One Reaction. Invariant Manifolds and A Priori Bounds

The time independent equations describing a system with the single 

The boundary condition (2.2.4) results from the symmetry of the system-spherical region with uniform surface conditions. The boundary condition (2.2.5) can be physically realized if the resistance for heat and mass transport between the spherical region and the surrounding fluid is much smaller than the resistance for transport in the interior of the region. Then the surface of the region is kept at very nearly the same state as that of the surrounding medium. 

These relations show that the family of curves 

It can be seen from Eqs. (2.2.10), (2.2.11) that r(uo) is a bounded arc under very general assumptions about the transport coefficients I i. Off, kc. Accordingly, one can obtain a priori bounds for the state variables u(r), depending on Uq and the transport coefficients but independent of the reaction rates. In the special case of constant Dj, 

Another type of boundary conditions arises when the resistance for heat and mass transfer between the surface of the pellet and the surrounding fluid is not much smaller than the resistance for transport within the pellet. In this case the surface state u(ro) is not the same with the state of the surrounding fluid and the boundary conditions can be formulated by means of transfer coefficients 1 as 

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