# 37 Energy Balances Review

## Important Equations

 Kinetic Energy $E_{k} = \frac{1}{2} mu^{2}$ $\dot{E}_{k} = \frac{1}{2} \dot{m} u^{2}$ Potential Energy $E_{p} = m g z$ $\dot{E}_{p} = \dot{m} g z$ $\Delta E_{p} = E_{p2} - E_{p1} = m g (z_{2} - z_{1})$ First Law of Thermodynamics $\Delta U + \Delta E_{k} + \Delta E_{p} = Q + W$ Flow Work $\dot{W}_{fl} = \dot{W}_{fl-in} - \dot{W}_{fl-out} = P_{in}\dot{V}_{in} - P_{out}\dot{V}_{out}$ Steady-state Open System Energy Balance $\dot{Q} + \dot{W} = \Sigma_{out} \dot{E}_{j} - \Sigma_{in} \dot{E}_{j}$ $\dot{Q} + \dot{W}_{s} = \Delta\dot{H} + \Delta\dot{E}_{k} + \Delta\dot{E}_{p}$ Enthalpy $\hat{H} = \hat{U} + P\hat{V}$ $\Delta\hat{H} = \Sigma_{i}\Delta\hat{H}_{i}$ Heat Capacity (closed system) $C_{V}(T) = \bigg(\frac{\delta\hat{U}}{\delta T}\bigg)_{V}$ Internal Energy (closed system) $d\hat{U} = C_{V}(T)dT$ $\Delta\hat{U} = \int^{T_{2}}_{T_{1}}C_{V}dT$ Heat Capacity (open system) $C_{P}(T) = \bigg(\frac{\delta\hat{H}}{\delta T}\bigg)_{P}$ Enthalpy (open system) $\Delta\hat{H} = \int^{T_{2}}_{T_{1}}C_{P}dT$ Heat of Reaction Method $\Delta\dot{H} = \xi\Delta\dot{H}_{r} + \Sigma\dot{n}_{out}*\int^{T_{out}}_{T_{ref}}C_{P}dT - \Sigma\dot{n}_{in}*\int^{T_{in}}_{T_{ref}}C_{P}dT$ Heat of Formation Method $\xi\Delta\dot{H}^{\circ}_{r} = \Sigma\dot{n}_{out}*\hat{H}^{\circ}_{f,i} - \Sigma\dot{n}_{in}*\hat{H}^{\circ}_{f,i}$