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\left( 1 -  \frac{\rho_s^2 \, q_s^2}{A^2} \cdot \frac{c(p)}{\rho}   \right)  \frac{dp}{dl} = \rho \, g \, \frac{dz}{dl}  - \frac{\rho_s^2 \, q_s^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho}



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qp(l=0) = \frac{\rhop_s \cdot q_s}{\rho}




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u(l) = \frac{\rho_s \cdot q_s}{\rho \cdot A}


where

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p0alignmentleft
p(l=0) = p_s

bodyq_s =

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anchorp0
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q(l=0)

= q_s

fluid flow rate at source

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body\rho_s =

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anchorp0
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\rho(T_s,

p_s)

= \rho_s

...

fluid density at source tempoerature and pressure

LaTeX Math Inline
bodyс(p)

Fluid Compressibility

LaTeX Math Inline
body--uriencoded--f(%7B\rm Re%7D, \, \epsilon)

Darcy friction factor

LaTeX Math Inline
body--uriencoded--\displaystyle %7B\rm Re%7D = \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7B4 \rho_s q_s%7D%7B\pi d%7D \frac%7B1%7D%7B\mu(T, p)%7D

Reynolds number

LaTeX Math Inline
body--uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D

characteristic linear dimension of the pipe

(or exactly a pipe diameter in case of a circular pipe)

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