Inlet Manifold Equations
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(eq-1)
\large
$$
\Delta H_{\exp ansion} = {{\left( {V_{in} - V_{out} } \right)V_{out} } \over g}
$$
$$
\sum\limits_{i = 1}^{n - 1} {\Delta H_{\exp ansion} = {{V_M ^2 } \over g}{{n - 1} \over {2n}} \to {\rm{Approaches}}\;} {{V_M ^2 } \over g}\;{\rm{for}}\;{\rm{large}}\;{\rm{n}}
$$
$$
{\rm{where }}V_M = {\rm{velocity}}\;{\rm{in}}\;{\rm{the}}\;{\rm{manifold}}
$$
$$
V_{in} \;{\rm{and}}\;V_{out} \;{\rm{are}}\;{\rm{the}}\;{\rm{velocities}}\;{\rm{before}}\;{\rm{and}}\;{\rm{after}}\;{\rm{the}}\;{\rm{expansion}}
$$
$$
f = {{0.25} \over {\left[ {\log \left( {{\varepsilon \over {3.7D}} + {{5.74} \over {{\mathop{\rm Re}\nolimits} ^{0.9} }}} \right)} \right]^2 }}
$$
$$
C_{long} = \left[ {f{{L_M } \over {D_M }}{{2n - 1} \over {6n}} + {{n - 1} \over n}} \right]
$$
$$
K_{control} = K_{e_P } \left( {{{D_M^2 } \over {nK_{vc} D_P^2 }}} \right)^2
$$
$$
\Pi _Q = \sqrt {{{C_{p_{short} } + K_{control} } \over {C_{p_{long} } + K_{control} }}}
$$
$$
D_M = \left( {{{8Q_M ^2 } \over {g\pi ^2 h_l }}{{C_{long} } \over {1 - \Pi _Q^2 }}} \right)
$$
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Energy Dissipation Constraint on Port Velocity
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(eq-2)
\large
$$
D_{Port} \cong \left[ {{1 \over {20\varepsilon _{Max} }}\left( {{{4Q_{Port} } \over {\pi K_{vc} }}} \right)^3 } \right]^{{1 \over 7}}
$$
$$
V_{Port} = {{4Q_{Port} ^{{1 \over 7}} } \over {\pi \left[ {{1 \over {20\varepsilon _{Max} }}\left( {{4 \over {\pi K_{vc} }}} \right)^3 } \right]^{{2 \over 7}} }}
$$
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Scour velocity
Design manifolds to have a flow velocity not less than 0.15 m/s nor greater than 0.45 m/s.
The minimum scour velocity is:
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\large
$$
\tau _{o_{Min} } = {2 \over 3}d_{Floc} \left( {\rho _{Floc} - \rho _{H2O} } \right)g\tan \theta
$$
$$
V_{Scour} = \sqrt {{{\tau _{o_{Min} } } \over {\rho _{H_2 O} }}{{\sqrt {500000} } \over {0.332}}}
$$
(eq-3)
$$
\varepsilon _{Max} = {1 \over {20D_{Port} }}\left( {{{V_{Scour} } \over {K_{vc} }}} \right)^3
$$
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