K-Factor formula for fire sprinklers and water mist

In this article, we examine the flow of water through an orifice, and we define an orifice as an opening (with a closed perimeter) in an element of a flow system. This orifice will be a fire sprinkler head or water mist nozzle in a fire protection system. We can use the k-factor formula for almost any rounded orifice.

Background and Theory

The concept of flow through an orifice dates back to 1644, when Italian physicist Evangelista Torricelli (a pupil of Galileo and also inventor of the barometer) established that flow velocity through an opening varies with the square root of the pressure head. This principle underpins the K-factor formula used in fire protection hydraulics.

Torricelli discovered that the flow through an orifice varied with the square root of the pressure and later determined the following basic relationship:

$q =av$

When:
q = flow from the orifice
a = cross-sectional area of the orifice
v = velocity

This has led to the accepted theorem for flow through a round orifice:

$q =a\sqrt{\left(2gh\right)}$

The formula above is theoretical, and once we account for friction, turbulence, and water-stream contraction, it can be simplified to the k-factor formula for fire protection systems by reducing its complexity to a single constant "k".

The K factor formula for fire protection  

When we start any hydraulic calculation for water-based fire protection systems, such as fire sprinklers and water mist systems, the k-factor formula is the first one we will need to use. As it is so fundamental, all fire protection engineers must understand how it works. The formula calculates the discharge flow from the nozzle (fire sprinkler, water mist or deluge nozzle) in its most common form. If we are given the head pressure and k-factor, we can also calculate the k-factor or the required pressure using this formula.

The discharge from a sprinkler head or water mist nozzle can be calculated from the formula below:

$q = k{p}^{0.5}$

When:
q = flow
k = nozzle discharge coefficient or k-factor for head 
p = pressure

We can rewrite the formula to give us the k-factor as follows: 

$k =\frac{q}{p^{0.5}}$

Or the Pressure as below:

$p =(\frac{q}{k}{)}^2$

The units which we use are essential and must not be mixed. You must also be very cautious with the k factor and ensure that you get the correct value for a metric or imperial calculation. The units for both are given below:

For metric calculations:

p = Pressure in bar
q = flow in litres per minute 
k = discharge constant Lpm/bar^0.5

And for imperial calculation: 

p = pressure in psi
q = flow in gpm
k = discharge constant gpm/psi^0.5 

We can also use K-factors for many other applications in fire hydraulics, such as flow from a fire hydrant, wet riser outlet, hose reel or foam monitor. The list is almost endless, so being familiar with the above formulas is essential.

Units and Conversions

It is critical to use consistent units when applying the K-factor formula. The two common unit systems are:

Metric: ( p ) in bar, ( q ) in litres per minute (L/min), ( k ) in L/min/bar(^{0.5})
Imperial: ( p ) in pounds per square inch (psi), ( q ) in gallons per minute (gpm), ( k ) in gpm/psi(^{0.5})
Mixing units can lead to significant errors and potentially unsafe designs.

Metric and Imperial K-factor and conversion

Often, K-factors are given as an imperial value in gpm/psi^½. This value cannot be entered into Canute FHC without first converting to its metric equivalent L/min/bar^½. To convert gpm/psi½ to L/min/bar^½, we need to multiply by 14.275 to ascertain an approximate value.

As an example, A sprinkler head has a discharge coefficient of 4.2 gpm/psi. What would be the metric equivalent valve: 4.2 x 14.275 = 59.955 Lpm/bar. We only need to use K-factors to one decimal place, so 59.955 would become 60.0 Lpm/bar^½. 

The table below shows the conversion of some typical imperial sprinkler head k-factor conventions to the metric equivalent.

K-factor gpm/psi½	k-factor L/min/bar½
2.8	40.0
4.2	60.0
5.6	80.0
8.0	114.0
11.2	160.0
14.0	200.0
25.2	360.0

We strongly advise obtaining the metric K-factor from the manufacturer wherever possible. 

Units and Conversions

It is critical to use consistent units when applying the K-factor formula. The two common unit systems are:

Metric: ( p ) in bar, ( q ) in litres per minute (L/min), ( k ) in L/min/bar(^{0.5})
Imperial: ( p ) in pounds per square inch (psi), ( q ) in gallons per minute (gpm), ( k ) in gpm/psi(^{0.5})
Mixing units can lead to significant errors and potentially unsafe designs.