{"id":4715,"date":"2024-10-05T18:04:05","date_gmt":"2024-10-05T18:04:05","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4715"},"modified":"2024-10-05T18:04:05","modified_gmt":"2024-10-05T18:04:05","slug":"real-gases-compressibility-charts","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/10\/05\/real-gases-compressibility-charts\/","title":{"rendered":"Real Gases: Compressibility Charts"},"content":{"rendered":"\n

Calculation of any of the variables\u00a0p<\/em>,\u00a0T<\/em>,\u00a0\"Image\", and\u00a0z<\/em>\u00a0using the generalized equation\u00a0p<\/em>\"Image\"\u00a0=\u00a0z<\/em>RT<\/em>\u00a0can be assisted by using graphs called\u00a0generalized compressibility\u00a0charts<\/strong>, or\u00a0z<\/em>\u00a0factor charts.<\/p>\n\n\n\n

Four parameters are displayed in\u00a0Figure 7.7. Any two values will fix a point from which you can determine the other two. For example, if\u00a0p<\/em>r<\/em><\/sub>\u00a0and\u00a0T<\/em>r<\/em><\/sub>\u00a0are known (point 1), the value of\u00a0\"Image\"\u00a0can be determined by interpolating\u00a0between the two closest curves of\u00a0\"Image\", and\u00a0z<\/em>\u00a0can be determined by drawing a horizontal line from point 1 to the\u00a0z<\/em>\u00a0axis.<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Figure 7.7. A compressibility chart involves four parameters: z<\/em>, pr<\/sub><\/em>, Tr<\/sub><\/em>, and \"Image\".<\/p>\n\n\n\n

Figures 7.8a\u00a0and\u00a07.8b\u00a0show two examples of the\u00a0generalized compressibility\u00a0factor<\/strong>\u00a0charts prepared by Nelson and Obert.2<\/sup>\u00a0These charts are based on data for 30 gases.\u00a0Figure 7.8a\u00a0represents\u00a0z<\/em>\u00a0for 26 gases (excluding H2<\/sub>, He, NH3<\/sub>, and H2<\/sub>O) with a maximum deviation of 1%, and H2<\/sub>\u00a0and H2<\/sub>O within a deviation of 1.5%.\u00a0Figure 7.8b\u00a0is for 9 gases and errors can be as high as 5%. Note that the vertical axis in\u00a0Figure 7.8b\u00a0is not\u00a0z<\/em>\u00a0but\u00a0zT<\/em>r<\/em><\/sub>. To use the charts for H2<\/sub>\u00a0and He (only), make corrections to the actual constants to get\u00a0pseudocritical<\/strong>\u00a0constants as follows:<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n
\"Image\"\/<\/figure>\n\n\n\n

Figure 7.8a. Generalized compressibility chart for lower pressures showing z as a function of pr<\/sub><\/em>, Tr<\/sub><\/em>, and \"Image\"<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Figure 7.8b. Generalized compressibility chart for higher values of pr<\/sub><\/em><\/p>\n\n\n\n

Then you can use\u00a0Figures 7.8a\u00a0and\u00a07.8b\u00a0for these two gases using the pseudocritical constants as replacements for their true values. You will find these two charts and additional charts for other ranges of\u00a0p<\/em>r<\/em><\/sub>\u00a0and\u00a0T<\/em>r<\/em><\/sub>\u00a0on the CD that\u00a0accompanies this book in a format that can be expanded to get better accuracy.<\/p>\n\n\n\n

Instead of the reduced specific volume, a third parameter shown on the charts is the dimensionless\u00a0ideal reduced volume\u00a0defined by<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

where \"Image\" is the ideal critical specific<\/a> volume<\/strong> (not the experimental value of the critical specific volume which yields poorer predictions) and is calculated from<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Both \"Image\" and \"Image\" are easy to calculate since T<\/em>c<\/sub> and p<\/em>c<\/sub> are presumed known or can be estimated for a compound. The development of the generalized compressibility charts is of considerable practical as well as pedagogical value because their existence enables you to make engineering calculations with considerable ease, and it also permits the development of thermodynamic functions for gases for which no experimental data are available.<\/p>\n\n\n\n

Frequently Asked Questions<\/h4>\n\n\n\n

1.<\/strong>\u00a0What is in the blank region in\u00a0Figure 7.8a\u00a0below the curves for\u00a0T<\/em>r<\/em><\/sub>\u00a0and\u00a0\"Image\"ri<\/em><\/sub>? The blank region corresponds to a different phase\u2014a liquid.<\/p>\n\n\n\n

2.<\/strong> Will p<\/em>\"Image\" = z<\/em>RT<\/em> work for a liquid phase? Yes, but relations to calculate z<\/em> accurately are more complex than those for the gas phase. Also, liquids are not very compressible, so at the moment we can bypass p<\/em>–\"Image\"–T<\/em> relations for liquids.<\/p>\n\n\n\n

3.<\/strong> Why should I use p<\/em>\"Image\" = z<\/em>RT<\/em> when I can look up the data needed in a handbook or on the Web? Although considerable data exists, you can use p<\/em>\"Image\" = z<\/em>RT<\/em> to evaluate the accuracy of the data and interpolate within data points. If you do not have data in the range you want, use of p<\/em>\"Image\" = z<\/em>RT<\/em> is the best method of extrapolation. Finally, you may not have any data for the gas of interest.<\/p>\n\n\n\n

\n\n\n\n

<\/a>Example 7.7. Use of the Compressibility Factor in Calculating a Specific Volume<\/p>\n\n\n\n

In spreading liquid ammonia fertilizer, the charges for the amount of NH3<\/sub> used are based on the time involved plus the pounds of NH3<\/sub> injected into the soil. After the liquid has been spread, there is still some ammonia left in the source tank (volume = 120 ft3<\/sup>), but in the form of a gas. Suppose that your weight tally, which is obtained by difference, shows a net weight of 125 lb of NH3<\/sub> left in the tank at 292 psig. Because the tank is sitting in the sun, the temperature in the tank is 125\u00b0F.<\/p>\n\n\n\n

Your boss complains that his calculations show that the specific volume of the NH3<\/sub>\u00a0gas is 1.20 ft3<\/sup>\/lb, and hence there are only 100 lb of NH3<\/sub>\u00a0in the tank. Could he be correct? See\u00a0Figure E7.7.<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Figure E7.7<\/p>\n\n\n\n

Solution<\/p>\n\n\n\n

The simplest calculation to make to get the specific volume of the ammonia in the tank is to select a pound or pound mole as a basis:<\/p>\n\n\n\n

Basis: 1 lb of NH3<\/sub><\/p>\n\n\n\n

Apparently, your boss used the ideal gas law (z<\/em> = 1) in getting the figure of 1.20 ft3<\/sup>\/lb of NH3<\/sub> gas:<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

What should you do? Ammonia probably does not behave like an ideal gas under the observed conditions of temperature and pressure. You can apply\u00a0pV<\/em>\u00a0=\u00a0zn<\/em>RT<\/em>\u00a0to calculate\u00a0n<\/em>\u00a0and determine the real amount of NH3<\/sub>\u00a0in the tank if you include the correct compressibility factor in the real gas law. Let\u2019s compute\u00a0z<\/em>;\u00a0z<\/em>\u00a0is a function of\u00a0T<\/em>r<\/em><\/sub>\u00a0and\u00a0p<\/em>r<\/em><\/sub>. You can look up all of the values of the necessary parameters in\u00a0Appendix F\u00a0or on the CD.<\/p>\n\n\n\n

T<\/em>c<\/em><\/sub> = 405.5 K \u21d2 729.9\u00b0R          p<\/em>c<\/em><\/sub> = 111.3 atm \u21d2 1636 psia<\/p>\n\n\n\n

Then since<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

From the Nelson and Obert (N&O) chart, Figure 7.8a<\/a>, you can read z<\/em> \u2245 0.855. The value may be somewhat in error because ammonia was not one of the gases included in the preparation of the figure. Rather than calculating the specific volume directly, let\u2019s calculate it from the ratio of pV<\/em>real<\/sub> = z<\/em>real<\/sub>n<\/em>RT<\/em> to pV<\/em>ideal<\/sub> = z<\/em>ideal<\/sub>n<\/em>RT<\/em>, the net result of which is<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

On the basis of 1 lb NH3<\/sub>,<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

On the basis of 120 ft3<\/sup> in the tank,<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Certainly 117 lb is a more realistic figure than 100 lb, but it still could be in error, considering that the residual weight of the NH3<\/sub> in the tank is determined by difference.<\/p>\n\n\n\n

As a matter of interest, as an alternative to making these calculations, you could look up the specific volume of NH3<\/sub> at the conditions in the tank in a handbook. You would find that \"Image\" = 0.973 ft3<\/sup>\/lb, equivalent to 123 lb of NH3<\/sub>, the correct value. Would you tell your boss to use the right compressibility factor, or state that you used the handbook value of \"Image\"?<\/p>\n\n\n\n

If you calculated\u00a0z<\/em>\u00a0from Equation (7.3), you would get<\/p>\n\n\n\n

z<\/em> = z<\/em>0<\/sup> + z<\/em>1<\/sup>\u03c9 = 0.864 \u2013 0.107(0.250) = 0.837<\/p>\n\n\n\n

What would the mass of ammonia in the tank be using z<\/em> = 0.873?<\/p>\n\n\n\n

\n\n\n\n
\n\n\n\n

<\/a>Example 7.8. Use of the Compressibility Factor in Calculating a Pressure<\/p>\n\n\n\n

Liquid oxygen is used in the steel industry, in the chemical industry, in hospitals, as rocket fuel oxidant, and for wastewater treatment as well as in many other applications. A tank sold to hospitals contains 0.0284 m3<\/sup> of volume filled with 3.500 kg of liquid O2<\/sub> that will vaporize at \u201325\u00b0C. After all of the O2<\/sub> in the tank vaporizes, will the pressure in the tank exceed the safety limit for the tank specified as 104 kPa?<\/p>\n\n\n\n

Solution<\/p>\n\n\n\n

Basis: 3.500 kg of O2<\/sub><\/p>\n\n\n\n

You can find from\u00a0Appendix F\u00a0on the CD that for oxygen<\/p>\n\n\n\n

T<\/em>c<\/em><\/sub> = 154.4 K                    p<\/em>c<\/em><\/sub> = 49.7 atm \u21d2 5035 kPa<\/p>\n\n\n\n

However, you cannot proceed to solve this problem in exactly the same way as the preceding problem because you do not know the pressure of the O2<\/sub> in the tank to begin with. But you can use the pseudoparameter, \"Image\", which is available as a parameter on the Nelson and Obert charts, as a second parameter to fix a point on the compressibility charts.<\/p>\n\n\n\n

First calculate<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Note that the specific molar volume<\/em> must be used in calculating \"Image\" since \"Image\" is the volume per mole.<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Then<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Now you know the values of two parameters, \"Image\" and<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

From the Nelson and Obert chart (Figure 7.8b<\/a>) you can read<\/p>\n\n\n\n

p<\/em>r<\/em><\/sub> = 1.43<\/p>\n\n\n\n

Then<\/p>\n\n\n\n

p<\/em> = p<\/em>r<\/em><\/sub>p<\/em>c<\/em><\/sub>
                                = 1.43(5035) = 7200 kPa<\/p>\n\n\n\n

The pressure of 104<\/sup> kPa will not be exceeded. Even at room temperature the pressure will be less than 104<\/sup> kPa.<\/p>\n\n\n\n

\n\n\n\n

To get one snapshot of the difference between estimates of\u00a0z<\/em>\u00a0by three of the methods discussed in this chapter,\u00a0Table 7.4\u00a0compares the experimental values of\u00a0z<\/em>\u00a0for ethylene with predictions by three methods: N&O charts, Pitzer\u2019s relation, and the ideal gas laws.<\/p>\n\n\n\n

<\/a>Table 7.4. A Comparison of Values of the Compressibility Factor z<\/em> for Ethylene*<\/a> Determined via Three Different Methods with the Associated Experimental Values<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

*\u00a0w<\/em>\u00a0= 0.089;\u00a0Tc<\/sub><\/em>\u00a0= 282.8 K;\u00a0pc<\/sub><\/em>\u00a0= 50.5 atm;\u00a0z<\/em>0<\/sup>\u00a0and\u00a0z<\/em>1<\/sup>\u00a0are from the tables in\u00a0Appendix C.<\/p>\n","protected":false},"excerpt":{"rendered":"

Calculation of any of the variables\u00a0p,\u00a0T,\u00a0, and\u00a0z\u00a0using the generalized equation\u00a0p\u00a0=\u00a0zRT\u00a0can be assisted by using graphs called\u00a0generalized compressibility\u00a0charts, or\u00a0z\u00a0factor charts. Four parameters are displayed in\u00a0Figure 7.7. Any two values will fix a point from which you can determine the other two. For example, if\u00a0pr\u00a0and\u00a0Tr\u00a0are known (point 1), the value of\u00a0\u00a0can be determined by interpolating\u00a0between the two […]<\/p>\n","protected":false},"author":1,"featured_media":4604,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[570],"tags":[],"class_list":["post-4715","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-ideal-and-real-gases"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/gases.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4715","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4715"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4715\/revisions"}],"predecessor-version":[{"id":4716,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4715\/revisions\/4716"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4604"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4715"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4715"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4715"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}