Simulations and analysis of transversal patterns in a homogeneous three-dimensional (3-D) model of adiabatic or cooled packed bed reactors (PBRs) catalyzing a first-order exothermic reaction were presented. In the adiabatic case the simulation verify previous criteria, claiming the emergence of such patterns when (?Tad/?Tm)/(PeC/PeT) surpasses a critical value larger than unity, where ?Tad and ?Tm are adiabatic and maximal temperature rise, respectively. The reactor radius required for such patterns should be larger than a bifurcation value, calculated here from the linear analysis. With increasing radius new patterned branches, corresponding to eigenfunction of the problem emerge, whereas other branches become unstable. The maximal temperature of the 3-D simulations may exceed the 1-D prediction, which may affect design procedures. Cooled reactor may exhibit patterns, usually axisymmetric ones that can be characterized by two anomalies: the peak temperature may exceed the corresponding value of an adiabatic reactor and may increase with wall heat-transfer coefficient, and the peak temperature in a sufficiently wide reactor need not lie at the center but rather on a ring away from it. In conclusions, we argue that transversal patterns are highly unlikely to emerge in practical adiabatic PBRs with a single exothermic reaction, as in practice PeC/PeT > 1. That eliminates patterns in stationary and downstream-moving fronts, whereas patterns may emerge in upstream-moving fronts, as shown here. This conclusion may not hold for microkinetic models, for which stationary modes may be established over a domain of parameters. This suggests that a 1-D model may be sufficient to analyze a single reaction in an adiabatic reactor and a 2-D axisymmetric model is sufficient for a cooled reactor. The predictions of a 2-D cylindrical thin reactor with those of a 3-D reactor were compared, to show many similarities but some notable differences. (c) 2012 American Institute of Chemical Engineers AIChE J, 2012