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Pressure profile along the pipe


LaTeX Math Block
anchorPressureProfile
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L = \frac{p_0}{2 \, G \, c^* \, \rho_0}  \cdot \ln \frac{G \, \rho_0^2(1+c^* p/p_0)-F}{G \, \rho_0^2(1+c^*)-F}
-\frac{j_m^2}{2} \, \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G}



LaTeX Math Block
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 \cos \theta \leq 0



LaTeX Math Block
anchorPressureProfile
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L = \frac{\rho_0}{j_m^2 \cdot f/(2d)} 
\left[ (p_0-p) + \frac{c^*}{2 p_0} (p_0^2 - p^2) \right]
 - j_m^2 \, \ln \frac{\rho_0}{\rho}



LaTeX Math Block
anchor1
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 \cos \theta = 0


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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