Generalized volumetric energy balance for directly irradiated bubbling fluidized beds

•A fluidized particle receiver for beam-down concentrating solar power is studied.•Open and closed bubbling fluidized bed receivers are irradiated to the bed surface.•A correlation to estimate the bed surface emissivity is proposed and validated.•The methodology to apply a simplified bed energy balance is explained.•The new approach simplifies the thermal design of the receiver at steady state. Beam-down concentrating solar towers are receiving much attention from the scientific and industry sectors to generate electricity, produce chemical fuels, and drive energy-intensive processing of materials. The easy thermal design of the solar receiver is essential for the deployment of this renewable energy. This work proposes the equations and methodology to solve the steady-state energy balance of directly irradiated bubbling fluidized beds. The analysis focuses on solving the heating of the bed surface assuming that the energy flux received at the bed surface is absorbed through a layer of particles. The proposed methodology considers a volumetric energy balance of the top-bed surface to calculate the bed temperature. The bed surface emissivity (εb) is a critical parameter in order to solve such an energy balance. To deal with the radiative losses, a correlation is proposed to estimate bed emissivity at the top-bed surface that depends on the dimensionless length number, dp*=Ar13. The results are validated against both new experimental and literature data for the open and closed fluidized bed receivers. The absolute relative error when estimating the steady state temperature of the systems studied is about 5.2 % (on average). The relative error follows a Gaussian distribution guaranteeing an unbiased estimation of the bed temperatures at steady-state operation.