First do the energy and material balance around the complete assembly when it is in steady operation. If there is a good agreement of input and output mass and total energy including reaction energy in agreement with chemical compositions,  then use any available information like temperatures, flow rates and compositions to do separate  energy and material balances of each part. It works when something like carbon or polymer Is not accumulating or being leached out of the beds. Series and parallel reactors follow the same principles but with different details for each configuration.

Modeling of TBR reactors has regarded occurrence of an axial, one-dimensional flowing continuum. In some cases, the radial flow effects were ignored (pseudo-homogeneous model) and the main gradients of concentration & temperature were considered to occur only in the axial direction. Then, the continuity, heat and momentum transfer equations give a set of ordinary differential equations. Afterwards, the set of equations were solved by suitable numerical methods. Upon ignoring the radial effects of all the transport mechanisms, the one-dimensional models predict constant temperature & conversions in the cross section. This is an oversimplification for highly temperature-dependent reactions. In the two-dimensional pseudo-homogeneous models, the radial & axial processes are lumped in a single entity: the effective transport parameters for mass & heat transfer. This treatment of TBR reactors led to lot of efforts on the understanding, prediction and interpretation of such effective parameters. Results of these efforts are many correlations that have limited success with data & have led to many discussions on the suitability of using such parameters.