A new charter high school designed to train nurses is expected to win final approval from the state Board of Regents for Elementary and Secondary Education Thursday afternoon. The school, which would be public and paid for by taxpayers, is scheduled to open in downtown Providence in September, starting with 136 10th and 11th graders.Supporters of the Rhode Island Nurses Institute Middle College say the school will serve a dual purpose: help the state address a growing nurse shortage and provide a high-quality education to hundreds of students, many of whom come from low-income families and are stuck in low-performing schools. Students from across the state can apply. The first year's class is already full and there is a wait list of more than 20 students, say the school's sponsors.