I
W. J. BREBACHl and GEORGE
THODOS
The Technological Institute, Northwestern University, Evanston, 111.
Viscosity-Reduced State Correlation for Diatomic Gases Correlations are presented which make possible the prediction of viscosity for gases and liquids
THEORETICAL Viscosity of Gases at
relationships defining the transport properties of gases in terms of intermolecular forces and a set of collision integrals have been developed by Chapman, Enskog, and others. For the Lennard-Jones potential, the collision integral, f 2 ( * j 2 ) * [ T N ]has been evaluated by Hirschfelder, Bird, and Spotz (20, 27) and can be used in the expression
for defining viscosity to the first approximation in terms of centipoises. The force constant, € / K , is used to produce the normalized temperature, T N =
T
I:,
which in turn defines the collision
integral, Q ( * J ~ ) *[ T N ] , from tabulated values (22). Values of u and E / K are presented for a number of diatomic gases in Table I. For the kth approximation, the viscosity coefficient is given by : [Plh
= fP' [PI 1
(2)
where ffi(k) is a viscosity correction factor. For the third approximation, values of f P ( * ) are presented elsewhere (22). 1
Present address, Visking Co., Chicago,
Ill.
Moderate Pressures
This investigation was limited to the viscosity of diatomic gases. Reliable high pressure viscosity data for gases d are not plentiful and are restricted to the more common substances; viscosity data at atmospheric pressure are more abundant and cover extended ranges of temperature. A review of experimental data for nitrogen (28, 3 7 , 43) indicates that viscosity is pressure-insensitive from 3 mm. of mercury up to 10 atm. This range of pressures has been arbitrarily designated as the moderate pressure region. Viscosity data at atmospheric pressure for 11 diatomic gases are plotted against reduced temperature in Figure 1 where essentially parallel relationships exist, particularly for hydrogen, nitrogen, oxygen, carbon monoxide, and nitric oxide. This behavior suggests that the theorem of corresponding states should apply to this family of substances. Following an approach analogous to that used for thermal conductivities (50), an arbitrary reduced temperature, TR = 1.0 'was selected to define pTc*, the viscosity at the critical temperature and atmospheric pressure. Viscosity ratios, p * / p ~ ~ *were , calculated for each sub-
stance and plotted against reduced temperature, TR, to produce a single relationship (Figure 2) which is linear for values of TR _< 1.0 and TR 2 3.5. The resulting straight lines can be expressed in equation form as : p*/pT,*
p*/p~,* =
T ~ 3 0 . 9 7 9for
=
T E 5 1.0
(3)
1.196 TRc.659 for T R 2 3.5
(4)
For the diatomic gases in Table I, viscosities at moderate pressures were calculated with Equation 2. With these values, viscosity ratios were plotted against reduced temperature (Figure 3). Values at the critical temperature, ,UT,*, were calculated with Equation 2 and are presented in Table I. With the exception of hydrogen, these calculated values are in good agreement with those resulting from the experimental data. The value for hydrogen calculated with Equations 1 and 2, pTC*= 156.5 X centipoise, disagrees with the value of pTe* = 178 X 10-5 centipoise resulting from the experimental data of Figure 1. Calculated ratios for the different gases (Figure 3) show excellent agreement up to TR = 3.0. Above this temperature, some scattering is noted. For hydrogen, these ratios are higher than those calculated for the other diatomic gases at the same temperature. This disparity may be attributed to the lower . . . . . .
,
I
,
r; - -
T.,OK.
4172 1181 Hydrogen Chloride,UCI Hydrogen led,d. HI
3245 4241 826
575 2 2900r 880 lO-$p 1330 417 2 1870
.
-
1181
,I' Reduced Ternpemivre, TR = T/Tc
Figure 1. Viscosity-reduced temperature relationships for diatomic gases at approximately atmospheric pressure are essentially parallel
870
33 3 178 3245 1580 4241 2620 826 3120 180 1245 126 2 865 1548 1170
.
I
I
2
I
I
3
.4 5 6
l.lllllli.lll
810
,
I
2
,
I
3
,,,,,,,,,,., 4 5 6 0 1 0
, I
,
,
20
1 -
-
,
,
I
, X,40
Reduced Temperature, T, = T/ Tc
Figure 2. Viscosity ratios produced from literature data and reduced temperature for some diatomic gases produce a unique relationship VOL. 50. NO. 7
JULY 1958
1095
30
I
I
I
,
,
,Illlillll,l,,l
,
,
I
-v-rmT
,
pFc* value calculated with Equation 2 and the force constants, u = 2.968 A. and E / K = 33.3' K. The curve of Figure 3 is similar to that in Figure 2 and was again linear for values of T R 5 1.0 and T R 2 3.5. For these regions, the calculated viscosity ratio of Figure 3 can be expressed in equation form as:
20 -
10? 6:
%I+&
-
:;
p*/p~,* =
3-
c +
B
2-
.t ::I
h
-
f
I 8I LLOO L .
Bromine, Br,
+
= - a! >
Carbon Monoxide, CO Chlorine, CI, Fluorine. F. Hydrogen, H, Hydrogen Chloride,HCI Chloride, HCI Hydrogen Iodide, HI Iodine I Nitric 'O:ide, NO NiIric'O:lde, Nitrogen. N, Oxygen, O w e n , 0,
A 7
5-
* x m A
3
ci
0 0
IV ,
I
.I
.2
I
.3
I
I I O I ~ ! ~ 1 , 1 , 1
4 5 6
8 10
,
,
2
,
I
3
, ,,,,,,,,,,,.,
4 5 6
,
8 10
2943x 10'6cp 874 1843 867 1565 1588 2622 3749 1245 860 1167
5752 1330 4172 1181
333 3245 424 I 4241 826 180 1262 1548 20
,
2_
1
-
,
, ,,,,,, 30 40 W d O O
Reduced Temperature, T. = T/T,
A 20,000 -
NITROGEN __-
,opoo
lwabah! ((/or) 0 Michelr and Gibson (gori d. Rudenko (liquid) v Rudenko and Shubnikow (liquid)
Figure 3. Viscosity ratios at moderate pressures YS. reduced temperature for some common diatomic gases show excellent agreement
0
8,000i L 6 000
$000
"
.-,x
a%
5
T Rfor T R 5 1.0
(5)
UT,* = 1.241 T R ~for .T R~2 ~3.5~ ~
-
I
Effect of Pressure om Viscosity
In previous thermal conductivity studies on gaseous and liquid argon (50), temperature and pressure were eliminated as variables through the correlation proposed by Abas-Zade ( 7 ) involving residual thermal conductivity, k - k*, and p , the density. The quantity k - k* represents the isothermal increase in thermal conduc-
0
I,OOO= _.____...._.._._..__.-.-....--
Points calculated w i t h Hirschfelder, Curtiss, a n d Bird equation
800; 600: 500:
(6)
Equations 5 and 6 differ slightly from the corresponding relationships of Equations 3 and 4 resulting from the experimental data. This slight difference results from the use of force constants (22) that are not completely consistent with experimental viscosity values utilized in these studies.
+
6
4poo
,g
2,000
Itferbeeh and Paernel
3,000
400-
4 Figure 4.
Residual viscosity-density relationship for nitrogen in gaseous and liquid states i s continuous
*I -
200 I
30 40 ,,,,,,,,,, 3 4 5 6 8 IO ;(,,t
.Ol
02
04 .06.08.1
p,
2
,
, 2
Density, grarns/cc.
HYDROGEN
,
] 3
F i g u r e 5. R e sidual viscositydensity relationship for oxygen in gaseous and liquid states i s also continuous
I
/
P
20
grom/cc
p,
Density, gf'OmS/CC.
NITROGEN
pc = 1825 X A
lttwbesh and Paomal
p , Density, grorns/cc. 1096
INDUSTRIAL AND ENGINEERING CHEMISTRY
4 Figure 6.
Residual viscosity-density relationship for hydrogen in gaseous and liquid states shows continuous behavior Figure 7. Reduced state viscosity correlation for nitrogen, from literature values, i s presented for sub a t m o s p h e r i c and elevated pressures and for liquid state b
d centipoisas
10 8
P
'f B 2
E2
6 54-
3: 2-
10:
8+
:-
543010.10.4
08
T.,
Reduced Temperoture
tivity due to pressure effects above 1 atmosphere. These studies showed that a single continuous relationship for the gaseous and liquid states resulted when k - k* was plotted against density on log-log coordinates. In this investigation, a similar method of correlation involving density and residual viscosity, p - p*, for both gases and liquids was adopted. Viscosity data for nitrogen, including values for the liquid state and for the gaseous state a t high pressures, were used to calculate residual viscosities which were plotted against corresponding densities of nitrogen to produce the continuous relationship in Figure 4. The gaseous data of Michels and Gibson (43) are in good agreement with those of Iwasaki (28) and the curve through them, when extended, passes through the viscosity points representing the liquid region of nitrogen (53, 54). From the relationship of Figure 4 and the critical density of nitrogen, p o = 0.311 gram per cc., the residual viscosity at the critical point is ( p - p * ) , = 1960 X centipoise. Using this value, along with the viscosity of nitrogen a t atmospheric pressure and TR = 1.0, p*To = 865 X 106 ' centipoise obtained from Figure 1, the viscosity at the critical point, po, is calculated to be 1825 X 10-5 centi1825 X 10-6 poise. The ratio of viscosities -__ 865 X 10-6 produces a value of 2.1 1 as the constant for the relationship pc
= 2.11 pTc*
(7)
Equation 7 can be assumed to apply to the diatomic gases. Following a similar approach, residual viscosities and densities for oxygen were correlated to produce the relationship of Figure 5. The viscosity data of Kiyama and Makita (39) have not been included because they scattered considerably and on the basis of this study were assumed to be unreliable. A critical density for oxygen, p c = 0.430 gram per cc., produces from Figure 5 a critical residual viscosity, ( p - p*)c = 1300 X 10-6 centipoise. With this value and pro* = 1170 X 10-6 centipoise, the viscosity for oxygen at the critical point is calculated to be pa = 2470 X 10-6 centipoise. Similarly,, the ratio of vis2470 X 10-6 produces the value cosities 1170 of 2.11 as was the case for nitrogen. This consistency infers the applicability of Equation 7 to diatomic gases exhibiting normal behavior. A similar treatment of the viscosity data for Xydrogen (77, 18, 27, 29, 44) produced the relationship in Figure 6, from which a critical residual viscosity, ( p - p * ) , = 132 X 10-6 centipoise, corresponds to the critical density, pr =
0.031 gram per cc. With this value and pFe* = 178 X 10-6 centipoise, the viscosity at the critical point was calculated to be, pc = 310 X centipoise. Unlike the values for nitrogen and oxygen, the viscosity ratio for hydrogen 178 X lo-' = 1.742, was found to be 310 lo-6
an inconsistency expected in view of the significant quantum deviations encountered with hydrogen. T o account for these deviations, de Boer and Bird (8) proposed the dimensionless quantum mechanical parameter, h* = h / v 2 / &
(8)
Hirschfelder, Curtiss, and Bird (22) showed that this parameter accounts for quantum deviations associated with hydrogen. The critical viscosity value, pc = 1825 X centipoise, has been used to reduce experimental values of nitrogen, which are correlated with reduced temperature and pressure (Figure 7). Both high pressure gaseous data (28, 43) and saturated liquid data (53,54) are included, as well as the subatmospheric data of Johnston, Mattox, and Powers (37), which extend down to absolute pressures of 0.01 mm. of mercury (PR= 0.004 X lo-*). Their data permit construction of a reduced state correlation of viscosity at subatmospheric pressures approaching rarefied gas conditions. The resulting subatmospheric isobars become nearly independent of temperature with decreasing pressure and systematically converge to the atmospheric isobar.
T.,
Reduced Temperature
Figure 8. Density temperature-pressure relationships for nitrogen Reduced State Correlation for Nitrogen
The residual viscosity-density correlation for nitrogen of Figure 4 and the moderate pressure viscosity plot presented in Figure 1 permit the calculation of viscosities under all conditions of temperature and pressure for both gaseous and liquid states. An enlarged plot of the density correlation for nitrogen (Figure 8) was the result of a compilation of data obtained from the litera-
) I +
'Q
8
5 p; T,-126ZoK.
Reduced Temperature, T.
1825110'5c~ntipo~se~ P , = 3 3 5 ~ 1 m . p,=O3ll g n m k
=L Tc
Figure 9. Reduced state viscosity correlation is specific for nitrogen but applicable to other diatomic gases VOL. 50, NO. 7 I
0
JULY 1958
1097
Application to Other Diatomic Gases
To apply Figure 9 to other diatomic gases, critical viscosities must be assigned to these substances. From the viscosity values, /.LT,*,in Figure 2, critical viscosities have been calculated with Equation 7 to produce the values in Table I. Figure 9 will not produce viscosities for hydrogen consistent with experimental values. T o bring hydrogen in line, it was necessary to use the values ,uc’ = 2.11 (178 X = 375.6 X centipoise, P,‘ = 15.68 atm. for the high pressure region (44) and P,’ = 20.6 a h . for subatmospheric pressures (26). Because adjusted critical values are required, hydrogen does not follow the theorem of corresponding states applicable to the other diatomic gases. This anomalous behavior can be attributed to quantum deviations of considerable extent for hydrogen (22). The critical viscosities calculated with Equation 7 have been used to produce the expression =
T,
,
Normalized Temperature
Figure 10. Normalized viscosity correlation for nitrogen i s equivalent to correlation of Figure 9 Hirschfelder, Curtiss, and Bird
ture (2, 4, 73, 15, 27, 40, 47, 45, 47, 49, 57). T o complete the regions of this plot, densities were calculated with the compressibility factors of Nelson and Obert (46). Densities corresponding to a specified temperature and pressure produced residual viscosities, p - p * , from Figure 4. For these conditions, a viscosity value for nitrogen is readily produced from the residual viscosity and the corresponding moderate pressure value, p * , obtained from Figure 1. These calculated values have been reduced with the critical viscosity, p c =
Gas Br2
co Cln
F2 H2
HCI
HI I2
NO N2 0 2
1098
Critical Constants Mol. Wt. T,,’ K. P,, atm.
method
1825 X 10-5 centipoise to produce the reduced state correlatiori for nitrogen (Figure 9). This correlation is similar to the generalized viscosity correlation developed by Uyehara and Watson (77). For pressures above 1 atm., Uyehara and ,Watson constructed their correlation from experimental viscosities of a number of substances. The reduced state correlation in Figure 9 is specific for nitrogen and includes the effect of pressure ranging from PE = 0.003 X IO-* to Pz = 40 and the effect of temperature, which extends u p to T R = 100.
575.2 133.0 417.2 118.1 33.3
102.0 34.5 76.1 25.0 12.80
4.268 3.590 4.115 3.653 2.968
520 110 357 112 33.3
2900 880 1870 870 178
2943 874 1843 867 156.5
6119 1857 3946 1836 375.6
36.46 127.93 253.84 30.01 28.02 32.00
324.5 424.1 826 180 126.2 154.8
81.5 81. 92.1 64. 33.5 50.1
3.305 4.123 4.982 3.470 3.681 3.433
360 324 550 119 91.5 113
1580 2620 3720 1245 865 I170
1588 2622 3749 1245 860 1167
3334 5528 7849 2627 1825 2470
I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY
x
10-4
%J;i73 (Pc)”/3 Tal’@
(9)
for the diatomic gases. Uyehara and Watson (77) presented a similar expression having a constant of 7.70 instead of 7.40. The critical viscosities resulting from Equation 9 and that of Uyehara and Watson produced average deviations of 2.4 and 4.27,, respectively, from values calculated with Equation 7. More exacting critical viscosities can be produced directly from a single viscosity value and the reduced state correlation of Figure 9. Normalized State Correlation for Nitrogen
Recent theoretical contributions of Hirschfelder, Curtiss, and Bird dealing with transport properties (22) in the dense phase region involve the use of dimensionless temperature and pressure ratios. To produce these quantities, they define the reference states for temperature. pressure, volume. and the transport properties in terms of the LennardJones potential force constants E and v .
Table I. Fundamental Constants for Diatomic Gases Lennard-Jones PTo* x loi, Potential Centipoises p c x 105, Parameters Exptl. Calcd. Centipoises a, A C/K, O K . (Fig. 1) (Eq. 2 ) (Eq. 7)
159.83 28.01 70.91 38.00 2.016
7.40
Literature Sources
(10, 62) (SO, 59, 6 3 ,
74,7 6 )
(11 , 65, 7 5 )
(3% (9, 19, 24, 26,92, 36,37, 48, 58-53, 56-68,70, 73-76)
(64) (66) (11)
(62, 76)
(9,26,92, 98, 66, 56,59>60,63, 70, 72, 74 76) (23,25, 32, 60, C S , 7 0 , ‘74> 75)
DIATOMIC OASES Following a dimensional analysis approach, it i s possible to show these reference states of temperature, pressure, and viscosity (6, 7, 22) to be
Table! II. Reference Temperatures, Pressures, and Viscosities Resulting from Lennard-Jones Potential Force Constants for Diatomic Gases Tc,u, Pep, p e , u x 105,
Pe>u= 0.9869 X 10'8 a,:
co Clz
atmospheres (11)
Fz Hz HCI HI 12 NO Nz
= 12.885 X
105
dMe centipoises u2 '
(12)
The reference temperature, T,,,, and and calculated reference values of P,,,, pe,ofor the substances considered in this study are presented in Table 11. With Equations 10, 11, and 12, it now becomes possible to define the normalized temperature, pressure, and viscosity as TN = T/T,.,, PN = P/Pf,u, and p N = p/p,,,. These dimensionless ratios have been used to produce the normalized viscosity correlation (Figure 10) which was constructed through transformation of values obtained from Figure 9 with Equations 10, 11, and 12. Therefore, Figure 10 is equivalent to Figure 9 and is presented to include these normalized states resulting from classical kinetic theory.
0 2
Table 111.
No. of
co Clr
Fz Hz HCl HI
'
*
e
Viscosities have been calculated for several substances with Figure 9 ; values are included for monoatomic and polyatomic gases (5, 74, 30, 42, 57, 75). Table I11 shows comparisons between experimental and calculated values. For gaseous and liquid nitrogen, the average deviation was 1.2370 for approximately 200 points investigated. For the other diatomic gases, the average deviation was 1.85%. Values of pGfor other gases were calculated from atmospheric viscosity values, and the corresponding reduced values were obtained from Figure 9. Using these values, viscosities obtained from Figure 9 were in good agreement with experimental values for gaseous helium, neon, and argon. The large deviation of 9.8y0 for carbon dioxide may be explained by the electron distribution which seems to cause two carbon dioxide molecules to have a preferred orientation on collision. Excessive deviations from the limited experimental data for liquid oxygen and hydrogen were found. These deviations may be due to the paramagnetic nature of liquid oxygen and the quantum deviations associated with hydrogen. I n addition, viscosities of several gaseous mixtures (28, 37, 59, 63) have been calculated using Figure 9 and the pseudocritical concept of Kay (34). Comparisons (Table IV) indicate a consistent excessive devigtion for mixtures containing hydrogen, helium, and
28 20 9 172 6 6
7
NO Nz
21 198 48
0 2
O K .
Atm.
Centipoises
520 110 357 112 33.3 360 324 550 119 91.5 113
911 323.8 698.0 313.0 173.5 135.9 630 606 388.0 249.9 380.6
23,953 6,520 14,225 7,401 1,408 15,874 18,129 22,794 7,513 5,656 7,726
Experimental and Calculated Viscosities Are in Good Agreement for Helium, Neon, and. Argon
Gas points Monoatomic 33 A 7 He Ne 9 Diatomic 27 Bre
I2
Application of Reduced State Correlation
u, A. 4.268 3.590 4.115 3.653 2.968 3.305 4.123 4.982 3.470 3.681 3.433
Gas Brz
(10)
~
Gaseous State Pressure, Temp., OK. atm.
Saturated Liauid No. of Temp., points O K.,
Deviation, % Gas Liquid
273-323 180-371 80-717
9-1800 1 1
4.00 1.47 1.47
286-867 81-524 289-772 87-273 14-1,098 294-523 293-523 379-796 119-371 90-1,695 79-1 9 102
1
1.90 2.34 2.61 2.93 2.97 2.62 0.86 0.59 0.39 0.99 1.35
1
1 1 0.000007-1899
7
18.2-20.6
6 6
69.1-90.1 83.6-90.2
1 1 1 1 0.00001 - 966 1-152
201
9.00 46.8
Triatomic
coz
Polyatomic CH4 CsHs
9
263-313
10-119
9.8
12 12
298-498 298-500
1-340 1-340
5.33 4.77
Table IV. Experimental and Calculated Viscosities for Gaseous Mixtures Show a Consistent Excessive Deviation for Hydrogen, Helium, and Neon No. of Temp., Pressure, Deviation, Mixture Points O K. Atm. % Hz-Nz HrCO Nz-CO Hz-02 Nz-02
co-02
He-Ar Ne-Ar He-Ne Air Air
8 5 4 6 6 6 6 6 6
25 44
1
195-423 195-373 300-550 300-550 300-550 300-500 293-473 293-473 293473 323-423 80-306
neon. Hydrogen-free diatomic mixtures showed good agreement. The excessive deviations encountered with hydrogen, helium, and neon can be explained by their respective quantum mechanical parameters. Using Equation 8, A* values were calculated for the inert and diatomic gases considered in these studies. The highest A* value was 2.74 for helium, followed by 1.80 for hydrogen, and 0.58 for neon. For the remaining substances, the calculated A* values ranged from 0.235 for nitrogen to 0.024 for iodine. T h e accuracy of Figure 9 in predicting viscosities in the high pressure region
7.41 8.82 0.49 9.28 0.45 0.61 18.6 7.12 7.08 1.04 2.51
1 1 1
1 1 1 1 1
22-196
0.000004-1
has been compared with the generalized reduced viscosity correlation of Uyehara and Watson (77) and the reference viscosity plot of Comings, Mayland, and Egly (72). In these comparisons, available high pressure viscosity data for nitrogen, oxygen, and hydrogen have been used to produce the following average percentage deviations : Deviation, % ' Present Hydrogen Nitrogen Oxygen
(71)
(12)
work
7.19 5.59 9.57
1.22 1.13
1.47 0.79 0.62
~~
VOL. 50,
NO. 7
JULY 1958
1099
Rev. Phys. Chem. Japan, 12, 49
An extension of these comparisons includes carbon dioxide (57), methane, and propane (5) and shows that the Uyehara-Watson correlation is capable of predicting viscosities a t high pressures more accurately than Figure 9 for these polyatomic gases. The normalized viscosity correlation of Figure 1 0 and the reference states presented in Table I1 have been used to calculate viscosities for the diatomic gases treated in this study. Values calculated for nitrogen from subatmospheric conditions to pressures as high as 966 atmospheres yielded a n average deviation of approximately 1yo when compared with experimental values. For hydrogen, a t atmospheric and high pressure conditions, the deviation was 1.59%; however, considerable deviations were noted for the subatmospheric region. For the remaining nine diatomic gases, deviations resulting from experimental data a t atmospheric pressure averaged O.86Y0. Because of these favorable comparisons, the normalized correlation of Figure 10 is presented in conjunction with the reduced state correlation of Figure 9.
References
Nomenclature
Physikalische Chemische Tabellen,” I11 Erg. Bd., S. 189, 1935. Golubev, I., Dissertation (Russ.) Moscow. Nitroeen Institute. 1940. (19) Gunther, P., Z.”fihysik. Cheh. 110,
fp(3)
= viscosity correction factor for the
h
= Planck’s constant, 6.624 X 10-27
third approximation
k
=
k*
=
m
M
P Pc PN
= = = =
erg second thermal conductivity. , , cal./sec. cm. OK. thermal conductivitv of Eases a t atmospheric pressure, cal.,/sec. cm. OK. mass of molecule, grams molecular weight pressure, atmospheres critical pressure, atmospheres ,
Y
P/ T3 reduced pressure, P/P,
= normalized pressure,
T T,
= = absolute temperature, O K. = critical temperature, O K.
T N
= normalized temperature, T /
PR
2 K
TR = reduced temperature, T / T , E
= maximum energy of attraction
for Lennard-Jones potential, ergs K = Boltzmann constant, 1.38047 X 10-’6 erg/ K. A* = quantum mechanical parameter, dimensionless p = viscosity, centipoises p* = viscosity of gases a t atmospheric pressure, centipoises pLc = viscosity a t the critical point, centipoises p~ = normalized viscosity, UT^*- viscosity of gases a t atmospheric pressure and critical temperature, centipoises p = density, grams per cc. pc = critical density, grams per cc. u = collision diameter for LennardJones potential, A. Q(’V’)*[TNJ = collision integral function
(19521.
(1) Abas-Zade, A. K., Zhur. Eksptl. i Teoret. Fiz.23, 60 (1952). (2) Baly, E. C. C., Donnan, F, G., J . Chem. Soc. 81, 907 (1902). (3) Becker, E. W., Misenta, R., Z. Physik 140. 535 11955). (4) Benedh, Manson, J. Am. Chem. Soc. 59, 2224 (1937). (5) Bicher, L. B., Katz, D. L., IND.ENC. CHEM.35, 754 (1943). (6) Bird, R. B., University of Wisconsin,
Madison, Wis.,. private communica_ tion, 1957. (7) Boer, J. de, Physica 14, 139 (1948). (8) Boer, J. de, Bird, R. B., Phys. Rev.
83, 1259 (1951). (9) Bop630i: J H., Jr., Zbid., 35, 1284
Braunei’H., Basch, R., Wentzel, W., Z. physik. Chem. A137, 176 (1928). Braune, H., Linke, R., Zbid., A148, 1- 0- 5- 1 1 9,. ~ Comings, E. W., Mayland, B. J., Egly, R. S., “Viscosity of Gases at High Pressures,” Univ. Illinois, Eng. Expt. Station, Bull. 354 (1944). Dodge, B. F., Davis, H. N., J . Am. \ - - - -
Chem. Soc. 49, 610 (1927). Edwards, R. S., Proc. Roy. SOC. (London) A119, 578 (1928). Friedman, A. S., White, David, J . Am. Chem. SOC. 72, 3931 (1950). Galkov, G. I., Gerf, S. F., J . Tech. Phys. (U.S.S.R.) 11, 613 (1941). Gibson, R. O., “Landolt-Bornstein,
626 (1924). (20) Hirschfelder, J. O., Bird, R. Spotz, E. L., Chem. Revs. 44, (1949). (21) Hirschfelder, J. O., Bird, R. Spotz, E. L., J . Chem. Phys. 968 (1948). (22) Hirschfelder, J. O., Curtis, C.
B., 205
B.,
I
,
(33) (34) (35) (36) (37) (38) . . (39)
1 100 INDUSTRIAL AND ENGINEERING CHEMISTRY
(1952).
Mathias, E., Onnes, H. K., Crommelin, C . A . , Verslag K. Akad. Wetenschappen23,983 (1914).
Michels, A., Botzen, A., Schuurman, W., Physica 20,1141 (1954). Michels, A., Gibson, R. O., Proc. Roy. Soc. (London) A134, 288 (1931).
Michels, A., Schipper, A. C . J., Rintoul, W. H., Physica 19, 1011 (1953).
Michels, A., Wassenaar, T., de Graaff, W Prins, Chr., Physica 19, 26 (lG3). Nelson, L. C., Obert, E. F., Trans. Am. SOC.Mech. Engrs. 76, 1057 (1954). Onnes, H. K., Dorsman, C., Holst, G., Verslag K. Akad. Wetenschappen 23, 982 (1914).
Onnes, H. K., Dorsman, C., Weber, S., Zbid., 22, 1375 (1913).
Otto, J., Michels, A,, Wouters, H., Physik. Z. 35, 97 (1934).
Owens, E. J., Thodos, George, A.Z. Ch.E. Journal. 3, 454 (1957).
Perry, J. H., “Chemical Engineers’ Handbook,” p. 206, McGraw-Hill, New York, 1950. Rankine, A. O., Proc. Roy. Soc. (London) 88, 575 (1913).
Rupenko, N. S., Zhur. Eksptl. i l’eoret. Fiz. 9, 1078 (1939). Rudenko, N. S., Shubnikow, L. W., Physik. 2. Sowjetunion 6, 470 (1934).
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RECEIVED for review June 1, 1957 ACCEPTED December 30, 1957 An 81/% X 11 inch reproduction of Figure 9 is available on request from the Chemical Engineering Department, Northwestern University, Evanston, 111.
