$$ Re = \frac {\rho VD}{\mu} $$
In its simplest form, the Reynolds Number expresses the ratio of inertial forces to viscous forces in fluid flow. Pipe flows exhibit distinctive behaviors based on their Reynolds Number, thus we can categorize them into three well-defined groups: Laminar flow, Transition flow, and Turbulent flow.
Laminar flow prevails when the viscous forces dominate, typically observed at Reynolds Numbers (Re) below 2100. Transition flow occurs in the intermediate range, with 2100 < Re < 4000, bridging the gap between Laminar and Turbulent Flow. Turbulent flow takes over when inertial forces dominate, typically with Reynolds Numbers exceeding 4000.
$$ \frac {l}{D} = 0.06 Re $$
$$ \frac {l}{D} = 4.4(Re)^{1/6} $$
The entrance length, $l$, is the distance along the pipe or duct where the flow conditions are still adjusting from the entrance of the pipe. The first equation above can be used under Laminar flow conditions. Under Turbulent flow conditions, the second equation should be used.
In Laminar flow, viscous effects must be taken into account across the entire length of the pipe. This results in a pressure drop across a specified length of pipe being a function of velocity, length, diameter, and the dynamic viscosity of the fluid.
During Turbulent flow, a relatively thin viscous sublayer is generated along the pipe wall, resulting in the surface roughness of the wall being significant. Thus, the pressure drop depends on velocity, diameter, length, surface roughness, dynamic viscosity, and density. Performing dimensional analysis on this relationship results in the following:
$$ \varDelta p = f \frac {l}{D} \frac {\rho V^2}{2} $$
Where $f$ is designated as the friction factor and is a function of the Reynolds Number and the surface roughness divided by the diameter. Experiments typically determine the friction factor and can be found using the Moody Chart. The turbulent portion of the Moody chart can be represented using the following relationship, known as the Haaland equation.
$$ \frac {1}{\sqrt f} = -1.8 \log[ (\frac {(\frac {\varepsilon}{D})} {3.7})^{1.11} + \frac {6.9}{Re}] $$
Using the relationship for mean velocity, the friction factor for fully developed laminar flow can be approximated as the following:
$$ f = \frac {64}{Re} $$
In any piping system, there will inevitably be occurrences of energy losses. In a typical system, there are two primary categories of loss: major and minor. Major head loss occurs in straight pipes and is a result of viscous effects. Minor head loss is fundamentally due to inertial effects, stemming from the geometry of the system such as elbows, valves, etc. The terms ‘major’ and ‘minor’ are not necessarily indicative of the magnitudes of their respective losses. A modified form of the Bernoulli Equation can be used to incorporate these losses for analysis:
$$ \frac {p_1}{\gamma} + \alpha_1 \frac {V_1^2}{2g}+z_1 = \frac {p_2}{\gamma} + \alpha_2 \frac {V_2^2}{2g}+z_2 + h_L $$
The $h_L$ term includes both major and minor head losses. To compensate for the velocity profile not being uniform across the pipe due to the no-slip condition, $\alpha_1$ and $\alpha_2$ (kinetic energy coefficients) are used. For any non-uniform flow, $\alpha$ will be greater than one, and in fully developed flow, $\alpha_1 = \alpha_2$. For the ideal and inviscid case, $\alpha_1 = \alpha_2 = 0$ and $h_L = 0$, reducing to the Bernoulli Equation. Major and minor losses can be found using the following formulas: