Simulation of Solar Radiation System

In this research work, a mathematical model for sim ulation of the solar radiation system has been developed by using a system dynamics methodolo gy. Bangladesh is used here for model validation. The simulated results obtained from thi s model have compared with the experimental results and found reasonably a good agreement. Ther efor the performance of the model is to be expected as satisfactory.


INTRODUCTION
The sun is an effective black body whose outer surface has a temperature of 6000 0 K [1] and is emitting an incredible amount of solar radiation. The earth intercepts only two-billionth of this radiation and saves all life on the planet from freezing to death [2] . Besides, the emitting radiation can be used as an alternative source of energy for mankind by using modern technology. Since in this century, we have virtually identified the whole successful functioning economic system with a steady increasing consumption of energy. When solar radiation/energy is used to generate electrical/mechanical energy for any specific place, it must be needed to forecast solar energy which will convert to electrical energy to recover the demand. Hence, the amount of solar energy for that place must be known.
Technology for measurements of solar radiation is costly and has instrumental hazard. So an alternative method for estimation of solar radiation is required. The system dynamics methodology is used here to simulate the solar radiation system, which is comparatively a new innovation, in comparison with the known methods of operation research. This method has been extensively used by Forrester [3,4] for industrial dynamics, Meadows [5] for commodity production cycle, Wadhawa [6] for regional planning, Syeed [7] for rural development and Alam [8,9] for rural energy system planning. In this paper a mathematical model is developed to simulate the availability of solar radiation in Bangladesh using system dynamics methodology.

Definitions of Solar Radiation Used in Mathematical
Model: For understanding the mathematical model, it is required to define some parameters of the solar system. These are as follows.

Irradiance:
The rate at which radiant energy is incident on a unit surface area. The symbol I is used with appropriate subscripts for the beam and diffuse radiation.

Beam Radiation (I b ):
The solar radiation received from the sun without being scattered by the atmosphere is called beam radiation. It is direct solar radiation.

Diffuse Radiation (I d ):
Solar radiation whose direction has been changed through scattering by the atmosphere is known diffuse radiation.

Global Radiation or Terrestrial/Total Solar Radiation (I h ):
The sum of beam and diffuse radiation in hourly on a surface is called global or total solar radiation, i.e. I h = I b + I d .
Solar Time: Solar time is the time based on the apparent angular motion of the sun across the sky. Solar noon is the time where the sun crosses the meridian of the observer. There is a difference between clock time and solar time because the solar time varies at any instant depending on the east-west displacement. Solar time is related to standard time by Duffie and Beckman [10] : Where: L st = Standard meridian for the local time zone, L loc = Longitude of the location in equation in degrees west, E = 229.2x(0.000075+0.001868B-0.03277SinB-0.014615CosB-0.04089Sin28 B = (n-1)360/365, n = Day of the year numbered from 1 st January. 1 ≤ n ≤ 365 Solar Geometry/Earth Angle: Earth angle and its components ( Fig. 1) are described in the following ways.
The latitude is the angular distance of the point on the earth measured north or south of the equator is latitude -90° ≤ φ ≤ 90°.
(ii) Longitude: Angular distance measured east and west of the prime meridian is longitude.
(iii) Declination Angle (δ δ δ δ): Angle made by the line joining the center of the sun and the earth with its projection on the equatorial plane, north positive is declination angle. It is zero at the autumnal and vernal equinoxes, is 23.45° at the summer solstice on June 21 and -23.45 0 at the winter solstice on December 21 in the northern hemisphere. The range of declination angle is given by -23.45 0 ≤ δ ≤ 23.45° According to Cooper [11] , at the intervening periods of the year δ can be approximated by a sinusoidal variation (2) (iv) Hour Angle (ω ω ω ω): Angular displacement of the sun east or west of the local meridian due to rotation of the earth on its axis at 15 0 per hour is hour angle. It expresses the time of the day with respect to the solar noon. It can be expressed by ω = 15(t-12). The solar beingnstant denoted by I sc is the energy from the sun per unit time received on a unit surface area perpendicular to the direction of propagation of radiation at the earth mean distance from the sun outside the atmosphere. The I sc value according to NASA/ASTM is 1353 Watts per square meter [10] .

Extraterrestrial Solar Radiation (E 0 ):
The radiation that would be received in the absence of the atmosphere in addition to the solar constant is called extraterrestrial solar radiation.
Angles to Describe the Position of the Sun in the Sky: Figure 2 represents the angles to describe the position of the sun in the sky. Angles are described in the following ways: (i) Solar Altitude Angle (α α α α s ): It is the angle between the projection of the sun's rays on the horizontal plane the sun's resurrection of the sun's rays.
(ii) Zenith Angle (θ θ θ θ Z ): It is the angle between the sky's rays and a line perpendicular to the plane through the point. Here, θ Z + α s = π/2. Angle Incidence on a Plane: Angles describing the position of a surface in relation to the sun's rays and the earth are defined in this section and described by the Fig. 3.
a. Slope of the surface β is the angle between the plane of the surface and the horizontal. b. Surface azimuth angle γ is the angle made in the horizontal plane between the line due south and formalojection of the formal to the surface of the horizontal plane.
c. Angle of incidence of radiation on a surface θ is the angle between the beam radiation on a surface and the normal to that surface.

Attenuation of Solar Radiation by Atmosphere:
Solar radiation at normal incidence received at the surface of the earth is subject to vary due to changes in the extraterrestrial radiation and to two additional more significant phenomena. These are as follows: * Atmospheric scattering by air molecules, water vapor and dust and * Atmospheric absorption by O 3 , H 2 O and CO 2 .
Atmospheric Scattering by Air Molecules, Watervapor and Dust: Air molecules are very small compared to the wavelengths of radiation in a solar energy spectrum. Water vapor scattering depends on the amount of perceptible water and a scattering coefficient which also be developed for water vapor that varies with λ −2 , where λ is the wavelength of the radiation. Dust scatters from particles those are much larger than air molecules and which vary in size and concentration, location, height and time, those are more difficult to assess.
The total effective scattering on the beam radiation is the product of the three exponential terms, each is a function of wavelength (for air mass λ −4 and dust λ −0.75 ) and of the amount of molecules, dust and perceptible moisture through which the radiation is transmitted.

Atmospheric Absorption by O 3 , H 2 O and CO 2 :
Absorption of radiation in the atmosphere in the solar energy spectrum is due largely to ozone in ultraviolet and water vapor in bands of the infrared. When the wavelength is below 0.29 micrometers, the radiation is almost completely absorbed by the atmosphere. When wavelength varies from 0.29 to 0.35 micrometers, ozone absorption is decreased provided that there is no absorption. There is also a weak ozone absorption band near λ = 0. 64 micrometers. Figure 4 shows the flow chart of the solar radiation system. Using this flow diagram, mathematical formulations have been presented here as a model for the solar radiation system.

Variation of Extraterrestrial Radiation:
The variation of extraterrestrial solar radiation with respect to time of the year is shown in Fig. 5. Two sources of variation are as follows.

Variation in Radiation Emitted by the Sun:
There are conflicting reports in the literature on periodic variations of intrinsic solar radiation. It has been suggested that there is a small variation with different periodicities and variation related to sunspot activities. Others consider the measurement to be included or not indicative of regular variability. Using Nimbus and Mariner satellites over the period of several months, it is shown that the variation is within limits of 0.2% over a time when sunspot activity is very low.
Variation of Earth-Sun Distance: Extraterrestrial radiation depends on earth-sun distance and varies 3% throughout the year. The dependence of extraterrestrial radiation on time of the year, developed by Duffie and Beckman [10] , is indicated by the following Eq. 3: 0 sc 360n E I 1 0.033Cos 365 Where: E 0 = Extraterrestrial radiation measured on in the plane of the nth day of the year I sc = Solar constant

Extraterrestrial Radiation on Horizontal Surface:
The solar radiation outside the atmosphere, that is, the hourly extraterrestrial radiation incident on horizontal plane can be written by the following way [10] : Cos 365 where, θ z = Zenith angle of the sun.
For a horizontal surface at any time between sunrise and sunset, according to Ref [17] , the cosine of zenith angle can be expressed by: Considering β = 0 and γ = 0, Eq. (5) Combining Eq. (4) and (6) The extraterrestrial daily solar radiation on a horizontal surface can be obtained by integrating Eq. (7) over the period from sunrise to sunset. Using ω = ω s , we have: Where: H 0 = Daily extraterrestrial solar radiation on a horizontal surface, ω s = Sunset hour angle in degrees.
Estimation of Average Solar Radiation: Radiation data are the best source of information for estimation average incident radiation. Lacking of these data from a nearby location of similar climate like Bangladesh, it is possible to use the empirical relationship to estimate radiation from hours of sunshine or cloudiness.
The original angstrom-type regression Equation related monthly average daily radiation to clear day radiation at any location is as follows:  [12] and others modified the method based on extraterrestrial radiation on a horizontal surface (obtained from Eq. (7), monthly daily average extraterrestrial radiation), rather than on clear day radiation: Hussain [14] has correlated this Eq. (11) for weather data in Bangladesh as well as Haider [13] and Hussain [14] reported the following angstrom-type regression Equation for Bangladesh: Prediction of Hourly Radiation from Daily Radiation Data: When hour-by-hour performance calculations are to be done, it may be necessary to start with daily data and estimated hourly values of the daily numbers (hourly values and daily numbers). In an investigation by Collres-Pereira and Rabl [15] developed an analytical expression for the ratio of hourly to daily global radiation:

Terrestrial Solar Radiation for Beam and Diffuse Radiation:
The correlation between hourly diffuse and global radiation developed by Muneer et al. [16] can be expressed by the following ways: ρ= Diffuse ground reflectance, ρ= 0.2 [16] , ρ= 0. 7 of fresh snow covered countries [16] .
Liu and Jordan [17] used the relation for ρ=0. 2 (1-c) +0.7c, Where, the c=fractional time of a month when the ground in covered in more than one inch of snow.

RESULTS AND DISCUSSION
In this section, the model was simulated during 7 A.M. to 5 P.M. using system dynamics methodology. The diffused solar radiation of the typical days 19 th August, hourly radiation on 4 th May, diffused radiation on 19 th January from horizontal surface are presented and compared with the measurement results [19] . The simulated and measurement results are shown in Fig. 6-8.

Validation of Simulated Results:
The model results of hourly and diffused radiation on a horizontal surface for the same above mentioned typical days are compared with the respective experimental data [19] and are shown in Fig. 6-8. Fig. 6: Simulated and Hussain [18]  Hussain [18] Simulated Hussain [18] Simulated Hussain [18]   Hussain [18] Simulated Hussain [18] Simulated Hussain [18]   Hussain [18] Simulated Hussain [18] Simulated Hussain [  Hussain [18] Simulated Hussain [18] Simulated Hussain [18]   Hussain [18] Simulated Hussain [18] Simulated Hussain [18] Id Ih  Id  Ih  For Id For Ih Id  Ih  Id  Ih  For Id For Ih  Id  Ih  Id  Ih  For Id For Ih  7 14  Hussain [18] Simulated Hussain [18] Simulated Hussain [ The simulated radiation is higher than that of experimental measured data due to instrumental error, the ambient air temperature, the thermal characteristics of the solar system and including the availability of solar energy. The agreement between simulated results and Ref [19] found satisfaction in the sets 'a' = 0.  [13] . The simulated values are compared with the reported values of [19] for the month of January, May and August shown in Fig. 6-8 for different sets of parameters [a=0. 18 and b=0. 39, a=0. 185 and b=0. 395, a=0. 175 and b=0. 375, a=0. 15 and b=0. 32, a=0. 14 and b=0. 34 and a=0. 14, b=0. 345]. The sensitivity analysis of the parameters is shown in Table 1-6. The best agreement was found between the simulated results and experimental results in the month of January, when a=0. 15 and b=0. 32, May, when a=0. 175 and b=0. 375 and August, when a=0. 18 and b=0. 39. Therefore it is concluded that the radiations usually vary with the weather variation and the constant a and b will be varied accordingly. It is also shown that the radiation depends on the other constants such as diffuse ground reflectance (ρ), ground to a collector angle (β) and range of latitude (φ). In case of Bangladesh, ρ = 0.2, β = 24.5° and φ=20 0 34 / -26 0 38 are used here along with the above set values of a and b for model validation. This model is also applicable to any other country with respect to availability of the said parameters for the concerned country.

CONCLUSION
The model for simulation of the solar radiation system is justified for different sets of parameters. The model has the best relation with hourly and diffuse radiation for clear sunny days. The performance of the model is found satisfactory as well as system dynamics methodology is applicable to simulate the solar