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Pressure profile

Pressure gradient profile

LaTeX Math Block

anchor	PPconst
alignment	left

j_m = j_m[p(l)] \rightarrow p = p(l)

LaTeX Math Block

anchor	gradP
alignment	left

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

Mass Flux

Mass Flowrate

LaTeX Math Block

anchor	MassFlux
alignment	left

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

anchor	MassFlowrate
alignment	left

\dot m = j_m \cdot A = \rho_s
A \cdot \sqrt{ 2 \cdot A\frac
{
g \, \cdotDelta z + \sqrt{int_p^{p_0} \frac{2 \, d}{f_s \, l }} \cdot \sqrt{g{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 \, \Delta z(l) + (p_s - p)/ \rho_s}
cdot \, l}{ 2 \, d}
}
}

Volumetric Flowrate

Intake Fluid velocity

LaTeX Math Block

anchor	PPconstVolumtericFlowrate
alignment	left

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

anchor	IntakeFluidVelocity

LaTeX Math Block

anchor	PPconst
alignment	left

u_s = j_m/ \rho_s =q_s / A = \sqrt{
\frac{2 \, d 1}{ f\rho_s} \,cdot l }}\sqrt{ 2 \cdot \sqrtfrac
{  
g \, \Delta z(l) + (p_s - p)/ \rho_s \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	j_m

Intake mass flux

LaTeX Math Inline

body	\dot m

mass flowrate

LaTeX Math Inline

body	u_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)

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