The Sensitivity of k-Values on Compressor Performance

One of the most important physical properties of a gas is the ratio of specific heats.  It is used in the design and evaluation of many processes.  For compressors, it is used in the design of components and determination of the overall performance of the machine.  Engineers are frequently asked to evaluate a compressor performance utilizing traditional equations of head, power and discharge temperature.  While these simplified equations may not give exact results, they give useful information needed to troubleshoot a machine, predict operating conditions, or a long-term trend analysis.  The accuracy of the performance information will depend on the proper selection of the ratio of specific heats.  This Tip of the Month (TOTM) will investigate the application of the ratio of specific heats to compressors, its sensitivity to the determination of machine performance and give recommendations for improved accuracy.

Background of k-value

The ratio of specific heats is a physical property of pure gases and gas mixtures and is known by many other names including: adiabatic exponent, isentropic exponent, and k-value.   It is used to define basic gas processes including adiabatic and polytropic compression.  It also appears in many of the traditional equations commonly used to determine a compressor head, gas discharge temperature, gas power, and polytropic exponent.  The k-value also influences the operating speed of a compressor, but we will simplify the present analysis by deleting speed from our evaluation.  The following commonly used compressor performance equations show how the k-value is utilized in the design and evaluation of compressors.

Note:    The actual Z-value will vary from the suction to discharge conditions.  Z_S is sometimes replaced with Z_AVE to approximate the variations in compressibility value [1, 5]. See the nomenclature at the end of this TOTM.

The above equations are written in terms of the adiabatic process with the exception of Equation 5, which refers to the polytropic process.  Both compression processes are similar and will give the same actual results.  The adiabatic and polytropic methods are extensively used by manufacturers to design compressors, and make use of k-values to calculate their performance.  However, as will be seen, the effect of the k-value and the calculated results will influence both compression processes alike.  For simplicity, this Tip of the Month will use the adiabatic process.
It can be seen from Equations 1-5 that the k-value has an effect on a compressor head, temperature, power, and polytropic exponent.  In order to determine how small changes in the k-value can influence a compressor performance, let us first define the k-value of a pure gas.  The thermodynamic definition of a gas k-value is given by Equation 6.  It shows the relationship to the specific heat at constant volume, C_V and specific heat at constant pressure, C_P.  Both values vary with temperature and pressure.

For a pure gas there are many references that give C_P and C_V values at various conditions.  One useful source is National Institute of Standards and Technology.  Their website is http://webbook.nist.gov/chemistry/fluid/

The method of determining the k-value for gas mixtures is more complex.  The major difference is that a gas mixture does not behave as any one of its components but as an “equivalent” gas.  Therefore, to determine the k-value of the mixture, we must know the mole fraction of each component, Y_i and the molar specific heat at constant pressure for each component, M C_Pi.   Equation 7 can be used to determine the k-value of an ideal gas mixture [1, 5].  Real gases may deviate from the calculated value.

While Equations 1-7 are applicable for manual calculations methods, it is important to note that process simulation packages determine the compressor head and discharge temperature utilizing equations of state.  The results are the same but the methods are very different.

K-value Sensitivity Analysis

In the compression process the temperature and pressure of the process gas both increase.  Not knowing what k-value to select for evaluating the compression process can lead to errors.  For example, a typical propane compressor may have a k-value at suction conditions of 1.195.  At the compressor discharge conditions the k-value is 1.254.  The difference in the two values varies by 4.94 percent and can have a significant influence in the performance evaluation.  The following example illustrates how minor changes in the k-value can influence the calculated compressor head, temperature, power and the polytropic coefficient.

Example 1: A natural gas compressor is operating at the conditions given below.  Only the k-value is varied from 1.20 to 1.28, all other given parameters remain constant.   Figure 1 illustrates how the “apparent” performance of a compressor can change by varying the k-value.

It can be seen from Figure 1 that the discharge temperature deviated over 18.8 percent by only changing the k-value by 6.7 percent.  In this case the k-value varied from a value of 1.20 to 1.28; which is the typical range for natural gas.  Similarly, the power changed by 2.5 percent, polytropic exponent by 9.5 percent, and adiabatic head by 2.5 percent for the same variation of the k-value.  The changes in compressor performance described in Figure 1 can be much larger depending on the gas composition and the operating temperature and pressure.

Corrected k-Value Recommendations

The k-value sensitivity for a single-stage machine is not nearly the problem as a multi-stage compressor.  For a single-stage machine, the pressure ratio is typically lower and the temperature and pressure changes are less.  As a result the changes in k-value are not as great and accurate results can be obtained by approximating the k-value at the suction conditions.  However, for multi-stage machines, where the pressure and temperature ratios are higher, the k-value sensitivity is more of a factor in evaluating compressor performance. Most compressor manufacturers calculate the k-value for each stage of compression and avoid errors introduced by utilizing an overall k-value. Without their software, we are left with a corrected k-value by empirical methods.

There are many useful approximations that will correct for changes in the k-value as the process gas passes through the compressor.  Normally the k-value will decrease during compression but not always.  Utilizing the suction conditions to estimate the k-value will generally give higher values of temperature, heat, and power.  The polytropic exponent generally decreases as the adiabatic exponent decreases.  To avoid potential discrepancies, a k-value correct may be warranted.  The following are six methods of determining the corrected k-value commonly used in industry.

1. At T_S and P_S:  This method determines the k-value at suction conditions and is useful for single stage compressors or applications where there is little change in the k-value.  The k-value is easy to determine and tends to overestimate results, especially if the temperature and pressure do not change significantly.  For greater values of R_P the results may become so conservative they become useless.k = k_s at suction conditions
2. At T_D and P_D:  This method determines the k-value at discharge conditions.  The k-value is less conservative and tends to underestimate results.  The k-value may be difficult to determine, especially if the discharge temperature is unknown.    For gases with highly variable k-values, an iterative solution may be required to estimate the discharge temperature and corrected k-value.k = k_D at discharge conditions
3. At T_AVE and P_STD [5]:  This method utilizes the average operating temperature at standard pressure and determines the k-value.  Numerous reference books propose this method.  Errors are introduced because the k-value at standard pressure may not accurately represent values at the operating pressure.k = at average operating temperature and standard pressure
4. At T_AVE and P_AVE:  This method utilizes the k-value at the average operating temperature and pressure.k = at average operating temperature and pressure
5. Average value [1, 3]:  This empirical method takes the average k-value at compressor inlet conditions and outlet conditions.  Utilizing the average k-value will result in performance values that are closer to the actual performance of the compressor.
6. Weighted average value [4]: This empirical method takes the weighted average of the suction, mid-point and discharge conditions.  Note that the mid-pressure is determined by equivalent pressure ratios, .  The mid-temperature is estimated from the mid-pressure.  This method considers the staged k-value to change with diverging isentropic and pressure lines shown on a Mollier chart.

Example 2 illustrates the various methods used to determine corrected k-values given above.  It also compares the range of the resulting values.

Example 2: A propane compressor is operating at the given conditions shown below.  Table 1 lists the k-values attributed to various operating and reference conditions [6].

Summary

This Tip of the Month has defined the physical property of process gases called the k-value or ratio of specific heats.  It has shown that small changes in the k-value can have a significant effect on the calculated values of head, power, gas discharge temperature, and polytropic exponent.  Recommendations were also given to improve the accuracy by utilizing different k-value methods.

To learn more about similar cases, we suggest attending our ME44 (Overview of Pumps and Compressors in Oil and Gas Facilities), ME46 (Compressor Systems – Mechanical Design and Specification), PL4 (Fundamental Pipeline Engineering), G40 (Process/Facility Fundamentals), G4 (Gas Conditioning and Processing), and PF4 (Oil Production and Processing Facilities) courses.

By: Joe Honeywell

References

1. Ronald P Lapina, Estimating Centrifugal Compressor Performance, Vol. 1, Gulf Publishing, 1982.
2. John M. Campbell, Gas Conditioning and Processing, Vol. 2, John M. Campbell & Co., 8th Edition.
3. Elliott Compressor Refresher Course,
4. John M. Schultz, “The Polytropic Analysis of Centrifugal Compressors”, Journal of Engineering for Power, January 1962.
5. Gas Processor Suppliers Association, Engineering Data Book, Section 13, 2004
6. National Institute of Standards and Technology, Web Site for Properties of Propane, Fluid Data.
7. ASME PTC10-1997, Performance Test Codes, “Compressors and Exhausters”, R2003