An improved lumped model for freezing of a freely suspended supercooled water droplet in air stream

• Published in: Journal of engineering mathematics Vol. 130; no. 1
• Main Authors: dos Anjos, Emerson B., Naveira-Cotta, Carolina P., Tiwari, Manish K., Cotta, Renato M., Carvalho, Igor S.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-10-2021; Springer Nature B.V
• Subjects: Accuracy; Air flow; Applications of Mathematics; Boundary conditions; Computational Mathematics and Numerical Analysis; Droplets; Exact solutions; Flow Dynamics; Freezing; Integral equations; Integral transforms; Mathematical and Computational Engineering; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Ordinary differential equations; Parametric analysis; Partial differential equations; Recalescence; Solidification; Supercooling; Systems analysis; Temperature distribution; Theoretical and Applied Mechanics; Water drops; Heat conduction; Nonlinear partial differential equations; Phase change; Integral transforms; Moving boundary; Droplet freezing; Lumped analysis
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-021-10161-z

This work deals with the mathematical modeling of the transient freezing process of a supercooled water droplet in a cold air stream. The aim is to develop a simple yet accurate lumped-differential model for the energy balance for a freely suspended water droplet undergoing solidification, that allows for cost effective computations of the temperatures and freezing front evolution along the whole process. The complete freezing process was described by four distinct stages, namely, supercooling, recalescence, solidification, and cooling. At each stage, the Coupled Integral Equations Approach (CIEA) is employed, which reduces the partial differential equation for the temperature distribution within the spherical droplet into coupled ordinary differential equations for dimensionless boundary temperatures and the moving interface position. The resulting lumped-differential model is expected to offer improved accuracy with respect to the classical lumped system analysis, since boundary conditions are accounted for in the averaging process through Hermite approximations for integrals. The results of the CIEA were verified using a recently advanced accurate hybrid numerical-analytical solution through the Generalized Integral Transform Technique (GITT), for the full partial differential formulation, and comparisons with numerical and experimental results from the literature. After verification and validation of the proposed model, a parametric analysis is implemented, for different conditions of airflow velocity and droplet radius, which lead to variations in the Biot numbers that allow to inspect for their influence on the accuracy of the improved lumped-differential formulation.