Indirect measurements of streamwise solid fraction variations of granular flows accelerating down a smooth rectangular chute

In this study, we detail a method for estimating the flux-averaged solid fraction of a steady granular flows moving down an inclined rectangular chute using velocity measurements from along the perimeter cross section, combined with knowledge of the mass flow rate through the cross section. The chute is 5 cm wide and 150 cm long with an adjustable inclination angle. Four inclination angles, from 27° to 36° at 3° intervals, are tested. This angle range overlaps the internal friction angle of the glass beads, which are 4 mm nominal in diameter. Two slender mirrors are installed at the top and the bottom of the transparent chute to reflect images of the flow down the chute of the two surfaces. This allows photographic recording of the flow with a PIV imaging system and measurement of the flow depth. The mass flow rate can be calibrated simultaneously by collecting the accumulated mass at the chute exit. A linear interpolation scheme is proposed to interpolate the volume flow rate in each section of the chute. Sensitivity analysis suggests that the relative standard deviation of this scheme is about ±6%, i.e., the resultant solid volume fraction is only moderately dependent on the interpolation scheme for the tested cases. This is further confirmed by a direct intercepting method. Compared to the sophisticated magnetic resonance imaging (MRI) or the radioactive positron emission particle tracking (PEPT) methods, the present method is verified as a cost-effective and nonhazardous alternative for ordinary laboratories. Two distinct groups of streamwise dependence of the solid fractions are found. They are separated by the inclination angle of the chute and agreed with the internal friction angle. In the experiments using the two smaller inclination angles, the solid fraction ratios are found to be linear functions of the streamwise distance, while for the two larger inclination angles, the ratios have a nonlinear concave shape. All decrease with growing downstream distance.