  (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.
