Minimum Mass Design of Insulation Made of Functionally Graded Material

Abstract

The problem of steady state heat conduction in a functionally graded open-cell metal foam thermal insulation is studied. The mass is minimized by varying the solidity profile for a given thickness. An optimality condition is derived and the optimization problem is reduced to that of an ordinary, nonlinear differential equation, which is solved numerically. The results include optimum cell size variation through the thickness of the insulation for given aerodynamic heating and the corresponding temperature distribution. It is shown that for a given thickness using a functionally graded insulation is predictably lighter than uniform one INTRODUCTION Metal foams [1] and advanced metallic thermal protection systems [2] are being investigated for use in multifunctional structures for reusable launch vehicles. Such multifunctional structures would insulate the vehicle interior from aerodynamic heating as well as carry primary vehicle loads. Varying the density, geometry, and/or material composition from point to point within the foam can produce functionally graded materials (FGM) that may be superior to uniform materials. To develop and test FGM for thermal protection systems, it is important to develop an understanding of what material property distributions offer significant efficiency gains. Satchi Venkataraman et al. [3] developed criterion for minimizing heat conduction through an open-cell titanium foam with variable cell size through its thickness. For a fixed inner wall temperature and foam thickness the outside wall temperature is maximized. Maximizing the outside wall temperature maximizes the heat radiated at the surface and therefore corresponds to minimizing the transmitted heat. The current study seeks to identify density profiles that may yield large improvements in weight efficiency compared to materials with uniform density. These results will be then used to direct research into improved modeling of FGM that will be used to refine the initial optimization. Finally, it is hoped that the results could be used to direct testing of promising configurations. The thermal protection systems (TPS) problem is inherently a transient one. However, we are first solving the simpler steady state problem to gain understanding of the effects of using functionally graded insulations. In this paper an optimality criterion is derived for minimizing the mass of an open-cell titanium foam with variable cell size through its thickness. The effective thermal conductivity of the foam is a function of temperature, pressure, properties of the foam material, and the foam geometry. The objective of optimization problem is to determine the density distribution that minimizes the mass of a titanium foam of given thickness for a fixed inner and outer wall temperatures. The optimality condition is developed and used to obtain the optimum density profile. The minimum mass obtained using a functionally graded foam and uniform density foam are compared to illustrate performance payoffs provided by optimization of graded foam properties. 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con 22-25 April 2002, Denver, Colorado AIAA 2002-1425 Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.