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\bigg( 1 -  \frac{c(p) \, \rho_0^2 \, q_0^2}{A^2}   \bigg )  \frac{dp}{dl} = \rho(p) \, g \, \frac{dz}{dl}  - \frac{\rho_0^2 \, q_0^2 }{2 A^2 d} \frac{f(
u
p)}{\rho(p)}



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u(l) = \frac{\rho_0 \cdot q_0}{\rho(p) \cdot A}



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q(l) = \frac{\rho_0 \cdot q_0}{\rho(p)}



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f(p


(see Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model )

The friction factor 

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body--uriencoded--f = f (%7B\rm Re%7D) = f(u) = f (\rho(p))

Approximations

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Incompressible pipe flow 
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body\rho(p) = \rho_0
with constant friction 
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bodyf(u) = f_0

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