T H E JOULE-THOMSON EFFECT OF METHANE, NITROGEN, AND MIXTURES OF THESE GASES JOHN H. PERRY AND CARL V. HERRMANN The Grasselli Chemical Company, Cleveland, Ohio Received December 18, 1934 INTRODUCTION
The experimental difficulties of measuring Joule-Thomson coefficients are great and, until fairly recently, too few thermodynamic data (P-V-T and specific heat) of mixtures of gases were available to calculate these coefficients. Although such calculations are tedious, there is little doubt but that the results obtained are certainly as accurate as engineering calculations of design require, and, in many cases, the accuracy of such calculations compares favorably with that of experimentally measured values. It is the purpose of this paper to present the results of some calculations of the Joule-Thomson coefficients of methane, nitrogen, and mixtures of these gases. The calculations were made originally because of the small amount of actual experimental data available in this field. For the same reason the method will become increasingly important as the liquefaction of gases plays a continually greater r81e in industry. This is particularly . true because of the large number of gases to which it may be applied. THEORETICAL
It is well known that the Joule-Thomson effect, or any other thermodynamic property of a substance, can be readily evaluated if, first, an accurate equation of state representing the pressure-volume-temperature data of the substance is available, and second, if the variation of a pertinent thermal property with temperature is known. Probably the most accurate and widely applicable equation of state proposed to date is that of Beattie and Bridgeman (3), which reproduces the experimental data of fourteen gases to within 0.1 to 0.2 per cent over a wide range of temperature and pressure. This equation is:
where A
=
(
AO 1 -
c), B = Bo(1
-
V
1189
:),
e =
-,C
and Ao,a, BO,b, and c
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JOHN H. PERRY AND CARL V. HERRMANN
are constants specific for each gas, V is volume, P is the pressure, T is the temperature, and R is the gas constant. I n this equation, as in the rest of this paper, the units used are: volume in liters per mole, pressure in atmospheres, and temperature in degrees Kelvin. Bridgeman (6) utilized this equation of state in deriving an expression for the Joule-Thomson effect based on the thermodynamic relationship pc, =
(g) - v
As a check of this equation he calculated the values for air, and showed that the agreement with the observed values was excellent. However, this equation was unwieldy and calculations with it were tedious. This led Beattie (2) to attempt several approximations which simplified considerably the final equation without impairing appreciably the accuracy, as was proved by recalculating the values of p for air. This latter equation is that used in this paper, since it was sufficiently accurate and was much more convenient to use. The equation is: PC, =
R2T2 - -k m 4 3Aoa 5B0c>
(- B0+?!.!+!?.)-(2RT~-
where p is the Joule-Thomson coefficient, C, is the specific heat at constant pressure, and Ao, a, Bo, b, c, R, T , and P are as in the Beattie-Bridgeman equation of state. CALCULATIONS
The constants Ao,a, BO,b, and c were calculated for methane and nitrogen by Beattie and Bridgeman and presented in one of their original articles (4). A method of obtaining these constants for mixtures of gases from the constants for the pure gases has been evolved by Beattie (1) and Beattie and Ikehara ( 5 ) . This is quite simple and merely involves the linear combination of the a’s, b’s, c’s, and Bo’s, and squaring the linear combination of the square root of the Ao’s. The constants calculated in this way are as given in table 1. The constants for the pure gases are those used by Beattie (1) when he showed that the above method of combining constants held for the nitrogen-methane mixtures. Beattie and Bridgeman had also proposed another set of values for nitrogen. Several calculations showed that either set would have been sufficiently exact for the present purpose. The equation of state for methane reproduced the observed pressures with an average difference of 0.05 per cent over a range of 0 to 200°C. and up to 243 atmospheres. That for nitrogen gave an average deviation of 0.04 per cent in the same temperature range and up to 213 atmospheres.
MEASUREMENT OF JOULE-THOMSON
1191
COEFFICIENTS
The sole remaining requirement for the calculation of the Joule-Thomson coefficients is a knowledge of the value of C, for the pure gases and for the mixture of gases. Eastman (7) has collected the specific heat data for a number of gases and formulated empirical equations of the form,
C, = a
+ p T + -yT2+ ....
to reproduce these data. For methane and nitrogen they are as follows: for methane, C, = 5.90 + 0.0096T; for nitrogen, C, = 6.76 + 0.000606T 0.00000013T2. The equation for methane is accurate to only about 5 per cent in the range 150 to 400'K.; and that for nitrogen to about 1.5 per cent in the range 300 to 2500°K.
+
TABLE 1 Constants calculated for mixtures of methane and nitrogen Mole per cent
CHI.. .........
Mole per cent Na.
A............ a,............
B............ b. . . . . . . . . . . . .
c. . . . . . . . . . . . .
R ............
75 25
100 0
2.2769 1,9575 0.01855 0,01838 0.05269 0.05587 -0.01587 0.01592 12.8 x 104 11.02 x 104 0.08206 0.08206
50 50
26 75
1,6622 0.01822 0.04951 0.01598 9.22 x 104 0.08206
1,3910 0.01805 0.04633 0.01603 7.41 x 104 0,08206
1 ~
'
14
1.1440 0.01788 0.04314 0.01608 5.60 x 104 0.08206
C,, calculated by thege equations, is expressed in calories per mole. The factor, 0.0413 liter-atmospheres per calorie, was used to express C, in liter-atmospheres. The final equations are then : Methane: C, = 0.244 +'0.000396T Nitrogen: C, = 0.279 0.0000250T
+
+ 0.0000000054T2
By means of linear combination of the coefficients in these equations similar equations for the gas mixtures can easily be obtained. The calculation of the Joule-Thomson coefficients at atmospheric pressure then becomes relatively simple. However, to calculate the coefficients at other pressures it is necessary to evaluate C, a t these pressures. This can be done by means of another equation derived by Beattie (la) from thermodynamic relations and the Beattie-Bridgeman equation of state. This equation is:
c,=
c:+[g+g]p
where C, is the specific heat a t absolute temperature T and a t pressure P, A o , c, and R are as in the Beattie-Bridgeman equation of state, and CE is a constant for the given temperature.
1192
JOHN H. PERRY AND CARL V. HERRMANN
CE must be calculated for each temperature at which it is desired to evaluate C,. This is easily done, since C, can be calculated from the usual equations a t 1 atmosphere, after which substitution will permit solving for Cg. With CE known for a certain temperature, C, can be evaluated for any pressure a t this temperature. This same method applies to gas mixtures if the constants for the Beattie-Bridgeman equation of state and the C , equations are known. These calculations of the Joule-Thomson coefficients can be systematized so that the process is not so long and tedious as it would a t first appear to be. DISCUSSION O F RESULTS
The results with the pure gases and the three mixtures are given in table 2. The lower temperatures are outside the range for which the original equations were derived, but it is believed that because of t8heexcellent agreement between the observed and the calculated P-V-T data in the given range the extrapolation does not introduce serious error when taking into account the accuracy of the calculations as a whole. The original calculations were made principally for engineering purposes, so that the accuracy of the calculated values did not have to be of an extremely high order. The accuracy in the calculations was limited by the accuracy of the available specific heat equations, which, in the case of methane, was not very great.
Pure gases For the purpose of comparison, the few available data on the JouleThomson effect for methane and nitrogen are given below. The agreement between the observed and calculated data for methane is quite good, but that for nitrogen is very poor. Inasmuch as the equation of state for nitrogen fitted the P-V-T data so well and gave such good results in the calculations of the equations of state of mixtures, it is believed that the older observed data are probably at fault. Because of the difficulty of the experimental determinations of Joule-Thomson coefficients this is not surprising. Gas mixtures An examination of the equations for the Joule-Thomson coefficients will reveal the fact that with the given method of combining constants the coefficients for gas mixtures should be obtainable for most purposes by a linear combination of the coefficients of the pure gases. An examination of table 2 will indicate that this is true within the accuracy of the calculations. USE O F FIGURES 1 AND 2 The isotherms for the pure gases are given in figures 1 and 2. To estimate the Joule-Thomson coefficients for any mixture, it is only necessary
1193
MEASUREMENT O F JOULE-THOMSON COEFFICIENTS
TABLE 2 Joule-Thomson coejicients of nitrogen-methane mixtures JOULE-THOMSON COEFFICIENTS
PRESSURE IN ATMOSPHERES
I
ZOOOR.
250°K.
I
I
300°K.
350°K.
I
400°K.
5 25 50 100
0.88 0.85 0.73 0.62 0.46
0.58 0.57 0.51 0.46 0.37
0.41 0.40 0.37 0.34 0.28
0.30 0.30 0.28 0.26 0.22
0.23 0.22 0.210.20 0.17
1 5 25 50 100
0.77 0.74 0.65 0.55 0.41
0.50 0.49 0.45 0.40 0.34
0.36 0.35 0.33 0.30 0.25
0.26 0.26 0.24 0.22 0.19
0.20 0.19 0.18 0.17 0.15
1
0.65 0.64 0.56 0.48 0.37
0.43 0.42 0.39 0.35 0.28
0.31 0.30 0.26 0.26 0.21
0.22 0.22 0.21 0.19 0.16
0.17 0.16 0.15 0.14 0.12
0.54 0.53 0.47 0.41 0.32
0 36 0.35 0.33 0.29 0.24
0.25 0 25 0.23 0.21 0.17
0.18 0.18 0 17 0.16 0.13
0.14 0.13 0.13 0.11 0.09
0.15 0.14 0.13 0.12 0.10
0.10 0.10 0.09 0.08 0.07
1
5 25 50 100 1
5 25 50 100
-
(e) 100 per cent NZ 1
5 25 50 100
c .. . .
L.,
.. .., .. , . . . . ./ ,
0.44 0.43 0.38 0.33 0.26
0
0.333
0.29 0.28 0.26 0.23 0.19
1
20 0.291
1
0.20 0.20 0.18 0.17 0.14
40 0.250
1
60
0.215
1
80 0.187
1
~
100
0.159
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JOHN H. PERRY AND CARL V. HERRMANN
FIQ.1. THE JOULE-THOMSON EFFECTFOR METHANE
FIQ.2. THE JOULE-THOMSON EFFECTFOR NITROGEN
MEASUREMENT OF JOULE-THOMSON COEFFICIENTS
1195
to combine linearly the values for the pure gases as obtained from these curves. Since p is the change of temperature with pressure, a positive sign indicates that a decrease in pressure will be accompanied by a decrease in temperature. CONCLUSIONS
The Joule-Thomson coefficients of methane and nitrogen have been calculated a t the temperatures 200, 250, 300, 350, and 400°K. and pressures of 1, 5, 25, 50, and 100 atmospheres, using the Beattie-Bridgeman equation of state. A method of obtaining the coefficients for mixtures of these gases from those of pure gases has been indicated, and data for three mixtures of these gases have been calculated and tabulated. REFERENCES
(1) BEATTIE:J. Am. Chem. SOC. 61, 19 (1929). (la) BEATTIE:Phys. Rev. 34,1615 (1929). (2) BEATTIE:Phys. Rev. 36, 643 (1930). (3) BEATTIEAND BRIDQEMAN: J. Am. Chem. SOC.49, 1665 (1927); 60, 3133, 3151 (1928); Proc. Am. Acad. Arts Sci. 63, 229 (1928). (4) BEATTIEAND BRIDQEMAN: Z. Physik 62, 95 (1930). (5) BEATTIEAND IKEHARA: Proc. Am. Acad. Arts Sci. 64, 127 (1930). Phys. Rev. 34, 527 (1929). (6) BRIDQEMAN: (7) EASTMAN: Bur. Mines Tech. Paper No. 445 (1929). a
TEE JOURNAL OF PEYBICAL OEEMIBTRY, VOL. XXXIX, NO. g
