Enthalpies and Densities of High-pressure Mixtures Containing Light Hydrocarbons and Related Fluids. Estimation by a Simple Graphical Method Ernest0 Valdes-Krieg,’ J. A. R. Renuncio,2 and J. M. Prausnitr’ Chemical Engineering Department, University of California, Berkeley, California 94 720
For preliminary process design, it is often necessary to estimate enthalpies and densities of gaseous and liquid mixtures at advanced pressures. Easy-to-use charts for this purpose are given here. These charts, based on Pitzer’s three-parameter theorem of corresponding states, cover acentric factors between zero and 0.3 and include recent results for low reduced temperatures. Binary constants are introduced using the method of Barner and Quinlan. Calculated results are in good agreement with experiment for typical single-phase and two-phase mixtures encountered in natural-gas and petroleum-related technology. While accuracy appears to be only slightly lower than that attained using more sophisticated methods, the calculations described here can be made easily, using only a slide rule or desk calculator.
Enthalpies and densities of fluid mixtures at high pressures are frequently needed in process design. Numerous authors have shown that for relatively simple fluids, these properties can be calculated from corresponding-states cor relations; both‘diagrams and tables for this purpose were first presented more than a generation ago. In recent years, however, as large computers became more widely available, workers in this field have tended to correlate thermodynamic properties by ever more complicated equations of state. While such equations may, in some cases, produce more accurate results, the necessary calculations, even with a computer, are often needlessly complex and expensive, especially for the oft-encountered situation where, for preliminary flow sheets, it is satisfactory to use reasonable approximations. This work presents a simple method for speedily obtaining such approximations. In principle this method is not new, but unlike previously published “simple” methods, this one incorporates recently developed correlations for low reduced temperatures and, instead of using tables (which are inconvenient for interpolation), it employs charts applicable to both gaseous and liquid phases. The purpose of this work, therefore, is not to report scientifically significant new research, but to present a convenient engineering tool. Application is directed to those design engineers who, for approximate or exploratory work, require only a few estimates but who want to obtain them rapidly, without the need of sophisticated computer programs.
Generalized Density and Enthalpy Charts In 1955 Pitzer showed that the configurational properties of normal fluids (gases and liquids) can be correlated within the framework of an expanded theorem of corresponding states. In this context, a normal fluid is one whose molecules are nonpolar (or slightly polar), do not deviate excessively from spherical symmetry, and obey classical (rather than quantum) statistical mechanics. For our purposes here normal fluids include the lighter hydrocarbons, nitrogen, oxygen, carbon monoxide, argon, hydrogen sulfide, and carbon dioxide. We define the reduced temperature and reduced pressure in the usual way T R = TIT,; P R = PIP, (1) Present address: Atteljas 576 Ermita Ixtapalapa, Universidad Nacional Autonoma de Mexico, Mexico DF, Mexico. * Present address: Dpto. de Quimica Fisica, Facultad de Quimicas, Ciudad Universitaria, Madrid-3, Spain.
where the subscript R denotes a reduced property and the subscript c denotes a critical property. The compressibility factor z is given as a linear function of acentric factor w z = Pu/RT = z(O)
+ wz(l)
(2) where u is the molar volume, R is the gas constant, and z ( O ) and z ( I ) are universal functions of T Rand PR. Finally, we define the reduced density p~ by (3) In eq 3, the numerator in the brackets is very close to (but not identical with) the compressibility factor a t the critical point. The reduced density defined by eq 3 is therefore very close to (but not identical with) the ratio of density to critical density. The latter is difficult to measure experimentally and often is not known with high accuracy. Figures 1, 2, 3, and 4 show reduced densities as a function of reduced pressure, reduced temperature, and acentric factor. These charts are based on generalized tables given by Lewis, Randall, Pitzer, and Brewer (1961) and extended to lower reduced temperatures by Lee and Kesler (1975). The work of Lee and Kesler (1975) appears to be the most reliable now available; it is in general agreement, but perhaps a slight improvement upon the correlations of Lu (1973) and Boyle and Reece (1971). For low densities, results are based on the equation of Pitzer and Curl (1957) for the second virial coefficient. Figures 1-4 apply to both gases and liquids. The short, dashed lines indicate slight extrapolations of liquid-phase isotherms into the two-phase region; these are sometimes necessary for estimation of liquid-mixture densities. The residual enthalpy AH is defined by
AH = H ( T , P ) - H ( T , P = 0 )
(4)
Equation 4 indicates that AH is the enthalpy of a fluid at temperature T and pressure P minus the enthalpy of that same fluid at the same temperature but in the ideal-gas state. Since AH is usually a negative quantity, we prefer to calculate
-AH. Figures 5 and 6 give the reduced residual enthalpy (multiplied by -1) as a function of reduced density, reduced temperature, and acentric factor. Expanded forms of these figures for the moderate-pressure region are shown in Figure 7. These generalized charts are based on the tables of Lewis, Randall, Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
429
32 28 24 PR
20 16
12 08 04 001 002
005 0 1 0 2
05
2
I
5
IO
001 002
005 0 1
PR
Figure 1. Reduced density as a function of reduced pressure and reduced temperature for acentric factor equal to zero.
05
02 PR
I
.2
5
10
Figure 4. Reduced density as a function of reduced pressure and reduced temperature for acentric factor equal to 0.3.
70
001 002
005 0 1 02
05
I
2
5
70
IO
PR
Figure 2. Reduced density as a function of reduced pressure and reduced temperature for acentric factor equal to 0.1.
0
04
08
16
12
20
24
28
32
PR
Figure 5. Reduced residual enthalpies as a function of reduced density and reduced temperature for acentric factor equal to zero (lower part) and for acentric factor equal t o 0.1 (upper part).
001 0 0 2
005 0 1
02 PR
05
I
2
5
IO
Figure 3. Reduced density of a function of reduced pressure and reduced temperature for acentric factor equal to 0 2.
Pitzer, and Brewer (1961), Carruth and Kobayashi (1972), and Lee and Kesler (1975). In the past, generalized charts for enthalpy have used reduced pressure (rather than reduced density) as the independent variable. When the reduced pressure is used, there is necessarily a discontinuity in proceeding from the liquid phase t o the vapor phase. However, as is evident in Figures 5 , 6, and 7, when the reduced density is chosen as the independent variable, there is no discontinuity. The reduced density (unlike the reduced pressure) is a continuous function 430
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
PR
Figure 6. Reduced residual enthalpies as a function of reduced density and reduced temperature for acentric factor equal to 0.2 (lower part) and for acentric factor equal to 0.3 (upper part).
as one proceeds from the liquid, through the two-phase region, into the gaseous phase. For practical calculations applicable t o both phases, the reduced density is a more convenient variable than the reduced pressure.
Table I. Reducing Parameters for Two Binary Mixtures
lo0
0 75 0 50
One-phase"
0 25
e e
147.1
291.1
46.15
8.25
Two-phase 2.74 Vapor 94.4 153.9 38.59 Liquid 98.1 180.2 , 43.62 1.69 " 49.4 mol % methane-50.6 mol % propane. 56.6 mol % methane-43.4 mol % nitrogen at -200 O F and 250 psia; y(methane) = 0.466; x(methane) = 0.855.
0
Figure 7. Reduced residual enthalpies in the low reduced-density region.
Equations 5 to 10 apply to a homogeneous phase, either gas or liquid. To find the enthalpy and density of a two-phase system, it is necessary t o establish the relative quantities as well as the compositions of the two equilibrium phases. T o do so, it is necessary to utilize phase-equilibrium (K-factor) data and material balances. If the overall composition of a twophase system is given by mole fractions 81,,32, . . , then the liquid phase mole fractions X I , x 2 , . . . and the vapor phase mole fractions y1, y2, . . are found by
.
Mixtures To estimate densities and enthalpies in homogeneous fluid mixtures, we use the one-fluid approximation; Le., we assume that the configurational properties of a homogeneous mixture are identical with those of a hypothetical pure fluid whose characteristic reducing parameters (T,, P,, and w ) are functions of the mixture's composition. There has been much discussion in the literature concerning what these functions (often called mixing rules) should be. Experience has shown that best results are obtained when a mixing rule is applied to the critical volume (instead of the critical pressure) and when a characteristic binary constant is introduced into the mixing rule for the critical temperature. Consider a gas mixture with mole fractions y1, y2, . . . etc. Let subscript M denote mixture. We propose
where
and (7) where
The binary constant K12 is not far removed from unity; its value must be found from some experimental data for the 1-2 binary. A large number of these constants has been presented by Barner and Quinlan (1969). Finally OM = zy;Wi
(9)
and
For a liquid mixture, eq 5 to 10 are also used but in that event liquid-phase mole fraction x replaces vapor-phase mole fraction y .
. 81 = x1 ( L + V K I ) ,where K1 = y J x 1 82 = X P ( L + V K P )where , K P= y z l x z
(11)
(12)
and similar equations for all other components. In these equations L is the liquid-phase molar fraction of the two-phase system and V is the vapor-phase molar fraction. For a two-phase mixture, the density and residual enthalpy are then found by (13)
lH=L ( A H L )+ V(AH")
(14)
where, for simpler notation, subscript M has been omitted. Liquid-phase properties p L and AHL are found from the generalized charts using the mixing rules given by eq 5 t o 10 with mole fraction x ; similarly, vapor-phase properties p u and AI+ are found from the generalized charts using mixing rules given by eq 5 to 10 with mole fraction y . When K-factor data are not available, estimates can be made using a variety of techniques, for example, those recommended by the Gas Processors' Association (808 Home Federal Building, Tulsa, Okla. 74103).
Illustrative Calculations T o illustrate the use of the charts presented here, we calculate first the enthalpy of a binary liquid mixture containing 49.4 mol % methane, 50.6 mol % propane a t -30 OF and 750 psia. Table I presents the reducing parameters for this mixture as calculated from eq 5-10. At -30 OF, the reduced temperature is 0.82 and the reduced pressure is 1.10. From Figure 1 (o= 0), we obtain by interpolation p~ = 2.18. From Figure 2 (w = O . l ) , we obtain by interpolation p~ = 2.20. To interpolate with respect to acentric factor, we use 1
1
PR
PR(w = 0 )
- (o= 0.0825) =
giving a reduced density of 2.20. We define the reduced residual enthalpy by ~ H = R -AH/RT,. From Figure 5, by interpolation a t w = 0, we obtain SR = 4.42; by interpolation a t w = 0.1,we obtain AHR = 4.87. Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
431
Table 11. Residual Enthalpies for Methane-Propane (a(CH4) = 0.494; a(CsH8) = 0.506) Temp, O F
-200 ( T R = 0.49)
Pressure, psia
-AH, Btullb Calcd Obsd
250 500 750
213 213 212
L L L
111
132 150 165
L+V L+V L L+V L+V L+V V L+V L+V
250 500 750
0
250 500 750
138
113 135 148
250 500 750
13 37 74
15 42 82
( T R= 0.88) 100
T R = 1.069)
135 165 98 118
Table 111. Residual Enthalpies for Methane-Nitrogen (a(CH4) = 0.566; a(N2) = 0.434) Temp, OF -270 ( T R = 0.66)
-230 ( T R= 0.80)
-200 ( T R = 0.90) -170 ( T R= 1.005)
200 ( T R= 2.3)
Pressure, psia
Phase
250 500 750 2000
140 139 139 135
135 135 134 131
L L L L
250 500 750 2000
125 125 124 123
122 123 122 121
L L L L
250
43 112
44 110 112 112
L+V L L L
14 40 91 102
V L+V L L
750 2000
115
250 500 750 2000
15 43
250 500 750 2000
114
88
102 2.8 5.0
7.3 16.8
2.5 5.0 7.2 17.8
V V V V
Finally, interpolating with respect to acentric factor according to
~ H R (= u 0.0825) = 1 H ~ ( w = 0)
we obtain AHR = 4.79. The molecular weight of the mixture is 30.236. For the residual enthalpy we find AH = -165 Btu/lb, in excellent agreement with the experimental value -165 reported by Cochran and Lenoir (1974). Second, we illustrate use of the charts for two two-phase binary mixtures. The first one has the overall composition 81 = 0.491 (methane) and 82 = 0.509 (n-heptane) a t -100 O F and 400 psia. At this temperature and pressure, Chang et al. (1966) report K1 = 3.22 and K2 = 7 X The material balance equations yield L = 0.738, V = 0.262, x 1 = 0.3105, and y1 = (essentially) 1. The critical constants for the vapor phase are 432
-200 ( T R= 0.45) -100 ( T R = 0.62) 100 ( T R = 0.97) 160 ( T R= 1.07) 200 ( T R= 1.13)
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
Pressure, psia 250 2000 250 2000 1000 2000 1000 1500 250 1500 2000
-W, Btullb Calcd Obsd 235 226 208 201 143 148 87 118
Phase
L L L L L L L L V L L
225 219 202 197 144 149 92
11.2
122 11.1
97.2 110
97.8 113
Table V. Residual Enthalpies for Methane-+Heptane (Amethane) = 0.491; g(heptane) = 0.509) Temp, OF ~
-AH,Btullb
Pressure, psia
Calcd
Obsd
400 1000 2500 2500
152 174 163 135
156 154 149 128
L+V L L L
2500
113
106
L
2500
87
86
L
Phase
~~
-100 ( T R = 0.49)
-AH, B t d b Calcd Obsd
500
Temp, OF
Phase
217 215 214
-30 ( T R = 0.82)
Table IV. Residual Enthalpies for Ethane-Propane Methane) = 0.764; a(propane) = 0.236)
100 ( T R= 0.76) 250 ( T R = 0.965) 400 ( T R= 1.17)
essentially those of pure methane. For the liquid phase they are T,= 466.2 K, P, = 33.63 atm, and w = 0.2467. The reduced density for the liquid phase is 3.08 and that for the vapor phase is 0.21. For the liquid phase ~ H isR8.04 and for the vapor phase it is 0.66. Finally, for the two-phase mixture, -AH = 152 Btu/lb, in good agreement with the experimental value 156. (See Table V.) For the second two-phase binary mixture we have overall composition 81 = 0.566 (methane) and 8 2 = 0.434 (nitrogen) a t -200 OF and 250 psia. At this temperature and pressure K1 = 0.545 and K2 = 3.68 (NGSMA 1957). The material balance equations yield L = 0.257, V = 0.743, x 1 = 0.855, and y1 = 0.466. Critical constants for the two phases are given in Table I. The reduced densities are 2.20 for the liquid and 0.16 for the vapor. For the liquid AHRis 4.58 and for the vapor it is 0.55. For the two-phase mixture, - l H is 42.8 Btu/lb, in excellent agreement with the experimental value, 43.9. (See Table 111.)
Comparison of Calculated and Experimental Residual Enthalpies T o obtain some estimate of the expected accuracy of the generalized enthalpy charts, calculations were performed for a few binary mixtures: methane-propane, methane-nitrogen, ethane-propane, and methane-heptane. Calculations include both single-phase and two-phase liquid and gaseous mixtures. Results are summarized in Tables 11-V. All experimental enthalpies are from Cochran and Lenoir (1974). K-factor data for methane-propane are from Reamer et al. (1950) and from Price and Kobayashi (1959). In Tables 11-V, the tabulated reduced temperatures T Rare for the overall composition, 81 and 82. In a two-phase system, as discussed above, it is necessary to find one reduced temperature for the liquid phase, based on composition X I and x p , and another reduced temperature for the vapor phase, based on composition 3 1 and y2.
Agreement between calculated and experimental enthalpies is satisfactory, in many cases within experimental uncertainty. With a little practice and a desk calculator, calculations can be performed quickly, provided that the number of components in the mixture is not too large. For accurate work, it is desirable to use charts larger than those published here. Such charts are available from the authors.
Acknowledgment For financial support, the authors are grateful to the National Science Foundation, to the Gas Processors' Association, and to the American Gas Association. One of us (J. A. R. Renuncio) acknowledges with thanks a grantfrom the cornmission of Educational Exchange between the United States and Spain. In the early stages of this work, we were assisted by Edwin L. Force.
Literature Cited Barner, H. E., Quinlan, C. W., i d . Eng. Chem., Process Des. Dev., 8,407 (1969). Boyle, G , Reece, D.1 oil 69 ( 3 , 56 Carruth. G. F., Kobayashi, R., Ind. Eng. Chem., Fundam., 11, 509 (1972). Chang, H.L., Hurt, L. T., Kobayashi, R., AiChE J., 12, 1212 (1966). Cochran, G. A,, Lenoir, J. M., Research Report 11, Gas Processors' Association, 1812 First Place, Tulsa, Okla. 74103 (Sept 1974). J,, 21, 510 (,975), Lee, I,, Kesler, M, G,, Lewis, G. N., Randall, M., "Thermodynamics," Revised by K. S. Pitzer and L. Brewer. 2nd ed. McGraw-Hill. New York. N.Y.. 1961 Lu, B. C-Y.. Hsi, C., Chang, S. D., Tsang, A,, AlChEJ., 19, 748 (1973). Natural Gasoline Supply Men's Association, Engineering Data Book, 7tt- ed,Tulsa, Okla., 1957. Pitzer, K . S., Lippmann. D. Z., Curl, R. F., Huggins, C. M ., Petersen, D. E., J. Am. Chem. Soc., 77, 3427 (1955). Pitzer. K. S..Curl. R. F.. J. Am. Chem. SOC.,79, 2369 (1957). Price, A. R., Kobayashi, R.. J. Chem. Eng. Data, 4, 40 (1959). Reamer, H. H., Sage, 5.H.. Lacey, W. N.. ind. Eng. Chem.. 42, 534 (1950). J.1
J.8
I
Receiued f o r reuiew November 11, 1975 Accepted March 15,1976
Evaporation Suppression through In-Situ Formation of Continuous Polymer Films Kenneth L. Moore. and Russell Reed, Jr. Propulsion Development Department, Naval Weapons Center, China Lake, California 93555
Continuous polymer films have been used successfully in small-scale tests to demonstrate the suppression of evaporation from water surfaces in a desert environment. It is postulated that these films are formed by air-activated cross-linking of polyfunctional prepolymers. Experimental results indicate that evaporation can be reduced by more than 80%. The films demonstrate good resistance to breakup by wind and wave action in shallow (60 X 120 cm) pans.
Introduction Various techniques have been devised for reducing water losses due to evaporation from storage reservoirs. These typically involve the formation of one or more monolayers of cetyl, stearyl, and behenyl alcohols (Dressler and Guinat, 1973; Dressler, 1959). Mixtures of some of these substances are available commercially for use as evaporation suppressants. Certain of these alcohols have been used in large-scale evaporation control work. Early experiments used metering dispensers placed in the water as means of dispersing the alcohols over the water surface. In the largest experiment t o date, two crop-dusting airplanes were used t o coat a 30 000acre reservoir with a 1:l mixture of cetyl and stearyl alcohols; this succeeded in saving an estimated 15 million gallons per day (Bester, 1967). Essential to the process of evaporation reduction by monolayer films is the ability to keep such films in a state of compression. Whenever the film is allowed to lapse from this condition, its ability to inhibit evaporation is severely degraded. Evaporation reductions claimed in large-scale efforts typically vary from 16 to 50%. In this paper a water conservation measure based on airactivated cross-linking of prepolymers in the molecular weight range of 3000-4000 is discussed. Films formed in this manner are continuous in nature and are characterized by a high degree of flexibility and excellent resistance to wind and wave action. Breakup in high velocity winds does occur, but the
films will repair themselves when wind velocity drops below 15 mph. Evaporation reductions in excess of 90% have been noted in both laboratory and field tests.
Theory In view of the mobility of the relatively low molecular weight alcohols such as octadecanol, they tend to accumulate on the shore in high wind conditions; therefore, relatively large amounts are required to maintain a hydrophobic monolayer film on the surface of a reservoir. Alcohols having longer hydrocarbon chains would function more efficiently since the area covered would generally be expected to increase as molecular weight increases; in addition, volatilization of the film would be reduced. Ideally, the in-situ formation of a polymeric film would represent the most desirable approach since a film of low volatility and with some degree of strength would resist breakup by wave motion. Film healing might also be possible since polymer bonds could be formed from unreacted sites. In order to achieve a relatively high molecular weight liquid hydrophobic alcohol, an unsaturated structure such as polybutadiene was chosen as a likely backbone. The presence of the carbon-carbon double bonds allow freer rotation, an effect which tends to result in liquids rather than solid waxes; liquids should spread more rapidly and uniformly over the surface than solids. Several commercially available hydroxyl terminated polybutadienes (HTPB) were chosen €or study; these included Polymer R, R-45M, and R-15. Physical properties Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
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