Mathematical Problems in Engineering

Volume 2015, Article ID 897357, 17 pages

http://dx.doi.org/10.1155/2015/897357

## On the Analytical Approach to Present Engineering Problems: Photovoltaic Systems Behavior, Wind Speed Sensors Performance, and High-Speed Train Pressure Wave Effects in Tunnels

Universidad Politécnica de Madrid, ETSI Aeronáutica y del Espacio, Instituto Universitario de Microgravedad “Ignacio Da Riva” (IDR/UPM), Plaza del Cardenal Cisneros 3, 28040 Madrid, Spain

Received 18 December 2014; Accepted 8 March 2015

Academic Editor: Bin Jiang

Copyright © 2015 Santiago Pindado et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

At present, engineering problems required quite a sophisticated calculation means. However, analytical models still can prove to be a useful tool for engineers and scientists when dealing with complex physical phenomena. The mathematical models developed to analyze three different engineering problems: photovoltaic devices analysis; cup anemometer performance; and high-speed train pressure wave effects in tunnels are described. In all cases, the results are quite accurate when compared to testing measurements.

#### 1. Introduction

The evolution of engineering has provided engineers with better tools, that is, better calculation possibilities (more sophisticated numerical approximations and computers) and better testing equipment. However, the classical approach, based on mathematical models directly linked to the physics of the analyzed phenomena, should not be left aside. Even in the rush to obtain a suitable solution to an engineering problem, the analytical approximations can state the limits to accept the solution from more complex calculations. Besides, a very good explanation for the need of analytical methods is offered by Simmonds and Mann [1] in relation to the use of perturbation methods, as they “produce analytic approximations that often reveal the essential dependence of the exact solution on the parameters in a more satisfactory way than does a numerical solution.” It cannot be more properly expressed.

At the IDR/UPM Institute different analytical models have been developed along the last decades to study quite complicated problems. As an example, the stability analysis of liquid bridges has been thoroughly studied since the late 70s, when the effect of microgravity on the liquid bridges was brought to Professor Da Riva’s and Professor Meseguer’s attention [2–9]. Another good example is the analysis of train-induced pressure effects on pedestrian and sign panels carried out by Professor Sanz-Andrés [10–12].

At present, several problems are analyzed at the IDR/UPM Institute under the guidance of Professor Meseguer. In the following sections the analytical models developed to study three different phenomena are described. In Section 2, the analytical models used to study solar cells/panels are described. These models have been developed to analyze the solar panels of the UPMSat-2 satellite and model its power subsystem. This 50 kg microsatellite is planned to be launched by the end of 2015/beginning of 2016. In the present work, a new and a bit more complex analytical approximation to solar panels behavior has been developed. This approximation is based on a 2-diode equivalent electrical circuit, which is obviously less simple than the one based on the 1-diode equivalent electrical circuit already used at the IDR/UPM to estimate photovoltaic power supply [13–16].

Additionally, a model used to study the rotor aerodynamics of the cup anemometer is described in Section 3. The cup anemometer, invented by J. T. R. Robinson in the 19th century, is today the standardized instrument regarding the wind power performance measurements [17]. It has been extensively studied and optimized during the 20th century (see in [18] a thorough review of the literature) and, at present, it is the reference instrument in relation to wind production forecasting in the field and wind turbine performance control. Since 2009 the performance of this instrument is studied at the IDR/UPM Institute [18–26]. As a result, a quite accurate analytical method has been developed to analyze the anemometer performance based on the cups’ aerodynamics, including the effect of perturbations. In the present work, the analytical model is used to study the equilibrium positions (hereinafter referred as “flag” positions) of a damaged anemometer with one cup missing. This is not a rare case, bearing in mind that 30% of mast-mounted anemometers return for recalibration in bad shape [27].

Finally, a mathematical model developed to study the effect of pressure waves in high-speed railway tunnels is included in Section 4. This model can be used in preliminary design of tunnel sections and to evaluate tunnel costs and also to find the optimal location for damper devices depending on the trains and the travel speeds during the pass through the tunnel. Conclusions are summarized in Section 5.

#### 2. Solar Cell/Panel Modeling

The behavior of a solar cell is normally modeled by a 1-diode and 2-resistor electric circuit (see Figure 1); this simple approximation captures the nature of the more important physical effects related to the photovoltaic conversion of the light to current. However, this model that fits quite correctly the behavior of monocrystalline and polycrystalline silicon cells has certain deviation when the solar cell is not operating at standard test conditions [28, 29]. It should also be fair to say that today this model can be considered a simplification from the more accurate 2-diode and 2-resistor electric circuit model (see Figure 1), which was proposed by Wolf and Rauschenbach in 1961 to introduce the effect of recombination of electrons and holes in the depletion region [30]. These two models have been successfully applied to model solar panels made of cells grouped in parallel series [31, 32]. In the following subsection, the 1-diode/2-resistor equivalent circuit model is firstly described as an introduction to the second subsection, in which the 2-diode/2-resistor equivalent circuit model developed in the present work is presented.