{"id":319,"date":"2021-05-29T17:23:54","date_gmt":"2021-05-29T21:23:54","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/thermo1\/?post_type=chapter&p=319"},"modified":"2022-09-02T15:27:09","modified_gmt":"2022-09-02T19:27:09","slug":"key-equations-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/thermo1\/chapter\/key-equations-2\/","title":{"raw":"2.6 Key Equations","rendered":"2.6 Key Equations"},"content":{"raw":"Pressure, temperature, and specific volume<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Pressure<\/td>\r\n[latex] P = F\/A [\/latex]<\/td>\r\n<\/tr>\r\n
Absolute and gauge pressures<\/td>\r\n[latex]P_{gauge}\\ =\\ P_{abs}\\ \u2013 P_{atm}[\/latex]<\/td>\r\n<\/tr>\r\n
Absolute and vacuum pressures<\/td>\r\n[latex]P_{vac}\\ =\\ P_{atm}\\ \u2013 P_{abs}[\/latex]<\/td>\r\n<\/tr>\r\n
Density<\/td>\r\n[latex]\\rho=m\/\\mathbb{V}[\/latex]<\/td>\r\n<\/tr>\r\n
Specific volume<\/td>\r\n[latex]v=\\mathbb{V}\/m=1\/\\rho[\/latex]<\/td>\r\n<\/tr>\r\n
Conversion of temperatures in Kelvin and Celsius degrees<\/td>\r\n[latex]T\\left(\\rm{K}\\right)=T(^\\rm{\\circ}C)+273.15[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nEnergy, enthalpy, and entropy<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Total stored energy in a system<\/td>\r\n[latex] \\begin{align*} E &=U+KE+PE\\\\ &=mu+\\dfrac{1}{2}\\ mV^2+mgz \\; \\; \\; (V: velocity) \\end{align*}[\/latex]<\/td>\r\n<\/tr>\r\n
Total stored specific energy in a system<\/td>\r\n[latex]e=\\displaystyle\\frac{E}{m}=u+\\frac{1}{2}\\ V^2+gz\r\n\r\n\\; \\; \\; \\; (V: velocity)[\/latex]<\/td>\r\n<\/tr>\r\n
Enthalpy<\/td>\r\n[latex]H=U+P\\mathbb{V}[\/latex]<\/td>\r\n<\/tr>\r\n
Specific internal energy \u00a0<\/em><\/td>\r\n[latex]u=U\/m[\/latex]<\/td>\r\n<\/tr>\r\n
Specific enthalpy\u00a0<\/em><\/td>\r\n[latex]h=H\/m [\/latex]\u00a0\u00a0 and \u00a0 [latex]h=u+Pv[\/latex]<\/td>\r\n<\/tr>\r\n
Specific entropy\u00a0<\/em><\/td>\r\n[latex]s=S\/m[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nSaturated liquid-vapour two-phase mixtures<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Quality<\/td>\r\n[latex]x=\\dfrac{m_g}{m_{mix}} [\/latex]<\/td>\r\n<\/tr>\r\n
Specific volume<\/td>\r\n[latex]v=v_f+x\\left(v_g-v_f\\right)=\\left(1-\\ x\\right)v_f+xv_g[\/latex]<\/td>\r\n<\/tr>\r\n
Specific internal energy \u00a0<\/em><\/td>\r\n[latex]u=u_f+x(u_g-u_f)=(1-\\ x)u_f+xu_g[\/latex]<\/td>\r\n<\/tr>\r\n
Specific enthalpy\u00a0<\/em><\/td>\r\n[latex]h=h_f+x(h_g-h_f)=(1-\\ x)h_f+xh_g[\/latex]<\/td>\r\n<\/tr>\r\n
Specific entropy\u00a0<\/em><\/td>\r\n[latex]s=s_f+x(s_g-s_f)=(1-\\ x)s_f+xs_g[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nCompressed liquid (when the compressed liquid tables are not available)<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Specific volume<\/td>\r\n[latex]v\\approx\\ v_{f@T}[\/latex]<\/td>\r\n<\/tr>\r\n
Specific internal energy \u00a0<\/em><\/td>\r\n[latex]u\\approx\\ u_{f@T} [\/latex]<\/td>\r\n<\/tr>\r\n
Specific enthalpy\u00a0<\/em><\/td>\r\n[latex]h\\approx\\ h_{f@T} [\/latex]<\/td>\r\n<\/tr>\r\n
Specific entropy\u00a0<\/em><\/td>\r\n[latex]s\\approx\\ s_{f@T}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"

Pressure, temperature, and specific volume<\/strong><\/p>\n\n\n\n\n\n\n\n\n
Pressure<\/td>\n[latex]P = F\/A[\/latex]<\/td>\n<\/tr>\n
Absolute and gauge pressures<\/td>\n[latex]P_{gauge}\\ =\\ P_{abs}\\ \u2013 P_{atm}[\/latex]<\/td>\n<\/tr>\n
Absolute and vacuum pressures<\/td>\n[latex]P_{vac}\\ =\\ P_{atm}\\ \u2013 P_{abs}[\/latex]<\/td>\n<\/tr>\n
Density<\/td>\n[latex]\\rho=m\/\\mathbb{V}[\/latex]<\/td>\n<\/tr>\n
Specific volume<\/td>\n[latex]v=\\mathbb{V}\/m=1\/\\rho[\/latex]<\/td>\n<\/tr>\n
Conversion of temperatures in Kelvin and Celsius degrees<\/td>\n[latex]T\\left(\\rm{K}\\right)=T(^\\rm{\\circ}C)+273.15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Energy, enthalpy, and entropy<\/strong><\/p>\n\n\n\n\n\n\n\n\n
Total stored energy in a system<\/td>\n[latex]\\begin{align*} E &=U+KE+PE\\\\ &=mu+\\dfrac{1}{2}\\ mV^2+mgz \\; \\; \\; (V: velocity) \\end{align*}[\/latex]<\/td>\n<\/tr>\n
Total stored specific energy in a system<\/td>\n[latex]e=\\displaystyle\\frac{E}{m}=u+\\frac{1}{2}\\ V^2+gz \\; \\; \\; \\; (V: velocity)[\/latex]<\/td>\n<\/tr>\n
Enthalpy<\/td>\n[latex]H=U+P\\mathbb{V}[\/latex]<\/td>\n<\/tr>\n
Specific internal energy \u00a0<\/em><\/td>\n[latex]u=U\/m[\/latex]<\/td>\n<\/tr>\n
Specific enthalpy\u00a0<\/em><\/td>\n[latex]h=H\/m[\/latex]\u00a0\u00a0 and \u00a0 [latex]h=u+Pv[\/latex]<\/td>\n<\/tr>\n
Specific entropy\u00a0<\/em><\/td>\n[latex]s=S\/m[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Saturated liquid-vapour two-phase mixtures<\/strong><\/p>\n\n\n\n\n\n\n\n
Quality<\/td>\n[latex]x=\\dfrac{m_g}{m_{mix}}[\/latex]<\/td>\n<\/tr>\n
Specific volume<\/td>\n[latex]v=v_f+x\\left(v_g-v_f\\right)=\\left(1-\\ x\\right)v_f+xv_g[\/latex]<\/td>\n<\/tr>\n
Specific internal energy \u00a0<\/em><\/td>\n[latex]u=u_f+x(u_g-u_f)=(1-\\ x)u_f+xu_g[\/latex]<\/td>\n<\/tr>\n
Specific enthalpy\u00a0<\/em><\/td>\n[latex]h=h_f+x(h_g-h_f)=(1-\\ x)h_f+xh_g[\/latex]<\/td>\n<\/tr>\n
Specific entropy\u00a0<\/em><\/td>\n[latex]s=s_f+x(s_g-s_f)=(1-\\ x)s_f+xs_g[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Compressed liquid (when the compressed liquid tables are not available)<\/strong><\/p>\n\n\n\n\n\n\n
Specific volume<\/td>\n[latex]v\\approx\\ v_{f@T}[\/latex]<\/td>\n<\/tr>\n
Specific internal energy \u00a0<\/em><\/td>\n[latex]u\\approx\\ u_{f@T}[\/latex]<\/td>\n<\/tr>\n
Specific enthalpy\u00a0<\/em><\/td>\n[latex]h\\approx\\ h_{f@T}[\/latex]<\/td>\n<\/tr>\n
Specific entropy\u00a0<\/em><\/td>\n[latex]s\\approx\\ s_{f@T}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":175,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[47],"contributor":[],"license":[],"class_list":["post-319","chapter","type-chapter","status-publish","hentry","chapter-type-standard"],"part":246,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/319","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\/319\/revisions"}],"predecessor-version":[{"id":4362,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapters\/319\/revisions\/4362"}],"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\/319\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/media?parent=319"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/pressbooks\/v2\/chapter-type?post=319"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/contributor?post=319"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/thermo1\/wp-json\/wp\/v2\/license?post=319"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}