Constructal-theory network of conducting paths for cooling a heat generating volume

Abstract This paper develops a solution to the fundamental problem of how to collect and "channel" to one point the heat generated volumetrically in a low conductivity volume of given size. The amount of high conductivity material that is available for building channels (high conductivity paths) through the volume is fixed. The total heat generation rate is also fixed. The solution is obtained as a sequence of optimization and organization steps. The sequence has a definite time direction, it begins with the smallest building block (elemental system) and proceeds toward larger building blocks (assemblies). Optimized in each assembly are the shape of the assembly and the width of the newest high conductivity path. It is shown that the paths form a tree-like network, in which every single geometric detail is determined theoretically. Furthermore, the tree network cannot be determined theoretically when the time direction is reversed, from large elements toward smaller elements. It is also shown that the present theory has far reaching implications in physics, biology and mathematics. Copyright 1996 Elsevier Science Ltd. 1. THE VOLUME-TO-POINT ACCESS PROBLEM This paper is about one of those fundamental prob- lems that suddenly appear 'obvious', but only after considerable technological progress has been made on pushing the frontier. The technology in this case is the cooling of electronics (components and packages), where the objective is to install a maximum amount of electronics (heat generation) in a fixed volume in such a way that the maximum temperature does not exceed a certain level. The work that has been done to devise cooling techniques to meet this objective is enormous and is continuing at an accelerated pace [1 7]. In brief, most of the cooling techniques that are in use today rely on convection or conjugate convection and conduction, where the coolant is either a single phase fluid or one that boils. The frontier is being pushed in the direction of smaller and smaller package dimensions. There comes a point where miniaturization makes convection cooling impractical, because the ducts through which the coolant must flow take up too much space. The only way to channel the generated heat out of the electronic package is by conduction. This conduction path will have to be very effective (of high thermal conductivity, kp), so that the temperature difference between the hot spot (the heart of the package) and the heat sink (on the side of the package) will not exceed a certain value. Conduction paths also take up space. Designs with fewer and smaller paths are better suited for the min- iaturization evolution. The fundamental problem addressed in this paper is this : consider a finite-size volume in which heat is being generated at every point and which is cooled through a small patch (heat sink) located on its boundary. A finite amount of high con- ductivity (kp) material is available. Determine the optimal distribution of kp material through the given volume such that the highest tem- perature is minimized. I will show that the solution to this problem is aston- ishingly simple and beautiful, with far reaching impli- cations in physics, mathematics and the natural evol- ution of living systems. 2. THE SMALLEST (ELEMENTAL) SYSTEM We begin with recognizing the function of the elec- tronic package or assembly of packages: the gen- eration of heat at a certain rate (q) in a fixed volume (V). The heat generation rate per unit volume is q'" = q/V and for simplicity, we assume that q" is constant (i.e. q is distributed uniformly), although this assumption can easily be abandoned in follow-up studies of the fundamental problem. The fraction of the volume V that is occupied by a(l the high conductivity paths is V~,. This fraction also fixed, although, as noted already, a smaller ratio Vr/V is better for miniaturization. Again, for simplicity, we assume that Vp << V. The thermal conductivity of the conducting paths is constant (kp) and much larger than the thermal conductivity of the electronic material (k~) that occupies the rest of the volume. Finally, since the fundamental problem leads 799