Inferring the cross-sectional area profile of an unseen solid in a draining tank from liquid level dynamics

• We integrate wet experiments, mathematical modeling, and Bayesian statistical inversion to solve an inverse problem of reconstruction with quantified uncertainty. • The inverse problem is to infer (a) the horizontal cross-sectional area profile of an unseen solid contained in a tank from (b) time series measurements of the liquid level in the tank as it drains, driven by gravity, through a small hole. • Because the solid displaces water in the tank, the rate at which the liquid level drops reflects the area of the solid’s horizontal cross-section; the top of the liquid “scans” the area profile of the solid as the tank drains. • In our experiment with a glass bottle inside a draining tank, we reconstructed the profile of the bottle with <10% error. • Our method offers a non-destructive way to gather information about the geometry of an unknown solid, or a bed of packed solid particles, inside of an opaque or inaccessible tank. We aim to reconstruct the horizontal cross-sectional area profile of an exogenous, heavy, unseen solid contained in a tank from measurements of the liquid level in the tank as it drains (driven by gravity) through a small orifice in its side. (Because the solid displaces liquid, the rate of decrease of the liquid level provides information about the cross-sectional area of the solid at that height; as the liquid level drops, it “scans” the solid’s area profile.) We combine mathematical modeling, Bayesian statistical inversion, Monte Carlo simulation, and wet experiments of a tank draining of water to demonstrate and test our ability to infer the area profile of an exogenous solid with quantified uncertainty. In our experiment, the posterior distribution over the area profile of the solid (a bottle) in our draining tank agreed reasonably well with our [held-out] length-measurements ( < 10% mean reconstruction error on the solid’s radius). Our approach may be practically useful to non-destructively gather information about the geometry of an unknown solid, or a packed bed of solid particles, contained in an opaque or inaccessible tank.