Centrifugal Compressor Polytropic Performance—Improved Rapid Calculation Results—Cubic Polynomial Methods

: This paper presents a new improved approach to calculation of polytropic performance of centrifugal compressors. This rapid solution technique is based upon a constant efﬁciency, temperature-entropy polytropic path represented by cubic polynomials. New thermodynamic path slope constraints have been developed that yield highly accurate results while requiring fewer computing resources and reducing computing elapsed time. Applying this thermodynamically sound cubic polynomial model would improve accuracy and shorten compressor performance test duration at a vendor’s shop. A broad range of example case results verify the accuracy and ease of use of the method. The example cases conﬁrm the cubic polynomial methods result in lower calculation uncertainty than other methods.


Introduction
A highly accurate centrifugal compressor polytropic performance approximation method has been developed that is easy to employ. This real gas method is based upon a constant efficiency, temperature-entropy polytropic path. The elegance of this method is its exceedingly simple way of calculating polytropic efficiency with sufficiently high precision as required for compressor performance testing, while providing the polytropic compression path on T-s and h-s diagrams to a high degree of accuracy. A constant efficiency polytropic path can be modeled as either a single or several sequential piecewise cubic polynomial segments affording solutions that allow for determining thermodynamic state variables along a continuous path [1,2]. New analytic terms have been developed for slope and curvature of temperature versus entropy along the constant efficiency polytropic path [1]. Furthermore, a new screening method has been developed to assist in determining how many cubic polynomial segments are required to provide sufficient accuracy for a given application. Example cases are reported that demonstrate accuracy and comparisons to other polytropic performance calculation methods as well as a review of results documented by Evans [3]. Cubic polynomial endpoint path methods are demonstrated to achieve better accuracy than any other endpoint polytropic efficiency calculation method. Cubic polynomial sequential segment path methods are shown to be superior to other multi-point numerical methods described in available literature.
The Taher-Evans Cubic Polynomial method (TE-CP) provides not only overall polytropic compression performance results but has the ability to predict fluid state parameters at any arbitrary point along the polytropic compression path. The analysis is based upon an inlet flange to discharge flange constant efficiency polytropic path for a single, uncooled, compressor section that may contain multiple impellers. Thermodynamic state parameters used in the calculations are based upon total conditions at inlet and discharge measurement locations. Typically, these are pressure and temperature measurements. An Equation of State (EOS) provides all other necessary thermodynamic state parameters based upon The polytropic compression process is defined as a reversible and non-adiabatic process, in which the ratio of useful compression work, , to the change of enthalpy, ℎ, remains constant along the polytropic compression path as defined by Equation (2) and illustrated in Figure 1. = ℎ Since the polytropic process is defined as a reversible process, the degraded energy can be represented by heat, which is equal to . The polytropic work, , as the measure of useful compression work, is the difference between the total enthalpy change and the area under the polytropic path on the T-s diagram between inlet and discharge, as illustrated in Figure 2. Substituting Equation (1) into Equation (2) yields a convenient form for compressor polytropic efficiency shown in Equation (3).
The shape of the polytropic path on the T-s diagram, which illustrates the gas behavior as temperature and entropy increase during the compression process, can significantly influence the value of degraded energy, . Additionally, the use of different equations of state to predict thermodynamic conditions along the polytropic compression path can greatly influence the results, as illustrated by example cases in Appendix D. Since the polytropic process is defined as a reversible process, the degraded energy can be represented by heat, which is equal to 2 1 Tds. The polytropic work, w p , as the measure of useful compression work, is the difference between the total enthalpy change and the area under the polytropic path on the T-s diagram between inlet and discharge, as illustrated in Figure 2. Substituting Equation (1) into Equation (2) yields a convenient form for compressor polytropic efficiency shown in Equation (3).
The shape of the polytropic path on the T-s diagram, which illustrates the gas behavior as temperature and entropy increase during the compression process, can significantly influence the value of degraded energy, Tds. Additionally, the use of different equations of state to predict thermodynamic conditions along the polytropic compression path can greatly influence the results, as illustrated by example cases in Appendix D.

Polytropic Path: Polynomial Approximation Methods
The constant efficiency polytropic path on the T-s diagram is a continuous real-valued function of entropy defined on a bounded interval , . This actual polytropic path can be approximated to a high degree of accuracy using a single polynomial or a series of sequential piecewise segment polynomials. The concept is to accurately estimate the actual unknown temperature function, T(s), by an approximating function that is simple but accurate enough (i.e., within acceptably small error tolerances) to calculate , which is the degraded part of energy transfer in the polytropic compression process. Taher [1], documented the mathematics of applying sequential piecewise cubic polynomials to approximate a constant efficiency temperature-entropy polytropic path.
The temperature-entropy relationship along the polytropic path can be approximated using a single polynomial for the entire range from inlet to discharge. This is referred to as an "Endpoint" method since the thermodynamic state variables at both compressor inlet and discharge are required inputs and are known from testing. For some applications, when a single polynomial cannot provide sufficient accuracy, a series of sequential steps or segments of polynomials can be employed. The following sections will discuss and compare first-and third-degree polynomial approximants as used to calculate . First degree polynomials are described as using "steps" while third degree polynomials are described as using "segments." This distinction is due to the discontinuity of path slope change for linear steps whereas sequential piecewise cubic segments possess a continuous slope change. It will be shown that this continuity of slope at the knots gives rise to increased accuracy while using fewer segments to achieve the desired accuracy.

Temperature-Entropy Polytropic Path Approximation: Linear Polynomial Endpoint Method
A linear straight-line approximation between inlet and discharge conditions has been used by Kent [10] and Sandberg [23] to approximate the actual polytopic path on the T-s diagram. Neither Kent nor Sandberg explicitly mentioned employing a linear approximation for the polytropic path on the T-s diagram, but in fact a straight-line approximation of temperature, T, as a function of entropy, s, is used in their linear approximation method. Since pressure and temperature are known at both endpoints, all other thermodynamic properties at inlet and discharge can be determined by an equation of state.
Equation (4) represents temperature as a function of entropy connecting compressor section inlet and discharge points on a T-s diagram. The form of the function is a classic straight line.

Polytropic Path: Polynomial Approximation Methods
The constant efficiency polytropic path on the T-s diagram is a continuous real-valued function of entropy defined on a bounded interval [s 1 , s 2 ]. This actual polytropic path can be approximated to a high degree of accuracy using a single polynomial or a series of sequential piecewise segment polynomials. The concept is to accurately estimate the actual unknown temperature function, T(s), by an approximating function that is simple but accurate enough (i.e., within acceptably small error tolerances) to calculate 2 1 Tds, which is the degraded part of energy transfer in the polytropic compression process. Taher [1], documented the mathematics of applying sequential piecewise cubic polynomials to approximate a constant efficiency temperature-entropy polytropic path.
The temperature-entropy relationship along the polytropic path can be approximated using a single polynomial for the entire range from inlet to discharge. This is referred to as an "Endpoint" method since the thermodynamic state variables at both compressor inlet and discharge are required inputs and are known from testing. For some applications, when a single polynomial cannot provide sufficient accuracy, a series of sequential steps or segments of polynomials can be employed. The following sections will discuss and compare first-and third-degree polynomial approximants as used to calculate 2 1 Tds. First degree polynomials are described as using "steps" while third degree polynomials are described as using "segments". This distinction is due to the discontinuity of path slope change for linear steps whereas sequential piecewise cubic segments possess a continuous slope change. It will be shown that this continuity of slope at the knots gives rise to increased accuracy while using fewer segments to achieve the desired accuracy.

Temperature-Entropy Polytropic Path Approximation: Linear Polynomial Endpoint Method
A linear straight-line approximation between inlet and discharge conditions has been used by Kent [10] and Sandberg [23] to approximate the actual polytopic path on the T-s diagram. Neither Kent nor Sandberg explicitly mentioned employing a linear approximation for the polytropic path on the T-s diagram, but in fact a straight-line approximation of temperature, T, as a function of entropy, s, is used in their linear approximation method. Since pressure and temperature are known at both endpoints, all other thermodynamic properties at inlet and discharge can be determined by an equation of state.
Equation (4) represents temperature as a function of entropy connecting compressor section inlet and discharge points on a T-s diagram. The form of the function is a classic straight line. The constraints applied to Equation (4) are shown in Equations (5) and (6).
Equations (7) and (8) show the analytically derived values for the slope and intercept of the straight-line approximant. These values are constant and only dependent upon the thermodynamic parameters at the compression endpoints.
Substituting Equations (7) and (8) into Equation (4) yields Equation (9), 1 T(s), which is a first-degree polynomial that approximates the actual polytropic path T(s). (1) The deviation of the approximated temperature from the actual temperature value using a linear approximant is represented by the error function (1) 1 ErT(S) in Equation (10).
As shown in Figure 3, temperature deviation varies along the polytropic path in the interval [s 1 , s 2 ] from the compressor section inlet to discharge. The actual polytropic path may largely deviate from a straight line connecting the endpoints.
Equations (7) and (8) show the analytically derived values for the slope and intercept of the straight-line approximant. These values are constant and only dependent upon the thermodynamic parameters at the compression endpoints.
The deviation of the approximated temperature from the actual temperature value using a linear approximant is represented by the error function ( ) ( ) in Equation (10).
As shown in Figure 3, temperature deviation varies along the polytropic path in the interval , from the compressor section inlet to discharge. The actual polytropic path may largely deviate from a straight line connecting the endpoints.
1 ErTDS = 0.1054 kJ/kg (0.698% relative deviation), which is represented by the area bounded between the two curves of (1) 1 T (s) and The term (1) 1 ErTDS in Equation (11) shows the error in calculating 2 1 Tds using a linear approximation for the temperature-entropy relationship between endpoints of the polytropic compression path.
The (1) 1 ErTDS error can only be estimated by comparing the result against a more accurate approximation for the polytropic path. In this paper, a 10-segment cubic polynomial method (i.e., using 10 T(s) approximants) is used as the basis of evaluation. (See Appendix B for confirmation of the accuracy of the multi-segment cubic polynomial method as applied with 10 segments).
Employing a straight line as the approximant to model the actual polytropic path on the T-s diagram overly simplifies the actual compression path resulting in reduced accuracy of the calculation for the 2 1 Tds and thus the resulting polytopic efficiency rendering the polytropic work values less accurate. For example, as illustrated in Figure 3, a linear approximation for the polytropic path on T-s diagram overestimates the degraded energy (i.e., 1 ErTDS = 0.1054 kJ/kg) by the bounded area between the linear approximant and the actual path as estimated with the cubic polynomial method. A linear approximant is inherently limited to a fixed constant slope along the entire polytropic path. This is a major factor in estimating the 2 1 Tds with large errors using a linear endpoint approximant.

Temperature-Entropy Polytropic Path Approximation: Cubic Polynomial Endpoint Method
The desire to improve the accuracy of approximating the actual polytropic path on the T-s diagram has led to employing the approximant function as a cubic polynomial. Polynomials of the first degree and second degree are limited since they cannot account for path slope changes and concavity of the actual polytropic path respectively. A polynomial of the third degree is the simplest form of a polynomial approximant, which allows the model to account for these two important features. As explained later in this paper (see Section 4), the slope and concavity of the polytropic path on the T-s diagram reveal significant thermodynamic insight about the behavior of fluids during the compression process and must be carefully studied.
Using Equation (12), the actual temperature that increases with entropy, T(s), along the polytropic path of a compressor section can be approximated using the third-degree 1 T(s) = As 3 + Bs 2 + Cs + D Since thermodynamic conditions at endpoints are known, the constraints applied to Equation (12) are shown in Equations (13)- (16).
Relationships (15) and (16) determine values of the slope of the temperature-entropy curve at endpoints of the polytropic path. For details on how these equations are derived from thermodynamic relationships, see Appendix A. 1 T(s) in Equation (12) are "analytically" derived from the known conditions above. The results are shown in Equations (17)- (20).
Coefficient A accounts for the deviation of the average value of the slope at the end- , from that of the straight line, which connects the endpoints, ( ∆T ∆s = T 2 −T 1 S 2 −S 1 ). Coefficient B accounts for the change of slope along the polytropic path. When the polytopic path on the T-s diagram approaches to a straight line that connects the endpoints, (e.g., in the case of compressing an ideal gas) coefficients A and B approach zero and coefficients C and D represent the slope and the intercept of the straight line as shown in relationships (7) and (8).
The deviation of the temperature approximated using a cubic polynomial approximant from the actual temperature value is shown by the error function 1 ErT(s) in Equation (21). The temperature deviation varies along the polytropic path in the interval [s 1 , s 2 ] from the compressor inlet to discharge.
As illustrated in Figure 4, the deviation of approximated temperature from the actual temperature shows significant improvement for the cubic polynomial approximation over the linear version, which is expected since Coefficients A, B, C and D of the third order polynomial ( ) ( ) in Equation (12) are "analytically" derived from the known conditions above. The results are shown in Equations (17)- (20).
Coefficient A accounts for the deviation of the average value of the slope at the endpoints, , from that of the straight line, which connects the endpoints, ( ∆ ∆ = ). Coefficient B accounts for the change of slope along the polytropic path. When the polytopic path on the T-s diagram approaches to a straight line that connects the endpoints, (e.g., in the case of compressing an ideal gas) coefficients A and B approach zero and coefficients C and D represent the slope and the intercept of the straight line as shown in relationships (7) and (8).
The deviation of the temperature approximated using a cubic polynomial approximant from the actual temperature value is shown by the error function ( ) ( ) in Equation (21).
The temperature deviation varies along the polytropic path in the interval , from the compressor inlet to discharge.
As illustrated in Figure 4, the deviation of approximated temperature from the actual temperature shows significant improvement for the cubic polynomial approximation over the linear version, which is expected since   1 ErTDS = 0.0237 kJ/kg (0.157% relative deviation), which is represented by the area bounded between the two curves of 1 T(s), and cubic polynomial, 1 T(s), endpoint approximants from the highly accurate polytropic path as approximated with a 10-segment piecewise cubic polynomials, 10 T(s), for the high pressure propane case with the endpoint conditions shown on the figure. The cubic polynomial endpoint approximant provides a maximum temperature deviation that is less than a third of that of the linear endpoint approximant. . Temperature deviation of the cubic polynomial endpoint method from the actual polytropic path as approximated with 10-segment piecewise cubic polynomial approximants is shown. The highest temperature deviation is 1.185 K (0.289% relative deviation), at s = 3.414 kJ/kg·K. The error in estimating the degraded energy is ( ) = 0.0237 kJ/kg (0.157% relative deviation), which is represented by the area bounded between the two curves of ( ) ( ) ( ) ( ). Figure 5 compares the temperature deviation of the linear, ( ) ( ), and cubic polynomial, ( ) ( ), endpoint approximants from the highly accurate polytropic path as approximated with a 10-segment piecewise cubic polynomials, ( ) ( ), for the high pressure propane case with the endpoint conditions shown on the figure. The cubic polynomial endpoint approximant provides a maximum temperature deviation that is less than a third of that of the linear endpoint approximant. The term ( ) in Equation (22) shows the error in calculating using the cubic polynomial endpoint approximation.
The improved accuracy of calculating using the cubic polynomial endpoint approximant, ( ) ( ), as compared to the linear approximant, ( ) ( ), can be calculated using Equation (23).
Equation (23) shows that the accuracy of the cubic polynomial approximant as compared to a linear approximant increases for compression applications with a large difference between the slopes at the endpoints. As explained in the next section of this paper, this analytical relationship conveniently provides a means to evaluate accuracy of the linear endpoint method.

Polytropic Efficiency Calculation
The improved accuracy of calculating 2 1 Tds using the cubic polynomial endpoint approximant, 1 T(s), as compared to the linear approximant, 1 T(s), can be calculated using Equation (23).
Equation (23) shows that the accuracy of the cubic polynomial approximant as compared to a linear approximant increases for compression applications with a large difference between the slopes at the endpoints. As explained in the next section of this paper, this analytical relationship conveniently provides a means to evaluate accuracy of the linear endpoint method.

Polytropic Efficiency Calculation
Since the late 19th century, various calculation methods have been proposed to estimate polytropic efficiency. The polytropic efficiency can only be estimated. The accuracy of the estimation depends upon the calculation method and the accuracy of thermodynamic properties used in the calculation. As shown in this paper, different equations of state for a fixed method can result in variation of polytropic efficiency values, which may be even larger than the effect of measurement uncertainties. It is important to ensure that the same equation of state as used for calculating expected performance is used when calculating polytropic efficiency based on equipment test results.
Regardless of which calculation method is used, usually five significant digits can well serve the purpose for any compressor performance evaluation. Accuracy of measured inlet and discharge conditions as well as of the equation of state and test data used to obtain thermodynamic properties for the calculation method may impose to reduce the number of significant digits. The identification of significant digits is only possible through knowledge of the circumstances [25]. However, in this paper, the intended precision of polytropic efficiency is chosen as five significant digits in order to evaluate different polytropic efficiency calculation methods.
A novel approach is used in this paper to differentiate between the actual polytropic efficiency (for a fixed calculation method and EOS) from the calculated polytropic efficiency by using an error function, Erη. This is believed to help clarify that regardless of the calculation method, the actual polytropic efficiency, η p , can only be estimated with some acceptable error, Erη, as compared to a more accurate reference method. In this paper, the Taher-Evans Cubic Polynomial 10-segment method is used as the basis to compare the accuracy of other methods.

Polytropic Efficiency Calculation: Linear Polynomial Endpoint Method
In the simplest case, the polytropic path can be approximated using a linear approximant, which connects the endpoints (i.e., one step) as follows: The degraded energy in the polytropic compression process, 2 1 Tds, is approximated using the linear endpoint approximant  (1) 1 T(s) ds using the relationship (9), the relationship (26) emerges: Using the relationships (2) and (26), the polytopic efficiency, 1 η, as approximated with the linear endpoint approximation method is developed as follows: (1) The deviation of the approximated endpoint polytropic efficiency, 1 η, from the actual polytropic efficiency, η p , using the linear endpoint approximant, 1 T(s), is represented by the error function (1) 1 Erη in Equation (28).
A similar relationship to (27) for calculating the polytropic efficiency was suggested by Stepanoff [26] in 1955.

Polytropic Efficiency Calculation: Taher-Evans Cubic Polynomial Endpoint Method
In a more accurate case, the polytropic path can be approximated using a cubic polynomial, which connects the endpoints (i.e., one segment) as follows: The degraded energy in the polytropic compression process, 2 1 Tds, can be approximated using the cubic polynomial endpoint approximant, 1 T(s): By replacing 1 T(s) from (29) and analytically performing the integral, 2 1 (3) 1 T(s) ds, using relationships (17)- (20), the relationship (31) emerges: Using the relationships (2) and (31), the polytopic efficiency, 1 η, as approximated with the cubic polynomial endpoint approximation method is developed as follows: (3) Equation (32) can be solved recursively by assuming an initial value for the efficiency to calculate endpoint slopes E 1 and E 2 , calculating the resulting efficiency from (32), and minimizing the difference between assumed and calculated efficiency.
By comparing relationships (27) and (32) the efficiency deviation between the linear and cubic polynomial endpoint methods can be developed as follows: As shown by the relationship (34), the deviation linearly increases with the change of slopes, (E 2 − E 1 ), at endpoints. This analytical relationship conveniently provides a means to evaluate accuracy of the linear methods as applied to endpoints.
Once the polytropic efficiency, 1 η is determined, coefficients A, B, C, and D are calculated, and the polytopic path model on T-s and h-s diagrams can be graphically represented using relationships (29) and (51). Figure 6 compares the efficiency deviation of the linear endpoint and cubic polynomial endpoint methods from the highly accurate 10-segment cubic polytropic efficiency, 10 η, using different equations of state for the high pressure propane case with the conditions shown on the figure. Evidently, the linear endpoint method involves significantly larger deviations for the polytropic efficiency as compared with the cubic polynomial endpoint method.
Using illustrative examples in Appendix B, it is shown that Taher-Evans cubic polynomial endpoint method provides the highest accuracy among all other endpoint methods.

Polytropic Efficiency Calculation: Linear Multistep Method
As illustrated in Figures 3 and 6, the linear approximant, ( ) ( ), when applied to endpoints may introduce large deviations from the actual T-s polytropic path resulting in reduced accuracy for the calculated polytropic efficiency, ( ) . By applying the composite trapezoidal rule, the integral is approximated by employing piecewise linear polynomial approximants that cover the entire path in multiple steps with smaller subintervals , .
In the linear multi-step method, temperature deviation along the polytropic path reduces as the number of steps, j, which approximate the entropy interval , increases.
where for the th step, As shown in Figure 7, maximum temperature deviation along the polytropic path reduces as the number of steps increases. As expected, linear approximation involves larger deviation as compared with cubic polynomial approximation for the same number of steps or segments. As reviewed in Appendix C, a maximum of five cubic polynomial segments has been documented to provide sufficient accuracy for a broad range of example cases by Evans [3].
The degraded energy in the polytropic compression process, , is approximated by dividing the path into j-steps and using a linear approximant for each step.
where ( ) shows the error in approximating the integral, ( ) , by using the linear approximant ( ) ( ) in the subinterval , +1 . By summing up errors of all j-steps, the total error for approximating the integral , with linear approximants using j-steps is developed: 1 η, and cubic polynomial, 1 η, endpoint methods from the highly accurate 10-segment cubic polynomials polytropic efficiency, 10 η, as calculated using different equation of states.

Polytropic Efficiency Calculation: Linear Multistep Method
As illustrated in Figures 3 and 6, the linear approximant, 1 η. By applying the composite trapezoidal rule, the integral 2 1 Tds is approximated by employing piecewise linear polynomial approximants that cover the entire path in multiple steps with smaller subin- In the linear multi-step method, temperature deviation along the polytropic path reduces as the number of steps, j, which approximate the entropy interval [s 1 , s 2 ] increases.
where for the ith step, Figure 7, maximum temperature deviation along the polytropic path reduces as the number of steps increases. As expected, linear approximation involves larger deviation as compared with cubic polynomial approximation for the same number of steps or segments. As reviewed in Appendix C, a maximum of five cubic polynomial segments has been documented to provide sufficient accuracy for a broad range of example cases by Evans [3].
The degraded energy in the polytropic compression process, 2 1 Tds , is approximated by dividing the path into j-steps and using a linear approximant for each step.
where (1) j Er i TDS shows the error in approximating the integral, i+1 i T(s) ds, by using the linear approximant By summing up errors of all j-steps, the total error for approximating the integral 2 1 Tds, with linear approximants using j-steps is developed:  Table A1 in Appendix B for gas compositions and compressor inlet and discharge conditions. By applying the relationship (26) for each step, and using the composite trapezoidal rule, the integral, , is approximated as follows: Using the relationships (2) and (38), the polytopic efficiency, ( ) , is developed as follows: The deviation of the approximated polytropic efficiency, ( ) , from the actual polytropic efficiency, using linear approximants, ( ) ( ), at each step is represented by the error function, ( ) , in Equation (40).
Because the linear method ignores the change in slope and concavity along the compression path, a large number of steps are needed to achieve the desired accuracy. Sandberg [23] provided no criteria to determine the required number of linear steps to approximate . The uncertainty about the required number of steps in the linear multi-step method could possibly be overcome by applying a large number of linear steps (50 to 100+ steps) to ensure sufficient accuracy. However, pre-selecting 50 or 100+ linear polynomial steps has been  Table A1 in Appendix B for gas compositions and compressor inlet and discharge conditions. By applying the relationship (26) for each step, and using the composite trapezoidal rule, the integral, 2 1 Tds, is approximated as follows: Using the relationships (2) and (38), the polytopic efficiency, j η, is developed as follows: (1) The deviation of the approximated polytropic efficiency, j η, from the actual polytropic efficiency, η p using linear approximants, (1) j T i (s), at each step is represented by the error function, (1) j Erη, in Equation (40).
Because the linear method ignores the change in slope and concavity along the compression path, a large number of steps are needed to achieve the desired accuracy. Sandberg [23] provided no criteria to determine the required number of linear steps to approximate 2 1 Tds. The uncertainty about the required number of steps in the linear multi-step method could possibly be overcome by applying a large number of linear steps (50 to 100+ steps) to ensure sufficient accuracy. However, pre-selecting 50 or 100+ linear polynomial steps has been shown to be inefficient, (see Figure 11). Utilizing a large number of linear steps significantly increases the calculation time due to the required nested iteration loops. Figure A3b documents a comparison of elapsed time measurements for the supercritical propane example case.

Polytropic Efficiency Calculation: Taher-Evans Cubic Polynomial Multi-Segment Method
The T-s polytropic path can be accurately approximated with a set of piecewise thermodynamically coupled cubic polynomials. The T-s polytropic path is divided into "j" segments over the entropy range, [s 1 , s 2 ], from compressor section inlet to discharge. Each subinterval [s i , s i+1 ] of the actual path, T(s), is approximated by a cubic polynomial approximant, j T i (s), which allows the temperature as well as T-s polytropic path slope, (dT/ds), to be matched at endpoints of adjacent segments, thus increasing accuracy while smoothing the overall path. The key factor in the accuracy of cubic polynomial multisegment method as compared with the linear multistep method is taking advantage of applying the thermodynamic constraint, dT ds , at each intermediate point (knot) to enable smooth transition from one to the following segment.
The relationship (41) shows the actual temperature-entropy path, T(s), as approximated with a set of "j" cubic polynomial approximants, j T i (s), and the temperature deviation at each segment, where (3) j T i (s) at each segment is determined by: The coefficients (A i , B i , C i , D i ), of each cubic polynomial approximant are calculated using the relationships (17) to (20), where subscripts 1 and 2 are replaced with endpoint conditions of the ith segment.
Similar to relationships (30) and (31), the degraded energy is calculated, j T i (s) ds and its related error, j Er i TDS, at each segment are defined. The total error for approximating the integral 2 1 Tds, using cubic polynomial approximants is determined by summing up errors of each segment.
(3) 2 1 Tds, is approximated as follows: Using the relationships (2) and (44), the polytopic efficiency j η is developed as follows: The deviation of the approximated polytropic efficiency (3) j η from the actual polytropic efficiency, η p using cubic polynomial approximants (3) j T i (s) at each step is represented by the error function (3) j Erη in Equation (46).

Polytropic Efficiency Calculation Procedure: Taher-Evans Cubic Polynomial Multi-Segment Method
The solution requires an initial estimate of efficiency and then recursive solutions for segment discharge temperature. Equation (45) is solved recursively by assuming an initial value for the efficiency, j η to calculate endpoint slopes E 1 and E 2 , calculating the resulting efficiency from (45), and minimizing the difference between assumed and calculated efficiency. The step-by-step procedure is explained as follows: Note: inlet and discharge pressures and temperatures discussed below are assumed to be for total conditions at a given compressor section.
(1) Determine an initial estimate of polytropic efficiency by assuming (E 2 = E 1 ) in the relationship (45) for the cubic polynomial endpoint method. Note: for the purpose of compressor performance testing, the initial value of the efficiency is the expected efficiency provided by the compressor manufacturer. (2) Select the number of segments (j).
There are different ways to select intermediate knots. In this paper, the overall pressure ratio is divided across the number of segments (j) to determine equal pressure ratio steps as shown in the next step.
(3) Use "j" number of equal pressure ratio segments. p r = j p 2 p 1 .
(4) Use Equation (29) to estimate the initial value of segment outlet temperature. (This requires a solution to the cubic polynomial endpoint method). Note: steps 1 through 4 are only required for the initialization of the solution algorithm.
Calculate the segment polytropic efficiency using the relationship (45). (7) Calculate efficiency deviation from the initially estimated polytropic efficiency from step 1. (8) Iterate segment outlet temperature to match assumed efficiency.
Note: different root finding algorithms may be used to optimize iterations for segment outlet temperature. Note: the acceptable efficiency convergence should be set less than 1E −6 . (9) Move to next segment using outlet conditions of previous segment as inlet conditions. (10) Repeat steps 5 through 9 for each segment. (11) Compare final discharge temperature with the given value, T d (12) If step 11 result is not within a small tolerance, iterate efficiency and determine new values for coefficients A, B, C and D. (13) Repeat steps 5 through 11 until agreement is reached within the tolerance.
Note: the acceptable convergence from the given discharge temperature should be set less than 1E −8 .
In the Taher-Evans Cubic Polynomial method, intermediate points are determined by setting one thermodynamic variable (such as pressure or entropy) at each end of a segment and calculating other thermodynamic variables to force the segment endpoint to lie on a constant efficiency path via nested recursive algorithms (within a small error tolerance). Using cubic polynomial approximants, the polytropic path slope can be accurately calculated since the first derivative of each segmental polynomial can be determined at each intermediate point (see E i and E i+1 in Equation (42)).
The difference between final segment calculated discharge temperature and the known discharge temperature, T d , can be minimized by iterating efficiency. These nested iterations of efficiency (inner loop) and the final segment discharge temperatures (outer loop) converge quickly.
When a final efficiency is determined, summations of enthalpy changes and Tds losses can be performed for the entire set of segments. The cubic polynomial multi-segment method provides intermediate points along the constant efficiency temperature-entropy path that have been forced to coincide with the actual path. Additionally, the path slope at each intermediate point is equal at the end of the ith segment and the beginning of the i+1 segment. See Table 1.

Distinction between Cubic Interpolation and Approximation
As the final point in this section, it should be noted that the approximation of functions is different from interpolation. In the case of interpolation, at certain intermediate points, the value of an unknown function is given (such as measured values at intermediate points), but the function itself is not known. Interpolating cubic splines are used to interpolate between known intermediate points, which is irrelevant in the problem of approximating an unknown function without any given intermediate point. Accordingly, the descriptor "spline" is not applied in this paper as it usually is thought of as being used to approximate a curve passing through previously known knots.  Table A1 in the Appendix B for gas compositions and compressor inlet and discharge conditions for Case 10.

Polytropic Path on Temperature-Entropy Diagram
For an uncooled compressor section, entropy increases along the polytropic path. The rate of change of temperature with entropy at each point along the polytropic path is determined with the relationship (A10) from Appendix A.
The change of temperature with entropy from the compressor section inlet to discharge varies as temperature, specific heat, c p , and compressibility function, X, change at each point along the compression path. At conditions near to the critical point, the rate of change of c p and X is usually large. This has significant impact on the change of temperature with entropy as the relationship (A10) suggests. The supercritical propane example case has inlet conditions extremely close to the critical point and the thermodynamic properties are varying rapidly as compression begins. Figures 8 and 9 illustrate the variation of specific heat, c p , and compressibility function, X, respectively, in the vicinity of the critical point. Each isobar has a distinct peak. The inlet conditions lie in an area on the flank of a peak while discharge conditions (not shown) are far removed from the critical area peaks. Equations (15) and (16) would show a large difference in slope of the T-s path between inlet, E 1 , and discharge, E 2 , as documented in Table A2. These large differences are one indicator of the difficulty of a compression case.
is different from interpolation. In the case of interpolation, at certain intermediate poin value of an unknown function is given (such as measured values at intermediate point the function itself is not known. Interpolating cubic splines are used to interpolate be known intermediate points, which is irrelevant in the problem of approximating an unk function without any given intermediate point. Accordingly, the descriptor "spline" applied in this paper as it usually is thought of as being used to approximate a curve p through previously known knots.

Polytropic Path on Temperature-Entropy Diagram
For an uncooled compressor section, entropy increases along the polytropic pat rate of change of temperature with entropy at each point along the polytropic path is mined with the relationship (A10) from Appendix A.
The change of temperature with entropy from the compressor section inlet to disc varies as temperature, specific heat, , and compressibility function, , change at each along the compression path. At conditions near to the critical point, the rate of change and is usually large. This has significant impact on the change of temperature wi tropy as the relationship (A10) suggests. The supercritical propane example case ha conditions extremely close to the critical point and the thermodynamic properties are v rapidly as compression begins. Figures 8 and 9 illustrate the variation of specific he and compressibility function, , respectively, in the vicinity of the critical point. Each has a distinct peak. The inlet conditions lie in an area on the flank of a peak while disc conditions (not shown) are far removed from the critical area peaks. Equations (15) an would show a large difference in slope of the T-s path between inlet, , and dischar as documented in Table A2. These large differences are one indicator of the difficult compression case.   The cubic polynomial endpoint method can be used to approximate the concavity actual T-s polytropic path. No other endpoint methods can provide such a mathematic thermodynamic insight about the behavior of the actual T-s polytropic path from k thermodynamic conditions at the inlet and discharge of the compressor section.
The concavity of the polynomial approximant can be examined using the second ative test of the path equation with the following criteria:  The cubic polynomial endpoint method can be used to approximate the concavity of the actual T-s polytropic path. No other endpoint methods can provide such a mathematical and thermodynamic insight about the behavior of the actual T-s polytropic path from known thermodynamic conditions at the inlet and discharge of the compressor section.
The concavity of the polynomial approximant can be examined using the second derivative test of the path equation with the following criteria: 1 T(s), changes at the inflection point The actual location of the inflection point on the T-s polytropic path can be more precisely approximated by applying the cubic polynomial multi-segment method.
The rate of change of dT ds with entropy can be considered as a screening criterion to determine the difficulty of a compression case.
The curve shape of the polytropic path as predicted by the cubic polynomial endpoint method provides an easy to apply criteria to determine how many multi-segments to employ to assure an acceptable relative deviation. Based on 115 cases covering 22 fluids (7 pure fluids and 15 mixtures reviewed in Appendix C) studied by Evans [3], the number of cubic polynomial segments required to achieve ≤0.001% relative deviation for the polytropic efficiency is three segments for Category I and five segments for Categories II and III. The actual location of the inflection point on the T-s polytropic path can be more precisely approximated by applying the cubic polynomial multi-segment method.
The rate of change of with entropy can be considered as a screening criterion to determine the difficulty of a compression case.
The curve shape of the polytropic path as predicted by the cubic polynomial endpoint method provides an easy to apply criteria to determine how many multi-segments to employ to assure an acceptable relative deviation. Based on 115 cases covering 22 fluids (7 pure fluids and 15 mixtures reviewed in Appendix C) studied by Evans [3], the number of cubic polynomial segments required to achieve ≤0.001% relative deviation for the polytropic efficiency is three segments for Category I and five segments for Categories II and III.

Enthalpy Change with Entropy along the Polytropic Path
The relationship (2) can be rearranged to As the efficiency of the polytropic process is constant, the slope of the tangent to the path increases as the temperature rises along the compression path ( > 0). Rearranging the relationship (47) and taking the derivative with respect to gives: By replacing with , the second derivative of the function h(s) along the polytropic compression path appears:

Enthalpy Change with Entropy along the Polytropic Path
The relationship (2) can be rearranged to As the efficiency of the polytropic process is constant, the slope of the tangent to the path increases as the temperature rises along the compression path ( dh ds > 0). Rearranging the relationship (47) and taking the derivative with respect to ds gives: By replacing T 1−η p with dh ds , the second derivative of the function h(s) along the polytropic compression path appears: For an uncooled compressor section, dT ds > 0. Therefore, d 2 h ds 2 > 0. As shown in Figure 1, the polytropic compression path defined with function h(s) is always concave upward along the compression path since the second derivative is positive in an uncooled compressor section. Equation (A11) in Appendix A provides the thermodynamic value of d 2 h ds 2 at any point along the polytropic compression path.
The change of enthalpy with entropy at any point along the polytropic path can be approximated by substituting the relationship of temperature from (12) to (47): By integrating (50), the relationship of enthalpy and entropy can be approximated at any point along the polytropic path: where, s represents any entropy value between compressor section inlet, s 1 and section discharge, s 2 .

Comparison of Cubic and Linear Polynomial Results
The Taher-Evans Cubic Polynomial 10 segment method is used as a comparison standard to determine relative deviations in polytropic efficiency for all other methods. See Equation (A12) for the relative deviation definition. The number of recommended cubic segments based upon T-s path curve shapes listed in Figure 10 was developed from results reported by Evans [3]. Figure 11 shows that determining a similar set of recommendations for the number of linear polynomial steps required was not possible. The wide scatter of the linear methods' results that would yield similar accuracy to the cubic methods was very random. The change of enthalpy with entropy at any point along the polytropic path can b proximated by substituting the relationship of temperature from (12) to (47): By integrating (50), the relationship of enthalpy and entropy can be approximated a point along the polytropic path: where, represents any entropy value between compressor section inlet, and section charge, .

Comparison of Cubic and Linear Polynomial Results
The Taher-Evans Cubic Polynomial 10 segment method is used as a compa standard to determine relative deviations in polytropic efficiency for all other methods Equation (A12) for the relative deviation definition. The number of recommended cubic ments based upon T-s path curve shapes listed in Figure 10 was developed from resul ported by Evans [3]. Figure 11 shows that determining a similar set of recommendation the number of linear polynomial steps required was not possible. The wide scatter o linear methods' results that would yield similar accuracy to the cubic methods was random.  Appendices B and C discuss example cases and compare various polytropic calculation methods. The 19 example cases in Appendix B were selected for illustration of a range of relatively easy and difficult polytropic efficiency calculation applications. Several of these are used to show various characteristics of the calculation results. Appendix C is a review of 115 example cases studied by Evans [3], and includes the 19 example cases from Appendix B. This larger population further confirms and documents the trends discovered when applying the multi-segment cubic and multi-step linear polynomial methods.

•
The Taher-Evans Cubic Polynomial method (TE-CP), defined, described and tested herein, illustrates that a highly accurate calculation method for real gas centrifugal compressor polytropic performance efficiency has been developed and implemented that employs a temperature-entropy cubic polynomial path function.
Both endpoint and sequential segment versions are described and tested. Previously published polytropic efficiency calculation methods including first degree linear polynomial methods are shown to be less accurate and/or slower to achieve a solution than Taher-Evans Cubic Polynomial methods. The TE-CP methods' superior results are due to the cubic polynomial path having additional thermodynamic constraints applied to more accurately approximate the actual path slope at any point along the path. Determination of path slope at the compression endpoints is independent of performance calculation method. T-s polytropic path curve shapes can be determined from derivatives of the cubic polynomial path equation and are indicative of calculation relative difficulty. The curve shape is used as a criterion to select the required number of cubic segments. No other polytropic method can provide such a mathematical and thermodynamic insight into the fluid compression process. The number of T-s path cubic polynomial segments required to achieve an acceptable relative deviation ≤0.001% for polytropic efficiency is three segments for category I and five segments for categories II and III curve shapes. As shown in Figure A10, the cubic polynomial methods provide low uncertainty with only a few cubic segments. Uncertainty is based upon polytropic efficiency relative deviation, and its magnitude is related to the size of the statistical population. Cubic polynomial methods provide continuous equations to plot the polytropic path on T-s and h-s diagrams. This is a very unique feature of the cubic polynomial method that enables visualizing the polytropic path. The cubic path coefficients (A, B, C, and D) in Equation (12)   Cubic polynomial endpoint path methods were demonstrated to achieve better accuracy than any other endpoint polytropic efficiency calculation method. Cubic polynomial sequential segment path methods were demonstrated to achieve high accuracy acceptable polytropic efficiency results significantly faster with fewer segments required than linear polynomial path methods based upon 115 compression example cases. For all 115 example cases, T-s polytropic paths agreed between 10 cubic segments and 100 linear steps which confirmed the validity of using the result of 10-segment cubic polynomial approximants as the standard for comparison for polytropic efficiency relative deviation and uncertainty. Cubic polynomial path methods show that the required number of segments to achieve an acceptable result remains within a maximum of five segments or less for all the cases studied. For the linear polynomial path method, the required number of steps did not follow a regular pattern as shown in Figure  11. No rule or algorithm can be determined for the required number of linear polynomial steps to achieve an acceptable result. While both cubic and linear polynomial methods can reach acceptable results, the cubic methods require fewer repetitive iterations and are completed much faster. Sequential segment cubic polynomial method calculations are typically at least an order of magnitude faster than sequential step linear polynomial methods. This is a clear advantage for cubic methods when processing compressor performance test data in real time. The cubic polynomial methods provide an order of magnitude lower uncertainty with significantly fewer cubic segments than linear polynomial methods. (See Figure A10). The cubic polynomial methods' uncertainty rapidly decreases as the number of segments increases while the linear polynomial methods illustrate a much more gradual decrease in uncertainty as they approach an asymptote. The linear polynomial methods uncertainty asymptote appears to be more than an order of magnitude greater than the cubic methods.
• Identical calculations for selected example cases using several different equations of state to obtain thermodynamic state point data for application of cubic and linear polynomial methods revealed significant differences in results for efficiency can occur based upon the chosen EOS. See Appendix D for selective examples.

Recommendations
• When rapid, highly accurate, low uncertainty, centrifugal compressor polytropic efficiency calculations are required, the Taher-Evans Cubic Polynomial methods (TE-CP) should be used. • Due to significant differences found between results when employing various equations of state, it is recommended that a project use the same equation of state throughout the project timeline to avoid potential discrepancies.
10 T 6 (s) indicates the cubic polynomial approximant, which is used to approximate the segment number 6 of a polytropic path that is divided into 10 segments. This symbology provides a clear way to compare different methods discussed in this paper. inf inflection p Polytropic, or pressure r ratio d compressor section discharge (measured) 1 compressor section inlet 2 compressor section discharge

Appendix A. Temperature Change with Entropy along the Polytropic Path
Derivation of the following two important relationships, dT ds and d 2 h ds 2 along the polytropic compression path is explained in this appendix.
Choose the entropy to be a function of temperature and pressure. Writing the differential of s = s(T, P) gives: The first partial derivative in (A1) can be written as follows: where ∂h ∂T P = c P and ∂h ∂s P = T from the first partial derivate of the Gibbs relationship dh = Tds + vdP.
The second partial derivative in (A1) is one of the Maxwell relations: Substituting the first and second partial derivatives in (A1) from (A2) and (A3) yields: Using the definition of isobaric expansivity β = 1 υ ∂v ∂T p for the second partial derivative gives: Employing the Gibbs relationship dh = Tds + vdP for a polytropic process and substituting dh with vdP η p gives: vdP = Tds Substituting vdP from (A5) to (A4) yields: This can be further simplified to From (A8) the slope of the polytropic path on a T-s diagram is determined: Substituting (βT − 1) with the Schultz compressibility function X gives: By replacing dT ds from (A10) to (47) gives: The relationships (A10) and (A11) were developed and presented for the first time by Taher [1].
The relationship (A11) shows that the polytropic path on the enthalpy-entropy diagram is concave upward since the second derivative is positive in an uncooled compressor section.

Appendix B. Comparative Numeric Examples of Application of the Taher-Evans Cubic Polynomial Methods
The Taher-Evans Cubic Polynomial method for calculation of centrifugal compressor polytropic performance has been applied to 19 compression example cases which are comprised of five pure fluids and four hydrocarbon fluid mixtures. Eight cases are considered low pressure while eleven cases are considered medium or high pressure. Input data for ten cases came from existing literature references while nine cases are new. All cases were chosen based upon the desire to explore a wide range of process conditions for fluids common to the hydrocarbon industry.
The fluids and process conditions selected represent a combination of relatively easy and difficult calculations for centrifugal compressor polytropic performance. Table A1 lists input data for the 19 cases. The "easy cases" are found by the results listed below to be low pressure versions for CO 2 , C 3 H 8 and CH 4 /C 3 H 8 /CO 2 . The most difficult case is supercritical propane due to inlet conditions.
Using results of the above referenced analyses, this appendix will review polytropic efficiency calculation method comparisons for following three categories: • Three point methods Cubic Polynomial two segment; Huntington's three point [13].
The Taher-Evans Cubic Polynomial 10 segment method is used as a comparison standard to determine relative deviations in polytropic efficiency for all other methods. The validity of using this standard is confirmed below. Equation (A12) documents the relative deviation calculation.
Relative Deviation, % = 100 × η TE-CP 10 Segment − η p η TE-CP 10 Segment (A12) Industry standardized compressor data sheets document operating conditions and performance characteristics with polytropic efficiency shown as a percentage, typically with two decimal places of accuracy, such as 68.25%. One criterion applied in the example cases below is that the deviation should be known to be less than 0.001% of an acceptable standard. Since all polytropic efficiency calculation methods are approximations, a standard determines the magnitude of the difference between two calculations.
As mentioned in the paper (see also Appendix D), polytropic performance results can be impacted by the choice of an equation of state (EOS). The REFPROP (version 10) software package [28] produced by the National Institute of Science and Technology (NIST) was employed for calculations in this appendix. REFPROP provides reference quality, multi parameter, real gas equations of state for thermodynamic and transport properties of individual fluids. REFPROP applies optimized binary interaction parameters via mixing rules to determine mixture properties.
When using the REFPROP software package, all example case inlet and discharge state point phases are single phase and either superheated or supercritical. For some example cases, the close proximity of inlet conditions to a fluid's critical point yielded two phase or liquid phases for some other equations of state used in Appendix D. The polytropic performance methods discussed in this paper are neither designed nor recommended for two phase flows. The impact of equation of state choices will be reviewed in Appendix D. Appendix B.1. Endpoint Methods Schultz [6] described several alternative versions of his methodology in his original ASME paper all of which applied a path equation of pv n = c. The variations involved calculating his "polytropic head factor", f , in various ways as well as multiple methods of averaging his "polytropic volume exponent", n. For the comparisons below, the following two variations of Schultz's method are included for examination since they were included in the 1997 version of ASME PTC10 [4]. Mallen and Saville [9] employed a path equation of Tds dT = c. Path equations are described above in this paper for the liner and cubic polynomial methods. Figure A1a-e illustrate polytropic efficiency relative deviation results for endpoint methods for the 19 example cases. The Taher-Evans endpoint method provides superior results compared to the other endpoint methods for all example cases. Example cases PTC10 CO 2 , LP CO 2 , LP propane and LP CH 4 /C 3 H 8 /CO 2 are considered easy cases since the Taher-Evans Cubic Polynomial endpoint method achieves deviations from the 10-segment method of less than 0.001%. Thus, using multi-segment analyses would not be required for these cases. The two Schultz based methods show significant errors for some example cases and do not exhibit a consistent pattern. Figure A1f directly compares two related cases in that the PTC10 CO 2 case represents a Type 2 test designed for the HPNG example case. The PTC10 CO 2 case is a low-pressure case and deviations are seen to be smaller than for the HPNG case as expected. However, the Taher-Evans endpoint method provides better accuracy for the PTC 10 HPNG case. The HPNG case requires more segments to achieve less than a 0.001% deviation level.
Notable comparisons for endpoint methods: • Figure A1a HP ethylene; Mallen and Saville [9] highlighted their improvements over Schultz's method for high pressure, but still have an approximate 1% deviation; LP and HP ethylene; typical and expected patterns for cubic and linear polynomials.
• Figure A1b,c CO 2 and propane; as pressure increases accuracy decreases is a typical pattern PTC10; the HPNG case will require more cubic segments to achieve sufficient accuracy than the Type 2 test design using CO 2  Figure A1.
(a-f) Summary comparison of relative deviation results for endpoint methods. Table A2 lists the T-s path inlet and discharge slope and curve category for the 19 example cases. Five cases show that an inflection point in curvature exists along the path. For these five cases, endpoint analyses do not provide sufficient polytropic efficiency accuracy and multi-segment analyses are recommended. Changes in path slope and curvature can be used as criteria for determining how many cubic segments are needed to provide sufficient accuracy (See Section 4, Figure 10 and Appendix C).  Table A2 lists the T-s path inlet and discharge slope and curve category for the 19 example cases. Five cases show that an inflection point in curvature exists along the path. For these five cases, endpoint analyses do not provide sufficient polytropic efficiency accuracy and multi-segment analyses are recommended. Changes in path slope and curvature can be used as criteria for determining how many cubic segments are needed to provide sufficient accuracy (See Section 4, Figure 10 and Appendix C).  Figure A2a-e provide a summary of the polytropic efficiency deviation results comparing the Taher-Evans Cubic Polynomial 2 segment method with Huntington's three-point method [13]. The two-segment cubic polynomial method shows sufficient accuracy for 13 of 19 example cases while Huntington's three-point method shows sufficient accuracy for only 7 of 19 example cases based upon the 0.001% requirement. In all example cases, the cubic polynomial method is more accurate than Huntington's method. Figure A2f compares PTC10 CO 2 and HPNG cases. Both cubic and the Huntington's three-point method are sufficient for the low-pressure test case but only the cubic polynomial is sufficient for the HPNG case. Table A3 lists the polytropic efficiency calculated for Taher-Evans Cubic Polynomial 2 through 10 multi-segment methods, and linear polynomial 10, 20, 50, 90 and 100 multi-step methods. The general trend is that using more segments or steps yields higher accuracy. In many example cases, as more segments or steps are applied, the efficiency only changes beyond the fourth decimal place as highlighted in the table by the outlined values. The results are shown to have "stabilized" to four decimal places for additional segments or steps. In all example cases, the highest accuracy shown agrees between cubic and linear methods, but the linear method requires many more steps before "stabilizing". The general trend of higher-pressure cases requiring more segments or steps to stabilize is confirmed.

Appendix B.3. Multi-Step and Multi-Segment Methods
For the supercritical ethane example case, the cubic method demonstrates a polytropic efficiency of 79.9519% for three segments and remains at that value for the four decimal places shown as the number of segments is increased. The linear method needs between 50 and 90 steps to reach the same accuracy for efficiency. In other words, the linear method approaches an asymptote much more slowly than the cubic method at the cost of increased computing resources and elapsed time to reach a solution. This illustrates the usefulness of the cubic method in being able to take advantage of matching the slope of the T-s path between sequential segments. Both the cubic and linear methods are approximants that can reach essentially the same level of approximation, but the cubic method employs more thermodynamic constraints to achieve the resulting polytropic efficiency much quicker.
It is not necessary to reach a "stabilized" polytropic efficiency to satisfy the maximum allowable deviation requirement of 0.001% as compared to the Taher-Evans Cubic Polynomial 10 segment results. Deviations are less than 0.001% for 13 of 19 example cases using 2 cubic segments while only 4 of 19 are less than 0.001% for 10 linear steps.  Table A3 lists the polytropic efficiency calculated for Taher-Evans Cubic Polynomial 2 through 10 multi-segment methods, and linear polynomial 10, 20, 50, 90 and 100 multi-step methods. The general trend is that using more segments or steps yields higher accuracy. In many example cases, as more segments or steps are applied, the efficiency only changes beyond the fourth decimal place as highlighted in the table by the outlined values. The results are shown to have "stabilized" to four decimal places for additional segments or steps. In all example cases, the highest accuracy shown agrees between cubic and linear methods, but the Figure A2. (a-f) Summary comparison of relative deviation results for three point methods.  Figure A3a illustrates polytropic efficiency relative deviation results for all cubic multisegment methods and linear multi-step methods for the supercritical propane example case. The more difficult an example case is, the more segments or steps it takes to achieve a relative deviation of 0.001% or less. This most difficult case achieves this target with 5 segments for the cubic polynomial method while it would take 44 steps for the linear polynomial method to reach the same deviation value as the 5 cubic segment method. The cubic polynomial method does not suffer from the requirement of a small step size for each segment as does the linear polynomial method. Figure A3b shows the elapsed calculation times for 5 cubic segments and 44 linear steps are approximately 3.5 s and 55.8 s, respectively. Thus, the linear method takes 16 times longer to reach the same level of accuracy for this very difficult example case. For the six example cases that required more than 2 cubic segments to reach a deviation less of than 0.001%, the elapsed time ratio varied from 5 to 16 times longer for the linear method.

Appendix B.3. Multi-Step and Multi-Segment Methods
The elapsed time to reach a solution is dependent upon the algorithm employed to resolve the inner and outer nested, recursive loops for discharge temperature and efficiency. The algorithm used for both cubic and linear method results in this appendix is identical. Thus, while another algorithm might reach solutions faster or slower, the ratio between cubic and linear method elapsed times should remain approximately the same for a given example case.
deviation of 0.001% or less. This most difficult case achieves this target with 5 segments for the cubic polynomial method while it would take 44 steps for the linear polynomial method to reach the same deviation value as the 5 cubic segment method. The cubic polynomial method does not suffer from the requirement of a small step size for each segment as does the linear polynomial method. Figure A3b shows the elapsed calculation times for 5 cubic segments and 44 linear steps are approximately 3.5 s and 55.8 s, respectively. Thus, the linear method takes 16 times longer to reach the same level of accuracy for this very difficult example case. For the six example cases that required more than 2 cubic segments to reach a deviation less of than 0.001%, the elapsed time ratio varied from 5 to 16 times longer for the linear method.
The elapsed time to reach a solution is dependent upon the algorithm employed to resolve the inner and outer nested, recursive loops for discharge temperature and efficiency. The algorithm used for both cubic and linear method results in this appendix is identical. Thus, while another algorithm might reach solutions faster or slower, the ratio between cubic and linear method elapsed times should remain approximately the same for a given example case. Figures A4a-e illustrate polytropic efficiency relative deviation results for all cubic multi-segment methods and linear multi-step methods for the 19 example cases. The 10 cubic segment results are used as the standard for comparison. For each case, fewer cubic segments are needed than linear steps to achieve the 0.001% required accuracy. Figure A4f directly compares the HPNG case and the PTC10 CO2 Type 2 performance test designed to prove the compressor should meet the design requirements within the tolerances allowed by the ASME  Figure A4a-e illustrate polytropic efficiency relative deviation results for all cubic multi-segment methods and linear multi-step methods for the 19 example cases. The 10 cubic segment results are used as the standard for comparison. For each case, fewer cubic segments are needed than linear steps to achieve the 0.001% required accuracy. Figure A4f directly compares the HPNG case and the PTC10 CO 2 Type 2 performance test designed to prove the compressor should meet the design requirements within the tolerances allowed by the ASME PTC10 code [4].

Appendix B.4. High Pressure Ethylene Example
This example was first created by Mallen and Saville [9] to demonstrate differences between their polytropic work calculation method and Schultz's method [6] at higher pressures. Since then, it has been used by many authors as a strong test of other methods due to its large pressure and temperature ranges. The high compression ratio and extreme discharge temperature for this case illustrate it would likely not be feasible to build a single section compressor to accomplish this application. However, its use as an example to test calculation methods has been documented.
Ethylene has been described as an unruly fluid by Brown, [29]. Evans and Huble, [22], postulated Brown's comment could be due to the proximity of ethylene's critical temperature to usual inlet temperature ranges for common applications causing rapid, large changes in Figure A4. (a-f) Summary comparison of results for linear multi-step and cubic polynomial multi-segment methods.

Appendix B.4. High Pressure Ethylene Example
This example was first created by Mallen and Saville [9] to demonstrate differences between their polytropic work calculation method and Schultz's method [6] at higher pressures. Since then, it has been used by many authors as a strong test of other methods due to its large pressure and temperature ranges. The high compression ratio and extreme discharge temperature for this case illustrate it would likely not be feasible to build a single section compressor to accomplish this application. However, its use as an example to test calculation methods has been documented.
Ethylene has been described as an unruly fluid by Brown, [29]. Evans and Huble, [22], postulated Brown's comment could be due to the proximity of ethylene's critical temperature to usual inlet temperature ranges for common applications causing rapid, large changes in specific heat, c p , and especially density with small inlet temperature changes. In the vicinity of the critical point, large changes in inlet density can occur due to small changes in inlet temperature causing a centrifugal compressor's operating point to move from one side of its performance map to the other without any process operator control changes.
The Taher-Evans Cubic Polynomial method provides the ability to calculate and graphically display the T-s values at any point along the constant polytropic efficiency path between inlet and discharge. The HP ethylene example case serves to demonstrate the method's usefulness to visualize the path, its curvature and an inflection point. Figure A5a shows paths for 1 and 10 segment cubic polynomial methods and 1 and 100 step linear polynomial methods. The 100-step linear method's orange markers are seen to lie on top of the solid blue 10-segment cubic path. The one segment cubic dotted blue line shows an inflection point near the center of the graph. The 10-segment cubic polynomial path begins with an upward concavity that transitions to downward at the inflection point. The 100 step linear markers follow this pattern illustrating the accuracy of the cubic polynomial method's path. Figure A5b shows an expanded view of the area outlined by a black box in Figure A5a. The inflection point of the 10-segment cubic polynomial path is shown to occur at a slightly higher temperature for the more refined calculation. The 10-segment cubic polynomial begins with 5 upward concave segments, and ends with 4 downward concave segments. The 6th segment is "s" shaped since it starts out concave upward and shifts at the inflection point to concave downward. The curvature of the cubic segments is determined by values of the derivatives of the T-s path equation. The point where the blue 10-segment curve crosses the orange dotted linear endpoint line is not an inflection point, but rather just a crossing of the upward concave segments of the 10-segment cubic method path. The Taher-Evans Cubic Polynomial method provides the ability to calculate and graphically display the T-s values at any point along the constant polytropic efficiency path between inlet and discharge. The HP ethylene example case serves to demonstrate the method's usefulness to visualize the path, its curvature and an inflection point. Figure A5a shows paths for 1 and 10 segment cubic polynomial methods and 1 and 100 step linear polynomial methods. The 100-step linear method's orange markers are seen to lie on top of the solid blue 10segment cubic path. The one segment cubic dotted blue line shows an inflection point near the center of the graph. The 10-segment cubic polynomial path begins with an upward concavity that transitions to downward at the inflection point. The 100 step linear markers follow this pattern illustrating the accuracy of the cubic polynomial method's path. Figure A5b shows an expanded view of the area outlined by a black box in Figure A5a. The inflection point of the 10-segment cubic polynomial path is shown to occur at a slightly higher temperature for the more refined calculation. The 10-segment cubic polynomial begins with 5 upward concave segments, and ends with 4 downward concave segments. The 6th segment is "s" shaped since it starts out concave upward and shifts at the inflection point

Appendix B.5. Supercritical Propane Example
The results shown in Table A3 illustrated that the supercritical propane example case is the most difficult. The following discussion uses this case to shed light upon reasons performance calculations are more difficult for certain applications. Figures A6a,b are pressuretemperature and pressure-enthalpy diagrams for this example. Figure A6a shows inlet conditions are very near the critical point such that the reduced pressure and reduced temperature are very near 1.0. While these inlet conditions might not be recommended in normal compression industry practice, the example serves as an "acid test" for polytropic calculation methods. Figure A6b shows isotherms in the dense phase region are nearly horizontal at inlet but are nearly vertical at discharge. For the cubic polynomial compression curve shown in the figure, near the inlet, pressure changes are more rapid than enthalpy changes while the opposite is true near discharge conditions giving the path its shape. Near discharge, one value of enthalpy could be associated with two values of pressure at the same temperature making an EOS root finding algorithm more difficult to converge. Figures 8 and 9 in the main body of the paper illustrate the rapid variation of thermodynamic properties and in the vicinity of the critical point for this application. This causes large differences of path slope between the inlet and discharge ( and ).

Appendix B.5. Supercritical Propane Example
The results shown in Table A4 illustrated that the supercritical propane example case is the most difficult. The following discussion uses this case to shed light upon reasons performance calculations are more difficult for certain applications. Figure A6a,b are pressure-temperature and pressure-enthalpy diagrams for this example. Figure A6a shows inlet conditions are very near the critical point such that the reduced pressure and reduced temperature are very near 1.0. While these inlet conditions might not be recommended in normal compression industry practice, the example serves as an "acid test" for polytropic calculation methods. Figure A6b shows isotherms in the dense phase region are nearly horizontal at inlet but are nearly vertical at discharge. For the cubic polynomial compression curve shown in the figure, near the inlet, pressure changes are more rapid than enthalpy changes while the opposite is true near discharge conditions giving the path its shape. Near discharge, one value of enthalpy could be associated with two values of pressure at the same temperature making an EOS root finding algorithm more difficult to converge. Figures 8 and 9 in the main body of the paper illustrate the rapid variation of thermodynamic properties c p and X in the vicinity of the critical point for this application. This causes large differences of path slope between the inlet and discharge (E 1 and E 2 ).   Figure A7 illustrates the T-s path determined by four methods for the supercritical propane example case. The cubic and linear polynomial endpoint methods are shown with dotted lines. The 10 segment Taher-Evans Cubic Polynomial method is shown with a solid blue line. The 100-step linear method is shown with markers which lay on top of the cubic polynomial line. The endpoint methods clearly do not provide a sufficiently accurate path for this example case. The 10-segment cubic polynomial and the 100-step linear polynomial agree on the path.  Figure A7 illustrates the T-s path determined by four methods for the supercritical propane example case. The cubic and linear polynomial endpoint methods are shown with dotted lines. The 10 segment Taher-Evans Cubic Polynomial method is shown with a solid blue line. The 100-step linear method is shown with markers which lay on top of the cubic polynomial line. The endpoint methods clearly do not provide a sufficiently accurate path for this example case. The 10-segment cubic polynomial and the 100-step linear polynomial agree on the path. The supercritical propane example case serves to illustrate the accuracy of the 10 segment Taher-Evans Cubic Polynomial method even when inlet conditions are near the critical point of a fluid.

Appendix C. Overview of 115 Example Case Results
Evans [3] provided a broad set of 115 example cases that compares endpoint and multistep linear polynomial solutions to endpoint and multi-segment cubic polynomial solutions. This set covers 22 fluids (7 pure fluids and 15 mixtures) with a wide range of mole weight and critical point properties. The 19 example cases discussed in Appendix B above were included in the 115 example cases. Seventeen polytropic efficiency calculation methods were included which yielded a total of 1955 independent calculations. This effort is sufficient to document the validity of the results for the polytropic performance methods employed and gives a robust comparison basis for the conclusions and recommendations included in this paper.
One consistent result for all 115 example cases matches the trend shown in Figure A7 above in that the 10 segment cubic polynomial temperature-entropy based polytropic path and the 100 step linear polynomial path are matched. If fact, these two methods agreed to four decimal places for polytropic efficiency. The figures and tables below summarize the The supercritical propane example case serves to illustrate the accuracy of the 10 segment Taher-Evans Cubic Polynomial method even when inlet conditions are near the critical point of a fluid.

Appendix C. Overview of 115 Example Case Results
Evans [3] provided a broad set of 115 example cases that compares endpoint and multi-step linear polynomial solutions to endpoint and multi-segment cubic polynomial solutions. This set covers 22 fluids (7 pure fluids and 15 mixtures) with a wide range of mole weight and critical point properties. The 19 example cases discussed in Appendix B above were included in the 115 example cases. Seventeen polytropic efficiency calculation methods were included which yielded a total of 1955 independent calculations. This effort is sufficient to document the validity of the results for the polytropic performance methods employed and gives a robust comparison basis for the conclusions and recommendations included in this paper.
One consistent result for all 115 example cases matches the trend shown in Figure A7 above in that the 10 segment cubic polynomial temperature-entropy based polytropic path and the 100 step linear polynomial path are matched. If fact, these two methods agreed to four decimal places for polytropic efficiency. The figures and tables below summarize the superior results provided by the cubic polynomial methods for all of the 115 example cases due to much more rapid solutions requiring fewer computing resources. Figure A8 illustrates that the cubic multi-segment and linear multi-step methods both approach an asymptote as the number of segments and steps increase. The cubic polynomial methods approach the asymptote much more rapidly yielding an acceptable solution in less than 10 segments while the linear polynomial methods would require more than 50 steps to provide an equivalent solution. The cubic and linear methods approach the asymptote from opposite directions for this methane, propane, and carbon dioxide mixture example case. superior results provided by the cubic polynomial methods for all of the 115 example cases due to much more rapid solutions requiring fewer computing resources. Figure A8 illustrates that the cubic multi-segment and linear multi-step methods both approach an asymptote as the number of segments and steps increase. The cubic polynomial methods approach the asymptote much more rapidly yielding an acceptable solution in less than 10 segments while the linear polynomial methods would require more than 50 steps to provide an equivalent solution. The cubic and linear methods approach the asymptote from opposite directions for this methane, propane, and carbon dioxide mixture example case. Figure A8. Asymptotic approach to a solution.
For the 115 example cases, 114 reached an acceptable relative deviation of ≤0.001% in three or less cubic polynomial segments as documented in Table A3. Only one example case required 5 cubic segments due to inlet conditions being extremely close to the critical point, which is unlikely to occur in reality. Based upon an analysis of these results, the recommended number of segments required for a particular application can be related to the path categories defined in Figure 10 above.
For the 115 example cases, 114 reached an acceptable relative deviation of ≤0.001% in three or less cubic polynomial segments as documented in Table A4. Only one example case required 5 cubic segments due to inlet conditions being extremely close to the critical point, which is unlikely to occur in reality. Based upon an analysis of these results, the recommended number of segments required for a particular application can be related to the path categories defined in Figure 10 above.
Recall that Figure 11 shows that determining a similar set of recommendations for the number of linear polynomial steps required was not possible. The wide scatter of the linear methods results that would yield similar accuracy results to the cubic methods was very random.  Figure A9 shows dissimilar patterns for average elapsed calculation time between cubic segments and linear steps. The cubic polynomial solutions average time per segment decreases with an increasing number of segments. However, the linear polynomial solutions pattern shows an increasing time per step as the number of steps increases. The averages are over all 115 example cases. Recall that Figure 11 shows that determining a similar set of recommendations for the number of linear polynomial steps required was not possible. The wide scatter of the linear methods results that would yield similar accuracy results to the cubic methods was very random. Figure A9 shows dissimilar patterns for average elapsed calculation time between cubic segments and linear steps. The cubic polynomial solutions average time per segment decreases with an increasing number of segments. However, the linear polynomial solutions pattern shows an increasing time per step as the number of steps increases. The averages are over all 115 example cases. Figures A8 and A9 combined show that the Taher-Evans Cubic Polynomial polytropic path methods are superior. While both cubic and linear polynomial methods can reach acceptable results, the cubic methods require fewer repetitive iterations and are completed much faster. This is a clear advantage when processing test data in real time. Figure A10 compares polytropic efficiency calculation uncertainty for cubic and linear polynomial methods. Both exhibit decreasing uncertainty with an increase in cubic segments or linear steps. However, the cubic polynomial methods uncertainty rapidly decreases as the number of segments increases while the linear polynomial methods illustrate a much more gradual decrease as they approach an asymptote. The linear methods uncertainty asymptote appears to be more than an order of magnitude greater than the uncertainty of the 9-segment cubic method. A comparison of uncertainty between 115 and 19 example case statistical populations revealed that uncertainty decreased with a larger population for both cubic and linear polynomial methods. However, cubic methods had lower uncertainty than linear methods for both population sizes. Figures A8 and A9 combined show that the Taher-Evans Cubic Polynomial polytropic path methods are superior. While both cubic and linear polynomial methods can reach acceptable results, the cubic methods require fewer repetitive iterations and are completed much faster. This is a clear advantage when processing test data in real time. Figure A10 compares polytropic efficiency calculation uncertainty for cubic and linear polynomial methods. Both exhibit decreasing uncertainty with an increase in cubic segments or linear steps. However, the cubic polynomial methods uncertainty rapidly decreases as the number of segments increases while the linear polynomial methods illustrate a much more gradual decrease as they approach an asymptote. The linear methods uncertainty asymptote appears to be more than an order of magnitude greater than the uncertainty of the 9-segment cubic method. A comparison of uncertainty between 115 and 19 example case statistical populations revealed that uncertainty decreased with a larger population for both cubic and linear polynomial methods. However, cubic methods had lower uncertainty than linear methods for both population sizes.

Appendix D. Equation of State Impact on Polytropic Efficiency
When a compression application is first designed by a user, it typically involves a cess simulator that has many equations of state available. One is selected from those avai within a specific simulator software package based upon the overall project Type. When ious vendors receive an inquiry for a compressor containing the necessary inlet and disch state point operating conditions and other pertinent details, each vendor may select a d ent equation of state for their detailed compressor design according to individual experi Using different equations of state can yield variations in predicted performance. The s holds true for analyzing compressor test data to verify conformance to testing codes an contractual guarantees.
A user is encouraged to require the use of a consistent equation of state througho project by all parties involved from inception to final testing and operation to minimi avoid EOS related discrepancies. It should be pointed out that variations exist in the im mentation of similarly named equations of state across various software platforms. Sh these variations cause issues, they should be investigated and documented.
All polytropic efficiency calculation results shown in this Appendix D were perfor with the Taher-Evans Cubic Polynomial endpoint method. These results are influence thermodynamic state point parameters calculated by the various equations of state. From main text of this paper, the required equations are (15), (16) and (32).
For given inputs of fluid composition, total pressure and total temperature at inlet discharge the parameters required to be calculated by an EOS to determine polytropic ciency are:

Appendix D. Equation of State Impact on Polytropic Efficiency
When a compression application is first designed by a user, it typically involves a process simulator that has many equations of state available. One is selected from those available within a specific simulator software package based upon the overall project Type. When various vendors receive an inquiry for a compressor containing the necessary inlet and discharge state point operating conditions and other pertinent details, each vendor may select a different equation of state for their detailed compressor design according to individual experience. Using different equations of state can yield variations in predicted performance. The same holds true for analyzing compressor test data to verify conformance to testing codes and/or contractual guarantees.
A user is encouraged to require the use of a consistent equation of state throughout a project by all parties involved from inception to final testing and operation to minimize or avoid EOS related discrepancies. It should be pointed out that variations exist in the implementation of similarly named equations of state across various software platforms. Should these variations cause issues, they should be investigated and documented.
All polytropic efficiency calculation results shown in this Appendix D were performed with the Taher-Evans Cubic Polynomial endpoint method. These results are influenced by thermodynamic state point parameters calculated by the various equations of state. From the main text of this paper, the required equations are (15), (16) and (32).
For given inputs of fluid composition, total pressure and total temperature at inlet and discharge the parameters required to be calculated by an EOS to determine polytropic efficiency are:
Even though various EOS may use different reference states for enthalpy and entropy, it is the change in these parameters that are used in Equation (32), therefore differences in individual reference states for different EOS is not an issue for the purpose of this study. REFPROP, NIST reference fluid thermodynamic and transport properties database [28,36,37] Employs individual, reference quality EOS for each pure fluid; Applies mixing rules to the Helmholtz energy of the mixture components; REFPROP is a program, not a database of measurements.
Results for the first four EOS above were derived from Aspen HYSYS V. 10 [27] while the REFPROP results were calculated via a spreadsheet with the EOS embedded. Figure A11 illustrates variations in polytropic efficiency for seven example cases from Appendix B using the five equations of state listed above. These EOS induced polytropic efficiency variations can exceed variation due to measurement uncertainty values in some cases. The supercritical propane example case results are misleading in that only REFPROP considered the inlet conditions to be superheated (see Figure A6a). The EOS used within the process simulator considered the fluid to be either two phase or liquid. This serves to illustrate the care that must be exercised in selecting an EOS when inlet conditions are near the critical point of a fluid.
Other than the supercritical propane example case, the largest polytropic efficiency variation of 12.75% is seen for the MP CH 4 /C 3 H 8 /CO 2 example case. The figure clearly demonstrates the need to use a consistent approach to selection of an equation of state for compression calculations. Even though various EOS may use different reference states for enthalpy and entropy, it is the change in these parameters that are used in Equation (32), therefore differences in individual reference states for different EOS is not an issue for the purpose of this study.
The equations of state investigated are: REFPROP, NIST reference fluid thermodynamic and transport properties database [28,36,37] o Employs individual, reference quality EOS for each pure fluid; o Applies mixing rules to the Helmholtz energy of the mixture components; o REFPROP is a program, not a database of measurements.
Results for the first four EOS above were derived from Aspen HYSYS V. 10 [27] while the REFPROP results were calculated via a spreadsheet with the EOS embedded. Figure A11 illustrates variations in polytropic efficiency for seven example cases from Appendix B using the five equations of state listed above. These EOS induced polytropic efficiency variations can exceed variation due to measurement uncertainty values in some cases. The supercritical propane example case results are misleading in that only REFPROP considered the inlet conditions to be superheated (see Figure A6a). The EOS used within the process simulator considered the fluid to be either two phase or liquid. This serves to illustrate the care that must be exercised in selecting an EOS when inlet conditions are near the critical point of a fluid.
Other than the supercritical propane example case, the largest polytropic efficiency variation of 12.75% is seen for the MP CH4/C3H8/CO2 example case. The figure clearly demonstrates the need to use a consistent approach to selection of an equation of state for compression calculations. Figure A11. Variations of the calculated polytropic efficiency using different equations of state for seven example cases. Taher-Evans Cubic Polynomial endpoint method is used for the calculations.