Improved apparatus for use in process systems which include exothermic chemical conversions of organic compounds to value added products is disclosed, more particularly, flow reactors for exothermic chemical conversions using a fixed heterogeneous catalyst with means for control of the exotherm. Flow reactors of the invention comprise a plurality of walled conduits each having an outer surface disposed for contact with a heat-transfer medium, an inlet distribution manifold adapted for flow communication with a downstream manifold through channels formed by heterogeneous catalytic material disposed within each conduit during operation in a sequence of zones for catalyst having the same or different length along the longitudinal coordinate of the conduit and within each zone essentially uniform cross-section of the conduit measured in a plane perpendicular to the longitudinal coordinate thereby defining volume of the zone, and the sequence of zones comprising of at least two zones such that each downstream zone has a larger or smaller cross-section than the contiguous upstream zone. Another aspect of the invention includes processes which use such flow reactors, for example the continuous manufacture of maleic anhydride.