Study of a zero-emission airship transport system based on the geostrophic flight concept

Titulació: ENGINYERIA AERONÀUTICA Alumne : FRANCESC BETORZ MARTÍNEZ Títol PFC: STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT Director del PFC: Dr. ROBERTO FLORES LE ROUX Convocatòria de lliurament del PFC: GENER 2011 Contingut d’aquest volum: -TECHNICAL REPORT- -ANNEX- STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 2 Acknowledgements The realization of this study has been possible thanks to many people who supported me throughout the last months: First of all I would like to thank my supervisor Dr. Roberto Flores for all his support. He showed such high enthusiasm in my field of study and thus provided me with extra motivation. Moreover, I want to point out that his broad knowledge of engineering was a significant key for the successful development of this study. Over all, without his inspiration and encouragement it would not have been possible. I wish to express my appreciation to the following members of the Servei Meteorològic de Catalunya: Mr. Jordi Cunillera, Mr. Abdel Sairouni and Mr. Manel Bravo. They kindly provided me with weather forecast data for the year 2008 without any costs for my academic research and further assisted me with explanations about technical aspects of weather forecasting. I want to thank them for their positive attitude towards my project and for always being willing to help me out when I had some questions. Additional data was provided by the appreciated expert balloonist and engineer Mr. Josep M. Lladó. Only thanks to his data of the balloon track I was able to perform the wind model data verification. Moreover, I would like to express my appreciation to the expert balloonist Uwe Schneider who enriched my work by giving good advice about balloon navigation and weather forecast data. Of course I would like to thank my family, including my parents and my brother and sisters, for all their love and for supporting me always. I want to mention especially my mother who I would like to thank for her love, for standing beside me at any time and encouraging me constantly with great patience. My sincere thanks go to my fiancée Linda Moser for her contribution due to her constant support and motivation, her interest in my study, and for proof-reading of my project. I want to particularly emphasize that she was the initial reason to inspire me to do this project because in summer 2009 she enabled me to fly in an airship for the very first time in my life. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 3 I also received a lot of motivation from my department colleagues who worked together with me at ETSEIAT where I was employed as a technician of the aerospace laboratory. Thank you for your help and support. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 4 Table of contents Acknowledgements ...................................................................................................... 2 Table of contents .......................................................................................................... 4 List of figures ................................................................................................................. 6 List of tables ................................................................................................................... 9 Glossary ......................................................................................................................... 10 1 Introduction .............................................................................................................. 12 1.1 Objective and motivation ............................................................................ 12 1.2 Geostrophic flight concept .......................................................................... 12 1.3 Airships: a new return? ................................................................................ 13 2. Antecedents ............................................................................................................ 15 2.1 Airship electric flight...................................................................................... 15 2.2 Geostrophic flight ........................................................................................... 18 3. Atmospheric characterization ........................................................................... 19 3.1 Planetary boundary layer and free atmosphere ................................. 19 3.2 Numeric weather forecast models ........................................................... 21 3.3 MM5 model ........................................................................................................ 22 3.4 Weather forecasting at Servei Meteorològic de Catalunya ............ 23 3.4.1 Data used in the study ......................................................................... 24 3.4.1.1 Data format .................................................................................... 26 3.4.1.2 Wind data validation ................................................................... 27 4. Solar power ............................................................................................................ 31 4.1 Solar angles ...................................................................................................... 31 4.2 Solar irradiance. Available power ............................................................ 33 4.3 Components of the solar power system for airship propulsion .... 37 4.3.1 Solar cells .................................................................................................. 37 4.3.2 Energy storage ........................................................................................ 39 4.3.3 Electric motor & power conditioner ................................................. 40 5. Solar-powered flight ............................................................................................ 42 STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 5 5.1 Fundamental equations ................................................................................ 42 5.2 Projected area calculation ........................................................................... 43 5.3 Conceptual design of “Zero”: a recreational solar airship .............. 45 5.3.1 Reference airship .................................................................................... 45 5.3.2 Requirements and configuration....................................................... 49 5.3.3 Size and weights ..................................................................................... 50 5.3.4 Performance ............................................................................................. 53 5.3.4.1 Solar speed ..................................................................................... 53 5.3.4.2 Emergency speed ......................................................................... 56 5.3.4.3 Range and autonomy.................................................................. 57 6. Geostrophic flight .................................................................................................. 58 6.1 General concept .............................................................................................. 58 6.2 Guaranteed covered area ............................................................................ 60 6.3 Trajectory calculation ................................................................................. 61 6.3.1 General considerations ......................................................................... 61 6.3.2 Vertical trajectory ................................................................................... 62 6.3.3 Horizontal trajectory ............................................................................. 69 6.3.4 Determination of the atmospheric parameters along the trajectory ................................................................................................................... 72 6.3.4.1 Horizontal airship coordinates in the grid ........................... 72 6.3.4.2 Horizontal coordinates not coinciding with the grid points .......................................................................................................................... 75 6.3.5 Matlab code ............................................................................................... 79 7. Case study ............................................................................................................... 81 7.1 GCA for the “Zero” airship .......................................................................... 81 8. Environmental implications ............................................................................... 95 9. Conclusions and further recommendations ................................................ 96 10. Budget .................................................................................................................... 98 11. References ............................................................................................................ 99 ANNEX: Matlab Code STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 6 List of figures Figure 1. Airship trajectory during a geostrophic flight in the northern hemisphere Figure 2. Tissandier brothers electric flight in 1883 [2] Figure 3. Airship “La France” in 1884 [2] Figure 4. Khoury’s Sunship. 1978 [3] Figure 5. Planetary boundary layer structure [6] Figure 6. Skill of the 36 hour and 72 hour 500 hPa forecasts produced at NCEP [8] Figure 7. Geographical coverage of the three domains [10] Figure 8. Covered area, 12-km SMC’s domain, mesh of 69x69 points Figure 9. 12-hour short term prediction Figure 10. Balloon trajectory [Google Earth] Figure 11. Flight altitude profile Figure 12. Real and predicted balloon trajectories Figure 13. Distance between real and computed final points Figure 14. Solar angles (altitude A and azimuth Z) [12] Figure 15. Latitude, hour angle and declination [12] Figure 16. Sun’s declination during solstices [12] Figure 17. Direct solar irradiance in Barcelona at 2000 meters altitude Figure 18. Direct solar irradiance in Barcelona at sea level Figure 19. Projected area normal to the solar flux Figure 20. Main components of the solar power system for airship propulsion [4] Figure 21. Solar cell efficiency evolution (NREL laboratory) [16] Figure 22. Amorphous silicon cells on a flexible polymer substrate [17] STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 7 Figure 23. Permanent magnet brushed DC motor D135RAG [22] Figure 24. Determination of solar cells’ projected area Figure 25. AU-12 during takeoff [24] Figure 26. Flight altitude as function of effective payload (ISA conditions) Figure 27. Performance comparison between airship AU-12 (blue) and first iteration for “Zero” (red). Figure 28. Performance comparison between airship AU-12 (blue) and second iteration for “Zero” (red). Figure 29. Solar cells’ projected area for airship “Zero” Figure 30. Maximum speed of airship “Zero” Figure 31. Airship geostrophic flight Figure 32. Mean wind speed at 850, 795 and 700 hPa (year 2008) Figure 33. GCA example: ZERO_SP_41.38_2.18_2000_02_1100_13102008 Figure 34. “Adapted” helium volume Figure 35. Constant helium volume Figure 36. Second order modified Euler method Figure 37. Flight altitude between geopotential altitudes Figure 38. Wind velocity for grid points at 2000 m. 12:00 h, 19th November 2008 Figure 39. Horizontal trajectory coordinates among grid points Figure 40. Element shape in local axes Figure 41. Isoparametric transformation Figure 42. Wind velocity interpolation Figure 43a. ZERO_SP_41.38_2.18_2000_03_1130_15012008 Figure 43b. Range as function of the airship’s azimuth Figure 44a. ZERO_SP_41.38_2.18_2000_03_1130_15022008 STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 8 Figure 44b. Range as function of the airship’s azimuth Figure 45a. ZERO_SP_41.38_2.18_2000_03_1130_15032008 Figure 45b. Range as function of the airship’s azimuth Figure 46a. ZERO_SP_41.38_2.18_2000_03_1130_15042008 Figure 46b. Range as function of the airship’s azimuth Figure 47a. ZERO_SP_41.38_2.18_2000_03_1130_15052008 Figure 47b. Range as function of the airship’s azimuth Figure 48a. ZERO_SP_41.38_2.18_2000_03_1130_15062008 Figure 48b. Range as function of the airship’s azimuth Figure 49a. ZERO_SP_41.38_2.18_2000_03_1130_15072008 Figure 49b. Range as function of the airship’s azimuth Figure 50a. ZERO_SP_41.38_2.18_2000_03_1130_15082008 Figure 50b. Range as function of the airship’s azimuth Figure 51a. ZERO_SP_41.38_2.18_2000_03_1130_15092008 Figure 51b. Range as function of the airship’s azimuth Figure 52a. ZERO_SP_41.38_2.18_2000_03_1130_15102008 Figure 52b. Range as function of the airship’s azimuth Figure 53a. ZERO_SP_41.38_2.18_2000_03_1130_15112008 Figure 53b. Range as function of the airship’s azimuth Figure 54a. ZERO_SP_41.38_2.18_2000_03_1130_15122008 Figure 54b. Range as function of the airship’s azimuth STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 9 List of tables Table 1. Comparison of planetary boundary layer and free atmosphere characteristics [6] Table 2. Comparative values of the real path and the predicted balloon trajectory Table 3. AU-12 airship technical data [24] Table 4. Au-12 net weight breakdown Table 5. “Zero” propulsion system weight. First iteration Table 6. OEW breakdown for airship “Zero”. Second iteration Table 7. “Zero” airship main dimensions and motorization Table 8. Maximum daily differences in T and P for each month (2008) Table 9. Values of and Table 10. GCAs’ mean range Table 11. PFC cost STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 16 This was the first recorded electric-powered flight, though it was not to be the last. One year later in 1884 Charles Renard and Arthur Krebs (inventors and military officers in the French Army Corps of Engineers) duplicated this feat with their electric-powered airship La France, which notably was also the world's first fully-controllable airship (figure 3). La France was the first airship that could return to its starting point in light wind conditions and represented a vast improvement over earlier models. It was 50.3 meters long, its maximum diameter was 8.2 meters, and it had an envelope volume of 1.869 cubic meters. Like the Tissandiers' airship, an electric battery-powered motor propelled La France, but this one produced 5.6 kilowatts, five times the available power in Tissandier’s airship. The first flight of La France took place on August 9, 1884. After a fully controlled flight of 8 kilometers and 23 minutes, Renard and Krebs landed successfully at the same point where they had taken off. Although the batteries limited its flying range, La France demonstrated that controlled flight was possible if the airship had a sufficiently powerful lightweight motor [1]. Figure 3. Airship “La France” in 1884 [2] STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 17 Despite the success achieved by airship La France, the rapid improvement of internal combustion engines in the late nineteenth century made them to become the most suitable for airship propulsion. These new engines offered much higher power to weight ratios than their equivalent heavy electric motors. Moreover, the range and autonomy using internal combustion engines were not limited as they were when using batteries. Thus, La France was the last manned electricpowered airship. Since then all manned airships incorporate internal combustion engines. Concerning solar powered manned airships the first theoretical proposal was made by Khoury and Mowforth in 1978 [3]. In their work the concept of the “Sunship” was introduced. The idea basically consisted in covering the envelope surface of a conventional helium airship with an array of flexible thin film solar cells (figure 4). In theory the solar cells generate enough power to feed the two main electric motors. Khoury justifies his idea with these words: “Solar energy is attractive in an environmental conscious age. Sunlight is a renewable, free, nonpolluting and non-inflammable fuel. A solar-powered airship would not require refuelling when operating in remote sunny areas of the world or at high altitudes. A solar-powered airship is defined as an airship that attains its power for propulsion primarily from solar energy, albeit in conjunction with on-board energy storage for operations at night and in windy conditions” [4]. Figure 4. Khoury’s Sunship. 1978 [3] STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 18 The proposed Sunship would be 80 meters long with a diameter of 23 meters and a total hull volume of about 22.000 cubic meters. Considering the maximum direct solar energy available at an altitude of 1000 meters at solar noon and using reasonable values for the different efficiencies involved, the theoretical maximum attainable speed would be 83 km/h. The maximum speed would increase to 101 km/h if diffuse and reflected solar radiation would also be considered for an airship totally covered with solar cells [4]. Besides Khoury’s theoretical work, no manned solar powered airships have been built up to date. 2.2 Geostrophic flight The use of the winds for air navigation goes back to the early era of balloons and airships. In 1910 the famous German meteorologist Hugh Hergesell already anticipated in his book " With Zeppelin to Spitzbergen" that for a hypothetical trip to the North Pole by airship "a special method of navigating should be developed where the track to follow was chosen by taking into account a weather forecast chart, in order to use the favorable winds instead of going against them” [5]. Subsequently and until the disappearance of the great airships in the 1940s the use of weather forecast data was common practice in airship route planning. However, the available weather forecast in the first half of the twentieth century was of poor quality. Until the advent of computers and numerical models it was not possible to make accurate short and medium term weather forecasts. Thus, a type of navigation like the one proposed in this study would have been impossible in the golden era of the airships due to the lack of reliable weather forecast data. Nowadays the situation is paradoxically the opposite. Meteorological numerical models are able to produce accurate weather predictions but, as mentioned in chapter 1, airships play a secondary role in the aviation industry and are no longer used for passenger or cargo transport. The aim of this project is to study the feasibility of a recreational two-seat zeroemission airship by combining the idea of the solar airship with a navigation based on accurate weather forecasting data. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 19 3. Atmospheric characterization 3.1 Planetary boundary layer and free atmosphere The troposphere is the first layer of the atmosphere and extends from the ground up to an average altitude of 11 kilometers. The troposphere itself can be divided in two zones: a planetary boundary layer (PBL) near the Earth’s surface and the free atmosphere above it. According to Stull’s definition, “the planetary boundary layer can be defined as the part of the troposphere that is directly influenced by the presence of the Earth’s surface, and responds to surface forcings with a timescale of about an hour or less” [6]. The planetary boundary has a complex structure (figure 5) and its thickness is variable in time and space, ranging from hundreds of meters to a few kilometers. One of the characteristics that make this boundary layer different from the rest of the atmosphere is its turbulent nature. The wind speed suffers from rapid fluctuations and the vertical mixing is strong. Table 1 summarizes the most important differences between the PBL and the free atmosphere. Table 1. Comparison of planetary boundary layer and free atmosphere characteristics [6] Above the PBL, in the free atmosphere, winds are almost geostrophic. Geostrophic wind is a theoretical wind which results from the equilibrium between Property Planetary boundary layer Free atmosphere Turbulence Almost continuously turbulent over its whole depth Sporadic clear air turbulence in thin layers of large horizontal extent Friction Strong drag against the Earth’s surface. Large energy dissipation Small viscous dissipation Dispersion Rapid turbulent mixing in the vertical and horizontal Small molecular diffusion. Often rapid horizontal transport by mean wind Winds Near logarithmic wind speed profile in the surface layer. Subgeostrophic, crossisobaric flow common Winds nearly geostrophic Thickness Varies between 100m to 3km in time and space. Diurnal oscillations over land Less variable, thickness ranging from 8 to 18km. Slow time variations STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 20 the pressure gradient force and the Coriolis force. Geostrophic wind is parallel to the isobars and assumes that there is no friction and that the isobars are perfectly straight. A better approximation of the real wind in the free atmosphere is the socalled gradient wind where, besides the geostrophic balance, also the centrifugal force due to the curvature of the isobars is taken into account. Figure 5. Planetary boundary layer structure [6] Weather conditions play a very important role in airships’ performance given the relative low speed and high volume of these aircrafts. Taking into account the characteristics of the atmospheric layers mentioned above, an airship flying in the free atmosphere should experience the following advantages with respect to flying in the PBL: - Less turbulence and wind gusts which guarantee a more comfortable and stable flight. - Better wind prediction since numerical weather forecast models better predict the winds in the free atmosphere due to the turbulent nature and friction forces present in the PBL. - Flying above the planetary boundary layer avoids fog and low clouds and allows better visibility. - Winds are more constant in intensity and direction in the free atmosphere and therefore, allow the implementation of effective flying strategies. In this study, an airship flight whose trajectory is determined in advance using accurate weather forecast data and that is performed mainly over the planetary boundary layer is defined as a geostrophic flight. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 21 3.2 Numeric weather forecast models Weather forecasting is the core element for the proposed geostrophic flight, where the idea is to take advantage of the predicted wind fields and use them favorably for navigation. More specifically (see chapter 6), the atmospheric variables that must be accurately predicted in order to forecast the trajectory of an airship are temperature, pressure distribution and the zonal and meridional winds. Zonal wind is the wind component along the local parallel whereas meridional wind is the wind component along the local meridian. Nowadays numerical weather models are the fundamental tool for accurate weather forecasting. Although the first efforts to accomplish regular weather forecasts were done almost a century ago, it wasn't until the advent of computer simulation that regular weather predictions became feasible [7]. Numerical weather prediction simulates the evolution of the main meteorological parameters that determine the state of the atmosphere by using mathematical models. These models are based on the physical laws that describe the evolution of the atmosphere and constitute a system of equations. Given the complexity of these equations and the huge volume of data involved in the calculations, they can only be solved approximately using computers. The maximum time range for which these equations are solved is known as the prediction horizon. Despite of the atmosphere being a continuous medium, these equations are solved only at certain points to reduce computation time. All these points where the equations are solved constitute a three-dimensional mesh. If the distance between these points (called grid) decreases, the model is capable of simulating meteorological phenomena of smaller scale and has therefore a higher resolution. However, it also means an increase of the computing time needed to run the simulation. On the other hand, the effects of physical processes that the model is unable to solve explicitly are estimated by using parameterizations. Parameterizations are a set of mathematical formulas that approximately describe atmospheric smaller scale phenomena (clouds, radiative fluxes, turbulence, convection, etc.). Numerical weather predictions have improved dramatically since their origin. The remarkable progress in forecasting over the past 50 years is illustrated by the record of skill of the 500 hPa forecasts produced at the National Centers for Environmental Prediction (NCEP) [8]. Forecast skill scores, expressed as percentages of perfect forecasts, have improved steadily over the past 50 years and each introduction of a new prediction model has resulted in further improvement (figure 6). For a 36 hour forecast the skill score was almost 80% in STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 22 2004. This percentage increases for very-short term predictions which are to ones to be used in geostrophic flight. Although all numerical weather forecast models are based on the same physical laws, there are differences in their mathematical formulation and also in the numerical techniques which are used to solve the equations. Moreover, a distinction is made between the models whose geographic area or domain covers the whole planet (global models) and the models that only cover a specific area (limited-area models). The MM5 model is a limited-area model. Figure 6. Skill of the 36 hour and 72 hour 500 hPa forecasts produced at NCEP [8] 3.3 MM5 model The PSU/NCAR (Pennsylvania State University/National Center for Atmospheric Research) model (known as MM5) is a limited-area, hydrostatic or nonhydrostatic, terrain-following sigma-coordinate model designed to simulate or predict mesoscale and regional-scale atmospheric circulation [9]. These include phenomena that occur at spatial scales ranging from a few to several hundred kilometers, such as storms, breezes, or frontal systems amongst others. The model is supported by several pre- and post-processing programs, which collectively form the MM5 modeling system. The MM5 modeling system software is provided free of charge and mostly written in FORTRAN. It is continuously STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 23 being improved by contributions from users at several universities, laboratories and weather forecast agencies like “Servei Meteorològic de Catalunya”. The MM5 model was initially developed at Pennsylvania State University in the early 70's. Since then, it has undergone many changes designed to broaden its usage. These include: (i) A multiple-nest capability. (ii) Non-hydrostatic dynamics, which allow the model to be used at a fewkilometer scale. (iii) Multitasking capability on shared- and distributed-memory machines. (iv) Four-dimensional data-assimilation capability (v) More physics options. Since the MM5 is a regional model, it requires an initial condition as well as lateral boundary conditions to run. The initial atmospheric state is determined by performing a process of data assimilation before the numerical models begin their calculations. This process consists in feeding the model with data from several meteorological observations (rawinsondes, surface stations, satellite, radar, etc.) that describe the initial state of the atmosphere. 3.4 Weather forecasting at Servei Meteorològic de Catalunya Servei Meteorològic de Catalunya (SMC) is the Catalan agency for weather forecasting [10]. Currently, SMC carries out several numerical simulations using the MM5 model every day in order to perform the weather forecast for Catalonia. The three different domains (limited areas) in which the model is run are shown in figure 7. The biggest domain has a grid of 36 km, the medium one of 12 km and the smallest one of 4 km. The model is run twice a day, at 00 and 12 UT, for the three different domains but only produces graphical outputs of the first two domains, the 36 and 12 km grid, which are distributed through the SMC website. First of all an initial simulation is carried out in the larger domain (36 km grid) and a 72-hour prognosis is completed. Due to the fact that this is a limited-area model, additional data from a global model is needed in order to determine the initial meteorological variables in the domain boundaries. The initial conditions are improved with the assimilation of meteorological observational data from rawinsondes and surface stations, in order to match the initial state as close as possible to reality. Another simulation is performed using the 12-km grid domain and a 48-hour prognosis. The initial and boundary conditions for this simulation are derived out of the output data obtained from the previous simulation on the large domain. As the grid is smaller in this case it is able to simulate atmospheric STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 24 phenomena that occur at smaller scales. In addition, short-term predictions (12- hour prognosis) are computed eight times a day (00, 03, 06, 09, 12, 15, 18 and 21 UTC) using the same 12-km grid domain. Figure 7. Geographical coverage of the three domains [10] 3.4.1 Data used in the study The weather forecast data used in this project to study the feasibility of the geostrophic flights was supplied by SMC. The supplied data corresponds to the 12-hour short-term prediction data in the 12-km grid domain and for the entire year 2008. Due to the dimensions of the covered area and the grid distance, the mesh is composed by 69x69 points (figure 8). These short-term forecast predictions are computed every 3 hours and predict the following 12 hours. It is assumed that the first 3 hours of each run represent the most accurate prediction, and therefore only data from the first 3 hours of each run is used (figure 9). The computation was carried out for the entire year 2008 and for 5 different pressure levels (on the ground, 950 hPa, 850 hPa, 700 hPa and 500 hPa). STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 25 Figure 8. Covered area, 12-km SMC’s domain, mesh of 69x69 points Thus, every 3 hours the temperature (T), zonal and meridional wind (u,v) and geopotential altitude (h) were computed at each of those pressure levels mentioned above. Figure 9. 12-hour short term prediction STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 32 The solar altitude A can be determined using the following formula [4]: where is the latitude of the observer P in degrees, is the hour angle time after solar noon and is the Sun’s declination which represents the seasonal variation in the Sun’s apparent motion (figure 15). Figure 15. Latitude, hour angle and declination [12] The hour angle can be calculated in degrees as: H is 0 at noon, where the solar altitude is at its maximum. The Sun’s declination N varies between +23.45º at summer solstice and -23.45º at winter solstice (figure 16). Measuring the time of the year in days from the spring equinox the declination can be calculated in degrees by: Finally, the solar azimuth Z can be calculated as [4]: STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 33 Figure 16. Sun’s declination during solstices [12] 4.2 Solar irradiance. Available power The solar energy reaching the upper limits of the Earth’s atmosphere has a yearly average value of 1.366 W/m2. This value is called the solar constant D0. Due to the fact that the Earth’s orbit is slightly elliptical the solar constant varies by ±3.4 percent throughout the year with the maximum irradiance occurring at the perihelion (Earth closest to the Sun) and the minimum at the aphelion. When travelling through the Earth’s atmosphere, direct solar radiation is progressively attenuated until it reaches the ground level. The degree of attenuation depends on the amount of air mass encountered by the solar rays in its path through the atmosphere, which mainly depends on the solar altitude. The longer the path length of the solar rays through the atmosphere the greater the attenuation. Likewise the lower the solar altitude the greater is the solar irradiance attenuation. Direct solar radiation at sea level and for clear sky conditions can be expressed as a function of the solar altitude as [13]: where D0 is the solar constant, A is the solar altitude and c=0.357 and s=0.678 are two empirical constants. This equation is valid only for solar altitudes greater than 20º. Direct solar flux increases with altitude above sea level and it reaches the solar constant outside the atmosphere. In order to take into account this effect, the previous equation has to be modified [14]: STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 34 where h is the altitude above sea level in kilometers and a=0.14 is another empirical constant. This formula is valid only for the first few kilometers above sea level. The total solar radiation that reaches a point on the Earth’s surface is the sum of the direct, the diffused and the reflected solar radiation. Diffuse solar radiation is the portion of solar radiation that is scattered downwards by the molecules of the atmosphere. During clear days, the magnitude of diffuse radiation is no greater than 10% of the total solar radiation received at the Earth’s surface. However, clouds have a significant influence on diffuse radiation and only this kind of radiation may reach the Earth's surface during extremely cloudy days. When the solar radiation irradiates upon a surface which is opaque like clouds and ground surface, a portion of the radiation is absorbed and the remaining portion is reflected. The amount of reflected radiation depends on the albedo or surface reflectance of the object. Taking into account the previous formulas, direct solar radiation for clear sky conditions can be calculated for a given latitude, day of the year, local solar time and altitude above sea level. Figure 17 and 18 show the daily direct solar radiation for a point located over Barcelona (latitude 41ºN) at 2000 meters of altitude and at sea level, respectively. The blue, green and red lines correspond respectively to the summer solstice, the winter solstice and equinoxes. Any other day of the year will be represented by a line which shall be between the lines of the summer and winter solstice. The maximum direct solar irradiance through the year is obtained at noon during the summer solstice. At 2000 meters altitude over Barcelona the maximum direct solar irradiance is 1.062 W/m2 whereas at sea level is 944 W/m2. It is also remarkable that at 2000 meters altitude the direct solar radiation at noon is always greater than 900 W/m2 in clear sky conditions. As seen in figures 17 and 18 and as mentioned above the altitude has a significant influence on the value of direct radiation. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 35 Figure 17. Direct solar irradiance in Barcelona at 2000 meters altitude Figure 18. Direct solar irradiance in Barcelona at sea level 500 600 700 800 900 1.000 1.100 46810 12 14 16 18 20 Direct solar irradiance (W/m2) Hour 150 250 350 450 550 650 750 850 950 4 6 8 10 12 14 16 18 20 Direct solar irradiance (W/m2) Hour STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 36 Finally, the available solar energy reaching a surface S can be obtained by multiplying the incident direct solar radiation Dh by the projected area Sp normal to the solar flux (figure 19). Figure 19. Projected area normal to the solar flux STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 37 4.3 Components of the solar power system for airship propulsion The main components of a solar power system for airship propulsion are the solar cells, the energy storage elements, the electric motor and the power conditioner [4] Figure 20. Main components of the solar power system for airship propulsion [4] 4.3.1 Solar cells Solar cells are the main element of the system since they are responsible for transforming the incident solar radiation into electric power. The photovoltaic effect is the basis for the conversion of solar radiation into electricity in solar cells, which are made of a semiconductor material. Conversion efficiency is the most important parameter of a solar cell and is defined as the ratio of the electrical power output of the cell to the solar energy input onto the cell: STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 38 As shown in figure 21 solar cell efficiencies have increased significantly since the 1970s. Although laboratory specimens have reached efficiencies around 40%, typical silicon commercial solar cells have an efficiency of around 12% and those using the much more expensive gallium arsenide have an efficiency not greater than 20% [15]. Figure 21. Solar cell efficiency evolution (NREL laboratory) [16] Flexible thin film solar cells are the ones suitable for electrical airship propulsion. Their mechanical flexibility and light weight make them ideal for covering the top surface of the airship’s envelope. Commercial amorphous silicon cells have a proven efficiency of 12% (figure 2) and a specific power of 1000 W/Kg [17]. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 39 Figure 22. Amorphous silicon cells on a flexible polymer substrate [17] 4.3.2 Energy storage Although electric power for solar flight is obtained mainly through the solar cells, it is essential to have an electric storage system on board not only to feed the instrumentation and navigation systems on board but also to overcome the following situations: - Failure of the solar cell system: Should a malfunction occur in the solar cell system the energy storage devices on board must be able to supply enough energy and during a sufficient period of time in order to guarantee a safe landing. - Temporary little incident solar radiation due to clouds or other causes: In this case it will be necessary to partially feed the engine with electrical energy from the batteries during the time period of low solar irradiation. - Occasional needs for maximum power: In general the highest power supplied by the solar cells will be less than the maximum power that can be delivered by the electric motors. Occasionally there may be situations where the maximum engine power is needed (e.g. strong wind gusts, collision avoidance, etc.). The additional energy to the solar power in order to reach maximum power comes from the energy storage system. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 40 The most suitable system for energy storage on an airship are rechargeable batteries. The fundamental parameter that determines the performance of a battery is its specific energy. Typical values of specific energy range from 40 Wh/kg for lead acid batteries to 350 Wh/kg for advanced lithium sulfur batteries [18] [19]. Batteries have been used successfully in recent manned and unmanned airplane electric flights. In 2007 the electric airplane “Electra” performed a 48-minutes manned flight using lithium polymer batteries with a specific energy of 200 Wh/Kg [20]. On the other hand, in 2008 the unmanned electric airplane “Zephyr-6” performed a 3-day flight using advanced lithium sulfur batteries with a specific energy of 350 Wh/kg [21]. Batteries are the heaviest element of the solar propulsion system and therefore they must be accurately adopted according to the operational requirements of the aircraft. 4.3.3 Electric motor & power conditioner The power conditioner is responsible for regulating the electrical power supplied to the engine under the pilot’s demand and for feeding the auxiliary navigation systems. It is also responsible for the global management of the variable direct current produced by the solar cells and the energy stored in the batteries. Finally, the electric motor drives the propeller producing the desired thrust. The characteristics required in an airship electric motor are: low weight, high electrical performance, minimal maintenance and low cost. DC electric motors are the most suitable given their easy regulation and due to the nature of the available power on board (direct current). The high weight of conventional DC motors has been dramatically reduced since steel of the frames has been replaced by aluminum. Additionally, permanent magnet motors have also significantly improved the performance of DC motors and are potential candidates for electric airship propulsion. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 41 Figure 23. Permanent magnet brushed DC motor D135RAG [22] Figure 23 shows the permanent magnet brushed DC motor D135RAG of the Lynch Motor Company. This engine, which was used successfully by the electric airplane “Electra” in 2007 [20] develops a rated power of 16.8kW, weights approximately 11 kg and has a peak efficiency of 91% [22]. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 48 altitude (payload 2) is obtained by completely filling the ballonets with air at sea level. In this case, as the airship ascends, the helium gas expands and the ballonets diminish their volume. The maximum flight level of payload 2 will be achieved when the ballonets are completely deflated and the helium gasbags are completely expanded occupying the entire envelope (this altitude is the so-called “pressure height”). Considering ISA conditions the values of payload 1 and 2 for the AU-12 airship are: The maximum flight level for payload 2 under ISA conditions can be obtained knowing that at the pressure height the helium has expanded completely: For any other payload having a mass between payload 1 and 2 the maximum flight altitude can be calculated by interpolating the altitudes obtained above for payload 1 and 2 (figure 26). Figure 26. Flight altitude as function of effective payload (ISA conditions) 0 500 1000 1500 2000 2500 3000 0100 200 300 400 500 Max. flight altitude (m) Payload (kg) STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 49 Additionally, it has to be mentioned that the “effective” payload of the airship can be a value lower than payload 2 (e.g. only the pilot as payload). However, as stated above, airships operate in almost neutral buoyancy conditions (with slightly positive static heaviness). It means that if the “effective” payload is lower than payload 2, ballast has to be added as additional payload in order to guarantee slightly positive static heaviness conditions. 5.3.2 Requirements and configuration The requirements for the “Zero” airship are the following: - Two-seat airship with at least the same payload capability as airship AU- 12. - Same or better flight altitude capability as airship AU-12. - Same length/diameter ratio as airship AU-12. - Electrical propulsion system with zero emissions of pollutants. - Back-up power system in case of malfunction of the primary feeding electric system. The back-up power system must allow the normal operation of the electric engines at its nominal rated power for at least 1 hour. Under these requirements the proposed configuration for the “Zero” airship is the following: - Solar powered airship based on the “Sunship” concept [3] with a grid of thin film flexible solar cells (acting as a primary feeding electric system) covering the whole upper surface on the airship’s envelope. The selected flexible cells are amorphous silicon cells mounted on a flexible polymer substrate. These cells have a proven efficiency of 12% and a specific power of 1000 W/kg [17]. - 2 permanent magnet brushed DC motors D135RAG of the Lynch Motor Company [22]. Each motor has a nominal rated power of 16.8 kW and a peak power of 34.3 kW. The weight of each motor is 11 kg. - Back-up power system based on secondary batteries. The selected batteries are lithium polymer batteries with a specific energy of 200 Wh/kg [26]. - Size and dimensions of the envelope and control surfaces are proportional to those of the airship AU-12. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 50 5.3.3 Size and weights The determination of the size and weight of the “Zero” airship given its requirements and general configuration is done by an iterative process. The starting point for the sizing of “Zero” is to consider an airship with the original size and hull dimensions of the AU-12 airship. In this scenario the weights of the hull group, tail group and gondola do not change. The weight of the solar propulsion system can be estimated taking into account the following considerations: - The weight of the solar cells is proportional to the upper surface of the airship’s envelope. The considered surface density for the solar cells is 0.13 kg/m2 [17]. - The weight of the electric motors and the power conditioner are provided by the manufacturers [22]. - The weight of the batteries is chosen to fit the design requirements given its specific energy of 200 Wh/kg. - The rest of the components are estimated taking into account the same criteria used for airship AU-12. Table 5 shows the weight breakdown for the propulsion system in the first iteration. Sub-component Mass (kg) Installed engine 24.0 Flexible solar cells 47.0 Collector grid network 14.1 Power conditioner 35.0 Ducted propeller 10.0 Duct 30.0 Transmission system 10.0 Vector system 4.8 Secondary batteries 100 TOTAL PROPULSION SYSTEM WEIGHT 274.9 Table 5. “Zero” propulsion system weight. First iteration In this case no fuel weight has to be added and therefore the operational empty weight is: STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 51 And payload 1 and 2 for the first iteration are: Both values are lower than the original ones obtained for airship AU-12. The performance in terms of payload and flight altitude is shown in figure 27. Figure 27. Performance comparison between airship AU-12 (blue) and first iteration for “Zero” (red). The requirement in terms of payload is not satisfied so another iteration has to be done. A bigger volume envelope for “Zero” has to be selected in order to generate more lift. In this second iteration we consider a hull length of 36 meters and a maximum envelope diameter of 9 meters. The new envelope volume is 1526 m3 and the volume of the ballonets is up to 378 m3. The weight of the main components changes except for the gondola. The operational empty weight (OEW) is obtained proceeding in a similar way as done previously. Table 6 shows the main component’s breakdown for the second iteration. 0 500 1000 1500 2000 2500 3000 0100 200 300 400 500 Max. flight altitude (m) Payload (kg) STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 52 Component Mass (kg) Hull group 476.3 Tail group 154.0 Gondola 123.5 Propulsion system 279.5 TOTAL OPERATIONAL EMPTY WEIGHT 1033 Table 6. OEW breakdown for airship “Zero”. Second iteration Payload 1 and 2 for the second iteration are: Both values are greater than the original ones obtained for airship AU-12. The maximum flight level for payload 2 under ISA conditions for the second iteration is: The performance in terms of payload and flight altitude for the second iteration is shown in figure 28. Figure 28. Performance comparison between airship AU-12 (blue) and second iteration for “Zero” (red). 0 500 1000 1500 2000 2500 3000 0100 200 300 400 500 Max. flight altitude (m) Payload (kg) STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 53 Iteration 2 fulfills the payload and flight altitude requirements and is therefore selected as the final configuration for airship “Zero”. Table 7 summarizes its main dimensions and its motorization. Table 7. “Zero” airship main dimensions and motorization 5.3.4 Performance 5.3.4.1 Solar speed The speed performance of the solar airship “Zero” is estimated using the equation for the speed obtained in 5.1 and assuming clear sky conditions. Unlike conventional powered airships, the maximum reachable speed of a solar powered airship in a given time depends on the latitude and altitude of the aircraft and also on the hour and day of the year. Without feeding the electric engines with the secondary batteries the available power for propulsion is entirely obtained from the solar cells. Therefore, it is function of the solar radiation reaching the top of the airship, and it is also function of the projected area of the solar cells. Solar radiation depends on the hour, day of the year, latitude and altitude as has been explained in chapter 4. The projected area of the solar cells depends on the azimuth of the airship, the solar altitude and the solar azimuth as discussed in this chapter. In order to save weight, the “Zero” airship has been designed with only the upper surface of the envelope covered by flexible solar cells. Due to this configuration only the direct solar radiation is considered when calculating the available solar radiation at any time. For clear sky conditions the reflected radiation corresponds Envelope volume 1526 m3 including air ballonets, up to 378 m3 Length/diameter ratio 4 Max. envelope diameter 9 m Envelope length 36 m Solar cells surface 411 m2 Batteries weight 100 kg Operational empty weight 1033 kg Engine type 2 x D135RAG (LMC) Max. engine power 68.6 kW STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 54 mainly to the radiation reflected by the ground and consequently this kind of radiation which is reaching the upper surface of the envelope is negligible. On the other hand during clear days, the contribution of diffuse radiation to the total solar radiation is no greater than 10% [4]. The exact contribution of diffuse radiation is difficult to predict in advance since it depends on air humidity conditions, suspension particles content, contamination, etc. Consequently only direct solar radiation is taken into account for calculating the speed performance of the solar airship. By neglecting the reflected and diffuse solar radiation the available power obtained at any moment is slightly underestimated. However, due to the cubic root relationship between available power and speed it can be easily found that for the “Zero” airship, on a clear day a maximum error of 3% is made by not considering the reflected and diffuse solar radiation, which is considered acceptable for the present study. The speed performance of the “Zero” airship will be analyzed at solar noon for a latitude of 41ºN (Barcelona) and 2000 m of altitude. The four most significant solar days of the year will be considered: summer solstice, autumn equinox, winter solstice and spring equinox. The following efficiencies and parameters are considered: - = 1 kg/m3 (ISA conditions) - = 0.0045 [27] [28] - = 0.12 [16] - = 0.95 [22] - = 0.9 [28] - = 0.8 [28] - = 822 m2 Daily direct solar radiation for a point located over Barcelona (latitude 41ºN) at 2000 meters of altitude has been calculated in chapter 4 for the selected days. The values at noon of the direct solar radiation to be considered are: - = 1062 W/m2 - = 1020 W/m2 - = 905 W/m2 Finally, the solar cells’ projected area is calculated applying the methodology explained in this chapter and for values of the airship azimuth (from southern direction) ranging from 0º to 180º. In order to do that a Matlab code has been implemented and the results are shown in figure 29. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 55 Figure 29. Solar cells’ projected area for airship “Zero” The blue, green and red lines correspond respectively to the summer solstice, the winter solstice and the equinoxes. Like for the direct solar radiation, any other day of the year will be represented by a line which shall be between the lines of the summer and winter solstice. The following considerations can be deduced from figure 29: - The higher the solar altitude the higher the projected area for a given airship azimuth. - As the solar altitude increases the variation of the projected area with the airship azimuth is less pronounced. - The maximum solar cells’ projected area for a given solar altitude is always obtained when the airship’s azimuth is perpendicular to the solar azimuth. Finally, the theoretical speed of the solar airship is shown in figure 30. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 56 Figure 30. Maximum speed of airship “Zero” The maximum attainable solar speed is 22.7 m/s (81.7 km/h) and corresponds to “Zero” flying with an azimuth of 90º at the noon of the summer solstice (blue line). On the other hand the lowest solar speed is 16.72 m/s (60.2 km/h) and corresponds to “Zero” flying with an azimuth of 0º at the noon of the winter solstice (green line). The following considerations can be deduced from figure 30: - The higher the solar altitude, the higher the maximum speed for a given airship azimuth. - As the solar altitude increases the variation of the maximum speed with the airship azimuth is less pronounced. Moreover, during half of the year, in summer and spring (between the blue and red line), the maximum speed is practically independent from the airship’s azimuth. - The maximum speed for a given solar altitude is always obtained when the airship’s azimuth is perpendicular to the solar azimuth. 5.3.4.2 Emergency speed In case of malfunction of the solar cells the secondary batteries must feed the electric motors in order to guarantee a normal airship operation for at least one hour. Assuming a maximum battery discharge of 70%, the available power is: This means an available power of 14000 W during 1 hour or 28000 W during half an hour. Considering the same efficiency values as the ones considered above STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 57 the emergency speed (or speed of the airship powered only by the batteries) is = 62.2 km/h during 1 hour or =78.4 km/h during half an hour. 5.3.4.3 Range and autonomy Like the maximum available speed, range and autonomy for a solar airship depend on the day of the year, latitude and altitude of flight. Considering a minimum value for the incoming solar irradiation of 750 W/m2 to be acceptable the autonomy of the “Zero” airship is 12 hours in the summer solstice and 6 hours in the winter solstice according to figure 17. For any other day of the year the autonomy of the “Zero” will be a value between 6 and 12 hours. For a solar airship the range not only depends on the parameters mentioned above but also on the airship’s azimuth and the wind conditions along the track. Consequently the traditional concept of range is not accurate enough to characterize the performance of a solar powered airship. As a result a new equivalent concept for range called the guaranteed covered area (GCA) will be introduced in chapter 6. A three-hour flight range characterization of the “Zero” airship using the GCA concept is presented in chapter 7. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 64 is the density of the helium contained in the gas envelope. and are the temperature and pressure of the air at altitude . and are the temperature and pressure of the helium. and are the gas constants for air and helium. is the mass of helium gas, which is constant. is the volume of the envelope containing the helium. The temperatures and are the same as assumed by hypothesis and in order to avoid differential of pressure across the envelope we impose that is equal to . It means that the helium is “adapted” and has the same static pressure as the atmospheric air at altitude . The gas envelope has an initial volume . It is obtained given the total weight of the airship and the initial altitude . When and are fixed we can easily obtain the mass of helium which remains constant along the flight (we assume no losses of gas). The needed volume of helium (painted yellow in figure 34) is obtained by adequately regulating the volume of the air ballonet. The initial volume of the ballonet is . The total volume of the envelope is the sum of the lifting gas volume and the ballonet volume. is fixed and constant. Now let’s consider an increment of time dt. The temperature and pressure values of the atmospheric air have changed to and . The density of the air has changed to . The density ratio is: The new helium temperature is equal to as assumed by hypothesis. The air ballonet is inflated or deflated as required in order to achieve . The new volume of helium is and the new volume of the ballonet . The density ratio for helium is: STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 65 The net static lift is: ; The net static lift does not change and neither does the airship weight. There is no net force acting upwards or downwards and the airship maintains its neutral buoyancy. As a result the altitude of the airship can be easily maintained constant. If the airship ascends or descends due to turbulence or wind gusts the pilot can incline the vectored thrust accordingly in order to compensate the perturbation and recover the desired flight altitude. Another option for the pilot is to slightly incline the airship’s hull upwards or downwards in order to produce an additional aerodynamic lift to compensate the perturbation. As a result, if the ballonet has enough capacity to adapt its volume to the changing values of air temperature and pressure the altitude of the airship along the trajectory can be easily remained constant. Option B. Pressure differential across the envelope. Constant helium volume In this case the volume of the helium envelope remains constant along the flight (figure 35). The envelope has to be able to resist the stress due to the pressure differential between inside and outside the volume. The altitude of the airship changes as the air temperature and pressure change with time. As in the previous case, at the initial position the net static lift is and the altitude is . STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 66 Figure 35. Constant helium volume After an increment of time dt, the density of the atmospheric air will change due to the variation of air temperature and pressure values. As already seen before the density ratio is: However, now the volume of the ballonet does not change and nor does the volume of helium. As a result the density of helium remains constant. The new net static lift is: STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 67 As a result there is a net vertical force acting on the vehicle. The airship will ascend or descend depending on the value of : airship ascends airship descends To summarize, the airship is in neutral buoyancy and therefore it can easily maintain a constant cruising altitude if the ballonet is capable to adapt its volume to the changing atmospheric conditions of pressure and temperature during the flight. Taking into account a cruising altitude of 2000 m, flight duration less than one day and the MM5 forecast weather data provided by SMC for the year 2008, we will estimate the required volume needed by the ballonets in order to perform a constant altitude flight. Then we will compare the required ballonet volume with the one available in the airship “Zero” in order to decide if we can assume option A in our vertical trajectory calculation. A Matlab code has been implemented to calculate the maximum daily differences (for each month) in temperature and pressure at 2000 m throughout the considered region as well as the monthly maximum and minimum pressure and temperature at the same altitude (Table 8). The calculation of has been done by using those maximum daily differences and the minimum monthly values for temperature and pressure (most unfavorable situation). The upper and lower values of sigma ( and ) have been computed as follows (Table 9): The required maximum and minimum helium volume will be: STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 68 ; And the required ballonet volume: Min T month (K) Max diff. T day (K) Min P month (Pa) Max diff. P day (Pa) January 265.3 14.0 77189.0 2208.2 February 266.1 11.4 78184.0 2083.8 March 259.4 18.5 77150.0 2551.1 April 267.0 16.5 76807.0 2414.0 May 270.0 17.1 78612.0 1281.7 June 272.0 14.9 78960.0 1204.4 July 274.8 20.2 79471.0 1233.8 August 274.0 22.3 79022.0 1457.2 September 271.9 19.1 78872.0 1715.7 October 265.4 15.9 77622.0 1861.7 November 261.8 18.0 76907.0 2218.8 December 260.4 14.1 77263.0 1993.0 Table 8. Maximum daily differences in T and P for each month (2008) January 1.029 1.053 1.083 0.972 0.950 0.923 February 1.027 1.043 1.071 0.974 0.959 0.934 March 1.033 1.071 1.107 0.968 0.933 0.907 April 1.031 1.062 1.095 0.970 0.942 0.913 May 1.016 1.063 1.081 0.984 0.940 0.925 June 1.015 1.055 1.071 0.985 0.948 0.934 July 1.016 1.074 1.090 0.985 0.931 0.917 August 1.018 1.081 1.101 0.982 0.925 0.908 September 1.022 1.070 1.094 0.979 0.934 0.914 October 1.024 1.060 1.085 0.977 0.943 0.921 November 1.029 1.069 1.100 0.972 0.935 0.909 December 1.026 1.054 1.081 0.975 0.949 0.925 Table 9. Values of and STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 69 According to Table 9 the maximum and minimum values of sigma (both in march) are and . The required helium volume for the airship “Zero” flying at 2000 m with 150 kg of effective payload and standard conditions is and therefore the volume of the ballonets is . The total envelope volume of airship “Zero” is 1526 m3. The required ballonet volumes to perform a constant 2000 m altitude flight are: The required maximum ballonet volume is smaller than the maximum available ballonet volume of airship “Zero” which is 378 m3. In his turn the required minimum ballonet volume is greater than 0 which is the lowest possible value. As a result we can consider option A for vertical trajectory calculation. There is no pressure differential across the envelope and the airship maintains the neutral condition buoyancy during all the flight. Besides other considerations this result is important because the pilot has an easy control over the vertical coordinate and is able to maintain the selected cruising altitude by means of the vectored thrust or by generating an appropriate aerodynamic lift by tilting the hull. When computing the predicted geostrophic flight trajectories of the airship, the “real” required changing volume of the ballonets along the track will also be calculated. The “real” ballonet volume is expected to be smaller than the maximum obtained before since the temperature and pressure conditions that the airship will encounter will be less extreme than the ones considered for estimating the required ballonet volume . 6.3.3 Horizontal trajectory The net force acting on the airship at any time is the resultant of the thrust provided by the airship engines and the drag force due to the relative velocity between the airship velocity vector and the wind velocity vector. The drag force is, more specifically: STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 70 Where: is the air density at cruising altitude Relative velocity vector Airship velocity vector Wind velocity vector Resistance coefficient Wet surface of the airship Relative velocity unitary vector The equation of motion is therefore: Or: Where is the total mass of the airship and its acceleration vector. The equation of motion is an ordinary differential equation and is solved by using a second order modified Euler method. This method uses an auxiliary intermediate point to better estimate velocities and accelerations at each time increment . STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 71 Figure 36. Second order modified Euler method The steps are: 1) At the initial point the horizontal coordinates , altitude , ground velocity and thrust of the airship are known. As shown later in this chapter by using accurate weather forecast data we can also obtain the vector wind velocity , air temperature , air pressure and thus air density for this initial point. Finally we can also determine the vector acceleration at point using the equation : Where 2) Next step is to consider a time increment and calculate the horizontal coordinates, velocity and acceleration of an intermediate auxiliary point . The two first are direct: In order to calculate we must proceed in a similar way as we did to calculate : STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 72 3) Finally, after a time increment we calculate the new position and velocity of the airship at the next point : Repeating the procedure we can advance the airship trajectory. The elapsed flight time is therefore . 6.3.4 Determination of the atmospheric parameters along the trajectory 6.3.4.1 Horizontal airship coordinates in the grid First, the airship is considered to have the same horizontal coordinates (latitude, longitude) as one of the MM5 grid points. Usually the cruising altitude of the airship at this location will not coincide with any of the geopotential altitudes corresponding to the available pressure levels. In general the airship altitude will be between two geopotential altitudes (figure 37). Furthermore, some kind of interpolation is needed in order to calculate the temperature T, pressure P and the wind speed (u,v) at the cruising altitude, which is assumed in this case to be 2000 m. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 73 Figure 37. Flight altitude between geopotential altitudes In order to calculate the temperature at the flight altitude, a linear variation of the temperature with altitude is considered [29] [30]: Where: - = Temperature of the grid point at pressure level 3 - = Temperature of the grid point at pressure level 4 - = Geopotential altitude of the grid point at pressure level 3 - = Geopotential altitude of the grid point at pressure level 4 Considering the atmospheric air as a perfect gas and assuming hydrostatic equilibrium, the pressure and density at the flight altitude are [29] [30]: STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 80 - Starting flight time, day and month. - Flight duration. - Coordinates of the starting point. - Flight altitude. - Constant airship azimuth. - Solar cells, electric components and mechanic efficiencies. - Propulsive efficiency. The main program calls the three functions mentioned above in order to perform specific repetitive computation tasks. The input/output values of these functions are: - projectarea2.m: computes the solar cells’ projected area taking as input the variables solar altitude, solar azimuth, airship’s azimuth and the solar cells surface. - solar3.m: computes the direct solar irradiation, the solar altitude and the solar azimuth taking as input variables the time, day, month, latitude and flight altitude. - wind6.m: computes the temperature, pressure and wind velocity values for a trajectory point taking as input variables the latitude, longitude and altitude of the corresponding trajectory point. This function calls the subfunctions isop.m and jaco.m which perform auxiliary tasks for calculating the isoparametric transformation and the bilinear interpolation. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 81 7. Case study 7.1 GCA for the “Zero” airship In order to study the range performance of the airship “Zero”, twelve guaranteed covered areas (GCA) are studied in this chapter. Each GCA has been calculated for the 15th day of each month of the year 2008, considering the following flight characteristics: - Airship: “Zero” - Energy source: Solar power - Starting point: Barcelona - Flight altitude: 2000 m - Flight duration: 3 hours - Starting time: 11:30 (UT) - Day: 15th of each month - MM: ranging from 01 to 12 And consequently, as explained in chapter 6.2 the notation is as follows: ZERO_SP_41.38_2.18_2000_03_1130_15MM2008 The payload is defined to be 200 kg and the following efficiencies and parameters have been taken into account: - = 0.0045 [27] [28] - = 0.12 [16] - = 0.95 [22] - = 0.9 [28] - = 0.8 [28] - = 822 m2 The GCAs have been obtained using the Matlab code developed in the framework of this study which is described in chapter 6 (annex 1). The trajectory corresponding to a flight with constant airship azimuth of 0º is painted in red in the GCA charts. A further graph shows the range (in km) as function of the azimuth of the airship (in degrees). STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 82 Figure 43a. ZERO_SP_41.38_2.18_2000_03_1130_15012008 Figure 43b. Range as function of the airship’s azimuth STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 83 Figure 44a. ZERO_SP_41.38_2.18_2000_03_1130_15022008 Figure 44b. Range as function of the airship’s azimuth STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 84 Figure 45a. ZERO_SP_41.38_2.18_2000_03_1130_15032008 Figure 45b. Range as function of the airship’s azimuth STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 85 Figure 46a. ZERO_SP_41.38_2.18_2000_03_1130_15042008 Figure 46b. Range as function of the airship’s azimuth STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 86 Figure 47a. ZERO_SP_41.38_2.18_2000_03_1130_15052008 Figure 47b. Range as function of the airship’s azimuth STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 87 Figure 48a. ZERO_SP_41.38_2.18_2000_03_1130_15062008 Figure 48b. Range as function of the airship’s azimuth STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 88 Figure 49a. ZERO_SP_41.38_2.18_2000_03_1130_15072008 Figure 49b. Range as function of the airship’s azimuth STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 89 Figure 50a. ZERO_SP_41.38_2.18_2000_03_1130_15082008 Figure 50b. Range as function of the airship’s azimuth STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 96 9. Conclusions and further recommendations This study has demonstrated that recreational airship solar geostrophic flight is feasible with current technology. The key element of geostrophic flight is the availability of accurate and reliable short term weather forecast data. In this sense the weather forecast data provided for this study by Servei Meteorològic de Catalunya has been able to accurately predict the flight trajectory of a hot-air balloon over the Pyrenees and therefore has been considered acceptable to be used for this study. In general, the dramatic improvement of numeric weather forecast models over the past 50 years may overcome one of the disadvantages of airship flight: its relatively low speed which makes airships more dependent on wind conditions. By knowing the weather conditions for the flight in advance, on the one hand potential dangerous wind situations can be avoided and on the other hand faster flight trajectories can be implemented in case of favorable wind conditions. The second main disadvantage of airships is its inherent big volume and envelope surface. Paradoxically this apparent disadvantage makes airships the ideal candidates for solar flight. Indeed, the still low efficiencies of thin film solar cells require the solar cells to be spread over a large surface, which is available on the upper side of the airship’s envelope. In this study a conceptual design of “Zero”, a recreational solar airship, has been made using already existing technology. Its performance in terms of payload and speed is similar to the one of the reference airship (AU-12) which is used for the design of “Zero”, with the additional advantage of producing zero contaminant gases during its flight. The range of the “Zero” airship has been studied for the year 2008 using the introduced technique of the Guaranteed Covered Area (GCA). As expected, during summer and spring the flight range of the airship is greater due to the greater available solar power. Because of the dependency of the GCA with the day of the year and location of the starting point, in the opinion of the author, nowadays the possible implementation of the solar geostrophic airship flight is restricted to the field of sportive recreational aviation. As a conclusion, given the large surface area inherent in airships and the potential to use this large area for the production of photovoltaic solar energy, the possible use of this lighter than air aircrafts as a complement to the traditional aviation industry should be considered positively, since the availability of accurate short- and midterm weather forecast data can make the 21st century airship flight safe and reliable. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 97 Finally, further recommendations for future studies include: - Consider not only the direct but also the reflected and diffuse radiation in the incoming solar radiation. Consider also other situations rather than clear sky conditions. - Study optimal airship envelope shapes in order to maximize the incoming solar power and static lift but minimize the drag. - Compare the results obtained for the geostrophic flight using the weather forecast data from the MM5 model with other regional and global weather prediction models. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 98 10. Budget This chapter contains the estimated budget for the realization of the "Study of a zero-emission airship transport system based on the geostrophic flight concept". The costs of this study are only related to the author’s working hours and to the cost of the educational license for the software Matlab. The weather forecast data of the year 2008 from SMC has been provided for free due to the academic background of this study. In order to determine the cost, the following considerations have been taken into account: - The duration of the project has been eight months and the author has invested a total of 600 hours. - An 8 €/hour tariff is assumed for the author. - The chosen currency is Euro. - Taxes are not included. The cost breakdown is shown in table 11. Table 11. PFC cost The total cost for the realization of this study is therefore 4887 €. Concept Quantity Units Unitary cost Overall Author working hours 600 hours 8 €/hour 4800 € Matlab license 1 - 87 € 87 € TOTAL 4887 € STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 99 11. References [1] Sprigg, C. The airship. University Press of the Pacific, USA. 2001 [2] Associazione Dirigibili Archimede. http://www.dirigibili-archimede.it (September 2010) [3] Khoury, G.A and Mowforth, E. A solar airship, more than a flight of fancy. New Scientist. July 1978 [4] Khoury, G.A. Airship technology. Cambridge Aerospace Series, UK. 1999 [5] Nobile, U. Alas sobre el Polo. Ed. Juventud, Spain. 1977 [6] Stull, R. B. An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers, The Netherlands. 1988 [7] Lynch, P. The origins of computer weather prediction and climate modeling. Journal of computational physics. March 2007 [8] National Centers for Environmental Prediction. http://www.ncep.noaa.gov/ (November 2010) [9] MM5. http://www.mmm.ucar.edu/mm5/mm5-home.html (November 2010) [10] Servei Meteorològic de Catalunya. http://www.meteo.cat/servmet/index.html (September 2010) [11] HYSPLIT. http://ready.arl.noaa.gov/HYSPLIT.php (August 2010) [12] http://www.itacanet.org/eng/elec.htm (October 2010) [13] Meinel, A.B. and Meinel, M.P. Applied solar energy, an introduction. Addison-Wesley Publishing, 1976 [14] Laue, E.G. Solar energy. Vol. 13, 1970 [15] Houston, A. and Rycroft, M. Keys to Space. ISU McGraw-Hill, USA. 1999 [16] National Renewable Energy Laboratory. http://www.nrel.gov (September 2010) [17] Beernink, K. et al. Lightweight, flexible solar cells on stainless steel foil and polymer for space and stratospheric applications. NASA GRC, 2007 STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 100 [18] http://batteryuniversity.com/ (September 2010) [19] Sion Power. http://www.sionpower.com/ (September 2010) [20] Electra. http://www.apame.eu/AA%20Projects.html (September 2010) [21] Zephir. http://www.qinetiq.com/home/products/zephyr.html (September 2010) [22] The Lynch Motor Company. http://www.lmcltd.net/ (October 2010) [23] Blakemore, T. Pressure Airships. University Press of the Pacific, USA. 2003 [24] Rosaerosystems. http://www.rosaerosystems.pbo.ru/ (October 2010) [25] Rotax engines. http://www.rotax-aircraft-engines.com (October 2010) [26] Kokam. http://www.kokam.com/ (October 2010) [27] Hoerner, S.F. Fluid-dynamic drag. Hoerner fluid dynamics, USA. 1965 [28] Dorrington, G.E. Performance of non-rigid airships operating in the neutral buoyancy condition. The Aeronautical Journal, February 2007 [29] McCormick, B. Aerodynamics, Aeronautics and Flight Mechanics. The Pennsylvania State Univ., USA. 1995 [30] Roskam, J. Airplane aerodynamics and performance. DARcorporation, USA. 1997 [31] ftp://ftp.eia.doe.gov/pub/oiaf/1605/cdrom/pdf/FormEIA-1605EZ_2005.pdf (December 2010) STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 101 Annex: Matlab Code STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 102 % % Programa dirig16.m % % Programa que determina les Guaranteed Covered Areas (GCA) del dirigible solar "Zero" %fixada un llei de l'alçada de vol % Predicció del camp de vent del 2008 amb model MM5 del Servei Meteorològic de Catalunya % % Novembre 2010 % Autor: Francesc Betorz Martínez % clear %inicialitza les variables % Lectura punts malla [lat,long] = textread('MM5latlonalt.txt','%f %f %*f'); %Lectura coordenades punts de la malla MM5 (latitut, longitut). 69x69 punts % Directori amb les dades de vent directoriTemp='D:\Aeronautica\PFC\Meteocat_Temp2008\'; %ruta a la carpeta de temperatures directoriGeop='D:\Aeronautica\PFC\Meteocat_Geopot2008\'; %ruta a la carpeta d'alçades geopotencials directoriVent='D:\Aeronautica\PFC\bones_smc_Vent_2008\'; %ruta a la carpeta de dades de vent % Dibuix de la zona de treball latlim = [37.2 44.2]; lonlim = [-3.3 6]; figure worldmap(latlim,lonlim) cont=0; %contador % Eficiències rend_cell=0.12; % Rendiment cel.lules solars rend_elec=0.95; % Rendiment elèctric motor rend_mec=0.9; % Rendiment mecànic rend_prop=0.8; % Rendiment propulsiu Area_disc=12.56 ; % Area discal en m2 % Definim elipsoide en posició horitzontal. Semieixos 18 i 4.5 STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 103 i=1; for phi=0:0.1:3.14 for teta=0:0.01:3.14 x(i)=18*sin(phi)*cos(teta); z(i)=4.5*sin(phi)*sin(teta); y(i)=4.5*cos(phi); i=i+1; end end for angw=0:30:330 % bucle amb diferents rumbs per trovar la GCA angw cont=cont+1 % Dades i paràmetres inicials dia='15'; % Dia del mes amb 2 dígits day=15; hora='10'; % Hora del dia amb 2 dígits (0 a 24) hour=10; minut='30'; % Minuts horaris, de 00 a 59 minute=30; mes='d'; % Mes de l'any segons conveni . P.e. d es desembre levels=[101300 95000 85000 70000 50000]; %Nivells de pressió del model MM5 (Pa) vector_uns=ones(4761,1); % Vector auxiliar m=1309; % massa dirigible en Kg 1000 cd=0.04; % coef de resistència volumètric (relatiu a V^2/3) swet=116; % V^2/3 angw=deg2rad(angw); %canvi a radians per poder entrar en els sinus i cosinus alvol=2000; % Alçada de vol en m, es considera constant. delta=10; %increment de temps en segons rterra=6371; % Radi de la Terra en km % Prediccions són cada tres hores. Hem d'adaptar les hores. STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 104 if hour>=00 & hour<03 minute=minute+((hour-0)*60); hour=0; end if hour>=03 & hour<06 minute=minute+((hour-3)*60); hour=3; end if hour>=06 & hour<09 minute=minute+((hour-6)*60); hour=6; end if hour>=09 & hour<12 minute=minute+((hour-9)*60); hour=9; end if hour>=12 & hour<15 minute=minute+((hour-12)*60); hour=12; end if hour>=15 & hour<18 minute=minute+((hour-15)*60); hour=15; end if hour>=18 & hour<21 minute=minute+((hour-18)*60); hour=18; end if hour>=21 & hour<24 minute=minute+((hour-21)*60); STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 105 hour=21; end % Càlculs inicials %coordenades del punt inicial on calcularem la trajectòria latini=41.38; %Lalitut inicial Barcelona 41.38 longini=2.18; %Longitut inicial Barcelona 2.18 %llei d'alçada de vol hvol=alvol*ones(7201,1); %Lei alçada de vol en metres. %determinació del punt de la malla mes pròxim al punt inicial distmin=100; %distància al punt de malla més pròxim posini=1; %inici contador de la posició. Posini es el punt de malla més pròxim al punt for k=1:4761 dist= distance(lat(k),long(k),latini,longini); %calcula distancia minima (ortodròmica) en graus entre els 2 punts distkm=deg2km(dist); if distkm<distmin distmin=distkm; %determina distància mínima posini=k; %determina punt de la malla més pròxim end end % Interpolació temporal inicial (suposem dades subministrades cada 3 hores) hores='00010203040506070809101112131415161718192021'; %variable alfanumèrica amb les hores de les prediccions nivell='12345'; %variable alfanumèrica amb els nivells de pressió hour1=hores((hour*2)+1:(hour*2)+2); %límit inferior horari hour2=hores((hour*2)+7:(hour*2)+8); %límit superior horari pes1=1-(minute/180); %pes sobre hour1 pes2=(minute/180); %pes sobre hour2 % Determinació dels nivells de pressió. Els nivells de pressió són lower i upper lower=3; %nivell inferior de pressió. Perque alçada constant de vol a 2000m upper=4; %nivell superior de pressió. Perque alçada constant de vol a 2000m STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 112 [h_low_2] = textread(nomfitxer_h_low_2,'%f','headerlines',5); %Alçada geopotencial en m del nivell inferior, hora 2 [h_up_2] = textread(nomfitxer_h_up_2,'%f','headerlines',5); %Alçada geopotencial en m del nivell superior, hora 2 [u_low_2] = textread(nomfitxer_u_low_2,'%f','headerlines',5); %Velocitat oest-est en m/s al nivell inferior, hora 2 [u_up_2] = textread(nomfitxer_u_up_2,'%f','headerlines',5); %Velocitat oest-est en m/s al nivel superior, hora 2 [v_low_2] = textread(nomfitxer_v_low_2,'%f','headerlines',5); %Velocitat nord-sud en m/s al nivell inferior, hora 2 [v_up_2] = textread(nomfitxer_v_up_2,'%f','headerlines',5); %Velocitat nord-sud en m/s al nivell superior, hora 2 end % Interpolació temporal t_low=(pes1*t_low_1)+(pes2*t_low_2); % Temperatura en K al nivell inferior de pressió a l'instant de càlcul t_up=(pes1*t_up_1)+(pes2*t_up_2); % Temperatura en K al nivell superior de pressió a l'instant de càlcul h_low=(pes1*h_low_1)+(pes2*h_low_2); % Alçada geopotencial en m al nivell inferior de pressió a l'instant de càlcul h_up=(pes1*h_up_1)+(pes2*h_up_2); % Alçada geopotencial en m al nivell superior de pressió a l'instant de càlcul u_low=(pes1*u_low_1)+(pes2*u_low_2); % Velocitat del vent zonal al nivell inferior de pressió en m/s a l'instant de càlcul u_up=(pes1*u_up_1)+(pes2*u_up_2); % Velocitat del vent zonal al nivell superior de pressió en m/s a l'instant de càlcul v_low=(pes1*v_low_1)+(pes2*v_low_2); % Velocitat del vent meridional al nivell inferior de pressió en m/s a l'instant de càlcul v_up=(pes1*v_up_1)+(pes2*v_up_2); % Velocitat del vent meriodinal al nivell superior de pressió en m/s a l'instant de càlcul % Càlcul de la temperatura de l'aire a l'alçada de vol. Es suposa variació lineal de la temperatura entre els dos nivells dis=h_up-h_low; %Distància en metres entre nivells inferior i superior a=(t_up-t_low)./dis; %Gradient de temperatura en K/m. El punt es perque la divisió es faci element a element tvol=t_low+a.*(hvol(i)-h_low); %Temperatura en K a alçada vol suposant variació lineal amb l'alçada. % Càlcul de la pressió de l'aire a l'alçada de vol. Es suposa relació hidrostàtica STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 113 preslevel_lower=levels(lower)*vector_uns; % Pressió al nivell inferior en tots el punts de malla (Pa) preslevel_upper=levels(upper)*vector_uns; % Pressió al nivell superior en tots el punts de malla (Pa) for k=1:4761 % per tots els punts de la malla, 69x69=4761 pvol(k)=levels(lower)*((tvol(k)/t_low(k))^(-g/(R*a(k)))); %Pressió en Pa a l'alçada de vol end pvol=pvol'; % el trasposem per poder operar amb vectors u3 i v3 % Càlcul velocitat del vent a l'alçada de vol. Suposem variació lineal amb la pressió kwindu=(u_up-u_low)./(preslevel_upper-preslevel_lower); u=(u_low+kwindu.*(pvol-preslevel_lower)); kwindv=(v_up-v_low)./(preslevel_upper-preslevel_lower); v=(v_low+kwindv.*(pvol-preslevel_lower)); pvol=pvol'; % el trasposem de nou per consistència amb el bucle % Posició i velocitat del punt intermig auxiliar lataux=latp(i-1)+km2deg(((delta/2)*vabslat(i-1))/1000); longaux=longp(i-1)+km2deg(((delta/2)*vabslong(i-1))/1000,rterra*cos(deg2rad(lataux))); longvaux=vabslong(i-1)+((delta/2)*along(i-1)); latvaux=vabslat(i-1)+((delta/2)*alat(i-1)); % Posició nou punt latp(i)=latp(i-1)+km2deg((delta*latvaux)/1000); longp(i)=longp(i-1)+km2deg((delta*longvaux)/1000,rterra*cos(deg2rad(latp(i)))); % Acceleració punt intermig auxiliar [upunt,vpunt,tpunt,ppunt,pos]=wind6(lataux,longaux,lat,long,u,v,tvol,pvol,posini); longvrelaux=longvaux-upunt; longvdragaux=-longvrelaux; latvrelaux=latvaux-vpunt; latvdragaux=-latvrelaux; tempaux=tpunt; presaux=ppunt; roaux=presaux/(R*tempaux); STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 114 posini=pos; vdrag=(longvdragaux^2+latvdragaux^2)^(1/2); [Irrad,solaraltd,solarazimd]=solar3(day,mes,hour,minute,lataux,alvol); %Calcul irradiació directa i altura i azimut solars [Sproj]=projectarea2(solaraltd,angw,solarazimd,x,y,z); %Càcul superfice cel.lules solars projectada power_av=Irrad*Sproj*rend_cell*rend_elec*rend_mec; % Potència disponible a l'eix hèlix thrust_zero=(power_av^2*(2*roaux*Area_disc))^(1/3); % Empenta a punt fixe vrel=(longvrelaux^2+latvrelaux^2)^(1/2); % Mòdul velocitat relativa if vrel<5 vrel=5; %Per evitar divisió per 0 i valors grans end thrust=(power_av*rend_prop)/vrel; % Mòdul de l'empenta if thrust>thrust_zero thrust=thrust_zero; % Límit del mòdul d'empenta és empenta a punt fixe end thrustlat=(cos(angw))*thrust; thrustlong=(sin(angw))*thrust; dragvalor=(0.5*roaux*(vdrag^2)*cd*swet); alfadrag=atan2(longvdragaux,latvdragaux); draglongaux=dragvalor*sin(alfadrag); longaaux=(thrustlong+draglongaux)/m; % Acceleració zonal punt auxiliar draglataux=dragvalor*cos(alfadrag); lataaux=(thrustlat+draglataux)/m; % Acceleració meridional punt auxiliar % Velocitat nou punt vabslong(i)=vabslong(i-1)+(delta*longaaux); % Velocitat zonal nou punt vabslat(i)=vabslat(i-1)+(delta*lataaux); % Velocitat meridional nou punt % Acceleració nou punt [upunt,vpunt,tpunt,ppunt,pos]=wind6(latp(i),longp(i),lat,long,u,v,tvol,pvol,posini); vrellong(i)=vabslong(i)-upunt; vdraglong(i)=-vrellong(i); STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 115 vrellat(i)=vabslat(i)-vpunt; vdraglat(i)=-vrellat(i); temptrack(i)=tpunt; prestrack(i)=ppunt; rotrack(i)=prestrack(i)/(R*temptrack(i)); posini=pos; vdrag=(vdraglong(i)^2+vdraglat(i)^2)^(1/2); [Irrad,solaraltd,solarazimd]=solar3(day,mes,hour,minute,latp(i),alvol); %Calcul irradiació directa i altura i azimut solars [Sproj]=projectarea2(solaraltd,angw,solarazimd,x,y,z); %Càcul superfice cel.lules solars projectada power_av=Irrad*Sproj*rend_cell*rend_elec*rend_mec; % Potència disponible a l'eix hèlix thrust_zero=(power_av^2*(2*rotrack(i)*Area_disc))^(1/3); % Empenta a punt fixe vrel=(vrellong(i)^2+vrellat(i)^2)^(1/2); % Mòdul velocitat relativa if vrel<5 vrel=5; %Per evitar divisió per 0 i valors grans end thrust=(power_av*rend_prop)/vrel; % Mòdul de l'empenta if thrust>thrust_zero thrust=thrust_zero; % Límit del mòdul d'empenta és empenta a punt fixe end thrustlat=(cos(angw))*thrust; thrustlong=(sin(angw))*thrust; dragvalor=(0.5*rotrack(i)*(vdrag^2)*cd*swet); alfadrag=atan2(vdraglong(i),vdraglat(i)); draglong(i)=dragvalor*sin(alfadrag); along(i)=(thrustlong+draglong(i))/m; % Acceleració zonal nou punt draglat(i)=dragvalor*cos(alfadrag); alat(i)=(thrustlat+draglat(i))/m; % Acceleració meridional nou punt i end STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 116 latpf(cont)=latp(1081); longpf(cont)=longp(1081); % Representació gràfica quiverm(lat,long,v,u,'r'); plotm(latp,longp); plotm(latp(1081),longp(1081),'*'); end plotm(latpf(1),longpf(1),'r*'); % Representació corva GCA [kp,vp]=convhull(latpf,longpf); plotm(latpf(kp),longpf(kp),'-'); STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 117 % % Function wind6.m % Function[upunt,vpunt,tpunt,ppunt,pos]=wind6(latpunt,longpunt,lat,long,u,v,t2000,p2000,posini) % Càlcul de la nova posició de pos, el punt de malla més proper al punt daux(1)=distance(lat(posini-70),long(posini-70),latpunt,longpunt); dauxkm(1)=deg2km(daux(1)); daux(2)=distance(lat(posini-69),long(posini-69),latpunt,longpunt); dauxkm(2)=deg2km(daux(2)); daux(3)=distance(lat(posini-68),long(posini-68),latpunt,longpunt); dauxkm(3)=deg2km(daux(3)); daux(4)=distance(lat(posini-1),long(posini-1),latpunt,longpunt); dauxkm(4)=deg2km(daux(4)); daux(5)=distance(lat(posini),long(posini),latpunt,longpunt); dauxkm(5)=deg2km(daux(5)); daux(6)=distance(lat(posini+1),long(posini+1),latpunt,longpunt); dauxkm(6)=deg2km(daux(6)); daux(7)=distance(lat(posini+68),long(posini+68),latpunt,longpunt); dauxkm(7)=deg2km(daux(7)); daux(8)=distance(lat(posini+69),long(posini+69),latpunt,longpunt); dauxkm(8)=deg2km(daux(8)); daux(9)=distance(lat(posini+70),long(posini+70),latpunt,longpunt); dauxkm(9)=deg2km(daux(9)); [dmin,ind]=min(dauxkm); %ind indica en quina posicio del vector dauxkm es troba el minim. dmin es aquest mínim. if ind==1 pos=posini-70; end if ind==2 pos=posini-69; STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 118 end if ind==3 pos=posini-68; end if ind==4 pos=posini-1; end if ind==5 pos=posini; end if ind==6 pos=posini+1; end if ind==7 pos=posini+68; end if ind==8 pos=posini+69; end if ind==9 pos=posini+70 end % determinació dels punts del quadrilàter dist1=distance(lat(pos+69),long(pos+69),latpunt,longpunt); dist1km=deg2km(dist1); dist2=distance(lat(pos-69),long(pos-69),latpunt,longpunt); dist2km=deg2km(dist2); dist3=distance(lat(pos+1),long(pos+1),latpunt,longpunt); dist3km=deg2km(dist3); dist4=distance(lat(pos-1),long(pos-1),latpunt,longpunt); STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 119 dist4km=deg2km(dist4); if (dist1km>=dist2km) & (dist4km>=dist3km) xcon(1)=long(pos-69); ycon(1)=lat(pos-69); xcon(2)=long(pos-68); ycon(2)=lat(pos-68); xcon(3)=long(pos+1); ycon(3)=lat(pos+1); xcon(4)=long(pos); ycon(4)=lat(pos); end if (dist1km>=dist2km) & (dist3km>dist4km) xcon(1)=long(pos-70); ycon(1)=lat(pos-70); xcon(2)=long(pos-69); ycon(2)=lat(pos-69); xcon(3)=long(pos); ycon(3)=lat(pos); xcon(4)=long(pos-1); ycon(4)=lat(pos-1); end if (dist2km>dist1km) & (dist4km>=dist3km) xcon(1)=long(pos); ycon(1)=lat(pos); xcon(2)=long(pos+1); ycon(2)=lat(pos+1); xcon(3)=long(pos+70); ycon(3)=lat(pos+70); xcon(4)=long(pos+69); ycon(4)=lat(pos+69); STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 120 end if (dist2km>dist1km) & (dist3km>dist4km) xcon(1)=long(pos-1); ycon(1)=lat(pos-1); xcon(2)=long(pos); ycon(2)=lat(pos); xcon(3)=long(pos+69); ycon(3)=lat(pos+69); xcon(4)=long(pos+68); ycon(4)=lat(pos+68); end %punt a posicionar xp=longpunt; yp=latpunt; %valors de chi i nu inicials chi=0; nu=0; for i=1:5 [xpos,ypos] = isop(chi,nu,xcon,ycon); [jac11, jac12, jac21, jac22]=jaco(chi,nu,xcon,ycon); A=[jac11 jac12; jac21 jac22]; delta=A\[xp-xpos;yp-ypos]; chi=chi+delta(1); nu=nu+delta(2); end N1=(1-chi)*(1-nu)/4; N2=(chi+1)*(1-nu)/4; N3=(chi+1)*(nu+1)/4; N4=(1-chi)*(1+nu)/4; if (dist1km>=dist2km) & (dist4km>=dist3km) STUDY OF A ZERO-EMISSION AIRSHIP TRANSPORT SYSTEM BASED ON THE GEOSTROPHIC FLIGHT CONCEPT 121 upunt=(N1*u(pos-69))+(N2*u(pos-68))+(N3*u(pos+1))+(N4*u(pos)); vpunt=(N1*v(pos-69))+(N2*v(pos-68))+(N3*v(pos+1))+(N4*v(pos)); tpunt=(N1*t2000(pos-69))+(N2*t2000(pos-68))+(N3*t2000(pos+1))+(N4*t2000(pos)); ppunt=(N1*p2000(pos-69))+(N2*p2000(pos-68))+(N3*p2000(pos+1))+(N4*p2000(pos)); end if (dist1km>=dist2km) & (dist3km>dist4km) upunt=(N1*u(pos-70))+(N2*u(pos-69))+(N3*u(pos))+(N4*u(pos-1)); vpunt=(N1*v(pos-70))+(N2*v(pos-69))+(N3*v(pos))+(N4*v(pos-1)); tpunt=(N1*t2000(pos-70))+(N2*t2000(pos-69))+(N3*t2000(pos))+(N4*t2000(pos-1)); ppunt=(N1*p2000(pos-70))+(N2*p2000(pos-69))+(N3*p2000(pos))+(N4*p2000(pos-1)); end if (dist2km>dist1km) & (dist4km>=dist3km) upunt=(N1*u(pos))+(N2*u(pos+1))+(N3*u(pos+70))+(N4*u(pos+69)); vpunt=(N1*v(pos))+(N2*v(pos+1))+(N3*v(pos+70))+(N4*v(pos+69)); tpunt=(N1*t2000(pos))+(N2*t2000(pos+1))+(N3*t2000(pos+70))+(N4*t2000(pos+69)); ppunt=(N1*p2000(pos))+(N2*p2000(pos+1))+(N3*p2000(pos+70))+(N4*p2000(pos+69)); end if (dist2km>dist1km) & (dist3km>dist4km) upunt=(N1*u(pos-1))+(N2*u(pos))+(N3*u(pos+69))+(N4*u(pos+68)); vpunt=(N1*v(pos-1))+(N2*v(pos))+(N3*v(pos+69))+(N4*v(pos+68)); tpunt=(N1*t2000(pos-1))+(N2*t2000(pos))+(N3*t2000(pos+69))+(N4*t2000(pos+68)); ppunt=(N1*p2000(pos-1))+(N2*p2000(pos))+(N3*p2000(pos+69))+(N4*p2000(pos+68)); end