{"id":1783,"date":"2020-07-02T15:50:36","date_gmt":"2020-07-02T19:50:36","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/chbe220\/?post_type=chapter&p=1783"},"modified":"2020-08-20T14:01:13","modified_gmt":"2020-08-20T18:01:13","slug":"energy-balances-review","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/chbe220\/chapter\/energy-balances-review\/","title":{"raw":"Energy Balances Review","rendered":"Energy Balances Review"},"content":{"raw":"

Important Equations<\/h2>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Kinetic Energy<\/td>\r\n\r\n

[latex]E_{k} = \\frac{1}{2} mu^{2}[\/latex]<\/p>\r\n

[latex]\\dot{E}_{k} = \\frac{1}{2} \\dot{m} u^{2}[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

Potential Energy<\/td>\r\n\r\n

[latex]E_{p} = m g z[\/latex]<\/p>\r\n

[latex]\\dot{E}_{p} = \\dot{m} g z[\/latex]<\/p>\r\n

[latex]\\Delta E_{p} = E_{p2} - E_{p1} = m g (z_{2} - z_{1})[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

First Law of Thermodynamics[latex][\/latex]<\/td>\r\n[latex]\\Delta U + \\Delta E_{k} + \\Delta E_{p} = Q + W[\/latex]<\/span><\/td>\r\n<\/tr>\r\n
Flow Work<\/td>\r\n[latex]\\dot{W}_{fl} = \\dot{W}_{fl-in} - \\dot{W}_{fl-out} = P_{in}\\dot{V}_{in} - P_{out}\\dot{V}_{out}[\/latex]<\/td>\r\n<\/tr>\r\n
Steady-state Open System Energy Balance<\/td>\r\n\r\n

[latex]\\dot{Q} + \\dot{W} = \\Sigma_{out} \\dot{E}_{j} - \\Sigma_{in} \\dot{E}_{j}[\/latex]<\/p>\r\n

[latex]\\dot{Q} + \\dot{W}_{s} = \\Delta\\dot{H} + \\Delta\\dot{E}_{k} + \\Delta\\dot{E}_{p}[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

Enthalpy<\/td>\r\n\r\n

[latex]\\hat{H} = \\hat{U} + P\\hat{V}[\/latex]<\/p>\r\n

[latex]\\Delta\\hat{H} = \\Sigma_{i}\\Delta\\hat{H}_{i}[\/latex]<\/span><\/p>\r\n<\/td>\r\n<\/tr>\r\n

Heat Capacity (closed system)<\/td>\r\n[latex]C_{V}(T) = \\bigg(\\frac{\\delta\\hat{U}}{\\delta T}\\bigg)_{V}[\/latex]<\/td>\r\n<\/tr>\r\n
Internal Energy (closed system)<\/td>\r\n[latex]d\\hat{U} = C_{V}(T)dT[\/latex]<\/span>\r\n\r\n[latex]\\Delta\\hat{U} = \\int^{T_{2}}_{T_{1}}C_{V}dT[\/latex]<\/span><\/td>\r\n<\/tr>\r\n
Heat Capacity (open system)<\/td>\r\n[latex]C_{P}(T) = \\bigg(\\frac{\\delta\\hat{H}}{\\delta T}\\bigg)_{P}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n
Enthalpy (open system)<\/td>\r\n[latex]\\Delta\\hat{H} = \\int^{T_{2}}_{T_{1}}C_{P}dT[\/latex]<\/span><\/td>\r\n<\/tr>\r\n
Heat of Reaction Method<\/td>\r\n[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>\r\n<\/tr>\r\n
Heat of Formation Method<\/td>\r\n[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>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Terms to Know<\/h2>\r\n[h5p id=\"5\"]\r\n\r\n ","rendered":"

Important Equations<\/h2>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Kinetic Energy<\/td>\n\n

[latex]E_{k} = \\frac{1}{2} mu^{2}[\/latex]<\/p>\n

[latex]\\dot{E}_{k} = \\frac{1}{2} \\dot{m} u^{2}[\/latex]<\/p>\n<\/td>\n<\/tr>\n

Potential Energy<\/td>\n\n

[latex]E_{p} = m g z[\/latex]<\/p>\n

[latex]\\dot{E}_{p} = \\dot{m} g z[\/latex]<\/p>\n

[latex]\\Delta E_{p} = E_{p2} - E_{p1} = m g (z_{2} - z_{1})[\/latex]<\/p>\n<\/td>\n<\/tr>\n

First Law of Thermodynamics[latex][\/latex]<\/td>\n[latex]\\Delta U + \\Delta E_{k} + \\Delta E_{p} = Q + W[\/latex]<\/span><\/td>\n<\/tr>\n
Flow Work<\/td>\n[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
Steady-state Open System Energy Balance<\/td>\n\n

[latex]\\dot{Q} + \\dot{W} = \\Sigma_{out} \\dot{E}_{j} - \\Sigma_{in} \\dot{E}_{j}[\/latex]<\/p>\n

[latex]\\dot{Q} + \\dot{W}_{s} = \\Delta\\dot{H} + \\Delta\\dot{E}_{k} + \\Delta\\dot{E}_{p}[\/latex]<\/p>\n<\/td>\n<\/tr>\n

Enthalpy<\/td>\n\n

[latex]\\hat{H} = \\hat{U} + P\\hat{V}[\/latex]<\/p>\n

[latex]\\Delta\\hat{H} = \\Sigma_{i}\\Delta\\hat{H}_{i}[\/latex]<\/span><\/p>\n<\/td>\n<\/tr>\n

Heat Capacity (closed system)<\/td>\n[latex]C_{V}(T) = \\bigg(\\frac{\\delta\\hat{U}}{\\delta T}\\bigg)_{V}[\/latex]<\/td>\n<\/tr>\n
Internal Energy (closed system)<\/td>\n[latex]d\\hat{U} = C_{V}(T)dT[\/latex]<\/span><\/p>\n

[latex]\\Delta\\hat{U} = \\int^{T_{2}}_{T_{1}}C_{V}dT[\/latex]<\/span><\/td>\n<\/tr>\n

Heat Capacity (open system)<\/td>\n[latex]C_{P}(T) = \\bigg(\\frac{\\delta\\hat{H}}{\\delta T}\\bigg)_{P}[\/latex]<\/span><\/td>\n<\/tr>\n
Enthalpy (open system)<\/td>\n[latex]\\Delta\\hat{H} = \\int^{T_{2}}_{T_{1}}C_{P}dT[\/latex]<\/span><\/td>\n<\/tr>\n
Heat of Reaction Method<\/td>\n[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
Heat of Formation Method<\/td>\n[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

Terms to Know<\/h2>\n
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