The reactive Graetz problem for a first order homogeneous decomposition reaction, and the limit of moderately large reaction constants

Abstract

The long term goal of the present study is to facilitate the inversion of experimental data on cluster evaporation in laminar flow inside tubes. This requires the determination of reactant and product fluxes as functions of three dimensionless governing parameters: the axial position x along the tube, the reaction rate coefficient K, and the ratio γ of product and reactant diffusivities. The reactant flux is first calculated by separation of variables, including the accurate determination of the first 35 eigenvalues for 13 discrete values of K in the range 0<K<500. This approach is far less efficient for the product flux because the eigenfunctions are not standard functions. A numerical solution is also implemented for the reactant and its product, though it can hardly cover the full range of the three relevant parameters. The most common experimental situation involves relatively large values of K, for which a three term asymptotic solution is developed in negative powers of K1/2, with errors of order K−2. These approximate results represent well the numerically calculated reactant fluxes for K > 10 in terms of two functions, H0(s) and H1(s), reported in tables. The product fluxes require two functions, I0(s,γ), I1(s,γ). These we compute for six values of γ, and report as simple analytical fits covering continuously the full physically relevant range 1<γ<1.5. The desired ability to analyze experimental data to yield kinetic information is therefore achieved without the need for additional numerical solutions to partial differential equations.