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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 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.

Outputs


p(l)

Pressure distribution along the pipe

q(l)

Flowrate distribution along the pipe

u(l)

Flow velocity distribution along the pipe

Inputs


T_0

Fluid temperature at inlet point (l=0)

T(l)

Along-pipe temperature profile 

p_0

Fluid pressure at inlet point (l=0)

\rho(T, p)

Fluid density 

q_0

Fluid flowrate  at inlet point (l=0)

\mu(T, p)

z(l)

Pipeline trajectory TVDss

A

Pipe cross-section area  
\theta (l)


Pipeline trajectory inclination, \displaystyle \cos \theta (l) = \frac{dz}{dl}

\epsilon

Inner pipe wall roughness

Assumptions


Steady-State flowQuasi-isothermal flow

\displaystyle \frac{\partial p}{\partial t} = 0

\displaystyle \frac{\partial T}{\partial t} =0 \rightarrow T(t,l) = T(l)

Homogenous flow

Constant cross-section pipe area A along hole

\displaystyle \frac{\partial p}{\partial \tau_x} =\frac{\partial p}{\partial \tau_y} =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l)

A(l) = A = \rm const


Equations


(1) \left( \rho(p) - j_m^2 \cdot c(p) \right) \cdot \frac{dp}{dl} = \rho^2(p) \, g \, \cos \theta(l) - \frac{ j_m^2 }{2 d} \cdot f(p)
(2) p(l=0) = p_0



(3) u(l) = \frac{j_m}{\rho(l)}
(4) q(l) =A \cdot u(l)

where

\displaystyle j_m =\frac{ \rho_0 \, q_0}{A}= \rm const

mass flux

q_0 = q(l=0)

Fluid flowrate at inlet point (l=0)

\rho_0 = \rho(T_0, p_0)

Fluid density at inlet point (l=0)

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

Fluid density at any point l

\displaystyle с(p) = \frac{1}{\rho} \left( \frac{\partial \rho}{\partial p} \right)_T

Fluid Compressibility

f(T, \rho) = f({\rm Re}(T, \rho), \, \epsilon)

Darcy friction factor

\displaystyle {\rm Re}(T,\rho) = \frac{j_m \cdot d}{\mu(T, \rho)}

Reynolds number in Pipe Flow

\mu(T,\rho)

dynamic viscosity as function of fluid temperature T and density \rho

\displaystyle d = \sqrt{ \frac{4 A}{\pi}}= \rm const

Characteristic linear dimension of the pipe

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



Alternative forms


(5) \frac{dp}{dl} = \left( \frac{dp}{dl} \right)_G + \left( \frac{dp}{dl} \right)_K + \left( \frac{dp}{dl} \right)_f

where

\displaystyle \left( \frac{dp}{dl} \right)_G = \rho \cdot g \cdot \cos \theta


gravity losses which represent  pressure losses for upward flow and pressure gain for downward flow

\displaystyle \left( \frac{dp}{dl} \right)_K = u^2 \cdot \frac{d \rho}{dl}


kinematic losses, which grow contribution at high velocities u = j_m / \rho and high fluid compressibility (like turbulent gas flow)

\displaystyle \left( \frac{dp}{dl} \right)_f = - \frac{ j_m^2}{2 d} \cdot \frac{f}{\rho}


friction losses which are always negative along the flow direction


Approximations


Pressure Profile in G-Proxy Pipe Flow @modelquadrature

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

Pressure Profile in GF-Proxy Pipe Flow @modelquadrature

\theta(l) = \theta_0 = \rm constf(T, p)=f_0 = \rm const

\theta(l) = \theta_0 = \rm constf(T, p)=f_0 = \rm const

\rho(T, p) = \rho(T) \cdot (1+ c^*(T) \cdot p/p_0)
Pressure Profile in Incompressible Quasi-Isothermal Proxy Pipe Flow @modelquadrature

\rho(p)=\rho_0 = \rm constT(t, l)=T(l)

Pressure Profile in Incompressible Isothermal Proxy Pipe Flow @modelclosed-form expression

\rho(p)=\rho_0 = \rm constT=T_0 = \rm const (isothermal)

Pressure Profile in GC-proxy static fluid column @modelclosed-form expression

\theta(l) = \theta_0 = \rm const\dot m = 0 (no flow)


See also




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