Propagation of Combustion in Solid Materials Part III. Temperature Distribution in a Smouldering Rod

Abstract

In Part II, it was reported that, when smouldering combustion propagates downward through a vertical rod of radius γ with a constant velocity υ, the distribution of temperature T inside the rod is given by
k(∂²T/∂x²)+cρυ(∂T/∂x)+2(1/γ+Δ/γ²)((q/T_i-T_a)-h)(T-T_a)=0    (1)
when referred to an ordinate Ox moving downward with the propagation. The constants k, c, ρ are respectively the thermal conductivity, the specific heat and the density of the rod ; Δ the thickness of a stagnant boundary layer assumed to exist around the rod ; T_i the ignition temperature ; T_a the ambient temperature ; h the heat transfer coefficient ; and q is a constant depending upon the chemical composition of the rod and the ambient air.
The solution of the equation(1) is, in general
T - T_a=θ=e^-αx(Ae^-βx+Be^βx)    (2)
where
α=cρυ/2k, β=1/2k √c²ρ²υ²-8k(1/γ+Δ/γ²)((q/T_i-T_a)-h)
and A and B are integration constants. Taking the fire front as the origin of Ox, the boundary condition is
θ=0,   ∂θ/∂x=0    at  x=0    (3)
As was discussed in Part I, there is no solution of (1) which satisfies (3) in a strict sense, but (3) is fulfilled approximately when
β=0    (4)
This gives us not only υ in terms of c, ρ, k, etc. as
υ²=8k/c²ρ²((1/γ)+(Δ/γ²))((q/T_i-T_a)-h)    (5)
but T in the form :
T-T_a=e^-αx(C＋Dx)    (6)
where C and D are integration constants.
It was reported in Part II that (5) showed remarkable coincidence with experiments. Now in this report, it will be shown that the equation (6), too, coincides satisfactorily with experimental results showing that our theory has got another strong evidence.