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Motivation

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One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the stationary steady-state fluid transport.

In many practical cases the stationary pressure distribution can be approximated by Isothermal or Quasi-isothermal homogenous fluid flow model.

Pipeline Flow Pressure Model is addressing this problem with account of the varying pipeline trajectory, gravity effects and fluid friction with pipeline walls.

Inputs & Outputs

...

Outputs

...

LaTeX Math Inline
bodyp_0

...

...

LaTeX Math Inline
bodyp(l)

Pressure distribution along the pipe

LaTeX Math Inline
bodyq

_0

(l)

Flowrate distribution along the pipe

Intake flowrate 

LaTeX Math Inline
bodyu(l)

Flow velocity distribution along the pipe

Inputs

...

LaTeX Math Inline
body

\theta (l)Pipeline trajectory inclination 

T_0

Fluid temperature at inlet point (

LaTeX Math Inline
body

--uriencoded--%7B\bf r%7D(l)

l=0
)

Pipeline trajectory

LaTeX Math Inline
bodyT(l)

Along-pipe temperature profile 

LaTeX Math Inline
bodyp_0

Fluid pressure at inlet point (

LaTeX Math Inline
bodyl=0
)

LaTeX Math Inline
body\rho(T, p)

Fluid density 

LaTeX Math Inline
bodyq_0

Fluid flowrate  at inlet point (

LaTeX Math Inline
bodyl=0
)

LaTeX Math Inline
body\mu(T, p)

Fluid

Dynamic fluid viscosity 

LaTeX Math Inline
bodyz(l)

Pipeline trajectory TVDss

LaTeX Math Inline
bodyA

Pipe cross-section area  
LaTeX Math Inline
body\theta (l)


Pipeline trajectory inclination,

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

LaTeX Math Inline
body\epsilon

Inner pipe wall roughness

Assumptions

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Stationary fluid
Steady-State flow
Homogenous fluid flowIsothermal or Quasi-isothermal conditions
Quasi-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


Equations

...

LaTeX Math Block
anchor
9QRCZ
PP
alignmentleft
\
bigg
left(
1
\rho(p) -  j_m^2 \
frac{
cdot c(p)
\,
\rho_0^2
 \
, q_0^2}{A^2} \bigg )
right) \cdot  \frac{dp}{dl} = \
rho
rho^2(p) \, g \, \cos \
frac{dz}{dl}
theta(l)  - \frac{
\rho_0^2 \, q_0^2
 j_m^2 }{2
A^2
 d} \
frac{
cdot  f(p)
}{\rho(p)}
 LaTeX Math Block
anchorp0
alignmentleft
p(l=0) = p_0



 LaTeX Math Block
anchor1
alignmentleft
u(l) = \frac{
\rho_0 \cdot q_0
j_m}{\rho(
p
l)
 \cdot A
}
 LaTeX Math Block
anchor1
alignmentleft
q(l) =A \cdot 
\frac{
u(l)
where
 LaTeX Math Inline
body--uriencoded--\displaystyle j_m =\frac%7B \rho_0 
 \cdot q_0}{\rho(p)}
 LaTeX Math Block
anchor1
alignmentleft
f(p
(see Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model )
...
\, q_0%7D%7BA%7D= \rm const
mass flux
 LaTeX Math Inline
bodyq_0 = q(l=0)
Fluid flowrate at inlet point (
 LaTeX Math Inline
bodyl=0
)
 LaTeX Math Inline
body\rho_0 = \rho(T_0, p_0)
Fluid density at inlet point (
 LaTeX Math Inline
bodyl=0
)
 LaTeX Math Inline
body\rho(l) = \rho(T(l), p(l))
Fluid density at any point 
 LaTeX Math Inline
bodyl
 LaTeX Math Inline
body--uriencoded--\displaystyle с(p) = \frac%7B1%7D%7B\rho%7D \left( \frac%7B\partial \rho%7D%7B\partial p%7D \right)_T
Fluid Compressibility
 LaTeX Math Inline
body--uriencoded--f(T, \rho) = f(%7B\rm Re%7D
...
(T, \rho), \, \epsilon)
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
Approximations
...
 LaTeX Math Inline
body\mu(T,\rho
...
 LaTeX Math Inline
bodyf(u) = f_0
)
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)



 Expand
titleDerivation
 Panel
borderColorwheat
bgColormintcream
borderWidth7
See Derivation of Pressure Profile in Steady-State Homogeneous Pipe Flow @model.
Alternative forms
...
...
 LaTeX Math Block
anchor
H8MPT LaTeX Math Block
anchorIFPGP
alignmentleft
PP
alignmentleft
p(l) = p_0 + \rho \, g \, z(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 \, l
  \frac{dp}{dl} = \left(   \frac{dp}{dl} \right)_G +  \left(   \frac{dp}{dl} \right)_K  +  \left(   \frac{dp}{dl} 
= \rho \, g \cos \theta(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 
 LaTeX Math Block
anchor1
alignmentleft
u(l) = \frac{q_0}{A}
 LaTeX Math Block
anchor1
alignmentleft
q(l) =q_0 = \rm const
where
...
 LaTeX Math Inline
body\displaystyle \cos \theta(l) = \frac{dz(l)}{dl}
...
correction factor for trajectory inclination
\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

Pressure Profile in G-Proxy Pipe Flow @modelquadrature
 LaTeX Math Inline
body\theta(l) = \theta_0 = \rm const
Pressure Profile in GF-Proxy Pipe Flow @modelquadrature
 LaTeX Math Inline
body\theta(l) = \theta_0 = \rm const
, 
 LaTeX Math Inline
bodyf(T, p)=f_0 = \rm const
Pressure Profile in GFC-Proxy Pipe Flow @model
algebraic equation
 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 @modelquadrature
 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)
The first term in 
 LaTeX Math Block Reference
anchorIFPGP
 defines the hydrostatic column of static fluid while the last term defines the friction losses under fluid movement:
...
 LaTeX Math Inline
body f(l) = f_s = \rm const
...

See also
Physics / Fluid Dynamics / Pipe Flow Dynamics / Pipe Flow Simulation 
[ Darcy friction factor ] [ Darcy friction factor @model ] [ Derivation of Stationary Isothermal Homogenous Reynolds number in Pipe Flow ] [ Pressure Profile in G-Proxy Pipe Flow @model ]
[ Temperature Profile in Homogenous Pipe Flow Temperature Profile @model ]
[ Fluid Compressibility ] [ Fluid Compressibility @model ]

References
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 Show If
grouparax
 Panel
bgColorpapayawhip
titleARAX
PipeFlowSimulator.xls
Pressure loss in pipe @ neutrium.net 
R. Shankar, Pipe Flow Calculations, Clarkson University [PDF]
Pressure loss in chokes @ Studopedia
...