Minor losses - Formulas

 Here is a list of formulas for head loss $h_L$ estimation in pipe systems. Some comments are also given for the sake of make the information as useful as possible.

Losses due to friction between pipe wall and fluid

As the fluid travels through a pipe, mechanical energy or momentum is lost due to friction. Friction occurs due to pipe wall roughness so that pipes with smooth walls such as PVC or copper pipes may produce low head loss. However, pipes with high wall roughness such as steel pipes, for example, will produce higher head losses.

You should notice that any pipe fitting would cause head loss due to friction but its contribution is considered in a separate fashion.

The way in which head losses $h_L$ due to friction is by using the Darcy equation.

The head loss $h_L$ in pipe sections

For this case you need the Darcy equation expressed as

$h_L=f_F\dfrac{L}{D}\dfrac{v^2}{2g}$        Eq. (01)

where

$f_F$ is the friction factor,

$L$ is the pipe length,

$D$ is the internal pipe diameter,

$v$ is the fuid velocity,

$g$ is the acceleration due to gravity.

The friction factor $f_F$ is determined from the Colebrook Equation once the Reynolds number $N_{Re}$ is calculated and the corresponding pipe roughness $\epsilon$ found in tables. 

Common values of $\epsilon$.

Pipe material	Roughness $\epsilon$ (m)	Roughness $\epsilon$ (ft)
Glass	Smooth	Smooth
Plastic	$3.0 \times 10^{-7}$	$9.8 \times 10^{-7}$
Drawn tubing; copper, brass, steel	$1.5 \times 10^{-6}$	$4.9 \times 10^{-6}$
Steel, commercial or welded	$4.6 \times 10^{-5}$	$1.5 \times 10^{-4}$
Galvanized iron	$1.5 \times 10^{-4}$	$5.0 \times 10^{-4}$
Ductile iron - coated	$1.2 \times 10^{-4}$	$4.0 \times 10^{-}$
Ductile iron - uncoated	$2.4 \times 10^{-4}$	$8.0 \times 10^{-4}$
Concrete, well made	$1.2 \times 10^{-4}$	$4.0 \times 10^{-4}$
Riveted steel	$1.8 \times 10^{-4}$	$6.0 \times 10^{-3}$

See the post Solving the Colebrook equation.

Equation (01) only applies for sections of straight pipes.

The head loss $h_L$ in fittings

Examples of fittings are

• valves
• socket or joints (pipe section connectors)
• elbows,
• filters,
• etc.

Each of these fittings have variations. For example, there are several types of valve designs

• butterfly valve,
• ball valve,
• globe valve,
• needle valve,
• check valve,
• etc.

The same occurs with elbows and other accesories.

Since head loss $h_L$ is different for every type of fitting a most complete  and trustable source of data is required.

The head loss $h_L$ for this cases is calculated from a variation of the Darcy Eq. (01). This is

$h_L=f_{F,T}\left( \dfrac{L}{D} \right)_E \dfrac{v^2}{2g}$    Eq. (02)

where the subscript $T$ stands for turbulent flow regime which means that the Colebrook equation can be easily reduced to

$\dfrac{1}{\sqrt{f_{F,T}}}=-2\log_{10}\left( \dfrac{\epsilon}{3.72D} \right)$    Eq. (03)

$\Rightarrow f_{F,T}=\dfrac{0.25}{\left[ \log_{10}\left( \dfrac{\epsilon}{3.72D} \right) \right]^2}$    Eq. (04)

As you can see the friction factor for fully turbulent regime can now be easily found from Eq. (04).

On the other hand, the subsript $E$, in EQ. (02), stands for equivalent. This concept is related to the equivalent head loss that may be produced by a section of straight pipe. Speaking of equivalent legnth $L_E/D_E$ help to understand the magnitud of the head loss caused by a given fitting.

The ratio $L_E/D_E$ is usually given in tables. Formulas can be provided as well.

Equation (02) is also written as

$h_L=K\dfrac{v^2}{2g}$      Eq. (05)

where

$K=f_{F,T}\left( \dfrac{L}{D} \right)_E$       Eq. (06)

Equation (06) and Eq. (02) are exactly the same. The only difference is that now $K$ needs to be determined from tables instead. $K$ is also known as the resistance coefficient.

For elbows

• $K=50\, F_{F,T}$ for 90$^\circ$ screwed elbow
• $K=30\, F_{F,T}$ for 90$^\circ$ standard elbow
• $K=20\, F_{F,T}$ for 90$^\circ$ long radius elbow
• $K=26\, F_{F,T}$ for 45$^\circ$ screwed elbow
• $K=16\, F_{F,T}$ for 45$^\circ$ standard elbow
• $K=60\, F_{F,T}$ for 90$^\circ$ square elbow (ell)

Typical elbows

For inlets

This is, the resistance due to pipe connection to a feeding reservoir.

• $K=0.5$ for square-edged inlets,
• $K=1.0$ for inward projecting pipes,
• for smoothed inlets (see table below)

Type of pipe inlets

Data table for K in smoothed inlets

$r/d$	$K$
0.00	0.50
0.02	0.28
0.04	0.24
0.06	0.15
0.1	0.09
0.15 & up	0.04

For valves

1. $K= 340\, f_{F,T}$ for fully open globe valve
2. $K= 150\, f_{F,T}$ for fully open angle (globe) valve
3. $K= 8\, f_{F,T}$ for fully open gate valve
4. $K= 35\, f_{F,T}$ for 3/4 open gate valve
5. $K= 160\, f_{F,T}$ for 1/2 open gate valve
6. $K= 900\, f_{F,T}$ for 1/4 open gate valve
7. $K= 45\, f_{F,T}$ for 2-8 in butterfly valve
8. $K= 35\, f_{F,T}$ for 10-14 in butterfly valve
9. $K= 25\, f_{F,T}$ for 16-24 in butterfly valve
10. $K= 420\, f_{F,T}$ for check valve (shaft type)
11. $K= 75\, f_{F,T}$ for check valve (disc type)

Any question? Write in the comments and I shall try to help.

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Ildebrando.