Page 326 - Dust Explosions in the Process Industries

Propagation of Flames in Dust Clouds 295

velocity S, due to the chemical change of number of molecules, and the gas expansion
velocity S, due to the heating of the gas.
The dependence of S, on pressure P and temperature Tuin the unburned mixture was
taken as

s, = Su,,(T,/7y(P,/P)P (4.51j
where the index r refers to the reference state of 300 K and atmospheric pressure; p is
an empirical constant that equals 0.5 or less for gases.
The problem was first simplified by treating the flame propagation as an “isothermal”
process, considering T, as a constant equal to the mixture temperature Tobefore igni-
tion, and Tbin the combustion products as a constant equal to the overall temperature
T, when all the mixture has burnt and the flame reaches the vessel wall.
The resulting analytical equation for the rate of pressure rise was

(4.52)

where R is the vessel radius and P, is the pressure when the flame reaches the vessel
wall. ‘Thisequation can be integrated analyticallyfor p = 0. If To= T,, Po=P,, S,,o =S,,,,
and p = 0, the equation reduces to

(4.53)

The maximum (dP/dt),, occurs when P = P,; that is,

(dPldt),, = &(Pm -p,)(P,/P,) (4.54)
R
Equation (4.54) shows that this idealized isothermal treatment predicts that (dP/dt),,, is
inversely proportional to R, that is, to the cube root of the vessel volume, in agreement
with the frequently quoted “cube root law.” However, this treatment also shows the
strict conditions under which the cube root law is valid. These conditions were explic-
itly pointed out by Eckhoff (1984/1985and 1987)in a simplified analysis.First, the thick-
ness of the reaction zone or flame must be negligible compared to R. Second, &(Tu,P)
must be independent of R. Under conditions of significant and unspecified turbulence,
which are typical of dust explosion experiments in closed vessels, neither of these
requirements is fulfilled (see Section 4.4.3.3 for further discussion).
Nagy et al. (1969) extended the isothermal treatment to the more realistic adiabatic
conditions for which T, and Tbare not constants but given by

(4.55)

(4.56)