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.\u00a0\u00a0 It is used to define basic gas processes including adiabatic and polytropic compression. \u00a0It also appears in many of the traditional equations commonly used to determine a compressor head, gas discharge temperature, gas power, and polytropic exponent.\u00a0 The k-value also influences the operating speed of a compressor, but we will simplify the present analysis by deleting speed from our evaluation.\u00a0 The following commonly used compressor performance equations show how the k-value is utilized in the design and evaluation of compressors.<\/p>\n
Note: \u00a0\u00a0 The actual Z-value will vary from the suction to discharge conditions.\u00a0 ZS<\/sub> is sometimes replaced with ZAVE<\/sub> to approximate the variations in compressibility value [1, 5]. See the nomenclature at the end of this TOTM.<\/p>\n The above equations are written in terms of the adiabatic process with the exception of Equation 5, which refers to the polytropic process.\u00a0 Both compression processes are similar and will give the same\u00a0actual<\/em> results.\u00a0 The adiabatic and polytropic methods are extensively used by manufacturers to design compressors, and make use of k-values to calculate their performance.\u00a0 However, as will be seen, the effect of the k-value and the calculated results will influence both compression processes alike.\u00a0 For simplicity, this Tip of the Month will use the adiabatic process. For a pure gas there are many references that give\u00a0CP<\/sub><\/em> and\u00a0CV<\/sub><\/em> values at various conditions.\u00a0 One useful source is National Institute of Standards and Technology.\u00a0 Their website is\u00a0http:\/\/webbook.nist.gov\/chemistry\/fluid\/<\/a><\/p>\n The method of determining the k-value for gas mixtures is more complex.\u00a0 The major difference is that a gas mixture does not behave as any one of its components but as an \u201cequivalent\u201d gas.\u00a0 Therefore, to determine the k-value of the mixture, we must know the mole fraction of each component,\u00a0Yi<\/sub><\/em> and the molar specific heat at constant pressure for each component, M CPi<\/sub><\/em>.\u00a0\u00a0 Equation 7 can be used to determine the k-value of an ideal gas mixture [1, 5].\u00a0 Real gases may deviate from the calculated value.<\/p>\n 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.\u00a0 The results are the same but the methods are very different.<\/p>\n K-value Sensitivity Analysis<\/strong><\/p>\n In the compression process the temperature and pressure of the process gas both increase.\u00a0 Not knowing what k-value to select for evaluating the compression process can lead to errors.\u00a0 For example, a typical propane compressor may have a k-value at suction conditions of 1.195.\u00a0 At the compressor discharge conditions the k-value is 1.254.\u00a0 The difference in the two values varies by 4.94 percent and can have a significant influence in the performance evaluation.\u00a0 The following example illustrates how minor changes in the k-value can influence the calculated compressor head, temperature, power and the polytropic coefficient.<\/p>\n Example 1<\/strong>: A natural gas compressor is operating at the conditions given below.\u00a0 Only the k-value is varied from 1.20 to 1.28, all other given parameters remain constant.\u00a0 \u00a0Figure 1 illustrates how the \u201capparent\u201d performance of a compressor can change by varying the k-value.<\/p><\/blockquote>\n 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.\u00a0 In this case the k-value varied from a value of 1.20 to 1.28; which is the typical range for natural gas.\u00a0 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.\u00a0 The changes in compressor performance described in Figure 1 can be much larger depending on the gas composition and the operating temperature and pressure.<\/p>\n Corrected k-Value Recommendations<\/strong><\/p>\n The k-value sensitivity for a single-stage machine is not nearly the problem as a multi-stage compressor.\u00a0 For a single-stage machine, the pressure ratio is typically lower and the temperature and pressure changes are less.\u00a0 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.\u00a0 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.<\/p>\n There are many useful approximations that will correct for changes in the k-value as the process gas passes through the compressor.\u00a0 Normally the k-value will decrease during compression but not always.\u00a0 Utilizing the suction conditions to estimate the k-value will generally give higher values of temperature, heat, and power.\u00a0 The polytropic exponent generally decreases as the adiabatic exponent decreases.\u00a0 To avoid potential discrepancies, a k-value correct may be warranted.\u00a0 The following are six methods of determining the corrected k-value commonly used in industry.<\/p>\n Example 2 illustrates the various methods used to determine corrected k-values given above.\u00a0 It also compares the range of the resulting values.<\/p>\n Example 2<\/strong>: A propane compressor is operating at the given conditions shown below.\u00a0 Table 1 lists the k-values attributed to various operating and reference conditions [6].<\/p><\/blockquote>\n Summary<\/strong><\/p>\n This Tip of the Month has defined the physical property of process gases called the k-value or ratio of specific heats.\u00a0 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.\u00a0 Recommendations were also given to improve the accuracy by utilizing different k-value methods.<\/p>\n To learn more about similar cases, we suggest attending our\u00a0ME44 (Overview of Pumps and Compressors in Oil and Gas Facilities)<\/a>,\u00a0ME46 (Compressor Systems – Mechanical Design and Specification)<\/a>,\u00a0PL4 (Fundamental Pipeline Engineering)<\/a>,\u00a0G40 (Process\/Facility Fundamentals)<\/a>,\u00a0G4 (Gas Conditioning and Processing)<\/a>, and PF4 (Oil Production and Processing Facilities)<\/a> courses.<\/p>\n![\"Equations\"](\"https:\/\/i0.wp.com\/www.jmcampbell.com\/tip-of-the-month\/wp-content\/uploads\/2011\/03\/12.gif\" "\\"1\\"")
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\nIt can be seen from Equations 1-5 that the k-value has an effect on a compressor head, temperature, power, and polytropic exponent.\u00a0 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.\u00a0 The thermodynamic definition of a gas k-value is given by Equation 6.\u00a0 It shows the relationship to the specific heat at constant volume,\u00a0CV<\/sub><\/em> and specific heat at constant pressure,\u00a0CP<\/sub><\/em>.\u00a0 Both values vary with temperature and pressure.<\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n\n
<\/a><\/li>\n<\/a>.\u00a0 The mid-temperature is estimated from the mid-pressure.\u00a0 This method considers the staged k-value to change with diverging isentropic and pressure lines shown on a Mollier chart.<\/li>\n<\/ol>\n<\/a><\/div>\n<\/a><\/p>\n