Ind. Eng. Chem. Res. 1989,28, 1103-1106
Table 111. Precipitation of Lanthanides by CaO Addition to the Leach Solution in the HCl System (50 mL of Leach Solution; 50 g of Rock) precipitate CaO
added, 1.0 2.0 3.0
final uH 1.2 1.8 5.0
wt,
g
1.1 4.9 9.1
LnZO3O content, % 0 0.2 1.0
Ln203
recovery. % 0 19 95
Ln = lanthanides.
(nearly 100%) from the leach solutions prior to P205recovery. The solutions obtained containing monocalcium phosphate can be treated as follows: (1)In the HNO, system, uranium is first extracted by a mixture of D2EHPA and TBP in hexane followed by extraction of lanthanides by TBP. (2) In the HCl system, uranium is first extracted by a mixture of D2EHPA and TBP in hexane followed by extraction of lanthanides by D2EHPA in toluene.
Acknowledgment We are grateful to H. J. Cheek of Albright & Wilson Inc. for kindly supplying us with a sample of bis(2-ethylhexyl) phosphoric acid. Registry No. DSEHPA, 298-07-7; TBP, 126-73-8;Ca(H2P04)2, 7758-23-8; Ca(N0&, 10124-37-5;CaC12, 10043-52-4;U, 7440-61-1.
References (1) Habashi, F.; Awadalla, F. T. In Situ and Dump Leaching of Phosphate Rock. Ind. Eng. Chem. Res. 1988, 27, 2165-69. (2) Habashi, F. Hydrometallurgy. In Principles of Extractive Metallurgy; Gordon & Breach Science Publishers: New York-London-Paris, 1970 (reprinted 1980); Vol. 2.
1103
(3) Habashi, F.; Awadalla, F. T.; Zailaf, M. The Recovery of Uranium and the Lanthanides from Phosphate Rock. J. Chem. Technol. Biotechnol. 1986, 36, 259-267. (4) Habashi, F.; Awadalla, F. T.; Yao, Xin-bao The Hydrochloric Acid Route to Phosphate Rock. J. Chem. Technol. Biotechnol. 1987, 37, 371-383. (5) Awadalla, F. T.; Habashi, F. The Removal of Radium During the Production of Nitrophosphate Fertilizer. Radiochim. Acta 1985,38, 207-210. (6) Habashi, F. The Recovery of Uranium from Phosphate Rock. Progress and Problems. In 2nd International Congress on Phosphorus Compounds Proceedings, Institut Mondial du Phosphate: Paris, 1980; pp 629-660. (7) Habashi, F. The Recovery of Lanthanides in Phosphate Rock. J. Chem. Technol. BiotechnoL 1985,35A(l), 5-14. (8) Awadalla, F. T.; Habashi, F. Determination of Uranium and Radium in Phosphate Rock and Technical Phosphoric Acid. 2. Anal. Chem. 1986,324, 33-36. (9) Habashi, F.; Zailaf, M.; Awadalla, F. T. Determination of the Lanthanides in Phosphate Rock. 2.Anal. Chem. 1986,325(1), 479-480. (10) Wilson, H. N. The Determination of Phosphate in the Presence of Soluble Silicates. Application to the Analysis of Basic Slag and Fertilizers. Analyst 1954, 75, 535-546. (11) Amer, S. Aplicaciones de la extraccion con disolventes a la hidrometalurgia. V Parte. Tierras raras, itrio y escandio. Rev. Metal. CENIM 1981, 17(4), 245-283.
* To whom all correspondence should be addressed. Farouk T. Awadalla, Fathi Habashi* Department of Mining & Metallurgy Lava1 University Quebec City, Canada G l K 7P4 Received for review October 24, 1988 Revised manuscript received March 17, 1989 Accepted April 11, 1989
Generalized Temperature-Dependent Parameters for the Peng-Robinson Equation of State for n -Alkanes Temperature-dependent parameters for the Peng-Robinson equation of state have been determined from saturated liquid and vapor volumes and vapor pressure data in the subcritical region and from P-V-T data in the supercritical region for a number of n-alkanes for which sufficient data were available. These parameters have been generalized in terms of reduced temperature and acentric factor, so they can be used for other hydrocarbons. While liquid volumes are generally poorly predicted with cubic equations of state and generalized parameters, we find that with the use of the Peng-Robinson equation of state with the proposed, generalized parameters the predicted liquid volumes are much better than those obtained with the original parameters, while other properties, such as vapor pressure and vapor volumes, are of comparable accuracy. Equations of state are frequently used for predicting the volumetric and thermodynamic properties of fluids and vapor-liquid equilibria. One method of improving the quality of equation of state predictions is to use temperature-dependent parameters obtained by fitting experimental data for the fluid under consideration (Harmens, 1975; Hsi and Lu, 1971; Joffe et al., 1970). However, correlations or generalizations of these fluid-specific parameters are needed for calculations of vapor-liquid equilibrium and volumetric properties in design and in process and reservoir simulation involving compounds for which limited data are available. The generalization of the equation of state parameters follows from the work of van der Waals more than a century ago and more recently from Soave (1972) and Peng and Robinson (1976). Haman et al. (1977) and Yarborough (1979) generalized the parameters of the SRK equation of state in terms of reduced temperature and acentric factor, but limited their corre0888-5885/89/2628-1103$01.50/0
lations to the subcritical region; they then used the values of the two equation of state parameters determined at the critical temperature for all higher temperatures. Morris and Turek (1986) allowed the parameters of the SRK equation of state to vary with temperature both above and below the critical temperature, but they did not propose the generalization of their parameters. Further, the correlations of Yarborough and of Morris and Turek result in a discontinuity in the temperature derivatives of the parameters at the critical point, which affects derivative properties, such as enthalpy and heat capacity. Xu and Sandler (1987a,b) correlated the parameters of the Peng-Robinson equation of state as a function of reduced temperature in the subcritical and supercritical region, and in the near-critical region, they used the cubic-spline functions at reduced temperature, which are continuous and smooth at the critical temperature. But their correlations are specific to each fluid. It should be pointed out 0 1989 American Chemical Society
1104 Ind. Eng. Chem. Res., Vol. 28, No. 7 , 1989 Table I. Literature Sources of comDonent methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane
Table 11. Values of the Coefficients a and j3 of Equations
Data
10-15
ref Augus et al., 1978 Vargaftik, 1983 Vargaftik, 1983 Das et al., 1973 Das et al., 1977 Vargaftik, 1983 Vargaftik, 1983 Vargaftik, 1983
2 3 4 5 6
that, once accurate parameters for the pure fluids have been obtained, they can be combined, using simple mixing rules, to get good predictions for the densities and phase equilibria of mixtures (Peng and Robinson, 1976;Xu and Sandler, 1987a,b). This investigation concerns the generalization of these temperature-dependent parameters of Xu and Sandler (1987a,b) for the Peng-Robinson equation of state. We obtained this generalization of the pure component parameters by using experimental data for vapor pressure, saturated liquid, and vapor volumes in the subcritical region and P-V-T data in the supercritical region. Indeed, the availability of such data limited the number of fluids we could consider. Accurate data of the type we need are only available for the n-alkanes. In Table I, we show the fluids considered and the literature sources for the data used in this work. Generalization of t h e Parameters $, and $b The parameters and +b are related to the quantities a and b of the Peng-Robinson equation of state P=RT/(V-b)-a/[V(V+ b)+b(V-b)] (1)
pyb
ayb
1
1
a,Bup
pyP
0.665 232 06 0.271 445 16 0.668 137 67 0.276 723 17 -0.031 05697 -0.015 358 53 0.069 142 90 0.008 774 81 0.15055515 -0.02433520 0.60699946 0.11630427 0.01834382 1.14837562 0.62472868 -0.11871977 -0.679 795 92 -0.152 717 58 4 , 2 3 1 280 95 0.002 79099 0.248 737 05 0.129 149 68 0.464 292 58 -0.087 503 72
The use of these objective functions was found to result in unique values of the parameters and +b in both the subcritical and supercritical regions. At this stage of the calculation,essentially exact agreement for vapor pressures was obtained, and excellent agreement was obtained for vapor and liquid densities. Most of the errors result from the parameter generalizations which follow. Next, the values of the parameters obtained from the above procedure were fit as functions of reduced temperature and acentric factor. Equations of the same form, but with different values of the coefficients in the generalized correlations of the parameters il., and +b, were used in the subcritical region 0.6 5 T , 5 0.985 and in the supercritical region T, 2 1.015 where T I = TIT,. They are = [ c I + cp(1 - T,’/’) + ~ g ( -l Tr1/2)2]2 (8)
+,
+,
Ic/b
= [d,
dp(1 - T I )+ d3(l - Tr)’I2
19)
with
+,
c1
=
CY1
+ a2w
(101
c2
=
CY3
+ CY4w
(11)
c3
=
(Y5
+ (Ygw
(12)
and
as follows: a = +,R2T,2/Pc
(2)
b = +t.RTC/PC (3) Because the two-phase region only exists below the critical temperature, the procedure to obtain the optimal values of the parameters and +b in the subcritical region is different from that in the supercritical region. We obtained the values of the parameters at each temperature in the subcritical region as follows: Step 1. Guess initial values of +2 and +bo. Step 2. Keeping the value of +bo unchanged, find the for which the exDerimenta1and Dredicted vaDor value pressures Psatagree (actukly for which IFs)satBf(’ - PsatdcI/ p..+exp < 10-4). Step 3. Keeping the value of +,l from step 2 unchanged, find the optimal value +bl by minimizing the objective function (4) F2 = IAVsa,L/VsatLI + IAVsatv/VsatvI where AV = Vexp - Vcalc, and the superscripts L and V denote the liquid and vapor phases, respectively. Step 4. If the criteria for convergence - +,ol I 10-4 (5) and
+,
-PI
I$bl
-
$bel 5
(6)
are satisfied, then +,I and $bl are the final values for the two parameters. Otherwise, return to step 2 with $2 = $,’ and $bo +bl: In the supercritical region, the parameters were obtained a t each temperature and a large range of pressures by minimizing the objective function F3 = IAV/Vl (7)
+ P2w + PP
(13) (14) d2 = P3 (15) d3 = P 5 + P S w The coefficients aiand Pi, determined by using the method of least squares, are given in Table 11. The generalized correlation in the subcritical region is based on the eight components considered in Table I for which data were available over a large range of temperatures and pressures. The correlation in the supercritical region does not include n-hexane, n-heptane, and n-octane because of the lack of supercritical volume data covering a sufficientlylarge range of conditions for these components. In the transition region, 0.985 5 TI 5 1.015, the parameters and +b can be calculated by the cubic-spline function, which is continuous and smooth at the critical point. This method, suggested by Xu and Sandler (1987a,b) avoids discontinuities in properties in the vicinity of the critical point. dl =
P1
+,
Results The average absolute deviations (AAD) of the predicted values from experimental volumes and vapor pressures at saturation using the generalized parameters we propose are shown in Table 111. For comparison, we also show the deviations in predictions that result from the use of the original parameters of the Peng-Robinson equation of state. In Table IV,we show the comparisons of the average absolute deviations in predictions that result from the proposed generalized and original parameters in the supercritical region. From Table 111, we see that the error in the predictions for the vapor pressure resulting from the proposed parameters is slightly greater than that from the original
Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 1105 Table 111. Deviations between Calculated and Experimental Values
AAD, % saturated liquid volume
vapor pressure component methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane av
no. of pts 36 16 15 17 16 20 22 23
orig 0.63 1.05 0.80 0.26 0.30 0.60 1.01 1.10 0.72
Prop 0.56 1.15 0.98 0.92 0.79 0.57 0.48 1.26 0.84
Table IV. Deviations between Calculated and Experimental Volumes in Supercritical Region" no. of Dts AAD. % Methane Prop 464 3.64 PR 464 4.16 PR* 464 1.87 ~~
orig 7.19 6.76 5.15 4.71 4.27 3.09 4.32 5.61 5.14
Prop PR PR*
2.27 2.93 2.25
Prop PR PR*
Propane 256 256 256
2.39 2.46 2.35
Prop PR
n-Butane 48 48
3.79 4.35
Prop PR
n-Pentane 34 34
4.94 5.36
Prop 0.64 1.47 0.43 0.61 0.18 0.35 0.49 0.36 0.54
orig 1.57 1.82 1.04 1.02 1.04 1.73 1.46 3.20 1.61
Prop 1.87 2.52 1.54 2.97 1.59 2.61 2.41 5.14 2.58
25
I
- PROPOSED
~
j
-
-
ORIGINAL
w
5> 1 0
l5
Ethane 47 1 471 471
saturated vapor volume
1
G
0
!+-2 5 w >
O
prop = PR EOS with proposed generalized parameters; PR = PR EOS with original parameters; PR* = PR EOS with temperature-dependent parameters proposed by Xu and Sandler (1987a,b). a
parameters, while the error in the predictions of the saturated liquid volumes calculated from the proposed parameters is almost an order of magnitude less than that with the original parameters of Peng and Robinson. This is an important improvement. For example, in petroleum reservoir simulation, the liquid-phase densities must be accurately known in order to obtain reasonably accurate viscosities and surface tensions needed in mobility calculations. However, reservoir simulations are so time consuming that only simple cubic equations of state can be used. Consequently, it is not uncommon for hybrid methods to be used (a thermodynamically inconsistent procedure of using one method for the phase equilibrium calculation and another for the phase densities). The method and parameters proposed here eliminate this and result in more accurate predictions. As an example, we show the percent deviation of saturated liquid volume of n-heptane as a function of temperature in the subcritical region in Figure 1. Similar results were obtained for the other components studied in this work. The data in Table IV show that the error in the predictions for the volume in the supercritical region resulting from the use of the proposed parameters is also less than that from the original parameters. It should be pointed out that, while there is a large improvement in liquid density predictions, the saturated vapor densities are slightly worse than with the usual Peng-Robinson parameters. This is a result of the fact that simple cubic equations of state, such as the Peng-
5 -
Z
I o
-5
-_
_-
-__
'-
l
\
I I 05
06
0 -
08
C O
c
R E D U C E D T E M PE R A T UR E Figure 1. Predicted liquid volumes for saturated n-heptane.
Robinson one, are not of high accuracy; for example, 2, = 0.307 for this equation. Therefore, by fitting the parameters as we have here, a compromise between the accuracy of the vapor and liquid predictions results. However, the increase in error in the vapor density is far smaller than the large decrease in error of the liquid density. Perhaps it is worth pointing out how we envision the correlation here to be used. We see the main application to be to petroleum reservoir simulation, and other situations in which good accuracy for both the phase equilibrium and phase densities are needed, but because of the long computation times, only simple equations of state can be used. In reservoir simulations, fluids are typically represented by real components up to the heptanes and a collection of pseudocomponentsfor the heavier fractions. The heavy pseudocomponents are usually taken to be n-alkanes and are generally below their critical temperatures at reservoir conditions. To obtain good accuracy for such calculations, the fluid-specific, temperature-dependent parameters proposed earlier (Xu and Sandler, 1987a,b) should be used for the defined components and the generalized parameters here used for heavy pseudocomponents. It should also be noted that reservoir simulations occur at isothermal conditions determined by local geology. Therefore, the slightly more complex computation of the equation of state parameters required here is of little importance, since this calculation must be done only once, at the beginning of a simulation. Finally, the method proposed here should be compared to the procedure of Peneloux et al. (1982) to reduce liquid
1106 Ind. Eng. Chem. Res., Vol. 28, No. 7, 1989 density errors when using cubic equations of state. In that procedure, a volume translation is introduced, which does not affect the vapor pressure or calculated phase equilibrium but does reduce the saturated liquid density error to 0 at a single temperature, for example at a reduced temperature of 0.7. As is evident from Figure 1,this would reduce the liquid density error in the range of reduced temperature from 0.6 to 0.75 but have only a small affect outside this temperature range where the errors are largest.
Conclusion Generalized correlations in the subcritical and supercritical regions have been obtained for the temperaturedependent parameters $= and $b of the Peng-Robinson equation of state in terms of reduced temperature and acentric factor using data for a number of n-alkanes. While the deviations of vapor pressure predictions which result from the proposed generalized parameters are slightly greater than that obtained with the original parameters of the Peng-Robinson equation of state, the use of the correlations presented here results in considerably more accurate predictions for volumes in the subcritical and supercritical regions, which is important in many applications, such as reservoir simulation, and has been a major shortcoming of cubic equations. Further, the generalized correlations we propose are simple and easy to use and do not result in discontinuities in any properties in the critical region.
Acknowledgment We acknowledge the financial support to Dake Wu from the Ministry of Education of the People’s Republic of China and a generous grant of computer time by the Department of Chemical Engineering of the University of Delaware.
Nomenclature a, b = parameters of the Peng-Robinson equation of state P = pressure T = absolute temperature V = molar volume
Greek Symbols A = difference between calculated and measured properties 0 = coefficients of eq 10-15 $ a , qb = temperature-dependent parameters of the equation of state w = acentric factor
Subscripts c = critical properties r = reduced sat = saturated condition
Literature Cited Angus, S., Armstrong, B., de Reuck, K. M., Eds. International Thermodynamic Tables of the Fluid State, Methane; Pergamon: Oxford, 1978. Das. T. R.: Reed. C. 0..Jr.: Eubank. P. T. PVT Surface and Thermcdynahic Properties of n-Butane. J. Chem. Eng. Data 1973,18, 244-252. Das, T. R.; Reed, C. O., Jr.; Eubank, P. T. PVT Surface and Thermodynamic Properties of n-Pentane. J . Chem. Eng. Data 1977, 22, 3-8. Haman, S. E. M.; Chung, W. K.; Elshayal, I. M.; Lu, B. C.-Y. Generalized Temperature-Dependent Parameters of the RedlichKwong Equation of State for Vapor-Liquid Equilibrium Calculations. Ind. Eng. Chem. Process Des. Dev. 1977, 16, 51-59. Harmens, A. Program for Low Temperature Equilibriums and Thermodynamic Properties. Cryogenics 1975, 15, 217-222. Hsi, C.; Lu, B. C.-Y. Prediction of Gas Solubilities and Phase Equilibria of Normal Fluid Mixtures with One Supercritical Component. Can. J . Chem. Eng. 1971,49,134-139. Joffe, J.; Schroeder, G. M.; Zudkevitch, D. Vapor-Liquid Equilibria with the Redlich-Kwong Equation of State. AIChE J . 1970,16, 496-498. Morris, R. W.; Turek, E. A. Optimal Temperature-Dependent Parameters for the Redlich-Kwong Equation of State. ACS Symp. Ser. 1986, 300, 398-405. Peng, D.-Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59-64. PBneloux, A.; Rauzy, E. A Consistent Correction for RedlichKwong-Soave Volumes. Fluid Phase Equilib. 1982, 8, 7-23. Soave, G. Equilibrium Constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972,27, 1197-1203. Vargaftik, N. B. Handbook of Physical Properties of Liquids and Gases; Hemisphere Publishing Corporation: Washington, DC, 1983. Xu, Zhong; Sandler, S. I. Temperature-Dependent Parameters and the Peng-Robinson Equation of State. Ind. Eng. Chem. Res. 1987a, 26, 601-606. Xu, Zhong; Sandler, S. I. Application to Mixtures of the PengRobinson Equation of State with Fluid-SpecificParameters. Ind. Eng. Chem. Res. 1987b, 26, 1234-1238. Yarborough, L. Application of a Generalized Equation of State to Petroleum Reservoir Fluids. In Equations of State in Engineering and Research; Chao, K. C., Robinson, R. L., Jr., Eds.; Advances in Chemistry 182; American Chemical Society: Washington, DC, 1979; pp 385-439.
cy,
Superscripts calc = calculated value exp = experimental value L = liquid sub = subcritical region sup = supercritical region V = vapor
* Author to whom correspondence should be addressed. +On leave from Guizhou Institute of Technology, People’s Republic of China. Dake WU,+ Stanley I. Sandler* Department of Chemical Engineering University of Delaware Newark, Delaware 19716 Received for review September 29, 1988 Accepted March 6 , 1989
