INDUSTRIAL A N D ENGINEERING CHEMISTRY
1508
(4) International Critical Tables. Sew York, hicGraw-Hili Hook
Co., 1928. (6) Keyes, D. B., IND.ENG.CHEM.,21, 998 (1929). (6) Perry, J. H., "Chemical Engineers' Handbook," 2nd ed., S e w York, McGraw-Hill Book Co., 1941. (7) Othmer, D. F., and Wentworth, T. O., Trans. Am. I n s t . C h m , Engrs., 36, 785 (1940). ( 5 ) Redlich, O., and Kister, A. T., ISD. ENG.C H E h f . . 40,345 (194s). (9) Robinson, C. S., and Gilliland, E. G., "Elements of Fractiomi Distillation," New York, MoGrawHill Book Co., 1939. (10) Scheibel, E. G., Chem. Eng. Progress, 44,681-90, 771-82 (194s). (11) Scheibel, E. G., U. S. Patent 2,493,265 (Jan. 3, 1950). (12) Smith, D. A., Kuong, J., Brown, G. G., and White, R. R . , Petroleum Refiner, 24,No. 5,296 (1946).
Voi. 42, No. 8
(13) Trimble, N.
M.,and F m a e r , G. L,1x0. Esc;. C H B A I21, ., 1 W : (1929). (14) Varteressian, K. A , and Fenske, bl. R., Ibid., 28,928 (1930). (15) Washburn. E. R., Beguin, A. E., arid Beckord. 0 . P.,.1. -1!ut., Chem. Sac., 61,1694-5 (1939). (16) Wontworth, T. O., Othmer, D. F., arid Pohler, G . lc,,T m r ~ s . A m . Inst. Chem. Engrs., 39,565 (1943).
(17) Wright, TI'. A , . J . Phgs. C l w n . ,37, 233 (1983). (18) Young, S.,German Patent 142,502 (1903). RECEIVEDOctober 3, 1049. Pre3ented before che Division oi Indu-itria! a n d Engineering Chemistry a t the 116th 3Ieeting of the .%hiemc.%sC H I U I C I T . SOCIETY, d t l u n t i c Ciry, S . ,J.
Thermal Conductivity of Gas d
Mixtures ALEXANDER L. LINDSAY AND LEROY A. BROIILEY Unicersity of Culiforniu, Berkeley, Calij.
Abequation is
developed for the thermal conductivity of gaseous mixtures which requires only a knowledge of the pure component conductivities, heat capacity o r iscosity, boiling points, and molecular weights. The equation reproduces 85 mixture conductivities from the li trrature with an average deviation of 1.9%.
type equation so successful for yiscosity when the -1's ;ire r x t i w r simple functions of the purc gns properties, N q u i i t i ~ ) n 1 14 as adopted.
ASSILJEWA ( I d ) in 1904 proposed the following rquation for the conductivity of a mixture of two gas?*:
x.
=
k _1 k? + _-___ XI 1 -I- 9 1 2 % 1 A,, XI
+
(1)
5 2
This was based on kinetic theory and is of the s:me form :is Butherland'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 viscosity and conductivity. Homever, he shows that this does not iit the experimental facta. It is further shown by Ch:qiman :md Cowling (3) that if high accuracy is desired it is necc:ssary to use :t somewhat complicated expression for the coriductivity of a gaseous mixture. However, as Hirschfelder et al. (6) point out, the Eucken (4) assumption (that the conductivit>-of B pure gas is a simple function of the viscosity and heat capacity) is riot rigorously proved as yet, and the best data indicate that it, m:ty be slightly in error. Because of this, Hirschfelder el ui. do not recommend the Chapman and Enskog ( 3 ) equation but instc:id recommend the use of the modified Eucken equation:
it, v a s felt that perhaps if the kinematic viseusit). u, (nionit~titui?i diffusivity) were replaced by the therrnal diffusivity t v L
the cquation
could be used toget,lier with Equation 1 to calculate niistuw inductivitv. This cquation was applied to 49 poitits :iriii 11ta.t value of the constant \vas fourid to bt. 1, I 14. Ttw i.c:sulti!iq equation lor binaries is 30.7 .
l -
0.06-
for gas mixtures as well as for the pure gases. HIT' is a function (6) of the gas and temperature, b u t under. all possible conditiolis it does not differ from unity by more than 0.5%. Although this is a fair approximation for pure gases it fails completely for it misture of gases, as can be seen in Figure 1. Kennard (8) and several other authors recommend the equation k m = ki ($1)'
+ K(ZIZ?)+
kp(~2)'
BtU
hr)(fl)('F)
-
0.04-
-
--I(IRSCHFEL0ER e t . att61 E q 2
(3)
where K is a constant t o be determined for the pair of gasvs in question. Even using a K determined by least squares from the data, the agreement is poor as can be seen in Figure 2. Equation 1 appears to be the most satisfactory and since Buddenberg and Wilke ( 2 ) and Sutherland (10) have found this
4UTHORS E q s 11-16
0
O
U
02 C"2
D A T A O F 188s HIRST"'
0 4 X
L
0 6
M O L E F R A C T I O N H,
O B
10 HZ
Figure 1. Thermal Conductivity o f I I y d r o g ~ l Carbon Dioxide ILixtures d t 0 " c:.
August
INDUSTRIAL AND ENGINEERING CHEMISTRY
19%
0.017
1509
I----I
0.016
AUTHORS E q a
o
11-16
DATA OF G R U S S AND S C H M I C ~ ”
0 012 0 0
A
Figure 2.
I
I
I
I
0 2
0 4
06
OB
X
M O L E FRACTION
I O
HELIUM
AIR
X
He
Figure 3.
Thermal Conductivity of Helium-Argon Mixtures at 0 ” C.
FRACTION AMMONIA
“3
A consideration of these data together with data for hydrogennitrogen, helium-argon, and hydrogen-ethylene indicated that the best agreement was obtained with a = 0.75 and b = 0. It is not implied that these are the best exponents but merely that they are satisfactory, and it does not seem possible to makr n better choice with the data available at the present time. If the final equation is generalized for a gas mixture of n coniponents, as Buddenberg and Wilke ( 2 ) have done for viscosity,
Although this equation reproduced the data with an average deviation of 3.5y0 and a maximum deviation of 11.7%, because of the difficulty of obtaining accurate mass diffusivity data, another equation for the A’s was developed. DEVELOPMENT O F FINAL EQUATION
MOLE
Thermal Conductivity of Air-Ammonia Mixtures at 20” C.
I)
From simple kinetic theory using Sutherland’s model, Sutherland (IO) showed that in Equation 1
or
He also showed that it was necessary t o multiply the righehand 2Mz 514 in order to make viscosity side of Equation 8 by
-4
(&xu;)
xi
L 4 j= 1
here when A i j = AIz
data agree with experiment. Accordingly the equation TVLLS rewritten:
r + C!a&?!(
--“-I
+ +$)
____ (1 “11’21’ (1
(I (1
+$) +$)
The subscripts are merely interchanged for other A values. The viscosity ratio, p ~ / p z ,was evaluated from the Euckcn equation rather than from viscosity data. This was done t o test the accuracy of the equation if viscosity data were not available.
An attempt was made to find the best values of the exponents, a and b. Table I summarizes the results of a comparison of the data of Ibbs and Hirst (7) on hydrogenTABLE I. COMPARISON OF EXPERIXENTAL A N D CALCULATED CONDUCTIVIT~ES carbon dioxide for various values of the exponents, Error, %
km oalod..
a and b, which gave close agreement with the experimental dat,a. All calculations were carried out with slide rule accuracy. A check of nine point.s indicated that the error due to slide rule calculation is about 0.1 %.
=0.75 km (7) b =0 0,01380 ,02420 0,01379 0,02362 0.03265 0.03254 0,0549 0,05533 0.0762 0.07719 (I
zH2 0o ., 1 3 45 2 8
0.60 0.78 0.901
a =0.76 b = 0 -- 20 .. 41 -0.3 4-0.8
+1.3
a =0.5 b = -1 -2.6 -5.8 -3.6 -0.2
+2.6
a =0.82 a =0.5 b 4-0.5 b = -0.76 -1.4 +0.8 -2.1 -0.6 f1.7 +1.8 f3.5 +3.6 +3.1 +3.9
-
b
a = 0.5 = -0.5 +ti. -4
+7.3
+9.9 +8 8 +3.8
INDUSTRIAL AND ENGINEERING CHEMISTRY
1510
TABLE11.
COMPARISON O F
EXPERIRIEKTAL AND C.4LCULATED
Except for hydrogen, deuterium, and helium which have values of the Sutherland constant of 79" K. = 142" R., the Sutherland constants of all other pure gases were taken as
GAS
MIXTURE CONDUCTIVITIES
(Using Equations 11 through 16)
Gas Pair and Lit. Cited
*yz
N2-COz ( 7 )
0
ki In 0,0978 0.00871
Rz-Np ( 7 )
0
0.0978
0.01331
Hz-Nz0 ( 7 )
0
0.W78
0,0092
Mole Fraction of First Gas
Conductivity. B.t.u./Hr./Ft./O F. Exptl. Calcd.
srror, 70 -0.1 -2.4 -0.3 f0.8 f1.3 f2.3 f3.9 +10.0 f8.9.
S = 1.5 T B
(14)
where T B is the boiling point a t 1 atmosphere pressure. This is perhaps not too good an a 5 sumption, but an error of 20% in a Sutherland constant only affects the calculated mixture conductivity about 1 %, hence the simplification is justified. For the collision of unlike molecules the Sutherland constant S12 may be taken as the geometric mean in all cases except where one of the molecules has a strong dipole. I n the latter case the procedure justified by Gruss and Schmick (5) was followed, and the geometric mean was multiplied by 0.733. Thus
0.142 0.355 0.50 0.75 0.901 0.159 0.390 0.652 0,803 0.209 0.386 0.599 0.812 0.2704 0.4537 0.8468 0.9461 0.163 0.272 0.566 0,634 0.794 0.18 0.40 0.60 0.802 0.1698 0,3140 0.5137 0.6110 0.8649
0.01380 0.02420 0.03266 0.0549 0.0762 0.01936 0.03073 0.0469 0.06215 0.01718
0,2038 0.3587 0.6108 0.7804 0.187 0.395 0.496 0.655 0.802 0,264 0.5879 0,7732
0.0101 0.01073 0.01185 0.01267 0.07820 0,0825 0.0848 0.0883 0.0925
0.00987
0.047 0.193 0.496 0,9059 0.9638
0.01072 0.01833 0.0366 0.0817 0.0973 0.0147 0.02503 0.0417 0.0677
0.01162 f8.4 0.01773 -3.2 0.03572 -2.4 0.08601 f0.4 0.09709 -0.2 0.01432 -2.6 -4.9 0.0238 0.04037 -3.2 0,06603 -2.5
0.576
0.01761 0.01808 0.01816 0.01809
-1.2 -0.9 -0.3
COMPARISON OF EQUATION WITH LITERATC'RE DATA
+0.5
Table I1 summarizes the experimental and calculated values of the thermal conductivities of gas mixtures. I n all cases the values of the conductivities of the pure gases were taken from the same authors that report the mixture conductivities.
He-A (11)
0
0,0820 0.00942
Hz-CO ( 7 )
0
0 0978
0 01283
0
0,0978 0.00944
26
0,1058 0.01277
Tz-A (1.9)
0
0.0137
0.00931
Hz-Dz ( 1 )
0
0.1012
0.07455
NHs-CzHa (9)
25
0.01525 0.01277
Hz-COz (9)
25
0.1058
Nz-COS (1.9)
0
0,1008 0.00821
0.1701 0.3698 0.6068 0.8346
NH3-Air (6)
80
0.01742 0.01659
0.216 0.410
0.02590
0.0411 0,0658
0.01795 0.02604 0,0661 0.0711 0.01936 0,02492 0.0436 0,0506 0.0653 0.01767 0.0305 0.04525 0,0663
0.01379 0.02362 0.03254 O.OS533 0.07719 0.01980 0.03192 0.05162 0.06768 0.01709 0.02592 0.04078 0,06370 0.01841 0.02688 0.06877 0.07255 0.01945 0.02467 0.04355 0,04929 0.06602 0.01730 0.02965 0.0448 0.06624
Vol. 42, No. 8
-0.5
0.0 -0.8 -3.2 +2.6 f3.2 f4.8 f2.0 fO.5
-0.1 -0.1 -2.6 f1.1 -2.1 -2.8 -1.0 4-1.4 0.02082 0.01934 -7.1 0.02778 0.02630 -5.3 0,0409 0,03897 -4.7 -5.5 0.04985 0.0471 -1.5 0.0796 0.0784 0.0102 0.01087 0.01198 0.01274 0.07922 0,0846 0.08725 0.09164 0.09561
f1.0 f1.4 fl.1
0.01393 0.01413 0.01482 0.01515 0.01515 0.01537
4-1.4 +2.2 f1.5
fO.5
f1.3 f2.5
f2.9 f3.8 f3.4
0.715
0.01783 0.01824 0.01822 0.01800
NHs-CO (6)
22
0.01344 0.01390
0,220 0.338 0.620 0.790
0.01445 0.01459 0.01438 0.01410
0.01455 0.01469 0.01462 0,01415
f0.7 f0.7 fl.O 4-0.4
NH3--4ir (6)
20
0.01330 0.01452
0.246 0.366 0.608 0.805
0.01527 0.01519 0.01474 0.01392
0.01501 0.01501 0.01465 0.01406
-1.7 -1.2 -0.6
HtO-Air (6)
80
0.01266 0.01659
0.197 0.306 0.444 0.519
0.01730 0.01708 0.01712 0.01697 0.01668 0.01653 0.01627 0.01619 0.01442 0.01430 0.01412 0.01397 0.0137 0.01356 0.01343 0.01339 0.01282 0.01284 0.01636 0.01610 0.01610 0.01586 0.01691 0.01567 0.01553 0.01544 0.01440 0.01437 0.01427 0.01422 0.01410 0.01404 0.01376 0.01376 0.01482 0.01484 0.01571 0.01578 0.01662 0.01665 0.01712 0.01747
-1.3 -0.9
GHz-Air (6)
20
0.01263 0.01452
0.141 0.320 0.536 0.630 0,900
O H r A i r (6)
65
0.01520 0.01621
0.211 0,464 0,646 0.821
GO-Air (6)
18
0.01374 0.01445
CHI-Air (6)
22
0.01747 0.0146
0.108 0.321 0.562 0.978 0.076 0.390 0,700 0.880
fl.O
-0.9
-0.5 -0.8 -1.1 -1.0 -0.3 f0.2 -1.6 -1.5 -1.5 -0.6 -0.2 -0.3 -0.4 0.0
f0.1 f0.4
$0.2 f2.0
813
=
dS,s,
(15)
except when one constituent is strongly polar, then 812
=
0.733 2-
(16)
This latter equation was used for mixtures containing steam or ammonia (Figure 3). Chapman and Cowling ( 3 ) showed that the Sutherland equation is not exactly true and Hirschfelder et al. (6) showed that the constant is not a true constant. However, as a considerable error may be introduced without influencing the mixture conductivity greatly and as the equation is simple to use, it is recommended. In the event it is desired to do so the ratio of the - groups may be replaced by bhe more (1 + 3
refined group, the W2(2)/V function of Hirschfelder (6). Since the equation has only been tested using the Sutherland form, it is recommended that it be used in that form.
CQNC LU SION S
Equation 11, which for binaries reduces t o Equation 1, together with Equations 12 through 16 have been used to calculate the conductivities of 85 different compositions of 16 gas pairs for which experimental data are available in the literature. The average deviation of the calculated from the experimental points is 1.9%. The use of the equations involves only a knowledge of the thermal conductivity of the pure gases a t the temperature together with the heat capacity or viscosity, normal boiling point, and molecular weight of the pure gases. These latter quantities are almost always readily available.
August 1950
I N D U S T R I A L AND E N G I N E E R I N G C H E M I S T R Y
Equation 11 which is the general form of Equation 1 should be tested more thoroughly for multicomponent mixtures. NOMENCLATURE
A = constant (Equation l),dimensionless c p = heat capacity a t constant pressure, B.t.u./lb. O F. D = mass diffusion coefficient, sq. ft./hr. H = function of molecular properties and temperature defined by Hirschfelder et al. (6) k thermal conductivity, B.t.u./hr./ft. ’F. M = molecularweight R = universal gas constant = 1.986, B.t.u./O F. lb. mole S = Sutherland constant (Equation 14). T = absolute temperature V = function of molecular properties and temperature defined by Hirschfelder et al. (6) z = mole fraction CY = thermal diffusivity (Equation 5 ) , sq. ft./hr. y = ratio of specific heat at constant pressure to that a t constant volume p = density, lb. mass/cu. f t . p = viscosity, lb. mass/hr. ft. i=
Subscnpts 1,2, i,j = components in mixture
1511
LITERATURE CITED
(1) Archer, C. T., Proe. Roy. SOC.(London), A165,474-85 (1938). (2) Buddenberg, J. W., and Wilke, C. R., IND.ENG.CHEM.,41,
1345 (1949). 13) . , Chaoman. S.. and Cowling. T. G.. “Mathematical Theory of Nbnuniform Gases,” Cambridge, England, The University Press, 1939. (4) Eucken, A.,Physilc. Z.,14,324(1913). (5) Gruss, H., and Schmick, H., Wiss. VerofentE. Siemens-Konzern., 7,202-24 (1928). (6) Hirschfelder, J. O., Bird, R. B., and Spotz, E. L., J . Chem. Phys., 16,968 (1948);Chem. Revs., 44,205 (1949). (7) Ibbs, T. L.,and Hirst, A. A., Proc. Roy. Xoc. (London),A123,13442 (1929). (8) Kennard, E. H.,“Kinetic Theory of Gases,” New York, McGraw-Hill Book Co.,1938. (9) Kornfeld, G., and Hilferding, K., 2. Physik. Chem. BodensteinFestband, 792-800 (1931). (IO) Sutherland, W., Phil. Mug., 40,421 (1895). (11) Wachsmuth, J., Phgsik Z., 7,235 (1908). (12) Wassiljewa,-4., Ibid., 5,737(1904). (13) Weber, S., Ann. Physik, 54,481-502 (1917). RECEIVED January 9, 1950. This work was performed under the auspiees of the Atomic Energy Commission.
HOT WIRE ANEMOMETRY Solution of Some Dificulties in Measurement of Low Water Velocities GEORGE B. MIDDLEBROOK’ AND EDGAR L. PIRET University of Minnesota, Minneapolis, M i n n .
I
T h e performance of a hot wire anemometer depends conditions, such as with water N A previous paper (7) upon the rate of cooling of an electrically heated fine wire from city mains, it is quickly data on the fundamental heat transfer characteristica placed in the flowing stream. A heated wire is capable found that difficulties arise of electrically heated fine of sensitivity at low rates of water flow and interferes but which do not ordinarily appear in 1 a b o r a t o r y work . little with the flow of the liquid. This type of anemometer wires in water have been preis particularly adaptable to studies of flow near walls and This paper describes some of sented and d i m e n s i o n l e s s for theoretical studies of turbulency and eddy diffusion. these difficulties and certain correlations of a generalized A previous paper has presented precise heat transfer data methods of reducing them nature have been developed. on fine wires. In practical usage as an anemometer, diffiwhich should be helpful to The present work presents culties arising from bubble formation on the wire have investigators considering the some practical aspects of the use of fine wires for velocity problem of using a n electrilimited the applications of the instrument. The investigation reported herein showed that many of these diffimeasurement purposes. cally heated fine wire for culties were caused by electrolysis. A thin coating of inmeasuring low velocity water sulation can be used to prevent this effect, but the use of a flow. PREVIOUS WORK F o r e x p e r i m e n t a l purshort wire is a simpler solution to the problem. Worthington and Malone poses, it is desirable to use (IO) obtained s o m e h e a t an instrument which can measure low velocities without interfering with the flow of transfer data by rotating the wire in a tank of water. Their the fluid. It is also desirable t o be able t o investigate flow results were erratic, below 0.2 foot per second, because of conditions 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 I n 1924, Davis (1) published considerable data on distilled water, paraffin oil, and transformer oil. The wire used was held velocities of 0.2 foot per second (6). Also, it is not particularly in a vertical position and rotated in a n annular trough. H e suited for memuring local velocities where the flow is disturbed. The hot wire anemometer can measure low velocity water flow relates t h a t the difficulties resulting from the accumulation of dirt on the wire were eliminated by filtering all liquids and below the range of the Pitot tube. Dryden has shown how useful fine wire anemometers can be in studies of turbulency cleaning the test wire frequently with another wire. Piret, James, and Stacy (7’) used a stationary horizontal wire and eddy diffusion (2). placed in front of a nozzle designed to give a uniform flow past In this work water velocities ranging from 0.4 t o 0.02 foot per the wire. It was found t h a t very erratic results were obtained second were readily measured with fine wires, Most previous unless distilled water was run through the apparatus. When work investigating the heat transfer characteristics of fine wires ordinary city water was used, poor results were obtained because in water has been done under ideal conditions; distilled water, free from suspended matter, and wires sufficiently long to reduce bubbles collected on the wire a t low wire temperatures of 80” to 100’ F. It was thought that the bubbles were due t o gases end effects are desirable for such work. However, when these hot wires are used to measure water flows under more practical being liberated from the water by the heating action of the wire. Recirculating the water in the apparatus caused the water to lPresent address, National Aniline Division, Allied Dye become cloudy and gave erratic readings. T o avoid these difCorporation, BuEalo, N. Y.
