Constructal distribution of multi-layer insulation Deok-Hong Kang 1,3 , Sylvie Lorente 2 and Adrian Bejan 1, * ,† 1 Department of Mechanical Engineering and Material Science, Duke University, Durham, NC 27708-0300, USA 2 UPS, INSA; LMDC (Laboratoire Matériaux et Durabilité des Constructions), Université de Toulouse, 135 avenue de Rangueil, F-31 077 Toulouse Cedex 04, France 3 Energy & Resources Research Department, Research Institute of Industrial Science & Technology, Pohang 790-600, South Korea SUMMARY Here, we show how to distribute multiple layers of insulation along a nonisothermal enclosure so that the total heat loss is minimal. The types and total amounts of insulation materials are fixed. Variables are the thicknesses of the insulation layers, their relative amounts, the temperature of the insulated surface, and the cross-sectional area of the enclosure. We show that, first, the structure of the multi-layer insulation must be such that the thicknesses of all the layers vary in the same way in the longitudinal direction x. Second, the x dependence of the enclosure cross-sectional area has a significant effect on the heat loss reduction associated with using the distributed insulation design. Greater reductions in heat loss are obtained when the enclosure is tapered such that it is narrower in the direction of the warm end. Third, the x dependence of the temperature distribution along the insulated wall has a significant effect on the reduction in heat loss through reduction in heat loss through the multi-layer insulation. Greater reductions are obtained when the wall temperature distribution is more convex. Even greater reductions in heat loss are possible when the three design features summarized previously are implemented simultaneously. Copyright © 2011 John Wiley & Sons, Ltd. KEY WORDS distributed energy systems; multilayer insulation; constructal theory; furnace design Correspondence *Adrian Bejan, Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300, USA. † E-mail: abejan@duke.edu Received 20 June 2011; Accepted 20 June 2011 1. INTRODUCTION In this paper, we consider the problem of distributing finite amounts of insulation on a heated enclosure. This is a fundamental problem, and the solution to it is rele- vant in the design of all heating processes where space and fuel come at premium. For example, in a modern reheating furnace, slabs of metal move through a long enclosure and are heated gradually by exposure to hot gasses produced by burners installed along the en- closure. The hot gasses flow against the direction of movement of the train of slabs. The objective of the installation is to deliver slabs that have a specified temperature, and to do it with minimal consumption of fuel. This objective drives the search for better heat- ing configurations that, among other features, must include better insulation architectures [1–6]. Our approach is based on constructal theory [7], accord- ing to which the furnace and all its streams is a flow system that is free to morph such that its global performance becomes better. The freedom to morph is represented by two architectural features that are not specified a priori, in this case, the distribution of insulation along the path traveled by the steel stream, and the width of the floor area. In other words, we will allow all ‘distributed designs’ to compete, and we will show that the better designs are not the traditional designs with uniformly distributed insula- tion and uniform floor width. The opportunity to distribute the insulation nonuni- formly was first noted in [8], with an example of single- layer insulation with one material of constant conductivity. The design and optimization of insulation systems was ex- plored further in [9–20]. In the present paper, we extend this exploratory re- search for design in three new directions: (1) Insulation with multiple layers, each layer being free to have a non- uniform thickness, (2) Enclosure cross-sections (or furnace floor areas) that vary in the longitudinal direction, and (3) Furnace temperature distributions that vary nonlinearly in the longitudinal direction. Features (2) and (3) are also recommended by the need to reduce the amount of fuel used for heating the solid stream that moves through the enclosure [21]. We show that features (1)–(3) are benefi- cial and in accord with the rest of the distributed-energy INTERNATIONAL JOURNAL OF ENERGY RESEARCH Int. J. Energy Res. 2013; 37:153–160 Published online 19 September 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/er.1895 Copyright © 2011 John Wiley & Sons, Ltd. 153