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1. ____________________________________________ Solar Energy System Design The largest solar electric generating plant in the world produces a maximum of 354 megawatts (MW) of electricity and is located at Kramer Junction, California. This solar energy generating facility, shown below, produces electricity for the Southern California Edison power grid supplying the greater Los Angeles area. The authors’ goal is to provide the necessary information to design such systems. The solar collectors concentrate sunlight to heat a heat transfer fluid to a high temperature.

The hot heat transfer fluid is then used to generate steam that drives the power conversion subsystem, producing electricity. Thermal energy storage provides heat for operation during periods without adequate sunshine. [pic] Figure 1. 1 One of nine solar electric energy generating systems at Kramer Junction, California, with a total output of 354 MWe. (photo courtesy Kramer Junction Operating Company) Another way to generate electricity from solar energy is to use photovoltaic cells; magic slivers of silicon that converts the solar energy falling on them directly into electricity.

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Large scale applications of photovoltaic for power generation, either on the rooftops of houses or in large fields connected to the utility grid are promising as well to provide clean, safe and strategically sound alternatives to current methods of electricity generation. [pic] Figure 1. 2   A 2-MW utility-scale photovoltaic power system co-located with a defunct nuclear power plant near Sacramento, California. (photo courtesy of DOE/NREL, Warren Gretz) The following chapters examine basic principles underlying the design and operation of solar energy conversion systems such as shown in Figure 1. 1 and 1. 2.

This includes collection of solar energy, either by a thermal or photovoltaic process, and integration with energy storage and thermal-to-electric energy conversion to meet a predefined load. Study of the interaction of these subsystems yields the important guidelines for the design of optimal solar energy systems. System design tools are provided to produce optimal sizing of both collector field and storage so that optimum system designs can be produced. Since our emphasis is on the design of entire solar energy conversion systems rather than design of its individual components, both thermal and photovoltaic systems are included.

This novel approach results from recognition of the commonality of most system design considerations for both types of solar energy systems. We will not dwell on the intricacies of individual component design, but instead encourage the designer to take experimental (or predicted) component input/output information and incorporate this into an overall system design. The system shown in Figure 1. 1 employs parabolic trough line-focus collectors. We will cover this and other types of collectors for capturing the sun’s energy including flat plate, parabolic dish, central receiver and photovoltaic collectors.

The purpose of a solar collector is to intercept and convert a reasonably large fraction of the available solar radiation. For solar thermal systems this energy is converted into thermal energy at some desired temperature and then, maybe, into electricity. For photovoltaic systems as shown in Figure 1. 2, intercepted solar energy is converted directly into low voltage direct current electricity. The engineering tradeoff between cost and performance of the components necessary to perform these processes has led to a wide variety of collector and system designs.

Reviews of solar collector designs representative of the different concepts that have been built and tested are presented here. The following sections serve as an overview of the solar energy system design process. They follow in a general manner, the flow of logic leading from the basic solar resource to the definition of an operating solar energy conversion system that meets a specified demand for either thermal or electrical energy. 1. 1 The Solar Energy Conversion System There are many different types of solar energy systems that will convert the solar resource into a useful form of energy.

A block diagram showing three of the most basic system types is shown as Figure 1. 3. In the first diagram , the solar resource is captured and converted into heat which is then supplied to a demand for thermal energy (thermal load) such as house heating, hot water heating or heat for industrial processes. This type of system may or may not include thermal storage, and usually include an auxiliary source of energy so that the demand may be met during long periods with no sunshine. [pic] Figure 1. 3 Diagram of a basic solar energy conversion systems. The AUX. box represents some auxiliary source of thermal or electrical energy.

If the demand (load) to be met is electricity (an electrical load) rather than heat, there are two common methods of converting solar energy into electricity. One method is by collecting solar energy as heat and converting it into electricity using a typical power plant or engine; the other method is by using photovoltaic cells to convert solar energy directly into electricity. Both methods are shown schematically in Figure 1. 3. In general, if solar energy conversion systems are connected to a large electrical transmission grid, no storage or auxiliary energy supply is needed.

If the solar energy conversion system is to be the only source of electricity, storage and auxiliary energy supply are usually both incorporated. If the thermal route is chosen, storage of heat rather than electricity may be used to extend the operating time of the system. Auxiliary energy may either be supplied either as heat before the power conversion system, or as electricity after it. If the photovoltaic route is chosen, extra electricity may be stored, usually in storage batteries, thereby extending the operating time of the system. For auxiliary power, an external electricity source is the only choice for photovoltaic systems. . 2 The Solar Resource The basic resource for all solar energy systems is the sun. Knowledge of the quantity and quality of solar energy available at a specific location is of prime importance for the design of any solar energy system. Although the solar radiation (insolation) is relatively constant outside the earth’s atmosphere, local climate influences can cause wide variations in available insolation on the earth’s surface from site to site. In addition, the relative motion of the sun with respect to the earth will allow surfaces with different orientations to intercept different amounts of solar energy.

Figure 1. 4 shows regions of high insolation where solar energy conversion systems will produce the maximum amount of energy from a specific collector field size. However, solar energy is available over the entire globe, and only the size of the collector field needs to be increased to provide the same amount of heat or electricity as in the shaded areas. It is the primary task of the solar energy system designer to determine the amount, quality and timing of the solar energy available at the site selected for installing a solar energy conversion system. [pic] Figure 1.    Areas of the world with high insolation. Just outside the earth’s atmosphere, the sun’s energy is continuously available at the rate of 1,367 Watts on every square meter facing the sun. Due to the earth’s rotation, asymmetric orbit about the sun, and the contents of its atmosphere, a large fraction of this energy does not reach the ground. In Chapter 2, we discuss the effects of the atmospheric processes that modify the incoming solar energy, how it is measured, and techniques used by designers to predict the amount of solar energy available at a particular location, both instantaneously and over a long term.

As an example of the importance of the material discussed in Chapter 2, Figure 1. 5 shows the variation of insolation over a full, clear day in March at Daggett, California, a meteorological measurement site close to the Kramer Junction solar power plant described previously. The outer curve, representing the greatest rate of incident energy, shows the energy coming directly from the sun (beam normal insolation) and falling on a square meter of surface area which is pointed toward the sun. The peak rate of incident solar energy occurs around 12:00 noon and is 1,030 Watts per square meter.

Over the full day, 10. 6 kilowatt-hours of energy has fallen on every square meter of surface area as represented by the area under this curve. [pic] Figure 1. 5   Insolation data from Daggett, California on a clear March day. The middle curve represents the rate of solar energy falling on a horizontal surface at the same location. For reasons to be discussed later this curve includes both the energy coming directly from the sun’s disc, and also that scattered by the molecules and particles in the atmosphere (total horizontal insolation). This scattered energy is shown as the bottom curve (diffuse insolation).

Over the entire day, 6. 7 kilowatt-hours of solar energy fall on every square meter of horizontal surface, of which 0. 7 kilowatt-hours comes from all directions other than directly from the sun. Techniques for estimating the temporal solar resource at any site on the face of the earth are presented in Chapter 2. In addition, the development and use of computerized meteorological data files is described. These data files based on long-term actual observations, form the time-dependent database of the computerized performance computations contained within this book and, indeed, much of the solar literature.

An example of a complete set of beam normal insolation data for a given location is shown in Figure 1. 6. Here we see hourly insolation data, summarized over a day, for each month of a year. With this type of data for a specific site, it is possible to predict accurately the output of a solar energy conversion system, whether it is a low temperature thermal system, a high temperature thermal system or a photovoltaic system. [pic] Figure 1. 6 Time and date description of the global, horizontal insolation solar resource for Cairo Egypt.

In addition to estimating the amount of energy coming from the sun, the solar designer must also be able to predict the position of the sun. The sun’s position must be known to predict the amount of energy falling on tilted surfaces, and to determine the direction toward which a tracking mechanism must point a collector. Chapter 3 discusses the computation of the position of the sun with respect to any given point on the face of the earth. Using only four parameters (latitude, longitude, date and local time), equations are derived to determine the location of the sun in the sky.

A characteristic fundamental to the capture of solar energy is that the amount of energy incident on a collector is reduced by a fraction equal to the cosine of the angle between the collector surface and the sun’s rays. Knowing the position of the collector (or any other surface for that matter) and the position of the sun equations in Chapter 3 may be used to predict the fraction of incoming solar energy that falls on the collector. These include situations where the collector is fixed or is tracked about a single axis, no matter what the orientation. 1. 3 Solar Collectors The solar collector is the key element in a solar energy system.

It is also the novel technology area that requires new understandings in order to make captured solar energy a viable energy source for the future. The function of a solar collector is simple; it intercepts incoming insolation and changes it into a useable form of energy that can be applied to meet a specific demand. In the following text, we will develop analytical understandings of flat-plate and concentrating collectors, as used to provide heat or electricity. Each type is introduced below. Flat-plate thermal solar collectors are the most commonly used type of solar collector. Their construction and operation are simple.

A large plate of blackened material is oriented in such a manner that the solar energy that falls on the plate is absorbed and converted to thermal energy thereby heating the plate. Tubes or ducting are provided to remove heat from the plate, transferring it to a liquid or gas, and carrying it away to the load. One (or more) transparent (glass or plastic) plates are often placed in front of the absorber plate to reduce heat loss to the atmosphere. Likewise, opaque insulation is placed around the backside of the absorber plate for the same purpose. Operating temperatures up to 125oC are typical.

Flat plate collectors have the advantage of absorbing not only the energy coming directly from the disc of the sun (beam normal insolation) but also the solar energy that has been diffused into the sky and that is reflected from the ground. Flat plate thermal collectors are seldom tracked to follow the sun’s daily path across the sky, however their fixed mounting usually provides a tilt toward the south to minimize the angle between the sun’s rays and the surface at noontime. Tilting flat-plate collectors toward the south provides a higher rate of energy at noontime and more total energy over the entire day. Figure 1. shows an installation of flat-plate thermal collectors. [pic] Figure 1. 7   Flat-plate thermal solar collectors for providing hot water. (photo courtesy of DOE/NREL, Warren Gretz) Flat-plate photovoltaic collectors contain an array of individual photovoltaic cells, connected in a series/parallel circuit, and encapsulated within a sandwich structure with the front surface being glass or plastic. Solar energy falls directly upon the photovoltaic cell front surface and produces a small direct current voltage, providing electrical energy to a load. Unlike thermal collectors however, the backside of the panel is not insulated.

Photovoltaic panels need to loose as much heat as possible to the atmosphere to optimize their performance. Like flat-plate thermal collectors, flat-plate photovoltaic collectors (panels) absorb both energy coming directly from the sun’s disc, and diffuse and reflected energy coming from other directions. In general, flat-plate photovoltaic panels are mounted in a fixed position and tilted toward the south to optimize noontime and daily energy production. However, it is common to see flat-plate photovoltaic panels mounted on mechanisms that track the sun about one tilted axis, thereby increasing the daily output of the panels. pic] [pic] [pic] Figure 1. 8   Flat-plate photovoltaic collector applications. (photos courtesy of DOE/NREL, Warren Gretz) When higher temperatures are required, concentrating solar collectors are used. Solar energy falling on a large reflective surface is reflected onto a smaller area before it is converted into heat. This is done so that the surface absorbing the concentrated energy is smaller than the surface capturing the energy and therefore can attain higher temperatures before heat loss due to radiation and convection wastes the energy that has been collected.

Most concentrating collectors can only concentrate the parallel insolation coming directly from the sun’s disk (beam normal insolation), and must follow (track) the sun’s path across the sky. Four types of solar concentrators are in common use; parabolic troughs (as used in the Kramer Junction, California solar energy electricity generating plant shown in Figure 1. 1), parabolic dishes, central receivers and Fresnel lenses. Figure 1. 9 shows these concepts schematically. [pic] Figure 1. 9   Three commonly used reflecting schemes for concentrating solar energy to attain high temperatures.

A parabolic trough concentrates incoming solar radiation onto a line running the length of the trough. A tube (receiver) carrying heat transfer fluid is placed along this line, absorbing concentrated solar radiation and heating the fluid inside. The trough must be tracked about one axis. Because the surface area of the receiver tube is small compared to the trough capture area (aperture), temperatures up to 400oC can be reached without major heat loss. Figure 1. 10c shows one parabolic trough from the Kramer Junction, California field shown in Figure 1. 1. A parabolic dish concentrates the incoming solar radiation to a point.

An insulated cavity containing tubes or some other heat transfer device is placed at this point absorbing the concentrated radiation and transferring it to a gas. Parabolic dishes must be tracked about two axes. Figure 1. 10b shows six 9kWe parabolic dish concentrators with Stirling engines attached to the receiver at the focus. A central receiver system consists of a large field of independently movable flat mirrors (heliostats) and a receiver located at the top of a tower. Each heliostat moves about two axes, throughout the day, to keep the sun’s image reflected onto the receiver at the top of the tower.

The receiver, typically a vertical bundle of tubes, is heated by the reflected insolation, thereby heating the heat transfer fluid passing through the tubes. Figure 1. 10a shows the 10 MWe Solar One central receiver generating plant at Daggett, California with its adjoining steam power plant. A Fresnel lens concentrator, such as shown in Figure 1. 10d uses refraction rather than reflection to concentrate the solar energy incident on the lens surface to a point. Usually molded out of inexpensive plastic, these lenses are used in photovoltaic concentrators.

Their use is not to increase the temperature, but to enable the use of smaller, higher efficiency photovoltaic cells. As with parabolic dishes, point-focus Fresnel lenses must track the sun about two axes. [pic] Figure 1. 10a A central receiver system. (courtesy of Sandia National Laboratories, Albuquerque) [pic] Figure 1. 10b Two-axis tracking parabolic dish collectors. (courtesy of Schlaich, Bergermann und Partner) [pic] Figure 1. 10c A single-axis tracking parabolic trough collector. (courtesy of Kramer Junction Operating Company) [pic] Figure 1. 0d A concentrating photovoltaic collector using Fresnel lenses. (courtesy of Amonix Corp. ) 1. 4 Need for Storage Like with any other power plant, solar power plant output must satisfy the demands of the utility market. During peak demand periods, kilowatt-hour prices are high and financial incentives are high for guaranteed supply. Solar plant input is limited by diurnal, seasonal and weather-related insolation changes. In order to cope with these fluctuations, the solar plant input may be backed up by fossil fuels, or the solar changes may be mitigated by a buffering storage system.

The choice depends on demands, system and site conditions, the relationship between storage capacity and collector area is discussed in Chapter 10. In thermal solar power plants, thermal storage and/or fossil backup act as: • an output management tool to prolong operation after sunset, to shift energy sales from low revenue off-peak hours to high revenue peak demand hours, and to contribute to guaranteed output • An internal plant buffer, smoothing out insolation charges for steadying cycle operation, and for operational requirements such as blanketing steam production, component pre-heating and freeze protection.

Photovoltaic plants in general need no internal buffer, and output management can be achieved with battery or other electrochemical storage, pumped hydroelectric storage, or with diesel-generator backup. The implications of battery storage are discussed in Chapter 10. [pic] Figure 1. 11   Stored solar energy provides a firm capacity of 31MW until midnight at which time fossil fuel backup us used. 1. 5   Integration with Power Cycles Because of their thermal nature, all the solar thermal technologies can be hybridized, or operated with fossil fuel as well as solar energy.

Hybridization has the potential to increase the value of concentrating solar thermal technology by increasing its availability and dispatchability, decreasing its cost (by making more effective use of power generation equipment), and reducing technological risk by allowing conventional fuel use when needed. Although an interconnected field of solar thermal collectors and thermal energy storage may be sufficient for providing high temperature heat directly to a thermal demand, a power generation subsystem must be incorporated into the system design if mechanical work or electrical power is to be an output from the system.

Chapter 11 reviews the technology for power generation with particular emphasis on power generation units suitable for interfacing with solar thermal energy collection subsystems. The inclusion of power generation in a solar thermal energy design presents a challenge in selecting the appropriate design conditions. The efficiency of a power generation unit usually increases with the operating temperature of the power generation cycle, whereas the efficiency of solar collectors decreases with temperature. A tradeoff must be performed to determine the best system design point. [pic]

Figure 1. 12   One of the steam cycle power cycles at the Kramer Junction solar energy generating system. (photo courtesy of DOE/NREL, Warren Gretz) 1. 6   Site Qualification Solar technologies using concentrating systems for electrical production require sufficient beam normal radiation, which is the beam radiation which comes from the sun and passes through the planet’s atmosphere without deviation and refraction. Consequently, appropriate site locations are normally situated in arid to semi-arid regions. On a global scale, the solar resource in such regions is very high.

More exactly, acceptable production costs of solar electricity typically occur where radiation levels exceed about 1,700 kWh/m? -yr, a radiation level found in many areas as illustrated in Figure 1. 4. Appropriate regions include the southwest United States, northern Mexico, the north African desert, the Arabian peninsula, major portions of India, central and western Australia, the high plateaus of the Andean states, and northeastern Brazil. Promising site locations in Europe are found in southern Spain and several Mediterranean islands. [pic] Figure 1. 3 A View of Kuraymat (Egypt), the envisaged site for a solar thermal power plant in the Egyptian desert with cooling water from the Nile and connections to the national high voltage grid. Solar electricity generation costs and feasibility of the project highly depend on the project site itself. A good site has to have a high annual beam insolation to obtain maximum solar electricity output. It must be reasonably flat to accommodate the solar field without prohibitive expensive earth works. It must also be close to the grid and a substation to avoid the need to build expensive electricity lines for evacuating the power.

It needs sufficient water supply to cover the demand for cooling water of its steam cycle. A backup fuel must be available for granting firm power during the times when no solar energy is available. Access roads must be suitable for transporting the heavy equipment like turbine generators to the site. Skilled personnel must be available to construct and operate the plants. Chapter 13 reviews the criteria, methodology and examples of site selection and qualification for solar plants. 1. 7    Economic and Environmental Considerations

The most important factor driving the solar energy system design process is whether the energy it produces is economical. Although there are factors other than economics that enter into a decision of when to use solar energy; i. e. no pollution, no greenhouse gas generation, security of the energy resource etc. , design decisions are almost exclusively dominated by the ‘levelized energy cost’. This or some similar economic parameter, gives the expected cost of the energy produced by the solar energy system, averaged over the lifetime of the system.

In the following chapters, we will provide tools to aid in evaluating the factors that go into this calculation. Commercial applications from a few kilowatts to hundreds of megawatts are now feasible, and plants totaling 354 MW have been in operation in California since the 1980s. Plants can function in dispatchable, grid-connected markets or in distributed, stand-alone applications. They are suitable for fossil-hybrid operation or can include cost-effective storage to meet dispatchability requirements.

They can operate worldwide in regions having high beam-normal insolation, including large areas of the southwestern United States, and Central and South America, Africa, Australia, China, India, the Mediterranean region, and the Middle East, . Commercial solar plants have achieved levelized energy costs of about 12-15? /kWh, and the potential for cost reduction are expected to ultimately lead to costs as low as 5? /kWh. [pic] Figure 1. 14  Projections of levelized electricity cost predictions for large scale solar thermal power plants.

Current costs are shown in blue with a 1-2 cent/kWh addition for ‘green’ power shown in green. 1. 8    Summary The authors’ overall objective is to illustrate the design of solar energy systems, both thermal and photovoltaic types. To do this, we examine the solar resource and the ability of various types of solar collectors to capture it effectively. Design tools are developed which integrate performance of isolated solar collectors, along with energy storage, into a larger system that delivers either electrical or thermal energy to a demand.

We show as many examples as possible, both graphic and photographic of these systems and their components. It is our hope that once the simplicity of solar energy system design is understood, engineers and manufacturers will provide new system designs that will expand the solar market worldwide and permit all to benefit from this clean, sustainable and distributed source of energy. [pic]3. The Sun’s Position In order to understand how to collect energy from the sun, one must first be able to predict the location of the sun relative to the collection device.

In this chapter we develop the necessary equations by use of a unique vector approach. This approach will be used in subsequent chapters to develop the equations for the sun’s position relative to a fixed or tracking solar collector, (Chapter 4) and the special case of a sun-tracking mirror reflecting sunlight onto a fixed point (Chapter 10). Once developed, the sun position expressions of this chapter are used to demonstrate how to determine the location of shadows and the design of simple sundials. In outline form, our development looks like this: o Earth-sun angles Time o Standard time zones o Daylight savings time o Sidereal time o Hour angle [pic][pic][pic] o Solar time o Equation of time o Time conversion o Declination angle [pic][pic][pic][pic][pic][pic] o Latitude angle [pic][pic][pic][pic][pic][pic] o Observer-Sun Angles Solar altitude, zenith and azimuth angles o Geometric view of sun’s path o Daily and seasonal events o Shadows and Sundials ? Simple shadows ? Sundials o Notes on Transformation of Vector Coordinates o Summary Although many intermediate steps of derivation used to obtain the equations described in this chapter have been omitted, it is hoped that there are adequate comments between steps to encourage the student to perform the derivation, thereby enhancing understanding of the materials presented.

Brief notes on the transformation of vector coordinates are included as Section 3. 5 and a summary of sign conventions for all of the angles used in this chapter is given in Table 3. 3 at the end of this chapter. Figures defining each angle and an equation to calculate it are also included. One objective in writing this chapter has been to present adequate analytical expressions so that the solar designer is able to develop simple computer algorithms for predicting relative sun and collector positions for exact design conditions and locations. This will eliminate the need to depend on charts and tables and simplified equations. 3. Earth-Sun Angles The earth revolves around the sun every 365. 25 days in an elliptical orbit, with a mean earth-sun distance of 1. 496 x 1011 m (92. 9 x 106 miles) defined as one astronomical unit (1 AU). This plane of this orbit is called the ecliptic plane. The earth’s orbit reaches a maximum distance from the sun, or aphelion, of 1. 52 ? 1011 m (94. 4 ? 106 miles) on about the third day of July. The minimum earth-sun distance, the perihelion, occurs on about January 2nd, when the earth is 1. 47 ? 1011 m (91. 3 ? 106 miles) from the sun. Figure 3. 1 depicts these variations in relation to the Northern Hemisphere seasons. [pic]

Figure 3. 1 The ecliptic plane showing variations in the earth-sun distance and the equinoxes and solstices. The dates and day numbers shown are for 1981 and may vary by 1 or 2 days. The earth rotates about its own polar axis, inclined to the ecliptic plane by 23. 45 degrees, in approximately 24-hour cycles. The direction in which the polar axis points is fixed in space and is aligned with the North Star (Polaris) to within about 45 minutes of arc (13 mrad). The earth’s rotation about its polar axis produces our days and nights; the tilt of this axis relative to the ecliptic plane produces our seasons as the earth revolves about the sun. . 1. 1 Time We measure the passage of time by measuring the rotation of the earth about its axis. The base for time (and longitude) measurement is the meridian that passes through Greenwich, England and both poles. It is known as the Prime Meridian. Today, the primary world time scale, Universal Time (previously called Greenwich Mean Time), is still measured at the Prime Meridian. This is a 24-hour time system, based on mean time, according to which the length of a day is 24 hours and midnight is 0 hours. Mean time is based on the length of an average day.

A mean second is l/86,400 of the average time between one complete transits of the sun, averaged over the entire year. In fact, the length of any one specific day, measured by the complete transit of the sun, can vary by up to 30 seconds during the year. The variable day length is due to four factors listed in order of decreasing importance (Jesperson and Fitz-Randolph, 1977): • The earth’s orbit around the sun is not a perfect circle but elliptical, so the earth travels faster when it is nearer the sun than when it is farther away. The earth’s axis is tilted to the plane containing its orbit around the sun. • The earth spins at an irregular rate around its axis of rotation. • The earth `wobbles? on its axis. Standard Time Zones – Since it is conventional to have 12:00 noon be approximately in the middle of the day regardless of the longitude, a system of time zones has been developed. See (Blaise 2000) for an interesting story about how this unification developed. These are geographic regions, approximately 15 degrees of longitude wide, centered about a meridian along which local standard time equals mean solar time.

Prior to about 1880, different cities (and even train stations) had their own time standards, most based on the sun being due south at 12:00 noon. Time is now generally measured about standard time zone meridians. These meridians are located every 15 degrees from the Prime Meridian so that local time changes in 1-hour increments from one standard time zone meridian to the next. The standard time zone meridians east of Greenwich have times later than Greenwich time, and the meridians to the west have earlier times.

Ideally, the meridians 7 [pic][pic][pic][pic][pic][pic]degrees on either side of the standard time zone meridian should define the time zone. However, boundaries separating time zones are not meridians but politically determined borders following rivers, county, state or national boundaries, or just arbitrary paths. Countries such as Spain choose to be on `European Time (15o E) when their longitudes are well within the adjacent Standard Time Meridian (0o). Figure 3. 2 shows these time zone boundaries within the United States and gives the standard time zone meridians (called longitudes of solar time). pic][pic] Figure 3. 2 Time zone boundaries within the United States. From Jesperson and Fitz-Randolph 1977. Daylight Savings Time -To complicate matters further in trying to correlate clock time with the movement of the sun, a concept known as daylight savings time was initiated in the United States in the spring of 1918 to “save fuel and promote other economies in a country at war” (Jesperson and Fitz-Randolph, 1977). According to this concept, the standard time is advanced by 1 hour, usually from 2:00 AM on the first Sunday in April until 2:00 AM on the last Sunday in October.

Although various attempts have been made to apply this concept uniformly within the country, it is suggested that the designer check locally to ascertain the commitment to this concept at any specific solar site. Sidereal Time – So far, and for the remainder of this text, all reference to time is to mean time, a time system based on the assumption that a day (86,400 seconds) is the average interval between two successive times when a given point on the earth faces the sun. In astronomy or orbital mechanics, however, the concept of sidereal time is often used.

This time system is based on the sidereal day, which is the length of time for the earth to make one complete rotation about its axis. The mean day is about 4 minutes longer than the sidereal day because the earth, during the time it is making one revolution about its axis, has moved some distance in its orbit around the sun. To be exact, the sidereal day contains 23 hours, 56 minutes, and 4. 09053 seconds of mean time. Since, by definition, there are 86,400 sidereal seconds in a sidereal day, the sidereal second is slightly shorter than the mean solar second is.

To be specific: 1 mean second = 1. 002737909 sidereal seconds. A detailed discussion of this and other time definitions is contained in another work (Anonymous, 1981, Section B). 3. 1. 2 The Hour Angle [pic][pic][pic] [pic]To describe the earth’s rotation about its polar axis, we use the concept of the hour angle[pic][pic][pic]. As shown in Figure 3. 3, the hour angle is the angular distance between the meridian of the observer and the meridian whose plane contains the sun. The hour angle is zero at solar noon (when the sun reaches its highest point in the sky).

At this time the sun is said to be ‘due south’ (or ‘due north’, in the Southern Hemisphere) since the meridian plane of the observer contains the sun. The hour angle increases by 15 degrees every hour. [pic] Figure 3. 3 The hour angle [pic][pic][pic] . This angle is defined as the angle between the meridian parallel to sun rays and the meridian containing the observer. [pic]Solar Time – Solar time is based on the 24-hour clock, with 12:00 as the time that the sun is exactly due south.

The concept of solar time is used in predicting the direction of sunrays relative to a point on the earth. Solar time is location (longitude) dependent and is generally different from local clock time, which is defined by politically defined time zones and other approximations. Solar time is used extensively in this text to define the rotation of the earth relative to the sun. An expression to calculate the hour angle from solar time is [pic][pic][pic][pic][pic][pic]                         (3. 1) [pic]where ts is the solar time in hours.

EXAMPLE: When it is 3 hours after solar noon, solar time is 15:00 and the hour angle has a value of 45 degrees. When it is 2 hours and 20 minutes before solar noon, solar time is 9:40 and the hour angle is 325 degrees (or [pic][pic][pic][pic][pic][pic][pic][pic][pic]35 degrees). [pic]The difference between solar time and local clock time can approach 2 hours at various locations and times in the United States, For most solar design purposes, clock time is of little concern, and it is appropriate to present data in terms of solar time.

Some situations, however, such as energy demand correlations, system performance correlations, determination of true south, and tracking algorithms require an accurate knowledge of the difference between solar time and the local clock time. Knowledge of solar time and Universal Time has traditionally been important to ship navigators. They would set their chronometers to an accurately adjusted tower clock visible as they left port. This was crucial for accurate navigation. At sea a ship’s latitude could be easily ascertained by determining the maximum altitude angle of the sun or the altitude angle of Polaris at night.

However, determining the ship’s longitude was more difficult and required that an accurate clock be carried onboard. If the correct time at Greenwich, England (or any other known location) was known, then the longitude of the ship could be found by measuring the solar time onboard the ship (through sun sightings) and subtracting from it the time at Greenwich. Since the earth rotates through 360 degrees of longitude every 24 hours, the ship then has traveled 1 degree of longitude away from the Prime Meridian (which passes through Greenwich) for every 4 minutes of time difference.

An interesting story about developing accurate longitude measurements may be found in Sobel, 1999. Equation of Time – The difference between mean solar time and true solar time on a given date is shown in Figure 3. 4. This difference is called the equation of time (EOT). Since solar time is based on the sun being due south at 12:00 noon on any specific day, the accumulated difference between mean solar time and true solar time can approach 17 minutes either ahead of or behind the mean, with an annual cycle. The level of accuracy required in determining the equation of time will depend on hether the designer is doing system performance or developing tracking equations. An approximation for calculating the equation of time in minutes is given by Woolf (1968) and is accurate to within about 30 seconds during daylight hours. [pic][pic][pic][pic][pic][pic](3. 2) [pic]where the angle x is defined as a function of the day number N [pic][pic][pic][pic][pic][pic][pic][pic][pic]                     (3. 3) [pic]with the day number, N being the number of days since January 1. Table 3. 1 has been prepared as an aid in rapid determination of values of N from calendar dates. Table 3. 1 Date-to-Day Number Conversion Month |Day Number, N |Notes | |January |d | | |February |d + 31 | | |March |d + 59 |Add 1 if leap year | |April |d + 90 |Add 1 if leap year | |May |d + 120 |Add 1 if leap year | |June |d + 151 |Add 1 if leap year | |July |d + 181 |Add 1 if leap year | |August |d + 212 |Add 1 if leap year | |September |d + 243 |Add 1 if leap year | |October |d + 273 |Add 1 if leap year | |November |d + 304 |Add 1 if leap year | |December |d + 334 |Add 1 if leap year | |Days of Special Solar Interest | |Solar Event |Date |Day Number, N | |Vernal equinox |March 21 |80 | |Summer solstice |June 21 |172 | |Autumnal equinox |September 23 |266 | |Winter solstice |December 21 |355 | |NOTES: | |d is the day of the month | |Leap years are 2000, 2004, 2008 etc. | |Solstice and equinox dates may vary by a day or two. Also, add 1 to the solstice and | |equinox day number for leap years. | [pic] Figure 3. 4 The equation of time (EOT).

This is the difference between the local apparent solar time and the local mean solar time. EXAMPLE: February 11 is the 42nd day of the year, therefore N = 42 and x is equal to 40. 41 degrees, and the equation of time as calculated above is [pic][pic][pic][pic][pic][pic][pic][pic][pic]14. 35 minutes. This compares with a very accurately calculated value of -14. 29 minutes reported elsewhere (Anonymous, 198l). This means that on this date, there is a difference between the mean time and the solar time of a little over 14 minutes or that the sun is “slow” relative to the clock by that amount. [pic][pic]To satisfy the control needs of concentrating collectors, a more accurate determination of the hour angle is often needed.

An approximation of the equation of time claimed to have an average error of 0. 63 seconds and a maximum absolute error of 2. 0 seconds is presented below as Equation (3. 4) taken from Lamm (1981). The resulting value is in minutes and is positive when the apparent solar time is ahead of mean solar time and negative when the apparent solar time is behind the mean solar time: [pic][pic][pic][pic][pic][pic]                     (3. 4) [pic]Here n is the number of days into a leap year cycle with n = 1 being January 1 of each leap year, and n =1461 corresponding to December 31 of the 4th year of the leap year cycle. The coefficients Ak and Bk are given in Table 3. 2 below.

Arguments for the cosine and sine functions are in degrees. Table 3. 2 Coefficients for Equation (3. 4) |k |Ak (hr) |Bk (hr) | |0 |2. 0870 ? 10-4 |0 | |1 |9. 2869 ? 10-3 |[pic][pic][pic][pic][pic][p| | | |ic][pic][pic][pic]1. 2229 ? | | | |10-1 | |2 |[pic][pic][pic][pic][pic][pic|[pic][pic][pic][pic][pic][p| | |][pic][pic][pic]5. 2258 ? 10-2|ic][pic][pic][pic]1. 5698 ? | | |10-1 | |3 |[pic][pic][pic][pic][pic][pic|[pic][pic][pic][pic][pic][p| | |][pic][pic][pic]1. 3077 ? l0-3|ic][pic][pic][pic]5. 1 602 ? | | | |10-3 | |4 |[pic][pic][pic][pic][pic][pic|[pic][pic][pic][pic][pic][p| | |][pic][pic][pic]2. 1867 ? l0-3|ic][pic][pic][pic]2. 9823 ? | | | |10-3 | |5 |[pic][pic][pic][pic][pic][pic|[pic][pic][pic][pic][pic][p| | |][pic][pic][pic]1. 5100 ? 10-4|ic][pic][pic][pic]2. 3463 ? | | |10-4 | [pic][pic][pic][pic][pic][pic][pic][pic][pic]Time Conversion – The conversion between solar time and clock time requires knowledge of the location, the day of the year, and the local standards to which local clocks are set. Conversion between solar time, ts and local clock time (LCT) (in 24-hour rather than AM/ PM format) takes the form [pic][pic][pic][pic][pic][pic]                     (3. 5) [pic]where EOT is the equation of time in minutes and LC is a longitude correction defined as follows: [pic][pic][pic][pic][pic][pic]                     (3. 6) [pic]and the parameter D in Equation (3. ) is equal to 1 (hour) if the location is in a region where daylight savings time is currently in effect, or zero otherwise. EXAMPLE: Let us find the clock time for solar noon at a location in Los Angeles, having a longitude of 118. 3 degrees on February 11. Since Los Angeles is on Pacific Standard Time and not on daylight savings time on this date, the local clock time will be: [pic][pic][pic][pic][pic][pic] [pic]3. 1. 3 The Declination Angle [pic][pic][pic] [pic]The plane that includes the earth’s equator is called the equatorial plane. If a line is drawn between the center of the earth and the sun, the angle between this line and the earth’s equatorial plane is called the declination angle[pic][pic][pic][pic], as depicted in Figure 3. 5.

At the time of year when the northern part of the earth’s rotational axis is inclined toward the sun, the earth’s equatorial plane is inclined 23. 45 degrees to the earth-sun line. At this time (about June 21), we observe that the noontime sun is at its highest point in the sky and the declination angle [pic][pic][pic][pic][pic][pic]= +23. 45 degrees. We call this condition the summer solstice, and it marks the beginning of summer in the Northern Hemisphere. [pic][pic]As the earth continues its yearly orbit about the sun, a point is reached about 3 months later where a line from the earth to the sun lies on the equatorial plane. At this point an observer on the equator would observe that the sun was directly overhead at noontime.

This condition is called an equinox since anywhere on the earth, the time during which the sun is visible (daytime) is exactly 12 hours and the time when it is not visible (nighttime) is 12 hours. There are two such conditions during a year; the autumnal equinox on about September 23, marking the start of the fall; and the vernal equinox on about March 22, marking the beginning of spring. At the equinoxes, the declination angle [pic][pic][pic][pic][pic][pic]is zero. [pic][pic] Figure 3. 5 The declination angle[pic][pic][pic][pic][pic][pic]. The earth is shown in the summer solstice position when [pic][pic][pic][pic][pic][pic]= +23. 45 degrees.

Note the definition of the tropics as the intersection of the earth-sun line with the surface of the earth at the solstices and the definition of the Arctic and Antarctic circles by extreme parallel sun rays. [pic][pic]The winter solstice occurs on about December 22 and marks the point where the equatorial plane is tilted relative to the earth-sun line such that the northern hemisphere is tilted away from the sun. We say that the noontime sun is at its “lowest point” in the sky, meaning that the declination angle is at its most negative value (i. e. , [pic][pic][pic][pic][pic][pic]= -23. 45 degrees). By convention, winter declination angles are negative. pic]Accurate knowledge of the declination angle is important in navigation and astronomy. Very accurate values are published annually in tabulated form in an ephemeris; an example being (Anonymous, 198l). For most solar design purposes, however, an approximation accurate to within about 1 degree is adequate. One such approximation for the declination angle is [pic][pic][pic][pic][pic][pic]                     (3. 7) [pic]where the argument of the cosine here is in degrees and N is the day number defined for Equation (3. 3) The annual variation of the declination angle is shown in Figure 3. 5. 3. 1. 4 Latitude Angle [pic][pic][pic][pic][pic][pic] pic]The latitude angle [pic][pic][pic][pic][pic][pic]is the angle between a line drawn from a point on the earth’s surface to the center of the earth, and the earth’s equatorial plane. The intersection of the equatorial plane with the surface of the earth forms the equator and is designated as 0 degrees latitude. The earth’s axis of rotation intersects the earth’s surface at 90 degrees latitude (North Pole) and -90 degrees latitude (South Pole). Any location on the surface of the earth then can be defined by the intersection of a longitude angle and a latitude angle. [pic]Other latitude angles of interest are the Tropic of Cancer (+23. 45 degrees latitude) and the Tropic of Capricorn (- 23. 45 degrees latitude).

These represent the maximum tilts of the north and south poles toward the sun. The other two latitudes of interest are the Arctic circle (66. 55 degrees latitude) and Antarctic circle (-66. 5 degrees latitude) representing the intersection of a perpendicular to the earth-sun line when the south and north poles are at their maximum tilts toward the sun. As will be seen below, the tropics represent the highest latitudes where the sun is directly overhead at solar noon, and the Arctic and Antarctic circles, the lowest latitudes where there are 24 hours of daylight or darkness. All of these events occur at either the summer or winter solstices. 3. 2 Observer-Sun Angles

When we observe the sun from an arbitrary position on the earth, we are interested in defining the sun position relative to a coordinate system based at the point of observation, not at the center of the earth. The conventional earth-surface based coordinates are a vertical line (straight up) and a horizontal plane containing a north-south line and an east-west line. The position of the sun relative to these coordinates can be described by two angles; the solar altitude angle and the solar zenith angle defined below. Since the sun appears not as a point in the sky, but as a disc of finite size, all angles discussed in the following sections are measured to the center of that disc, that is, relative to the “central ray” from the sun. 3. 2. 1 Solar Altitude, Zenith, and Azimuth Angles

The solar altitude angle [pic][pic][pic][pic][pic][pic]is defined as the angle between the central ray from the sun, and a horizontal plane containing the observer, as shown in Figure 3. 6. As an alternative, the sun’s altitude may be described in terms of the solar zenith angle [pic][pic][pic][pic][pic][pic]which is simply the complement of the solar altitude angle or [pic][pic][pic][pic][pic][pic][pic][pic]                     (3. 8) [pic]The other angle defining the position of the sun is the solar azimuth angle (A). It is the angle, measured clockwise on the horizontal plane, from the north-pointing coordinate axis to the projection of the sun’s central ray. [pic] Figure 3. Earth surface coordinate system for observer at Q showing the solar azimuth angle ? , the solar altitude angle [pic][pic][pic][pic][pic][pic]and the solar zenith angle [pic][pic][pic][pic][pic][pic]for a central sun ray along direction vector S. Also shown are unit vectors i, j, k along their respective axes. [pic][pic]The reader should be warned that there are other conventions for the solar azimuth angle in use in the solar literature. One of the more common conventions is to measure the azimuth angle from the south-pointing coordinate rather than from the north-pointing coordinate. Another is to consider the counterclockwise direction positive rather than clockwise.

The information in Table 3. 3 at the end of this chapter will be an aid in recognizing these differences when necessary. It is of the greatest importance in solar energy systems design, to be able to calculate the solar altitude and azimuth angles at any time for any location on the earth. This can be done using the three angles defined in Section 3. 1 above; latitude [pic][pic][pic][pic][pic][pic], hour angle [pic][pic][pic][pic][pic][pic], and declination [pic][pic][pic][pic][pic][pic]. If the reader is not interested in the details of this derivation, they are invited to skip directly to the results; Equations (3. 17), (3. 18) and (3. 19). pic][pic][pic]For this derivation, we will define a sun-pointing vector at the surface of the earth and then mathematically translate it to the center of the earth with a different coordinate system. Using Figure 3. 6 as a guide, define a unit direction vector S pointing toward the sun from the observer location Q: [pic][pic][pic][pic][pic][pic]                     (3. 9) [pic]where i, j, and k are unit vectors along the z, e, and n axes respectively. The direction cosines of S relative to the z, e, and n axes are Sz, Se and Sn, respectively. These may be written in terms of solar altitude and azimuth as [pic][pic][pic][pic][pic][pic]                     (3. 10) [pic]Similarly, a direction vector pointing to the sun can be described at the center of the earth as shown in Figure 3. 7.

If the origin of a new set of coordinates is defined at the earth’s center, the m axis can be a line from the origin intersecting the equator at the point where the meridian of the observer at Q crosses. The e axis is perpendicular to the m axis and is also in the equatorial plane. The third orthogonal axis p may then be aligned with the earth’s axis of rotation. A new direction vector S? pointing to the sun may be described in terms of its direction cosines S? m , S’e and S’p relative to the m, e, and p axes, respectively. Writing these in terms of the declination and hour angles, we have [pic][pic][pic][pic][pic][pic]                     (3. 11) [pic]Note that S? e is negative in the quadrant shown in Figure 3. 7. [pic] Figure 3. Earth center coordinate system for the sun ray direction vector S defined in terms of hour angle [pic][pic][pic][pic][pic][pic]and the declination angle [pic][pic][pic][pic][pic][pic]. [pic][pic]These two sets of coordinates are interrelated by a rotation about the e axis through the latitude angle [pic][pic][pic][pic][pic][pic]and translation along the earth radius QC. We will neglect the translation along the earth’s radius since this is about 1 / 23,525 of the distance from the earth to the sun, and thus the difference between the direction vectors S and S? is negligible. The rotation from the m, e, p coordinates to the z, e, n coordinates, about the e axis is depicted in Figure 3. 8. Both sets of coordinates are summarized in Figure 3. 9.

Note that this rotation about the e axis is in the negative sense based on the right-hand rule. In matrix notation, this takes the form [pic][pic][pic][pic][pic][pic][pic]                     (3. 12) [pic][pic] Figure 3. 8 Earth surface coordinates after translation from the observer at Q to the earth center C. [pic] Figure 3. 9 Composite view of Figures 3. 6, 3. 7 and 3. 8 showing parallel sun ray vectors S and S’ relative to the earth surface and the earth center coordinates. Solving, we have [pic][pic][pic][pic][pic][pic]                     (3. 13) [pic]Substituting Equations (3. 10) and (3. 1)(3. 11) for the direction cosine gives us our three important results [pic][pic][pic][pic][pic][pic]                     (3. 4) [pic][pic][pic][pic][pic][pic][pic]                     (3. 15) [pic][pic][pic][pic][pic][pic][pic]                     (3. 16) [pic]Equation (3. 14) is an expression for the solar altitude angle in terms of the observer’s latitude (location), the hour angle (time), and the sun’s declination (date). Solving for the solar altitude angle [pic][pic][pic][pic][pic][pic], we have [pic][pic][pic][pic][pic][pic][pic]                     (3. 17) [pic]Two equivalent expressions result for the solar azimuth angle (A) from either Equation (3. 15) or (3. 16). To reduce the number of variables, we could substitute Equation (3. 14) into either Equation (3. 15) or (3. 6); however, this substitution results in additional terms and is often omitted to enhance computational speed in computer codes. The solar azimuth angle can be in any of the four trigonometric quadrants depending on location, time of day, and the season. Since the arc sine and arc cosine functions have two possible quadrants for their result, both Equations (3. 15) and (3. 16) require a test to ascertain the proper quadrant. No such test is required for the solar altitude angle, since this angle exists in only one quadrant. The appropriate procedure for solving Equation (3. 15) is to test the result to determine whether the time is before or after solar noon. For Equation (3. 5), a test must be made to determine whether the solar azimuth is north or south of the east-west line. Two methods for calculating the solar azimuth (A), including the appropriate tests, are given by the following equations. Again, these are written for the azimuth angle sign convention used in this text, that is, that the solar azimuth angle is measured from due north in a clockwise direction, as with compass directions. Solving Equation (3. 15) the untested result, A’ then becomes [pic][pic][pic][pic][pic][pic]                     (3. 18) [pic]A graphical description of this test will follow in the next section. Solving Equation (3. 16), the untested result, A” becomes pic][pic][pic][pic][pic][pic]                     (3. 19) [pic]In summary, we now have equations for both the sun’s altitude angle and azimuth angle written in terms of the latitude, declination and hour angles. This now permits us to calculate the sun’s position in the sky, as a function of date, time and location (N, [pic][pic][pic][pic][pic][pic],[pic][pic][pic][pic][pic][pic]). [pic][pic]EXAMPLE: For a site in Miami, Florida (25 degrees, 48 minutes north latitude/ 80 degrees, 16 minutes west longitude) at 10:00 AM solar time on August 1 (not a leap year), find the sun’s altitude, zenith and azimuth angles….. For these conditions, the declination angle is calculated to be 17. 0 degrees, the hour angle -30 degrees and the sun’s altitude angle is then 61. 13 degrees, the zenith angle 28. 87 degrees and the azimuth angle 99. 77 degrees. 3. 2. 2  A Geometric View of Sun’s Path The path of the sun across the sky can be viewed as being on a disc displaced from the observer. This “geometric” view of the sun’s path can be helpful in visualizing sun movements and in deriving expressions for testing the sun angles as needed for Equation (3. 18) to ascertain whether the sun is in the northern sky. The sun may be viewed as traveling about a disc having a radius R at a constant rate of 15 degrees per hour. As shown in Figure 3. 0, the center of this disc appears at different seasonal locations along the polar axis, which passes through the observer at Q and is inclined to the horizon by the latitude angle [pic][pic][pic][pic][pic][pic]pointing toward the North Star (Polaris). [pic][pic] Figure 3. 10 A geometric view of the sun’s path as seen by an observer at Q. Each disc has radius R. The center of the disc is coincident with the observer Q at the equinoxes and is displaced from the observer by a distance of [pic][pic][pic][pic][pic][pic]at other times of the year. The extremes of this travel are at the solstices when the disc is displaced by ± 0. 424 R along the polar axis. It can be seen that in the winter, much of the disc is “submerged” below the horizon, giving rise to fewer hours of daylight and low sun elevations as viewed from Q. pic]At the equinoxes, the sun rises exactly due east at exactly 6:00 AM (solar time) and appears to the observer to travel at a constant rate across the sky along a path inclined from the vertical by the local latitude angle. Exactly one-half of the disc is above the horizon, giving the day length as 12 hours. At noon, the observer notes that the solar zenith angle is the same as the local latitude. The sun sets at exactly 6:00 PM, at a solar azimuth angle of exactly 270 degrees or due west. In the summer, the center of the disc is above the observer, giving rise to more hours of daylight and higher solar altitude angles, with the sun appearing in the northern part of the sky in the mornings and afternoons.

Since the inclination of the polar axis varies with latitude it can be visualized that there are some latitudes where the summer solstice disc is completely above the horizon surface. It can be shown that this occurs for latitudes greater than 66. 55 degrees, that is, above the Arctic Circle. At the equator, the polar axis is horizontal and exactly half of any disc appears above the horizon surface, which means that the length of day and night is 12 hours throughout the year. A test to determine whether the sun is in the northern part of the sky may be developed by use of this geometry. Figure 3. 11 is a side view of the sun’s disc looking from the east. [pic]

Figure 3. 11 Side view of sun path disc during the summer when the disc center Y is above the observer at Q. In the summer the sun path disc of radius R has its center Y displaced above the observer Q. Point X is defined by a perpendicular from Q. In the n – z plane, the projection of the position S onto the line containing X and Y will be [pic][pic][pic][pic][pic][pic]where [pic][pic][pic][pic][pic][pic]is the hour angle. The appropriate test for the sun being in the northern sky is then [pic][pic][pic][pic][pic][pic][pic][pic]                     (3. 20) [pic]The distance XY can be found by geometry arguments. Substituting into Equation (3. 0), we have [pic][pic][pic][pic][pic][pic][pic]                     (3. 21) [pic]This is the test applied to Equation (3. 18) to ensure that computed solar azimuth angles are in the proper quadrant. 3. 2. 3 Daily and Seasonal Events Often the solar designer will want to predict the time and location of sunrise and sunset, the length of day, and the maximum solar altitude. Expressions for these are easily obtained by substitutions into expressions developed in Section 3. 2. l. The hour angle for sunset (and sunrise) may be obtained from Equation (3. 17) by substituting the condition that the solar altitude at sunset equals the angle to the horizon.

If the local horizon is flat, the solar altitude is zero at sunset and the hour angle at sunset[pic][pic][pic][pic][pic][pic]becomes [pic][pic][pic][pic][pic][pic][pic]                     (3. 22) [pic]The above two tests relate only to latitudes beyond ± 66. 55 degrees, i. e. above the Arctic Circle or below the Antarctic Circle. [pic][pic][pic][pic][pic][pic][pic][pic][pic][pic] [pic]If the hour angle at sunset is known, this may be substituted into Equation (3. 18) or (3. 19) to ascertain the solar azimuth at sunrise or sunset. The hours of daylight, sometimes of interest to the solar designer, may be calculated as [pic][pic][pic][pic][pic][pic]                     (3. 23) [pic]where ? s is in degrees.

It is of interest to note here that although the hours of daylight vary from month to month except at the equator (where [pic][pic][pic][pic][pic][pic]), there are always 4,380 hours of daylight in a year (non leap year) at any location on the earth. [pic]Another limit that may be obtained is the maximum and minimum noontime solar altitude angle. Substitution of a value of [pic][pic][pic][pic][pic][pic]into Equation (3. 17) gives [pic][pic][pic][pic][pic][pic][pic]                     (3. 24) [pic]where [pic][pic][pic][pic][pic][pic]denotes the absolute value of this difference. An interpretation of Equation (3. 24) shows that at solar noon, the solar zenith angle (the complement of the solar altitude angle) is equal to the latitude angle at the equinoxes and varies by ±23. 5 degrees from summer solstice to winter solstice. [pic]EXAMPLE: At a latitude of 35 degrees on the summer solstice (June 21), find the solar time of sunrise and sunset, the hours of daylight, the maximum solar altitude and the compass direction of the sun at sunrise and sunset….. On the summer solstice, the declination angle is +23. 45 degrees, giving an hour angle at sunset of 107. 68 degrees. Therefore, the time of sunrise is 4:49:17 and of sunset, 19:10:43. There are 14. 36 hours of daylight that day, and the sun reaches a maximum altitude angle of 78. 45 degrees. The sun rises at an azimuth angle of 60. 94 degrees (north of east) and sets at an azimuth angle of 299. 6 degrees (north of west).

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