Analytical solution for multi-component pyrolysis simulations of thermal protection materials

This paper provides the analytical solution to a one-parameter Šesták–Berggren kinetic model for the thermoanalytical study of pyrolysis reactions involving multiple independent parallel reactions. Such multiple independent parallel reactions are widely used, for instance, in the modeling of the pyrolysis of thermal protection materials used in heatshields for spacecraft during the atmospheric entry phase. Solving inverse problems to infer parameters of the kinetic model through optimization techniques or Bayesian inference methods for uncertainty quantification may require a large number of evaluations of the response and its sensitivities (derivatives with respect to parameters). Moreover, in the case of kinetic equations, the Arrhenius parameters can exhibit strong dependence that can require further model evaluations for an accurate parameter calibration. The interest of this analytical solution is to reduce computation cost while having high accuracy to perform parameter calibration from experiments and sensitivity analysis. We propose to use exponential–integral functions to express the solution of the temperature integral, and we derive the analytical solution and its sensitivities for the parallel reaction model both for constant temperature (isothermal) and for constant heating rate conditions. The solution is validated on a six-equation model using parameters inferred in a previous work from the experimental data of the pyrolysis of a phenolic-impregnated carbon ablator material, and we compare the computational cost and accuracy of the implemented analytical solution with a numerical solution. Our results show that the use of such analytical solution with an accurate computation of the exponential–integral function significantly reduces the computational cost compared to the numerical solution