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The viscosity of liquefied gases
Boon, ]. P. Thomaes, G.
Physica 29
208-214
1963
THE VISCOSITY OF LIQUEFIED GASES by J. P. BOON and G. THOMAES Faculte des Sciences, Universite libre de Bruxelles, Bruxelles, Belgique
Synopsis The kinematic viscosities of liquid argon, krypton, oxygen and methane have been measured in a large range of temperature. The molecular parameters B* and r* are deduced from the experimental results by a method based on the principle of corresponding states and are compared with the values obtained from other viscosity data and from second virial coefficients measurements.
1. Introduction. At present, there subsists a large part of uncertainty in the study of transport phenomena in the liquid phase. For the case of the viscosity, the reasons are the following: 1°, there exists no exact theory of the liquid state; 2°. there are too little satisfactory experimental data. The most suitable approach to the problem would therefore seem to be a phenomenological one, based on the experimental study of the viscosity of substances with simple structure and for which the potential energy of a pair of molecules can be represented by means of the same universal function 1) : e(1') = e*q,(1'/1'*) (1)
where e* and r* are the coordinates of the interaction curve minimum. We have developed a method based on the principle of corresponding states, which permits us to deduce from the experimental results the relative values of the molecular parameters: (2)
and (3)
the substance A being taken as reference. These values are used to determine the viscosity of binary mixtures, from the knowledge of the viscosity of the pure components, as described in another papers). -
208-
THE VISCOSITY OF LIQUEFIED GASES
209
2, Determination of the molecular parameters. To avoid density measurements, we consider the kinematic viscosity: (4)
where:
'YJ = p =
viscosity coefficient density
and we apply the method, developed by Thomaes and Van It t er b ee k''), to the case of the kinematic viscosity, which has the dimension M-iEtL. By means of the parameters e* and r*, and the mass m, we reduce the function v to obtain the dimensionless quantity y independent of the chemical nature of the molecule:
v = Z-lv with: Z = m-i(e*' r*2)!
(5)
In the case of liquids with simple molecular structure, we can suppose that: ii =
v(P, v, T)
p=
pr*3/e*,)
(6)
with: V = vlr*3,
(7)
T = kTJe*. Assuming that the liquids are described by an equation of state:
v = v(p,
T)
(8)
and the experiments being carried out at saturation pressure:
p = p(T)
(9)
ii = v(T)
(10)
relation (6) becomes:
As can be seen on figure 1., the following relation obtained from Eyring's theory 4) , fits the experimental points very accurately: loge v = A
+
LIP'" RT
(11 )
where: LlP* is the activation energy, and A appears experimentally as a characteristic constant of the substance. Thus: dlnv=B (12) d IJT where the constant B is equal to (LlP"'JR) ,
210
J.
P. BOON AND G. THOMAES
---------
Taking into account the relations (5) and (12), we obtain: 1 dlnv d 1jT
T
1 dlnv d II T'
(13)
=T
or: (14)
where
B is a universal constant: B = Bkle".
(15)
The application of this relation to two liquids, A and B, determines the relative values of the molecular parameters of the two substances: BBB s* --=~=1+0 BAA
(16)
sA..!!
Relation (5) applied to the two substances, A and B, at the same reduced temperature: (17)
gives:
vBB[TAA(1 + Cl)] ZB (s~wr~~)~ mB-~ VAA[T~]-- = ZA = (sAAr~~)l m [ l
(18)
or: 1 +p=
r~B = ('JIBB[TAA(I +o)])(,mB)!r(1 +0)-1
r"l.A
'VAA[TAA]
(19)
mA
It may be remarked that the determination of (1 choice of (1 + 0).
+ p) depends on the
3. Experimental. The viscosimeter consists of a capillary pyrex tube (length = 15 em, interior diameter = 0.03 em) topped by a glass bulb (volume = 1.2 cm''). These dimensions were calculated to give a laminar flow with small quantities of liquid. The apparatus was designed so as to avoid surface effects and thermal expansion corrections in the liquid state range (7S 0 K - 1 2 5 ° K ) of the substances studied. The experiments are carried out at saturation pressure. The viscosimeter is placed in a cryostatic chamber, which is maintained at constant temperature during the measurements with a precision of ljIOO°K. The precision on final results is about 1 %. To avoid density measurements, the relative values of the kinematic viscosity (j'R) are directly obtained by comparing the flow times (t) of a given constant volume (1.2 ems) of the liquids studied; indeed, from
211
THE VISCOSITY OF LIQUEFIED GASES
Poiseuille's law, we obtain: 'V t 'VR=-=-
'Va
(20)
to
where 'Va is the reference value taken as unit. A complete description of the experimental method and apparatus used in this research will be published shortly. The gases were obtained from the company "L'Air Liquide", Liege, Belgium, and were purified by fractional distillation at low temperature in this laboratory. They were tested by mass spectrometry and by triple point pressure measurement and were shown to be about 99.98% pure. 4. Results. The values of the relative kinematic viscosities of 02, AT, Kr and CH4 are given in table I; the value for argon at 88.98°K was taken as reference (vo). TABLE I Ar
02 T(ol{)
I
75.35 78.01 79.90 82,30 84.60 86.88 89.14 91.57
Vn
1.378 1.278 1.209 1.128 1.068 1.017 0.970 0.925
T(ol{)
91.01 93.77 97.34 100.88 103.84 107.05 109.86 114.15
I
VR
1.157 1.135 1.084 1.052 1.035 1.000
86.04
87.14 8B.15 88.98
Kr
CH4 T(OK)
I
83.94 84.69
VR
2.520 2.320 2.084 1.902 1.810 1.681 1.602 1.522
T(OK)
116.09 117,03 117.91 117.93 118.86 119.92 120.43 1.21.57 122.55 123.01
I
V1l
1.026 0.994 0.986 0.979 0.960 0.948 0.936 0.917 0.896 O.BBB
Figure 1. shows the variation of the logarithm of the kinematic viscosity as a function of the reciprocal temperature, as described by equation (11). The straight lines were obtained by a least squares calculation fit to the experimental data. 5. Calculation of the molecular parameters. The molecular parameters were calculated from the experimental results, given in table I, by means of
J.
212
P. BOON AND G. THOMAES
relations (16) and (19). The values obtained are given in tables II and III. For comparison purposes, the values calculated by Thomaes and Van Lt t e r be ek t) from the viscosity measurements of Rudenka and Schubnikav 5 ) are reported in table II, column 2 and in table III, columns 3 and 4; those obtained from the second virial coefficients data II) 7) are given in table II, column 3 and in table III, column 5,
/ 0,7~
o.~o
0.25
0 , 0 0 / Kr
0.010
0.001
0:011
0.012
0.01J
Fig. 1. The logarithm of the relative kinematic viscosity as a function of the reciprocal temperature. TABLE II 1
+8
Argon *) Oxygene Krypton Methane
I
from (16)
I 1.00 0.81
from T'h o m a es and Van Itterbeek
I
from second virial coefficient B(T)
1.00 0.83
1.34
-
1.09
1.01
1.00 0.97 7 ) 1.38 6) 1.20 6)
TABLE III
1
+p
Argon*) Oxygene Krypton Methane
from (16) and (19)
from (16) and (1 + 8) from B(T)
1.00 1.51 1.20 1.29
1.00 1.39 1.18 1.23
from Thomaes and Van It t e r b e e k
from Thomaes and Van Itterbeek and (1 + 8) from critical data
from second vlrial coefficient B(T)
1.00 0.85 -
1.00 1.04
-
-
1.00 1.04 7) 1.08 8) 1.16 6)
-
*) Argon = reference substance
Table IV gives the values of the factors A and B, introduced in relations (11) and (12), and calculated from the least squares equations, and also the
213
THE VISCOSITY OF LIQUEFIED GASES
TABLE IV
I
A
Argon Oxygene Krypton Methane
I
B 10- 2
I
2,109 1.707 2.835 2.304
-2.366 -1.947 -2.422 -1.622
B 1.729 1.727 1.735 1.732
constant 13 calculated with the values of e*jk obtained from our measurements; the absolute value for argon is that calculated from the second virial coefficients data 6) : e".Ajk
= 122.0oK
We calculated the reduced kinematic viscosities (VR) by means of the molecular parameters obtained from our measurements: the values of VR are plotted against the reduced temperature on figure 2. ~R 1.21--
•a a
1.11--
ta
JlJ
...
1.0-
~
X
~
r
0.• I--
•.,. 0.8l-
I'
•.,. 1
0.71-
.,.
.,.
0.6e-
0.51--
.,.
a Ar 1/ ICr p
02
• eN, I
I
I
I
I
I
0.5
0.6
0.7
0.8
O.g
1.0
T
Fig. 2. The reduced kinematic viscosity as a function of the reduced temperature.
6. Conclusions. As can be seen on table II and III, the agreement between the values of the molecular parameters obtained by different methods is qualitatively satisfactory: it is particularly good for krypton and rather poor in the case of oxygen.
214
THE VISCOSITY OF LIQUEFIED GASES
However, it may be remarked that we obtain for 13 (table IV) a value which remains practically constant for the different substances. This, as also the single curve obtained for the function :;(1"), seems to indicate the validity of the principle of corresponding states as applied to the viscosity of liquids. Acknowledgements. We wish to thank Professor 1. Prigogine for his constant inretest during the course of this work. We are also indebted to Professor P. Goldfinger, Dr. G. Verbeke, Mr. G. Huybrechts and Miss L. Meyer for the mass spectrometric analysis of the gases, to Mr. Ch. Lefebvre and J. L. Guisset for the krypton distillation and to Mr. R. Lecoq for the oxygen distillation. This work has been supported by grants from the "Institut pour l'Encouragement a. la Recherge Scientifique dans l'Iridustrie et l'Agriculture", Brussels, Belgium. Received 8-8-62
REFERENCES I) Prigogine, 1., Be Il e ma n s, A. and Mathot, V., The molecular Theory of Solutions, Amsterdam, 1957. 2) Boon, J. P. and Thomaes, G., Physica 28 (1962) 1074. 3) Thomaes, G, and Van Itterbeek, J., Mol. Phys. 2 (1959) 372. 4) Glasstone, S., Laidler, K. J. and Eyring, H., The Theory of Rate' Processes, New York, 1941. 5) Rudenko, N. S, and Schubnikov, L. W., Phys. Z.. Sowjet, a (1935) 179. 6) Thomaes, G., Van S teenwinkel, R. and Stone, W., Mol. Phys. Ii (1962) 301. 7) Hirschfelder, J. 0., Curtiss, C. F. and Bird, R. B., The Molecular Theory of Gases and Liquids, New-York, 1956.
Physica 29
208-214
1963
THE VISCOSITY OF LIQUEFIED GASES by J. P. BOON and G. THOMAES Faculte des Sciences, Universite libre de Bruxelles, Bruxelles, Belgique
Synopsis The kinematic viscosities of liquid argon, krypton, oxygen and methane have been measured in a large range of temperature. The molecular parameters B* and r* are deduced from the experimental results by a method based on the principle of corresponding states and are compared with the values obtained from other viscosity data and from second virial coefficients measurements.
1. Introduction. At present, there subsists a large part of uncertainty in the study of transport phenomena in the liquid phase. For the case of the viscosity, the reasons are the following: 1°, there exists no exact theory of the liquid state; 2°. there are too little satisfactory experimental data. The most suitable approach to the problem would therefore seem to be a phenomenological one, based on the experimental study of the viscosity of substances with simple structure and for which the potential energy of a pair of molecules can be represented by means of the same universal function 1) : e(1') = e*q,(1'/1'*) (1)
where e* and r* are the coordinates of the interaction curve minimum. We have developed a method based on the principle of corresponding states, which permits us to deduce from the experimental results the relative values of the molecular parameters: (2)
and (3)
the substance A being taken as reference. These values are used to determine the viscosity of binary mixtures, from the knowledge of the viscosity of the pure components, as described in another papers). -
208-
THE VISCOSITY OF LIQUEFIED GASES
209
2, Determination of the molecular parameters. To avoid density measurements, we consider the kinematic viscosity: (4)
where:
'YJ = p =
viscosity coefficient density
and we apply the method, developed by Thomaes and Van It t er b ee k''), to the case of the kinematic viscosity, which has the dimension M-iEtL. By means of the parameters e* and r*, and the mass m, we reduce the function v to obtain the dimensionless quantity y independent of the chemical nature of the molecule:
v = Z-lv with: Z = m-i(e*' r*2)!
(5)
In the case of liquids with simple molecular structure, we can suppose that: ii =
v(P, v, T)
p=
pr*3/e*,)
(6)
with: V = vlr*3,
(7)
T = kTJe*. Assuming that the liquids are described by an equation of state:
v = v(p,
T)
(8)
and the experiments being carried out at saturation pressure:
p = p(T)
(9)
ii = v(T)
(10)
relation (6) becomes:
As can be seen on figure 1., the following relation obtained from Eyring's theory 4) , fits the experimental points very accurately: loge v = A
+
LIP'" RT
(11 )
where: LlP* is the activation energy, and A appears experimentally as a characteristic constant of the substance. Thus: dlnv=B (12) d IJT where the constant B is equal to (LlP"'JR) ,
210
J.
P. BOON AND G. THOMAES
---------
Taking into account the relations (5) and (12), we obtain: 1 dlnv d 1jT
T
1 dlnv d II T'
(13)
=T
or: (14)
where
B is a universal constant: B = Bkle".
(15)
The application of this relation to two liquids, A and B, determines the relative values of the molecular parameters of the two substances: BBB s* --=~=1+0 BAA
(16)
sA..!!
Relation (5) applied to the two substances, A and B, at the same reduced temperature: (17)
gives:
vBB[TAA(1 + Cl)] ZB (s~wr~~)~ mB-~ VAA[T~]-- = ZA = (sAAr~~)l m [ l
(18)
or: 1 +p=
r~B = ('JIBB[TAA(I +o)])(,mB)!r(1 +0)-1
r"l.A
'VAA[TAA]
(19)
mA
It may be remarked that the determination of (1 choice of (1 + 0).
+ p) depends on the
3. Experimental. The viscosimeter consists of a capillary pyrex tube (length = 15 em, interior diameter = 0.03 em) topped by a glass bulb (volume = 1.2 cm''). These dimensions were calculated to give a laminar flow with small quantities of liquid. The apparatus was designed so as to avoid surface effects and thermal expansion corrections in the liquid state range (7S 0 K - 1 2 5 ° K ) of the substances studied. The experiments are carried out at saturation pressure. The viscosimeter is placed in a cryostatic chamber, which is maintained at constant temperature during the measurements with a precision of ljIOO°K. The precision on final results is about 1 %. To avoid density measurements, the relative values of the kinematic viscosity (j'R) are directly obtained by comparing the flow times (t) of a given constant volume (1.2 ems) of the liquids studied; indeed, from
211
THE VISCOSITY OF LIQUEFIED GASES
Poiseuille's law, we obtain: 'V t 'VR=-=-
'Va
(20)
to
where 'Va is the reference value taken as unit. A complete description of the experimental method and apparatus used in this research will be published shortly. The gases were obtained from the company "L'Air Liquide", Liege, Belgium, and were purified by fractional distillation at low temperature in this laboratory. They were tested by mass spectrometry and by triple point pressure measurement and were shown to be about 99.98% pure. 4. Results. The values of the relative kinematic viscosities of 02, AT, Kr and CH4 are given in table I; the value for argon at 88.98°K was taken as reference (vo). TABLE I Ar
02 T(ol{)
I
75.35 78.01 79.90 82,30 84.60 86.88 89.14 91.57
Vn
1.378 1.278 1.209 1.128 1.068 1.017 0.970 0.925
T(ol{)
91.01 93.77 97.34 100.88 103.84 107.05 109.86 114.15
I
VR
1.157 1.135 1.084 1.052 1.035 1.000
86.04
87.14 8B.15 88.98
Kr
CH4 T(OK)
I
83.94 84.69
VR
2.520 2.320 2.084 1.902 1.810 1.681 1.602 1.522
T(OK)
116.09 117,03 117.91 117.93 118.86 119.92 120.43 1.21.57 122.55 123.01
I
V1l
1.026 0.994 0.986 0.979 0.960 0.948 0.936 0.917 0.896 O.BBB
Figure 1. shows the variation of the logarithm of the kinematic viscosity as a function of the reciprocal temperature, as described by equation (11). The straight lines were obtained by a least squares calculation fit to the experimental data. 5. Calculation of the molecular parameters. The molecular parameters were calculated from the experimental results, given in table I, by means of
J.
212
P. BOON AND G. THOMAES
relations (16) and (19). The values obtained are given in tables II and III. For comparison purposes, the values calculated by Thomaes and Van Lt t e r be ek t) from the viscosity measurements of Rudenka and Schubnikav 5 ) are reported in table II, column 2 and in table III, columns 3 and 4; those obtained from the second virial coefficients data II) 7) are given in table II, column 3 and in table III, column 5,
/ 0,7~
o.~o
0.25
0 , 0 0 / Kr
0.010
0.001
0:011
0.012
0.01J
Fig. 1. The logarithm of the relative kinematic viscosity as a function of the reciprocal temperature. TABLE II 1
+8
Argon *) Oxygene Krypton Methane
I
from (16)
I 1.00 0.81
from T'h o m a es and Van Itterbeek
I
from second virial coefficient B(T)
1.00 0.83
1.34
-
1.09
1.01
1.00 0.97 7 ) 1.38 6) 1.20 6)
TABLE III
1
+p
Argon*) Oxygene Krypton Methane
from (16) and (19)
from (16) and (1 + 8) from B(T)
1.00 1.51 1.20 1.29
1.00 1.39 1.18 1.23
from Thomaes and Van It t e r b e e k
from Thomaes and Van Itterbeek and (1 + 8) from critical data
from second vlrial coefficient B(T)
1.00 0.85 -
1.00 1.04
-
-
1.00 1.04 7) 1.08 8) 1.16 6)
-
*) Argon = reference substance
Table IV gives the values of the factors A and B, introduced in relations (11) and (12), and calculated from the least squares equations, and also the
213
THE VISCOSITY OF LIQUEFIED GASES
TABLE IV
I
A
Argon Oxygene Krypton Methane
I
B 10- 2
I
2,109 1.707 2.835 2.304
-2.366 -1.947 -2.422 -1.622
B 1.729 1.727 1.735 1.732
constant 13 calculated with the values of e*jk obtained from our measurements; the absolute value for argon is that calculated from the second virial coefficients data 6) : e".Ajk
= 122.0oK
We calculated the reduced kinematic viscosities (VR) by means of the molecular parameters obtained from our measurements: the values of VR are plotted against the reduced temperature on figure 2. ~R 1.21--
•a a
1.11--
ta
JlJ
...
1.0-
~
X
~
r
0.• I--
•.,. 0.8l-
I'
•.,. 1
0.71-
.,.
.,.
0.6e-
0.51--
.,.
a Ar 1/ ICr p
02
• eN, I
I
I
I
I
I
0.5
0.6
0.7
0.8
O.g
1.0
T
Fig. 2. The reduced kinematic viscosity as a function of the reduced temperature.
6. Conclusions. As can be seen on table II and III, the agreement between the values of the molecular parameters obtained by different methods is qualitatively satisfactory: it is particularly good for krypton and rather poor in the case of oxygen.
214
THE VISCOSITY OF LIQUEFIED GASES
However, it may be remarked that we obtain for 13 (table IV) a value which remains practically constant for the different substances. This, as also the single curve obtained for the function :;(1"), seems to indicate the validity of the principle of corresponding states as applied to the viscosity of liquids. Acknowledgements. We wish to thank Professor 1. Prigogine for his constant inretest during the course of this work. We are also indebted to Professor P. Goldfinger, Dr. G. Verbeke, Mr. G. Huybrechts and Miss L. Meyer for the mass spectrometric analysis of the gases, to Mr. Ch. Lefebvre and J. L. Guisset for the krypton distillation and to Mr. R. Lecoq for the oxygen distillation. This work has been supported by grants from the "Institut pour l'Encouragement a. la Recherge Scientifique dans l'Iridustrie et l'Agriculture", Brussels, Belgium. Received 8-8-62
REFERENCES I) Prigogine, 1., Be Il e ma n s, A. and Mathot, V., The molecular Theory of Solutions, Amsterdam, 1957. 2) Boon, J. P. and Thomaes, G., Physica 28 (1962) 1074. 3) Thomaes, G, and Van Itterbeek, J., Mol. Phys. 2 (1959) 372. 4) Glasstone, S., Laidler, K. J. and Eyring, H., The Theory of Rate' Processes, New York, 1941. 5) Rudenko, N. S, and Schubnikov, L. W., Phys. Z.. Sowjet, a (1935) 179. 6) Thomaes, G., Van S teenwinkel, R. and Stone, W., Mol. Phys. Ii (1962) 301. 7) Hirschfelder, J. 0., Curtiss, C. F. and Bird, R. B., The Molecular Theory of Gases and Liquids, New-York, 1956.