This study extends ongoing work to identify difficult concepts in thermal and transport science and measure students' understanding of those concepts via a concept inventory. Two research questions provided the focal point: "What important concepts in electric circuits and engineering mechanics do students find difficult to learn?" and "How can we describe students' mental models of the concepts identified in the previous question?" The research was conducted in two parts. The team first determined what concepts are difficult as judged by engineering faculty and content experts and second, focused on measuring students' understanding of these difficult concepts. In the first step, a Delphi methodology was used to determine which concepts in electrical circuits and in engineering mechanics were important, and yet difficult to learn. The Delphi method elicits and refines ideas about difficult concepts from a panel of engineering faculty experts. The method draws upon the experts' knowledge and collective opinion through a series of questionnaires interspersed with controlled opinion feedback. In the next part of the experiment, the team gathered data about the degree of understanding of these concepts by seniors in Mechanical/Civil and Electrical Engineering. Seniors at Colorado School of Mines with a specialty in either electrical, civil, or mechanical engineering were recruited in two rounds via an e-mail to the engineering seniors' e-mail list. Results suggest that concepts rated as important and well-understood by the Delphi participants are NOT understood by students. Analysis of the data indicated that these students do not fully understand many fundamental concepts.