Estimation of the flow rate of free falling granular particles using a Poisson model in time

A generic method to estimate the flow rate (number of particles passing per unit of time) of free falling granular particles in a tube is presented. This principle could be applied in pneumatic planters, fertiliser distribution, yield monitoring of grains and fruits and various industrial applications. The flow of particles in a tube was observed as an intermittent succession of clumps of particles separated by spacings among them. Since the clumping effect prohibits counting individual particles, the hypothesis was that the flow constitutes a random arrival process in time. In other words, the process of particle arrivals at the sensor was assumed equivalent to classical Poisson driven arrival processes from queueing theory, such as telephone calls arriving independently at a helpdesk. The assumption of a Poisson process allows for simple flow rate estimation, since according to theory, the flow density of such a process is equal to the reciprocal value of the mean of the spacing time intervals, which can be measured. An optical single interruption plane sensor was used to measure the time intervals during which clumps and spacings pass. This sensor suffers from inherent errors such as defocus and uncertain optical switching behaviour. Therefore, the sensor was characterised by equalising measured quantities with their theoretical equivalents. This implies that the estimated mean flow rate must be equal to the theoretical mean flow rate, however, the variability and extreme values among experiments indicate the usefulness and appropriateness of the method. To test the validity of the Poisson model assumption, 30 experiments were conducted in which 4000, 4.5 mm identical spherical particles were dropped from a funnel into a fall tube. To assess the performance of the method, it is not possible to compare measured flow rates with reference counterparts, since there are none. However, the initial number of particles per experiment is known and, therefore, this number was estimated using measurements. After characterisation, the original number of 4000 particles per experiment was estimated at 4000 with a standard deviation of 44 (1.1% coefficient of variation) among 30 datasets. The extreme values of the estimations were 4092 (+2.3% error) and 3930 (-1.8% error), respectively.