Thermal Analysis and Optimization of Orthotropic Pin Fins: A Closed-Form Analytical Solution

Analytical solutions for temperature distribution, heat transfer rate, and fin efficiency and fin effectiveness are derived and presented for orthotropic two-dimensional pin fins subject to convective-tip boundary condition. The generalized results are presented and discussed in terms of dimensionless variables such as radial and axial Biot numbers $(Bir,Biz)$⁠, fin aspect ratio, $L/R$⁠, and radial-to-axial conductivity ratio $k∗$⁠. Several special cases are derived from the general solution, which includes the insulated-tip boundary condition. It is also demonstrated that the classical temperature distribution and heat transfer rate from the two-dimensional isotropic pin fin introduced earlier in literature can easily be recovered from the general solutions presented in this paper. Furthermore, dimensionless optimization results are presented for orthotropic pin fins that can help to solve many natural and forced convection pin fin problems.