A second-law analysis-based analytical method for insulation thickness optimization in a space with constant insulation volume

Abstract

This paper develops the first closed-form analytical model for optimizing insulation thickness distribution in enclosures with non-uniform thermal boundaries (e.g., 10 degrees C and-5 degrees C), under a fixed insulation volume or cost constraint. Unlike prior second-law-based studies (e.g., Ke & ccedil;ebas,, Ucar, Arslan) that required iterative or numerical optimization, our approach derives explicit algebraic relations by equating marginal exergy destruction rates across all walls. This contribution fills a clear research gap by providing a non-iterative solution applicable to refrigerators, building envelopes, and industrial systems where unequal boundary conditions exist. A case study shows that the second-law-based optimum differs significantly from the first-law: the colder wall requires 150 mm insulation versus 124 mm by the first-law. This reduces exergy destruction by 15 %, and although total heat transfer rises slightly (7.84 vs. 7.4 W/m2), the overall system energy consumption decreases by 3.3 % thanks to improved COP. These results highlight the practical importance of exergy-based allocation, which prioritizes preserving work potential over merely reducing heat loss. The proposed closed-form formulas also provide a benchmark for validating numerical/CFD models and can be extended in future work to include life-cycle cost and environmental impact.