I
JAMES F. ROSS' and GEORGE MARTIN BROWN Northwestern University, Evanston, 111.
Viscosities of Gases at High Pressures The viscosities of methane, nitrogen, and helium have been determined in a new type of viscometer using instrument bellows and a capillary tube in the range 25" to -50" C. and 500 to 10,000 pounds per square inch gage.
VALUES
for the viscosities of gaseous substances have long been of great theoretical and practical importance. Theoretically, gas viscosity serves to establish molecular dimensions and the correct mathematical models to describe molecular reactions. Practically, viscosity is a parameter in most correlations of friction, diffusion, and heat transfer in fluid streams, entering directly into the Reynolds, Schmidt, and Prandtl dimensionless groups. T o date,' relatively little work has been carried out on the effect of high pressure on gas viscosity and that at around and above room temperature. The Enskog relationship is the best rational equation for representing the effect of pressure on gas viscosities. I t fits the best experimental high pressure viscosities better than any other nonempirical relationship and rests on a fairly secure theoretical foundation. It is unwieldy and requires a very accurate determination of ( , ~ / p ) ~ ~ ~ , By dimensional analysis it can be shown that W P O ) i'~/'~c)
(1 1
= F(P/Pc, T / T d = F'(P/pc,
TIT,)
(2)
Equation 1 was plotted graphically by Comings and Egly ( 5 ) , Comings, Mayland, and Egly (6), and Carr ( 3 ) . Equation 2 was used by Uyehara and * Present address, Esso Research Laboratories, Baton Rouge 1, La.
Watson ( 7 8 ) . As the viscosity of the gas at low pressures is usually known or can be estimated, while the critical viscosity is nebulous at best in actual practice, Equation 1 is to be preferred in correlating the effect of pressure on gas viscosity. The graphs based on this equation are far more general than any empirical relation and are usually accurate enough for practical purposes, but their greatest appeal, especially to engineers, is the ease with which they are used. Apparatus
Viscosities may be measured in several ways, classified in two categories (7). In general, either (1) a body is forced through a fluid and the viscous drag is measured or (2) the fluid is transpired through a restriction and the force-flow relationship is measured. The former method includes the falling sphere, rolling ball, oscillating disk, and rotating cylinder. The latter includes flow through a capillary tube. All these methods have been used for very accurate determination of gaseous viscosities at elevated pressures. The choice is largely one of convenience. Except for the rolling ball method, the viscous dragtype viscometers are impractical to construct for pressures much exceeding 1000 pounds per square inch. The rolling ball viscometer is ideally suited for viscosity measurements a t high pressure, but it is not an absolute viscometer, as its theory has not been adequately developed. I t has given very accurate results, but confidence has come only when determinations were confirmed in an absolute type viscometer. On several occasions, the fluid velocity around the rolling ball was high enough to make flow turbulent and fictitiously high values of viscosity were reported. The errors in the data were discovered
Background Information Maxwell (11) Warburg and Von Babo ($0) Phillips (15) Enskog (4, 7) Michels and Gibson (12) Leipunskil (10) Michels, Schipper, Rintoul (13
2026
Showed theoretically and demonstrated that gas viscosity at constant temperature is independent of pressure at and below I-atm. pressure Showed that viscosity of COSincreases with pressure Confirmed work of Warburg and Von Babo Extended kinetic theory to dense or compressed gas and derived first adequate expression of effect of pressure on gas viscosity Accurately measured viscosity of nitrogen up to 1000 atm. Showed Enskog relation represented effect of pressure on viscosities of N, a", COz, CO, CH4, HzO, SOz, CHaCl Viscosities of H and D do not follow Enskog equation up to 2000 atm. at 2 5 O to 125' C.
INDUSTRIAL AND ENGINEERING CHEMISTRY
orily when they could not be reproduced on a viscometer whose theory was more completely developed ( 6 ) . The capillary flow Viscometers are relatively easy to construct and use a t very high pressures and properly calibrated, give absolute measurements of viscosity based on Poiseuille's law. The earliest capillary flow viscometers for use a t elevated pressures were merely heavier models of atmospheric equipment. A real advance in pressure viscometry was made when Rankine developed a totally enclosed viscometer (76). With the Rankine viscometer a relatively large number of determinations have to be made for every viscosity measurement. The results obtained however, are among the best high pressure viscosity data that have been measured. -4high prtssure transpiration viscometer, developed by Michels and others (72, 73), eliminated the need for multiple determinations at each point, but required a long time between runs for depressurizing and repressurizing. Both the Michels and the Rankine apparatus depend upon liquid mercury to supply the driving force which causes flow of fluid through the capillary. As a result, they cannot be used at temperatures below the freezing point of mercury nor where mercury has an appreciable vapor pressure. Their use is restricted to substances which do not attack mercury. A new type of viscometer, developed to overcome these difficulties, depends upon the elasticity of metal bellows to provide a reproducible, known driving force to cause gas flow through a capillary. The same bellows measure gas flow, so that both driving force and flow rate through the capillary are measured simultaneously. This instrument (Figures 1 and 2) consists of a rigid framework, within which two matched metal bellows are mounted, in turn connected axially by a capillary tube. The capillary tube is free to move axially within the framework by compressing one bellows and expanding the other. Any nonaxial movement of the tube is prevented by careful alignment of all components. Qualitatively the principle of operation of this viscometer is simple. When the capillary is displaced from its neutral position, one bellows is compressed and the other expanded. The elasticity of the bellows sets up a force proportional to the displacement, to
BOMB HEAD
EOW BELLOWS
CLP LLARY ELECTRiCLL LEAD ELECTRS4.L CONIKTS
REiRlCTOR
BELLO S
RETRACTOR
HANDLE
Figures 1 and 2.
Bellows viscometer
try to restore the system to a neutral position. On the other hand, if the gas in one bellows is under a greater pressure than in the other, the pressure difference exerted across the bellows area forces the assembly away from the neutral position, to try to equalize the pressure. As long as the capillary tube is free to float between the two bellows, the pressure differential will exactly balance the elastic force exerted by the bellows when the capillary has been displaced from its neutral position by a distance proportional to the pressure differential across the capillary. If the gas is allowed to flow slowly from one bellows to the other through the capillary tube, the whole assembly will slowly move toward the neutral position as the pressure differential is dissipated, but a t every instant the displacement from the neutral position is determined by the pressure difference across the capillary (acceleration and initial effects of the assembly can be neglected because of the slowness with which the differential pressure is dissipated). The gas flow rate can be found, because the displacements are known as a function of time and the average area of the bellows is known. Driving force and flow rate are simultaneously measured by the displacement of the movable assembly from its neutral position. Only the time required for the capillary tube to pass two known points must be known, in addition to the instrument's constants, because Poiseuille's law can be integrated between these two limits to yield an equation for the viscosity of the gas as a function of time. The component parts of the viscometer are the bellows, capillary, framework, retractor, and electrical contacts, all mounted inside a pressure bomb.
Bellows. The bellows are a matched pair of beryllium-copper instrument bellows, Type U-l02AA, made by the Foxboro Instrument Co., Foxboro, Mass.; for use in the standard d/p cells. Instrument bellows were chosen because of their high reproducibility and general availability. Type U-l02AA bellows were the least stiff of the standard size bellows that could fit into the pressure bomb. Their minimum stiffness ensured that a t any given pressure drop across the bellows, the deflection would be a maximum. To incorporate these bellows in the viscometer, a '/s-inch hole was drilled in each end axially through the blind end. The bellows were washed thoroughly to remove all traces of soldering flux, grease, or metal chips, and then dried. Next the bellows were worked several hundred cycles of compression and expansion, to eliminate a tendency of the zero to shift. Capillary. A capillary tube of convenient size was available commercially -an Eimer & Amend dropping electrode capillary tube. A large number of these tubes were examined visually under a microscope to find one with a smooth, round, and uniform bore. The chosen capillary possessed a slightly elliptical bore; both ends were ellipses of almost identical eccentricity. The correction factor for elliptical bore capillary tubes is easily calculated (7), and for this capillary tube was not over 0.1%. The capillary tube was attached to the bellows by a pair of aluminum connectors. These connectors compressed Teflon packing against the capillary tube and a Teflon gasket against the bellows, to form a rigid leakproof connection between the capillary and the bellows. Framework. The rigid framework was constructed from two l/d-inch brass plates turned to a diameter slightly under 2 inches, so as to fit easily into a bomb 2 inches in inside diameter. A l / 4 28NF thread was tapped in the center of each plate to receive the bellows. Holes were drilled near the edge of these plates to receive the four l/d-inch brass connecting rods. The connecting rods were threaded a t each end to receive the nuts which hold the end plates in
place. This framework was rigidly connected to the head of the bomb by two screws which press the end of the connecting rods tightly against the head of the bomb. Construction of the framework is shown in Figures 1 and 2. Retractor. The retractor used to pull the bellows away from the neutral position was made from I/d-inch brass plates separated by four rods (Figures 1 and 2). The inner plate was slotted to clear the bellows mechanism when not in use but to make firm contact with the end of the bellows during retraction. The outer plate was slotted to hold the shaft that moved the whole retractor back and forth. This shaft was threaded to fit inside a threaded nut in the end of the bomb; turning the retractor causing the shaft to move back and forth. Electrical Contacts. The form of the electrical contacts proved very critical and a suitable assembly was found after much trial and error. These contacts were used to determine the precise moment during a run when the moving capillary tube passed two fixed reference points. The moving contact was 'attached to the capillary tube by a split Lucite block and consisted of a 3/4-inch 14-32 brass bolt machined for half its length to a diameter slightly smaller than the root diameter of the threads and then polished. The head of this bolt was connected electrically to a coil of fine enameled wire which led out of the head of the bomb. This wire was long, light, and limp enough to offer no measurable resistance to the movement of the capillary during a run. The fixed contacts consisted of two pieces of '/la-inch brass sheet (Figures 1 and 2). The edges that could touch the movable contact were filed to a knifeedge, then polished. These knife-edges were necessary because an earlier design using rounded edges caused excessive chattering when the contact was made or broken. The two fixed contacts were mounted on a small, Lucite block by pins, to pivot freely about the pins. The two contacts were held together by an exceedingly weak beryllium-copper M spring and held apart by a brass pin mounted on another part of the framework ~
Table I.
Viscometer Constants at
Capillary radius, cm. Capillary length, cm. Bellows constant, cm.S/g. Bellows assembly volume, cc. Low pressure side volume, cc. Total viscometer volume, cc. Initial differential pressure, g./sq. cm. Final differential pressure, g./sq. in. Average differential pressure, g./sq. cm. Viscometer donstants Micropoise/sec. Lb./sq. inch abs. Micropoise/sec./cc./g.
~
~
~~~
25" C.
T
L S
Vi
Y'
VT
API APz
ap Ki
K2 KZ
VOL. 49, NO. 12
0.002695 i= 0.000001 11.349 f 0.003 0.002386 0.000004 11.15 i 0 . 0 1 537.1 i 0.5 548.4 i 0.5 137.688 0.010 28.904 f 0.010 64.475 rt 0.007
*
*
0.4446 f 0.0009 67.42 f 0.21 1063 i 5
DECEMBER 1957
2027
(Figures 1 and 2). This separator pin was machined to such a diameter that the two knife-edges, when not displaced by the capillary contact, were held perfectly parallel to one another. The separator pin was grounded electrically. This allowed the capillary tube to move beyond the points where its contact touched the fixed contacts. Because the M spring was so weak, the push of the fixed contact against the capillary's contact did not interfere significantly with the force exerted by the bellows, and no abrupt change in capillary flow occurred when the contacts parted. Bomb. The viscometer was enclosed by a standard Type 406-2OJ3 15,000 pounds per square inch Aminco high pressure stainless steel bomb 2 inches in inside diameter by 10 inches. The head was machined to receive one superpressure electrical connector and two '/d-inch superpressure tubing connectors. The bottom of the bomb was modified to allow the reactor shaft to pass through it. The bomb was threaded to engage a heavy nut which contained the threads for the retractor shaft and a 30,000 pounds per square inch type packing gland. The heavy nut was sealed to the bomb by a copper gasket. Temperature. The viscometer and bomb were maintained at constant temperature during a run inside an Aminco Type 4-3352 subzero constant temperature test cabinet. Although a thermostat could not control the temperature inside the cold box any closer than & l o F., the large heat capacity of the bomb prevented the viscometer temperature from oscillating more than 0 . 2 O F. Temperatures were measured by copper-Constantan thermocouples accurate to f0.2' F. These thermocouples were calibrated against the dry ice point, freezing point of mercury, freezing point of carbon tetrachloride, and ice point, and compared with a Bureau of Standards calibrated thermometer a t room temperature. Plots for each couple correlated the differences from the 1938 Bureau of Standards calibration against temperature. Pressure. The pressure in the viscometer was measured primarily on an Aminco Type 460-440 dead weight gage, which was sensitive to less than 1 pound per square inch and claimed by the
Precision of Gages Range, Sensitivity, Accuracy, Lb./Sq. Lb./Sq. Lb./Sq.
Type Inch Dead weight 500-15,000 Bourdon 0-1,000 0-5,000 0-15,000
Inch
1 1 1 2 i 5 110
Inch
..
. c
1.01
,
' I
0.99
-60
I
I
I
l
I
I
1
-20
0 'C
I
l
-40
!
Temperature,
manufacturer to be accurate to within 0.1% of the pressure indicated. Three Bourdon gages were also used, carefully calibrated against the dead weight gage and adjusted to agree with it. Other apparatus included an Aminco Type 406-194 pressure booster, an electronic relay, and an electric clock reading to 0.1 second. A snubber consisting of 16 feet of 20-gage hypodermic needle tubing prevented too rapid pressure changes during compression and decompression. A dry ice trap was used to remove water and carbon dioxide from the gas. All lines were constructed from '/d-inch 16-gage stainless steel tubing. All fittings were Aminco 15,000 pounds per square inch series superpressure fittings. All values were Aminco 25,000 pounds per square inch superpressure values. Density. The densities of the several gases were obtained from the literature. For methane and nitrogen the values of Perry (74) and for helium those of Weibe, Gaddy, and Heins (27) were used.
!
,
4
I
40
20
Assuming further that P remains essentially constant during a run, or that P >> AP, and integrating,
where
The total volume of the system is constant,
v, =
V + V'
(10)
and at constant temperature and conipressibility factor, PiV, = P V + P'V'
(11)
Defining the bellows constant s = J7 =
(dV/dAP) Vi s AP
(12)
+
(13)
Viscometer Equation
Combining Equations 10, 11, and 13
The equation which describes the behavior of this viscometer can be derived as follows. In a properly designed viscometer, fluid flow will always be laminar, and Poiseuille's law will be followed ( I ) .
f'ivi = p V + ( p - A p )
(3)
As the density of the gas flowing through the capillary will vary from one end of the capillary to the other, a velocity gradient will exist along the capillary at any given time. But the mass velocitv will be constant or
Assuming isothermal flow and a constant compressibility factor in the capillary,
f l to 110 1 2 * 5
* 10 Since
2028
I."
tion of bellows constants
INDUSTRIAL A N D ENGINEERING CHEMISTRY
=
P ( V + V ' ) - V ' A P = PV, P = Pi -t- ( V ' / V , ) AP
-
V'AP (14)
Substituting Equations 13 and 14 in Equation 8, (Pi (Pi
~ A P I (Vi ) f
s
APi)
V'
4-yiAP,) ( V , f APz)
(7
Tr4P
z e
= ___ 8 ,uL
+ $ p i ) ( A p l - A p 2 )+
Rearranging and solving for p
-
(1.51
HIGH PRESSURE ~ e l i o v scodstant,
s,
mM
c m 5 / ~ m x 104
8
T
(L
+ nr)
A(PV) =
m (8 r ( L f nr)
Coefficient m was evaluated theoretically by Bousinesq (2) to be to a first approximation, m = 1.12. Very accurate experiments gave a more precise value for the flow of gases in capillary tubes having plain, sharp ends, m = 1.124 f 0.006 (77). Modifying Equation 17 to account for the Couette and kinetic energy corrected yields
0
2
1
3
4
5
6
Tlrne, Seconds x
Figure 4.
Viscometer calibration
where n = 1.146 and m = 1.124. As for this viscometer everything in Equation 19 is constant except 1.1: 8, Pi, and pi, it can be written as I n deriving Equation 17 two factors have been neglected. The first is the Couette correction, which compensates for the additional flow resistance of the gas in the bellows (7). As the gas approaches the capillary inlet, it does so along a family of trumpet-shaped streamlines, which offer a flow resistance equivalent to a lengthening of the capillary tube by a quantity, nr, so that L in Equation 17 should be replaced by (L nr). I t has been shown experimentally that, to a first approximation, n = 1.146. The second factor is a so-called kinetic energy correction, necessary because it was assumed in arriving at Equation 17 that the entire pressure drop was used to overcome viscous resistance. Actually, part is converted into kinetic energy to be dissipated in the exit bellows, and part is consumed in establishing a parabolic velocity gradient in the capillary tube. Much previous work has shown that these factors cause the viscosity calculated by Poiseuille's law to be too high by the term (72)
+
mW 8
T
(L
+ nr)
For the present viscometer the kinetic correction may be modified as follows :
where K I , K2, and K B are defined by Equation 19. As all factors entering in K1, K2, and K3 can be evaluated from the physical dimensions of the apparatus, this type of viscometer yields absolute rather than relative values for viscosities. When this viscometer is used to evaluate the viscosity of liquids, the pressure correction. term in Equations 19 and 20 drops out-that is, in effect, Pi = a. Recalling the definition of $ = -d . .V . - E -AV
d LIP- AP Equation 19 reduces to r r 4 APB
' = 8 (L + nr) AV
mp AV 8
T
(L
+ nr) 0
Calibration Calibrations summarized in Table I define the numerical values of constants K1, Kz, and K3 of Equation 20. Capillary Length, L. The length of the capillary tube was measured in a large precision micrometer. The length, when adjusted to 25' C., was 4.468 i 0.001 inches or 11.349 & 0.003 cm. Capillary Radius, t. The average radius of the capillary was measured in three ways:
1 . Microscopic examination of capillary
ends
2. Resistance of mercury 3. Viscosity of water
These methods gave values of 27.0, 26.96, and 26.94 microns, respectively. Microscopic examination, the least accurate method employed, involved measuring the average diameter a t each end of the capillary and assuming that the capillary bore was uniformly elliptical and tapering between the two ends. The electrical resistance of the capillary when filled with triply distilled mercury served as an accurate measure of its radius. The specific resistance of the mercury was determined a t the same time a t the same temperature on the same precision Wheatstone bridge, using a larger capillary with radius accurately known. Although this assumes that the effective radius for electrical flow, (Sr2u!L)lI2/L, is the same as the effective radius for fluid flow, (Sr4dL)'/4/L, due to the relative uniformness of the capillary's bore, the error was negligible (7). The viscosity of distilled water was also used to measure the effective radius of the capillary. By incorporating the capillary in an Ostwqld-type viscometer and taking the usual precautions, replicate measurements gave 26.94 microns a t 25' C. based on 0.8727 centipoise for the viscosity of water at 79.0' F. An average value of 26.95 f 0.01 microns was used as the most probdble value for the capillary's effective radius. Volume of Bellows Assembly, Vi. The volume of the low pressure part of the viscometer was determined by weighing the bellows system after assembly but before connection to the bomb head both empty and full of water. The volume of the rest of the high pressure side of the bellows assembly (bomb head, bypass line, and valve) was determined by filling it with mercury and weighing. In both cases, it was filled with liquid under vacuum to eliminate air bubbles. Volume at the neutral point, Vj, calculated as 11.15 f 0.01 cc. a t 25' C. Total Volume of Viscometer, V,. This volume was obtained by filling the assembled unit with air at pressures of 1.5 to 2 atm., then carefully bleeding off the gas into a gas buret. System pressures were measured on a mercury manometer to the nearest 0.02 inch. Careful control and replication of the measurements gave a total volume corrected for the volume of air in the manometer and lead lines of 548.4 f 0.5 cc. at 25O C . As a check, Vt was calculated from the dimensions of the apparatus as 546 cc. T h e value of 548.4 f 0.5 cc. was chosen as the more accurate estimate of V,. By difference, V' was 537.2 cc.; therefore, V' becomes 537.1 cc. when corrected for the average displacement of the bellows assembly VOL. 49, NO. 12
DECEMBER 1957
2029
500
during an actual run. Bellows Constant, s. The bellows constant, s = ( d V / d A P ) , was measured by rigidly connecting the bellows to a precision pipet. The bellows and pipet were in turn connected to the short end of a J-shaped open manometer such that the bellows and pipet lay horizontally. The entire assembly was evacuated and sufficient water was introduced to fill the bellows and part of the pipet. Mercury was introduced to seal the bottom of the J. All bubbles were then carefully worked out of the system. Sufficient mercury was added to move the mercury-water interface to the zero mark of the pipet, and the height of the mercury in the open part of the manometer was noted. Sufficient mercury was then added to move exactly 0.05 cc. of water into the bellows and the mercury level was again noted. Mercury was now withdrawn back to the zero mark on the pipet to check for hysteresis. Readings were taken a t 0, 0.05, 0.10, 0,0.15,0, etc., until the first sign of hysteresis was noted a t about 0.7 cc. The bellows were then recalibrated up to 0.5 cc., and thereafter no bellows were allowed to be stressed beyond an increase in volume of 0.5 cc. Plots were prepared relating change in volume to applied pressure. The slopes of these lines were the bellows constants, s. All the bellows used gave the same V - AP relationship within experimental accuracy. s is not a linear function of the applied pressure (Figure 3). The value of s used in the
450
400
350
300
250
Figure 5. Viscosity of nitrogen
200
150
0
5
6
7
10-
9
8
10-3
and stainless steel. Changes in the elasticity of the bellows can be estimated with sufficient accuracy from the temperature coefficient of Young's modulus for beryllium copFer, because, to a first approximation, the bellows constant, s, is inversely proportional to Young's modulus in any given system. Pressure would be expected to alter only the length and radius of the capillary tube in the total volume system. These pressure effects were estimated by the Lam6 formulas for thick-walled cylinders to be about 0.01 to 0.02y0 a t 10,000 pounds per square inch. Hence, K1, Kz, and Ks may be assumed independent of pressure. Numerical Values for K,,K,, and K,. The numerical values for K1, K z , and K 3 of Equation 20 are obtained from the calibrations described. At 25' C. these constants are: for p in micropoises, Pi in pounds per square inch absolute, pi in grams per cc., and 0 in seconds. K1 = 0.4446, Kz = 67.42, K I == 1063; thus, a t 25' C.
capillary. The initial differential pressure, AP1) was measured as the electric clock was started by the relay connected to the electrical contacts of the viscometer. Differential pressure was measured periodically until the clock stopped, when final differential pressure, APz, was measured. The average differential pressure across the bellows during a run, D,was determined graphically from a plot of differential pressure against time (Figure 4). Replicate determinations of AP1 and APz were then made to tie down the numerical values of these quantities more precisely. Average values were: AP1 = 137.688 grams per sq. cm.
APz = 28.904 grams per sq. cm. A P = 64.475 grams per sq. cm.
Effect of Temperature and Pressure on Viscometer Calibration. Constants K,, Kz, and K3 of Equation 20 vary with temperature, as shown in Figure 3, because of changes in the physical dimensions of the viscometer and the elasticity of the bellows. Dimensional changes with temperature can be estimated with sufficient accuracy from recorded thermal expansion coefficients for borosilicate glass, brass, beryllium copper,
0.4446 6
' = (1 + 67.42/P,)
-
1063
D;
(1 -t 67.42/PL)
e Operating Procedure. In operating this viscometer over a range of tempera-
Open Points: This InveStigBtion
I
600
450
" 400
tc
v1
I
d
350 1-
1
0
a $300
I
1
! I
500
0
6000
1 I
6
*
I
I
400
s I
1
2
1
+
300
4
VI
z
.2
4
3
700
500
2
2
pressure, PSIG
ds. was calviscometer equation , :AS culated by numerical integration to be 0.002386 cm.5 per gram. Differential Pressures, AP1, APz, and .G. The differential pressures were obtained by hooking up a water manometer across the assembly and applying a small excess pressure to the high pressure side of the capillary and bellows assembly. This pressure was allowed to dissipate itself through the
n
1
0
2
250
100
200
150
200
c
0 -50
-25
0
25
50
75
0
I
2
3
4
5
6
7
pressure, PSIG x
Temperature, *C
Figure 6 . Viscosity of nitrogen
2030
INDUSTRIAL AND ENGINEERING CHEMISTRY
Figure 7. Viscosity of methane
8
9
10
HIGH PRESSURE 7QQ
3
2
L
d $
0
0
E
-2 0
100 -50
-25
.
0
15
Temperature
.
so00
10000
Pressure, PSIG
.C
Figure 9. Agreement with other data at 25" C. contact left the first fixed contact, Results the electronic relay tripped, starting the timer. When the capillary's contact Nitrogen* The viscosity Of touched the second fixed contact, the was measured from -'0° to +250 c. relay again tripped, stopping the clock and from 500 to io,ooo pounds per and terminating a run. Pressure and 'quare inch gage and Figures and 6). Figure shows isotherms temperature were again determined, and Plotted data Obtained in this replicate rung made to verify the original data. Pressure was raised to the next investigation and to meet higher level by the pressure booster the data Of Johnston and McC1oskey pump and the run procedure repeated. (g), also the 25' C. isotherm for nitrogen obtained by Michels and Gibson (72). The two sets of data are in substantial agreement, the average deMaterials viation being about 0.5%. These deviaThe three gases whose viscosities were tions, plotted in Figure 9, indicate that measured were obtained commercially the bellows viscometer may give abin cylinders. No further purification solute values of viscosity about 1% low. was attempted, other than freezing out Figure 6 shows isobars drawn through Of dioxide and water* the present data from -50° to 25" C. and through Michels and Gibson's data from 25' to 75" C. The data are all consistent and probably highly accurate, Methane, Methane. The viscosity of 99% Mass Gas Nitrogena Heliuma Spectr. methane was determined in the interval of -50' to +25 ' C. and 1000 to 10,000 Nz 99.7 0.5 pounds per square inch gage (Table 111 0 2 0 99.5 and Figures 7 and 8). Figure 7 plots iso98.9 therms through the data obtained in this CnH8 1.1 investigation and the low pressure 602 0 0 Trace viscosity data of van Itterbeek (79). CZH4 Trace Hz 0 I t shows for comparison the 25' C. HzO 0 0 0 isotherm calculated from the viscosity a Manufacturers' bulletins. data on 99% methane determined by Carr. Carr's data determined a t 24" C. were adjusted to 25' C. using values of (b,~/UiaT)~ calculated from Figure 8 and
Figure 8. Viscosity of methane tures and pressures, it was much easier to vary pressure than temperature; one set of data was usually obtained a t one temperature in increasing increments of pressure and the next a t the next lower temperature in decreasing increments of pressure. I t was thus often possible to determine one isotherm in one day. Before a series of tests, the bomo was cooled to within 0.2' F. of the desired temperature, and the thermostat on the cold box was set a t this value. Closer control was not possible nor necessary. Foreign gases in the unit were removed by several evacuations and flushings with the gas whose viscosity was to be 1 were opened, determined. ~ 1 valves and gas was slowly admitted from the until the desired pressure had been reached. The unit was then allowed to reach temperature equilibrium and the pressure was carefully adjusted a t the dead weight gage to within 1 pound per square inch of the desired pressure. T o start a run the valve isolating the bomb from the rest of the system was closed. The bellows' retractor was pulled back and the bypass valve between the two bellows was closed. The timer used to measure the duration of the run was set to zero, and the retractor was quickly moved back to its original position, out of the way of the capillary and bellows assembly. After several minutes, when the capillary's
r
-1
... ... ... ... ... ...
:g4
\ 220
... ... ... ... ...
... ...
210
170
160
0
1
2
4
3
5
6
7
8
9
10
pressure, PSIG x
Figure 10.
Viscosity of helium
Figure 11. Viscosity of helium
Tcmpsraturs,
VOL. 49,
NO. 12
.C
DECEMBER 1957
2031
Table II.
Viscosity of Nitrogen
Press., Lb./Sq.
Temp., O
c.
25.0
0.0
Inch Gage
Run
Time, No. Density, Viscosity, See. Detns. G./Cc. ppoise
500 675 1,000 1,500 2,000 3,000 4,000 5,000 6,000 8,000 10,000
470.8 459.4 456.9 472.4 485.3 535.2 587.5 653.2 707.1 825.6 94L.5
10 6 8
500 1,000 1,500 2,000 3,000 4,000 5,000 6,000 8,000
435.6 434.5 443.0 469.3 526.8 591.0 661.0 723.8 856.8 993.0
10,000
4 2 2 2 2 2 2
0.0402 0.0539 0.0792 0,1177 0.1541 0.2212 0.2794 0.3304 0.3700 0.4371 0.4875
185.0 186.0 190.3 200.8 208.5 232.3 256.4 286.0 310.3 363.4 415.2
2 2 2 2 2 2 2 2 2 2
0.0445 0.0884 0,1315 0.1731 0.2468 0.3082 0.3615 0.4032 0.4718 0.5217
172.7 182.5 190.1 203.5 230.8 260.4 292.3 320.7 380.8 442.2
4
extrapolated to 10,000 pounds per square inch gage from the maximum pressure reached in the Carr experiments, 8000 pounds per square inch gage. Again, as with nitrogen, the 25' C. isotherm of this and previous investigations agrees closely. As shown in Figure 9, the average difference is the bellows viscometer about -0.5%, reading low. Figure 8 shows isobars through the present data. Helium. The viscosity of helium was measured over the same range of temperature and pressure as methane (Table IV and Figures 10 and 11). The isotherms of Figure 10 were extrapolated to 0 pounds per square inch gage, to meet as closely as possible the low pressure viscosities determined by Johnston and Grilly ( 8 ) . The present data appear to be the first high pressure measurements made on the viscosity of helium above 1000 pounds per square inch. These data show, as in Figure 10, that the pressure coefficient of helium viscosity is low. Whereas at 25" C. nitrogen shows an average viscosity increase of 5 to 8% per 1000 pounds per square inch, helium shows an increase of less than 1%. This diminished pressure effect is not too surprising because helium should, more than any other gas, approach the assumptions of the elementary kinetic theory-that the molecules are hard, nonattracting spheres of negligibly small volume-and come close to its conclusions-that viscosity is independent of pressure at constant temperature. Generalized Correlation. Since the publishing of Carr's generalized correlation ( 3 ) , which plots the ratio ( p j p ~ ) as a function of the reduced temperature
2032
Maximum N RX ~ 10-2
2 2
Press,. Temp., C.
-.25.1
3
5 6
7 7 7 6 6 5 2 4 6
-49.8
7 8 8 7 7 6 5
RUR
Lb./Sq.
Inch Gage
Time. Sec,
Detns.
500 1,000 1,500 2,000 3,000 4,000 5,000 6,000 8,000 1LO, 000
405.0 408.7 420.2 450.6 523.1 590.6 667.7 744.0 892.5 1041.5
2 2 2 2 2 2 2 2 2 2
0.0497 0.0992 0.1481 0.1945 0.2770 0.3455 0.3996 0.4435 0.5071 0.5497
161.9 173.3 181.8 197.1 231.3 262.7 298.1 332.9 400.6 468.3
500 1,000 1,500 2,000 3,000 4,000 5,000 6,000 8,000 10,000
379.0 390.8 405.5 438.4 522.4 604.6 684.8 769.3 933.5 1101.7
2 2 2 2 2 2
0.0558 0.1139 0.1737 0.2288 0.3237 0.3837 0.4471 0.4901 0.5516 0,5952
152.7 167.1 176.9 193.4 232.9 271.2 308.3 347.3 422.8 500.0
and pressure, the present data on helium and the high pressure data of Michels, Schipper, and Rintoul (73) on hydrogen and deuterium have made it possible to extend the generalized correlation beyond its previous limits of T , = 3, P, = 50, to beyond T , = 50, P, = 300. In this extreme pressuretemperature range, effect of pressure and temperature is notably less than near the critical point. To make an easily readable plot, the data have been replotted. Figure 12 shows clearly that the effect of pressure on viscosity is greatest near the critical temperature of a gas, and that as the temperature increases above the critical, the effect diminishes rapidly. At temperatures sufficiently removed from the critical, the viscosity of a gas is almost independent of pressure 12), a t a temperature ten -(Figure
No.
2
2 2 2
Density, Viscosity, G./Cc. ppoise
Maximum N R X~ IO-* 3
5
7 8 9 8 8
7 5 4
4
7 9 10 10
9 8
7 5 4
times the critical, viscosity is still within 1yo of atmospheric viscosity at a pressure tweIve times the critical, and at 350 times the critical temperature and I000 times the critical pressure, the viscosity is within 1% of atmospheric. Conclusions
With a new type of viscometer the viscosities of methane, nitrogen, and helium, determined between - 50' and 25' C. and at pressures up to 10,000 pounds, per square inch gage, show good agreement with previously published data where comparison is possible. The accuracy appears to be within 1%, the precision within 0.5%* These, with literature data were used to extend a plot of the pressure coefficient of gas viscosity as a function of reduced temperature and
Figure 12. Generalized viscosity correlation for high reduced temperatures and pressures
INDUSTRIAL AND ENGINEERING CHEMISTRY
1
2
4
6
10
20
40
60
100
Reduced P r e s s u r e , PIPc
200
400 600 1000
HIGH PRESSURE
Temp.,
c.
25.0
0.0
-25.0
-50.2
Table 111. Viscosity of Methane Press., MaxiVismum Run Lb./ Time, No. Density, Sq. Inch cosity, N R X~ Sec. Detns. G./Cc. ppoise 10-2 Gage 320.6 372.9 453.9 536.1 691.8 823.2 908.1
2
0.0509 0.1087 0.1621 0.2016 0.2525 0.2835 0.3063
133.5 160.1 197.0 234.0 303.8 359.7 400.7
5 7 7 6 5 4 3
25.0
2 2 2 2 2 2
1,000 2,000 3,000 4,000 6,000 8,000 10,000
309.0 398.7 503.0 602.8 781.9 930.6 1044.8
2 2 2 2 2 2 2
0.0587 0.1315 0.1919 0.2317 0.2757 0.3052 0.3257
129.8 172.8 220.4 265.7 346.7 413.9 465.5
6 7 7 6 4 3 3
0.0
1,000 2,000 3,000 4,000 6,000 8,000 10,000
292.2 436.5 590.8 716.5 913.1 1085.3 1220.1
2 2 2 2 2 2 2
0.0712 0.1696 0.2296 0.2637 0.3021 0.3268 0.3453
123.8 190.9 261.4 318.9 408,.8 487.4 548.9
7 8 6 4 3 2 2
-24.8
1,000 2,000 3,000 4,000 6,000 8,000 10,000
285.4 580.6 769.3 897.7 1109.9 1325.1 1415.3
2 2 2 2 2 2 2
0.0987 0.2298 0.2761 0.3005 0.3280 0.3506 0.3648
121.9 256.4 343.7 403.4 501.7 600.9 642.8
11 6 4 3 2 2 2
Nomenclature
Kl, Kz, Kg = viscometer constants defined by Equations 19 and 20 L = length of capillary tube M = molecular weight kinetic energy constant Couette approach constant = flow rate through capillary, moles per unit time P = pressure; static pressure on high pressure side of viscometer P‘ = static pressure on low pressure side of viscometer = average static pressure in viscometer during run = critical pressure = initial static pressure in viscometer before run = static pressure a t start of run = static pressure a t end of run = reduced pressure = P/P, 0 = volumetric flow rate through capillary, volume per unit time R = gas constant r = average effective radius of capillary tube s = bellows constant m
= =
c.
1,000 2,000 3,000 4,000 6,000 8,000 10,000
pressure to reduced temperatures of 50 and reduced pressures of 300. The new viscometer is an absolute rather than a relative instrument, is simply and easily constructed, uses readily available and inexpensive parts, and produces accurate, precise data speedily and easily. I t will operate a t any temperature where the bellows retain their elasticity, and a t any pressure above atmospheric, but is most suited for much higher pressures.
I
Temp.,
Table IV. Viscosity of Helium Press., RUIl Lb./ Vi5 Sq. Inch Time, No. Density, cosity, Gage Seo. Detns. G./Cc. ppoise
Maximum NRe
10-2
x
1,000 2,000 3,000 4,000 6,000 8,000 10,000
477.4 461.0 461.7 461.1 464.7 481.2 488.1
4 4 4 4 2 2 2
0.0109 0.0211 0.0306 0.0396 0.0563 0.0715 0.0853
199.0 198.3 200.7 201.5 204.2 210.3 215.4
