Cool bottom processes on the thermally-pulsing AGB and the isotopic composition of circumstellar dust grains

(Abridged) We examine the effects of cool bottom processing (CBP) on several isotopic ratios in the convective envelope during the TP-AGB phase of evolution in a 1.5 M_sun initial-mass star of solar initial composition. We use a parametric model which treats extra mixing by introducing mass flow between the convective envelope and the underlying radiative zone. The parameters of this model are the mass circulation rate (Mdot) and the maximum temperature (T_P) experienced by the circulating material. The effects of nuclear reactions in the flowing matter were calculated using a set of structures of the radiative zone selected from a complete stellar evolution calculation. The compositions of the flowing material were obtained and the resulting changes in the envelope determined. Abundant ^26Al was produced by CBP for log T_P > 7.65. While ^26Al/^27Al depends on T_P, the isotopic ratios in CNO elements depend dominantly on the circulation rate. The correspondence is shown between models of CBP as parameterized by a diffusion formalism within the stellar evolution model and those using the mass-flow formalism employed here. The isotopic ratios are compared with the data on circumstellar dust grains. It is found that the ratios ^{18}O/^{16}O, ^{17}O/^{16}O, and ^26Al/^27Al observed for oxide grains formed at C/O < 1 are reasonably well-understood. However, the ^15N/^14N, ^12C/^13C, and ^26Al/^27Al in carbide grains (C/O > 1) require many stellar sources with ^14N/^15N at least a factor of 4 below solar. The rare grains with ^12C/^13C < 10 cannot be produced by any red-giant or AGB source.