An operator-centered framework for conceptual understanding of transport equations

Transport phenomena are a cornerstone of engineering education, yet students often struggle to connect the mathematical structure of governing equations with their underlying physical meaning. This work proposes an operator-centered pedagogical framework aimed at improving conceptual understanding of transport equations. Rather than presenting governing equations as the outcome of formal mathematical derivations, the approach constructs and interprets them through the physical meaning of differential operators (gradient, divergence, and Laplacian), viewed as descriptors of local inhomogeneity, flux imbalance, and diffusion within continuous fields. Within this framework, mass, momentum and energy transport are presented through a unified interpretation that highlights their common structure and underlying physical principles. The approach was implemented through instructional seminars involving 72 engineering students whohad previously completed standard transport courses. Learning outcomes were evaluated through a conceptual pre-test/post-test assessment. The percentage of correct responses increased from 27% to 79%, indicating a substantial improvement in students’ ability to interpret differentialoperators and transport terms. These results were also supported by positive student feedback collected through a post-session questionnaire. The findings suggest that the proposed framework can effectively complement traditional instruction by bridging the gap between mathematical formalism and physical intuition