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Hydraulic calculations for fire protection engineers

In fire sprinkler design, one of the most important concepts is the principle of water density, yet many fire sprinkler system design engineers do not fully understand the concept, this sort of instruction will hopefully full fill this requirement.

We often refer to Design Density in lazy preference to Design Density of Discharge which, in turn, is a short way of saying Density of Application of Water. This is an unusual use of the word 'density' since we know, of course, that the density of water is 1. However, the density of application means how much water we apply over a certain area, much the same as pressure is a force applied over a unit area.

We are, therefore, talking about a volume of water spread over a certain area in a unit of time.

Volume can be measured in litres

The area can be measured in m^{2}

Time can be measured in minutes

The density of water application would be measured thus:

**Volume / Area x Time or Litre / m ^{3} x min**

It is necessary to bring this formula to a manageable state by changing the units. A Litre of water is defined as a cubic decimetre which is 10 centimetres × 10 centimetres × 10 centimetres (1 L ≡ 1 dm^{3} ≡ 1000 cm^{3}). Hence 1 L ≡ 0.001 m^{3} ≡ 1000 cm^{3} and 1 m^{3} (i.e. a cubic metre, which is the S.I. unit for volume) is exactly 1000 L.

Therefore, we can rewrite the formula:

**dm x dm x dm / 10dm x 10dm x min**

This can be simplified by cancelling out

**dm / 100min = 100mm / 100min = 1mm / 1min = mm/min**

** **

The density of application can, therefore, be measured in millimetres per minute (**mm/min**).

Whichever route you take, it is important to realise that when we use this strange, apparently linear unit, we are talking about a volume of water discharged over an area of 1m^{2} in 1 min.

In the case of Ordinary Hazard Installations (EN 12845) with a Density of Discharge of 5.0 mm/min bearing in mind that this really means 5 L/min^{2}/min then we are applying less than half a bucket full of water on every square metre each minute.

The art, of course, is how you tell it or, in this case, how you apply it. The sprinkler head distributes the water in an even pattern so that in the case of OH3, each of the 12m^{2} covered by the head receives its share of water. When testing a sprinkler head the floor is covered with 1m^{2} trays and after a discharge for 1 minute, there should be water in each tray to a depth of 5mm. The volume of water in the tray would be 5mm x 1000mm x 1000mm = 5,000,000mm^{3}. Since there are (100 x 100 x 100) ie. 1,000,000mm^{3} in 1dm^{3} the volume of water will, of course, be 5dm^{3} or 5 litres.

Taking OH3 as an example, if we design for a maximum of 18 sprinkler heads operating each capable of covering 12m^{2}, then the maximum area of operation will be (18 x 12)m^{2} = 216m^{2}. If each of the 18 sprinkler heads is discharging 5dm^{3}/m^{2} every minute then we will require a flow of (5 x 18 x 12)dm^{3}/min = 1080dm^{3}/min. In calculating pipe sizes this is approximated to 1000dm^{3}/min.

Now we know the theory of 'Design Density' we can use it in fire sprinkler hydraulic calculations to find the quantity of water required to flow from a fire sprinkler. If we know the area a sprinkler head is covering and the required design density then we can use the following formula:

**Area x Density = Quantity**

Therefore if we have a fire sprinkler head which is covering 8m2 and we require 12.5 mm/min

**8m ^{2} x 12.5 mm/min = 100 Litres/min**

This would be the minimum flow rate required for the sprinkler head to prove the correct Design Density. The specific design density to be used for design purposes is determined by reference to the occupancy fire hazard of the building once this is known the applicable design standard such as EN 12845, BS 9251 or NFPA 13 will have tables of occupancies from which you can find the required design density.

Let’s take an example, if we are designing a fire sprinkler system and we know that we required a design density of 15.0 mm/min over the design area then this is the starting point for our fire sprinkler hydraulic calculation.

We now need to take the first most remote fire sprinkler on the branch line (range pipe) and find the actual area the fire sprinkler is covering let’s assume it’s 7.5m^{2} for this example. We can now calculate the minimum flow rate from the fire sprinkler to provide the required design density, this can be found from the equation:

Q = D x A

When:

Q = flow rate from the fire sprinkler

D = Design density

A = Area of coverage for the fire sprinkler

For our example:

15 mm/min x 7.5m^{2} = 112.50 L/min

We now know that the most remote fire sprinkler must have a flow rate that is equal to or greater than 112.50 L/min. The next step in the calculation is to check that we can get this flow rate from the fire sprinkler and what pressure will be required.

We hope you have found this short introduction to water design density to be informative and that it will help you understand one of the fundamental concepts of sprinkler system design.

The litre is the SI derived unit for volume which is the volume of a cube with 10 cm sides and has the symbol L or l. A decimeter (dm^{3}) is 1,000 cubic centimetre(cm^{3}) or 1/1000 of a cubic metre.

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Leeds, West Yorkshire

United Kingdom LS2 8AJ

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