A numerical model based on the fundamental continuum transport equations is developed to simulate the Fischer-Tropsch synthesis in a slender packed column. Internal and external mass transport limitations are considered in a fully resolved manner where a two-way coupling between mass and heat transport is accounted for. A Langmuir-Hinshelwood type kinetic model is used for the reaction rate in the catalyst phase which is then intrinsically coupled to the fluid phase transport by enforcing the appropriate boundary condition using the Immersed Boundary Method. The fixed bed reactor consists of 220 particles packed in a random manner using the Discrete Element Method where cooling from the wall to the bed is also considered. The influence of temperature and concentration on conversion and selectivity is investigated. The Direct Numerical Simulation of the Fischer-Tropsch reactor is then compared with an equivalent 1-D heterogeneous reactor model which employs empirical closures.