Thermodynamic Properties of Gases CARBON DIOXIDE R. L. SWEIGERT, PAUL WEBER, AND R . L. ALLEN State Engineering Experiment Station, Georgia School of Technology, Atlanta, Ga.
T h e preparation of a large (30 X 80 inch) thermodynamic chart of the properties of carbon dioxide is described. Specific volume, enthalpy, and entropy values for carbon dioxide are plotted at temperatures -75' to 1800' F., over a range of specific volumes 0.1 to 1000 cubic feet per pound for pressures up to 3000 pounds per square inch. All available experimental data were gathered and used as a basis for the chart. These data, as well as their use, are discussed. The calculation and approximation methods employed to obtain values over ranges for which there exist no experimental data are described. Values for specific volume, enthalpy, and entropy are tabulated.
S
OME time ago the writers prepared for the Design Research Section of the Bureau of Ships, U. S. Navy, large scale charts covering thermodynamic properties of carbon dioxide, air, argon, oxygen, and steam. For each gas a 30 X 80 inch chart was prepared, on which were plotted enthalpy and entropy values over the temperature range -75' to 1800' F. and over a range of specific volumes from 0.1 to 1000 cubic feet per pound for pressures up to 3000 pounds per square inch absolute. For oxygen a necond chart was prepared covering these properties in the low-temperature range (including the liquid region). Available experimental data were gathered and used as a basis for making the charts. Because of the urgent need for these charts, no time was available for experimental determination of any of the lacking data. Extrapolation, interpolation, and approximation methods were used in filling out values in the temperature and pressure regions where experimental data were not available. The need for additional experimental determinations is evident. The most extensive work to date on the compilation of thermodynamic data of carbon dioxide, together with a temperatureentropy diagram, is that of Plank and Kuprianoff (16). Their diagram was prepared primarily for use in the refrigeration industry; therefore, the temperature and pressure range covered does not go beyond 300' F. and 1700 pounds per square inch. Since the chart needed for the present work was to cover temperatures from -75" to 1800" F. and pressures up t o 3000 pounds per square inch, that of Plank and Kuprianoff was inadequate. For the additional range needed, the only experimental data available for carbon dioxide are: compressibility data up t o 500" F. and 3000 pounds; Joule-Thomson data up to 575 F. and 3000 pounds; specific heat data up to 1800' F. a t atmospheric pressure, or a t zero pressure, calculated from spectroscopic data. The procedures followed in extending these data to the ranges desired are described later. After all the data were compiled, a 30 X 80 inch chart was drawn, upon which specific volume values, from 0.1 to 1000 cubic feet per pound were plotted as abscissas and temperatures were plotted as ordinates from -75" to 1800" F., in the form of constant pressure, enthalpy, and entropy lines. Figure 1 is a small
scale drawing of this chart, which contains only a few lines to indicate the form followed. I n the temperature and pressure range for which experimental data were available, the properties can be read from the large chart with an accuracy within the limits of error of the data. The various properties can be read from the chart over the temperature and pressure ranges for which experimental data are lacking, with amaximum error of not over 5% a t the highest pressures; the error is less than 2% a t pressures below 1500 pounds per square inch. The specific volume, enthalpy, and entropy values for the entire temperature and pressure range covered by the chart are presented in Tables I to VII. SPECIFIC VOLUME CALCULATIONS
Various equations of state, which are modifications of the familiar equation PV = NRT,have been derived for representing P-V-T relations of gases under conditions a t which their behavior deviates from that of a perfect gas. I n general, these equations are complicated and necessitate laborious calculations i n practical use. Furthermore, for accurate work they are generally limited to the pressure and temperature range in which experimental compressibility data exist so that the needed constants oan be obtained. Dodge (4) reviews the various general equations of state which have been proposed thus far. * Plank and Kuprianoff (16) derived a fairly simple equation of state for calculating specific volumes of carbon dioxide which follows the experimental data in the superheated vapor region with an average deviation of less than 1%:
T 1.32 P -
v = 0.2437-
+ 1.3794 X
10-8P
(1)
A general equation of state may be written
Pu
=
ZRT
(2)
where 2,the compressibility factor, is a function of pressure, temperature, and the nature of the gas. Values of 2 a t any temperature and pressure can be calculated from compressibility (Pv)data and substituted in Equation 2 for the desired P-V-I' relation calculations. However, values of 2 for different gases are most readily correlated by plotting 2 for various gases a t the same reduced pressures and temperatures. Compressibility factor charts (graphs of Z = Pv/RT against reduced pressures for various values of reduced temperatures) have been drawn for many gases (6,8,12,20).
Experimental compressibility data for carbon dioxide are available up to approximately 500' F. over the entire pressure range covered by this work (up to 3000 pounds per square inch] (1, 10,18, 14, 16, 17).
185
I. 86
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 38, No. 2
Specific volume values for saturated carbon dioxide vapor based on data by Plank and ~ by Kuprianoff (16) T V C tahulat,ed Quinn and Jones (17) ovcr Ihc temperature range from -70" F. up to the criiical point. ~
Q
~ FACTOR ~ CHART
ibilily E x : orb irr carbon dioxide werc cxlciil
lines represent vdues 11:ised on experimental d:ita t i v ~ i l t ~ h lfor c carbon dioxide.
a
4
bon moiioxide, cthyicne, methalie,
U F
2
dotted line in 1:igure 2 s h o w
t '
d
&
U
dioxide) based on these averaged data. Similar lines €or the other tempcratures above 500' F.were drawn on thc large scale chart,. Compressibility fsct,ors ~ v e r c also plotted against tcmpcratures for various pressures (Figure 3). The solid lines represent expcrimental data for carbon dioxide, and the dotted lines slio~vZ values based on averaged data from the other gases at corresponding reduced conditions. Extrapolation was used along with the Z value averages for other gases in connecting the dotted lines with the solid lines. From the large scale compressibility factor charts (Figures 2 and 3), values \yere obt'aincd for use in calculating the specific volumes for carbon dioxide in the superheated vapor region. The maximum values of reduced temperature and reduced pressure for carbon dioxide covered by this work are T R = 4.1287 and PR = 2.8. At these same maximum reduced conditions the experimental compressibility factors for all the other gases thus far reported are less than 1.05* The value for carbon dioxide at these same conditions, as read from the charts represented by Figures 2 and 3, is approximated t o be 1.01. Thus it is likely that the maximum error in the estimated values for compressibilit,y factors for carbon dioxide, upon which specific vol-
~
~
February, 1946
INDUSTRIAL AND ENGINEERING CHEMISTRY
ume calculations were based, is within 4y0a t 3000 pounds per square inch and 1800" F. At lower temperatures and/or lower pressures the probable errors are less (Figure 3). HEAT CAPACITY AND LATENT HEAT
A critical survey of the literature on heat capacity data for carbon dioxide was made by Leduc (11). Quinn and Jones (17) discussed these data as well as later work published through 1935. Sweigert and Beardsley (19) derived Equation 3 based on spectroscopic data for calculating the specific heat of carbon dioxide at zero pressure: C, = 16.2
-
1.41 X IO6
6.53 X IO3
T
+
TZ
(3)
Comparison of values obtained by Equation 3 with experimentally determined Cp values as reported in the literature ( 7 ) shows that the former are somewhat higher than those calculated by Leduc (11) from available experimental data. Only the experimental Cp data by Eucken and Lude (6) are in fairly good agreement with values obtained by Equation 3. However, Eucken and Lude did not report data above 400" F.
Figure 3.
187
It was believed that heat capacity values based on spectroscopic data would be more accurate than the experimentally determined Cp data because of more precise measurements possible with the former measurements which are independent of high temperatures. Therefore, Equation 3 was used in this work where a heat capacity relation at zero pressure was needed. Only fragmentary data on experimental heat capacities are available for carbon dioxide at pressures above 1 atmosphere. Jenkin and Pye (9) determined heat capacities a t various temperatures up to 90" F. and pressures up to 47 atmospheres. Eucken and Mucke ( 7 ) determined Cp values at 780" F. and 5.8 atmospheres and also at 1100" F. and 8.6 atmospheres. Workman (21) determined heat capacities a t 63 atmospheres for temperatures up to 212' F. Thus, except for purposes of checking, these experimental heat capacity data a t higher pressures were of
Compressibility Factor Chart for Carbon Dioxide
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188
Vol. 38, No. 2
2 and 3, Equation 1 was found to agree within 1% with Equation 2 over the following ranges in the superheated vapor region: from 32" to 1800" F. at all pressures up to 750 pounds per square inch; from 800" at all pressures up to 1500 pounds; from 1000" to 1800" at all pressures up to 3000 pounds. For calculation of enthalpy If in the superheated region over the temperature and pressure ranges just mentioned, Equation 9, developed from Equation 4, was used: dR =
c,dT
-A
- v] dP
[T(&)p
where c, =
(g) n
(4)
(5)
Equation 7 was used to calculate heat capacities in these same tcmperature and pressure ranges:
2)
(7)
= -AT($)
Solving Equation 7 and substituting values for cpo for carbon dioxide obtained from Equation 3, Equation 8 xas obtained: en ,I
-100,
I 3 3
0
0
0
0
? .
0
il
a
0
c-loo
0 0
0
N
m
41
0.3682 -
0.1484 X IO3 ?-
T
0.0320 X IO6 -t T2
0
P R C S S J R E - POUNDS PER S Q U A R E I N C H
F i g a r e 4.
3'5279~0,3(1,0000 -t- 0.5225 X 10-3P) (8)
Constant Enthalpy Lines for Carbon Dioxide
minor usefuiiiesa i n this work. Plank and Kuprianofl (16) tabulated the most important experimental and calculated values for the latent heat of vaporization of liquid carbon dioxide between the critical point and the triple point. Their values for heat of vaparixation mere used in this Lvork. EXTHALPY CALCULATIOKS
Calculation of enthalpy H and entropy S is facilitated if the relation between pressure, volume, and temperature is known. Values for the specific volume of carbon dioxide were calculated by Equation 1 for various temperatures and pressures outside the range of experimental compressibility data. These specific volume values were then compared with those obtained by Equatisn 2 (Po = ZRT), using 2 values tsken from Figures
Substituting in Equation 4 the relation for e, obtained from Equation 8 and the relation for (dIfjdP)T obtained from Equation 6 gave Equation 9:
H = 0.36822' - 341.71 log T
1'058y58 (1.0000
($1
- 0.0320Tx
106
-
f 0.5228 X 10-3P) f 1108.50
(9)
The constant term (1108.5) in Equation 9 was obtained by assigning a value of 180 B.t.u. per pound to the enthalpy of ssturated liquid at 32' F. Use of Equation 9 was limited to the temperature and pressure range referred to in the above discussion (in which Equation 1agrees with Equation 2 within 1%).
INDUSTRIAL AND ENGINEERING CHEMISTRY
February, 1946
189
Figure 5. TemperatureEntropy Chart for Carbon Dioxide
G' 12c
INTROPY
- B.T.U.
PER POUND DEGREE.
APPLICATION OF JOULE-THOMSON DATA
To obtain enthalpy values at temperatures and pressures outside the range in which Equation 9 was applicable, Joule-Thomson data and extrapolation methods were used. Extensive work on the Joule-Thomson effect in carbon dioxide has been done by Eurnett (2, 3) and more recently by Roebuck and eo-workers (18). The carbon dioxide used by Roebuck contained less air impurity than that by Burnett; therefore the Roebuck Joule-
Thomson data were used in calculating enthalpy values. Roebuck reported Joule-Thomson data at small temperature and pressure intervals from -75" to 575" F. for pressures up to 200 atmospheres. Based on the Joule-Thomson data of Roebuck and eo-workers (19), lines of constant enthalpy were plotted on a large scale as indicated by the dotted lines on Figure 4. Values of enthalpy along the zero pressure line were then calculated by Equation 9;
INDUSTRIAL A N D E N G I N E E R I N G C H E M I S T R Y
190
definite values were thus assigned t o the constant enthalpy lines already plotted. Remembering that along a constant enthalpy line
(a
(lo)
= A!
and along a constant pressure line (11)
and using a mean value of c p for small temperature intervals, enthalpy values were obtained from Joule-Thomson data at all temperatures in the superheated vapor region up t o 575' E'. at all pressures up to 3000 pounds per square inch. In this same temperature range a t pressures up to 750 pounds the enthalpy values calculated by Equation 9 chccked within 0.5% the values obtained using the Joule-Thomson data. Enthalpy values calculated by Equation 9 and those obtained with Joule-Thomson data cover the entire temperature and pressure range of the thermodynamic chart except in the following regions: betneen 575" and 800" F. at pressures from 750 to 1500 pounds; between 575" and 1000" F. at pressures from 1500 to 3000 pounds. Interpolation methods were uwd to obtain enthalpy values in these two regions. ENTROPY CALCULATIOKS
Using the expression dS =
($)dT - A ();:, where 3 = ($,)
dp
P
and substituting in Equation 12 the value (bv/aT), as determined from Equation 1 and the value for r mas represented by Equation 8, it is found that for carbon dioxiae:
s =0.8478 log T 4- 0.1484 X lo3 - 0.01BOTZX lo6- o.10388 log ___-__ 0'8136p
T
(
(1.0000
0.5225 X 10-3P)
- 0.9938
= dH/T
lines were calculated, with the aid of the graph, over the entire temperature and pressure range represented in Figure 4 by the area below the dotted boundary line. Values of S for t,he temperature and pressure ranges represented in Figure 4 by the area above the dotted boundary line, and obtained by Equation 13, were plotted on a large scale graph similar to Figure 4. Knowing the entropy value, for cxarnple, a t 800" F. and 1000 pounds per square inch, and having calculated changes of S at 1000 pounds per square inch by Equation 14, values of S at 1000 pounds below 800" F,were found by subtraction. I n this way S values Fere obtained all the way down to -75" F. This procedure was repeated for each of the other pressures in t,he range represented in Figure 4 by the areit below the dot'ted boundary line. After S vas determined for various temperatures and preswas sures, const,ant S lines were drawn; and since B large scal~? used, deviations could be noted, checkcd, and corrected. Otlicr methods of p l o t h g S values w r e also used. For examplc, constant S liiicv \\-ere drawn with ternpera,turee and vdumes as coordinates, and deviations noted, checked, and correct>cd. In addition, A S values for 5 B.t.u. intervals of H along coi pressure lines werc tabulated. Then the A(AS) valucs for these intervals ivere compared; these comparisons helped t,o reduce the size of possible errors. On the final thermodynamic properties chart 11-herethe abscissa is log v, the distances along a constant temperature line betwecn points of equal increments of entropy must be equal as long as the gas behaves like a perfect gas. Over a large part of the chart carbon dioxide \vas found t o behave pract'ically as a perfect gas. This behavior was extremely helpful in checking for errors in the calculations. By means of the methods described, enough values of specific volume, enthalpy, and entropy a t selected pressures and temperatures were determined to plot these properties on the large scale chart of thermodynamic properties (the temperature-log specific volume chart indicated in Figure 1). Some of these properties are listed in the tables. A temperature-entropy chart (30 X 40 inches) \vas also drawn from these values. The form of this chart is indicated in Figure 5, which shows only a few of the lines actually d r a m on t'he full scale chart1. ACKNOWLEDGMZh-T
The writers wish to acknowledge the assistance rendered by
R. A. Trotter, Department of Mechanical Engineering, who made (13)
The constant term (0.9938) in Equation 13 was obtained by assigning a value of 1 B.t.u./(pound) ( " R.) to the entropy of saturated liquid a t 32" F. The use of Equation 13 for calculation of entropy values n-as limited to the temperature and pressure range in Rhich specific volumes calculated by Equation 1 mere n-ithin 1% of those calculated by Equation 2. A combined graphical-computation method was used for obtaining entropy values in the low temperature-high pressure range (the range not covered by Equation 13) and represented on Figure 4 by the area below the dotted boundary line. Enthalpy values had already been calculated and plotted on a large scale graph (Figure 4). Along a constant pressure path,
ds
Vol. 38, No. 2
(14)
Small increments (0.1 B.t.u./pound) for dH were marked Off along constant pressure ordinates in Figure 4 and the corresponding average temperatures for these small increments were reado Substitution in Equation 14 provided corresponding values for dS. I n this way small increments Of dX along constant pressure
the final charts. The help rendered by various assistants who made and checked many of the calculations is also acknowlcdged. Thanks are extended to J. R. Roebuck, University of Wisconsin, and to Arthur B. Lamb, editor of the Journal of the American Chemical Society, for their kindness in furnishing us with a copy of the article by Roebuck and co-workers (18)prior to its publication. The advice and assistance rendered by Captain R. V. Xleinschmidt and Lieutenant Commander EX. 0. Croft, of the Design Research Section, Bureau of Ships, while the thermodynamic properties chart was in preparation, are gratefully acknowledged. NOMENCLATURE
A = B.t.u./ft.-lb. cp = heat capacity a t constant pressure, B.t.u./(lh.) ( ' R.) Cp = heat capacity a t constant pressure, B.t.u./(lb. mole) ( R.) 1 A limited number of full scale blue-line prints of t i e tliermodynamic properties chart (Figure 1) and of the temperature-entropy chart (Figure 5 ) are available and can be obtained a t S2.50 each and $1.80 each, respectively, by addressing the State Engineering Experiment Station, Georgia School of Technology, iltlanta, Ga. The quality of the prints of the thermodynamic properties chart is not quite so good as that of the smaller temperatureentropy chart prints because the original tracing was damaaed slightly. However, the prints of the temperature-entropy chart are excellent.
INDUSTRIAL AND ENGINEERING CHEMISTRY
February, 1946
H ,U
N P PE R
= = = = = =
S
=
t~
77
= =
Z
=
T = TR =
enthalpy, B.t.u./lb. Joule-Thomson coefficient of a gas number of moles of a gas pressure, lb./sq. in. abs. reduced pressure (pressure/critical pressure) gas constant = 1544 ft.-lb./(lb. mole) ( ' R.) entropy, B.t.u./(lb.) (",R.) absolute temperature, R. reduced temperature (temperature/critical temperature) specific volume, cu. ft./lb. total volume compressibility factor, Pv/RT
Subscripts p = constant V = constant T = constant H = constant
Hougen, C. A., and Watson, K. M., "Industrial Chemical Calculations", 2nd ed., New York, John Wiley & Sons, 1936. Jenkin, C. F., and Pye, D. R., T r a n s . R o y . SOC.(London), A215, 353 (1915). Keesom, W. H., Verhandel. A k a d . Wetenschappen Amsterdam, 12, 391, 544, 616, 821 (1903); Commun. P h y s . Lab. Univ. Leiden 88 (1903). Leduc, A., in International Critical Tables, Vol. V , p. 83, New, York, McGraw-Hill Book Co., 1929. Maron, S. H., and Turnbull, D., IND.ENG. CHEX, 34, 544 (1942). Michels, A., Bijl, A,, and Michels, C., Proc. Royal SOC.(London), A160, 376 (1937). Michels, A., Blaisee, V., and Michels, C., Ibid., A160, 358 (1937). Michels, A., and Michels, C., Ibid., A153, 201 (1935). Plank, R., and Kuprianoff, J., Z. ges. Kalte-Ind., 1, 1 (1929); 2. tech. P h y s i k , 10, 99 (1929). Quinn, E. L., and Jones, C. L., "Carbon Dioxide", A.C.S., MonoeraDh 72. New York. Reinhold Pub. Cora.. 1936. Roebuck, J.-R., Murrell, T. A., and Miller, E. E.; J . Am. Chem. SOC.,64, 400 (1942). Sweigert, R. L., and Beardsley, M. W., State Eng. Expt. Sta., Ga. School Tech., Bull. 2 (1938). Weber, H. C.. "Thermodynamics for Chemical Engineers", New York, John Wiley &-Sons, 1939. Workman, E. J., Phys. Rev.,38, 587 (1931).
pressure volume temperature enthalpy LITERATURE CITED
(1) Amagat, E. C., Compt. rend., 114, 1093 (1892). (2) Burnett, E. S., BUZZ. Gniv. Wis., 9, h'o. 6 (1926).
(3) Burnett, E. S., P h y s . Rev., [2] 22, 590 (1923). (4) Dodge, B. F., "Chemical Engineering Thermodynamics", New York, McGraw-Hill Book Co., 1944. ESG.CHEM..24. 1353 (1932). (5) Dodge. B. I?.. IND.
