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Proxy model of Pressure Profile in Homogeneous Steady-State Pipe Flow @model in the form of algebraic equation for the fast computation.


Outputs

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LaTeX Math Inline
bodyp(l)

Pressure distribution along the pipe

LaTeX Math Inline
bodyq(l)

Flowrate distribution along the pipe

LaTeX Math Inline
bodyu(l)

Flow velocity distribution along the pipe

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Steady-State flowQuasi-isothermal flow

LaTeX Math Inline
body--uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0

LaTeX Math Inline
body--uriencoded--\displaystyle \frac%7B\partial T%7D%7B\partial t%7D =0 \rightarrow T(t,l) = T(l)

Homogenous flow

Constant cross-section pipe area

LaTeX Math Inline
bodyA
along hole

LaTeX Math Inline
body--uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial \tau_x%7D =\frac%7B\partial p%7D%7B\partial \tau_y%7D =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l)

LaTeX Math Inline
bodyA(l) = A = \rm const

Constant inclinationConstant friction along hole

LaTeX Math Inline
body--uriencoded--\displaystyle \theta(l) = \theta = %7B\rm const%7D \rightarrow \cos \theta = \frac%7Bdz%7D%7Bdl%7D = %7B\rm const%7D

LaTeX Math Inline
bodyf(l) = f = \rm const

Linear density

LaTeX Math Inline
body--uriencoded--\rho = \rho%5e* \cdot ( 1 + c%5e* \cdot p)
  which leads to  
LaTeX Math Inline
body--uriencoded--\displaystyle c(p) = \frac%7Bc%5e*%7D%7B1 + c%5e* \cdot p %7D
  and  
LaTeX Math Inline
body--uriencoded--c%5e* \, \rho%5e* = c_0 \, \rho_0


Equations

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Pressure profile along the pipe
LaTeX Math Block
anchorPressureProfile
alignmentleft
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
alignmentleft
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
borderWidth7

See Derivation of Pressure Profile in GF-Proxy Pipe Flow @model


Alternative forms

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\cos \theta \neq

 Volumetric Flowrate in inclined pipe: 

LaTeX Math Inline
body\cos \theta \neq 0

LaTeX Math Block
anchorq_0G
alignmentleft
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], \quad n = \frac{f}{2 \, d \, G \, c^* \, \rho^*}
LaTeX Math Block
anchor1
alignmentleft
Volumetric Flowrate in horizontal pipe: 
 LaTeX Math Inline
body\cos \theta = 
 0
 LaTeX Math Block
anchorq0_G0
alignmentleft
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
alignmentleft
n = \frac{f \, L^*}{d}

 LaTeX Math Block
anchor
1
L*
alignmentleft
L^* = \frac{1}{2 \, G \, c^* \
cos
, \
theta
rho^*} = \frac{1}{2 \, G \, c_0
where
 \, \rho_0}

 LaTeX Math Block
anchorrho_rho0
alignmentleft
\
 LaTeX Math Inlinebody--uriencoded--\displaystyle \
rho_0/\rho = \
frac%7B1
frac{1+
c%5e
c^* p_
0%7D%7B1+c%5e* p%7D
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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 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
alignmentleft
q_0^2 = 
\frac{2 \, d \, A^2 \, G}{f} \cdot \left [
1 + \frac{(\rho/\rho_0)^2-1}{1- \exp (2 \, c_0 \, \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
alignmentleft
\dot m = \rho_0 \, q_0
Pressure profile in static fluid column (no flow)
 LaTeX Math Block
anchorstatic
alignmentleft
p(L) =
\rho =
\rho_0 \, \exp(c_0 \, \rho_0 \, G \, L) \, \sqrt{1 - \frac{f \, q_0^2}{2 \, d \, A^2} \cdot \frac{1
}{c^*} \cdot \left[ -1 + (1+c^* \, p_0) \cdot \exp(c^*  \rho^*  G \, L)  \right]
- \exp(-2 \, c _0 \, \rho_0 \, G \, L)}{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
alignmentleft
p(L) = p_0 + \frac{\rho/\rho_0 -1}{c_0} 
Pressure Profile in GC-proxy static fluid column @model
 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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