We propose a model for the self-propulsion of a small motor particle that generates a nonuniform concentration distribution of solute in the surrounding fluid via a constant solute flux asymmetrically from the motor surface. The net osmotic driving force and motor speed are investigated in the limits of slow and fast product particle flux (relative to the diffusive flux of the product species). When the only solute species in solution is that produced by the motor, the motor's speed is shown to be proportional to the solute flux for slow flux rates and to the square root of the solute flux for large flux rates. When solute species are already present in solution at concentration high compared to that generated by the motor, the motor speed at high flux rates saturates and scales as the diffusivity of the solute divided by the motor size. The analytical results compare well with Brownian dynamics simulations. Full hydrodynamic interactions are taken into account in the theoretical analysis.