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Dense Gas Dynamics studies the flow of gases in the thermodynamic region near the liquid-vapor critical point. In such a region the perfect gas law is invalid and has to be replaced with noticeably more complex equations of state. In the present work, a numerical solver is proposed for inviscid flows governed by arbitrary equations of state. In par...
Citations
... The caloric behaviour is modelled by assuming that the isocoric specific heat in the dense gas limit follows a simple power law. The governing equations are discretized by using a cellcentered finite volume scheme for structured multi-block meshes of third-order accuracy [5]. ...
... The turbulence is taken into account by means of the Spalart-Allmaras transport equation. The accuracy of the numerical solver has been already demonstrated in previous works [5,7]. ...
An efficient Robust Optimization (RO) strategy is developed for the design of 2D supersonic Organic Rankine Cycle turbine expanders. The dense gas effects are not-negligible for this application and they are taken into account describing the thermodynamics by means of the Peng-Robinson-Stryjek-Vera equation of state. The design methodology combines an Uncertainty Quantification (UQ) loop 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, a CFD solver, and a multi-objective non-dominated sorting algorithm (NSGA), also based on a Kriging surrogate of the multi-objective fitness function, along with an adaptive infill strategy for surrogate enrichment at each generation of the NSGA. The objective functions are the average and variance of the isentropic efficiency. The blade shape is parametrized by means of a Free Form Deformation (FFD) approach. The robust optimal blades are compared to the baseline design (based on the Method of Characteristics) and to a blade obtained by means of a deterministic CFD-based optimization.
... In the numerical simulations presented in this work, the viscous governing equations are discretized using a cell-centered finite volume scheme for structured multi-block meshes of third-order accuracy, which allows the computation of fluids modelled by the Peng-Robinson-Stryjek-Vera cubic equation of state in order to take into account the real gas effects [7]. The closure of the RANS equations system is achieved by implementing the Spalart-Allmaras turbulence model. ...
... Local time stepping, implicit residual smoothing and multi-grid acceleration are used in order to drive the solution to the steady state. The accuracy of the numerical solver has been already demonstrated in previous works [7][8][9][10]. ...
This work aims at assessing different Uncertainty Quantification (UQ) methodologies for the stochastic analysis and robust design of Organic Rankine Cycle (ORC) turbines under multiple uncertainties. Precisely, we investigate the capability of several state-of-the art UQ methods to efficiently and accurately compute the average and standard deviation of the aerodynamic performance of supersonic ORC turbine expanders, whose geometry is preliminarily designed by means of a generalized Method Of Characteristics (MOC). Stochastic solutions provided by the adaptive Simplex Stochastic Collocation method, a Kriging-based response surface method, and a second-order accurate Method of Moments are compared to a reference solution obtained by running a full-factorial Probabilistic Collocation Method (PCM). The computational cost required to estimate the average adiabatic efficiency, Mach number and pressure coefficient, as well as their standard deviations, to within a given tolerance level is compared, and conclusions are drawn about the more suitable method for the robust design of ORC turbines.
... The results showed that the impeller's aerodynamics seemed not to be influenced by the thermodynamic model adopted because the operating condition was well below its critical value. Cinnella and Congedo [15] [16] also developed a numerical method for real gas flow, and applied it to simulate the real gas effects on the aerodynamic performance of the NACA0012 airfoil. As the literature studied the conditions in the dense gas thermodynamic region, there existed strong real gas effects on the evaluation of shock wave generation between real gas and perfect gas models. ...
This paper presents a quantitative comparison of the flow fields of a radial turbine between real gas and perfect gas models for the internal combustion engine (ICE) organic Rankine Cycle (ORC) application. Three-dimensional turbulent Navier-Stokes simulations are carried out using CFD code NUMECA FINE TM /TURBO, which is linked to an accurate thermodynamic model for organic working fluid R123 in the form of thermodynamic tables. Four turbine operating conditions including the design point and three part-load points, the inlet compressibility factors of which are 0.82-0.89, are analyzed to discuss the differences of flow fields. Obvious derivations of thermodynamic parameters are investigated in the turbine flow fields. The derivations of speed of sound and density at the nozzle inlet are about 15-20%. There exist about 10m/s value differences in the nozzle outlet velocity evaluation, and furthermore a difference of 10 degrees in the rotor inlet incidence angle comparison. The derivations of relative Mach number are about 20-35% in the rotor outlet near the shroud surface. More than 30% differences are shown in the comparison of turbine total temperature drops. Other thermodynamic parameters show much smaller derivations. The differences of thermodynamic parameters lead to a 1-3% larger in mas flow rate, 1-2% larger in isentropic efficiency and 6-8% smaller in specific power comparison. However, there do not exist obvious differences on thermodynamic parameters distributions in the flow fields. The similar flow fields provide a suggestion that perfect gas model may be an acceptable model for turbine preliminary design and one-dimensional analysis in this gas thermodynamic region, and also the real gas flow fields simulated can be used as a start point to refine the turbine design. NOMENCLATURE R specific gas constant RM universal gas constant M molecular weight Cp specific heat at constant pressure Cv specific heat at constant volume γ specific heat ratio Kt turbulent heat conductivity μt turbulent viscosity Prt turbulent Prandl number Z compressibility factor Subscripts t total parameter 0 nozzle inlet parameter 3 rotor outlet parameter INTRODUCTION
A study about convergence of Genetic Algorithms (GAs) applied to aerodynamic optimization problems for transonic flows of dilute and dense gases is presented. Specific attention is devoted to working fluids of the Bethe—Zel'dovich—Thompson (BZT) type, which exhibit non classical dynamic behaviors in the transonic/supersonic regime, such as the disintegration of compression shocks. A reference, single-objective optimization problem, namely, wave drag minimization for a non-lifting transonic flow past a symmetric airfoil is considered. Several optimizations runs are performed for perfect and BZT gases at different flow conditions using a GA coupled with a flow solver. For each case, GA-hardness, i.e. the capability of converging more or less easily toward the global optimum for a given problem, is measured by means of statistical tools. For GA-hard problems, reduced convergence rate and high sensitivity to the choice of the starting population are observed. Results show that GA-hardness is greater for flow problems characterized by very weak shocks, and is strongly affected by numerical inaccuracies in the evaluation of the objective function. Then, some possible cures to GA-hardness are proposed and numerically verified. An efficient objective-function evaluation procedure based on Richardson extrapolation is proposed, allowing to drastically reduce GA-hardness with a very moderate increase (and sometimes a slight decrease) in computational cost of optimization runs. Finally, an application of the proposed strategy to a multi-objective optimization problem is provided, clearly demonstrating the advantages deriving by the use of the proposed technique.
The present work investigates the feasibility of a methodolody for the shape optimization of wet-steam nozzles. This is done by means of a wet-steam flow solver coupled with a multi-objective genetic algorithm. A moment method is used to describe the evolution of liquid droplets. The droplet size distribution is partially modelled by means of transport equations for the lowest-order moments of the droplet spectrum, which allows evaluating the wetness fraction and the mean radius of the droplets. These additial equations are coupled with the Euler equations that govern the motion of the two-phase mixture. The system of the governing equations is solved numerically through an uncoupled procedure: the main equations for the mixture are solved first, then the main flow properties are frozen and used to solve the additional equations. The numerical solver is then coupled with a genetic algorithm to obtain a proper design for the shape of nozzles in which steam and liquid droplets coexist. Several optimization strategies are investigated for both low-pressure and high pressure nozzles, and the pros and cons of each are pointed out.
The present work provides a numerical method for the simulation of wet-steam flows
with polydispersed spectra. The so-called moment method is used to represent the liquid
droplet evolution. This approach is based on a partial modeling of the droplet size dis-
tribution, through the resolution of transport equations for the lowest-order moments of
the droplet spectrum, which allows evaluating the wetness fraction and the mean radius of
the droplets. These transport equations are coupled with the Euler equations that govern
the motion of the two-phase mixture. Several equations of state are adopted to model the
thermodynamic behavior of the vapor phase. The system of the governing equations is
solved through an uncoupled procedure: the main equations for the mixture are solved
first, then the main flow properties are frozen and used to solve the additional equations.
All of the equations are discretized by means of a third-order accurate centered scheme.
The paper investigates the sensitivity of numerically computed flow fields to uncertainties in thermodynamic models for complex organic fluids. Precisely, the focus is on the propagation of uncertainties introduced by some popular thermodynamic models to the numerical results of a computational fluid dynamics solver for flows of molecularly complex gases close to saturation conditions (dense gas flows). A tensorial-expanded chaos collocation method is used to perform both a priori and a posteriori tests on the output data generated by thermodynamic models for dense gases with uncertain input parameters. A priori tests check the sensitivity of each equation of state to uncertain input data via some reference thermodynamic outputs, such as the saturation curve and the critical isotherm. A posteriori tests investigate how the uncertainties propagate to the computed field properties and aerodynamic coefficients for a flow around an airfoil placed into a transonic dense gas stream. V C 2011 American Institute of Physics. [doi:
Bethe–Zel’dovich–Thompson fluids (BZT) are characterized by negative values of the fundamental derivative of gasdynamics for a range of temperatures and pressures in the vapor phase, which leads to non-classical gasdynamic behaviors such as the disintegration of compression shocks. These non-classical phenomena can be exploited, when using these fluids in Organic Rankine Cycles (ORCs), to increase isentropic efficiency. A predictive numerical simulation of these flows must account for two main sources of physical uncertainties: the BZT fluid properties often difficult to measure accurately and the usually fluctuating turbine inlet conditions. For taking full advantage of the BZT properties, the turbine geometry must also be specifically designed, keeping in mind the geometry achieved in practice after machining always slightly differs from the theoretical shape. This paper investigates some efficient procedures to perform shape optimization in a 2D BZT flow with multiple-source uncertainties (thermodynamic model, operating conditions and geometry). To demonstrate the feasibility of the proposed efficient strategies for shape optimization in the presence of multiple-source uncertainties, a zero incidence symmetric airfoil wave-drag minimization problem is retained as a case-study. This simplified configuration encompasses most of the features associated with a turbine design problem, as far the uncertainty quantification is concerned. A preliminary analysis of the contributions to the variance of the wave-drag allows to select the most significant sources of uncertainties using a reduced number of flow computations. The resulting mean value and variance of the objective are next turned into metamodels. The optimal Pareto sets corresponding to the minimization of various substitute functions are obtained using a genetic algorithm as optimizer and their differences are discussed.
