{"id":291,"date":"2021-05-21T01:51:13","date_gmt":"2021-05-21T05:51:13","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/thermo1\/?post_type=chapter&p=291"},"modified":"2022-09-02T15:09:26","modified_gmt":"2022-09-02T19:09:26","slug":"key-equations-6","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/key-equations-6\/","title":{"raw":"6.12 Key equations","rendered":"6.12 Key equations"},"content":{"raw":"

Heat engine<\/strong><\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Net work output<\/td>\r\n[latex]{\\dot{W}}_{net,\\ out}={\\dot{Q}}_H-{\\dot{Q}}_L[\/latex]<\/td>\r\n<\/tr>\r\n
Thermal efficiency of any heat engine<\/td>\r\n[latex]\\eta_{th}=\\displaystyle\\frac{desired\\ output}{required\\ input}=\\frac{{\\dot{W}}_{net,\\ out}}{{\\dot{Q}}_H}=1-\\frac{{\\dot{Q}}_L}{{\\dot{Q}}_H}[\/latex]<\/td>\r\n<\/tr>\r\n
Thermal efficiency of Carnot heat engine<\/td>\r\n[latex]\\eta_{th,\\ rev}=1-\\displaystyle\\frac{T_L}{T_H}[\/latex]\r\n\r\n <\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n

Refrigerator and heat pump<\/strong><\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Net work input<\/td>\r\n[latex]{\\dot{W}}_{net,\\ in}={\\dot{Q}}_H-{\\dot{Q}}_L[\/latex]<\/td>\r\n<\/tr>\r\n
COP of any refrigerator<\/td>\r\n[latex]\\begin{align*} {COP}_R &=\\displaystyle\\frac{desired\\ output}{required\\ input} \\\\&=\\frac{{\\dot{Q}}_L}{{\\dot{W}}_{net,\\ in}} =\\frac{{\\dot{Q}}_L}{{\\dot{Q}}_H-{\\dot{Q}}_L} =\\frac{1}{{\\dot{Q}}_H\/{\\dot{Q}}_L-1} \\end{align*}[\/latex]<\/td>\r\n<\/tr>\r\n
COP of Carnot refrigerator<\/td>\r\n[latex]{COP}_{R,\\ rev}=\\displaystyle\\frac{T_L}{T_H-T_L}=\\frac{1}{T_H\/T_L-1}[\/latex]<\/td>\r\n<\/tr>\r\n
COP of any heat pump<\/td>\r\n[latex]\\begin{align*} {COP}_{HP} &=\\displaystyle\\frac{desired\\ output}{required\\ input} \\\\&=\\frac{{\\dot{Q}}_H}{{\\dot{W}}_{net,\\ in}}=\\frac{{\\dot{Q}}_H}{{\\dot{Q}}_H-{\\dot{Q}}_L}=\\frac{1}{{1-\\dot{Q}}_L\/{\\dot{Q}}_H} \\end{align*}[\/latex]<\/td>\r\n<\/tr>\r\n
COP of Carnot heat pump<\/td>\r\n[latex]{COP}_{HP,\\ rev}=\\displaystyle\\frac{T_H}{T_H-T_L}=\\displaystyle\\frac{1}{{1-T}_L\/T_H}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n

Entropy and entropy generation<\/strong><\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
The inequality of Clausius<\/td>\r\n[latex]\\displaystyle\\oint\\displaystyle\\frac{\\delta Q}{T}\\le0 \\ \\rm{(= for \\ reversible \\ cycles; \\ < for \\ irreversible \\ cycles)} [\/latex]<\/td>\r\n<\/tr>\r\n
Definition of entropy<\/td>\r\n[latex]\\begin{align*} \\rm{Infinitesimal \\ \\ form:} & \\ dS =\\left(\\displaystyle\\frac{\\delta Q}{T}\\right)_{rev} \\\\ \\rm{Integral \\ \\ form:} & \\ \\Delta S = S_2-S_1=\\displaystyle\\int_{1}^{2}\\left(\\displaystyle\\frac{\\delta Q}{T}\\right)_{rev} \\end{align*}[\/latex]<\/td>\r\n<\/tr>\r\n
Definition of entropy generation<\/td>\r\n[latex] {\\rm{Infinitesimal \\ \\ form:}} \\ dS =\\displaystyle\\frac{\\delta Q}{T}+\\delta S_{gen} \\\\ {\\rm{where}} \\ \\delta S_{gen}\\geq0 \\\\ \\rm{(= for \\ reversible \\ processes; \\ > for \\ irreversible \\ process)} [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n

The second law of thermodynamics<\/strong><\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
For closed systems (control mass)<\/td>\r\n[latex] \\begin {align*} S_2-S_1 &=\\displaystyle\\int_{1}^{2}\\displaystyle\\frac{\\delta Q}{T}+S_{gen} \\\\& \\cong\\sum\\frac{Q_k}{T_k}+S_{gen}\\ \\ \\ \\ \\ (S_{gen}\\geq0)\r\n\r\n\\end {align*} [\/latex]\r\nwhere [latex]T_k[\/latex] is the absolute temperature of the system boundary, in Kelvin.<\/td>\r\n<\/tr>\r\n
For steady-state, steady flow in a control volume (open systems)<\/td>\r\n[latex]\\sum{{\\dot{m}}_es_e}-\\sum{{\\dot{m}}_is_i}=\\sum\\displaystyle\\frac{{\\dot{Q}}_{c.v.}}{T}+{\\dot{S}}_{gen}\\ \\ \\ \\ \\ \\ \\left({\\dot{S}}_{gen}\\geq0\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
For steady and isentropic flow<\/td>\r\n[latex]\\sum{{\\dot{m}}_es_e}=\\sum{{\\dot{m}}_is_i}[\/latex]<\/td>\r\n<\/tr>\r\n
Change of specific entropy between two states of a solid or liquid<\/td>\r\n[latex]s_2-s_1=C_pln\\displaystyle\\frac{T_2}{T_1}[\/latex]<\/td>\r\n<\/tr>\r\n
Change of specific entropy between two states of an ideal gas<\/td>\r\nAssume constant\u00a0 [latex]C_p[\/latex] and [latex]C_v[\/latex] in the temperature range,\u00a0 [latex]s_2-s_1=C_pln\\displaystyle\\frac{T_2}{T_1}-Rln\\frac{P_2}{P_1}[\/latex]\r\n\r\n[latex]s_2-s_1=C_vln\\displaystyle\\frac{T_2}{T_1}+Rln\\frac{v_2}{v_1}[\/latex]<\/td>\r\n<\/tr>\r\n
Isentropic relations for ideal gases<\/td>\r\n[latex]Pv^k= \\rm {constant}[\/latex]\r\n\r\n[latex]\\displaystyle\\frac{P_2}{P_1}=\\displaystyle\\left(\\displaystyle\\frac{v_1}{v_2}\\right)^k=\\left(\\frac{T_2}{T_1}\\right)^{k\/(k-1)}[\/latex]\r\nwhere\u00a0 [latex]k=\\displaystyle\\frac{C_p}{C_v}[\/latex]\u00a0 and [latex]T[\/latex] is in Kelvin<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"

Heat engine<\/strong><\/p>\n\n\n\n\n\n
Net work output<\/td>\n[latex]{\\dot{W}}_{net,\\ out}={\\dot{Q}}_H-{\\dot{Q}}_L[\/latex]<\/td>\n<\/tr>\n
Thermal efficiency of any heat engine<\/td>\n[latex]\\eta_{th}=\\displaystyle\\frac{desired\\ output}{required\\ input}=\\frac{{\\dot{W}}_{net,\\ out}}{{\\dot{Q}}_H}=1-\\frac{{\\dot{Q}}_L}{{\\dot{Q}}_H}[\/latex]<\/td>\n<\/tr>\n
Thermal efficiency of Carnot heat engine<\/td>\n[latex]\\eta_{th,\\ rev}=1-\\displaystyle\\frac{T_L}{T_H}[\/latex]<\/p>\n

 <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Refrigerator and heat pump<\/strong><\/p>\n\n\n\n\n\n\n\n
Net work input<\/td>\n[latex]{\\dot{W}}_{net,\\ in}={\\dot{Q}}_H-{\\dot{Q}}_L[\/latex]<\/td>\n<\/tr>\n
COP of any refrigerator<\/td>\n[latex]\\begin{align*} {COP}_R &=\\displaystyle\\frac{desired\\ output}{required\\ input} \\\\&=\\frac{{\\dot{Q}}_L}{{\\dot{W}}_{net,\\ in}} =\\frac{{\\dot{Q}}_L}{{\\dot{Q}}_H-{\\dot{Q}}_L} =\\frac{1}{{\\dot{Q}}_H\/{\\dot{Q}}_L-1} \\end{align*}[\/latex]<\/td>\n<\/tr>\n
COP of Carnot refrigerator<\/td>\n[latex]{COP}_{R,\\ rev}=\\displaystyle\\frac{T_L}{T_H-T_L}=\\frac{1}{T_H\/T_L-1}[\/latex]<\/td>\n<\/tr>\n
COP of any heat pump<\/td>\n[latex]\\begin{align*} {COP}_{HP} &=\\displaystyle\\frac{desired\\ output}{required\\ input} \\\\&=\\frac{{\\dot{Q}}_H}{{\\dot{W}}_{net,\\ in}}=\\frac{{\\dot{Q}}_H}{{\\dot{Q}}_H-{\\dot{Q}}_L}=\\frac{1}{{1-\\dot{Q}}_L\/{\\dot{Q}}_H} \\end{align*}[\/latex]<\/td>\n<\/tr>\n
COP of Carnot heat pump<\/td>\n[latex]{COP}_{HP,\\ rev}=\\displaystyle\\frac{T_H}{T_H-T_L}=\\displaystyle\\frac{1}{{1-T}_L\/T_H}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Entropy and entropy generation<\/strong><\/p>\n\n\n\n\n\n
The inequality of Clausius<\/td>\n[latex]\\displaystyle\\oint\\displaystyle\\frac{\\delta Q}{T}\\le0 \\ \\rm{(= for \\ reversible \\ cycles; \\ < for \\ irreversible \\ cycles)}[\/latex]<\/td>\n<\/tr>\n
Definition of entropy<\/td>\n[latex]\\begin{align*} \\rm{Infinitesimal \\ \\ form:} & \\ dS =\\left(\\displaystyle\\frac{\\delta Q}{T}\\right)_{rev} \\\\ \\rm{Integral \\ \\ form:} & \\ \\Delta S = S_2-S_1=\\displaystyle\\int_{1}^{2}\\left(\\displaystyle\\frac{\\delta Q}{T}\\right)_{rev} \\end{align*}[\/latex]<\/td>\n<\/tr>\n
Definition of entropy generation<\/td>\n[latex]{\\rm{Infinitesimal \\ \\ form:}} \\ dS =\\displaystyle\\frac{\\delta Q}{T}+\\delta S_{gen} \\\\ {\\rm{where}} \\ \\delta S_{gen}\\geq0 \\\\ \\rm{(= for \\ reversible \\ processes; \\ > for \\ irreversible \\ process)}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

The second law of thermodynamics<\/strong><\/p>\n\n\n\n\n\n\n\n\n
For closed systems (control mass)<\/td>\n[latex]\\begin {align*} S_2-S_1 &=\\displaystyle\\int_{1}^{2}\\displaystyle\\frac{\\delta Q}{T}+S_{gen} \\\\& \\cong\\sum\\frac{Q_k}{T_k}+S_{gen}\\ \\ \\ \\ \\ (S_{gen}\\geq0) \\end {align*}[\/latex]
\nwhere [latex]T_k[\/latex] is the absolute temperature of the system boundary, in Kelvin.<\/td>\n<\/tr>\n
For steady-state, steady flow in a control volume (open systems)<\/td>\n[latex]\\sum{{\\dot{m}}_es_e}-\\sum{{\\dot{m}}_is_i}=\\sum\\displaystyle\\frac{{\\dot{Q}}_{c.v.}}{T}+{\\dot{S}}_{gen}\\ \\ \\ \\ \\ \\ \\left({\\dot{S}}_{gen}\\geq0\\right)[\/latex]<\/td>\n<\/tr>\n
For steady and isentropic flow<\/td>\n[latex]\\sum{{\\dot{m}}_es_e}=\\sum{{\\dot{m}}_is_i}[\/latex]<\/td>\n<\/tr>\n
Change of specific entropy between two states of a solid or liquid<\/td>\n[latex]s_2-s_1=C_pln\\displaystyle\\frac{T_2}{T_1}[\/latex]<\/td>\n<\/tr>\n
Change of specific entropy between two states of an ideal gas<\/td>\nAssume constant\u00a0 [latex]C_p[\/latex] and [latex]C_v[\/latex] in the temperature range,\u00a0 [latex]s_2-s_1=C_pln\\displaystyle\\frac{T_2}{T_1}-Rln\\frac{P_2}{P_1}[\/latex]<\/p>\n

[latex]s_2-s_1=C_vln\\displaystyle\\frac{T_2}{T_1}+Rln\\frac{v_2}{v_1}[\/latex]<\/td>\n<\/tr>\n

Isentropic relations for ideal gases<\/td>\n[latex]Pv^k= \\rm {constant}[\/latex]<\/p>\n

[latex]\\displaystyle\\frac{P_2}{P_1}=\\displaystyle\\left(\\displaystyle\\frac{v_1}{v_2}\\right)^k=\\left(\\frac{T_2}{T_1}\\right)^{k\/(k-1)}[\/latex]
\nwhere\u00a0 [latex]k=\\displaystyle\\frac{C_p}{C_v}[\/latex]\u00a0 and [latex]T[\/latex] is in Kelvin<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":175,"menu_order":13,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[47],"contributor":[],"license":[],"class_list":["post-291","chapter","type-chapter","status-publish","hentry","chapter-type-standard"],"part":286,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/291","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":23,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/291\/revisions"}],"predecessor-version":[{"id":4357,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/291\/revisions\/4357"}],"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\/291\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/media?parent=291"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapter-type?post=291"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/contributor?post=291"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/license?post=291"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}