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. 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 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 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 essential 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, so being familiar with the above formulas is essential.

### 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 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^{½} what would be the metric equivalent valve. 4.2 x 14.275 = 59.955 Lpm/bar^{½}. 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 imperil sprinkler head k-factor conventions to the metric equivalent.

K-factor gpm/psi^{½} | k-factor L/min/bar^{½} |

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 would strongly advise that you obtain the metric K-factor from the manufacturer where ever possible.

## Typical k-factor values for sprinkler and water mist heads

For many standard types of sprinklers, the design standards such as EN 12845 & NFPA 13 specify the standard k-factors and minimum pressure, which should be used for different Hazard classifications and design densities. For all other types of sprinkler heads, the manufactures data sheet should be referred to for the k-factor and minimum head pressure.

Hazard Class | Design Density (mm/min) | K-factor | Minimum Pressure (bar) | |

1 | Light Hazard | 2.25 | 57 | 0.70 |

2 | Ordinary Hazard | 5.00 | 80 | 0.35 or 0.50 |

As a designer, you must check the k-factor value for the nozzle or head manufacturer and ensure its application is correct. You should also seek guidance from the design standard which is applicable.

## The relationship between the k-factor, pressure and flow

The graph below shows the relationship between the k-factor, pressure and flow. You can clearly see from this that the flow from the head or nozzle increases for the same pressure with a high k-factor nozzle. We can sometimes use this to our advantage by selecting the correct sprinkler head k-factor to provide the design density required with the minimum energy requirement (water pressure).

As an example, if we have the pressure of 1.50 Bar and a k-factor of 50 the flow rate would be 61.20 L/min for a k-factor of 100 it would be 122.50 L/min and with a k-factor of 150, the flow rate would be 183.70 L/min. You can see that the flow has insecure by about 61 L/min each time we changed the k-factor by 50, this is because the pressure has remained the same at 1.50 Bar and if you look at the k-factor formula above the pressure is squared which will give us 1.225, this is then multiplied by the head k-factor in this case 50, 100 and 150 in our example. As the k-factor has increased by 50 each time the increase in flow has also increased by the same amount.

If you like this article, you may also be interested in our online K-factor calculator.