Page 419 - Dust Explosions in the Process Industries

386 Dust Explosions in the Process Industries

In general, the chemical rate of a combustion reaction may be written
R, = kCf"C:R (5.3)

wherep + q =m is the order of the reaction, and C,and CORare the concentration of fuel
and oxygen in the reaction zone. Where the fuel is nondepleting and q = 1, one gets

R, = kCOR (5.4)

The rate of diffusion of oxygen from the surroundings into the reaction zone is

where D is the thermal diffusion rate constant and COSis the oxygen concentration in
the surroundings.
As the temperaturein the reaction zone increases, the thermal reaction rate increases
according to equations (5.2) and (5.4), and apoint is reached where the rate is controlled
by the diffusional supply of oxygen to the reaction zone. Then, Rc = R,, and the right-
hand sides of equations (5.4) and (5.5) are equal. Here,

where

p = kD/(k+ D)
is called the Frank-Kamenetskii's overall rate constant, and k is as defined in equation
(5.2). By introducing the heat of reaction Q, the rate of heat generation can, according
to equation (5.6), be expressed as

By inserting equation (5.2) into (5.7) and substituting for pin (5.8), one gets

QCOsDf exp(-EIRT)
RG= (5.9)
D+ f exp(-EIRT)

The general expression for the heat loss from the system considered is
RL = U(T- To)", n 2 1 (5.10)
where U and IZ are characteristicconstants for the system, Tis the temperaturein the reac-
tion zone, and Tothe ambient temperature.
Figure 5.1 illustrates the stability and instability conditions in a system that behaves
according to equations (5.9) and (5.10). Figure 5.1 reveals three intersections between
the S-shapedRGcurve and the heat loss curve RL.In the figure,RLis a straight line, cor-
responding to n = 1, which applies to heat loss by conduction only. For convection, n is
5/4 and for radiation 4. The upper (3) and lower (1) intersectionsare stable, whereas the
intermediate one (2) is unstable. A perturbation in T at this point leads to either cooling
to the lower intersection(1)or a temperaturerise to the upper intersection(3). If the heat
loss decreases due to changes of the constants in equation (5.10), the heat loss curve RL