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Thermal conductivity measurements of methane in the dense gaseous state
. Physica 32 885-899
Misic, D. Thodos, G. 1966
THERMAL CONDUCTIVITY MEASUREMENTS OF METHANE IN THE DENSE GASEOUS STATE by D. MISIC Northwestern
University,
and G. THODOS Evanston,
Illinois,
U.S.A.
Synopsis The thermal conductivity of methane has been measured at 1.9, 24.6, 43.6 and 75.1 “C and pressures up to 580 atm with a coaxial cylindrical cell. For each of these temperatures, the thermal conductivity - reduced density relationships have been developed and have been used to establish the corresponding (k/p~)~l~ values. These values have been utilized to calculate thermal conductivities predicted from the Enskog theory. The measured thermal conductivity values also have been used to produce the relationships of k - k* versus pi and k - k* versus (LJPR/XR),,~ for methane. These relationships also include the thermal conductivity values of other investigators which were found in good agreement with the results of the present study. The experimental results of this investigation for the dense gaseous state assist to establish a continuous unique relationship between the gaseous and liquid states of metahne.
Intro&&on. To establish the thermal conductivity behavior of gases and liquids, reliable experimental data are essential. Such data become very valuable in practical applications and in the testing of theoretical models. In the present expertiental study, methane was arbitrarily selected to be investigated since for this substance thermal conductivity measurements are meager. Experimental values by Lenoir and Comingsr7) are restricted to 41.1X and pressures up to 200 atm, while those of Lenoir, Junk, and Comings 18) were taken at 52.8% and also pressures up to 200 atm. The early work of Keyesrs)
was conducted
at 50, 150, 250 and 300°C and
pressures only up to 60 atm, while his later work14) includes the low temperature region of - 152.7, - 101.9 and 0°C and pressures no higher than 9 atm. In 1950, Stolyarov, Ipatiev and Teodorovichs*) reported experimental thermal conductivities for methane between 20 and 200°C and pressures up to 500 atm. Their data do not exhibit good internal consistency indicating the presence of experimental difficulties in their study. The latest experimental thermal conductivity measurements of Ikenberry and Ricera) for methane in the dense gaseous and liquid regions are valuable because they extend the experimental data of this study from the dense gaseous region into the liquid region for this substance. -
885 -
886
D. MISIC AND G. THODOS
Experimental
eqtiipment and procedure.
The experimental
unit consisted
of a coaxial cylindrical cell having an annular gap of 0.0254 cm. and an outside diameter of the inner cylinder of 1.8898 cm. The total length of the assembled
cell was 35.56 cm and the outside diameter
was 8.57 cm. This cell was designed to withstand A detailed
description
of the cell and associated
elsewhere 21) 22). However,
of the outer cylinder
pressures up to 1700 atm. equipment
a minor change in the pressurizing
is presented procedure was
followed since the methane was available in quart size cylinders only at a pressure of not more than 80 atm. In this study, a prescribed amount of methane was condensed in a thermal expansion cell immersed in liquid nitrogen. This cell was constructed of 316 stainless steel which had been tested for pressures up to 1200 atm. After allowing the cell to reach room temperature, the gaseous methane, now available at pressures up to 700 atm. was slowly transferred directly either to the thermal conductivity cell or to the pressurizing vessel connected to the Ruska pump. The procedure then followed was identical to that outlined in the parent articlesa).
Fig. 1. Relationships
between
and 75.1 “C for methane
thermal
resulting
conductivity
from the experimental
and pressure
at
measurements
1.9, 24.6, 43.6 of this study.
THERMAL
Analysis
CONDUCTIVITY
The procedure
of data.
887
OF METHANE
used to calculate
thermal
conductivities
for methane was the same as that outlined previously for a similar study with nitrogenss) The contributions due to natural convection and radiation were kept perimental
to a minimum measurements.
and were well within A pressure correction
the accuracy of the exwas applied to the raw
thermal conductivity values, ka, to account for the pressure cell dimensions. This correction was calculated to be k = ka[ 1 + 2.464
effect
on the
x lo-s] P
(1)
where P is the pressure in psia. The resulting experimental thermal conductivities for methane are tabulated in tables I, II, III and IV for temperatures of 1.9. 24.6, 43.6 and 751°C respectively, and for pressures up to 577.7 atm. These values are plotted against pressure as shown in figure 1. The atmospheric thermal conductivity measurements of this study and those resulting from the values reported in the literature 4, 8, 11, 13, 14, 15, 17, 18, 1’9, 26, 27, 28, 30, 31, 32,33) are in good agreement. The atmospheric thermal conductivities resulting from these references are also presented in table I. In 1922, Enskogs), using principles of kinetic theory and a rigid spherical model, developed relationships for the transport properties of dense gases and liquids. For viscosity and thermal conductivity, these relationships are Application
of Ertskog
-= P
iu*
theory.
1
6
k -= k* To test his theory,
4
bP [ bPX + s + s
Enskog
+ +
0.7614
bpx
0.7574 bpx
1*I 1
(2)
.
was able to utilize only equation
that time, high pressure thermal conductivity
measurements
(2) because
at
were not availa-
ble. Using the high pressure viscosity measurements of W arburg and Baboss), Enskog obtained reasonable agreement between experimental values and viscosities calculated from his theory. For real gases Enskog proposed the following equation of state: P + up2 = %(I If a and b are assumed to be constant, with respect to temperature at constant
*) For the meaning
of the different
+bpx) equation (4) can be differentiated volume to produce the relationship
letters see the list at the end of this article.
0.313
0.440 0.555
0.625
0.675
0.718 0.795
0.960
1.06
1.26
1.26
1.27
1.40
1.53
1.84
1.90
82.2 99.3
109.9
118.7
124.5 137.1
164.3
183.4
229.6
230.0
232.4
273.7
328.3
517.9
572.6
0.190 0.280
40.1 56.5
0.313
0.165
35.0
62.3
0.091
20.0
62.3
0.0755
16.7
2.38 1
2.193
1.484
1.270
1.083
1.071
1.071
0.8353
0.7354
0.5234 0.5875
0.4887
0.450
0.3133 0.3985
0.2225
0.2225
0.1349 0.1988
0.1172
0.0646
0.0536
22.23
20.33
13.15
11.00
9.153
9.028
9.028
6.732
5.765
3.785 4.367
3.477
3.136
2.000 2.686
1.324
1.324
0.7462 1.162
0.6377
0.3353
0.2753
25.47
23.85
18.69
17.61
15.96
16.04
15.81
14.66
13.89
11.37 12.67
11.31
11.00
10.05 10.68
9.22
9.27
8.38 9.06
8.42
7.91
7.70
0.13
k’
17.95
16.33
11.17
10.09
8.44
8.52
8.29
7.14
6.37
3.85 5.15
3.79
3.48
2.53 3.16
1.70
1.75
0.86 1.54
0.90
0.39
0.26
I
1
useof
41.02
38.60
29.16
25.80
23.08
22.9 1
22.91
19.64
18.20
15.49 16.38
15.13
14.54
12.81 13.82
11.93
11.93
11.14 11.70
11.00
10.62
10.56
10.37
10.28
_._-.
10.28 X 10-s 10.28
1
data
33.88
32.02
24.56
21.80
19.50
19.36
19.32
16.53
15.25
12.78 13.60
12.44
11.89
10.25 11.22
9.36
9.36
8.54 9.13
8.50
7.96
7.08
7.64 x 10-b
11.46
33.00
34.26
31.39
23.80
22.17
20.70
22.18
12.77
9.82
12.43 7.32
10.01
8.07
2.00 5.05
1.52
0.97
1.94 0.78
0.99
0.57
1.26
-0.12
( % dev I
27.304
25.618
19.252
17.330
15.609
15.576
15.576
13.479
12.558
10.571 11.169
10.244
9.870
8.550 9.345
7.703
7.703
6.950 7.490
6.164
6.390
6.306
6.045
% 10-s
9.42
+
+
+
-
-
-
-
-
11.54
7.20
7.41
3.01
1.59
1.70
2.89
1.48
8.06
-11.75
- 7.03 - 11.85
-
- 10.27
- 14.92 - 12.50
- lb.45
- 16.90
- 17.06 - 17.24
- 26.79
- 19.22
- 18.95
- 20.98
use of Wp)mtn kcalc ( % dev
10-s cal ems/g s”K
at 1.9”C
12.34 x
kcnle
viscosity
=
for methane
(k/p)tin
quantities
I*, poises 5)
and related
0.00 x 10-s 0.00 0.00
0.0845
IO-6
k -
7.65
0.0170
kex,
7.57
0.024
1
1.02
1 (m&~R)pR
Cal/s cm OK
5.4
bPX
conductivities
10-s Cal/s cm°K
thermal
7.52 x
\
I
k’ = 7.52 x
and calculated
7.60
PR
Experimental
1.02
(
1.44
1.02
P, atm.
Tfi =
TABLE
889
THERMAL CONDUCTIVITY OF METHANE
By the substitutions,
p = pep~t P = PcP~
and T = T,TR,
can be expressed in terms of reduced parameters
equation
(5)
as follows:
Thus, in order to obtain bp~ values for a substance from equation
(6), PVT
data must be available for a substance over the range of interest. The reduced density correlation resulting from all available PVT measurements on methane (20) was used to produce PR versus TR relationships for reduced densities ranging from pR = 0.010 to pR = 3.00. The slopes resulting from these relationships were then used to calculate WRPTR),, the quantity 1 + bp~ using equation (6). The details of evaluating this quantity are identical to a similar study concerned with argon (2). The values of 1 + bp~ obtained for methane are plotted against reduced pressure, PR, to produce the isothermal relationships presented in figure 2. In this figure the range of reduced pressures extends up to PR = 20 and covers the temperature range 0.60 I TR 5 4.0.
METHANE (zO-0.289) T,=191.1 ‘K P,=45.6atm
Reduced
Pressure, e = $C
Fig. 2. Isothermal relationships between 1 + bpx and reduced pressure for methane.
Reliable values of bp~ for moderate pressures and pressures slightly above one atmosphere are obtained by plotting bpX/pR versus pR. For methane in the high density region, all the isotherms produce a single well defined relationship. However, in the low density region, considerable random scatter is encountered which results from the limitations to obtain dependable values of bpx from figure 2 at these conditions. An extension of the bpX/pR versus pR’relationship from the high density region to pR = 0, establishes the intercept bpX/pR = 0.71. The quantity bpx/pR = bpcX. At
1.34 1.42
1.54
1.63
1.69
1.78
306.5 341.0
405.2
454.2
499.1
567.1
0.845 1.00
171.1 205.1
1.13
0.750
152.4
1.24
0.697
139.7
272.5
0.585
120.0
238.5
0.335
0.395
73.2
0.312
69.1
85.4
0.226
51.6
2.0256
1.8066
1.6773
1.5030
1.1805 1.3007
1.0428
0.9108
0.6287 0.7730
0.5482
0.5067
0.4212
0.2808
0.2378
0.2215
0.1605
0.1037
18.630
16.413
15.100
13.335
10.109 11.306
0.764
7.470
4.762 6.135
4.018
3.634
2.877
1.751
1.435
1.319
0.9073
0.5577
22.98
21.84
20.87
19.85
17.69 18.67
16.81
15.96
13.41 14.74
12.22
12.48
11.12
9.99
9.69
9.78
9.12
8.64
8.41
0.146
0.1633 0.2524
34.4
0.0324
0.0695
0.0493
0.0457
16.8
/
8.18 8.49
/ (aPdaT&, 8.24
bpx
1.02 11.0
)
fl
x 10-s
kexp
Cal/s cm”K
1.02
pi
14.77
35.9 1
34.60
30.42
25.60 27.26
23.82
21.99
17.97 20.00
16.78
15.99
14.84
13.28
12.80
12.70
12.17
11.71
11.40
11.18 11.30
11.18
11.18
x 10-S
)
data
=
30.010
28.993
27.497
25.713
21.696 23.098
20.207
18.620
15.014 16.861
13.938
13.220
12.137
10.602
10.124
10.007
9.441
8.922
a.533
x 10-a
kealc
8.408
use of viscosity
12.77
for methane
WpJmtn
quantities
p, poisesb)
.13.63
I
related
32.65
x 10-j
k’
and
II
12.66
11.64
9.40 10.46
8.60
7.75
5.20 6.53
4.01
3.27
2.91
1.70
1.48
1.57
0.9 1
0.43
0.20
0.00 0.28
0.00
0.00
k -
conductivities
lo-5caljscm”K
thermal x
calculated k’ = 8.21
8.21
)
1.56
and
1.02
p, atm
TR =
Experimental
TABLE
14.27
30.59
32.75
31.75
29.54
22.65 23.72
20.21
16.67
11.96 14.39
14.06
5.93
9.14
6.13
4.48
2.32
3.52
3.26
1.46
-0.96
s”K
24.958
22.856
21.712
20.092
17.118 18.220
15.882
14.664
11.991 13.377
11.205
10.770
9.892
8.535
8.126
7.971
7.413
6.926
6.496
6.369
x 10-S
keal,
8.31
f
+
+
_
-
-
-
11.31
8.61
4.65
4.03
1.22
3.23 2.41
5.52
8.12
_ 10.58 _ 9.25
_
_ 13.70
-11.04
- 14.56
- 16.14
- 18.50
_ 18.72
- 19.84
- 22.76
- 24.98
j % dew
use of (klp),t.
10-G cal ems/g
/ % dev
x
at 24.6”C
THERMAL
CONDUCTIVITY
891
OF METHANE
pR = 0, x = 1, and therefore $JX/PR becomes the dimensionless group bp,. Since the critical molar density is pc = 0.162/l 6.042 = 0.010098 g-moles/cm3, the second virial coefficient becomes b = 0.7 1lo.01 0098 = 70.31 cm”/g-mole. This value produces the collision diameter (r = 3.82 x IO-8 cm, from the relationship b = 23zNa3/3. Hirschfelder, Curtiss, and Birds) present values of b = 70.16 cm3/g-mole and u. = 3.817 x 10-s cm.
Methane (z.=O289)
.
Fig. 3. Relationship between bpX/pR and pi for methane at 1.9, 24.6, 43.6 and 75.1 “C.
One approach to test the Enskog theory involves the use of experimental viscosity measurements corresponding to the temperature and pressure conditions of the thermal conductivity measurements. In this study the high pressure viscosity measurements of methane recently obtained by Giddingss) were used in conjunction with Equation (2) to obtain values of bp. These values were then applied to equation (3) to produce the ratio K/k*. The same results follow by dividing equation (3) by equation (2). From the k/k* values, the calculated high pressure thermal conductivities were obtained and were compared with the corresponding experimental values. These comparisons areincluded in tables I, II, III and IV for 1,9, 24.6, 43.6 and 75.1 “C, respectively. A second approach for the calculation of thermal conductivities at elevated pressures follows from the Enskog theory. By rearranging equation (3) becomes -
k
= bk*
P
Differentiating
equation
W/d ____ Wpx)
-
1
6
+ s
bPX
+
1
(7)
1
(8)
0.7574 bpx
*
(7) with respect to bpx gives = bk*
-
1
___
(bPX)2
+ or7574
.
Letting d(k/p)/d(bpx) b e zero, the bpx value at the minimum becomes bpx = 1.149. When this value is substituted into equation (7), it follows
1.69
573.9
1.02 1.13
1.54 1.62
0.755
171.1
239.2 273.2
463.7 518.1
0.685
155.8
1.44
0.632
143.9
1.41
0.540
123.5
407.6
0.521
388.8
0.363
86.6
120.1
1.23
0.287
69.1
1.32
0.258
62.6
341.2
0.084
21.4
307.2
0.0451 0.084
11.5 21.4
I .a066
1.5030 1.6540
1.329
1.2845
1.1524
1.0320
0.7946 0.9130
0.5512
0.4959
0.4557
0.3872
0.3725
I
0.258 1
0.2038
0.1832
0.0596
0.0320 0.0596
16.413
13.335 14.876
11.605
11.144
9.833
0.647
6.335 7.478
4.053
3.546
3.184
2.592
2.474
1.580
1.195
1.056
0.3080
0.1611 0.3080
22.74
20.72 21.70
19.40
19.35
18.33
17.48
15.40 16.53
12.98
12.70
12.43
11.84
11.74
10.67
10.10
9.77
9.12
8.88 9.13
8.82
1 WRW&~
thermal
x IO-5
keap
cal/s 1
III
13.88
11.86 12.84
10.54
10.49
9.47
8.62
6.54 7.66
4.12
3.84
3.57
2.98
2.88
1.81
1.24
0.91
0.26
0.02 0.27
0.00
0.00
0.00
x to-5
k - k'
cm OK
cm”k
conductivities
10-b Cal/s cmjs
8.97
bpx
x
1.02
/
calculated
k’ = a.86
and
1.02
PR
Experimental
a.92
/
1.66
I .02
P, atm.
7-R =
TABLE and
34.80
31.02 33.03
28.65
27.83
25.65
24.07
20.73 22.42
17.36
16.60
16.04
15.17
15.02
13.71
13.17
12.99
12.09
11.93 12.09
11.78
11.78
11.78
5)
x 10-S
1
29.972
27.154 28.520
24.866
24.150
22.278
20.907
17.917 19.440
14.770
14.044
13.503
12.644
12.494
11.159
10.579
10.378
9.299
9.083 9.299
data
13.32
x IO-5
k es,e
=
for methane
use of viscosity
(k/p),,,
quantities
~1,poises
related
I
14.51
31.80
31.05 31.43
28.18
24.8 1
21.54
19.60
16.34 17.68
13.79
10.58
8.63
6.79
6.42
4.58
4.74
6.22
1.96
2.29 1.85
s”K
23.904
20.957 22.440
19.305
18.854
17.585
16.444
14.139 15.291
11.739
11.145
10.705
9.971
9.834
8.668
8.140
7.945
6.860
6.634 6.860
x
10-S
j
7
+ +
-
-
-
-
-
-
_
11.89
5.12
1.14 3.41
0.49
2.56
4.06
5.93
8.19 7.44
9.56
- 12.24
- 13.88
- 15.78
- 16.24
- 18.76
- 19.41
- 18.68
- 24.78
- 25.29 - 24.86
% dev
use of (Wp)min kealc
1O-5 cal cm2jg
at 43.6”C
1 % dew )
x
THERMAL
CONDUCTIVITY
893
OF METHANE
that min and therefore,
equation
= 2.940 bk*
(7) can be expressed
(9)
in the alternate
form,
k -=
(10)
P
To apply equation (lo), isothermal experimental thermal conductivities should be available at sufficiently high pressures in order to establish the (k/P)nG, value from a k/p versus bpx or k/p versus p plot. The pressures used in this investigation did not go high enough to establish this minimum, except at 1.9”C. A plot of k/pR versus pi for this temperature is given in
Fig. 4. Kelationship
between
k/pR and pi resulting
measurements
of methane
from
the thermal
conductivity
at 1.9 “C
figure 4. Since for the other three temperatures, no minimum was observed, the following approach applicable to all four temperatures was used. The experimental thermal conductivities were plotted against PR, the reduced density, as shown in fig. 5. The following analytical expressions resulted from a least squares approach using an IBM 709 digital computer: 1.9”C k X 105 = 7.5801
+ 3.3058~~
+ 7.0778~:
-
+ 2.2411 pi
(11)
4.2029 pi + 1.1765 pi
(12)
6.3124~:
24.6”C k x
105 = 8.2399
+ 2.3775 pR + 7.0636 p; -
894
D. MISIC AND G. THODOS
43.6”C k x lo5 = 8.8248 + 3.8699 PR + 2.528 ,D;
(13)
75.1% k x
lo5 = 10.2626 + 3.1742 PR + 2.4854 p; + 0.1402 &
(14)
and are presented with the experimental values in fig. 5. Equations (ll), (12), (13) and (14) were divided by PR and were then differentiated with respect to PR to obtain relationships which when set to zero, produced the following PR and k/pR values at the minimum of each isotherm : Calculated minima “C 1.9 24.6 43.6 75.1
Fig. 5. Relationships
between
Pfl 1so9 1.683 1.868
1.848
k//m
I
12.34 x 10-S 12.77 13.32 13.80
k and pi for methane
at 1.9, 24.6, 43.6 and 75.1 “C
,
I
0.5674
0.7633 0.8615
0.773
0.990
205.1 273.2
0.9582
1.4140
1.17
1.49
1.56
344.0
523.2
577.7
1.5397
0.9582
341.2
1.085 1.17
307.2
-
0.4679
0.0788
0.648
bpx
0.111
I
k’ =
-
0.4143
13.713
7.9279 12.443
7.9279
6.9876
6.0408
4.1932
3.2920
) /
I
21.57
17.64 21.17
17.82
17.01
15.59
14.02
13.53
10.98
10.01 x 10-s
11.07 11.47
7.54
7.72
6.91
5.49
3.92
3.43
0.00 x IO-5 0.88
k -
cm”K
-
30.41 31.71
24.07
22.52 23.96
21.15
18.48
17.18
13.25
12.81 x 10-S
p, poises 5)
28.747
27.646
21.900
20.455 21.796
19.143
16.513
15.183
10.757 x 10-S
kesle
-
I
data
13.80
for methane
use of viscositv
quantities
ca
_-
and related Cklo)min ,... ~~=
k’
conductivities
IV
lo-scal/scm”K
thermal
ker,
10.10 x
and calculated
L W’RI~TR)~R
Experimental
171.1
PR
1.82
1.02 31.0
P, atm.
TR =
TABLE
x
20.60
33.27
30.59
22.3 1 24.15
20.25
17.78 22.79
- 2.03 12.22
o/odev
-
_7.268
22.085
20.828
16.310
16.310
15.348
14.350
12.305
11.229
x 10-S
use of
10-S cal cma/g s”K
at 75.1 “C
+
-
-
~ -
-
11.20
2.39
1.62
7.54
9.77 8.45
7.95
- 12.23
- 17.01
-33.81
0/0 dev
P) mln
-I 1
896
D. MISIC AND G. THODOS
Using the values of (k/~&~~, thermal with Equation IV. Application
conductivity
(lo), to produce the results presented
of semi-theoretical
approaches.
values were calculated in tables I, II, III and
The approach
suggested
by
Abas-zadel) that the residual thermal conductivity, k - k*, when plotted against density produces a single relationship for all temperatures, has been successfully “applied for argonz3), nitrogen 25), carbon dioxide 12), ammonia7), and methane24). The experimental data of this study have been subjected to a
0
2
I
3
P*
Fig. 6. lielationship between k -
k* and pi for methane in the gaseous and liquid
regions.
similar treatment which also includes the recent thermal conductivity values of Ikenberry and Ricelo) for liquid methane. A plot of k - k* versus PR is presented in fig. 6 which, in addition, includes the thermal conductivity values reported for methane in the dense gaseous state by KeyesIs). Lenoir,andComings17),Lenoir, Junk,andComingsl*),andStolyarov, Ipatiev, and Teodorovich28). These values produce the single unique relationship presented in fig. 6. Except for the few points of Ikenberry and Ricelo) taken near the critical temperature (TR = 1.048), the remaining values of his investigation and those of the other investigators, produce the unique relationship which is continuous throughout the gaseous and liquid states. The values reported by Stolyarov e.a. 28)“exhibit some inconsistencies and deviate somewhat from the established relationship of figure 6. The experimental thermal conductivity values of this study are consistent and assist to establish the relationship of fig. 6 in the dense gaseous region where reliable data were not available. Golubev6) has introduced the thermodynamic quantity, (WjaT),, to replace the density in the corklation of p - p*, the residual viscosity. The resulting ,U - ,LL*versus (aP/BT) v relationships were linear when plotted on
THERMAL
log-log coordinates. to produce
CONDUCTIVITY
This concept has been used by Lenner
generalized
relationships
897
OF METHANE
of viscosity
t and Thodos16)
which were linear
for
argon, krypton, and xenon. A logical extension to thermal conductivity appears appropriate in which the residual thermal conductivity, k L K*, in reduced form. Thus, is related to (aP/aT),, expressed for convenience the quantity
(aP,/aT,),,
of bpx given in tables of this study.
follows directly
I, II, III
In addition,
from equation
and IV which correspond
(6) and the values to the conditions
these values were produced for conditions
corre-
0 lkenberry and Rice 9 0 + 9
Fig. 7. Relationship
Lencmr and Comings Lenar. Junk, and Comings Siolyorov, Ipatw. and Teodorovich This invesiigatlon
between k - k’ and (aP,/aT& for methane in the dense gaseous and liquid regions.
sponding to the data of the other investigators 10, 13, 14, 17, 18, 28) and are presented in fig. 7. The thermal conductivity data of methane produce in fig. 7 a relationship which is linear on log-log coordinates and exhibits a scatter in the low density region. The resulting relationship can be expressed analytically as follows :
.
(15)
This relationship enables the calculation of thermal conductivities in the gaseous and liquid regions from the PVT behavior of the substance and its corresponding atmospheric thermal conductivity value. This type of correlation deserves further attention and should be tested with substances other than methane.
898
D. MISIC AND G. THODOS
Acknowledgement. support of the Petroleum
The authors Research
gratefully
acknowledge
Fund of the American
the financial
Chemical Society
through Grant 329-A. Also due thanks are extended to the Phillips Petroleum Company
for the methane
used in this study.
Nomenclature: a, b constants, equation (4)
thermal conductivity, Cal/s cm “K apparent thermal conductivity, Cal/s cm “K thermal conductivity of gas at atmospheric pressure, Cal/s cm”K molecular weight M N Avogadro’s number, 6.0228 x loss molecules/gram mole P pressure, atm PC critical pressure, atm reduced pressure, P/P, PR gas constant, 82.055 ems atm/g-mole”K H T absolute temperature, OK TC critical temperature, OK reduced temperature, T/Tc TR critical compressibility factor, P,v,/RT, zc k
ko k*
Greek viscosity, g/cm s viscosity of gas at atmospheric pressure, g/cm s 3.14159 2% density, g/ems P PC critical density, g/cm3 PR reduced density, p/pe collision diameter, cm B probability of nearess x P
P*
Received l-6-65
REFERENCES 1) Abas-zade, A. K., Doklady Akad. Nauk Azerbaidzhan SSR 3 (1947) 3. 2) Gediminas, Damasius and Thodos, George, Industr. engng Chem., Fundamentals 2 (1963) 73. 3) Enskog, David, Svensk. Akad. Handl. 63 No. 4 (1922). 4) Eucken, A., Phys. Z. 12 (1911) 1101. J. G., Ph. D. dissertation, Rice University, Houston, Texas (1963). 5) Giddings, I. F., “Viscosity of Gases and Gaseous Mixtures,” Fizmatgiz, Moscow (1959). 6) Golubev, 7) Groenier, W. S. and Thodos, George, J. them. Eng. Data 6 (1961) 240. 8) Hercus, E. 0. and Laby, T. H., Proc. roy. Sot. (London) 95 (1919) 190. J. O., Curtiss, C. F. and Bird, R. B., ‘*Molecular Theory of Gases and Liquids 9) Hirschfelder, Wiley, New York (1954). L. D. and Rice, S. A., J. them. Phys. 39 (1963) 1561. 10) Ikenberry, H. L. and Grilly, E. R., J. them. Phys. 14 (1946) 233. 11) Johnston, 12) Kennedy, J. T. and Thodos, George, A.I. Ch. E. Journal 7 (1961) 625.
THERMAL
CONDUCTIVITY
OF METHANE
899
13) Keyes, F. G., Trans. Am. Sot. Mech. Engrs. 76 (1954) 809. 14) Keyes, F. G., Trans. Am. Sot. Mech. Engrs. 77 (1955) 1395. M. W., Robinson, A. M., Scrivins, J., 15) Lambert, J. D., Cotten, K. J., Pailthorpe, Vale, W. R. F. and Young, R. M., Proc. roy. Sot. (London), A%31 (1955) 280. 16) Lennert, D. A. and Thodos, George, A.I. Ch. E. Journal 11 (1965) 155. 17) Lenoir, J. M. and Comings, E. W., Chem. Eng. Progr. 47 (1951) 223. 18) Lenoir, J. M., Junk, W. A. andcomings, E. W., Chem. Eng. Progr.dt) (1953) 539. 19) Mann, W. B. and Dickins, B. G., Proc. ray. Sot. (London), Al34 (1931) 77. 20) Matschke, D. E. and Thodos, George, J. Petroleum Technol. 12 (1960) 67. 21) Misic, Dragoslav, Ph. D. dissertation, Northwestern University (1965). 22) Misic, Dragoslav and Thodos, George, A.1.Ch.E. Journal 11 (1965)650. 23) Owens, E. J. and Thodos, George, A.1.Ch.E. Journal 3 (1957) 454. 24) Owens, E. J. and Thodos, George, Proc. Conf. Therm6d. Prop. Fluids, London, p. 163, July lo-12 (1957). 25) Schaefer, C. A. and Thodos, George, A. I. Ch. E. Journal 5 (1959) 367. 26) Schottky, W. F., 2. Elektrochem. 56 (1952) 889. 27) Stefan, J., Berichte der Wiener Akademie 65 (1872) II, 45. 28) Stolyarov, E. A., Ipatiev, V. V. and Teodorovich, V. P., Zhur. Fiz. Khim. “4 (1950) 166. 29) Warberg, E. and von Babo, L., Wied. Ann. 17 (1882) 390. 30) Weber, S., Ann Physik (4) 54 (1917) 437. 31) Weber, S., Ann. Physik (4) 54 (1917) 481. 32) Winkelmann, A., Ann. Physik 156 (1875) 497. 33) Ziegler, E., Dissertation, Universitat zu Halle, Germany 1904.
Misic, D. Thodos, G. 1966
THERMAL CONDUCTIVITY MEASUREMENTS OF METHANE IN THE DENSE GASEOUS STATE by D. MISIC Northwestern
University,
and G. THODOS Evanston,
Illinois,
U.S.A.
Synopsis The thermal conductivity of methane has been measured at 1.9, 24.6, 43.6 and 75.1 “C and pressures up to 580 atm with a coaxial cylindrical cell. For each of these temperatures, the thermal conductivity - reduced density relationships have been developed and have been used to establish the corresponding (k/p~)~l~ values. These values have been utilized to calculate thermal conductivities predicted from the Enskog theory. The measured thermal conductivity values also have been used to produce the relationships of k - k* versus pi and k - k* versus (LJPR/XR),,~ for methane. These relationships also include the thermal conductivity values of other investigators which were found in good agreement with the results of the present study. The experimental results of this investigation for the dense gaseous state assist to establish a continuous unique relationship between the gaseous and liquid states of metahne.
Intro&&on. To establish the thermal conductivity behavior of gases and liquids, reliable experimental data are essential. Such data become very valuable in practical applications and in the testing of theoretical models. In the present expertiental study, methane was arbitrarily selected to be investigated since for this substance thermal conductivity measurements are meager. Experimental values by Lenoir and Comingsr7) are restricted to 41.1X and pressures up to 200 atm, while those of Lenoir, Junk, and Comings 18) were taken at 52.8% and also pressures up to 200 atm. The early work of Keyesrs)
was conducted
at 50, 150, 250 and 300°C and
pressures only up to 60 atm, while his later work14) includes the low temperature region of - 152.7, - 101.9 and 0°C and pressures no higher than 9 atm. In 1950, Stolyarov, Ipatiev and Teodorovichs*) reported experimental thermal conductivities for methane between 20 and 200°C and pressures up to 500 atm. Their data do not exhibit good internal consistency indicating the presence of experimental difficulties in their study. The latest experimental thermal conductivity measurements of Ikenberry and Ricera) for methane in the dense gaseous and liquid regions are valuable because they extend the experimental data of this study from the dense gaseous region into the liquid region for this substance. -
885 -
886
D. MISIC AND G. THODOS
Experimental
eqtiipment and procedure.
The experimental
unit consisted
of a coaxial cylindrical cell having an annular gap of 0.0254 cm. and an outside diameter of the inner cylinder of 1.8898 cm. The total length of the assembled
cell was 35.56 cm and the outside diameter
was 8.57 cm. This cell was designed to withstand A detailed
description
of the cell and associated
elsewhere 21) 22). However,
of the outer cylinder
pressures up to 1700 atm. equipment
a minor change in the pressurizing
is presented procedure was
followed since the methane was available in quart size cylinders only at a pressure of not more than 80 atm. In this study, a prescribed amount of methane was condensed in a thermal expansion cell immersed in liquid nitrogen. This cell was constructed of 316 stainless steel which had been tested for pressures up to 1200 atm. After allowing the cell to reach room temperature, the gaseous methane, now available at pressures up to 700 atm. was slowly transferred directly either to the thermal conductivity cell or to the pressurizing vessel connected to the Ruska pump. The procedure then followed was identical to that outlined in the parent articlesa).
Fig. 1. Relationships
between
and 75.1 “C for methane
thermal
resulting
conductivity
from the experimental
and pressure
at
measurements
1.9, 24.6, 43.6 of this study.
THERMAL
Analysis
CONDUCTIVITY
The procedure
of data.
887
OF METHANE
used to calculate
thermal
conductivities
for methane was the same as that outlined previously for a similar study with nitrogenss) The contributions due to natural convection and radiation were kept perimental
to a minimum measurements.
and were well within A pressure correction
the accuracy of the exwas applied to the raw
thermal conductivity values, ka, to account for the pressure cell dimensions. This correction was calculated to be k = ka[ 1 + 2.464
effect
on the
x lo-s] P
(1)
where P is the pressure in psia. The resulting experimental thermal conductivities for methane are tabulated in tables I, II, III and IV for temperatures of 1.9. 24.6, 43.6 and 751°C respectively, and for pressures up to 577.7 atm. These values are plotted against pressure as shown in figure 1. The atmospheric thermal conductivity measurements of this study and those resulting from the values reported in the literature 4, 8, 11, 13, 14, 15, 17, 18, 1’9, 26, 27, 28, 30, 31, 32,33) are in good agreement. The atmospheric thermal conductivities resulting from these references are also presented in table I. In 1922, Enskogs), using principles of kinetic theory and a rigid spherical model, developed relationships for the transport properties of dense gases and liquids. For viscosity and thermal conductivity, these relationships are Application
of Ertskog
-= P
iu*
theory.
1
6
k -= k* To test his theory,
4
bP [ bPX + s + s
Enskog
+ +
0.7614
bpx
0.7574 bpx
1*I 1
(2)
.
was able to utilize only equation
that time, high pressure thermal conductivity
measurements
(2) because
at
were not availa-
ble. Using the high pressure viscosity measurements of W arburg and Baboss), Enskog obtained reasonable agreement between experimental values and viscosities calculated from his theory. For real gases Enskog proposed the following equation of state: P + up2 = %(I If a and b are assumed to be constant, with respect to temperature at constant
*) For the meaning
of the different
+bpx) equation (4) can be differentiated volume to produce the relationship
letters see the list at the end of this article.
0.313
0.440 0.555
0.625
0.675
0.718 0.795
0.960
1.06
1.26
1.26
1.27
1.40
1.53
1.84
1.90
82.2 99.3
109.9
118.7
124.5 137.1
164.3
183.4
229.6
230.0
232.4
273.7
328.3
517.9
572.6
0.190 0.280
40.1 56.5
0.313
0.165
35.0
62.3
0.091
20.0
62.3
0.0755
16.7
2.38 1
2.193
1.484
1.270
1.083
1.071
1.071
0.8353
0.7354
0.5234 0.5875
0.4887
0.450
0.3133 0.3985
0.2225
0.2225
0.1349 0.1988
0.1172
0.0646
0.0536
22.23
20.33
13.15
11.00
9.153
9.028
9.028
6.732
5.765
3.785 4.367
3.477
3.136
2.000 2.686
1.324
1.324
0.7462 1.162
0.6377
0.3353
0.2753
25.47
23.85
18.69
17.61
15.96
16.04
15.81
14.66
13.89
11.37 12.67
11.31
11.00
10.05 10.68
9.22
9.27
8.38 9.06
8.42
7.91
7.70
0.13
k’
17.95
16.33
11.17
10.09
8.44
8.52
8.29
7.14
6.37
3.85 5.15
3.79
3.48
2.53 3.16
1.70
1.75
0.86 1.54
0.90
0.39
0.26
I
1
useof
41.02
38.60
29.16
25.80
23.08
22.9 1
22.91
19.64
18.20
15.49 16.38
15.13
14.54
12.81 13.82
11.93
11.93
11.14 11.70
11.00
10.62
10.56
10.37
10.28
_._-.
10.28 X 10-s 10.28
1
data
33.88
32.02
24.56
21.80
19.50
19.36
19.32
16.53
15.25
12.78 13.60
12.44
11.89
10.25 11.22
9.36
9.36
8.54 9.13
8.50
7.96
7.08
7.64 x 10-b
11.46
33.00
34.26
31.39
23.80
22.17
20.70
22.18
12.77
9.82
12.43 7.32
10.01
8.07
2.00 5.05
1.52
0.97
1.94 0.78
0.99
0.57
1.26
-0.12
( % dev I
27.304
25.618
19.252
17.330
15.609
15.576
15.576
13.479
12.558
10.571 11.169
10.244
9.870
8.550 9.345
7.703
7.703
6.950 7.490
6.164
6.390
6.306
6.045
% 10-s
9.42
+
+
+
-
-
-
-
-
11.54
7.20
7.41
3.01
1.59
1.70
2.89
1.48
8.06
-11.75
- 7.03 - 11.85
-
- 10.27
- 14.92 - 12.50
- lb.45
- 16.90
- 17.06 - 17.24
- 26.79
- 19.22
- 18.95
- 20.98
use of Wp)mtn kcalc ( % dev
10-s cal ems/g s”K
at 1.9”C
12.34 x
kcnle
viscosity
=
for methane
(k/p)tin
quantities
I*, poises 5)
and related
0.00 x 10-s 0.00 0.00
0.0845
IO-6
k -
7.65
0.0170
kex,
7.57
0.024
1
1.02
1 (m&~R)pR
Cal/s cm OK
5.4
bPX
conductivities
10-s Cal/s cm°K
thermal
7.52 x
\
I
k’ = 7.52 x
and calculated
7.60
PR
Experimental
1.02
(
1.44
1.02
P, atm.
Tfi =
TABLE
889
THERMAL CONDUCTIVITY OF METHANE
By the substitutions,
p = pep~t P = PcP~
and T = T,TR,
can be expressed in terms of reduced parameters
equation
(5)
as follows:
Thus, in order to obtain bp~ values for a substance from equation
(6), PVT
data must be available for a substance over the range of interest. The reduced density correlation resulting from all available PVT measurements on methane (20) was used to produce PR versus TR relationships for reduced densities ranging from pR = 0.010 to pR = 3.00. The slopes resulting from these relationships were then used to calculate WRPTR),, the quantity 1 + bp~ using equation (6). The details of evaluating this quantity are identical to a similar study concerned with argon (2). The values of 1 + bp~ obtained for methane are plotted against reduced pressure, PR, to produce the isothermal relationships presented in figure 2. In this figure the range of reduced pressures extends up to PR = 20 and covers the temperature range 0.60 I TR 5 4.0.
METHANE (zO-0.289) T,=191.1 ‘K P,=45.6atm
Reduced
Pressure, e = $C
Fig. 2. Isothermal relationships between 1 + bpx and reduced pressure for methane.
Reliable values of bp~ for moderate pressures and pressures slightly above one atmosphere are obtained by plotting bpX/pR versus pR. For methane in the high density region, all the isotherms produce a single well defined relationship. However, in the low density region, considerable random scatter is encountered which results from the limitations to obtain dependable values of bpx from figure 2 at these conditions. An extension of the bpX/pR versus pR’relationship from the high density region to pR = 0, establishes the intercept bpX/pR = 0.71. The quantity bpx/pR = bpcX. At
1.34 1.42
1.54
1.63
1.69
1.78
306.5 341.0
405.2
454.2
499.1
567.1
0.845 1.00
171.1 205.1
1.13
0.750
152.4
1.24
0.697
139.7
272.5
0.585
120.0
238.5
0.335
0.395
73.2
0.312
69.1
85.4
0.226
51.6
2.0256
1.8066
1.6773
1.5030
1.1805 1.3007
1.0428
0.9108
0.6287 0.7730
0.5482
0.5067
0.4212
0.2808
0.2378
0.2215
0.1605
0.1037
18.630
16.413
15.100
13.335
10.109 11.306
0.764
7.470
4.762 6.135
4.018
3.634
2.877
1.751
1.435
1.319
0.9073
0.5577
22.98
21.84
20.87
19.85
17.69 18.67
16.81
15.96
13.41 14.74
12.22
12.48
11.12
9.99
9.69
9.78
9.12
8.64
8.41
0.146
0.1633 0.2524
34.4
0.0324
0.0695
0.0493
0.0457
16.8
/
8.18 8.49
/ (aPdaT&, 8.24
bpx
1.02 11.0
)
fl
x 10-s
kexp
Cal/s cm”K
1.02
pi
14.77
35.9 1
34.60
30.42
25.60 27.26
23.82
21.99
17.97 20.00
16.78
15.99
14.84
13.28
12.80
12.70
12.17
11.71
11.40
11.18 11.30
11.18
11.18
x 10-S
)
data
=
30.010
28.993
27.497
25.713
21.696 23.098
20.207
18.620
15.014 16.861
13.938
13.220
12.137
10.602
10.124
10.007
9.441
8.922
a.533
x 10-a
kealc
8.408
use of viscosity
12.77
for methane
WpJmtn
quantities
p, poisesb)
.13.63
I
related
32.65
x 10-j
k’
and
II
12.66
11.64
9.40 10.46
8.60
7.75
5.20 6.53
4.01
3.27
2.91
1.70
1.48
1.57
0.9 1
0.43
0.20
0.00 0.28
0.00
0.00
k -
conductivities
lo-5caljscm”K
thermal x
calculated k’ = 8.21
8.21
)
1.56
and
1.02
p, atm
TR =
Experimental
TABLE
14.27
30.59
32.75
31.75
29.54
22.65 23.72
20.21
16.67
11.96 14.39
14.06
5.93
9.14
6.13
4.48
2.32
3.52
3.26
1.46
-0.96
s”K
24.958
22.856
21.712
20.092
17.118 18.220
15.882
14.664
11.991 13.377
11.205
10.770
9.892
8.535
8.126
7.971
7.413
6.926
6.496
6.369
x 10-S
keal,
8.31
f
+
+
_
-
-
-
11.31
8.61
4.65
4.03
1.22
3.23 2.41
5.52
8.12
_ 10.58 _ 9.25
_
_ 13.70
-11.04
- 14.56
- 16.14
- 18.50
_ 18.72
- 19.84
- 22.76
- 24.98
j % dew
use of (klp),t.
10-G cal ems/g
/ % dev
x
at 24.6”C
THERMAL
CONDUCTIVITY
891
OF METHANE
pR = 0, x = 1, and therefore $JX/PR becomes the dimensionless group bp,. Since the critical molar density is pc = 0.162/l 6.042 = 0.010098 g-moles/cm3, the second virial coefficient becomes b = 0.7 1lo.01 0098 = 70.31 cm”/g-mole. This value produces the collision diameter (r = 3.82 x IO-8 cm, from the relationship b = 23zNa3/3. Hirschfelder, Curtiss, and Birds) present values of b = 70.16 cm3/g-mole and u. = 3.817 x 10-s cm.
Methane (z.=O289)
.
Fig. 3. Relationship between bpX/pR and pi for methane at 1.9, 24.6, 43.6 and 75.1 “C.
One approach to test the Enskog theory involves the use of experimental viscosity measurements corresponding to the temperature and pressure conditions of the thermal conductivity measurements. In this study the high pressure viscosity measurements of methane recently obtained by Giddingss) were used in conjunction with Equation (2) to obtain values of bp. These values were then applied to equation (3) to produce the ratio K/k*. The same results follow by dividing equation (3) by equation (2). From the k/k* values, the calculated high pressure thermal conductivities were obtained and were compared with the corresponding experimental values. These comparisons areincluded in tables I, II, III and IV for 1,9, 24.6, 43.6 and 75.1 “C, respectively. A second approach for the calculation of thermal conductivities at elevated pressures follows from the Enskog theory. By rearranging equation (3) becomes -
k
= bk*
P
Differentiating
equation
W/d ____ Wpx)
-
1
6
+ s
bPX
+
1
(7)
1
(8)
0.7574 bpx
*
(7) with respect to bpx gives = bk*
-
1
___
(bPX)2
+ or7574
.
Letting d(k/p)/d(bpx) b e zero, the bpx value at the minimum becomes bpx = 1.149. When this value is substituted into equation (7), it follows
1.69
573.9
1.02 1.13
1.54 1.62
0.755
171.1
239.2 273.2
463.7 518.1
0.685
155.8
1.44
0.632
143.9
1.41
0.540
123.5
407.6
0.521
388.8
0.363
86.6
120.1
1.23
0.287
69.1
1.32
0.258
62.6
341.2
0.084
21.4
307.2
0.0451 0.084
11.5 21.4
I .a066
1.5030 1.6540
1.329
1.2845
1.1524
1.0320
0.7946 0.9130
0.5512
0.4959
0.4557
0.3872
0.3725
I
0.258 1
0.2038
0.1832
0.0596
0.0320 0.0596
16.413
13.335 14.876
11.605
11.144
9.833
0.647
6.335 7.478
4.053
3.546
3.184
2.592
2.474
1.580
1.195
1.056
0.3080
0.1611 0.3080
22.74
20.72 21.70
19.40
19.35
18.33
17.48
15.40 16.53
12.98
12.70
12.43
11.84
11.74
10.67
10.10
9.77
9.12
8.88 9.13
8.82
1 WRW&~
thermal
x IO-5
keap
cal/s 1
III
13.88
11.86 12.84
10.54
10.49
9.47
8.62
6.54 7.66
4.12
3.84
3.57
2.98
2.88
1.81
1.24
0.91
0.26
0.02 0.27
0.00
0.00
0.00
x to-5
k - k'
cm OK
cm”k
conductivities
10-b Cal/s cmjs
8.97
bpx
x
1.02
/
calculated
k’ = a.86
and
1.02
PR
Experimental
a.92
/
1.66
I .02
P, atm.
7-R =
TABLE and
34.80
31.02 33.03
28.65
27.83
25.65
24.07
20.73 22.42
17.36
16.60
16.04
15.17
15.02
13.71
13.17
12.99
12.09
11.93 12.09
11.78
11.78
11.78
5)
x 10-S
1
29.972
27.154 28.520
24.866
24.150
22.278
20.907
17.917 19.440
14.770
14.044
13.503
12.644
12.494
11.159
10.579
10.378
9.299
9.083 9.299
data
13.32
x IO-5
k es,e
=
for methane
use of viscosity
(k/p),,,
quantities
~1,poises
related
I
14.51
31.80
31.05 31.43
28.18
24.8 1
21.54
19.60
16.34 17.68
13.79
10.58
8.63
6.79
6.42
4.58
4.74
6.22
1.96
2.29 1.85
s”K
23.904
20.957 22.440
19.305
18.854
17.585
16.444
14.139 15.291
11.739
11.145
10.705
9.971
9.834
8.668
8.140
7.945
6.860
6.634 6.860
x
10-S
j
7
+ +
-
-
-
-
-
-
_
11.89
5.12
1.14 3.41
0.49
2.56
4.06
5.93
8.19 7.44
9.56
- 12.24
- 13.88
- 15.78
- 16.24
- 18.76
- 19.41
- 18.68
- 24.78
- 25.29 - 24.86
% dev
use of (Wp)min kealc
1O-5 cal cm2jg
at 43.6”C
1 % dew )
x
THERMAL
CONDUCTIVITY
893
OF METHANE
that min and therefore,
equation
= 2.940 bk*
(7) can be expressed
(9)
in the alternate
form,
k -=
(10)
P
To apply equation (lo), isothermal experimental thermal conductivities should be available at sufficiently high pressures in order to establish the (k/P)nG, value from a k/p versus bpx or k/p versus p plot. The pressures used in this investigation did not go high enough to establish this minimum, except at 1.9”C. A plot of k/pR versus pi for this temperature is given in
Fig. 4. Kelationship
between
k/pR and pi resulting
measurements
of methane
from
the thermal
conductivity
at 1.9 “C
figure 4. Since for the other three temperatures, no minimum was observed, the following approach applicable to all four temperatures was used. The experimental thermal conductivities were plotted against PR, the reduced density, as shown in fig. 5. The following analytical expressions resulted from a least squares approach using an IBM 709 digital computer: 1.9”C k X 105 = 7.5801
+ 3.3058~~
+ 7.0778~:
-
+ 2.2411 pi
(11)
4.2029 pi + 1.1765 pi
(12)
6.3124~:
24.6”C k x
105 = 8.2399
+ 2.3775 pR + 7.0636 p; -
894
D. MISIC AND G. THODOS
43.6”C k x lo5 = 8.8248 + 3.8699 PR + 2.528 ,D;
(13)
75.1% k x
lo5 = 10.2626 + 3.1742 PR + 2.4854 p; + 0.1402 &
(14)
and are presented with the experimental values in fig. 5. Equations (ll), (12), (13) and (14) were divided by PR and were then differentiated with respect to PR to obtain relationships which when set to zero, produced the following PR and k/pR values at the minimum of each isotherm : Calculated minima “C 1.9 24.6 43.6 75.1
Fig. 5. Relationships
between
Pfl 1so9 1.683 1.868
1.848
k//m
I
12.34 x 10-S 12.77 13.32 13.80
k and pi for methane
at 1.9, 24.6, 43.6 and 75.1 “C
,
I
0.5674
0.7633 0.8615
0.773
0.990
205.1 273.2
0.9582
1.4140
1.17
1.49
1.56
344.0
523.2
577.7
1.5397
0.9582
341.2
1.085 1.17
307.2
-
0.4679
0.0788
0.648
bpx
0.111
I
k’ =
-
0.4143
13.713
7.9279 12.443
7.9279
6.9876
6.0408
4.1932
3.2920
) /
I
21.57
17.64 21.17
17.82
17.01
15.59
14.02
13.53
10.98
10.01 x 10-s
11.07 11.47
7.54
7.72
6.91
5.49
3.92
3.43
0.00 x IO-5 0.88
k -
cm”K
-
30.41 31.71
24.07
22.52 23.96
21.15
18.48
17.18
13.25
12.81 x 10-S
p, poises 5)
28.747
27.646
21.900
20.455 21.796
19.143
16.513
15.183
10.757 x 10-S
kesle
-
I
data
13.80
for methane
use of viscositv
quantities
ca
_-
and related Cklo)min ,... ~~=
k’
conductivities
IV
lo-scal/scm”K
thermal
ker,
10.10 x
and calculated
L W’RI~TR)~R
Experimental
171.1
PR
1.82
1.02 31.0
P, atm.
TR =
TABLE
x
20.60
33.27
30.59
22.3 1 24.15
20.25
17.78 22.79
- 2.03 12.22
o/odev
-
_7.268
22.085
20.828
16.310
16.310
15.348
14.350
12.305
11.229
x 10-S
use of
10-S cal cma/g s”K
at 75.1 “C
+
-
-
~ -
-
11.20
2.39
1.62
7.54
9.77 8.45
7.95
- 12.23
- 17.01
-33.81
0/0 dev
P) mln
-I 1
896
D. MISIC AND G. THODOS
Using the values of (k/~&~~, thermal with Equation IV. Application
conductivity
(lo), to produce the results presented
of semi-theoretical
approaches.
values were calculated in tables I, II, III and
The approach
suggested
by
Abas-zadel) that the residual thermal conductivity, k - k*, when plotted against density produces a single relationship for all temperatures, has been successfully “applied for argonz3), nitrogen 25), carbon dioxide 12), ammonia7), and methane24). The experimental data of this study have been subjected to a
0
2
I
3
P*
Fig. 6. lielationship between k -
k* and pi for methane in the gaseous and liquid
regions.
similar treatment which also includes the recent thermal conductivity values of Ikenberry and Ricelo) for liquid methane. A plot of k - k* versus PR is presented in fig. 6 which, in addition, includes the thermal conductivity values reported for methane in the dense gaseous state by KeyesIs). Lenoir,andComings17),Lenoir, Junk,andComingsl*),andStolyarov, Ipatiev, and Teodorovich28). These values produce the single unique relationship presented in fig. 6. Except for the few points of Ikenberry and Ricelo) taken near the critical temperature (TR = 1.048), the remaining values of his investigation and those of the other investigators, produce the unique relationship which is continuous throughout the gaseous and liquid states. The values reported by Stolyarov e.a. 28)“exhibit some inconsistencies and deviate somewhat from the established relationship of figure 6. The experimental thermal conductivity values of this study are consistent and assist to establish the relationship of fig. 6 in the dense gaseous region where reliable data were not available. Golubev6) has introduced the thermodynamic quantity, (WjaT),, to replace the density in the corklation of p - p*, the residual viscosity. The resulting ,U - ,LL*versus (aP/BT) v relationships were linear when plotted on
THERMAL
log-log coordinates. to produce
CONDUCTIVITY
This concept has been used by Lenner
generalized
relationships
897
OF METHANE
of viscosity
t and Thodos16)
which were linear
for
argon, krypton, and xenon. A logical extension to thermal conductivity appears appropriate in which the residual thermal conductivity, k L K*, in reduced form. Thus, is related to (aP/aT),, expressed for convenience the quantity
(aP,/aT,),,
of bpx given in tables of this study.
follows directly
I, II, III
In addition,
from equation
and IV which correspond
(6) and the values to the conditions
these values were produced for conditions
corre-
0 lkenberry and Rice 9 0 + 9
Fig. 7. Relationship
Lencmr and Comings Lenar. Junk, and Comings Siolyorov, Ipatw. and Teodorovich This invesiigatlon
between k - k’ and (aP,/aT& for methane in the dense gaseous and liquid regions.
sponding to the data of the other investigators 10, 13, 14, 17, 18, 28) and are presented in fig. 7. The thermal conductivity data of methane produce in fig. 7 a relationship which is linear on log-log coordinates and exhibits a scatter in the low density region. The resulting relationship can be expressed analytically as follows :
.
(15)
This relationship enables the calculation of thermal conductivities in the gaseous and liquid regions from the PVT behavior of the substance and its corresponding atmospheric thermal conductivity value. This type of correlation deserves further attention and should be tested with substances other than methane.
898
D. MISIC AND G. THODOS
Acknowledgement. support of the Petroleum
The authors Research
gratefully
acknowledge
Fund of the American
the financial
Chemical Society
through Grant 329-A. Also due thanks are extended to the Phillips Petroleum Company
for the methane
used in this study.
Nomenclature: a, b constants, equation (4)
thermal conductivity, Cal/s cm “K apparent thermal conductivity, Cal/s cm “K thermal conductivity of gas at atmospheric pressure, Cal/s cm”K molecular weight M N Avogadro’s number, 6.0228 x loss molecules/gram mole P pressure, atm PC critical pressure, atm reduced pressure, P/P, PR gas constant, 82.055 ems atm/g-mole”K H T absolute temperature, OK TC critical temperature, OK reduced temperature, T/Tc TR critical compressibility factor, P,v,/RT, zc k
ko k*
Greek viscosity, g/cm s viscosity of gas at atmospheric pressure, g/cm s 3.14159 2% density, g/ems P PC critical density, g/cm3 PR reduced density, p/pe collision diameter, cm B probability of nearess x P
P*
Received l-6-65
REFERENCES 1) Abas-zade, A. K., Doklady Akad. Nauk Azerbaidzhan SSR 3 (1947) 3. 2) Gediminas, Damasius and Thodos, George, Industr. engng Chem., Fundamentals 2 (1963) 73. 3) Enskog, David, Svensk. Akad. Handl. 63 No. 4 (1922). 4) Eucken, A., Phys. Z. 12 (1911) 1101. J. G., Ph. D. dissertation, Rice University, Houston, Texas (1963). 5) Giddings, I. F., “Viscosity of Gases and Gaseous Mixtures,” Fizmatgiz, Moscow (1959). 6) Golubev, 7) Groenier, W. S. and Thodos, George, J. them. Eng. Data 6 (1961) 240. 8) Hercus, E. 0. and Laby, T. H., Proc. roy. Sot. (London) 95 (1919) 190. J. O., Curtiss, C. F. and Bird, R. B., ‘*Molecular Theory of Gases and Liquids 9) Hirschfelder, Wiley, New York (1954). L. D. and Rice, S. A., J. them. Phys. 39 (1963) 1561. 10) Ikenberry, H. L. and Grilly, E. R., J. them. Phys. 14 (1946) 233. 11) Johnston, 12) Kennedy, J. T. and Thodos, George, A.I. Ch. E. Journal 7 (1961) 625.
THERMAL
CONDUCTIVITY
OF METHANE
899
13) Keyes, F. G., Trans. Am. Sot. Mech. Engrs. 76 (1954) 809. 14) Keyes, F. G., Trans. Am. Sot. Mech. Engrs. 77 (1955) 1395. M. W., Robinson, A. M., Scrivins, J., 15) Lambert, J. D., Cotten, K. J., Pailthorpe, Vale, W. R. F. and Young, R. M., Proc. roy. Sot. (London), A%31 (1955) 280. 16) Lennert, D. A. and Thodos, George, A.I. Ch. E. Journal 11 (1965) 155. 17) Lenoir, J. M. and Comings, E. W., Chem. Eng. Progr. 47 (1951) 223. 18) Lenoir, J. M., Junk, W. A. andcomings, E. W., Chem. Eng. Progr.dt) (1953) 539. 19) Mann, W. B. and Dickins, B. G., Proc. ray. Sot. (London), Al34 (1931) 77. 20) Matschke, D. E. and Thodos, George, J. Petroleum Technol. 12 (1960) 67. 21) Misic, Dragoslav, Ph. D. dissertation, Northwestern University (1965). 22) Misic, Dragoslav and Thodos, George, A.1.Ch.E. Journal 11 (1965)650. 23) Owens, E. J. and Thodos, George, A.1.Ch.E. Journal 3 (1957) 454. 24) Owens, E. J. and Thodos, George, Proc. Conf. Therm6d. Prop. Fluids, London, p. 163, July lo-12 (1957). 25) Schaefer, C. A. and Thodos, George, A. I. Ch. E. Journal 5 (1959) 367. 26) Schottky, W. F., 2. Elektrochem. 56 (1952) 889. 27) Stefan, J., Berichte der Wiener Akademie 65 (1872) II, 45. 28) Stolyarov, E. A., Ipatiev, V. V. and Teodorovich, V. P., Zhur. Fiz. Khim. “4 (1950) 166. 29) Warberg, E. and von Babo, L., Wied. Ann. 17 (1882) 390. 30) Weber, S., Ann Physik (4) 54 (1917) 437. 31) Weber, S., Ann. Physik (4) 54 (1917) 481. 32) Winkelmann, A., Ann. Physik 156 (1875) 497. 33) Ziegler, E., Dissertation, Universitat zu Halle, Germany 1904.