Background and Motivation
Since 2008, the rising price of crude oil has intensified the focus on renewable energy, highlighting the urgent need to balance fossil fuel conservation with the pursuit of alternative resources The selection of these alternatives hinges on economic, environmental, and user-related factors Solar energy stands out as a sustainable and cost-effective solution, being both abundant and easily accessible across various geographical locations The primary challenge remains in developing efficient methods for harnessing and storing this free energy.
Solar thermal devices convert solar radiation into thermal energy, with the efficiency of this conversion into mechanical energy largely dependent on the thermal energy's temperature Due to the relatively low energy density of solar radiation, surfaces exposed to unfocused sunlight tend to reach temperatures close to the surrounding environment To enhance energy density, solar radiation can be concentrated; higher concentration leads to increased energy density and, consequently, higher temperatures in solar energy systems This elevated temperature improves the efficiency of converting thermal energy into mechanical energy Additionally, since the sun moves in relation to the Earth, concentrating devices must be adjusted to follow its path.
This study focuses on the application of solar trackers, which enhance the efficiency of energy collection by concentrating sunlight onto the targeted plane of a solar tower.
Solar tracking application
Solar trackers are essential devices that adjust the alignment of solar collection systems to optimize energy production, taking into account the sun's daily and seasonal movement By integrating various optical devices like panels, reflectors, and lenses, these trackers enhance the efficiency of solar energy systems.
When deciding on the implementation of solar trackers, it's essential to evaluate various factors such as the type of solar technology utilized, the level of direct solar irradiation available, regional feed-in tariffs, and the installation and maintenance costs associated with the trackers.
There are various types of solar trackers depending on costs, sophistication, and performance However, the two principle categories of solar energy tracking system are single axis and dual axis [43]
Single-axis solar trackers, available in horizontal and vertical designs, optimize solar energy capture based on geographic location Horizontal trackers are ideal for tropical regions with high noon sun and shorter days, while vertical trackers suit high latitudes where the sun remains lower but summer days are extended These trackers are commonly utilized in concentrated solar power systems, particularly with parabolic and linear Fresnel mirror configurations.
Figure 1-1 is an example of single axis tracker which is used for solar parabolic trough
Figure 1-1: Single axis tracker use for solar parabolic trough [43]
Dual-axis solar trackers feature both horizontal and vertical axes, enabling them to follow the sun's movement across the sky anywhere globally These trackers are crucial for Concentrated Solar Power (CSP) systems, particularly in solar power towers and dish Stirling engine setups Their ability to optimize solar energy capture significantly enhances the efficiency of solar tower applications.
3 to the angle errors resulting from longer distances between the mirror and the central receiver located in the tower structure
Dual axis trackers are widely used in conventional solar photovoltaic systems as they optimize power output by keeping the panels aligned with direct sunlight for the maximum number of hours each day.
Dual-axis tracking systems play a crucial role in solar power applications, particularly in mitigating angle errors caused by increased distances between mirrors and central receivers in tower structures These systems are integral to Concentrated Solar Power (CSP) technologies, including solar power towers and dish systems utilizing Stirling engines.
Figure 1-2 show a dual axis tracking system which is applied for solar thermal disk [43]
Figure 1-2: Dual axis tracker of solar thermal disk [43]
Solar tower systems utilize reflectors, known as heliostats, arranged around a central tower These heliostats are computer-controlled to accurately track the sun and direct sunlight to a central receiver at the tower's peak The effectiveness of this system hinges on the precision of the tracking to ensure that the reflected sunlight reaches its focal point.
Objective of the Thesis
This study aims to reach three fundamental objectives:
To develop effective tracking system calculations, new equations must be formulated that account for realistic conditions, including coordinate errors, heliostat shape inaccuracies, flexible tilt angles, and the rotation angle of the target plane, based on established formulas.
Secondly, to simulate trackers for solar tower by applying MATLAB software
Finally, to make an evaluation of simulation and a comparison of the study with other published researches.
Thesis Organization
The dissertation is composed of three main parts:
Part 1: Background knowledge and related work are presented in Chapter 2
Part 2: Research and simulation calculation process tracking heliostat mirrors in Chapters
Part 3: Calculating and simulating of heat and temperature distribution obtained in chapter
The chapters are organized as follows:
Chapter 2: Presentation materials on the fundamentals and work related to conducting the approach
Chapter 3: Establishment of formulas, equations to calculate activities and application simulation for tracking heliostat process of solar towers system
Chapter 4: Calculation of heliostats model's errors
Chapter 5: Improve the accuracy of sun tracking systems
Chapter 6: Calculation and simulation of energy, temperature distribution obtained on the target surface and the performance system
Chapter 7: Summary of the main results and contributions of the thesis
Structure of the sun
The sun, a spherical body with a diameter of 1.39 million kilometers—over 110 times that of Earth—lies approximately 150 million kilometers away, with sunlight taking about 8 minutes to reach our planet Its mass is around 2 x 10^30 kg, and the core temperature ranges from 10 to 20 million Kelvin, averaging about 15.6 million Kelvin At these extreme temperatures, materials transition into a plasma state, where atomic nuclei move independently of their electrons, leading to thermonuclear explosions when collisions occur Observations of the cooler surface features of the sun indicate that fusion reactions are actively taking place within it.
Figure 2-1: Surface of the sun [21]
The sun is structurally divided into four regions, forming a massive gas sphere The core, or central area, is where convective movements and fusion reactions generate solar energy, boasting a radius of approximately 175,000 km, a specific weight of 160 kg/dm³, temperatures ranging from 14 million K to 20 million K, and immense pressures of hundreds of billions of atmospheres Surrounding the core is the radiation zone, also known as the convective zone, where energy transitions from the inner layers to the outer layers, primarily composed of materials like iron (Fe).
The sun's composition includes elements such as Calcium (Ca), Sodium (Na), Strontium (Sr), Chromium (Cr), Nickel (Ni), Carbon (C), Silicon (Si), and gases like Hydrogen (H) and Helium (He) The radiation zone extends approximately 400,000 kilometers in thickness, followed by the convection zone, which measures around 125,000 kilometers Above these layers lies the Chromosphere, characterized by a temperature of 6,000 K and a thickness of 1,000 kilometers The outermost layer, known as the transition region, forms the sun's atmosphere.
Figure 2-2: Structure of the Sun [21]
The sun's surface temperature reaches approximately 5762K, which is sufficiently high to excite atoms while still allowing for the presence of normal atoms and molecules Analysis of the sun's absorbed radiation spectrum reveals that it contains about two-thirds of the elements found on Earth, with hydrogen being the most abundant at 73.46%, followed by helium at 24.85% Other elements present in smaller quantities include oxygen (0.77%), carbon (0.29%), iron (0.16%), neon (0.12%), nitrogen (0.09%), silicon (0.07%), magnesium (0.05%), and sulfur (0.04%).
Solar radiation's primary energy originates from the nuclear fusion of hydrogen, resulting in the formation of helium This process involves hydrogen nuclei, which consist of positively charged protons Typically, like charges repel each other, but at elevated temperatures, the increased speed of these particles enables them to overcome this repulsion When combined with high pressure, they can approach each other closely enough to fuse.
7 combine together Thus each four hydrogen nuclei can produce a helium nucleus, two neutrinos and radiation amounts [21],[51]
Neutrinos are very sustainable non-electrical particles and have great penetrating ability After a reaction, neutrinos immediately leave the solar scope and do not participate in other events later
During fusion reactions, the sun converts solar material into energy, resulting in a mass loss of approximately 4 million tons per second Despite this significant reduction, the sun's state remains stable over billions of years Each day, the sun generates an astounding 9x10^24 kWh of energy, which is equivalent to the total electricity produced on Earth in an entire year.
The earth
The Earth formed approximately 5 billion years ago from a ring of dust and gas orbiting the sun, eventually becoming a self-rotating, porous sphere As gravity compressed the sphere, its temperature soared, melting it and allowing heavy elements like iron and nickel to sink to the core, surrounded by liquid magma The outer layer of the Earth features an atmosphere composed of hydrogen, helium, water vapor, methane, ammonia, and sulfuric acid Over time, as the Earth continued to cool and rotate, its temperature dropped enough for silicates to rise from the magma and solidify, creating a crust approximately 25 km thick, which includes mountains, plains, and deep valleys The interplay of internal radiation and solar energy further transformed the atmosphere, leading to the formation of water, nitrogen, oxygen, and carbon dioxide Eventually, the cooling atmosphere allowed water vapor to condense, resulting in rain and the formation of lakes, seas, and oceans over millions of years.
Nearly 2 billion years ago, the very first creatures appeared in water and grew into higher creatures, thenceforward evolved into human beings
The Earth's surface uniquely showcases two states of matter—solid and liquid—existing simultaneously The boundary between land and sea is the only location in the universe where matter exists in three forms: solid, liquid, and gas.
The relevant geometry between Earth and sun
Distance earth-sun
The Earth follows an elliptical orbit around the Sun, leading to variations in its distance throughout the year As a result, the Earth's position can change, bringing it closer to or farther from the Sun at different times.
The Earth orbits the Sun in approximately 365.24 days, maintaining an average distance of about 1.495 x 10^11 meters, known as one Astronomical Unit (AU) The maximum distance, or aphelion, occurs on July 3rd at 1.59 x 10^11 meters, while the minimum distance, or perihelion, takes place on January 2nd at 1.47 x 10^11 meters The Earth's rotation on its polar axis causes the daily cycle of day and night, while seasonal changes result from the tilt of the Earth's axis in relation to its orbital path around the Sun.
Figure 2-3: Variation of the earth-sun distance [21]
Simulation variation distance between the earth and the sun can be expressed as:
Figure 2-4: Distance between sun and earth as a function of time
The geometric relationship of the solar radiation rays
The geometric relationship between a plane layout and solar radiation can be defined by the apparent solar azimuth angle (γs) and the zenith angle (θz) as perceived by an observer on Earth The solar azimuth angle indicates the angle between the observer's south direction and the sun's projection onto their horizontal plane, while the zenith angle is the complement of the solar elevation angle (αs) Both angles are determined by factors such as the local hour angle (ω), declination angle (δ), and latitude (φ).
The local hour angle is directly related to local solar time (t s), which is defined as 12:00:00 when the sun reaches its highest point in the sky Due to Earth's elliptical orbit and variations in its rotational speed, true solar time can differ from mean solar time by up to 15 minutes throughout the year.
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1.46
E a rt h -s u n d is ta n c e / m discrete dataES-distance
10 time is defined as the length of an average day [10],[26] The local hour angle can be expressed as: o hr 12
Figure 2-5 shows the local hour angle of a day
Figure 2-5: Local hour angle of a day
The angular displacement of the sun relative to the local meridian is influenced by the Earth's rotation on its axis at a rate of 15 degrees per hour, with morning values considered negative and afternoon values positive.
The angle of declination refers to the angle formed between the Earth's polar axis and the elliptic plane of its orbit This angle can be estimated using various empirical relationships that offer different levels of precision The angle of declination is approximately 23.45 degrees.
The angle can be calculated according to the equation of cooper:
Relation between the time and local hour angle on a day
Or more accurately (error < 0.035 o ) by
With: and 1n365 n, the day number in a year (n = 1 for January the 1 st and n 365 for December 31 st )
The declination angle ( ) is described by Figure 2-6
Figure 2-6:Declination angle of the year [15]
Declination angle change throughout a year
The declination angle (δ) experiences seasonal variations due to the Earth's axial tilt and its orbit around the sun If the Earth were not tilted, the declination angle would consistently be 0° However, with a tilt of 23.45°, the declination angle fluctuates by this amount throughout the year Notably, during the spring and fall equinoxes, the declination angle is precisely 0°.
The equation of time (EoT) (in minutes) is an empirical equation that corrects for the eccentricity of the Earth's orbit and the Earth's axial tilt [13],[15],[16]
The difference between the mean solar time and true solar time is called the equation of time, given as:
To convert the local standard time to solar time, the following relationship is used,
Where: t s is the solar time (h) t l is the standard time (h)
L st is the standard meridian for the local time zone ( o )
L loc is the local meridian of the location ( o ) The factor of 15 o per hour since the earth rotates 1 o per 4 minutes
The time correction EoT is plotted in the Figure 2-7
Figure 2-7: Variation equation of the time of the year [15]
2.3.2.4 Solar Elevation ( s ) and Azimuth ( s ) angles:
The solar elevation angle (αs) and solar azimuth angle (γs) can be derived through algebraic expressions The solar zenith angle, which is the angle between the sun vector (S) and the observer's zenith axis, is illustrated in Figure 2-8 Notably, the zenith angle (θz) is the complement of the solar elevation angle (αs).
Figure 2-8: Relations of the solar angles [18]
Day of the year / day
E qu at io n of t im e / m in
True solar time variation throughout a year s(South)
In vector characters, S can be shown as follows:
In the equation Eq 2-8 i, j, and k are the unit vectors along the Zenith (z), the East (e), and the South (s) axes
In terms of solar azimuth angle (s) and solar elevation angle (s), ( S ) can be written as:
The new incident solar vector ( S ’ ) is expressed by solar altitude and azimuth angle, which is shown in Figure 2-9
Figure 2-9: Geocentric of sun angles [18]
Solar noon meridian p(polar axis)
The geocentric coordinate system, illustrated in Figure 2-9, features an m axis that intersects the Earth's center and the intersection point of the local meridian with the equatorial plane The e axis, which is perpendicular to the m axis, also resides within the equatorial plane Meanwhile, the p axis acts as the normal to the equatorial plane, intersecting at the South Pole The new solar vector (S') can be defined based on solar declination and the hour angle.
The top centric and geocentric coordinates are interrelated by a rotation about the e axis through the latitude ()
Figure 2-10: Geocentric to top centric coordinate transform φ p Polar axis e axis
Combining the equations Eq 2-9, Eq 2-11, and Eq 2-12 yield in,
Solving the equation Eq 2-13 for s and s angle yields,
Controller theory
To calculate the solar angle in Vienna on July 1, 2014, we utilize the solar altitude angle, solar azimuth angle, and zenith angle as outlined in Table 2-1 The solar azimuth angle measures the angle between the south direction and the projection of beam radiation on the horizontal plane, with negative values in the morning (east to south) and positive values in the afternoon (west to south) The solar altitude angle, which is the angle between the horizontal plane and the line to the sun, increases in the morning and decreases in the afternoon Additionally, the zenith angle, which represents the angle between the incident beam radiation and the normal axis of a surface, is the complement of the solar altitude angle.
In Figure 2-11 the azimuth angle and the elevation angle of the sun are expressed
Figure 2-11: Azimuth angle and elevation angle of the sun
Table 2-1 presents the hourly values for key solar angles in Vienna on July 1st, including the local hour angle (ω), solar altitude angle (αs), solar azimuth angle (γs), and zenith angle (θz).
Table 2-1 Solar angle at Vienna on 1/7/2014
Solar azimuth angle Zenith angle h 0 0 0 0
Figure 2-12 shows the solar altitude angle, the solar azimuth angle(s), and the zenith angle(z) as a function of time
Figure 2-12: Azimuth angle and elevation angle of the sun
Solar azimuth, altitude, zenith angle
Solar alitude angleSolar azimuth angle zenith angle
Chapter 3 SOLAR TOWER SYSTEMS AND TRACKING
Solar tower systems harness solar energy through a network of movable mirrors known as heliostats, which direct sunlight to a heat exchanger located at the top of a tower This heat exchanger utilizes concentrated solar energy to heat various heat transfer media, including water, helium, air, liquid sodium, or molten nitrate salt An illustrative example of a molten salt system features heliostats, a tower, hot and cold storage tanks, a steam generator, a steam turbine, and an air-cooled condenser.
Figure 3-1: Molten salt power tower system
Heliostats capture and direct sunlight onto a target plane, where molten salt from a cold storage tank is heated in the heat exchanger This heated salt is then moved to a hot salt storage tank, enabling efficient energy storage for power generation.
Hot salt is pumped to the steam generator, where it cools down while heating, evaporating, and superheating water from the air-cooled condenser The cold salt is then moved to a storage tank, and the superheated steam is sent to the steam turbine, where its energy is converted to power the generator After expansion in the turbine, the steam is cooled in the air-cooled condenser, releasing waste energy into the environment A tracking system effectively intercepts and concentrates sunlight onto the target plane, enabling solar tower systems with integrated storage to operate even at night.
All heliostats are arranged around the tower, as shown in Figure 3-2
Figure 3-2: Schematic of a solar tower system
To enhance system efficiency, the strategic placement of heliostats is crucial The optical performance of a concentration system is influenced by factors such as the cosine effect, blocking, shadowing, atmospheric transmission, and mirror reflectivity.
Tracking methods
Active tracking
Active tracking systems utilize gears, motors, and actuators to continuously direct sunlight to a target plane by leveraging solar orbit information and sensors for precise sun positioning The control system processes this data to operate the motors and actuators that adjust the heliostat's position Alternatively, some trackers rely solely on solar maps, which provide location-specific data about the sun's position throughout the year, allowing the control system to adjust the heliostats without the need for additional sensors.
Figure 3-3: Active tracking use sensors to operate the tracker
Some trackers utilize both a solar map and sensors to optimize solar energy collection While sensors effectively track the sun during clear weather, their accuracy diminishes in cloudy conditions In such instances, the solar map takes over to ensure reliable tracking This dual-system approach allows for efficient operation even in less-than-ideal weather, enhancing overall performance.
Passive tracking
In a passive tracking system, heliostats utilize compressed gas for solar tracking, where sunlight heats the gas containers, increasing their pressure This pressure difference arises from the varying amounts of solar energy received by the two containers, leading to a discrepancy that enables effective tracking of the sun's position.
Figure 3-4: Passive tracking compressed gas to operate the tracker
Figure 3-4 illustrates the operation of a tracker utilizing the passive tracking method During sunrise, the second container receives more solar energy due to the shade trough, resulting in higher pressure compared to the first container This pressure difference causes liquid to flow from the second container to the first, increasing the mass of the first container Consequently, the moment_M1 becomes stronger than moment_M2, leading to the rotation of the heliostat A counterweight generates moment_C to balance the total moments, ensuring the heliostat aligns with the sun for optimal energy capture The formula for moment_C is provided for further analysis.
At midday, both containers receive equal solar energy, with the counterweight positioned vertically along the heliostat's zenith axis, resulting in a horizontal equilibrium position for the heliostat The moment_C can be expressed using a specific formula.
In the afternoon, the first container absorbs more solar energy, resulting in higher pressure compared to the second container This pressure difference causes the liquid from the first container to flow into the second, which has a greater mass Consequently, the moment_M2 becomes stronger than moment_M1, leading to the rotation of the heliostat The counterweight's position creates a balancing moment_C that stabilizes the system, allowing the heliostat to continuously adjust and maintain its focus on the sun until sunset The relationship governing moment_C can be expressed through a specific formula.
After sunset, both containers cease to receive solar energy, resulting in equal pressure and mass between them The heliostat adjusts to a horizontal position, and this process will continue in the following days The moment_C can be calculated using a specific formula.
A control system is not required for a passive tracking system; however, it exhibits slow responsiveness to information and is susceptible to damage in high-speed winds.
Methods description
Two primary methods for heliostat tracking are the azimuth-elevation method and the spinning-elevation method, each with unique tracking angles and characteristics, as illustrated in Figures 3-5.
Figure 3-5:Two kinds of sun tracking with heliostats
The azimuth-elevation tracking method employs two axes for optimal orientation: the azimuth axis, which points towards the zenith, and the elevation axis, which is tangent to the heliostat frame and perpendicular to the azimuth axis.
The spinning-elevation tracking method utilizes two axes to optimize heliostat alignment The spinning axis directs towards the center of the target plane, while the elevation axis, which is perpendicular to the spinning axis, maintains the heliostat's normal within the tangential plane This configuration ensures that the heliostat effectively bisects the incident sunrays vector and the spinning vector, enhancing solar tracking efficiency.
Both methods aim to direct sunlight towards a designated target plane, despite the independent positioning of the sun and the target The key distinction lies in the use of different mechanisms to operate the heliostat frame.
Equations for tracking systems
Equations for tracking azimuth-elevation method
A heliostat prepared for the azimuth-elevation tracking method has two axes; it is shown in Figure 3-6:
Figure 3-6: Heliostat with two axes, azimuth and elevation
The tracking system's azimuth axis is perpendicular to the ground, while the elevation axis is perpendicular to the azimuth axis and situated on the heliostat surface Utilizing the azimuth-elevation tracking method, the heliostat's tracking angles are referenced to its normal vector As the sun moves, the heliostat adjusts its azimuth angle (γH) around the azimuth axis and its elevation angle (αH) around the elevation axis These angles can be determined based on the reflection law, which connects the sun position vector, the target position vector, and the heliostat's normal vector on Earth's surface The heliostat's tracking angles can be calculated by identifying the solar time, heliostat position, and target position on the surface.
Figure 3-7: Tracking angles by use of the azimuth-elevation method
3.4.1.1 Formulas for the calculation of the tracking angles, using the azimuth-elevation method
To establish the formula of tracking angles, the cartesian right handed coordinate system has to be established as shown in Figure 3-8
- The ground coordinate system is the basic coordinate system It is characterized by
In the context of the tower, the center point G on the ground surface serves as the origin The horizontal axis XG extends southward, while the horizontal axis YG points eastward Additionally, the vertical axis ZG is oriented towards the zenith, perpendicular to the horizontal plane.
The incidence coordinate system is defined by the axes X I, Y I, and Z I, with the heliostat's center point M serving as the origin In this system, Z I aligns with the incident ray and points directly towards the sun, while X I is situated in the horizontal plane and is perpendicular to Z I Additionally, Y I is oriented vertically, normal to X I, and points upwards.
The heliostat coordinate system is defined by the axes XH, YH, and ZH, with the center point M serving as the origin In this system, ZH aligns with the normal vector of the heliostat's plane, originating from point M The XH axis represents the horizontal direction of the heliostat, while YH is oriented perpendicularly to XH, extending upward Together, the XH and YH axes define the surface of the heliostat.
Figure 3-8: Coordinate system by use of azimuth-elevation tracking method
The target coordinate system is defined by the axes X T, Y T, and Z T, with the center point T of the target surface serving as the origin The Z T axis represents the normal vector of the target plane, oriented upwards, while X T lies within the horizontal plane of the target Additionally, Y T is perpendicular to X T and also points upwards, together forming the surface of the target plane.
The reflection coordinate system is defined by the axes X R, Y R, and Z R, with the heliostat's center point M serving as the origin In this system, Z R represents the line connecting M to T, while X R lies in the horizontal plane Additionally, Y R is oriented perpendicularly to X R and extends upwards.
Figure 3-9: Incident vector I ae in the ground coordinate system by use of the azimuth- elevation method
The incident vector ( I ae ) represented by the center point on the heliostat surface and towards the sun in the ground coordinate system is expressed in Figure 3-9
- The direction cosine factors of the incident vector ( I ae ) are cosi , cosi , cosi
The solar altitude angle (αs) and the solar azimuth angle (γs) are essential measurements that depend on the solar time and the location of the heliostat on Earth's surface.
Figure 3-10:Reflection vector in ground coordinate system with azimuth-elevation method
The reflection vector (R ae), which is determined by the center point on the heliostat surface and the center point of the target plane within the ground coordinate system, is illustrated in Figure 3-10.
- The direction cosine factors of the reflection vector ( R ae ) are cosr , cosr, cosr
- The target angle () is created by the line from the heliostat centre point to the target centre point and the axis centre of the tower
The heliostat facing angle (θH) is defined on the horizontal plane, formed by the line extending from the tower to the south and the line connecting the tower to the heliostat's position.
The target angle () and the facing angle (H)depend on the location of the heliostat and the target plane on the surface of the earth
The incident angle (θ) is defined as half of the angle formed between the incident vector and the reflection vector Utilizing the reflection law of vector angle cosine along with equations Eq 3-5 and Eq 3-6, we can derive the necessary results.
The incident angle ( ), from equation Eq 3-7 can be expressed to:
The normal vector of the heliostat which is illustrated in Figure 3-11 by the azimuth angle and the elevation angle can be expressed by:
According to the reflection law, the normal vector of the heliostat is expressed as:
The elevation angle (αH) and the azimuth angle (γH) of the heliostat's normal vector at its center point are crucial for the elevation-azimuth tracking method These angles are defined by the equations Eq 3-12 and Eq 3-13, which outline their respective calculations.
The tracking azimuth angle (H) and the tracking elevation angle (H) are shown in Figure 3-11
Figure 3-11: Coordinate system for azimuth-elevation sun tracking heliostat
3.4.1.2 Calculation of the tracking angles, using the azimuth-elevation method
The heights of the heliostat and the tower are:
The relationships between the heliostat and the tower are:
- Distance from the heliostat to the tower: 10m
Table 3-1 presents the hourly values for the solar altitude angle (αs), solar azimuth angle (γs), tracking azimuth angle (γH), and tracking elevation angle (αH) for Vienna on July 1, 2014.
Table 3-1 Tracking angle at Vienna on 1/7/2014 calculated with the azimuth-elevation method
Figure 3-12 depicts the corresponding graph
Figure 3-12: Graph of azimuth angle and elevation angle of a heliostat in Vienna at 1 st July 2014
Time of day in hours /h
Calculate alpha s , gama s , alpha H , gama H angle variable with time alpha s gamma s alpha H gama H
The tracking azimuth angle (H) is shown in Figure 3-12 In the morning, the tracking azimuth angle (H) is negative and it is located between the east and the south directions
In the afternoon, the positive angle is positioned between the west and south directions, with the tracking altitude angle (αH) rising during sunrise and falling during sunset.
Equations for the tracking spinning-elevation method
A heliostat prepared for the spinning-elevation tracking method has two axes; it is shown Figure 3-13
Figure 3-13: Two axes spinning and elevation
The tracking system's fixed spinning axis connects the heliostat's center point with the target plane's center point The elevation axis, which is perpendicular to the spinning axis, is situated on the heliostat's surface As the sun moves, the heliostat rotates around the spinning axis to align the incident surface of the sunrays with its meridian surface Simultaneously, it rotates around the elevation axis to direct the reflected sunrays towards the target surface The tracking angles of the heliostat can be determined by identifying the solar time, heliostat position, and target position on the Earth's surface, as illustrated in Figure 3-14.
Figure 3-14: Tracking angles by use of the spinning-elevation method
3.4.2.1 Formulas for the calculation of the tracking angles, using the spinning-elevation method
To establish the formula of tracking angles, the cartesian right handed coordinate system has to be established as shown in Figure 3-15
- The ground coordinate system is the basic coordinate system It is characterized by
In the context of a tower's coordinate system, the origin is defined at point G on the ground surface The axis XG extends horizontally towards the south, while the axis YG also lies in the horizontal plane, oriented towards the east Additionally, the axis ZG is perpendicular to the horizontal plane, directed upwards towards the zenith.
The incidence coordinate system is defined by the axes X I, Y I, and Z I, with the heliostat's center point M as the origin In this system, Z I aligns with the incident ray and points directly toward the sun, while X I is situated in the horizontal plane, perpendicular to Z I Additionally, Y I is oriented vertically, normal to X I, and points upward.
Figure 3-15: Coordinate system by use of spinning-elevation tracking method
The heliostat coordinate system is defined by the axes X_H, Y_H, and Z_H, with the center point M of the heliostat serving as the origin In this system, Z_H aligns with the normal vector of the heliostat's plane, originating from point M The X_H axis represents the horizontal orientation of the heliostat, while Y_H, which is perpendicular to X_H, points upwards, collectively forming the surface of the heliostat.
The target coordinate system is defined by the axes X_T, Y_T, and Z_T, with the center point T of the target surface serving as the origin The Z_T axis represents the normal vector of the target plane, oriented upwards, while X_T lies within the horizontal plane of the target Additionally, Y_T is perpendicular to X_T, together forming the surface of the target plane.
The reflection coordinate system is defined by the axes X R, Y R, and Z R, with the heliostat's center point M serving as the origin In this system, Z R represents the line connecting point M to point T, while X R lies in the horizontal plane Additionally, Y R is oriented vertically, perpendicular to X R, and points upward.
Figure 3-16: Incidence vector ( I se ) in the ground coordinate system by use of the spinning-elevation method
The incident vector ( I se ) represented by the center point on the heliostat surface and the direction towards the sun in the ground coordinate system is expressed in Figure 3-16
- The direction cosine factors of the incident vector ( I se ) are cosi , cosi, cosi
The solar altitude angle (αs) and the solar azimuth angle (γs) are essential parameters that are determined based on the solar time and the position of a heliostat on the Earth's surface.
Figure 3-17:Reflection vector in ground-coordinates by use of the spinning-elevation method
R_se is the vector that connects the center point of the heliostat surface to the center point of the target plane surface, directed towards the target plane within the ground coordinate system, as illustrated in Figure 3-17.
- The direction cosine factors of the reflection vector ( R se ) are cosr, cosr, cosr
- The target angle () is created by the line from the heliostat centre point to the target centre point and the axis centre of the tower θ se
The heliostat facing angle (θH) is determined by the vector formed between the intersection point of the tower's axis with the horizontal plane and the southern direction, as well as the intersection point and the center of the heliostat.
The target angle () and the facing angle (H)depend on the location of the heliostat and the target plane on the surface of the earth
The half of the angle between the incident vector ( I se ) and the reflection vector ( R se ) is the incident angle ()
From the equations Eq 3-14, the Eq 3-15, and the reflection law, the relationship between the normal vector, the incident vector, and the reflection vector is expressed as follow:
The incident angle (), from the equation Eq 3-16 can be expressed to:
The calculation of the tracking spinning angle (H) and the tracking elevation angle (E H) of the heliostat is solved as below:
The X R Y R Z R coordinate system is centered at the heliostat's origin point M, with Z R representing the line that connects M to the target center point T In this system, X R is situated in the horizontal plane, while Y R is oriented vertically, perpendicular to X R and directed upward.
T is the crossing point between Z R axis and Z G axis
The heliostat reaches its zero position when the normal at its center is aligned vertically with the normal of the meridian plane In this configuration, the surface (Y R Z R) perfectly coincides with the heliostat's meridian plane.
After rotating a spinning angle (H) around Z R axis, the true position of the heliostat plane coincides with the coordinate system XYZ It is expressed in Figure 3-18
Figure 3-18: Position of the heliostat after rotating about spinning angle
To find the direction cosine factors of the incident vector cosαi, cosβi, and cosγi in the new coordinate system XωYωZω, it is essential to transform the original ground coordinate system XGYGZG into the new system This transformation process is crucial for accurate calculations and is represented mathematically.
The ground coordinate system XGYGZG undergoes a rotation of (θH - π/2) around the ZG axis, followed by an additional rotation of (λ) around the XG axis, aligning its axes with the reflection coordinate system XRYRZR Subsequently, the system rotates by (-ωH) around the ZR axis, resulting in the axes of the original system aligning with the new coordinate system XωYωZω The transformation matrices for these rotations are detailed below.
(The ground coordinate system rotates by an angle of (H - /2) around the ZG axis)
(The ground coordinate system continuous to rotate by an angle of () around the X G axis)
(The new coordinate system rotates by an angle of (–ω H ) around the ZR axis)
After transforming, the direction cosine of the incident sunrays is written as: (0, sin2θ, cos2θ)
Therefore, the matrices transformation can be written:
Solving the equation Eq 3-18, we have:
Solving the Eq 3-19, the spinning angle is calculated as follow:
Because the half of the angle between the incident vector and the reflection vector is the incident angle ( ), the elevation angle (E H ) of the heliostat equals the incident angle ()
The coordinate system for spinning-elevation method of the tracking heliostat is expressed in Figure 3-19:
Figure 3-19: Coordinate system for spinning-elevation sun tracking heliostat
3.4.2.2 Calculation of the tracking angles, using the spinning-elevation method
The heights of the heliostat and the tower are:
The relation between the heliostat and the tower are:
- Distance from the heliostat to the tower: 10m
On July 1, 2014, Table 3-2 presents the hourly values for key solar angles in Vienna, including the solar altitude angle (αs), solar azimuth angle (γs), tracking spinning angle (ωH), and tracking elevation angle (EH).
Table 3-2 Tracking angle at Vienna on 1/7/2014 calculated by use of the spinning-elevation method
Figure 3-20 depicts the corresponding graph
Figure 3-20: Graph of spinning angle and elevation angle of a heliostat in Vienna at 1 st July
Time of day in hours /h
Calculate alpha s , gama s , elevation H , omega H angle variable with time alpha s gamma s elevation H omega H
Equations for tracking the sunrays of tower systems
Tracking sunrays by use of the azimuth-elevation method
3.5.1.1 Equations, describing the tracking of sunrays by use of the azimuth- elevation method
The incident beam from the sun appears as a cone on the heliostat surface, with a vertex angle of approximately 9.3 mrad In this setup, the ZI axis of the incident coordinate system aligns with the center axis of the beam, while the solar disc plane establishes a polar coordinate system centered at the solar disc's origin The polar coordinate points on the solar disc are defined by the radius (R sun) and angle (ϕ sun), where the radius ranges from 0 to ε/2 and the angle varies from 0 to 2 These parameters are illustrated in Figure 3-23.
Figure 3-23: Polar coordinate components of the incident beam by use of the azimuth- elevation method
The factors of the unit vector of the incident sunrays in the incident coordinate system X I
YI ZI are ix , iy , iz Theycan be written in the equation Eq 3-26:
To change the incident coordinate system XI YI ZI to the heliostat coordinate system XH YH
Z H , the cosine direction of the incident sunrays cosi , cosi, cosi have to be transformed in three steps:
To align the incident coordinate system (X I, Y I, Z I) with the ground coordinate system (X G, Y G, Z G), the incident coordinate system must first be rotated around the XI axis by an angle of (αs - /2) Subsequently, it should be rotated around the ZI axis by an angle of (γs - /2), as illustrated in Figure 3-24.
Figure 3-24: The transformation of incident sunrays from the incident coordinate system to the ground coordinate system by use of the azimuth-elevation method
The transformation is expressed by the matrices below:
(The coordinate system rotates by an angle of
(The coordinate system rotates by an angle of (s - π/2) around the ZI axis)
To align the ground coordinate system (XG, YG, ZG) with the heliostat coordinate system (XH, YH, ZH), it is necessary to rotate the ground coordinate system around the ZG axis by an angle of (θH – /2) This transformation is illustrated in Figure 3-25.
Figure 3-25: The transformation from the ground coordinate system to the reflection coordinate system by use of the azimuth-elevation method
The matrices of the transformation are written as below:
(The coordinate system rotates by an angle of (H – π/2) around the Z G axis)
After the rotation, it's essential to shift the coordinate system from point G in the ground coordinate system (XG, YG, ZG) to point M in the heliostat coordinate system (XH).
Y H Z H The translation is expressed in Figure 3-25
The matrices of the translation are written as below:
(The coordinate system translation from G to M)
The matrices to express the transformation of the cosine direction of the incident sunrays from the incident coordinate system X I Y I Z I to the heliostat coordinate system X H Y H Z H are re-written as below:
A point on the heliostat surface is defined by the coordinates Xh, Yh, and Zh In the heliostat coordinate system (XH, YH, ZH), the components of the unit normal vector at that point are represented by θnx, θny, and θnz, which can be expressed as follows.
- Xh and Yh are the angular factors along X H axis and Y H axis correspondingly to a perfect heliostat surface shape
- ε Xh and ε Yh are the angular factors along X H axis and Y H axis correspondingly to the heliostat surface shape errors
The cosine direction factors of the normal vector of a point on the heliostat coordinate system X H Y H Z H are cosn , cosn , cosn They can be written as below:
Due to the relationship between the normal vector and the incident vector in the reflection law of a vector angular cosine, the incident angle (θ h ) in the heliostat coordinate system
X H Y H Z H of all sunrays from the equation Eq 3-27 to the equation Eq 3-29 can be re- written as below:
In the heliostat coordinate system (XH, YH, ZH), the relationship between the incident angle (θh) and the reflection angle (θhr) as described by Snell's law allows us to express the direction cosine factors of the reflected sunrays, namely cosαhr, cosβhr, and cosγhr.
To determine the direction cosine factors (cosαr, cosβr, cosγr) of reflected sunlight and the coordinates (Xhr, Yhr, Zhr) of a point on the heliostat in the reflection coordinate system (XR, YR, ZR), a transformation from the heliostat coordinate system (XH, YH, ZH) to the reflection coordinate system is necessary This transformation involves a two-step rotation of the heliostat coordinate system (XH, YH, ZH).
To align the heliostat coordinate system (XH, YH, ZH) with the ground coordinate system (XG, YG, ZG), the first step involves rotating the heliostat system around the XH axis by an angle of (π/2 - αH) Subsequently, a rotation around the ZH axis by an angle of (π/2 - γH) is required This transformation process is illustrated in Figure 3-25.
The matrices of the transformation above are written as below:
(The coordinate system rotates by an angle of (/2 - H ) around the X H axis)
(The coordinate system rotates by an angle of (/2-H) around the ZH axis)
- Second step: In order to make the three axes of the new coordinate system X H1 Y H1 Z H1 and the reflection coordinate system XR YR ZR coincident, the new coordinate system XH1
The Y H1 Z H1 coordinate system must be rotated about the Z H1 axis by an angle of (π/2 - θH), followed by a rotation around the X H1 axis by an angle of (-λ) This transformation process is illustrated in Figure 3-25.
The matrices of the transformation above are written as follow:
(The coordinate system rotates by an angle of (/2- H) around the Z H1 axis)
(The coordinate system rotates by an angle of
The direction cosine factors cosr , cosr , cosr of the reflected sunrays in the reflection coordinate system X R Y R Z R are identified by:
The coordinates X hr , Y hr , Z hr of a point on the heliostat in the reflection coordinate system
XR YR ZR are identified by:
From the equations Eq 3-32 and Eq 3-33, the equations of the reflected sunrays are re- written to:
To determine the intersection points of reflected sunrays with the target plane in the reflection coordinate system (XR, YR, ZR), it is essential to convert the coordinates from the reflection system to the target coordinate system (XT, YT, ZT) This transformation of the reflection coordinate system (XR, YR, ZR) occurs in two stages.
In the first step, the reflection coordinate system (XR, YR, ZR) is shifted along the ZR axis by a distance of Lo, moving from the heliostat's center point M to the target plane's center point T This transformation is illustrated in Figure 3-26, where Lo represents the distance between the origin point M of the heliostat coordinate system (XH, YH, ZH) and the origin point T of the target coordinate system.
X T Y T Z T ) In the new coordinate system X nr Y nr Z nr , the position of the intersection point between the reflected sunrays and the surface (X nr , Y nr ) are identified as X r , Y r , 0
Figure 3-26: The translation of the reflection coordinate system along the Z r axis by a distance of L o from the heliostat to the target plane by use of the azimuth-elevation method
From the equation Eq 3-34, the coordinates of the intersection points between the reflected sunrays and the surface (X nr , Y nr ) in the reflection coordinate system X R Y R Z R are identified as below:
In the second step, we identify the direction cosine factors of the reflected sunrays, specifically cosαtr, cosβtr, and cosγtr, alongside the coordinate factors Xt, Yt, and Zt at the intersection points within the target coordinate system XT, YT, ZT This involves transforming the new reflection coordinate system Xnr, Ynr, Znr to the target system Initially, the new system undergoes a rotation of (-λ) around the Xnr axis, followed by a rotation of (π/2 – θH) around the Znr axis, aligning it with the parallel coordinate system Xpg, Ypg, Zpg of the ground coordinate system XG, YG, ZG Finally, an additional rotation of (δT) around the appropriate axis is performed to complete the transformation.
Xpg axis Then, the coordinate system rotates by an angle of (ω T) around Ypg axis Next,
The coordinate system undergoes a rotation of (π/2) around the Z-axis, aligning its directions with the target coordinate system X_T, Y_T, and Z_T The transformation matrices for this operation are outlined as follows.
(The coordinate system Xnr Y nr Z nr rotates by an angle of (- ) around the X nr axis)
(The coordinate system Xnr Y nr Z nr rotates by an angle of (π/2 – H) around the Znr axis)
(The coordinate system X nr Y nr Z nr rotates by an angle of (T) around the X pg axis)
(The coordinate system Xnr Y nr Z nr rotates by an angle of (ω T ) around the Y pg axis)
(The coordinate system X nr Y nr Z nr rotates by an angle (π/2) around Z pg axis)
From the transformed matrices above, the coordinate factors costr , cos tr , costr of the reflected sunrays in the target coordinate system X T Y T Z T are calculated as below:
And the coordinate factors X t , Y t , Z t of the intersection points in the target coordinate system X T Y T Z T are calculated as:
In the target coordinate system (L 0 = z T = 0), the coordinates of the intersection point between the reflected sunrays and target plane are calculated as below:
Using equations Eq 3-26 to Eq 3-38, we can determine the coordinates of all sunrays as well as the intersection points where the reflected sunrays meet the target plane.
3.5.1.2 Applying the equations of the azimuth-elevation method
To determine the sun's image location on the target plane, MATLAB is utilized to compute the intersection points of reflected sunrays with the target plane This calculation requires specific data inputs, including the day number (n), solar time (t s), longitude (Lg), latitude (La), heliostat height (h), tower height (H), distance from the heliostat to the tower (d H), heliostat facing angle (θH), target rotation angle (δT) around the X pg axis, and facing rotation angle (ωT) around the Y pg axis.
Example data are expressed in the table below:
Table 3-3 Data at 10h in Vienna on 1/7/2014 by use of tracking azimuth-elevation method n t s Lg La h H d H H T T h o o m m m o o o
Figure 3-27 shows the image of the sun by use of the equations for the tracking azimuth- elevation method on the target surface
Figure 3-27: Image of the sun on the target surface by use of the azimuth-elevation method
Tracking sunrays by use of the spinning-elevation method
3.5.2.1 Equations, describing the tracking of sunrays by use of the spinning- elevation method
The incident beam from the sun forms a cone with a vertex angle of approximately 9.3 mrad on the heliostat surface In this context, the ZI axis of the incident coordinate system aligns with the center axis of the beam, while the solar disc plane serves as the basis for the polar coordinate system, which originates at the center of the solar disc.
Image of the sun by use azimuth-elevation
The solar disc is characterized by 57 points, defined by a radius \( R_{sun} \) and an angle \( \phi_{sun} \) The radius \( R_{sun} \) ranges from 0 to \( \epsilon/2 \), while the angle \( \phi_{sun} \) varies from 0 to 2, as illustrated in Figure 3-28.
Figure 3-28: Polar coordinate components of the incident beam by use of the spinning- elevation method
The factors of the unit vector of the incident sunrays in the incident coordinate system X I
YI ZI are ix , iy , iz Theycan be written in Eq 3-39:
To change the incident coordinate system XI YI ZI to the heliostat coordinate system XH YH
Z H , the cosine direction of the incident sunrays cosi , cosi, cosi have to be transformed in four steps:
To align the incident coordinate system (XI, YI, ZI) with the ground coordinate system (XG, YG, ZG), the first step involves rotating the incident coordinate system around the XI axis by an angle of (αs - /2) Subsequently, a rotation around the ZI axis by an angle of (γs - /2) is required This transformation process is illustrated in Figure 3-29.
Figure 3-29: The transformation of incident sunrays from the incident coordinate system to the ground coordinate system by use of the spinning-elevation method
The transformation is expressed by the matrices below:
(The coordinate system rotates by an angle of
(The coordinate system rotates by an angle of (s - π/2) around the Z I axis)
- Step two: In order to make the three axes of the ground coordinate system X G Y G Z G and the reflection coordinate system X R Y R Z R coincident, the ground coordinate system
The coordinate system X G Y G Z G must first be rotated around the Z G axis by an angle of (θH – /2), followed by a rotation around the X G axis by an angle of (λ) This transformation process is illustrated in Figure 3-30.
Figure 3-30: The transformation from the ground coordinate system to the reflection coordinate system by use of the spinning-elevation method
The matrices of the transformation are written as follow:
(The coordinate system rotates by an angle of (H – π/2) around the Z G axis)
(The coordinate system rotates by an angle of () around the X G axis)
To align the reflection coordinate system XR YR ZR with the heliostat coordinate system XH YH ZH, the reflection system must first be rotated around the ZR axis by an angle of (–ωH) Subsequently, it should be rotated around the XR axis by an angle of (–EH) This transformation process is crucial for achieving parallelism between the two coordinate systems.
The matrices of the transformations are written as follow:
(The coordinate system rotates by an angle of (-H ) around the Z R axis)
(The coordinate system rotates by an angle of (-E H ) around the X R axis)
- Step four: After rotating, the parallel coordinate system of the heliostat coordinate system X H Y H Z H has to be shifted from the point G of the ground coordinate system X G Y G
ZG to the point M of the heliostat coordinate system XH YH ZH The translation is expressed in Figure 3-30
The matrices of the translations are written as below:
(The coordinate system translation from G to M)
The transformation matrices for converting incident sunray coordinates from the system XI YI ZI to the heliostat coordinate system XH YH ZH are presented below.
A point on the heliostat surface is defined by its coordinates Xh, Yh, and Zh In the heliostat coordinate system (XH, YH, ZH), the components of the unit normal vector at that point are represented by δnx, δny, and δnz, which can be mathematically expressed as follows.
- Xh and Yh are the angular factors along X H axis and Y H axis correspondingly to a perfect heliostat surface shape
- ε Xh and ε Yh are the angular factors along X H axis and Y H axis correspondingly to heliostat surface shape errors
The cosine direction factors of the normal vector of a point on the heliostat coordinate system XH YH ZH are cosn , cosn , cosn They can be given as below:
In the heliostat coordinate system (XH, YH, ZH), the incident angle (θh) of sunrays can be reformulated based on the relationship between the normal vector and the incident vector in the reflection law, as detailed in equations Eq 3-40 to Eq 3-42.
According to Snell's law, the relationship between the incident angle (θ h) and the reflection angle (θ hr) allows us to express the direction cosine factors—cos(α hr), cos(β hr), and cos(γ hr)—of the unit vector for reflected sunrays within the heliostat coordinate system (XH, YH, ZH).
In order to identify the direction cosine factors cosr , cosr , cosr of the reflected sunrays and the coordinates X r , Y r , Z r of a point on the heliostat in the reflection coordinate system
X R Y R Z R , the coordinates have to be transformed from the heliostat coordinate system X H
Y H Z H to the reflection coordinate system X R Y R Z R Therefore, the heliostat coordinate system X H Y H Z H rotates in two steps:
- First step, the heliostat coordinate system X H Y H Z H rotates around the X H axis by an angle of (E H ) The transformation is described in Figure 3-30
The matrix of the transformation above is written:
(The coordinate system rotates by an angle of (E H ) around the X H axis)
- Second step: The coordinate system rotates around the Z H axis by an angle of (H) The transformation is described in Figure 3-30
The matrix of the transformation of the second step is written by:
(The coordinate system rotates by an angle of (H ) around Z H axis)
The direction cosine factors cosr , cosr , cosr of the unit vector of the reflected sunrays in the reflection coordinate system X R Y R Z R is identified by:
The coordinates X hr , Y hr , Z hr of a point on the heliostat surface in the reflection coordinate system X R Y R Z R is indentified by:
From equations Eq 3-45 and Eq 3-46, the equations of the reflected sunrays are written to:
To determine the intersection points between reflected sunrays and the target plane within the reflection coordinate system (XR, YR, ZR), the transformation process is illustrated in Figure 3-31.
Figure 3-31: The shift of the reflection coordinate system along the Z R axis by a distance of L o from the heliostat to the target plane by use of the spinning-elevation method
The reflection coordinate system XR YR ZR transforms in two steps:
- Step one: Shifting the reflection coordinate system X R Y R Z R along the Z R axis by a distance of Lo from the centre point M of the heliostat to the centre point T of the target
64 plane This is shown in Figure 3-31 (L o is the distance from the origin point M of the heliostat coordinate system X H Y H Z H to the origin point T of the target coordinate system
X T Y T Z T ) In the new coordinate system X nr Y nr Z nr , the position of the intersection point between the reflected sunrays and the surface (X nr , Y nr ) are identified as X r , Y r , 0
In the equation Eq 3-47, the coordinates of the intersection points between the reflected sunrays and the surface (X nr , Y nr ) in the reflection coordinate system X R Y R Z R are identified as below:
In step two, we identify the direction cosine factors of the reflected sunrays, specifically cosαtr, cosβtr, and cosγtr, along with the coordinate factors Xt, Yt, and Zt at the intersection points within the target coordinate system XT YT ZT This transformation begins by rotating the new reflection coordinate system Xnr Ynr Znr by an angle of (-λ) around the Xnr axis, followed by a rotation of (π/2 – θH) around the Znr axis, aligning it with the ground coordinate system Xpg Ypg Zpg Subsequently, the parallel coordinate system Xpg Ypg Zpg undergoes a rotation of (δT) around the Ypg axis, and then an additional rotation of (ωT) around the Xpg axis, culminating in the necessary transformations to achieve the desired orientation.
Z pg axis Hence, the directions of the coordinate system coincide with the target coordinate system X T Y T Z T The matrices of the transformation are written as follow:
(The coordinate system X nr Y nr Z nr rotates by an angle of (- ) around the Xnr axis)
(The coordinate system Xnr Y nr Z nr rotates by an angle of (π/2 – H ) around the Z nr axis)
(The coordinate system Xnr Ynr Znr rotates by an angle of (T ) around the Y pg axis)
(The coordinate system X nr Y nr Z nr rotates by an angle of (ω T ) around the Xpg axis)
(The coordinate system Xnr Y nr Z nr rotates by an angle of (π/2) around the Z pg axis)
From the matrices transformations above, the coordinate factors cos tr , cos tr , cos tr of the reflected sunrays in the target coordinate system X T Y T Z T are calculated as below:
The coordinate factors X t , Y t , Z t of the intersection points in the target coordinate system
XT YT ZT are calculated as:
In the target coordinate system XT YT ZT (L 0 = zT = 0), the coordinates of the intersection points between the reflected sunrays and the target plane are calculated as below:
Using equations Eq 3-39 to Eq 3-51, we can determine the coordinates of all sunrays and identify where the reflected sunrays intersect with the target plane.
3.5.2.2 Applying the equations of the spinning-elevation method
To determine the sun's image location on the target plane, MATLAB is utilized to compute the intersection points of reflected sunrays with the target plane This calculation incorporates various parameters, including the day number (n), solar time (t s), longitude (Lg), latitude (La), heliostat height (h), tower height (H), distance from heliostat to tower (d H), heliostat facing angle (θH), target rotation angle (δT) around the X pg axis, and target facing rotation angle (ωT) around the Y pg axis.
Example data are expressed in the table below:
Table 3-4 Data at 10h in Vienna on 1/7/2014 by use of the tracking spinning-elevation method n t s Lg La h H d H H T T h o o m m m o o o
Figure 3-32 shows the image of the sun by use of the equations for the spinning-elevation method on the target surface
Figure 3-32: Image of the sun on the target surface calculated with equations for the spinning- elevation method
Image of the sun by use spinning-elevation
Chapter 4 ERRORS OF THE HELIOSTAT SYSTEMS
The accuracy of tracking systems is closely linked to the design of the heliostat, particularly in solar tower systems where the distance to the target plane is greater than in other concentrating systems To ensure optimal performance in concentrating solar energy on the target, a tracking accuracy of just a few milliradians (mrad) is required on the axes of the sun tracking system.
This chapter aims to explore the errors associated with heliostat systems to accurately describe their positioning The difference between ideal and actual sun tracking arises from various errors, including heliostat location errors, solar angle inaccuracies, and misalignment of the target center point To ensure the precision of the sun tracking system, it is essential to calculate the offsets of the image on the target plane.