![\"Equation](\"https:\/\/i0.wp.com\/www.jmcampbell.com\/tip-of-the-month\/wp-content\/uploads\/2011\/03\/13.gif\" "\\"1\\"")
<\/a> (1)<\/p>\nThe second term on the right hand side of this equation is generally not convenient to solve manually. However, it is trivial or zero for the following cases: (1) ideal gases, (2) constant pressure,\u00a0dP<\/em> = 0, and (3) for a liquid considered incompressible. For all three cases enthalpy is a mathematical function only of temperature.\u00a0Cp<\/em> is commonly expressed by equations of the form:<\/p>\n Where A, B, and C are constants that depend on system composition and\u00a0T<\/em> is the absolute temperature. In most instances it is sufficiently accurate to find a\u00a0Cp<\/em> at the average temperature\u00a0TAvg<\/em>, where:<\/p>\n CPAvg<\/em> is then found at this average temperature and<\/p>\n This approximate solution to the first integral, although not exact, is satisfactory for most applications. Heat capacity values for pure substances are readily available from many handbooks and similar reference material. As noted in Chapter 7 of Volume 1, Gas Conditioning and Processing [1], values of heat capacity can be found from the slope of\u00a0h<\/em>vs.\u00a0T<\/em> plots at a given pressure. The\u00a0CP<\/em> for hydrocarbon liquid mixtures may be estimated from the equations presented in Volume 1 [1]. In this Tip of The Month (TOTM), the variation of heat capacity of natural gases with temperature, pressure, and relative density (composition) will be demonstrated. Then an empirical correlation will be presented to account for these variations. This correlation will be used to estimate natural gas heat capacity for wide ranges of pressure, temperature, and relative density. Finally, the accuracy of the proposed correlation will be discussed.<\/p>\n Development of a Generalized CP Correlation: The derivative on the right hand side of Eq (6) may be obtained from an equation of state (EOS) but it is too tedious for hand calculations. Therefore, the PR EOS option in ProMax [4] was used to generate\u00a0CP<\/em> values for various values of pressure, temperature, and relative density. The total number of\u00a0CP<\/em> values calculated was 715. Table 1 presents the composition of five different natural gas mixtures used in this study. Figures 1 through 5 present variations of\u00a0CP<\/em> with pressure, temperature and gas relative density. The red highlighted regions in Figures 3, 4, and 5 identify the two phase region of gas and liquid where the\u00a0CP<\/em> concept is not valid. It should be noted that the isobar of 20 MPa represents a single phase even at low temperatures. However, at low temperature, the fluid is dense phase. Where T is temperature, P is pressure and\u00a0CP<\/em> is heat capacity. A non-linear regression algorithm was used to determine the optimum values of parameters \u201ca\u201d through \u201cf\u201d. First,\u00a0CP<\/em> values of each gas in Table 1 were used to determine \u201ca\u201d through \u201cf\u201d. Then all of the generated\u00a0CP<\/em> values were used to determine a set of generalized parameters. These parameters were tuned and rounded to best represent all five gases covering a wide range of relative density from 0.60 to 0.80. For each case, the parameters and the summary of statistical error analysis are presented in Table 2. Note that the\u00a0CP<\/em> values of the two phase region were not used for the regression process. The general range of this correlation is from 20 to 200 \u00b0C (68 to 392 \u00b0F) and from 0.10 to 20 MPa (14.5 to 2900 Psia).<\/p>\n Discussion and Conclusions<\/strong> Table 2. Parameters for the proposed correlation; Eq. (7) in SI and FPS system<\/p>\n AAPD= Average Absolute Percent Deviation and It should be noted that the concept of heat capacity is valid only for the single phase region.<\/p>\n Figures 3 through 5 indicate that for low temperatures, liquid forms and irregular behavior of\u00a0CP<\/em> is observed. By: Dr. Mahmood Moshfeghian<\/em><\/p>\n Reference:<\/strong><\/p>\n The change in enthalpy for a fluid where no phase change occurs between Points (1) and (2) can be expressed as: (1) The second term on the right hand side of this equation is generally not convenient to solve manually. However, it is trivial or zero for the following cases: (1) ideal gases, (2) constant […]<\/p>\n","protected":false},"author":23,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[3],"tags":[],"coauthors":[],"class_list":["post-276","post","type-post","status-publish","format-standard","hentry","category-gas-processing"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p1pQc4-4s","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"http:\/\/www.jmcampbell.com\/tip-of-the-month\/wp-json\/wp\/v2\/posts\/276","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.jmcampbell.com\/tip-of-the-month\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.jmcampbell.com\/tip-of-the-month\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.jmcampbell.com\/tip-of-the-month\/wp-json\/wp\/v2\/users\/23"}],"replies":[{"embeddable":true,"href":"http:\/\/www.jmcampbell.com\/tip-of-the-month\/wp-json\/wp\/v2\/comments?post=276"}],"version-history":[{"count":7,"href":"http:\/\/www.jmcampbell.com\/tip-of-the-month\/wp-json\/wp\/v2\/posts\/276\/revisions"}],"predecessor-version":[{"id":1103,"href":"http:\/\/www.jmcampbell.com\/tip-of-the-month\/wp-json\/wp\/v2\/posts\/276\/revisions\/1103"}],"wp:attachment":[{"href":"http:\/\/www.jmcampbell.com\/tip-of-the-month\/wp-json\/wp\/v2\/media?parent=276"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.jmcampbell.com\/tip-of-the-month\/wp-json\/wp\/v2\/categories?post=276"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.jmcampbell.com\/tip-of-the-month\/wp-json\/wp\/v2\/tags?post=276"},{"taxonomy":"author","embeddable":true,"href":"http:\/\/www.jmcampbell.com\/tip-of-the-month\/wp-json\/wp\/v2\/coauthors?post=276"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}<\/a> (2)<\/p>\n<\/a> (3)<\/p>\n<\/a> (4)<\/p>\n
\nFor a non-ideal, compressible fluid like natural gas, the second term on the right hand side of Eq.(1) can\u2019t be ignored. Therefore, in process simulation software, an equation of state like Soave-Redlich-Kwong (SRK) [2] or Peng-Robinson (PR) [3] is used to calculate\u00a0<\/a>h<\/em>. For many calculations involving the heat capacity of natural gas, Figure 8.3 in Volume 1 is appropriate. Heat capacity at system pressure and average temperature is read off the graph and multiplied by gas mass flow rate and\u00a0<\/a>T<\/em> to obtain the heat load, <\/a>.<\/p>\n<\/a> (5)<\/p>\n
\n<\/strong>
\nAs mentioned earlier,\u00a0CP <\/em>can be defined from the slope of\u00a0h<\/em> vs.\u00a0T<\/em> plots at constant pressure. Mathematically, this is expressed by:<\/p>\n<\/a> (6)<\/p>\n
\nTable 1. Gas compositions used for generating\u00a0CP <\/em>values<\/p>\n<\/a><\/p>\n
\nIn order to correlate all the curves shown in Figures 1-5 by a single equation, the following expression is proposed.<\/p>\n<\/a> (7)<\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n<\/a><\/p>\n
\nA single and relatively simple correlation has been developed to estimate heat capacity of natural gases as a function of pressure, temperature, and relative density (composition). This correlation covers wide ranges of pressure (0.10 to 20 MPa, 14.5 to 2900 Psia), temperature (20 to 200 \u00b0C, 68 to 392 \u00b0F), and relative density (0.60 to 0.80). A generalized set of parameters in addition to an individual set of parameters have been determined and reported in Table 2. The error analysis reported in Table 2 indicates that the accuracy of this equation is quite good and can be used for natural gas heat duty calculations. For the generalized set of parameters, the average absolute percentage error (AAPD) and the maximum absolute percent deviations (MAPD) for the total of 715 points are 4.34 and 23.61, respectively. The applicable ranges of the proposed correlation are shown in Table 2.<\/p>\n<\/a><\/p>\n
\nMAPD= Maximum Absolute Percent Deviation
\nNPT= Number of Points and
\nSG = Relative Density (Specific Gravity)
\nNote: Below the above the temperature ranges for pressures 2, 5, 7, and 10 MPa (14.5 to 1450 Psia), the gas mixture may be in two phase (gas and liquid) region.<\/p>\n
\nTo learn more about similar cases and how to minimize operational problems, we suggest attending our G40(Process\/Facility Fundamentals)<\/a>, G4 (Gas Conditioning and Processing)<\/a> and G5 (Gas Conditioning and Processing – Special)<\/a> courses.<\/p>\n\n