# Formulas For Head Loss Due To Friction in Pipe ( Major Loss )

When a fluid flows from one section to another section, there occurs a reduction in its total energy due to friction between the pipe wall and the flowing fluid. This energy loss is represented as head loss (in terms of the water column in m).

Following are some formulas for calculating head loss in fluid mechanics given by different scientists.

## Darcy Weisbach Equation

• Consider fluid of unit weight γ is flowing in a pipe as shown. The static pressure head is measured at sections (1) and (2). Let hf is the head loss of flowing fluid from sections (1) to (2).

Applying energy equation at sections (1) and (2)

$\huge \frac{p_1}{\gamma}+\frac{V_1^2}{2g}+z_1\ =\ \frac{p_2}{\gamma}+\frac{V_2^2}{2g}+z_2+h_f$

Where, hf= head loss between sections (1) and (2)

due to friction

Since the flow is steady and uniform

V1=V2

$\huge \left(\frac{p_1}{\gamma}+z_1\right)\ -\ \left(\frac{p_2}{\gamma}+z_2\right)=h_L$

As per Darcy-Weisbach equation

$\huge h_f=\frac{4f^{\prime}LV^2}{2gD}$

Where, L= Lenth of the pipeline ,   f’ = Friction Coefficient

D= Diameter of the pipe,     V= Average Velocity

• f’ is a dimensionless quantity whose value depends on the roughness coefficient of pipe surface and Reynolds number of the flow

f’ = 16/Re  for laminar flow in pipes (Re<2000)

= 0.079/(Re)0.25  fro turbulent flow in pipes (Re>4000)

## Modified equation of Dracy – Weisbach

Darcy Weisbach formula is the most commonly used formula in the case of pipe flow. This formula is not applied to open channels for the calculation of head loss.

$\huge h_f=\frac{fLV^2}{2gD}$

Where, f= friction factor = 4f’

= 16/Re  for laminar flow in pipes (Re<2000)

= 0.316/(Re)0.25  for turbulent flow in smooth pipes (4000<Re<10^5)

## Chezy’s Formula

Chezy’s formula is applicable to both pipe flow as well as open channel flow.

V= C√RS

Where, C = Chezy’s coefficient, V= Average velocity. R= Hydraulic radius

S= hL/L = Slope of energy line (Hydraulic gradient line)

In the case of pipe flow,

$\huge h_L=\frac{4LV^2}{C^2D}$

By comparing Darcy Weisbach equation and Chezy’s equation

$\huge \frac{fLV^2}{2gD}=\frac{4LV^2}{C^2D}$

$\huge C=\ \sqrt{\frac{8g}{f}}$

## Manning’s Formula

Mannings formula is most common for analysis of open channel flow but often used to analyze pipe problems too.

$\huge V=\frac{1}{n}R^{\frac{2}{3}}S^{\frac{1}{2}}$

Where,
n = Manning’s roughness (or rugosity) coefficient, R = Hydraulic mean radius, S = Slope of H.G.L. (Hydraulic gradient line)

By comparing Manning’s formula with Chezy’s formula, we get

$\huge C=\frac{1}{n}R^{\frac{1}{6}}$

The value of ‘n’ depends on the boundary and the hydraulic gradient line.

## Hazen William’s Formula

This formula is mostly used for designing water supply systems.

$\huge V=0.85C_1R^{0.63}S^{0.54}$

where C1 = Coefficient whose value depends upon the type of boundary