The main goal of this work is to obtain semi-analytical solutions for the geometric optimization of fins considering the presence of a primary surface. Optimized fin dimensions devoted to finned surfaces are already available in the literature; however, these solutions have as assumption a known fin base temperature. In contrast, in many practical situations, employed fins are attached to a primary surface that has restricted dimensions due to design requirements. In these cases, the primary surface dimensions are kept fixed and the known condition is a temperature or convective condition prescribed on the primary surface boundary. For such scenarios, to the best of our knowledge, there are no simple analytical expressions derived in the literature. Thus, reasoning on this main problem, in the present work, analytical optimum expressions are derived in order to deal with this kind of applications. The obtained analytical equations are subsequently compared with optimum results acquired from numerical calculations. In summary, both analytical and numerical results show a good agreement. The main contribution of this work is to provide closed solutions for the mentioned optimization problem, thus allowing the construction of simple charts, facilitating the corresponding fin design process.

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