thermal conductivity of mixture

7/28/2019 Thermal Conductivity of mixture

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1508 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Voi. 42, No. 8(4) International Critical Tables. Sew York, hicGraw-Hili Hook(6) Keyes,D.B., IND.ENG. H E M . ,1, 998 (1929).(6) Perry, J . H., "Chemical Engineers' Handbook," 2nd ed.,Sew(7) Othmer, D. F.,and Wentworth, T. O., Trans. Am . Inst. C h m ,(5) Redlich, O., and Kister, A. T., SD.ENG. HEhf.. 40,345 (194s).(9) Robinson, C. S., and Gilliland, E. G., "Elements of Fractiomi

(10) Scheibel,E. G.,Chem. Eng. Progress, 44, 681-90, 771-82 (194s).(11) Scheibel,E. G., U. S.Patent 2,493,265 (J an. 3, 1950).(12) Smith, D. A., Kuong, J ., Brown, G. G., and White, R. R.,

Co., 1928.York, McGraw-Hill Book Co., 1941.Engrs., 36, 785 (1940).Distil lation," NewYork,MoGrawHill Book Co., 1939.

Petroleum Refiner,24, No. 5, 296 (1946).

(13) Trimble, N. M. , nd Fmaer, G. L , x0. Esc;. C H B A I . ,1, 1 W:(14) Varteressian, K.A , andFenske, bl . R., Ibid., 28, 928 (1930).(15) Washburn. E. R., Beguin, A . E., arid Beckord. 0 . P. , 1. -1 ut.,(16) Wontworth, T. O., Othmer, D. F., arid Pohler, G. l c , , Tmr~s(17) Wright, TI ' . A,. .Phgs. Clwn. ,37, 233 (1983).(18) Y oung,S. ,German Patent 142,502 (1903).

(1929).Chem. Sac . ,61,1694-5 (1939).Am. nst. Chem.Engrs., 39, 565 (1943).

RE CE IV E DOctober 3, 1049. Pre3ented before che Division oi Indu-itria!and Engineering Chemistry atthe116th 3Ieeting o the .%hiemc.%s H I U I C I TSOCIETY,tluntic Ciry, S . J .

Thermal Conductivity of GasdMixtures

ALEXANDER L . LINDSAY AND LEROY A. B R O I I L E YUnicersityof Culiforniu, Berkeley, Calij.

Abquation is developed fo r the thermal conductivityof gaseous mixtures which requires only a knowledge ofthe pure component conductivities, heat capacity or i s-cosity, boiling points, and molecular weights. Theequa-tion reproduces85mixture conductivities from the li trra-ture with an average deviation of 1.9%.ASSILJEWA ( I d ) n 1904 proposed the following rqua-tion for the conductivity of a mixture of two gas?*:

k k? (1)=- _1-_-___1+A ,, -XIX 5 21 -I- 9 1 2 %

This was based on kinetic theory and is of the s:me form :isButherland's (10)equation for the viscosity of a gaseous mixture.The simplified kinetic theory of gases leads, according to Weber( I S ) , to the conclusion that the A's are the same for viscosityand conductivity. Homever, he shows that this does not iitthe experimental facta. It is further shown by Ch:qiman :mdCowling (3) that if high accuracy is desired it is necc:ssary touse :t somewhat complicated expression for the coriductivity ofa gaseous mixture. However, as Hirschfelder et al. (6) pointout, the Eucken (4)assumption (that the conductivit>-of B puregas is a simple function of the viscosity and heat capacity) is riotrigorously proved as yet, and the best data indicate that it, m:tybe slightly in error. Because of this, Hirschfelder el ui . do notrecommend the Chapman and Enskog (3) equation but instc:idrecommend the use of the modified Eucken equation:

for gas mixtures as well as for the pure gases. HIT' is a function(6) of the gas and temperature, but under. all possible conditiolisit does not differ from unity by more than 0.5%. Although thisis a fair approximation for pure gases it fails completely for itmisture of gases, as can be seen in Figure 1.

Kennard (8)and several other authors recommend the equationkm =ki ($1)' +K(ZIZ?)+kp(~2)' (3)

where K is a constant to be determined for the pair of gasvs inquestion. Even using a K determined by least squares from thedata, the agreement is poor as can be seen in Figure2.

Equation 1 appears to be the most satisfactory and sinceBuddenberg and Wilke (2) and Sutherland (10) have found this

type equation so successful for yiscosity when the -1's ;ire rx t wrsimple functions of the purc gns properties, Nqui it i~)n 1 14 asadopted.

it,vas felt that perhaps if the kinematic viseusit). u , (nionit~titui?diffusivity) were replaced by the therrnal diffusivity tvL

the cquation

could be used toget,lier with Equation 1 to calculate niistuw in-ductivitv. This cquation was applied to 49 poitits :iriii 11tatvalue of the constant \vas fourid to bt. 1 I 14. Ttw i.c:sulti!iqequation lor binaries is30. .

l0. 06- -B t U

h r ) ( f l ) ( ' F )0. 04 -- U T H O R S

E q s 11-16- - I ( IRSCHFEL0ER

e t . a t t 6 1 E q 20 D A T A O F 188s

H I R S T " 'O U LC "2 X M O L E F R A C T I O N H, H ZFigure 1. Thermal Conductivity of I I ydrog~l -Carbon Dioxide I L ixtures dt 0" c:.

0 2 0 4 0 6 O B 10


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August 19% I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 1509

0 I I I I0 0 2 0 4 0 6 O B I OA X MOL E F R A C T I O N H E L I U M H eThermal Conductivity of Helium-Argonigure 2. Mixtures at 0 C .

I-----.017I.016A U T H O R S E q a 11-16

o DATA O F G R U S S A ND S C H M I C ~

X M O L E F R A C T I O N A M M O N I A 3

Mixtures at 20 C.

0 012A IR

Figure 3. Thermal Conductivity of Air-Ammonia

Although this equation reproduced the data with an averagedeviation of 3.5y0and a maximum deviation of 11. 7% becauseof the difficulty of obtaining accurate mass diffusivity data,another equation for theAswas developed.

A consideration of these data together with data for hydrogen-nitrogen, helium-argon, and hydrogen-ethylene indicated thatthe best agreement was obtained with a = 0.75and b = 0. I tis not implied that these are the best exponents but merely thatthey are satisfactory, and it does not seem possible to makr nbetter choice with the data available at the present time.If the final equation is generalized for a gas mixture of n coni-ponents, as Buddenberg and Wilke (2) have done for viscosity,

DEVELOPMENT OF FI NAL EQUATION I)From simple kinetic theory using Sutherlands model, Suther-

land ( I O) showed that in Equation 1

Healso showed that it was necessary to multiply the righehandside of Equation 8by(&xu;)n order to make viscositydata agree with experiment.

2Mz 5 1 4

Accordingly the equation TVLLS rewritten:r

+ C ( ____1 + 1 1 2 1 (1 +$)

(I +$)(1 +$)An attempt was made to find the best values of the exponents,aandb.

or

-4 - - - I xi L 4j = 1here whenA i j =A Iz

The subscripts are merely interchanged for other A values. Theviscosity ratio, p ~ / p z ,was evaluated from the Euckcn equationrather than from viscosity data. This was done to test theaccuracy of the equation if viscosity data were not available.

Table I summarizes the results of a compari-son of the dataof Ibbs and Hirst (7)on hydrogen-carbon dioxide for various values of the exponents, TABLE. COMPARISONF EXPERIXENTALND CALCULATEDONDUCTIV IT~--moalod.. Error, %a and b, which gave close agreement with the (I =0.75 a =0.76 a = 0 . 5 a =0.82 a = 0 . 5 a = 0. 5experimental dat,a. zH2 km (7) b =0 b = 0 b = - 1 b 4- 0 . 5 b = -0.76 b = -0 .5

0.142 0,01380 0,01379 -0. 1 -2. 6 -1. 4 +0.8 +ti . -4-5.8 -2.1 -0.6 +7.3ll calculations were carried out with slide rule o,358 ,02420 0,02362 -2.40.78 0,0549 0,05533 4-0.8 -0.2 f3.5 +3.6 +8 8that the error due to slide rule calculation is 0.901 0.0762 0.07719 +1.3 +2.6 +3.1 +3.9 + 3 . 8

accuracy. A check of nine point.s indicated 0.60 0.03265 0.03254 -0. 3 -3.6 f 1. 7 +1.8 +9.9about 0.1%.


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1510 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol. 42, No. 8

TABLE1. COMPARISON O F EXPERIRIEKTALND C.4LCULATED GA SMIXTURE CONDUCTIVITIES(Using Equations 11 through 16)

Gas Pairand Lit. CitedN2-COz (7)

Rz-Np (7)

Hz-Nz0 (7)

He-A (1 )

Hz-CO (7)

Tz-A (1.9)

Hz-Dz ( 1 )

NHs-CzHa ( 9)

Hz-COz ( 9)

Nz-COS (1.9)

NH3-Air (6 )

NHs-CO ( 6)

NH3--4ir (6)

HtO-Air (6)

GHz-Air ( 6)

OHrAir ( 6)

GO-Air (6)

CHI-Air (6 )

*yz0

0

0

0

0

0

26

0

0

25

25

0

80

22

20

80

20

65

18

22

ki I n0, 0978 0.00871

0.0978 0.01331

0. W78 0, 0092

0, 0820 0. 00942

0 0978 0 01283

0, 0978 0. 00944

0, 1058 0. 01277

0. 0137 0.00931

0. 1012 0.07455

0.01525 0. 01277

0. 1058 0. 00987

0, 1008 0. 00821

0. 01742 0. 01659

0.01344 0. 01390

0.01330 0. 01452

0.01266 0. 01659

0.01263 0.01452

0. 01520 0. 01621

0.01374 0. 01445

0.01747 0.0146

MoleFractionof FirstGas0. 1420. 3550.500. 750. 9010. 1590. 3900. 6520, 8030. 2090. 3860. 5990. 8120. 27040. 45370. 84680.94610. 1630. 2720. 5660, 6340. 7940. 180. 400. 600. 8020. 16980, 31400. 51370. 61100. 86490, 20380. 35870.61080. 78040.1870.3950. 4960. 6550.8020, 2640. 58790, 77320. 0470. 1930. 4960, 90590. 96380. 17010. 36980. 60680.83460.2160. 4100.5760.7150, 2200. 3380.6200.7900. 2460. 3660.6080.8050.1970. 3060.4440. 5190. 1410. 3200. 5360. 6300, 9000. 2110, 4640, 6460. 8210.1080. 3210. 5620. 9780. 0760.3900, 7000. 880

Conductivity.B.t.u./Hr./Ft./O F. srror,Exptl. Calcd. 70

0. 01380 0. 013790. 02420 0. 023620. 03266 0. 032540. 0549 O OS5330. 0762 0. 077190. 01936 0. 019800. 03073 0. 031920.0469 0.051620. 06215 0. 067680. 01718 0. 017090.02590 0. 025920. 0411 0. 040780,0658 0, 063700. 01795 0. 018410. 02604 0. 026880, 0661 0. 068770. 0711 0.072550. 01936 0. 019450, 02492 0. 024670.0436 0.043550, 0506 0, 049290.0653 0. 066020. 01767 0. 017300. 0305 0.029650.04525 0. 04480, 0663 0. 066240. 02082 0. 019340. 02778 0. 026300, 0409 0, 038970. 04985 0. 04710.0796 0. 07840.0101 0. 01020. 01073 0. 010870. 01185 0.011980.01267 0. 012740. 07820 0.079220, 0825 0, 08460. 0848 0. 087250.0883 0. 091640. 0925 0. 095610. 01393 0. 014130. 01482 0. 015150. 01515 0. 015370. 01072 0. 011620. 01833 0. 017730. 0366 0. 035720. 0817 0. 086010. 0973 0.097090. 0147 0. 014320. 02503 0.02380. 0417 0.040370. 0677 0, 066030.01783 0.017610. 01824 0.018080. 01822 0.018160. 01800 0. 018090. 01445 0. 014550. 01459 0. 014690. 01438 0. 014620. 01410 0, 014150. 01527 0. 015010. 01519 0. 015010. 01474 0. 014650.01392 0.014060. 01730 0. 017080. 01712 0.016970. 01668 0. 016530. 01627 0. 016190. 01442 0.014300. 01412 0.013970. 0137 0. 013560. 01343 0. 013390. 01282 0.012840. 01636 0. 016100. 01610 0. 015860. 01691 0.015670. 01553 0. 015440. 01440 0. 014370. 01427 0.014220. 01410 0. 014040. 01376 0.013760. 01482 0. 014840. 01571 0.015780. 01662 0. 016650. 01712 0.01747

- 0. 1- 2. 4-0. 3f0.8f 1. 3f2. 3f 3. 9+10. 0f 8. 9.- 0 . 50. 0- 0. 8- 3. 2+2.6f 3. 2f 4. 8f 2. 0f O . 5- 0. 1-0. 1- 2. 6f 1 . 1- 2. 1- 2. 8- 1. 04- 1. 4- 7. 1- 5. 3- 4. 7- 5. 5- 1. 5f 1. 0f1. 4f l . 1fO.5f1. 3f 2 . 5f 2. 9f3. 8f 3. 44-1. 4+2. 2f 1. 5f 8. 4- 3. 2- 2. 4f 0. 4-0. 2- 2. 6- 4. 9- 3. 2- 2. 5- 1. 2- 0. 9- 0. 3+ 0 . 5f 0 . 7f 0. 7f l . O4- 0. 4-1. 7- 1. 2- 0. 6f l . O-1. 3-0. 9-0.9-0. 5- 0. 8- 1 . 1-1. 0-0. 3f0. 2- 1. 6-1. 5- 1. 5- 0. 6- 0. 2- 0. 3- 0. 40.0f0 .1f0 .4$0.2f 2. 0

Except for hydrogen, deuterium, and heliumwhich have values of the Sutherland constant of79" K. = 142" R., the Sutherland constantsofall other pure gases were taken asS = 1.5TB (14)

where TB s the boiling point at 1 atmospherepressure. This is perhaps not too good an a5sumption, but an error of 20% in aSutherlandconstant only affects the calculated mixtureconductivity about 1%, hence the simplificationis justified.For the collision of unlike molecules theSutherland constant S12may be taken as thegeometric mean in all cases except where one ofthe molecules has a strong dipole. I n thelatter case the procedure justified by Gruss andSchmick (5) was followed, and the geometricmean was multiplied by0.733. Thus

8 1 3 = dS,s, (15)except when one constituent is strongly polar,then

8 1 2 = 0.733 - (16)This latter equation was used for mixtures con-taining steam or ammonia (Figure3).Chapman and Cowling (3) showed that theSutherland equation is not exactly true andHirschfelder e t al. (6) showed that the constant snot a true constant. However, asaconsiderableerror may be introduced without influencing themixture conductivity greatly and as the equa-tion is simple to use, it is recommended. Inthe event i t is desired to do so the ratio of the1+- groups may be replaced by bhe more

refined group, the W2(2)/V function of Hirsch-felder (6).Since the equation has only been tested usingthe Sutherland form, it is recommended that itbe used in that form.

( 3COMPARISON OF EQUATION WITHLITERATC'RE DATA

Table I1 summarizes the experimental andcalculated values of the thermal conductivitiesof gas mixtures. I n all cases the values of theconductivities of the pure gases were taken fromthe same authors that report the mixture con-ductivities.

CQNCLUSIONSEquation 11, which for binaries reduces toEquation 1,together with Equations 12 through16 have been used to calculate the conductivi-ties of 85 different compositions of 16 gas pairsfor which experimental data are available inthe literature. The average deviation of thecalculated from the experimental points is1.9%. The use of the equations involves onlya knowledge of the thermal conductivity ofthe pure gases at the temperature togetherwith the heat capacity or viscosity, normalboiling point, and molecular weight of the puregases. These latter quantities are almost al-ways readily available.


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August 1950 I N D U S T R I A L A ND E N G I N E E R I N G C H E M I S T R Y 1511LITERATURE CITEDEquation 11which is the general form of Equation 1should

be tested more thoroughly for multicomponent mixtures. (1) Archer, C.T., Proe. Roy. SOC. London),A165, 474- 85 (1938).( 2) Buddenberg, J . W., and Wilke, C. R., IND. NG. HEM. ,41,NOMENCLATUREA = constant (Equation l ), imensionlesscp = heat capacity at constant pressure, B.t.u./lb. O F.D = mass diffusion coefficient,sq . ft./hr.H = function of molecular properties and temperature definedk i= thermal conductivity, B.t.u./hr./ft. .M = molecularweightR = universal gas constant =1.986,B.t.u./O F.lb. moleS = Sutherland constant (Equation 14).T = absolute temperatureV = function of molecular properties and temperature definedz = mole fractionCY = thermal diffusivity (Equation5 ) ,sq. ft./hr.y = ratio of specific heat at constant pressure to that at con-p = density, lb. mass/cu. f t.p = viscosity, lb. mass/hr. ft.Subscnpts

by Hirschfelder etal. (6)

by Hirschfelder etal. (6)

stant volume

1,2, ,j =components in mixture

1345 (1949).13) Chaoman. S.. and Cowling. T. G.. Mathematical Theory of, Nbnuniform Gases, Cambridge, England, The UniversityPress, 1939.( 4) Eucken, A. , Physilc.Z. , 4, 324 1913).(5) Gruss, H., and Schmick, H., Wiss. VerofentE. Siemens-Kon-(6) Hirschfelder, J . O., Bird, R. B., and Spotz, E. L., J . Chem( 7) Ibbs,T.L.,and Hirst, A. A. , Proc. Roy. Xoc. (London),A123, 134-(8) Kennard, E. H. , Kinetic Theory o Gases, New York, Mc-( 9) Kornfeld, G., and Hilferding, K., 2. P hysik. C hem. Bodenstein-

zern., 7, 202-24 (1928).Phys., 16, 968 ( 1948) ; Chem Revs., 44, 205 (1949).42 (1929).Graw-Hill BookCo.,1938.Festband, 792-800 ( 1931) .( I O) Sutherland,W., Phil. Mug., 40, 421 (1895).(11) Wachsmuth, J. , Phgsik Z., 7, 235 (1908) .(12) Wassiljewa, 4., bid., 5, 737 1904) .(13) Weber, S. , Ann.Physik, 54, 481- 502 (1917).

RECEIVEDanuary 9, 1950.of the Atomic Energy Commission.This work was performed under the auspiees

HOT WIRE ANEMOMETRYSolution of Some Dificulties in Measurement of Low Water Velocities

GEORGE B. MIDDLEBROOK AND EDGAR L . PIR ETUniversityof Minnesota, Minneapolis, M inn.

N A previous paper (7) T he performance of a hot wire anemometer depends conditions, such as with waterI ata on the fundamental upon the rate of cooling of an electrically heated fine wire from city mains, i t is quicklyheat transfer characteristica placed in the flowing stream. A heated wire is capable found that difficulties ariseof electrically heated fine of sensitivity at low rates of water flow and interferes but which do not ordinarily ap-wires in water have been pre- li ttle with theflowof the liquid. This type of anemometer pear in 1abora tory work .sented and dimensionl ess is particularly adaptable to studies of flow near walls and This paper describes some ofcorrelations of a generalized for theoretical studies of turbulency and eddy diffusion. these difficulties and certainnature have been developed. A previous paper has presented precise heat transfer data methods of reducing themThe present work presents on fine wires. In practical usage as an anemometer, diff i- which should be helpful tosome practical aspects of the culties arising from bubble formation on the wire have investigators considering theproblem of using an electri- limited the applications of the instrument. The investi- use of fine wires for velocitycally heated fine wire for gation reported herein showed that many of these diffi- measurement purposes.measuring low velocity water culties were caused by electrolysis. A thin coating of in-flow. sulation can be used to prevent this effect, but the use of aFor experi mental pur- short wire is a simpler solution to the problem.poses, it is desirable to use Worthington and Malonean instrument which can ( I O) obtained some heatmeasure low velocities without interfering with the flow of transfer data by rotating the wire in a tank of water. Theirthe fluid. It is also desirable to be able to investigate flow results were erratic, below 0.2 foot per second, because ofconditions very close to a wall. The Pitot tube has been swirling currents set up by the wire and holder.used frequently for such purposes, but its use is difficult below In 1924, Davis ( 1) published considerable data on distilledvelocities of 0.2foot per second (6). Also, it is not particularly water, paraffin oil, and transformer oil. The wire used was heldsuited for memuring local velocities where the flow is disturbed. in a vertical position and rotated in an annular trough. HeThe hot wire anemometer can measure low velocity water flow relates that the difficulties resulting from the accumulation ofbelow the range of the Pitot tube. Dryden has shown how dirt on the wire were eliminated by filtering all liquids anduseful fine wire anemometers can be in studies of turbulency cleaning the test wire frequently with another wire.and eddy diffusion (2). Piret, James, and Stacy (7) used a stationary horizontal wireIn this work water velocities ranging from 0.4to 0.02foot per placed in front of a nozzle designed to give auniform flow pastsecond were readily measured with fine wires, Most previous the wire. It was found that very erratic results were obtainedwork investigating the heat transfer characteristics of fine wires unless distilled water was run through the apparatus. Whenin water has been done under ideal conditions; distilled water, ordinary city water was used, poor results were obtained becausefree from suspended matter, and wires sufficiently long to reduce bubbles collected on the wire at low wire temperatures of 80end effects are desirable for such work. However, when these to 100 F. It was thought that the bubbles were due to gaseshot wires are used to measure water flows under more practical being liberated from the water by the heating action of the wire.

Dye Recirculating the water in the apparatus caused the water toCorporation, BuEalo, N. Y. become cloudy and gave erratic readings. To avoid these dif-

PREVI OUS WORK

lPresent address, National Aniline Division, Allied

thermal conductivity of mixture

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