{"id":253,"date":"2020-07-02T15:50:36","date_gmt":"2020-07-02T15:50:36","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/chbe220\/chapter\/energy-balances-review\/"},"modified":"2026-03-16T13:52:28","modified_gmt":"2026-03-16T13:52:28","slug":"energy-balances-review","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/chbe220\/chapter\/energy-balances-review\/","title":{"raw":"Energy Balances Review","rendered":"Energy Balances Review"},"content":{"raw":"<h2>Important Equations<\/h2>\n<table class=\"grid\" style=\"border-collapse: collapse;width: 100%;height: 269px\" border=\"0\">\n<tbody>\n<tr style=\"height: 30px\">\n<td style=\"width: 50%;height: 30px\">Kinetic Energy<\/td>\n<td style=\"width: 50%;height: 30px\">\n<p style=\"text-align: center\">[latex]E_{k} = \\frac{1}{2} mu^{2}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\dot{E}_{k} = \\frac{1}{2} \\dot{m} u^{2}[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 30px\">\n<td style=\"width: 50%;height: 30px\">Potential Energy<\/td>\n<td style=\"width: 50%;height: 30px\">\n<p style=\"text-align: center\">[latex]E_{p} = m g z[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\dot{E}_{p} = \\dot{m} g z[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta E_{p} = E_{p2} - E_{p1} = m g (z_{2} - z_{1})[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">First Law of Thermodynamics[latex][\/latex]<\/td>\n<td style=\"width: 50%;height: 15px\"><span style=\"color: #000000;font-size: 16px\">[latex]\\Delta U + \\Delta E_{k} + \\Delta E_{p} = Q + W[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">Flow Work<\/td>\n<td style=\"width: 50%;height: 15px;text-align: center\">[latex]\\dot{W}_{fl} = \\dot{W}_{fl-in} - \\dot{W}_{fl-out} = P_{in}\\dot{V}_{in} - P_{out}\\dot{V}_{out}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 10px\">Steady-state Open System Energy Balance<\/td>\n<td style=\"width: 50%;height: 10px\">\n<p style=\"text-align: center\">[latex]\\dot{Q} + \\dot{W} = \\Sigma_{out} \\dot{E}_{j} - \\Sigma_{in} \\dot{E}_{j}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\dot{Q} + \\dot{W}_{s} = \\Delta\\dot{H} + \\Delta\\dot{E}_{k} + \\Delta\\dot{E}_{p}[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">Enthalpy<\/td>\n<td style=\"width: 50%;height: 15px\">\n<p style=\"text-align: center\">[latex]\\hat{H} = \\hat{U} + P\\hat{V}[\/latex]<\/p>\n<p style=\"text-align: center\"><span style=\"font-size: 16px\">[latex]\\Delta\\hat{H} = \\Sigma_{i}\\Delta\\hat{H}_{i}[\/latex]<\/span><\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">Heat Capacity (closed system)<\/td>\n<td style=\"width: 50%;height: 15px;text-align: center\">[latex]C_{V}(T) = \\bigg(\\frac{\\delta\\hat{U}}{\\delta T}\\bigg)_{V}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">Internal Energy (closed system)<\/td>\n<td style=\"width: 50%;height: 15px;text-align: center\"><span style=\"font-size: 16px\">[latex]d\\hat{U} = C_{V}(T)dT[\/latex]<\/span>\n\n<span style=\"font-size: 16px\">[latex]\\Delta\\hat{U} = \\int^{T_{2}}_{T_{1}}C_{V}dT[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">Heat Capacity (open system)<\/td>\n<td style=\"width: 50%;height: 15px;text-align: center\"><span style=\"font-size: 16px\">[latex]C_{P}(T) = \\bigg(\\frac{\\delta\\hat{H}}{\\delta T}\\bigg)_{P}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">Enthalpy (open system)<\/td>\n<td style=\"width: 50%;height: 15px;text-align: center\"><span style=\"font-size: 16px\">[latex]\\Delta\\hat{H} = \\int^{T_{2}}_{T_{1}}C_{P}dT[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 47px\">\n<td style=\"width: 50%;height: 47px\">Heat of Reaction Method<\/td>\n<td style=\"width: 50%;height: 47px;text-align: center\"><span style=\"font-size: 16px\">[latex]\\Delta\\dot{H} = \\xi\\Delta\\dot{H}_{r} + \\Sigma\\dot{n}_{out}*\\int^{T_{out}}_{T_{ref}}C_{P}dT - \\Sigma\\dot{n}_{in}*\\int^{T_{in}}_{T_{ref}}C_{P}dT[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 47px\">\n<td style=\"width: 50%;height: 47px\">Heat of Formation Method<\/td>\n<td style=\"width: 50%;height: 47px;text-align: center\"><span style=\"font-size: 16px\">[latex]\\xi\\Delta\\dot{H}^{\\circ}_{r} = \\Sigma\\dot{n}_{out}*\\hat{H}^{\\circ}_{f,i} - \\Sigma\\dot{n}_{in}*\\hat{H}^{\\circ}_{f,i}[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Terms to Know<\/h2>\nThe original version of this chapter contained H5P content. You may want to remove or replace this element.\n\n&nbsp;","rendered":"<h2>Important Equations<\/h2>\n<table class=\"grid\" style=\"border-collapse: collapse;width: 100%;height: 269px\">\n<tbody>\n<tr style=\"height: 30px\">\n<td style=\"width: 50%;height: 30px\">Kinetic Energy<\/td>\n<td style=\"width: 50%;height: 30px\">\n<p style=\"text-align: center\">[latex]E_{k} = \\frac{1}{2} mu^{2}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\dot{E}_{k} = \\frac{1}{2} \\dot{m} u^{2}[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 30px\">\n<td style=\"width: 50%;height: 30px\">Potential Energy<\/td>\n<td style=\"width: 50%;height: 30px\">\n<p style=\"text-align: center\">[latex]E_{p} = m g z[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\dot{E}_{p} = \\dot{m} g z[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta E_{p} = E_{p2} - E_{p1} = m g (z_{2} - z_{1})[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">First Law of Thermodynamics[latex][\/latex]<\/td>\n<td style=\"width: 50%;height: 15px\"><span style=\"color: #000000;font-size: 16px\">[latex]\\Delta U + \\Delta E_{k} + \\Delta E_{p} = Q + W[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">Flow Work<\/td>\n<td style=\"width: 50%;height: 15px;text-align: center\">[latex]\\dot{W}_{fl} = \\dot{W}_{fl-in} - \\dot{W}_{fl-out} = P_{in}\\dot{V}_{in} - P_{out}\\dot{V}_{out}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 10px\">Steady-state Open System Energy Balance<\/td>\n<td style=\"width: 50%;height: 10px\">\n<p style=\"text-align: center\">[latex]\\dot{Q} + \\dot{W} = \\Sigma_{out} \\dot{E}_{j} - \\Sigma_{in} \\dot{E}_{j}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\dot{Q} + \\dot{W}_{s} = \\Delta\\dot{H} + \\Delta\\dot{E}_{k} + \\Delta\\dot{E}_{p}[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">Enthalpy<\/td>\n<td style=\"width: 50%;height: 15px\">\n<p style=\"text-align: center\">[latex]\\hat{H} = \\hat{U} + P\\hat{V}[\/latex]<\/p>\n<p style=\"text-align: center\"><span style=\"font-size: 16px\">[latex]\\Delta\\hat{H} = \\Sigma_{i}\\Delta\\hat{H}_{i}[\/latex]<\/span><\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">Heat Capacity (closed system)<\/td>\n<td style=\"width: 50%;height: 15px;text-align: center\">[latex]C_{V}(T) = \\bigg(\\frac{\\delta\\hat{U}}{\\delta T}\\bigg)_{V}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">Internal Energy (closed system)<\/td>\n<td style=\"width: 50%;height: 15px;text-align: center\"><span style=\"font-size: 16px\">[latex]d\\hat{U} = C_{V}(T)dT[\/latex]<\/span><\/p>\n<p><span style=\"font-size: 16px\">[latex]\\Delta\\hat{U} = \\int^{T_{2}}_{T_{1}}C_{V}dT[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">Heat Capacity (open system)<\/td>\n<td style=\"width: 50%;height: 15px;text-align: center\"><span style=\"font-size: 16px\">[latex]C_{P}(T) = \\bigg(\\frac{\\delta\\hat{H}}{\\delta T}\\bigg)_{P}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 50%;height: 15px\">Enthalpy (open system)<\/td>\n<td style=\"width: 50%;height: 15px;text-align: center\"><span style=\"font-size: 16px\">[latex]\\Delta\\hat{H} = \\int^{T_{2}}_{T_{1}}C_{P}dT[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 47px\">\n<td style=\"width: 50%;height: 47px\">Heat of Reaction Method<\/td>\n<td style=\"width: 50%;height: 47px;text-align: center\"><span style=\"font-size: 16px\">[latex]\\Delta\\dot{H} = \\xi\\Delta\\dot{H}_{r} + \\Sigma\\dot{n}_{out}*\\int^{T_{out}}_{T_{ref}}C_{P}dT - \\Sigma\\dot{n}_{in}*\\int^{T_{in}}_{T_{ref}}C_{P}dT[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 47px\">\n<td style=\"width: 50%;height: 47px\">Heat of Formation Method<\/td>\n<td style=\"width: 50%;height: 47px;text-align: center\"><span style=\"font-size: 16px\">[latex]\\xi\\Delta\\dot{H}^{\\circ}_{r} = \\Sigma\\dot{n}_{out}*\\hat{H}^{\\circ}_{f,i} - \\Sigma\\dot{n}_{in}*\\hat{H}^{\\circ}_{f,i}[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Terms to Know<\/h2>\n<p>The original version of this chapter contained H5P content. You may want to remove or replace this element.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"author":1,"menu_order":22,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-253","chapter","type-chapter","status-publish","hentry"],"part":238,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/pressbooks\/v2\/chapters\/253","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":1,"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/pressbooks\/v2\/chapters\/253\/revisions"}],"predecessor-version":[{"id":254,"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/pressbooks\/v2\/chapters\/253\/revisions\/254"}],"part":[{"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/pressbooks\/v2\/parts\/238"}],"metadata":[{"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/pressbooks\/v2\/chapters\/253\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/wp\/v2\/media?parent=253"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/pressbooks\/v2\/chapter-type?post=253"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/wp\/v2\/contributor?post=253"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/wp\/v2\/license?post=253"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}