The convection heat transfer coefficient h is a concept that is heavily used by all thermal engineers to solve practical problems. It allows them to approximately analyze complicated systems, without having to systematically perform detailed 3D simulations of all parts of any real system, which is always highly complicated. At least in the first design phase, engineers would appreciate a `handy? approximation, like that of h, which belongs to a larger category of what is called compact models. However, defining and using this quantity h for system simulation suffers from fundamental and conceptual problems that will be elucidated in this paper. Experienced thermal engineers know how to use it judiciously avoiding thus these problems, almost unconsciously. However as systems become more and more complicated, in terms of the number of its elements, automated calculations are becoming the rule. Hence, a `black box? model of convection is needed for system level simulation and design, which would always give meaningful results in any situation. In this paper a solution is proposed based on the general theory of compact models, which has known recent breakthroughs leading to a general and rigorous theory. This theory has been applied so far for thermal modeling of conduction problems. It will be generalized here to convection problems. The result is a new concept that generalizes h such as to be able to correctly handle situations that were not adequately modeled before.