Designing experiments for estimating an appropriate outlet size for a silo type problem

TL;DR

This paper develops a methodology for designing experiments to accurately estimate the minimum outlet size in a silo to prevent jamming, using optimal design theory and sensitivity analysis.

Contribution

It introduces a novel approach for optimal experimental design in non-linear models, specifically applied to silo jamming problems, with a comprehensive sensitivity and simulation study.

Findings

Optimal $c$-designs are derived using the Elfving method.

Sensitivity analysis shows how designs depend on parameter estimates.

Simulation confirms the effectiveness of the design methodology.

Abstract

The problem of jam formation during the discharge by gravity of granular material through a two-dimensional silo has a number of practical applications. In many problems the estimation of the minimum outlet size which guarantees that the time to the next jamming event is long enough is crucial. Assuming that the time is modeled by an exponential distribution with two unknown parameters, this goal translates to the optimal estimation of a non-linear transformation of the parameters. We obtain $c$-optimum experimental designs with that purpose, applying the graphic Elfving method. Since the optimal designs depend on the nominal values of the parameters, a sensitivity study is additionally provided. Finally, a simulation study checks the performance of the approximations made, first with the Fisher Information matrix, then with the linearization of the function to be estimated. The results are useful for experimenting in a laboratory and translating then the results to a larger scenario. Apart from the application a general methodology is developed in the paper for the problem of precise estimation of a one-dimensional parametric transformation in a non-linear model.