 |
Brighton Webs Ltd. statistical and data services for industry |
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
Home Index Feedback |
Simple Model for Clear Sky Solar Radiation The amount of solar radiation reaching the Earth's surface is mainly determined by:
This page sets out to describe a simple model of clear sky solar radiation. The finer points, such as refraction, precise astronomical coordinates and long term changes in Sun-Earth geometry are ignored. There is the caveat, that the reader should determine if the approach is appropriate for a specific application. Geometry The main elements of Earth-Sun geometry are:
These factors combine to give us day, night, the seasons and a load of 3D geometry. Energy Reaching a Point on the Earth's Surface Solar radiation reaches a point on the Earth from three sources which add up to the total which is referred to here as ITotal.
The author's observation of small amorphous silicon PV panels and a prototype solar thermal device in South East England suggests that they function well when the level of direct radiation is high (i.e. on a cloudless sunny day) and less well when the available radiation is composed mainly of diffuse radiation (i.e. a dull, overcast day in winter). Thus this model is based mainly on direct solar radiation. Reflected radiation is ignored. Lambert's Cosine Law The energy received by a point on the Earth's surface from beam radiation is determined by the angle of the beam to the surface. I is determined by Lamberts cosine law: When q is zero (sun overhead), cos(q) is 1, the point receives maximum radiation, when it 90 deg. (sunrise and sunset), cos(q) is zero and the point receives no radiation. Thus to derive the solar radiation received, we need two parameters, IB and q. Angle of Incidence Benford and Bock derived a formula for q which is shown below: The arguments to this function divide into two groups, those that we specify and those which are a consequence of Earth-Sun geometry. Those that we define are:
The Earth-Sun parameters are:
Before looking at declination and hour angle, it will be helpful to look at the Earth's orbit around the Sun and its rotation around its own axis. The Earth's Orbit around the Sun The Earth's orbit around the Sun is described by Kepler's laws of planetary motion. The orbit is elliptical with the Sun located at one of the foci. The diagram shows geometry and events of the orbit: Whilst for the purposes of this model, the tilt of the Earth's axis relative to the orbital plane can be assumed to be constant, then the angle of incidence of the Sun's rays during a year will be determined by it. This is illustrated in the diagrams below which show the angle of incidence at solar noon at the solstices and equinoxes:
Declination A formula which gives an approximation of declination in degrees based on the day number (e.g. 01-Jan=1, 02-Jan=2 etc.) is shown below: A more convenient form for use in computer models provides the declination in radians: The declination for the events of the orbit, given by this formula are tabulated below:
Solar Noon, Sunrise and Sunset Solar noon is when the point on the earth's surface where the observer is located, aligns with the centres of the Earth and Sun. At this point, the observer will see that the altitude of the Sun is at its maximum value. Solar noon provides the reference for the hour angle. At Solar Noon, the hour angle is zero. Sunrise and Sunset take place when the altitude of the Sun is zero. Whilst this simplistic definition suits the needs of the model, it is important to realise that a more sophisticated computation is used for values which appear in newspapers and astronomical tables. These take into account factors such a refraction due to the Earth's atmosphere and the relationship between solar time and clock time (defined by the Equation of Time). Hour Angle Viewed from above the North Pole, the Earth is rotating anti-clockwise around its axis. The Hour Angle defines the point in the rotation relative to Solar Noon. The Earth rotates through 360 degrees in 24 hours, thus it is rotating at the rate of 15 degrees per hour. At 09:00 Solar time, solar noon is three hours away, thus the hour angle is -45 degrees. By convention, angles in advance of solar noon are negative and those after it positive. Hour Angle at Sunrise and Sunset The hour angle at sunrise is obtained from this formula: As the time difference (and same angular rotation) between sunrise and solar noon is the same as that between solar noon and sunset, we now have all the geometry needed to estimate the angle of incidence on our surface during the day and throughout the year. The Solar Constant The solar energy reaching a satellite orbiting the Earth, would receive about 1370 W/m2 from the sun spread over a spectrum ranging from 100 nm to 4,000 nm, with the peak energy being in the visible light band (380 to 770 nm).
Processes such as absorption by water and CO2 reduce the energy received at the Earth's surface by approximately 30% to around 1000W/m2. The spectrum is also modified with parts of it being attenuated or extinguished. Inverse Square Law and Orbital Position As an object moves away from a point source of radiation, the power it receives is inversely proportional to the square of the distance. This is the inverse square law which is often expressed in the formula below. Take an example, at 1 metre the radiation from a light bulb is 2 W/m2, move the sensor until it is 2 metres from the bulb and the radiation is 0.5 W/m2. The average distance between the earth and the sun 149.6 million km. The closest it comes to the sun is 147.1 million km at Perihelion (around 04-January) and the furthest it gets is 152.1 million km at Aphelion (around 04 July). The table below shows how solar radiation changes from the average at these two points.
A formula which provides an approximate relationship between insolation and day number is shown below: The argument to the cosine function is in radians. Air Mass The term Air Mass is used to describe the relative length of the path of the Sun's rays through the Earth's atmosphere. When the sun is directly overhead (the altitude is 90 degrees and the zenith angle 0 degrees), it has the shortest path, this is termed air mass one. As the altitude decreases, the amount of the atmosphere through which the rays must travel increases. This distance is expressed as a ratio of the minimum distance. The diagram below shows the formula for Air Mass (AM) using the plane-parallel model which assumes the Earth to be flat. For zenith angles of less than 80 degrees this may be a reasonable assumption. The weakness of the plane-parallel model is that AM goes to infinity when the zenith angle is 90 degrees (at sunrise and sunset when the sun is on the horizon). At these times, the air mass is approximately 40 due to the curvature of the earth as shown below: An empirical formula derived by Kasten & Young (1989) allows values of air mass to be derived for zenith angles up to 90 degrees: The table shows the values of AM derived from both formulas:
The author's experience of observing small PV panels and solar thermal devices in an urban environment in the South East of England is that at high zenith angles/low altitudes is that direct radiation is weak and except in open locations, the device will probably be in shadow from buildings, trees etc.. Thus the plane parallel model offers a trade-off between accuracy and simplicity. Air Mass and Direct Solar Radiation On a sunny autumn day, if you take an early morning walk in the park, even though the sun is shining you may need some warm clothes to ward off the chill. Return at midday and you will be comfortable in a shirt and shorts. Around seven in the morning the sun is low in the sky (say, altitude around 10 deg. and zenith around 80 deg.) at this time the air mass is approximately 5.0 and the solar radiation on your face will be approximately 500 w/m2. At solar noon the sun will be higher in the sky (lower than summer, but you will still have to look upwards to see it, the altitude will be around 35 degrees and the zenith 55 degrees), the air mass will have dropped to approximately 1.7 and the sunshine will be a warming 1,000 w/m2. An empirical formula derived by Meinel provides a relationship between direct solar radiation and air mass: This allows us to estimate the direct component of solar radiation at different times of the day. It is assumed that observations are made a sea level. If you take to the mountains, you decrease the thickness of the atmosphere between you and the sun and more complex relationship is needed. Diffuse Solar Radiation The importance of diffuse radiation depends on the response of the device receiving it (It's essential to the human eye on a cloudy day when all the available light is diffuse). For the applications for which this model is developed, direct radiation is most important, thus a simple solution has been used which that diffuse radiation is 10% of the direct component. This almost certainly leads to an under-estimate at high zenith angles. The Model At this point we have the angle of incidence of the sun's rays to a surface and an estimate of the magnitude of the radiation, thus we can use Lambert's cosine law to estimate the clear sky radiation received by a solar device. The model is outlined in pseudo code below. In the model, Q is the energy produce by the device, this function will vary with application, at its simplest it could by Itotal multiplied by an appropriate coefficient of efficiency. A interval length of 12 minutes is shown below, this is an arbitrary value.
For any region which is subject to significant cloud cover throughout the year, the value of Q will be an overestimate and the model must be evolved to handle cloud cover. Reality Check The reality check for the model is the comparison of the output of a horizontally mounted pyranometer located in a similar latitude and weather system. A pyranometer is a device which measure full spectrum solar radiation. It is logically equivalent to a horizontally mounted solar device with efficiency of 100%. The horizontal mounting ensures that it will receive only direct and diffuse radiation and zero reflected radiation. The pyranometer output is the minimum and maximum values on days where the morning cloud cover was reported as zero. Whilst cloud cover might be zero in the morning, it is possible/probable that it will change as the day progresses, hence the fluctuations. The model appears to give a reasonable average maximum value of direct and diffuse radiation on a given day of the year. Page Modified: 06-January-2010 |
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
For more information: info@brighton-webs.co.uk |
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
