Mass transfer and wall reaction

In the case of monoliths operating in laminar flow, transverse mixing is assured only by molecular diffusion, which has a 

The dimensionless velocity profiles are given in Section 8.2.1 and the shape factor (S) takes the values of 0 or 1, for parallel plates or circular channels, respectively. The two dimensionless parameters in Equation 8.16 are the transverse mass Peclet number 

The cases where axial diffusion cannot be neglected lead to the formulation of the extended Graetz problem for which analytical solutions have been given. If axial diffusion is indeed negligible (and this should be true for long channels), the only two mechanisms at play are axial convection and transverse diffusion. From this interaction, two different regimes appear which describe the behavior for axial distances of the same order of the channel length z = 0 L)  

This parameter can take a wide range of values. For long microchannels, the inlet Danckwerts boundary condition is simplified to uniform inlet concentration for Pcm a  

In the transverse direction, the symmetry boundary condition is appropriate at r = 0. Additionally, a condition at the wall is required. In the case of a microchannel reactor where a first-order reaction is occurring in the catalytic layer, this is given by 

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