We use numerical modeling to study the flow patterns of an active nematic confined in a cylindrical capillary, considering both planar and homeotropic boundary conditions. We find that active flow emerges not only along the capillary axis but also within the plane of the capillary, where radial vortices are formed. If topological defects are imposed by the boundary conditions, they act as local pumps driving the flow. At higher activity, we demonstrate escape of the active defects and flow into the third dimension, indicating the importance of dimensionality in active materials. We argue that measuring the magnitude of the active flow as a function of the capillary radius allows determination of a value for the activity coefficient.