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The viscosity of carbon dioxide between 0°C and 75°C and at pressures up to 2000 atmospheres
Michels, A. B o t z e n , A. S c h u u r m a n , W. 1957
PhysicaXXlII 95-102
T H E V I S C O S I T Y O F C A R B O N D I O X I D E B E T W E E N 0°C A N D 75°C A N D A T P R E S S U R E S U P TO 2000 A T M O S P H E R E S by A. MICHELS, A. BOTZEN and W. SCHUURMAN 155th publication of the Van der Waals Fund Van der Waals-laboratorium, Gemeente Universiteit, Amsterdam, Nederland Synopsis
T h e v i s c o s i t y of c a r b o n d i o x i d e h a s b e e n m e a s u r e d w i t h a t r a n s p i r a t i o n m e t h o d a t t e m p e r a t u r e s b e t w e e n 0 ° a n d 75°C a t p r e s s u r e s u p t o 2000 a t m o s p h e r e s . P r e v i o u s m e a s u r e m e n t s d e s c r i b e d earlier r e v e a l e d a n u n e x p e c t e d i n c r e a s e of t h e t h e r m a l c o n d u c t i v i t y of a r g o n r o u n d t h e c r i t i c a l d e n s i t y . T h e r e f o r e special a t t e n t i o n h a s b e e n g i v e n t o t h e c r i t i c a l r e g i o n a l t h o u g h t h e t r a n s p i r a t i o n m e t h o d is n o t p a r t i c u l a r s u i t a b l e for t h i s region. T h e v i s c o s i t y of CO2 a b o v e 40°C s h o w s t h e n o r m a l t r e n d w~.th pressure, b u t d e v i a t e s considerably from Enskog's theory. In the critical region the viscosity shows a similar a b n o r m a l b e h a v i o u r as f o u n d for t h e h e a t c o n d u c t i v i t y of argon. V i s c o s i t y m e a s u r e m e n t s of a r g o n d i d n o t s h o w t h i s effect b u t in this case t h e t e m p e r a t u l e s r e l a t i v e t o t h e critical t e m p e r a t u r e were h i g h e r t h a n i n t h e p r e s e n t m e a s u r e n m n t s of COs. T h e r e s u l t s a r e c o m p a r e d w i t h t h o s e of o t h e r a u t h o r s .
§ 1. Introduction. The viscosity of carbon dioxide was measured by the transpiration method, which has been described previously 1)2)3)4), at temperatures of 0 °, 25 °, 29.90 °, 31.10 °, 32.20 °, 34.60 °, 40", 50 ° and 75°C. The highest measured pressure was about 2000 atmospheres attained at 75°C. The density corresponding to each temperature and pressure was calculated using the compressibility isotherms of M i c h e l s , M i c h e l s , W o u t e r s and B l a i s s e 5) 6) 7) 8). The C09 used was prepared from commercial gas by a three-stage distillation process 9). § 2. The results. The results of the measurements are given in tables I to IX. The pressure P is expressed in international atmospheres, the density, ~, both in Amagat units and in g/cm 8 and the coefficient of viscosity, 7, in g/cm sec. TABLE I The viscosity of CO2 at 0°C •P
int. atm. 9.148 10.43 24.62 27.26 30.24 32.32
~
~
Am. 9.712 11.20 30.41 34.82 40.35 44.63
g/cm 3 0.01920 0.02214 0.06012 0.06884 0.07977 0.08823
-- 95 --
r / . 105 g/cm sec. 13.85 13.86 14.21 14.34 14.49 14.65
96
A. M I C H E L S , A. BOTZEN A N D W . S C H U U R M A N
T A B L E II
TABLE III
The visco sity of CO2 at 25°C
The viscosity of CO2 at 29,90°C
P
Q
~
int. arm.
Am.
g/cm3
10;44 31.27 44.17 57.21 78.51 98.26 111.6
0.02064 0.06182 0.08732 0.1131 0.1552 0.1943 0.2206
366 373 387 422.2 457.0 491.7 527.3 563.3 596.3
0.724 0.737 0.765 0.8347 0.9035 0.9721 1.042 I.I14 1.179
10.82 28.74 37.60 44.98 54.04 59.60 62.05 65.06 67,32 74.59 109.6 180.5 307.3 522.9 864.3 1323
~] . 105 g/cm sec.
15.07 15.51 15.90 16.38 17.54 18.94
20.13 61.3 60.6 64.8 76.06 92.06 112.1 138.6 173.9 218.8
P int. atm. I 1.22
30.23 38.56 47.59 56.84 62.88 65.57 69.13 70.67 71.03 77.73 95.41 136.1 214.7 350.0 574.6 926.6
Q Am. 10.64 32.32 44.05 59.42 80.28 99.7 III.6 136.3 159 171 352 386.9 422. I 457.2 491.8 527.3 563.2
~ g/cm 3
~] • lOs g/era sec.
0.02104 0.06390 0.08709 0.1.175 0.1587 0.1971 0.2207 0.2694 0.314 0.338 0.696 0.7649 0.8345 0.9039 0.9723 1.042 1.113
15.32 15.79 16.14 16.78 17.87 19.20 20.12 22.55 27.1 29. l 55.6 65.12 76.56 92.49 12.4 38.9 73.7
T A B L E IV
TABLE V
The viscosity of CO2 at 31.10°C
The viscosity of CO2 at 32.20°C.
P
~
O
int. atm.
Am.
g/cma
I0.59 32.78 44.27 58.74 78.79 98.53 Ill.4 132.6 154 170 185 199 204 212
0.02094 0.06481 0.08752 0.1161 0.1.558 0.1948 0.2202 0.2622 0.3045 0.336 0.366 0.393 0.403 0.419 0.437 0.439 .0.461 0.484 0.492 0.504 0.520 0.528 0.530 0.563 0,593 0..696 0.7645 0.8339 0.9033 0.9725 1.042 1.114 1.179
I 1.22
30.77 38.96 47.60 56.84 63.31 66.34 69.77 70.70 72.42 72.80 72.91 72.92 72.95 72.95 72.95 72.96 72.96 72.96 72.96 72.97 72.97 72.97 73.20 73.67 81.55 100.3 141.8 222.0 360.5 586.9 942.8 1416
221 222 233 245 249 255 263 267 268 285 3OO 352 386.7 421.8 456.9 491.9 527.3 563.3 596.3
~] • 105 g/cm sec.
15.36 15.86 16.22 16.81 17.84 19.08 20.15 22.12 25.5 28.7 32.3 35.2 49 53 59 6O 68 71 74 76 78 8O 8O 66 5O 55.3 65.07 76.53 92.45 112.5
138.7 173.4 216.9
P
~
~
int. atm.
Am.
g/cm 8
I 1.22
30.97 39.16 47.60 56.84 64.06 66.24 66.90 70.91 70.92 72.56 73.65 74.05 74.37 74.45 74.60 74.69 74.74 74.77 74.84 74.94 75.12 75.14 75.60 76.59 85.17 104.9 148.9 229.9 369.7 597.5 955.6
10.54 32.84 44.23 58.15 77.60 98.90 107.5 I I0.3
134.0 134.1 150 168 178 192 196 208 220 225 230 241 255 269 271 288 307 352 386.6 422.3 457. I 491.9 527.2 563.4
0.02084 0.06492 0.08744 0.1150 0.1534 0.1955 0.2125 0.2181 0.2649 0.2651 0.297 0.332 0.352 0.380 0.387 0.411 0.435 0.445 0.455 0.476 0.504 0.532 0.536 0.569 0.607 0.696 0.7643 0.8349 0.9037 0.9725 1.042 I.I14
~] . 105 g/cm see.
15.42 15.92 16.27 16.86 17.83
19.23 19.96 20.07 22.25 22.32 24.8 27.5
29.5 32.3 33.1 42.5 47.3 49.0 50.3 52.2 55.8 56.8 56.5 47.7 47.2 55.4 65.05 76.75 92.55 12.4 38.6 73.5
VISCOSITY
OF CARBON
TABLE
DIOXIDE
AT PRESSURES
TABLE
VI
P int. atm.
Q Am.
0 g/cm 3
'q" 105 g / c m sec.
76.78 77.62 78.30 78.91 79.53 80.76 83.21
172 190 212 236 260 287 318
0.340 0.376 0.419 0.467 0.514 0.567 0.629
28.4 31.4 34.7 42.3 45.4 46.7 48.4
VII
T h e v i s c o s i t y of COo. a t 4 0 . 0 0 ° C P int. a r m . II.22 32.12 40.82 49.88 60.94 68.65 72.28 77.72 81.07 83.16 84.69 86.09 87.06 89,01 90.53 93.04 96.46 102.0 108.7 III.2 138.5 193.1 284.3 436.7 437.2 680.8 1056.5 1551
@ Am.
@ g/cm 3
~1" 105 g / c m sec.
10.23 32.85 44.28 58.12 79.31 99.03 II0,8 134.0 154.8 172.5 188.7 206.7 225.9 248.9 267.1 289.8 308.1 329.3 346.9 351.7 386.8 423.1 457.3 492.0 492.1 527.4 563.4 596.5
0.02022 0.06494 0.08754 0.1149 0.1568 0.1958 0.2191 0.2649 0.3060 0.3410 0.3731 0.4086 0.4466 0.4921 0.5281 0.5729 0.6091 0.6510 0,6858 0.6953 0.7647 0.8365 0.9041 0.9727 0.9729 1.043 I.I14 1.179
15.79 16.33 16.69 17.24 18.32 19.59 20.42 22.33 24.46 26.54 28.53 31.05 33.42 36.60 39.39 43.00 45.69 49.80 53.84 55.36 64.99 76.07 91.12 111.5 112.3 137.7 170.7 216,1
Physica XXIII
2000
97
ATM
VIII
T h e v i s c o s i t y of CO 2 a t 50°C
T h e v i s c o s i t y of CO2 a t 3 4 . 6 0 ° C
TABLE
UP TO
P int. a t m . 11.22 32.88 42.96 51.82 65.16 75.20 79.07 85.74 90.98 94.31 97.42 99.70 103.3 103.8 107.2 110.9 115.9 122.5 131.7 143.7 146.3 182.0 246.8 350.0 521.4 785.0 1183 1699
0 Am.
@ g/cm s
~]" lOs g / c m sec.
9.863 32.05 44.35 56.69 79.17 101.1 111.3 132.5 153.6 169.9 187.3 201.0 223.9 226.8 247.2 266.5 287.8 308.5 329.2 348.8 352.2 386.6 422.3 456.4 491,9 527.4 563.4 596.3
0.01950 0.06336 0.08768 0.1121 0.1565 0.1999 0.2200 0.2620 0.3037 0.3354 0.3703 0.3974 0.4427 0.4484 0.4887 0.5269 0.5690 0.6099 0.6508 0.6896 0.6963 0.7643 0.8349 0.9023 0.9725 1.043 1.114 1.179
16.26 16.79 17.21 17.67 18.81 20.17 20.90 22.54 24.45 25.91 27.99 29.47 32.41 32.92 35.65 38.48 41.95 45,64 50.11 54.58 55.65 65.27 77.11 92.30 I12.9 137.8 172.0 216.9
TABLE
IX
T h e v i s c o s i t y of CO z a t 75°C
P int. a t l n .
0 Anl.
"11 g/cm s
12.16 37.99 49.00 61.73 76.28 89.16 95.82 106.9 I15.8 123.5 129.4 136.1 136.7 144.9 145.7 154.9 163.3 175.5 189.9 208.4 230.1 293.1 386.1 544.3 733.5 I046 1498 2070
9.843 33.69 45.43 60.53 80.35 100.9 112.8 135.0 155.0 173.7 188.7 205.7 207.1 227.5 229.3 250.2 267.1 288.0 308.5 329.1 348.3 386.8 422.2 460.8 491.8 527.5 563.4 596.5
0.01946 0.06659 0.08982 0.1197 0.1589 0.1995 0.2230 0.2689 0.3064 0.3434 0.3731 0.4067 0.4094 0.4498 0.4533 0.4946 0.5281 0.5694 0.6099 0.6506 0.6886 0.7647 0.8347 0.9110 0.9723 1.043 1.114 1.179
Q " 105 g / e r a see. 17.43 18.08 18.50 19.14 20.18 21.43 22.25 23.94 25.66 27.44 28.92 30.87 31.12 33.61 33.87 36.76 39.29 42.81 46.67 51.01 55.56 66.32 78.34 95.09 112.8 138,4 171.5 217.0 7
A. MICHELS, A. BOTZEN AND W. SCHUURMAN
98
Fig. 1 shows ~ p l o t t e d as a function of density at different t e m p e r a t u r e s and Fig. 9. gives the same d a t a as a function of pressure. I t can be seen f r o m Fig. 1 t h a t at t e m p e r a t u r e s just a b o v e the critical t e m p e r a t u r e (To = 31.04 °C) the curves h a v e a m a x i m u m n e a r the critical density (oc = 237 Amagat). 225 9
~elO s
~11 s¢¢
~OC
A
O°C
13
25 °
X
Z9.9 °
0 •
31.1 ° 3Z.Z °
@
34.b °
-I-
40 °
t i
SO ° 175
"e,
7s °
I,~l
All ttmptmturcs
1501
125 I . ~ -
C02 75t--
501-------
251-----
0
I
I
I
I
I
IOO 2OO 300 400 500 Am Fig. 1 The coefficient of viscosity vs. density.
This is evident at 31.10 °, 32.20 °, 34.60 ° and less p r o n o u n c e d at 400C. Similar b e h a v i o u r has been r e p o r t e d for the t h e r m a l c o n d u c t i v i t y - d e n s i t y curves of argon 10) n e a r the critical density. I n the critical region, however, o n l y little precision can be ascribed to
V I S C O S I T Y OF CARBON D I O X I D E AT P R E S S U R E S UP TO 2 0 0 0 ATM
99
the viscosity data. This is a consequence of the large density gradient in the capillary during the measurement (the pressure difference causing the gas flow through the capillary is of the order of ½ atm. and varies during a measurement. In the worst case at 31.10°C such a pressure difference corresponds to a density difference of 100 Amagat). ._~¢m s ¢ c |
"lOs
,ooc
/
o 3,.,o
-
.
~
!
2001
0 25°
C02
x zg.e°
.+
3 .2o
J
®t ,oo 34.60
.
J J
J
J
J
'
~ J J J~
J
~
~/
3C
50
..----~ p
400
0
800
1200
1600otto
F i g . 2. T h e c o e f f i c i e n t of v i s c o s i t y v s . p r e s s u r e .
Except near the critical pressure the viscosity-pressure curves of Fig. 2 show the same general character as the corresponding curves for nitrogenl) 2), hydrogen 3) 4), deuterium 3) 4) and argon 11). TABLE
X
author
method
pressure range and temperatures
W a r b u r g and v o n B a b o la) P h i l l i p s x4) S t a k e l b e c k is)
transpiration method transpiration method falling ball method
S c h r S e r and B e c k e r xe) N a l d r e t t and M a a s s 17)
rolling baU method oscillating disc method
C o m i n g s and E g l y xs) K i y a m a and M a k i t a xg)
transpiration method rolling baU method
45-115 atm., 32 ° and 40°C. 1-120 atm., 20 °, 30 °, 32 °, 35 ° and 40°C. 1-100 atm., - - 1 5 °, 0% 10% 20 °, 30 ° and 40°C. 1-120 atm., 20°C. around the critical pressure: 31.10 °, 31.14 °, 31.2 °, 31.5 ° and 32.0°C. 1-171 atm., 40°C. 1-100 atm., 50 ° up to 300°C in steps of 500C.
It was a t t e m p t e d to compare the experimental results with Enskog's theory of dense gases, following a procedure described previously 12). The discrepancy between theory and experiment, however, is so large for CO2
100
A~ MICHELS, A. BOTZEN AND W. SCHUURMAN
that it is of no interest to tabulate Enskog's values in detail. At a density as low as 40 Amagat the theoretical values are already double the experimental ones; the disagreement becomes still more striking at higher densities.
§ 3. Comparision with measurements o/ other authors. In the literature, viscosity data on C09. are found only at comparatively low pressures. Table X gives a survey of these measurements. A large scale plot of some of the viscosity-density curves and of the viscosity-pressure curves of the present paper is shown in figures 3 and 4 8 0 ~ g c a i¢C
Cr~
"~. 105
7S°C
II 7/ //?oo
~z
t 70 1
t I
31.1 iI iI iI
60 n iI
32.2 °
/ ' 5 0 ~
iiiII
/i /
t32o
4 0 ~
,ooo
3C
,I
1/i//,'.,"
soOi#11./
Phillips
-- .
Naldrett & Iviaass Kiyama & Makita
Warbur 9 &yon Babo Mlch¢|~lBotzcn & Sciluurmoll
20
---~Q
I0 0
%° 11 I00
I i I
200
300
i I i 4 0 0 Am
Fig. 3. Viscosity-density curves in comparison with literature.
VISCOSITY OF CARBON DIOXIDE AT PRESSURES UP TO 2 0 0 0 ATM
101
respectively. Some data of other authors are plotted in Fig. 3, others in Fig. 4. 1C
cmse~
-
-
coz
~7.lOs
10C
xX~ xxx x xX20oc x~ x
9C
xx xx xx x
x~
8C
/200 .gl aj
7C
6C
5C Jl.l c
4G
3(]-**"
--
2{3-.
75o(: ~
,,°"
.~
II
x."lt
/
~
~
oOC
10--
.... Warb~rg&yon Bobo ,ee.e.. Schrotr&B~:kcr +++++ Stak¢lbcck ~ Comings& Egly Midw~,pBot.z¢n&Schuurmon
> P
--,I,i,l,lllfllllllll
0 20 40 60 80 100 120 140 160 180 atm Fig, 4. V i s c o s i t y - p r e s s u r e c u r v e s in c o m p a r i s o n w i t h l i t e r a t u r e .
These figures show, that the values of this paper are lower than those of K i y a m a and M a k i t a and of S c h r 6 e r and B e c k e r. There is a reasonable agreement with the data of S t a k e l b e c k , P h i l l i p s and of W a r b u r g and v o n B a b o at low pressures, and with the data of C o m i n g s and E g l y above 80 atm. Instead of the maxima in the viscosity-density curves in the critical region, N a l d r e t t and M a a s s find only a small relative increase. This disagreement m a y be due to the different experimental methods used. Received 7- I 1-56.
102
VISCOSITY O F CARBON D I O X I D E AT P R E S S U R E S U P TO 2 0 0 0 ATM REFERENCES
I) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
M i c h e l s , A. and G i b s o n , R. O., Proc. roy. Soc. A 134 (1931) 288. G i b s o n , R. 0., The viscosity o/gases at high pressures. Thesis, H. J. Paris, Amsterdam (1933). M i c h e l s , A., S c h i p p e r , A. C. J. and R i n t o u l , W. H., Physica 19 (1953) I011. S c h i p p e r , A. C. J., D e viscositeit va~ gecomprimeerde watersto/ en deuterium. Thesis, Van Goreum en Comp. N.V., Assen (1954). M i e h e l s , A. and M i e h e l s , C., Proc roy. Soc. A 153 (1935) 201. M i c h e l s , A., M i c h e l s , C. and W o u t e r s , H., Proc. roy Soe. A 153 (1935) 214. M i e h e l s , A. and M i c h e l s , C., Proc. roy. Soe. A 160 (1937) 348. M i c h e l s , A., B l a i s s e , B. and M i c h e l s , C., Proc. roy Soc. A 160 (1937) 358. M i c h e l s , A. and M i c h e l s , C., Phil. Trans. A ,°31 (1933) 409. M i c h e l s , A., B o t z e n , A., F r i e d m a n , A. S. and S e n g e r s , J. V., Physica 22 (1956) 121. M i c h e l s , A., B o t z e n , A. and S c h u u r m a n , W., Physica 20 (1954) 1141. M i c h e l s , A. and B o t z e n , A., Physica 19 (1953) 585. W a r b u r g , E. and B a b o , L. y o n , Wied. Ann. 17 (1882) 390. P h i l l i p s , P., Proc. roy Soc. A 8"/ (1912) 48. S t a k e l b e e k , H., Z. ges. K~ilte-Ind. 40 (1933) 33. S c h r 6 e r , E. and B e c k e t , G., Z. phys. Chem. A 173 (1935) 178. N a l d r e t t , S. N. and M a a s s , O., Can. J. Research 18B (1940) 322. C o m i n g s , E. W. and E g l y , R. S., ind. eng. Chem. 33 (1941) 1224. K i y a m a , R. and M a k i t a , T. Rev. phys. Chem. Japan 21 (1951) 63; 22 (1952) 49.
PhysicaXXlII 95-102
T H E V I S C O S I T Y O F C A R B O N D I O X I D E B E T W E E N 0°C A N D 75°C A N D A T P R E S S U R E S U P TO 2000 A T M O S P H E R E S by A. MICHELS, A. BOTZEN and W. SCHUURMAN 155th publication of the Van der Waals Fund Van der Waals-laboratorium, Gemeente Universiteit, Amsterdam, Nederland Synopsis
T h e v i s c o s i t y of c a r b o n d i o x i d e h a s b e e n m e a s u r e d w i t h a t r a n s p i r a t i o n m e t h o d a t t e m p e r a t u r e s b e t w e e n 0 ° a n d 75°C a t p r e s s u r e s u p t o 2000 a t m o s p h e r e s . P r e v i o u s m e a s u r e m e n t s d e s c r i b e d earlier r e v e a l e d a n u n e x p e c t e d i n c r e a s e of t h e t h e r m a l c o n d u c t i v i t y of a r g o n r o u n d t h e c r i t i c a l d e n s i t y . T h e r e f o r e special a t t e n t i o n h a s b e e n g i v e n t o t h e c r i t i c a l r e g i o n a l t h o u g h t h e t r a n s p i r a t i o n m e t h o d is n o t p a r t i c u l a r s u i t a b l e for t h i s region. T h e v i s c o s i t y of CO2 a b o v e 40°C s h o w s t h e n o r m a l t r e n d w~.th pressure, b u t d e v i a t e s considerably from Enskog's theory. In the critical region the viscosity shows a similar a b n o r m a l b e h a v i o u r as f o u n d for t h e h e a t c o n d u c t i v i t y of argon. V i s c o s i t y m e a s u r e m e n t s of a r g o n d i d n o t s h o w t h i s effect b u t in this case t h e t e m p e r a t u l e s r e l a t i v e t o t h e critical t e m p e r a t u r e were h i g h e r t h a n i n t h e p r e s e n t m e a s u r e n m n t s of COs. T h e r e s u l t s a r e c o m p a r e d w i t h t h o s e of o t h e r a u t h o r s .
§ 1. Introduction. The viscosity of carbon dioxide was measured by the transpiration method, which has been described previously 1)2)3)4), at temperatures of 0 °, 25 °, 29.90 °, 31.10 °, 32.20 °, 34.60 °, 40", 50 ° and 75°C. The highest measured pressure was about 2000 atmospheres attained at 75°C. The density corresponding to each temperature and pressure was calculated using the compressibility isotherms of M i c h e l s , M i c h e l s , W o u t e r s and B l a i s s e 5) 6) 7) 8). The C09 used was prepared from commercial gas by a three-stage distillation process 9). § 2. The results. The results of the measurements are given in tables I to IX. The pressure P is expressed in international atmospheres, the density, ~, both in Amagat units and in g/cm 8 and the coefficient of viscosity, 7, in g/cm sec. TABLE I The viscosity of CO2 at 0°C •P
int. atm. 9.148 10.43 24.62 27.26 30.24 32.32
~
~
Am. 9.712 11.20 30.41 34.82 40.35 44.63
g/cm 3 0.01920 0.02214 0.06012 0.06884 0.07977 0.08823
-- 95 --
r / . 105 g/cm sec. 13.85 13.86 14.21 14.34 14.49 14.65
96
A. M I C H E L S , A. BOTZEN A N D W . S C H U U R M A N
T A B L E II
TABLE III
The visco sity of CO2 at 25°C
The viscosity of CO2 at 29,90°C
P
Q
~
int. arm.
Am.
g/cm3
10;44 31.27 44.17 57.21 78.51 98.26 111.6
0.02064 0.06182 0.08732 0.1131 0.1552 0.1943 0.2206
366 373 387 422.2 457.0 491.7 527.3 563.3 596.3
0.724 0.737 0.765 0.8347 0.9035 0.9721 1.042 I.I14 1.179
10.82 28.74 37.60 44.98 54.04 59.60 62.05 65.06 67,32 74.59 109.6 180.5 307.3 522.9 864.3 1323
~] . 105 g/cm sec.
15.07 15.51 15.90 16.38 17.54 18.94
20.13 61.3 60.6 64.8 76.06 92.06 112.1 138.6 173.9 218.8
P int. atm. I 1.22
30.23 38.56 47.59 56.84 62.88 65.57 69.13 70.67 71.03 77.73 95.41 136.1 214.7 350.0 574.6 926.6
Q Am. 10.64 32.32 44.05 59.42 80.28 99.7 III.6 136.3 159 171 352 386.9 422. I 457.2 491.8 527.3 563.2
~ g/cm 3
~] • lOs g/era sec.
0.02104 0.06390 0.08709 0.1.175 0.1587 0.1971 0.2207 0.2694 0.314 0.338 0.696 0.7649 0.8345 0.9039 0.9723 1.042 1.113
15.32 15.79 16.14 16.78 17.87 19.20 20.12 22.55 27.1 29. l 55.6 65.12 76.56 92.49 12.4 38.9 73.7
T A B L E IV
TABLE V
The viscosity of CO2 at 31.10°C
The viscosity of CO2 at 32.20°C.
P
~
O
int. atm.
Am.
g/cma
I0.59 32.78 44.27 58.74 78.79 98.53 Ill.4 132.6 154 170 185 199 204 212
0.02094 0.06481 0.08752 0.1161 0.1.558 0.1948 0.2202 0.2622 0.3045 0.336 0.366 0.393 0.403 0.419 0.437 0.439 .0.461 0.484 0.492 0.504 0.520 0.528 0.530 0.563 0,593 0..696 0.7645 0.8339 0.9033 0.9725 1.042 1.114 1.179
I 1.22
30.77 38.96 47.60 56.84 63.31 66.34 69.77 70.70 72.42 72.80 72.91 72.92 72.95 72.95 72.95 72.96 72.96 72.96 72.96 72.97 72.97 72.97 73.20 73.67 81.55 100.3 141.8 222.0 360.5 586.9 942.8 1416
221 222 233 245 249 255 263 267 268 285 3OO 352 386.7 421.8 456.9 491.9 527.3 563.3 596.3
~] • 105 g/cm sec.
15.36 15.86 16.22 16.81 17.84 19.08 20.15 22.12 25.5 28.7 32.3 35.2 49 53 59 6O 68 71 74 76 78 8O 8O 66 5O 55.3 65.07 76.53 92.45 112.5
138.7 173.4 216.9
P
~
~
int. atm.
Am.
g/cm 8
I 1.22
30.97 39.16 47.60 56.84 64.06 66.24 66.90 70.91 70.92 72.56 73.65 74.05 74.37 74.45 74.60 74.69 74.74 74.77 74.84 74.94 75.12 75.14 75.60 76.59 85.17 104.9 148.9 229.9 369.7 597.5 955.6
10.54 32.84 44.23 58.15 77.60 98.90 107.5 I I0.3
134.0 134.1 150 168 178 192 196 208 220 225 230 241 255 269 271 288 307 352 386.6 422.3 457. I 491.9 527.2 563.4
0.02084 0.06492 0.08744 0.1150 0.1534 0.1955 0.2125 0.2181 0.2649 0.2651 0.297 0.332 0.352 0.380 0.387 0.411 0.435 0.445 0.455 0.476 0.504 0.532 0.536 0.569 0.607 0.696 0.7643 0.8349 0.9037 0.9725 1.042 I.I14
~] . 105 g/cm see.
15.42 15.92 16.27 16.86 17.83
19.23 19.96 20.07 22.25 22.32 24.8 27.5
29.5 32.3 33.1 42.5 47.3 49.0 50.3 52.2 55.8 56.8 56.5 47.7 47.2 55.4 65.05 76.75 92.55 12.4 38.6 73.5
VISCOSITY
OF CARBON
TABLE
DIOXIDE
AT PRESSURES
TABLE
VI
P int. atm.
Q Am.
0 g/cm 3
'q" 105 g / c m sec.
76.78 77.62 78.30 78.91 79.53 80.76 83.21
172 190 212 236 260 287 318
0.340 0.376 0.419 0.467 0.514 0.567 0.629
28.4 31.4 34.7 42.3 45.4 46.7 48.4
VII
T h e v i s c o s i t y of COo. a t 4 0 . 0 0 ° C P int. a r m . II.22 32.12 40.82 49.88 60.94 68.65 72.28 77.72 81.07 83.16 84.69 86.09 87.06 89,01 90.53 93.04 96.46 102.0 108.7 III.2 138.5 193.1 284.3 436.7 437.2 680.8 1056.5 1551
@ Am.
@ g/cm 3
~1" 105 g / c m sec.
10.23 32.85 44.28 58.12 79.31 99.03 II0,8 134.0 154.8 172.5 188.7 206.7 225.9 248.9 267.1 289.8 308.1 329.3 346.9 351.7 386.8 423.1 457.3 492.0 492.1 527.4 563.4 596.5
0.02022 0.06494 0.08754 0.1149 0.1568 0.1958 0.2191 0.2649 0.3060 0.3410 0.3731 0.4086 0.4466 0.4921 0.5281 0.5729 0.6091 0.6510 0,6858 0.6953 0.7647 0.8365 0.9041 0.9727 0.9729 1.043 I.I14 1.179
15.79 16.33 16.69 17.24 18.32 19.59 20.42 22.33 24.46 26.54 28.53 31.05 33.42 36.60 39.39 43.00 45.69 49.80 53.84 55.36 64.99 76.07 91.12 111.5 112.3 137.7 170.7 216,1
Physica XXIII
2000
97
ATM
VIII
T h e v i s c o s i t y of CO 2 a t 50°C
T h e v i s c o s i t y of CO2 a t 3 4 . 6 0 ° C
TABLE
UP TO
P int. a t m . 11.22 32.88 42.96 51.82 65.16 75.20 79.07 85.74 90.98 94.31 97.42 99.70 103.3 103.8 107.2 110.9 115.9 122.5 131.7 143.7 146.3 182.0 246.8 350.0 521.4 785.0 1183 1699
0 Am.
@ g/cm s
~]" lOs g / c m sec.
9.863 32.05 44.35 56.69 79.17 101.1 111.3 132.5 153.6 169.9 187.3 201.0 223.9 226.8 247.2 266.5 287.8 308.5 329.2 348.8 352.2 386.6 422.3 456.4 491,9 527.4 563.4 596.3
0.01950 0.06336 0.08768 0.1121 0.1565 0.1999 0.2200 0.2620 0.3037 0.3354 0.3703 0.3974 0.4427 0.4484 0.4887 0.5269 0.5690 0.6099 0.6508 0.6896 0.6963 0.7643 0.8349 0.9023 0.9725 1.043 1.114 1.179
16.26 16.79 17.21 17.67 18.81 20.17 20.90 22.54 24.45 25.91 27.99 29.47 32.41 32.92 35.65 38.48 41.95 45,64 50.11 54.58 55.65 65.27 77.11 92.30 I12.9 137.8 172.0 216.9
TABLE
IX
T h e v i s c o s i t y of CO z a t 75°C
P int. a t l n .
0 Anl.
"11 g/cm s
12.16 37.99 49.00 61.73 76.28 89.16 95.82 106.9 I15.8 123.5 129.4 136.1 136.7 144.9 145.7 154.9 163.3 175.5 189.9 208.4 230.1 293.1 386.1 544.3 733.5 I046 1498 2070
9.843 33.69 45.43 60.53 80.35 100.9 112.8 135.0 155.0 173.7 188.7 205.7 207.1 227.5 229.3 250.2 267.1 288.0 308.5 329.1 348.3 386.8 422.2 460.8 491.8 527.5 563.4 596.5
0.01946 0.06659 0.08982 0.1197 0.1589 0.1995 0.2230 0.2689 0.3064 0.3434 0.3731 0.4067 0.4094 0.4498 0.4533 0.4946 0.5281 0.5694 0.6099 0.6506 0.6886 0.7647 0.8347 0.9110 0.9723 1.043 1.114 1.179
Q " 105 g / e r a see. 17.43 18.08 18.50 19.14 20.18 21.43 22.25 23.94 25.66 27.44 28.92 30.87 31.12 33.61 33.87 36.76 39.29 42.81 46.67 51.01 55.56 66.32 78.34 95.09 112.8 138,4 171.5 217.0 7
A. MICHELS, A. BOTZEN AND W. SCHUURMAN
98
Fig. 1 shows ~ p l o t t e d as a function of density at different t e m p e r a t u r e s and Fig. 9. gives the same d a t a as a function of pressure. I t can be seen f r o m Fig. 1 t h a t at t e m p e r a t u r e s just a b o v e the critical t e m p e r a t u r e (To = 31.04 °C) the curves h a v e a m a x i m u m n e a r the critical density (oc = 237 Amagat). 225 9
~elO s
~11 s¢¢
~OC
A
O°C
13
25 °
X
Z9.9 °
0 •
31.1 ° 3Z.Z °
@
34.b °
-I-
40 °
t i
SO ° 175
"e,
7s °
I,~l
All ttmptmturcs
1501
125 I . ~ -
C02 75t--
501-------
251-----
0
I
I
I
I
I
IOO 2OO 300 400 500 Am Fig. 1 The coefficient of viscosity vs. density.
This is evident at 31.10 °, 32.20 °, 34.60 ° and less p r o n o u n c e d at 400C. Similar b e h a v i o u r has been r e p o r t e d for the t h e r m a l c o n d u c t i v i t y - d e n s i t y curves of argon 10) n e a r the critical density. I n the critical region, however, o n l y little precision can be ascribed to
V I S C O S I T Y OF CARBON D I O X I D E AT P R E S S U R E S UP TO 2 0 0 0 ATM
99
the viscosity data. This is a consequence of the large density gradient in the capillary during the measurement (the pressure difference causing the gas flow through the capillary is of the order of ½ atm. and varies during a measurement. In the worst case at 31.10°C such a pressure difference corresponds to a density difference of 100 Amagat). ._~¢m s ¢ c |
"lOs
,ooc
/
o 3,.,o
-
.
~
!
2001
0 25°
C02
x zg.e°
.+
3 .2o
J
®t ,oo 34.60
.
J J
J
J
J
'
~ J J J~
J
~
~/
3C
50
..----~ p
400
0
800
1200
1600otto
F i g . 2. T h e c o e f f i c i e n t of v i s c o s i t y v s . p r e s s u r e .
Except near the critical pressure the viscosity-pressure curves of Fig. 2 show the same general character as the corresponding curves for nitrogenl) 2), hydrogen 3) 4), deuterium 3) 4) and argon 11). TABLE
X
author
method
pressure range and temperatures
W a r b u r g and v o n B a b o la) P h i l l i p s x4) S t a k e l b e c k is)
transpiration method transpiration method falling ball method
S c h r S e r and B e c k e r xe) N a l d r e t t and M a a s s 17)
rolling baU method oscillating disc method
C o m i n g s and E g l y xs) K i y a m a and M a k i t a xg)
transpiration method rolling baU method
45-115 atm., 32 ° and 40°C. 1-120 atm., 20 °, 30 °, 32 °, 35 ° and 40°C. 1-100 atm., - - 1 5 °, 0% 10% 20 °, 30 ° and 40°C. 1-120 atm., 20°C. around the critical pressure: 31.10 °, 31.14 °, 31.2 °, 31.5 ° and 32.0°C. 1-171 atm., 40°C. 1-100 atm., 50 ° up to 300°C in steps of 500C.
It was a t t e m p t e d to compare the experimental results with Enskog's theory of dense gases, following a procedure described previously 12). The discrepancy between theory and experiment, however, is so large for CO2
100
A~ MICHELS, A. BOTZEN AND W. SCHUURMAN
that it is of no interest to tabulate Enskog's values in detail. At a density as low as 40 Amagat the theoretical values are already double the experimental ones; the disagreement becomes still more striking at higher densities.
§ 3. Comparision with measurements o/ other authors. In the literature, viscosity data on C09. are found only at comparatively low pressures. Table X gives a survey of these measurements. A large scale plot of some of the viscosity-density curves and of the viscosity-pressure curves of the present paper is shown in figures 3 and 4 8 0 ~ g c a i¢C
Cr~
"~. 105
7S°C
II 7/ //?oo
~z
t 70 1
t I
31.1 iI iI iI
60 n iI
32.2 °
/ ' 5 0 ~
iiiII
/i /
t32o
4 0 ~
,ooo
3C
,I
1/i//,'.,"
soOi#11./
Phillips
-- .
Naldrett & Iviaass Kiyama & Makita
Warbur 9 &yon Babo Mlch¢|~lBotzcn & Sciluurmoll
20
---~Q
I0 0
%° 11 I00
I i I
200
300
i I i 4 0 0 Am
Fig. 3. Viscosity-density curves in comparison with literature.
VISCOSITY OF CARBON DIOXIDE AT PRESSURES UP TO 2 0 0 0 ATM
101
respectively. Some data of other authors are plotted in Fig. 3, others in Fig. 4. 1C
cmse~
-
-
coz
~7.lOs
10C
xX~ xxx x xX20oc x~ x
9C
xx xx xx x
x~
8C
/200 .gl aj
7C
6C
5C Jl.l c
4G
3(]-**"
--
2{3-.
75o(: ~
,,°"
.~
II
x."lt
/
~
~
oOC
10--
.... Warb~rg&yon Bobo ,ee.e.. Schrotr&B~:kcr +++++ Stak¢lbcck ~ Comings& Egly Midw~,pBot.z¢n&Schuurmon
> P
--,I,i,l,lllfllllllll
0 20 40 60 80 100 120 140 160 180 atm Fig, 4. V i s c o s i t y - p r e s s u r e c u r v e s in c o m p a r i s o n w i t h l i t e r a t u r e .
These figures show, that the values of this paper are lower than those of K i y a m a and M a k i t a and of S c h r 6 e r and B e c k e r. There is a reasonable agreement with the data of S t a k e l b e c k , P h i l l i p s and of W a r b u r g and v o n B a b o at low pressures, and with the data of C o m i n g s and E g l y above 80 atm. Instead of the maxima in the viscosity-density curves in the critical region, N a l d r e t t and M a a s s find only a small relative increase. This disagreement m a y be due to the different experimental methods used. Received 7- I 1-56.
102
VISCOSITY O F CARBON D I O X I D E AT P R E S S U R E S U P TO 2 0 0 0 ATM REFERENCES
I) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
M i c h e l s , A. and G i b s o n , R. O., Proc. roy. Soc. A 134 (1931) 288. G i b s o n , R. 0., The viscosity o/gases at high pressures. Thesis, H. J. Paris, Amsterdam (1933). M i c h e l s , A., S c h i p p e r , A. C. J. and R i n t o u l , W. H., Physica 19 (1953) I011. S c h i p p e r , A. C. J., D e viscositeit va~ gecomprimeerde watersto/ en deuterium. Thesis, Van Goreum en Comp. N.V., Assen (1954). M i e h e l s , A. and M i e h e l s , C., Proc roy. Soc. A 153 (1935) 201. M i c h e l s , A., M i c h e l s , C. and W o u t e r s , H., Proc. roy Soe. A 153 (1935) 214. M i e h e l s , A. and M i c h e l s , C., Proc. roy. Soe. A 160 (1937) 348. M i c h e l s , A., B l a i s s e , B. and M i c h e l s , C., Proc. roy Soc. A 160 (1937) 358. M i c h e l s , A. and M i c h e l s , C., Phil. Trans. A ,°31 (1933) 409. M i c h e l s , A., B o t z e n , A., F r i e d m a n , A. S. and S e n g e r s , J. V., Physica 22 (1956) 121. M i c h e l s , A., B o t z e n , A. and S c h u u r m a n , W., Physica 20 (1954) 1141. M i c h e l s , A. and B o t z e n , A., Physica 19 (1953) 585. W a r b u r g , E. and B a b o , L. y o n , Wied. Ann. 17 (1882) 390. P h i l l i p s , P., Proc. roy Soc. A 8"/ (1912) 48. S t a k e l b e e k , H., Z. ges. K~ilte-Ind. 40 (1933) 33. S c h r 6 e r , E. and B e c k e t , G., Z. phys. Chem. A 173 (1935) 178. N a l d r e t t , S. N. and M a a s s , O., Can. J. Research 18B (1940) 322. C o m i n g s , E. W. and E g l y , R. S., ind. eng. Chem. 33 (1941) 1224. K i y a m a , R. and M a k i t a , T. Rev. phys. Chem. Japan 21 (1951) 63; 22 (1952) 49.