4
Mathematics
for the
Driver/
Operator
4
Knowledge Objectives
• List the elements needed to calculate
pump discharge pressure.
• Describe the concepts underlying
theoretical hydraulic calculations.
• Describe the concepts underlying fireground hydraulic calculations.
• Describe how to utilize fire-ground
hydraulic calculations during an incident.
4
Skills Objectives
•
•
•
•
•
•
(1 of 5)
Calculate the smooth-bore nozzle flow.
Calculate the friction loss in single hose lines.
Calculate the friction loss in multiple hose lines.
Calculate the elevation pressure loss and gain.
Calculate the friction loss in an appliance.
Use a Pitot gauge to test the friction loss in a
portable master stream appliance through 3”
(76-mm) hose.
4
Skills Objectives
(2 of 5)
• Use in-line gauges to test the friction loss in a
specific hose.
• Determine the pump discharge pressure in a
wye scenario with equal lines.
• Determine the pump discharge pressure in a
wye scenario with unequal lines.
• Determine the pump pressure for a Siamese
connection line by the split flow method.
4
Skills Objectives
(3 of 5)
• Calculate the friction loss in Siamese
connection lines by coefficient.
• Calculate the friction loss in Siamese
connection lines by percentage.
• Calculate the pump discharge pressure for a
pre-piped elevated master stream device.
• Calculate the pump discharge pressure for an
elevated master stream.
4
Skills Objectives
(4 of 5)
• Calculate the pump discharge pressure for a
standpipe during preplanning.
• Calculate the pump discharge pressure for a
standpipe.
• Calculate the flow rate (GPM) for a given
hose size and nozzle pressure using a slide
rule calculator.
• Calculate the friction loss for a given hose
size and nozzle pressure using a slide rule
calculator.
4
Skills Objectives
(5 of 5)
• Perform calculations by hand.
• Perform the hand method of calculation for
2½” (51-mm) hose.
• Perform the hand method of calculation for
1¾” (44-mm) hose.
• Perform the subtract 10 method of
calculation.
• Perform the condensed Q method of
calculation.
4
Introduction
(1 of 5)
• Fire service hydraulic calculations
determine the required pump discharge
pressure (PDP) for fire-ground operations.
– Driver/operator has responsibilities at each
incident.
4
Introduction
(2 of 5)
• Driver/operator is critical to the success of
the firefighting attack.
– Actions ensure the safety of the attack team,
occupants within the fire-involved structure,
personnel on scene, and any exposures.
– Decisions are often a matter of life and death
4
Introduction
(3 of 5)
• Fire service hydraulics
– Study of the characteristics and movement of
water pertaining to calculations for fire
streams and fire-ground operations
– Two categories
• Theoretical hydraulics: scientific or more exact
calculations
• Fire-ground hydraulics: less exact but more userfriendly calculation methods
4
Introduction
(4 of 5)
• Hydraulics is part scientific and part inexact.
– Requires balance
– Many numerical values influence the
calculations.
• Manufacturers offer nozzles, appliances, and
hose with individual specs that vary from
those on traditional charts.
– Flow tests with department equipment confirm
pressures to use.
4
Introduction
(5 of 5)
• Driver/operator applies his or her
knowledge of specific equipment on the
apparatus to the methodology underlying
hydraulic theory.
– Driver/operator must adjust for variables in
scenarios.
– Practice and experience with hydraulics
calculations are imperative.
4
Pump Discharge Pressure
(1 of 4)
• Fighting fire with water: water flow (GPM
or L/min) versus heat generation (Btu or
kcal)
– Critical rate of flow is the flow required to
overcome heat generated by the fire.
– Proper hose and nozzle selection is one of
the most important attack decisions after
securing a water supply.
4
Pump Discharge Pressure
(2 of 4)
• Driver/operator supplies hose lines with
optimal PDP.
– Pressure too great: stream breaks up,
lessening its effectiveness
– Inadequate pressure: insufficient flow to
overcome the fire, could endanger the safety
of the team
4
Pump Discharge Pressure
(3 of 4)
• PDP is the total pressure needed to overcome
all friction, appliance, and elevation loss while
maintaining adequate nozzle pressure to
deliver effective fire streams.
• Several factors
–
–
–
–
Nozzle pressure
Friction loss in hose lines
Elevation gain or loss
Friction loss in appliances
4
Pump Discharge Pressure
(4 of 4)
• Calculations become more complex based
on the fire attack operation.
• Start with a basic PDP formula:
PDP = NP + FL
– PDP = pump discharge pressure
– NP = nozzle pressure
– FL = friction loss
4
Nozzle Pressures
(1 of 3)
• Establish the nozzle pressure from the
scenario to use in the formula.
• Pressure required at the nozzle to deliver
a fire stream and flow rate for the nozzle’s
design
– Defined by the nozzle manufacturer
– Determined through testing
4
Nozzle Pressures
(2 of 3)
4
Nozzle Pressures
(3 of 3)
• Manufacturer’s specs for max water flow from
an aerial device is listed with the nozzle specs.
• Broken-stream nozzles are available in fog and
smooth-bore varieties.
• Three standard nozzle pressures (SNPs)
sufficient for fire-ground operations:
– 100 psi (700 kPa) for all fog nozzles
– 50 psi (350 kPa) for smooth-bore handline
– 80 psi (560 kPa) for smooth-bore master stream
4
Determining Nozzle Flow
• Flow rate is the volume of water moving
through a nozzle, measured in GPM or
L/min.
• Fog nozzles are designed with
predetermined flow rates based on a set
nozzle pressure.
• Flow and pressure differ by design and the
purchaser’s selection.
4
Friction Loss
(1 of 2)
• After determining the nozzle pressure,
determine the friction loss.
– Pressure lost from turbulence as water
passes through pipes, hoses, fittings,
adapters, and appliances
– Measured in psi or kPa
4
Friction Loss
(2 of 2)
• Modern FL equation: FL = C Q2 L
– Widely accepted as fire service standard
– Success of equation is due to its ability to
adapt to changes in hose diameter, water
flow, and hose length
4
Calculating Friction Loss
(1 of 3)
• FL = C Q2 L
– FL = friction loss
– C = coefficient, numerical measure constant
for each specific hose diameter
– Q = quantity of water flowing (GPM or L/min)
divided by 100
– L = length of hose in feet or meters, divided by
100
4
Calculating Friction Loss
(2 of 3)
• May express FL formula as
FL = C × (Q/100)2 × L/100
– Since driver/operators use the formula
frequently, Q and L are divided by 100
mentally to simplify and shorten the written
form to FL = C Q2 L.
4
Calculating Friction Loss
(3 of 3)
• Multiple hose lines of different sizes and
lengths
– On the fire ground, you may need multiple
water pressures from one pumper.
– Under normal conditions, fully open the valve
to prevent excess turbulence and friction loss
through the valve.
– Situations may occur where partially closing a
valve is necessary.
4
Elevation Pressure (EP)
(1 of 4)
• Adjust calculations for EP: distance the
nozzle is above or below the pump
– Elevation loss: pressure lost when the nozzle
is above the pump
– Elevation gain: pressure gained when the
nozzle is below the pump
– After calculating the EP, it must be added to or
subtracted from the PDP.
4
Elevation Pressure (EP)
(2 of 4)
• Elevation is relative to grade; altitude is
relative to sea level.
• Water exerts pressure of 0.434 psi per 1’
(9.817 kPa/m) of water column.
– For water discharging below the center line of
the pump, subtract.
– For water discharging above the pump, add.
4
Elevation Pressure (EP)
(3 of 4)
• To speed the calculation process further,
determine the elevation change in 10’ (3
m) increments and multiply by 5 psi (5 psi
per 10’) [(10 kPa per 3 m)].
– 5 psi per 30 kPa of grain or loss for each floor
of elevation change in a residential structure
where floor spacing is commonly 10’ (3 m)
– Not all buildings have floors spaced every 10’
(3 m)
4
Elevation Pressure (EP)
© haveseen/ShutterStock, Inc.
(4 of 4)
4
Appliance Loss (AL)
(1 of 4)
• Appliances are devices that connect and
adapt hoses and direct and control water
flow in hose layouts.
– Adapters and reducers
– Gated wyes
– Siamese connections
– Water thieves
– Monitors and manifolds
– Elevated master stream devices
4
Appliance Loss (AL)
Courtesy of Akron Brass Company
Courtesy of Akron Brass Company
(2 of 4)
4
Appliance Loss (AL)
(3 of 4)
• Determining friction loss in appliances
– Need two in-line pressure gauges for testing
hose
– Friction loss is minimal until the pump is
flowing 350 GPM (1300 L/min) or greater.
– Conduct tests on level ground and complete
the process at various flows to see how the
friction loss changes.
4
Appliance Loss (AL)
(4 of 4)
4
Total Pressure Loss (TPL)
• Insert AL and EP into the friction loss
formula.
– Start with PDP = NP + FL and determine NP.
– Determine the TPL.
– Expanded FL formula becomes:
TPL = C  Q2  L + AL + EP
– After calculating the TPL, apply it to the PDP
formula, now expressed as PDP = NP + TPL.
4
Wyed Hose Lines
• A wye splits a single line into two.
– Requires calculations to find the final FL
4
Siamese Hose Lines
(1 of 2)
• A Siamese connection is a device allowing
multiple hose lines to converge into one.
– Often used on the intake side of a pump,
allowing multiple lines to supply the pumper
– Used by departments that do not have a
large-diameter hose
– May be used to bring two or three lines into
one attack line
4
Siamese Hose Lines
(2 of 2)
• Methods to calculate FL in lines to a
Siamese connection when lines are of
equal size and length
– Split flow method
– Coefficient method
– Percentage method
4
Calculating Elevated Master Streams
(1 of 3)
• Prepiped elevated master stream
– Aerial fire apparatus (ladder truck) with fixed
waterway attached to the underside of the
ladder with a water inlet at the base supplying
a master stream device at the end
– No single FL amount can adequately
encompass all prepiped elevated master
stream devices.
4
Calculating Elevated Master Streams
(2 of 3)
© Kurt Hegre, The Fresno Bee/AP Photos
4
Calculating Elevated Master Streams
(3 of 3)
• Water comes into the intake, splits in a “T”,
turns with an “L”, goes toward the front of
the truck, turns again with an “L”, and
again with another “L” into the pump.
– All of these turns increase the FL.
– Use at least 25 psi (175 kPa) FL for all master
stream appliances.
4
Standpipe Systems
(1 of 4)
• Standpipe operations pose a challenge.
• Fire fighters must know buildings, occupancy,
fire load, special hazards, and hydrant
locations.
• For these buildings, water supply is critical.
• Standpipe systems require the greatest
number of calculations and very demanding
pump pressures.
4
Standpipe Systems
(2 of 4)
• Driver/operator controls only one pressure
to the standpipe: the highest pressure
needed at any one discharge (regardless
of the floor it is on)
– Operator must pump to the highest pressure
required.
• In commercial buildings and multistory
structures, attack hose should be 2½”
lines (64 mm) with smooth-bore nozzles.
4
Standpipe Systems
(3 of 4)
• Fire fighter needs to know the required
operating pressure of attack hose lines to
control the pressure from the outlet.
– Attack hose line must be flowing water if the
crew member is to obtain an accurate gauge
reading.
• Using an in-line gauge is preferred for
standpipe operations.
4
Standpipe Systems
(4 of 4)
• PDP for standpipe operations requires that
calculations be made for supply lines to
the standpipe, FL within the standpipe
system, FL for the attack handlines, and
pressure loss due to an increase in
elevation.
• If there is no preincident plan and the riser
size is unknown, add 25 psi (175 kPa) to
account for the standpipe riser.
4
Pressure-Regulating Valves (PRV) (1 of 3)
• Installed on standpipe risers where static
pressures exceed 175 psi (1225 kPa) per
NFPA 13
– If pressures while flowing exceed 100 psi (700
kPa), NFPA 14 requires the installation of a
device at the outlet to restrict or reduce flow
pressure to a maximum of 100 psi (700 kPa).
4
Pressure-Regulating Valves (PRV) (2 of 3)
• To determine the pressure
– Head: height of the column of water above the
discharge
– Head pressure: pressure in the column of water
– Divide the number of feet (height) by 2.304.
– For metric, divide the number of meters by 9.812.
• The water column is in a standpipe riser.
– As the riser height increases, the head pressure
could exceed the burst pressure of the fire hose.
4
Pressure-Regulating Valves (PRV) (3 of 3)
• Standpipe test documentation should be
on file in the building’s maintenance office.
– Design pressure of the building’s standpipe
system should meet the minimum PRV
pressure.
4
Net Pump Discharge Pressure
(NPDP)
• Static water sources require pulling a draft.
• Dynamic sources are positive-pressure
sources.
• Incoming pressure: pump does not have to
work as hard to achieve proper PDPs
• NPDPpps: pressure created by the pump
after receiving pressure from a hydrant or
another pump
4
Fire Service Hydraulic Calculations
(1 of 3)
• Theoretical hydraulic calculations present
more exact calculations, require more
mathematical skills, and take more time to
compute.
– Away from the fire ground, they may be
computed on paper, calculator, or computer.
– Used on the fire ground by more experienced
driver/operators
4
Fire Service Hydraulic Calculations
(2 of 3)
• Fire-ground hydraulic calculations estimate
calculations more quickly due to incident
urgency.
– Used as a backup, to check results, or to ensure
reasonable accuracy
– Reinforce learning as the driver/operator
computes theoretical hydraulics calculations in
the field
– Methods yield working approximations but are not
as accurate as theoretical computation.
4
Fire Service Hydraulic Calculations
(3 of 3)
• Formulas for determining PDP
– PDP = NP + FL
– FL = C Q2 L + AL + EP
• Most fire-ground hydraulic methods account
for FL only.
– Acquired FL added to formula to determine PDP.
– Consider all essential components of hydraulics
when using fire-ground hydraulic methods.
– Each reduces the math required
4
Charts
(1 of 3)
• Created by fire fighters for handline and
master stream calculations for hoses,
nozzles, and devices specific to the
apparatus within the department
– List common pressures for attack hose and
nozzle combinations at different lengths
– Calculations must be accurate for the chart to
be reliable.
4
Charts
(2 of 3)
4
Charts
(3 of 3)
• May be prepared for complex problems
• Charts for high-rise buildings suggest a
pressure for a range of floors.
• Attack team should know in advance
what pressure is needed from a riser for
the particular attack line and nozzle
combination in their high-rise/standpipe
bag.
4
Hydraulic Calculators
• May be manual (mechanical) or electronic
– Manual hydraulic calculator may consist of a
sliding card or slide rule.
– Can handle calculations involving a variety of
nozzle pressures, flow rates, and hose diameters
• Electronic calculators may be mounted or
hand-held devices.
– Allows the calculation of engine pressure, FL,
application rate, flow rate, and reaction force
4
Hand Method
(1 of 2)
• Purpose of the hand method (counting
fingers method) is to quickly determine the
FL per 100’ (30 m) of hose
– Add AL, EP, and NP to provide the PDP.
• Hand calculation methods for almost every
hose size
– Methods for 1¾” and 2½” hose are most
common.
4
Hand Method
(2 of 2)
• Hand method for 2½” hose works nicely
because the coefficient for 2½” hose is 2
when using imperial units.
4
Subtract 10 Method
(GPM Flowing Method)
• Determines FL in 2½” hose only
– For flows of 160 GPM or greater
• Useful for either fog or smooth-bore
nozzles
– Simplicity of method is its strength
– Can determine FL in 2½” hose very quickly
4
Condensed Q Method
(1 of 9)
• Quick method for calculating FL per 100’ in
3” to 5” hose line only
– Useful when the apparatus is part of a relay
and supplying another pumper
– There is no simple metric equivalent to this
method.
4
Condensed Q Method
• Condensed Q formulas
– 3” hose FL = Q2
– 3½” hose FL = Q2/3
– 4” hose FL = Q2/5
– 5” hose FL = Q2/15
(2 of 9)
4
Condensed Q Method
(3 of 9)
• Condensed Q method in a relay operation
– When two pumpers arrive at the scene at the
same time, the first pumper positions for attack
while the other lays a line from the attack pumper
to a hydrant to establish a water supply.
– The operator of the supply pumper must quickly
calculate the PDP for the supply line to the attack
pumper.
– If pressure is too high or too low, the operator of
the attack pumper will radio the supply operator.
4
Condensed Q Method
(4 of 9)
• Flow meters and electronic pump
controllers
– Flow meter: straight section of pipe, threaded on
each end, and placed between two sections of
hose
– Sensing device on top of the meter is designed to
measure the flow (GPM) through the hose.
– Indicate the actual volume of water discharged
through each line
4
Condensed Q Method
(5 of 9)
• Using a flow meter lessens the
calculations to determine the PDP.
• Many flow meters are available from a
variety of manufacturers.
– Flow meters are generally accurate within 1 to
3 percent but require periodic calibration to
maintain accuracy.
4
Condensed Q Method
(6 of 9)
• Flow meter can deliver a desired flow
without the operator knowing the length of
the hose lines, FL, or EP.
– Useful in multistory buildings
– Eliminate math needed for relay operation
4
Condensed Q Method
(7 of 9)
• Electronic pump controllers and pressure
governors
– Technology has reduced the calculations that
the driver/operator must perform.
– Multiplexing: combining information from
several different sources
– Trend in modern apparatus is to use
electronic control and monitoring equipment.
4
Condensed Q Method
(8 of 9)
• Electronic components are designed by
independent contractors and integrated
into a manufacturer’s apparatus.
• Technological innovation assists the
driver/operator but causes a possibility of
electrical failure disabling the apparatus.
– The threat of electrical malfunction increases.
4
Condensed Q Method
(9 of 9)
• Preincident plan
– Conducted with the owner’s permission at a
convenient time
– The survey yields valuable information.
– Gather building blueprints or diagrams and
relevant data.
4
Summary
• Fire service hydraulic calculations are used to
determine the required PDP for fire-ground
operations.
• PDP = NP (nozzle pressure) + FL (friction loss).
• Fire service hydraulic calculations may
generally be categorized as either theoretical
hydraulics or fire-ground hydraulics.
• Fire-ground hydraulic calculations reinforce the
learning process.