An Exact Solution for Fluid Flow and Heat Convection through Triangular Ducts Considering the Viscous Dissipation

Today, the study of flow and heat transfer in non-circular ducts are of increasing importance in various industries and applications such as microfluidics, where lithographic methods typically produce channels of square or triangular cross-section. Also, heat transfer in non-circular ducts is important in designing the compact heat exchangers to enhance the heat transfer. In the current study, an EXACT analytical solution for the convective heat transfer in conduits with equilateral triangle cross-section is presented for the first time. The effect of viscous dissipation on heat transfer and temperature distribution through the duct is investigated in detail. This effect is of great importance especially in flow of high viscous fluids in micro-channels. In order to study the effect of viscous dissipation in both cooling and heating cases, the Brinkman number is employed. The exact solution is found by calculating the particular solution which satisfies the thermal boundary conditions. Based on the finite expansion method, an exact analytical solution for temperature distribution and a correlation for dimensionless Nusselt number is obtained as functions of the Brinkman number. The maximum temperature and Nusselt number at the centroid of the conduit for the specific case of Brinkman number equal to zero is calculated equal to 5/9 and 28/9, respectively. The proposed method of solution could be used to find the exact solution for similar problems such as analysis the heat convection in non-circular geometries.