|
In fire safety engineering, the cost of
experiments is very high owing to the fact that they are until now realised
only on a real scale.
A dimensionless approach will allow the determination of
the parameters and numbers that pilot the governing phenomena. This will be
used as in the similarity method, which allows realisation of the experiments
in the laboratory on a small scale.
In the present investigation dimensionless numbers as Ri, Gr and Pr which
govern the phenomena are highlighted.
It is known that the worst killer in fires
is the smoke inhalation because of the toxic gaseous species which cause a
substantial threat to life and property. In a context of world-wide concern
about fire safety, engineers and legislators pay more and more attention to
the smoke management in buildings. When focusing on how to conserve safety in
buildings during a fire catastrophe, the first thing that comes to mind is to
evacuate the smoke and heat released by the fire; therefore research is being
done on optimisation of the smoke evacuation design.
Predicting the smoke characteristics in a
building subject to a fire is a complex undertaking. It depends on the
geometry of the fire compartment, type of fire, the ambient conditions outside
the buildings,…etc [1].
An engineering analysis is always needed to
assess the ability of a smoke management system to satisfy stipulated
performance criteria concerning the assessment of fire effluent flow within a
building, and the design of smoke control and venting systems.
Engineering methods for the design of smoke
control systems have been available for a long time in the form of nomograms;
it has been actively studied during recent decades. Calculation methods and
computer codes have been developed to make the necessary evaluations.
Openings like doors and windows are the
principal means to allow the fire and the combustion products to spread
outside the room of origin.
The vent also allows air to reach the
combustion zone and thus influence the size of the fire.
In the present study we give a technical
dimensionless approach to study the heat transfer in a fire compartment.
|
2. Mathematical formulation and the fire compartment
modelling |
The mass conservation in a building requires
that through all openings, and in the steady state, the mass flows entering
and leaving the compartment are the same.
The conservation of the mechanical energy (kinetic
and buoyant energy)
Finally, from the upper vent, the heat
transferred outside the compartment is:
is the heat flux lost by conduction and radiation to the
outside.
Equations 1, 2 and 3 are the simplified form
of the conservation of mass, momentum and energy.
The following figures, represent the
phenomena with the zone model approach, in order to have a discrete
calculation and to make equations easy to solve by hand.
Figure 1 : Compartment fire geometry
Figure 2 : Pressure evolution with the
height
The pressure dependence on the height inside
and outside the compartment is given by figure 2.
When conceiving the design of the natural smoke control, the smoke mass flow
to be evacuated depends on the pressure difference between inside and outside
at the vent position, which creates the exhaust velocity:
Since the position of the neutral plan is
not known a priori, following Bjorn Karlsson and James G. Quintiere [2], it is
better to express this pressure difference across the vent in terms of the
pressure difference across the inlet opening
Then the smoke mass flow across the vent
will be:
The pressure drop across the inlet is given
by:
so that
h1 is also unknown, but equating the mass flow
entering and outgoing from the fire compartment, in the stationary regime give:
Then the mass flow of smoke through the vent is:
With the following ideal gas law formula:
|
 (11) |
we obtain:
It is not accurate to give the traditional value of ~0.6 to
the discharge coefficients (Ci and Cv ) for every design. Most authors give
the same value to both coefficients; this is not justified.
Note that Cv and Ci are for a particular vent design and
would generally vary from one design to another, depending mainly on the
Reynolds number and the vent geometry.
Another way to express this outgoing smoky mass flow, is to
traduce density in term of temperature via the dilatation expression:
When using the first order approximation of density.
Equation 10 becomes:
|
3. Dimensionless approach |
In the smoke evacuation field, several calculations have
been carried out in dimensionless form: Cooper[3], and Yamada and Cooper[4]
have investigated smoke evacuation through a ceiling vent. Another
dimensionless formulation is given in this investigation to lead to a global
similarity approach.
Assuming the following variables,
|
 ,
 |
the momentum equation changes to
The mass flow expression becomes
where Gr and Ri, are respectively the Grashof and the
Richardson numbers.
The Richardson number represents the ratio of the buoyancy
over the inertia; if Ri approaches zero there is no outflow of smoke by
natural ventilation and recourse to a mechanical exhaust is necessary.
If the Richardson number is very high, the mass flow tends
toward the value
The energy equation takes the form
The Nusselt number, Nu, is the ratio of the convective flux
from the plume, to the conductive flux transferred to the ceiling through the
smoke layer.
The conductive flux transferred to the ceiling is defined as:
The heat transferred through the upper vent, grows with the
Grashof number, and it decreases with the Richardson number.
Particular case : where Ai>>Av
When the inlet area is much bigger than the outlet area,
the neutral plane merges with the lower smoke boundary (according to relation
9), and the relation (16) is reduced then to its maximal value:
The situation is as if the temperature of smoke is very
high (or its density very low). It is very important to note that Ai/Av and
ro/r, play the same role
in the neutral plane position. It is this position, which determines the smoky
mass flow quantity.
The heat transfer relation (18) becomes in this case:
Three dimensionless expressions: the smoky
mass flow, the density ratio and the transferred energy to the ambient
surrounding, are given and are expressed with the Grashof, Prandtl, and
Richardson numbers, the ratio of the aerodynamic opening areas, and with the
ratio of the vent area over the base surface of the compartment.
The conversion of densities to temperatures
is done with the gas dilatation formula, which doesn’t assume smoke is an
ideal gas.
With this formulation, the passage to the
small scale is allowed. So a limited number of experiments are required for
the study of a given problem (value to the discharge coefficient, validation
of the dilatation law to the hot smoke…). It is very economical in comparison
to the dimensional formulation, which needs lots of experiment.
[1] Dhimdi S. and Vandevelde P. Electrical
analogy in heat and smoke evacuation, Int. J. on Architectural
Sc.Vol.2.pp1-6,(2001)
[2] Bjorn Karlsson and James G. Quintiere.
Enclosure Fire Dynamics book, CRC PRESS LLC, 2000
[3] Cooper L.Y. Combined Buoyancy and
pressure Driven Flow through a Horizontal Vent, NISTIR 5384 (1994)
[4] Yamada T. and Cooper L.Y. Experimental
study of the exchange flow Through a Horizontal Ceiling vent in Atrium Fires,
Building research institute, 12th Meeting of the
Fire research and Safety. Tokyo (1992)
| 
smoke
dimensionless
Thermal
expansion coefficient
