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Inputs

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LaTeX Math Inline
bodyT_s0

Intake temperature 

LaTeX Math Inline
bodyT(l)

Along-pipe temperature profile 

LaTeX Math Inline
bodyp_s0

Intake pressure 

LaTeX Math Inline
body\rho(T, p)


Fluid density 

LaTeX Math Inline
bodyq_s0

Intake flowrate 

LaTeX Math Inline
body\mu(T, p)


Fluid viscosity 

LaTeX Math Inline
bodyz(l)

Pipeline trajectory TVDss

LaTeX Math Inline
bodyA

Pipe cross-section area  
LaTeX Math Inline
body\theta (l)


Pipeline trajectory inclination,

LaTeX Math Inline
body--uriencoded--\displaystyle \cos \theta (l) = \frac%7Bdz%7D%7Bdl%7D

LaTeX Math Inline
body\epsilon

Inner pipe wall roughness

...

Isoviscous flow  bodyConstant cross-section pipe area
Stationary flowHomogenous flowIsothermal or Quasi-isothermal conditions

Incompressible fluid  

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

LaTeX Math Inline



\mu(T, p) = \mu_s = \rm const
LaTeX Math Inline
bodyA
along hole

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Pressure profilePressure gradient profile



LaTeX Math Block
anchorPressureProfile
alignmentleft
F(p, l)=\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}  - (2/j_m^2) \,  \int_p^{p_0} \frac{dp}{\rho} - (2/j_m^2) \, g \, \Delta z(l) = 0



LaTeX Math Block
anchorgradP
alignmentleft
\frac{dp}{dl} = {\rm Numerical} \ {\rm Derivative}


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 =  
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{1}{\rho^2} + \frac{1}{\rho_0^2} \right) 
\cdot \frac{f \, \cdot \, l}{ 2 \, d}
}
}


 Volumetric Flowrate

Intake Fluid velocity


LaTeX Math Block
anchorVolumtericFlowrate
alignmentleft
q_s0 =  
\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{1}{\rho^2} + \frac{1}{\rho_0^2} \right) 
\cdot \frac{f \, \cdot \, l}{ 2 \, d}
}
}



LaTeX Math Block
anchorIntakeFluidVelocity
alignmentleft
u_s0 = 
\frac{1}{\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{1}{\rho^2} + \frac{1}{\rho_0^2} \right) 
\cdot \frac{f \, \cdot \, 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_s 0 = \frac%7BdV_s%7D%7Bdt%7D 0%7D%7Bdt%7D = \dot m / \rho_s0

Intake flowrate 

LaTeX Math Inline
bodyu_s 0 = u(l=0) = q_s 0 / A = j_m / \rho_s0

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 0 q_s%7D%7B0%7D%7B\pi d%7D \frac%7B1%7D%7B\mu_s%7Dmu%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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