High-speed supersonic radial compressors are a critical enabling technology for meeting the requirements of future aviation-propulsion and thermal-management systems. These turbomachines must be designed to be both efficient and robust on the widest possible operating range. Flow instabilities in the form of rotating stall and surge are therefore phenomena that must be accurately predicted early in the design process. Unsteady full-Annulus computational fluid dynamics (CFD) can be used to get accurate information about the onset of instabilities, but at the expense of costly simulations. As a result, the design of new compressors continues to rely on existing correlations for the prediction of the critical mass flowrate. This approach, however, leads to suboptimal compressor designs. This article provides a review of the numerical methodologies that can be used for the accurate prediction of the critical mass flowrate in high-speed centrifugal compressors. Methods of different fidelity level and computational cost are described. Two particularly promising models, namely, those proposed by Spakovszky and Sun, are subsequently examined in more detail. Exemplary applications of these two models are finally discussed.

Small-scale turbomachinery operating at high rotational speed is a key technology for increasing the power density of energy and propulsion systems. A notable example is the turbine of an organic Rankine cycle turbogenerator for thermal recuperation from prime engines and industrial processes. Such systems typically operate with organic compounds characterized by complex molecular structures to allow the design of efficient fluid machinery and flexibility in matching the heat source and sink temperature profiles. Gas-lubricated bearings are considered advantageous compared to traditional oil-lubricated rolling element bearings for supporting the turbine rotor, enabling greater machine compactness and reduced complexity, and avoiding contamination of the working fluid. In certain operating conditions, however, the lubricant of the gas bearing is in thermodynamic states near the saturated vapor line or in the vicinity of the fluid critical point, whereby nonideal effects are relevant and may affect bearing performance. This work investigates the physics of thin film flows in gas bearings operating with fluids made by complex molecules. The influence of nonideal thermodynamic effects on gas bearing performance is discussed by analysis of the fluid bulk modulus. Reduced values of the nondimensional bulk modulus near the critical point or saturated vapor line decrease bearing performance. The main parameter characterizing the influence of molecular complexity on bearing performance is shown to be the acentric factor. For complex fluids with large acentric factors, the impact of nonideal thermodynamic effects on nondimensional bearing load capacity and rotor-dynamic characteristics is less pronounced.

A supersonic inlet turbine can extract substantial energy from the highly fluctuating and transonic flow delivered by a rotating detonation combustor (RDC). However, a transition duct is necessary to achieve the supersonic inlet conditions required by the turbine. In this work, the supersonic transition duct is designed with the method of characteristics (MOC). A generalized implementation of the MOC is proposed for the generation of annular ducts with asymmetric and rotated hub and shroud walls. The model is extended to deal with ideal and non-ideal flows, namely flows characterized by non-ideal thermodynamic effects, and its accuracy has been verified through comparison with results obtained with computational fluid dynamics (CFD) simulations. In addition, boundary layer flow equations are combined with the MOC to predict viscous losses on the endwalls and to adjust duct geometry by accounting for the boundary layer thickness. Furthermore, it is essential to predict the effects of the large unsteadiness generated by the detonation combustor for an efficient operation of the turbine. The maximum incidence angle at the turbine inlet is predicted with a one-dimensional annular duct model. Supersonic duct flow behavior to unsteady inlet conditions is characterized through two-dimensional inviscid axisymmetric unsteady CFD simulations. The accuracy of the reduced order models is finally verified with a three-dimensional unsteady viscous simulation assuming inlet flow conditions representative of RDC operation.

The radial-inflow turbine (RIT) is a widely adopted turbo-expander in power and propulsion systems of low-to-medium power capacity due to its high efficiency and compactness. Compared to conventional radial turbines for gas turbines and air cycle machines, the design of expanders for high-tem-perature organic Rankine cycle power systems involves additional chal-lenges, as these machines operate with very high expansion or volumetric flow ratio and partly or entirely in the nonideal compressible fluid dynamic regime. This study examines the impact of the working fluid, of the volumetric flow ratio, and of the nonideal thermodynamic effects on the design guidelines for RIT. To this purpose, a reduced-order modeling framework for turbine fluid-dynamic design encompassing a loss model based on first principles is developed and verified against results from uRANS. Results highlighted that at the geometrical scale of interest the impact of the working fluid molecular complexity on the efficiency is marginal. Moreover, it is shown that the average isentropic pressure-volume exponent [Entity In Abstract]) can be used to predict the magnitude of nonideal thermodynamic effects on the stage efficiency, whose variation depends on the value of the volumetric flow ratio and of the work and flow coefficients. Design guidelines that can be used for preliminary turbine design in system-level calculations are presented in graphical form, and illustrate the relation between the optimal set of stage duty coefficients, i.e., the work and flow coefficients that maximize the efficiency, the stage efficiency, the volumetric flow ratio, and the similarity parameter [Entity In Abstract] .

Heat exchangers are key components of thermal energy conversion systems, however, their optimal design is still based on reduced order models relying on semi-empirical heat transfer correlations. CFD-based design optimization emerged as a viable method to provide a significant improvement in performance at an affordable cost. This study presents a framework to optimize multiple heat transfer surfaces concurrently using the adjoint method. The heat transfer surfaces are parametrized using a CAD-based parametrization method, and their performance is evaluated using a RANS solver complemented by its discrete adjoint counterpart for gradient computation. The optimization framework is applied to minimize the pressure drop across a bare-tube heat exchanger while constraining the heat transfer rate. Two variants of the same optimization problem are formulated: in the first one, the sensitivities are averaged and the tubes are constrained to maintain the same shape, while in the second variant, the shape of the tubes can vary, resulting in an optimum solution with non-identical tube shapes. The results show that the optimized geometry reduces the pressure drop by 19% if the tube shapes are identical, and by 25% in the case of non-identical shapes, compared to the baseline. To identify the physical mechanisms contributing to the fluid-dynamic losses, entropy generation along the flow path was investigated. The results reveal that the major loss reduction observed for the case of non-identical tube shapes is due to the better thermo-hydraulic performance of the first and last tubes.

The blade solidity, namely the blade chordtopitch ratio, largely affects the fluiddynamic performance of turbomachinery. For turbomachines operating with air or steam, the optimal value of the solidity which maximizes the efficiency is estimated with empirical correlations such as the ones proposed by Zweifel (1945) and Traupel (1966). However, if the turbomachine operates with unconventional fluids, the accuracy of these correlations becomes questionable. Examples of such working fluids are the nonideal (dense) vapors of organic compounds (e.g., hydrocarbons, siloxanes) used to operate organic Rankine cycle (ORC) power systems. This study investigates the effect of both the working fluid and the flow compressibility on the optimum pitchtochord ratio of turbine stages. A first principle model for the profile losses is developed for this purpose. Charts providing the optimal pitchtochord ratio for unconventional turbine stages are then provided. Numerical simulations of the flow over a turbine stator cascade have been conducted to validate the model results and evaluate the influence of both working fluid, flow compressibility, and solidity value on the loss breakdown. The results show that the optimal solidity of turbine cascades value significantly increases with the flow compressibility. Therefore, models providing the optimal solidity based on the estimate of passage loss only are not suited for unconventional turbines.

The gas dynamics of single-phase nonreacting fluids whose thermodynamic states are close to vapor-liquid saturation, close to the vapor-liquid critical point, or in supercritical conditions differs quantitatively and qualitatively from the textbook gas dynamics of dilute, ideal gases. Due to nonideal fluid thermodynamic properties, unconventional gas dynamic effects are possible, including nonclassical rarefaction shock waves and the nonmonotonic variation of the Mach number along steady isentropic expansions. This review provides a comprehensive theoretical framework of the fundamentals of nonideal compressible fluid dynamics (NICFD). The relation between nonideal gas dynamics and the complexity of the fluid molecules is clarified. The theoretical, numerical, and experimental tools currently employed to investigate NICFD flows and related applications are reviewed, followed by an overview of industrial processes involving NICFD, ranging from organic Rankine and supercritical CO2 cycle power systems to supercritical processes. The future challenges facing researchers in the field are briefly outlined.

High-speed supersonic radial compressors are a critical enabling technology for meeting the requirements of future aviation-propulsion and thermal-management systems. These turbomachines must be designed to be both efficient and robust on the widest possible operating range. Flow instabilities in the form of rotating stall and surge are therefore phenomena that must be accurately predicted early in the design process. Unsteady full-annulus computational fluid dynamics can be used to get accurate information about the onset of instabilities, but at the expense of costly simulations. As a result, the design of new compressors continues to rely on existing correlations for the prediction of the critical mass flow rate. This approach, however, leads to sub-optimal compressor designs. This article provides a review of the numerical methodologies that can be used for the accurate prediction of the critical mass flow rate in high-speed centrifugal compressors. Methods of different fidelity level and computational cost are described. Two particularly promising models, namely those proposed by Spakovszky and Sun, are subsequently examined in more detail. Exemplary applications of these two models are finally discussed.

The blade solidity, namely the blade chord-to-pitch ratio, largely affects the fluid-dynamic performance of turbomachinery and its cost. For turbomachines operating with air or steam, the optimal value of the solidity which maximizes the efficiency is estimated with empirical correlations such as the ones proposed by Zweifel (1945) and Traupel (1966). However, if the turbomachine operates with unconventional fluids, the accuracy of these correlations is questionable. Examples of such fluids are the organic compounds (e.g., hydrocarbons, siloxanes) used in organic Rankine cycle (ORC) power systems. This study concerns an investigation on how the working fluid, its thermodynamic state, and flow compressibility influence the optimum pitch-to-chord ratio of turbine stages. A first principle reduced-order model (ROM) for the computation of profile losses was developed for this purpose. The ROM results are compared with those obtained from numerical simulations of the flow over two axial turbine cascade geometries. The influence of both the working fluid, the flow compressibility, and the solidity value on both the boundary layer state at the blade trailing edge and the base pressure are evaluated. Models to compute mixing and passage losses in the compressible regime as a function of the axial solidity are proposed and discussed. Results show that the value of the optimal solidity of turbine cascades significantly increases with the flow compressibility, and mildly increases if the fluid is in the dense vapor state. Moreover, the optimal solidity value is strongly affected by the mixing process occurring downstream of the blade trailing edge. Therefore, currently available models for its estimation, which are solely based on the minimization of the profile losses, are inaccurate.

In this work, we present an investigation about the sources of dissipation in adiabatic boundary layers of non-ideal compressible fluid flows. Direct numerical simulations (DNS) of transitional, zero-pressure gradient boundary layer flows are performed for two fluids characterized by different complexity of the fluid molecules, namely, “air” and siloxane MM. Different sets of thermodynamic free-stream boundary conditions are selected to evaluate the influence of the fluid state on both the frictional loss and the dissipation mechanisms. The thermophysical properties of siloxane MM are calculated with a state-of-the-art equation of state. Results show that the dissipation due to both time-mean strain field, irreversible heat transfer, and turbulent dissipation differs significantly depending on both the molecular complexity of the fluid and its thermodynamic state. The dissipation coefficient calculated from the DNS results is then compared against the one obtained using a reduced-order model (ROM), which solves the two-dimensional boundary layer flow equations for an arbitrary fluid [M. Pini and C. De Servi, “Entropy generation in laminar boundary layers of non-ideal fluid flows,” in 2nd International Seminar on Non-Ideal Compressible Fluid Dynamics for Propulsion and Power (Springer, 2020), pp. 104-117]. Results from both the DNS and the ROM show that low values of the overall dissipation are observed in the case of fluids made of simple molecules, e.g., air, and if the fluid is at a thermodynamic state in the proximity of that of the vapor-liquid critical point.

Small-scale turbomachinery operating at high rotational speed is a key technology for increasing the power density of energy and propulsion systems. A notable example is the turbine of an organic Rankine cycle turbogenerator for thermal recuperation from prime engines and industrial processes. Such systems typically operate with organic compounds characterized by complex molecular structures, to allow the design of efficient fluid machinery and flexibility in matching the heat source and sink temperature profiles. Gas lubricated bearings are considered advantageous compared to traditional oil-lubricated rolling element bearings for supporting the turbine rotor, enabling greater machine compactness and reduced complexity, and to avoid contamination of the working fluid. In certain operating conditions, however, the lubricant of the gas bearing is in thermodynamic states near the saturated vapor line or in the vicinity of the fluid critical point whereby non-ideal effects are relevant and may affect bearing performance. This work investigates the physics of thin film flows in gas bearings operating with fluids made by complex molecules. The influence of non-ideal thermodynamic effects on gas bearing performance is discussed by analysis of the fluid bulk modulus. Reduced values of the non-dimensional bulk modulus near the critical point or saturated vapor line decrease bearing performance. The main parameter characterizing the influence of molecular complexity on bearing performance is shown to be the acentric factor. For complex fluids with large acentric factors, the impact of non-ideal thermodynamic effects on non-dimensional bearing load capacity and rotor-dynamic characteristics is less pronounced.

This work assessed the accuracy of the SU2 flow solver in predicting the isentropic expansion of Siloxane MM through the converging-diverging nozzle test section of the Organic Rankine Cycle Hybrid Integrated Device (ORCHID) [9]. The expansion is modeled using compressible Euler equations, and assuming adiabatic flow, while the fluid thermodynamic properties are estimated using the Peng-Robinson equation of state. The boundary conditions for the experiment and simulations correspond to a stagnation temperature and pressure of T¯0=253.7∘C and P¯0=18.36bar. At these inlet conditions the compressibility factor of the fluid is Z₀= 0.58. The back pressure was equal to P¯b=2.21bar. The Mach number along the centreline, and static pressure along the nozzle surface were used as the system response quantities for the validation exercise. The studied SU2 model provides valid predictions for Mach number and static pressure. The largest deviation observed in the Mach number comparison between the simulation and experiment is in the uniform flow region of the nozzle and is equal to E_Mach= 0.045. Regarding the pressure trend, the largest discrepancy occurs in the kernel region and is equal to E_pressure= 9 kPa. At the same time, the simulated Mach number and static pressure reach a maximum absolute uncertainty of ± 0.015 and of ± 20 kPa, respectively. For both quantities, these values are reached in the region close to the throat. All the uncertainties calculated for the simulated pressure profile were larger than those of the experiments. The static pressure is particularly sensitive to the geometrical uncertainties of the nozzle profile, especially inside the kernel region. A proper characterisation of the nozzle geometry was therefore required to perform a meaningful validation of the fluid dynamic solver. The developed infrastructure can be used in the future for the validation of SU2 in different operating conditions and flow cases.

Modeling non-ideal compressible flows in the context of computational fluid-dynamics (CFD) requires the calculation of thermodynamic state properties at each step of the iterative solution process. To this purpose, the use of a built-in fundamental equation of state (EoS) in entropic form, i.e., s= s(e, ρ), can be particularly cost-effective, as all state properties can be explicitly calculated from the conservative variables of the flow solver. This approach can be especially advantageous for massively parallel computations, in which look-up table (LuT) methods can become prohibitively expensive in terms of memory usage. The goal of this research is to: i) develop a fundamental relation based on the entropy potential; ii) create a data-driven model of entropy and its first and second-order derivatives, expressed as a function of density and internal energy; iii) test the performance of the data-driven thermodynamic model on a CFD case study. Notably, two Multi-Layer Perceptron (MLP) models are trained on a synthetic dataset comprising 500k thermodynamic state points, obtained by means of the Span-Wagner EoS. The thermodynamic properties are calculated by differentiating the fundamental equation, thus ensuring thermodynamic consistency. Conversely, thermodynamic stability is properly enforced during the regression process. Albeit the method is applicable to the development of equation of state models for arbitrary fluids and thermodynamic conditions, the present work only considers siloxane MM in the single phase region. The MLP model is implemented in the open-source SU2 software [8] and is used for the numerical simulation of non-ideal compressible flows in a planar converging-diverging nozzle. Finally, the accuracy and the computational performance of the data-driven thermodynamic model are assessed by comparing the resulting flow field, the wall time and the memory requirements with those obtained with direct calls to a cubic EoS, and with a LuT method.

The Environmental Control System (ECS) is the main consumer of non-propulsive power onboard aircraft. The use of an electrically-driven Vapor Compression Cycle (VCC) system, in place of the conventional air cycle machine, can lead to a substantial increase of the coefficient of performance. This work documents the development of an integrated design optimization method for VCC-based aircraft ECS, where the sizing of the system is performed along with the conceptual design of the compact heat exchangers and the high-speed centrifugal compressor. A data-driven model of the compressor has been developed to reduce the complexity of the VCC system model and the computational cost of the associated optimization problem. The model is based on artificial neural networks and has been trained on a synthetic dataset of 165k centrifugal compressor designs, generated with an in-house tool. The case study selected to demonstrate the capabilities of the proposed methodology is the multi-objective design optimization of an electrically-driven VCC system for the ECS of a single-aisle, short-haul aircraft, flying at cruise conditions. The results show that the number of function evaluations needed to identify the Pareto front reduces by a factor of three when using the data-driven model, in place of a meanline method. At the same time, the robustness of the numerical solver is improved, leading to the identification of optimal solutions covering a wider design space. Finally, the proposed methodology enables the analysis of the trends established between the system performance metrics and the design of the individual components.

In this work, we investigate the sources of dissipation in adiabatic boundary layers of non-ideal compressible fluid flows. Direct numerical simulations of transitional, zero-pressure gradient boundary layers are performed with an in-house solver considering two fluids characterized by different complexity of the fluid molecules, namely air and siloxane MM. Different sets of thermodynamic free stream boundary conditions are selected to evaluate the influence of the fluid state on the frictional loss and dissipation mechanisms. The thermo-physical properties of siloxane MM are obtained with a state-of-the-art equation of state. Results show that the dissipation due to both time-mean strain field and irreversible heat transfer, and the turbulent dissipation are significantly affected by both the molecular complexity of the fluid and its thermodynamic state. The dissipation coefficient calculated from the DNS is then compared against the one obtained from a reduced-order boundary layer CFD model [1] which has been extended to treat fluids modeled with arbitrary equations of state [7].

The environmental control system (ECS) is the largest auxiliary power consumer, i.e, around 75% of non-propulsive power, among the aircraft subsystems. The adoption of a novel ECS architecture, based on an electrically-driven vapor compression cycle system, can enable a twofold increase of coefficient of performance (COP), as compared to the conventional air cycle machine (ACM). The core of this technology is a high-speed, miniature centrifugal compressor, consisting of two impellers mounted in back-to-back configuration, and running on gas bearings operating with refrigerant. The fluid dynamic design optimization of the twin-stage compressor, to be installed in the vapor compression cycle test rig under realization at Delft University of Technology, is documented in this paper. First, the scaling analysis for centrifugal compressor is extended to provide guidelines for the design of twin-stage machines. Then, a multi-objective conceptual design optimization is performed by resorting to an in-house reduced-order model (ROM), coupled to a genetic algorithm. The fluid dynamic performance and the structural integrity of the optimal design are assessed by means of a hybrid framework, encompassing CFD and ROMs, and by FEA. The results show that it is possible to design a twin-stage compressor for the target application, featuring an average efficiency higher than 70%, a maximum compression ratio exceeding 9, and an operating range of 0.27 at the design rotational speed, despite the detrimental effects of motor cooling and miniature size.

This paper documents a thermo-fluid-dynamic mean-line model for the preliminary design of multistage axial turbines with blade cooling applicable to supercritical CO₂ turbines. Given the working fluid and coolant inlet thermodynamic conditions, blade geometry, number of stages and load criterion, the model computes the stage-by-stage design along with the cooling requirement and ultimately provides an estimate of turbine efficiency via a semi-empirical loss model. Different cooling modes are available and can be selected by the user (stand-alone or combination): convective cooling, film cooling, and thermal barrier coating. The model is applied to attain the preliminary aero-thermal design of the 600 MW cooled axial supercritical CO₂ turbine of the Allam cycle. Results show that a load coefficient varying from 3 to 1 throughout the machine, and a reaction degree ranging from 0.1 to 0.5 lead to the maximum total-to-static turbine efficiency of about 85%. Consequently, as opposed to uncooled CO₂ turbines, a repeated stage configuration is an unsuited design choice for cooled sCO₂ machines. Moreover, the study highlights that film cooling is considerably less effective compared to conventional gas turbines, while increasing the number of stages from 5 to 6 and adopting higher rotational speeds leads to an increased efficiency.

In this work we examine the flow deviation and its relationship to critical choking, i.e., choking of the meridional component of velocity, in transonic turbine cascades operating with non-ideal compressible flows. To this purpose, a generalized expression of the corrected flow per unit area as a function of both the thermodynamic state and the molecular complexity of the working fluid, the Mach number, and the amount of swirl is derived. The trends of the corrected flow with respect to these quantities are used to infer physical insights on the flow deviation and on the operability of transonic turbine cascades in off-design conditions. Moreover, reduced-order models for the estimation of the flow deviation and the preliminary assessment of the losses have been developed and validated against the results of CFD simulations performed on a representative transonic turbine stator. Results suggest that flows of dense organic vapors exhibit larger deviations than those pertaining to compounds made of simple molecules, e.g., air. Furthermore, transonic turbines expanding dense vapors reach critical choking conditions at lower Mach numbers than the ones operating with simple molecules, and are affected by larger dissipation due to viscous mixing.

Non-ideal compressible flows exhibit physical behaviors that are quantitatively and qualitatively different than those of a perfect or ideal gas. As such, the classical gas dynamic relationships that can be found on fluid-mechanics textbooks cannot be directly applied to characterize this type of flows. NiceProp is a tool for interactively learning the fundamentals of non-ideal compressible fluid dynamics and the design implications for fluid machinery components. The software is written in Python and features a highly modular structure to ease code readability and to facilitate its further development. The target audience of the software is represented by students, researchers and industry professionals interested to get started or deepen the comprehension of non-ideal compressible flow phenomena.