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\left(\rho(p) 1 -  \frac{\rho_0^2 \, q_0^2}{A^2} \cdot \frac{j_m^2 \cdot c(p)}{\rho}   \right)  \frac{dp}{dl} = \rhorho^2(p) \, g \, \cos \theta(l)  - \frac{\rho_0^2 \, q_0^2 j_m^2 }{2 A^2 d} \frac{cdot  f({\rm Re}, \, \epsilon)}{\rho}



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p(l=0) = p_0




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



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


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body--uriencoded--\displaystyle j_m =\frac%7B \rho_0 \, q_0%7D%7BA%7D

mass flux

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bodyq_0 = q(l=0)

fluid flow rate at pipe intake

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body\rho_0 = \rho(T_0, p_0)

fluid density at intake temperature and pressure

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bodyс(p)

Fluid Compressibility

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

Darcy friction factor

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body--uriencoded--\displaystyle %7B\rm Re%7D = \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7B4 \rho_0 q_0%7D%7B\pi d%7D \frac%7B1%7D%7B\mu(T, p)%7D

Reynolds number in Pipe Flow

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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