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Prepared by

Prof. Dr. A. R. El-Ghalban

Department of Mechanical Engineering

University of Engineering and Technology

Taxila, Pakistan

The Sun: Earth’s Energy Source

- The Sun is located about 150x109 m from the Earth at the center of the Solar System.
- The Sun generates a large amount of energy due to a continuous thermonuclear fusion reaction occurring in its interior.
- In this interaction Hydrogen combine to form Helium and the excess energy is released in the form of electromagnetic radiation.

The Sun: Earth’s Energy Source

- The total energy emitted by the Sun per unit time (Solar luminosity) is L0 = 3.9x1026 Watts. The energy flux at the surface of the Sun is approximately 64 x 106 W/m2 .
- The average solar energy flux at the Sun’s surface, a distance of r0 from its center, is given by the Solar luminosity (L0) divided by the area of a sphere with a radius r0:
I0 = L0/4πr02

- Sun’s surface temperature is about 5780 K.

The Sun: Earth’s Energy Source

- Due to the location of the Earth in the solar system , a range of temperatures exists close to its surface makes the Earth a habitable planet.

- This temperature range is determined through an energy balance between the solar radiation absorbed by the Earth and the energy the Earth sends back into space.

The Sun: Earth’s Energy Source

- This process is known as the Earth energy (or radiation) balance.

- Earth’s internal source of energy, due to radioactive decay of various elements in its mantel and due to its warm core, is much smaller (~3x10-5 times) than the amount received from the sun.

- The energy flux emitted from the Sun spreads over an increasing spherical surface as it moves into space.
- Because the area of a sphere increases in proportion to the square of its radius, the radiative energy flux from the sun decreases as the inverse of the square of the distance from the Sun.
- The solar fluxes at two different distances from the Sun, I1 and I2, relate to one another as the inverse square of their distances from it, r1 and r2, that is:
I1/ I2 = (r2/r1)2

Electromagnetic Energy Transfer

- Solar radiation is energy, traveling through space as electromagnetic (EM) wave radiation.
- Radiation is a form of energy transfer that does not require mass exchange or direct contact between the heat exchanging bodies.
- Radiation involves the propagation of EM energy at the speed of light c* = 3x1010 cm/s.
- The speed of light c*, the frequency of the EM waves ν, and its wavelength λ are linked through the following relationship: c* = λν

- A body that emits energy over all frequencies in a continuous manner is called a blackbody.
- Blackbody radiation is a function of temperature and wavelength.
- This dependence is described in Planck’s law of radiation, which relates the EM energy flux emitted by a blackbody per unit wavelength to the wavelength and the temperature:
E(T,λ) = C1 /(λ5[ exp(C2 /λT) − 1] )

Where C1 and C2 are constants λ is the wavelength in m, and T is the absolute temperature in K

- Planck's law states a complex relationship between the energy flux per unit wavelength, the wavelength, and the temperature. From it we can derive two more simplified relationship.
- Wien law, stating the relationship between the wavelength corresponding to the maximum energy flux output by a blackbody λmax (in μm) and its absolute temperature T (in K): .
λmax = 2898/T

- Using Wien law and the Earth and Sun average temperatures 288 and 5780 K, respectively we find that their λmax correspond to about 10 and 0.5 μm.
- Stefan-Boltzman law stating the relationship between absolute temperature and the total energy flux emitted by a blackbody, over the entire wavelength range Ib(in W/m2)
Ib = σT4

where σ is referred to as the Stefan-Boltzman constant = 5.67 x 10−8 W/m2 K4

Solar Energy and the Climate System

- The planets rotate around the Sun in elliptically shaped orbits with the sun in one of its foci. Aphelion is the orbit position farthest from the sun and perihelion closest.
- Each orbit is defined by its mean distance from the Sun (d), by its eccentricity (e) and by its orientation in space.
- Each planet rotates around its axis, which in generally inclined with the respect to the orbital plane as measured by the obliquity angle

Solar Energy and the Climate System

- The rotation rate around the axis determine the length of the day and,
- The planet’s orbital rotation rate determine the length of its year.
- Eccentricity results in relatively small variations in incoming radiation, which are not the main reason for the seasonality.
- Obliquity (Φ) is the main reason for seasonality. If Φ is different from zero, the lengths of day and night over most of the planet’s surface are not equal but for two times during the year, the equinox times.

Solar Energy and the Climate System

- The difference between the lengths of day and night is zero on the planet’s equator and changes poleward.
- The days are longer than the night on the hemisphere tilting towards the Sun leading to more incoming Solar energy than in the other hemisphere.
- The times of year when the difference between the lengths of day and night reach their extreme values are called solstices.

- Latitude lines run horizontally, parallel and equally distant from each other.
- Degrees latitude are numbered from 0° to 90° north and south.
- Zero degrees is the equator, the imaginary line which divides our planet into the northern and southern hemispheres.
- North Pole is 90° north and South Pole and 90° south.
- Each degree of latitude is approximately 69 miles (111 km) apart.

- Longitude lines (meridians) are vertical, converge at the poles and are widest at the equator (about 69 miles or 111 km apart).
- Zero degrees longitude is located at Greenwich, England (0°).
- The degrees continue 180° east and 180° west where they meet and form the International Date Line in the Pacific Ocean.

Extraterrestrial Radiation (Solar constant)

- Solar constant ( Io ), is the radiation incident outside the earth's atmosphere. On average, it is 1367 W/m2. This value varies by ±3% as the earth orbits the sun.
Io = 1367 * (Rav / R)2 W/m2

- where (Rav) is the mean sun-earth distance and (R ) is the actual sun-earth distance depending on the day of the year
- Where β = 2 π n / 365 and n is the day of the year. For example, January 15 is year day 15 and February 15 is year day 46. There are 365 or 366 days in a year depending if the year is a leap year.

- Solar Declination is the angle between the Sun's rays and Earth's equatorial plane.(Technically, it is the angle between the Earth-Sun vector and the equatorial plane.)

- The Declination angle is 23.5° during the Northern Summer Solstice, and –23.5° during the Southern Summer Solstice. It is between ±23.5° the rest of the year.
- Following equations could be used for calculating solar declination angle δ

Where N is the day in the year

Solar Elevation (Sun height) Angle ( θ )

- The solar elevation angle is the elevation angle of the sun. That is, the angle between the direction of the sun and the (idealized) horizon.
- It can be calculated, to a good approximation, using the following formula:
Where

θs is the solar elevation angle,

h is the hour angle of the present time ,

δis the current sun declination and

Φ is the local latitude

The system of standard time is based on two facts:

Solar Time and Local Standard Time

- The Earth completes a total rotation on its axis once every twenty-four hours.

- There are 360° of longitude all the way around the Earth.

- The Earth turns 360° in 24 hours, or at a rate of 15° an hour.

(360° in a day÷24 hours = 15° an hour)

- Each standard meridian is the center of a time zone.

- Each time zone is 15° wide.

The Greenwich Time Zone, for example, is centered on the Prime Meridian

Solar Time and Local Standard Time

- This time zone is supposed to be 15° wide and extends from 7½° W to 7½°E.

- However, the boundaries of standard time don’t exactly run along meridians. The boundaries have been changed to fit the borders of countries and even smaller areas.

The relationship between solar time and local standard time is required to describe the position of the sun in local standard time.

Solar Time and Local Standard Time

- Local standard time is the same in the entire time zone whereas solar time relates to the position of the sun with respect to the observer.

- That difference depends on the exact longitude where solar time is calculated.

As the earth moves around the sun, solar time changes slightly with respect to local standard time.

Solar Time and Local Standard Time

- This is mainly related to the conservation of angular momentum as the earth moves around the sun.

- This time difference is called the equation of time and can be an important factor when determining the position of the sun for solar energy calculations.

- An approximate formula for the equation of time (Eqt) in minutes depending upon the location of earth in its orbit as following;

E slightly with respect to local standard time. qt = - 14.2 sin [π (n + 7) / 111] for year day n between 1 and 106

Solar Time and Local Standard Time

- Eqt = 4.0 sin [π (n - 106) / 59) for year day n between 107 and 166

- Eqt = - 6.5 sin [π( n - 166) / 80) for year day n between 167 and 365

Solar Time and Local Standard Time slightly with respect to local standard time.

- To adjust solar time for a longitude one have to add the value resulted from the time equation and to add or subtract the amount that the local time is ahead or behind the clock time for the time zone to the local time.

Tsolar = Tls + Eqt/ 60 ± (Longlocal – Longsm)/15 hours

Where Tsolar is the local solar time,

Tls is the local standard time,

Longlocal is the longitude of the observer in degrees and

Longsm is the longitude for the standard meridian for the observer's time

zone.

Solar hour angle ( slightly with respect to local standard time. h)

- Since the earth rotates approximately once every 24 hours, the hour angle changes by 15 degrees per hour and moves through 360 degrees over the day.

- Typically, the hour angle is defined to be zero at solar noon, when the sun is highest in the sky.

h = π * (12 - Tsolar) / 12 , radians

Where Tsolar is the local solar time

Solar zenith angle ( slightly with respect to local standard time. ωs)

- The zenith angle is the opposite angle to the sun height θs.
ωs = ( 90° – θs).

- At a sun height of 90°, the sun is at the zenith and the zenith angle is therefore zero.

Sun azimuth (α slightly with respect to local standard time. S)

- The sun azimuth (αS ) is the angle, measured clockwise, between geographical North and the point on the horizon directly below the sun.

Sun azimuth (α slightly with respect to local standard time. S)

- (Another definition is sometimes used, whereby the definition of the sun height remains the same but the sun azimuth is counted as zero when the sun is in the South and measured anticlockwise. Sometimes the symbols of azimuth and sun height are also interchanged.)

Solar Radiation on Earth Surface slightly with respect to local standard time.

- The amount of direct radiation on a horizontal surface can be calculated by multiplying the direct normal irradiance times the cosine of the zenith angle (ω).

- On a surface tilted (T) degrees from the horizontal and rotated ( γ ) degrees from the north-south axis, the direct component on the tilted surface is determined by multiplying the direct normal irradiance by the following value for the cosine of the incidence angle (θ ) ;

Solar Radiation on Earth Surface slightly with respect to local standard time.

cos (θ) = sin(δ)sin(λ)cos(T) - sin(δ)cos(λ)sin(T)cos(γ)

+cos(δ)cos(l)cos(T)cos(h)

+cos(δ)sin(λ)sin(T)cos(γ)cos(h)

+cos(δ)sin(T)sin(γ)sin(h)

where

λ is the latitude of the location of interest,

δ is the sun declination and

h is the hour angle .

Thank you slightly with respect to local standard time.

Solar Energy Flux slightly with respect to local standard time.

Earth energy (or radiation) balance. slightly with respect to local standard time.

The Sun: Earth’s Energy Source slightly with respect to local standard time.

Solar Flux in Space slightly with respect to local standard time.

Latitude and Longitude slightly with respect to local standard time.

Latitude and Longitude slightly with respect to local standard time.

Earth in Orbits slightly with respect to local standard time.

Distance from Sun d = 150x109 m,

eccentricity e = (a-b)/(a+b) = 0.017,

axis tilt Φ = 23.5°, Solar Flux (I0) = 1367 W/m2

perihelion (147 million km), aphelion (152 million km).

Earth in Orbits slightly with respect to local standard time.

Earth in Orb slightly with respect to local standard time. its

Electromagnetic Energy Transfer slightly with respect to local standard time.

Electromagnetic Energy Transfer slightly with respect to local standard time.

Blackbody Radiation slightly with respect to local standard time.

Solar Declination ( slightly with respect to local standard time. δ), Elevation (γ ) and Zenith (ω)