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Ignition Ignition is defined as:
There are two types of ignition. Piloted Ignition (insert a picture of a spark plug sparking or a small flame) This particular kind of ignition consists of a spark or a small flame providing the initial energy for a fire to ignite Spontaneous Ignition (insert a picture that looks like an explosion???) This kind of ignition is the result of a flame developing spontaneously in a flammable mixture of gases. As you have seen in the Science section, a flame can release a tremendous amount of energy into the ambient environment. The key to sustaining and propagating a flame after ignition is the rate of energy release. If the rate of energy release is less than or equal to the rate of heat loss by the flammable mixture to the environment, then the flame will not propagate. If the rate of heat release is greater than the rate of heat loss, then the reaction will become unstable and proceed to a higher temperature. The figure below can be used to qualitatively understand this idea. When the temperature of the mixture is greater than T2, the heat release rate is greater than the heat loss. The mixture becomes unstable and begins to heat up, until the temperature T3 is reached. T3 is the higher temperature normally associated with combustion.
While this concept is not rigorously valid for spark piloted ignition, it does provide a useful schematic for thinking about how ignition is achieved and a flame propagated. There are two models for finding the critical ignition temperature. The first was proposed by Semenov. He assumed a uniform temperature in the reacting mixture and that the heat losses could be described by the product of a overall heat transfer coefficient and the difference in temperature between the walls of the reacting vessel and the mixture. Because this method assumes a lumped model for the reacting mixture, it is called the “low Biot number model”. To see the equations for this model, follow this link. (link to equations) The second model was developed by Frank Kamenetskii and takes into account the temperature variations across the mixture. This is known as the “high Biot number” model. To see the equations for this model, follow this link (link to equations) The values for the auto-ignition temperatures for several fuels have been tabulated and some common fuels are shown on a table from the 1997 NFPA Handbook (link to table 1) These values for auto-ignition temperature are relevant in that they represent the temperature that must be achieved in a flammable mixture at a distance greater than the quench distance above a surface, whether it be a solid or liquid, for spontaneous ignition to occur. The flammability of a liquid is related to this through the flashpoint. The flashpoint of a liquid fuel is the minimum temperature that is needed to insure a flammable mixture above the liquid surface. This is found using our knowledge of the lower flammability limit of the fuel (link to science) and the Clausius-Calpyron equation to find the partial pressure of the liquid (link to science). Flashpoints of certain fluids are listed in Table 2 (link). This idea of a flashpoint is also valid in a solid, but in this case it refers to the minimum temperature that will cause the solid to undergo pyrolysis. Unlike the liquids, this involves a chemical change in the substance of the solid, and so is a destructive reaction. Part of the solid is decomposed and flammable vapors are released. In this case, the establishment of a flame is not necessarily more “difficult” but does require more conditions to be met. First, a sufficient heat flux is necessary to pyrolize the solid at a high enough rate to provide a flammable mixture to the atmosphere above the solid. Second, a pilot ignition source must be near and, third, the conditions must be suitable for flame propagation. There are two possibilities for a piloted ignition of a solid, a continuous heat flux and a discountinous heat flux. A continuous heat flux is defined as being separate from the ignition source and of a necessary level that the temperature of the solid surface is high enough to cause a flow a pyrolized material to sustain a flame. In this case, the flashpoint temperature is the temperature that will cause the flow of volatiles in sufficient amount to sustain a flame. To see the equations for this condition, click hear.(link) A discontinuous heat flux is one that is applied and then removed after the flame becomes self-supporting. In this case, a flux is applied to a solid until the flow of volatiles is high enough that the heat release from the flame is sufficient to keep the pyrolysis going and sustain the flame. To see the equations for this condition, click here (link). Spark-piloted ignition: The phenomenon of spark piloted ignition does not rely on a heat flux from a flame to produce the heat flux necessary to ignite a flammable mixture of gasses. Instead, a high energy spark ionizes the materials in the gap the spark jumps over. This ionization results in the formation of highly reactive chemical species, or ‘radicals.’ These radicals the attack the fuel and start the chemical reactions that lead to a flame and flame propagation. Semenov Model Semenov assumes that the gas is at a uniform temperature, T, and that DT is the temperature difference between the gas and the wall of the vessel enclosing it. If this is true, then the heat loss to the walls of this vessel can be described mathematically by: 
where h is the coefficient of heat transfer between the gas and the walls of the vessel and S is the surface area. The critical ambient temperature, Ta,crit, is defined as the temperature at which the heat production is equal to the heat lost. In this situation, 
and, taking the derivative with respect to the gas temperature,
The heat released by combustion is expressed by the following formula: Tus, Equation 2 becomes:
and Equation 3 becomes:
dividing Equation 5 by Equation 6 yields 
by assuming EA>>RT, Equation 7 is solved by binomial expansion for values of the equilibrium gas temperature. This solution, however, is not very realistic because it assumes a uniform gas temperature. Thus, Frank-Kaminetskii (link) developed a model that takes into account the temperature differences in the gas. Frank-Kaminetskii (presented in Drysdale)Assuming a one-dimensional container with uniform symmetrical heating, the heat equation becomes: 
for this equation, T is the gas temperature, r is the position inside the gas, Q is the volumetric rate of heat generation, k is the thermal conductivity, a is the thermal diffusivity and k takes a different value depending on geometery (0 for an infinite slab, 1 for an infinite cylinder and 2 for a sphere). In addition, the reaction rate is assumed to be in a single Arrhenius expression. Also, there is no reactant consumption , the Biot number is large enough for conduction to determine the rate of heat loss (Bi > 10) and the thermal properties are independent of temperature. Thus, the boundary conditions are now:
Tro=To at t≤0 (surface) If a few non-dimensional variables are defined:
Assuming e<<1, Equation 8 becomes:
Declaring a new variable, d, such that
Equation 9 becomes:
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at r=0 and
t≥0 (center)
and
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