Motivation

One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the stationary 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

Assumptions

Equations

\bigg( 1 -  \frac{c(p) \, \rho_0^2 \, q_0^2}{A^2}   \bigg )  \frac{dp}{dl} = \rho \, g \, \frac{dz}{dl}  - \frac{\rho_0^2 \, q_0^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho}

q(l) = \frac{\rho_0 \cdot q_0}{\rho}

u(l) = \frac{\rho_0 \cdot q_0}{\rho \cdot A}

p(l=0) = p_0

q(l=0) = q_0

\rho(T_0, p_0) = \rho_0

where

See Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model.

Approximations

Incompressible pipe flow  with constant viscosity 

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

\frac{dp}{dl} = \rho_0 \, g \cos \theta(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 

q(l) =q_0 = \rm const

u(l) = u_0 = \frac{q_0}{A} = \rm const

where

\frac{dp}{dl} = \rho_0 \, g \, \frac{dz}{dl}  - \frac{\rho_0 \, q_0^2 }{2 A^2 d} f_0

The first term in  defines the hydrostatic column of static fluid while the last term defines the friction losses under fluid movement:

In most practical applications in water producing or water injecting wells the water can be considered as incompressible and friction factor  an be assumed constant  along-hole ( see  Darcy friction factor in water producing/injecting wells ).

References