Fire Sprinklers.

Hydraulics and Fluid Mechanics The Bernoulli Equation The Hydrant Flow Equation Hydraulic Simulation Models Hydraulic Calculations and Fire Sprinkler Design Computers as Hydraulic Models Glossary

Hydraulics and Fluid Mechanics

Hydraulics and Fluid Mechanics is that branch of applied mechanics dealing with the behavior of fluids at rest and in motion. Here is a synopsis of important concepts and ideas:

Water exerts hydrostatic pressure because it weighs 62.4 lbs per cubic foot i.e. a container 1 foot long by 1 foot wide and 1 foot deep filled with water exerts 62.4/144 = 0.4333 psi at the base of the container. Does that sound familiar ? So a 100 foot column of water exerts a pressure of 43.3 psi at it's base. Conversely a pressure of 43.3 psi is equivalent to 43.3·(1/0.43333) = 2.31·43.31 = 100 ft head of water. More on this later in connection with the Toricelli equation.

Water exerts elevation pressure by virtue of it's height above a given datum e.g. sea level. A 50 foot deep tank filled with water and located on a hill 50 foot above sea level, would exert the same pressure as the 100 foot column of water i.e. 43.3 psi. That makes hill tops ideal for locating water tanks.

Water flows in response to velocity pressure (v ²/2g) where v is the velocity in ft/sec and g is 32.2 ft/sec² or the constant of gravitational acceleration. The pitot pressure from a hydrant flow test is a direct measure of the velocity pressure as expressed by Toricelli's Law: h = n²/2g = 2.31·P ft where:
h is a height of water, expressed in ft of head.
v is the velocity of flow, expressed in ft/sec.
g is the constant of gravitational acceleration, equal to 32.2 ft/sec².
P is the pitot pressure, expressed in psi.
Using the Bernoulli equation and Toricelli's Law, the flow may be expressed as a function of pressure P and outlet diameter D as shown in Figure 1.

The Bernoulli equation ties all the different elements of pressure together from the source to the most remote sections of the sprinkler system. We must assume that the equations for continuity, steady state and impulse momentum apply i.e. ideal conditions that are not complicated by turbulence.

Friction losses occur as water flows through fire sprinkler piping. Part of the friction loss is from direct contact with the interior of the pipe, fittings and devices such as valves. Part of it is due to turbulence within the fluid. The smoother the pipe the lower the friction loss. The Hazen-Williams equation is simple and works quite well in estimating friction loss under conditions of smooth or laminar flow. The Darcy-Weisbach equation is more complicated but it is much better at estimating friction losses when flow is not smooth i.e. turbulent flow.

The flow from a discharging sprinkler is estimated by multiplying the sprinkler nozzle coefficient k by the square root of the pressure at the nozzle i.e. Q = kÖP, where Q is the flow expressed in gpm, k is the sprinkler nozzle coefficient expressed in dimensionless units and P is the sprinkler nozzle pressure expressed in psi.

A hydraulic calculation is a physical accounting system, very much like a checkbook, that balances energy income and expenditure. The Bernoulli equation is a mathematical expression of that balance.
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The Bernoulli Equation

        The physical principles governing flow in pressure conduits was developed by Daniel Bernoulli (1700-1782) and published in his book Hydrodynamica in 1738. The Bernoulli equation is used to analyze fluid flow along a streamline from a location 1 to a location 2 and is expressed as:
z₁ + r₁/g + n₁²/2g = z₂ + r₂/g + n₂²/2g + h_L where:
z₁ & z₂ are the elevations at locations 1 and 2, also called the elevation head, expressed in ft of water.
r₁ & r₂ are the pressures at locations 1 and 2, expressed in lbs/ft².
g  is the specific density of water, equal to 62.4 lbs/ft³.
n₁ & n₂ are the velocities at locations 1 and 2, expressed as ft/sec.
g is the constant of gravitational acceleration, equal to 32.2 ft/sec².
        The last term, h_L, is the Darcy-Weisbach estimate of headloss due to friction: h_L = f·(L/D)·n²/2g where:
h_L is the headloss due to friction, expressed in ft.
f  is the Moody friction factor, expressed in dimensionless units.
n  is the flow velocity, expressed in ft/sec.
L is the length of pipe connecting locations 1 and 2, expressed in ft.
D is the pipe internal diameter, expressed in ft.
         The Moody f is a function of kinematic viscosity, relative roughness and the Reynolds Number. While this method includes the effects of turbulence at high flow velocities, it is more complicated to use than the popular Hazen-Williams equation.
        Bernoulli was a friend and intellectual rival of the legendary physicist Sir Isaac Newton. J.L. Lagrange (1736-1813) was the first to present the modern form of the equation in his book Mechanique Analytique published in 1788. Lagrange also listed the assumptions required for the Bernoulli equation to work. The Bernoulli equation assumes that the fluid and device meet four criteria:
1. The fluid is incompressible
2. The fluid is inviscid
3. The flow is steady
4. The flow is along a streamline
        Most liquids meet the incompressible assumption and many gases can even be treated as incompressible if their density varies only slightly from 1 to 2. The steady flow requirement is usually not too hard to achieve for situations typically analyzed by the Bernoulli equation. Steady flow means that the flow rate (i.e. discharge) does not vary with time. The inviscid fluid requirement implies that the fluid has no viscosity. All fluids have viscosity although viscous effects are negligible over short distances. The Bernoulli equation is important because it incorporates the major parameteric components used to describe and analyze the flow of any fluid through pipes or conduits under pressure.
        In the Bernoulli equation, fluid flow occurs in response to differences in pressure between two points in a flow stream. Since our experience and intuition often confirms that water tends to flow downhill, the difference in elevation z (ft), between two points in the flow stream is the first component. The fluid column pressure r (lbs/ft²) is the second component. It is divided by the specific weight of the fluid g (62.4 lbs/ft³), to make it consistent with the linear units used in expressing the elevation component. The sum of the two is referred to as the piezometric head or the hydraulic grade line. The third component in the Bernoulli equation is velocity pressure n²/2g. The sum of the three components, the total head available to the fluid, is called the energy grade line. There is also a head loss component h_L which appears on the right side of the equation. We looked at how the Darcy-Weisbach method is used to estimate the headloss. We next look at the Hazen-Williams equation.
        The Hazen-Williams equation is expressed as:

p = 4.52·Q^1.85/C^1.85·d^4.87
where:
p is the friction loss, expressed in psi/ft.
Q is flow, expressed in gpm.
C is the Hazen-Williams C-factor, expressed in dimensionless units.
d is the pipe diameter, expressed in inches.
        The C-factor measures the internal smoothness of the pipe. The higher the C-factor, the lower the headloss due to friction. The C-factor for new pipes is published in product brochures.

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The Hydrant Flow Equation

The procedure for deriving the equation of flow from a hydrant is similar to the one used to determine the flow from an orifice of diameter D ft located at a depth of h ft below the water surface in the water container shown in Figure 1. We will use a modified Bernoulli equation to derive the equation for estimating the flow from a fire hydrant. We begin by writing the Bernoulli Equation between the water surface located at 1 and the orifice located at 2. We then replace z with h and since the water in Figure 1 does not flow through any piping, the pipe length L equals 0 and the headloss term reverts to 0:

Figure 1: The Hydrant Flow Equation

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Hydraulic Simulation Models

A hydraulic simulation model defines and formalizes the linkages between variables that affect the distribution of flow and pressure in a fire sprinkler piping network. It represents a compromise between theoretical complexity and practical reality to the point where the model can be used as a tool for predicting how changes in the design variables will affect the performance and viability of the fire sprinkler system. The model must recognize and accept three major types of data :

Hydrant flow test results are probably the single most important type of data affecting the model since most fire sprinkler systems rely on public water supplies. Water storage tanks and pumps may be required in areas where hydrant tests indicate low flows and pressures.
Design criteria have an important impact on the model. An Ordinary Hazard Type II design specifies a much higher density and size of design area than one based on Light Hazard. The design criterion seeks to ensure that the fire sprinkler system will deliver the water flow and pressure required to suppress or extinguish the fire. There may also be special requirements specified by insurance underwriters or local authorities having jurisdiction e.g. restrictions on allowable maximum velocity.
Piping Network Geometry and Composition i.e. "Nodes" and "Pipes" play a very important role. The model must precisely define all the nodes and pipes to be included in the hydraulic calculation. Node numbers describe specific points throughout the piping system to be calculated and carry information on elevations, sprinkler K-Factors and inside hose stream. Pipe numbers describe beginning and ending nodes, pipe diameters, pipe lengths, fittings, equivalent lengths and C-Factors. The beginning and ending nodes may define trees, loops, grids, compound grids and hybrid piping systems.

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Hydraulic Calculations and Fire Sprinkler Design

Hydraulic calculations are based on the concepts and principles of hydraulics and fluid mechanics expressed in the form of a hydraulic model that incorporates a sequence of pre-defined physical accounting procedures :

Determine the type of hazard and select values for density and area per sprinkler that are applicable to the design area. Density multiplied by area per sprinkler gives the flow required at the most remote sprinkler. Use this value along with the sprinkler K-Factor to check the minimum starting pressure, which should exceed 7 psi.
Select the next point upstream, calculate the pressure due to elevation and friction loss and add this to the minimum starting pressure at the most remote sprinkler. Extend this process to each and every node all the way back to the source node.
If there are loops and grids along the way, route the flow through all the pipe segments in accordance with the principles of hydraulics and fluid mechanics.
Add fixed pressure devices such as detector checks and valves to derive the total sprinkler pressure and flow at the source node.
Compare the total sprinkler pressure and flow with what is available on the water supply graph as determined by a hydrant flow test. If the total sprinkler pressure exceeds what is available on the graph, modify the design by increasing pipe sizes or changing the geometry of the piping network and repeat the calculation until the total sprinkler pressure is accomodated by the graph.
A hydraulic calculation may well be the single most important activity associated with designing a fire sprinkler system. The success of a design hinges almost entirely on the extent to which discharging sprinklers can meet the density and flow requirements specified in the design criteria. If the calculations fail, or are in error, the design must be rejected. Hydraulic calculations are usually performed at the bid or preliminary stage of a fire sprinkler project. The results could determine both the physical and economic viability of the sprinkler project. Hydraulic calculations must be done "in house" to ensure flexibility and control over the design process.

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Computers as Hydraulic Models

In the days when hydraulic calculations were done "by hand", piping geometry was a critical element. Complex loops, grids and compound grid systems were often beyond the capacity of most humans, even those armed with calculators. In fact grid systems had to wait until Professor Hardy Cross ( University of Colorado ) developed his famous technique for iteratively balancing flows in gridded piping systems. The advent of modern high speed digital computers heralded a new age in hydraulic calculations. Mathematical techniques such as finite element, finite difference and Newton-Raphson made it possible to simulate a virtually limitless combination of design parameters. The effect of countless changes in design parameters can be calculated and displayed in a matter of seconds. Developments in computer software such as Microsoft Windows and Apple Macintosh have made it easier to enter and edit data and view the results of hydraulic calculations on the screen or sent to a printer. Hydraulic calculation software have come to symbolize the computerized aspect of a fire sprinkler hydraulic model.

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Glossary

Static Pressure (psi) - measured by a pressure gage installed on a hydrant during a hydrant flow test.
Residual Pressure (psi) - measured by a pressure gage on a hydrant flowing full. When the hydrant is flowed the gage registers a drop in pressure. Sometimes multiple hydrants on the same water main must be flowed in order to register an appreciable drop in gage pressure.
Flow (gpm) - measured by a pitot gage held in the center of the flow stream of a hydrant flowing full. Static, Residual and Flow are usually measured as part of a hydrant flow test conducted by fire departments and water utilities. The resulting curve usually plots as a flow of zero gpm at a positive value of Static pressure and slopes downward to the right.
Density (gpm/sq ft) - a design criterion for flow required in the most demanding part of the fire sprinkler system.
Area Per Sprinkler (sq ft) - a design criterion for the maximum area covered by a single sprinkler in the design area.
Design Flow (gpm) - the flow obtained by multiplying Density by the Area Per Sprinkler.
Design Area (sq ft) - the total area covered by sprinklers that must be included in the hydraulic calculation. The hydraulic calculation must include all sprinklers located in the design area.
K-Factor ( ) - a dimensionless number (no units) expressing a sprinkler flow coefficient. The flow from a given sprinkler can be determined by multiplying the K-Factor by the square root of the pressure at the sprinkler.
Starting Pressure (psi) - obtained by solving for pressure based on design flow and K-Factor. It is the pressure expected at the most remote sprinkler based on a given design flow and sprinkler K-Factor.
Required Pressure (psi) - the total pressure required to operate the sprinkler system so that it equals the starting pressure, and required flow, at the most remote sprinkler.
Available Pressure (psi) - the total pressure on the water supply curve corresponding to the required sprinkler flow at the most remote sprinkler. The difference between Available Pressure (AP) and Required Pressure (RP) is commonly referred to as the "pressure cushion". When RP exceeds AP we have a negative pressure cushion.

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