# Canute

## Bernoullis Theorem

Bernoulli's theorem is a method of expressing the law of conservation of energy to the flow of fluids.

Bernoulli's principle stats that, in the flow of fluid (a liquid or gas), an increase in velocity occurs simultaneously with decrease in pressure. That statement is a simplification of Bernoulli's equation (below) which plots the situation at any point on a streamline of the fluid flow and applies the law of conservation of energy to flow. Put another way, the total energy of the flow at any point along a horizontal pipe is equal to the sum of the pressure head, the velocity head and the elevation in the absence of friction. This is a principle of considerable importance to those concerned with flow in sprinkler pipework

When

z = Potential head or elevation
p = Pressure
v = Velocity
g = Acceleration of gravity
d = Density of fluid

If friction losses are ignored and no energy is added or removed from the pipe the total head h, in the above equation will be constant for any point in the fluid. However in practice energy will increase and decrease with the effect of pumps and friction loss and this must be included in the Bernoulli's equation. All practical formulas for the flow of fluids are derived from the Bernoulli's theorem with modifications to account for losses due to friction.

## K-Factor formula for fire sprinklers

When we start any hydraulic calculation for a water based fire protection systems such as fire sprinklers, water mist, hose reel and deluge systems the k-factor formula is one formula which all fire protection engineers must know and understand. It allows us to calculate the discharge flow from any type of nozzle (fire sprinkler, water mist or a deluge nozzle) for which we have a k-factor. We can also calculate the k-factor for any nozzle if we have not been given one, however you must check with the manufacture that this is acceptable.

The k-factor formula is the start of all hydraulic calculation for fire protection systems for both manual and computerized calculations and is also required for the checking of both types.

The discharge from a sprinkler head or nozzle can be calculated from the formula bellow:

q = kp0.5

when q = flow in L/min
k = nozzle discharge coefficient or k-factor for head in Lpm/bar0.5
p = pressure in bar

This formula can be rewritten to give us:

k = q / p0.5 and p = ( q / k )2

Our Hcalc Hydraulic Calculator will allow you to explore the K Factor formula in more detail and will allow you to calculate the flow, pressure or find the k factor for a nozzle or fire sprinkler. You can freely download and use our Hcalc software.

For standard type sprinkler heads the many design standards specify standard k-factors and minimum pressure, which can 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.

We 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. In fact 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/psi½ this value cannot be entered into FHC without first converting to its metric equivalent Lpm/bar½. To convert gpm/psi½ to Lpm/bar½ we need to multiply by 14.4 (Approximate)

Example: A sprinkler head has a discharge coefficient of 4.2 gpm/psi½ what is its metric equivalent valve. 4.2 x 14.4 = 60.48 Lpm/bar½.

We only need to use K-factors to one decimal place so 60.48 becomes 60.5 Lpm/bar½.

## Velocity in pipe

Some fire sprinkler design authorities such as EN 12845 limit the velocity through pipes and valves in fire sprinkler systems; this is the case with EN 12845 however NFPA and FM do not have any restriction. The case for limiting velocity is that the Hazen-Williams formula is less accurate outside its normal range and equivalent pipe lengths for fittings, which are generally used, start to lose their validity. Some authorities believe that velocity is self-limiting as pressure losses increase exponentially as velocities increase, so pipe sizes must be increased to make use of available water supply pressure.

EN 12845 limits velocity to 6 m/s through valves and flow switches and 10 m/s at any other point in the system.
Velocity in pipe can be calculated using the following formula:-

The following table lists the maximum flows in litres per minute which can be obtained through steel pipework to EN 10255 specifications for both 6 m/s and 10 m/s.

Maximum flows through pipes for EN 12845 fire sprinkler systems

## The Hazen Williams formula for use in fire sprinkler systems

The Hazen Williams formula is an empirical equation and has long been used for calculating the friction loss in pipework for water based fire sprinkler protection systems. This equation uses the coefficient C to specify the pipes roughness, which is not based on a function of the Reynolds number, as in other pressure loss equations. This however has the disadvantage that the equation can only be used when water is flowing within the 'turbulent' flow range. If the system is outside the normal pressure and flow range or the system is to use additives, or will be subject to unusual temperature conditions then the Darcy Weisbach equation may be more appropriate.

The Hazen Williams formula has the advantage of been simple to calculate by using a scientific calculator where as the Darcy Weisbach equation requires the use 'f' friction factor and this can only be calculated by an number of iterations as 'f' is on both sides of the equation. You can use a Moody diagram to find the value of 'f' however this is both time consuming and almost certainly and inaccurate method.

The Hazen William formula has now become adopted through the world as the pressure loss formula to use for the hydraulic design of fire sprinkler systems and in almost all cases the use of the hazen william formula will provide adequate answers. The Hazen William formula can also be used for the calculation of water mist systems where the system pressure does not exceed 12 bar (low pressure water mist systems) or the water velocity does not exceed 7.6 m/s and the minimum pipe size is 20mm in the case of intermediate and high pressure water mist systems.

You can use Canute Hcalc software (hydraulic calculator) to visually explore the relationship between the flow, pipe diameter and the pipe c-factor in the Hazen Williams formula which will give you a good understanding of the formula. The Hcalc software is free to download and use.

when p = pressure loss in bar per meter
Q = flow through the pipe in L/min
C = friction loss coefficient
d = internal diameter of the pipe in mm

## Value of C for use in the Hazen-Williams formula

Listed in the table bellow are typical values for the coefficient C, which can be used in the Hazen-Williams formula for different fire sprinkler design standards. The value of C represents the pipes roughness with higher values of C giving lower friction losses. The values given in the design standards allow for degradation of the pipe, for instance new cast iron pipe has a C coefficient of 130 and EN 12845 gives the value of 100, this is equivalent to a pipe, which is about 20 years old

## Glossary of hydraulics for fire protection

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

B

• Barometer: An instrument used for the measurement of atmospheric pressure
• Bar: Is a unit of pressure 1 bar = 10000 pascal (Pa).  The bar is used through Europe for the measurement of pressure in fire sprinkler systems.
• Booster pump: A fire pump used to boot the pressure of the existing water supply
• Bourdon gauge: The most common device used to measure system pressures.

C

•  Centrifugal pump: Modern industrial fire pumps are centrifugal pumps.  Pressure is added to the water by the centrifugal force created by a rotating wheel (impeller) or the rotating vanes of a turbine.
• Certified shop test valve: Before a pump is shipped by the manufacturer, it will be tested in the shop.  The results of this test will be plotted on graph paper.
• Circulation relief valve: A small relief valve that opens up and provides enough water flow into and out of the pump to prevent the pump from overheating when it is operating at churn against a closed system.
• Complex loop:  A piping system that is sometimes called a "grid" and is characterised by one or more of the following:  more than one inflow point, more than one outflow point, and/ or more than two paths between inflow and outflow points.
• Controller:  The electric control panel used to switch pump on and off and to control its operation.

D

• Darcy -Weisbach:  Technique used to establish the pressure lost to friction in a piping system.
• Differential manometer: A device whose primary application is to reflect the differences in pressures between two points in a system.

F

•  Flow hydrant: The hydrant from which the water is discharged during a hydrant flow test.
• Flow test:  Tests conducted to establish the capabilities of water supply systems and referred to as flow tests because they involve flowing fire hydrants.  The objective of a flow test is to establish quantity ( gallons per minute) and pressures available at a specific location on a particular water supply system.
• Fluid:  Any substance that can flow; a substance which has definite mass and volume at constant temperature and pressure, but no definite shape; and with the inability to sustain shear stresses.
•  Fluid mechanics: In the general terms of physics, force is that which causes motion.

• Force: In the general terms of Physics, force is that which causes motion.

G

H

•  Hardy cross method: An interactive technique used for solving the complicated problems involving gridded water supply systems.
• Hazen-Williams formula: An empirical formula for calculating friction loss in water systems that is the fire protection industry standard.  To comply with the most nationally recognised standards, the Hazen-Williams formula must be used.
• Head: Pressure expressed in units of feet of water.
• Horizontal split -case pump: A centrifugal pump with the impeller shaft installed horizontally and often referred to as a split-case pump.  This is because the case in which the shaft and impeller rotates is split in the middle and can be separated exposing the shaft, bearings and impeller.
• Hydraulics: The branch of fluid mechanics dealing with the mechanical properties of liquids (in the text water) and the application of these properties in engineering.
• Hydrokinetics: A branch of hydraulics having to do with liquids (water) in motion, particularly in relation to forces created by or applied to the liquid in motion.
• Hydrostatics: A branch of hydraulics dealing with the properties of liquids (water) at rest, particularly in relation to pressures resulting from or applied to the static liquid.

J

• Jockey pump: A jockey pump is a small capacity, high pressure pump used to maintain constant pressures on the fire protection system.  A jockey pump is often used to prevent the main pump from starting unnecessarily.

K

•  Kinematic viscosity:  The kinematic viscosity of a fluid is the ration of its absolute viscosity (lb sec/ft2) to its mass density (lb sec2/ft4).
• Kinetic energy: The energy which a body possesses because of its motion.

L

• Laminar flow:  A fluid is in the state of laminar flow if its Reynolds number is 2,100 or less; laminar flow is related to very low liquid velocities.
• Liquid:  A fluid having a definite volume, unlike gases, which will expand to fill the vessel containing it.

M

• Moody diagram: A Diagram used with the Darcy-Weisbach friction loss computation technique to relate the Reynolds number, pipe size, and roughness to a friction factor.

N

• Net pressure:  The net pressure is the pressure added to the system  by the pump.

O

• Orifice plate meter:  An orifice plate meter is a device used for measuring water flow and is similar in principle to a Venturi meter.  The change of water velocity is accomplished by using a plate with an orifice that is smaller than the diameter of the pipe in which it is placed.

P

•  Pascal's law:  Principle 1, known as Pascal's law, points out that pressure acts in all directions and not simply downward.
• Pascal: The SI unit for pressure is the pascal (Pa) which is equal to one Newton per square meter (N/m2).  For fire protection this measurement of pressure is small so the unit Bar or kPa is used in most part of the world.

• Piezometer tube: This device uses the heights of liquid columns to illustrate the pressures existing in hydraulic systems.
• Pitot tube:  Common device used to measure velocity pressure and thus fluid velocity.  The pitot tube consists of a small diameter tube, usually about one-sixteenth inch in internal diameter which is connected to a pressure gauge.
• Potential energy: Stored energy which has the ability to perform work once released.
• Pressure: is the force per unit area (symbol P).
• Pressure head: Is a term used in fluid mechanics to represent the internal energy of a fluid due to the pressure exerted on its container. It may also be called static pressure head or simply static head.
• PSI: In fire protection, pressure is most often dealt within units of pounds per square inch (psi).

R

• Relief valve: The relief valve is provided to open up and discharge water to a drain should the pressure become excessive.  This valve is located between the pump and the discharge check valve and is required with pumps driven by variable speed drivers.
• Residual pressure:  The pressure at the test hydrant while water is flowing.  It represents the pressure remaining in the system while the test water is flowing.
• Reynolds number: is a dimensionless number that state if the flow is in a laminar or turbulent stat (Symbol Re) .

S

• Simple loop:  A loop in which there is exactly one inflow point and one outflow point, and exactly two paths between the inflow and outflow points.
• Specific gravity: The specific gravity (Sg) of a substance may be defined generally as the ratio of the weight density of the substance to the weight density of another substance, usually water.
• Static pressure:   The normal pressure existing on a system before the flow hydrant is opened.

T

• Total energy:  The total energy (TE) at any point in a system might be defined as the sum of the potential energy and kinetic energy at that point.
• Turbulent state:  Fluid flow is in the turbulent state higher velocities where there is no definite pattern to the direction of the water particles.  Turbulent flow is reflected by a calculated Reynolds number in excess of 2,100.

V

• Venturi meter:  When coupled with a differential manometer, a venture meter may be used to measure water velocity.  The device consists essentially of a piece of pipe in which the cross-sectional area has been constricted.

W

• Water hammer:  Stopping any flowing stream too rapidly can cause a phenomenon called water hammer.  Water hammer is a violent increase in pressure which can be large enough to rupture the piping.