ABSTRACT. The possibility that rarefaction as well as compression shocks may form in single phase vapours was envisaged first by Bethe [1]. However calculations based on the Van der Waals equation of state indicated that the former type of shock is possible only if the ratio of specific heat at constant volume c_v to universal gas constant R is larger than about 17.5 which he considered too large to be satisfied by real fluids. This conclusion was contested by Thompson [2] who showed that the type of shock capable of forming in arbitrary fluids is determined by the sign of the thermodynamic quantity
        

ABSTRACT. Turbompumps deliver liquid fuel and oxidizer into the main combustion chamber in liquid rocket engines and are usually composed of radial pumps driven by axial turbines.  Turbopumps operate under extreme environments characterized by high speeds, high pressures, cryogenic temperatures, and steep temperature gradients. Therefore, turbopumps are the critical devices which determine the success of liquid rocket systems.  To prevent cavitation in the pumps, turbopumps deploy inducers, and, yet, the inducers themselves undergo cavitation.  This cavitation can have thermodynamic, hydrodynamic, and even rotordynamic consequences.  This talk will present such issues which arise in turbopump development efforts.  

ABSTRACT. A significant improvement in the economy-of-scale of small-scale organic Rankine cycle (ORC) systems can arise from the appropriate design of components that can be manufactured in large volumes and implemented flexibly into a wide range of systems and potential applications. This, in turn, requires accurate predictions of component performance that can capture variations in the cycle conditions, parameters or changes to the working fluid. In this paper previous work investigating a modified similitude theory used to predict the performance of subsonic ORC turbines is extended to analyse the supersonic flow of organic fluids within 2D converging-diverging nozzles. Two nozzles are developed using a minimum length method of characteristics design model coupled to REFPROP. These are designed for R245fa and Toluene as working fluids with nozzle exit Mach numbers of 1.4 and 1.7 respectively. First, the nozzle performance is confirmed using CFD simulations, and then further CFD simulations are performed to evaluate the performance of the same nozzles over a range of different inlet conditions and with different working fluids. The CFD simulations are compared to predictions made using the original and modified similitude theories, and also to predictions made by conserving the Prandtl-Meyer function for the different operating conditions. The results indicate that whilst the modified similitude model does not accurately predict nozzle performance, conserving the Prandtl-Meyer function allows to predict the nozzle outlet Mach number to within 2% providing there is not a significant change in the polytropic index. Finally, the effect of working fluid replacement on the ORC system is discussed, and preliminary results demonstrate the possibility of matching a particular turbine to a heat source through optimal working fluid selection.

ABSTRACT. The non-monotone dependence of the speed of sound along adiabatic transformation is demonstrated to result in the admissibility of non-ideal increase of the flow Mach number across oblique shock waves, for pre-shock states in the close proximity of the liquid-vapour saturation curve. This non-ideal behaviour is not admissible for dilute, ideal gases and it associated to a less-than-unity value of the fundamental derivative of gasdynamics. Therefore, non-ideal shock waves are expected to be observed also in fluids with relatively low molecular complexity, including methane and ammonia. The simple yet qualitatively sound van der Waals model is used to confirm the present findings and to provide exemplary non-ideal shock waves.

ABSTRACT. The calculation of isentropic flow and normal shock waves of real gases are important, especially in the preliminary design of turbo-machinery and test rigs. In an ideal gas, the relations for one-dimensional isentropic flow and normal shock waves are well known and can be found in standard textbooks. However, for fluids exhibiting strong deviations from the ideal gas assumption universal relations do not exist due to complex equations of state. This paper presents a analytical method for the prediction of isentropic real gas flows and normal shock waves, based on the Redlich-Kwong (RK) equation of state. Explicit expressions based on a series expansion for describing isentropic flow of Novec 649 are compared to Refprop data and ideal gas equations. For moderate pressures the RK method is in very good agreement with the Refprop data, while the ideal gas equations fail to predict the real gas behaviour. The same observations are made for normal shock calculations, where both real gas methods yield very close results. Especially the predicted stagnation pressure losses across a shock wave are in excellent agreement.

ABSTRACT. The dynamic response of pressure probes for unsteady flow measurements in turbomachinery is studied for fluids operating in non-ideal thermodynamic conditions. In the last decade the scientific community witnessed a significant growth of the interest towards turbomachinery systems based on Organic Rankine Cycles (ORC). Such devices exploit key features of fluids characterized by high molecular complexity. Under certain conditions of pressure and temperature, typically in the close proximity of the saturation curve and the critical point, the behaviour of these fluids is known to significantly depart from the one of the ideal gas model. The application of advanced and accurate Equations of State (EoS) is crucial for the reliable analysis of ORC turbines, not only for modeling issues or theoretical studies, but also for the design and the application of proper measurement techniques for non-ideal compressible flows. Among the most relevant instrumentations currently applied for the experimental analysis of the flow in turbomachinery, aerodynamic pressure probes stand out for their capability to measure both flow properties and pressure field, thus allowing to estimate the performance of the stationary cascades. With the aim to measure the time-resolved flow and pressure field downstream of turbomachinery rotors, where significant unsteady flows are found, the concept of fast-response aerodynamic pressure probe was intensively developed in the last decade and has been recently demonstrated as a reliable and extremely relevant technique for the experimental analysis of the flow downstream of turbine stages. To extend the application of such a measurement device to ORC turbines, a shortcoming in the available analytical and experimental data calls for a careful investigation of the physics related to such devices in the limit of non-ideal flows. In particular, the evolution of the flow inside the line-cavity system connecting the pressure tap on the probe head with the sub-surface mounted pressure sensor is expected to be highly effected by non-ideal fluid effects. To this end, the step response of a fast-reponse pressure probe, initially designed for unsteady measurements in air turbines/compressors, is investigated numerically in order to assess the expected time response when dealing with a flow in a non-ideal fluid regime. Numerical simulation are carried out exploiting the Non-Ideal Compressible Fluid-Dynamics (NICFD) solver embedded in SU2. SU2 is an open-source collection of software tools for performing Partial Differential Equation analysis originally conceived at Stanford university and currently developed by an joint international team of researchers from different countries. At first, the analysis is carried out assuming ideal gas thermodynamic behavior, to validate the computational framework against experimental dynamic calibration in the low-pressure shock tube of Politecnico di Milano. Then, the Peng-Robinson EoS is applied to accurately represent the expected thermodynamic behaviour of the organic fluid with an ORC turbine. The step-reponse of the probe predicted using the two classes of EoS is then compared to assess the impact of non-ideal effects on the dynamic system in terms of settling time, characteristic frequencies and fluid-dynamic damping.

ABSTRACT. Calculation of tail probabilities remains very challenging even with the most recent and performing uncertainty quantification techniques. To tackle this problem, this work aims at proposing an efficient method basing on Monte Carlo method, importance sampling and interpolation methods making use of Adjoint states. Several approaches are proposed in literature for using adjoint evaluations for accelerating Monte Carlo convergence. Generally, they rely on the building of a linear approximation in order to compute the small probability that the objective function exceeds a certain critical value, usually representing the probability that a system fails or suffers catastrophic losses. The adjoint calculation is then used to calculate the sensitivity gradient, which can be used to approximate the objective function as a linear function of the random variables describing the sources of uncertainty. The information provided by this linear approximation permits to reduce the variance of the Monte Carlo method using control variate and importance sampling. The method proposed here is based on a continuous update of the interpolation function via the gradient computation, which is then used with the importance sampling technique. The advantages of such an approach is to improve and retain the good convergence properties of MC methods when treating a very high number of dimension, by further accelerating the convergence by using the interpolation on functional evaluations and gradients. The interest of this kind of approach for predicting non-classical gasdynamics phenomena is then demonstrated with several examples, such as the computation of a rarefaction shock wave (RSW) in a dense-gas shock tube. In this case, since a RSW is relatively weak and that the prediction of its occurrence and intensity are highly sensitive to uncertainties on the initial flow conditions and on the fluid thermodynamic model, the objective is to obtain a reliable estimate for the RSW probability of occurrence.

ABSTRACT. The aim of this work is  the development of a fully explicit scheme in the framework of time dependent hyperbolic problems with strong interacting discontinuities to retain high order accuracy in the context of compressible multiphase flows. A new methodology is presented to compute compressible two-fluid problems based on the five equation model given in Kapila et al. (Physics of Fluids 2001), which takes the formal limit (see Murrone and Guillard, J. Comp. Physics 2005) of the Baer and Nunziato model (Int. J. of Multiphase flow 1986) when the relaxation parameters tend simultaneously to infinity. We present a predictor-corrector scheme, following the concept of residual distribution in Ricchiuto and Abgrall (J. Comp. Physics 2010). The advantage of this formalism is that the time step is dictated by a CFL- like condition, contrarily to the original Baer- and Nunziato-like system, and, moreover, the system is much smaller. The drawback is that the PDE is written in a non- conservation form. Testing the predictor-corrector scheme on second order accuracy, the study considers one prediction and one correction step, adopting for the space discretization a residual based distribution scheme. This formulation allows seeing the component of the discretization as an error between two different approximations. In particular, the residual is computed considering a Lax Friedrich's scheme (Ricchiuto et al., J. Comp. Physics 2010). In order to construct a stable and non-oscillating approximation of the discontinuities, the residual, maintaining the consistency requirement, is distributed to each node belonging to a cell via a limiter. The last is designed similarly to a positive stream-wise invariant with a filtering term, using instead of the residual its eigenforms. We compare this scheme to a hybrid one, where we couple the presented explicit residual distribution with a classical Glimm’s scheme (J. Sci. Stat. Comp. 1982). In particular, we take advantage of Glimm's ability to approximate the front of shocks, while retaining the advantages of the explicit residual distribution in the other areas. The switch from one scheme to the other is performed by means of a shock detector, capturing large momentum jumps. This numerical methodology can be easily extended to unstructured meshes. With respect to other contributions in that area, we investigate a method that provides for mesh convergence the exact solutions, where the studied non-conservative system is associated to consistent jump relations. Test cases on a stiffened gas for a two phase compressible flow in one dimension for a Riemann problem have verified that the approximation converges to its exact solution. The results have been compared with the pure Glimm’s scheme and the expected exact solution, finding a good overlap and proving the method to be robust.

ABSTRACT. The accurate modelling and simulation of two-phase flows is a key issue for many engineering applications such as condensing steam within turbines and nozzles, sprays in combustion, fluid mixing systems. Currently, the majority of two-phase models for homogeneous condensation are calibrated on the large amount of experimental data available for water and steam at low reduced pressures. However, the direct application of such models to non-ideal thermodynamic conditions, i.e. in regions where the vapour thermo-physical behaviour departs from that of a perfect gas, is rather questionable and worth of being investigated for application in the proximity of the critical point.

Recently, an increasing number of researches have proposed the use of the method of moments to describe the droplet spectra of the second (discrete) phase. The great advantage of this approach is the unrestricted applicability to any fluid and the relatively low computational cost as compared with the well-established lagrangian and eulerian-eulerian methods.

This work carried out a numerical investigation of quasi-1D condensing flows in supersonic nozzles at high reduced pressures. Two different set of conservation laws, referred to as mixture and continuum phase, were considered and supplemented by transport relations for the dispersed phase formulated through the method of moments. Both models were integrated in time using a segregated approach allowing for metastable conditions by the use of cubic equations of state of increasing complexity.

Numerical calculations were performed on several nozzle test cases for which experimental observations are made available [1]. The results show significant deviations between the numerical models and the experiments while augmenting the inlet total pressure of the nozzle, suggesting that classical models amply validated at low reduced pressures need to be corrected when condensing in non-ideal thermodynamic regions.

References

1. Gyarmathy, G. (2005). Nucleation of steam in high-pressure nozzle experiments. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 219(6), 511–521. doi:10.1243/095765005x31388

ABSTRACT. Cavitation is characterized by vapor bubbles creation in the liquid phase as a consequence of a pressure drop. This phenomenon can be reproduced by means of several two-phase models. An equation of state is commonly used in order to define the thermophysical properties of the two fluids and to close the model. The aim of this work is to study how the uncertain parameters of the equation of state (EOS) can influence the prediction of the cavitation structures. These uncertainties are propagated through a two-phase numerical solver for evaluating the impact on the predictive character of the numerical solution. The variability of the mixture velocity and the mixture pressure are analyzed.

ABSTRACT. In the present contribution, we study high-pressure combustion of oxygen and methane using Large-Eddy Simulation (LES). The work is motivated by liquid rocket propulsion engines (LRE) in which the chamber pressure often exceeds the critical pressure of the injected fluid and the temperature of one or both propellants is smaller or close to the pseudo-critical temperature. The propellants are thus in a trans- or supercritical state in which the thermodynamic and transport properties are highly non-linear functions of temperature and pressure. Real-gas effects have a significant influence on the operation of LREs and numerous experimental and numerical studies contributed to a better understanding of such flows. Many valuable coaxial single-injector experiments were carried out on the Mascotte test bench to study trans- and supercritical combustion of LOx/GH2 as well as of LOx/GCH4. Among them is the work of Singla et al. who investigated methane combustion at different operating conditions.

This configuration is an interesting test case for numerical tools and serves as a reference for our real-gas LES methods for turbulent combustion presented herein. The implementation is based on the open-source software OpenFOAM. The publicly available code has been extended by a real-gas thermodynamics model based on the Peng-Robinson equation of state together with the volume-translation method of Abudour et al. This method significantly improves the accuracy of the density prediction with respect to the uncorrected version at minimal extra computational cost. The real-gas thermodynamics models are incorporated into a flamelet combustion model to allow for simulations at supercritical pressures. Chemical reactions on the subgrid scales are described using a presumed Beta-shape Probability Density Function.

Preliminary results are encouraging. The figure shows an iso-surface of the stoichiometric mixture fraction colored with the temperature. The characteristic flame shape that has been observed in the experiments is well reproduced. In the final paper we will present the real-gas flamelet approach in greater detail. In particular, the validity of the underling assumptions will be addressed and the necessity of solving an additional energy equation will be discussed. The averaged simulation results will then be compared with the available OH* measurements.

ABSTRACT. The LE-7A engine is the first-stage engine of the Japanese-made H-IIA launch vehicle. This engine has been developed by improving and reducing the price of the LE-7 engine used in the H-II launch vehicle. In the qualification combustion tests, the original designed LE-7A (LE-7A-OR) engine experienced two major problems, a large side load in the transient state of engine start and stop and melt on nozzle generative cooling tubes. The reason for the troubles of the LE-7A-OR engine was investigated by conducting experimental and numerical studies. In actual engine conditions, the main hot gas stream is a heated steam. Furthermore, the main stream temperature in the nozzle changes from approximately 3500 K at the throat to 500 K at the exit. In such a case, the specific heat ratio changes depending on the temperature. A similarity of the Mach number should be considered when conducting a model flow test with a similar flow condition of the Mach number between an actual engine combustion test and a model flow test. High-speed flow tests were conducted using CO2 gas heated up to 673 K as a working fluid and a 1:12 sub-scaled model nozzle of the LE-7A-OR engine configuration. The problems of the side force and the conducted form of the shock waves generated in the nozzle of the LE-7A-OR engine during engine start and stop were reproduced by the model tests of experimental and numerical investigations. This study presented that the model flow test using heated CO2 gas is useful and effective in verifying the numerical analysis and the design verification before actual engine combustion tests.

ABSTRACT. This paper illustrates an innovative approach for improving the prediction of a piston expander for exhaust heat recovery. While nearly 30 percent of the fuel energy is lost as waste heat in the form of hot exhaust gases, exhaust heat recovery promises one of the biggest fuel economy potential regarding the technologies available in the next decade. Applied to heavy commercial vehicles (HCVs), buses or off road vehicles, a bottoming Rankine Cycle (RC) on exhaust heat shows a great potential in recovering the exhaust gases energy, even for part loads. Nowadays it seems to be clear that the heavy duty industry will implement RC on their long haul trucks in the 2020s as an answer to future stringent regulation and the still increasing customers request for operating cost reduction. A 2 to 5% fuel economy is achievable on such vehicles. An experimental and numerical characterization of a piston expander is presented here for assessing the prediction of the proposed physical model. Experimental measurements are characterized and their variability is propagated through the numerical code by means of state-of-art uncertainty quantification techniques. Several sources of uncertainties are taken into account at the same time, thus yielding various indications concerning the most predominant parameters, and their influence on several outputs of interest. Numerical results are validated with respect to the experimental measurements, by comparing numerical and experimental error bars. Additional analysis are then performed by using both ANOVA techniques and surrogate modelling.

ABSTRACT. The capabilities of the open-source SU2 software suite for the numerical simulation of viscous flows over unstructured grid are extended to non-ideal compressible-fluid dynamics (NICFD). A built-in thermodynamic library is incorporated to account for the non-ideal ther- modynamic characteristics of fluid flows evolving in the close proximity of the liquid-vapour saturation curve and critical point. The numerical methods, namely the Approximate Riemann Solvers (ARS), viscous fluxes and boundary conditions are generalised to non-ideal fluid prop- erties. Non-reflecting boundary conditions are furthermore implemented to enhance the fidelity of turbomachinery simulations. A variety of test cases are carried out to assess the performance of the solver. At first, numer- ical methods are verified against analytical solution of reference NICFD test cases, including steady shock reflection in two spatial dimensions. Then, non-ideal gas effects past turbine cascades, typically encountered in Organic Rankine Cycle applications, are investigated and debated. The obtained results demonstrate that SU2 is highly suited for the analysis and the automatic design of internal flow devices operating in the non-ideal compressible-fluid regime.

ABSTRACT. The use of dense gases as working media in turbomachinery, referred to as Organic Rankine Cycle (ORC) turbines, is proposed as a method of recovery of variable energy sources such as waste heat from industrial processes. Whereas a traditional Rankine Cycle operates with steam as the working fluid, ORC turbines use an organic fluid such as hydrocarbons, silicon oils or other organic refrigerants. Proposed heat sources for ORC turbines typically include variable energy sources such as solar thermal collectors or waste heat from industrial processes. As a result, to improve the feasibility of this technology, the resistance to variable input conditions must be taken into account at an early stage of the development process. Robust design has been developed to improve the product quality and reliability in industrial Engineering. 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. The robustness is determined by a measure of insensitivity of the design with respect to variations of the design parameters, like geometrical tolerances or fluctuations of the operating conditions. 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 an accurate design methodology for 2D supersonic ORC expanders based on a method of characteristics (MOC) generalized to complex equations of state, 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 (mean and variance) of the uncertain system output, coupled to a multi-objective non-dominated sorting genetic algorithm (NSGA). We search for an optimal shape that maximizes the mean and minimizes the variance of the expander isentropic efficiency. 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 Bezier curves and the design variables are Bezier control points. 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 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 finally compared, in terms of isentropic efficiency and power output, to the MOC baseline shape and to an injector designed by means of a standard deterministic optimizer.

ABSTRACT. This paper discusses the shape-optimization of non-conventional centrifugal turbine nozzles for Organic Rankine Cycle (ORC) applications. The optimal aerodynamic design is supported by the use of a non-intrusive, gradient-free technique specifically developed for shape optimization of turbomachinery profiles. The method is constructed as a combination of a geometrical parametrization technique based on B-Splines, a high-fidelity and experimentally validated Computational Fluid Dynamic (CFD) solver, and a surrogate-based evolutionary strategy. The non-ideal gas behaviour featuring the flow of organic fluids in the cascades of interest is introduced via a look-up-table approach, which is rigorously applied throughout the whole optimization process thanks to the non-intrusive character of the evolutionary strategy. Two transonic centrifugal nozzles are considered, featuring very different loading and radial extension. The use of a systematic and automatic design method to such a non-conventional configuration highlights the character of centrifugal cascades, which require a specific and non-trivial definition of the shape, especially in the rear part, to avoid the onset of shock waves. It is shown that the optimization acts in similar way for the two cascades, identifying an optimal curvature of the blade that both provides a relevant increase of cascade performance and a reduction of downstream gradients.

ABSTRACT. The efficiency of expanders is of prime importance in determining the overall performance of a variety of thermodynamic power systems, with reciprocating-piston expanders favoured at intermediate-scales of application (typically 10-100 kW). Once the mechanical losses in reciprocating machines are minimized (e.g. through careful valve design and operation), losses due to the unsteady thermal-energy exchange between the working fluid and the solid walls of the containing device can become the dominant loss mechanism. In this work, gas-spring devices are investigated numerically in order to focus explicitly on the thermodynamic losses that arise due to this unsteady heat transfer. The specific aim of the study is to investigate the behaviour of real gases in gas springs and compare this to that of ideal gases in order to attain a better understanding of the impact of real-gas effects on the thermally induced losses in reciprocating expanders and compressors. A CFD-model of a gas spring is developed in OpenFOAM. Three different fluid models are compared: an ideal-gas model with constant thermodynamic and transport properties; an ideal-gas model with temperature-dependent properties; and a real-gas model using the Peng-Robinson equation-of-state with temperature and pressure-dependent properties. Results indicate that, for simple, mono- and diatomic gases, like helium or nitrogen, there is a negligible difference in the pressure and temperature oscillations over a cycle between the ideal and real-gas models. However, when considering heavier (organic) molecules, such as propane, the ideal-gas model tends to overestimate the temperature and pressure compared to the real-gas model, especially if the temperature dependency of the thermodynamic properties is not taken into account. In fact, the ideal-gas model predicts higher pressures by as much as 25% compared to the real-gas model. Additionally, the ideal-gas models (both alternatives) underestimate the thermally induced loss compared to the real-gas model for heavier gases. This discrepancy is most pronounced at rotational speeds where the losses are highest. The real-gas model predicts a peak loss of 8.9% of the compression work, while the ideal-gas model predicts a peak loss of 5.7%. These differences in the work loss are due to the fact that the gas behaves less ideal during expansion than during compression, with the compressibility factor being lower during compression. This behaviour cannot be captured with the ideal-gas law. It is concluded that real-gas effects must be taken into account in order to predict accurately the thermally induced loss mechanism when using heavy fluid molecules in such devices.