105 Subscript L = Liquid m = At melting point Comment This equation is good for liquid argon and for liquid metals. It is not good for antimony and bismuth. This equation is best applied for monomolecular (atomic) liquids at freezing point. Accuracy Ho calculated values compared with experimental values are available. Accuracy claimed to be t 8,4$. Arrhenius liquation (16) A Lb= o-27S (fLb) 5-8 nomenclature = Viscosity, eentipoise f = Density, Srams/ec Subscript L s Liquid b = At boiling point Comment This most simple relation appears to be more accurate than any other correlation at normal boiling point. 106 Accuracy Sambill^10^ tested 18 random organic liquids and the average and maximum errors of calculated values and of /^Ib with respect to the experimental values were 21.2^ and 51.6$ which compares favorably with the 20$ average error claimed by Arrhenius *(1^) Sambill. Wallace R. (8) nos V ' 333h M..- 'L <b 5-9 M /333aA/-/ Vt a = ( M A H V b / l7, ssr) - t-8° 5-9a- Nomenclature = Viscosity, centipoises f = Density gr^s/cc at T°K Normal boiling point °K M = Molecular weight latent heat of vaporization at Btu/lb, T = Temperature °K a s ' Defined by equation 5f-9a. Subscript L = liquid b s Boiling point Comment According to Gambill, it is noted that this equation is accurate for organic liquids, with j^tup to 30 Cp. 107 Equation might be most accurate in the 10-80°C range. He derived this equation with combining Dinbigh's equation^ and Palmer's equation.^^ Accuracy For twelve very diverse organic liquids tested, the average and maximum deviations between experimental and calculated values were 33$ and 94$. Data covered the temperature range 0-40°C. II. EFFECT OF TEMPERATURE OH LIQUID VISCOSITY Andrade's Equation (2) - A e 6/r OR. 5-10 nomenclature JU = Viscosity, centipoise. T = Abs.Temperature °K A s Constant of equation B s Constant of equation Subscript L 2 Liquid Comment For solving this equation, one must know two or more values of viscosity at different temperatures. Vis cosity is plotted against i* on send-logarithmic paper* 108 Straight line is obtained for short temperature intervals, Jig. 5-2 shows such plot for Acetic acid, Ethanol and Acetone, This same equation also derived by Reynolds^20^, de Guzman£12^and by Eyiing, et al^^HH) All concluded that this early equation is most accurate for small change in temperature. Accuracy It has been amply demonstrated that the accuracy of equation is as good as that of most experimental data. Correlation within 1-2^ has been obtained for many inor ganic, organic liquids, for fused salts and glasses and many ferric salt solutions. The error^is greatest for highly polar liquids. Bingham and Stookev Equation (4) A + 6 T 5 - 1 , Nomenclature c f ? = Fluidity = Ja = Viscosity, Centipoise T = Absolute Temp. °K A s Characteristic constant for individual compound B s Characteristic constant for individual compound Comment Bingham and Stookey^proposed that this equation 109 pe tlSGOSITI IS’A FUlCfiOM OF TEMPERATURE 110 for an individual compound the fluidity may be best re lated to the temperature. The constant A is usually found to be the same for all members of a series, that is, for a homologous series lines on a graph of ~ plotted against T are straight and have a common intersection (13) point at absolute Zero. Pig. 5-3 illustrates plot of f j i against T for two homologous series,, normal hydro carbons and esters of acetic acid. Once the intersection points have been established by such a plot, the viscosity line of any other member of the same series can be drawn- in based on one point. In this way the curve for decane was drawn-in based on one known viscosity of 0,77 centipoise at 22°C. Modified Andrade* s Equation (1) yUtv'/ 3 = 6eC/vT 5-12 Nomenclature Viscosity, centipoises V = Specific Volume, ee/gram T = i Absolute temp. °K B = Constant of equation C = Constant of equation Subscrint L = Liquid 111 112 Comment To determine value of B and 0 one must know at least two or more values of J U . u and specific volume at different temperature. Andrade claimed this equation is better for large ranges of viscosity. Shrinivasan and Prasad^22^ found the equation is no more accurate than original (2 ) Andrade' ' equation and concluded that its additional complication is unnessary. Accuracy This equation has been thoroughly and successfully tested with data for many organic liquids, for fused salts and mineral oils and for liquid metal, including Ha, K, Uak. It did not give good results for water and tertiary alcohols. Thomas1 Equation^2^ 5-13 v .03 * C+ nomenclature = Viscosity, centipoises \ J - = Specific volume cc/gram T = Absolute temperature, °K C = Constant of equation B = Viscosity constant Subscript c = At critical 113 L = Liquid Comment (24) After examining 123 compound, Thomas; ' found C is equal to 0*0670 for non-associate liquids and B can be calculated from Table 3-1♦ This equation claimed to be most accurate among all equations* Accuracy Thomas1 equation is as accurate as either of Andrades’ equations.123 liquids were tested and variation was only 1$. ( 1 cl When one Value is KnownN When only one value of viscosity is known for a substance, the easiest way to obtain approximate viscos ities at other temperatures is to use the fig* 5-4. Comment (13) fig.5-^- is based on data for a large number of very diverse liquids, mainly organic, including water. Mercury is completely abnormal in relation to this curve, and some pure organics are anomolous. Lewis, W.K. and L. Squires, said deviations, generally do not exceed 20$ and average error less, especially at high viscosities. How to use chart. Locate the known value of viscosity scale and then 114 Ft*7.5 '4 ■ V iscosity, cent i poise. loo loo^—4* — |oo*c— |00*C— — |oo°c VISGO^'lIl VARIES WITH tM pERITURE 1t5 follow the curve the necessary amount— as indicated on the abscissa to reach the temperature at whichIs desired. Eaeh major division on the abscissa scale represents a change 100°C. Pronosed Line Chart (13) / ml = • •5 p ..»1 2 o .0 170 ---- ------ -p^©*l6'7 nomenclature At = Viscosity, Centipoises M a Molecular weight P = Pressure, atmosphere T = Temperature ° K pt _ Vapor pressure mm of Eg Subscriot c = At critical point L = Liquid Comment Pig. 5-5 is a line chart suggested in private com munications between hr. Johnson and hr. Haung from the University of Toronto with J. Czerny. This Fig. 5-5^1 **) serves two purposes, one is to find liquid viscosity and second is to find viscosity changes with temperature. This equation yields viscosity at any temperature. Use of Chart Pseudo Line F»» Alcohol With t^oie Thnn One Cor bon Atom looo 5oo 50 o oK'oi/i o'in 1 0 tnuw k > t o < M az yU, Liquid ViScoSHV - CentipoiS£ PROPOSED AtlQiNMENT chart M - Molecular Weight 117 a* The viscosity of benzene at 26°0. is obtained by the following line ABODE from molecular weight 78.1, to the critical pressure 48*6 atm*, to the critical temperature 562°K, to the vapor pressure at 26°0 is 100 mm Hg and finally down to the estimated viscosity 0.56 centipolse* The observed value at this tempera ture is 0*59 cp* b* The second application is to predict the viscosity of a liquid at a different temperature based on one known value. Thus, suppose the viscosity of methyl accetate at its boiling point is required knowing that at 24®C. When the vapor pressure is 200 mm % the viscosity is 0.37 cp. The sequence of operation is shown by fol lowing the line FGHJ. The predicted boiling point viscosity is 0.26 cp. Observed value at this tempera ture was 0.27 cp. 118 RIFERMCM 1. Andrade, E.N. da O*, Endeavour, X, (195^*-) 2. Andrade, E.N. da 0. Nature, 12£ 309, 582 (1930) 3. Banerji, Current Science, YS 283* (194-7) 4* Bingham, E.G. and S.D.Stookey, J. American Chemical Society 61, 1625, (1939) 5. Dinbigh. K.G. Journal Society Chemical Ind. 65 61-63, {1946) 6. Friend, J.N. Nature, 432, (1942) 7. Friend, J.N., Hargreaves, Fhil. Mag. j§4 643, (1943) 8. Gambill, Wallace R. Chemical Engineering, Jan* 12 128, (1959 9. Gambill, Wallace R. Chemieal Engineering, April 7, 149, (1958) 10. Gambill, Wallace R. Chemical Engineering, Jan. 130, (1959) 11. Grunberg, 1. and A.H. Nissan Nature, 161 170 (1948) 12. Guzman, Jide. Anales Soc. Espan. Fis. Quim, H 353 (1913) 13. Haung, C.J. and A, I, Johnson. Petroleum Refinery, May, 224-5 (1959) 14. Kincaid, J, E., Eyrlng, H. and Steam, A. E. Chemical reviews 28 338-43 (1941) 15. lewis, W. K. and Squires, 1. Oil and Gas Journal Nov. 15, 91-96, (1934) 16. Partington. J. "An Advanced Treatise on Physical Chemistry," Vol.II, Longmans, Green and Co., London (1949) 17. Palmer, G. Industrial and Engineering Chemistry 40 89-92 (1948) 18. Perry, J.H. "Chemical Engineers hand book," 3rd ed. 119 MeGraw Hill Book Go*, Inc*, Hew York (1950) 19* Powell, E. 1*, W. 1* Eoseveare, and H. Byring. Ind. Sagg. Chemistry 33 430-35, (1941) 20* Eeynolds, Phil. Trans., JJI 157, (1886) 21,. Eeid, E. C. and T. K. Sherwood, wThe Properties of Gases and liquids.* 210-13, McGraw Hill Book Co., Inc. New York, (1958) 22. Shrinivasan and Prasad. Phil. Mag., 22 258 (1942) 23* Souders, M. Jacs 60 154 (1938) 24. Thomas, 1. H. J. Chemical Society. 573-79 (1946) 25. Trouton and H. Byring. Ind. and Engg. Chemistry 21 (1941) SUMMARY 120 The object of this work was to survey the recent methods of the prediction of viscosity of mixture of gases, mixture of liquids, and liquid substances* A survey of the literature showed that there are a significant number of equations, and some viscosity data available on gas mixtures, but fewer data on liquid mix tures. Chapter 3, contains 13 different methods for the prediction of viscosity of gas mixtures. Pigs. 3-1 to 3-20 represent the calculated and observed values. It was concluded that Wilke and Herning-Zipperer equations are simple to use. Johnson and Hirschfelder equations are good when no experimental data are available. Chapter 4, contains 15 equations for the prediction of viscosity of liquid mixtures. Mgs, 4-1 to 4-7 repre sent the calculated and observed values. Experimental data are shown in Mgs. 4-8 to 4-23. Finally, it was concluded that the cubic equation is good for non-ideal liquid mixtures. Tamura and Kurata equation is good for general use i.e. for ideal and non-ideal mixtures. Chapter 5 represents 15 different methods for the prediction of viscosity of liquid substanees and the effect of temperature on viscosity. Mgs. 5-1 to 5-5 121 represent the change of viscosity with temperature. It was concluded that Andrade equation is good for the esti mation of viscosity as a function of temperature provided several experimental data points are known. 122 CONCLUSION AND RECOMMENDATIONS I, Gas Mixtures Wilke equation Is the simplest equation for binary, Ideal or non-ideal gas mixtures, of this equation is the function of pure component molecular weights, viscosi ties, mole fractions and densities. Gas viscosity data for mixtures both ideal and non-ideal can be estimated using Wilke equation. Its accuracy is contingent on the accuracy of the density at mixture temperature. Heming and Zipperer’s Correlation often called the “Square Root*' rule provides the easiest way to estimate viscosity of multicomponent mixtures. An average error was 1,9#» therefore this correlation can be used for practical purposes. Gas viscosity data for mixtures at both low and high pressures can be estimated using either Johnson’s equation or Hirschfelder*s equation. In many cases, no experimental data are available, therefore Hirschfelder’s equation can be used to calculate the viscosities of binary mixtures. The computations are somewhat tedious but have been shown to be quite reliable. Johnson’s method saves considerable time of computation but results are somewhat lower accuracy. 123 II* Liquid Mixtures For non-ideal mixtures none of these equations is good. The only equation that gives fair results is the McAllister's cutic equation, which is rather tedious, and even by this equation the maximum error may reach as high as 15*8$. The cubie equation agrees with experi mental data better than any other equations found in the literature. However, it involves complicated thermo dynamic functions of the mixture and pure components, therefore it is not used for rapid estimations. For ideal mixture, most of the equations are good for prediction of liquidimixture viscosity, Kendall- Monroe and Arrhenius equations, however, are simpler to use. The Tamura and Kurata correlation is the best for general use of ideal and non-ideal mixtures. An average error is below 5$ and maximum error is below 10$, there fore this equation can be used for practical purposes. Ill, Liquid Substances Effect of,temperature on liquid viscosity can be estimated by Andrade's equation. Smaller the range of temperature, accuracy will be better. The drawback of this equation is to know viscosities of liquid at any two temperatures. When one value of viscosity is known at any 124 temperature, the desired value at any temperature can he found by using Pig. 5-4. Maximum error does not exceed 20$ and an average erroriis less. for most practical purposes this figure is the best. Prom the recent survey, no formula that has been reported so far is both simple enough for practical purpose and accurate enough to fit all kinds of mixtures, mainly due to the fact that the viscosity of certain types of mixture exhibit a strange behavior which cannot be represented by a simple relationship. This subject demands further investigation. „Uni7ersity_of_.g.ou.t]isrn^Ca3Ul.fornia.

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Creator Gandhi, Shami (author)

Core Title Prediction of viscosity of pure liquids, mixture of gases and mixture of liquids

Degree Master of Science

Degree Program Chemical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

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Advisor Rebert, Charles J. (committee chair), Beeson, Carrol M. (committee member), Lockhart, Frank J. (committee member)

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