Non-Newtonian fluid flow in a pipe system is considered and a third grade non-Newtonian fluid is employed in the analysis. The velocity and temperature distributions across the pipe are presented. Entropy generation number due to heat transfer and fluid friction is formulated. The influences of non-Newtonian parameter and Brinkman number on entropy generation number are examined. It is found that increasing the non-Newtonian parameter reduces the fluid friction in the region close to the pipe wall. This in turn results in low entropy generation with increasing non-Newtonian parameter. Increasing Brinkman number enhances the fluid friction and heat transfer rates; in which case, entropy number increases with increasing Brinkman number.