Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
Table VII. Comparison of Specific Heats (C, ) of Methane at 50.0 Bar
A. Jones et al. B. Keesom et al. temp, C,: J K -c!125 3.47 135 3.56 145 3.68 155 3.93 165 4.19 175 4.94 185 7.04 195 19.65 205 5.73 215 4.02 225 3.35 235 3.09 245 2.89
C,: J
-g-
3.39 3.47 3.52 3.89 4.14 4.77 7.11 21.1 4.85 3.77 3.35 2.97 2.80
B -A, 7% -2.3 -2.5 -4.3 -1.0 -1.2 -3.5 +1.0 +7.5 -15 -6.2 0.0 -4.0 -3.0
K and at pressures up to 70 bar with an absolute error a t any temperature of less than 1%. Literature Cited
C. This work
C,:J -Q 3.46 3.56 3.70 3.90 4.22 4.87 6.99 19.97 5.59 3.82 3.31 4.05 2.87
339
Clark, R. G., McKinley, C., Paper 11 in "Proceedings of the Forty-Sixth Annual Convention of the Natural Gas Processors Associatm", Houston, March 1967. Dolan, J. P., Eakln, B. E., Bukacek, R. F., Ind. Eng. Chem. Fundam., 7 , 647 (1968). Jenkins, A. C., Berwaldt, 0. E., Ind. Eng. Chem. Process Des. D e v . , 2 , 193 (1963). Jones, M. L., Mage, D. T., Faulkner, R. C., Kak, D. L., Chem. Eng. hog. Symp. Ser., No. 4 4 , 59, 52 (1962). Keesom, W. H., Bijl, A., Monte, L. A. J., Appl. Sci. Res. A , 3 , 261 (1952). Manker, E. A., Mage, D.T., Mather, A. E., Powers, J. E., Katz, D.L., Paper 3 In "Proceedings of the Forty-Third Annual Convention of the Natural Gas Processors Association", New Orleans, March 1964. Mather, A. E., Yesavage, V. F., Powers, J. E., Kak, D. L., Paper 12 in " P r d n g s of the Forty-Fifth Annual Convention of the Natural Gas Processors Association", 1966. Sa@al, P. N., Wt,J. M., Jambhekar, A., Wkm, G. M.. in "International Advances in Cryogenic Engineering. Proceedings of the 1964 Cryogenic Engineering Conference held at the University of Pennsylvania", Vol. 10, p 224, K. D. Timmerhaus, Ed., Plenum Press, New York, N.Y., 1965. Wilson, G. M., Barton, S. T., Paper 18 in "Proceedings of the Forty-Sixth Annual Conference of the Natural Gas Processors Association", Houston, March 1967.
C-A %
-0.3 0.0 +0.6 -0.7 +0.7 -1.4 -0.7 +1.5 -2.4 -5.0 -1.2 -1.3 -0.7
Conclusion
A flow calorimeter has been developed which can measure enthalpy-temperature relations of small amounts of condensable gases at temperatures between 100 and 300
Received for review July 17, 1978 Accepted July 23, 1979
A Flow Calorimeter for Condensable Gases at Low Temperatures and High Pressures. 2. Compilation of Experimental Results and Comparison with Predictions Based on a Modified Redlich-Kwong Equation of State Plet H. G. van Kasteren' and Hans Zeldenrust Koninklijke/Shell-Laboratorium,Amsterdam (Shell Research B. V.) Amsterdam-Noord, The Netherlands
The enthalpy-temperature relationships of methane, ethane, propane, nitrogen, and binary mixtures thereof have been determined. These basic thermodynamic data have been used to improve the predictions of a computer program based on a modified Redlich-Kwong equation of state. The agreement between experiments and final predictions is in general good except for the liquid phase of some binaries containing nitrogen. This discrepancy is postulated to be due to the existence of two liquid phases.
Introduction
For the economic design of liquefaction and regasification plants (for natural gas transportation purposes) accurate enthalpy data are required. The total enthalpy change from room temperature down to 110 K and also the enthalpies at intermediate temperatures are of interest. Therefore detailed isobaric enthalpy-temperature reiationships are needed. These cooling curves depend on the composition of the gas mixture and on the pressure. In view of the great difference in composition of natural gases, the number of cryogenic refrigerants and the variety of operating pressures encountered in liquefaction plants, a calculation procedure for the prediction of cooling curves is indispensable. For this prediction of thermodynamic properties a large number of equations of state have been proposed in the literature and a survey of these was made by Van Aken et al. (1976). We have used a computer program to predict 0019-7874/79/10 18-Q339$01.QQ/Q
cooling curves which is based on a modified RedlichKwong equation of state (MRK). Judging from the small amount of experimental data available on systems at low temperatures, we suspected that the enthalpies predicted using this program would diverge seriously from experiment, particularly for multicomponent systems a t temperatures below 170 K. However, the accuracy of experimentally determined enthalpy changes a t low temperatures is also doubtful, since they are often derived from the difference between two large enthalpy changes over very large temperature intervals. These uncertainties with respect to both the available experimental data and the predicted values led us to carry out a program in which a number of accurate, detailed cooling curves were determined with the aid of the lowtemperature flow calorimeter and the data so obtained used to improve the predictions of the MRK-based computer program. The results of this program are @ 1979 American Chemical Society
340
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
Table I. Heat Capacities and Enthalpies of Nitrogen at 50.7 Bar (735 psia) C,,Jg-' K-' H, J g - ' temp, K exptl IMRKa exptl IMRKa
100 105 110 115 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 155 160 170 180 190 200 210 2 20 230 240 250 260 270
2.08 2.17 2.28 2.44 2.70 2.85 3.06 3.35 3.77 4.44 5.44 6.78 7.03 5.65 4.64 3.82 3.25 2.86 2.59 2.39 2.06 1.82 1.59 1.46 1.36 1.29 1.24 1.22 1.19 1.17 1.15 1.14 1.13
2.15 2.26 2.43 2.67 3.04 3.26 3.54 3.90 4.39 5.44 5.90 6.44 6.07 5.02 4.14 3.51 3.06 2.74 2.51 2.33 2.03 1.83 1.60 1.47 1.39 1.33 1.29 1.26 1.23 1.21 1.19 1.18 1.17
0.0 10.5 21.7 33.3 46.0 51.6 57.3 63.5 70.7 79.5 89.5 101.0 113.5 128.5 138.5 146.5 153.5 159.5 165.0 170.0 181.5 190.5 207.5 224.7 236.4 249.8 262.5 274.7 286.7 298.6 310.3 321.8 333.1
0.0 11.1 22.8 35.3 49.7 56.0 62.7 70.2 78.3 87.4 98.5 111.1 123.6 134.8 144.1 151.5 158.3 164.1 169.2 174.1 184.8 194.5 211.5 226.8 241.1 254.7 267.5 280.0 292.4 304.7 316.9 329.0 340.7
IMRK = improved MRK-based prediction.
Table 11. Heat Capacities and Enthalpies of Methane (99.8 mol % ) temp, K
C,
J g-' K-'
exptl
IMRK
I
110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 2 60 270 140 150 160 170 175 185 190 200 210
3.37 3.43 3.49 3.62 3.79 4.04 4.48 5.44 4.48 3.60 3.18 2.95 2.80
3.68 3.87 4.18 4.95 5.70 4.60 3.92 3.21
H,J g-' exptl
50.0 bar ( 7 2 5 psia) 3.31 0 3.39 34.0 3.49 68.6 104.2 3.65 141.0 3.88 4.27 179.2 4.85 221.7 6.23 270.6 14.1 341.0 6.57 540.5 4.18 599.5 3.49 635.0 3.11 668.5 2.97 699.1 2.82 727.7 2.74 2.67 32.0 bar (464 psia) 3.73 0 4.02 37.0 4.52 77.5 5.56 122.5 6.74 151.0 4.00 456.0 3.54 477.8 3.08 511.5 2.85 542.5
IMRK
0 33.4 67.7 103.4 140.9 181.4 226.7 281.2 366.6 546.1 596.7 634.6 667.6 698.1
726.9 754.6 781.5 0 38.6 81.1 130.6 160.8 468.1 486.9 519.7
549.2
presented here; they have also been presented in part by Lammers et al. (1978).
Table 111. Heat Capacities and Enthalpies of Ethane (99.5 mol %) te mP, K
C,, J g-' K-' exptl
110 120 130 140 150 160 170 180 190 200 210 220 2 30 240 250 260 270
2.30 2.31 2.33 2.34 2.35 2.38 2.41 2.45 2.48 2.54 2.57 2.64 2.72 2.80 2.93
110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 2 60 270
2.34 2.35 2.35 2.36 2.39 2.41 2.44 2.47 2.51 2.54 2.59 2.67 2.74 2.84 2.96
IMRK
H,J exptl
50 .7 bar ( 7 3 5 psia) 2.31 0 2.35 23.3 2.36 46.1 2.36 69.5 2.36 92.7 2.36 116.5 2.37 140.7 2.38 165.3 2.42 190.2 2.46 215.4 2.52 241.1 2.59 267.2 2.69 293.8 2.80 321.4 2.96 350.1 3.16 3.44 32.0 bar (464 psia) 2.31 0 2.35 23.2 2.36 46.8 2.36 70.3 2.36 94.2 2.31 118.2 2.38 142.4 2.40 167.1 2.44 192.1 2.48 217.3 2.54 242.9 2.63 269.3 2.73 296.2 2.87 324.0 3.06 353.0 3.33 3.74
g-'
IMRK
0 23.3 46.9 70.4 93.9 117.5 141.1 164.8 188.9 213.2 238.1 263.6 290.0 317.4
346.2 376.7 409.6 0 23.3 46.7 70.4 93.9 117.6 141.3 165.3 189.4 213.8 238.9 264.7 291.7 320.6 ~~
349.1 380.9 416.3
Experimental Program Using the low-temperature flow calorimeter described in part 1we measured the heat capacities and enthalpies of the pure forms of the major components usually occurring in natural gas, viz. methane, ethane, propane, and nitrogen. These accurate experimental data were then used to improve the values predicted by the MRK-based computer program. Subsequently, cooling curves of binary mixtures of these components and of two multicomponent systems were determined to check the ability of the computer program to handle these more complex systems. Results and Analysis The experimental data together with the improved MRK (see Appendix) predictions for the pure samples are given in Tables I-IV; for comparison, the experimental data for isopentane at atmospheric pressure are given in Table V. The results for the binary mixtures are given in Tables VI-XI. Binary mixtures may show two-phase behavior over a large temperature range. It should therefore be noted that, when this is the case, the tabulated values for the average heat capacities in the two-phase temperature region include the heat of vaporization corresponding to the change in the amount of liquid present. Two of the binaries showed unexpected behavior and are therefore discussed separately below. Binary Ethane/Nitrogen. The results for a mixture with 40.9 mol 7'0 ethane at pressures of 50.7 and 30.4 bar are given in Table IX. This mixture exhibits two-phase behavior over a temperature range of more than-100'. Although the liquid state exists only over a very small
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979 341 Table IV. Heat Capacities and Enthalpies of ProDane (99.8 mol % 1 H,J g - ’ C,, J g-’ K-’ temp, K exptl IMRK exptl IMRK 50.7 bar (735 psia) 0.0 1.91 1.98 19.2 2.01 38.5 2.03 58.1 2.03 77.8 2.03 97.6 2.04 117.7 2.05 138.0 2.06 158.6 2.08 179.3 200.3 2.10 2.13 221.6 2.16 243.2 265.2 2.21 287.5 2.26 310.3 2.31 2.38 333.6
110 120 130 140 150 160 170 180 190 200 210 220 2 30 240 250 260 270
1.91 1.92 1.94 1.96 1.98 2.00 2.02 2.04 2.06 2.08 2.11 2.15 2.18 2.21 2.26 2.30 2.35
115 135 155 175 195 225 255
1.91 1.95 1.97 2.02 2.07 2.16 2.30
0.0 19.5 39.5 59.7 80.0 100.3 120.6 141.0 161.5 182.1 203.0 224.2 245.6 267.4 289.7 312.5
336.0
25.3 bar (367 psia) 1.95 2.02 2.03 2.04 2.07 2.16 2.32
Table V. Heat Capacities and Enthalpies of Isopentane (99.9 mol %) H,J g - ’ C,, J g-’ K-’ temp, K exptl IMRK exptl IMRK 1.0 bar (14.7 Dsia) 1.76 0.0 1.76 17.1 1.80 34.5 52.1 1.81 69.9 1.82 87.9 1.82 106.2 1.83 124.7 1.84 143.6 1.85 162.7 1.87 182.1 1.90 1.92 201.9 222.0 1.95 242.5 1.99 263.3 2.03 284.6 2.06 -
120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270
1.71 1.73 1.75 1.77 1.80 1.82 1.85 1.87 1.90 1.93 1.96 1.99 2.03 2.06 2.10 2.14
I
0.0 17.9 35.8 53.9 72.0 90.2 108.5 126.9 145.3 163.8 182.9 202.0 221.3 241.0 261.0
281.4
temperature interval, it is clear that the experimentally determined heat capacity within this region is about 10% higher than that predicted. This is surprising in view of our experience with other binaries and the predicted heat capacities at 110 K for pure ethane and pure nitrogen. It is likely that this discrepancy is due to the existence of two liquid phases with this binary at low temperatures, as reported by Yu et al. (1969) and Chang and Lu (1963). At low temperatures there are three co-existing phases. The top liquid layer, which is in equilibrium with the vapor, is rich in nitrogen (about 95%), while the bottom liquid layer contains only 30% nitrogen. At temperatures above about 133 K the top liquid layer becomes identical with the vapor. This “critical” temperature is lower than the upper critical solution temperature (UCST) of the two liquids. Heating a mixture of two liquids in the range below their UCST will bring them closer together, in terms of energy, and the extra energy required will appear as an
Table VI. Heat Capacities and Enthalpies of Methane/Nitrogen (53.3/46.7 mol h) temp, K 110 120 130 135 140 145 150 152 154 156 158 160 162 164 166 168 170 17 5 180 185 190 195 205 215 225 235 245 255 265 27 5
110 115 120 125 130 135 140 142 144 146 148 150 152 154 156 158 160 165 170 175 180 185 190 195 205 215 225 235 245 255 265 27 5
C,, J g - ’ K-’ exptl
IMRK
H, J g - ’ exptl
50.7 bar (7 35 p i a ) 2.56 0.0 2.71 26.2 2.95 53.7 3.13 68.5 3.37 83.3 3.72 99.7 4.27 117.6 4.60 125.4 5.06 5.64 6.57 39.4 169.0 10.7 13.4 10.7 231.5 7.41 5.73 263.3 3.88 288.1 3.15 306.5 3.30 2.83 2.77 321.6 2.57 2.52 2.38 2.36 347.3 2.14 369.4 2.10 1.96 2.00 389.6 1.85 1.92 408.6 1.79 1.85 426.7 1.73 1.80 444.3 1.68 1.77 461.3 1.65 1.74 477.8 1.72 494.1
2.56 2.68 2.83 2.96 3.15 3.40 3.86 4.18 4.60
30.4 bar ( 4 4 1 psia) 2.62 0.0 2.68 2.70 2.81 26.8 2.82 2.95 3.14 55.0 3.18 3.41 3.85 86.7 12.60 11.90 11.56 11.60 12.00 188.4 12.80 14.08
2.35 2.05 1.89 1.83 1.75 1.69 1.65 1.61 1.58 1.56 1.54 1.53
2.71 2.41 2.23 2.10 2.01 1.95 1.89 1.85 1.78 1.74 1.70 1.68 1.66 1.64 1.64 1.63
301.1 315.0 338.5 358.9 377.8 395.7 412.8 429.4 445.6 461.6 477.3 492.8 508.1
IMRK 0.0 26.3 54.5 69.7 85.9 103.6 123.5 132.4 142.0 152.7 164.8 162.3 197.5 222.0 247.2 265.0 277.9 300.9 318.3 333.0 346.2 358.4 380.7 401.4 420.9 439.7 458.0 475.9 493.4 510.7
0.0 13.3 27.0 41.4 56.6 72.9 91.0 102.8 127.3 150.5 173.5 197.0 221.7 248.5 278.2 309.2 314.8 327.6 339.1 349.9 360.2 370.1 379.7 389.0 407.1 424.6 441.8 458.7 475.3 491.8 508.2 524.5
increased average specific heat. The computer program, on the other hand, only calculated the enthalpy and specific heat for a single-phase liquid. Support for this mechanism as the cause of the observed discrepancy is found in the average specific heats at temperatures slightly above the “critical” temperature (133 K) where one of the liquid phases has disappeared. For
342
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
Table VII. Heat Capacities and Enthalpies of Methane/Ethane (71.4/28.6 mol %) temp, K 110 120 130 140 150 160 170 180 190 200 210 212 214 216 218 220 222 224 226 2 28 230 232 234 2 36 2 38 2 40 242 244 246 2 48 250 2 60 270
115 125 135 145 155 165 180 208 210 212 214 216 218 220 222 224 226 228
C,
exptl
J
g-I
K-' IMRK
H,J exptl
50.7 bar (735 psia) 2.91 2.84 0.0 30.7 2.93 2.88 58.5 2.92 2.96 3.00 88.4 2.96 3.06 3.03 118.6 3.14 3.12 149.6 3.25 181.5 3.25 3.38 3.43 214.5 3.58 249.1 3.70 4.12 3.83 286.1 4.90 327 5.15 336 350 9.60 367 388 410 10.32 9.90 432 9.77 452 9.56 9.42 9.26 47 1 9.22 490 9.12 9.1 2 509 9.03 9.10 527 9.00 9.1 1 544 9.03 9.14 9.14 562 58 1 9.20 9.34 9.29 9.54 599 9.75 617 635 3.84 654 3.67 660 3.99 3.53 668 3.24 3.10 703 2.84 733 2.96
Table VIII. Heat Capacities and Enthalpies of Methane/Propane (91.0/9.0 mol %)
g-I
IMRK 0.0 28.6 57.4 87.0 116.9 147.6 179.4 212.7 248.2 287.1 331.7 341.7 353.9 379.8 403.2 425.0 445.3 464.7 483.5 501.8 519.9 537.9 555.9 574.0 592.4 611.2 630.4 650.2 663.1 670.6
temp, K 110 120 130 140 150 160 170 180 190 192 194 196 198 202 204 206 2 08 210 212 214 216 218 220 2 30 2 40 250 2 60 270
C,, J
exptl 3.02 3.05 3.10 3.16 3.25 3.36 3.53 3.80 4.30
30.4 bar ( 4 4 1 psia) 2.95 2.87 2.98 2.92 3.01 2.96 3.06 3.03 3.13 3.13 3.22 3.26 3.48 3.56 8.78 8.56 8.54 8.40 8.41 8.33 8.38 8.32 8.40 8.40 8.48 8.54 8.63 8.74 8.86 9.04 9.16 9.36 9.50 9.75 9.88 10.16
example, at 30.4 bar and 140 K 50% of the mixture is still in the liquid state and yet the agreement between experiment and predictions is good. The predicted heat capacities for the vapor phase agree very well with the experimental ones, in contrast with the results for the first binary mixture of methane and nitrogen. Binary Propane/Nitrogen. Enthalpy measurements have been carried out on a binary mixture with 10.8 mol % propane and 89.2 mol % nitrogen at pressures of 50.7 and 30.4 bar. The experimental results together with the predictions are given in Table XI. Again, as with the binary ethane/nitrogen, the experimentally determined liquid-phase specific heats are much higher than the predicted ones. In the table only one figure is given for the lower temperature region, viz. at 120 K; the observed values at 50.7 bar were: an average specific heat of 2.31
H , J g-I IMRK
K-' IMRK
exptl
50.7 bar (7 35 psia) 2.97 0 .o 3.03 30.3 3.09 61.O 92.3 3.17 3.28 124.3 157.2 3.44 3.67 191.5 4.02 227.9 4.60 267.8 4.81 5.02 5.27 294.5 5.56
0.0 30.0 60.5 91.8 124.0 157.5 193.0 231.2 274.1 303.6
- - _ _ _ _ _ ._- -_- -_- -. -
2.82 2.65
2.83 2.71
0.0 31.1 57.6 79.1 95.6 110.1 124.1 136.1 147.6 158.1 208.0 254.3 301.9 335.8 363.2
0.0 31.6 54.9 73.4 89.1 103.1 115.7 127.5 138.6 149.3 198.5 245.7 294.5 330.7 358.4
30.4 bar (441 psia)
710.5
739.9
g-I
110 120 130 140 150 160 170 175 180 181 182 183 184 185 190 200 210 2 20 2 30 240 250 2 60 27 0
3.01 3.05 3.10 3.18 3.29 3.45 3.69 3.85 4.11
2.99 3.05 3.12 3.21 3.35 3.54 3.85 4.06 4.35
4.95 4.1 7 4.1 1 4.33
4.82 4.19 4.17 4.49
2.35 2.27
2.43 2.39 2.37
0 .o 30.4 61.3 92.7 124.9 158.4 194.1 213.2 232.6
444.0 508.5 553.0 594.5 636.3 683.0 726.0 750.4 ___ 773.6
0.0 30.2 6 1.O 92.6 125.4 159.7 196.5 216.3 237.3 241.7 302.8 345.1 374.3 395.9 456.4 516.3 560.2 601.6 644.7 692.0 735.9 760.0
783.8
between 110 and 118.0 K and 2.57 for the range of 117.0 to 124.6 K. Similarly, at 30.4 bar an average value of 2.52 holds for the range between 110.0 and 118.0 K and one of 3.32 between 118.0 to 123.0 K. The latter interval might include part of the two-phase vapor-liquid region. Therefore the experimental value of 3.20 at 120 K for the lower pressure has been placed between parentheses. A t higher temperatures and in the vapor phases the agreement between experiment and prediction is very good. This behavior is similar to that of the ethane/ nitrogen binary. We therefore assume that the discrepancies between experiment and predictions for the liquid phase are also due to a phase separation at low temperatures.
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
Table X. Heat Capacities and Enthalpies of Ethane/Propane (61.0/39.0 mol % )
Table IX. Heat Capacities and Enthalpies of Ethane/Nitrogen (40.9/59.1 mol %) temp, K 110 115 120 125 130 131 132 133 134 135 145 155 165 175 185 195 205 215 225 235 245 255 257 259 261 263 2 65 110 120 122 123 124 130 140 150 160 170 180 190 200 210 215 2 20 225 230 235 237 2 39 241 243 2 50 2 60 270
C,, J g-’ ___ exptl 2.24 2.27 2.30 2.32 2.47
3.08 2.46 2.27 2.27 2.38 2.50 2.7 3 3.06 3.56 2.45 2.06 1.88 1.81 1.78 1.76 2.30
2.29 2.13 2.1 3 2.20 2.37 2.62 3.02 3.61 3.98 4.41 4.98 5.68
1.57 1.52 1.48
K-’ IMRK
H , J g-’ exptl
50.7 bar (7 35 psia) 2.02 0.0 2.04 11.4 22.8 2.06 2.09 45.9 2.11 2.12
2.90 2.44 2.30 2.30 2.41 2.56 2.81 3.16 3.70
1.81 1.79 1.78 1.76
59.1 99.6 126.2 149.6 172.1 195.3 219.7 245.7 274.5 307.3 345.8 392.6 441.8
460.9
30.4 bar ( 4 4 1 psia) . 2.03 0 .o 2.08 23.3 2.09 2.10 2.29 2.15 2.14 2.20 2.38 2.65 3.07 3.66 4.05 4.56 5.18 6.00
1.56 1.54 1.52 1.50
87 .O 112.5 134.5 155.7 177.1 200.0 224.7 2 52.8 285.3 325.1 348.4 374.9 405.7
467.4 482.8
498.0
343
IMRK 0.0 10.1 20.4 30.7 41.2 43.3 51.2 54.8 58.4 62.9 99.8 125.7 149.0 171.9 195.2 219.8 246.5 276.2 310.2 350.3 399.6 429.0 442.0 450.9 454.5 458.0
461.6 0.0 20.5 24.7 26.8 62.1 83.7 108.9 131.0 152.2 173.9 196.7 221.8 250.2 283.5 302.7 324.1 348.3 37 6.1 408.4 423.0 438.7 455.6 459.6 470.4 485.6
500.7
Multicomponent Systems. The results for a multicomponent mixture are given in Table XII. Effect of Small Amounts of Helium. From the literature (Gonzales et al., 1968) it can be derived that small amounts of helium have a pronounced influence on the thermodynamic behavior of gas mixtures, as expressed by the following citation: “Small amounts of helium in multicomponent mixtures cause an increase of the bubble point a t temperatures below 144 K. The process that causes this increase appears to be preferential condensation of the hydrocarbons with an increase in the percentage of helium in the gaseous phase”. In view of these observations we investigated the effect of helium on a cooling curve, in particular since our computer program does not
temp, K 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 2 60 27 0
C,, J g-I K-I exptl
H J g-’ exptl
IMRK
50.7 bar (7 35 psia) 2.12 0 .o 2.08 2.17 20.3 2.14 2.19 41.5 2.17 2.19 63.1 2.19 2.19 84.8 2.21 2.19 106.7 128.7 2.23 2.20 2.25 151.3 2.21 2.27 173.5 2.23 2.30 196.2 2.26 2.29 219.4 2.33 243.0 2.36 2.34 2.41 2.40 266.6 291.1 2.46 2.46 2.54 2.55 316.1 341.9 2.64 2.65 2.7 2 2.78 368.5
0.0 21.5 43.3 65.1 87.0 109.0 130.9 152.9 175.1 197.6 220.3 243.5 267.1 291.4 316.5 342.5 369.6
IMRK
Table XI. Heat Capacities and Enthalpies of Propane/Nitrogen (10.8/89.2 mol %) temp, K 110 120 122 124 126 128 130 140 150 160 170 180 190 200 210 220 230 240 2 50 2 60 270
110 120 122 124 126 128 130 140 150 160 170 180 190 200 210 220 230 240 250 2 60 270
C,, J g-’ K-’ exptl
IMRK
H , J g-’ exptl IMRK
50.7 bar (7 35 psia) 2.00 0 2.50 2.13 23.2 2.17 2.21 2.26 39.2 3.76 4.30 2.34 1.88 1.73 1.65 1.65 1.68 1.77 1.90 2.08 2.29 2.57
2.33 1.89 1.74 1.66 1.66 1.69 1.76 1.88 2.07 2.29 2.53
1.30
1.31
52.5 105.5 135.6 156.2 174.2 191.0 207.4 224.0 241.1 259.3 279.2 300.9 325.0 353.0 367.6
30.4 bar (441 psia) 2.05 0 (3.20) 2.23 26.5 2.28 33.2
1.86 1.61 1.52 1.49 1.51 1.56 1.66 1.84 2.06 2.29 2.7 3
1.86 1.57 1.51 1.51 1.52 1.57 1.67 1.85 2.05 2.29 2.74
1.22 1.22
1.23 1.23
140.1 157.2 172.7 187.7 202.6 217.9 234.0 251.3 270.9 292.5 317.7 347.2 365.1 377.3
0 20.59 24.89 29.26 33.72 46.53 54.55 105.37 133.89 154.52 172.58 189.6 206.1 222.8 240.0 258.3 277.9 299.4 323.5 350.8 367.8 0 21.3 25.8 98.0 107.1 113.4 118.6 139.2 156.2 172.0 187.2 202.3 217.7 233.9 251.4 270.6 292.4 317.6 347.3 365.5 377.8
incorporate helium in the calculations. For the calculations helium was taken to be nitrogen. The results at a pressure of 50.7 bar are given in Table XIII. It is seen that for the
344
Ind. Eng. Chem. Fundam., Vol. 18, No. 4, 1979
Table XII. Heat Capacities and Enthalpies of Multicomponent Mixture 1" temp, K 105 115 125 135 145 155 165 175 185 189 195 199 201 205 215 221 2 30 2 40 2 50 255 2 60 265 270
C,, J g-I K-l exptl IMRK
H , J g-' exptl IMRK
50.7 bar (735 psia) 2.98 0.0 3.12 3.03 31.1 3.16 3.09 62.4 3.21 94.1 3.16 3.28 126.6 3.26 3.40 3.41 160.0 3.56 194.7 3.62 3.80 231.4 3.93 4.18 271.3 4.46 4.42 288.3 4.78 5.16 5.52 316.2 360.0 445.5 532.9 568.9 613.8 658.4 700.3 2.81
2.85 2.80 2.72 2.68
Table XIII. Heat Capacities and Enthalpies of Multicomponent Mixture 2"
7 34.6 762.5
0.0 30.1 60.7 92.0 124.1 157.4 192.5 230.1 271.8 290.3 321.0 398.5 461.7 542.0 575.3 618.5 662.0 703.0 722.2 736.3 750.0 763.5
a Composition, mol %: C, = 89.94; C, = 4.40; C,,- = 0.12; C, = 3.35; n-C, = 0.63; i-C, = 0.74; n-C,= 0.03; i-C, = 0.01; N, = 0.78.
mixture with 0.3% helium added the liquid heat capacities at low temperatures are about 5% higher than those of the mixture without helium. The heat capacity of this small amount of helium, on a weight basis, is negligible for the total specific heat of the mixture. Thus helium does have a peculiar influence on the cooling curve, but only in multicomponent mixtures, as additional experiments have shown that the heat capacities of pure methane and pure nitrogen are not affected by the addition of small amounts of helium. Moreover, it affects the liquid region only, since the difference amounts to 7% for the range from 105 to 190 K, whereas it is only 1% for the range from 190 to 250 K, which covers the two-phase and gaseous regions. Therefore measurements at the lower pressure of 27.6 bar have been carried out in the low-temperature region. The results are given in Table XIV. No comparisons with the original mixture could be made, as no measurements were carried out at this pressure before the addition of helium. The bubble point at 50.7 bar, as calculated with the program, is about 198 K. From the temperature dependence of the experimentally determined liquid-phase specific heats for both mixtures at 50.7 bar (see Table XIII) one would expect the bubble point to be not very far from the predicted value of 198 K. The increased temperature dependence between 180 and 190 K indicates that the bubble point of the mixture with helium will be somewhat lower than that of the mixture without helium, but definitely above 190 K. Thus both mixtures are completely in the liquid state at the lowest temperatures and therefore the increase in the specific heats cannot be caused by a lower bubble point. We are therefore convinced that the effect of helium is not merely an increase of the bubble-point pressure. In analogy with the low-temperature behavior of the binaries ethanelnitrogen and propane/ nitrogen, possibly small amounts of helium induce a phase separation in the liquid state for a multicomponent mixture. The existence of two liquid phases could explain the increased heat capacity of the liuqid. Upon heating
H , J g-'
C,, J g-, K-I
exptl temp, K mixt 105 115 125 135 145 155 165 170 175 180 185 190 192 194 196 198 200 202 204 206 208 210 212 214 216 218 220 230 240 250
3.18 3.20 3.25 3.32 3.43 3.57 3.76 3.90 4.10 4.40 4.82 5.44
5.00 4.66 4.40 3.56 3.15 2.94
exptl
IMRK + 0.3% -
He
50.7 3.33 3.35 3.40 3.47 3.58 3.74 3.95 4.10 4.31 4.64 5.15 6.20
4.68 4.43 4.20 3.49 3.15 2.94
(mixt) mixt
+ 0.3% He
bar (735 psia) 3.13 0.0 3.18 32.0 3.26 64.0 3.35 96.6 3.49 130.8 3.69 166.2 4.00 202.8 4.22 221.2 4.51 242.2 4.91 263.2 5.52 285.2 6.54 309.2 7.21 8.18 9.67
4.71 4.39 4.14 3.95 3.34 3.03 2.86
0.0 33.5 67.2 101.4 136.9 173.3 211.8 232.0 254.0 277.5 303.0 331.0
410.2
465.0
592.7
622.5
611.7 650.2 683.9 714.3
640.0 677.7 711.0 741.0
IMRK (mixt)
0.0 30.8 63.8 96.8 131.0 166.9 205.3 225.9 247.7 271.2 297.1 326.9 340.6 355.9 373.6 396.4 458.3 497.4 524.9 546.2 563.8 579.1 592.7 604.5 613.6 622.1 630.2 666.1 697.9 727.3
Composition, mol %: C, = 93.8; C, = 4.1; C,'- = 0.3; C, = 1.2; N, = 0.3. Table XIV. Heat Capacities and Enthalpies of Multicomponent Mixture 2 with Helium C,
J g-' K-'
exptl ( + He)
IMRK
temp, K
27.6 bar
50.7 bar
27.6bar
50.7 bar
115 125 135 145 155 165
3.36 3.45 3.54 3.64 3.85 4.19
3.35 3.40 3.47 3.58 3.74 3.95
3.22 3.30 3.42 3.59 3.86 4.30
3.19 3.26 3.36 3.50 3.70 4.01
such a mixture the top liquid layer becomes identical with the vapor phase and this temperature can be interpreted as the bubble point. Although it is not possible to conclude from our experiments what the exact mechanism is, it is worthwhile to keep the possibility of two liquid phases in mind, if helium recovery is being considered. Concluding Remarks A number of cooling curves have been determined experimentally and used as basic data for a computer program, based on a modified Redlich-Kwong equation of state, to enable predictions to be made for multicomponent systems. The predicted total enthalpy changes when going from 110 to 270 K, although not always within experimental error, are in fair agreement with experiment (see Table XV). Furthermore, some peculiarities observed call for special attention. (a) The liquid-phase heat capacities of the binaries ethanelnitrogen and propane/nitrogen were found
No. 4, 1979 345
Ind. Eng. Chem. Fundam., Vol. 18,
Table XV. Comparison of Total Enthalpy Changes from Experiment with Predictions enthalpy change pressure, exptl, IMRK, compn, temp %dev bar J g-' mol % range,K N*
c, c, C, c; C,
100 100 100 100 100 100 100 7 1.4128.6 91.0/9.0 91.019.0 91.019.0 61.0139.0 53.3146.7 40.9/59.1 40.9159.1 10.8189.2 10.8189.2 see Table XI1 see Table XI11
100-270 140-210 110-250 110-250 110-250 110-270 120-270 110-270 110-27 0 110-1 9 6 202-270 110-270 110-275 110-270 110-265 110-270 110-270 105-270 105-250
50.7 32.0 50.0 32.0 50.7 50.7 1.o 50.7 30.4 50.7 50.7 50.7 50.7 30.4 50.7 30.4 50.7 50.7 50.7
333.1 542.5 727.7 353.0 350.1 333.6 284.6 733.0 77 3.6 294.5 363.2 368.5 494.1 498.0 460.0 377.3 367.6 762.5 714.3
2.3 1.2 0 .o -1.1 -1.1 0.7 -1.1 0.9 1.3 3.1 -1.3 0.3 3.3 0.5 0.3 0.2 0.1 0.1 1.8
to be much higher than the predictions. It is argued that this is caused by a liquid-liquid phase separation a t low temperatures. (b) Small amounts of helium increase the liquid-phase heat capacity of a multicomponent mixture, but not that of pure methane and pure nitrogen. Although we are not sure of the exact mechanism involved, a liquid-liquid phase separation a t low temperatures might explain the increased heat capacity.
the parameter values used are given below -P= - -
RT
1
a
V-b
V(V+b)
+ 1.62.~).
1.0 + (1.45
a1 = 1.0
+ kl(
1-
kl = -0.127 k2 = 0.015
-
);
+ k2[1 -
(q)
+ 5.45*10-4~~-'
0.355.~- 0.65.10-4*~-'
b = 0 . 0 8 6 P7C 5 11.0 - O.ZO.u(
2
- 1.0))
P = pressure; T = temperature; Tc = critical temperature; w
= acentric factor; and V = molar volume.
Literature Cited Chacg, S.D., Lu, B.C.-Y., Chem. Eng. Frog. Symp. Ser., No. 81,63,16 (1963). Gonzales, M. H., Subrernaniam, T. K., Kao, R. L., Lee, A. L., Paper 21 in "Proceedings of the First International Conference on LiquefiedNatural Gas", Chicago, April 1968, Institute of Gas Technology, Chicago, IN., 1966. Lammers, J. N. J. J., van Kasteren, P. H. G., Kroon, G. F., Zeidenrust, H., in "Proceedings of the Fifty-Seventh Annual Convention of the Gas Processors Association", New Orleans, March 1978. Van Aken, A. B., Lammers, J. N. J. J., Simon, M. M., in "Chemical Engineering in a Changing World. Proceedings of the Plenary Sessions of the First World Conference on Chemical Engineering", Amsterdam, June-Juiy 1976, p 51 1, W. T. Koetsier, Ed., Eisevier, Amsterdam, 1976. Yu, P., Eishayal, I.M.,Lu, B. C.-Y., Can. J. Chem. Eng., 47, 495 (1969).
Appendix
A modified Redlich-Kwong equation of state has been used for the prediction of enthalpies. This equation and
Receiued for reuiew July 17, 1978 Accepted July 23, 1979
The Gravity Flow of Gases, Liquids, and Bulk Solids Frederick A. Zenz" and Fredrlc E. Zenz Manhattan College, Rlverdale, Bronx, New York 10471
I t is demonstrated that the gravity flow of bulk solids issuing from an opening in the bottom of a bin, the flow of liquid over a weir, and the flow of gas under the slots of a bubble cap or an inverted weir, can all be represented by a common equation. This equation has been tested by observing solids lighter and heavier than water flowing out of bins submeraed in a tank of water and by observing the flow of water over a weir submerged in a tank of oil.
-
of Bulk S o l i d s It has been demonstrated (Zenz, 1975) that the efflux of freely flowing bulk solids from a bin bearing a round hole or a slot in its bottom and situated in dry ambient air can be represented by the dimensionally consistant relationship
T h e Gravity F l o w
W , = g1J2pBh,'Jz/(tan a)lJ2 (1) where W , = solids efflux rate in lb/s X ft2 of hole area, g = acceleration field, ft/s2, pB = solids bulk density, lb/ft3, a = bulk solids angle of internal friction, deg, and h, = narrowest dimension of the opening through which the bulk solids are flowing, ft. In instances where the solid particles are large relative to the opening, an area cor0019-7874/79/1018-0345$01 .OO/O
rection which consists of reducing the length and width, or the diameter, of the opening by 1.5 times the particle diameter (Zenz, 1962) must be applied. Equation 1is also only valid for openings whose narrowest corrected dimension is a t least 16 times the particle diameter. For reasons which might not be obvious at the moment, consider the addition of a term to the right-hand side of eq 1 which at least would not be in conflict with the data on which it was founded
where pf = density of the surrounding medium, lb/ft3. To
0 1979 American
Chemical Society
