{"id":247,"date":"2020-06-23T16:27:05","date_gmt":"2020-06-23T16:27:05","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/chbe220\/chapter\/phase-change-and-heat-capacity\/"},"modified":"2026-03-16T13:52:23","modified_gmt":"2026-03-16T13:52:23","slug":"phase-change-and-heat-capacity","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/chbe220\/chapter\/phase-change-and-heat-capacity\/","title":{"raw":"Phase Change and Heat Capacity","rendered":"Phase Change and Heat Capacity"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nBy the end of this section, you should be able to:\n<p id=\"Evaluate:\"><strong>Evaluate <\/strong>the cost of utilities in processes<\/p>\n<p id=\"Characterize:\"><strong>Characterize<\/strong>\u00a0energy changes in a system due to changes in temperature<\/p>\n<p id=\"Analyze:\"><strong>Analyze<\/strong>\u00a0energy balances on processes involving phase changes<\/p>\n\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h2 id=\"Energy-Balance-Equipment\">Energy Balance Equipment<\/h2>\n<ul>\n \t<li>Almost all energy exchange equipment is either a heat exchanger or jacket<\/li>\n \t<li>Heat exchangers and jackets often work by transferring heat with a stream of fluid<\/li>\n \t<li>Electrical heating elements are also available, but their high cost per unit energy may limit their use in industry<\/li>\n<\/ul>\n<blockquote>The figure below is a shell-and-tube heat exchange:\n\n<img class=\" wp-image-243 aligncenter\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/8\/2020\/06\/DP-HX-300x188.png\" alt=\"\" width=\"449\" height=\"281\">\n<p style=\"text-align: center\">Image from <a title=\"User:Turbojet\" href=\"Turbojet\">Turbojet<\/a> \/<a title=\"Creative Commons Attribution-Share Alike 4.0\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\">CC BY-SA 4.0<\/a><\/p>\n<\/blockquote>\n<strong>Utilities<\/strong>: Utilities are services at a site such as water, electricity, and gas.\n\n&nbsp;\n<div>Choosing the correct utility can make a big difference in the cost of heating or cooling. For example, evaporating toluene at [latex]190^{\\circ}C[\/latex] with a heat duty of 3,000,000 kJ\/hr can be heated using the following utility options:<\/div>\n<div><\/div>\n<div>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 97.2899%;height: 132px\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 14.2857%;text-align: center\"><strong>Utility<\/strong><\/td>\n<td style=\"width: 14.2857%;text-align: center\"><strong>Inlet [latex]T (^{\\circ}C)[\/latex]\u00a0<\/strong><\/td>\n<td style=\"width: 14.2857%;text-align: center\"><strong>Outlet [latex]T (^{\\circ}C)[\/latex]<\/strong><\/td>\n<td style=\"width: 14.2857%;text-align: center\"><strong>P ([latex]psia[\/latex])<\/strong><\/td>\n<td style=\"width: 14.2857%;text-align: center\"><strong>Cost (dollars\/GJ)<\/strong><\/td>\n<td style=\"width: 14.2857%;text-align: center\"><strong>Cost (dollars\/hr)<\/strong><\/td>\n<td style=\"width: 14.2857%;text-align: center\"><strong>Cost dollars\/yr @ 8,000 hrs<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.2857%;text-align: center\">High-pressure steam<\/td>\n<td style=\"width: 14.2857%;text-align: center\">250<\/td>\n<td style=\"width: 14.2857%;text-align: center\">249<\/td>\n<td style=\"width: 14.2857%;text-align: center\">575<\/td>\n<td style=\"width: 14.2857%;text-align: center\">5.66<\/td>\n<td style=\"width: 14.2857%;text-align: center\">16.98<\/td>\n<td style=\"width: 14.2857%;text-align: center\">135,840<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.2857%;text-align: center\">Electricity<\/td>\n<td style=\"width: 14.2857%;text-align: center\">N\/A<\/td>\n<td style=\"width: 14.2857%;text-align: center\">N\/A<\/td>\n<td style=\"width: 14.2857%;text-align: center\">N\/A<\/td>\n<td style=\"width: 14.2857%;text-align: center\">18.72<\/td>\n<td style=\"width: 14.2857%;text-align: center\">56.16<\/td>\n<td style=\"width: 14.2857%;text-align: center\">449,280<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n&nbsp;\n\n<em>The high-pressure steam is a much cheaper option here<\/em>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h2 id=\"Phase-Changes\">Phase Changes<\/h2>\nPhase changes take place when a compound or mixture of compounds undergoes a change in their state of matter. The enthalpy changes for phase changes occur at a constant temperature and pressure and are determined experimentally.\n<table class=\"grid aligncenter\">\n<thead>\n<tr>\n<th>Parameter<\/th>\n<th>Phase Transition<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Liquid \u2192 Gas<\/td>\n<td>Heat of Vapourization ([latex]\\Delta\\hat{H}_{vap}[\/latex])<\/td>\n<\/tr>\n<tr>\n<td>Solid \u2192 Liquid<\/td>\n<td>Heat of Fusion ([latex]\\Delta\\hat{H}_{fus}[\/latex])<\/td>\n<\/tr>\n<tr>\n<td>Gas \u2192 Liquid<\/td>\n<td>(-) Heat of Vapourization ([latex]-\\Delta\\hat{H}_{vap}[\/latex])<\/td>\n<\/tr>\n<tr>\n<td>Liquid \u2192 Solid<\/td>\n<td>(-) Heat of Fusion ([latex]-\\Delta\\hat{H}_{fus}[\/latex])<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<em>Note that heat of vapourization is also referred to as latent heat<\/em>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h2 id=\"Heat-Capacities\">Heat Capacities<\/h2>\nHeat capacities are physical properties that describe how much heat is needed to increase the temperature of a compound for a unit temperature per unit mass of a compound.\n\n<strong>For a closed system<\/strong> where the only changing system variable is temperature:\n<p style=\"text-align: center\">[latex] Q = \\Delta\\hat{U} [\/latex]<\/p>\n<span style=\"text-align: initial;font-size: 1em\">The heat capacity at constant volume can be described as follows, where [latex]C_{V}[\/latex] is a function of temperature [latex]T[\/latex]:<\/span>\n<p style=\"text-align: center\">[latex]C_{V}(T) = \\bigg(\\frac{\\delta\\hat{U}}{\\delta T}\\bigg)_{V}[\/latex]<\/p>\nNotation: [latex]\\big(\\;\\;\\big)_{V}[\/latex] means that the volume is kept constant during the process\n<p style=\"text-align: left\">This expression can be rearranged and integrated to obtain the following:<\/p>\n\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 63.9196%;height: 81px\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 100%;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<\/tbody>\n<\/table>\nIf the [latex]C_{V}[\/latex] is constant:\n<p style=\"text-align: center\">[latex]\\Delta\\hat{U} = C_{V}*(T_{2}-T_{1})[\/latex]<\/p>\n<p style=\"text-align: left\"><strong>For an open system<\/strong>, the heat capacity is defined under constant pressure conditions:<\/p>\n\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 58.1833%;height: 97px\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 100%\">\n<p style=\"text-align: center\"><span style=\"font-size: 16px\">[latex]C_{P}(T) = \\bigg(\\frac{\\delta\\hat{H}}{\\delta T}\\bigg)_{P}[\/latex]<\/span><\/p>\n<p style=\"text-align: center\"><span style=\"font-size: 16px\">[latex]\\Delta\\hat{H} = \\int^{T_{2}}_{T_{1}}C_{P}dT[\/latex]<\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span style=\"text-align: initial;font-size: 1em\">If the [latex]C_{P}[\/latex] is constant:<\/span>\n<p style=\"text-align: center\"><span style=\"text-align: initial;font-size: 1em\">[latex]\\Delta\\hat{H} = C_{P}*(T_{2}-T_{1})[\/latex]<\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"prompt input_prompt\"><\/div>\n<div>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Example: Phase Changes and Heat Capacity in Energy Balances<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div class=\"textbox__content\" style=\"text-align: center\">\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<div style=\"text-align: left\">Consider a system where a mixture of dichloroethane ([latex]EDC[\/latex]), hydrogen chloride ([latex]H\\!Cl[\/latex]), and vinyl chloride ([latex]V\\!C[\/latex]) is entering a distillation column for separation. Distillation columns work based on vapor-liquid equilibrium, and therefore the column will generally operate at a temperature between the boiling points of all compounds.<\/div>\n<div>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 91.6844%;height: 117px\" border=\"0\">\n<tbody>\n<tr style=\"height: 28px\">\n<td style=\"width: 25%;height: 28px;text-align: center\"><strong>Compound<\/strong><\/td>\n<td style=\"width: 25%;height: 28px;text-align: center\"><strong>Formula<\/strong><\/td>\n<td style=\"width: 25%;height: 28px;text-align: center\"><strong>Boiling Point at 1 atm ([latex]^{\\circ}C[\/latex])<\/strong><\/td>\n<td style=\"width: 25%;height: 28px;text-align: center\"><strong>Heat of Vapourization at 1 atm ([latex]kJ\/mol[\/latex])<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 25%;height: 15px;text-align: center\">EDC<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">[latex]C_{2}H_{4}Cl_{2}[\/latex]<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">84<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">35<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 25%;height: 15px;text-align: center\">HCl<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">[latex]HCl[\/latex]<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">-85<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">16<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 25%;height: 15px;text-align: center\">VC<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">[latex]C_{2}H_{3}Cl[\/latex]<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">-13<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">21<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n&nbsp;\n\nThe mixture exits a reactor at [latex]200^{\\circ}C[\/latex] and has the following properties:\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 86.1307%;height: 137px\" border=\"0\">\n<tbody>\n<tr class=\"border\" style=\"height: 30px\">\n<td style=\"width: 20%;height: 30px;text-align: center\"><strong>Compound<\/strong><\/td>\n<td style=\"width: 20%;height: 30px;text-align: center\"><strong>Formula<\/strong><\/td>\n<td style=\"width: 20%;height: 30px;text-align: center\"><strong>Flow to Separate (tonne\/hr)<\/strong><\/td>\n<td style=\"width: 20%;height: 30px;text-align: center\"><strong>Flow to Separate (tonne-mol\/hr)<\/strong><\/td>\n<td style=\"width: 20%;height: 30px;text-align: center\"><strong>[latex]y_{i}[\/latex] (gas mole fraction)<\/strong><\/td>\n<\/tr>\n<tr class=\"border\" style=\"height: 15px\">\n<td style=\"width: 20%;height: 15px;text-align: center\">EDC<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">[latex]C_{2}H_{4}Cl_{2}[\/latex]<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">47.5008<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">0.48<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">0.25<\/td>\n<\/tr>\n<tr class=\"border\" style=\"height: 15px\">\n<td style=\"width: 20%;height: 15px;text-align: center\">HCl<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">[latex]HCl[\/latex]<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">26.2512<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">0.72<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">0.375<\/td>\n<\/tr>\n<tr class=\"border\" style=\"height: 15px\">\n<td style=\"width: 20%;height: 15px;text-align: center\">VC<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">[latex]C_{2}H_{3}Cl[\/latex]<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">45<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">0.72<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">0.375<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox__content\">\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n\n&nbsp;\n\n<img class=\"wp-image-244 aligncenter\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange1-300x84.jpg\" alt=\"\" width=\"600\" height=\"168\">\n<blockquote>Recall <strong>Raoult's Law<\/strong>:\n<p style=\"text-align: center\">[latex]x_{i} = \\frac{p_{i}*}{y_{i}\u00d7P }[\/latex]<\/p>\n\n<table class=\"grid\" style=\"border-collapse: collapse;width: 100%;height: 90px\" border=\"0\">\n<tbody>\n<tr style=\"height: 45px\">\n<td style=\"width: 16.6667%;height: 45px;text-align: center\"><strong>Compound<\/strong><\/td>\n<td style=\"width: 16.6667%;height: 45px;text-align: center\"><strong>Formula<\/strong><\/td>\n<td style=\"width: 16.6667%;height: 45px;text-align: center\"><strong>Flow to Separate (tonne\/hr)<\/strong><\/td>\n<td style=\"width: 16.6667%;height: 45px;text-align: center\"><strong>Flow to Separate (tonne-mol\/hr)<\/strong><\/td>\n<td style=\"width: 16.6667%;height: 45px;text-align: center\"><strong>[latex]y_{i}[\/latex] (gas mole fraction)<\/strong><\/td>\n<td style=\"width: 16.6667%;height: 45px;text-align: center\"><strong>[latex]x_{i}[\/latex] (liquid mole fraction)<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">EDC<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">[latex]C_{2}H_{4}Cl_{2}[\/latex]<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">47.5008<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.48<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.25<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.94<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">HCl<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">[latex]HCl[\/latex]<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">26.2512<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.72<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.375<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.005<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">VC<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">[latex]C_{2}H_{3}Cl[\/latex]<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">45<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.72<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.375<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.053<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n&nbsp;<\/blockquote>\nSuppose we operate the distillation column at the dew point of the system to separate benzene. For this system, the dew-point temperature is [latex]46.4^{\\circ}C[\/latex], therefore we must use a heat exchanger to bring the mixture from [latex]200^{\\circ}C[\/latex] to [latex]46.4^{\\circ}C[\/latex]\n\n<img class=\"wp-image-245 aligncenter\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange2-300x102.jpg\" alt=\"\" width=\"582\" height=\"197\">\n\n<em>How much energy is removed to cool this stream from from [latex]200^{\\circ}C[\/latex] to [latex]46.4^{\\circ}C[\/latex]?<\/em>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n\n&nbsp;\n<div>The <em>ideal gas<\/em> heat capacities for the mixture described above are listed below:<\/div>\n<div><\/div>\n<div>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 100%\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 20%;text-align: center\"><strong>Compound<\/strong><\/td>\n<td style=\"width: 20%;text-align: center\"><strong>Formula<\/strong><\/td>\n<td style=\"width: 20%;text-align: center\"><strong>Flow to Separate (tonne\/hr)<\/strong><\/td>\n<td style=\"width: 20%;text-align: center\"><strong>Flow to Separate (tonne-mol\/hr)<\/strong><\/td>\n<td style=\"width: 20%;text-align: center\"><strong>Cp (J\/mol-K)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;text-align: center\">EDC<\/td>\n<td style=\"width: 20%;text-align: center\">[latex]C_{2}H_{4}Cl_{2}[\/latex]<\/td>\n<td style=\"width: 20%;text-align: center\">47.5008<\/td>\n<td style=\"width: 20%;text-align: center\">0.48<\/td>\n<td style=\"width: 20%;text-align: center\">29<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;text-align: center\">HCl<\/td>\n<td style=\"width: 20%;text-align: center\">[latex]HCl[\/latex]<\/td>\n<td style=\"width: 20%;text-align: center\">26.2512<\/td>\n<td style=\"width: 20%;text-align: center\">0.72<\/td>\n<td style=\"width: 20%;text-align: center\">29<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;text-align: center\">VC<\/td>\n<td style=\"width: 20%;text-align: center\">[latex]C_{2}H_{3}Cl[\/latex]<\/td>\n<td style=\"width: 20%;text-align: center\">45<\/td>\n<td style=\"width: 20%;text-align: center\">0.72<\/td>\n<td style=\"width: 20%;text-align: center\">29<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div><\/div>\n<ul>\n \t<li>EDC:<\/li>\n<\/ul>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = \\dot{n}\\int^{T_{2}}_{T_{1}} C_{P}(T)dT[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 0.48\\frac{tonne-mol}{h}*1,000,000\\frac{mol}{tonne-mol}*29\\frac{J}{mol-K}*(200^{\\circ}C-46.4^{\\circ})*\\frac{1K}{1^{\\circ}}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 2.14 x 10^{9} \\frac{J}{h}[\/latex]<\/p>\n\n<ul>\n \t<li>HCl and VC:<\/li>\n<\/ul>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = \\dot{n}\\int^{T_{2}}_{T_{1}} C_{P}(T)dT[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 0.72\\frac{tonne-mol}{h}*1,000,000\\frac{mol}{tonne-mol}*29\\frac{J}{mol-K}*(200^{\\circ}C-46.4^{\\circ})*\\frac{1K}{1^{\\circ}}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 3.21 x 10^{9} \\frac{J}{h}[\/latex]<\/p>\nThere is no exchange of energy in the form of work in a heat exchanger. Therefore, the total heat removed [latex]\\dot{Q}=\\Sigma\\Delta\\dot{H}[\/latex] is the sum of the change in enthalpy for each species:\n<p style=\"text-align: center\">[latex]\\dot{Q}=\\Sigma\\Delta\\dot{H}=8.56 x 10^{9} \\frac{J}{h}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise: Energy Balance for a Heat Exchanger<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div class=\"textbox textbox--exercises\">\n<div class=\"textbox__content\">\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<div>\n\nConsider an equimolar binary mixture of n-hexane and n-heptane at a constant pressure of 1 atm flowing at 1 kmol\/h. This mixture is originally at [latex]150 ^{\\circ}C[\/latex] and needs to be cooled to [latex]85^{\\circ}C[\/latex] for a process in order to ensure vapor-liquid equilibrium is satisfied. The process uses a heat exchanger to achieve this cooling. The heat capacities for both compounds can be described by the following expression:\n<p style=\"text-align: center\">[latex]C_{P} = A + BT + CT^{2} + DT^{3}[\/latex]<\/p>\n<p style=\"text-align: left\">where [latex]C_{P}[\/latex] is in J\/mol-K and the constants A, B, C, and D are listed below:<\/p>\n\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 76.7744%;height: 86px\" border=\"0\">\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"width: 20%;height: 15px;text-align: center\"><strong>Compound<\/strong><\/td>\n<td style=\"width: 16.7548%;height: 15px;text-align: center\"><strong>A<\/strong><\/td>\n<td style=\"width: 18.1775%;height: 15px;text-align: center\"><strong>B<\/strong><\/td>\n<td style=\"width: 19.7832%;height: 15px;text-align: center\"><strong>C<\/strong><\/td>\n<td style=\"width: 19.6342%;height: 15px;text-align: center\"><strong>D<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 20%;height: 15px;text-align: center\">n-Hexane<\/td>\n<td style=\"width: 16.7548%;height: 15px;text-align: center\">-4.413<\/td>\n<td style=\"width: 18.1775%;height: 15px;text-align: center\">0.528<\/td>\n<td style=\"width: 19.7832%;height: 15px;text-align: center\">-3.119E-04<\/td>\n<td style=\"width: 19.6342%;height: 15px;text-align: center\">6.494E-8<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 20%;height: 15px;text-align: center\">n-Heptane<\/td>\n<td style=\"width: 16.7548%;height: 15px;text-align: center\">-5.146<\/td>\n<td style=\"width: 18.1775%;height: 15px;text-align: center\">0.6762<\/td>\n<td style=\"width: 19.7832%;height: 15px;text-align: center\">-3.651E-04<\/td>\n<td style=\"width: 19.6342%;height: 15px;text-align: center\">7.658E-08<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nHow much heat needs to be removed by the heat exchanger to reach the required temperature for the process?\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Solution<\/h3>\n<strong>Step 1: <\/strong>Calculate the change in enthalpy for each compound using the heat capacities taking 1 kmol\/hr as the molar flow.\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = \\dot{n}\\int^{T_{2}}_{T_{1}} C_{P}(T)dT[\/latex]<\/p>\nwhere [latex]T_{1} = 150 ^{\\circ}C=423K[\/latex] and [latex]T_{2} = 85 ^{\\circ}C=358 K[\/latex]\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = \\dot{n}\\int^{T_{2}}_{T_{1}} (A + BT + CT^{2} + DT^{3})dT[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = \\dot{n}*(AT + \\frac{B}{2}T^{2} + \\frac{C}{3}T^{3} + \\frac{D}{4}T^{4})\\bigg|^{T_{2}}_{T_{1}}[\/latex]<\/p>\nFor n-hexane:\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 0.5\\frac{kmol}{h}*(-4.413*T + \\frac{0.528}{2}T^{2} + \\frac{-3.119E-04}{3}T^{3} + \\frac{6.494E-8}{4}T^{4})\\bigg|^{358 K}_{423 K}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 0.5\\frac{kmol}{h}*1000 \\frac{mol}{kmol}*1859\\frac{J}{mol}*\\frac{kJ}{1000 J}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 929.5 \\frac{kJ}{h}[\/latex]<\/p>\nSimilarly for n-heptane:\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 0.5\\frac{kmol}{h}*1000 \\frac{mol}{kmol}*2424\\frac{J}{mol}*\\frac{kJ}{1000 J}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 1212 \\frac{kJ}{h}[\/latex]<\/p>\n<strong>Step 2:<\/strong> Sum up the enthalpy changes for the components.\n\n\\begin{align*}\n\\dot{Q} &amp;= \\Sigma\\Delta\\dot{H}\\\\\n\\dot{Q} &amp; = (929.5+1212)\\frac{kJ}{h}\\\\\n\\dot{Q} &amp;= 2142\\frac{kJ}{h}\n\\end{align*}\n\n<\/div>\n&nbsp;\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"Choosing-Utilities-for-System\">Choosing Utilities for System<\/h3>\n&nbsp;\n<div>For the VCM distillation system described above, we have the following utility options for cooling the stream from [latex]200^{\\circ}C[\/latex] to [latex]46.4^{\\circ}C[\/latex]:<\/div>\n<div><\/div>\n<div>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 95.1758%;height: 141px\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 16.6667%;text-align: center\"><strong>Option<\/strong><\/td>\n<td style=\"width: 16.6667%;text-align: center\"><strong>Utility<\/strong><\/td>\n<td style=\"width: 16.6667%;text-align: center\"><strong>Inlet T ([latex]^{\\circ}C[\/latex])<\/strong><\/td>\n<td style=\"width: 16.6667%;text-align: center\"><strong>Outlet T ([latex]^{\\circ}C[\/latex])<\/strong><\/td>\n<td style=\"width: 16.6667%;text-align: center\"><strong>P<\/strong><\/td>\n<td style=\"width: 16.6667%;text-align: center\"><strong>Cost (dollars\/GJ)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;text-align: center\">A<\/td>\n<td style=\"width: 16.6667%;text-align: center\">Cooling Water<\/td>\n<td style=\"width: 16.6667%;text-align: center\">20<\/td>\n<td style=\"width: 16.6667%;text-align: center\">25<\/td>\n<td style=\"width: 16.6667%;text-align: center\">N\/A<\/td>\n<td style=\"width: 16.6667%;text-align: center\">0.378<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;text-align: center\">B<\/td>\n<td style=\"width: 16.6667%;text-align: center\">Refrigerated Water<\/td>\n<td style=\"width: 16.6667%;text-align: center\">5<\/td>\n<td style=\"width: 16.6667%;text-align: center\">15<\/td>\n<td style=\"width: 16.6667%;text-align: center\">N\/A<\/td>\n<td style=\"width: 16.6667%;text-align: center\">4.77<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;text-align: center\">C<\/td>\n<td style=\"width: 16.6667%;text-align: center\">Low T Refrigerant<\/td>\n<td style=\"width: 16.6667%;text-align: center\">-20<\/td>\n<td style=\"width: 16.6667%;text-align: center\">-5<\/td>\n<td style=\"width: 16.6667%;text-align: center\">N\/A<\/td>\n<td style=\"width: 16.6667%;text-align: center\">8.49<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div><\/div>\n<div>A rule of thumb is that we want a stream we are transferring energy to have a [latex]10^{\\circ}C[\/latex] temperature difference with the process stream. This will ensure our heat transfer occurs at an effective rate and that our heat exchanger does not need to be too big.<\/div>\n<div>Since our outlet temperature is only [latex]46.4^{\\circ}C[\/latex], all options have an inlet temperature at least [latex]10^{\\circ}C[\/latex] below this desired outlet temperature. We could pick any but will select the cheapest option, which is cooling water.<\/div>\n<div><\/div>\n<div>\n\n<em>How much will cooling this stream cost?<\/em>\n\n\\begin{align*}\nCost \\Big(\\frac{dollars}{h}\\Big) &amp;= \\dot{Q}*cost\\Big(\\frac{dollars}{GJ}\\Big)\\\\\n&amp; = 8.56\\frac{GJ}{h}*0.378\\frac{dollars}{GJ}\\\\\n&amp; = 3.24\\frac{dollars}{h}\n\\end{align*}\n\n&nbsp;\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h2 id=\"Process-Paths\">Process Paths<\/h2>\nRecall that [latex]\\hat{U}[\/latex] and [latex]\\hat{H}[\/latex] are state properties. These properties depend on the state and not the path to that state.\n\nIt is easier to calculate the enthalpy change by changing one variable at a time through a hypothetical process path.\n<blockquote>1 - Calculate change in enthalpy by only changing the temperature at the same state\n\n2 - Calculate change in enthalpy by only changing the pressure at the same state (in the course, we usually neglect the effect of pressure on enthalpy change)\n\n3 - Calculate change in enthalpy by only changing the phase<\/blockquote>\n<p style=\"text-align: left\">To obtain the desired change in enthalpy, add each enthalpy change where only 1 state property changes:<\/p>\n\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 40.8819%;height: 63px\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 100%;text-align: center\"><span style=\"font-size: 16px\">[latex]\\Delta\\hat{H} = \\Sigma_{i}\\Delta\\hat{H}_{i}[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Example: EDC Process Path<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\n<span style=\"text-align: initial;font-size: 1em\">Consider the enthalpy change of [latex]EDC[\/latex] transforming from vapour at [latex]200^{\\circ}C[\/latex] and 3 atm to liquid at [latex]25^{\\circ}C[\/latex] and 1 atm:<\/span>\n\n&nbsp;\n<div>\n\n<img class=\"wp-image-246 aligncenter\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/8\/2026\/03\/ProcessPaths-300x100.jpg\" alt=\"\" width=\"459\" height=\"153\">\n\nIn this process, there are 3 variables changing:\n<ul>\n \t<li>pressure<\/li>\n \t<li>temperature<\/li>\n \t<li>phase (vapour to liquid)<\/li>\n<\/ul>\nThe following steps are taken in the process path and added together to get the overall change:\n<blockquote>[latex]\\Delta\\hat{H}_{1}[\/latex] is the enthalpy change from going from 3 atm to 1 atm at a constant temperature of 200\u00b0C in the vapour phase ([latex]v[\/latex])\n\n[latex]\\Delta\\hat{H}_{2}[\/latex] is the enthalpy change from going from 200\u00b0C to 84\u00b0C at a constant pressure of 1 atm in the vapour phase ([latex]v[\/latex])\n\n[latex]\\Delta\\hat{H}_{3}[\/latex] is the enthalpy change for a phase change, going from the vapour phase ([latex]v[\/latex]) to the liquid phase ([latex]l[\/latex]) at a constant pressure and temperature of 1 atm and 84\u00b0C\n\n[latex]\\Delta\\hat{H}_{4}[\/latex] is the enthalpy change from going from 84\u00b0C to 25\u00b0C at a constant pressure of 1 atm in the liquid phase ([latex]l[\/latex])<\/blockquote>\nWhen we add these changes together, we can get the overall change in enthalpy:\n<p style=\"text-align: center\">[latex]\\Delta\\hat{H} = \\Delta\\hat{H}_{1} + \\Delta\\hat{H}_{2} + \\Delta\\hat{H}_{3} + \\Delta\\hat{H}_{4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise: Process Path<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\n<span style=\"color: #333333;font-size: 1em\">What process paths can be taken to calculate the change in enthalpy for acetone going from [latex]25^{\\circ}C[\/latex] in the liquid phase to [latex]60^{\\circ}C[\/latex] in the vapour phase?<\/span>\n\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Solution<\/h3>\n<strong>Step 1:<\/strong> Bring the acetone to the<span style=\"font-size: 1em\">\u00a0<\/span><span style=\"font-size: 1em\">boiling point temperature from [latex]25^{\\circ}C[\/latex] without changing the phase using the [latex]C_{P}[\/latex] in the liquid phase.<\/span>\n\n<strong style=\"font-size: 1em\">Step 2:<\/strong><span style=\"font-size: 1em\">\u00a0<\/span><span style=\"font-size: 1em\">Use the latent heat (or heat of<\/span><span style=\"font-size: 1em\">\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">vaporization) to calculate the enthalpy of changing the phase from liquid to vapour.<\/span>\n\n<strong>Step 3:<\/strong> Bring the acetone to [latex]60^{\\circ}C[\/latex] from the boiling point temperature using the [latex]C_{P}[\/latex] in the vapour phase.\n\n<\/div>\n&nbsp;\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section, you should be able to:<\/p>\n<p id=\"Evaluate:\"><strong>Evaluate <\/strong>the cost of utilities in processes<\/p>\n<p id=\"Characterize:\"><strong>Characterize<\/strong>\u00a0energy changes in a system due to changes in temperature<\/p>\n<p id=\"Analyze:\"><strong>Analyze<\/strong>\u00a0energy balances on processes involving phase changes<\/p>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h2 id=\"Energy-Balance-Equipment\">Energy Balance Equipment<\/h2>\n<ul>\n<li>Almost all energy exchange equipment is either a heat exchanger or jacket<\/li>\n<li>Heat exchangers and jackets often work by transferring heat with a stream of fluid<\/li>\n<li>Electrical heating elements are also available, but their high cost per unit energy may limit their use in industry<\/li>\n<\/ul>\n<blockquote><p>The figure below is a shell-and-tube heat exchange:<\/p>\n<p><img decoding=\"async\" class=\"wp-image-243 aligncenter\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/8\/2020\/06\/DP-HX-300x188.png\" alt=\"\" width=\"449\" height=\"281\" srcset=\"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2020\/06\/DP-HX-300x188.png 300w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2020\/06\/DP-HX-65x41.png 65w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2020\/06\/DP-HX-225x141.png 225w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2020\/06\/DP-HX-350x219.png 350w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2020\/06\/DP-HX.png 640w\" sizes=\"(max-width: 449px) 100vw, 449px\" \/><\/p>\n<p style=\"text-align: center\">Image from <a title=\"User:Turbojet\" href=\"Turbojet\">Turbojet<\/a> \/<a title=\"Creative Commons Attribution-Share Alike 4.0\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\">CC BY-SA 4.0<\/a><\/p>\n<\/blockquote>\n<p><strong>Utilities<\/strong>: Utilities are services at a site such as water, electricity, and gas.<\/p>\n<p>&nbsp;<\/p>\n<div>Choosing the correct utility can make a big difference in the cost of heating or cooling. For example, evaporating toluene at [latex]190^{\\circ}C[\/latex] with a heat duty of 3,000,000 kJ\/hr can be heated using the following utility options:<\/div>\n<div><\/div>\n<div>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 97.2899%;height: 132px\">\n<tbody>\n<tr>\n<td style=\"width: 14.2857%;text-align: center\"><strong>Utility<\/strong><\/td>\n<td style=\"width: 14.2857%;text-align: center\"><strong>Inlet [latex]T (^{\\circ}C)[\/latex]\u00a0<\/strong><\/td>\n<td style=\"width: 14.2857%;text-align: center\"><strong>Outlet [latex]T (^{\\circ}C)[\/latex]<\/strong><\/td>\n<td style=\"width: 14.2857%;text-align: center\"><strong>P ([latex]psia[\/latex])<\/strong><\/td>\n<td style=\"width: 14.2857%;text-align: center\"><strong>Cost (dollars\/GJ)<\/strong><\/td>\n<td style=\"width: 14.2857%;text-align: center\"><strong>Cost (dollars\/hr)<\/strong><\/td>\n<td style=\"width: 14.2857%;text-align: center\"><strong>Cost dollars\/yr @ 8,000 hrs<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.2857%;text-align: center\">High-pressure steam<\/td>\n<td style=\"width: 14.2857%;text-align: center\">250<\/td>\n<td style=\"width: 14.2857%;text-align: center\">249<\/td>\n<td style=\"width: 14.2857%;text-align: center\">575<\/td>\n<td style=\"width: 14.2857%;text-align: center\">5.66<\/td>\n<td style=\"width: 14.2857%;text-align: center\">16.98<\/td>\n<td style=\"width: 14.2857%;text-align: center\">135,840<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 14.2857%;text-align: center\">Electricity<\/td>\n<td style=\"width: 14.2857%;text-align: center\">N\/A<\/td>\n<td style=\"width: 14.2857%;text-align: center\">N\/A<\/td>\n<td style=\"width: 14.2857%;text-align: center\">N\/A<\/td>\n<td style=\"width: 14.2857%;text-align: center\">18.72<\/td>\n<td style=\"width: 14.2857%;text-align: center\">56.16<\/td>\n<td style=\"width: 14.2857%;text-align: center\">449,280<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p><em>The high-pressure steam is a much cheaper option here<\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h2 id=\"Phase-Changes\">Phase Changes<\/h2>\n<p>Phase changes take place when a compound or mixture of compounds undergoes a change in their state of matter. The enthalpy changes for phase changes occur at a constant temperature and pressure and are determined experimentally.<\/p>\n<table class=\"grid aligncenter\">\n<thead>\n<tr>\n<th>Parameter<\/th>\n<th>Phase Transition<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Liquid \u2192 Gas<\/td>\n<td>Heat of Vapourization ([latex]\\Delta\\hat{H}_{vap}[\/latex])<\/td>\n<\/tr>\n<tr>\n<td>Solid \u2192 Liquid<\/td>\n<td>Heat of Fusion ([latex]\\Delta\\hat{H}_{fus}[\/latex])<\/td>\n<\/tr>\n<tr>\n<td>Gas \u2192 Liquid<\/td>\n<td>(-) Heat of Vapourization ([latex]-\\Delta\\hat{H}_{vap}[\/latex])<\/td>\n<\/tr>\n<tr>\n<td>Liquid \u2192 Solid<\/td>\n<td>(-) Heat of Fusion ([latex]-\\Delta\\hat{H}_{fus}[\/latex])<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><em>Note that heat of vapourization is also referred to as latent heat<\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h2 id=\"Heat-Capacities\">Heat Capacities<\/h2>\n<p>Heat capacities are physical properties that describe how much heat is needed to increase the temperature of a compound for a unit temperature per unit mass of a compound.<\/p>\n<p><strong>For a closed system<\/strong> where the only changing system variable is temperature:<\/p>\n<p style=\"text-align: center\">[latex]Q = \\Delta\\hat{U}[\/latex]<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">The heat capacity at constant volume can be described as follows, where [latex]C_{V}[\/latex] is a function of temperature [latex]T[\/latex]:<\/span><\/p>\n<p style=\"text-align: center\">[latex]C_{V}(T) = \\bigg(\\frac{\\delta\\hat{U}}{\\delta T}\\bigg)_{V}[\/latex]<\/p>\n<p>Notation: [latex]\\big(\\;\\;\\big)_{V}[\/latex] means that the volume is kept constant during the process<\/p>\n<p style=\"text-align: left\">This expression can be rearranged and integrated to obtain the following:<\/p>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 63.9196%;height: 81px\">\n<tbody>\n<tr>\n<td style=\"width: 100%;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<\/tbody>\n<\/table>\n<p>If the [latex]C_{V}[\/latex] is constant:<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\hat{U} = C_{V}*(T_{2}-T_{1})[\/latex]<\/p>\n<p style=\"text-align: left\"><strong>For an open system<\/strong>, the heat capacity is defined under constant pressure conditions:<\/p>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 58.1833%;height: 97px\">\n<tbody>\n<tr>\n<td style=\"width: 100%\">\n<p style=\"text-align: center\"><span style=\"font-size: 16px\">[latex]C_{P}(T) = \\bigg(\\frac{\\delta\\hat{H}}{\\delta T}\\bigg)_{P}[\/latex]<\/span><\/p>\n<p style=\"text-align: center\"><span style=\"font-size: 16px\">[latex]\\Delta\\hat{H} = \\int^{T_{2}}_{T_{1}}C_{P}dT[\/latex]<\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"text-align: initial;font-size: 1em\">If the [latex]C_{P}[\/latex] is constant:<\/span><\/p>\n<p style=\"text-align: center\"><span style=\"text-align: initial;font-size: 1em\">[latex]\\Delta\\hat{H} = C_{P}*(T_{2}-T_{1})[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"prompt input_prompt\"><\/div>\n<div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example: Phase Changes and Heat Capacity in Energy Balances<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div class=\"textbox__content\" style=\"text-align: center\">\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<div style=\"text-align: left\">Consider a system where a mixture of dichloroethane ([latex]EDC[\/latex]), hydrogen chloride ([latex]H\\!Cl[\/latex]), and vinyl chloride ([latex]V\\!C[\/latex]) is entering a distillation column for separation. Distillation columns work based on vapor-liquid equilibrium, and therefore the column will generally operate at a temperature between the boiling points of all compounds.<\/div>\n<div>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 91.6844%;height: 117px\">\n<tbody>\n<tr style=\"height: 28px\">\n<td style=\"width: 25%;height: 28px;text-align: center\"><strong>Compound<\/strong><\/td>\n<td style=\"width: 25%;height: 28px;text-align: center\"><strong>Formula<\/strong><\/td>\n<td style=\"width: 25%;height: 28px;text-align: center\"><strong>Boiling Point at 1 atm ([latex]^{\\circ}C[\/latex])<\/strong><\/td>\n<td style=\"width: 25%;height: 28px;text-align: center\"><strong>Heat of Vapourization at 1 atm ([latex]kJ\/mol[\/latex])<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 25%;height: 15px;text-align: center\">EDC<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">[latex]C_{2}H_{4}Cl_{2}[\/latex]<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">84<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">35<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 25%;height: 15px;text-align: center\">HCl<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">[latex]HCl[\/latex]<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">-85<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">16<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 25%;height: 15px;text-align: center\">VC<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">[latex]C_{2}H_{3}Cl[\/latex]<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">-13<\/td>\n<td style=\"width: 25%;height: 15px;text-align: center\">21<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>The mixture exits a reactor at [latex]200^{\\circ}C[\/latex] and has the following properties:<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 86.1307%;height: 137px\">\n<tbody>\n<tr class=\"border\" style=\"height: 30px\">\n<td style=\"width: 20%;height: 30px;text-align: center\"><strong>Compound<\/strong><\/td>\n<td style=\"width: 20%;height: 30px;text-align: center\"><strong>Formula<\/strong><\/td>\n<td style=\"width: 20%;height: 30px;text-align: center\"><strong>Flow to Separate (tonne\/hr)<\/strong><\/td>\n<td style=\"width: 20%;height: 30px;text-align: center\"><strong>Flow to Separate (tonne-mol\/hr)<\/strong><\/td>\n<td style=\"width: 20%;height: 30px;text-align: center\"><strong>[latex]y_{i}[\/latex] (gas mole fraction)<\/strong><\/td>\n<\/tr>\n<tr class=\"border\" style=\"height: 15px\">\n<td style=\"width: 20%;height: 15px;text-align: center\">EDC<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">[latex]C_{2}H_{4}Cl_{2}[\/latex]<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">47.5008<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">0.48<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">0.25<\/td>\n<\/tr>\n<tr class=\"border\" style=\"height: 15px\">\n<td style=\"width: 20%;height: 15px;text-align: center\">HCl<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">[latex]HCl[\/latex]<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">26.2512<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">0.72<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">0.375<\/td>\n<\/tr>\n<tr class=\"border\" style=\"height: 15px\">\n<td style=\"width: 20%;height: 15px;text-align: center\">VC<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">[latex]C_{2}H_{3}Cl[\/latex]<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">45<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">0.72<\/td>\n<td style=\"width: 20%;height: 15px;text-align: center\">0.375<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox__content\">\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>&nbsp;<\/p>\n<p><img decoding=\"async\" class=\"wp-image-244 aligncenter\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange1-300x84.jpg\" alt=\"\" width=\"600\" height=\"168\" srcset=\"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange1-300x84.jpg 300w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange1-768x216.jpg 768w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange1-65x18.jpg 65w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange1-225x63.jpg 225w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange1-350x99.jpg 350w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange1.jpg 806w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/p>\n<blockquote><p>Recall <strong>Raoult&#8217;s Law<\/strong>:<\/p>\n<p style=\"text-align: center\">[latex]x_{i} = \\frac{p_{i}*}{y_{i}\u00d7P }[\/latex]<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse;width: 100%;height: 90px\">\n<tbody>\n<tr style=\"height: 45px\">\n<td style=\"width: 16.6667%;height: 45px;text-align: center\"><strong>Compound<\/strong><\/td>\n<td style=\"width: 16.6667%;height: 45px;text-align: center\"><strong>Formula<\/strong><\/td>\n<td style=\"width: 16.6667%;height: 45px;text-align: center\"><strong>Flow to Separate (tonne\/hr)<\/strong><\/td>\n<td style=\"width: 16.6667%;height: 45px;text-align: center\"><strong>Flow to Separate (tonne-mol\/hr)<\/strong><\/td>\n<td style=\"width: 16.6667%;height: 45px;text-align: center\"><strong>[latex]y_{i}[\/latex] (gas mole fraction)<\/strong><\/td>\n<td style=\"width: 16.6667%;height: 45px;text-align: center\"><strong>[latex]x_{i}[\/latex] (liquid mole fraction)<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">EDC<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">[latex]C_{2}H_{4}Cl_{2}[\/latex]<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">47.5008<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.48<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.25<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.94<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">HCl<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">[latex]HCl[\/latex]<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">26.2512<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.72<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.375<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.005<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">VC<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">[latex]C_{2}H_{3}Cl[\/latex]<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">45<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.72<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.375<\/td>\n<td style=\"width: 16.6667%;height: 15px;text-align: center\">0.053<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p><\/blockquote>\n<p>Suppose we operate the distillation column at the dew point of the system to separate benzene. For this system, the dew-point temperature is [latex]46.4^{\\circ}C[\/latex], therefore we must use a heat exchanger to bring the mixture from [latex]200^{\\circ}C[\/latex] to [latex]46.4^{\\circ}C[\/latex]<\/p>\n<p><img decoding=\"async\" class=\"wp-image-245 aligncenter\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange2-300x102.jpg\" alt=\"\" width=\"582\" height=\"197\" srcset=\"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange2-300x102.jpg 300w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange2-65x22.jpg 65w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange2-225x77.jpg 225w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange2-350x119.jpg 350w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/PhaseChange2.jpg 825w\" sizes=\"(max-width: 582px) 100vw, 582px\" \/><\/p>\n<p><em>How much energy is removed to cool this stream from from [latex]200^{\\circ}C[\/latex] to [latex]46.4^{\\circ}C[\/latex]?<\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>&nbsp;<\/p>\n<div>The <em>ideal gas<\/em> heat capacities for the mixture described above are listed below:<\/div>\n<div><\/div>\n<div>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr>\n<td style=\"width: 20%;text-align: center\"><strong>Compound<\/strong><\/td>\n<td style=\"width: 20%;text-align: center\"><strong>Formula<\/strong><\/td>\n<td style=\"width: 20%;text-align: center\"><strong>Flow to Separate (tonne\/hr)<\/strong><\/td>\n<td style=\"width: 20%;text-align: center\"><strong>Flow to Separate (tonne-mol\/hr)<\/strong><\/td>\n<td style=\"width: 20%;text-align: center\"><strong>Cp (J\/mol-K)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;text-align: center\">EDC<\/td>\n<td style=\"width: 20%;text-align: center\">[latex]C_{2}H_{4}Cl_{2}[\/latex]<\/td>\n<td style=\"width: 20%;text-align: center\">47.5008<\/td>\n<td style=\"width: 20%;text-align: center\">0.48<\/td>\n<td style=\"width: 20%;text-align: center\">29<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;text-align: center\">HCl<\/td>\n<td style=\"width: 20%;text-align: center\">[latex]HCl[\/latex]<\/td>\n<td style=\"width: 20%;text-align: center\">26.2512<\/td>\n<td style=\"width: 20%;text-align: center\">0.72<\/td>\n<td style=\"width: 20%;text-align: center\">29<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;text-align: center\">VC<\/td>\n<td style=\"width: 20%;text-align: center\">[latex]C_{2}H_{3}Cl[\/latex]<\/td>\n<td style=\"width: 20%;text-align: center\">45<\/td>\n<td style=\"width: 20%;text-align: center\">0.72<\/td>\n<td style=\"width: 20%;text-align: center\">29<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div><\/div>\n<ul>\n<li>EDC:<\/li>\n<\/ul>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = \\dot{n}\\int^{T_{2}}_{T_{1}} C_{P}(T)dT[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 0.48\\frac{tonne-mol}{h}*1,000,000\\frac{mol}{tonne-mol}*29\\frac{J}{mol-K}*(200^{\\circ}C-46.4^{\\circ})*\\frac{1K}{1^{\\circ}}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 2.14 x 10^{9} \\frac{J}{h}[\/latex]<\/p>\n<ul>\n<li>HCl and VC:<\/li>\n<\/ul>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = \\dot{n}\\int^{T_{2}}_{T_{1}} C_{P}(T)dT[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 0.72\\frac{tonne-mol}{h}*1,000,000\\frac{mol}{tonne-mol}*29\\frac{J}{mol-K}*(200^{\\circ}C-46.4^{\\circ})*\\frac{1K}{1^{\\circ}}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 3.21 x 10^{9} \\frac{J}{h}[\/latex]<\/p>\n<p>There is no exchange of energy in the form of work in a heat exchanger. Therefore, the total heat removed [latex]\\dot{Q}=\\Sigma\\Delta\\dot{H}[\/latex] is the sum of the change in enthalpy for each species:<\/p>\n<p style=\"text-align: center\">[latex]\\dot{Q}=\\Sigma\\Delta\\dot{H}=8.56 x 10^{9} \\frac{J}{h}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise: Energy Balance for a Heat Exchanger<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div class=\"textbox textbox--exercises\">\n<div class=\"textbox__content\">\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<div>\n<p>Consider an equimolar binary mixture of n-hexane and n-heptane at a constant pressure of 1 atm flowing at 1 kmol\/h. This mixture is originally at [latex]150 ^{\\circ}C[\/latex] and needs to be cooled to [latex]85^{\\circ}C[\/latex] for a process in order to ensure vapor-liquid equilibrium is satisfied. The process uses a heat exchanger to achieve this cooling. The heat capacities for both compounds can be described by the following expression:<\/p>\n<p style=\"text-align: center\">[latex]C_{P} = A + BT + CT^{2} + DT^{3}[\/latex]<\/p>\n<p style=\"text-align: left\">where [latex]C_{P}[\/latex] is in J\/mol-K and the constants A, B, C, and D are listed below:<\/p>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 76.7744%;height: 86px\">\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"width: 20%;height: 15px;text-align: center\"><strong>Compound<\/strong><\/td>\n<td style=\"width: 16.7548%;height: 15px;text-align: center\"><strong>A<\/strong><\/td>\n<td style=\"width: 18.1775%;height: 15px;text-align: center\"><strong>B<\/strong><\/td>\n<td style=\"width: 19.7832%;height: 15px;text-align: center\"><strong>C<\/strong><\/td>\n<td style=\"width: 19.6342%;height: 15px;text-align: center\"><strong>D<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 20%;height: 15px;text-align: center\">n-Hexane<\/td>\n<td style=\"width: 16.7548%;height: 15px;text-align: center\">-4.413<\/td>\n<td style=\"width: 18.1775%;height: 15px;text-align: center\">0.528<\/td>\n<td style=\"width: 19.7832%;height: 15px;text-align: center\">-3.119E-04<\/td>\n<td style=\"width: 19.6342%;height: 15px;text-align: center\">6.494E-8<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"width: 20%;height: 15px;text-align: center\">n-Heptane<\/td>\n<td style=\"width: 16.7548%;height: 15px;text-align: center\">-5.146<\/td>\n<td style=\"width: 18.1775%;height: 15px;text-align: center\">0.6762<\/td>\n<td style=\"width: 19.7832%;height: 15px;text-align: center\">-3.651E-04<\/td>\n<td style=\"width: 19.6342%;height: 15px;text-align: center\">7.658E-08<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>How much heat needs to be removed by the heat exchanger to reach the required temperature for the process?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Solution<\/h3>\n<p><strong>Step 1: <\/strong>Calculate the change in enthalpy for each compound using the heat capacities taking 1 kmol\/hr as the molar flow.<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = \\dot{n}\\int^{T_{2}}_{T_{1}} C_{P}(T)dT[\/latex]<\/p>\n<p>where [latex]T_{1} = 150 ^{\\circ}C=423K[\/latex] and [latex]T_{2} = 85 ^{\\circ}C=358 K[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = \\dot{n}\\int^{T_{2}}_{T_{1}} (A + BT + CT^{2} + DT^{3})dT[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = \\dot{n}*(AT + \\frac{B}{2}T^{2} + \\frac{C}{3}T^{3} + \\frac{D}{4}T^{4})\\bigg|^{T_{2}}_{T_{1}}[\/latex]<\/p>\n<p>For n-hexane:<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 0.5\\frac{kmol}{h}*(-4.413*T + \\frac{0.528}{2}T^{2} + \\frac{-3.119E-04}{3}T^{3} + \\frac{6.494E-8}{4}T^{4})\\bigg|^{358 K}_{423 K}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 0.5\\frac{kmol}{h}*1000 \\frac{mol}{kmol}*1859\\frac{J}{mol}*\\frac{kJ}{1000 J}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 929.5 \\frac{kJ}{h}[\/latex]<\/p>\n<p>Similarly for n-heptane:<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 0.5\\frac{kmol}{h}*1000 \\frac{mol}{kmol}*2424\\frac{J}{mol}*\\frac{kJ}{1000 J}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\dot{H} = 1212 \\frac{kJ}{h}[\/latex]<\/p>\n<p><strong>Step 2:<\/strong> Sum up the enthalpy changes for the components.<\/p>\n<p>\\begin{align*}<br \/>\n\\dot{Q} &amp;= \\Sigma\\Delta\\dot{H}\\\\<br \/>\n\\dot{Q} &amp; = (929.5+1212)\\frac{kJ}{h}\\\\<br \/>\n\\dot{Q} &amp;= 2142\\frac{kJ}{h}<br \/>\n\\end{align*}<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"Choosing-Utilities-for-System\">Choosing Utilities for System<\/h3>\n<p>&nbsp;<\/p>\n<div>For the VCM distillation system described above, we have the following utility options for cooling the stream from [latex]200^{\\circ}C[\/latex] to [latex]46.4^{\\circ}C[\/latex]:<\/div>\n<div><\/div>\n<div>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 95.1758%;height: 141px\">\n<tbody>\n<tr>\n<td style=\"width: 16.6667%;text-align: center\"><strong>Option<\/strong><\/td>\n<td style=\"width: 16.6667%;text-align: center\"><strong>Utility<\/strong><\/td>\n<td style=\"width: 16.6667%;text-align: center\"><strong>Inlet T ([latex]^{\\circ}C[\/latex])<\/strong><\/td>\n<td style=\"width: 16.6667%;text-align: center\"><strong>Outlet T ([latex]^{\\circ}C[\/latex])<\/strong><\/td>\n<td style=\"width: 16.6667%;text-align: center\"><strong>P<\/strong><\/td>\n<td style=\"width: 16.6667%;text-align: center\"><strong>Cost (dollars\/GJ)<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;text-align: center\">A<\/td>\n<td style=\"width: 16.6667%;text-align: center\">Cooling Water<\/td>\n<td style=\"width: 16.6667%;text-align: center\">20<\/td>\n<td style=\"width: 16.6667%;text-align: center\">25<\/td>\n<td style=\"width: 16.6667%;text-align: center\">N\/A<\/td>\n<td style=\"width: 16.6667%;text-align: center\">0.378<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;text-align: center\">B<\/td>\n<td style=\"width: 16.6667%;text-align: center\">Refrigerated Water<\/td>\n<td style=\"width: 16.6667%;text-align: center\">5<\/td>\n<td style=\"width: 16.6667%;text-align: center\">15<\/td>\n<td style=\"width: 16.6667%;text-align: center\">N\/A<\/td>\n<td style=\"width: 16.6667%;text-align: center\">4.77<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;text-align: center\">C<\/td>\n<td style=\"width: 16.6667%;text-align: center\">Low T Refrigerant<\/td>\n<td style=\"width: 16.6667%;text-align: center\">-20<\/td>\n<td style=\"width: 16.6667%;text-align: center\">-5<\/td>\n<td style=\"width: 16.6667%;text-align: center\">N\/A<\/td>\n<td style=\"width: 16.6667%;text-align: center\">8.49<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div><\/div>\n<div>A rule of thumb is that we want a stream we are transferring energy to have a [latex]10^{\\circ}C[\/latex] temperature difference with the process stream. This will ensure our heat transfer occurs at an effective rate and that our heat exchanger does not need to be too big.<\/div>\n<div>Since our outlet temperature is only [latex]46.4^{\\circ}C[\/latex], all options have an inlet temperature at least [latex]10^{\\circ}C[\/latex] below this desired outlet temperature. We could pick any but will select the cheapest option, which is cooling water.<\/div>\n<div><\/div>\n<div>\n<p><em>How much will cooling this stream cost?<\/em><\/p>\n<p>\\begin{align*}<br \/>\nCost \\Big(\\frac{dollars}{h}\\Big) &amp;= \\dot{Q}*cost\\Big(\\frac{dollars}{GJ}\\Big)\\\\<br \/>\n&amp; = 8.56\\frac{GJ}{h}*0.378\\frac{dollars}{GJ}\\\\<br \/>\n&amp; = 3.24\\frac{dollars}{h}<br \/>\n\\end{align*}<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h2 id=\"Process-Paths\">Process Paths<\/h2>\n<p>Recall that [latex]\\hat{U}[\/latex] and [latex]\\hat{H}[\/latex] are state properties. These properties depend on the state and not the path to that state.<\/p>\n<p>It is easier to calculate the enthalpy change by changing one variable at a time through a hypothetical process path.<\/p>\n<blockquote><p>1 &#8211; Calculate change in enthalpy by only changing the temperature at the same state<\/p>\n<p>2 &#8211; Calculate change in enthalpy by only changing the pressure at the same state (in the course, we usually neglect the effect of pressure on enthalpy change)<\/p>\n<p>3 &#8211; Calculate change in enthalpy by only changing the phase<\/p><\/blockquote>\n<p style=\"text-align: left\">To obtain the desired change in enthalpy, add each enthalpy change where only 1 state property changes:<\/p>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 40.8819%;height: 63px\">\n<tbody>\n<tr>\n<td style=\"width: 100%;text-align: center\"><span style=\"font-size: 16px\">[latex]\\Delta\\hat{H} = \\Sigma_{i}\\Delta\\hat{H}_{i}[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example: EDC Process Path<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><span style=\"text-align: initial;font-size: 1em\">Consider the enthalpy change of [latex]EDC[\/latex] transforming from vapour at [latex]200^{\\circ}C[\/latex] and 3 atm to liquid at [latex]25^{\\circ}C[\/latex] and 1 atm:<\/span><\/p>\n<p>&nbsp;<\/p>\n<div>\n<p><img decoding=\"async\" class=\"wp-image-246 aligncenter\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/8\/2026\/03\/ProcessPaths-300x100.jpg\" alt=\"\" width=\"459\" height=\"153\" srcset=\"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/ProcessPaths-300x100.jpg 300w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/ProcessPaths-65x22.jpg 65w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/ProcessPaths-225x75.jpg 225w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/ProcessPaths-350x116.jpg 350w, https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-content\/uploads\/sites\/8\/2026\/03\/ProcessPaths.jpg 515w\" sizes=\"(max-width: 459px) 100vw, 459px\" \/><\/p>\n<p>In this process, there are 3 variables changing:<\/p>\n<ul>\n<li>pressure<\/li>\n<li>temperature<\/li>\n<li>phase (vapour to liquid)<\/li>\n<\/ul>\n<p>The following steps are taken in the process path and added together to get the overall change:<\/p>\n<blockquote><p>[latex]\\Delta\\hat{H}_{1}[\/latex] is the enthalpy change from going from 3 atm to 1 atm at a constant temperature of 200\u00b0C in the vapour phase ([latex]v[\/latex])<\/p>\n<p>[latex]\\Delta\\hat{H}_{2}[\/latex] is the enthalpy change from going from 200\u00b0C to 84\u00b0C at a constant pressure of 1 atm in the vapour phase ([latex]v[\/latex])<\/p>\n<p>[latex]\\Delta\\hat{H}_{3}[\/latex] is the enthalpy change for a phase change, going from the vapour phase ([latex]v[\/latex]) to the liquid phase ([latex]l[\/latex]) at a constant pressure and temperature of 1 atm and 84\u00b0C<\/p>\n<p>[latex]\\Delta\\hat{H}_{4}[\/latex] is the enthalpy change from going from 84\u00b0C to 25\u00b0C at a constant pressure of 1 atm in the liquid phase ([latex]l[\/latex])<\/p><\/blockquote>\n<p>When we add these changes together, we can get the overall change in enthalpy:<\/p>\n<p style=\"text-align: center\">[latex]\\Delta\\hat{H} = \\Delta\\hat{H}_{1} + \\Delta\\hat{H}_{2} + \\Delta\\hat{H}_{3} + \\Delta\\hat{H}_{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise: Process Path<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><span style=\"color: #333333;font-size: 1em\">What process paths can be taken to calculate the change in enthalpy for acetone going from [latex]25^{\\circ}C[\/latex] in the liquid phase to [latex]60^{\\circ}C[\/latex] in the vapour phase?<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Solution<\/h3>\n<p><strong>Step 1:<\/strong> Bring the acetone to the<span style=\"font-size: 1em\">\u00a0<\/span><span style=\"font-size: 1em\">boiling point temperature from [latex]25^{\\circ}C[\/latex] without changing the phase using the [latex]C_{P}[\/latex] in the liquid phase.<\/span><\/p>\n<p><strong style=\"font-size: 1em\">Step 2:<\/strong><span style=\"font-size: 1em\">\u00a0<\/span><span style=\"font-size: 1em\">Use the latent heat (or heat of<\/span><span style=\"font-size: 1em\">\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">vaporization) to calculate the enthalpy of changing the phase from liquid to vapour.<\/span><\/p>\n<p><strong>Step 3:<\/strong> Bring the acetone to [latex]60^{\\circ}C[\/latex] from the boiling point temperature using the [latex]C_{P}[\/latex] in the vapour phase.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":1,"menu_order":15,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-247","chapter","type-chapter","status-publish","hentry"],"part":238,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/pressbooks\/v2\/chapters\/247","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\/247\/revisions"}],"predecessor-version":[{"id":248,"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/pressbooks\/v2\/chapters\/247\/revisions\/248"}],"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\/247\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/wp\/v2\/media?parent=247"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/pressbooks\/v2\/chapter-type?post=247"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/wp\/v2\/contributor?post=247"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/chbe220\/wp-json\/wp\/v2\/license?post=247"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}