What is the formula of the most economical relation between D and L?

The most economical relation between D (diameter) and L (length) in various engineering applications, especially in the design of beams, columns, or pipes, often depends on minimizing the cost, weight, or material usage while meeting structural or functional requirements. However, without specifying the context (e.g., structural engineering, fluid mechanics, etc.), it’s difficult to provide a precise formula, as the “most economical” relation can vary significantly across different disciplines and applications.

A general concept in structural engineering for optimizing beams under certain loads might use the principle that the material cost is usually minimized when the stress in the material is uniformly distributed. In the case of cylindrical beams or shafts subject to torsion, for example, an optimal relationship might be derived considering material strength, the moment of inertia, and the specific loading conditions, but a universally applicable “most economical” formula without further context is not feasible.

If we assume a context of fluid flow through pipes, where D represents the diameter of the pipe and L represents the length of the pipe, the economic relation might involve minimizing the total cost, which includes both the cost of the pipe and the cost of pumping the fluid through it over its operational lifetime. This could lead to an optimization problem where the goal is to find the values of D and L that minimize the total cost. The Darcy-Weisbach equation, among others, might come into play, but again, specifying an exact formula requires more details about the particular application, including the