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Propagation of Flames in Dust Clouds 371

detonation front velocities, temperatures, pressures, and concentrations of reaction prod-
ucts. Davis (1987) discussed the slightly different approaches taken by Chapman and
Jouguet. Chapman simply postulated that a detonation front is a shock wave precipitat-
ing in, its wake chemical reactions that supply the energy required for maintenance of
the steady propagation of the shock wave through the explosible gas. In that case, the
theory of shock wave propagation through a gas could be used to describe detonation
by replacing the unreacted gas behind the shock by the products of the combustion reac-
tion and adding the heat of reaction. The resulting theory predicted a specific minimum
velocity for self-sustained detonation for any given explosible mixture, which Chapman
found to be in excellent agreement with the velocities measured in the gas explosion
experiments conducted by Dixon (1893). Chapman therefore simply postulated that
the minimum velocity predicted by his theory was the detonation velocity of the system
considered.
Jouguet (1905, 1906) had been working along similar lines, but his slightly different
approach revealed the important additional conclusion that the detonation wave veloc-
ity equals the velocity of sound in the hot, compressed reaction products immediately
behind the shock front. The C-J theory is concerned with only the simple system, con-
sisting of a homogeneous unreacted gas at a set of initial conditions, and the correspon-
ding shocked reaction products, separated by an infinitely thin, plane of discontinuity.
The theory results from the three equations for conservation of mass, momentum, and
energy across the discontinuity. and the equation of state, as shown by, for example, Jost
and Wagner (in Freytag, 1965), Glassman (1977), and Nettleton (1987). Nettleton refers
to computer codes that can be used for calculating C-J parameters for various explosi-
ble gas mixtures.
As pointed out by Kuchta (1985) the detonation peak pressure for gaseous mixtures
is approximately twice the maximum pressure for adiabatic constant-volume combus-
tion of the same mixture (absolute pressures). Kuchta also gave the following equation,
which relates the C-J detonation front pressure ratio to the detonation front velocity VD:

(4.91)

Here, P, is the detonation front pressure, PI is the pressure in the unreacted gas ahead of
the detonation front, y1 and yz are the specific heat ratios of the unreacted gas mixture
and the reaction products, and C1is the sound velocity in the unreacted gas mixture.
As long as the reaction zone is very thin, as it is for many explosible gadair mixtures
if the composition does not deviate too much from the stoichiometric one, the predicted
C-J parameters agree with experiments within a few percent. However, when the com-
position approaches the limits of detonation, where the thickness of the reaction zone
becomes significant, this is no longer the case. The C-J theory is concerned with only
the initial and final states of the gas and not with the route from the one state to the other.
Nettleton (1987) pointed out that, close to the limits of ability to detonate, the predicted
C-J detonation velocities are significantly higher, by 20% or more, than those actually
measiired. The discrepancies between predicted and measured pressures and densities
of the flow just behind the shock front are also correspondingly large in such mixtures.
Therefore, more-refined theories were required.