THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS Three Park Avenue, New York, N.Y EQUATIONS OF STATE FOR GAS COMPRESSOR DESIGN AND TESTING

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1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS Three Park Avenue, New York, N.Y CT-12 The Society shall not be responsible for statements or opinions advanced in papers or discussion at meetings of the Society or of Its Divisions or Sections, or printed in its publications. Discussion Is printed only if the paper is published In an ASME Journal. Authorization to photocopy for internal or personal use is granted to libraries and other users registered with the Copyright Clearance Center (CCC) provided S3/article is paid to CCC, 222 Rosewood Dr., Danvers, MA Requests for special permission or bulk reproduction should be addressed to the ASME Technical Publishing Department. Copyright by ASME M Rights Reserved Printed in U.S.A. EQUATIONS OF STATE FOR GAS COMPRESSOR DESIGN AND TESTING ) III III Sumit K. Kumar" Rainer Kurz' John P. O'Connell' 'Solar Turbines Incorporated San Diego, CA Department of Chemical Engineering University of Virginia, Charlottesville, VA ABSTRACT In the design and testing of gas compressors, the correct determination of the thermodynamic properties of the gas, such as enthalpy, entropy and density from pressure, temperature and composition, plays an important role. Due to the wide range of conditions encountered, pressure, specific volume and temperature (pv-t) equations of state (LOS) and ideal gas heat capacities, along with measured data, are used to determine the isentropic efficiency of a compressor configuration and to model the actual behavior of real gases and compressors. There are many possible model choices. The final selection should depend on the applicability of the EOS to the gas and the temperature dependence of the heat capacities, as well as the particular process of interest along with the range of pressures and temperatures encountered. This paper compares the thermodynamic properties from five commonly used equations in the gas compressor industry the Redlich-Kwong (RK), Redlich-Kwong-Soave (RI(S), Peng-Robinson (PR), Benedict-Webb-Rubin-Starling (BWRS), and Lte-Kesler- Plocker (LICP) models. It also compares them with a high accuracy LOS for methane from Wagner and Setzmann in the common range for gas compressors. The validity of a linear temperature dependence for ideal gas heat capacities is also evaluated. The objective was to determine if the models give significant differences in their predicted efficiencies. It was found that different LOS gave somewhat different enthalpy changes for methane, ethane and nitrogen for real compressions. This appeared to be connected to the different densities given by the models. Interestingly, the isentropic enthalpy changes are quite similar, suggesting that the effect is canceled out when two properties are involved. However, since the efficiency is the ratio of isentropic enthalpy change to actual enthalpy change, the LOS yield different efficiencies. These differences are on the same order as the typical tolerances allowed for prediction and testing of industrial gas compressors (3 to 5%) and comparisons with the highly accurate equation of state for pure methane from Wagner and Setzmann (1991) showed similar differences. Commonly, the ideal gas heat rapatity is assumed linear in temperature from 10 to 150 C (50 to 300 F). Comparison of this form with a quadratic expression from the literature and the highly accurate equation of Wagner and Setzmann for methane, showed insignificant differences among the methods for temperatures up to 600 K (1080 R). NOMENCLATURE A,B,C,D a,b cro 4 MW PCT PCP RG SG Constants Cubic LOS Parameters Ideal Gas Specific Heat at Constant Pressure Head Enthalpy Binary Parameters Molecular Weight Pressure Power Pseudo Critical Temperature Pseudo Critical Pressure Volumetric Flow Specific Gas Constant Specific Gravity Entropy Temperature Specific Volume Presented at the International Gas Turbine & Aeroengine Congress & Exhibition Indianapolis, Indiana June 7-June 10, 1999

2 X,Y Molar Volume Mass Flow Mole Fraction LOS Function Compressibility Factor Density lsentropic Efficiency Pitzer Acentric Factor Subscripts Critical Mixture Reference Isentropic INTRODUCTION The operating conditions for gas compressors are typically defined in terms of pressures, temperatures and mass or standard flows (Fig. 1). The data that can be used to describe the aerothermodynamic behavior of the compressor are head, efficiency and actual flow. To construct a relationship between the former and the latter set of data, LOS can be used. A study was conducted to determine how LOS may differ in predicting actual and isentropic enthalpy changes as well as entropy changes between common states for typical natural gas mixtures. The goal was to assess the extent and cause of any differences in the LOS results that could be of great importance for the design, performance prediction, and testing of gas compressors. The focus is on the description of enthalpy and entropy differences which seem not to have been systematically examined even though there are many studies treating the compressibility factor (Beinecke and Luedtke, 1983) and phase equilibria (Sandler, 1993). The reliability of assuming linear ideal gas heat capacities was also investigated. The five LOS investigated in this study are the original Redlich- Kwong (1948), Redlich-Kwong-Soave (Soave. 1972), the Peng- Robinson(1976), Lee-Kesler-Ploecker (Ploecker et al., 1978) and Benedict-Webb-Rubin-Starling (Starling. 1973). Comparisons were made among the models as well as against highly accurate results from the equations of Wagner and Setzmann (1991). The reliability of assuming linear ideal gas heat capacities also was evaluated. Finally, we address the implementation of the EOS, because it seems that differences in certain details may explain different results even when apparently the same EOS was used. Fig. 1 Natural Gas Compressor on Offshore Platform The enthalpy (h) is a function of pressure, temperature and gas composition defined through a set of mole fractions (y). The actual absorbed power (Pni) involves the mass flow rate (W): PgaS = W H The mass flow rate is obtained from the actual or volumetric flow rate (Q) and the gas density (p): The density is found from the temperature (7) and pressure (p) with the compressibility factor (Z): (2) W = PQ (3) When Z differs from unity, the gas is not ideal and its value is a p = p I ZRT (4) function of T, p. and gas composition. The result of these definitions is that Pia, is found from: THE USE OF EOS IN THE DESIGN, PERFORMANCE PREDICTION AND TESTING OF GAS COMPRESSORS While the operating conditions for gas compressors are typically defined in terms of pressures, temperatures and mass or standard flows, the relevant data that describe the behavior of a compressor are the head (H), which is related to the work input, the volumetric flow (Q) and efficiency (n), which compares the real process to an isenuopic process between the same inlet state and outlet pressure. The head, or specific enthalpy difference between two states(e.g., inlet and discharge side of the compressor), is defined by: = pqh - P QH L.K1 In order to define the quality of the compression process. H is usually compared to the head for an ideal compression process which is defined as compression between the same inlet T 1 and p l and outlet p 2. with the outlet temperature being fictitious T2s. as = s(p2,t2 (y) ) s(p,t [y] )= (6) (5) H = h(p 2> T 2, W h(p >T1> (Y ) (1) 2

3 This isentropic change of state defines an isentropic head, such that: =h(p2>t23, fyi ) - 41, 71,fyi ) (7) The quality or efficiency of the compression process is defined by: (8) H Compressor characteristics in terms of head versus flow and efficiency versus flow are found by comparing test data, taken with test gases such as Nitrogen, with results obtained from the thermodynamic calculations above. The characteristics can later be used to calculate the performance of the compressor under arbitrary conditions of pressures, temperatures and gas compositions. As long as the same EOS is used for obtaining compressor performance predictions and data Suction, errors are minimized. An EOS is a relation among variables of a fully specified system: T, p, p and the N-1 component mole fractions y; (Alberty and Silbey, 1997). This is usually expressed in the form: Z = Z(p,T,{y}) (0) since in a multiphase region, multiple values of p give the same value of p. Thermodynamics gives rigorous relations for enthalpy and entropy differences from derivatives and integrals of 2 from any EoS and ideal gas specific heat,. A gas is said to be in a specified state if it has zero degrees of freedom. The degrees of freedom are the number of properties that can be arbitrarily set before all other properties become specified. The formula for the degrees of freedom of N nonreacting gases is: DF = N tt phases + 2 (10) In gas compressor design calculations, only one phase exists and the gas composition is usually specified, so two more degrees of freedom must be chosen. Generally, p and T are specified and the number of phases is always one. Then, all other thermodynamic properties are fixed and calculated via an EOS. Since real gas behavior commonly plays a role in gas compressors, knowledge of the relationships between pressures and temperatures, on one hand, and enthalpies, entropies and densities, on the other hand, is of great importance in compressor design, their performance under arbitrary operating conditions, and test data Suction. Especially during gas compressor performance tests, the selection of a particular LOS can have an important effect on the apparent efficiency and absorbed gas power. THERMODYNAMIC PROBLEM DEFINITION In order to decide on the most appropriate (LOS) to be used for designing and testing gas compressors for natural gas applications, five frequently applied LOS were studied: original Redlich-Kwong, Redlich-Kwong-Soave, Peng-Robinson (Reid et al., 1986), Lee- Kesler-Ploecker (Ploecker et al., 1978) and Starling version of the Benedict-Webb-Rubin model (Starling, 1973). The variation in entropy or enthalpy between two states of a gas or gas mixture, each defined by a temperature and pressure, is independent from the path chosen from one state the other (Reid et al., 1986). A convenient path involving three steps of changing the real gas to an ideal gas at T 1, changing the ideal gas from T1 to 71 and changing the ideal gas back to the real gas at 1'2 (Fig. 2): h= f (p,t) dh=(ahlapkdp+ohlan,dt p2 h, = johlap) r dp+ johlatldt TI T2 k-k = (h -h p,),,+ 7c dt (h h p2) T2 TI T2 The terms in the parentheses of Eq. II are called departure functions, real gas contributions or residual properties, which relate the enthalpy at some p and T to that at an ideal gas reference state at T, H. These departure functions can be calculated solely from the LOS. The same approach can be used for the entropy. The ideal gas law is based on the assumption that the molecules of the gas do not interact with each other or that there is no attractive or repulsive forces between two molecules. The heat capacity of a gas is the amount of energy which the gas needs to absorb before its temperature increases one unit. For an ideal gas, the heat capacity 4 is a function only of 7'. An empirical equation for the ideal gas heat capacity can be stated as a polynomial, e.g., third order polynomial: C; = A + BT + CT 2 + DT 3. (12) A, B. C, and D are empirical parameters or constants based on the type of gas being analyzed. Once an equation for 4 is found, the ideal gas enthalpy change, which is the change in total energy in the gas as it goes from state one to state two, can be found by: Ah = 1, 2 C dt P Fig. 2. Calculation Path for Equations of State (13) 3

4 Even for an ideal gas, the entropy change depends upon the initial and final temperatures and pressures. The entropy change is expressed by rn C AS = j --LdT- T (14) When calculating the enthalpy or entropy of a given state, an arbitrary reference state must be selected whose enthalpy and entropy are set to zero. The enthalpy and entropy for a given state is calculated relative to this reference. Therefore, any absolute value of the enthalpy or entropy of a gas at a given state has no real meaning, given its dependence on the reference state. However, when the enthalpy difference between two states is calculated, the reference state cancels out, so an enthalpy or entropy difference is an actual value that does not depend on the reference state. Since the ideal gas contribution is the same for any EOS, a program was used to study how the predictions in thermodynamic properties affected the departure functions. First, five equations of state were used to analyze methane. The results of this analysis were compared to experimental data well described by Wagner and Setzmann (1991) to understand how the simpler equations compare. RATIONALE FOR AND SCOPE OF THE PROJECT Cubic and modified BWR EOS are the ones most often used in industry because of their relative simplicity, reasonable accuracy, and applicability to mixtures (Peng and Robinson, 1976). Individual users base theft choice of an equation on factors such as available data, past experience or personal preference. Even within a certain industry, such as for compressor design, there is often disagreement on which equation of state gives the most accurate predictions so different models are used in practice. Therefore, it is important to know how one equation's predictions behave relative to another's as well as to real data. Unfortunately, though much literature exists that explains these trends for physically measurable properties, such as density and vaporliquid equilibrium concentrations, only a few references could be found on properties such as enthalpy and entropy. Thus, we have tried.to describe what can be found for the properties of gas compressor systems when different EOS are used. Several criteria can be used to select an EOS. The first is the conditions that are being studied. The principal concerns are that sufficient accuracy is obtained over the range of conditions encountered while minimizing complexity in expressions and calculations for coding and speed. An equation's behavior for all properties should be known over the range of actual conditions for the components and mixtures. For the present application, where the temperature of the gas is generally above its critical temperature, where it cannot be condensed into a liquid, and at relatively low pressures, any of the chosen equations will be reasonably accurate (Alberty and Silbey, 1997). However, their differences might be significant for compressor performance predictions. This may indicate that some EOS are better for certain applications but any possible generalizations have not been found. However, the solution procedure for p from p. T, and {y} must be relatively simple and robust. If the EOS is not cubic, iterative solutions are needed. In multiphase computations, these can require excessive iterations and may converge to a wrong value. Fortunately for compressor states, there is only a single solution for p but convergence can be slow. While computation time is usually not an issue for evaluating a single operating point of a compressor, many design and prediction programs require a large number of iterations involving the calculation of gas properties. In these cases simplicity of root finding can be quite important. Finally, EOS involve parameters which must be obtained from data. Whether the parameter values are empirically fitted or related via corresponding states concepts to other pure component properties such as to 7; and pa the properties of mixtures involve mixing and combining rules that characterize pairs of components in the mixture. If some, of these parameters are not available or insufficient data exist for fitting, then the equation is of limited use. It has been the authors experience that different EOS yield efficiencies that can be 2 to 3% different from each other. Recognizing that the tolerance on performance guarantees on compressors is generally within 3 to 5 %, these deviations are significant. This study examines the different equations of state to determine performance characteristics of gas compressors. It focuses on natural gas compression, where gas mixtures generally contain 90% or more methane. This and the fact that the work of Wagner and Setzmann (1991) provides a complete basis for the properties of methane made this substance a suitable focus for our analysis. In addition, ethane was studied in order to see if a more complicated molecule had a major effect on the results, nitrogen was considered since it is one of the standard test gases of the industry and a few computations were made with a model natural gas mixture. These comparisons are presented in graphs and analyzed in this research. REVIEW The first significant attempt to describe the thermodynamic properties of fluids by a single function was in 1873 by van der Weals. Much work has been done since then to improve the quantitative aspects especially for phase equilibria in mixed systems (Peng and Robinson, 1976; T'svu et al., 1991, 1992, 1995ab. Although the simplest mathematical form is a cubic in density which allows an analytic solution for p, however, this form may not be accurate enough in some cases. More complex forms have been developed (e.g., Starling, 1973; Ploecker et al., 1978; Wagner and Setzmarm, 1991). Modern cubic EOS are more accurate than van der Weals' equation, so they are widely used in industry (Twu et al., 1992). The Redlich- Kwong (RIC) equation was introduced in 1948 and modified by Soave (1972) and Peng and Robinson (PR) (1976) to increase the accuracy of vapor-liquid equilibrium predictions. Since then, Twu et al. (1995ab) have introduced modifications for both the RIC and PR equations. The next level of complexity involves the Benedict-Webb-Rubin equation and its variations, such as the Benedict-Webb-Rubin-Starling (BWRS) (Starling, 1973) and Lee-Kesler-Ploecker (LKP) (Ploecker et al., 1978) forms. These equations contain a large set of parameters. II in the BWRS case and the p value must be found iteratively. L1CP simplifies the BWRS by assuming the principle of three-parameter corresponding states is reliable for natural gas mixtures; thus, requiring only the critical temperature and pressure (7; and pc) and Pitzer's acentric factor that is readily available for most pure components in literature (Ploecker et al., 1978). Since many of the model improvements were intended to yield better predictions of vapor-liquid systems, which are avoided in gas compressors, it is not 4

5 clear if these improvements improve EOS accuracy for gas compressor systems and conditions. The highest level of complexity and the most accurate are multiproperty correlations of single component data with elaborate mathematical functions for the T and p dependence, such as the work of Wagner and Setzrnann (1991). These equations contain large numbers of parameters (30 to 100) and have many terms for computing the desired information. Equations such as these cannot be used for substances other than the one for which they were developed and not for mixtures. They are intended to provide benchmark information to compare more accessible models. Wagner and Setzmann (1991) equation, for example, is used only for methane but it can give all of the properties of methane within the uncertainty of current experimental information. To understand how the EOS compare with each other and how all of them compare with a highly accurate analysis, we studied results from five frequently applied EOS. These include the original Redlich- Kwong (1948), Redlich-Kwong-Soave (1974), Peng-Robinson (1976), Lee-Kesler-Ploecker (Ploecker et al., 1978) and the Benedict-Webb- Rubin-Starling model (Starling, 1973). Their results were compared to properties from Wagner and Setzrnann's tables (1991). DESCRIPTION OF THE EQUATIONS OF STATE The departure functions for enthalpy and entropy for each of the five EOS can be found in the literature (Reid et al., 1986; Peng and Robinson, 1976: Ploecker et al., 1978; and Starling, 1973). Herein, the RIC, RKS and PR EOS are referred to as cubics. In Eq. 15, Z represents the compressibility factor of the gas, defined as: The quantities X and Y are two other types of compressibility (15) The linear function for ideal CV/R is calculated using the Cp /R values at 10 and 149 C (50 and 300 F). These two points are used to find the slope of a straight line on a Cp /R versus temperature plot. This slope is used to solve for the y intercept of the following simple linear equation: = CT + B (18) Finally, the specific gravity (SG) and the real gas parameter (RG) are calculated. SG is calculated relative to the molecular weight of air SG The RG parameter is given by. Re = 0.287k/ / kgk SG Cubics EOS For the Redlich-Kviong EOS: ' a Z = a b RT I 's (v b) ( 1 9) (20) (21) Two parameters a and b found from matching the critical conditions: factors used in compressor design. The formulas for each: a.42748r b.08664rt, (22) PC P r T dz Z ar Z dp The calculation of the molecular weight and the heat capacity at given temperatures of the gas mixture is done by using the following mixing rules: T, is the critical temperature, p, is the critical pressure, and R is the universal gas constant. The Soave form of the RK EOS (RICS) has the same b parameter, but a is chosen to give better vapor pressures for (16) fluids described by three-parameter corresponding states: where:.42748r 2 T2 r Pc (23) 2w.Ey 1 MW, ap = E yjcp, (17) fir/ = w 0.176w 2 Here, T, is the reduced temperature (T/T c). (24) MWi and y are the molecular weight and mole fraction of each component in the mixture. The heat capacities are divided by R to make them dimensionless so, when the linear function is found, the result for a given temperature need only be multiplied by R in the desired units to get the heat capacity in those units. For the Peng-Robinson model:.45724r a [1+ fw(1 T,Y2 )r PC b =.07780RTc (25) Pc 5

6 where: flu = w w 2 (26) PCT Ic"4 (32) However, when 7', 21, this part of the a parameter is no longer a function of T For mixtures, the pure component parameters are combined and mixed. The mixing rule for the b parameter is a simple mole fraction average of the pure component b values: b, =Ey "); (27) where y, is the mole fraction of the component in the gas. The mixing rule for a is a quadratic mole fraction expression: PCP where: R( wm )(PCT) Vcff, 1 ( I/3 1/3 ) 3 Vai = 8 V d +V c a.= EE yori (cl1a1 f2(1--kii ) (28) where 14 is an empirical binary parameter. If Ick, is equal to 0, the rule simplifies to: a m = (E y ia," )2 (29) The values of X and Y require the temperature and pressure derivatives da/dt and da/dp. Lee-Kesler-Ploecker The LKP equation is like the BWRS, a modification of the original BWR EOS. The compressibility factor (Z) is calculated by inter or extrapolation from the compressibility factor (Z (1))) of a simple fluid (Methane, w 3) =0) and the compressibility factor Z ) of a reference fluid (N-Octane): Z =z (0 + -6) (Z (r) Z (13) ) co 09 (30) The expressions for both fo and Z(0 are known (Ploecker et al, 1978). The LICP LOS has mixing rules that are very different from the cubics. The mixing rules for this equation are used to find the pseudocritical pressure and temperature (PCP and PCT) and acentric factor. The rule for the accentric factor is a simple mole fraction average: y,w, (31) where wa is the accentric factor for the gas mixture and w, is the accentric factor for a component. The mixing rules for PCP and PCT are: = (Td Ted 112 k ii ; (33) BWRS The Starling version of the original BWR EOS added three extra parameters for improving the temperature dependence of the eight parameter form. These parameters must be found for each pure gas. There also are mixing rules for the 11 parameters (Starling, 1973). SOLVING FOR THE COMPRESSIBILITY FACTOR (2) For the cubic EOS, an analytical method can be used to solve for, the three roots of p, thus yielding Z. There are three roots to any cubic equation; however, when Tr >1 only the largest real root has any physical significance. After Z is calculated, the X and Y compressibility factors along with c p are calculated. For the LICP and BWRS models, Z is found by an iterative method. If the EOS is written in the form G = plprt-z(t, p) as a function of p, the behavior of the LICP equation at lower Tr and Pr is as shown in Fig. 3. The iterations follow the secant method with an initial guess for Z to be 0.8. Convergence is assumed when the change in Z is less then 0.01%. From the graph, it can be seen that as long as the initial guess is to the right of the middle root, the solution will converge to the largest one. At higher Tr and pr, the function is simpler as shown in Fig. 4. Since there is only one root, the solution will converge to it no matter where the initial guess starts off. For mixtures near phase boundaries, the relationship is more complex since more than one phase may appear. In any case, once Z is found, c,, X and Y are computed. The procedure for solving for the compressibility factor using the BWRS equation is the same as what was outlined for the LICP 6

7 Fig. 3. Behavior of LKP EOS, at Lower Tr and Pr Fig. 4. Behavior of LKP EOS at Higher Tr and Pr equation. The only difference is that the function 0 is different since it is derived from the specific EOS being used. The function also tends to behave as outlined in Figs. 3 and 4. COMPARISONS OF EOS MODELS The first stage of this study was to establish general trends in the prediction of the change in enthalpy, when a gas is compressed from state one to state two using two arbitrarily selected states. State one was always at a temperature of 38 C (560 R) and a pressure of bar (14.7 psia). State two was always defined at a temperature of C (1080 R), but the pressure was systematically increased in order to generate data of the change in the head or enthalpy difference (h2-111) versus the change in pressure (p 2 - p 1). Firstly, this information was collected for pure methane, ethane and nitrogen. Methane was chosen because it is the simplest hydrocarbon and is the largest single component in natural gas compression. Also, it is a substance for which an excellent description of all properties is available. Ethane was used to see the effect of a somewhat more complicated molecule, while nitrogen was used because it is a frequently used gas for performance tests on gas compressors. Additionally, a complicated gas mixture of hydrocarbons and nonhydrocarbons was tested. Secondly, models for the ideal gas heat capacity (c p) were examined. The accuracy of all computed results depend greatly on the accuracy of c p since ideal gas contributions dominate all changes of state in devices such as compressors. In order to study the validity of the linear heat capacity model, the calculation method was changed so that the second order heat capacity equation was used to explicitly calculate the heat capacity of a gas at a given temperature. Since efficiency (ii) is one of the most important parameters taken under consideration by compressor users, determination of whether different EOS predict different efficiencies and the cause of these differences is important to know. Therefore, all EOS were used both for pure gases and for gas mixtures, to predict the efficiency for a variety of compression processes. For comparison purposes, a baseline efficiency of around 80% for the typical conditions is assumed. While computation time is usually not an issue for evaluating a single operating point of a compressor, many design and prediction programs require a large number of iterations involving the calculation of gas properties. In these cases, the cubic EOS that allow a direct solution, typically offer advantages in computing time and stability of the program. RESULTS The plots of Ah versus a for the three pure gases studied are shown in Figs. 5 through 7. The smallest pressure difference is 0.49 bar (7.08 psia). These plots clearly show that different EOS predict different heads for the same change of state. These plots clearly show that different EOS predict different heads for the same change of state and that there is some pattern to the results but variations occur. The methane case shows that the order of lahl for smaller Ap is PR < RK < BWRS < LICP < SW, but for larger Ap the SW and BWRS values become lower than the others. On the other hand, the order of Idyl is about the same at all Ap: RK < PR < LICE' < SW < BWRS. With ethane, enthalpy changes are larger than methane for the same pressure change but the LOS have about the same differences except for LKP and BWRS which are in closer agreement in particular. H for PR is still consistently lower than the others. Finally, for nitrogen, it is seen that the general decreasing trend of H is not followed though the order of the LOS is the same and again the BWR models are close together. Since the LICP is a variation of the BWRS equation, the fact that their predictions are the same for nitrogen and ethane is an expected result. However, the fact that they differ for methane shows that the simplified application of the BWR as proposed by Lee, Kesler and Ploecicer does have an effect on the results generated by the equation. Since methane was tested as a pure gas, mixing coefficients are not the cause of this difference. All three of the plots show that the RK and RICS equations give essentially the same results since all of the calculations are above the critical temperature where the T dependence of the a value disappears. Finally, for methane the work of Setzmann and Wagner (1991) provides direct comparisons enthalpy and entropy data for methane (Fig. 5). Since this work, due to experimental evidence is a good benchmark for methane, their equation was used to generate data for methane as another point of comparison. For all pressures below 138 bar (2000 psia), all equations except PR show a very good agreement with the benchmark results. At higher pressures, BWRS begins to deviate more from the benchmark results, but gets closer again once the pressures get above 276 bar (4000 psia). RIC and LICP are still very close to the benchmark. At very high pressures above this, BWRS actually becomes the most accurate equation. It should be noted that Wagner and Setzmann's (1991) ideal gas expression was replaced by the ideal gas expression used for other EOS. As shown in Fig. 8, this made only a slight difference. 7

8 266 8 c 262 a < 250 Delta Enthalpy vs. Delta P: Ti = 560 R, P1 = 14.7 psia, T2 = 1080 R Linearized CP, Methane AK PR UCP BWRS Wagner 'Cc DELTA P, psia 109=01001M Fig. 5. Delta Enthalpy vs Delta Pressure, Methane 240 a 235 c C. < 220 Lu Z Delta Enthalpy vs. Delta P: T1 = 560 R, P1 = 14.7 psia, T2 = 1080 R Linearized CP, Ethane PR DELTA P. psia BVVRS RKS M Fig. 6. Delta Enthalpy vs Delta Pressure, Ethane Since the ideal gas ah is calculated using the ideal gas heat capacity model, two other heat capacity models were compared to the one in use. Figure 8 shows a comparison of the linearized model (Cp Old) with both a quadratic T dependence (Cp New) and Wagner's heat capacity model (Cp Wagner) for methane. As Fig. 8 shows, the linearized heat capacity model is accurate over the range of temperatures that are of interest since the heat capacity of methane exhibits linear behavior over these range of temperatures. Little error is introduced by using the linear model for the specific heat between 10 and 149 C (50 and 300 F) for methane and nitrogen and natural gas mixtures of 90%+ methane. The difference can be seen in Table 1 for the lowest ap, essentially only ideal gas contributions, where the SW value is slightly higher than the EOS. Therefore, no error seems to be introduced by using a linear model for the heat capacity between 10 and 149 C (50 and 300 F). The next step was to see if differences were also present in the isentropic efficiency predictions of each EOS. For this example, a realistic natural gas mixture with a specific gravity of was used (Table 1). Qualitatively similar observations as described in Table I were made with natural gas mixtures with a specific gravity between 0.58 and 0.67, pure methane and pure nitrogen. D TEMPERATURE 1018.o 1014.o >: a_ 1010 t- Lu ow Delta Enthalpy vs. Delta P: T1 = 560 R, P1 = 14.7 psia, T2 = 1080 R Linearized CP, Nitrogen LKP Fig. 8. Cp vs Temperature Table I. Gas Properties for Test Gas Gas Composition Mole, % Methane 90 Ethane 5 Propane 2 Nitrogen 2 Carbon Dioxide DELTA P. psia GO:03403M Specific Gravity Pseudo Critical Temperature K (362.3 R) Pseudo Critical Pressure bar (679.5 psia) Fig. 7. Delta Enthalpy vs Delta Pressure, Nitrogen 8

9 For the calculations in the example, the followin g conditions were used. Suction condition was always at Tj - 20 C (68 F) and p 2-50 bar (725 psia). The gas was compressed to varying end pressures (Pi) with T2 such that the reference EOS (RK) gives 80% efficienc y. The results are shown in Fi g. 9. Differences as hi gh as 2% exist among the LOS models. Clearl y, it cannot be concluded that a certain LOS will always lead to hi gher efficienc y than other LOS. Examination of the calculations indicates that this is mainl y due to differences in the actual head as opposed to the isenttt mic head, which means that there will be a similar effect on the calculated gas power. The distinction among the LOS results is the hi gh pressure density. The effect of this may cancel out between the enthalp y and entropy computations of the isentropic case, but remain in the actual case. The RK and RKS equations yield virtually identical results ; therefore, they are not plotted separatel y in Fig. 9. This result was expected, since the corrections of Soave (1972) onl y have a si gnificant impact closer to the critical point. Comparin g the results in Fi gs. 5 and 9, it seems that a significant part of the differences between the LOS has to be attributed to the difference in the entrop y calculation rather than the enthalpy calculation. Similar calculations were made for pure methane so that comparisons could be made over wider ran ges of conditions and with the data computed by Wagner and Setzmann (SW). Table 2 gives the results for chan ges of state that were easil y interpolated from the SW tables. A gain the pattern of Fi g. 5 is seen, but interestin gly, all of the LOS efficiencies are less than the accurate Wa gner and Setzmann (SW) values. The differences as shown in Table 2 onl y become significant for elevated pressures. Another example shows the results of an actual compressor test (Fig. 10). Identical confi gurations were tested with nitro gen and during a field performance test with a natural gas mixture containin g about 95% methane (Kurz and Brun. 1998). The RIC EOS was used to reduce the data. The close correlation between both sets of data is an indicator for the general validity and accuracy of the approach usin g LOS. CONCLUSIONS Firstly, EOS models lead to different results for thermod ynamic properties for substances at the states commonl y encountered in gas compressor systems. These can lead to noticeable differences in the calculation of absorbed power due to discrepancies in the predicted actual head. They also can lead to noticeable differences in the calculated efficiencies for the same chan ge in state, due to both discrepancies in the calculated actual and isentropic head. Secondl y, over the ran ge of temperatures of interest for gas compression applications, a linear heat capacit y model is probabl y adequate, especiall y for systems dominated by methane. More complex expressions could be used, but since the industr y is accustomed to this description there is little incentive to use them. 0.9 t uj 0.8 Ca Lu 0.7 ta U. o s a: 1.0 cc z 03 CA Ore INLET FLOW COEFFICIENT U Fig. 9. Isentropic Efficiency Differences among EOS for a Natural Gas Mixture (when p, = 50 bar (725 psia), T,= 20 C (68 F) and varying p T, chosen to give trz 80% for RK EOS) Fig. 10. Comparison of Test Results from Test with Nitrogen and Field Test with Natural Gas Table 2. Comparison of EOS and SW Values of Isentropic Efficiency for Pure Methane and Inlet of p, = 80 bar (1160 psia), 7', = 310 K (558 R) and pl. p2, bar (psia) RIC T2, R i, RK 1, PR 11, LKP 11, BWRS n, SW 200(2900) (7290)

10 For the design and testing of gas compressors, the EOS models studied give results within the tolerance of compressor specifications for natural gas mixtures over the common pressure and temperature ranges. It is not possible to always conclude that any equation of state is always more accurate than others, but consistent use of the same model for design, performance prediction and testing will yield reliable results. REFERENCES Alberty, RA., and Silbey, RJ., 1997, Physical Chemistry, Second Edition, John Wiley & Sons, New York. Beinecke, D., and Luedtke, IC, 1983, "Die Auslegung von Turboverdichtem unter Beruecicsichtigung des realen Gasverhaltens, VDI Berichte, No. 487, pp Kurz, R., and Brun, K., 1997, Wield Testing of Gas Turbine Driven Centrifugal Compressor Packages-Test Procedures and Measurement Uncertainties," Proceedings of the 26th Texas A&M Turbomachinety Symposium. Kurz, R., and Brun, K., 1998, "Field Performance Testing of Gas Turbine Driven Compressor Sets," Proceedings of the Pipeline Simulation Interest Group, 30th Annual Meeting. Peng, D.Y., and Robinson, D.B., 1976, "A New Two-Constant Equation of State," Ind. Eng. Chem. Fundam., Vol. 15, pp Ploecker, U., Knapp, H., and Prausnitz, J., 1978, "Calculation of High Pressure Vapor-Liquid Equilibria from a Corresponding-States Correlation with Emphasis on Asymmetric Mixtures." hid. Eng. Chem. Process Des. Dev Vol. 17, pp Reid, R.C., Prausnitz, J.M., and Poling. B.E , The Properties of Gases and Liquids. Fourth Edition, McGraw-Hill Book Company, New York. Sandler, Si., Ed., 1993, "Models for Thermodynamic and Phase Equilibria Calculations," Marcel Dekker, Inc., New York. Soave, G., 1972, Chem. Eng.Sci., pp Starling, K.E., 1973, "Fluid Thermodynamic Properties for Light Petroleum Systems," Gulf Publishing Company, Houston, Texas. Twu, CM., Muck, D., Cunningham, J.R., and Coon, I.E., 1991, "A Cubic Equation of State with a New Alpha Function and a New Mixing Rule,"Fluid Phase Equilibria, Vol. 69, pp TW11, C.H., Coon, I.E., and Cunningham, J.R., 1992, "A New Cubic Equation of State," Fluid Phase Equilibria. Vol. 75, pp Twu, C.H., Coon, LE., and Cunningham, J.R., 1995a, "A New Generalized Alpha Function for a Cubic Equation of State Part 1, Peng-Robinson Equation," Fluid Phase Equilibria, Vol. 105, pp ' Twu, C.H., Coon, I.E., and Cunningham, IR., 1995b, "A New Generalized Alpha Function for a Cubic Equation of State Part 2. Redlich-Kwong Equation," Fluid Phase Equilibria, Vol. 105, pp Wagner, W., and Setzmann, U., 1991, "A New Equation of State and Tables of Thermodynamic Properties for Methane Covering the Range from the Melting Line to 625 K at Pressures up to 1000 Mpa," J. Phys. Chem. Ref Data, Vol. 20, pp