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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (20 August 2020) | Viewed by 40196

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Prof. Dr. Arturo Hidalgo

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Departmento de Ingeniería Geológica y Minera, E.T.S.I. de Minas y Energía, Universidad Politécnica de Madrid, Ríos Rosas, 21, 28003 Madrid, Spain
Interests: numerical simulation; finite volume methods; environmental applications; computational fluid dynamics
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Special Issue Information

Dear Colleagues,

Mathematical modelling and numerical simulation are fundamental tools in the description of a wide variety of physical and technical phenomena. The capability of mathematical models to represent real situations and therefore, to make predictions on the behaviour of these systems makes their use essential when dealing with real-world applications.

This Special Issue mainly focuses on new research related to the mathematical modelling of energy and environmental processes and its numerical analysis. Some applications of interest may include the mathematical modelling of heat transfer in architecture so as to get efficient heating and cooling in buildings. Moreover, in the context of both energy and the environment, research on mathematical models for the study of non-conventional energy sources, such as solar energy (both photovoltaic and thermosolar), wind turbines or mathematical modelling for biomass formation can be relevant. In addition, Magnetohydrodynamics modelling for fusion plasmas and its influence on nuclear fusion may represent relevant research for this Special Issue. Other applications to be considered are based on shallow water models in the context of environmental hydrodynamics, with applications to different hydrodynamic situations which may include, for instance, dam breaks involving shock propagation and wet-dry fonts. In addition, environmental flows based on multi-phase flow model and flow in porous media can be part of this Special Issue.

A very relevant feature of this Special Issue is the numerical resolution of the mathematical models under study, based on a wide variety of numerical schemes such as finite volumes, finite elements, finite differences or Discontinuous Galerkin, to name a few. Relevant research may include new and very efficient numerical methods which represent a step forward in the numerical resolution of mathematical models in the context of energy and environment applications.

Rigorous analytical theory of the mathematical models under study is also very relevant for this Special Issue.

To sum up, this Special Issue intends to gather new research on mathematical models mainly focused on energy and environmental applications. Particular emphasis will be placed on the numerical resolution of the models, using very efficient numerical schemes, and also on a rigorous theoretical analysis of the models under study.

Prof. Dr. Arturo Hidalgo
Guest Editor

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Keywords

Mathematical modelling in Energy and Environment
Efficient numerical schemes
Heat transfer in industry and buildings
Aerodynamics
Fluid dynamics
Shallow water models, dam-break
Multiphase flows, flow in porous media

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Published Papers (11 papers)

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Research

19 pages, 6761 KiB

Open AccessFeature PaperArticle

A Spatial-Temporal Model for the Evolution of the COVID-19 Pandemic in Spain Including Mobility

by Francesc Aràndiga, Antonio Baeza, Isabel Cordero-Carrión, Rosa Donat, M. Carmen Martí, Pep Mulet and Dionisio F. Yáñez

Mathematics 2020, 8(10), 1677; https://doi.org/10.3390/math8101677 - 1 Oct 2020

Cited by 28 | Viewed by 4574

Abstract

In this work, a model for the simulation of infectious disease outbreaks including mobility data is presented. The model is based on the SAIR compartmental model and includes mobility data terms that model the flow of people between different regions. The aim of the model is to analyze the influence of mobility on the evolution of a disease after a lockdown period and to study the appearance of small epidemic outbreaks due to the so-called imported cases. We apply the model to the simulation of the COVID-19 in the various areas of Spain, for which the authorities made available mobility data based on the position of cell phones. We also introduce a method for the estimation of incomplete mobility data. Some numerical experiments show the importance of data completion and indicate that the model is able to qualitatively simulate the spread tendencies of small outbreaks. This work was motivated by an open call made to the mathematical community in Spain to help predict the spread of the epidemic. Full article

(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment)

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20 pages, 1668 KiB

Open AccessFeature PaperArticle

Exploring a Convection–Diffusion–Reaction Model of the Propagation of Forest Fires: Computation of Risk Maps for Heterogeneous Environments

by Raimund Bürger, Elvis Gavilán, Daniel Inzunza, Pep Mulet and Luis Miguel Villada

Mathematics 2020, 8(10), 1674; https://doi.org/10.3390/math8101674 - 1 Oct 2020

Cited by 6 | Viewed by 3375

Abstract

The propagation of a forest fire can be described by a convection–diffusion–reaction problem in two spatial dimensions, where the unknowns are the local temperature and the portion of fuel consumed as functions of spatial position and time. This model can be solved numerically in an efficient way by a linearly implicit–explicit (IMEX) method to discretize the convection and nonlinear diffusion terms combined with a Strang-type operator splitting to handle the reaction term. This method is applied to several variants of the model with variable, nonlinear diffusion functions, where it turns out that increasing diffusivity (with respect to a given base case) significantly enlarges the portion of fuel burnt within a given time while choosing an equivalent constant diffusivity or a degenerate one produces comparable results for that quantity. In addition, the effect of spatial heterogeneity as described by a variable topography is studied. The variability of topography influences the local velocity and direction of wind. It is demonstrated how this variability affects the direction and speed of propagation of the wildfire and the location and size of the area of fuel consumed. The possibility to solve the base model efficiently is utilized for the computation of so-called risk maps. Here the risk associated with a given position in a sub-area of the computational domain is quantified by the rapidity of consumption of a given amount of fuel by a fire starting in that position. As a result, we obtain that, in comparison with the planar case and under the same wind conditions, the model predicts a higher risk for those areas where both the variability of topography (as expressed by the gradient of its height function) and the wind velocity are influential. In general, numerical simulations show that in all cases the risk map with for a non-planar topography includes areas with a reduced risk as well as such with an enhanced risk as compared to the planar case. Full article

(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment)

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17 pages, 259 KiB

Open AccessArticle

On the Existence of Solutions of a Two-Layer Green Roof Mathematical Model

by J. Ignacio Tello, Lourdes Tello and María Luisa Vilar

Mathematics 2020, 8(9), 1608; https://doi.org/10.3390/math8091608 - 18 Sep 2020

Cited by 4 | Viewed by 2047

Abstract

The aim of this article is to fill part of the existing gap between the mathematical modeling of a green roof and its computational treatment, focusing on the mathematical analysis. We first introduce a two-dimensional mathematical model of the thermal behavior of an extensive green roof based on previous models and secondly we analyze such a system of partial differential equations. The model is based on an energy balance for buildings with vegetation cover and it is presented for general shapes of roofs. The model considers a vegetable layer and the substratum and the energy exchange between them. The unknowns of the problem are the temperature of each layer described by a coupled system of two partial differential equations of parabolic type. The equation modeling the evolution of the temperature of the substratum also considers the change of phase of water described by a maximal monotone graph. The main result of the article is the proof of the existence of solutions of the system which is given in detail by using a regularization of the maximal monotone graph. Appropriate estimates are obtained to pass to the limit in a weak formulation of the problem. The result goes one step further from modeling to validate future numerical results. Full article

(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment)

22 pages, 2010 KiB

Open AccessFeature PaperArticle

Numerical Approach of the Equilibrium Solutions of a Global Climate Model

by Arturo Hidalgo and Lourdes Tello

Mathematics 2020, 8(9), 1542; https://doi.org/10.3390/math8091542 - 9 Sep 2020

Cited by 5 | Viewed by 2426

Abstract

We consider a coupled model surface-deep ocean effect, where an Energy Balance Model (EBM) is used for modelling the surface temperature and a two-dimensional heat equation represents the evolution of the temperature of the deep ocean. Although the model under study is based on that proposed by Watts & Morantine (1990), here we consider a modified model that incorporates other processes, such as the nonlinear diffusion and the action of coalbedo, depending on the temperature. The stationary states of the model under study, taking the solar constant as the parameter, are numerically attained. The results of the simulation are depicted in a

{(Q, u)}

plot where u is the temperature in the surface and Q is the solar constant. The numerical solution is achieved by means of a finite volume scheme with Weighted Essentially Non-Oscillatory (WENO) reconstruction in space and third order Runge-Kutta scheme, which verifies the Total Variation Diminishing (TVD) property, for time integration. The equilibrium states are accomplished by evolving in time the numerical solution until the stationary solutions are reached. The main novel results of this work concern the numerical obtention of the stationary solutions of both the EBM and the coupled model EBM-deep ocean and the agreement of these results with the theoretically obtained in previous works, where an interval of values of the solar constant Q was obtained with the existence of at least three stationary solutions. In this work, we have numerically obtained more than three stationary solutions for such interval of Q. Full article

(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment)

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15 pages, 2490 KiB

Open AccessArticle

Mathematical Model of Decomposition of Methane Hydrate during the Injection of Liquid Carbon Dioxide into a Reservoir Saturated with Methane and Its Hydrate

by Marat K. Khasanov, Nail G. Musakaev, Maxim V. Stolpovsky and Svetlana R. Kildibaeva

Mathematics 2020, 8(9), 1482; https://doi.org/10.3390/math8091482 - 2 Sep 2020

Cited by 10 | Viewed by 2634

Abstract

The article describes a mathematical model of pumping of heated liquid carbon dioxide into a reservoir of finite extent, the pores of which in the initial state contain methane and methane gas hydrate. This model takes into account the existence in the reservoir of three characteristic regions. We call the first region “near”, the second “intermediate”, and the third “far”. According to the problem statement, the first region contains liquid CO₂ and hydrate, the second region is saturated with methane and water, the third contains methane and hydrate. The main features of mathematical models that provide a consistent description of the considered processes are investigated. It was found that at sufficiently high injection pressures and low pressures at the right reservoir boundary, the boundary of carbon dioxide hydrate formation can come up with the boundary of methane gas hydrate decomposition. It is also shown that at sufficiently low values of pressure of injection of carbon dioxide and pressure at the right boundary of the reservoir, the pressure at the boundary of hydrate formation of carbon dioxide drops below the boiling pressure of carbon dioxide. In this case, for a consistent description of the considered processes, it is necessary to correct the mathematical model in order to take into account the boiling of carbon dioxide. Maps of possible solutions have been built, which show in what ranges of parameters one or another mathematical model is consistent. Full article

(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment)

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18 pages, 1553 KiB

Open AccessEditor’s ChoiceArticle

Analyzing the Impact of the Renewable Energy Sources on Economic Growth at the EU Level Using an ARDL Model

by Mihail Busu

Mathematics 2020, 8(8), 1367; https://doi.org/10.3390/math8081367 - 14 Aug 2020

Cited by 32 | Viewed by 5825

Abstract

Energy is one of the most important drivers of economic growth, but as the population is increasing, in normal circumstances, in all countries of the world, there is a demand for energy produced from conventional resources. Increasing prices of conventional energy and the negative impact on the environment are two of the main reasons for switching to renewable energy sources (RESs). The aim of the paper is to quantify the impact of the RESs, by type, on the sustainable economic growth at the European Union (EU) level. The research was performed for all 28 EU member states, for a time frame from 2004 to 2017, through a panel autoregressive distributed lag (ARDL) approach and causality analysis. Furthermore, Hausman test was performed on the regression model. By estimating the panel data regression model with random effects, we reveal through our results that RESs, namely wind, solar, biomass, geothermal, and hydropower energy, have a positive influence on economic growth at EU level. Moreover, biomass has the highest impact on economic growth among all RES. In fact, a 1% increase in biomass primary production would impact the economic growth by 0.15%. Based on econometric analysis, our findings suggest that public policies at the EU level should be focused on investment in RESs. Full article

(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment)

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15 pages, 5955 KiB

Open AccessArticle

Effects of the Angled Blades of Extremely Small Wind Turbines on Energy Harvesting Performance

by Junseon Park, Seungjin Lee and Joong Yull Park

Mathematics 2020, 8(8), 1295; https://doi.org/10.3390/math8081295 - 5 Aug 2020

Cited by 2 | Viewed by 2970

Abstract

Low-intensity winds can be useful power sources in the context of energy harvesting. This study aims to enhance the power generation capacity of a super micro wind turbine (SMWT) in low-intensity winds by modifying the blade geometry, which cannot be realized in conventional wind turbines owing to the stress concentration. By controlling the curved angle (θ) in the middle of the blade, the rotor performance can be improved, and the rotor diameter can be reduced to increase installation density. Experimental results indicated that the optimal θ value was 105°, at which the AC voltage was improved by 7.4% compared to that in the case of the basic model with θ = 0°. The maximum electric power output was 9.333 μW and the load resistance was 47.62 kΩ. Moreover, a computational fluid dynamics analysis was performed to clarify the pressure field and streamlines on and around the blade to demonstrate the aerodynamic performance of the SMWT. The proposed blade geometry is one of many possible designs that can enhance extremely small wind turbines for energy harvesting. Full article

(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment)

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21 pages, 4250 KiB

Open AccessArticle

Implicit-Explicit Methods for a Convection-Diffusion-Reaction Model of the Propagation of Forest Fires

by Raimund Bürger, Elvis Gavilán, Daniel Inzunza, Pep Mulet and Luis Miguel Villada

Mathematics 2020, 8(6), 1034; https://doi.org/10.3390/math8061034 - 24 Jun 2020

Cited by 12 | Viewed by 3790

Abstract

Numerical techniques for approximate solution of a system of reaction-diffusion-convection partial differential equations modeling the evolution of temperature and fuel density in a wildfire are proposed. These schemes combine linearly implicit-explicit Runge–Kutta (IMEX-RK) methods and Strang-type splitting technique to adequately handle the non-linear parabolic term and the stiffness in the reactive part. Weighted essentially non-oscillatory (WENO) reconstructions are applied to the discretization of the nonlinear convection term. Examples are focused on the applicative problem of determining the width of a firebreak to prevent the propagation of forest fires. Results illustrate that the model and numerical scheme provide an effective tool for defining that width and the parameters for control strategies of wildland fires. Full article

(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment)

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18 pages, 2849 KiB

Open AccessFeature PaperArticle

Kharitonov Theorem Based Robust Stability Analysis of a Wind Turbine Pitch Control System

by Aitor Saenz-Aguirre, Ekaitz Zulueta, Unai Fernandez-Gamiz, Daniel Teso-Fz-Betoño and Javier Olarte

Mathematics 2020, 8(6), 964; https://doi.org/10.3390/math8060964 - 12 Jun 2020

Cited by 10 | Viewed by 4333

Abstract

Wind energy has recently become one of the most prominent technologies among electrical energy generation systems. As a result, wind-based renewable energy generation systems are incessantly growing, and wind turbines of different characteristics are being installed in many locations around the world. One drawback associated with different characteristics of the wind turbines is that controllers have to be designed individually for each of them. Additionally, stable performance of the wind turbines needs to be ensured in the whole range of their operating conditions. Nowadays, there are many causes for uncertainties in the actual performance of a horizontal axis wind turbine, such as variations in the characteristics of the wind turbine, fabrication tolerances of its elements or non-linearities related to different operating-points. Hence, in order to respond to these uncertainties and ensure the stability of the wind turbine, robust control and stability theories have been gaining importance during recent years. Nevertheless, the use of robust stability analyses with complex wind turbine models still needs to be faced and remarkably improved. In this paper, a stability analysis of the pitch system control of a horizontal axis wind turbine based on the Kharitonov robust stability method is proposed. The objective was to assess the robust stability of a pitch controller in response to uncertainties arising from varying operating conditions of the National Renewable Energies Laboratory (NREL) 5 MW class IIA wind turbine. According to the results, the proposed method could satisfactorily respond to limited variations in the characteristics of the model, but could lack accuracy in cases of bigger variations or employment of high order complex mathematical models. Full article

(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment)

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15 pages, 41300 KiB

Open AccessArticle

Sliding Modes Control for Heat Transfer in Geodesic Domes

by Frank Florez, Pedro Fernández de Cordoba, John Taborda, Miguel Polo, Juan Carlos Castro-Palacio and María Jezabel Pérez-Quiles

Mathematics 2020, 8(6), 902; https://doi.org/10.3390/math8060902 - 3 Jun 2020

Cited by 4 | Viewed by 3442

Abstract

The analysis and modeling of unconventional thermal zones is a first step for the inclusion of low-cost spaces and for the assessment of the environmental impact among areas of human use in warm climates. In this paper, the heat transfer in a geodesic dome located at the University of Magdalena (Colombia) is modeled and simulated. The simulator is calibrated against experimental measurements and used to study the effect of different loads which are regulated by a controller in sliding modes explicitly designed for this case. The closed-loop system is used together with ASHRAE Standard 55 to characterize comfort conditions within the dome and the effect on the overall thermal sensation with increasing the number of occupants. Full article

(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment)

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42 pages, 11353 KiB

Open AccessFeature PaperArticle

A Second-Order Well-Balanced Finite Volume Scheme for the Multilayer Shallow Water Model with Variable Density

by Ernesto Guerrero Fernández, Manuel Jesús Castro-Díaz and Tomás Morales de Luna

Mathematics 2020, 8(5), 848; https://doi.org/10.3390/math8050848 - 23 May 2020

Cited by 11 | Viewed by 3098

Abstract

In this work, we consider a multilayer shallow water model with variable density. It consists of a system of hyperbolic equations with non-conservative products that takes into account the pressure variations due to density fluctuations in a stratified fluid. A second-order finite volume method that combines a hydrostatic reconstruction technique with a MUSCL second order reconstruction operator is developed. The scheme is well-balanced for the lake-at-rest steady state solutions. Additionally, hints on how to preserve a general class of stationary solutions corresponding to a stratified density profile are also provided. Some numerical results are presented, including validation with laboratory data that show the efficiency and accuracy of the approach introduced here. Finally, a comparison between two different parallelization strategies on GPU is presented. Full article

(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment)

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