Hoppers and silos are widely used in process industries for handling everything from grains to building materials to pharmaceutical materials. Nonetheless, the prediction of the outflow characteristics from hoppers is still challenging owing to the complex rheological characteristics of granular materials. The mass flow rate from hoppers are known to depend on the hopper angle, the outlet size, and the material properties themselves. Several phenomenological correlations have been proposed to predict flow rates based on the hopper angle and the outlet diameter. With some empirical calibration, these correlations are able to predict the mass flow rate [1–3]. Investigating these models and their underlying assumptions requires quantitative, spatially resolved measurements of the velocity and solid fraction, which together determine the local flow rate.
Here we report on recently developed MRI methods [4] to provide quantitative measurements of the solid fraction and velocity in three-dimensional (3D) hoppers each with different hopper angle and outlet size [5]. These quantitative measurements provide the basic requirements for addressing and evaluating phenomenological models that have been proposed to predict the flow rate out of a hopper. We show that the solid fraction decreases smoothly from the bulk value above the outlet, indicating that the assumption of a “free-fall arch” used in some mass flow correlations is invalid. Furthermore, we show that the solid fraction, velocity and vertical evolution of the acceleration are all self-similar when normalised by the value at the centre of the outlet in a 3D hopper, in agreement with recent studies of 2D hoppers [3].