We analyze dynamics of stationary nonuniform patterns, traveling waves, and spatiotemporal chaos in a simple model of a tubular cross-flow reactor. The reactant is supplied continuously via convective flow and/or by diffusion through permeable walls of the reactor. First order exothermic reaction kinetics is assumed and the system is described by mass and energy balances forming coupled reaction-diffusion-convection equations. Dynamical regimes of the reaction-diffusion subsystem range from pulses and fronts to periodic waves and complex chaotic behavior. Two distinct types of chaotic patterns are identified and characterized by Lyapunov dimension. Next we examine the effects of convection on various types of the reaction-diffusion regimes. Remarkable zigzag fronts and steady state patterns are found despite the absence of differential flow. We employ continuation techniques to elucidate the existence and form of these patterns.