A numerical investigation of laminar natural double diffusive convection in an open ended vertical cylindrical annulus with unheated entry and unheated exit is performed. Both boundary conditions of uniform wall temperature/uniform wall concentration (UWT/UWC) and uniform heat flux/uniform mass flux (UHF/UMF) are considered. Results of dimensionless induced volume rate $Q$, average Nusselt number $\overline{Nu}$ and Sherwood number $\overline{Sh}$ are obtained for air flow under various buoyancy ratio $N$, Grashof numbers due to heat and mass transfer $Gr_T$ and $Gr_M$, Schmidt number $Sc$ and combinations of unheated entry, heated section and unheated exit length. Since the flow under consideration is a boundary layer type, the governing partial differential equations was discretized to a linear system of equations by the use of an implicit finite difference method. The nonlinear convective terms are approximated by second upwind difference method for the numerical stability. The numerical results reveal that the presence of unheated entry and unheated exit severely affects the heat and mass transfer rates. The numerical solutions are found to approach asymptotically the closed form solutions for fully developed flow. Further, the present numerical results are validated with the existing solutions for pure thermal convection and are found to be in good agreement.