# Influence of Thermodynamic Property Perturbations on Nozzle Design and Non-Ideal Compressible Flow Phenomena

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## Abstract

In recent years, the scientific community has shown increased interest in the use of uncertainty quantification techniques for complex thermodynamic systems. One such complex system is the Organic Rankine Cycle (ORC) power system for which a turbine is often the prime mover. These systems operate in a region close to the critical conditions of the fluid which makes their thermodynamic behaviour complex. Additionally, the organic fluids are commonly characterized by low speed of sound which induces compressibility effects such as shock waves as the fluids expand through a turbine. In order to design optimum turbine blades and to accurately estimate the performance of these machines, it is imperative to gain a better understanding of these compressible flow effects. The flow through a de-Laval nozzle is a simple representation of flow through a turbine. Thus the objective of this research is to gain an improved understanding of the influence of uncertainty in the thermodynamic properties on nozzle design and compressible flow phenomena. The study is realized by performing an uncertainty quantification analysis using DAKOTA which is coupled with Matlab. Quite often a model such as a wedge can be placed in the nozzle to induce the required fluid dynamic phenomena. In this study, these phenomena are characterized in terms of the angle and intensity of the shock waves generated. An Euler shock wave simulator code is used to determine the shock wave angle and shock wave intensity. Two set of simulations are performed depending on the position of the wedge in the de-Laval nozzle. In the first set of simulations, the wedge is placed at the exit of the nozzle in a region which predominantly shows ideal gas behaviour. In the second set of simulations, the wedge is placed at 7.8 mm from the nozzle throat to capture the influence of real gas effects on the computed flow dynamic quantities. In addition, the deviation in the nozzle profile due to the uncertainties in the input parameters is also estimated. The two sets are further subdivided into two cases depending on the choice of the uncertain input variables. In the first case the fluid dependant parameters and the geometric parameters are considered to be uncertain. The fluid dependant parameters are the critical temperature, critical pressure, acentric factor, $\kappa_1$ parameter used in the iPRSV equation of state and the four coefficients of the ideal gas isobaric heat capacity of MM (hexamethyldisiloxane) while the geometric parameter considered is the wedge angle of the geometry. In the second case the total pressure and total temperature which are the operating conditions are also uncertain in addition to the fluid dependant and geometric parameters. The results of the two set of simulations are presented in terms of deviation in the shock wave properties from the nominal. For both the cases with the wedge placed at the exit of the nozzle, the shock wave angle and the intensity is found to vary by 0.17\% and 0.06\% respectively. With the wedge near the throat, for the first case the deviations are computed as 2.2\% and 2.7\% in the angle and intensity of the shock waves respectively while for the second case the properties of the shock wave vary by 2.6\% and 3.1\% respectively. In addition the average deviation in the nozzle profile for the both the cases is found to be 0.29 mm. These results indicate that in the real gas region, the effect of the uncertainty in the input parameters is amplified which causes a large deviation in the compressibility effects. The deviation in the shock wave angle is verified by performing CFD simulations in SU2 for the cases that yield maximum deviation. The results are found to be in good agreement with an average deviation of 0.33\% with the wedge at the exit and 1.2\% for the wedge at 7.8 mm from the throat. Finally the analysis performed on the nozzle is extended to a radial outflow turbine. Nine cases are simulated by varying the critical temperature and pressure of MM to estimate the deviation in the static enthalpy loss. The simulation results indicate a large deviation in the loss which denotes that the critical point values have a significant effect on the losses in an ORC turbine. The results obtained from this study thus provide a deeper insight into the thermodynamic properties that influence the behaviour of shock waves and nozzle design.