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  The only parameter which characterizes the explosion is the Damköhler number . When  is very high, conduction time is longer than the chemical reaction time and the system explodes with high temperature since there is not enough time for conduction to remove the heat. On the other hand, when  is very low, heat conduction time is much faster than the chemical reaction time, such that all the heat produced by the chemical reaction is immediately conducted to the wall, thus there is no explosion, it goes to an almost steady state, Amable Liñán coined this mode as slowly reacting mode. At a critical Damköhler number  the system goes from slowly reacting mode to explosive mode. Therefore, , the system is in steady state. Instead of solving the full problem to find this , Frank-Kamenetskii solved the steady state problem for various Damköhler number until the critical value, beyond which no steady solution exists. So the problem to be solved is
  with boundary conditions
  the second condition is due to the symmetry of the vessel. The above equation is special case of Liouville–Bratu–Gelfand equation in mathematics.