# Expansions in the Non-Ideal Compressible Flow Regime

### A Numerical and Experimental Study

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## Abstract

Organic Rankine Cycle systems are gaining increasing attention as a modular, cost-effective and decentralised thermal energy recovery solution. The design of an efficient expander in the ORC system is crucial to its rapid uptake by the industry. Compared to volumetric expanders, a radial turbine provides advantages in terms of flexibility and better part-load performances. The design of ORC radial turbines is faced with two challenges. Firstly, the expansion in the stator occurs close to the vapour-liquid dome and the critical point, where the ideal gas assumption is no longer valid. Such expansions fall under the Non-ideal thermodynamic regime. Secondly, due to the lower speed of sound in the working fluid, the expansions are supersonic and are accompanied by compressible flow features such as shock waves, expansion fans which generate entropy and reduce the performance of the expander. There exists significant validated design methodologies and empirical relations for conventional turbomachinery. However this is not the case for unconventional turbomachinery where established loss correlations are not applicable. Hence, there is a knowledge gap in validated design tools for unconventional turbomachinery like ORC radial turbines. The thesis forms a part of the validation campaign of the open-source flow solver, SU2 and approaches the exercise by studying expansions in two paradigmatic test cases, namely (i) a linear stator cascade and (ii) a converging - diverging nozzle.

The study of expansions through a linear cascade is a preceding step to the study of a rotating radial turbine. Consequently, a RANS simulation on the single channel blade passage is studied under the validated assumption of flow periodicity. The expansion takes place from inlet total conditions of 18.4 bara and 525 K (Z =0.558) to a static pressure of 1.95 bara (Z = 0.951) and exit flow of Mach 2. Two types of response quantities are studied namely direct and system response quantities. Direct response quantities are those that can be measured directly such as the pressure, Mach, density. System response quantities are derived from direct measurements and provide information to characterise the performance of the stator. The selected system response quantities are: (i) Pressure loss coefficient (C_{p}= 0.074), (ii) Kinetic energy loss coefficient (ζ_{KE}= 0.115), (iii) Entropy loss coefficient (ζs=0.118), (iv) Base pressure loss coefficient (C_{pb}= -0.065), (v) Standard deviation of exit flow angle (σ_{β2} = 1.244) and (vi) Standard deviation of exit Mach (σ_{M2}= 0.033). Experimentally, it is possible to measure the pressure loss coefficient, base pressure loss coefficient and the flow uniformity at the outlet using a combination of pressure and direct velocity measurements. Through a Design of Experiments approach, the sensitivity of the flow to input uncertainties was studied through a stochastic collocation based forward propagation of the uncertainty. The input uncertainties considered are the inlet total pressure, fluid viscosity and critical point properties. The Sobol indices from the uncertainty quantification indicate the more dominant influence of the critical point properties over other inputs considered. The results also validate the use of a constant viscosity assumption for the RANS simulation. The subsequent planned unit test case was to characterise the expansion through an optimised stator blade row. To this end, a deterministic adjoint based optimisation was performed with the objective function of minimising entropy generation. The resulting geometry was studied and resulted in a pressure loss coefficient improvement of around 4%, although stator flow uniformity at the exit was compromised. The uncertainty quantification performed on the optimised blade geometry yielded robustness improvements on the pressure loss coefficient and reduction in uncertainties associated with direct response quantities, although no strong correlations can be drawn between the improvements in uncertainty and the deterministic optimisation. Given current machining tolerances, the realisation of such a negligible geometry change is not viable from an experimental

perspective.

The second section of the thesis deals with experimental investigation of expansions in a converging-diverging nozzle through a matrix of isobars with increasing degree on non-ideality. Two isentropes with inlet pressure of 2.73 bara (Z = 0.9526) and 6 bara (Z = 0.901) with pressure ratio of 8.76 were performed with recording of pressure, temperature, flowrate and density measurements along the ORCHID. The flow field was visualised using the schlieren method using a z-type layout. The thesis reports the first measurements of total pressure before the nozzle inlet and vapour density and flowrate measurements. The experimental data was post-processed to identify steady-state and quantify the Type A and Type B uncertainties. The schlieren images were used to extract information on the Mach distribution along the nozzle mid-plane using an inhouse line extraction tool. Lastly, a 2.5^{o} wedge at the exit of the nozzle is used to generate oblique shock waves that are then manually measured. The flow field at the exit of the nozzle is in the ideal region where the shock angle only depends on upstream Mach number. Hence, the experimentally observed oblique shock angles for the off-design case are close to on-design (Inlet pressure = 18.4 bara) predictions.