Prediction of the Enthalpy of Petroleum Fractions. The Pseudocompound Method Peir K. Huang and Thomas E. Daubert* Department of Chemical Engineering, 165 Fenske Laboratory, The Pennsylvania State University, University Park, Pa. 16802
Undefined hydrocarbon mixtures or petroleum fractions can be characterized by considering them as mixtures of pure hydrocarbons called pseudocompounds for each molecular type present and for which data are available in the boiling range of the fraction. These pseudocompounds for each molecular type can then be used to predict enthalpies using methods available for defined mixtures. The proposed method calculates total enthalpies by combining the ideal gas enthalpies with the effects of pressure as predicted by the corresponding states method of Pitzer and reproduces the experimental data tested to within 3 BTU/lb for both liquid and vapor enthalpy. When compared with existing methods this method shows errors less than half as large. Preliminary additional work has shown that the method is also applicable to the prediction of heat capacities of undefined mixtures.
Introduction Accurate enthalpy values for petroleum fractions are important in designing petroleum refining and related processes. As experimental data are not available, a reliable prediction method is needed. A recent survey of existing methods for predicting petroleum fraction enthalpies culminating in the publication of American Petroleum Institute Technical Data Book-Petroleum Refining (1970) recommends that the Johnson-Grayson (1961) method was probably the best available method for estimating enthalpy values. Johnson and Grayson used Watson K and API gravity as correlating parameters and presented a set of four charts (each a t a different Watson K ) to correlate enthalpies. The Watson K was defined by
K=
(Me ABP)* / 3 SpGr, 6 0 F / 6 0 F
drocarbon types (n-paraffins, n-alkylcyclopentanes, n - l olefins, and n-alkylbenzenes) a t different temperatures. From this figure a linear relationship can be derived for each type at definite temperature if the boiling point range is not too large. This relationship can be expressed by
H" = ffTb
Me ABP =
+
CABP
=
C
(OTbi
+
B)Xwi
(3
i.1
2
2
(2)
n
Homix
U
MABP =
/3
where H" is the ideal gas enthalpy of a pure hydrocarbon in Btu/lb (reference point: H" = 0 Btu/lb a t O"R), T b is the boiling point of a pure hydrocarbon in O F , and a and fl are coefficients, depend on the hydrocarbon type and temperature. The ideal gas enthalpy of a mixture of pure hydrocarbons of the same type a t definite temperature assuming no enthalpy of mixing will be
where MABP
+
xmiTbi
i=l
where W m i x is the ideal gas enthalpy of the mixture in Btu/lb, Tbi is the boiling point of component i in O F , x w L is the weight fraction of component i, and n is the number of components. Equation 3 can be rearranged to
/
and x m L = mole fraction of component i, x u & = volume fraction of component i, and TbL= normal boiling point of component i. The average boiling points were correlated in an empirical plot based on the ASTM D86 distillation curve. Enthalpy values predicted from this method were no better than * l O Btu/lb, and in the pseudocritical range, errors exceeding 20 Btu/lb resulted. The object of this work (Huang, 1973) was to develop a method of improved accuracy. It follows the basic idea that if a petroleum fraction can be properly described by pure compounds (called pseudocompounds) which characterize each of the molecular types present in an undefined mixture, prediction methods for defined mixtures can be applied directly to petroleum fractions. Development of t h e Pseudocompound Method T h e Ideal G a s Enthalpies of Petroleum Fractions. A compilation of ideal gas enthalpy data of hydrocarbons was made by American Petroleum Institute Research Project 44 (1972). Figure 1 shows the relationship between the ideal gas enthalpy and the boiling point for four hy-
where the term
is the weight average boiling point (WABP) of the mixture. If eq 4 is expressed in terms of WABP, then
Homix = a(WABP)
+
p
(5)
Equation 5 indicates that mixtures of the same molecu'lar type will have the same ITrnlx a t the same temperature so long as they have the same WABP. In other words, the ideal gas enthalpy of a mixture which was composed of pure compounds of the same type can be calculated by combining two pure compounds of the same type as long as the combination leads to the same WABP. As these two pure compounds do not necessarily exist in the actual mixture, they are called pseudocompounds. As was mentioned before, this approach is applicable when the boiling point range is not too large. Thus, the pseudocompounds should be chosen as close as possible to the WABP of the mixture-one higher and one lower. Ind. Eng. Chem., Process Des. Develop., Vol. 13,No. 4 , 1974
359
e ' 800 O O
parture from ideal gas (Btu/lb mole), and R is the ideal gas constant (Btu/lb mol O F ) . The acentric factor is defined as
0 r - - - - - 7
P* w = - log- - 1.0 PC
l
5 400-
!
v)
2-
200
/
. ?/--
-E
Oo F
C
I
I
200
400
BOILING POINT,
I
600
800
(7)
where P* is the vapor pressure (psia) and Pc is the critical pressure (psia). To obtain the "zero" and "one" terms of eq 6 a double interpolation is required with respect to the reduced temperature and reduced pressure. For reduced pressures smaller than 0.2, the generalized virial equation given by Pitzer can be employed.
c " D -
O
(T, = 0.7)
(p%H)
= - P,[(0.1445
+
O F
Figure 1. Ideal gas enthalpies of hydrocarbons us. boiling points: A, n-paraffins ( C 6 - C z 0 ) ; B, n-alkylcyclopentanes (C6-cZO); C, n1-olefins (C6-Czo); D, n-alkylbenzenes(Cs-Czo).
0 . 0 7 3 ~ )- (0.660.- 0 . 9 2 ~ ) T , - ' (0.4155
+ 1.50~)T,-'
- (0.0484
+ 3 8 8 ~ ) T , ' ~-
0 . 0 6 5 7 ~ T , - ~ ] (8) Since the boiling point of each pseudocompound is known, the weight fraction of each pseudocompound is also defined by the requirement of the same WABP. The development to this point is limited to a single hydrocarbon type although petroleum fractions usually contain different types. It can be assumed that the WABP of each type will be the same as the WABP of the fraction. A further assumption is made that the WABP, VABP, and 50% boiling point (ASTM D-86) of the fraction are approximately the same. Thus, the characteristic boiling point, WABP, can be replaced by VABP or directly by the 50% boiling point on a n ASTM distillation curve. If the composition of each type in the fraction is known, then the pseudocompounds and their compositions are completely defined. By the combination of these pseudocompounds the ideal gas enthalpy of the fraction is easily calculated. For practical applications, the four molecular types in Figure 1 can be chosen to represent paraffins, naphthenes, olefins, and aromatics. Although the composition of each pseudocompound should be in weight per cent to calculate H"mix, Btu/lb, little difference is expected if liquid volume per cent is used instead of weight per cent. The Isothermal Effects of Pressure on Enthalpies of Petroleum Fractions. The pseudocompounds for a petroleum fraction were defined in the calculation of ideal gas enthalpies. In order to calculate total enthalpy the effect of pressure must be correlated simultaneously. The Pitzer (Curl and Pitzer, 1958) approach was employed for this part of the work. For pure compounds, Pitzer, basing his work on the principle of corresponding states, treats the enthalpy departure from the ideal gas by tabulating
as functions of the reduced temperature and the reduced pressures. The following relationship defines the enthalpy function of a substance.
(at a given Tr and P,) where w is the acentric factor for the material, T , is the reduced temperature ( T / T c ) ,T and Tc are the temperature and critical temperature ( O R ) , Pr is the reduced pressure (PiPC),P and Pc are the pressure and critical pressure (psia), A" - H is the enthalpy de360
Ind. Eng. Chern., Process Des. Develop., Vol. 13, No. 4, 1974
The enthalpy departure from an ideal gas on a unit mass basis is
where M is the molecular weight of the material and H" H i s the enthalpy departure from ideal gas (Btu/lb). For defined mixtures, the Pitzer correlation is valid if Tc, Pc, and w are replaced by pseudoproperties, TPC,Ppc, and up.
where Tpc is the pseudocritical temperature Ppc is the pseudocritical pressure (psia), w p is the pseudoacentric factor, Xmi is the mole fraction of component i, T o is the critical temperature of component i ( O R ) , P,, is the critical pressure of component i (psia), oL is the acentric factor of component i. If a specific quantity is needed, M in eq 9 will be the molecular weight of the mixture. For petroleum fractions defined by several pseudocompounds, the same procedure can be applied as for defined mixtures. However, in this case, M will not be the molecular weight of the actual fraction. Since the pseudocompound method was developed based on specific quantities rather than on molar quantities, only the specific enthalpy of the pseudomixture is equivalent to that of actual petroleum fraction. There is no reason to say that the molar quantity has to be the same. Therefore, M should be the molecular weight of the pseudomixture. In the calculation of ideal gas enthalpies for petroleum fractions, weight fraction or liquid volume fraction is used. In the calculation of pressure effects, mole fraction is used. Thus, the following conversions are required ( O R ) ,
where xwi is the weight fraction of component i, xv, is the liquid volume fraction of component i, Xmi is the mole fraction of component i, pi is the liquid density of component i, M , is the molecular weight of component i, p is the liquid density of the mixture, and M is the molecular weight of the mixture. In practical applications, the method can often be simplified using volume fraction throughout the course of calculation. For example, if liquid volume fraction is the only information available
Equations 18-21 are used in the Pitzer correlation for the isothermal effect of pressure on the enthalpies of petroleum fractions. The original Pitzer table ranged from 0.8 to 4.0 in T,. The lower reduced temperature limit has been extended to 0.35 using the work of Chao and Greenkorn (1971). The maximum reduced pressure is 9.0 when Tr > 0.8 and 3.0 when Tr < 0.8. Calculation Procedure. The calculating procedures using the pseudocompound method are outlined below. 1. A data bank which includes boiling point, molecular weight, liquid density, critical temperature, critical pressure, and acentric factor for four hydrocarbon types (nparaffins, n-alkylcyclopentanes, n-1-olefins and n-alkylbenzenes) is prepared. 2. The ASTM distillation data and molecular type analysis (paraffins, naphthenes, olefins, and aromatics) of the petroleum fraction are obtained. 3. The 50% boiling point from ASTM distillation data is taken as the characteristic boiling point. Two pseudocompounds are defined for each molecular type, one having a boiling point higher than the 50% boiling point, the other having a boiling point lower than the 50% boiling point. 4. The compositions of these two pseudocompounds in each homologous mixture are determined by defining a characteristic boiling point equal to the 50% boiling point. The compositions so determined are in weight per cent. It is assumed that weight per cent and liquid volume per cent are equal. 5 . Multiplying the above compositions by the compositions of corresponding molecular type determined from molecular type analysis, the amount of each pseudocompound in this mixture is defined. 6. If rigorous composition conversion is needed, eq 13, 14, 15, and 16 are applied. Otherwise volume per cent can be used for simplicity throughout the course of calculation wherever composition factors are needed. 7. The ideal gas enthalpy for each pseudocompound is obtained from API 44 tables or calculated from the Passut-Danner (1972) polynomials extended by Huang and
Daubert (1974). Using the composition the ideal gas enthalpy of the petroleum fraction is determined. 8. The isothermal pressure effect on the enthalpy of the petroleum fraction is calculated by the Pitzer correlation. The parameters needed in the Pitzer correlation are calculated rigorously by eq 10, 11, and 12 or 18, 19, and 20 for simplicity. Since specific enthalpy is required eq 9 is used after obtaining the molar enthalpy from the Pitzer table. Molecular weight can be calculated by mole composition or simply by volume composition (eq 21). For vapors, if Pr is smaller than 0.2 eq 8 is used. 9. Combining ideal gas enthalpy value with the pressure effect, the total enthalpy of the petroleum fraction is obtained. The enthalpy is always a difference relative to some reference state. Therefore, the same procedure is required to obtain the value a t the reference state. The difference between actual state and reference state is the value of interest. A sample calculation is given in the supplementary material included in the microfilm edition of the journal. See the paragraph at the end of the paper regarding this supplementary material.
Results and Discussion The Lenoir and Hipkin (1973) enthalpy data set was used to test the pseudocompound method. The data comprised six naphthas, one kerosene, one fuel oil, and one gas oil with compositions ranging from 0 to 60 vol % paraffins, 0 to 70 vol % naphthenes, 0 to 17 vol % olefins, and 5 to 83 vol 70 aromatics. Detailed characterization of the petroleum fractions is given in Table 1 of the supplementary material available in the microfilm edition. In this data set, most of the enthalpy values are based on a reference state of 75°F saturated liquid although some are based on a reference state of 75°F and the pressure o f ' measurement. In the latter case data have been corrected to the former reference state. Thus, all calculated values will be based on a 75°F saturated liquid reference. The results of calculation by the pseudocompound method using simple composition for the liquid and vapor enthalpy show that for liquid enthalpy the average error for simple composition is 2.6 Btu/lb us. 3.1 Btu/lb for converted composition while for vapor enthalpy 2.8 and 2.9 Btu/lb errors, respectively, were obtained. Thus, the results using converted composition have been found to be equivalent or even slightly inferior, and the simple composition (volume per cent) can be used for most practical applications. For liquids the method correlated 633 data points, of the total of 729, with a n average error of 2.6 Btu/lb. Most of the data points which cannot be correlated are due to the range of applicability of the Pitzer table (0.1 5 P, 5 3.0 when Tr < 0.8) which limits its usefulness in the highpressure regions. For vapors the method correlated almost all of the 332 data points with an average error of 2.8 Btu/ lb. The data points which cannot be correlated fall into the liquid region in the Pitzer table. For comparison purposes, the same data set was taken to test the Johnson-Grayson method which correlated 551 data points (of the 729) with an average error of 6.0 Btu/ lb for liquids and all of the 332 data points for vapors with a n average error of 9.6 Btu/lb. Tables 2 and 3 of the supplementary material list for each petroleum fraction the number of data points both available and evaluated and the average, maximum, bias, and root-mean-square errors for each fraction and the total data set for the simple composition method for liqInd. Eng. Chem., Process Des. Develop., Vol. 13, No. 4, 1974
361
uids and vapors, respectively. Similar data for use of the Johnson-Grayson method are given for liquids and vapors in Tables 4 and 5, respectively. As shown, the pseudocompound method gave marked improvement over the Johnson-Grayson method for predicting the enthalpy of petroleum fractions. Further work has also shown its superiority in predicting the heat capacity of petroleum fraction vapors. This leads to the potential conclusion that the pseudocompound approach can be extended to correlate other thermodyfiamic properties of petroleum fractions, such as heat capacity, entropy and fugacit.y, and may also be applicable to prediction of the transport properties-viscosity and thermal conductivity. Literature Cited API Division of Refining, "Technical Data Book-Petroleum Refining," Chapter 7 , 2 n d ed, New York. N. Y . , 1970. API Research Project 44, "Selected Values of Properties of Hydrocarbons and Related Compounds," Tables of Physical and Thermodynamic Properties of Hydrocarbons, A and M Press, College Station, Tex., 1972.
Chao, K. C., Greenkorn. R. A., Proceedings 50th Annual Convention, NGPA. pp 42-46, Tulsa, Okla.. 1971. Cur!, R. F., Jr.. Pitzer. K. S., lnd. €ng. Chem., 50, 265 (1958). Huang, P. K., M. S. Thesis, T h e Pennsylvania State University, University Park, Pa., 1973. Huang, P. K.. Daubert, 1. E., lnd. Eng. Chem., Process Des. Develop., 13, 193 (1974). Johnson, R. L., Grayson, H. G., Petrol. Rehner. 40 ( 2 ) , 123 (1961). Lenoir, J . M., Hipkin, H . G..J. Chem. Eng. Data, 18, 195 (1973). Passut, C. A,, Danner, R. P., Ind. Eng Chem.. Process Des. Develop., 11. 543 (1972).
Receiuedfor review December 31, 1973 Accepted April 22,1974 Supplementary Material Available. A sample calculation using the pseudocompound method will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 X 148 mm, 24X reduction, negatives) containing all of the supplementary material for the papers in this issue may be obtained from the Journals Department, American Chemical Society, 1155 16th St., N.W., Washington, D. C. 20036. Remit check or money order for $3.00 for photocopy or $2.00 for microfiche, referring to code number PROC-74-359.
Activity Coefficients at High Concentrations in the Hydrochloric Acid-Sodium Chloride-Water System Edward W. Funk' Universidad
Tecnica del Estado, Santiago. Chile
Semiempirical equations are proposed to describe the activity coefficients at high concentrations in the HCI-NaCI-H20 system; these equations complement the Bronsted-Guggenheim treatment of dilute solutions. The proposed equations for the activity coefficients of the electrolytes contain two parameters determined using activity coefficient data for the ternary system. One parameter is independent of the total molality and the other decreases systematically with molality; both are only weak functions of temperature. The activity coefficient of water in the ternary system is well described using only data for the binary subsystems. The activity coefficients have been defined similarly to those of nonelectrolytes; however, they are easily related to the familiar mean ionic activity coefficients often used for electrolyte solutions. An important advantage of these new activity coefficients is that they allow a simple description of phase equilibria involving electrolytes
Introduction Activity coefficients in concentrated aqueous electrolyte solutions are necessary for a great variety of chemical, metallurgical, and geological problems. Although there is a large number of experimental data (Harned and Owen, 1958; Robinson and Stokes, 1970; Seidell, 1965) for ternary aqueous electrolyte systems, few equations are available to correlate the activity coefficients of these systems in the concentrated region. The most successful present methods are those discussed by Meissner and coworkers (Meissner and Kusik, 1972, 1973; Meissner and Tester, 1972; Meissner, et al., 1972) and Bromley (1973). For the development of a thermodynamic framework especially suited to concentrated systems, there are complete and precise thermodynamic data for the HC1-NaC1HzO system and its constituent binaries from very low molality up to highly concentrated solutions. Figure 1 shows the mean ionic activity coefficients of HC1 and NaCl (Robinson and Stokes, 1970) as functions of the mo-
' Corporate
Research Laboratory, Exxon Research and Engineering,
Linden, N. J. 07036 362
Ind. Eng. Chern., Process
Des. Develop., Vol. 13, No. 4 , 1974
lality at 25°C in their respective binary solutions, and also the activity coefficient of each electrolyte at infinite dilution in the ternary solution. In dilute solutions, the activity coefficients of the electrolytes change only slightly with composition at constant total molality; however, this change becomes large in concentrated solutions. For numerous applications, it is important to describe the composition dependence of these activity coefficients using a minimum of data for the ternary system. It is very difficult to describe accurately the activity coefficients in the above system from low concentrations up to highly concentrated or saturated solutions. Fortunately, the region below total molalities of 0.2 is already well described by the Bronsted-Guggenheim theory. Harned's rule is usually used for the concentrated region although, unfortunately, the ternary parameters in Harned's rule depend on both temperature and total molality; often it is difficult to extrapolate the parameters to new conditions. The proposed equations for the activity coefficients of the electrolytes contain ternary parameters that are easily extrapolated to unstudied conditions of temperature and
