In this article, we describe a method for finding congruent number elliptic curves with high ranks. The method involves an algorithm based on the Monsky's formula for computing 2-Selmer rank of congruent number elliptic curves, and Mestre-Nagao's sum which is used in sieving curves with potentially large ranks. We apply this method for positive squarefree integers in two families of congruent numbers and find some new congruent number elliptic curves with rank 6.