A numerical model is proposed to perform CFD simulations of biomass boilers working in different operating conditions and analyse the results with low computational effort. The model is based on steady fluxes that represent the biomass thermal conversion stages through the conservation of mass, energy, and chemical species in the packed bed region. The conversion reactions are combined with heat and mass transfer submodels that release the combustion products to the gas flow. The gas flow is calculated through classical finite volume techniques to model the transport and reaction phenomena. The overall process is calculated in a steady state with a fast, efficient, and reasonably accurate method, which allows the results to converge without long computation times. The modelling is applied to the simulation of a 30 kW domestic boiler, and the results are compared with experimental tests with reasonably good results for such a simple model. The model is also applied to study the effect of air enrichment in boiler performance and gas emissions. The boiler operation is simulated using different oxygen concentrations that range from 21% to 90% in the feeding air, and parameters such as the heat transferred, fume temperatures, and emissions of CO, CO2, and NOx are analysed. The results show that with a moderated air enrichment of 40% oxygen, the energy performance can be increased by 8%, CO emissions are noticeably reduced, and NOx remains practically stable.