In the present work, the study of alumina-water nanofluid heat transfer between two concentric vertical cylinders has been done by modified Buongiorno’ s model (BM) to examine the impacts of TEMPERATURE JUMP and slip velocity boundary conditions for a wide range of Knudsen number. Runge-Kutta-Fehlberg method, as a standard integration scheme along with a shooting method, has been chosen for solving nonlinear ordinary differential equations (ODEs) along with boundary conditions. The main concentration of this paper is on the TEMPERATURE JUMP since the slip velocity has been extensively examined in many studies. The presence of TEMPERATURE JUMP boundary condition by varying Knudsen number was considered to investigate the effects of the bulk mean nanoparticle volume fraction ϕ B, mixed convection parameter Nr, buoyancy parameter Ng, and heat flux ratio ε on the total dimensionless heat transfer coefficient HTC and the dimensionless pressure gradient Ndp. The obtained results indicate that TEMPERATURE JUMP boundary condition plays a pivotal role in TEMPERATURE profile, heat transfer coefficient and pressure drop; for instance, the negligence of TEMPERATURE JUMP near walls causes to undervalue heat transfer coefficient in continuum flow regime and overestimate it in slip flow regime