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.

Outputs

	Pressure distribution along the pipe
	Flowrate distribution along the pipe
	Flow velocity distribution along the pipe

	Fluid temperature at inlet point ()		Along-pipe temperature profile
	Fluid pressure at inlet point ()		Fluid density
	Fluid flowrate at inlet point ()		Dynamic fluid viscosity
	Pipeline trajectory TVDss		Pipe cross-section area
	Pipeline trajectory inclination,		Inner pipe wall roughness

Steady-State flow	Quasi-isothermal flow

Homogenous flow	Constant cross-section pipe area along hole

\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)

p(l=0) = p_0

u(l) = \frac{j_m}{\rho(l)}

q(l) =A \cdot u(l)

where

	mass flux
	Fluid flowrate at inlet point ()
	Fluid density at inlet point ()
	Fluid density at any point
	Fluid Compressibility
	Darcy friction factor
	Reynolds number in Pipe Flow
	dynamic viscosity as function of fluid temperature and density
	Characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe)

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

where

	gravity losses which represent pressure losses for upward flow and pressure gain for downward flow
	kinematic losses, which grow contribution at high velocities and high fluid compressibility (like turbulent gas flow)
	friction losses which are always negative along the flow direction

Approximations

Pressure Profile in G-Proxy Pipe Flow @model	quadrature
Pressure Profile in GF-Proxy Pipe Flow @model	quadrature	,
Pressure Profile in GFC-Proxy Pipe Flow @model	algebraic equation	,
Pressure Profile in Incompressible Quasi-Isothermal Proxy Pipe Flow @model	quadrature	,
Pressure Profile in Incompressible Isothermal Proxy Pipe Flow @model	closed-form expression	, (isothermal)
Pressure Profile in GC-proxy static fluid column @model	closed-form expression	, (no flow)