# Darcy Friction Factor

Darcy-Weisbach Equation
In fluid dynamics, the Darcy–Weisbach equation is a phenomenological equation, which relates the major head loss, or pressure loss, due to fluid friction along a given length of pipe to the average velocity. This equation is valid for fully developed, steady, incompressible single-phase flow.

The Darcy–Weisbach equation can be written in two forms (pressure loss form or head loss form). The head loss form can be written as:

where:

• Δh = the head loss due to friction (m)
• fD = the Darcy friction factor (unitless)
• L = the pipe length (m)
• D = the hydraulic diameter of the pipe D (m)
• g = the gravitational constant (m/s2)
• V = the mean flow velocity V (m/s)

Evaluating the Darcy-Weisbach equation provides insight into factors affecting head loss in a pipeline.

• Consider that the length of the pipe or channel is doubled, the resulting frictional head loss will double.
• At constant flow rate and pipe length, the head loss is inversely proportional to the 4th power of diameter (for laminar flow). Thus, reducing the pipe diameter by half increases the head loss by a factor of 16. This is a significant increase in head loss and shows why larger diameter pipes lead to much smaller pumping power requirements.
• Since the head loss is roughly proportional to the square of the flow rate, then if the flow rate is doubled, the head loss increases by a factor of four.
• The head loss is reduced by half (for laminar flow) when the fluid’s viscosity is reduced by half.

Except for the Darcy friction factor, each of these terms (the flow velocity, the hydraulic diameter, the length of a pipe) can be easily measured. The Darcy friction factor takes the fluid properties of density and viscosity into account, along with the pipe roughness. This factor may be evaluated using various empirical relations, or it may be read from published charts (e.g.,, Moody chart).

## Darcy Friction Factor

There are two common friction factors in use, the Darcy and the Fanning friction factors.

Fanning Friction Factor
The Fanning friction factor, named after John Thomas Fanning, is a dimensionless number that is one-fourth of the Darcy friction factor, so attention must be paid to note which one of these is used as the friction factor. This is the only difference between these two factors. In all other aspects, they are identical, and by applying the conversion factor of 4, the friction factors may be used interchangeably.
fD = 4.fF
The Darcy friction factor is a dimensionless quantity used in the Darcy–Weisbach equation, for the description of frictional losses in pipe or duct as well as for open-channel flow. This is also called the Darcy–Weisbach friction factor, resistance coefficient, or simply friction factor.

The friction factor has been determined to depend on the Reynolds number for the flow and the degree of roughness of the pipe’s inner surface (especially for turbulent flow). The friction factor of laminar flow is independent of the roughness of the pipe’s inner surface.

The pipe cross-section is also important, as deviations from circular cross-sections will cause secondary flows that increase the head loss. Non-circular pipes and ducts are generally treated by using the hydraulic diameter.

## Relative Roughness

The quantity used to measure the roughness of the pipe’s inner surface is called the relative roughness, and it is equal to the average height of surface irregularities (ε) divided by the pipe diameter (D).

where both the average height surface irregularities and the pipe diameter are in millimeters.

If we know the relative roughness of the pipe’s inner surface, then we can obtain the value of the friction factor from the Moody Chart.

The Moody chart (also known as the Moody diagram) is a graph in the non-dimensional form that relates the Darcy friction factor, Reynolds number, and the relative roughness for fully developed flow in a circular pipe.

## Summary:

• Head loss of the hydraulic system is divided into two main categories:
• Major Head Loss – due to friction in straight pipes
• Minor Head Loss – due to components as valves, bends…
• Darcy’s equation can be used to calculate major losses.
• The friction factor for fluid flow can be determined using a Moody chart.
• The friction factor for laminar flow is independent of the roughness of the pipe’s inner surface. f = 64/Re
• The friction factor for turbulent flow depends strongly on the relative roughness. The Colebrook equation determines it. It must be noted that the friction factor is independent of the Reynolds number at very large Reynolds numbers.

## Why is head loss very important?

As can be seen from the picture, the head loss is formed key characteristic of any hydraulic system. In systems in which some certain flowrate must be maintained (e.g.,, to provide sufficient cooling or heat transfer from a reactor core), the equilibrium of the head loss and the head added by a pump determine the flow rate through the system.

Example: Moody Chart
Determine the friction factor (fD) for fluid flow in a pipe of 700mm in diameter that has the Reynolds number of 50 000 000 and an absolute roughness of 0.035 mm.

Solution:

The relative roughness is equal to ε = 0.035 / 700 = 5 x 10-5. Using the Moody Chart, a Reynolds number of 50 000 000 intersects the curve corresponding to a relative roughness of 5 x 10-5 at a friction factor of 0.011.

## Darcy Friction Factor for various flow regime

The most common classification of flow regimes is according to the Reynolds number. The Reynolds number is a dimensionless number comprised of the physical characteristics of the flow, and it determines whether the flow is laminar or turbulent. An increasing Reynolds number indicates increasing turbulence of flow. As can be seen from the Moody chart, the Darcy friction factor also depends on the flow regime (i.e., on the Reynolds number).

Darcy Friction Factor for Laminar Flow
For practical purposes, if the Reynolds number is less than 2000, the flow is laminar. The accepted transition Reynolds number for flow in a circular pipe is Red,crit = 2300. For laminar flow, the head loss is proportional to velocity rather than velocity squared. Thus the friction factor is inversely proportional to velocity.

The Darcy friction factor for laminar (slow) flows is a consequence of Poiseuille’s law that and it is given by the following equations:

Darcy Friction Factor for Transitional Flow
At Reynolds numbers between about 2000 and 4000, the flow is unstable as a result of the onset of turbulence. These flows are sometimes referred to as transitional flows. The Darcy friction factor contains large uncertainties in this flow regime and is not well understood.
Darcy Friction Factor for Turbulent Flow
If the Reynolds number is greater than 3500, the flow is turbulent. Most fluid systems in nuclear facilities operate with turbulent flow. In this flow regime, the resistance to flow follows the Darcy–Weisbach equation: it is proportional to the square of the mean flow velocity. The Darcy friction factor depends strongly on the relative roughness of the pipe’s inner surface.

The most common method to determine a friction factor for turbulent flow is to use the Moody chart. The Moody chart (also known as the Moody diagram) is a log-log plot of the Colebrook correlation that relates the Darcy friction factor, Reynolds number, and the relative roughness for fully developed flow in a circular pipe. The Colebrook–White equation:

which is also known as the Colebrook equation, expresses the Darcy friction factor f as a function of pipe relative roughness ε / Dh and Reynolds number.

In 1939, Colebrook found an implicit correlation for the friction factor in round pipes by fitting the data of experimental studies of turbulent flow in smooth and rough pipes.

For hydraulically smooth pipe and the turbulent flow (Re < 105), the friction factor can be approximated by the Blasius formula:

f = (100.Re)

It must be noted that the friction factor is independent of the Reynolds number at very large Reynolds numbers. This is because the thickness of the laminar sublayer (viscous sublayer) decreases with increasing Reynolds number. For very large Reynolds numbers, the thickness of the laminar sublayer is comparable to the surface roughness, and it directly influences the flow. The laminar sublayer becomes so thin that the surface roughness protrudes into the flow. The frictional losses, in this case, are produced in the main flow primarily by the protruding roughness elements, and the contribution of the laminar sublayer is negligible.

References:
Reactor Physics and Thermal Hydraulics:
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Major Loss