{"id":289,"date":"2021-05-21T01:50:43","date_gmt":"2021-05-21T05:50:43","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/thermo1\/?post_type=chapter&p=289"},"modified":"2022-09-02T15:17:01","modified_gmt":"2022-09-02T19:17:01","slug":"key-equations-5","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/key-equations-5\/","title":{"raw":"5.5 Key equations","rendered":"5.5 Key equations"},"content":{"raw":"

Constant-pressure and constant-volume specific heats<\/strong><\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Constant-pressure specific heat<\/td>\r\n[latex]C_p=\\left(\\displaystyle\\frac{\\partial\\ h}{\\partial\\ T}\\right)_p[\/latex]<\/td>\r\n<\/tr>\r\n
Constant-volume specific heat<\/td>\r\n[latex]C_v=\\left(\\displaystyle\\frac{\\partial\\ u}{\\partial\\ T}\\right)_v[\/latex]<\/td>\r\n<\/tr>\r\n
Relations between [latex]C_p [\/latex]\u00a0 and\u00a0 [latex]C_v[\/latex] for ideal gases<\/strong><\/td>\r\n[latex]k=\\displaystyle\\frac{C_p}{C_v} \\qquad \\qquad C_p=C_v+R [\/latex]\r\n\r\n[latex]C_v=\\displaystyle\\frac{R}{k-1}\u00a0 \\qquad\u00a0 C_p=\\displaystyle\\frac{kR}{k-1}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nSpecific enthalpy<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n
Change in specific enthalpy<\/td>\r\n[latex]\\Delta h = h_2-h_1[\/latex]<\/td>\r\n<\/tr>\r\n
Change in specific enthalpy for i<\/strong>deal gases<\/strong><\/td>\r\n[latex]\\Delta h = h_2-h_1 = C_p(T_2-T_1)[\/latex]\r\n(assuming constant [latex]C_p[\/latex] in the temperature range)<\/td>\r\n<\/tr>\r\n
Relation between [latex]\\Delta h [\/latex]\u00a0 and\u00a0 [latex]\\Delta u[\/latex] for solids and liquids<\/strong><\/td>\r\n[latex]\\Delta h \\approx\\Delta u\\approx C_p(T_2-T_1)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nMass conservation equations in a control volume<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Volume flow rate<\/td>\r\n[latex]\\dot{\\mathbb{V}}=\\displaystyle\\frac{d\\mathbb{V}}{dt}=V_{avg,\\ n}A=\\dot{m}v[\/latex]<\/td>\r\n<\/tr>\r\n
Mass flow rate<\/td>\r\n[latex]\\dot{m}=\\displaystyle\\frac{dm}{dt}=\\rho\\ V_{avg,\\ n}A=\\rho\\dot{\\mathbb{V}}[\/latex]<\/td>\r\n<\/tr>\r\n
Transient flow<\/strong><\/td>\r\n[latex]\\displaystyle\\frac{dm_{CV}}{dt}=\\sum{\\dot{m}}_i-\\sum{\\dot{m}}_e\\neq0[\/latex]<\/td>\r\n<\/tr>\r\n
Steady flow<\/strong><\/td>\r\n[latex]\\displaystyle\\frac{dm_{CV}}{dt}=\\sum{\\dot{m}}_i-\\sum{\\dot{m}}_e=0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n

Energy conservation equations in a control volume<\/strong><\/p>\r\n\r\n\r\n\r\n\r\n\r\n
Transient flow<\/strong><\/td>\r\n[latex]\\displaystyle\\frac{dE_{CV}}{dt}\\neq0[\/latex]\r\n\r\n[latex]\\begin{align*} \\displaystyle\\frac{dE_{CV}}{dt}={\\dot{Q}}_{cv}-{\\dot{W}}_{cv} &+\\sum{{\\dot{m}}_i(h_i+\\frac{1}{2}V_i^2+gz_i)} \\\\&-\\sum{{\\dot{m}}_e(h_e+\\frac{1}{2}V_e^2+gz_e)} \\end{align*}[\/latex]<\/td>\r\n<\/tr>\r\n
Steady flow<\/strong><\/td>\r\n[latex]\\displaystyle\\frac{dE_{CV}}{dt}=0[\/latex]\r\n\r\n[latex] {\\dot{Q}}_{cv}\u00a0 +\\sum{{\\dot{m}}_i\\left(h_i+\\displaystyle\\frac{1}{2}V_i^2+gz_i\\right) \\\\={\\dot{W}}_{cv} +\\sum{{\\dot{m}}_e\\left(h_e+\\displaystyle\\frac{1}{2}V_e^2+gz_e\\right)}} [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n

Mass and energy conservation equations for steady-state, steady-flow (SSSF) devices<\/strong><\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
SSSF device<\/td>\r\nAssumptions<\/td>\r\nMass conservation<\/td>\r\nEnergy conservation<\/td>\r\n<\/tr>\r\n
Expansion device<\/strong><\/td>\r\nAdiabatic flow; Negligible work transfer with the surroundings; Negligible changes in kinetic and potential energies<\/td>\r\n[latex]{\\dot{m}}_i={\\dot{m}}_e[\/latex]<\/td>\r\n[latex]h_i=h_e[\/latex]<\/td>\r\n<\/tr>\r\n
Nozzle and diffuser<\/strong><\/td>\r\nAdiabatic flow; Negligible work transfer with the surroundings; Negligible change in potential energy<\/td>\r\n[latex]{\\dot{m}}_i={\\dot{m}}_e[\/latex]<\/td>\r\n[latex]h_i+\\displaystyle\\frac{1}{2}V_i^2=h_e+\\displaystyle\\frac{1}{2}V_e^2[\/latex]<\/td>\r\n<\/tr>\r\n
Mixing chamber<\/strong><\/td>\r\nNegligible work transfer with the surroundings; Negligible changes in kinetic and potential energies<\/td>\r\n[latex]\\sum{{\\dot{m}}_i=\\sum{\\dot{m}}_e}[\/latex]<\/td>\r\n[latex]{\\dot{Q}}_{cv}+\\sum{{\\dot{m}}_ih_i=\\sum{{\\dot{m}}_eh_e}}[\/latex]<\/td>\r\n<\/tr>\r\n
Heat exchanger<\/strong><\/td>\r\nNegligible work transfer with the surroundings; Negligible changes in kinetic and potential energies<\/td>\r\n[latex]{\\dot{m}}_i={\\dot{m}}_e[\/latex]\r\n(for each of the hot and cold streams, separately)<\/td>\r\n[latex]{\\dot{Q}}_{cv}+\\sum{{\\dot{m}}_ih_i=\\sum{{\\dot{m}}_eh_e}}[\/latex]<\/td>\r\n<\/tr>\r\n
Turbine<\/strong><\/td>\r\nAdiabatic flow; Negligible changes in kinetic and potential energies<\/td>\r\n[latex]{\\dot{m}}_i={\\dot{m}}_e=\\dot{m}[\/latex]<\/td>\r\n[latex]{\\dot{W}}_{shaft}=\\dot{m}(h_i-h_e)[\/latex]<\/td>\r\n<\/tr>\r\n
Compressor<\/strong><\/td>\r\nAdiabatic flow; Negligible changes in kinetic and potential energies<\/td>\r\n[latex]{\\dot{m}}_i={\\dot{m}}_e=\\dot{m}[\/latex]<\/td>\r\n[latex]{\\dot{W}}_{shaft}=\\dot{m}(h_e-h_i)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"

Constant-pressure and constant-volume specific heats<\/strong><\/p>\n\n\n\n\n\n
Constant-pressure specific heat<\/td>\n[latex]C_p=\\left(\\displaystyle\\frac{\\partial\\ h}{\\partial\\ T}\\right)_p[\/latex]<\/td>\n<\/tr>\n
Constant-volume specific heat<\/td>\n[latex]C_v=\\left(\\displaystyle\\frac{\\partial\\ u}{\\partial\\ T}\\right)_v[\/latex]<\/td>\n<\/tr>\n
Relations between [latex]C_p[\/latex]\u00a0 and\u00a0 [latex]C_v[\/latex] for ideal gases<\/strong><\/td>\n[latex]k=\\displaystyle\\frac{C_p}{C_v} \\qquad \\qquad C_p=C_v+R[\/latex]<\/p>\n

[latex]C_v=\\displaystyle\\frac{R}{k-1}\u00a0 \\qquad\u00a0 C_p=\\displaystyle\\frac{kR}{k-1}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Specific enthalpy<\/strong><\/p>\n\n\n\n\n\n
Change in specific enthalpy<\/td>\n[latex]\\Delta h = h_2-h_1[\/latex]<\/td>\n<\/tr>\n
Change in specific enthalpy for i<\/strong>deal gases<\/strong><\/td>\n[latex]\\Delta h = h_2-h_1 = C_p(T_2-T_1)[\/latex]
\n(assuming constant [latex]C_p[\/latex] in the temperature range)<\/td>\n<\/tr>\n
Relation between [latex]\\Delta h[\/latex]\u00a0 and\u00a0 [latex]\\Delta u[\/latex] for solids and liquids<\/strong><\/td>\n[latex]\\Delta h \\approx\\Delta u\\approx C_p(T_2-T_1)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Mass conservation equations in a control volume<\/strong><\/p>\n\n\n\n\n\n\n
Volume flow rate<\/td>\n[latex]\\dot{\\mathbb{V}}=\\displaystyle\\frac{d\\mathbb{V}}{dt}=V_{avg,\\ n}A=\\dot{m}v[\/latex]<\/td>\n<\/tr>\n
Mass flow rate<\/td>\n[latex]\\dot{m}=\\displaystyle\\frac{dm}{dt}=\\rho\\ V_{avg,\\ n}A=\\rho\\dot{\\mathbb{V}}[\/latex]<\/td>\n<\/tr>\n
Transient flow<\/strong><\/td>\n[latex]\\displaystyle\\frac{dm_{CV}}{dt}=\\sum{\\dot{m}}_i-\\sum{\\dot{m}}_e\\neq0[\/latex]<\/td>\n<\/tr>\n
Steady flow<\/strong><\/td>\n[latex]\\displaystyle\\frac{dm_{CV}}{dt}=\\sum{\\dot{m}}_i-\\sum{\\dot{m}}_e=0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Energy conservation equations in a control volume<\/strong><\/p>\n\n\n\n\n
Transient flow<\/strong><\/td>\n[latex]\\displaystyle\\frac{dE_{CV}}{dt}\\neq0[\/latex]<\/p>\n

[latex]\\begin{align*} \\displaystyle\\frac{dE_{CV}}{dt}={\\dot{Q}}_{cv}-{\\dot{W}}_{cv} &+\\sum{{\\dot{m}}_i(h_i+\\frac{1}{2}V_i^2+gz_i)} \\\\&-\\sum{{\\dot{m}}_e(h_e+\\frac{1}{2}V_e^2+gz_e)} \\end{align*}[\/latex]<\/td>\n<\/tr>\n

Steady flow<\/strong><\/td>\n[latex]\\displaystyle\\frac{dE_{CV}}{dt}=0[\/latex]<\/p>\n

[latex]{\\dot{Q}}_{cv}\u00a0 +\\sum{{\\dot{m}}_i\\left(h_i+\\displaystyle\\frac{1}{2}V_i^2+gz_i\\right) \\\\={\\dot{W}}_{cv} +\\sum{{\\dot{m}}_e\\left(h_e+\\displaystyle\\frac{1}{2}V_e^2+gz_e\\right)}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Mass and energy conservation equations for steady-state, steady-flow (SSSF) devices<\/strong><\/p>\n\n\n\n\n\n\n\n\n\n
SSSF device<\/td>\nAssumptions<\/td>\nMass conservation<\/td>\nEnergy conservation<\/td>\n<\/tr>\n
Expansion device<\/strong><\/td>\nAdiabatic flow; Negligible work transfer with the surroundings; Negligible changes in kinetic and potential energies<\/td>\n[latex]{\\dot{m}}_i={\\dot{m}}_e[\/latex]<\/td>\n[latex]h_i=h_e[\/latex]<\/td>\n<\/tr>\n
Nozzle and diffuser<\/strong><\/td>\nAdiabatic flow; Negligible work transfer with the surroundings; Negligible change in potential energy<\/td>\n[latex]{\\dot{m}}_i={\\dot{m}}_e[\/latex]<\/td>\n[latex]h_i+\\displaystyle\\frac{1}{2}V_i^2=h_e+\\displaystyle\\frac{1}{2}V_e^2[\/latex]<\/td>\n<\/tr>\n
Mixing chamber<\/strong><\/td>\nNegligible work transfer with the surroundings; Negligible changes in kinetic and potential energies<\/td>\n[latex]\\sum{{\\dot{m}}_i=\\sum{\\dot{m}}_e}[\/latex]<\/td>\n[latex]{\\dot{Q}}_{cv}+\\sum{{\\dot{m}}_ih_i=\\sum{{\\dot{m}}_eh_e}}[\/latex]<\/td>\n<\/tr>\n
Heat exchanger<\/strong><\/td>\nNegligible work transfer with the surroundings; Negligible changes in kinetic and potential energies<\/td>\n[latex]{\\dot{m}}_i={\\dot{m}}_e[\/latex]
\n(for each of the hot and cold streams, separately)<\/td>\n		[latex]{\\dot{Q}}_{cv}+\\sum{{\\dot{m}}_ih_i=\\sum{{\\dot{m}}_eh_e}}[\/latex]<\/td>\n<\/tr>\n
Turbine<\/strong><\/td>\nAdiabatic flow; Negligible changes in kinetic and potential energies<\/td>\n[latex]{\\dot{m}}_i={\\dot{m}}_e=\\dot{m}[\/latex]<\/td>\n[latex]{\\dot{W}}_{shaft}=\\dot{m}(h_i-h_e)[\/latex]<\/td>\n<\/tr>\n
Compressor<\/strong><\/td>\nAdiabatic flow; Negligible changes in kinetic and potential energies<\/td>\n[latex]{\\dot{m}}_i={\\dot{m}}_e=\\dot{m}[\/latex]<\/td>\n[latex]{\\dot{W}}_{shaft}=\\dot{m}(h_e-h_i)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":175,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[47],"contributor":[],"license":[],"class_list":["post-289","chapter","type-chapter","status-publish","hentry","chapter-type-standard"],"part":284,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/289","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\/289\/revisions"}],"predecessor-version":[{"id":4034,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/289\/revisions\/4034"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/parts\/284"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/289\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/media?parent=289"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapter-type?post=289"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/contributor?post=289"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/license?post=289"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}