Thermal optimization of the pin-fin heat sink with variable fin density cooled by natural convection

Highlights: • Introducing the concept of a variable fin density for the pin-fin heat sink cooled by natural convection. • Developing the compact modeling method with the new model for the effective thermal conductivity. • Optimizing the thermal performance of the pin-fin heat sink under the constraint of fixed base plate and fin height dimensions. • Improving the thermal performance (>10%) and reducing the weight of the heat sink (>30%) simultaneously. Thermal performance of the pin-fin heat sink cooled by natural convection is numerically investigated when the fin density is allowed to vary along the flow direction. The distribution of the fin density is thermally optimized by using a compact modeling method, in which the pin-fin heat sink is modeled as a fluid-saturated porous medium. In the porous region, the volume-averaged governing equations are used; the Brinkman-Forchheimer-extended Darcy equation for fluid flow and the one-equation model for heat transfer. Based on the analogy between convective heat transfer between the fin and air and that carried by the fluid flowing through the porous medium, a new model for the effective thermal conductivity is proposed. The proposed model is validated by the experimental data available in literature and the numerical results obtained from a direct numerical simulation. For minimal thermal resistance, optimization is conducted using the Kriging method under the constraint of fixed base plate and fin height dimensions. As a result, the optimal distribution of the fin density is obtained as a function of the porosity $\left(\varphi \right)$ along the flow direction. The optimal distribution of the fin density is found to be monotonically increasing from $\varphi =0.898$ to $\varphi =0.946$, which indicates the densely-populated fins in the lower part and the coarsely-populated fins in the upper part. Consequently, the optimized pin-fin heat sink with variable fin density has 11% lower thermal resistance and 30% less weight than those of the uniformly-populated pin-fin heat sink with $\varphi =0.891$ optimized using an existing correlation. This study suggests that varying the fin density is very useful to not only improving the thermal performance but also reducing the weight of the pin-fin heat sink cooled by natural convection.