Robust optimization of ORC turbine expanders

In recent years, the Organic Rankine Cycle (ORC) technology has received great interest from the scientific and technical community because of its capability to recover energy from low-grade heat sources. In some applications, as the Waste Heat Recovery (WHR), ORC plants need to be as compact as possible because of geometrical and weight constraints. Recently, these issues have been studied in order to promote the ORC technology for Internal Combustion Engine (ICE) applications. Since proposed heat sources for ORC turbines typically include variable energy sources such as WHR from industrial processes or automotive applications, as a result, to improve the feasibility of this technology, the resistance to variable input conditions is taken into account. The numerical optimization under uncertainties is called Robust Optimization (RO) and it overcomes the limitation of deterministic optimization that neglects the effect of uncertainties in design variables and/or design parameters. To measure the robustness of a new design, statistics such as mean and variance (or standard deviation) of a response are calculated in the RO process. In this work, the method of characteristics (MOC) design of supersonic ORC nozzle blade vanes is used to create a baseline injector shape. Subsequently, this is optimized through a RO loop. The stochastic optimizer is based on a Bayesian Kriging model of the system response to the uncertain parameters, used to approximate statistics of the uncertain system output, coupled to a multi-objective non-dominated sorting genetic algorithm (NSGA). An optimal shape that maximizes the mean and minimizes the variance of the expander isentropic efficiency is searched. The isentropic efficiency is evaluated by means of RANS (Reynolds Average Navier-Stokes) simulations of the injector. The fluid thermodynamic behavior is modelled by means of the well-known Peng- Robinson-Stryjek-Vera equation of state. The blade shape is parametrized by means of a Free Form Deformation approach. In order to speed-up the RO process, an additional Kriging model is built to approximate the multi-objective fitness function and an adaptive infill strategy based on the Multi Objective Expected Improvement for the individuals is proposed in order to improve the surrogate accuracy at each generation of the NSGA. The robustly optimized ORC expander shape is compared to the results provided by the MOC baseline shape and the injector designed by means of a standard deterministic optimizer.