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6.4. MODIFICATION OF THE CONDITIONS AT THE “COLD BOUNDARY” 139
of absorbing a certain amount of heat Q from the flame and of acting as a filter which
allows the unburnt gases to reach the flame but prevents the combustion products from
diffusing towards the unburnt gases. By assuming that the holder is located at point
x = 0, the system of Eqs. (6.2), (6.3) and (6.5a) will still hold, but conditions (6.13)
at the cold boundary will have to be substituted by the following
x = 0 : T = T 0 , ε = 0, λ dT
dx
≠ 0. (6.20)
To the contrary, for x < 0 the composition and state of the mixture are uniform
x < 0 : Y = ε = 0, T = T 0 . (6.21)
The value of Y at x = 0 remains undetermined, but different from zero. Therefore
through x = 0 there exists a discontinuity in the composition of the mixture which
becomes possible due to the existence of the filter.
It happens here, as in the case of ignition temperature, that the structure and
propagation velocity of the flame are independent from the value of parameter Q,
provided it is not close to heat of reaction q or very close to zero, which justifies the
practical value of the proposed model.
T f
T f
T T 0 i 1−ε T 0
1−Y
Karman ´ ´ boundary condition
1−Y
Hirschfelder boundary condition
1−ε
Figure 6.1: Boundary conditions for ignition temperature and flame holder models.
Figure 6.1 summarizes and compares the conditions at the cold boundary for
the two solutions proposed. Since both lead to the same results, in the following we
will use only the assumption that an ignition temperature exists.
Figure 6.2 shows in a qualitatively way the form of the solutions obtained for
the propagation velocity of the flame in a given mixture when the ignition temperature,
assumed for it, is changed. It is seen that there is a wide interval AB of ignition
temperatures within which the propagation velocity of the flame is independent from
T i . The quantitative solutions for some typical cases will be included further on.