Assessment of Deterministic Shape Optimizations Within a Stochastic Framework for Supersonic Organic Rankine Cycle Nozzle Cascades

Abstract

The design of converging–diverging blades for organic Rankine cycle (ORC) applications widely relies on automated shape-optimization processes. As a result, the optimization produces an adapted-nozzle cascade at the design conditions. However, only few works account for the uncertainties in those conditions and their consequences on cascade performance. The proposed solution, i.e., including uncertainties within the optimization routine, demands an overall huge computational cost to estimate the target output statistic at each iteration of the optimization algorithm. With the aim of understanding if this computational cost is avoidable, we study the impact of uncertainties in the design conditions on the robustness of deterministically optimized profiles. Several optimized blades, obtained with different objective functions, constraints, and design variables, are considered in the present numerical analysis, which features a turbulent compressible flow solver and a state-of-the-art uncertainty-quantification (UQ) method. By including measured field variations in the formulation of the UQ problem, we show that a deterministic shape optimization already improves the robustness of the profile with respect to the baseline configuration. Guidelines about objective functions and blade parametrizations for deterministic optimizations are also provided. Finally, a novel methodology to estimate the mass-flow-rate probability density function (PDF) for choked supersonic turbines is proposed, along with a robust reformulation of the constraint problem without increasing the computational cost.

Impact of geometric, operational, and model uncertainties on the non-ideal flow through a supersonic ORC turbine cascade

“…In absence of geometric variability, the PDFs of Y have a complex non-gaussian shape with different mean values and support; in particular, the support is narrow enough to make the full-load conditions always outperforming the part-load one. Conversely, the PFDs ofṁ are of gaussian type with very narrow support; this result had been already observed and discussed in detail in [38], and is motivated by the combination of inlet pressure and temperature uncertainties. As the cascade is chocked in the two conditions, and the uncertainties in the inlet conditions are the same, the PDFs appear identical.…”

Non-ideal compressible flows in supersonic turbine cascades

“…The employed turbulence model is SST, whose boundary conditions are set as turbulence intensity ( for all simulations) and eddy viscosity ratio ( for all simulations). Note that these boundary conditions have very limited influence on the cascade performance for moderate values of the upstream total pressure ( up to ), see Romei, Congedo & Persico (2019); we expected that the same considerations hold for a larger range of total pressures. The cascade Reynolds number, based on the blade chord and computed using the mass-flow-averaged density, dynamic viscosity and velocity, is for the simulations involving an expansion far from the thermodynamic critical point.…”

Quantile-based robust optimization of a supersonic nozzle for organic rankine cycle turbines

“…Finally considering the flow rate, all the PDFs are qualitatively similar and exhibit a nearly identical support. As for the entire range of variability considered the cascades is always in choked-flow conditions, the shape and the support of the PDFs directly depend of the variability in the inlet total conditions and the PDF takes a trapezoidal shape, as discussed in detail in [48]. Some of the PDFs are slightly shifted, with the RO-E-q 95 blade featuring a slightly smaller mean and quantiles and the O-NS blade exhibiting a slightly higher mean and quantiles, but most of the realizations of all the optimal blades are within the acceptability range of probabilistic constraint.…”