## 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**). In the head loss form can be written as:

where:

- Δh = the head loss due to friction (m)
*f*= the Darcy friction factor (unitless)_{D}- L = the pipe length (m)
- D = the hydraulic diameter of the pipe D (m)
- g = the gravitational constant (m/s
^{2}) - V = the mean flow velocity V (m/s)

where:

- Δp = the pressure loss due to friction (Pa)
*f*= the Darcy friction factor (unitless)_{D}- L = the pipe length (m)
- D = the hydraulic diameter of the pipe D (m)
- g = the gravitational constant (m/s
^{2}) - V = the mean flow velocity V (m/s)

___________

**Darcy-Weisbach equation**provides insight into factors affecting the 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), and thus reducing the pipe diameter by half increases the head loss by a factor of 16. This is a very 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**viscosity of the fluid is reduced by half**.

With the exception of 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 by the use of various empirical relations, or it may be read from published charts (e.g. **Moody chart**).

## Summary:

- Head loss of 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 roughness**of the pipe’s inner surface.**f = 64/Re****The friction factor**for turbulent flow depends strongly on the**relative roughness.**It is determined by the Colebrook equation. It must be noted,**at very large Reynolds numbers**, the friction factor is independent of the Reynolds number.

## Why the head loss is very important?

As can be seen from the picture, the head loss is forms **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 determines the flowrate through the system.

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