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Stationary flowHomogenous flowIsothermal or Quasi-isothermal conditions



Constant cross-section pipe area

LaTeX Math Inline
bodyA
along hole


Equations

...

FluxMass  Volumetric FlowrateVolumtericFlowrateq0 frac{A}{\rho_s} \cdot \
Pressure profile


LaTeX Math Block
anchorPressureProfile
alignmentleft
F(p, l)=\left(  \frac{1}{\rho^2} - \frac{1}{\rho_0^2}   \right)  
+ \left(  \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2}   \right)  
\cdot \frac{l}{ 2 \, d}  - (2/j_m^2) \,  \int_p^{p_0} \frac{dp}{\rho} - (2/j_m^2) \, g \, \Delta z(l) = 0


Mass Flowrate


LaTeX Math Block
anchor
MassFlux
MassFlowrate
alignmentleft
j_
\dot m =  
A \cdot \sqrt{ 2 \cdot \frac
{
g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho}
}
{
\left(
 \frac{1}{\rho^2} - \frac{1}{\rho_0^2} 
\right) 

+ \left( 
\frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} 
\right) 

\cdot \frac{l}{ 2 \, d}
}
}

 Volumetric Flowrate

 LaTeX Math Block
anchor
MassFlowrate
VolumtericFlowrate
alignmentleft
\dot m
q_0 =  
\frac{A}{\rho_s} \cdot \sqrt{ 2 \cdot \frac
{
g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho}
}
{
\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right) 
+ \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right) 
\cdot \frac{l}{ 2 \, d}
}
}


Mass Flux

 LaTeX Math Block
anchor
MassFlux
alignmentleft
j_
m =  
\
sqrt{ 2 \cdot \frac
{
g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho}
}
{
\left(  \frac{1}{\rho^2} - \frac{1}{\rho_0^2}   \right)  
+ \left(  \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2}   \right)  
\cdot \frac{l}{ 2 \, d}
}
}


where
 LaTeX Math Inline
body\rho_0 = \rho(T_0, p_0)
Intake fluid density 
 LaTeX Math Inline
bodyj_m = \dot m / A
Intake mass flux
 LaTeX Math Inline
body--uriencoded--\displaystyle \dot m = \frac%7Bdm %7D%7B dt%7D
mass flowrate
 LaTeX Math Inline
body--uriencoded--\displaystyle q_0 = \frac%7BdV_0%7D%7Bdt%7D = \dot m / \rho_0
Intake flowrate 
 LaTeX Math Inline
bodyu_0 = u(l=0) = q_0 / A = j_m / \rho_0
Intake Fluid velocity
 LaTeX Math Inline
body\Delta z(l) = z(l)-z(0)
elevation drop along pipe trajectory
 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 = \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%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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