Steady-state Formulation Research Articles

Correlation functions from tensor network influence functionals: The case of the spin-boson model.

We investigate the application of matrix product state (MPS) representations of the influence functionals (IFs) for the calculation of real-time equilibrium correlation functions in open quantum systems. Focusing specifically on the unbiased spin-boson model, we explore the use of IF-MPSs for complex time propagation, as well as IF-MPSs for constructing correlation functions in the steady state. We examine three different IF approaches: one based on the Kadanoff-Baym contour targeting correlation functions at all times, one based on a complex contour targeting the correlation function at a single time, and a steady state formulation, which avoids imaginary or complex times, while providing access to correlation functions at all times. We show that within the IF language, the steady state formulation provides a powerful approach to evaluate equilibrium correlation functions.

Performance enhancement of a fin and tube heat exchanger with the novel arrangement of curved winglets using a multi-objective optimization approach

The present study considers performing multi-objective optimization of a fin and tube heat exchanger to enhance its performance. For this, a group of three curved rectangular winglets, placed on alternate sides of the tube, are considered. Two different geometrical arrangements are analyzed. In the first case – referred to as case A – the distance between the three curved blades is varied in a way that the relative gap between the curved blades is the same. In the second case – referred to as case B – the distance between the three curved blades is held fixed at the entry section, but the relative gap is varied at the exit of the curved blades. The experiments are designed using the Latin hypercube sampling technique, and the responses against each set of design variables are calculated using computational fluid dynamics. The conservation equation for mass, momentum and energy are solved along with transition SST (a three equation) turbulence model considering a steady state formulation. The resulting data is used to train the artificial neural network that serves as a surrogate model for GA to generate the Pareto front points. Conclusive results indicate that the optimized models from both cases largely outperform compared to the previously reported studies. Furthermore, the performance of the case B variant is slightly better than for case A.

Mathematical Analysis of Nonlinear Differential Equations in Polymer Coated Microelectrodes

A theoretical analysis is conducted on the electrochemical behaviour of micro disk electrodes coated with thin coatings of electroactive polymers. The main focus of efforts to characterize the planar diffusion chemical reaction within polymer-modified ultra-microelectrodes is the creation of a theoretical model that explicitly takes into account the potential for the polymer film to cover the inlaid micro disc support surface. Approximate formulae for the steady-state amperometric response and formulation of the boundary value problem are given. The impact of substrate concentration, mediator concentration and current responsiveness in the solution besides the polymer film is also investigated. Akbari-Ganji method (AGM) and differential transformation method (DTM), two adequate and widely available analytical methods, were utilized to determine the steady-state non-linear diffusion equation. The approximate analytical solution for the substrate concentration and mediator concentration and the current for the small experimental kinetic values and diffusion coefficients are presented. We additionally determine the problem’s numerical solution using the MATLAB tool. A satisfactory agreement can be seen when the numerical outcomes verify with the analytical findings.

Are steady-state wake models and lookup tables sufficient to design profitable wake steering strategies? A Large Eddy Simulation investigation

We leverage Large Eddy Simulation (LES) of wind farms to study the performances of an open-loop wake steering controller. This global controller is elaborated based on Lookup Table (LUT) generated using FLORIS. Our objectives are to highlight the potential problems that can arise from model errors due to its steady-state formulation and from difficulties to properly define the ambient flow state. We consider two wind farm configurations: a first theoretical layout of six machines, and a second layout representing a commercial wind farm that is currently studied for wake steering strategy, the King Plains wind farm. We represent our wind farm environment using LES within which the turbines are modeled as actuator disks. In addition to the standard local controller, the wind farm utilizes a LUT to compute the optimal yaw offset angles. Before its integration into the controller, this LUT is first populated using the yaw optimizer integrated into FLORIS. The primary input parameters for the LUT are derived from the freestream atmospheric conditions, which are parametrized in terms of TI, hub height wind speed, and wind direction. In this work, we investigate different strategies for estimating the ambient wind direction fed as input to the LUT during the control and their impact on the dynamics of the yaw actuation. While results confirm the potential of this type of controller, they also highlight the importance of a proper sensing to determine the freestream state.

Effects on the resolution of the near-wake by the use of different meshes using RANS and URANS

This work compares the wake resolution for several meshes in order to find an accurate yet computationally efficient grid. The mesh is identified through a parametric study where the wake resolution is gradually increased using up to 55 million cells. The mesh is designed in order to support CFD-based shape design, considering not only the wind turbine power but also an improved wake recovery. Furthermore, a comparison between steady-state and transient RANS formulations is presented. The proposed meshing should allow for extending a recently published CFD-based shape design focusing on wind turbine power, to also include wake behavior in an optimization context.

Experimental analysis of a modified scheme for supplying single-phase remote loads from micro-hydro based three-phase self-excited induction generator

Induction machine (IM) plays significant role in rural electrification, acting as a generator for small kilowatt to large megawatt applications. These IMs are therefore used for power generation, gaining a huge acceptance within the researchers because of its brushless system, robust design and cheap market rate. In off-grid operation, the induction generator (IG) also known as self-excited IG (SEIG), requires supply of external reactive power (RP) for building up its terminal voltage and satisfying the electrical load demand. A small kilowatt level renewable-driven three-phase SEIG is generally deployed for generating single-phase power in off-grid mode for remote areas. Generation of single-phase power and supplying the load from a three-phase SEIG in off-grid operation, results in the imbalance of the SEIG's stator current and phase voltage. In this regard, various capacitor-based excitation (CBE) schemes have been reported for supplying the required RP to the three-phase SEIG and generating single-phase power, while reducing the mentioned difficulties. The current paper emphasizes on the steady-state mathematical formulation of a proposed modified CBE scheme for generating single-phase power and supplying remote hilly load from a hydro-driven three-phase SEIG under the fixed speed of the generator. Further, a state-of-the-art experimental test-bench of the proposed scheme, where a hydro-driven three-phase SEIG supplying single-phase resistive-inductive (RL) load has been showcased with the same being simulated in the MATLAB environment. The result indicates that the proposed CBE scheme can effectively supply single-phase RL load of different power factors (p.f.), having promising voltage regulation with balanced SEIG operation.

Solution of OECD/NEA PWR MOX/UO2 benchmark with a high-performance pin-by-pin core calculation code

Expanding upon the framework of the steady-state pin-by-pin 2D/1D decoupling method, a novel and high-performance pin-by-pin transient calculation method has been introduced. This transient method, consistent to the steady-state formulation, is designed for time-dependent calculations utilizing a 3D diffusion-based finite difference method (FDM). The inherent complexity of the large 3D problem is effectively managed by decoupling it into a series of planar (2D) and axial (1D) problems. In addition, tens of thousands of pin-cells are grouped into hundreds of boxes to reduce the computing burden for the 1D calculations without essential loss of the accuracy. Two-level coarse mesh finite difference (CMFD) formulation comprising multigroup nodewise CMFD and two-group assemblywise CMFD is employed as well to accelerate the convergence. Errors originating from the pin-level homogenization, energy group condensation, and the use of lower order calculation methods are simultaneously corrected by the pinwise super homogenization (SPH) equivalence factor.The transient method is evaluated with OECD/NEA PWR MOX/UO2 benchmark. Code-to-code comparison with the nTRACER direct whole core calculation code yielded highly satisfactory results for the transient scenario as well as the steady-state problems. Furthermore, comparative analyses with conventional nodal calculations show superiority of the pin-by-pin calculation.

Steady-state approximations for Hodgkin-Huxley cell models: Reduction of order for uterine smooth muscle cell model.

Multi-scale mathematical bioelectrical models of organs such as the uterus, stomach or heart present challenges both for accuracy and computational tractability. These multi-scale models are typically founded on models of biological cells derived from the classic Hodkgin-Huxley (HH) formalism. Ion channel behaviour is tracked with dynamical variables representing activation or inactivation of currents that relax to steady-state dependencies on cellular membrane voltage. Timescales for relaxation may be orders of magnitude faster than companion ion channel variables or phenomena of physiological interest for the entire cell (such as bursting sequences of action potentials) or the entire organ (such as electromechanical coordination). Exploiting these time scales with steady-state approximations for relatively fast-acting systems is a well-known but often overlooked approach as evidenced by recent published models. We thus investigate feasibility of an extensive reduction of order for an HH-type cell model with steady-state approximations to the full dynamical activation and inactivation ion channel variables. Our effort utilises a published comprehensive uterine smooth muscle cell model that encompasses 19 ordinary differential equations and 105 formulations overall. The numerous ion channel submodels in the published model exhibit relaxation times ranging from order 10-1 to 105 milliseconds. Substitution of the faster dynamic variables with steady-state formulations demonstrates both an accurate reproduction of the full model and substantial improvements in time-to-solve, for test cases performed. Our demonstration here of an effective and relatively straightforward reduction method underlines the particular importance of considering time scales for model simplification before embarking on large-scale computations or parameter sweeps. As a preliminary complement to more intensive reduction of order methods such as parameter sensitivity and bifurcation analysis, this approach can rapidly and accurately improve computational tractability for challenging multi-scale organ modelling efforts.

Stochastic expansion planning of integrated energy system: A benders-based decomposition approach

This paper introduces a two-stage stochastic optimization model for the renewables-transmission expansion planning (RTEP) problem to obtain the optimal size and site of the renewable energy generating unit, as well as the investment decisions related to transmission lines and gas pipelines. The uncertainties associated with demand growth and stochastic renewable sources production levels are considered via a series of scenarios, and then, the scenario-reduction approach, i.e., the k-medoids method, is applied to reduce the computation burden. In addition, the gas flow-pressure difference steady-state formulations in the expansion planning model are reconstructed convexly using second-order cone constraints. A Benders-based decomposition method is adopted as a solution methodology in the co-planning problem. The proposed model was simulated on a modified IEEE 39-bus and Belgian gas-20 energy test network, and the simulation results validate the correctness and applicability of the presented approaches.

Considerations Concerning Numerical Modelling of Flow and Heat Transfer Phenomena in Continuous Casting of Steel

Previous studies on the mathematical modelling of fluid flow-heat transfer phenomena in continuous casting, reported in this journal and elsewhere, are examined and, analyzed computationally. It is shown that mathematical model of continuous casting based on a constant, in lieu of a time varying inlet temperature, can produce different estimates of temperature field and solidified shell thickness in the mold. In contrast, thermal free convection phenomenon is found to be unimportant to the precise modelling of heat transfer in the mold. It is argued that a transient, rather than a steady state formulation, should be the preferred choice as the former ensures easy integration of additional process features as well as speedy convergence.

Steady state performance of molten salt natural circulation loop with different orientations of heater and cooler

Molten salts are proposed as a coolant and sensible heat storage medium for Solar Thermal Power Plant (STPP) as well as fuel and coolant for Molten Salt Reactor (MSR). The objective of the present study is to investigate the steady state behavior of molten salt in natural circulation flow condition along with effect of different orientations of heater and cooler on the steady state mass flow rate. For which a Molten Salt Natural Circulation Loop (MSNCL) has been setup with molten nitrate salt (mixture of Sodium Nitrate and Potassium Nitrate in 60:40 ratio by wt.) as working fluid. Steady state experiments in four different orientations of heater and cooler [Horizontal Heater Horizontal Cooler (HHHC), Horizontal Heater Vertical Cooler (HHVC), Vertical Heater Horizontal Cooler (VHHC) and Vertical Heater Vertical Cooler (VHVC)] have been performed and data generated are compared with the steady state correlations available in open literature which are derived considering either standard friction factor correlations or friction factor correlations deduced from their experiments and theoretical studies. Based on the deviations observed in the experimental values and correlation predictions, an improved formulation has been derived. The steady state formulation has been improved by incorporating Wall Thermal Inertia (WTI) effect and Heat Loss (HL) effect considering the importance of WTI & HL in dealing with high temperature molten salt loop. The improved formulation has been compared with the experimental results for different orientations of heater and cooler. Further, a new friction factor correlation has also been deduced from the experiments for vertical heater orientation. Description of MSNCL along with validation studies performed for different steady state correlation against experimental data generated in MSNCL, are reported in this paper.

Modeling the Gradual RESET of Phase Change Memory With Confined Geometry

This work develops a model of the temperature and resistance for the RESET of confined phase change memory (PCM). The confined PCM, unlike the mushroom type, has heating inside the cell and higher resistance sensitivity to active region size. The ideal temperature profile is derived from a steady-state heat conduction formulation based on a quasi-3D approach. A thermal subcircuit is then proposed to implement the heating module, considering the thermoelectric effect (TEE) and temperature dependence of phase change materials, to obtain the active region size. The resistance of the irregular active region is then calculated using an integral method. Calibrated TCAD simulations and experimental data are used to verify the model with desired scalability and accuracy.

Short-Circuit Analysis of AC Distribution Systems Dominated by Voltage Source Converters Considering Converter Limitations

This paper deals with the short-circuit analysis of distribution systems dominated by Voltage Source Converters (VSCs). A novel methodology has been proposed in this paper for short-circuit calculation of power systems populated with VSCs where the control modes and potential current-saturated operation of power converters are considered. The studied distribution system has been modeled using an element-based steady-state formulation that includes the converter equations in both normal operation and fault scenarios. Then, equilibrium points of the studied distribution system are identified by solving the established systems of equations for different potential current-saturation states of the VSCs. The methodology identifies possible solutions involving different saturation states of different converters. The proposed methodology is applied to a specific case study of a distribution system penetrated with VSCs in both grid-forming and grid-following control. The short-circuit analysis results indicate that VSCs operation shows a significant impact on the short-circuit equilibrium points of the studied system. In addition, multiple equilibrium points may exist depending on whether the grid-forming converter is saturated or not. The existence of multiple equilibrium points is validated through dynamic simulations.

A novel methodology for effective short-circuit calculation in offshore wind power plants considering converter limitations

This paper deals with the short-circuit calculation of offshore wind power plants (OWPPs). The short-circuit calculation presented in this paper assumes the studied system achieves a new equilibrium state during the fault. The studied system is modeled with an element-based steady-state formulation where the grid-support control and potential current-saturated operation of wind turbine (WT) converters are included. In particular, the current-saturation modifies the converter control and represents various potential operation states of the converter. Then, an iterative methodology is proposed to identify the short-circuit equilibrium point that satisfies converters’ limitations and to obtain the current-saturation states of all power converters with a specific fault scenario in the studied system. The numerical case studies indicate that the WT converters operation has a significant influence to the system equilibrium point during the fault. In addition, the presented methodology is efficient for short-circuit calculation of a large-scale OWPP.

PERFORMANCE ANALYSIS AND OPTIMIZATION OF A HELICAL CROSS-FLOW WATER TURBINE

In this study, a Computational Fluid Dynamics model is designed to investigate the performance analysis of a helical cross-flow water turbine by using Comsol Multiphysics. In order to predict the main performance characteristics of turbine such as power output and torque, a numerical model is developed which is accurate, fast and quite simple to be used for a simulation. The flow field around turbine is solved with the Rotating Machinery feature in the Comsol CFD Module using a k-ω turbulence model and a steady state formulation. The Navier-Stokes equations are used in the model which are arranged in a rotating frame in the inner domain and in fixed coordinates in the outer domain. The boundary between the inner and the outer domain is a continuity boundary condition that transfers momentum to the fluid in the inner domain. This model also uses the Frozen Rotor study type which speeds up the computation time. Then, to increase the performance of the cross-flow water turbine, a new angular velocity profile is explored with Comsol Optimization Module. Thus, a variable speed turbine control method is developed. Compared to the constant velocity control method, new angular velocity control method yielded a 3% increase in the efficiency of turbine.

Turbulent Non-Stationary Reactive Flow in a Cement Kiln

The reduction of emissions from large industrial furnaces critically relies on insights gained from numerical models of turbulent non-premixed combustion. In the article Mitigating Thermal NOx by Changing the Secondary Air Injection Channel: A Case Study in the Cement Industry, the authors present the use of the open-source OpenFoam software environment for the modeling of the combustion of Dutch natural gas in a cement kiln operated by our industrial partner. In this paper, various model enhancements are discussed. The steady-state Reynolds-Averaged Navier-Stokes formulation is replaced by an unsteady variant to capture the time variation of the averaged quantities. The infinitely fast eddy-dissipation combustion model is exchanged with the eddy-dissipation concept for combustion to account for the finite-rate chemistry of the combustion reactions. The injection of the gaseous fuel through the nozzles occurs at such a high velocity that a comprehensive flow formulation is required. Unlike in Mitigating Thermal NOx by Changing the Secondary Air Injection Channel: A Case Study in the Cement Industry, wave transmissive boundary conditions are imposed to avoid spurious reflections from the outlet patch. These model enhancements result in stable convergence of the time-stepping iteration. This in turn increases the resolution of the flow, combustion, and radiative heat transfer in the kiln. This resolution allows for a more accurate assessment of the thermal NO-formation in the kiln. Results of a test case of academic interest are presented. In this test case, the combustion air is injected at a low-mass flow rate. Numerical results show that the flow in the vicinity of the hot end of the kiln is unsteady. A vortex intermittently transports a fraction of methane into the air stream and a spurious reaction front is formed. This front causes a transient peak in the top wall temperature. The simulated combustion process is fuel-rich. All the oxygen is depleted after traveling a few diameters into the kiln. The thermal nitric oxide is formed near the burner and diluted before reaching the outlet. At the outlet, the simulated thermal NO concentration is equal to 1 ppm. The model is shown to be sufficiently mature to capture a more realistic mass inflow rate in the next stage of the work.

A Monolithic Finite Element Formulation for Magnetohydrodynamics Involving a Compressible Fluid

This work develops a new monolithic finite-element-based strategy for magnetohydrodynamics (MHD) involving a compressible fluid based on a continuous velocity–pressure formulation. The entire formulation is within a nodal finite element framework, and is directly in terms of physical variables. The exact linearization of the variational formulation ensures a quadratic rate of convergence in the vicinity of the solution. Both steady-state and transient formulations are presented for two- and three-dimensional flows. Several benchmark problems are presented, and comparisons are carried out against analytical solutions, experimental data, or against other numerical schemes for MHD. We show a good coarse-mesh accuracy and robustness of the proposed strategy, even at high Hartmann numbers.

Propeller Load Modelling In The Calculations Of Marine Shafting Torsional Vibrations

Technical rules of IACS classification societies require that calculations of torsional vibrations for all propulsion shafting systems, as well as for shafting systems of auxiliary machinery above certain power, shall be prepared, submitted and validated for the vessels requesting the class certificate. These calculations may be approached either by conventional analytical models based upon systems of ordinary differential equations describing the actual dynamic system, or by simulation modelling of the same dynamic system. The research team of the authors has been established within the R&D Centre of the Faculty of Maritime Studies at Split. The aim of the team is to investigate possibilities and constraints for implementation of the SimulationX software to the simulation modelling of torsional vibrations of dynamic systems, such as marine propulsion and auxiliary shafting. In general, results of these calculations depend upon the concept of the system (two-stroke vs. four-stroke Diesel engine, torsional vibration damper present or not, flexible coupling present or not, fixed pitch or controllable pitch propeller, etc.), dimensions of components, their material properties and service loading. Propeller load modelling within the service loading is very important. Even the preliminary research results show that its simplest part, i.e. the steady-state propeller loading of the system without taking excitations into account, may have a significant influence on the results of torsional vibrations calculation and consequently even to their acceptability by the class society. For this reason, the present paper describes two different ways to express propeller steady-state loading formulations, as prescribed by different sources and authorities, as well as their influence on the torsional vibrations calculation results and meeting the criteria of the IACS Unified Requirements, presenting them in a real verified and validated case study.

Bend stiffener linear viscoelastic thermo-mechanical analysis, Part II: Numerical solution and case study

The viscoelastic energy dissipated by the bend stiffener when subjected to cyclic loading increases the polyurethane temperature and may affect the system curvature distribution in a coupled manner. The steady-state thermo-mechanical mathematical formulation and material experimental characterization has been presented in a companion paper (Part I). In this work (Part II), the rate of viscoelastic mechanical energy dissipation is obtained from the steady-state mechanical formulation employing the shooting method combined with the Runge–Kutta method for the two-point boundary value problem numerical solution. The temperature field distribution within the bend stiffener volume is then estimated with the three-dimensional steady-state thermal model employing the finite difference technique with a boundary-fitted coordinate system that transforms the physical domain into a structured computational domain grid. The partial differential equations governing heat transfer are solved in the computational domain and transformed back into the physical domain by transformation relations. A detailed description of the iterative thermo-mechanical numerical solution procedure is presented and a case study carried out demonstrating the loading frequency and oscillation amplitude influence on the bend stiffener temperature field distribution. It is shown that the bend stiffener may be exposed to high internal temperatures for some loading conditions.

Bend stiffener linear viscoelastic thermo-mechanical analysis. Part I — Experimental characterization and mathematical formulation

Bend stiffeners are subjected to cyclic loading during offshore operation or when subjected to a controlled full-scale qualification test. Due to the viscoelastic nature of the polyurethane, energy is dissipated within the material volume and the structure may experience a temperature increase, a phenomenon known as self-heating. The top connection is a flexible riser critical region in terms of fatigue, being the bend stiffener the main responsible for curvature control. As the curvature distribution is highly affected by the nonlinear time–temperature bend stiffener response, a detailed thermo-mechanical assessment may become relevant for riser lifetime and polyurethane material failure assessment, specially during accelerated full-scale tests. In the present paper (Part I), the polyurethane experimental characterization and steady-state thermo-mechanical mathematical formulation are presented for the bend stiffener self-heating assessment. A steady-state formulation is derived for a temperature dependent linear viscoelastic large deflection beam model to estimate the heat generation during harmonic tip loading. The temperature field distribution is calculated through a three-dimensional steady-state thermal model considering the viscoelastic heat calculated from the mechanical model with an iterative scheme. Stress relaxation tests are performed at different temperatures to determine the viscoelastic properties followed by thermal properties characterization through differential scanning calorimetry and by the Flash method to determine the specific heat, thermal conductivity and diffusivity, respectively. In a companion paper (Part II) the iterative numerical scheme is detailed and a case study presented.