A Reduced Three-Phase Model for Solidification of Liquid in Large Tanks

A liquid water-urea mixture is stored onboard diesel vehicles and used for exhaust aftertreatment. In cold weather conditions, the mixture may freeze and the freezing process may span over a day. In the first part of our study (Journal of Thermal Science and Engineering Applications — Transactions of the ASME, Vol. 13, p. 011008, 2021) it was shown that traditional computational methods are impractical for modeling such large-duration freezing processes because of restrictions in the time-step size posed by numerical stability and physical time-scale considerations. A model, in which natural convection driven thermal transport is treated as a diffusive process, was developed and demonstrated. Since the flow field was not computed in this model, the computations were found to be orders of magnitude more efficient than traditional methods. This preliminary model did not account for the expansion of ice. Here, a new model that accounts for the expansion of ice and the consequent rise of the initial air-water/ice interface (ice dome formation) is presented. An additional conservation equation for excess volume fraction is introduced to this end and is solved using the unstructured finite-volume procedure and sub-time-stepping. Since the flow field is not computed, the air-water/ice interface is tracked using a new algorithm similar to the traditional volume-of-fluid method, but one that constructs fluxes using a diffusive formulation rather than an advective one. Validation studies in full-scale three-dimensional tanks show good agreement with measured temperature-time data. It is found that the air-water/ice interface first evolves to a concave shape before finally becoming a convex ice dome after full solidification.