Pressure Loss Along a Full Pipe

Introduction

The pressure loss, or 'headloss', of a turbulent flow through a pipeline is made up of two components:

Friction losses, referred to as major losses
Valves and fittings resistance losses, referred to as minor losses

Friction Losses (Darcy-Weisbach Equation)

The accepted method of computing pressure loss due to friction is the Darcy Weisbach Equation. This equation can be expressed as a head loss:

\begin{matrix} h_{f} = f_{D} \frac{L}{D} \frac{V^{2}}{2 g} \\ (1) \end{matrix}

Where:

$h_{f}$	is the head loss due to friction (m);
$L$	is the length of the pipe (m);
$D$	is the inside diameter the pipe (m);
$V$	is the fluid velocity (m/s);
$g$	is gravity;
$f_{D}$	is the friction factor, a dimensionless coefficient

Or as pressure loss:

\begin{matrix} Δ p = f_{D} \frac{L}{D} \frac{ρ V^{2}}{2} \\ (2) \end{matrix}

Where:

$ρ$	is the fluid density (kg/m³);
$Δ p$	is the pressure drop (Pa);

Note that all the variables in equationsandare straight-forward system parameters except $f_{D}$ , the friction factor.

The friction factor, known as the Darcy friction factor, or the Moody friction factor, is a dimensionless factor to take into account the roughness of the pipe and the turbulence of the fluid. The turbulence of the fluid is measured by the dimensionless Reynolds number, defined by the equation:

\begin{matrix} R e = \frac{ρ V D}{μ} = \frac{v D}{ν} \\ (3) \end{matrix}

Where:

$μ$	is the dynamic viscosity of the fluid (Pa·s);
$ν$	is the kinematic viscosity (m²/s).

To calculate the friction factor for a laminar flow (Reynolds number of less than 2300) the following explicit formula is used:

\begin{matrix} f_{D} = \frac{64}{R e} \\ (4) \end{matrix}

For a fully turbulent flow (Reynolds number of over 4000) the following implicit equation describes the friction factor:

\begin{matrix} \frac{1}{\sqrt{f_{D}}} = - 2 {log}_{10} (\frac{ϵ}{3.7 D} + \frac{2.51}{R e \sqrt{f_{D}}}) \\ (5) \end{matrix}

Where:

$ϵ$	is the absolute roughness of the pipe's internal surface (m);

Note that this must be solved numerically but an approximation such as the Swamee-Jain equation can be used as a starting value:

\begin{matrix} f_{D} = \frac{0.25}{{[{log}_{10} (\frac{ϵ}{3.7 D} + \frac{5.74}{{R e}^{0.9}})]}^{2}} \\ (6) \end{matrix}

Fittings Losses

Losses due to fittings, termed minor losses, are dependent on some or all of the following:

Bore Size
Flow Velocity

Head loss for most fittings are described by the follow equation:

\begin{matrix} h_{L} = K \frac{v^{2}}{2 g} \\ (7) \end{matrix}

Where K is the resistance coefficient of the fitting. The symbols used below are:

$d_{1}$	is the internal diameter of the smaller pipe;
$d_{2}$	is the internal diameter of the larger pipe;
$β$	is the diameter ratio $= d_{1} / d_{2}$

To convert K from one pipe size basis to another, the following formula is used:

\begin{matrix} K_{a} = K_{b} {(\frac{d_{a}}{d_{b}})}^{4} = \frac{K_{b}}{β^{4}} \\ (8) \end{matrix}

Sudden Bore Size Change

TODO: figure

Resistance coefficient for sudden enlargements:

K_{1} = {(1 - \frac{d_{1}^{2}}{d_{2}^{2}})}^{2}

Resistance coefficient for sudden contractions:

K_{1} = 0.5 {(1 - \frac{d_{1}^{2}}{d_{2}^{2}})}^{2}