The combustion of solid materials which contain no oxydant in themselves takes place only on their surface supported by the air supply from outside. For the first step of the study of the propagation of combustion of this kind, the downward propagation of smouldering (combustion without flame) along a vertical sheet of cardboard was investigated, because this proved to be the simplest case both experimentally and theoretically.
The fire front proceeds downwards with a velocity
V (cm/s) making an angle
θ with the vertical. The propagation velocity
υ perpendicular to the fire front was observed in a vessel of temperature
Ta with rectangular pieces of cardboard of various breadth and thickness, and the results for
υ2 were summarized as follows:
υ2=0.55 × 10
-2{1/
b+1/
d+0.53/
bd}(1/(
Ti-
Ta)-1.55 × 10
-3)
where
Ti is the vessel temperature which makes the cardboard ignite instantaneously and was found, in an electric furnace, to be 460°C.
In the theoretical part, the authors considered the heat balance in a part of length
dX in the strip just in front of the fire front with the following assumptions.
(1) When a solid material in the open air ignites, combustion gases envelop it and prevent fresh air from getting to the burning surface. Combustion takes place when fresh air reaches the surface by natural convection. The heat
dQ1 produced in the part
dX in the time
dt is considered to be proportional to the velocity of the ascending convection current which was assumed to vary as the temperature difference (
T-
Ta) between
dX and the ambient air.
(2)
dQ1 is also proportional to the velocity of the combustion reaction which depends upon the ambient temperature. Since
υ should be infinite when
Ta→
Ti ,
dQ1 was assumed to be proportional to 1/(
Ti-
Ta).
(3)
dQ1 is proportional, too, to the surface area
dS of the part
dX. When burning, however, the surface of the solid material is covered with a stagnant boundary layer of combustion gases and the fresh air is supplied by diffusion through this layer from outside. Therefore, the surface area of
dX was assumed to be 2(
b+
d+
ε)
dX instead of (
b+
d)
dX, where
ε is about four times the thickness of this layer.
(4) Heat is dissipated through this stagnant layer of temperature
T into the ambient field of temperature
Ta. Thus the heat dissipated
dQ2 from
dX in
dt was assumed to be 2
h(
b+
d+
ε)
dX(
T-
Ta)
dt, where
h is the heat transmission coefficient.
From these considerations, Fourier’s equation was derived as follows :
cρ (
∂T/
∂t)=
k(
∂2T/
∂X2)cosec
2θ+2(1/
b+1/
d+
ε/
bd)(
q/(
Ti-
Ta)-
h)(
T-
Ta),
where
q is a constant depending upon the chemical properties of the cardboard.
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