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Pressure profilePressure gradient profile


LaTeX Math Block
anchorPPconst
alignmentleft
p(l) = p_s + \rho_s \, g \, z(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s \, l



LaTeX Math Block
anchorgradP
alignmentleft
\frac{dp}{dl} = \rho_s \, g \cos \theta(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s


Mass Flowrate


LaTeX Math Block
anchor
PPconst
alignmentleft
q(l) =q_s = \rm const
LaTeX Math Block
anchor1
alignmentleft
u(l) = u_s = \frac{q_s}{A} = \rm const
\dot m = A \cdot \sqrt{\frac{2 \, d \, \rho_s}{f_s \, l }} \cdot \sqrt{ \rho_s \, g \, \Delta z(l) + p_s - p }


Volumetric

Mass

Flowrate


LaTeX Math Block
anchorPPconst
alignmentleft
\dot m
q_s = A \cdot \sqrt{\frac{2 \, d }{\
, \
rho_s
}{
 \, f_s \, l }} \cdot \sqrt{ \rho_s \, g \, \Delta z(l) + p_s - p }


Volumetric Flowrate

Fluid velocity


LaTeX Math Block
anchorPPconst
alignmentleft
q
u_s =
A \cdot 
\sqrt{\frac{2 \, d }{\rho_s \, f_s \, l }} \cdot \sqrt{ \rho_s \, g \, \Delta z(l) + p_s - p }
Fluid velocity

where
 LaTeX Math Inline
body\dot m
Mass Flowrate
 LaTeX Math Inline
body--uriencoded--f_s = f(%7B\rm Re%7D_s, \, \epsilon)
Darcy friction factor at intake point
 LaTeX Math Inline
body--uriencoded--\displaystyle %7B\rm Re%7D_s = \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_s%7D
Reynolds number at intake point
 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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