Differential model, regression problems

The regression for integral kinetic analysis is generally non-linear. Differential equations may include unobservable variables, which may produce some additional problems. For instance, heterogeneous catalytic models include concentrations of species inside particles, while these are not measured. The concentration distributions, however, can affect the overall performance of the catalyst/reactor. 

Temperature profiles can be determined from the transient heat conduction equation or, in integral models, by assuming some functional form of the temperature profile a priori. With the former, numerical solution of partial differential equations is required. With the latter, the problem is reduced to a set of coupled ordinary differential equations, but numerical solution is still required. The following equations embody a simple heat transfer limited pyrolysis model for a noncharring polymer that is opaque to thermal radiation and has a density that does not depend on temperature. For simplicity, surface regression (which gives rise to convective terms) is not explicitly included. 

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