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4.4: Energy Balances Review

  • Page ID
    101171

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    Important Equations

    Kinetic Energy

    \[E_{k} = \frac{1}{2} mu^{2}\]

    \[\dot{E}_{k} = \frac{1}{2} \dot{m} u^{2}\]

    Potential Energy

    \[E_{p} = m g z\]

    \[\dot{E}_{p} = \dot{m} g z\]

    \(\Delta U + \Delta E_{k} + \Delta E_{p} = Q + W\)
    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 \(\xi\Delta\dot{H}^{\circ}_{r} = \Sigma\dot{n}_{out}*\hat{H}^{\circ}_{f,i} - \Sigma\dot{n}_{in}*\hat{H}^{\circ}_{f,i}\)

    Terms to Know

    An interactive or media element has been excluded from this version of the text. You can view it online here:
    https://pressbooks.bccampus.ca/chbe220/?p=1783


    This page titled 4.4: Energy Balances Review is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by .

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