K-Factor formula for fire sprinklers and water mist

In this article, we are looking at the flow of water through an orifice, and we will define the orifice as an opening (with a closed perimeter) in an element of a flow system. For us 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.

In 1644 an Italian physicist Torricelli (a pupil of Galileo and also invented the barometer), discovered that the flow through an orifice varied to the 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√(2gh)

The formula above is theoretical and once we take into account the effects of friction, turbulence and the contraction of the water stream the formula can be simplified to what we know as 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 which we will need to use and as it is so fundamental all fire protection engineers must have a good understanding of how it works. In its most common form, the formula allows us to calculate the discharge flow from the nozzle (fire sprinkler, water mist or a deluge nozzle) if we are given the head pressure and k-factor, we can also calculate the k-factor or the pressure required with this formula.

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

q = kp^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 below: 

k = q / p^0.5

or the pressure as below:

p = ( q / k )²

The units which we use are important and much not be mixed. you much 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 litre 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, and this is why it is important to be familiar with the above formulas. Often K-factors are given as an imperial value in gpm/psi^½ this value cannot be entered into FHC without first converting to its metric equivalent Lpm/bar^½. To convert gpm/psi½ to Lpm/bar^½ we need to multiply by 14.4 (Approximate) Example: A sprinkler head has a discharge coefficient of 4.2 gpm/psi^½ what is its metric equivalent valve. 4.2 x 14.4 = 60.48 Lpm/bar^½.We only need to use K-factors to one decimal place, so 60.48 becomes 60.5 Lpm/bar^½.