Particle deposition on the vertical wall in a dilute turbulent pipe flow is modeled. The different mechanisms of particle transport to the wall are considered; i.e., Brownian motion, turbulent diffusion and turbophoresis. The Saffman lift force, the electrostatic force, the virtual mass effect and the wall surface roughness are taken into account in the model developed. The boundary condition for accounting for the probability of particle adhesion to the wall (p) is suggested. An analytical solution for deposition of small Brownian particles is obtained. The range of the particle relaxation times, within which the suggested boundary condition is reliable, is evaluated. The numerical code developed is validated by comparison of the computational results with the experimental data found in the open literature (see the 1st figure below). Various regimes of particle deposition at different particle-wall adhesion probabilities are illustrated numerically and explained. We clearly show that the degree of the deposition rate reduction with a decrease in the particle adhesion probability strongly depends on the particle relaxation time (see the 2nd figure below).