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Pressure profile along the pipe


LaTeX Math Block
anchorPressureProfile
alignmentleft
L =\int_{\rhop_0}^{\rhop} \frac{ \rho - j_m^2 \, dpc(p) }{G \, \rho^2 - F} -\frac{j_m^2}{2}(\rho)} \, \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G}dp


where

LaTeX Math Inline
body--uriencoded--\displaystyle j_m = \frac%7B \dot m %7D%7B A%7D

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 = \frac%7B \dot m %7D%7B \rho_0%7D

Intake volumetric flowrate

LaTeX Math Inline
body\rho_0 = \rho(T_0, p_0)

Intake fluid density 

LaTeX Math Inline
body\Delta z(l) = z(l)-z(0)

elevation drop along pipe trajectory

LaTeX Math Inline
body--uriencoded--f(T,p) = f(%7B\rm Re%7D(T,p), \, \epsilon)

Darcy friction factor 

LaTeX Math Inline
body--uriencoded--\displaystyle %7B\rm Re%7D(T,p) = \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7Bj_m \cdot d%7D%7B\mu(T,p)%7D

Reynolds number in Pipe Flow

LaTeX Math Inline
body\mu(T,p)

dynamic viscosity as function of fluid temperature 

LaTeX Math Inline
bodyT
 and pressure 
LaTeX Math Inline
bodyp

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)

LaTeX Math Inline
bodyG = g \, \cos \theta


LaTeX Math Inline
body--uriencoded--F = j_m%5e2 \cdot f/(2d)


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