M a y 1950
INDUSTRIAL AND ENGINEERING CHEMISTRY ACKNOWLEDGMENT
The authors wish to express their appreciation for the work of Mary Humphries, who was responsible for the distillation of the azeotropes herein reported. LITERATURE CITED
Deanesly, R. ht. ($0Shell Developmellt CO.),u. S. Patent 2,290,636 (July 21, 1942). Ibid., 2,360,655 (Oct. 17, 1944). Denton, W. I., and Bishop, R.B. (to Socony-Vacuum Oil Co., Inc.), Ibid., 2,450,637 (Oct. 5, 1948).
885
(4) Dijck, W. J. van (to Shell Development Co.), Ibid., 2,023,375 (Dee. 3,1935). * (5) Doss, M. P., ”Physical Constants of Hydrocarbons,” 4th ed., Texas Co., ,May 14, 1943. (6) Lecat, &I. M., Compt. rend., 222, 733 11946). (7) Mair, B. J., and Streiff, A . J., J . Eesearch A’atl. B w .Standards, 24, 395 (1940). (8) Mair, B. J., and Willingham, C. D., Ibid., 21, 535 (1938). (9) Pratt. H. R. C. (to Imoerial Chemical Industries). U. S. Patent 2,305,106 (Dee. 15, i942). ~ ~ ~F. D ~ , , ~i ~~B i ~ standards, , ~ 1 . . cirC, 461 (1947). (11) Teter, J. W., and Merwin, W. J. (to Sinclair Refining Co.), U. S. Patent 2,411,346 (Nov. 19, 1946). RECEIVED Xovember 21, 1949.
Viscosity of Highly
&
Compressed Fluids CORRELATION WITH CRITICAL CONSTANTS L. GRUNBERG Dr. Rosin Industrial Research Company, Ltd., Wembley, Middlesex, England
A. H. NISSAN‘ The Bowater Paper Corporation, Ltd., Northfleet, Kent, England T h e viscosity of highly compressed fluids is correlated with critical constants in the form of a nomograph on which the ratio of the viscosity of the compressed fluid, ~ p to , the viscosity of the fluid at moderate (atmospheric) pressure, T*, is given as a function of the reduced temperature, T, = TIT, and of the reduced density, pr = p , p c . The density was chosen as the second parameter in order to make interpolation in the critical region possible. The nomograph is applicable to highly compressed gases and to liquids near the critical point. The accuracy obtainable is of the order of *lo%. A summary of the theoretical
T
HE experimental determination of the viscosity of highly
compressed gases and of liquids near the critical point is very difficult, and comparatively few results have so far been published in the literature. Comings and Egly ( 3 ) correlated the available data and plotted a chart on which the ratio of the viscosity of the compressed gas to the viscosity of the gas at atmospheric pressure is shown as functions of the reduced pressure and the reduced temperature. Cyehara and Watson (26) also correlated the available data, expressing the results in the form of a generalized chart on which the redured viscosity-that is, the viscosity divided by the viscosity at the critical pointis shown as functions of the reduced temperature and the reduced pressure. Both these charts suffer from the same defect. For conditions under which experimental results are most difficult to obtain, namely in the critical region, the charts are both inaccurate and difficult t o interpolate. This disadvantage derives from the fact that the pressure was chosen as one of the parametera. There are several indications that by choosing the density instead of the pressure as the second parameter, this difficulty n-ould not arise. Foremost among these is the fact t h a t the viscositpdensity isotherms in the critical region are not as steep as the viscosity-pressure isotherms. Phillips (f2), for example, deduced from his experimental results on carbon di1 Present address, Bowaters Development and Research Ltd., Northfleet, Kent, England.
considerations leading to the nomograph are given. It is assumed that in a highly compressed fluid transfer of momentum may occur by two mechanisms, one translational, gas-type, and the other vibrational, liquid-type. The following equation is obtained:
is a function independent of the nature of the substance and dependent only on the reduced density.
f(p,)
oxide, that in the critical region the viscosity varies with the square of the density. Watson, Wien, and Murphy ( 1 7 ) found that in the critical region the kinematic viscosity of petroleum fractions vaiies little with the pressure, whereas the absolute viscosity varies considerably. Investigations on the relation between the viscosity and the density showed this latter property to be of great importance particularly for highly compressed gases and for liquids near the critical point. On the basis of these investigations, which are reported in this paper, the authors were able to obtain a general equation in which the ratio of the viscosity of the compressed gas or of the liquids near the critical point to the viscosity of the gas or vapor at moderate pressure and at the same temperature is given as a function of the reduced density and of the reduced temperature. Stated in general terms
where qp = ?A
the viscosity under pressure
= the viscosity of the vapor (not liquid)
pheric) pressure
P r = P _ = pc
density under test critical density
at moderate (atmos-
886
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 42, No. 5
- /e2
- 1.3
- 1.4
-le5
T -
Tc
-1.6
- 1.7 -1.8 -I*9
--2.0
-3 I -2.2 -2.3 -28 4
435i,3
-2.5 -2.6 -2.7 -2.8 -2.9 --3* 0
-3.5
--4*0 -4.5 --5#0
-5.5 Figure 1. Nomograph for Evaluating qPIq.4 from T , and
P,
May 1950
INDUSTRIAL AND ENGINEERING CHEMISTRY
a87
EXAMPLE 3. h'itronen a t 25" C. and
Carbon dioxide Carbon monoxide Ethylene Methane
Oxygen Sitrogen
4~, 12 12' 9 3 3 6 5 5. 5 4
Liquid and gas
Gas
1 ".,
20 30 32 35 100 200 40 25 75 125 225 50 25 50
75
8
0.41-1.80 0.39-1.73 0.37-1.72 0.35-1.64 0.54-1.44 0.75-1.21 0.41-1.45 0.50-1.42 0.52-1.16 0.42-0.98 0.48-0.76 0.44 0.48-1.74 0.67-1.74 0.61-1.66
C 4.1
++ 88.. 62
+f 30 .. 51 ++ 3.7 2.9
-18.1 -20.3 4-17.6 4-17.9 f14.3 f 7.6 4.5 2.6 -24.2 -24.2 -24.1
-11.3 -15.9 +10.2 f13.8 +10.7 f 6.5 3.2 2.6 9.0 -13.7 -13.1
f 5.7
-
fore T, = 2.36 and pr = 1.15. From the nomograph, ?C = 1.85. Since the TA viscosity of nitrogen a t 25' C. and a t atmospheric pressure is 1.76 x 10-4 'poise, the viscosity a t 387.6 atms., according to the nomograph, is 3.26 X poise. The observed value of the viscosit'y is 3.04 X 10-4 poise
--
(10).
- -
"I
by comparing experimental results available in the literature with values obtained from the nomograph. Table I summarizes the results. The percentage of error was calculated according to
absolute temperature of test absolute critical temperature
Equation 1 can be expressed by loglo
[E - (1 - 0.5~,)"~]
- 46.08T,
= 1.855
-I- loglof(p,) (2)
wherej(p,) is a function which is independent of the nature of the substance and can be obtained empirically from the experimental results available in the literature. Basically, Equation 1 differs little from the equation used by Comings and Egly ( 3 ) : = ?A
f (P,,Tr)
(3)
where p , = absolute pressure/absolute critical pressure. Equation 3, however, byas not evaluated numerically and consequently the existing data could only be reproduced on a chart, which, as has been pointed out, in some regions is difficult to interpolate. Equation 2 can be evaluated numerically and since it is applicable to densities greater than the critical it was applicable, within certain limits of accuracy, both to compressed gases and to liquids near the critical point. THE NOMOGRAPH
Provided f(p,) in Equation 2 is known, this equation can be evaluated numerically and expressed in the form of a nomograph (Figure 1). The nomograph consists of three axes: the reduced density axis; the reduced temperature axis; and the axis representing the ratio, ?p/qil. Any straight line passing through appropriate values of pr and T , cuts the axis a t the appropriate value of q p / v A . Three examples will suffice to explain the use of the nomograph for the evaluation of the viscosity of compressed gases and of liquids near the critical point.
(E)
[(E)
Experimental
-
x
100
Nomograph
Although the results suggest that only a random error is involved, with some substances there is a definite tendency to deviate in one direction. With nitrogen, for example, the error increases steadily from -2.5 to -24.2 as the reduced density increases from 0.48 to 1.74. Another substance with a predominantly negative error is carbon monoxide. With carbon dioxide the error is negative at low densities and becomes positive as the density increases. On the whole, the investigation showed that an error of * 10% can be expected and that errors of the order of 20% may occur under some circumstances. Critical Isotherm of Viscosity. By means of the nomograph the critical isotherm of viscosity can be obtained by determining the values of qp/?A a t various values of pr and a t T , = 1. Figure 2 shows this curve; on the same graph are plotted experimental results available for this region. The results shown are for carbon dioxide a t 32' C. ( l a ) , carbon dioxide a t 31.1" C. ( l l ) , ethylene at 10" C. (Q),and water at 370" C. (14). As mentioned the available viscosity correlations (3, 1 6 ) are difficult to use in the critical region. In Table I1 the experimental viscosities for carbon dioxide (fb) are compared with values obtained from the nomograph and from the chart published by Uyehara and Watson (16). The values obtained from the nomograph agree within less
EXAMPLE 1. Ethylene a t 40' C. and 55.5 atms has a density of 0.091 gram per ml. The critical temperature of ethylene is 9.7" C. and the critical density is 0.220 gram per ml. Therefore, 0 414. From the nomograph, E = 1.26. ?A Since the viscosity of ethylene a t 40" C. and atmospheric pressure poise, the viscosity a t 55.5 atms., according to the is 1 093 X nomograph, is 1.38 X l o W 4 The observed value of the , viscosity is 1.41 X poise EXAMPLE 2. Liquid carbon dioxide at 20" C. and 72 atms. has a density of 0.812 gram per ml. The critical temperature of carbon dioxide is 31.1 ' C. and the critical density is 0.460 gram per ml. Therefore T , = 0.963 and pF = 1.765. From the nomograph, _ "- 5.10. Since the viscosity of carbon dioxide vapor a t 20" C. ?A and atmospheric pressure is 1.48 X lo-' poise the viscosity of liquid carbon dioxide a t 72 atms., according to the nomograph, is 7.55 X 10-4 poise. The observed value of the viscosity 13 7.71 X poise (12).
T,
= 1.106 and p r =
67
Figure 2.
Critical Isotherm of Viscosity
INDUSTRIAL AND ENGINEERING CHEMISTRY
888
TABLE11. COMPARISOXOF EXPERIMENTAL RESULTS FOR CARBONDIOXIDE( l a ) WITH PRESENT CORRELATIOS AXD WITH UYEHARA AND WATSOS
(16)
1.104, pr
1 793 1.765 I . 669 0.413
1.728 1.697 1.652 1.615 1,656 1.534 1.478 1.443 1.419 1.381 0.624 0.385
1.514\ 1.425 1.315 1.233 I . 123 1.096' 1.041 1.014
7.42 7.04 6.73 6.35 5.81 Liquid 5.66 5.65 5.23 5.29 5.05 4.95 4.90 4.78 4.59 4.58 Vapor 2.29 2.30 1.87 1.91 Temp, 3Z0 C. ( 7 ' ~ = 1.004) 7.44 7.88 6.97 7.41 6.66 6.95 6.04 6.27 5.58 5.86 5.35 5.60 Gas 4.96 5.28 4.14 4.48 3.78 4.06 2.59 2.54 2.20 2.14 1.92 1.87 Temp. 35' C. ( T r 1.013) 6.55 6.93 6.32 6.60 5.62 5.86 4.84 5.11 4.45 4.56 Gas 3.32 3.61 2.32 2.37 2.12 2.14 1.92 1.78
8.37 7.93 7.70
1.000
0.987,
1.635 1.426 1,274 1.191 1.151 1.096' 1.041 1.027 1.014 0.959 0.822,
1.641 1.611 1,513 1.419 1.361 1.074 0.628 0.493 0.353
1.560' 1.493 1.315 1.206 1.164, 1,096 1.027 0.959 0.822,
(2'7
2.13
= 0.997)
7.70 7.33 6.93 6.43 5.92
nisms of transfer of momentum may be repreacnted by different expressions. Viscosity of Gases at Moderate Pressure. If v m and A, the usual expressions derived from the kinetic theory of gases, x c introduced, then for an ideal gas Equation 4 becomes
(16)
9.80 9.09
Vapor Temp. 30' C.
8.23 7.71 6.97 1.86
q.104,
7.77 7.55 6.66 1.89
Liquid
1.717 1.689 1.652 1,584 1.521 1.483 1.423 1.297 1.206 0.782 0.554 0.369
a
v.104,
ExperiThis State mental ( 1 2 ) Work Temp. 20' C. (Tr = 0.963)
Vol. 42, No. 5
9=---
0.0847 (2rnlcT)l l 2
(5)
IT2
where IC = Boltzniann's coiiatant and u = the collision diainetw. To correct for the nonideal behavior of the gas it is necesbaiv to introduce a correction, such as Sutherland's ( l j ) , or Rcinganum's (19). The latter proposed an equation of the type
i i
i i
i 3.09 2.78 2.02 8.23 8.13 7.99 7.61 i i
! i i
2.64 1.96
where IC and D are constants characteristic of the gas. Constlint D may be represented by the ratio of two energies, say E y l R , where R is the gas constant. Viscositv can be envisaged as a single degree of freedom phenomenon, and because at the critical point the average potential energy in any direction equals the kinetic energy in that direction, Eg should be of the order of 0.5 RT,, the energy corresponding t o one degree of freedom at the critical temperature. h iiuinber of gases were investigated by plotting log q/T1/2 against 1/T and calculating Eg from the slope of the curve. I n Table I11 the values of Eg obtained in this way are compared with 0.5 RT,. ED
8.23
If the factor e - T i represents the fraction of the molecules
7.72
which will transfer momentum according to the kinetic theory of gases, the viscosity of a real gas is given by
s.oq z i
i
i
'/sRTC
2.61 1.98
0.0847 (21r1kT)'/~ e___-___ 17=
RT
( 7)
U2
i = Impossible t o interpolate with any reasonable degree of accuracy.
than 10% with the experimental results. The chart by Cyehara and Watson was difficult to use, owing to the steepness of the viscosity-isobars.
Many equations of state stipulate that the actual volume of the molecules should be a definite fraction of the molecular volume at the critical point-in other words u should be proportional to
(s) 113
, where M
is the inolccular Tyeight and N is Avogadro's
number.
BASIS OF NOMOGRAPH
The basis of the nomograph viae an attempt to evaluate the viscosity of liquids and gases from fundamental characteristics of the molecules. Like many workers the authors found that some empirical constants could not be eliminated from their equations. The results of these investigations, therefore, necessarily remained semiempirical. The following is a summary of the m-ork. The viscosity of liquids and of gases a t moderate pressures is usually explained by different theories, and it is consequently difficult to describe the two phenomena by the same mechanism. It is more difficult to describe the state of matter intermediate between the liquid and the gaseous state-namely, the critical liquid and the highly compressed gas. If, however, it is assumed that the frictional loss in viscous flow may be due to different mechanisms which may even function simultaneously, it is possible to set up a general equation, such as the following 9 = 2Jrn~7Y1~X
(4)
where D = the coefficient of viscosity v m = the frequency of transfer of momentum across a plane in
the fluid a t right angles to the direction of flow m = the mass of a single molecule X = the mean intermolecular distance or the mean free path
The key factor in Equation 4 is vm, which in different mecha-
Acetylene Benzene Bromine n-Butane Isobutane Carbon dioxide Carbon monoxide Chloroform Ethane Ethyl acetate Ethyl ether Ethylene Nethane Methyl chloride Nitrogen Oxygen Propane
259 307 412 358 375 302
307 458 572 425 404 302 133 542 303 520 463 281 190 417 125
130
432 303 411 4 : 7 290 205
395 127 1.50 340
T.4BI.E
153
366
Iv Ratio,
Substance Carbon dioxide Carbon monoxide Chloroform Ethane Ethyl ether Ethylene Methane Methyl chloride Nitrogen Oxygen
(,>,'
2.000 2.012 1.918 1.960 2.015 1.950 2,000 2.012 2.012 2.000
INDUSTRIAL AND ENGINEERING CHEMISTRY
May 1950
889
Applying these corrections it may be postu1at)edthat AND CALCULATED VISCOSITIES OF EIGHT TABLE V. OBSERVED GASESAT 0 ” C.
Gas Carbon dioxide Carbon monoxide Chloroform Ethane Ethyl ether Methane Nitrogen Methane
Tc OK. 304.3 134.2 536.2 305.3 467.0 190.7 126.1 154.4
G.YM~. 0.460 0,311 0.516 0,210 0.263 0.162 0.311 0.430
TABLE VI. CONSTANTS p, p s , Substance Carbon tetrachloride 1 2-Dibromoethane 1:2-Diohloroethane Ethyl ether Ethyl iodide Benzene Methyl ethyl ketone Methyl Diethyl propyl ketone ketone
M 44.0 28.0 119.4 30.1 74.1 16.0 28.0 32.0
AND p~ FOR A
P
ps)
(
113
Eviso
q.104 Obsyd., Poises 1.39 1.66
0.94 0.85
.
0.68 1.03 1.66 1.89
NUMBER OF LIQUIDS PF
G.?MI. 1.677 2.28 1.394 0.849 2.12 1.00 0.909 0.887 0.895
2.18 2.22 2.02 2.07 2.09 2.09 1.93 2.03 2.09
Table IV gives the ratio of
q.104 Calcd., Poises 1.50 1.76 0.93 0.98 0.73 1.13 1.78 2.03
G./MI. 1.678 2.21 1.371 0.853 2.23 0.900 0.936 0.878 0.898
urn =
v,.C.e
RT
2*,hTl 13V2 / 3 = v,:C.e RT
(12)
where C = the number of holes or unoccupied equilibrium positions per molecule; Eviec = the average bond energy between two molecules given by 2yN1’3VZ/3; and y = the surface energy in calories per square em. for nonpolar and polar unassociated liquids. It is a more complex function for associated liquids (6). Introducing these relations, the viscosity of liquids is given by 4 =
5 .c . ($)
&
1/2
1 13
3* aa
I
e
(2,nkT)
RT
(13)
I t is further postulated that:
c
as =
1IP
p($)
113
.pa
.(;
-
1)
(14
where P = proportionality constant equal to approximately 2; = the density of the closest packing of the molecules, approximately equal to the density a t the melting point, P F . Equation 14 was deduced in the following way: C, the number of holes per molecule will be proportional to the free volume per molecule ps
to as obtained from (I
Equation 7 . It thus seems that the relation between u and p c is given by
(8) where
a!
is a proportionality constant.
Similarly the amplitude
a,will be proportional to the cube root of the molecular volume:
Introducing Equation 8 into Equation 7 (9)
where a’ is a proportionality constant. a!
gives an equation in which the viscosity of a gas a t moderate pressure is expressed in terms of the temperature, the molecular weight, the critical density, and the critical temperature, “Reduced equations” for the viscosity of gases a t moderate pressures have been proposed by previous workers, in particular Licht and Stechert (8). Table V gives the observed values of the viscosity of eight gases at 0’ C. compared with values calculated according to Equation 9. The calculated values agree within 10% with the observed viscosities. Viscosity of Liquids. Andrade ( I ) showed that a t the melting point, V, in Equation 4 can be expressed by v,:
where V = the molar volume; and quency, given by
UL
= the Lindemann fre-
relations and equating 7 = CY
Combining these two
p gives Equation 14.
Table VI shows the values of p, pal and PF for a number of liquids. The viscosity of liquids can therefore be represented by:
p e representing the density of closest packing stands in a definite relation to p o the critical density, and for several liquids, 2, the
ratio @varies between 3.0 and 3.4. Po
Taking 2 ~ 3 . 2Equation , 15 can now be simplified to
where Evisacan be calculat,ed from the surface energy for “simple” liquids according to (6) and
Other values of 2,of course, yield equations similar to Equation where aa is the amplitude of vibration when the molecules just “touch” each other, acting as linear oscillators. In order to apply Equations 10 and 11 t o temperatures other than the melting point, it is necessary to consider: the existence of unoccupied “holes” ( 5 ) in the liquid; and the fact that the viscosity of liquids varies exponentially with the reciprocal of the absolute ternperature.
TABLE VII. EXPERIMENTAL AND CALCULATED VISCOSITIES OF SEVENLIQUIDS. 4 20’
c.
EW80
Liquids
M
86.2 %%~~~etrachloride 153.8 119.4 109.0 Ethyl ether 74.1 Methyl formate 60.1 Methyl acetate 74.1
gt,!$‘kL$ide
G.,%fI. 0.659 1,594 1.489 1.460 0.714 0.974 0.934
G.YM1. 0.234 0.558 0.516 0.513 0.263 0.349 0.326
Kg. C h . / Mole 1.92 2.29 2.04 1.73 1.52 1.55 1.80
qa- 8.2,
Poises 0.001906 0.005220 0.003082 0.002092 0.001381 0.001523 0.001717
?Is-
3.5,
Poises 0.003932 0.01076 0.005625 0.005370 0.002724 0.003070 0.003620
7 obsvd.,
€ oises 0,003258 0.009750 0,005712 0.004020 0.002258 0,003548 0.003880
~
INDUSTRIAL AND ENGINEERING CHEMISTRY
E90
Vol. 42, No. 5
The first assumption can be tested by extrapolating Equations
9 and 16. At the critical temperature Equation 9 becomes:
From the relation between E,,,, and the surface energy or the energy of vaporization it ran be deduced that a t the c r i t i d point E,,,,=O, so that in extrapolating Equation 10 to the critical point
The form of Equations 17 and 18 leads t,o the deduction that the cube of the critical temperature, T,3, wax essentially an aciditive propert'y for many series of compounds ( 7 ) . The same conclusion cannot be drawn from other equations proposed for the viscosity at the critical point ( 1 6 ) . Except for the numerical coefficient the two equations are identical and it would seein that the viscosity a t the critical point is about twelve times the viscosity at the critical tempeyiiture and moderate pressure. If the experiniental data. of viscosity at the critical point are examined, Equation 18 gives values which are too large by a factor of about 6. These discrepancies indicate that at the critical point' the transfer of momentum is neither entirely translational nor entirely vibrational in nature. One must' assume, therefore, that in t,he critical region and also in highly cornpressed gases the mechanism of transfer of rnomentuin Equation 4 can then lie must be the summation of tn-o effe written: Figure 3.
(pi)r against
p,
€or Uiimber of Subctaiires
V,,iL)'nL'h
2'(VmT
(1 9 )
+
16, and in Table 1-11experiniental viscosities for seven liquids a t 20" C. are compared n-ith calculated values assuming Z = 3.2 and Z = 3.3. Table VI1 shows that the viscosity ralculated by Lquation 16 is very sensitive to the value of 2. There is also some evidence that 2 may not be an invariant constant but ma!- be a more complex function of the critical density. Instead of evaluating the exact form of this funct,ion, the authors halve chosen, as will be seen later, to 'evaluate empirically the contribution of the critical density to the viscosity equation used in t'he nomograph by a study of a large number of liquids and gases under pressure. It' is also a fortunate circumstance that the contribution of Equation 16 to the final equation of the noniograph is multiplied by a fraction less than unity which, for example, a t the critical temperature is of the order of 0.1. I n this way the final equation used in the nomograph gives tolerably close agreement with practice, despite the inaccuracies involved in evaluating Z and the other constant, 8. Viscosity near Critical Point and Viscosity of Highly Compressed Gases. Equation 4 represents a general equation for viscosity which leads to: Equation 9 for gases at, moderate pressure, assumipg translational transfer of niomeiituni; and Equation 16 for liquids, assuming a type of vibrational traiisfer of momentum, similar to that proposed hy Aiidrade (I). The gaseous and the liquid states of matter approach each other in t,he critical region; hence the viscosity at this point should be predictable by one of two systems of equations:
1. If the change in the nature of
7 =
from the translational to the vibrational type is abrupt, the two equat'ions extrapolated to the critical region should be identical. 2. If the change is cont,inual and gradual, then it must be assumed that each equation gives only t'hat part contributed to the total transfer of momentum by the type of vin assumed and that in reality a t all the points on the curve of the equat'ion of state, the equation of viscosity is a weighted sum of the two effects. The two equations, extrapo1at)ed to the crit'ical point, should be noncoincident, the difference between the two values so calcuated being a measure of neglecting eit,her effect on the sum total. u,
v , , ~ ~vrrLT ~ ; = the trmsfer u t momentum by so that Y,, = vAiT translation; and vi,zL = the transfer of momentum of the type encountered in liquids. It iollows froni Equation I O that in the critical region ant1 for highly compressed gases 'I = rtra'ri
+
T , it,r
(20)
that, is, the viscosity can be resolved int,o t w o component vi+ cosities, one translational and the other or' the liquid type, vibrational. [Born and Green ( 2 ) arrive a t the conclusion, h y a niathematiea1 analysis far superior to the present attempt, that the viscosity of fluids is a mixture of two t'ypes of viscosit>ies. As practically the whole of the work was done before and in ignorance of Born's and Green's conclusions, the possible usefulness of the nomograph encouraged t'he authors t o publish the present account.] Equation 20 can be restated in the form 1)
=
hlEq.9
f X7Es.16
(21)
x are numerical coefficients depmding on the degree n and therefore are functions of the density.
Factor
1~ can be evaluated approximately by coneideririg that th.e arerage velocity of molecules is transmit,ted x-ith infinite velocity t'hrongh t,he diameter of the molecules and also that the number of moIecules free to follow translational movements is reducrd in proportion to the free space in the fluid. These t4.o effects can he corrected for by equat,ing
+
I
(I
- Kpr)2/3
where K e 0 . 5 . The viscosity in the critical region and of a compressed gag is then given by
if it i3 assumed that E,,,, -0-that
E-~ vtsc is, e R T * l o
May 1950
INDUSTRIAL AND ENGINEERING CHEMISTRY
It was not possible to evaluate factor x a priori but only from experimental results. Accprding to Equation 20 qvlbr can be obtained as the difference between q experimental and 9 translational. By plotting qvlbr/TIP against p, curves are obtained for nitrogen (gas), carbon dioxide (gas and liquid), methane (gas), ethylene (gas), and n-pentane (gas and liquid). For each substance a single curve independent of temperature was obtained, showing that qvibr/T1/2 is only a function of the density. The results of these calculations are shown in reduced form on Figure 3. This figure shows that if qv,br/T1l 2 is divided by the value of q v l b r / T ' / 2 a t the critical density and these values * are plotted against the reduced density, a unique curve results. This indicates that x is dependent only on the reduced density, but is independent of the nature of the substance. At the critical density x was found to be equal to 0.10.
89 1
If the transfer of momentum at the critical point is entirely of the liquid type, then (Equation 18) pc2 1 3 .
T,l
/2
=
3.7
x
10-4
=
3.7
x
p c 2 13 Tcl 1 2 i o + ___ 311
Mi 16
Since xc= 0.10 Swbrc
At any
Other
temperature,
T j
and density,
(26)
pJ
Using Equations 9,23,24, and 25
Rearranging Equation 28
or in logarithmic form
Equation 30 is the same as Equation 5 given when the nomograph was described. I n order to determine f ( p , ) a plot was made of
for several gases for which data are available in the literature. A mean curve was drawn through the experimental results. This curve is shown on Figure 4. From the mean curve values of Iogiof(pr)were read off and the nomograph was calculated. LITERATURE CITED
Figure 4.
s
Function Loglo f(p,) against Loglop,
Basic Equation. Let q p = the viscosity of the compressed gay, and qA = the viscosity a t moderate (say, atmospheric) pressure.
According to Equation 20 VP
E
+
Strani
t
(23)
ltvibr
KOW 'jtrsns
=
-
V A ( ~
0.5~r)~'~
where (Equation 9) 213
SA
5.17 x 10-5%e
From Figure 3 it follows that
TlI2
-3 2T
(24)
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RECEIVED March 16, 1949.
