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Propagation velocity estimation for condensed phase combustion. (English) Zbl 0791.34022

The equations for 1D steady condensed phase combustion may be reduced to \[ da/dt=-(\kappa_ 2/v^ 2) \varphi (a)k(T)/(T-T_ b+qa), \quad T_ c<T<T_ b, \] with boundary conditions \(T=T_ c:a=1\), \(T=T_ b:a=0\). Here \(T\) is the temperature, \(a\) is the reactant concentration, \(\kappa\) is the thermal diffusivity, \(v\) is the propagation speed, \(\varphi (a)\) is the kinetic law, and \(q=Q/c\), where \(Q\) is the heat released by the reaction and \(c\) is the specific heat. Further, \(k(T)=k_ 0 \exp (- E/RT)\) for \(T>T_ c\), and \(k(T)=0\) for \(T_ i<T<T_ c\), where \(k_ 0\) is the pre-exponential factor, \(E\) is the activation energy and \(R\) is the universal gas constant. The equation may be written as \(v^ 2=B(a(T),T)\) with \(B(\rho(T),T)=-\kappa \varphi (\rho)k(T)/(T-T_ b+q \rho)\) \((d \rho/dT)\). For any test function \(\rho(T)\) satisfying the above boundary conditions, the maximal and minimal values of \(B(\rho(T),T)\) provide upper and lower estimates for \(v^ 2\). The proximity of the bounds shows how good a particular test function is, which may be considered as an approximate solution.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
49R50 Variational methods for eigenvalues of operators (MSC2000)
80A25 Combustion
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