Mass Burning Rate – Energy Release Rate
1. INTRODUCTION
Once a flame is ignited and spreads over a surface, the area under the flames
undergoes degradation supported by the energy released by the flame. This process
undergoes a transient period but, if the fuel is thick enough, eventually reaches steady
state. The fuel emerging from the surface will react at the flame releasing energy; this is
called the energy release rate. The entire process is described in Figure 1. Details on
how to evaluate the burning rate are given by Drysdale [1] and Quintiere [2] and on the
energy release rate and its estimation by Babrauskas and Grayson [3].
&
Q
C
Control Volume
&
Q
H
Flame
&
Q
R
&
Q
F
q& ′e′
q& ′v′
q& ′l′
q& ′c′
Fuel
Area (A)
Figure 1.
Repartition of energy within a diffusion flame.
1
&)
2. ENERGY RELEASE RATE ( Q
The energy release rate is the amount of energy produced by the flame per unit time.
In its simplest form is given by equation (1):
& = A ΔH m
& ′F′
Q
C,F
(1)
& ′F′
Where “A” is the area, ΔH C , F the heat of combustion per unit mass of fuel burnt and m
the mass of fuel produced at the surface (mass burning rate). An alternate way to express
the energy release rate is by means of the heat of combustion per unit mass of oxygen
& O)
consumed ( m
& = ΔH m
&O
Q
C ,O
(2)
Where ΔH C ,O is the heat of combustion per unit mass of oxygen consumed (i.e. 13.1
& O ) is not expressed per unit area, this
MJ/kg). Note that the mass of oxygen consumed ( m
is because the area through which the oxygen reaches the flame is difficult to determine.
Equations (1) and (2) are equivalent and which one to use depends mostly on practical
issues related to the quantification of the different values.
The total energy released by the flame can be divided into the different parts that
are transferred by different modes of heat transfer or towards different directions:
& =Q
& +Q
& +Q
& +Q
&
Q
C
R
H
F
(3)
&
Q
C
Is the energy that is convected towards the plume above of the flame. Natural
convection will drive the hot products (and consequently the energy) above the
flame.
&
Q
R
Is the energy that is radiated away from the flame towards external targets. The
fraction of the energy radiated is generally given by a radiative fraction ( χ ) which
& ≈ χQ
&.
is generally taken as approximately 30%, χ ≈ 0.3 [4] and thus Q
R
&
Q
H
Is the energy that remains within the control volume of the flame. Once flames
& tends towards zero.
have attained steady state conditions Q
H
&
Q
F
Is the energy transmitted by the flame towards the surface of the fuel. This energy
includes convection, conduction and radiation. Convection and conduction will
be positive inputs towards the fuel but convection will vary depending on the
temperature of the gases over the material at a specific location within the surface.
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3. ENERGY BALANCE AT THE SURFACE
Conducting an energy balance per unit area allows a closer analysis of the fuel
surface. This implies the assumption that the energy is distributed homogeneously across
the surface. This assumption is justifiable for most of the surface area of the fuel. The
energy input to the surface is given by:
& = Aq& ′′
Q
F
e
(4)
q& ′e′
T∞
TP
T
q& ′l′
q& ′v′
Gas
0
Fuel
q& ′c′
t
x
Figure 2.
Energy balance at the fuel surface
The energy necessary to vaporize a unit mass of fuel varies depending on the
nature of the gasification process. Most liquid fuels (light hydrocarbons) undergo
gasification without any chemical decomposition. The energy necessary for the phase
change of a unit mass of fuel is called the latent heat of vaporization ( ΔH V ). For most
solids and liquids formed by heavier hydrocarbons, the decomposition process implies a
chemical breakdown of the molecules. This process is generally endothermic and the
energy necessary to gasify a unit mass of fuel is called the heat of pyrolysis ( ΔH P ). For
the purposes of this analysis the distinctions in the decomposition pathway are not
relevant, what is important is the endothermicity of the process, therefore a generic heat
of gasification will be used ( ΔH G ). The energy necessary to vaporize a unit mass of fuel
per unit are is thus given by:
& ′F′ ,G
q& ′V′ = ΔH G m
(5)
& ′F′ ,G is the mass of fuel generated per unit surface area.
Where m
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The energy lost from the fuel surface includes convection and re-radiation and can
be described by any of the formulations below:
′ + q& ′S′,R = h C (TP − T∞ ) + εσ(TP4 − T∞4 ) = h C (TP − T∞ ) + h R (TP − T∞ )
q& ′L′ = q& ′CV
It is a common practice to simplify this term to a simple linearized total heat transfer
coefficient
q& ′L′ = h T (TP − T∞ )
(6)
Finally, in-depth conduction is given by the temperature gradient at the surface
q& ′C′ = −k
∂T
∂x
(7)
x =0
This term varies significantly with the nature of the fuel. Liquid fuels are characterized
by the presence of re-circulation currents induced by buoyancy. These currents
homogenize the temperature distributions reducing in-depth conduction. Nevertheless,
convective motion transfers heat to the interior of the pool leading to an additional term
of in-depth convection.
The energy balance at the surface is then given by:
aq& ′e′ = q& ′V′ + q& ′C′ + q& ′L′
(8)
For convenience, the right hand side of the equation is generally presented in a different
form:
⎛ q& ′′ + q& ′L′
& ′F′ ,G + ⎜ C
aq& ′e′ = ΔH G m
⎜ m
⎝ & ′F′,G
⎞
⎟m
& ′′
& ′′
⎟ F , G = ( ΔH G + Q L ) m F , G
⎠
(9)
Where QL represent the total heat not used to vaporize the fuel normalized per unit mass
of fuel generated. In other words, of the total heat available a fraction goes to gasification
( ΔH G ) and the rest goes away to the solid or the gas (QL). Assuming the absortivity to
be unity (a=1) the mass of fuel generated is give by
& ′F′,G =
m
q& ′e′
( ΔH G + Q L )
(10)
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4. THE MASS TRANSFER NUMBER (B)
From Equation (1) the energy generated by a molecule of fuel per unit area is given
by the following expression:
q& ′g′ =
&
Q
& ′F′, B
= ΔH C ,F m
A
(11)
& ′F′,B is the mass of fuel burnt. As shown before, the energy generated can be
Where m
divided into the energy fed back to the fuel ( q& ′e′ ), the energy necessary to heat the gases
that then will be lost due to convection
q& ′L′ ,C =
&
Q
C
& ′F′, B + m
& ′O′ ,B )Cp(TF − T∞ )
= (m
A
and the energy lost due to radiation which can be given by the radiative fraction ( χ )
q& ′L′ ,r =
&
Q
R
& ′F′, B
= χΔH C,F m
A
Therefore, an energy balance in the gas phase is given by:
& ′F′ ,B = χΔH C , F m
& ′F′ ,B + ( m
& ′F′ ,B + m
& ′O′ ,B )Cp (TF − T∞ ) + q& ′e′
q& ′g′ = ΔH C ,F m
(12)
Since diffusion flames burn stoichiometric then:
& ′O′ , B
m
=φ
& ′F′,B
m
where φ is the stoichiometric coefficient. Substituting the above expression in equation
(12) and re-arranging an expression for the energy feedback per unit area can be
obtained:
& ′F′, B + m
& ′F′, B (1 + φ)Cp (TF − T∞ )
q& ′e′ = (1 − χ) ΔH C ,F m
(13)
And substituting into equation (10) leads to a phenomenological definition of the mass
transfer number:
B=
& ′F′,G (1 − χ) ΔH C ,F + (1 + φ)Cp(TF − T∞ )
m
=
& ′F′,B
m
ΔH G + Q L
(14)
As can be noted, the mass transfer number represents the mass of fuel
generated per unit mass of fuel burnt.
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Although there are many definitions of the mass transfer number, mostly
generated for specific conditions such as droplet burning or boundary layer burning. All
retain the same physical concept, which is the capability of the flame to self-sustain by
generating more fuel. If B>1 the flame will produce more fuel than the necessary to
sustain burning.
As observed from equation (14) the “B” has a number of unknown or difficult to
determine parameters, the first of them are the losses (QL). Since the losses depend on
the particular experimental conditions, there is a tendency to neglect them when
tabulating values for the “B” number. This is appropriate as a reference value but losses
need to be incorporated when using the “B” number to try to calculate the mass burning
rate.
Table 1
Fuel
B Number
n-Pentane
8.1
n-Hexane
6.7
n-Heptane
5.8
n-Octane
5.2
n-Decane
4.3
Benzene
6.1
Toluene
6.1
Xylene
5.8
Methanol
2.7
Ethanol
3.3
Acetone
5.1
Kerosene
3.9
Diesel Oil
3.9
Examples of the “B Number” for different liquids
5. REFERENCES
1. “An Introduction to Fire Dynamics,” D. Drysdale, John Wiley and Sons, 1998.
2. “Principles of Fire Behavior,” J. G. Quintiere, Delmar Publishers, 1998.
3. “Heat Release in Fires,” Babrauskas, V. and Grayson, S.J. Editors, Elsevier Applied
Science, 1992.
4. “Combustion Fundamentals of Fire,” G. Cox, Academic press, 1995.
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