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Pressure profile along the pipe
LaTeX Math Block
anchorPressureProfile
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L = \frac{1}{2 \, G \, c^*  \rho^*}  \cdot \ln \frac{G \, \rho^2-F}{G \, \rho_0^2-F}
-\frac{d}{f} \cdot \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G}
LaTeX Math Block
anchor1
alignmentleft
 \cos \theta \neq 0
LaTeX Math Block
anchorG0
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L = \frac{1}{2F\, c^* \rho^*} \cdot (\rho_0^2 - \rho^2)
 -+ \frac{2d}{f} \cdot \ln \frac{\rho_0}{\rho}
LaTeX Math Block
anchor1
alignmentleft
 \cos \theta = 0

...

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 = f(%7B\rm Re%7D(T,\rho), \, \epsilon) = \rm const

Darcy friction factor 

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

Reynolds number in Pipe Flow

LaTeX Math Inline
body\mu(T,\rho)

dynamic viscosity as function of fluid temperature 

LaTeX Math Inline
bodyT
 and density 
LaTeX Math Inline
body\rho

LaTeX Math Inline
body--uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D = \rm const

characteristic linear dimension of the pipe

(or exactly a pipe diameter in case of a circular pipe)

LaTeX Math Inline
bodyG = g \, \cos \theta = \Delta Z/L = \rm const

gravity acceleration along pipe 

LaTeX Math Inline
body--uriencoded--\Delta Z = Z_%7Bout%7D - Z_%7Bin%7D

altitude drop in downwards direction (positive if descending)

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


Expand
titleDerivation
Panel
borderColorwheat
bgColormintcream
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See Derivation of Pressure Profile in GF-Proxy Pipe Flow @model


Alternative forms

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 Volumetric Flowrate in inclined pipe: 

LaTeX Math Inline
body\cos \theta \neq 0

LaTeX Math Block
anchorq_0G
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q_0^2 = \frac{2 d A^2 G}{f} \cdot \left[ 

1 + \frac{ (\rho/\rho_0)^2 -1}{1- (\rho_0/\rho)^{\frac{2}{n-1}} \cdot 
\exp \left( \frac{fL/d}{ n-1}  \right)}
\right]

Volumetric Flowrate in horizontal pipe: 

LaTeX Math Inline
body\cos \theta = 0

LaTeX Math Block
anchorq0_G0
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q_0^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{1 - (\rho/\rho_0)^2}{2 \ln (\rho_0/\rho) + fL/d}

where

LaTeX Math Block
anchorn
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n = \frac{f \, L^*}{
2 \, 
d}
LaTeX Math Block
anchorL*
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L^* = \frac{1}{2 \, G \, c^* \, \rho^*} = \frac{1}{2 \, G \, c_0 \, \rho_0}
LaTeX Math Block
anchorrho_rho0
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\rho_0/\rho = \frac{1+c^* p_0}{1+c^* p}

with the following asymptotes:

Low compressible fluids:

LaTeX Math Inline
body--uriencoded--c%5e* p \ll 1, \, \, c%5e* p_0 \ll 1

High compressible fluids:

LaTeX Math Inline
body--uriencoded--c%5e* p \gg 1, \, \, c%5e* p_0 \gg 1

LaTeX Math Inline
body--uriencoded--\displaystyle \rho_0/\rho = c%5e* \cdot (p_0-p)

 

LaTeX Math Inline
body\displaystyle \rho_0/\rho = p_0/p


Approximations

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1 to (< 1 m ) (c^* \rho^*cdot }{2d}cdot \frac{j_m^2Grho_0^2( c^* \rho^*)p(L) = \frac{1}{c^*} \cdot \left[ -1 + (1+c^* p_0) \cdot \exp (c^* \rho^* \, G \, L f2d j_m^2Grho}big( c^*rho^* G\big) } \right]L/L^*-1 +1+c^*p)cdotexp(L/L^*)}{c^*}

LaTeX Math Inline
bodyn \geq 1
 which is equivalent to
LaTeX Math Inline
body--uriencoded--L%5e* \geq

d
 and holds true for

the most of practical tube diameters

, as the lowest practical values of 

LaTeX Math Inline
body--uriencoded--L%5e* \geq d
are 
LaTeX Math Inline
body--uriencoded--L%5e* \geq 7,000 \, %7B\rm m%7D
 

LaTeX Math Block
anchorq0_G
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q_0^2 = 
\frac{2 \, d \, A^2 \, G}{c^* \rho^*f} \cdot \frac{left [
1 -+ \frac{(\rho/\rho_0)^2-1}{1- \cdotexp (2 \exp, c_0 \left(  -L/ L^* \right)}{2 \ln (\rho_0/\rho) + fL/d \cdot (1- \exp \left(  - L/ L^* \right))/(L/L^*)}, \rho_0 \, G \, L)}
 \right]
=
\frac{2 \, d \, A^2  \, g}{f \, L} \cdot \left [ 
\Delta Z + ((\rho/\rho_0)^2 -1) \cdot  \frac{ \Delta Z}{1 - \exp(2 \, c_0 \, \rho_0 \, g \,  \Delta Z)}
\right]
LaTeX Math Block
anchorq0_G
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\dot m = \rho_0 \, q_0
LaTeX Math Block
anchorq0_G
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\dot m^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{\rho_0^2 - \rho^2 \cdot \exp \left(  -L/ L^* \right)}{2 \ln (\rho_0/\rho) + fL/d \cdot (1- \exp \left(  - L/ L^* \right))/(L/L^*)}
LaTeX Math Block
anchorstatic
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\rho =

\rho_0 \, \exp
(c_0 \, \rho_0 \, G \, L) \
, \sqrt{
1 - \frac{f
 \
, q_0^2}{
2 \, d \
, A^2} \cdot 
\frac{1
- \exp(-2 \,
 c _0 \, \rho_0 \, G \, L
)}
LaTeX Math Block
anchorstatic
alignmentleft
{G}} 
=\rho_0 \, \exp (с_0 \, \rho_0 \, g \, \Delta Z) \cdot \sqrt{
 1 - \frac{
8}{
\pi^2}
 \cdot   \frac{
f \, L}{
d^5} \
cdot q_0^2
 \cdot \
frac{1 - \exp
(- 2 \, c_0 \, 
\rho_0 \
, g \,
 \Delta Z) } { g \, \Delta Z}}
LaTeX Math Block
anchor1
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p(L) = p_0 + 
\frac{\rho/\rho_0 -1}{c_0}
Pressure Profile in GC-proxy static fluid column @model
LaTeX Math Inline
body\dot m = 0, \, q_0 = 0
 (no flow)
LaTeX Math Block
anchorstatic
alignmentleft
\rho = \rho_0 \, \exp (
c_0 \, \rho_0 \, g \, \Delta Z)
LaTeX Math Block
anchorstatic
alignmentleft
p(L) =  p_0 + \frac{
\exp (
c_0 \, 
\rho_0
 \, g \
, \
Delta Z) -1}{c_0}


See also

Physics / Fluid Dynamics / Pipe Flow Dynamics / Pipe Flow Simulation / Pressure Profile in Homogeneous Quasi-Isothermal Steady-State Pipe Flow @model

[ Pressure Profile in G-Proxy Pipe Flow @model / Pressure Profile in GF-Proxy Pipe Flow @model ] 

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