Swirling flows inside a pipe have been used widely for enhancement of heat and mass transfer in energy conversion devices. Predicting the swirl decay rate is important associated with engineering problems such as gas turbine combustors, compressors, mixers, separators and so on. In the present study, the swirl decay rate inside a straight annular pipe and its primary parameters are investigated with flow simulations and mathematical modeling. A ODE-based one-dimensional model of the swirl number is derived. In order to include the effects of viscous diffusion and turbulent dispersion, the wall shear stress and the turbulent viscosity are modeled by adapting empirical correlations. In order for validation with CFD simulations for various geometric conditions, we revised the previous wall shear stress model via an assumption on the azimuthal velocity profile. The swirl number profiles and decay rates from the model show reasonable agreement with those of CFD simulations. The developed model introduce three primary parameters that can affect the swirl decay rate - the inlet Reynolds number, the ratio of the inner and outer radii (referred to as gamma) and the inlet swirl intensity. The swirl decay rate decreases as the inlet Reynolds number increases or inlet swirl intensities decreases. A geometric parameter gamma shows a trend of the swirl decay rate distinguished from the other parameters. From the one-dimensional model, it is found that there exists an optimal gamma value that can minimize the swirl decay rate. We analyze the characteristics of the optimal gamma value. A reason for the existence of the optimal gamma values is examined from the ODE-based model and a balance between the wall shear stress and the turbulent diffusion in the radial direction. The present analyses are supported by accurate flow simulations. The optimal gamma values found here can be useful for designing the inner and outer radii that can minimize the momentum loss in a device such as gas turbine combustors.