In this paper we discuss the numerical solution of the Helmholtz equation with mixed boundary conditions on a 2D physical domain Ω. The so called radiation problem depends on the constant wavenumber k, that in some medical applications can be of order of thousands. For these values of k the classical Finite Element Method (FEM) faces up several numerical difficulties. To mitigate these limitations we apply the Isogeometric Analysis (IgA) to compute the approximated solution uh. Main steps of IgA are discussed and specific proposals for their fulfillment are addressed, with focus on some aspects not covered in available publications. In particular, we introduce a low distortion quadratic NURBS parametrization of Ω that represents exactly its boundary and contributes to the accuracy of uh. Our approach is non-isoparametric since uh is a bicubic tensor product polynomial B-spline function on Ω. This allows to improve the numerical solution refining the approximation space and keeping the coarser parametrization of the domain. Moreover, we discuss the role of the number of degrees of freedom in the directions perpendicular and longitudinal to wave front and its relationship with the noise and the shift in amplitude and phase of uh. The linear system derived from IgA discretization of the radiation problem is solved using GMRES and we show through experiments that the incomplete factorization of the Complex Shifted Laplacian provides a very good preconditioner. To solve the radiation problem, we have implemented IgA approach using the open source package GeoPDEs. A comparison with FEM is included, to provide evidence that IgA approach is superior since it is able to reduce significantly the pollution error, especially for high values of k, producing additionally smoother solutions which depend on less degrees of freedom.@en

In this paper we use the Isogeometric Analysis (IgA) to solve the Helmholtz equation with Dirichlet boundary condition over a bounded physical 2D domain. Starting from the variational formulation of the problem, we show how to apply IgA to obtain an approximated solution based on biquadratic B-spline functions. We focus the attention on problems where the physical domain has very irregular boundary. To solve these problems successfully a high quality parametrization of the domain must be constructed. This parametrization is also a biquadratic tensor product B-spline function, with control points computed as the vertices of a quadrilateral mesh with optimal geometric properties. We study experimentally the influence of the wave number and the parametrization of the physical domain in the accuracy of the approximated solution. A comparison with classical Finite Element Method is also included. The power of IgA is shown solving several difficult model problems, which are particular cases of the Helmholtz equation and where the solution has discontinuous gradient in some points, or it is highly oscillatory. For all model problems we explain how to select the knots of B-spline quadratic functions and how to insert knew knots in order to obtain good approximations. The results obtained with our imple-mentation of the method prove that IgA approach is successful, even on regions with irregular boundary, since it is able to offer smooth solutions having at the same time some singular points and high number of oscillations.

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The operation of large industrial furnaces will continue to rely on hydrocarbon fuels in the near foreseeable future. Mathematical modeling and numerical simulation is expected to deliver key insights to implement measures to further reduce pollutant emissions. These measures include the design optimization of the burners, the dilution of oxidizer with exhaust gasses, and the mixing of natural gas with hydrogen. In this paper, we target the numerical simulation of non-premixed turbulent combustion of natural gas in a single heating section of a ring pit anode baking furnace. In previous work, we performed combustion simulations using a commercial flow simulator combined with an open-source package for the three-dimensional mesh generation. This motivates switching to a fully open-source software stack. In this paper, we develop a Reynolds-Averaged Navier-Stokes model for the turbulent flow combined with an infinitely fast mixed-is-burnt model for the non-premixed combustion and a participating media model for the radiative heat transfer in OpenFoam. The heat transfer to the refractory brick lining is taken into account by a conjugate heat transfer model. Numerical simulations provide valuable insight into the heat release and chemical species distribution in the staged combustion process using two burners. Results show that at the operating conditions implemented, higher peak temperatures are formed at the burner closest to the air inlet. This results in a larger thermal nitric-oxide concentration. The inclusion of the heat absorption in the refractory bricks results in a more uniform temperature on the symmetry plane at the center of the section. The peak in thermal nitric-oxides is reduced by a factor of four compared to the model with adiabatic walls@en

No alternatives are currently available to operate industrial furnaces, except for hydrocarbon fuels. Plant managers, therefore, face at least two challenges. First, environmental legislation demands emission reduction. Second, changes in the origin of the fuel might cause unforeseen changes in the heat release. This paper develops the hypothesis for the detailed control of the combustion process using computational fluid dynamic models. A full-scale mock-up of a rotary cement kiln is selected as a case study. The kiln is fired by the non-premixed combustion of Dutch natural gas. The gas is injected at Mach (Formula presented.) via a multi-nozzle burner located at the outlet of an axially mounted fuel pipe. The preheated combustion air is fed in (co-flow) through a rectangular inlet situated above the attachment of the fuel pipe. The multi-jet nozzle burner enhances the entrainment of the air in the fuel jet. A diffusion flame is formed by thin reaction zones where the fuel and oxidizer meet. The heat formed is transported through the freeboard, mainly via radiation in a participating medium. This turbulent combustion process is modeled using unsteady Favre-averaged compressible Navier–Stokes equations. The standard k- (Formula presented.) equations and standard wall functions close the turbulent flow description. The eddy dissipation concept model is used to describe the combustion process. Here, only the presence of methane in the composition of the fuel is accounted for. Furthermore, the single-step reaction mechanism is chosen. The heat released radiates throughout the freeboard space. This process is described using a P1-radiation model with a constant thermal absorption coefficient. The flow, combustion, and radiative heat transfer are solved numerically using the OpenFoam simulation software. The equations for flow, combustion, and radiant heat transfer are discretized on a mesh locally refined near the burner outlet and solved numerically using the OpenFoam simulation software. The main results are as follows. The meticulously crafted mesh combined with the outlet condition that avoids pressure reflections cause the solver to converge in a stable manner. Predictions for velocity, pressure, temperature, and species distribution are now closer to manufacturing conditions. Computed temperate and species values are key to deducing the flame length and shape. The radiative heat flux to the wall peaks at the tip of the flame. This should allow us to measure the flame length indirectly from exterior wall temperature values. The amount of thermal nitric oxide formed in the flame is quantified. The main implication of this study is that the numerical model developed in this paper reveals valuable information on the combustion process in the kiln that otherwise would not be available. This information can be used to increase fuel efficiency, reduce spurious peak temperatures, and reduce pollutant emissions. The impact of the unsteady nature of the flow on the chemical species concentration and temperature distribution is illustrated in an accompanying video.

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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.@en

This paper verifies a mathematical model that is developed for the open source CFD-toolbox OpenFOAM, which couples turbulent combustion with conjugate heat transfer. This feature already exists in well-known commercial codes. It permits the prediction of the flame’s characteristics, its emissions, and the consequent heat transfer between fluids and solids via radiation, convection, and conduction. The verification is based on a simplified 2D axisymmetric cylindrical reactor. In the first step, the combustion part of the solver is compared against experimental data for an open turbulent flame. This shows good agreement when using the full GRI 3.0 reaction mechanism. Afterwards, the flame is confined by a cylindrical wall and simultaneously conjugate heat transfer is activated and analysed. It is shown that the combustion and conjugate heat transfer are successfully coupled.

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Thermal nitric-oxide (NOx) formation in industrial furnaces due to local overheating is a  widely known problem. Various industries made significant investments to reduce thermal NOx by varying the operating conditions and designs of the furnace. Finding optimal operating conditions or design parameters by experimenting in the furnace, however, is difficult. Numerical modeling can provide significant information in such cases. In this paper, a three dimensional steady state finite element model for the anode baking industrial furnace is discussed. The COMSOL Multiphysics software is used for modeling the non-premixed turbulent combustion and the conjugate heat transfer to the insulation lining. The mesh generation using the cfMesh software allows to increase the spatial resolution locally at the outlet of the fuel nozzles while maintaining the overall quality of the mesh. The temperature and species mass fraction obtained from the finite element model are calibrated by adjusting the amount of artificial diffusion in the transport equations for the species. The simulated temperature agrees well with the measured data from our industrial partner in regions distant from the flames. The model underestimates the measured oxygen mass fraction. The spatial gradients in oxygen mass fraction, however, are captured well by the model. The effects of variation of the fuel mass flow rate and the fuel pipe diameter on the NOx generation are studied. The results show that by decreasing the fuel mass flow rate and increasing the fuel pipe diameter by 45%, the peak in thermal NOx ppm generated in the furnace decreases by 42%.@en

Thermal nitric-oxide (NOx) formation in industrial furnaces due to local overheating is a widely known problem. Various industries made significant investments to reduce thermal NOx by varying the operating conditions and designs of the furnace. It is difficult to find the optimal operating conditions that minimize NOx formation in the furnace by trial and error methods. The high temperature in the furnace complicates performing experiments in the furnace. Numerical modeling can provide significant information in such cases. Therefore, the objective of this paper is to obtain a numerical model of the furnace in such a way that the operating conditions can be varied and examined. In this paper, a three-dimensional steady-state finite element model for the anode baking industrial furnace is discussed. The COMSOL Multiphysics software is used for modeling the non-premixed turbulent combustion and the conjugate heat transfer to the insulation lining. The cfMesh software is used for obtaining the mesh. The results show that the simulated temperature agrees well with the measured data from our industrial partner in regions distant from the flames. The analysis shows that by decreasing the fuel mass flow rate and increasing the fuel pipe diameter by 45%, the peak in thermal NOx ppm generated in the furnace decreases by 42%. The model is limited by the use of a single-step chemistry mechanism with an eddy dissipation combustion model and a simplified approach for radiation, such as the P1 approximation model. The model can be further improved by considering a detailed chemistry mechanism model for combustion and a discrete ordinate model for radiation.

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This work studies how non-premixed turbulent combustion in a rotary kiln depends on the
geometry of the secondary air inlet channel. We target a kiln in which temperatures can reach values above 1800 degrees Kelvin. Monitoring and possible mitigation of the thermal nitric-oxide (NOx) formation is of utmost importance. The performed reactive flow simulations result in detailed maps of the spatial distribution of the flow, thermodynamics and chemical conditions of the kiln. These maps provide valuable information to the operator of the kiln. The simulations show the difference between the existing and the newly proposed geometry of the secondary air inlet. In the existing configuration, the secondary air inlet is rectangular and located above the base of the burner pipe. The secondary air flows into the furnace from the top of the flame. The heat release by combustion is unevenly distributed throughout the flame. In the new geometry, the secondary air inlet is an annular ring placed around the burner pipe. The secondary air flows circumferentially around the burner pipe. The new secondary air inlet geometry is shown to result in a more homogeneous spatial distribution of the heat release throughout the flame. The peak temperatures of the flame and the production of thermal NOx are significantly reduced. Further research is required to resolve limitations of various choices in our modeling approach.
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One of the quickest ways to influence both the wall temperature and thermal NOx emissions in rotary kilns is to change the air–fuel ratio (AFR). The normalized counterpart of the AFR, the equivalence ratio, is usually associated with premixed flames and studies of its influence on diffusion flames are inconsistent, depending on the application. In this paper, the influence of the AFR is investigated numerically for rotary kilns by conducting steady-state simulations. We first conduct three-dimensional simulations where we encounter statistically unstable flow at high inflow conditions, which may be caused by vortex stretching. As vortex stretching vanishes in two-dimensional flow, the 2D simulations no longer encounter convergence problems. The impact of this simplification is shown to be acceptable for the thermal behaviour. It is shown that both the wall temperature and thermal NOx emissions peak at the fuel-rich and fuel-lean side of the stoichiometric AFR, respectively. If the AFR continues to increase, the wall temperature decreases significantly and thermal NOx emissions drop dramatically. The NOx validation, however, shows different results and indicates that the simulation model is simplified too much, as the measured NOx formation peaks at significantly fuel-lean conditions.@en

We consider a fundamental integer programming (IP) model for cost–benefit analysis and flood protection through dike building in the Netherlands, due to Zwaneveld and Verweij (2017). Experimental analysis with data for the IJsselmeer shows that the solution of the linear programming relaxation of the IP model is integral. This naturally leads to question whether the polytope associated to the IP is always integral. In this paper we first give a negative answer to this question by proving the non-integrality of the polytope. Secondly, we establish natural conditions that guarantee the linear programming relaxation of the IP model is integral. We show that these conditions are indeed satisfied by the recent data on flood probabilities, damage and investment costs of IJsselmeer. Finally, we show that the IP model can be solved in polynomial time when the number of dike segments, or the number of feasible barrier heights, are bounded.@en

The emissions from the industrial furnaces impact the environment. Among the various factories, those having anode baking furnaces are working on reducing the pollutant emissions. The aerodynamics in the furnace influences the emissions due to the high dependence of combustion and radiation phenomena on the mixing characteristics. Therefore, this paper aims to establish the numerical simulation results for the three-dimensional turbulent flow in a single section of an anode baking furnace with a high rate of fuel injection. The stabilized non-linear finite element approach on the Reynolds-averaged Navier-Stokes (RANS) equation is used with COMSOLMultiphysics. The turbulent viscosity ratio is highly sensitive to the mesh for the standard k-ε model. The requirements of the Cartesian and refined mesh near the jet development region is explained. The comparison of meshes generated by two meshing tools namely cfMesh and COMSOL Multiphysics default Mesher is carried out. The high numerical diffusion in the flow models due to the coarser mesh leads to convergence but deficit the precision in the results. This paper shows that the mesh generated by cfMesh with flow aligned refinement combined with the non-linear finite element solver in COMSOL Multiphysics proves to provide accurate results of turbulent quantities.

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The load flow equations express the balance of power in an electrical power system. The power generated must equal the power consumed. In the AC time-harmonic case, the load flow equations are non-linear in the voltage phasors associated with the nodes in the network. The development of future power systems urgently requires new, highly efficient and robust load flow solvers. In this contribution we aim at making the following three scientific contributions. We first show that the use of a globalization procedure is required to ensure the convergence of a Newton load flow simulation of a stressed network. Such operational conditions are more likely to occur in the future. We subsequently show that the use of an inexact Newton–Krylov method results in faster computations. We employ Quotient Minimal Degree (QMD) as a matrix reordering method, incomplete LU factorization (ILU) as a preconditioner, Generalized Minimal Residual (GMRES) as a Krylov acceleration, and the Dembo-Steihaus strategy to defined the accuracy of the linear solver at each Newton iteration. We finally show the results of iterative solution algorithms that allow to exploit the decomposition of a network into subnetworks. Decompositions with and without overlapping nodes are tested.@en

The upward trends in renewable energy penetration, cross-border flow volatility and electricity actors’ proliferation pose new challenges in the power system management. Electricity and market operators need to increase collaboration, also in terms of more frequent and detailed system analyses, so as to ensure adequate levels of quality and security of supply. This work proposes a novel distributed load flow solver enabling for better cross border flow analysis and fulfilling possible data ownership and confidentiality arrangements in place among the actors. The model exploits an Inexact Newton Method, the Newton–Krylov–Schwarz method, available in the portable, extensible toolkit for scientific computation (PETSc) libraries. A case-study illustrates a real application of the model for the TSO–TSO (transmission system operator) cross-border operation, analyzing the specific policy context and proposing a test case for a coordinated power flow simulation. The results show the feasibility of performing the distributed calculation remotely, keeping the overall simulation times only a few times slower than locally.@en

The wish to reduce the environmental footprint and to enhance economic gains of rotary kilns pushes the numerical simulation of combustion to include conjugate heat transfer. In this paper we study the influence of the refractory wall and radiative heat loss to the ambient, on the combustion process of a reference kiln model. Numerical results show that the inclusion of the refractory lining and the external radiative heat loss allows the inner wall temperature distribution to vary, with 60 % difference between its minimum and maximum. This is in sharp contrast with models that assume a fixed temperature at the wall. Consequently the maximum inner wall temperature increases by more than 200 %, the maximum flame temperature by nearly 13 % and maximum freeboard gas temperature by up to 90 %. It is thus important to account for these effects when modeling rotary kilns.@en

The anode baking process is developed and improved since the 1980s due to its importance in Aluminium industry. The process is characterized by multiple physical phenomena including turbulent flow, combustion process, conjugate heat transfer, and radiation. In order to obtain an efficient process with regards to quality of anodes, soot-free combustion, reduction of NOx and minimization of energy, a mathematical model can be developed. A mathematical model describes the physical phenomena and provides a deeper understanding of the process. Turbulent flow is one of the important physical phenomena in an anode baking process. In the present work, isothermal turbulent flow is studied in detail with respect to two turbulence models in COMSOL Multiphysics software. The difference between wall boundary conditions for these models and their sensitivity towards the boundary layer mesh is investigated. A dimensionless distance in viscous scale units is used as a parameter for comparison of models with and without a boundary layer mesh. The investigation suggests that the boundary layer mesh for both turbulence models increase the accuracy of flow field near walls. Moreover, it is observed that along with the accuracy, the numerical convergence of Spalart-Allmaras turbulence model in COMSOL Multiphysics is highly sensitive to the boundary layer mesh. Therefore, development of converged Spalart-Allmaras model for the complete geometry is difficult due to the necessity of refined mesh. Whereas, the numerical convergence of k-ε model in COMSOL Multiphysics is less sensitive to the dimensionless viscous scale unit distance. A converged solution of the complete geometry k-ε model is feasible to obtain even with less refined mesh at the boundary. However, a comparison of a developed solution of k-ε model with another simulation environment indicates differences which enhance the requirement of having converged Spalart-Allmaras model for complete geometry.@en

The brushless doubly-fed induction machine (DFIM) has great potential for wind turbine applications. However, it has not yet been commercialized due to its complicated operating principle. Previously, a computationally efficient FE model has been developed. Some design guidelines for the stator pole-pair combinations and the nested-loop rotors have been gained from the previous work. This paper brings the model and design guidelines together to optimize the design of a 3.2MW brushless DFIM. Both the active material cost and the efficiency are optimized. The results show that the magnetic loading of the brushless DFIM is increased for a better design by using the FE based optimization tool. The optimized designs increase the efficiency and the shear stress while reducing the torque ripple and the THD level of the stator voltages. However, the optimized designs result in a high electric loading which would be a challenge for cooling.

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In recent work we showed that the performance of the complex shifted Laplace preconditioner for the discretized Helmholtz equation can be significantly improved by combining it multiplicatively with a deflation procedure that employs multigrid vectors. In this chapter we argue that in this combination the preconditioner improves the convergence of the outer Krylov acceleration through a new mechanism. This mechanism allows for a much larger damping and facilitates the approximate solve with the preconditioner. The convergence of the outer Krylov acceleration is not significantly delayed and occasionally even accelerated. To provide a basis for these claims, we analyze for a one-dimensional problem a two-level variant of the method in which the preconditioner is applied after deflation and in which both the preconditioner and the coarse grid problem are inverted exactly. We show that in case that the mesh is sufficiently fine to resolve the wave length, the spectrum after deflation consists of a cluster surrounded by two tails that extend in both directions along the real axis. The action of the inverse of the preconditioner is to shrink the length of the tails while at the same time rotating them and shifting the center of the cluster towards the origin. A much larger damping parameter than in algorithms without deflation can be used.@en

Brushless Doubly-fed Induction Machines (BDFIMs) have been envisaged as a potential candidate for generators in offshore wind farms.@en