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Motivation


Numerical quadrature solution of  Pressure Profile in Homogeneous Steady-State Pipe Flow @model


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

Intake temperature 

T(l)

Along-pipe temperature profile 

p_0

Intake pressure 

\rho(T, p)

q_0

Intake flowrate 

\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

Constant inclination

\displaystyle \theta(l) = \theta = {\rm const} \rightarrow \cos \theta = \frac{dz}{dl} = {\rm const}



Equations


Pressure profile along the pipe
(1) L = L(p) = \int_{p_0}^{p} \frac{ \rho(p) - j_m^2 \, c(p) }{G \, \rho^2(p) - F(\rho(p))} \, dp

where

\displaystyle j_m = \frac{ \dot m }{ A}= \rm const

mass flux

\displaystyle \dot m = \frac{dm }{ dt}= \rm const

mass flowrate

\displaystyle q_0 = \frac{dV_0}{dt} = \frac{ \dot m }{ \rho_0}

Intake volumetric flowrate

\rho_0 = \rho(T_0, p_0)

Intake fluid density 

\Delta z(l) = z(l)-z(0)

elevation drop along pipe trajectory

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 c(T,p) = \frac{1}{\rho} \left( \frac{\partial \rho}{\partial p} \right)_T

fluid compressibility

\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)

G = g \, \cos \theta= \rm const

gravity acceleration along pipe 

F(T, \rho) = j_m^2 \cdot f(T,\rho)/(2d)




Alternative forms


Density form
(2) L = L(\rho) =\int_{\rho_0}^{\rho} \frac{ 1/c(\rho) - j_m^2/\rho }{G \, \rho^2 - F(\rho)} \, d\rho
Pressure-Density form
(3) L = \int_{p_0}^{p} \frac{ \rho \, dp}{G \, \rho^2 - F(\rho)} - j_m^2 \cdot \int_{\rho_0}^{\rho} \frac{1}{\rho} \, \frac{d \rho}{G \, \rho^2 - F(\rho)}

This form is useful for derivation of Pressure Profile in GF-Proxy Pipe Flow @model 


Approximations



See also

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