The convective-diffusive mass transfer problem with chemical reaction in a Couette planar flow has been analyzed in terms of the integral-spectral methods originally introduced by Arce et al. (Comput. Chem. Eng. 2 (11) (1988) 1103). The problem is solved by inverting the differential model into an integral equation of a Volterra type, in the axial variable, and of the Fredholm type, in the radial coordinate, The kernel of such an integral equation is given by the Green function which does not contain any of the kinetic parameters of the (homogeneous and/or heterogeneous) reaction term, This Green function is computed in terms of the eigenfunctions and eigenvalues of the Sturm-Liouville problem associated with the radial variable. The Sturm-Liouville problem is solved (analytically) by using Airy functions and the final integral equation must be solved by an iteration procedure. Several of the mathematical formulation details are discussed and many numerical examples are presented to illustrate the technique: for example, concentration profiles for systems with heterogeneous (wall) catalytic reactions, homogenous (global) reactions and simultaneous (global and wall catalytic) reactions with kinetics of a general form, i.e. power-law and Langmuir-Hinshelwood types of functions are investigated. The solutions to the class of problems considered here are obtained as particular cases of the general integral equation solution of the differential model discussed in the article. The effects of relevant parameters in the system on the computational algorithm with respect to convergence and (numerical) stability characteristics are discussed.