Two Constructal Routes to Minimal Heat Flow Resistance via Greater Internal Complexity

Abstract

This paper shows that the geometry of the heat flow path between a volume and one point can be optimized in two fundamentally different ways. In the “growth” method of the original constructal theory the structure is optimized starting from the smallest volume element of fixed size. Growth, or optimal numbers of constituents assembled into larger volumes, is one route to resistance minimization. In the “design” method the overall volume is fixed, and the designer works “inward” by optimizing the internal features of the heat flow path. The design method is new. It is shown analytically that the two methods produce comparable geometric results in which the high-conductivity channels form constructal tree networks, and where the low-conductivity material fills the interstices. For simplicity, it is assumed that the high-conductivity channels and their tributaries make 90-deg angles. In both approaches, the overall resistance decreases as the internal complexity of the conductive composite increases. In the growth method the number of constituents in each assembly can be optimized. In the design method, some of the constituent numbers cannot be optimized: these numbers assume the roles of weak parameters. The growth method is the simplest, and provides a useful approximation of the design and performance that can be achieved using the design method. Numerical solutions of the volume-to-point optimization problem confirm the results obtained analytically, and show that the geometric features of the optimal design are robust.