Circular closed conduits with free surface have many applications in nature. Two-dimensional flow hydraulic simulation studies in these channels are limited and generally one-dimensional relations and equations are used for modeling and designing these channels. In this study, a quasi-two-dimensional mathematical model (Shiono and Knight, SKM) based on the Navier-Stokes equations is used to calculate the transverse distribution of flow velocity and boundary shear stress in a deposited pipe (or pipe with a flat bed). To solve the SKM model, three key coefficients of this model must be determined which play a decisive role in simulating the transverse velocity distribution and boundary shear stress. In this study, the Rameshwaran and Shiono (2007) equation was used to calculate the friction coefficient (f) and a numerical optimization process based on laboratory data was used to calibrate the two coefficients of eddy viscosity (λ) and the secondary flow (β). The results indicate the different capabilities of the Shiono and Knight models in simulating the transverse velocity and shear stress distribution. The results indicate the different capabilities of the Shiono and Knight model in simulating the lateral distribution of velocity and shear stress. Although this model estimates the velocity distribution with reasonable accuracy (with an average and maximum error of 6 and 7.1 percent, respectively), the boundary shear stress distribution (with an average and maximum error of 18.5 and 22 percent, respectively) is far from acceptable values.