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

Abstract

This paper provides the analytical solution to a one-parameterŠesták-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 (PICA)

Bayesian Identification of Pyrolysis Model Parameters for Thermal Protection Materials Using an Adaptive Gradient-Informed Sampling Algorithm With Application to a Mars Atmospheric Entry

“…When the temperature is a linear function of time (constant heating rate), an analytical solution can be found for the multi-component single-step pyrolysis model (Eq. ( 6)) and its gradients [12]. We will use the analytical expressions in the gradient-informed sampling method which will significantly speed up the computational time taken by each evaluation of the model and its gradients and avoid discretization errors in the solution of the system of ODEs.…”

“…where 1 S is the indicator function of S ⊆ R. The activation energy is strictly positive, but in the case where several pyrolysis reactions are considered, the values of E i are further limited to the bounded domain [L Ei , U Ei ] in order to avoid the different pyrolysis reactions to mix together, which would make the potential function more complex to explore. The maximum values for r i are limited to 10, as higher values were not found in previous similar pyrolysis models ([13]), and first order reactions (r i = 1), for which the analytical solution is different, are not considered here ( [12]).…”

“…Due to the higher dimensionality, the algorithm needs more iterations to learn the local scales and correlations and the adaptation phase is therefore longer. The solution to the pyrolysis model and its derivatives with respect to the parameters are all computed analytically [12].…”

“…The objective of this paper is to infer from the new experimental data of Bessire et al [10] an uncertaintyquantified pyrolysis model. The class of pyrolysis models that we adopt is the class of multi-component single-step models [11,12], which represent the decomposition mechanisms in terms of a system of chemical kinetic equations with reaction rates that obey an Arrhenius-like generic model. In previous work by Torres-Herrador et al [13], the new experimental data of Bessire et al [10] have already been used to estimate parameters of such a multi-component single-step pyrolysis model, but the model identification was formulated as a deterministic inverse problem in [13],…”