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Stationary flowHomogenous flowIsothermal or Quasi-isothermal conditions

Incompressible fluid  

LaTeX Math Inline
body\rho(T, p)=\rho_s = \rm const

Isoviscous flow  

LaTeX Math Inline
body\mu(T, p) = \mu_s = \rm const

Constant cross-section pipe area

LaTeX Math Inline
bodyA
along hole


Equations

...

Pressure profilePressure gradient profile


LaTeX Math Block
anchorPPconst
alignmentleft
j_m = j_m[p(l)] \rightarrow p = p(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 FluxMass Flowrate


LaTeX Math Block
anchorMassFlux
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{1}{\rho^2} + \frac{1}{\rho_0^2}   \right)  
\cdot \frac{f \, \cdot \, l}{ 2 \, d}
}

}



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


 Volumetric Flowrate

Intake Fluid velocity


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



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


where

LaTeX Math Inline
bodyj_m

Intake mass flux

LaTeX Math Inline
body\dot m

mass flowrate

LaTeX Math Inline
bodyu_s = u(l=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_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)



Expand
titleDerivation


Panel
borderColorwheat
bgColormintcream
borderWidth7

See Pressure Profile in Stationary Quasi-Isothermal Homogenous Pipe Flow @model


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