{"id":881,"date":"2021-06-07T00:07:50","date_gmt":"2021-06-07T04:07:50","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/thermodynamic-tables\/"},"modified":"2022-12-26T23:34:11","modified_gmt":"2022-12-27T04:34:11","slug":"thermodynamic-tables","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/thermodynamic-tables\/","title":{"raw":"2.4 Thermodynamic tables","rendered":"2.4 Thermodynamic tables"},"content":{"raw":"<div class=\"thermodynamic-tables\">\r\n\r\n<span lang=\"en-US\" xml:lang=\"en-US\">Thermodynamic tables are <\/span><span lang=\"en-US\" xml:lang=\"en-US\">commonly used to determine the properties of a substance at a given state. <\/span><span lang=\"en-US\" xml:lang=\"en-US\">This book includes<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> t<\/span><span lang=\"en-US\" xml:lang=\"en-US\">he tables for <\/span><span lang=\"en-US\" xml:lang=\"en-US\">four<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">pure <\/span><span lang=\"en-US\" xml:lang=\"en-US\">substances<\/span><span lang=\"en-US\" xml:lang=\"en-US\">: water, ammonia, R134a, and carbon dioxide. <\/span> <span lang=\"en-US\" xml:lang=\"en-US\">The data in these tables are obtained from <\/span><a href=\"https:\/\/webbook.nist.gov\/chemistry\/fluid\/\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">NIST Chemistry <\/span><span lang=\"en-US\" xml:lang=\"en-US\">WebBook<\/span><span lang=\"en-US\" xml:lang=\"en-US\">, SRD 69<\/span><\/a>, which consists of the thermophysical properties of various common fluids.\r\n<p class=\"indent no-indent\"><span lang=\"en-US\" xml:lang=\"en-US\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-a-thermodynamic-properties-of-water\/\" target=\"_blank\" rel=\"noopener\">Appendix A<\/a>: Thermodynamic Properties of Water<\/span><\/p>\r\n\r\n<ul>\r\n \t<li><span lang=\"en-US\" xml:lang=\"en-US\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-a-thermodynamic-properties-of-water#TA1\" target=\"_blank\" rel=\"noopener\">Table A1<\/a>: Saturated water<\/span><\/li>\r\n \t<li><span lang=\"en-US\" xml:lang=\"en-US\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-a-thermodynamic-properties-of-water#TA2\" target=\"_blank\" rel=\"noopener\">Table A2<\/a>: Superheated vapour, water<\/span><\/li>\r\n \t<li><span lang=\"en-US\" xml:lang=\"en-US\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-a-thermodynamic-properties-of-water#TA3\" target=\"_blank\" rel=\"noopener\">Table A3<\/a>: Compressed liquid water<\/span><\/li>\r\n<\/ul>\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-b-thermodynamic-properties-of-ammonia\/\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Appendix <\/span><span lang=\"en-US\" xml:lang=\"en-US\">B<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Thermodynamic Properties of <\/span><span lang=\"en-US\" xml:lang=\"en-US\">Ammonia<\/span>\r\n<ul>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-b-thermodynamic-properties-of-ammonia#TB1\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Table <\/span><span lang=\"en-US\" xml:lang=\"en-US\">B1<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Saturated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">ammonia<\/span><\/li>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-b-thermodynamic-properties-of-ammonia#TB2\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Table <\/span><span lang=\"en-US\" xml:lang=\"en-US\">B2<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Superheated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">ammonia<\/span><\/li>\r\n<\/ul>\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-c-thermodynamic-properties-of-r134a\/\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Appendix <\/span><span lang=\"en-US\" xml:lang=\"en-US\">C<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Thermodynamic Properties of <\/span><span lang=\"en-US\" xml:lang=\"en-US\">R134a<\/span>\r\n<ul>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-c-thermodynamic-properties-of-r134a#TC1\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Table <\/span><span lang=\"en-US\" xml:lang=\"en-US\">C1<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Saturated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">R134a<\/span><\/li>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-c-thermodynamic-properties-of-r134a#TC2\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Table <\/span><span lang=\"en-US\" xml:lang=\"en-US\">C2<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Superheated<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> R134a<\/span><\/li>\r\n<\/ul>\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-d-thermodynamic-properties-of-co2\/\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Appendix <\/span><span lang=\"en-US\" xml:lang=\"en-US\">D<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Thermodynamic Properties of <\/span><span lang=\"en-US\" xml:lang=\"en-US\">Carbon Dioxide<\/span>\r\n<ul>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-d-thermodynamic-properties-of-co2#TD1\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Table <\/span><span lang=\"en-US\" xml:lang=\"en-US\">D1<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Saturated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">CO<\/span><sub>2<\/sub><\/li>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-d-thermodynamic-properties-of-co2#TD2\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Table <\/span><span lang=\"en-US\" xml:lang=\"en-US\">D2<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Superheated<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">CO<\/span><sub>2<\/sub><\/li>\r\n<\/ul>\r\n<p class=\"import-Normal\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">Tables A1, B1, C1, and D1 are the tables for the saturated fluids. <\/span><span lang=\"en-US\" xml:lang=\"en-US\">They are <\/span><span lang=\"en-US\" xml:lang=\"en-US\">used to find the properties of the <\/span><span lang=\"en-US\" xml:lang=\"en-US\">corresponding <\/span><span lang=\"en-US\" xml:lang=\"en-US\">fluid<\/span><span lang=\"en-US\" xml:lang=\"en-US\">s<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> in saturated liquid, saturated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">vapour,<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> and two-phase regions. Tables A2, B2, C2, and D2 are the <\/span><span lang=\"en-US\" xml:lang=\"en-US\">superheated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">vapour<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> tables<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> for <\/span><span lang=\"en-US\" xml:lang=\"en-US\">find<\/span><span lang=\"en-US\" xml:lang=\"en-US\">ing<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> the properties of the fluid<\/span><span lang=\"en-US\" xml:lang=\"en-US\">s<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> in the superheated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">vapour<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> region. Table A3 is the compressed liquid table<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> for water. <\/span><\/p>\r\n&nbsp;\r\n<p class=\"import-Normal\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">In these tables, the specific volume, specific internal energy, specific enthalpy, and specific entropy are tabulated as functions of the pressure and temperature. <\/span><span lang=\"en-US\" xml:lang=\"en-US\">Among those thermodynamic properties, [latex]P[\/latex]<\/span>, [latex]T[\/latex], and [latex]v[\/latex] <span lang=\"en-US\" xml:lang=\"en-US\">are measurable properties<\/span>, <span lang=\"en-US\" xml:lang=\"en-US\">and [latex]u[\/latex], [latex]h[\/latex], and [latex]s[\/latex] <\/span><span lang=\"en-US\" xml:lang=\"en-US\">cannot be measured directly. T<\/span><span lang=\"en-US\" xml:lang=\"en-US\">hey<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">are calculated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">with respect to <\/span><span lang=\"en-US\" xml:lang=\"en-US\">predefined <\/span><span lang=\"en-US\" xml:lang=\"en-US\">reference states.<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> The reference state<\/span><span lang=\"en-US\" xml:lang=\"en-US\">s<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> used for <\/span><span lang=\"en-US\" xml:lang=\"en-US\">different tables in this book are stated in <\/span><span lang=\"en-US\" xml:lang=\"en-US\">A<\/span><span lang=\"en-US\" xml:lang=\"en-US\">ppendices A to D. <\/span><\/p>\r\n&nbsp;\r\n<p class=\"import-Normal\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">It is important to be aware that thermodynamic tables can be generated with respect to different reference states, which means that <\/span><span lang=\"en-US\" xml:lang=\"en-US\">the values of specific internal energy<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">[latex]u[\/latex], specific enthalpy [latex]h[\/latex], and specific entropy [latex]s[\/latex]<\/span> can vary <span lang=\"en-US\" xml:lang=\"en-US\">depending on the source of the tables<\/span><span lang=\"en-US\" xml:lang=\"en-US\">. Additionally, in most thermodynamic analyses, we are concerned with the <em lang=\"en-US\" xml:lang=\"en-US\">changes<\/em> in these specific properties, such as [latex]\\Delta u[\/latex], [latex]\\Delta h[\/latex], and [latex]\\Delta s[\/latex]. Therefore, it is important to use tables from a single source in order to ensure that consistent values of [latex]u[\/latex], [latex]h[\/latex], and [latex]s[\/latex] are used in the analysis. Using tables from different sources in calculations can lead to incorrect or misleading conclusions.<\/span><\/p>\r\n&nbsp;\r\n<p class=\"import-Normal\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">When using thermodynamic tables, we will need to consider the following two questions: (1) how many independent variables are required to fix the state of a fluid? and (2) how the phase of a fluid is determined? is it a compressed liquid, superheated vapour, or two-phase liquid-vapour mixture?<\/span><\/p>\r\n&nbsp;\r\n<p class=\"import-Normal\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">By examining the tables in Appendices A to D, you probably have already noticed that all properties in these tables are intensive properties. For a non-reactive system in thermodynamic equilibrium, the Gibbs phase rule indicates that the number of independent, intensive properties of the system can be found as <\/span><\/p>\r\n<p style=\"text-align: center\"><span lang=\"en-US\" xml:lang=\"en-US\">[latex]f=c+2-p[\/latex]<\/span><\/p>\r\nwhere\r\n<p style=\"padding-left: 40px\"><span lang=\"en-US\" xml:lang=\"en-US\">[latex]f[\/latex]:\u00a0 the number of independent, intensive properties of the system that are needed to fix the state of the system. It is also called the degree of freedom of the system.\r\n<\/span><\/p>\r\n<p style=\"padding-left: 40px\"><span lang=\"en-US\" xml:lang=\"en-US\">[latex]c[\/latex]: the number of components of the system. For a pure substance, [latex]c=1[\/latex].\r\n<\/span><\/p>\r\n<p style=\"padding-left: 40px\"><span lang=\"en-US\" xml:lang=\"en-US\">[latex]p[\/latex]: the number of phases of the system. For a single-phase\u00a0 system, [latex]p=1[\/latex]. If a system is in a two-phase region, [latex]p=2[\/latex].\r\n<\/span><\/p>\r\n&nbsp;\r\n\r\n<span lang=\"en-US\" xml:lang=\"en-US\">For a single-phase system of a pure substance, [latex]c=1[\/latex], [latex]p=1[\/latex]; therefore, [latex]f=2[\/latex]. This means that two independent, intensive properties are required to fix the state of the system. In other words, if any TWO independent, intensive properties from this list, [latex]P[\/latex], [latex]T[\/latex], [latex]v[\/latex], [latex]u[\/latex], [latex]h[\/latex], [latex]s[\/latex], are known, then the remaining intensive properties can be determined from the thermodynamic tables.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span lang=\"en-US\" xml:lang=\"en-US\">For a pure substance in a two-phase, saturated region, such as a mixture of saturated liquid water and saturated water vapour, [latex]c=1[\/latex], [latex]p=2[\/latex]; therefore, [latex]f=1[\/latex]. This means that the system is of one-degree of freedom; if one independent, intensive property, such as the saturated temperature, [latex]T_{sat}[\/latex], or the saturated pressure, [latex]P_{sat}[\/latex] of the mixture is fixed, all other intensive properties of the saturated mixture can be found from the thermodynamic tables, which include [latex]P_{sat}[\/latex] (or [latex]T_{sat}[\/latex]), [latex]v_f[\/latex], [latex]v_g[\/latex], [latex]u_f[\/latex], [latex]u_g[\/latex], [latex]h_f[\/latex], [latex]h_g[\/latex], [latex]s_f[\/latex], and [latex]s_g[\/latex]. With the quality [latex]x[\/latex], we can calculate the specific properties,\u00a0 [latex]v[\/latex], [latex]u[\/latex], [latex]h[\/latex], and [latex]s[\/latex] of the saturated mixture. In summary, for a pure substance in a two-phase, saturated region, if any TWO independent, intensive properties from this list, [latex]P[\/latex], [latex]T[\/latex], [latex]v[\/latex], [latex]u[\/latex], [latex]h[\/latex], [latex]s[\/latex], and [latex]x[\/latex], are known, then the remaining intensive properties can be calculated from the thermodynamic tables.\r\n<\/span>\r\n\r\n<\/div>\r\n<div class=\"thermodynamic-tables\">\r\n<p class=\"import-Normal no-indent\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">The following flow charts demonstrate four common scenarios and t<\/span>he corresponding procedures of finding thermodynamic properties.<\/p>\r\n&nbsp;\r\n\r\nCase 1: <strong>both <em>T<\/em> and<em> P <\/em>are given<\/strong>. You may draw a [latex]P-T[\/latex] diagram (see <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/phase-diagrams#2.3.2\" target=\"_blank\" rel=\"noopener\">Figure 2.3.2<\/a>) to help you better understand the flow chart in Figure 2.4.1.\r\n\r\n[caption id=\"attachment_894\" align=\"aligncenter\" width=\"800\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_P-T.png\" target=\"_blank\" rel=\"noopener\"><img class=\"wp-image-894\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_P-T-1024x890.png\" alt=\"Flow chart for determining fluid properties from thermodynamic tables if P and T are known. \" width=\"800\" height=\"696\" \/><\/a> <em><strong>Figure 2.4.1<\/strong><\/em>\u00a0<em>Flow chart for determining fluid properties from thermodynamic tables if P and T are known<\/em>[\/caption]\r\n\r\nCase 2: <strong>both <em>T<\/em> and <em>x<\/em> are given<\/strong>. You may draw a [latex]T-v[\/latex] diagram (see <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/phase-diagrams#2.3.4\" target=\"_blank\" rel=\"noopener\">Figure 2.3.4<\/a>) to help you better understand the flow chart in Figure 2.4.2.\r\n\r\n[caption id=\"attachment_896\" align=\"aligncenter\" width=\"800\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-x.png\" target=\"_blank\" rel=\"noopener\"><img class=\"wp-image-896\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-x-1024x505.png\" alt=\"Flow chart for determining fluid properties from thermodynamic tables if T and x are known\" width=\"800\" height=\"395\" \/><\/a> <em><strong>Figure 2.4.2<\/strong><\/em>\u00a0<em>Flow chart for determining fluid properties from thermodynamic tables if T and x are known<\/em>[\/caption]\r\n\r\n<\/div>\r\n<div class=\"thermodynamic-tables\">\r\n\r\nCase 3: <strong>both <em>T<\/em> and <em>v<\/em> are given<\/strong>. You may draw a [latex]T-v[\/latex] diagram (see <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/phase-diagrams#2.3.4\" target=\"_blank\" rel=\"noopener\">Figure 2.3.4<\/a>) to help you better understand the flow <a id=\"2.4.3\"><\/a>chart in Figure 2.4.3.\r\n\r\n[caption id=\"attachment_898\" align=\"aligncenter\" width=\"900\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-v.png\" target=\"_blank\" rel=\"noopener\"><img class=\"wp-image-898\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-v-1024x726.png\" alt=\"Flow chart for determining fluid properties from thermodynamics tables if T and v are known\" width=\"900\" height=\"638\" \/><\/a> <em><strong>Figure 2.4.3<\/strong><\/em>\u00a0<em>Flow chart for determining fluid properties from thermodynamics tables if T and v are known<\/em>[\/caption]\r\n\r\nCase 4: <strong>temperature <em>T<\/em> and one of<em> u, h <\/em>and<em> s <\/em>are given<\/strong>. The procedure is exactly the same as for case 3. Replace [latex]v[\/latex] with the given <span lang=\"en-US\" xml:lang=\"en-US\">[latex]u[\/latex], [latex]h[\/latex], or [latex]s[\/latex]<\/span> in the flow chart, <a href=\"#2.4.3\">Figure 2.4.3<\/a>.\r\n\r\n&nbsp;\r\n<p class=\"import-Normal no-indent\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">T<\/span><span lang=\"en-US\" xml:lang=\"en-US\">he compressed liquid table is presented only for water in <\/span><span lang=\"en-US\" xml:lang=\"en-US\">the pressure range of 0.5-50 MPa<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> i<\/span><span lang=\"en-US\" xml:lang=\"en-US\">n this book<\/span><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> When the compressed liquid tables are not available for a specific fluid or in a specific range, the <\/span><span lang=\"en-US\" xml:lang=\"en-US\">saturated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">liquid <\/span><span lang=\"en-US\" xml:lang=\"en-US\">properties at <\/span><span lang=\"en-US\" xml:lang=\"en-US\">the same temperature<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> may be used as an approximation, i.e., [latex]v\\approx\\ v_{f@T}[\/latex], [latex]u\\approx\\ u_{f@T} [\/latex], [latex]h\\approx\\ h_{f@T} [\/latex], and [latex]s\\approx\\ s_{f@T}[\/latex]<\/span>.<\/p>\r\n&nbsp;\r\n<p class=\"import-Normal no-indent\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">The tables in Appendices A to D are presented <\/span><span lang=\"en-US\" xml:lang=\"en-US\">with <\/span><span lang=\"en-US\" xml:lang=\"en-US\">a<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> small<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> temperature increment<\/span><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> L<\/span><span lang=\"en-US\" xml:lang=\"en-US\">inear interpolations, see <a href=\"#2.4.4\">Figure 2.4.4<\/a>, <\/span><span lang=\"en-US\" xml:lang=\"en-US\">are often used if the <\/span><span lang=\"en-US\" xml:lang=\"en-US\">given temperature or other properties cannot be found directly from these tables.\r\n<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: center\">[latex]y=y_0+\\left( x-x_0 \\right) \\dfrac{y_1-y_0}{x_1-x_0} [\/latex]<\/p>\r\n&nbsp;\r\n\r\nwhere [latex]\\left( x, y \\right) [\/latex] is the state, at which the property [latex]x[\/latex] is known and the property [latex]y[\/latex] is to be found. [latex]\\left( x_0, y_0 \\right) [\/latex] and [latex]\\left( x_1, y_1 \\right) [\/latex] indicate the properties of two known states, between which the unknown state [latex]\\left( x, y \\right) [\/latex] is located. To improve the accuracy, the two states should be selected as close as possible to the unknown <a id=\"2.4.4\"><\/a>state.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_2978\" align=\"aligncenter\" width=\"300\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1.png\" target=\"_blank\" rel=\"noopener\"><img class=\"wp-image-2978 size-medium\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1-300x300.png\" alt=\"Three points on a linear line, illustrating the principle of linear Interpolation\" width=\"300\" height=\"300\" \/><\/a> <em><strong>Figure 2.4.4<\/strong> Linear interpolation<\/em>[\/caption]\r\n\r\n&nbsp;\r\n<p class=\"import-Normal no-indent\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">The following examples demonstrate how to <\/span><span lang=\"en-US\" xml:lang=\"en-US\">use these tables to\u00a0<\/span><span lang=\"en-US\" xml:lang=\"en-US\">find the properties of <\/span><span lang=\"en-US\" xml:lang=\"en-US\">a <\/span><span lang=\"en-US\" xml:lang=\"en-US\">compressed liquid, superheated vapour, and liquid-vapour mixture. <\/span><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p class=\"import-Normal\"><span lang=\"en-US\" xml:lang=\"en-US\">Determine the properties of water at <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=<\/span><span lang=\"en-US\" xml:lang=\"en-US\">1<\/span><span lang=\"en-US\" xml:lang=\"en-US\">5<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=<\/span><span lang=\"en-US\" xml:lang=\"en-US\">100 kPa. <\/span><\/p>\r\n&nbsp;\r\n<p class=\"import-Normal\"><span lang=\"en-US\" xml:lang=\"en-US\"><span style=\"text-decoration: underline\"><em>Solution<\/em><\/span>:<\/span><\/p>\r\n\r\n<ol>\r\n \t<li>\r\n<p class=\"no-indent indent\"><span lang=\"en-US\" xml:lang=\"en-US\">Both <\/span>temperature <span lang=\"en-US\" xml:lang=\"en-US\">and <\/span>pressure<span lang=\"en-US\" xml:lang=\"en-US\"> are given for water. Use the flow chart for case 1, <a href=\"#2.4.1\">Figure 2.4.1<\/a>.<\/span><\/p>\r\n<\/li>\r\n \t<li>\r\n<p class=\"no-indent indent\"><span lang=\"en-US\" xml:lang=\"en-US\">From <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-a-thermodynamic-properties-of-water#TA1\" target=\"_blank\" rel=\"noopener\">Table A1<\/a>: <\/span><span lang=\"en-US\" xml:lang=\"en-US\">at<\/span> <em>T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=150<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C, <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><sub><em>sa<\/em><\/sub><sub>t<\/sub><span lang=\"en-US\" xml:lang=\"en-US\"> = <\/span><span lang=\"en-US\" xml:lang=\"en-US\">0.47617 MPa = 476.17 kPa. <\/span><\/p>\r\n<\/li>\r\n \t<li>\r\n<p class=\"no-indent indent\"><span lang=\"en-US\" xml:lang=\"en-US\">Because <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=100 kPa <\/span><span lang=\"en-US\" xml:lang=\"en-US\">&lt; <\/span><span lang=\"en-US\" xml:lang=\"en-US\">476.17 kPa<\/span><span lang=\"en-US\" xml:lang=\"en-US\">, or <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P <\/em><span lang=\"en-US\" xml:lang=\"en-US\">&lt; <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><sub><em>sat<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">,<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> water <\/span><span lang=\"en-US\" xml:lang=\"en-US\">at this state <\/span><span lang=\"en-US\" xml:lang=\"en-US\">is <\/span><span lang=\"en-US\" xml:lang=\"en-US\">a superheated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">vapour<\/span><span lang=\"en-US\" xml:lang=\"en-US\">. <\/span><\/p>\r\n<\/li>\r\n \t<li>\r\n<p class=\"no-indent indent\"><span lang=\"en-US\" xml:lang=\"en-US\">From <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-a-thermodynamic-properties-of-water#TA2\" target=\"_blank\" rel=\"noopener\">Table A2<\/a><\/span><span lang=\"en-US\" xml:lang=\"en-US\">:<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">at <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=150<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=100 kPa<\/span><em lang=\"en-US\" xml:lang=\"en-US\">, <\/em><\/p>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"no-indent indent\" style=\"padding-left: 40px\"><em lang=\"en-US\" xml:lang=\"en-US\">v <\/em><span lang=\"en-US\" xml:lang=\"en-US\">= <\/span>1.93665 m<sup>3<\/sup>\/kg,\u00a0\u00a0 <em>u<\/em> = 2582.94 kJ\/kg<\/p>\r\n<p class=\"no-indent indent\" style=\"padding-left: 40px\"><em>h<\/em> = 2776.60 kJ\/kg,\u00a0\u00a0\u00a0\u00a0 <em>s<\/em> = 7.6148 kJ\/kgK<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p class=\"import-Normal\"><span lang=\"en-US\" xml:lang=\"en-US\">Determine the properties of a<\/span><span lang=\"en-US\" xml:lang=\"en-US\">mmonia<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> at <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">v<\/em> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0.2<\/span> m<sup>3<\/sup>\/kg<span lang=\"en-US\" xml:lang=\"en-US\">. <\/span><\/p>\r\n&nbsp;\r\n<p class=\"import-Normal\"><span lang=\"en-US\" xml:lang=\"en-US\"><em><span style=\"text-decoration: underline\">Solution<\/span><\/em>:<\/span><\/p>\r\n\r\n<ol>\r\n \t<li>\r\n<p class=\"no-indent indent\"><span lang=\"en-US\" xml:lang=\"en-US\">Both <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\"> and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><span lang=\"en-US\" xml:lang=\"en-US\"> are given for<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> ammonia<\/span><span lang=\"en-US\" xml:lang=\"en-US\">. Use the flow chart for case <\/span><span lang=\"en-US\" xml:lang=\"en-US\">3, <a href=\"#2.4.3\">Figure 2.4.3<\/a><\/span><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span><\/p>\r\n<\/li>\r\n \t<li>\r\n<p class=\"no-indent indent\"><span lang=\"en-US\" xml:lang=\"en-US\">From <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-b-thermodynamic-properties-of-ammonia#TB1\" target=\"_blank\" rel=\"noopener\">Table B1<\/a><\/span><span lang=\"en-US\" xml:lang=\"en-US\">: at <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C, <\/span><em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><sub><em>f<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span>0.001566 m<sup>3<\/sup>\/kg and <em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><sub><em>g<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 0.289297 m<sup>3<\/sup>\/kg.<\/p>\r\n<\/li>\r\n \t<li>\r\n<p class=\"no-indent indent\">Because <em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><sub><em>f<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">&lt; <em>v <\/em>&lt;<\/span> <em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><sub><em>g<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">,<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> ammonia at this state is <\/span><span lang=\"en-US\" xml:lang=\"en-US\">a <\/span><span lang=\"en-US\" xml:lang=\"en-US\">liquid-<\/span><span lang=\"en-US\" xml:lang=\"en-US\">vapour<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> two<\/span><span lang=\"en-US\" xml:lang=\"en-US\">-<\/span><span lang=\"en-US\" xml:lang=\"en-US\">phase mixture. Its pressure and quality are\r\n<\/span><\/p>\r\n<\/li>\r\n<\/ol>\r\n<p style=\"padding-left: 80px\">[latex]P=P_{sat}=0.42939 \\ \\rm{MPa} =429.39 \\ \\rm{kPa}[\/latex]<\/p>\r\n<p style=\"padding-left: 80px\">[latex]x=\\dfrac{v-v_f}{v_g-v_f} =\\dfrac{0.2-0.001566}{0.289297-0.001566}=0.68965[\/latex]<\/p>\r\n&nbsp;\r\n<p class=\"indent no-indent\" style=\"padding-left: 40px\"><span lang=\"en-US\" xml:lang=\"en-US\">From <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-b-thermodynamic-properties-of-ammonia#TB1\" target=\"_blank\" rel=\"noopener\">Table B1<\/a><\/span><span lang=\"en-US\" xml:lang=\"en-US\">: at <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C, <\/span><\/p>\r\n<p style=\"padding-left: 80px\"><em lang=\"en-US\" xml:lang=\"en-US\">u<\/em><sub><em>f<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span>342.48 kJ\/kg\u00a0\u00a0\u00a0 and\u00a0 \u00a0 <em lang=\"en-US\" xml:lang=\"en-US\">u<\/em><sub><em>g<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 1481.17 kJ\/kg<\/p>\r\n<p style=\"padding-left: 80px\"><em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><sub><em>f<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 343.16 kJ\/kg\u00a0\u00a0\u00a0 and\u00a0 \u00a0 <em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><sub><em>g<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 1605.39 kJ\/kg<\/p>\r\n<p style=\"padding-left: 80px\"><em lang=\"en-US\" xml:lang=\"en-US\">s<\/em><sub><em>f<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 1.4716 kJ\/kgK\u00a0\u00a0 and \u00a0\u00a0 <em lang=\"en-US\" xml:lang=\"en-US\">s<\/em><sub><em>g<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 6.0926 kJ\/kgK<\/p>\r\n&nbsp;\r\n<p class=\"no-indent indent\" style=\"padding-left: 40px\"><span lang=\"en-US\" xml:lang=\"en-US\">T<\/span><span lang=\"en-US\" xml:lang=\"en-US\">herefore, the specific internal energy,\u00a0 specific enthalpy, and specific entropy of this two-phase mixture are <\/span><\/p>\r\n&nbsp;\r\n<p style=\"padding-left: 80px\">[latex]\\begin{align*} u &amp;=u_f+x(u_g-u_f) \\\\&amp;=342.48+0.68965 \\times (1481.17-342.48)=1127.78 \\ \\rm{kJ\/kg} \\end{align*}[\/latex]<\/p>\r\n<p style=\"padding-left: 80px\">[latex]\\begin{align*} h &amp;=h_f+x(h_g-h_f) \\\\&amp;= 343.16+0.68965 \\times (1605.39-343.16)=1213.66 \\ \\rm{kJ\/kg} \\end{align*}[\/latex]<\/p>\r\n<p style=\"padding-left: 80px\">[latex]\\begin{align*} s &amp;=s_f+x(s_g-s_f) \\\\&amp;=1.4716+0.68965 \\times (6.0926-1.4716)=4.6585 \\ \\rm{kJ\/kgK} \\end{align*}[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p class=\"import-Normal no-indent\" style=\"text-align: left\"><span lang=\"en-US\" xml:lang=\"en-US\">Refrigerant R134a has a specific enthalpy <\/span><em lang=\"en-US\" xml:lang=\"en-US\">h <\/em><span lang=\"en-US\" xml:lang=\"en-US\">= <\/span><span lang=\"en-US\" xml:lang=\"en-US\">420<\/span> kJ\/kg<span lang=\"en-US\" xml:lang=\"en-US\"> at <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=<\/span><span lang=\"en-US\" xml:lang=\"en-US\">2<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C<\/span><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">Determine <\/span><span lang=\"en-US\" xml:lang=\"en-US\">the pressure <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><span lang=\"en-US\" xml:lang=\"en-US\"> and specific volume <\/span><em lang=\"en-US\" xml:lang=\"en-US\">v<\/em> <span lang=\"en-US\" xml:lang=\"en-US\">of <\/span><span lang=\"en-US\" xml:lang=\"en-US\">R134a<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> at<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> this state. <\/span><\/p>\r\n&nbsp;\r\n<p class=\"import-Normal hanging-indent\" style=\"text-align: left\"><span lang=\"en-US\" xml:lang=\"en-US\"><em><span style=\"text-decoration: underline\">Solution<\/span><\/em>:<\/span><\/p>\r\n\r\n<ol style=\"text-align: left\">\r\n \t<li class=\"no-indent\"><span lang=\"en-US\" xml:lang=\"en-US\">Refer to case 4 as both <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\"> and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><span lang=\"en-US\" xml:lang=\"en-US\"> are given for<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">R134a<\/span><span lang=\"en-US\" xml:lang=\"en-US\">. B<\/span><span lang=\"en-US\" xml:lang=\"en-US\">ecause the <\/span><span lang=\"en-US\" xml:lang=\"en-US\">p<\/span><span lang=\"en-US\" xml:lang=\"en-US\">rocedures for cases 3 and 4 are the same, <\/span><span lang=\"en-US\" xml:lang=\"en-US\">the flow chart for case <\/span><span lang=\"en-US\" xml:lang=\"en-US\">3, <a href=\"#2.4.3\">Figure 2.4.3<\/a>,<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> is used by replacing <\/span><em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><span lang=\"en-US\" xml:lang=\"en-US\"> with <\/span><em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span><\/li>\r\n \t<li class=\"no-indent\"><span lang=\"en-US\" xml:lang=\"en-US\">From <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-c-thermodynamic-properties-of-r134a#TC1\" target=\"_blank\" rel=\"noopener\">Table C1<\/a><\/span><span lang=\"en-US\" xml:lang=\"en-US\">:\u00a0\u00a0\u00a0 <\/span><span lang=\"en-US\" xml:lang=\"en-US\">a<\/span><span lang=\"en-US\" xml:lang=\"en-US\">t <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=<\/span><span lang=\"en-US\" xml:lang=\"en-US\">2<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C, <\/span><em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><sub><em>g<\/em><\/sub><span lang=\"en-US\" xml:lang=\"en-US\">=<\/span>409.75 kJ\/kg. Because <em lang=\"en-US\" xml:lang=\"en-US\">h <\/em><span lang=\"en-US\" xml:lang=\"en-US\">= <\/span><span lang=\"en-US\" xml:lang=\"en-US\">420<\/span> kJ\/kg <span lang=\"en-US\" xml:lang=\"en-US\">&gt; <\/span><em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><sub><em>g<\/em><\/sub><span lang=\"en-US\" xml:lang=\"en-US\"> ,<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">R134a<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> at this state is a <\/span><span lang=\"en-US\" xml:lang=\"en-US\">superheated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">vapour<\/span><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span><\/li>\r\n \t<li class=\"no-indent\"><span lang=\"en-US\" xml:lang=\"en-US\">From <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-c-thermodynamic-properties-of-r134a#TC2\" target=\"_blank\" rel=\"noopener\">Table C2<\/a><\/span><span lang=\"en-US\" xml:lang=\"en-US\">:<\/span><\/li>\r\n<\/ol>\r\n<p style=\"padding-left: 80px;text-align: left\"><span lang=\"en-US\" xml:lang=\"en-US\">A<\/span><span lang=\"en-US\" xml:lang=\"en-US\">t <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=<\/span><span lang=\"en-US\" xml:lang=\"en-US\">2<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><sub><em>1<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">= 1<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0 kPa:<\/span> <em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><sub><em>1<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">= <\/span><span lang=\"en-US\" xml:lang=\"en-US\">420.31<\/span> kJ\/kg, <em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><sub><em>1<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 0.233731 m<sup>3<\/sup>\/kg<\/p>\r\n<p style=\"padding-left: 80px;text-align: left\"><span lang=\"en-US\" xml:lang=\"en-US\">A<\/span><span lang=\"en-US\" xml:lang=\"en-US\">t <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=20<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><sub><em>2<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">= 150 kPa:<\/span> <em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><sub><em>2<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">= 419.33<\/span> kJ\/kg, <em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><sub><em>2<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 0.154053 m<sup>3<\/sup>\/kg<\/p>\r\n<p class=\"no-indent\" style=\"padding-left: 40px;text-align: left\"><span lang=\"en-US\" xml:lang=\"en-US\">Because <\/span><span lang=\"en-US\" xml:lang=\"en-US\">419.33<\/span> kJ\/kg<span lang=\"en-US\" xml:lang=\"en-US\"> &lt; <\/span><span lang=\"en-US\" xml:lang=\"en-US\">420<\/span> kJ\/kg &lt; <span lang=\"en-US\" xml:lang=\"en-US\">420.31 <\/span>kJ\/kg, <span lang=\"en-US\" xml:lang=\"en-US\">t<\/span><span lang=\"en-US\" xml:lang=\"en-US\">he pressure of R134a <\/span><span lang=\"en-US\" xml:lang=\"en-US\">at the given state must be between 100 kPa and 150 kPa. <\/span><span lang=\"en-US\" xml:lang=\"en-US\">Use linear interpolation to calculate the pressure and specific volume<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> at the given state<\/span><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span><\/p>\r\n<p style=\"padding-left: 40px;text-align: left\"><span style=\"text-decoration: underline\"><em lang=\"en-US\" xml:lang=\"en-US\">Pressure <\/em><\/span><\/p>\r\n<p style=\"padding-left: 80px\">[latex]\\because\\dfrac{P-P_1}{P_2-P_1} =\\dfrac{h-h_1}{h_2-h_1}[\/latex]<\/p>\r\n<p style=\"padding-left: 80px\">[latex]\\therefore\\dfrac{P-100}{150-100} =\\dfrac{420-420.31}{419.33-420.31}[\/latex]<\/p>\r\n<p style=\"padding-left: 80px;text-align: left\">[latex] \\therefore P = 115.82 \\ \\rm{kPa}[\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"padding-left: 40px\"><span style=\"text-decoration: underline\"><em lang=\"en-US\" xml:lang=\"en-US\">Specific volume<\/em><\/span><\/p>\r\n<p style=\"padding-left: 80px\">[latex]\\because\\dfrac{v-v_1}{v_2-v_1} =\\dfrac{h-h_1}{h_2-h_1}[\/latex]<\/p>\r\n<p style=\"padding-left: 80px\">[latex]\\therefore\\dfrac{v-0.233731}{0.154053-0.233731} =\\dfrac{420-420.31}{419.33-420.31}[\/latex]<\/p>\r\n<p style=\"padding-left: 80px\">[latex] \\therefore v = 0.208527 \\ \\rm{m^3\/kg}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"thermodynamic-tables\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Practice Problems<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"52\"]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"thermodynamic-tables\">\n<p><span lang=\"en-US\" xml:lang=\"en-US\">Thermodynamic tables are <\/span><span lang=\"en-US\" xml:lang=\"en-US\">commonly used to determine the properties of a substance at a given state. <\/span><span lang=\"en-US\" xml:lang=\"en-US\">This book includes<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> t<\/span><span lang=\"en-US\" xml:lang=\"en-US\">he tables for <\/span><span lang=\"en-US\" xml:lang=\"en-US\">four<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">pure <\/span><span lang=\"en-US\" xml:lang=\"en-US\">substances<\/span><span lang=\"en-US\" xml:lang=\"en-US\">: water, ammonia, R134a, and carbon dioxide. <\/span> <span lang=\"en-US\" xml:lang=\"en-US\">The data in these tables are obtained from <\/span><a href=\"https:\/\/webbook.nist.gov\/chemistry\/fluid\/\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">NIST Chemistry <\/span><span lang=\"en-US\" xml:lang=\"en-US\">WebBook<\/span><span lang=\"en-US\" xml:lang=\"en-US\">, SRD 69<\/span><\/a>, which consists of the thermophysical properties of various common fluids.<\/p>\n<p class=\"indent no-indent\"><span lang=\"en-US\" xml:lang=\"en-US\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-a-thermodynamic-properties-of-water\/\" target=\"_blank\" rel=\"noopener\">Appendix A<\/a>: Thermodynamic Properties of Water<\/span><\/p>\n<ul>\n<li><span lang=\"en-US\" xml:lang=\"en-US\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-a-thermodynamic-properties-of-water#TA1\" target=\"_blank\" rel=\"noopener\">Table A1<\/a>: Saturated water<\/span><\/li>\n<li><span lang=\"en-US\" xml:lang=\"en-US\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-a-thermodynamic-properties-of-water#TA2\" target=\"_blank\" rel=\"noopener\">Table A2<\/a>: Superheated vapour, water<\/span><\/li>\n<li><span lang=\"en-US\" xml:lang=\"en-US\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-a-thermodynamic-properties-of-water#TA3\" target=\"_blank\" rel=\"noopener\">Table A3<\/a>: Compressed liquid water<\/span><\/li>\n<\/ul>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-b-thermodynamic-properties-of-ammonia\/\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Appendix <\/span><span lang=\"en-US\" xml:lang=\"en-US\">B<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Thermodynamic Properties of <\/span><span lang=\"en-US\" xml:lang=\"en-US\">Ammonia<\/span><\/p>\n<ul>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-b-thermodynamic-properties-of-ammonia#TB1\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Table <\/span><span lang=\"en-US\" xml:lang=\"en-US\">B1<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Saturated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">ammonia<\/span><\/li>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-b-thermodynamic-properties-of-ammonia#TB2\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Table <\/span><span lang=\"en-US\" xml:lang=\"en-US\">B2<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Superheated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">ammonia<\/span><\/li>\n<\/ul>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-c-thermodynamic-properties-of-r134a\/\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Appendix <\/span><span lang=\"en-US\" xml:lang=\"en-US\">C<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Thermodynamic Properties of <\/span><span lang=\"en-US\" xml:lang=\"en-US\">R134a<\/span><\/p>\n<ul>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-c-thermodynamic-properties-of-r134a#TC1\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Table <\/span><span lang=\"en-US\" xml:lang=\"en-US\">C1<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Saturated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">R134a<\/span><\/li>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-c-thermodynamic-properties-of-r134a#TC2\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Table <\/span><span lang=\"en-US\" xml:lang=\"en-US\">C2<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Superheated<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> R134a<\/span><\/li>\n<\/ul>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-d-thermodynamic-properties-of-co2\/\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Appendix <\/span><span lang=\"en-US\" xml:lang=\"en-US\">D<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Thermodynamic Properties of <\/span><span lang=\"en-US\" xml:lang=\"en-US\">Carbon Dioxide<\/span><\/p>\n<ul>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-d-thermodynamic-properties-of-co2#TD1\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Table <\/span><span lang=\"en-US\" xml:lang=\"en-US\">D1<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Saturated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">CO<\/span><sub>2<\/sub><\/li>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-d-thermodynamic-properties-of-co2#TD2\" target=\"_blank\" rel=\"noopener\"><span lang=\"en-US\" xml:lang=\"en-US\">Table <\/span><span lang=\"en-US\" xml:lang=\"en-US\">D2<\/span><\/a><span lang=\"en-US\" xml:lang=\"en-US\">: Superheated<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">CO<\/span><sub>2<\/sub><\/li>\n<\/ul>\n<p class=\"import-Normal\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">Tables A1, B1, C1, and D1 are the tables for the saturated fluids. <\/span><span lang=\"en-US\" xml:lang=\"en-US\">They are <\/span><span lang=\"en-US\" xml:lang=\"en-US\">used to find the properties of the <\/span><span lang=\"en-US\" xml:lang=\"en-US\">corresponding <\/span><span lang=\"en-US\" xml:lang=\"en-US\">fluid<\/span><span lang=\"en-US\" xml:lang=\"en-US\">s<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> in saturated liquid, saturated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">vapour,<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> and two-phase regions. Tables A2, B2, C2, and D2 are the <\/span><span lang=\"en-US\" xml:lang=\"en-US\">superheated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">vapour<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> tables<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> for <\/span><span lang=\"en-US\" xml:lang=\"en-US\">find<\/span><span lang=\"en-US\" xml:lang=\"en-US\">ing<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> the properties of the fluid<\/span><span lang=\"en-US\" xml:lang=\"en-US\">s<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> in the superheated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">vapour<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> region. Table A3 is the compressed liquid table<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> for water. <\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"import-Normal\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">In these tables, the specific volume, specific internal energy, specific enthalpy, and specific entropy are tabulated as functions of the pressure and temperature. <\/span><span lang=\"en-US\" xml:lang=\"en-US\">Among those thermodynamic properties, [latex]P[\/latex]<\/span>, [latex]T[\/latex], and [latex]v[\/latex] <span lang=\"en-US\" xml:lang=\"en-US\">are measurable properties<\/span>, <span lang=\"en-US\" xml:lang=\"en-US\">and [latex]u[\/latex], [latex]h[\/latex], and [latex]s[\/latex] <\/span><span lang=\"en-US\" xml:lang=\"en-US\">cannot be measured directly. T<\/span><span lang=\"en-US\" xml:lang=\"en-US\">hey<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">are calculated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">with respect to <\/span><span lang=\"en-US\" xml:lang=\"en-US\">predefined <\/span><span lang=\"en-US\" xml:lang=\"en-US\">reference states.<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> The reference state<\/span><span lang=\"en-US\" xml:lang=\"en-US\">s<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> used for <\/span><span lang=\"en-US\" xml:lang=\"en-US\">different tables in this book are stated in <\/span><span lang=\"en-US\" xml:lang=\"en-US\">A<\/span><span lang=\"en-US\" xml:lang=\"en-US\">ppendices A to D. <\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"import-Normal\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">It is important to be aware that thermodynamic tables can be generated with respect to different reference states, which means that <\/span><span lang=\"en-US\" xml:lang=\"en-US\">the values of specific internal energy<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">[latex]u[\/latex], specific enthalpy [latex]h[\/latex], and specific entropy [latex]s[\/latex]<\/span> can vary <span lang=\"en-US\" xml:lang=\"en-US\">depending on the source of the tables<\/span><span lang=\"en-US\" xml:lang=\"en-US\">. Additionally, in most thermodynamic analyses, we are concerned with the <em lang=\"en-US\" xml:lang=\"en-US\">changes<\/em> in these specific properties, such as [latex]\\Delta u[\/latex], [latex]\\Delta h[\/latex], and [latex]\\Delta s[\/latex]. Therefore, it is important to use tables from a single source in order to ensure that consistent values of [latex]u[\/latex], [latex]h[\/latex], and [latex]s[\/latex] are used in the analysis. Using tables from different sources in calculations can lead to incorrect or misleading conclusions.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"import-Normal\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">When using thermodynamic tables, we will need to consider the following two questions: (1) how many independent variables are required to fix the state of a fluid? and (2) how the phase of a fluid is determined? is it a compressed liquid, superheated vapour, or two-phase liquid-vapour mixture?<\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"import-Normal\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">By examining the tables in Appendices A to D, you probably have already noticed that all properties in these tables are intensive properties. For a non-reactive system in thermodynamic equilibrium, the Gibbs phase rule indicates that the number of independent, intensive properties of the system can be found as <\/span><\/p>\n<p style=\"text-align: center\"><span lang=\"en-US\" xml:lang=\"en-US\">[latex]f=c+2-p[\/latex]<\/span><\/p>\n<p>where<\/p>\n<p style=\"padding-left: 40px\"><span lang=\"en-US\" xml:lang=\"en-US\">[latex]f[\/latex]:\u00a0 the number of independent, intensive properties of the system that are needed to fix the state of the system. It is also called the degree of freedom of the system.<br \/>\n<\/span><\/p>\n<p style=\"padding-left: 40px\"><span lang=\"en-US\" xml:lang=\"en-US\">[latex]c[\/latex]: the number of components of the system. For a pure substance, [latex]c=1[\/latex].<br \/>\n<\/span><\/p>\n<p style=\"padding-left: 40px\"><span lang=\"en-US\" xml:lang=\"en-US\">[latex]p[\/latex]: the number of phases of the system. For a single-phase\u00a0 system, [latex]p=1[\/latex]. If a system is in a two-phase region, [latex]p=2[\/latex].<br \/>\n<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span lang=\"en-US\" xml:lang=\"en-US\">For a single-phase system of a pure substance, [latex]c=1[\/latex], [latex]p=1[\/latex]; therefore, [latex]f=2[\/latex]. This means that two independent, intensive properties are required to fix the state of the system. In other words, if any TWO independent, intensive properties from this list, [latex]P[\/latex], [latex]T[\/latex], [latex]v[\/latex], [latex]u[\/latex], [latex]h[\/latex], [latex]s[\/latex], are known, then the remaining intensive properties can be determined from the thermodynamic tables.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span lang=\"en-US\" xml:lang=\"en-US\">For a pure substance in a two-phase, saturated region, such as a mixture of saturated liquid water and saturated water vapour, [latex]c=1[\/latex], [latex]p=2[\/latex]; therefore, [latex]f=1[\/latex]. This means that the system is of one-degree of freedom; if one independent, intensive property, such as the saturated temperature, [latex]T_{sat}[\/latex], or the saturated pressure, [latex]P_{sat}[\/latex] of the mixture is fixed, all other intensive properties of the saturated mixture can be found from the thermodynamic tables, which include [latex]P_{sat}[\/latex] (or [latex]T_{sat}[\/latex]), [latex]v_f[\/latex], [latex]v_g[\/latex], [latex]u_f[\/latex], [latex]u_g[\/latex], [latex]h_f[\/latex], [latex]h_g[\/latex], [latex]s_f[\/latex], and [latex]s_g[\/latex]. With the quality [latex]x[\/latex], we can calculate the specific properties,\u00a0 [latex]v[\/latex], [latex]u[\/latex], [latex]h[\/latex], and [latex]s[\/latex] of the saturated mixture. In summary, for a pure substance in a two-phase, saturated region, if any TWO independent, intensive properties from this list, [latex]P[\/latex], [latex]T[\/latex], [latex]v[\/latex], [latex]u[\/latex], [latex]h[\/latex], [latex]s[\/latex], and [latex]x[\/latex], are known, then the remaining intensive properties can be calculated from the thermodynamic tables.<br \/>\n<\/span><\/p>\n<\/div>\n<div class=\"thermodynamic-tables\">\n<p class=\"import-Normal no-indent\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">The following flow charts demonstrate four common scenarios and t<\/span>he corresponding procedures of finding thermodynamic properties.<\/p>\n<p>&nbsp;<\/p>\n<p>Case 1: <strong>both <em>T<\/em> and<em> P <\/em>are given<\/strong>. You may draw a [latex]P-T[\/latex] diagram (see <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/phase-diagrams#2.3.2\" target=\"_blank\" rel=\"noopener\">Figure 2.3.2<\/a>) to help you better understand the flow chart in Figure 2.4.1.<\/p>\n<figure id=\"attachment_894\" aria-describedby=\"caption-attachment-894\" style=\"width: 800px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_P-T.png\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-894\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_P-T-1024x890.png\" alt=\"Flow chart for determining fluid properties from thermodynamic tables if P and T are known.\" width=\"800\" height=\"696\" srcset=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_P-T-1024x890.png 1024w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_P-T-300x261.png 300w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_P-T-768x668.png 768w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_P-T-65x57.png 65w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_P-T-225x196.png 225w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_P-T-350x304.png 350w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_P-T.png 1380w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/a><figcaption id=\"caption-attachment-894\" class=\"wp-caption-text\"><em><strong>Figure 2.4.1<\/strong><\/em>\u00a0<em>Flow chart for determining fluid properties from thermodynamic tables if P and T are known<\/em><\/figcaption><\/figure>\n<p>Case 2: <strong>both <em>T<\/em> and <em>x<\/em> are given<\/strong>. You may draw a [latex]T-v[\/latex] diagram (see <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/phase-diagrams#2.3.4\" target=\"_blank\" rel=\"noopener\">Figure 2.3.4<\/a>) to help you better understand the flow chart in Figure 2.4.2.<\/p>\n<figure id=\"attachment_896\" aria-describedby=\"caption-attachment-896\" style=\"width: 800px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-x.png\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-896\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-x-1024x505.png\" alt=\"Flow chart for determining fluid properties from thermodynamic tables if T and x are known\" width=\"800\" height=\"395\" srcset=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-x-1024x505.png 1024w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-x-300x148.png 300w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-x-768x379.png 768w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-x-65x32.png 65w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-x-225x111.png 225w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-x-350x173.png 350w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-x.png 1460w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/a><figcaption id=\"caption-attachment-896\" class=\"wp-caption-text\"><em><strong>Figure 2.4.2<\/strong><\/em>\u00a0<em>Flow chart for determining fluid properties from thermodynamic tables if T and x are known<\/em><\/figcaption><\/figure>\n<\/div>\n<div class=\"thermodynamic-tables\">\n<p>Case 3: <strong>both <em>T<\/em> and <em>v<\/em> are given<\/strong>. You may draw a [latex]T-v[\/latex] diagram (see <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/phase-diagrams#2.3.4\" target=\"_blank\" rel=\"noopener\">Figure 2.3.4<\/a>) to help you better understand the flow <a id=\"2.4.3\"><\/a>chart in Figure 2.4.3.<\/p>\n<figure id=\"attachment_898\" aria-describedby=\"caption-attachment-898\" style=\"width: 900px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-v.png\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-898\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-v-1024x726.png\" alt=\"Flow chart for determining fluid properties from thermodynamics tables if T and v are known\" width=\"900\" height=\"638\" srcset=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-v-1024x726.png 1024w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-v-300x213.png 300w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-v-768x545.png 768w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-v-1536x1089.png 1536w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-v-65x46.png 65w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-v-225x160.png 225w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-v-350x248.png 350w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Flowchart_known_T-v.png 1640w\" sizes=\"auto, (max-width: 900px) 100vw, 900px\" \/><\/a><figcaption id=\"caption-attachment-898\" class=\"wp-caption-text\"><em><strong>Figure 2.4.3<\/strong><\/em>\u00a0<em>Flow chart for determining fluid properties from thermodynamics tables if T and v are known<\/em><\/figcaption><\/figure>\n<p>Case 4: <strong>temperature <em>T<\/em> and one of<em> u, h <\/em>and<em> s <\/em>are given<\/strong>. The procedure is exactly the same as for case 3. Replace [latex]v[\/latex] with the given <span lang=\"en-US\" xml:lang=\"en-US\">[latex]u[\/latex], [latex]h[\/latex], or [latex]s[\/latex]<\/span> in the flow chart, <a href=\"#2.4.3\">Figure 2.4.3<\/a>.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"import-Normal no-indent\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">T<\/span><span lang=\"en-US\" xml:lang=\"en-US\">he compressed liquid table is presented only for water in <\/span><span lang=\"en-US\" xml:lang=\"en-US\">the pressure range of 0.5-50 MPa<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> i<\/span><span lang=\"en-US\" xml:lang=\"en-US\">n this book<\/span><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> When the compressed liquid tables are not available for a specific fluid or in a specific range, the <\/span><span lang=\"en-US\" xml:lang=\"en-US\">saturated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">liquid <\/span><span lang=\"en-US\" xml:lang=\"en-US\">properties at <\/span><span lang=\"en-US\" xml:lang=\"en-US\">the same temperature<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> may be used as an approximation, i.e., [latex]v\\approx\\ v_{f@T}[\/latex], [latex]u\\approx\\ u_{f@T}[\/latex], [latex]h\\approx\\ h_{f@T}[\/latex], and [latex]s\\approx\\ s_{f@T}[\/latex]<\/span>.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"import-Normal no-indent\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">The tables in Appendices A to D are presented <\/span><span lang=\"en-US\" xml:lang=\"en-US\">with <\/span><span lang=\"en-US\" xml:lang=\"en-US\">a<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> small<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> temperature increment<\/span><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> L<\/span><span lang=\"en-US\" xml:lang=\"en-US\">inear interpolations, see <a href=\"#2.4.4\">Figure 2.4.4<\/a>, <\/span><span lang=\"en-US\" xml:lang=\"en-US\">are often used if the <\/span><span lang=\"en-US\" xml:lang=\"en-US\">given temperature or other properties cannot be found directly from these tables.<br \/>\n<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">[latex]y=y_0+\\left( x-x_0 \\right) \\dfrac{y_1-y_0}{x_1-x_0}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>where [latex]\\left( x, y \\right)[\/latex] is the state, at which the property [latex]x[\/latex] is known and the property [latex]y[\/latex] is to be found. [latex]\\left( x_0, y_0 \\right)[\/latex] and [latex]\\left( x_1, y_1 \\right)[\/latex] indicate the properties of two known states, between which the unknown state [latex]\\left( x, y \\right)[\/latex] is located. To improve the accuracy, the two states should be selected as close as possible to the unknown <a id=\"2.4.4\"><\/a>state.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_2978\" aria-describedby=\"caption-attachment-2978\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1.png\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2978 size-medium\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1-300x300.png\" alt=\"Three points on a linear line, illustrating the principle of linear Interpolation\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1-300x300.png 300w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1-1024x1024.png 1024w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1-150x150.png 150w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1-768x768.png 768w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1-1536x1536.png 1536w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1-65x65.png 65w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1-225x225.png 225w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1-350x350.png 350w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/06\/Fig.-2-11-1.png 2048w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-2978\" class=\"wp-caption-text\"><em><strong>Figure 2.4.4<\/strong> Linear interpolation<\/em><\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p class=\"import-Normal no-indent\" style=\"text-align: justify\"><span lang=\"en-US\" xml:lang=\"en-US\">The following examples demonstrate how to <\/span><span lang=\"en-US\" xml:lang=\"en-US\">use these tables to\u00a0<\/span><span lang=\"en-US\" xml:lang=\"en-US\">find the properties of <\/span><span lang=\"en-US\" xml:lang=\"en-US\">a <\/span><span lang=\"en-US\" xml:lang=\"en-US\">compressed liquid, superheated vapour, and liquid-vapour mixture. <\/span><\/p>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p class=\"import-Normal\"><span lang=\"en-US\" xml:lang=\"en-US\">Determine the properties of water at <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=<\/span><span lang=\"en-US\" xml:lang=\"en-US\">1<\/span><span lang=\"en-US\" xml:lang=\"en-US\">5<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=<\/span><span lang=\"en-US\" xml:lang=\"en-US\">100 kPa. <\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"import-Normal\"><span lang=\"en-US\" xml:lang=\"en-US\"><span style=\"text-decoration: underline\"><em>Solution<\/em><\/span>:<\/span><\/p>\n<ol>\n<li>\n<p class=\"no-indent indent\"><span lang=\"en-US\" xml:lang=\"en-US\">Both <\/span>temperature <span lang=\"en-US\" xml:lang=\"en-US\">and <\/span>pressure<span lang=\"en-US\" xml:lang=\"en-US\"> are given for water. Use the flow chart for case 1, <a href=\"#2.4.1\">Figure 2.4.1<\/a>.<\/span><\/p>\n<\/li>\n<li>\n<p class=\"no-indent indent\"><span lang=\"en-US\" xml:lang=\"en-US\">From <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-a-thermodynamic-properties-of-water#TA1\" target=\"_blank\" rel=\"noopener\">Table A1<\/a>: <\/span><span lang=\"en-US\" xml:lang=\"en-US\">at<\/span> <em>T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=150<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C, <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><sub><em>sa<\/em><\/sub><sub>t<\/sub><span lang=\"en-US\" xml:lang=\"en-US\"> = <\/span><span lang=\"en-US\" xml:lang=\"en-US\">0.47617 MPa = 476.17 kPa. <\/span><\/p>\n<\/li>\n<li>\n<p class=\"no-indent indent\"><span lang=\"en-US\" xml:lang=\"en-US\">Because <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=100 kPa <\/span><span lang=\"en-US\" xml:lang=\"en-US\">&lt; <\/span><span lang=\"en-US\" xml:lang=\"en-US\">476.17 kPa<\/span><span lang=\"en-US\" xml:lang=\"en-US\">, or <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P <\/em><span lang=\"en-US\" xml:lang=\"en-US\">&lt; <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><sub><em>sat<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">,<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> water <\/span><span lang=\"en-US\" xml:lang=\"en-US\">at this state <\/span><span lang=\"en-US\" xml:lang=\"en-US\">is <\/span><span lang=\"en-US\" xml:lang=\"en-US\">a superheated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">vapour<\/span><span lang=\"en-US\" xml:lang=\"en-US\">. <\/span><\/p>\n<\/li>\n<li>\n<p class=\"no-indent indent\"><span lang=\"en-US\" xml:lang=\"en-US\">From <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-a-thermodynamic-properties-of-water#TA2\" target=\"_blank\" rel=\"noopener\">Table A2<\/a><\/span><span lang=\"en-US\" xml:lang=\"en-US\">:<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">at <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=150<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=100 kPa<\/span><em lang=\"en-US\" xml:lang=\"en-US\">, <\/em><\/p>\n<\/li>\n<\/ol>\n<p class=\"no-indent indent\" style=\"padding-left: 40px\"><em lang=\"en-US\" xml:lang=\"en-US\">v <\/em><span lang=\"en-US\" xml:lang=\"en-US\">= <\/span>1.93665 m<sup>3<\/sup>\/kg,\u00a0\u00a0 <em>u<\/em> = 2582.94 kJ\/kg<\/p>\n<p class=\"no-indent indent\" style=\"padding-left: 40px\"><em>h<\/em> = 2776.60 kJ\/kg,\u00a0\u00a0\u00a0\u00a0 <em>s<\/em> = 7.6148 kJ\/kgK<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p class=\"import-Normal\"><span lang=\"en-US\" xml:lang=\"en-US\">Determine the properties of a<\/span><span lang=\"en-US\" xml:lang=\"en-US\">mmonia<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> at <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">v<\/em> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0.2<\/span> m<sup>3<\/sup>\/kg<span lang=\"en-US\" xml:lang=\"en-US\">. <\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"import-Normal\"><span lang=\"en-US\" xml:lang=\"en-US\"><em><span style=\"text-decoration: underline\">Solution<\/span><\/em>:<\/span><\/p>\n<ol>\n<li>\n<p class=\"no-indent indent\"><span lang=\"en-US\" xml:lang=\"en-US\">Both <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\"> and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><span lang=\"en-US\" xml:lang=\"en-US\"> are given for<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> ammonia<\/span><span lang=\"en-US\" xml:lang=\"en-US\">. Use the flow chart for case <\/span><span lang=\"en-US\" xml:lang=\"en-US\">3, <a href=\"#2.4.3\">Figure 2.4.3<\/a><\/span><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span><\/p>\n<\/li>\n<li>\n<p class=\"no-indent indent\"><span lang=\"en-US\" xml:lang=\"en-US\">From <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-b-thermodynamic-properties-of-ammonia#TB1\" target=\"_blank\" rel=\"noopener\">Table B1<\/a><\/span><span lang=\"en-US\" xml:lang=\"en-US\">: at <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C, <\/span><em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><sub><em>f<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span>0.001566 m<sup>3<\/sup>\/kg and <em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><sub><em>g<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 0.289297 m<sup>3<\/sup>\/kg.<\/p>\n<\/li>\n<li>\n<p class=\"no-indent indent\">Because <em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><sub><em>f<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">&lt; <em>v <\/em>&lt;<\/span> <em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><sub><em>g<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">,<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> ammonia at this state is <\/span><span lang=\"en-US\" xml:lang=\"en-US\">a <\/span><span lang=\"en-US\" xml:lang=\"en-US\">liquid-<\/span><span lang=\"en-US\" xml:lang=\"en-US\">vapour<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> two<\/span><span lang=\"en-US\" xml:lang=\"en-US\">&#8211;<\/span><span lang=\"en-US\" xml:lang=\"en-US\">phase mixture. Its pressure and quality are<br \/>\n<\/span><\/p>\n<\/li>\n<\/ol>\n<p style=\"padding-left: 80px\">[latex]P=P_{sat}=0.42939 \\ \\rm{MPa} =429.39 \\ \\rm{kPa}[\/latex]<\/p>\n<p style=\"padding-left: 80px\">[latex]x=\\dfrac{v-v_f}{v_g-v_f} =\\dfrac{0.2-0.001566}{0.289297-0.001566}=0.68965[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p class=\"indent no-indent\" style=\"padding-left: 40px\"><span lang=\"en-US\" xml:lang=\"en-US\">From <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-b-thermodynamic-properties-of-ammonia#TB1\" target=\"_blank\" rel=\"noopener\">Table B1<\/a><\/span><span lang=\"en-US\" xml:lang=\"en-US\">: at <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C, <\/span><\/p>\n<p style=\"padding-left: 80px\"><em lang=\"en-US\" xml:lang=\"en-US\">u<\/em><sub><em>f<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span>342.48 kJ\/kg\u00a0\u00a0\u00a0 and\u00a0 \u00a0 <em lang=\"en-US\" xml:lang=\"en-US\">u<\/em><sub><em>g<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 1481.17 kJ\/kg<\/p>\n<p style=\"padding-left: 80px\"><em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><sub><em>f<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 343.16 kJ\/kg\u00a0\u00a0\u00a0 and\u00a0 \u00a0 <em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><sub><em>g<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 1605.39 kJ\/kg<\/p>\n<p style=\"padding-left: 80px\"><em lang=\"en-US\" xml:lang=\"en-US\">s<\/em><sub><em>f<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 1.4716 kJ\/kgK\u00a0\u00a0 and \u00a0\u00a0 <em lang=\"en-US\" xml:lang=\"en-US\">s<\/em><sub><em>g<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 6.0926 kJ\/kgK<\/p>\n<p>&nbsp;<\/p>\n<p class=\"no-indent indent\" style=\"padding-left: 40px\"><span lang=\"en-US\" xml:lang=\"en-US\">T<\/span><span lang=\"en-US\" xml:lang=\"en-US\">herefore, the specific internal energy,\u00a0 specific enthalpy, and specific entropy of this two-phase mixture are <\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 80px\">[latex]\\begin{align*} u &=u_f+x(u_g-u_f) \\\\&=342.48+0.68965 \\times (1481.17-342.48)=1127.78 \\ \\rm{kJ\/kg} \\end{align*}[\/latex]<\/p>\n<p style=\"padding-left: 80px\">[latex]\\begin{align*} h &=h_f+x(h_g-h_f) \\\\&= 343.16+0.68965 \\times (1605.39-343.16)=1213.66 \\ \\rm{kJ\/kg} \\end{align*}[\/latex]<\/p>\n<p style=\"padding-left: 80px\">[latex]\\begin{align*} s &=s_f+x(s_g-s_f) \\\\&=1.4716+0.68965 \\times (6.0926-1.4716)=4.6585 \\ \\rm{kJ\/kgK} \\end{align*}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p class=\"import-Normal no-indent\" style=\"text-align: left\"><span lang=\"en-US\" xml:lang=\"en-US\">Refrigerant R134a has a specific enthalpy <\/span><em lang=\"en-US\" xml:lang=\"en-US\">h <\/em><span lang=\"en-US\" xml:lang=\"en-US\">= <\/span><span lang=\"en-US\" xml:lang=\"en-US\">420<\/span> kJ\/kg<span lang=\"en-US\" xml:lang=\"en-US\"> at <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=<\/span><span lang=\"en-US\" xml:lang=\"en-US\">2<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C<\/span><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">Determine <\/span><span lang=\"en-US\" xml:lang=\"en-US\">the pressure <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><span lang=\"en-US\" xml:lang=\"en-US\"> and specific volume <\/span><em lang=\"en-US\" xml:lang=\"en-US\">v<\/em> <span lang=\"en-US\" xml:lang=\"en-US\">of <\/span><span lang=\"en-US\" xml:lang=\"en-US\">R134a<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> at<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> this state. <\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"import-Normal hanging-indent\" style=\"text-align: left\"><span lang=\"en-US\" xml:lang=\"en-US\"><em><span style=\"text-decoration: underline\">Solution<\/span><\/em>:<\/span><\/p>\n<ol style=\"text-align: left\">\n<li class=\"no-indent\"><span lang=\"en-US\" xml:lang=\"en-US\">Refer to case 4 as both <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\"> and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><span lang=\"en-US\" xml:lang=\"en-US\"> are given for<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">R134a<\/span><span lang=\"en-US\" xml:lang=\"en-US\">. B<\/span><span lang=\"en-US\" xml:lang=\"en-US\">ecause the <\/span><span lang=\"en-US\" xml:lang=\"en-US\">p<\/span><span lang=\"en-US\" xml:lang=\"en-US\">rocedures for cases 3 and 4 are the same, <\/span><span lang=\"en-US\" xml:lang=\"en-US\">the flow chart for case <\/span><span lang=\"en-US\" xml:lang=\"en-US\">3, <a href=\"#2.4.3\">Figure 2.4.3<\/a>,<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> is used by replacing <\/span><em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><span lang=\"en-US\" xml:lang=\"en-US\"> with <\/span><em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span><\/li>\n<li class=\"no-indent\"><span lang=\"en-US\" xml:lang=\"en-US\">From <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-c-thermodynamic-properties-of-r134a#TC1\" target=\"_blank\" rel=\"noopener\">Table C1<\/a><\/span><span lang=\"en-US\" xml:lang=\"en-US\">:\u00a0\u00a0\u00a0 <\/span><span lang=\"en-US\" xml:lang=\"en-US\">a<\/span><span lang=\"en-US\" xml:lang=\"en-US\">t <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=<\/span><span lang=\"en-US\" xml:lang=\"en-US\">2<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C, <\/span><em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><sub><em>g<\/em><\/sub><span lang=\"en-US\" xml:lang=\"en-US\">=<\/span>409.75 kJ\/kg. Because <em lang=\"en-US\" xml:lang=\"en-US\">h <\/em><span lang=\"en-US\" xml:lang=\"en-US\">= <\/span><span lang=\"en-US\" xml:lang=\"en-US\">420<\/span> kJ\/kg <span lang=\"en-US\" xml:lang=\"en-US\">&gt; <\/span><em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><sub><em>g<\/em><\/sub><span lang=\"en-US\" xml:lang=\"en-US\"> ,<\/span> <span lang=\"en-US\" xml:lang=\"en-US\">R134a<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> at this state is a <\/span><span lang=\"en-US\" xml:lang=\"en-US\">superheated <\/span><span lang=\"en-US\" xml:lang=\"en-US\">vapour<\/span><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span><\/li>\n<li class=\"no-indent\"><span lang=\"en-US\" xml:lang=\"en-US\">From <a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/back-matter\/appendix-c-thermodynamic-properties-of-r134a#TC2\" target=\"_blank\" rel=\"noopener\">Table C2<\/a><\/span><span lang=\"en-US\" xml:lang=\"en-US\">:<\/span><\/li>\n<\/ol>\n<p style=\"padding-left: 80px;text-align: left\"><span lang=\"en-US\" xml:lang=\"en-US\">A<\/span><span lang=\"en-US\" xml:lang=\"en-US\">t <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=<\/span><span lang=\"en-US\" xml:lang=\"en-US\">2<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><sub><em>1<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">= 1<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0<\/span><span lang=\"en-US\" xml:lang=\"en-US\">0 kPa:<\/span> <em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><sub><em>1<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">= <\/span><span lang=\"en-US\" xml:lang=\"en-US\">420.31<\/span> kJ\/kg, <em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><sub><em>1<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 0.233731 m<sup>3<\/sup>\/kg<\/p>\n<p style=\"padding-left: 80px;text-align: left\"><span lang=\"en-US\" xml:lang=\"en-US\">A<\/span><span lang=\"en-US\" xml:lang=\"en-US\">t <\/span><em lang=\"en-US\" xml:lang=\"en-US\">T<\/em><span lang=\"en-US\" xml:lang=\"en-US\">=20<\/span><sup>o<\/sup><span lang=\"en-US\" xml:lang=\"en-US\">C<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> and <\/span><em lang=\"en-US\" xml:lang=\"en-US\">P<\/em><sub><em>2<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">= 150 kPa:<\/span> <em lang=\"en-US\" xml:lang=\"en-US\">h<\/em><sub><em>2<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">= 419.33<\/span> kJ\/kg, <em lang=\"en-US\" xml:lang=\"en-US\">v<\/em><sub><em>2<\/em><\/sub> <span lang=\"en-US\" xml:lang=\"en-US\">=<\/span> 0.154053 m<sup>3<\/sup>\/kg<\/p>\n<p class=\"no-indent\" style=\"padding-left: 40px;text-align: left\"><span lang=\"en-US\" xml:lang=\"en-US\">Because <\/span><span lang=\"en-US\" xml:lang=\"en-US\">419.33<\/span> kJ\/kg<span lang=\"en-US\" xml:lang=\"en-US\"> &lt; <\/span><span lang=\"en-US\" xml:lang=\"en-US\">420<\/span> kJ\/kg &lt; <span lang=\"en-US\" xml:lang=\"en-US\">420.31 <\/span>kJ\/kg, <span lang=\"en-US\" xml:lang=\"en-US\">t<\/span><span lang=\"en-US\" xml:lang=\"en-US\">he pressure of R134a <\/span><span lang=\"en-US\" xml:lang=\"en-US\">at the given state must be between 100 kPa and 150 kPa. <\/span><span lang=\"en-US\" xml:lang=\"en-US\">Use linear interpolation to calculate the pressure and specific volume<\/span><span lang=\"en-US\" xml:lang=\"en-US\"> at the given state<\/span><span lang=\"en-US\" xml:lang=\"en-US\">.<\/span><\/p>\n<p style=\"padding-left: 40px;text-align: left\"><span style=\"text-decoration: underline\"><em lang=\"en-US\" xml:lang=\"en-US\">Pressure <\/em><\/span><\/p>\n<p style=\"padding-left: 80px\">[latex]\\because\\dfrac{P-P_1}{P_2-P_1} =\\dfrac{h-h_1}{h_2-h_1}[\/latex]<\/p>\n<p style=\"padding-left: 80px\">[latex]\\therefore\\dfrac{P-100}{150-100} =\\dfrac{420-420.31}{419.33-420.31}[\/latex]<\/p>\n<p style=\"padding-left: 80px;text-align: left\">[latex]\\therefore P = 115.82 \\ \\rm{kPa}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 40px\"><span style=\"text-decoration: underline\"><em lang=\"en-US\" xml:lang=\"en-US\">Specific volume<\/em><\/span><\/p>\n<p style=\"padding-left: 80px\">[latex]\\because\\dfrac{v-v_1}{v_2-v_1} =\\dfrac{h-h_1}{h_2-h_1}[\/latex]<\/p>\n<p style=\"padding-left: 80px\">[latex]\\therefore\\dfrac{v-0.233731}{0.154053-0.233731} =\\dfrac{420-420.31}{419.33-420.31}[\/latex]<\/p>\n<p style=\"padding-left: 80px\">[latex]\\therefore v = 0.208527 \\ \\rm{m^3\/kg}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"thermodynamic-tables\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Practice Problems<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"h5p-52\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-52\" class=\"h5p-iframe\" data-content-id=\"52\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"S_2.4_Q\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<div class=\"media-attributions clear\" prefix:cc=\"http:\/\/creativecommons.org\/ns#\" prefix:dc=\"http:\/\/purl.org\/dc\/terms\/\"><h2>Media Attributions<\/h2><ul><li about=\"https:\/\/commons.wikimedia.org\/wiki\/File:LinearInterpolation.svg\"><a rel=\"cc:attributionURL\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File:LinearInterpolation.svg\" property=\"dc:title\">Linear interpolation<\/a>  &copy;  ElectroKid    is licensed under a  <a rel=\"license\" href=\"https:\/\/creativecommons.org\/publicdomain\/mark\/1.0\/\">Public Domain<\/a> license<\/li><\/ul><\/div>","protected":false},"author":175,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[47],"contributor":[],"license":[],"class_list":["post-881","chapter","type-chapter","status-publish","hentry","chapter-type-standard"],"part":246,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/881","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/users\/175"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/881\/revisions"}],"predecessor-version":[{"id":3440,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/881\/revisions\/3440"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/parts\/246"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/881\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/media?parent=881"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapter-type?post=881"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/contributor?post=881"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/license?post=881"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}