Simple Algorithms

1
Simple Algorithms
2
Introduction

• The purpose of this lesson is to present a set
  tools that can be used for quick estimations fire
  characteristics or fire generated conditions.
• Prior to using these algorithms or equations it
  is important to realize that they are simplified
  methods that have assumptions and limitations,
  which must be considered to assure that the
  equations are being used appropriately.
• The concept and terminology of each equation will
  be described and applications for each will be
  provided.
• Time to Ignition
• Burning rate
• Flame height
• Heat release rate for flashover
• Time to Flashover

3
Time to Ignition

• In many investigations, the time line is a
  critical link between the ignition and
  development of the fire and other events, such
  as
• location of the suspect,
• location of the victim,
• status of fire detection or suppression systems
  etc.
• Therefore it may be important to estimate, for a
  given fire or heating (heat flux) condition, how
  long it would take for an object to reach its
  ignition temperature, Tig.
• The time to ignition calculations must be divided
  into two categories
• Thermally-thin solids and
• Thermally-thick solids.

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Time to Ignition

• A thermally thin object is assumed to be the same
  temperature throughout its thickness or width.
• The physical limit on thermally thin is
  approximately 2 mm or less than a 1/16th of an
  inch.
• Typically objects like clothing, curtains, or
  sheets of paper could be considered as thermally
  thin.
• A thermally thick object will have a temperature
  gradient throughout its thickness or width.

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Time to Ignition

• Conversely, objects such as wood furniture, piles
  of paper, and composite items such as wallpaper
  over drywall would be considered as thermally
  thick.
• In some cases, composites can delaminate prior to
  ignition.
• For example, paint or wall coverings may pull
  away from a thermally thick wall due to the
  expansion of air bubbles as the material heats.
• Once the thin material is not in contact with the
  wall, it will heat up faster and behave in a
  thermally thin manner.

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Thermally-Thin Case
The time to ignition is equal to the amount of
heat that the object can store per unit area and
degrees Kelvin multiplied by the temperature
increase required to reach the ignition
temperature of the object divided by the energy
input rate per unit area.

• where
• tig time to ignition (s)
• Tig Ignition temperature of the object (?C)
• T? Initial temperature of object (?C)
• density of object (kg/m3)
• c specific heat of object (kJ/kg K)
• l thickness of object (m)
• heat flux per unit area (kW/m2)

• Exercise
• Thin oak veneer use the properties from PFB, p238
  Table 3-1
• The oak veneer has a thickness of 1 mm, an
  ignition temperature of 390 ?C and initial
  temperature of 20 ?C.
• The heat flux per unit area is 20 kW/m2.
• Note Watch your units.
• In this case we are cheating a little.
• One degree Celsius is equal to one degree Kelvin
  however they have a different base, i.e. 0 ?C
  273.15 K. Hence differences in temperature can be
  represented in ?C even though the specific heat
  is given terms of K.
• Make sure that the units you input into the
  equation match the units required.
• For example 1 mm needs to be converted to m for
  use in the equation. 1 mm is equal to 0.001 m.

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Thermally-Thick Case
This case assumes that the solid is infinitely
thick. The time to ignition is equal to a heat
loss constant multiplied by the thermal inertia,
??c (kay rho cee) by the square of the quantity
of the temperature increase required to reach the
ignition temperature of the object divided by the
energy input rate per unit area.

• where
• tig time to ignition (s)
• Tig Ignition temperature of the object (?C)
• T? Initial temperature of object (?C)
• C heat loss constant
• k thermal conductivity (kW/m-K)
• density of object (kg/m3)
• c specific heat of object (kJ/kg K)
• heat flux per unit area (kW/m2)

• In the thick case we have to consider not only
  how the material will store the energy (specific
  heat) but also how the energy will move or flow
  through the material using the thermal
  conductivity.
• The product of the specific heat, density and
  thermal conductivity is referred to as the
  thermal inertia of the material.
• The thermal inertia is used to enable us to
  predict the rate of rise of a materials
  temperature.
• Exercise
• Given a 0.1 m thick piece of oak, and the same
  initial conditions as in the thin case, estimate
  the time to ignition.
• How does it compare with the thin case?
• Does that make sense?

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Burning Rate

• The heat release rate of an object or a room is
  important to understanding how the energy
  released from that item would change the thermal
  conditions in a room or how it might ignite other
  nearby fuels.
• The heat release rate or burning rate is equal to
  the mass loss rate of the burning object
  multiplied by the effective heat of combustion.
• This method would typically be used for a
  specific burning object like a piece of
  furniture.
• Basically anything that can be burned on a load
  cell.

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Burning Rate
Where heat release rate (kW) mass loss rate
(g/s) effective heat of combustion (kJ/g)
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Burning Rate

• When calculating the heat release rate of liquid
  pool fires, such as gasoline or liquid propane it
  is desirable to use another form of the equation
  because the mass loss rate data is usually
  provided in mass loss rate per unit area.
• This form gives the investigator more flexibility
  to address the specific scenario and the fire
  area involved.
• This can be done for liquid hydrocarbon fires as
  well as for well documented fuel loads (PFB Table
  6-5).

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Burning Rate
Where heat release rate (kW) mass loss rate
per unit area (g/s m2) effective heat of
combustion (kJ/g) burning area (m2)
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Flame Height

• Several relationships have been developed between
  the heat release rate and the flame height.
• Hence if you know the heat release rate of a
  burning item, an estimation of the flame height
  can be made.
• This may be useful to determine if the flame
  impinged on the ceiling.
• It may help explain why the fire did not spread.
• It may be used to compare with a witness
  statement about how large the flames were.

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Flame Height
In the equation provided in NFPA 921 Ch 3.5.5, we
have the flame height is equal to the heat
release rate to the 4/10ths power multiplied
times a coefficient and a ventilation factor
which is dependent the location of the fire
relative to walls.

• where
• flame height (m)
• ventilation factor
• 1 open burning
• 2 burning next to a wall
• 4 burning in a corner
• heat release rate (kW)

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Flame Height
Another way to use the equation is to rearrange
it to solve for heat release rate. It can be
used as a check to determine if the remains of
the fuel would have provided an adequate heat
release rate relative to the damage found at the
scene. If a witness tells you the flames were 3
m high and you have marks on the wall, which
support the witness statements, you can estimate
the heat release rate of the fire.

• where
• flame height (m)
• ventilation factor
• 1 open burning
• 2 burning next to a wall
• 4 burning in a corner
• heat release rate (kW)

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Flame Height

• In PFB, Quintiere presents a flame height
  equation developed by Heskestad.
• The equation, indicates that the flame height is
  equal to a constant multiplied by the heat
  release rate to the 2/5th power and subtracting
  an adjustment factor based on the diameter of the
  fire.
• Both equations are similar.
• The multiplication constants, 0.174 and 0.23
  could be rounded to 0.2.
• It is important to note that 0.4 is equal to
  2/5ths.
• The significant differences are that the first
  equation has a factor to account for flame
  location relative to walls or corners and the
  other has a factor to account for the diameter of
  the fire.

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Flame Height
Flame location relative to walls or corners
Heat Release Rate
Height of Flame (m)
Diameter of the fire
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Heat Release Rate Required for Flashover

• Previously we have learned that a number of
  variables affect the growth of a fire in a
  compartment
• volume of room,
• ceiling height,
• fuel load,
• fuel geometry,
• location of ignition fire,
• and last but not least ventilation.
• The equation is presented in NFPA 921, CH
  3.5.4.1.
• This equation was developed to estimate the
  minimum heat release rate required to flashover a
  room based solely on the ventilation limit of the
  compartment.

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Heat Release Rate Required for Flashover
The minimum heat release rate required to create
a flashover in a compartment is equal to the area
of the opening multiplied by the height of the
opening to the one half power (or the square
root of the opening) multiplied by a constant,
750.

• Where
• HRRmin minimum heat release rate (kW)
• Ao area of opening (m2)
• ho height of opening (m)

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Estimating Time to Flashover
Now the we have a method for estimating the
minimum heat release rate required to flashover a
room, we can use our knowledge of t-squared
fires to get a sense of the time to flashover for
a given fire growth rate. t-squared fires are
ranges of fire growth rates based on the time to
reach 1,000 kW or 1 MW in heat release rate. The
fire growth curves show some of the
representative fuel packages that can generate
slow, medium, fast, or ultra-fast fire
development.
The boundaries of these design fires are as
follows
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Estimating Time to Flashover
This equation shows that the heat release rate is
equal to the fire growth rate coefficient
multiplied by the square of the time in seconds.
Where ? fire growth rate coefficient
(kW/s1/2) t2 time (s)
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CONCLUSION

• The tools given above are used for generating
  quick estimates and testing your hypothesis.
• The equations are simplifications, which are
  assuming a wide range of conditions.
• A complete description of the theory behind each
  algorithm, underlying assumptions and its
  empirical basis or validation are beyond the
  scope of this course.
• However these issues are very important when
  applying the equations and need to be considered
  by a person with appropriate experience if these
  are to be used on case.
• The more tools that you can use on a given
  problem to check, double check and test your
  hypothesis, the better.
• Even though these equations are simplified, they
  can provide significant insight into a fire,
  while you are still at the scene.