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Motivation

...

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

...

Intake temperature  LaTeX Math Inlinebody

LaTeX Math Inline
body

T_0

p(l)

Pressure distribution along the pipe

LaTeX Math Inline
body

p_0Intake pressure 

q(l)

Flowrate distribution along the pipe

LaTeX Math Inline
body

q

u(l)

Flow velocity distribution along the pipe

Inputs

...

LaTeX Math Inline
body

q

T_0

Intake flowrate 

Fluid temperature at inlet point (

LaTeX Math Inline
body

u(

l=0
)

Flow velocity distribution along the pipe

LaTeX Math Inline
body

\theta

T(l)

Pipeline trajectory inclination 

Along-pipe temperature profile 

LaTeX Math Inline
body

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

Pipeline trajectory

p_0

Fluid pressure at inlet point (

LaTeX Math Inline
body

T(

l=0
)

Along-pipe temperature profile 

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

...

Stationary fluid flowHomogenous fluid
Steady-State flow
Isothermal or
Quasi-isothermal
conditions
 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
anchorPP
alignmentleft
\
bigg
left(
1
\rho(p) -
\frac{
j_m^2 \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{f({\rm Re}, \, \epsilon)}{\rho}
cdot  f(p)
LaTeX Math Block
anchor
1
p0
alignmentleft
q
p(l=0) =
\frac{\rho
p_0
\cdot q_0}{\rho}



 LaTeX Math Block
anchor1
alignmentleft
u(l) = \frac{
\rho_0 \cdot q_0
j_m}{\rho
 \cdot A
(l)}
 LaTeX Math Block
anchor
p0
1
alignmentleft
p
q(l
=0
) =
 p_0 LaTeX Math Block
anchorp0
alignmentleft
A \cdot u(l)
where
 LaTeX Math Inline
body--uriencoded--\displaystyle j_m =\frac%7B \rho_0 \, q_0%7D%7BA%7D= \rm const
mass flux
 LaTeX Math Inline
bodyq_0 = q(l=0)
 = q_0 LaTeX Math Block
anchorp0
alignmentleft
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
) = \rho_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
 = \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7B4 \rho_0 q_0%7D%7B\pi d%7D \frac%7B1%7D%7B
(T,\rho) = \frac%7Bj_m \cdot d%7D%7B\mu(T, 
 p
\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
Characteristic linear dimension of the pipe
(or exactly a pipe diameter in case of a circular pipe)
See Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model.
Approximations
Incompressible pipe flow 
 LaTeX Math Inline
body\rho(p) = \rho_0
 with constant viscosity 
 LaTeX Math Inline
body\mu(T, p) = \mu_0



 Expand
titleDerivation
 Panel
borderColorwheat
bgColormintcream
borderWidth7
See Derivation of Pressure Profile in Steady-State Homogeneous Pipe Flow @model.
Alternative forms
...
 LaTeX Math Block
anchorPP
Pressure profilePressure gradient profileFluid velocityFluid rate LaTeX Math BlockanchorPPconst
alignmentleft
p(l) =
 
p_0
 
+ \rho_0 \, g \, z(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 \, l LaTeX Math Block
anchorIFPGP
alignmentleft
\frac{dp}{dl} = \left(   \frac{dp}{dl} \right)_G +  \left(   \frac{dp}{dl}
 =
 \
rho
right)_
0
K 
\,
 
g
+ 
\cos
 \
theta
left(
l)
  
-
 \frac{
\rho_0 \, q_0^2 }{2 A^2 d
dp}{dl} \
, f_0 
 LaTeX Math Block
anchor1
alignmentleft
q(l) =q_0 = \rm const
 LaTeX Math Block
anchor1
alignmentleft
u(l) = u_0 = \frac{q_0}{A} = \rm const
where
...
 LaTeX Math Inline
body\displaystyle \cos \theta(l) = \frac{dz(l)}{dl}
...
correction factor for trajectory inclination
...
titleDerivation
...
borderColorwheat
bgColormintcream
borderWidth7
...
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
body
...
f(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
body
...
f(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)
Equation 
 LaTeX Math Block Reference
anchorPP
 becomes:
 LaTeX Math Block
anchorPP
alignmentleft
\frac{dp}{dl} = \rho_0 \, g \, \frac{dz}{dl}  - \frac{\rho_0 \, q_0^2 }{2 A^2 d} f_0
and can be explicitly integrated leading to 
 LaTeX Math Block Reference
anchorPPconst
.
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
...
 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
...