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#### 5.8.2.3 Energy balance of phase k

Time derivative term

Let us define the mass-weighted phase space average enthalpy: ${\stackrel{⌣}{{H}_{k}}}^{k}=\frac{{〈{\rho }_{k}{h}_{k}〉}_{k}^{S}}{{\stackrel{ˆ}{{\rho }_{k}}}^{k}}$ ${\int }_{{A}_{k}}\frac{\partial \rho \left(h+\frac{{v}^{2}}{2}\right)}{\partial t}dA=\frac{\partial {A}_{k}<{\rho }_{k}\left({h}_{k}+\frac{{{v}_{k}}^{2}}{2}\right){>}_{k}}{\partial t}=\frac{\partial A{R}_{k}{\stackrel{ˆ}{{\rho }_{k}}}^{k}\left({\stackrel{⌣}{{H}_{k}}}^{k}+\frac{{{\stackrel{⌣}{{V}_{k}}}^{k}}^{2}}{2}\right)}{\partial t}$ An approximation was necessary for the kinetic energy term:

$<{\rho }_{k}\frac{{{v}_{k}}^{2}}{2}{>}_{k}\cong {\stackrel{ˆ}{{\rho }_{k}}}^{k}\frac{{{\stackrel{⌣}{{V}_{k}}}^{k}}^{2}}{2}$ Advection term

$\begin{array}{l}{\int }_{{A}_{k}}\nabla ·\rho \stackrel{\to }{v}\left(h+\frac{{v}^{2}}{2}\right)dA=\frac{\partial A{R}_{k}{〈{\rho }_{k},{v}_{kz},{h}_{k}〉}_{k}^{S}}{\partial z}+\frac{\partial A{R}_{k}{〈{\rho }_{k},{v}_{kz},\frac{{{v}_{k}}^{2}}{2}〉}_{k}^{S}}{\partial z}+M{H}_{ik}+E{C}_{ik}\\ \cong \frac{\partial A{R}_{k}{\stackrel{ˆ}{{\rho }_{k}}}^{k}{\stackrel{⌣}{{V}_{k}}}^{k}\left({\stackrel{⌣}{{H}_{k}}}^{k}+\frac{{{\stackrel{⌣}{{V}_{k}}}^{k}}^{2}}{2}\right)}{\partial z}+\frac{\partial A{R}_{k}{q}_{kz}^{d}}{\partial z}+M{H}_{ik}+E{C}_{ik}\end{array}$

MHik and ECik are the volumetric energy transfer through ...

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