{"id":1839,"date":"2021-07-27T20:21:57","date_gmt":"2021-07-28T00:21:57","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/6-9-the-second-law-of-thermodynamics-for-open-systems\/"},"modified":"2022-08-12T01:02:02","modified_gmt":"2022-08-12T05:02:02","slug":"6-9-the-second-law-of-thermodynamics-for-open-systems","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/6-9-the-second-law-of-thermodynamics-for-open-systems\/","title":{"raw":"6.9 The second law of thermodynamics for open systems","rendered":"6.9 The second law of thermodynamics for open systems"},"content":{"raw":"<div class=\"6.9-the-second-law-of-thermodynamics-for-open-systems\">\r\n\r\n&nbsp;\r\n<p class=\"import-Normal\">Entropy can be transferred to a system via two mechanisms: (1) heat transfer and (2) mass transfer. For open systems, the second law of thermodynamics is often written in the rate form; therefore, we are interested in the time rate of entropy transfer due to heat transfer and mass transfer.<\/p>\r\n&nbsp;\r\n<p style=\"padding-left: 160px\">[latex] \\dot{S}_{heat} =\\dfrac{dS_{heat}}{dt} \\cong \\displaystyle\\sum\\dfrac{\\dot{Q}_k}{T_k}[\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"padding-left: 160px\">[latex] \\dot{S}_{mass} = \\displaystyle\\sum\\dfrac{dS_{mass}}{dt} = \\displaystyle \\sum \\dot{m}_{k}s_{k}[\/latex]<\/p>\r\nwhere\r\n<p style=\"padding-left: 40px\">[latex]\\dot{m}[\/latex]: rate of mass transfer<\/p>\r\n<p style=\"padding-left: 40px\">[latex]\\dot{Q}_k [\/latex]: rate of heat transfer via the location [latex]k[\/latex] of the system boundary, which is at a temperature of [latex]T_k[\/latex] in Kelvin<\/p>\r\n<p style=\"padding-left: 40px\">[latex]\\dot{S}_{heat} [\/latex]: time rate of entropy transfer due to heat transfer<\/p>\r\n<p style=\"padding-left: 40px\">[latex]\\dot{S}_{mass} [\/latex]: time rate of entropy transfer that accompanies the mass transfer into or out of a control volume<\/p>\r\n<p style=\"padding-left: 40px\">[latex]s_k[\/latex]: specific entropy of the fluid<\/p>\r\n\r\n<\/div>\r\n<div class=\"6.9-the-second-law-of-thermodynamics-for-open-systems\">\r\n\r\n&nbsp;\r\n<p class=\"import-Normal\">Applying the entropy balance equation, [latex]\\Delta \\rm {entropy= + in - out + gen}[\/latex], to a control volume, see <a href=\"#6.9.1\">Figure 6.9.1<\/a>, we can write the following equations:<\/p>\r\n\r\n<ul>\r\n \t<li>General equation for both steady and transient flow devices<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p style=\"padding-left: 80px\">[latex]\\dfrac{{{dS}}_{{c}.{v}.}}{{dt}}=\\displaystyle\\left(\\sum{{\\dot{{m}}}_{i}{s}_{i}}+\\displaystyle\\sum\\frac{{\\dot{{Q}}}_{{c}.{v}.}}{{T}}\\right)-\\displaystyle\\left(\\sum{{\\dot{{m}}}_{e}{s}_{e}}\\right)+\\displaystyle{\\dot{{S}}}_{{gen}}\\ \\ \\ \\ \\ \\ ({\\dot{{S}}}_{{gen}} \\ge 0)[\/latex]<\/p>\r\n\r\n<div class=\"6.9-the-second-law-of-thermodynamics-for-open-systems\">\r\n<ul>\r\n \t<li class=\"import-Normal\">For steady-state, steady-flow devices, [latex]\\dfrac{{dS}_{c.v.}}{dt}=0[\/latex]; therefore,<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 80px;text-align: left\">[latex]\\displaystyle \\sum{{\\dot{{m}}}_{e}{s}_{e}}-\\sum{{\\dot{{m}}}_{i}{s}_{i}}=\\displaystyle \\sum\\dfrac{{\\dot{{Q}}}_{{c}.{v}.}}{{T}}+{\\dot{{S}}}_{{gen}}\\ \\ \\ \\ \\ \\ ({\\dot{{S}}}_{{gen}} \\ge\u00a0 0)[\/latex]<\/p>\r\n\r\n<ul>\r\n \t<li class=\"import-Normal\">For steady and [pb_glossary id=\"674\"]isentropic[\/pb_glossary] flow devices, [latex]\\dot{Q}_{c.v.}=0 [\/latex] and [latex]\\dot S_{gen}=0[\/latex]; therefore,<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center\">[latex]\\displaystyle\\sum{{\\dot{{m}}}_{e}{s}_{e}}=\\displaystyle\\sum{{\\dot{{m}}}_{i}{s}_{i}}[\/latex]<\/p>\r\n\r\n<\/div>\r\nwhere\r\n<p style=\"padding-left: 40px\">[latex]\\dot{m}[\/latex]: rate of mass transfer of the fluid entering or leaving the control volume via the inlet [latex]i[\/latex] or exit [latex]e[\/latex], in kg\/s<\/p>\r\n<p style=\"padding-left: 40px\">[latex]\\dot{Q}_{c.v.} [\/latex]: rate of heat transfer into the control volume via the system boundary (at a constant [latex]T[\/latex]), in kW<\/p>\r\n<p style=\"padding-left: 40px\">[latex]S_{c.v.}[\/latex]: entropy in the control volume, in kJ\/K<\/p>\r\n<p style=\"padding-left: 40px\">[latex]\\dfrac{{dS}_{c.v.}}{dt}[\/latex]: time rate of change of entropy in the control volume, in kW\/K<\/p>\r\n<p style=\"padding-left: 40px\">[latex]\\dot S_{gen}[\/latex]: time rate of entropy generation in the process, in kW\/K<\/p>\r\n<p style=\"padding-left: 40px\">[latex]s[\/latex]: specific entropy of the fluid entering or leaving the control volume via the inlet [latex]i[\/latex] or exit [latex]e[\/latex], in kJ\/kgK<\/p>\r\n<p style=\"padding-left: 40px\">[latex]T[\/latex]: <a id=\"6.9.1\"><\/a> absolute temperature of the system boundary, in Kelvin<\/p>\r\n\r\n\r\n[caption id=\"attachment_2736\" align=\"aligncenter\" width=\"500\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/Second_law_open_system.png\" target=\"_blank\" rel=\"noopener\"><img class=\"wp-image-2736\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/Second_law_open_system-300x206.png\" alt=\"Flow through a control volume, showing the entropy transfers and entropy generation\" width=\"500\" height=\"344\" \/><\/a> <em><strong>Figure 6.9.1<\/strong><\/em>\u00a0<em>Flow through a control volume, showing the entropy transfers and entropy generation<\/em>[\/caption]\r\n\r\n<div class=\"6.9-the-second-law-of-thermodynamics-for-open-systems\">\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\r\nThe diagrams in <a href=\"#6.9.e1\">Figure 6.9.e1<\/a> show a reversible process in a steady-state, single flow of air. The letters <em>i<\/em> and <em>e<\/em> represent the initial and final states, respectively. Treat air as an ideal gas and assume \u0394<em lang=\"en-US\" xml:lang=\"en-US\">KE<\/em>=\u0394<em lang=\"en-US\" xml:lang=\"en-US\">PE<\/em>=0<em lang=\"en-US\" xml:lang=\"en-US\">.<\/em> Are the change in specific enthalpy \u0394<em lang=\"en-US\" xml:lang=\"en-US\">h<\/em>=<span style=\"border: none windowtext 0pt;padding: 0\"><em>h<\/em><sub><em>e<\/em><\/sub>\u2212<em>h<\/em><sub><em>i<\/em><\/sub><\/span>, specific work <span style=\"border: none windowtext 0pt;padding: 0\"><em>w, <\/em><\/span>and specific heat transfer<span style=\"border: none windowtext 0pt;padding: 0\"><em> q\u00a0 <\/em><\/span>positive, zero, or negative values? What is the relation between<a id=\"6.9.e1\"><\/a> <span style=\"border: none windowtext 0pt;padding: 0\"><em>w <\/em><\/span>and <span style=\"border: none windowtext 0pt;padding: 0\"><em>q<\/em><\/span><span class=\"import-apple-converted-space\">?<\/span>\r\n\r\n[caption id=\"attachment_3371\" align=\"aligncenter\" width=\"500\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2022\/07\/Fig.-6-18.png\" target=\"_blank\" rel=\"noopener\"><img class=\"wp-image-3371\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2022\/07\/Fig.-6-18-300x140.png\" alt=\"T-s and P-v diagrams of a reversible process\" width=\"500\" height=\"233\" \/><\/a> <em><strong>Figure 6.9.e1 <\/strong>T-s and P-v diagrams of a reversible process for an ideal gas<\/em>[\/caption]\r\n\r\n<em><span style=\"text-decoration: underline\">Solution:<\/span><\/em>\r\n\r\nThe specific work can be evaluated mathematically and graphically.\r\n<p style=\"padding-left: 40px\">(1) Mathematically,<\/p>\r\n<p style=\"padding-left: 80px\">[latex] \\because v_{e} &gt; v_{i} [\/latex]<\/p>\r\n<p style=\"padding-left: 80px\">[latex] \\therefore w = \\displaystyle\\int_{i}^{e}{Pd{v}\\ } &gt;\\ 0[\/latex]<\/p>\r\n<p style=\"padding-left: 40px\">(2) Graphically, the specific work is the area under the process curve in the [latex]P-v[\/latex] diagram; therefore [latex]w[\/latex] is positive, see <a href=\"#6.9.e2\">Figure 6.9.e2<\/a>.<\/p>\r\n&nbsp;\r\n\r\nIn a similar fashion, the specific heat transfer can also be evaluated graphically and mathematically.\r\n<p style=\"padding-left: 40px\">(1) Graphically,<\/p>\r\n<p style=\"padding-left: 80px\">[latex]\\because ds=\\left(\\displaystyle\\frac{\\delta q}{T}\\right)_{rev}[\/latex]<\/p>\r\n<p style=\"padding-left: 80px\">[latex] \\therefore q_{rev} = \\displaystyle \\int_{i}^{e}{Tds} = T(s_e-s_i)\\\u00a0 &gt;\\ 0 [\/latex]<\/p>\r\n<p style=\"padding-left: 40px\">For a reversible process, the area under the process curve in the [latex]T-s[\/latex] diagram represents the specific heat transfer of the reversible process; therefore [latex] q=q_{rev} [\/latex] is positive, see <a href=\"#6.9.e2\">Figure 6.9.e2<\/a>.<\/p>\r\n&nbsp;\r\n<p style=\"padding-left: 40px\">(2) The same conclusion, [latex] q_{rev}&gt;0 [\/latex], can also be derived from the second law of thermodynamics mathematically, as follows.<\/p>\r\n<p style=\"padding-left: 80px\">[latex]\\dot{m}(s_e-s_i)=\\displaystyle\\sum\\frac{\\dot{Q}}{T_{surr}}+\\dot{S}_{gen}[\/latex]<\/p>\r\n<p style=\"padding-left: 40px\">For a reversible process, [latex] \\dot{S}_{gen} [\/latex]= 0, and the fluid is assumed to be always in thermal equilibrium with the system boundary, or [latex]T = T_{surr}[\/latex]; therefore,<\/p>\r\n<p style=\"padding-left: 80px\">[latex]q_{rev} = \\dfrac{\\dot{Q}}{\\dot{m}} = T(s_e-s_i) &gt; 0[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThe change in specific enthalpy can then be evaluated. For an ideal gas,\r\n<p style=\"padding-left: 80px\">[latex] \\Delta h = h_e - h_i = C_p(T_e - T_i) [\/latex]<\/p>\r\n<p style=\"padding-left: 80px\">[latex] \\because T_e = T_i [\/latex]<\/p>\r\n<p style=\"padding-left: 80px\">[latex]\\therefore h_e = h_i [\/latex]\u00a0\u00a0 and\u00a0\u00a0 [latex]\\Delta h =0[\/latex]<\/p>\r\n&nbsp;\r\n\r\nNow, we can determine the relation between [latex]w[\/latex] and [latex] q_{rev} [\/latex] from the first law of thermodynamics for control volumes.\r\n<p style=\"padding-left: 80px\">[latex] \\because \\dot{m}( h_e - h_i ) = \\dot{Q}_{rev} - \\dot{W} = 0 [\/latex]<\/p>\r\n<p style=\"padding-left: 80px\">[latex]\\therefore \\dot{Q}_{rev} = \\dot{W} [\/latex]<\/p>\r\n<p style=\"padding-left: 80px\">[latex]\\therefore q_{rev} = w [\/latex]<\/p>\r\nIn this reversible process, the specific heat transfer and specific work must be the same. Graphically, the two areas under the [latex]P-v[\/latex] and [latex]T-s[\/latex] diagrams <a id=\"6.9.e2\"><\/a>must be the same.\r\n\r\n[caption id=\"attachment_2188\" align=\"aligncenter\" width=\"500\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/6.9.1.png\" target=\"_blank\" rel=\"noopener\"><img class=\"wp-image-2188\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/6.9.1-300x128.png\" alt=\"T-s and P-v diagrams, showing the solutions for a reversible process for an ideal gas\" width=\"500\" height=\"214\" \/><\/a> <em><strong>Figure 6.9.e2<\/strong> T-s and P-v diagrams, showing the solutions for a reversible process of an ideal gas<\/em>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\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=\"48\"]\r\n\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"6.9-the-second-law-of-thermodynamics-for-open-systems\">\n<p>&nbsp;<\/p>\n<p class=\"import-Normal\">Entropy can be transferred to a system via two mechanisms: (1) heat transfer and (2) mass transfer. For open systems, the second law of thermodynamics is often written in the rate form; therefore, we are interested in the time rate of entropy transfer due to heat transfer and mass transfer.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 160px\">[latex]\\dot{S}_{heat} =\\dfrac{dS_{heat}}{dt} \\cong \\displaystyle\\sum\\dfrac{\\dot{Q}_k}{T_k}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 160px\">[latex]\\dot{S}_{mass} = \\displaystyle\\sum\\dfrac{dS_{mass}}{dt} = \\displaystyle \\sum \\dot{m}_{k}s_{k}[\/latex]<\/p>\n<p>where<\/p>\n<p style=\"padding-left: 40px\">[latex]\\dot{m}[\/latex]: rate of mass transfer<\/p>\n<p style=\"padding-left: 40px\">[latex]\\dot{Q}_k[\/latex]: rate of heat transfer via the location [latex]k[\/latex] of the system boundary, which is at a temperature of [latex]T_k[\/latex] in Kelvin<\/p>\n<p style=\"padding-left: 40px\">[latex]\\dot{S}_{heat}[\/latex]: time rate of entropy transfer due to heat transfer<\/p>\n<p style=\"padding-left: 40px\">[latex]\\dot{S}_{mass}[\/latex]: time rate of entropy transfer that accompanies the mass transfer into or out of a control volume<\/p>\n<p style=\"padding-left: 40px\">[latex]s_k[\/latex]: specific entropy of the fluid<\/p>\n<\/div>\n<div class=\"6.9-the-second-law-of-thermodynamics-for-open-systems\">\n<p>&nbsp;<\/p>\n<p class=\"import-Normal\">Applying the entropy balance equation, [latex]\\Delta \\rm {entropy= + in - out + gen}[\/latex], to a control volume, see <a href=\"#6.9.1\">Figure 6.9.1<\/a>, we can write the following equations:<\/p>\n<ul>\n<li>General equation for both steady and transient flow devices<\/li>\n<\/ul>\n<\/div>\n<p style=\"padding-left: 80px\">[latex]\\dfrac{{{dS}}_{{c}.{v}.}}{{dt}}=\\displaystyle\\left(\\sum{{\\dot{{m}}}_{i}{s}_{i}}+\\displaystyle\\sum\\frac{{\\dot{{Q}}}_{{c}.{v}.}}{{T}}\\right)-\\displaystyle\\left(\\sum{{\\dot{{m}}}_{e}{s}_{e}}\\right)+\\displaystyle{\\dot{{S}}}_{{gen}}\\ \\ \\ \\ \\ \\ ({\\dot{{S}}}_{{gen}} \\ge 0)[\/latex]<\/p>\n<div class=\"6.9-the-second-law-of-thermodynamics-for-open-systems\">\n<ul>\n<li class=\"import-Normal\">For steady-state, steady-flow devices, [latex]\\dfrac{{dS}_{c.v.}}{dt}=0[\/latex]; therefore,<\/li>\n<\/ul>\n<p style=\"padding-left: 80px;text-align: left\">[latex]\\displaystyle \\sum{{\\dot{{m}}}_{e}{s}_{e}}-\\sum{{\\dot{{m}}}_{i}{s}_{i}}=\\displaystyle \\sum\\dfrac{{\\dot{{Q}}}_{{c}.{v}.}}{{T}}+{\\dot{{S}}}_{{gen}}\\ \\ \\ \\ \\ \\ ({\\dot{{S}}}_{{gen}} \\ge\u00a0 0)[\/latex]<\/p>\n<ul>\n<li class=\"import-Normal\">For steady and <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_1839_674\">isentropic<\/a> flow devices, [latex]\\dot{Q}_{c.v.}=0[\/latex] and [latex]\\dot S_{gen}=0[\/latex]; therefore,<\/li>\n<\/ul>\n<p style=\"text-align: center\">[latex]\\displaystyle\\sum{{\\dot{{m}}}_{e}{s}_{e}}=\\displaystyle\\sum{{\\dot{{m}}}_{i}{s}_{i}}[\/latex]<\/p>\n<\/div>\n<p>where<\/p>\n<p style=\"padding-left: 40px\">[latex]\\dot{m}[\/latex]: rate of mass transfer of the fluid entering or leaving the control volume via the inlet [latex]i[\/latex] or exit [latex]e[\/latex], in kg\/s<\/p>\n<p style=\"padding-left: 40px\">[latex]\\dot{Q}_{c.v.}[\/latex]: rate of heat transfer into the control volume via the system boundary (at a constant [latex]T[\/latex]), in kW<\/p>\n<p style=\"padding-left: 40px\">[latex]S_{c.v.}[\/latex]: entropy in the control volume, in kJ\/K<\/p>\n<p style=\"padding-left: 40px\">[latex]\\dfrac{{dS}_{c.v.}}{dt}[\/latex]: time rate of change of entropy in the control volume, in kW\/K<\/p>\n<p style=\"padding-left: 40px\">[latex]\\dot S_{gen}[\/latex]: time rate of entropy generation in the process, in kW\/K<\/p>\n<p style=\"padding-left: 40px\">[latex]s[\/latex]: specific entropy of the fluid entering or leaving the control volume via the inlet [latex]i[\/latex] or exit [latex]e[\/latex], in kJ\/kgK<\/p>\n<p style=\"padding-left: 40px\">[latex]T[\/latex]: <a id=\"6.9.1\"><\/a> absolute temperature of the system boundary, in Kelvin<\/p>\n<figure id=\"attachment_2736\" aria-describedby=\"caption-attachment-2736\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/Second_law_open_system.png\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2736\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/Second_law_open_system-300x206.png\" alt=\"Flow through a control volume, showing the entropy transfers and entropy generation\" width=\"500\" height=\"344\" srcset=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/Second_law_open_system-300x206.png 300w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/Second_law_open_system-1024x704.png 1024w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/Second_law_open_system-768x528.png 768w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/Second_law_open_system-1536x1055.png 1536w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/Second_law_open_system-2048x1407.png 2048w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/Second_law_open_system-65x45.png 65w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/Second_law_open_system-225x155.png 225w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/Second_law_open_system-350x240.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><figcaption id=\"caption-attachment-2736\" class=\"wp-caption-text\"><em><strong>Figure 6.9.1<\/strong><\/em>\u00a0<em>Flow through a control volume, showing the entropy transfers and entropy generation<\/em><\/figcaption><\/figure>\n<div class=\"6.9-the-second-law-of-thermodynamics-for-open-systems\">\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>The diagrams in <a href=\"#6.9.e1\">Figure 6.9.e1<\/a> show a reversible process in a steady-state, single flow of air. The letters <em>i<\/em> and <em>e<\/em> represent the initial and final states, respectively. Treat air as an ideal gas and assume \u0394<em lang=\"en-US\" xml:lang=\"en-US\">KE<\/em>=\u0394<em lang=\"en-US\" xml:lang=\"en-US\">PE<\/em>=0<em lang=\"en-US\" xml:lang=\"en-US\">.<\/em> Are the change in specific enthalpy \u0394<em lang=\"en-US\" xml:lang=\"en-US\">h<\/em>=<span style=\"border: none windowtext 0pt;padding: 0\"><em>h<\/em><sub><em>e<\/em><\/sub>\u2212<em>h<\/em><sub><em>i<\/em><\/sub><\/span>, specific work <span style=\"border: none windowtext 0pt;padding: 0\"><em>w, <\/em><\/span>and specific heat transfer<span style=\"border: none windowtext 0pt;padding: 0\"><em> q\u00a0 <\/em><\/span>positive, zero, or negative values? What is the relation between<a id=\"6.9.e1\"><\/a> <span style=\"border: none windowtext 0pt;padding: 0\"><em>w <\/em><\/span>and <span style=\"border: none windowtext 0pt;padding: 0\"><em>q<\/em><\/span><span class=\"import-apple-converted-space\">?<\/span><\/p>\n<figure id=\"attachment_3371\" aria-describedby=\"caption-attachment-3371\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2022\/07\/Fig.-6-18.png\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3371\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2022\/07\/Fig.-6-18-300x140.png\" alt=\"T-s and P-v diagrams of a reversible process\" width=\"500\" height=\"233\" srcset=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2022\/07\/Fig.-6-18-300x140.png 300w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2022\/07\/Fig.-6-18-1024x477.png 1024w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2022\/07\/Fig.-6-18-768x358.png 768w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2022\/07\/Fig.-6-18-1536x716.png 1536w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2022\/07\/Fig.-6-18-65x30.png 65w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2022\/07\/Fig.-6-18-225x105.png 225w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2022\/07\/Fig.-6-18-350x163.png 350w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2022\/07\/Fig.-6-18.png 1750w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><figcaption id=\"caption-attachment-3371\" class=\"wp-caption-text\"><em><strong>Figure 6.9.e1 <\/strong>T-s and P-v diagrams of a reversible process for an ideal gas<\/em><\/figcaption><\/figure>\n<p><em><span style=\"text-decoration: underline\">Solution:<\/span><\/em><\/p>\n<p>The specific work can be evaluated mathematically and graphically.<\/p>\n<p style=\"padding-left: 40px\">(1) Mathematically,<\/p>\n<p style=\"padding-left: 80px\">[latex]\\because v_{e} > v_{i}[\/latex]<\/p>\n<p style=\"padding-left: 80px\">[latex]\\therefore w = \\displaystyle\\int_{i}^{e}{Pd{v}\\ } >\\ 0[\/latex]<\/p>\n<p style=\"padding-left: 40px\">(2) Graphically, the specific work is the area under the process curve in the [latex]P-v[\/latex] diagram; therefore [latex]w[\/latex] is positive, see <a href=\"#6.9.e2\">Figure 6.9.e2<\/a>.<\/p>\n<p>&nbsp;<\/p>\n<p>In a similar fashion, the specific heat transfer can also be evaluated graphically and mathematically.<\/p>\n<p style=\"padding-left: 40px\">(1) Graphically,<\/p>\n<p style=\"padding-left: 80px\">[latex]\\because ds=\\left(\\displaystyle\\frac{\\delta q}{T}\\right)_{rev}[\/latex]<\/p>\n<p style=\"padding-left: 80px\">[latex]\\therefore q_{rev} = \\displaystyle \\int_{i}^{e}{Tds} = T(s_e-s_i)\\\u00a0 >\\ 0[\/latex]<\/p>\n<p style=\"padding-left: 40px\">For a reversible process, the area under the process curve in the [latex]T-s[\/latex] diagram represents the specific heat transfer of the reversible process; therefore [latex]q=q_{rev}[\/latex] is positive, see <a href=\"#6.9.e2\">Figure 6.9.e2<\/a>.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 40px\">(2) The same conclusion, [latex]q_{rev}>0[\/latex], can also be derived from the second law of thermodynamics mathematically, as follows.<\/p>\n<p style=\"padding-left: 80px\">[latex]\\dot{m}(s_e-s_i)=\\displaystyle\\sum\\frac{\\dot{Q}}{T_{surr}}+\\dot{S}_{gen}[\/latex]<\/p>\n<p style=\"padding-left: 40px\">For a reversible process, [latex]\\dot{S}_{gen}[\/latex]= 0, and the fluid is assumed to be always in thermal equilibrium with the system boundary, or [latex]T = T_{surr}[\/latex]; therefore,<\/p>\n<p style=\"padding-left: 80px\">[latex]q_{rev} = \\dfrac{\\dot{Q}}{\\dot{m}} = T(s_e-s_i) > 0[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The change in specific enthalpy can then be evaluated. For an ideal gas,<\/p>\n<p style=\"padding-left: 80px\">[latex]\\Delta h = h_e - h_i = C_p(T_e - T_i)[\/latex]<\/p>\n<p style=\"padding-left: 80px\">[latex]\\because T_e = T_i[\/latex]<\/p>\n<p style=\"padding-left: 80px\">[latex]\\therefore h_e = h_i[\/latex]\u00a0\u00a0 and\u00a0\u00a0 [latex]\\Delta h =0[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Now, we can determine the relation between [latex]w[\/latex] and [latex]q_{rev}[\/latex] from the first law of thermodynamics for control volumes.<\/p>\n<p style=\"padding-left: 80px\">[latex]\\because \\dot{m}( h_e - h_i ) = \\dot{Q}_{rev} - \\dot{W} = 0[\/latex]<\/p>\n<p style=\"padding-left: 80px\">[latex]\\therefore \\dot{Q}_{rev} = \\dot{W}[\/latex]<\/p>\n<p style=\"padding-left: 80px\">[latex]\\therefore q_{rev} = w[\/latex]<\/p>\n<p>In this reversible process, the specific heat transfer and specific work must be the same. Graphically, the two areas under the [latex]P-v[\/latex] and [latex]T-s[\/latex] diagrams <a id=\"6.9.e2\"><\/a>must be the same.<\/p>\n<figure id=\"attachment_2188\" aria-describedby=\"caption-attachment-2188\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/6.9.1.png\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2188\" src=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/6.9.1-300x128.png\" alt=\"T-s and P-v diagrams, showing the solutions for a reversible process for an ideal gas\" width=\"500\" height=\"214\" srcset=\"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/6.9.1-300x128.png 300w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/6.9.1-1024x438.png 1024w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/6.9.1-768x329.png 768w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/6.9.1-1536x657.png 1536w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/6.9.1-65x28.png 65w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/6.9.1-225x96.png 225w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/6.9.1-350x150.png 350w, https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-content\/uploads\/sites\/499\/2021\/07\/6.9.1.png 1655w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><figcaption id=\"caption-attachment-2188\" class=\"wp-caption-text\"><em><strong>Figure 6.9.e2<\/strong> T-s and P-v diagrams, showing the solutions for a reversible process of an ideal gas<\/em><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\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-48\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-48\" class=\"h5p-iframe\" data-content-id=\"48\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"S_6.9_Q\"><\/iframe><\/div>\n<\/div>\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:\/\/upload.wikimedia.org\/wikipedia\/commons\/8\/86\/First_law_open_system.svg\"><a rel=\"cc:attributionURL\" href=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/8\/86\/First_law_open_system.svg\" property=\"dc:title\">Entropy transfers and entropy generation through a C.V.<\/a>  &copy;  Pbroks13    is licensed under a  <a rel=\"license\" href=\"https:\/\/creativecommons.org\/publicdomain\/mark\/1.0\/\">Public Domain<\/a> license<\/li><\/ul><\/div><div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_1839_674\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_1839_674\"><div tabindex=\"-1\"><p>An isentropic process refers to a process that is reversible and adiabatic. The entropy remains constant in an isentropic process.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":175,"menu_order":10,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1839","chapter","type-chapter","status-publish","hentry"],"part":286,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/1839","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":27,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/1839\/revisions"}],"predecessor-version":[{"id":4227,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/1839\/revisions\/4227"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/parts\/286"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/1839\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/media?parent=1839"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapter-type?post=1839"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/contributor?post=1839"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/license?post=1839"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}