Ignition time equation

Applying Equations (5.21) to the adiabatic time corresponding to the critical Damkohler number, and realizing for a three-dimensional pile of effective radius, r,. A 3 (e.g. Sc = 3.32 for a sphere for Bi —> oo), then we estimate a typical ignition time at 6 = Sc 3 of... 

To calculate ATa( from equations 7.2, 7.6, and 7.8, it is therefore necessary to determine tu tf, 7), Tf, k, and T. The ignition time, tu is dehnedby the operator. The value of ff is empirically obtained, based on the criterion that A d becomes constant for t > tf. The smallest tf value that gives the limiting ATad should be chosen. 

Tlie ignition time delay can be calculated from tlie equation in die problem statement ... 

Consideration of the history (to calculate quantities like ignition times) necessitates retention of time derivatives in the conservation equations. Just as in the previous section, to achieve the greatest simplicity we adopt a thermal theory, although in various applications that have been cited the full set of conservation equations has been considered. Let a reactive material occupy the region x > 0, and to avoid complications assume that the material remains at rest and has a constant density p, although coordinate transformations readily enable this assumption to be removed. Let the material, initially at temperature Tq, be exposed to a constant heat flux q — — A 5T/5x at X = 0 for all time t > 0, where A is the constant thermal conductivity of the material. The time-dependent equation for conservation of energy for the material, analogous to equation (9), is... 

A first approximation to the ignition time may be obtained by equating the heat-release term in equation (31) to either of the other two terms, as calculated from the inert solution, evaluated at x = 0. In view of equations (32) and (33), this gives... 

Investigation of the properties of equation (45) provides the improved formula for the ignition time. 

The product Kpc is a quantity called the thermal inertia that emerges from the transient heat transfer analysis of ignition time (see eq. 40). The temperature dependence of k, p, and c in equations 26-28 suggests that the product of these terms (ie, the thermal inertia) should have the approximate temperature dependence ... 

Fig. 8. Numerical solution of the Langevin equation (eq. (18)) in the region of thermal explosion. Left part Individual realizations of the process of explosion featuring the considerable dispersion of ignition times. Right part Probability distribution of ignition times, illustrating further the random character of explosion...

Residue Hea.tup. Equations 27—30 can be used to estimate the time for residue heatup, by replacing the Hquid properties, such as density and heat capacity, with residue properties, and considering the now smaller particle in evaluating the expressions for ( ), and T. In the denominator of T, 0is replaced by and is replaced by T the ignition temperature of the residue. 

The fundamental parameters in the two main methods of achieving ignition are basically the same. Recent advances in the field of combustion have been in the development of mathematical definitions for some of these parameters. For instance, consider the case of ignition achieved by means of an electric spark, where electrical energy released between electrodes results in the formation of a plasma in which the ionized gas acts as a conductor of electricity. The electrical energy Hberated by the spark is given by equation 2 (1), where V = the potential, V 7 = the current. A 0 = the spark duration, s and t = time, s. 

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