In this study, the performance of micro heat sink with nanofluid of water-Al2O3 of 1% and the optimum geometry dimensions are evaluated in the trapezoidal microchannel by using constructal theory and the two-phase Eulerian-Lagrangian method. Also, the optimal microchannel diameter obtained with the intersection of asymptotes method, was compared with the numerical state. The results show that for a constant pressure drop, microchannel with a 70 degrees side angle in the optimum state, has the maximum total thermal conductivity in the dimensionless state. In addition, by reducing the side angle of the trapezoidal cross-section, the total thermal conductivity would be reduced in a dimensionless state. The maximum thermal conductivity in a trapezoidal microchannel will happen in the volume fraction range of 0.4 <=eta <= 0.5. The maximum and minimum error between numerical and analytical study for the optimum hydraulic diameter is 47.54% and 32%, respectively. (C) 2017 Elsevier B.V. All rights reserved.