Pipe Friction Loss Calculation

Calculating the friction loss in a given pipe system includes two things. Calculation of friction loss in straight pipes and calculating the friction loss in pipe fittings e.g. bends, fittings, valves etc. and equipment. The article will describe how to calculate friction loss in straight pipes and fittings including examples at the end.

The procedure for calculating friction loss in straight pipes described on this page using the Darcy-Weisbach equation is implemented in the Friction Loss Calculator on this page.

Flow regime - Reynolds number

The flow regime in the pipe shall be known in order to calculate the friction loss. This is done by calculating the Reynolds number. The dimensionless Reynolds number is the ratio of inertial forces to viscous forces see more on Wikipedia.

Reynolds number is for flow in pipes defined as:

\begin{align} Re=\frac{wD_H}{\nu} \end{align}

where

\begin{align} D_h & = \text{Hydraulic diameter in }m\\ w & = \text{Mean velocity in } m/s\\ \nu & = \text{Kinematic viscosity in }m^2/s \end{align}

The flow in pipes is normally considered laminar when the Reynolds Number (Re) is below 2300 and turbulent above 2900.

Hydraulic diameter

The purpose of the hydraulic diameter is to make it possible to use the same equations regardless of pipe shape. It is defined as the 4 times the cross sectional area A divided by the perimeter P.

\begin{align} D_h = \frac{4A}{P} \end{align}

For a circular pipe the hydraulic diameter will equal the inner diameter of the pipe

\begin{align} D_h = \frac{4\frac{\pi}{4} D_i^2}{\pi D_i}=D_i \end{align}
Table 1 - Hydraulic diameter for different pipe shapes
Cross section shape Hydraulic diameter Note
Circular Dh=Di The inner diameter
Square Dh=a a is the length of a side
Rectangular duct (Completely filled) Dh=2ab/(a+b) with a being the height and b the width
Annulus Dh=Dout-Din Dout = outer diameter and Din=Inner diameter

Friction loss in straight pipes

Friction Coefficient - Moody Diagram

The friction coefficient shall be determined in order to calculate the friction loss in the straight pipes. It is a function of surface roughness and flow type. One way to determine it is by using The Moody Diagram. It show the Darcy-Weisbach friction factor as function of roughness and Reynolds number and is a quick way to quickly determine the friction factor.

Another possibility is to calculate the friction coefficient using the equations in the next section. This procedure has also been included in an easy to use Friction Coefficient Calculator.

Moody diagram
Moody diagram

Laminar Flow Re<2300

The friction coefficient is for laminar flow i.e. Reynolds number Re<2300 defined as:

\begin{align} f_D = \frac{64}{Re} \end{align}

Smooth Pipes - Blasius Equation

Blasius equation for smooth pipes with no roughness and turbulent flow i.e. ks=0

\begin{align} f_D = 0.316Re^{-0.25} \end{align}

Rough Pipes - Colebrook Equation

Colebrooks equation for rough straight pipes and turbulent flow i.e. ks>0

\begin{align} \frac{1}{\sqrt{f_D }}= 1.74-0.87\ln \left( \frac{2k_s}{D_i}+\frac{18.7}{Re\sqrt{f_D }} \right) \end{align}

This equation has the disadvantage that it shall be solved numerically. See this page on how to solve it including examples or even easier use the calculator on this page.

Typical roughness for pipes

Table 2 - Pipe roughness
Material Roughness ks mm
Welded steel pipe, new 0.03-0.15
Welded steel pipe, with fouling 0.15-3.0
Galvanized steel pipe, new 0.1-0.2
Galvanized steel pipe, with fouling 0.5-1.0
Concrete, coarse 0.25
Concrete, new smooth 0.025
Drawn tubing 0.0025
Glass, Plastic Perspex 0.0025
Iron Cast 0.15
Sewers, old 3
Steel, mortar lined 0.1
Steel, rusted 0.5
Steel, structural or forged 0.025
Water mains, old 1.0

Friction Loss Straight Pipes - Darcy-Weisbach Equation

The friction loss in straight pipes is with the friction coefficient calculated using the Darcy-Weisbach equation:

\begin{align} \Delta p_{fp} = f_D\frac{1}{2} \frac{L}{D_i}\rho w^2 \end{align}

It is also implemented in the Friction Loss Calculator on this page.

Alternative procedure for Water only - Hazen-Williams Equation

An alternative way to determine friction loss in straight pipes is to use the Hazen-Williams equation. It has the advantage that the factor C is independent of the Reynolds Number and consequently the cumbersome procedure of determining the friction coefficient is avoided. The disadvantage is that the procedure is only valid for water, at room temperature and at conventional flow velocities.

\begin{align} S = \frac{h_f}{L} = \frac{10.67Q^{1.852}}{C^{1.852}D_i^{4.8704}} \end{align}

Where:

\begin{align} S & = \text{Hydraulic slope}\\ h_f & = \text{Head loss mWc}\\ L & = \text{Pipe length}\\ Q & = \text{Volumetric flow rate }m^3/s\\ C & = \text{Pipe roughness coefficient}\\ D_i & = \text{Pipe inside diameter} \end{align}

Pipe roughness coefficient

Table 3 - Pipe roughness coefficient to be used together with Hazen-Williams Equation
Material C Factor low C Factor high
Cast iron new 130 130
Cast iron 10 years 107 113
Cast iron 20 years 89 100
Cast iron 30 years 75 90
Cast iron 40 years 64 83
Cement-Mortar Lined Ductile Iron Pipe 140 140
Concrete 100 140
Copper 130 140
Steel 90 110
Galvanized iron 120 120
Polyethylene 140 140
Polyvinyl chloride (PVC) 150 150
Fibre-reinforced plastic (FRP) 150 150

Friction loss in fittings, valves, equipment etc.

There are several ways to calculate the friction loss in fittings i.e. bends, valves, size changes, etc. One way is to treat it as added equivalent length to the straight pipe friction loss calculation and another to calculate the friction loss for each fitting. The first approach will be described here.

\begin{align} \Delta p_{ff} = \zeta _x \frac{1}{2}\rho w_x^2 \end{align}

Where: The x denotes up- or downstream with 1 being upstream and 2 downstream of the fitting.

\begin{align} \zeta & = \text{Resistance Coefficient}\\ \rho & = \text{Density } kg/m^3\\ w_x & = \text{Average velocity } m/s \end{align}
Table 4 - Resistance coefficient for various fittings
Fitting Parameter Resistance Coefficient ζ Note
Bend 90° R/D=1 0.40 ζβ = ζ90°β/90°
R/D=2 0.30
R/D=4 0.28
R/D=6 0.33
Reducer gradual D1/D2=1.2 0.02 ζ refer to w2
D1=inlet diameter
D2=outlet diameter
D1/D2=1.4 0.04
D1/D2=1.6 0.04
D1/D2=1.8 0.05
D1/D2=2.0 0.06
Reducer gradual D2/D1=1.2 0.1 ζ refer to w1
D1=inlet diameter
D2=outlet diameter
D2/D1=1.4 0.2
D2/D1=1.6 0.5
D2/D1=1.8 1.5
D2/D1=2.0 2.5
Ball valve 0.1-0.2 Full bore
Butterfly valve 0.2 Completely open
Gate valve 0.1-0.3 Without flow restrictions
Gate valve 0.3-1.2 With flow restrictions
Gate valve 0.2-2.5 High pressure
Globe valve 2-10 Straight
Globe valve 1-2 Wye type
Globe valve 3-12 Angled
Check valve (swing type) 0.4-1.0
Screw down non-return valve 1-8
Check valve (ball type) 0.5-2

Flow Factor kv and Flow Coefficient Cv

The Flow Factor kv and Flow Coefficient Cv are very similar. The only difference being the units used. The flow factor uses SI-units and is used throughout the world whereas the Flow Coefficient uses imperial units and is mainly used in the United States.

The Flow Factor kv defines the friction loss for water at a density of ρ=1000kg/m3.

\begin{align} k_v & = \frac{q_v}{\sqrt{\Delta p}}\\ \Delta p & =\left( \frac{q_v}{k_v} \right)^2 \\ q_v & = k_v\sqrt{\Delta p} \end{align}

Where (The units are very important in this!!)

\begin{align} k_v & = \text{The } k_v \text{ value with the unit } m^3/h\\ q_v & = \text{The volumetric flow rate } m^3/h\\ \Delta p & = \text{Friction loss in bar} \end{align}

For liquids other than water or for water at a different temperature and consequently different density the equation is changed to:

\begin{align} q_v & = k_v\sqrt{\frac{\Delta p}{\left( \frac{\rho }{\rho _{water}} \right)}} \end{align}

The use of the Flow Coefficient CV (or flow-capacity rating of valve) is identical to the procedure for the Flow Factor only the units are different.

\begin{align} q_v & = C_v\sqrt{\frac{\Delta p}{\left( \frac{\rho }{\rho _{water}} \right)}} \end{align}

Where (The units are very important in this!!)

\begin{align} C_v & = \text{The Flow Coefficient }\\ q_v & = \text{The volumetric flow rate in US gallons per minute}\\ \Delta p & = \text{Friction loss in psi} \end{align}

Conversion is possible between the Form Factor and the Flow Coefficient and vice versa

\begin{align} k_v&=0.865C_v\\ C_v&=\frac{k_v}{0.865} \end{align}

Example

The pipe in this example has an inner diameter of 50mm, is 10m long, it has 10 90° bends with R/D=2 the flow rate is 15m3/h and the water in the pipe has a temperature of 30°C.

Step 1 Determine thermodynamic properties for the fluid in the pipe. In this example the calculator for water is used.

Table 5 - Thermodynamic properties used in the example
Symbol Property Value
ρ Density 996.7kg/m3
ν Kinematic viscosity 0.8 mm2/s=0.8×10-6m2/s

Step 2 is to calculate the Reynolds Number.

The average velocity:

\begin{align} w = \frac{q}{\frac{\pi }{4}D_i^2}= 2.1m/s \end{align}

and the Reynolds number:

\begin{align} Re=\frac{wD_H}{\nu} = \frac{2.1m/s\cdot 0.05m}{0.8\cdot 10^{-6}m^2/s} = 132629 \end{align}

Step 3 is to determine the friction coefficient by either using the Moody Diagram or by calculating it using the Colebrook equation. Steel pipe with some fouling is used in the example and the roughness is estimated to ks=0.5mm based on table 2.

\begin{align} \frac{1}{\sqrt{f_D }}= 1.74-0.87\ln \left( \frac{2k_s}{D_i}+\frac{18.7}{Re\sqrt{f_D }} \right) \end{align}

The friction coefficient is determined to fD=0.0383 by solving the Colebrook equation using one of the methods described here

Step 4 Calculate the friction loss using Darcy-Weisbach Equation in the straight pipe.

\begin{align} \Delta p_{fp} = f_D\frac{1}{2} \frac{L}{D_i}\rho w^2 = 17171Pa = 0.172Bar \end{align}

Step 5 The next thing after having determined the friction loss of the straight pipe is to determine the friction loss in fittings. The 90° bends has a radius of 100mm hence R/D=2 giving a resistance coefficient of λ=0.30 (see table 4). The friction loss for each bend is:

\begin{align} \Delta p_{ff} = \zeta _x \frac{1}{2}\rho w_x^2 = 673.2Pa \end{align}

The total friction loss for the 10 bends is

\begin{align} \Delta p_{ff} = 10\cdot 673.2Pa = 6732Pa = 0.067 Bar \end{align}

Step 6 Calculate the entire friction loss for the pipe including the fittings in this case only 90° bends but normally it also includes valves, reducers, equipment etc.

\begin{align} p_{loss}=0.172bar+0.067bar = 0.239bar \end{align}