Energy Balances Review

37 Energy Balances Review

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

Δ U + Δ E_{k} + Δ E_{p} = Q + W

$\Delta U + \Delta E_{k} + \Delta E_{p} = Q + W$

Flow Work

{\dot{W}}_{f l} = {\dot{W}}_{f l - i n} - {\dot{W}}_{f l - o u t} = P_{i n} {\dot{V}}_{i n} - P_{o u t} {\dot{V}}_{o u t}

$\dot{W}_{fl} = \dot{W}_{fl-in} - \dot{W}_{fl-out} = P_{in}\dot{V}_{in} - P_{out}\dot{V}_{out}$

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) = (\frac{δ \hat{U}}{δ T})_{V}

$C_{V}(T) = \bigg(\frac{\delta\hat{U}}{\delta T}\bigg)_{V}$

Internal Energy (closed system)

d \hat{U} = C_{V} (T) d T

$d\hat{U} = C_{V}(T)dT$

$\Delta\hat{U} = \int^{T_{2}}_{T_{1}}C_{V}dT$

Heat Capacity (open system)

C_{P} (T) = (\frac{δ \hat{H}}{δ T})_{P}

$C_{P}(T) = \bigg(\frac{\delta\hat{H}}{\delta T}\bigg)_{P}$

Enthalpy (open system)

Δ \hat{H} = \int_{T_{1}}^{T_{2}} C_{P} d T

$\Delta\hat{H} = \int^{T_{2}}_{T_{1}}C_{P}dT$

Heat of Reaction Method

Δ \dot{H} = ξ Δ {\dot{H}}_{r} + Σ {\dot{n}}_{o u t} * \int_{T_{r e f}}^{T_{o u t}} C_{P} d T - Σ {\dot{n}}_{i n} * \int_{T_{r e f}}^{T_{i n}} C_{P} d T

$\Delta\dot{H} = \xi\Delta\dot{H}_{r} + \Sigma\dot{n}_{out}*\int^{T_{out}}_{T_{ref}}C_{P}dT - \Sigma\dot{n}_{in}*\int^{T_{in}}_{T_{ref}}C_{P}dT$

Heat of Formation Method

ξ Δ {\dot{H}}_{r}^{\circ} = Σ {\dot{n}}_{o u t} * {\hat{H}}_{f, i}^{\circ} - Σ {\dot{n}}_{i n} * {\hat{H}}_{f, i}^{\circ}

$\xi\Delta\dot{H}^{\circ}_{r} = \Sigma\dot{n}_{out}*\hat{H}^{\circ}_{f,i} - \Sigma\dot{n}_{in}*\hat{H}^{\circ}_{f,i}$

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