Motivation

...

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

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.

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Outputs

...

LaTeX Math Inline

body	p(l)

Pressure distribution along the pipe

InputsOutputsPipeline trajectory

LaTeX Math Inline

body

{\bf r} = {\bf r}(l) = \{ x

q(l)

, \, y(l), \, z(l) \}along-pipe stabilized pressure distribution

Flowrate distribution along the pipe

LaTeX Math Inline

body

p

u(l)

Pipe cross-section area

Flow velocity distribution along the pipe

Inputs

...

LaTeX Math Inline

body

Aalong-pipe stabilized flowrate distribution

T_0

Fluid temperature at inlet point (

LaTeX Math Inline

body

q(

l=0

)

Along-pipe temperature profile

LaTeX Math Inline

body	T(l)

along

Along-pipe

stabiliszed average flow velocity distribution Fluid density

temperature profile

LaTeX Math Inline

body

u(l)

p_0

Fluid pressure at inlet point (

LaTeX Math Inline

body	l=0

)

LaTeX Math Inline

body	\rho(T, p)

Fluid density

LaTeX Math Inline

body	q_0

Fluid flowrate at inlet point (

LaTeX Math Inline

body

\rho

l=0

)

LaTeX Math Inline

body	\mu(T, p)

and

Dynamic fluid viscosity

LaTeX Math Inline

body

\mu(T, p)Intake pressure

z(l)

Pipeline trajectory TVDss

LaTeX Math Inline

body	A

Pipe cross-section area

 LaTeX Math Inline
body

p_0, Intake rate

\theta (l)

Pipeline trajectory inclination,

LaTeX Math Inline

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

LaTeX Math Inline

body

q_0

\epsilon

Inner pipe wall roughness

Assumptions

...

Stationary fluid flow Homogenous fluid flow Isothermal or Quasi-isothermal conditions

Steady-State flow

Quasi-isothermal flow

LaTeX Math Inline

body

\epsilon

Assumptions

--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

body	A

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

body	A(l) = A = \rm const

Equations

...

LaTeX Math Block

anchor

9QRCZ

PP
alignment	left

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

anchor	p0
alignment	left

p(l=0) = p_0

LaTeX Math Block

anchor	1
alignment	left

u(l) = \frac{

\rho_0 \cdot q_0

j_m}{\rho(

p) \cdot A(

l)}

LaTeX Math Block

anchor	1
alignment	left

q(l) =

\frac{

A \cdot u(l)

where

LaTeX Math Inline

body	--uriencoded--\displaystyle j_m =\frac%7B \rho_0

\cdot q_0}{\rho(p)}

(see Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model )

Approximations

Incompressible pipe flow with constant friction

\, q_0%7D%7BA%7D= \rm const

mass flux

LaTeX Math Inline

body	q_0 = q(l=0)

Fluid flowrate at inlet point (

LaTeX Math Inline

body	l=0

)

LaTeX Math Inline

body	\rho_0 = \rho(T_0, p_0)

Fluid density at inlet point (

LaTeX Math Inline

body	l=0

)

LaTeX Math Inline

body	\rho(l) = \rho(T(l), p(l))

Fluid density at any point

LaTeX Math Inline

body	l

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

LaTeX Math Inline

body	\mu(T,\rho)

dynamic viscosity as function of fluid temperature

LaTeX Math Inline

body	T

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

title	Derivation

Panel

borderColor	wheat
bgColor	mintcream
borderWidth	7

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

Alternative forms

...

LaTeX Math Block

anchor	PP
alignment	left

Pressure profilePressure gradient profileFluid velocityFluid rate

LaTeX Math Block

anchor	H8MPT
alignment	left

p(l) = p_0 + \rho \, g \, z(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 \, l

LaTeX Math Block

anchor	IFPGP
alignment	left

\frac{dp}{dl} = \

rho \, g \cos \theta(l) -

left(   \frac{dp}{dl} \right)_G +  \left(   \frac{

\rho_0 \, q_0^2 }{2 A^2 d} \, f_0

LaTeX Math Block

anchor	1
alignment	left

u(l) = \frac{q_0}{A(l)}

LaTeX Math Block

anchor	1
alignment	left

q(l) =q_0 = \rm const

where

...

LaTeX Math Inline

body	\displaystyle \cos \theta(l) = \frac{dz(l)}{dl}

...

correction factor for trajectory deviation

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

body	u = 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 @model

quadrature

LaTeX Math Inline

body	\theta(l) = \theta_0 = \rm const

Pressure Profile in GF-Proxy Pipe Flow @model

quadrature

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 @model

quadrature

LaTeX Math Inline

body	\rho(p)=\rho_0 = \rm const

,

LaTeX Math Inline

body	T(t, l)=T(l)

Pressure Profile in Incompressible Isothermal Proxy Pipe Flow @model

closed-form expression

LaTeX Math Inline

body	\rho(p)=\rho_0 = \rm const

,

LaTeX Math Inline

body	T=T_0 = \rm const

(isothermal)

Pressure Profile in GC-proxy static fluid column @model

closed-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

anchor	IFPGP

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

...

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Versions Compared

Old Version 57

New Version Current

Key

Motivation

Outputs

Inputs

Assumptions

Assumptions

Equations

Approximations

Incompressible pipe flow with constant friction

Alternative forms

Page tree

Page History

Versions Compared

Old Version 57

New Version Current

Key

Motivation

Outputs

Inputs

Assumptions

Assumptions

Equations

Approximations

Incompressible pipe flow with constant friction

Alternative forms