I
C. J. PINGS, Jr.l,
and B. H. SAGE
California Institute of Technology, Pasadena, Calif.
Equations of State Volumetric behavior in liquid and gas phases and enthalpy changes can be predicted by use of a new equation of state. The second article shows the practical application of this equation to propane
An Application of Orthogonal Polynomials
MANY
z
equations of state have been proposed and it is with some hesitation that still another modification of existing expressions is suggested. The advent of automatic digital computers opened the possibility of using analytical expressions with many more coefficients than was feasible in the past. For practical purposes, it is desirable that an equation of state describe the volumetric behavior of a fluid in both the gas and liquid phases throughout the ranges of pressures and temperatures of interest. I n addition, the description of the chemical potential should be such that a reasonable prediction can be made of vapor pressure of pure substances and of the composition of coexisting phases in mixtures. The Benedict equation of state (9-72) approaches this objective for hydrocarbons and their mixtures somewhat better than other expressions previously proposed. However, this equation does not describe accurately the volumetric and phase behavior of hydrocarbons in many regions of industrial interest. Furthermore, it employs an exponential function which adds much to the difficulties of machine computation. A somewhat different approach has been undertaken by the
Present address, Department of Chem-
istry and Chemical Engineering, Stanford
University, Stanford, Calif.
authors to develop an analytical expression for describing the volumetric and phase behavior of hydrocarbons. A discussion of equations of state can conveniently be based on the exhaustive review by Beattie and Stockmayer (8). Pickering (37) has presented an extensive bibliography for work completed before 1925. Hirschfelder and coworkers (75) reviewed the features of a virial equation of state and presented the relations between the virial coefficients and the transfer characteristics of gases and liquids. The early work of van der Waals (45, 46) has been subjected to much refinement. Dieterici (20, 27) and Berthelot (73) supplemented the work of van der Waals, and their three expressions are usually considered the classical examples of equations of state. Onnes (33) proposed the application of polynominals and truncated series expansions to describe the volumetric behavior of liquids and gases. From early work reviewed by Pickering (37), Beattie and Bridgeman (2) developed a most useful equation of state which has been applied to many different gases with good success. The evaluation of the coefficients for ethane (#), n-butane (7), 1-butene (6),neopentane (3), and n-pentane (5) are examples of the application of the Beattie-Bridgeman equation of $ate to hydrocarbons. Beattie and Stockmayer (8) summarized the applications of this equation, which apparently is satisfactory for nearly all nonpolar gases a t
pressures up to about 4000 pounds per square inch. Redlich and Kwong (39) proposed a relatively simple equation of state primarily directed for evaluating the chemical potentials of the components of gaseous mixtures. The latter expression is simple but it does not permit as accurate a description of the volumetric behavior of gases as that obtained with the Beattie-Bridgeman equation of state. Peek (36) and Zwanzig (48) proposed equations of state limited in application to gases at high temperatures. Such equations have particular utility in describing the behavior of gases in guns and in shock waves. Martin and Hou (30) also suggested an equation of state for gases, and Himpan proposed an expression to describe the volumetric behavior of gas and liquid phases (24, 25). Benedict and coworkers (9-72) contributed an equation of state which was effective in describing the properties of hydrocarbons. They developed an expression based in part upon the earlier work of Beattie and Bridgeman (2) and presented a method of evaluating the empirical coefficients. This equation with Benedict’s coefficients (9-72) describes both the dependence of vapor pressure on temperature and the volumetric behavior of hydrocarbon gases at pressures u p to approximately 4000 pounds per square inch. I t does not quantitatively describe the volumetric behavior of condensed liquids nor does VOL. 49, NO. 8
AUGUST 1957
13 1 5
it yield an accurate representation in the vicinity of the critical state. Application of the Benedict equation to the prediction of volumetric behavior of liquids and gases has been extended to pressures of 10,000 pounds per square inch by Opfell and others (35, 47). However, when it was extended to this pressure, the equation would not predict the phase behavior and at the same time describe the volumetric behavior of the liquid and gas phases. An equation with a larger number of empirical coefficients than that employed by Benedict is necessary to describe adequately the volumetric and phase behavior of hydrocarbons and their mixtures throughout the range of temperatures and pressures of interest to the petroleum industry. In developing the application of statistical mechanics to the prediction of the general form of an equation of state, Ursell (44) made one of the more fruitful of the early efforts. His work was revived by Mayer (37), and since that time many workers have devoted substantial effort to this field. Lee and Yang (28, 47) contributed to the application of statistical mechanics to equations of state by their treatment of the two-phase region. The work of van Dranen (22) and de Boer (76) are examples of the widespread recent interest in this field.
bility factor Z within the area about the origin of 1 / v which is free of singularities of Z (28, 47). A number of the existing equations of state may be the modified or truncated forms of the virial equation. The van der Waals equation (45, 46) bears a. similarity to a viral expansion truncated after two terms.
The Virial Equation of State
I n this expression, terms involving the third and fourth powers of reciprocal volume were omitted and an exponential term was used. There does not appear to be any theoretical reason for these modifications which apparently were made in the interest of economy of coefficients. The virial coefficients in Equation 4 are markedly interdependent. The Benedict equation of state appears to describe the volumetric and phase behavior of pure hydrocarbons and their mixtures with greater accuracy than other equations. Martin and Hou (30) proposed a modified form of a virial equation which is set forth in Equation 5.
Early work involving the application of statistical mechanics for developing a n equation of state was summarized by Beattie and Stockmayer ( 8 ) . Much effort has been expended IO evaluate the phase integral utilizing the LennardJones (29) potential. Sufficient progress has been made to establish the effect of temperature upon the coefficients of the so-called "virial" equation of state. At present, the mathematical analysis of such temperature dependence cannot be extended to include a quantitative description of the phenomenon of condensation. However, Mayer (32) suggested some approximate methods of approach to that problem. The qualitative effect of temperature on the condensation phenomenon was predicted by Lee and Yang (28, 47) by an extension of Mayer's analysis to the complex plane. For present purposes, the virial equation of state may be written in the following form in which the coefficients B, C, and D,are pure temperature functions :
-_ ''
- 2
RT
=
1
+ B ( T ) Y1- + C ( T ) P -1 f . . . (1)
The first virial coefficient is unity. The second is identified as B, the third as C, and so forth. This infinite series converges uniformly to the compressi-
13 1 6
The Beattie-Bridgeman equation of state (2) may be written in the following form:
I t involves four virial coefficients \vhich are interdependent, since all are made from combinations of the five parameters in the equation. Benedict, Webb, and Rubin (9-72; proposed an equation of state which may be written in the following virial form :
cients will not correspond to the true virial coefficients. The arbitrary form and the truncation of these expressions largely preclude direct association Ivith the virial equation of state. In view of the interest in virial coefficients for the prediction of volumetric behavior (8) and transport characteristics (74, 75, 26, 27), few have been evaluated experimentally. However, from four to eight virial coefficients are required to describe accurately the volumetric behavior of a pure substance in the liquid, gas, and critical regions. In addition, at least three parameters are necessary to describe the temperature dependence of each virial coefficient. Under these circumstances at least 12 coefficients must be evaluated from experimental data and probablv as many as 30 are required to obtain d satisfactory description of experimental volumetric data in the liquid and gas phases. For an equation with seven virial coefficients with three constants to describe the temperature dependence ot each coefficient, the solution of a matrix involving 841 elements is required to establish the optimum value of each of 21 temperature constants in the least squares sense (79). Experience in establishing coefficients by least squares methods for the Benedict equation (77. 34,47) indicates that many of the matrices are ill-conditioned (23), thus imposing practical difficulties in the accurate evaluation of the coefficients. In using polynominal approximations of the virial series, the values obtained for the coefficients depend on the point of truncation of the series. Two different truncations will in general result In two completely different sets of coefficients. The object of the present investigation was to evolve an equation of state which was an open-ended series in temperature and reciprocal mola! volume, but Lvhich would eliminate the bulk of the computational difficulties encountered with the virial equation of state or a n orthodox power series expansion. The results were such that the equation of state can be regrouped into the virial form, thus maintaining the advantages of association with developed theory. Variables
The equation, expressed in terms of the variable - b rather than can be expanded and expressed in the true virial form but the coefficients become complicated. Martin and Hou (30) used exponential functions to describe temperature dependence. Although the foregoing equations of state bear a formal resemblance to the virial equation, nevertheless the numerical values obtained for the coeffi-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
v
v,
The choice of variables to use in an equation of state is often difficult. If the conventional variables of pressure. temperature, and reciprocal molal volume are employed. the equation may br more complicated than required. Experience (7, 43) with expressions involving fugacity, enthalpy, and entropy indicates the utility of a derived variable in simplifying the integral expressions associated with determining the variation in these thermodynamic properties with
E O U A T I O N S O F STATE changes in state. In the present instance, a derived function, defined in the following way, was useful.as the independent variable of the equation of state:
P
The function L is related to pressure by the following expression : P
R T [ p L ( a ,T )
+ 11.RTLU' + RTU =
(7)
In developing an equation of state, L was conveniently expressed as a function of the reciprocal molal volume and temperature. The pressure of compressibility factor can be established from the function L by application of Equation 6 or 7. This function is finite at infinite attenuation in much the same fashion as is the residual volume (40). The function L has finite values in the two-phase region and it may exhibit a minimum at twice the vapor pressure. Form of Equation of State
The possible simplification in treatment resulting from the use of orthogonal polynomials has been recognized (78, 42). They are limited in their application to volumetric data since the range of reciprocal molal volumes must be the same for each temperature investigated. I n other words, the data must cover a rectangular domain in the plane of the interdependent variables, temperature and reciprocal molal volume. I n some instances the experimental data are obtained in this form, but usually such is not the case. However, by extrapolation the data can be arranged in a form satisfying this requirement. Such an extrapolation in no way vitiates the applicability of the equation to the region of temperature and molal volume where experimental data are available, and should be regarded only as an aid LO the analysis. The Tchebichef polynomials here identified as T,(x) are usually recorded for values of the argument between -1 and + l . For this reason a normalized reciprocal molal volume function was defined in terms of the maximum reciprocal molal volume.
X
Figure 1 .
Tchebichef polynomials as functions of their argument
For many situations in which experimental data are available a t evenly spaced temperature intervals and the experimental uncertainty in reconciling the measured temperatures to the international platinum scale can be neglected, the Gram polynomials (78, 42) can be used to describe the temperature dependence of L(p, T ) . For the case of h equally spaced isotherms, the appro-
priate normalizing function expressed (38) as
can
be
In this equation of state the derived function L ( u , T ) was expressed in terms of the reciprocal molal volume and temperature. Two types of expansions were considered. I n both, the depend-
75
50
25 h
E 9
0
-25
-50
I n one form of the equation of state, a series of Tchebichef polynomials was employed to describe the temperature dependence of the function L (u, T ) . The normalized temperature function was written as
-75
-4
-2
0
4
2
E Figure 2.
Gram polynomials as functions of their argument VOL. 49, NO. 8
0
AUGU5T 1957-
7377
ence of L(fr, T ) on molal volume was expressed in a series of Tchebichef polynomials. The temperature dependence was expressed in one case by a series of Tchebichef polynomials and in the second case by a series of Gram polynomials. The designations TchebichefTchebichef and Tchebichef-Gram were selected for these two forms of the equation of state. If the equation of state is to be applicable to the heterogeneous region. then the general relations of thermodynamic equilibrium must be satisfied. From the resulting equality of the chemical potential, pressure, and temperature in the coexisting liquid and gas phases it follows that:
Equation 11 is a restraint which must be satisfied at all temperatures if the resulting equation is to predict vapor pressure satisfactorily. Orthogonal Polynomial Form of Equation of State The preceding brief discussion has been supplemented by more complete mathematical considerations (38). Several of the Tchebichef and Gram polynomials are shown as functions of their arguments in Figures 1 and 2. In applying these polynomials to the equation of state it was assumed that the independent variables were normalized as described and that the derived function L ( u , T ) was adjusted so that the functional surface is continuous and smooth and satisfies Equation 11 in the two-phase region. For one form of the equation, the reciprocal molal volume dependence was expressed in Tchebichef polynomials of ascending order and the temperature dependence in Gram polynomials of ascending order. The normalized reciprocal molal volume, x , was the argument of the Tchebichef polynomials and the normalized temperature, e, was the argument of the Gram polynomials. Under these circumstances the equation of state assumed the form
In a similar way the equation may be written entirely in terms of the Tchebichef polynomials. Then the normalized reciprocal molal volume, x , and the normalized temperature, y, are the arguments of the polynomials. Under these conditions the equation assumed the form UP, T) =
z-1
- = cooTo(y)To(n) P
-t- co1To(y)T1(x)
+
Pressure :
P S RTU
+ RTuZc * i j T & ) T r ( ~ )(16)
Fugacity :
+ + .. .
cloTl(y)To(x) CIlTl(Y)Tl(X)
+ coaTo(y)Tz(x) -t. . . +...
-t-
..
~
Enthalpy :
(13)
This equation is an infinite series with respect to both the normalized reciprocal molal volume and the temperature. Equation 12 or 13 may be truncated to yield any requisite degree of accuracy of description of the experimental data. The number of terms required can be established only by applying the experimental data for a particular substance. Slightly different nomenclature was used for the coefficient of *the Tchebichef-Tchebichef and the Tchebichef-Gram form. This differentiation was employed to avoid confusion in referring to the two equations. For convenience, a summation convention has permitted the use of abbreviated forms of Equations 12 and 13. If the range of each of the subscripts, i and j , is taken from zero to infinity, and repeated subscripw utilized to indicate summation range it follows that: L(P,
and temperature the following expressions apply:
T)=
~ i j V < ((€)Ti ~ ) (x) 2 a*,;Vi(")(e)T,(x) (14)
and L ( u , T ) = cijTi(y)Tj(x) G C*ijTiCy) T j ( X )
(15)
Equations 14 and 15 are abbreviations of Equations 12 and 13. The second approximate equality indicated in Equations 14 and 15 represents the approximation realized with a finite number of terms chosen for a particular application. I n this instance the asterisks on the coefficient tensors indicate that the number of terms in the polynomials with respect to reciprocal molal volume and temperature is finite. The order, h, of the grid spacing for the Gram polynomials must be greater than the number of terms employed (38)
Entropy:
In Equations 16 to 19 the normalized reciprocal molal volume and temperature as defined by Equations 8 and 3 were employed. The maximum values of reciprocal molal volume and the maximum and minimum temperaturfs must be precisely those used in the evaluation of the coefficients. In other words, the same normalization function must be employed in applying the equation as was used in the evaluation of the coefficients. Similar relations can be derivfd for the Tchebichef-Gram form of the equation of state. Expressions for other thermod! namic quantities than the four presented in Equations 16 to 19 can be evaluated. Several of the foregoing expressions involve integrals and derivatives of the polynomials. In the caSe of the Gram polynomials the only convenient method appears to involve a direct operation upon the polynomials, the details of which are available (
