Correlating Viscosities of Gases with Temperature and Pressure DONALD F. OTHMER AND SAMUFL JOSEFOWITZ Polytechnic Institute of Brooklyn, N. Y.
At constant pressure, viscosities of
gases and of gaseous mixtures may be plotted directly to give straight lines on logarithmic paper against a temperature scale calibrated from the vapor pressure of a reference liquid. A t constant temperature, gas viscosities may be plotted to give straight lines on logarithmic paper against the ratio of the kinematic pressure and the gas density. Methods of correlating gas viscosities, temperatures, and pressures are
expanded to allow the estimation of the viscosity of gases under different conditions where their PVT data and the viscosity at one set of conditions are known. A nomograph has been constructed to allow ready determination of viscosities; it is believed that values so determined fall well within the precision of published experimental data. Thermodynamic derivations indicate the theoretical basis of these plots and the resulting nomograph.
I
in this manner as straight lines. Any deviations are well within the probable experimental error. The equation of the straight lines in Figure 1 is:
N PREVIOUS papers many physical properties of pure
liquids and solutions have been correlated by simple l o g a rithmic plots for which a special temperature scale was developed. Straight-line plots may be simply made on log paper for vapor pressures, viscosities, surface tensions, densities, and other properties. Thus a n algebraic equation may be wiitten for vapor pressures: L
log P' =
log u = A log P +*c (2) where u = viscosity P = vapor pressure a t temperature of reference substance A, C = constants
2L' log P' + c
Thus, i t is necessary only t o evaluate the viscosity a t a given pressure for two different temperatures i n order t b establish the whole range of viscosities almost to the temperature of liquefaction. A third intermediary point will check the continuity of the line. Since most gases which were examined plotted with almost identical slopes at atmospheric pressure, the viscosity over a moderate range of temperature could be approximated for any other gas if only one viscosity measurement were available. The method also checks with various mixtures of gases investigated ($0, 83). Figure 2 shows that the isobaric viscosity lines for the mixtures of hydrogen and propane do not fall between the viscosity lines of the pure compounds; therefore this system has, at atmospheric pressure, compositions of maximum viscosity which are similar t o the familiar compositions of maximum vapor pressure-Le., minimum constant-boiling mixtures ( 1 7 ) . The composition of maximum viscosity, furthermore, changes not only with pressure, but also with temperature, since the slopes of the various constant-compositionlines arenot parallel. Figures 1, 2, and 3 indicate that, if the isobaric viscosities of any 'two gases are straight-line functions of the corresponding vapor pressures, then the isobaric viscosity of either one may be plotted against the isobaric viscosity of the other at the same temperatures t o give straight lines. Such a plot is also of interest; but normally i t would be regarded as simpler t o make the temperature calibration from the vapor pressures of a reference substance than from the corresponding viscosities.
Here vapor pressure, P , and latent heat, L,of one material are functions of the corresponding pressure, I",and latent heat, L',of another material at the same temperature and a constant of integration, C. Similar equations involving a ratio of energies were obtained between vapor composition, viscosities, densities, and other properties of one liquid (13-19), and a property of a reference liquid. By taking advantage of the fact t h a t the ratio of energies resulting from a consideration of a reference substance is much more constant than the individual energies related t o various physical properties, simple correlations were obtained based on straight-line plots. Thus, interpolation and extrapolation are easily possible and accurate enough for engineering uses, even from the small amount of data usually available. APPLICATION T O ISOBARIC VISCOSITY
Comings, Mayland, and Egly (1, I , 3)reviewed and correlated available viscosity data of gases a t elevated pressures; they showed that by the use of dimensional analysis t h e viscosities of various gases may be related t o one another and to reduced temperatures and pressures. It was thought t h a t the relations presented by them could be simplified, from the families of more or less empirical curves which they developed, by the graphical and thermodynamic technique previously used for correlating other properties of liquids and gases. Based on the general methods described in other articles, two different plots of viscosity data for gases were found to give straight lines. The first (Figure 1) indicates a plot of viscosities of different gases at constant pressure along the vertical axis. A temperature scale is plotted on standard log paper along the horizontal axis by indicating a t the appropriate values of the vapor pressures of water, the corresponding temperatures. The ordinates representing these temperatures were drawn vertically, and on them were plotted the viscosities at these temperatures taken from standard reference sources. The correlation is excellent, and the viscosities of any of these gases may be plotted
APPLICATION TO ISOTHERMAL VISCOSITY
Theoretically, the viscosity of a perfect gas should be independent of the pressure and vary only with temperature. Owing t o deviations from the perfect gas laws, however, wide variations of gas viscosities are observed with change in pressure. It has been found t h a t a plot of the viscosity of a gas a t constant temperature against the ratio of the kinematic pressure t o the density of the gas gives straight line on log paper. These are drawn on two graphs (Figure 4) because the lines are very close for the same compound at different temperatures. The equation of these lines is thus: 111
INDUSTRIAL AND ENGINEERING CHEMISTRY
112
Vol. 38, No. 1
Figure 1. Log Plot of Viscosity of Various Gases at Atmospheric Pressure against a Temperature Scale Obtained from the Vapor Pressure of a Reference Liquid 1. Xenon 2. Argon 3. Oxygen 4. Helium 5. Air 6. Carbon dioxide 7. Nitrous oxide 8. Hydrogen 9. Propane 10. Ethylene
(3) where Pk = kinematic pressure d = density K , 6' = constants The slopes of these lines are equal to K , which varies with the temperature of the gas. Since IC was found to be almost the same for many gases a t the same reduced temperature, a general correlation of theslope of the constant-temperature lines with temperature was worked out by plotting the slope of these lines against reduced temperature (Figure 5 ) . I n working out this correlation, viscosity data were taken from Comings, Mayland, and Egly ( 3 ) for carbon dioxide, ethylene, and methane, from llichels and Gibson ( 1 2 ) for nitrogen, and from Smith and Brown ( g 2 ) for propane. NOMOGRAPH FOR GASEOUS VISCOSITY
By consideration of the relation indicated in Figure 5 , i t is plain that K is a function of reduced temperature, and Equation 3 can be written: P log u = !(a,) log C' (4)
2+
where C' is a function of the temperature and of the particular
Diethyl ether h'-Butane Ieobutane Hydrobromic acid Hydriodic acid Hydrochloric acid Chlorine Bromine Iodine Methyl chloride
gas bcing considered. This equation can conveniently be represented by a nornograph (Figure 6) This nomograph consists of a combination of a 2 chart and a point-coordinate chart. The two charts are interconnected by one common line. The right-hand scale of the 2 chart is calibrated for reduced temperatures from T, = 1.0 t o T , = 4.0. This scale corresponds t o term K in Equation 3 and t o term f(T,) in Equation 4. The diagonal scale is calibrated for the term P,ld in liter atmospheres per gram over the range 0.1 to 8.0. The third line of the 2 chart is the reference line iyhich corresponds t o the term K log (P,/d) of Equation 3; since this term in itself docs not give the viscosity of the gas, the calibrations of this line (which were used in the development of the chart) are omitted for the sLke of simplicity. The point-coordinate scale a t the left of Figure 6 is used to add (multiply since logarithms are involved) the value C' to the value of the terra K log (P,/d). Since C' varies for different compounds and different tempcratures, and is constant only with respect to pressure, i t is necessary to locate on the central gridnork of the nomograph a point which corresponds to this constant for each compound and each temperature separately. Table I lists these values for individual gases and the corresponding coordinates to be used in locating the particular point foP each gas a t different temperatures. To use the nomograph, the viscosity a t a given pressure can be read off directly by connecting the reduced tenperature reading on the right-hand scale with the corresponding value of P,/d to get a point on the reference line. This point is connected to the point on the gridwork corresponding to the X and Y coordinates of the compound and the temperature in question, This straight line intersects the left-hand scale a t the viscosity reading for the given conditions. As viscosity data become available for other gases a t any pressure, the corresponding points can be located on the grid t o allow the use of the nomograph for these gases. For each gas the value of X (the horizontal distance on the coordinate chart) is constant with respect t o temperature; and only Y , the vertical distance, changes in locating the point. Furthermore, compounds with equal molecular weights have almost identical X values. This makes it possible to find the approximate location of the point when only one viscosity value for that gas is available, by working this single point of data backward through the chart until the construction line intersects the ordinate or value of X on the coordinate chart. When the Y values at two temperatures are known, the viscosities a t these two temperatures and any pressure can be de~
Vapor Pressure Of Water (mm]
11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
INDUSTRIAL AND ENGINEERING CHEMISTRY
January, 1946
113
where a/V2 is the cohesive pressure, and b is a measure of the volume occupied by the molecules. The values of a and b are tabulated for most gases, and the kinematic pressure can therefore be calculated from the expression
Pk=P+-
a
(6)
V2.
where V = specific volume a = van der Waals constant P = observed pressure
It is not recommended that the correlation of the slopes of the isotherms be used when accurate data are available, since the reduced temperature scale involves the assumption t h a t the slopes of the isotherms in Figure 4 are exactly ,equal for all compounds at the same reduced temperatures. The nomograph can be used, however, to great advantage when high accuracy is not required and few data are available. Figure 2. Log Plot of Viscosity of Mixtures of Xenon with Hydrogen (above) and Propylene with Hydrogen (below)at Atmospheric Pressure against a Temperature Scale Obtained from the Vapor Pressures of a Reference Liquid
THERMODYNAMIC BACKGROUND FOR ISOBARIC PLOT
Guzman (8)showed that for a liquid Clog T
log+ = a - - . -
termined directly from the nomograph. The viscosities a t any other temperatures can then easily be intrapolated and extrapolated by straight-line isobaric correlation described earlier.
(7)
from which he derived the equation:
To find the viscosity of ethylene a t 30' C. and a EXAMPLE. pressure of 137 atmospheres: density of gas = 355 gramslliter specific volume = 0.0789 literlmole van der Waals a = 4.471 litera atm./molee
Pk = P
3 = d
+2
V2
= 137
4.471 + (0.0789) %-
where 4 = liquid fluidity T = absolute temperature E = energy of activation of liquid a, b = constants
= 854atm.
Reinganum (81) showed that the viscosity, u, of a gas is proportional t o the absolute temperature and t o
= 2.405 liter atm./gram
reduced temperature
X = 3.0, Y
=
u = aTe-D/T
1.072
= 10.0
Connect 1.072 on scale c t o 2.405 on scale b, and connect the point of intersection on scale T wlth point x = 3, y = 10 on the grid; the viscosity is then read as 455 micropoises on scale a. This value compares favorably with the experimental value of 435 micropoises obtained by Comings, Mayland, and Egly (3).
(9)
where a and D are constants similar to the Southerland constant for a gas, as shown by Loeb (11). Equation 9 can be written:
Temperature
,200 u1
30
50 I
OC.
70 I
95 I
i
USE OF CORRELATION
Few data are available on the viscosity of gases at high pressures, and the experimental technique t o obtain such data is difficult. With this correlation i t is possible to estimate the viscosity of gases a t high pressures to a n accuracy of a few per cent for most gases. Only the viscosity of the gas at one pressure must be known along with PVT data, so that the kinematic pressure can be calculated. The kinematic pressure of a gas is the observed pressure plus the cohesive pressure. It can be obtained, within sufficient accuracy for most engineering calculations, from equations of state such as the van der Waals equation:
Figure 3. Log Plot of Viscosity of Methane at Various Pressures against a Temperature Scale Obtained from the Vapor Pressure6 of a Refezence Liquid
INDUSTRIAL AND ENGINEERING CHEMISTRY
114
-*
substances this may be regardcd as substantially constant, depending upon the molecular structure and related properties. Thus, the slope of the line obtained by plotting log u against log P is shown t o be constant and indicates why the lines obtained are straight. PLOTAT CONSTANT TEMPERATURE. The viscosity of a perfect gas can be calculated from the kinetic theory of gases ( 7 ) :
1
Carbon Dioxide
A Nitrogen
Vol. 38, No. 1
Ethylene Methane A Propane
0
1 u = -mnd 3
(16)
1 u =3 - dcl
0.Y
1.0
1.5
2.0
3.0
where m = weight per molecule n = number of molecules per cc. d = density, 1 = mean free path E = mdecular velocity
(liter atrn. per gram)
K'nernotic Denslty
Figure 4. Log Plot of Gas Viscosity at Constant Temperature against the Ratio of Kinematic Pressure to Density of the Gas ( T R = Reduced Temperature)
(18) and C = 3 d P / d where y
or in terms of fluidities
=
(1%
molecular diameter ( 7 ) .
Substituting Equations 18 and 19 in Equation 17, 1 u = - d3 3 @ ( &d ) d 2 rdy2
which is similar t o Equation 7 and from which a similar viscosity expression (Equation 8) can be obtained for a gas, or the following: d_l _ n u= _- E dT RT2
(12)
The Clausius-Clapeyron equation relates P (vapor pressure), T (absolute temperature), and L (molar latent heat of vaporization) of a liquid:
L
l n_P -d =
(13)
RT2
dT
This equation, however, is derived for perfect gases, and several modifications must be made to make it applicable to real gases. Combining constants in Equation 21 to a single constant, C", and taking the logarithms, 2 log u = log C"
+ logzP
(22)
which gives a straight line on log paper if u is plotted against Pld. The slope of this line is 1/~, and i t should be independent of temperature for an ideal gas. From Equation 21,
If Equation 12 is divided by equation 13 at the same temperature, d h u dlnP
-
-E For a n actual gas this cannot be assumed, however, to be con-
L
On integration,
1.5
-E
log U = -log L
P -C
stant, since actual molecules are not incompressible spheres of constant radius. They are complex planetary systems of electrical charges whose effective diameter varies with temperature (9,10). Jaeger (9) showed t h a t the van der Waals constant, b, which is a direct measure of the molecular volume and, hence, diameter, varies as
1
(15)
This derivation is the same as t h a t previously given for liquid viscosities (16) and indicates that a logarithmic plot of viscosities of a gas against the vapor pressures of a reference liquid a t the same temperature would be a line TT ith a slope equal t o the ratio of the energy of activation of the gas and the latent heat of the reference liquid. It wa,>previously shown t h a t this ratio is constant for a given substance (4-7, 16). For large groups of
L
z0.6
8
I
i
bf =
l where b'
Figure 5. Log Plot of Slopes of Isothermal Viscosity Plots from Figure 4 against Reduced Temperature
b&IT
(24)
van der Waals term corrected for temperature b = theoretical van der Waals term C = a constant, =
INDUSTRIAL AND ENGINEERING CHEMISTRY
January, 1946
115
*
r
. I20 130 150 I60
le0
14
13
.
I I IO
Q)
220
240 260 280 300 32 0
40 2, c
.v .a
23tf $-
8
Y 7 6
5 4 3
600
20
600
2 I
0 700
Is*
0
I
e
3
4
5
6
7
X
12>so0
Figure 6.
General Nomograph for Gas Viscosity
TABLE I. X
The nomograph m a y be uaed as follow.: 1. Connect the reduced temperature value on malo c with a straight line to the value of ? k / d . Connect thia reference point to the point on the gridwork found from Table K, correspondingto the gas and temperaturein question.
4.
Extend this last line to aut soale 0.
5.
Read off from this scala the vhcosity of fluidity of the gas at the given conditions of temperature and pressure.
Y
VALVESFOR VISCOSITY NOMOQRAPH
x .
2. Extend this line to give a point on referenas lino r. 3.
AND
Ethylene
3.1 3.2 4.6 3.7 3.1 3.0
Nitrous oxide Oxygen Xenon
3.0 4.6 3.4 6.2
Acetylene
6
Air Carbon dioxide Carbon monoxide Ethane
Methane Nitrogen
0.6
0'
C.
...
5.6 8.0 5.2
...
10.8 11.4 5.6
...
4.8 3.6
30° C. 10.7 5.2 7.4 4.8 10.7 10.0 10.9 5.0 6.0 4.3 3.0
Y
50° C. 9.4 5.0 7.0 4.6 10.1 9.7 10.6 4.8 9.8 4.1 2.8
100' C. 8.5 4.7 6.1 4.3 Q.4 9.0 10.0 4.7 5.4 3.5 2.2
116
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 38, N o 1
ARITHMETIC RELATION OF VISCOSITY TO KIP,-EMMATIC PRESSURE AND DENSITY
Another interesting correlat,ion is that of viscosity us. kinematic pressure divided by density. As pointed out by Corriirlgs In a private cornmunicatian, this correlation gives straight Lines on arit’hmetic paper (Figure 7). He developed a relation frorn previous work ( 1 , d , 3) in which it was assumed that
Corbon Dioxide Nitrogen o Ethylene Methane A Propane 9
4
600 h
(31)
500 where
viscosity a t low pressure kinematic pressure P; = ideal gas pressure a t same temperature and density as actual gas 7*1
=
P k =
400
but Pi= RTd
,300
.-c
132)
Therefore, substituting Equation 32 in 31,
.E 200
(33)
>
IO0
At constant temperature proportional to Pk/d.
0 0
1.0 Kinematic Pressure Density
and R T are constants so that u k
I n plotting u against Pk/d, straight lines are obtained. d o not pass through the origin and have the equation,
3,O (liter atm. per gram) 2.0
u = m ‘Pk -+a d
Figure 7. Arithmetic Plot of Viscosity of Gases at Constant Temperature against the Ratio of Kinematic Pressure to Density of Gas
Furthermore, the mean free path has been shown to be a function of temperature (11):
I‘ = 1/(1
u1
+ ;>
These
(t34)
where nz and a are const’ants. K O similar correlation with reduced temperature could be found, however, for const,ant m in Equation 34 as was found for the constant K in Equation 3 Hence, it would appear t h a t a more correct expression might involve some power other than unity for the viscosity in i t a relation t o the ratio of kinematic pressure and density. ACKNOWLEDGMENT
where 1’ = mean free path corrected for temperature I = ideal mean free path D = a constant which, for many gases (10) is:
D
= To/1.14
(26)
Thanks are due to Edward W. Comings for helpful suggestions in the preparation of this paper, and particularly for calling a t tention to the proportionality between the viscosity and the ratio of kinematic pressure to density, as indicated in Figure 7 .
Substituting Equations 26 and 25 into 17: . /
LITERBTURE CITED
1
Comings, E. W., and Egly, R. S., IND. ENG.CHBX.,32, 714 (1940).
Comings, E. W., and Mayland, B. J ~Chem. , & M e t . Eng , 52, No. 3, 116 (1945). Comings, E. W., Mayland, B.J., and Egly, R. S., Univ. Ill. B U A ~
where T, = reduced temperature
42,No. 15 (1944).
Equation 21, therefore, is more correctly expressed as log u
=
log C“
+ K log Pd- + log 1 +1.14T, 1.14T,
(28)
The third modification to Equation 22 is that the expression
31/mfor the velocity of the molecules was derived on the assumption that no cohesive pressure exists in a perfect gas (11) and that P = Pk. I n an actual gas, the cohesive pressure must be added to the observed P t o obtain a true velocity; therefore, E = 3 m d
(29)
When these factors are taken into consideration, Equation 27 a t constant temperature becomes: r*
log u = log C ”
1.14 T, + K log PI, - f log 1 + 1.14 T, d
(30)
Since the temperature is constant, the last term is a constant which may be combined with the other constant, log C”, t o give Equation 3; as already shown, Equation 3 gives a straight line with slope K when plotted a t constant temperature.
Ewell, R. H., J. Chem. Phys., 5, 571, 967 (1937). Ewell, R. H., and Eyring, H., Ibid., 5, 726 (1937). Eyring, H., Ibid., 4, 283 (1936). Glasstone, S., “Textbook of Physical Chemistry”, Kew York, D. Van Nostrand Co., 1940. Guzmbn, J. de, Anales SOC. espaa. j%, quirn., 11, 353 (1913). Jaeger, G., “Die Fortschritte der Kinetischen Gas Theorie’ Braunschweig, F. Vieweg und Sohn, 1919. Jellinek, K., “Lehrbuoh der physikalisohen Chemie”, Voi, 1 Part 1, p. 391, Stuttgart, F. Enke, 1914. Loeb, L. B., “Kinetic Theory of Gases”, New York, McGrawHill Book Co., 1934. Michels and Gibson, Proc. Roy Soe. (London), A134, 288 (193 1) Othmer, D. F.,INB. ENQ.CHmf., 32, 841 (1940). Ibid., 34, 1072 (1942). Ibid., 36, 669 (1944). Othmer, D. F., and Conwell, 9. W., Ibid., 37, 1112 (1948). Othmer, D. F., and Gilmont, R., Ibid., 36, 858 (1944). Othmer, D. F., and Sawyer, F. Ibid.,35, 1269 (1943). Othmer, D. F., and White, R. E., Ibid., 34, 962 (1942). Rankine, 0. A., PYOC. Roll. SOC.(London), A84, 181(1911) Reinganum, Physik. Z., 2, 242 (1900). Smith, A. S., and Brown, G. G., IND.ENG.CHEM.,35, 705 (1943), Trautz, M., and Ruf, F., Ann. Physik, 151 20,131 (1934).
e.,
