Singular Perturbation Solution of Time Dependent Mass-Transfer with Non-linear Chemical Reaction

In this work the author presents two analyses of time dependentprocesses in which diffusion and non-linear reaction take placesimultaneously. The first is an analysis of the distributionof a single reactant in a stagnant medium when the reactionrate is proportional to the concentration squared. In the secondanalysis the time dependent distributions of two reactants aretraced, when the reaction rate is porportional to the productof the concentrations. The processes under consideration areassumed to take place in unbounded containers and attentionis focused on the cases in which the initial concentrationshave non-zero space average. The analyses are carried out assumingthat the space variations of the initial distributions and thediffusivities are large while the reaction constant(s) is small.Therefore, initially the changes in the distribution due todiffusion are much faster than the rate of absorption in thereaction. However, eventually the processes become reactiondominated. The two distinct behaviours are represented by correspondingasymptotic expansions which are matched in the usual manner.On the other hand, if the initial average concentrations vanish,then, in as much as the processes are diffusion dominated initially,they are diffusion dominated thoughout. The distributions canthen be solved-for by regular perturbations. The treatment of the single component case shows that when theinitial average concentration does not vanish, the residualamount of reactant is inversely proportional to the time thatelapsed since the beginning of the process multiplied by thereaction constant. This residual amount is altogether independentof the bulk quantity supplied initially. The decay of the binaryprocess is similar provided the bulk quantities supplied initiallyare well proportioned. If there is a disproportion, the decayis exponential rather than algebraic. In all events, late inthe process the residual amount(s) is independent of the diffusivity.