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titleDerivation
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See Derivation of Pressure Profile in Steady-State Homogeneous Pipe Flow @model.

Alternative forms

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LaTeX Math Block
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  \frac{dp}{dl} = \left(   \frac{dp}{dl} \right)_G +  \left(   \frac{dp}{dl} \right)_K  +  \left(   \frac{dp}{dl} \right)_f

where

LaTeX Math Inline
body--uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_G = \rho \cdot g \cdot \cos \theta


gravity losses which represent  pressure losses for upward flow and pressure gain for downward flow

LaTeX Math Inline
body--uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_K = u%5e2 \cdot \frac%7Bd \rho%7D%7Bdl%7D


kinematic losses, which grow contribution at high velocities 

LaTeX Math Inline
bodyu = j_m / \rho
 and high fluid compressibility (like turbulent gas flow)

LaTeX Math Inline
body--uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_f = - \frac%7B j_m%5e2%7D%7B2 d%7D \cdot \frac%7Bf%7D%7B\rho%7D


friction losses which are always negative along the flow direction


Approximations


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Pressure Profile in GFC-Proxy Pipe Flow @model

algebraic equation

Pressure Profile in G-Proxy Pipe Flow @modelquardraturequadrature

LaTeX Math Inline
body\theta(l) = \theta_0 = \rm const

Pressure Profile in GF-Proxy Pipe Flow @modelquardraturequadrature

LaTeX Math Inline
body\theta(l) = \theta_0 = \rm const
, 
LaTeX Math Inline
bodyf(T, p)=f_0 = \rm const

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Show If
grouparax
Show If
grouparax
LaTeX Math Inline
body\theta(l) = \theta_0 = \rm const
, 
LaTeX Math Inline
bodyf(T, p)=f_0 = \rm const

, 

LaTeX Math Inline
body--uriencoded--\rho(T, p) = \rho(T) \cdot (1+ c%5e*(T) \cdot p/p_0)
Pressure Profile in Incompressible Quasi-Isothermal Proxy Pipe Flow @modelquardraturequadrature

LaTeX Math Inline
body\rho(p)=\rho_0 = \rm const
, 
LaTeX Math Inline
bodyT(t, l)=T(l)

Pressure Profile in Incompressible Isothermal Proxy Pipe Flow @modelclosed-form expression

LaTeX Math Inline
body\rho(p)=\rho_0 = \rm const
, 
LaTeX Math Inline
bodyT=T_0 = \rm const
 (isothermal)

Pressure Profile in GC-proxy static fluid column @modelclosed-form expression

LaTeX Math Inline
body\theta(l) = \theta_0 = \rm const
, 
LaTeX Math Inline
body\dot m = 0
 (no flow)

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