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A study of the radiative parameters for design of a solar pond Das, Aurobindo Kenneth 1985

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A STUDY OF THE RADIATIVE PARAMETERS FOR DESIGN OF A SOLAR POND By AUROBINDO KENNETH DAS B.E.(Hons) Birla Institute of Technology and Science Pilani, India 1979 M.Engg. The Asian Institute of Technology Bangkok, Thailand 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE D E G R E E OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1985 © Aurobindo Kenneth Das, 1985 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at The University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of British Columbia 2324 Main Mall Vancouver, B.C., Canada V6T 1W5 Date: September, 1985 ABSTRACT This research presents the development of a transmittance- absorptance parameter for a solar pond. Such a parameter represents, directly, the fraction of the incident solar radiation which is absorbed at the bottom of the solar pond. It can be used to represent pond performance through an equation analogous to the Hottel-Whillier-Bliss Equation for a flat-plate solar collector. The above parameter is called the transmittance - absorptance product and is an energy-weighted quantity. Monthly values of the proposed parameter are developed from an hour-by-hour simulation. The simulation utilizes hourly values of spectral solar radiation reaching the earth's surface which are computed from a state-of-the-art algorithm that has been slightly modified to better estimate diffuse spectral radiation at large solar zenith angles; the modification is also presented. Thermal conductive losses through the water layers and the surrounding earth together with evaporative and convective losses, are usually the only loss mechanisms considered for a solar pond. Under clear skies and to a lesser extent under cloudy skies, a longwave radiation heat loss also occurs from the pond surface. The estimation of radiative loss from any terrestrial surface requires detailed computations and atmospheric data. The procedure has been greatly simplified through a correlation which yields spectral atmospheric emissivity from the amounts of absorbing gases present in the atmosphere. It is recommended for further study that the performance of a solar pond be estimated using the proposed transmittance-absorptance product to compute the solar energy absorbed in the pond, and that the longwave radiative loss from the pond be included in the analysis. A comparison with data obtained from an existing solar pond is recommended to validate the results obtained in this study. ii LIST OF TABLES 2.1 Seasonal variation of atmospheric ozone 79 2.2 Rayleigh spectral attenuation coefficient 79 2.3 Spectral absorption coefficients for water vapor 80 2.4 Spectral absorption coefficients for mixed gases 80 2.5 Spectral absorption coefficients for ozone 81 2.6 F ratio for an aerosol 81 c 2.7 Diffuse correction factors for various amounts of aerosol 82 3.1(a) Coefficients (a^) to compute spectral atmospheric emissivity in the range 5.25<X<8.92 and 10.24<X<42.54 nm 97 3.1(b) Coefficients (b^) to compute spectral atmospheric emissivity in the range 8.92 <X< 10.24 ym 98 3.2 Comparison of measured values of atmospheric radiation (R) with values computed from the detailed technique 99 iii LIST OF FIGURES 2.1 Solar spectral radiation outside the earth's atmosphere and at ground level under typical conditions 72 2.2 Variation of air density with altitude 72 2.3 Path of a solar ray through the earth's atmosphere 73 2.4 Variation of ozone concentration with altitude 73 2.5(a) Vertical distribution of particulate concentration 74 2.5(b) Particle size distribution for three haze models 74 2.6 Rayleigh spectral transmittance as a function of airmass 74 2.7(a) Spectral aerosol attenuation coefficient (kfl^ ) using the methods of Angstrom and Bird 75 N2.7(b) Effect of airmass on spectral aerosol transmittance 75 2.8 Spectral transmittance due to water vapor absorption 76 2.9 Spectral transmittance of uniformly mixed gases 76 2.10 Diffuse spectral irradiance as a function of airmass 77 2.11 Diffuse spectral irradiance as a function of Angstrom's turbidity coefficient, |3 77 2.12 the various components of spectral diffuse irradiance on a horizontal surface 78 3.1 Absorption spectra for atmospheric gases and absorption spectrum of the atmosphere 83 3.2(a) Fundamental vibrations of the water vapor molecule 83 3.2(b) Structure of the water vapor molecule : 83 3.3 Fundamental vibrations of the CO2 molecule 84 3.4 Typical atmospheric transmittance due to molecular absorption in a bandwidth of 0.012 microns 84 3.5 Schematic layered atmosphere for transmission calculations 84 iv 3.6(a) Temperature variations for standard atmospheres 85 3.6(b) Ozone distribution for standard atmospheres 85 3.7(a) Spectral emissivity of the mid-latitude summer atmosphere 86 3.7(b) Spectral atmospheric radiation for the mid-latitude summer atmosphere .86 3.8(a) Spectral emissivity of the mid-latitude winter atmosphere 87 3.8(b) Spectral atmospheric radiation for the mid-latitude winter atmosphere 87 3.9(a) Spectral emissivity of the tropical atmosphere 88 3.9(b) Spectral atmospheric radiation for the tropical atmosphere 88 3.10(a) Spectral emissivity of the sub-arctic summer atmosphere 89 3.10(b) Spectral atmospheric radiation for the sub-arctic summer atmosphere 89 3.11(a) Spectral emissivity of the sub-arctic winter atmosphere 90 3.11(b) Spectral atmospheric radiation for the sub-arctic winter atmosphere ... 90 3.12 Typical variation of spectral emissivity with water vapor content (w) at various wavelengths 91 3.13(a) Comparison of e ^  values from the detailed technique and the simplified procedure for the MSA 92 3.13(b) Comparison of values from the detailed technique and the simplified procedure for the MSA 92 3.14(a) Comparison of e ^  values from the detailed technique and the simplified procedure for the MWA 93 3.14(b) Comparison of R^  values from the detailed technique and the simplified procedure for the MWA 93 3.15(a) Comparison of values from the detailed technique and the simplified procedure for the Tropical atmosphere 94 3.15(b) Comparison of R^  values from the detailed technique and the simplified procedure for the Tropical atmosphere 94 v 3.16(a) Comparison of e ^  values from the detailed technique and the simplified procedure for the SAS atmosphere 95 3.16(b) Comparison of values from the detailed technique and the simplified procedure for the SAS atmosphere 95 3.17(a) Comparison of e &^ values from the detailed technique and the simplified procedure for the SAW atmosphere 96 3.17(b) Comparison of R^  values from the detailed technique and the simplified procedure for the SAW atmosphere 96 4.1 Schematic of a typical salt-gradient solar pond 100 4.2 Schematic to estimate the hourly solar radiation absorbed in a solar pondlOOi 4.3 Schematic to estimate the hourly value of (ra) for any solar position ez .0 101 ^ 4.4 Typical hourly variation of (ra) for various latitudes 101 4.5(a) Hourly variation of direct beam radiation for a typical case at 0 = 30 N in June 102 4.5(b) Hourly variation of the (ra) value for a typical case at 0=30 N in June 102 4.6 Dependence of (TO) on pond absorptance (ap) 103 4.7(a) Monthly variation of (7a) at 0 N latitude for kp=0.4 nr1 104 4.7(b) Monthly variation of (7a) at 0 N latitude for kp = 0.6 nr1 105 4.7(c) Monthly variation of (7a) at 0 N latitude for kp=0.8 nr1 106 4.8(a) Monthly variation of (7a) at 15 N latitude for kp = 0.4 nr1 107 4.8(b) Monthly variation of (7a) at 15 N latitude for kp=0.6 nr1 108 4.8(c) Monthly variation of (7a) at 15 N latitude for kp=0.8 nr1 109 4.9(a) Monthly variation of (7a) at 30 N latitude for kp=0.4 nr1 110 4.9(b) Monthly variation of (7a) at 30 N latitude for kp=0.6 nr1 Ill vi 4.9(c) Monthly variation of (TO) at 30 N latitude for kp=0.8 nr1 112 4.10(a) Monthly variauon of (7a) at 45 N latitude for kp = 0.4 nr1 113 4.10(b) Monthly variation of (7a) at 45 N latitude for kp=0.6 nr1 114 4.10(c) Monthly variation of (7a) at 45 N latitude for kp = 0.8 nr1 115 4.11(a) Monthly variauon of (7a) at 60 N latitude for kp=0.4 nr1 116 4.11(b) Monthly variation of (7a) at 60 N latitude for kp=0.6 nr1 117 4.11(c) Monthly variation of (7a) at 60 N latitude for kp = 0.8 nr1 118 4.12 Variation of (7a), for diffuse radiation with k and D 119 vii NOMENCLATURE a^, bj^ Coefficients for spectral atmospheric emissivity in the simplified algorithm B(T,X) Spectral emissive power of a blackbody at temperature (T) and wavelength (X) [W nr2 unr1] C^ Correction factor for solar spectral diffuse radiation D Depth of solar pond [m] A, B, C, D, E Empirical coefficients for total atmospheric emissivity Fc Ratio of energy scattered in the forward direction to total energy scattered by an aerosol h Local altitude [m] h' Variable of integration [m] h Height of ozone layer [km] 3 H Thickness of the atmosphere [km] H Monthly-average daily global solar radiation incident on the solar pond [kJ nr2 day1] H, Monthly-average daily beam solar radiation incident on b the solar pond [kJ nrJ day"1] Hj Monthly-average daily diffuse solar radiation incident on the solar pond [kJ nr2 day1] i Running index I Instantaneous global solar radiation incident on a flat plate solar collector [W nr2] Aerosol-scattered spectral diffuse irradiance arriving on a horizontal surface [W nrJ nmr1] viii Multiply reflected spectral diffuse irradiance arriving on a horizontal surface [W m~2 Mm"1] I ^  Direct beam spectral irradiance on a surface normal to the solar rays [W nr2 Mm-1] i Q n ^ Extraterrestrial spectral irradiance at mean sun-earth distance on a surface normal to the solar rays [W nr2 um1] j Running index k Running index k^ Coefficient of attenuation due to scattering and absorption by aerosols kg^  Coefficient of attenuation due to absorption by mixed gases kQ^ Coefficient of attenuation due to absorption by ozone [cm-1] kp Extinction coefficient of the brine solution [nr1] k^ Coefficient of attenuation due to Rayleigh scattering by air molecules kw^ Coefficient of attenuation due to absorption by water vapor [cm-1] k^  Absorption coefficient for an atmospheric absorber in the infrared wavelengths [m2 kg"1] k^ Effective infrared absorption coefficient for a given wavelength band of width AX [mJ kg"1] L Length of solar pond [m] ma Airmass at actual pressure m_ Ozone mass ix Relative optical mass Number of layers into which the atmosphere is divided Refractive index of the brine solution in the solar pond Number of reflections Partial pressure of water vapor [mbar] Atmospheric pressure at any altitude (h) [mbar] Atmospheric pressure at ground level [mbar] Optical depth between ground level and altitude (h) or between ground level and layer (i) Effective optical depth between ground level and layer (i) for a given wavelength band of width AX Useful heat gain from a flat plate solar collector [W nr2] Monthly-average daily energy absorbed in a solar pond [kJ nr2 day1] Monthly-average daily useful energy available from a solar pond [kJ nr2 day1] Radius of the earth [km] Reflectance of direct solar radiation and diffuse solar radiation from the pond surface, respectively Total (spectrally integrated) longwave radiation from the atmosphere [W nr2] Relative humidity Spectral longwave radiation received at ground level [W nr2 /im-1] Spectral longwave radiation received at ground level from layer (i) of the N-layered atmosphere. [W mJ Mm-1] Spectral longwave radiation lying in the wavelength band centered at Xj which is received at ground level from N layers of the atmosphere [W nr2 /inr1] Length of slant path traversed by a solar ray in the earth's atmosphere [m] Variable of integration Ambient temperature, temperature of the surface layer of the atmosphere or screen-height temperature [K] Monthly-average value of the ambient temperature [K] Dew-point temperature [K] Fluid temperature at inlet of solar collector [K] Monthly-average temperature of the circulating brine solution measured at the pond inlet [K] Effective sky temperature [K] Heat loss coefficient for a flat plate soiar collector [W nr2 K"1] Heat loss coefficient for a solar pond based on monthly-average values [W nr2 K"1] Precipitable water vapor contained in the atmosphere in the vertical direction [cm] Width of the solar pond [m] Wavelength exponent in Angstrom's turbidity formula for aerosol attenuation Wavelength exponent in a modified turbidity formula for aerosol attenuation Solar absorptance of the bottom surface in a solar pond xi Angstrom's turbidity parameter Turbidity parameter in a modified formula for attenuation due to aerosols Total emissivity of the atmosphere Spectral emissivity of the atmosphere Solar zenith angle [degrees] Wavelength [Mm] Center of wavelength band (j) [jim] Density of attenuating species [kg nr3] Spectral albedo of the cloudless atmosphere Spectral albedo of the ground Stefan-Boltzmann constant [W nr2 K"4] Instantaneous value of the transmittance absorptance product for a solar device, the fraction of incident energy which is absorbed Monthly-average value of the daily transmittance absorptance product for a solar pond for global radiation when the diffuse component is negligible Monthly average value of the daily transmittance absorptance product for a solar pond for diffuse radiation Transmittance due to aerosol attenuation Transmittance due to mixed-gas absorption Transmittance due to ozone absorption Transmittance due to Rayleigh scattering by air molecules Transmittance due to water-vapor absorption Transmittance of beam radiation for it's first traverse through brine solution in a solar pond xii Transmittance of diffuse radiation for each successive traverse through brine solution in a solar pond 7 ^ Spectral transmittance of the atmosphere to longwave radiation 0 Geographic latitude [degrees] \p Azimuth angle of the sun [degrees] a> Single-scattering albedo of an aerosol 0 xiii ACKNOWLEDGEMENTS The author wishes to express his deepest gratitude to his supervisor Prof. M.Iqbal for his invaluable guidance, suggestions and encouragement throughout the course of this study. He would also like to thank Prof. P.G.Hill and Prof. R.Cole for their serving as Thesis Committee members. This research was supported through funding from the Natural Sciences and Engineering Research Council of Canada which is gratefully acknowledged. xiv T a b l e n f C o n t e n t s ABSTRACT ii LIST OF TABLES iii LIST OF FIGURES iv NOMENCLATURE viii ACKNOWLEDGEMENTS xiv 1. INTRODUCTION 1 2. SOLAR SPECTRAL RADIATION UNDER CLEAR SKIES 6 2.1 Introductory Remarks 6 2.2 The Atmosphere of the Earth 7 2.3 Scattering and Absorption in • the Atmosphere 11 2.3.1 Rayleigh scattering by air molecules 11 2.3.2 Aerosol attenuation 12 2.3.3 Absorption by water vapor, mixed gases and ozone 14 2.3.4 Direct spectral irradiation on the ground 17 2.4 An Improved Algorithm to Estimate Spectral Diffuse Irradiation on the Ground 17 2.4.1 Rayleigh-scattered spectral diffuse radiation 19 2.4.2 Aerosol-scattered spectral diffuse radiaition 19 2.4.3 Multiply reflected spectral diffuse radiation 20 2.4.4 The improved algorithm for spectral diffuse radiation on the ground 21 2.5 Conclusions 23 3. A SEMPLJTEED TECHNIQUE TO COMPUTE SPECTRAL ATMOSPHERIC RADIATION 25 3.1 Introductory Remarks 25 3.2 Origin of Longwave Radiation in the Atmosphere 27 3.2.1 Water vapor absorption 28 3.2.2 Carbon dioxide absorption 29 xv 3.2.3 Ozone absorption 30 3.3 Presently Available Methods to Compute Incoming Atmospheric Radiation 31 3.3.1 Empirical relations to compute atmospheric radiation 31 3.3.2 A detailed method to compute spectral atmospheric radiation 35 3.4 Simplified Technique to Compute Spectral Atmospheric Radiation 43 3.5 Comparison of Results from the Detailed Method and the Simplified Technique 48 3.6 Conclusions 50 4. MONTHLY AVERAGE TRANSMITTANCE-ABSORPTANCE PRODUCT FOR A SOLAR POND 51 4.1 Introductory Remarks 51 4.2 Development of the Transmittance-Absorptance Product 52 4.2.1 Development of the hourly transmittance-absorptance product (ra) for a solar pond 54 4.2.2 Development of the daily transmittance-absorptance product (ra) for a solar pond 59 4.3 Discussion of the Daily Transmittance-Absorptance Results 61 4.4 Conclusions 64 CONLUSIONS 66 RECOMMENDATIONS FOR FURTHER STUDY 67 REFERENCES 68 APPENDIX A 120 APPENDIX B 1 2 1 APPENDIX C 1 2 2 APPENDIX D 1 2 5 xvi Chapter 1 INTRODUCTION In order to evaluate the performance of any solar thermal device, it is necessary to estimate the input energy to the device and its thermal losses to the surroundings. A solar pond is a solar collector comprising of a shallow pond of brine solution in which an artificially maintained salt concentration gradient prevents convection. Such a device has an inbuilt storage in the form of a layer of hot brine at the bottom of the pond. The brine is circulated through heat exchangers to recover the stored energy. The large thermal storage capacity in a solar pond offers the prospect of seasonal energy supply to a community or large consumer in a location which has high incident solar radiation and clear, sunny skies. This is usually the case in dry tropical and mid-latitude regions. Among many solar energy devices, a solar pond is particularly attractive for certain applications like power generation and large-scale domestic heating. Temperatures upto 90° C can be achieved under conditions of high solar irradiation. Unlike other solar devices a solar pond is simple in design and cheaper to construct and maintain. Typical comparative costs are $50 nr2 and $200 nr2 for a solar pond and a flat plate solar collector, respectively. A solar pond operates best under conditions of high incident solar radiation and clear skies. Other factors are: deep ground water table to minimize heat loss, low evaporation rate, dry soil for good thermal insulation. To understand the working of a solar pond, compare the behaviour under sunlight of two shallow ponds one of which is filled with homogenous water while the other contains fresh water on top and heavier salt water at the bottom. A large fraction of the solar energy penetrates to the bottom of the pond and is absorbed there. In the homogenous pond the bottom layer 1 2 warms up and heat is transported upwards by natural convection to be lost to the surroundings. In the other pond, however, convection is suppressed by the density gradient Therefore, heat can be lost only by conduction upwards to the air and laterally into the surrounding soil. Solar pond research orignated in Israel nearly thirty years ago. Weinbergerfl] presented one of the earliest papers on solar pond simulation to predict the temperature of the pond. Weinberger considered only the upward thermal conduction loss through the brine solution. The effect of energy withdrawl on pond efficiency and pond stability were first investigated by him. Several investigations have been done on solar-pond based power plants - Tabor[2], Sonn and Letan[3], Wright[4] and others. Fluorocarbons are used in the Rankine cycle for such an application. Space heating is an increasingly common application for solar ponds. Rabl and Nielsen[5] have done a thermal and economic analysis for a solar pond facility for large scale space heating. They derived equations for the pond temperatures during year round operation, taking into account the heat that can be stored in the ground underneath the pond. Only conductive losses were considered by them, infrared radiation to the atmosphere was neglected[5]. They found that solar ponds can supply adequate heating even in regions near the arctic circle, and the cost estimates showed that solar ponds for large scale community heating are competitive with conventional heating. Results gathered over a two year period from the Miamisburg solar pond - the largest (2020 m2) in the U.S. - have also been published. The data observed from this pond has been found to agree well with predicted values and the cost of the delivered energy is lower than the cost of heating with fuel oil. Studies on specific aspects of solar ponds such as the temperature and salt concentration gradients for steady state operation[6] and modelling of the 3 upper convection zone[7] have also been done. In order to estimate the instantaneous and long term performance of a solar pond, one needs to know the energy incident on the pond surface and the fraction of it which is absorbed by the pond bottom. Also required is an estimate of the thermal losses from the pond. The conductive, convective losses from the pond have been extensively studied, however, less attention appears to have been devoted to the radiative loss from the pond surface into the sky. A simple approach can be taken to predict the instantaneous or long-term performance of the pond. Analogous to a flat-plate solar collector, the useful energy from a solar pond could be expressed in terms of a transmittance-absorptance product which determines, directly, the fraction of daily radiation which is absorbed by the pond bottom. Then, knowing the heat loss coefficient for the pond, the useful energy can be easily determined. The loss coefficient can be estimated from available literature and the incident radiation at a given location is available either as measurements, from radiation maps, or can be estimated using a broadband or spectral solar radiation algorithm[13]. As part of this study the algorithm to compute spectral diffuse solar radiation has been improved and, to be consistent, the spectral solar radiation algorithm has been employed to compute incident radiation. The total solar radiation (i.e. spectrally integrated) can be easily obtained from the spectral values which is then used to compute the energy-averaged transmittance-absorptance parameter. The following equation expresses the useful energy available from a 4 solar pond: U s e f u l e n e r g y A b s o r b e d s o l a r e n e r g y C o n d u c t i o n , c o n v e c t i o n , e v a p o r a t i o n l o s s e s + A t m o s p h e r i c r a d i a t i o n i n c i d e n t on pond I n f r a r e d r a d i a t i o n f r o m pond t o s k y The first term on the right hand side can be computed knowing the incident solar radiation and the transmittance-absorptance product of the pond. The second term represents the thermal losses which are usually considered in pond analysis. The third and fourth terms are ignored in presently available literature, and represent the net infrared loss from the surface of a solar pond. This loss is typically 100 W nr2 and compared to the first term (approx. 500 W nr2) is an appreciably large quantity which should be included in the analysis. The objective of this research is to develop monthly-average daily values of the transmittance-absorptance product ( 7 a ) for a variety of pond parameters upto latitudes of 60°. The monthly-average values are developed from hourly values using the computed hourly incident solar radiation. Also presented is a simplified technique to estimate the infrared radiative loss from any terrestrial surface; this technique can be employed to calculate the radiative loss from the pond surface to the sky as another component of the total thermal loss. The research is divided into three main chapters. Chapter 2 presents a state-of-the-art algorithm developed by Leckner[8], Brine and Iqbal[9]. In this study, the above algorithm has been revised in part and is used to calculate the hourly incident solar energy. Chapter 3 presents a simplified technique to compute longwave spectral atmospheric radiation. Such radiation originates from certain gases in the atmosphere and its estimate is required to predict the net radiative loss to the sky from any terrestrial surface. This simple algorithm is developed in response to the presently used method which requires a great deal 5 of computation and input data. It yields the radiative loss from the pond surface through simple calculations. Chapter 4 develops, first, the hourly values of the transmittance-absorptance product and then, using the revised algorithm of Chapter 2, the daily values. Using the results of this study, the energy absorbed in a solar pond can be conveniently determined. The longwave radiation loss from the pond together with other thermal losses usually considered, can be used to estimate pond temperatures. This latter aspect involves the physics of a solar pond. Since this study deals only with the radiative parameters for a pond, the determination of pond temperatures incorporating infrared loss is recommended for further investigation. Chapter 2 SOLAR SPECTRAL RADIATION UNDER CLEAR SKIES 2.1 INTRODUCTORY REMARKS In order to determine the solar radiation incident on a solar pond, one requires a means to calculate the attenuation of extraterrestrial solar radiation as it propagates through the earth's atmosphere. This chapter presents an improved state-of-the-art algorithm to estimate the total solar radiation incident on the earth's surface under clear skies. The sun radiates as a blackbody at an equivalent temperature of 5777 K.. Nearly 99% of its emitted energy is contained in wavelengths less than 4 iim and is called shortwave radiation. The spectral distribution of this energy is different outside the earth's atmosphere from that at ground level. When solar radiaton enters the atmosphere it undergoes two types of attenuation processes, namely, scattering and absorption. Both processes are wavelength dependent and the former generates diffuse radiation, of which only a portion reaches the ground. The other component of solar radiation arriving on the ground comes directly in line from the solar disk and is called direct or beam radiation. Figure 2.1 compares solar spectral radiation received outside the earth's atmosphere and on the ground under typical conditions. This chapter presents a recent simple algorithm to estimate direct spectral irradiance at ground level(Eq. 2.16). An improved method is proposed to estimate diffuse spectral irradiance on a horizontal surface at ground level. Together they constitute a simple state-of-the-art algorithm to compute spectral beam and diffuse solar radiation. 6 7 Rigorous methods to compute spectral attenuaton of the solar beam by the atmosphere were developed at the U.S. Air Force Research Laboratories. The results are used in the computer code LOWTRAN presented by McClatchey et al.[ 10,11,12]. Leckner[8] presented a simple algorithm, based on the above work, to compute beam spectral irradiance on the ground. Brine and Iqbal[9] extended the method to compute diffuse solar spectral radiation under clear skies applicable upto zenith angles of 60°. The complete algorithm is described in detail by Iqbal[13]. In this chapter the above algorithm is slightly modified to extend its applicability to zenith angles of upto 80°. At such large solar zenith angles the sun is close to the horizon and the radiation received at the earth's surface is almost entirely diffuse radiation. This modified formulation will be used to estimate the solar spectral energy incident on a solar pond under varying atmospheric conditions. 2.2 THE ATMOSPHERE OF THE EARTH The earth's atmosphere consists mainly of molecular nitrogen and molecular oxygen. The approximate composition is: 78% nitrogen, 21% oxygen, 1% argon and 0.033% carbon dioxide by volume. Upto about 100 km altitude, the concentration of these gases is nearly constant in time and space. In addition to these gases, the atmosphere contains water vapor and particulate matter (aerosols) which are highly variable. Ozone is present in small amounts in the stratosphere and its concentration changes slowly with latitude and season. All gaseous molecules in air deplete solar radiation by scattering which occurs at all wavelengths and is called a continuum process. However, absorption by gaseous molecules occurs at discrete wavelengths only. The absorbing gases are mainly water vapor, ozone, carbon dioxide, oxygen and 8 nitrogen. Absorption by water vapor and - ozone are treated seperately while the remaining 'uniformly mixed gases' are regarded as a single absorber. The extent to which solar radiation is attenuated depends on the type and number of molecules in the path of a solar ray. The mass of an atmospheric constituent in a column of unit cross sectional area is called optical mass given by: m = / p ds (2.1) s act where (ds) is the differential path length and p is the density at that point in the atmosphere. Figure 2.2 shows the typical variation of air density with altitude in an atmosphere. The continuously varying density causes the ray to follow a curved path and the integration in Eq. (2.1) must be performed along the actual slant path of the ray. More useful for calculations is the non-dimensional quantity called relative optical mass (mr) which is defined by: / p ds J P dh H where the denominator is the mass of the attenuating gas in the vertical direction. The situation is described in Fig. 2.3. Using the density distribution for dry air, Kasten[14] obtained the following expression for relative optical mass of air, simply called air mass: m = [cos 6 + 0.15(93.885 - 6 ) - 1 - 2 5 3 ] - 1 ( 2 < 3 ) r z z where 6 is the solar zenith angle. For stations not at sea level the air mass must be pressure corrected: 9 P m s m r 1013.25 (2.4) a Using the vertical distribution of water vapor a similar expression for the relative water vapor mass was obtained by Kasten. However, it is common to utilize the value of mr itself for attenuation by water vapor. Ozone has a characteristic distribution in the atmosphere. It is found mainly between 10 and 35 km altitude as seen in Fig. 2.4. Robinson[15] proposed the following expression for the relative ozone mass mo assuming that all ozone was concentrated at height h3 and h3 = 22 km : Evidently, this is an oversimplification. It has been shown[13] that Kasten's formula, Eq. (2.3), for the relative optical mass of air gives results close to those obtained from Eq. (2.5) except at zenith angles greater than 85°. In this study, Eq. (2.3) will be used also for relative ozone mass. The total amount of the attenuating species present in the vertical direction in the atmosphere is also needed to compute spectral attenuation of the solar beam. For example, the water vapor content of the atmosphere is measured as the height of a column of liquid water (having unit cross sectional area) that would be formed if all the water vapor in the zenith direction were condensed at the earth's surface. In the absence of data, the precipitable water (w) can be estimated from the relative humidity (RJJ) as given by Leckner[8]: mo = (1 + h 3/r e)[ cos2 6 + 2(h_/r )] -0.5 (2.5) *" = 0.493 T (2.6a) 10 where py is the partial pressure of water vapor in saturated air, and T& is the ambient temperature. The value of py is read from a psychrometric chart or calculated from the semi-empirical equation: P y = exp(26.23 - "^-) (2.6b) a The value of water vapor content (w) can vary from a few millimeters to several centimeters depending upon season and location, rarely exceeding 5 cm. The ozone amount (z) in the atmosphere is less variable than water vapor and can be obtained from tabulated seasonal values given in [15], and reproduced in Table 2.1. It is the height of a gaseous column of ozone obtained when all ozone in the vertical direction (in a column of unit cross sectional area) is brought to normal temperature and surface pressure(NTP). At the Equator total ozone averages 0.24 cm(NTP) while in the Polar regions it could be as much as 0.46 cm(NTP). Suspended particles in the atmosphere are called aerosols. They originate from land as smoke, pollen, dust etc. or from the sea as salt crystals. The amount of aerosol in the atmosphere varies greatly in time and space. Figure 2.5(a) shows a profile of aerosol concentration with height which was employed by Elterman[16]. Furthermore, aerosols vary greatly in size and optical properties. Figure 2.5(b), from Deirmendjian[17], illustrates the particle size distribution for three types of aerosols. Aerosols attenuate both by scattering and absorption but the effects are difficult to separate. Given the uncertainities in the amount, size distribution and properties of aerosols, approximate formulae are used. The Angstrom's turbidity formula utilizes two parameters a and B to characterize an aerosol. Beta (|3) is called the Angstrom's turbidity coefficient and represents, as an 11 index, the amount of aerosol present in the vertical direction. In natural atmospheres /3 can vary from 0.5 to a very small number. Alpha (a) is called the wavelength exponent and is related to the average particle size of the aerosol. It can vary from 4 for very small particles of the order of air molecules to smaller values for larger particles. A value of a = 1.3 is commonly employed since it was originally suggested by Angstrom. Knowing the various relative optical masses and the amount of each attenuating component in the atmosphere, the spectral transmittances can be calculated as detailed in the following sections. 2.3 SCATTERING AND ABSORPTION IN THE ATMOSPHERE 2.3.1 RAYLEIGH SCATTERING BY AIR MOLECULES As mentioned earlier, scattering by air molecules is a continuum process. When the particles are smaller than one-tenth the wavelength of the incident radiation, as are air molecules under solar radiation, the energy scattered in the forward and backward direction is equal. Such scattering is referred to as Rayleigh scattering and the scattered energy appears as diffuse radiation. Rayleigh attenuation depends on the particle size, number of particles per unit volume and their refractive index. For the density and concentration profiles of the U.S. Standard Atmosphere of 1962, Elterman[9] computed the Rayleigh attenuation coefficients (kr^ ). Leckner[8] uses the following expression for the same coefficient: krA = °-008735 A~ 4 , 0 8 (2.7) Yet another expression has been proposed by Kneizys et a/.[18]: 12 k - [A A (115.6406 -rX 1.335T-1 (2.8) In Table 2.2 a comparison of the attenuation coefficients obtained from Eq. (2.7) and Eq. (2.8) is given. Values of k^ computed by Elterman and considered quite accurate are also given for comparison. In the ultraviolet region some difference is observed but this can be ignored since ozone absorbs strongly in this region compared to Rayleigh attenuation. Consistent with the algorithm of Iqbal[13], equation (2.7) will be employed in this study. The complete expression for transmittance due to Rayleigh attenuation ( T ^ ) is then written as: T r A = exp(-0.008735 A ~ 4 ' 0 8 m ) (2.9) where m f l is the pressure-corrected relative air mass. The effect of.air mass on Rayleigh transmittance is shown in Fig. 2.6 obtained by using Eq. (2.9). We next study attenuation due to atmospheric aerosols. 2.3.2 AEROSOL ATTENUATION As mentioned earlier, particulate matter in the atmosphere absorbs as well as scatters solar radiation. Atmospheric aerosols vary in size from 0.01 Mm to 100 Mm. In general, scattering by aerosols is much greater than their absorption and is called Mie scattering. Unlike Rayleigh scattering by air molecules, Mie scattering produces more forward than backward scattered radiation. As particle size increases so does the energy scattered in the forward direction. 13 The following formula called Angstrom's turbidity formula is widely used: k , = M " a (2.10) aX where k&^ is the aerosol attenuation coefficient and a and /3 are characteristics of the aerosol. Equation (2.10) assumes that a plot k^ versus X on a log-log scale is linear with a slope of (-)a. Bird[19] observed a curvature in such a plot for rural aerosols and proposed the following expression for ka^ in which the coefficients are wavelength dependent: -a kaX = 3 n X (2.H) W h e r e a ± = 1.0274 for X < 0.5 urn a 2 = 1.2060 for X > 0.5 urn and /3j and are tabulated as functions of the aerosol attenuation coefficient. Equation (2.10) applies to a general atmosphere while Eq. (2.11) applies to rural aerosol only. A detailed comparison has been made of the two equations in an earlier investigation which also forms part of this study. Figure 2.7(a) shows that the results are very close to each other. The slope of the line from Eq. (2.10) remains invariant while that from Bird's equation changes at 0.50 am. Once the aerosol attenuation coefficient is determined, the spectral aerosol transmittance (?"a^ ) is obtained from: TaX = «P< - k a X »a> ( 2 < 1 2 ) 14 Figure 2.7(b) compares T ^  computed using the methods of Angstrom and Bird. It can be seen that at smaller air masses (m =2, 6 60°) both methods give results close to each other and even at large air mass, say m =6, a 0^80°, the results are in reasonable agreement. It is known[13] that for typical conditions the magnitude of aerosol scattered diffuse radiation arriving at the earth's surface at X>1 nm is very small. Hence the larger differences observed in Fig. 2.7(b) at longer wavelengths should not be too important Furthermore, in view of the large uncertainity in specifying aerosol characteristics in a real atmosphere and the desirability for a more general method, it is felt that Angstrom's formulation, Eq. (2.10), should be adequate. Consequently, Angstrom's turbidity formula along with Eq. (2.12) will be used in this study. 2.3.3 ABSORPTION BY WATER VAPOR, MIXED GASES AND OZONE As mentioned earlier, absorption of solar radiation by gases in the atmosphere occurs at discrete wavelengths. The principal absorbing gases are atomic oxygen(O), atomic nitrogen(N), molecular oxygen(02), molecular nitroge^^), ozone(Oj), water vapo^ HjO) and carbon dioxide(C02). They absorb to varying degrees in the solar spectrum(X<4 Mm). Atomic oxygen and nitrogen occur in the upper atmosphere and absorb strongly upto 0.085 Mm. Diatomic oxygen and nitrogen along with ozone absorb upto 0.20 Mm. Molecular oxygen and nitrogen together with water vapor, carbon dioxide and ozone absorb in the visible and infrared region of the solar spectrum. Water vapor and ozone are treated individually since their amounts in the atmosphere vary in time and space. The remaining absorbers, together with some minor constituents, are fairly evenly distributed in the earth's atmosphere and are treated collectively as 'mixed gases'. 15 a. Absorption by water vapor Leckner[8] developed an expression for absorption by water vapor from the work of McClatchey et al.[ 10,11,12] who employed detailed methods. Leckner's study presented the absorption coefficients (kw^) at wavelengths identical to those in the extraterrestrial solar spectrum (Appendix A). Values of k w ^ given by Leckner[8] and presented in Table 2.3, are used in the following expression developed by him to calculate spectral transmittance due to water vapor absorption ( T W ^ ) : TwA = exp[-0.2385 k w A w 1^/(1 + 20.07 ^ * , n ^ ) 0 ' 4 5 ] (2.13) In this equation H>(cm) is the amount of precipitable water in the atmosphere in the vertical direction and (nrnp gives the water vapor amount in the actual path of the ray. Figure 2.8 shows a typical variation of T w ^ with wavelength for w=l cm. As expected, the transmittance varies markedly in narrow spectral regions. Transmittance due to mixed-gas absorption is presented next b. Absorption by mixed gases Using the detailed results of McClatchey et al., Leckner[8] presented the following expression to compute transmittance due to mixed gases (Tg^) ; T . = exp[-1.41 k , m /(1+118.93 k m )°* 4 5] (2.14) gX L gA a gA a where is the effective absorption coefficient presented in Table 2.4. Since mixed gases are considered to be uniformly distributed at all geographic locations, the air mass itself is a measure of the amount of the, absorber in 16 the path of a ray. Figure 2.9 illustrates the spectral transmittance of uniformly mixed gases at m =1 . 3 . Equations (2.13) and (2.14) both have a similar form. A theoretical variation of air density, water vapor concentration, temperature and pressure are used to obtain the equations. Leckner determined a transmittance function for each wavelength interval, separately for water vapor and mixed gases. The various transmittance functions were then adjusted to a generalized transmittance function by determining the appropriate values of the spectral absorption coefficients. We next present the simple formula for absorption due to ozone. c. Absorption by ozone As mentioned earlier, ozone absorbs strongly in the ultraviolet and also in the visible region of the solar spectrum. Vigroux[20] presented the ozone absorption coefficients (kQ^ ) at wavelength intervals which did not correspond to those in the extraterrestrial solar spectrum. Leckner[8] derived interpolated values at matching wavelengths and these are presented in Table 2.5. These values are suitable for use in the following expression: ToA = e x P ( _ k o X z V (2-15) where z is the amount of ozone in the vertical direction and mr is substituted for the relative mass of ozone. The term (zmp gives the actual amount of ozone in the slant path traversed by the refracted ray. The attenuation processes occuring in the atmosphere have been treated in a simple manner in the foregoing sections. The direct spectral irradiance and the diffuse spectral irradiance at ground level can now be computed using the 17 various transmittances obtained so far. 2.3.4 DIRECT SPECTRAL IRRADIATION ON THE GROUND The formulation presented in the foregoing sections is used to compute the spectral radiation perpendicular to the solar beam, called direct spectral radiation. Solar radiation received outside the earth's atmosphere has a measured spectral distribution as given in Appendix A and is in the form of beam radiation. This beam is progressively attenuated as it traverses the atmosphere. The spectral irradiation of this beam on a surface at ground level at the mean sun-earth distance and perpendicular to its direction (Jny) is then given by[13]: 1nX = ^ n X ^ r X T a X T W X TgX T DX^ (2-16) where i Q n ^ is the spectral beam irradiance outside the atmosphere at any desired wavelength (from Appendix A) and the terms within brackets yield the overall transmittance due to the five attenuation processes discussed earlier. The effects of atmospheric parameters such as m , p\ w and z on the spectral s solar beam radiation is presented in Ref[13]. Knowing the atmospheric transmittance due to the five attenuating processes and direct normal radiation at ground level, one can estimate the diffuse radiation reaching the earth's surface as described in the next section. 2.4 AN IMPROVED ALGORITHM TO ESTIMATE SPECTRAL DIFFUSE IRRADIATION ON THE GROUND Diffuse radiation originates from the scattering effects of aerosols and air molecules. The direct solar beam strikes an air molecule (or aerosol particle) 18 and produces primary scattered radiation. This radiation in turn impinges on neighbouring molecules and generates secondary scattered radiation. The process continues and is referred to as multiple scattering; contribution from which is small and the process is complex. Brine and Iqbal[9] have developed a simple algorithm to compute diffuse spectral irradiance by considering primary scattering only. Their results have been shown to agree well with those from a rigorous model upto zenith angles of 60° (m ^ 2). This simple, yet reasonably accurate, algorithm was slightly modified by Bird[19] to extend its validity to zenith angles upto 80° (m =6). This was done by employing a diffuse correction factor (C^) obtained by matching the computational results from the simple algorithm with those from a rigorous calculation. It has been shown through another investigation that the correction factors derived by Bird needed to be revised since they were obtained using slightly different formulae for aerosol attenuation coefficient ozone relative optical mass (mQ) and in the case of water vapor absorption, unfortunately, a formula with a typing error (of Brine and Iqbal[9]) was employed-. Hence a revised set of values of are presented here. There are three components of diffuse radiation arriving at the earth's surface. Rayleigh scattering and aerosol scattering generate two of the above components denoted as ( i ^ ) and ( i ^ ) respectively. These two components are reflected upward by the ground and then re-reflected by the atmosphere. This process continues ad infinitum. Such multiple reflections of diffuse radiation constitute the third component and can also be calculated by the simple algorithm presented in [13] which has been modified in this study. 19 2.4.1 RAYLEIGH- SCATTERED SPECTRAL DIFFUSE RADIATION The spectral intensity of the solar beam on a horizontal plane outside the earth's atmosphere is i o n^ c o s(^ z) where i Q n ^ is the extraterrestrial normal solar spectral intensity. The scattered diffuse radiation reaching the earth's surface after scattering by air molecules, aerosols and absorption by gases is given by[13]: f d r A = c o s \ T w A T g A T D A I 0 ' A ) T a A J ( 2 - ^ ) It has been assumed that half of the scattered radiation is directed towards ground. As observed in Figure 2.6 Rayleigh scattering is confined to the shorter wavelengths (X<1 um approx). This will be illustrated in the diagrams which follow. 2.4.2 AEROSOL-SCATTERED SPECTRAL DIFFUSE RADIAITION As mentioned earlier, aerosol scattered diffuse radiation (1^^) is difficult to calculate accurately since aerosol characteristics are poorly specified. Approximate values of a and Q are used to first obtain T ^ from Eq. (2.12). To calculate two additional parameters are used. The single scattering albedo of an aerosol (OJ ) is the fraction of the energy scattered by an aerosol 0 to the total attenuation (scattering plus absorption). This fraction depends on the material, shape and size of the aerosol; it is assumed independent of wavelength and usually lies between 0.6 and 1.0[13] . Aerosols in urban areas contain more carbon and a> —0.60 can be used. Rural aerosols absorb to a 0 smaller extent and w =*0.90. Unlike Rayleigh scattering, forward and back 0 scattering are unequal for an aerosol. The portion scattered forward depends on the zenith angle. Table 2.6 presents values of F defined as the ratio of the 20 energy scattered in the forward direction to the total energy scattered, proposed by Robinson[15]. The expression for can be written in the simple form[13]: haX * KnX c o s 9 Z \ X T gA T oX K V ^ a X ^ r J < 2- 1 8> A comparison of the various components of diffuse radiation is made in the Figures that follow. 2.4.3 MULTIPLY REFLECTED SPECTRAL DIFFUSE RADIATION The spectral diffuse radiation resulting from multiple reflections between ground and the atmosphere is obtained as a product of the total downcoming solar spectral radiation (after the first pass through the atmosphere) and a series of terms containing ground and atmospheric reflectances. Ground reflectance is considered wavelength independent and values of 0.7 and 0.2 with and without snow are commonly used. Iqbal[13] has compiled values of ground albedo for various ground conditions. From first principles it can be shown[13] that is: * * * ' t fdr> + * i n X cos 6 J fr,"^, ygX P a X where p'^ and are the ground, and atmospheric spectral reflectances respectively. The atmospheric reflectance for diffuse radiation is calculated from[13]: p a x " Kx T g x T o x [ O . 5 ( I - T ; X ) X ; a + ( i - F . ) « - O ( I - T ; X ) T ; J (2.20) 21 I I drA + I daA + I dmA (2.21) Results from the approximate method shown above agree with rigorously obtained results upto solar zenith angles of 60 degrees[9]. 2.4.4 THE IMPROVED ALGORITHM FOR SPECTRAL DIFFUSE RADIATION ON THE GROUND As explained earlier, Bird[19] modified the above formulation[9] slightly. He proposed a correction factor C^  which adjusts the Rayleigh and aerosol scattered diffuse radiation at ground level, to match results from the BRITE code[21]. This rigorous computation procedure (BRITE) has been developed and periodically revised to give agreeable results with observed data. Bird and Riordan[22] compare the spectral values of diffuse radiation obtained from their correction factors with those from the BRITE computations. The agreement is good and justifies the simpler method using correction factors[22]. Bird and Riordan[22] proposed the following correction. The term ( i ^ + is replaced by C. (i. » + i , .) throughout to obtain: (2.22) and then = CA< fdrA + !daA> + 1 dmA (2.23) 22 Bird's values of correction factors were tabulated as functions of wavelength and zenith angle for a number of values of k&^ . It can be inferred that larger values of will be required at high zenith angles, higher aerosol content in the atmosphere and shorter wavelengths. The new set of correction factors, consistent with Iqbal's algorithm[13], is presented in Table 2.7. In the neighbourhood of 0.30 jim the correction factors obtained were very large due to computational problems. Since spectral intensity at ground level is negligble at such small wavelengths[13], correction factors from 0.35 Mm upwards only have been presented. Thus the simple algorithm presented here is a revision of the one described in detail by lqbal[13]. Using the proposed correction factors and the simple algorithm of Iqbal, good agreement is obtained between values computed herein and those obtained from the rigorous code upto zenith angles of 80°. The following plots show the influence of various parameters on diffuse spectral radiation, obtained using the modifications presented in this section. Figure 2.10 shows the effect of increasing air mass on i ^ ; the increased pathlength at higher air mass produces greater scattering, yet the diffuse radiation reaching the ground decreases on account of the cosine term in I, . and I, . . daX dmA The amount of aerosol in the atmosphere strongly influences spectral diffuse radiation. The turbidity parameter j3 measures aerosol content and Fig. 2.11 shows increasing with increasing p\ It can be seen that aerosol attenuation dominates in the region of longer wavelengths i.e. above 0.45 Mm. In order to illustrate the relative magnitudes of 1 ^ , i^mx m <* Fig. 2.12 has been presented for a set of typical conditions at m&=l . The wavelength integrated values of the above three components of diffuse radiation in Fig. 2.12 are 60, 100, and 40 W nrJ respectively. 23 Summarizing, values of spectral diffuse radiation from the revised Iqbal's algorithm agree with those of Bird[19]. Bird's values of 1^ match those from the detailed BRITE code[21]. Consequently, it is expected that the modified formulation presented here yields results in accordance with the detailed procedure of BRITE even at large air mass. In a recent study Bird and Riordan[22] present formulae proposed by Justus and Paris[23] which do not require tabulated correction factors. However, Bird and Riordan maintain that the correction factors may still be the most accurate approach. 2.5 CONCLUSIONS The simple algorithm of Iqbal[13] has been revised in light of a study by Bird[19] and is presented in this chapter. The modification pertains to the calculation of diffuse solar spectral radiation at ground level. Hence, the updated algorithm provides* a simple procedure to calculate solar spectral radiation — both direct and diffuse — upto zenith angles of 80° at 144 wavelengths between 0.30 and 4.0 um. Results from this simple formulation agree with those of Bird who has shown his own results to match a rigorous code. His results also agree with spectral measurements at a single location under varying atmospheric conditions. The results obtained from the simple, revised algorithm presented herein are, at best, as accurate as the rigorious codes such as BRITE These codes are believed to be within ± 5 % for beam radiation and ±15% for diffuse radiation. The revised algorithm presented in this chapter computes the instantaneous solar radiation incident on the earth's surface. These instantaneous spectral values of direct and diffuse solar radiation can be integrated to obtain 24 the total (i.e. spectrally integrated) direct and difuse radiation reaching the earth's surface under a given set of atmospheric conditions. Assuming this radiation to be constant during each hour, the energy-averaged parameter (fa) is computed for each hour as explained in Chapter 4. Chapter 3 A SIMPLIFIED TECHNIQUE TO COMPUTE SPECTRAL ATMOSPHERIC RADIATION 3.1 INTRODUCTORY REMARKS A qualitative knowledge of downward atmospheric radiation is of importance in certain applications like radiative cooling of terrestrial surfaces, solar collector analysis, performance of plastic covered greenhouses and in plant leaf temperature studies. The atmosphere is transparent to the longwave radiation emitted by the earth's surface in certain wavelength intervals; this allows the earth to maintain an equilibrium temperature by losing enormous quantities of heat gained each day from the sun. Under clear skies an object can cool below ambient air temperature by radiative heat loss to the sky, a process governed by downcoming atmospheric radiation. Accurate estimation of this radiation along with a knowledge of surface properties can be used to predict the cooling obtained in buildings without mechanical conditioners. The analysis of this section applies to clear sky conditions. Under cloudy skies, the downcoming atmospheric radiation increases since clouds are blackbody radiators of infrared radiation. As seen in Chapter 2 the sun radiates as a blackbody at an equivalent temperature of 5777 K. Nearly 99% of its emitted energy is contained in wavelengths less than 4 Mm and is called shortwave radiation. On the other hand, the equivalent radiant temperature of the earth's surface is about 275 K. Over 99% of this energy is emitted at wavelengths greater than 4 Mm and is called longwave or infrared radiation. 25 26 In this chapter a simplified method has been developed to estimate the incoming atmospheric radiation under clear skies. The presently used technique is complex and requires a large amount of data. The simplified method presented here can be employed to predict the net radiative heat loss from a solar pond. Water vapor, carbon dioxide and ozone are the main emitters of longwave radiation in the atmosphere. The thermal radiation emitted by the earth's surface is absorbed by these atmospheric gases in certain wavelength intervals, and the resulting re-emission appears as longwave atmospheric radiation. The remaining unabsorbed portion of the earth's radiation escapes into the outer space. Available methods to estimate the thermal radiance of the atmosphere under clear skies, fall into two classes. The first consists of empirical methods based on direct measurements of the broadband, i.e. spectrally integrated, atmospheric radiation. The measurements are correlated to a routinely measured meteorological parameter like the partial pressure of water vapor, surface-air temperature or dewpoint temperature. Since they have adjustable coefficients, they are able to achieve reasonable agreement with experimental results. None of them, however, provide spectral information. The second class of methods utilizes detailed concentration profiles of atmospheric constituents together with a knowledge of their radiative properties, to compute spectral values of atmospheric radiation. Such methods are based on the radiative exchange theory of the atmosphere and are preferred in principle, over the broadband methods. They can be applied to any given atmospheric condition. In practice, however, they suffer from the necessity of detailed input information on the state of the atmosphere: temperature, pressure and density variations in addition to the concentration profiles and the spectral absorption 27 coefficients of the radiatively active gases. The simplified method developed in this chapter bridges the gap between the two class of methods. It calculates spectral longwave atmospheric radiation using two commonly available meteorological parameters. Hence, the proposed technique overcomes the shortcoming of the presently used algorithm and is used, in this study, to estimate the net radiative loss from a solar pond. Before presenting the simplified algorithm, the origin of longwave radiation and the currently used methods to compute it are discussed. 3.2 ORIGIN OF LONGWAVE RADIATION IN THE ATMOSPHERE Longwave atmospheric radiation originates from certain gases in the atmosphere which absorb and emit radiation in the infrared region of the electromagnetic spectrum. The 4 to 40 Mm range of wavelengths contain nearly 95% of radiation emitted by bodies at ambient temperature. The absorption of this radiation by atmospheric gases leads to corresponding emission. Although oxygen and nitrogen account for 99% (by volume) of the gases in the atmosphere, they do not absorb or emit radiation in the infrared. Asymmetric molecules like water vapor, carbon dioxide, ozone and the oxides of nitrogen are the sources of longwave radiation. Contributions from nitrogen oxides, aerosols and hydrocarbons are minor and are usually ignored. Water vapor, carbon dioxide and ozone account for most of the absorption in the cloudless atmosphere. A low resolution absorption spectra[24] is shown in Fig. 3.1. Strong water vapor absorption bands fall below 8 Mm. Weak absorption between 8 and 13 Mm results in a region of higher transparency, called the atmospheric window, through which the earth looses 28 heat into space. Above 20 Mm water vapor behaves as a perfect absorber. Carbon dioxide is highly absorptive from 14 to 16 Mm. It is the second most important atmospheric absorber even though it constitutes only 0.03% by volume in the atmosphere. Ozone absorbs strongly between 9 and 10 Mm which corresponds to a region of minimum absorption by water vapor and carbon dioxide. A superposition of the above spectra reveals several gaps where the atmosphere is rather transmissive. A brief description of the absorption properties of the three main absorbers follows next 3.2.1 WATER VAPOR ABSORPTION Water vapor is the single most important absorber and emitter of longwave radiation in the atmosphere. The structure of the molecule has been found to be as shown in Fig. 3.2(a). In the unexcited state it forms an isosceles triangle with side lengths S Q H = 0.958*10"4 Mm and an apex angle of 104° 30". Absorption by water vapor occurs due to vibrational and rotational transitions of the molecule. Below 8 Mm the absorption is caused by vibrational transitions whereas beyond 13 Mm purely rotational transitions occur. Figure 3.2(a) shows the observed types of normal vibrations of the water vapor molecule[25] corresponding to the following wavelengths: X,= 5.97 Mm X2 = 2.72 Mm X3= 2.64 Mm The water vapor spectrum above 13 Mm is associated with rotational transitions shown in Fig. 3.2(b). The resulting absorption becomes stronger above 20 Mm. Between 8 and 13 Mm the absorption due to water vapor is rather weak, as seen in Fig. 3.1 . The absorption spectra in this region does not possess the line structure of the molecule observed at other wavelengths. Three 29 mechanisms have been proposed to explain such 'continuum absorption' of the water vapor molecule. Varanasi[26] suggested that the continuum may be due to the presence of dimers (bound states of two water molecules) formed due to weak hydrogen bonding, making available many different transitions when the molecules break apart Given many different bound-free transitions, a continuous absorption spectrum would result A second hypothesis attributes the continuous spectra to the distant wings of strong water vapor absorption lines located at neighbouring wavelengths[27]. Yet a third explanation attributes the phenomenon to clusters of hydrogen-bonded water vapor molecules generated by atmospheric ions[28]. The overall transparency of the atmosphere to longwave radiation is largely determined by the transparency in this range of wavelengths called the atmospheric window. It is mainly through this window that terrestrial surfaces can lose longwave radiation into space. Absorption in this window is highly sensitive to the amount of water vapor present in the atmosphere. Consequently, water vapor content of the atmosphere becomes a critical parameter to determine longwave atmospheric radiation. Carbon dioxide is another important absorbing gas and its characteristics are presented next 3.2.2 CARBON DIOXIDE ABSORPTION The proportion of carbon dioxide in the atmosphere is relatively constant (0.03% by volume). Infrared absorption by carbon dioxide is a result of vibrational transitions in the linear molecule which has a carbon-oxygen distance of SQQ = 1.15*10"" um. Figure 3.3 from Ref[25] shows two types of normal vibrations of the CO2 molecule which correspond to the following wavelengths: Xj = 14.8 am X2=4.2 Lim 30 As explained earlier, longwave radiation emitted by bodies at terrestrial temperatures lies above 4 nm, so only the absorption band at 14.8 Mm need be considered for our purpose. In practice, absorption by gases occurs at many closely spaced wavelengths; referred to as a band. In the case of carbon dioxide, the band is centered at 15 Mm and extends from 13 to 17 Mm. As an approximation, its effect is incorporated by appropriate adjustment of the water vapor absorption coefficients. Ozone is the third important absorber of longwave radiation in the atmosphere. Ozone absorbs in a narrow region within the atmospheric window. Its characteristics are presented next 3.2.3 OZONE ABSORPTION Ozone is formed by photochemical reactions in the upper atmosphere and absorbs intensely in the ultraviolet & visible portion of the solar spectrum. The ozone molecule is linear with S Q Q = 1.278*10~4 Mm. In the infrared, ozone absorbs strongly between 9 and 10 Mm. Though there exists another absorption band at 14.1 Mm, and it is overlapped by more intensive HjO and COj bands so that only the 9 to 10 Mm band need be considered. As explained in Chapter 2 the ozone concentration in the atmosphere changes with altitude, season and geographic latitude. Having briefly seen the absorption characteristics of the three main absorbers in the atmosphere, an overview of the presently available methods to compute longwave radiation from the atmosphere is presented. The simplified procedure developed in this research is presented in Section 3.4 31 3.3 PRESENTLY AVAILABLE METHODS TO COMPUTE INCOMING ATMOSPHERIC RADIATION As mentioned in Sec. 3.1, the methods for computing the thermal radiation from clear skies fall into two categories. The first consists of empirical techniques based on direct measurements of atmospheric radiation. The measurements are then correlated to a meteorological parameter. However, such techniques provide no spectral information. Detailed profiles of atmospheric constituents, together with their radiative properties are used to estimate atmospheric radiation in the second class of methods. The more detailed techniques are preferred in principle because they consider the actual radiative exchange between each atmospheric layer and the ground. They also yield the spectral distribution of longwave atmospheric radiation. However, the detailed input data required for such a method and its comlex computation is a disadvantage. Some of the empirical relations of the first category are summarized next. 3.3.1 EMPIRICAL RELATIONS TO COMPLnE ATMOSPHERIC RADIATION In order to estimate the total — i.e. spectrally integrated — longwave radiation from the atmosphere it is customary to specify either of the following: (i) uniform 'effective' sky temperature T sky defined by R = o T A sky (3.1) where a is the Stefan-Boltzmann constant and R is the measured longwave atmospheric radiation, the atmosphere being regarded as a blackbody at T, ; 32 or (ii) a total atmospheric emittance e defined by, e a T 4 (3.2) R = a a where T& is the temperature of ambient air at ground level. The earliest correlation to predict atmospheric emissivity was proposed in 1916 by Angstrbm[29]. It related the total emissivity to the partial pressure of water vapor(p ): where A, B and C are empirical constants which have been assigned values in the ranges (0.75-0.82), (0.15-0.32) and (0.09-0.13) respectively, and py is expressed in millibars. The empirical constants in the above equation are determined from numerous experiments and as shown vary over a wide range. This variation was explained to be the result of inadequate consideration of all the factors that effect atmospheric radiation and also due to the differences in methods of measurements. According to Kondratyev[25], the most frequently used values are A-0.806, 5=0.236 and C-0.069 . Another empirical equation proposed by Brunt[30] in 1932 is: e = A - B exp(- c P„,) a v (3.3) e a " D + E P v 0.5 (3.4) where D and E are empirical constants which lie in the range (0.34-0.71) and (0.023-0.110) respectively. Based on data collected by Dines and Dines[31], Brunt obtained Z)=0.52 and £=0.065 which have been verified by some other 33 studies. The wide variation in the constants of the above equation prompted Swinbank[32] to re-examine the data of Dines and Dines. Swinbank argued that the relationship between e & and py results basically from a correlation between temperature and humidity. The nature of the correlation between e& and pv would therefore depend on the temperature humidity regime of the location. Using the same data as used by Brunt — that of Dines and Dines[31] measured at Benson, Oxfordshire — Swinbank correlated the values of radiation (R) with surface air temperature (T ) rather than vapor pressure and arrived at the relation: Swinbank then used his own measurements made at two stations in Australia and at various locations in the Indian Ocean to arrive at: Both the above expresions yield coefficients within 2% of each other, even though the data is from widely different locations. This close agreement is in sharp contrast to the location-dependent coefficients in the earlier formulations. Idso and Jackson[33] developed another empirical correlation based on a = 0.92 x 10 (3.6) e f l = 1 - 0.261 exp[-7.77 x 10~4(273-T ) 2] (3.7) The above equation was based on data gathered in Alaska, Arizona, Australia 34 and the Indian Ocean. Clark and Allen[34] have employed the dewpoint temperature to summarize the results of 800 clear sky measurements made at San Antonio, Texas: E = 0.787 + 0.764 Zn (T /273) (3.8) a <ip Berdhal and Fromberg[24] reported results of 2945 clear-sky measurements made at Tucson(AZ), Gaithersburg(MD), and StLouis(MO): e = 0.741 + 0.0062 T. (3.9) a dp All the above empirical expressions are based on measured longwave radiation and its correlation with a suitable meteorological parameter. Since they all have adjustable coefficients, they are able to achieve good agreement with experimental results. Their coefficients are location dependent, they ignore the structure of the atmosphere and, most importantly, they do not provide any spectral information. The following subsection of the present section presents a detailed method which is used to estimate spectral atmospheric radiation. In the next section, this method is employed to achieve the main objective of this chapter, viz. to develop a location-independent correlation to compute spectral atmospheric radiation using the total amount of absorbing gases as the only input 35 3.3.2 A DETAILED METHOD TO COMPUTE SPECTRAL ATMOSPHERIC RADIATION The empirical correlations presented earlier express the total atmospheric emissivity as a function of partial pressure of water vapor, ambient air temperature or dewpoint temperature. Thus knowing either py, Ta or T^  one can compute e . Using Eq. (3.2) and the measured value of T one can 2. S calculate the total — i.e. spectrally integrated — longwave radiation(R) incident on the earth's surface. Apart from the shortcoming that these equations are purely empirical, location-dependent and do not consider the absorption properties nor the amount of the absorbing gases present in the atmosphere, a knowledge of the total atmospheric radiation has limited usefulness. In order to estimate the amount of this incident radiation which is absorbed by a terrestrial surface, one needs to know its infrared absorptance which is a wavelength dependent surface property in most practical cases. This requires the spectral distribution of R . An example is the plastic covered flat plate solar collector in which the absorber plate loses heat to the sky by radiation through the partially transparent cover unlike a glass cover which is opaque to infrared radiation. A water surface, as in a solar pond, behaves as a blackbody absorber/radiator for infrared radiation. A detailed method to compute longwave radiation is preferred in principle, since it treats each absorbing gas separately by considering its measured spectral absorption coefficient, its content in the atmosphere, and the variation of air temperature, pressure and absorbing gas concentration with altitude. The method yields spectral atmospheric radiation (R^ ) from which the spectral atmospheric emissivity (*a^ ) can be obtained as: R, 36 where B(T ,X) is the spectral emissive power of a blackbody (Planck's 3. distribution[38]) at the ambient air temperature (T ) and wavelength X. It is 3. explained earlier that the range of wavelengths that are of interest lie between 4 and 40 ym in the infrared. Absorption properties of the atmospheric constituents have been obtained by various researchers. Kondratyev[35] presented a set of values in 1965 which have been improved upon by Ramsey et al.[36] and are used in this study. Wolfe and Zissis[37] have presented an exhaustive collection of experimental data from many researchers. Figure 3.4 from Ref[21] shows an example of the possible complexity of determining absorption coefficients. The total extent of the curve in Fig. 3.4 represents a region of only about 0.012 Mm. Since molecular absorption varies greatly with wavelength, it is impractical to consider each line in the absorption spectra individually. Hence an averaging technique is used over each wavelength interval to compute an average absorption coefficient in that interval. In the present study, the 105 unequal wavelength intervals proposed by Ramsey et a/.[36] are used to cover the range from 5 to 43 Mm. The absorption coefficients are determined by averaging the transmittance of infrared radiation in each band. Their values are presented in Appendix B; the wavelength intervals used for averaging purposes are also listed. The effect of carbon dioxide has been incorporated in the adjusted absorption coefficients. Ozone absorption in the infrared occurs between 9 and 10 Mm and only such values are considered by Ramsey et al. The following material of this section is adapted from Ramsey et al. Using their detailed technique with a variety of atmospheric conditions, sets of emissivities are obtained. These are then used to develop the simple spectral correlation presented in Section 3.4. 37 a. Governing equations Consider Fig. 3.5. Let the atmosphere be divided into N horizontal layers labelled 12, 13 etc. Each layer is considered isothermal, homogenous in composition and at a uniform pressure. Layer thicknesses are chosen such that the above assumptions are approximately satisfied; this requires layers to be thin upto an altitude of several kilometers and thicker layers can be used thereafter. The spectral distribution of infrared energy arriving at the earth's surface is composed of emissions from each layer. These emissions undergo successive absorption/emission as they propagate downward through intervening layers. The total infrared radiation (R) received on the surface of the earth can be obtained by an integration of the spectral values (R^ ): R = / R dX (3.11) A o From the one-dimensional transfer equation between two infinite parallel surfaces as given by Sparrow and Cess[38], the above equation can be rewritten as: <x> qH R - 2/ / B(T,X) E 2 ( t ) dt dX (3-12; o o where q^ is total optical depth of the atmosphere and E 2 is given by 1 E 2 ( t ) = / exp(—) dy o where n is cost^ . ~tx (3.13) In general, the optical depth for any absorbing species from ground level to an 38 elevation (h) is defined by h q h = / kx p(h') dh' ( 3 . 1 4 ) o where is the spectral absorption coefficient, p is the density of the absorbing species and h' is the local altitude. It is observed that k^  depends on pressure i.e. altitude as well. Also, is a spectral quantity through its dependence on k^  in Eq. (3.14). Equation (3.12) is evaluated for using the layer model of the atmosphere with N layers, illustrated in Fig. 3.5. The equation can then be written as: N N C q i + l p=l I 2B(T.,X) / / exp(—)dy dt (3.15) i=l 1 V t=q. p=0 where R. . is the contribution of the i * layer of the spectral radiation received at ground level, q^  is the optical depth between ground level and the layer (i) evaluated at the wavelength being considered. For simplicity, the subscript X is dropped from (q^ ). Ramsey et al. rewrote Eq. (3.15) as: N Rx = Z 2B(Ti,X )[E 3(q.) - E 3 ( q . + 1 ) ] i=l where i -q, E 3(q i) = / u exp(-^)dp (3.16) o As explained earlier, it is impractical to consider each absorption line individually. Hence computations are performed for a number of wavelength bands varying from 0.025 urn to 1.0 Mm in width(AX), 105 such bands are 39 used in this study. Consider a wavelength interval (j) centered at Xj and having a bandwidth AX. The spectral radiation between Xj-AX/2 and Xj+AX/2 from the N layers of the atmosphere can be written from Eq. (3.16) as: X .+AX/2 N j Rx = I 2B(T 1,X j) * / [ E 3 ( q i ) - E 3(q 1 + 1)]dX (3.17) j 1 = 1 X -AX/2 Rather than evaluate qj at each spectral line an effective optical thickness is computed for each band from: «i,AX = hx ^ PO»')dh' (3.18) o s where £ ^ is the effective absorption coefficient determined separately by a line-by-line averaging technique within each band. The k^ values first proposed by Kondratyev[35] have been revised by Ramsey et a/.[36] and used in their study as well as this research. These are the values which are reported in Appendix B for HjO and O^ . The exponent n is determined experimentally; values of n=0.9 for water vapor and n=0.4 for ozone are recommended by McClatchey et a/.[10]. Incorporating Eq. (3.18) into Eq. (3.17) we have the following: N R x I 2B(T.,X.)[E 3(q i ) A X) - E 3 ( q . + 1 > A X ) ] A X (3.19) j i = 1 The effect of all absorbing gases are given by the sum of individual effective optical thicknesses, and Ramsey et al. wrote the final form of Eq. (3.19) as : \ = j x 2B(T i,X.)[E 3(Z q i ) A X ) - E^Z ^ l . A X ^ ^ ( 3 ' 2 0 ) 40 where £ means summation of q^^ over all absorbing gases for layer (i) of the atmosphere. In order to use Eq. (3.20) we need to know the atmospheric conditions in great detail. b. Model atmospheres The detailed layer-by-layer method being presented here requires the spectral absorption coefficients of water vapor, ozone and carbon dioxide. In addition to these the method requires, as input, the temperature, pressure variations with altitude, and a distribution of the absorbing gases in the atmosphere upto a height of many kilometers. McClatchey et al.[W] have compiled Model atmospheres. Such prescribed atmospheres are based on balloon, rocket and satellite observations. The model atmospheres are widely used and form a part of the well known LOWTRAN computer codes for solar radiation estimation. The five model atmospheres proposed by them in Ref[10] are: (i) Tropical(TRP) (ii) Midlatitude summer(MSA) (iii) Midlatitude winter(MWA) (iv) Subarctic summer(SAS) (v) Subarctic winter(SAW) Each model atmosphere presents in tabular form, the pressure, temperature, air density, water vapor and ozone concentration at one kilometer intervals upto 25 km and larger intervals thereafter upto a maximum altitude of 100 km. Appendix C presents the five model atmospheres. As an example, the variation of temperature and ozone concentration with altitude is illustrated in Fig. 3.6(a) and 3.6(b) for four different model atmospheres. The information presented in 41 Appendix B and C is utilized to obtain the results computed from the detailed method which is presented next c. Results of detailed computations Using the equations presented earlier in this section together with the absorption properties from Appendix B and atmospheric data from Appendix C, computations were done to obtain spectral emissivity of the atmosphere for each of the five model atmospheres. For this purpose a computer program was developed. The program utilizes, as input, the following information: (i) atmospheric pressure, temperature and air density variation with altitude; (ii) water vapor and ozone concentration with altitude; (iii) absorption coefficients of water vapor and ozone in 105 wavelength bands between 5.25 and 42.54 Mm. Results from the detailed computations are presented in Figs. 3.7 to 3.11 for the five model atmospheres. Each of these figures shows the variation of, first; the atmospheric spectral emissivity (^ a^ ) with wavelength, and second; the spectral atmospheric radiation (R^ ) received at ground level. Had the atmosphere behaved as a blackbody at a uniform surface-air temperature (T ) its spectral emissive power would have varied as shown by the smooth curves in Figs. 3.7(b) to 3.11(b). This curve is simply the Planck's distribution B(T,X) evaluated at the surface air temperature corresponding to each atmosphere. Each of the above figures states the precipitable water vapor and ozone amount obtained from the atmospheric concentration profiles. A comparison of Figs. 3.7(a) to 3.11(a) shows that the atmosphere has wavelength regions in the infrared where its emissivity is small. The corresponding absorption in these regions is also small, thus enabling the earth's 42 surface to radiate through the atmosphere into space. Radiation outside such windows is absorbed - to varying degrees by the atmosphere and re-radiated as infrared atmospheric radiation. Another striking feature of Figs. 3.7 to 3.11 is the influence of water vapor amount on spectral and total emissivity. For the tropical atmosphere which contains the greatest amount of precipitable water(H>), only the primary atmospheric window between 8 and 12 nm is observed. At other wavelengths the atmosphere approximates a blackbody. As the water vapor content decreases successively for MSA, MWA, SAS and SAW atmospheres, several secondary windows begin to appear. These regions of low e ^  increase transparency and reduce the downcoming infrared radiation from the atmosphere (R) as noted in Figs. 3.7(b) to 3.11(b). The above results indicate that it should be possible to relate the total and spectral atmospheric emissivity directly to the amount of water vapor and ozone contained in the atmosphere. Since these two gases are the main absorbers whose content in the atmosphere is variable, it is evident that their amounts could, through a suitable correlation, be used to predict incoming longwave radiation. This is a departure from other empirical relations that use temperature or vapor pressure without any consideration of the radiative mechanism. The objective of the next section is to present the development of a correlation which relates atmospheric emissivity to the radiating-gas amount present in the atmosphere. The method yields spectral values of atmospheric emissivity and requires only w and z as input, both of which can be estimated easily as explained below. 43 3.4 S I M P L I F I E D T E C H N I Q U E T O C O M P U T E S P E C T R A L A T M O S P H E R I C R A D I A T I O N In Sec. 3.3 it has been shown that the detailed method, though preferred in principle, requires a large amount of data as input In reality it is impractical to specify the variation of temperature, pressure and absorbing gas concentration with altitude at any given moment at a location. However, the amount of precipitable water vapor can be determined from 'surface' meteorological parameters such as surface air temperature (T ) and relative humidity ( R ^ ) which are easily measured. Using E q . (2.6) the value of w can be obtained as[8]: w = 0.493 ^ < 2' 6 a> a Thus, the amount of water vapor can be quite easily determined from standard measurements. The amount of ozone(z) in the atmosphere does not undergo short term changes like water vapor. Monthly average values at different latitudes have been monitored and are presented in Table 2.1 . Ozone amount undergoes only slow seasonal variations and so average monthly values used from Table 2.1 should be sufficiently accurate. The values of w and z can be easily obtained as shown above. Since carbon dioxide has a fixed concentration i n the atmosphere, one need not consider it explicitly; and the amounts of water vapor and ozone become appropriate variables for a suitable correlation. It is logical to expect that the total (and spectral) atmospheric radiation depends in some way on the amount o f radiating gases present in the atmosphere. 44 Of the five Standard atmospheres described in Sec. 3.2.2 and tabulated in the Appendix, the midlatitude summer atmosphere (MSA) has characteristics that fall between the extremes of the tropical and the subarctic atmosphere. Since it represents average characteristics the MSA was selected to develop the correlation between R^ , and w and z. Once R^  is computed, the spectral emissivity is given by: a* B(T ,A) (3.10) The appropriate units are: R^  (W nr2 Mm-1), B (W nr2 Mm1) and (dimensionless). Knowing R^  from the detailed method presented in Sec. 3.3, the total atmospheric emissivity can be witten as: 00 / R^ dX £ a = ~ ~ (3.21a) a T a R (3.21b) a that is, as a ratio of the total downward atmospheric radiation R(W nr2) to that from a perfectly (blackbody) absorbing/emitting atmosphere at the uniform temperature of its surface layer T , In order to simplify the detailed algorithm through a correlation, an investigation is made into the variation of e ^  with the total water vapor and ozone amounts within each wavelength band between 5 and 43 Mm. It has been explained earlier that strong ozone absorption overlaps the weak water vapor band between 9 and 10 Mm; consequently the effects can be separated. Thus, it is reasonable to correlate computed from the detailed method with ozone amount (z) in the range 9 to 10 Mm and with water vapor amount (w) outside this range of wavelengths. 45 In order to study the variation of e ^  with w and z it is necessary to vary the independent parameters w and z over a wide range and to obtain e ^  values for each case by using the detailed algorithm. A curve-fitting is then performed within each wavelength band with the computed values of e^and the corresponding values of w or z. The following discussion refers to water vapor absorption, viz., to wavelengths outside the 9 to 10 Mm range. The procedure to develop the correlation between e ^  and z, the ozone amount is analogous. The MSA has a specific profile for water vapor concentration with altitude. This yields a single value of precipitable water, w. To obtain the concentration profiles that will yield larger and smaller values of w, the MSA concentration at every altitude is multiplied by a constant factor. Hence a number of profiles, each corresponding to a different value of w, could be generated from the MSA. These profiles are then used as input for the detailed computation technique to yield sets of e ^ values (105 values in each set from 5 to 43 Mm) with each set corresponding to a different value of w. Six profiles are obtained from the basic MSA profile corresponding to 0.58, 0.87, 1.16, 1.74, 2.32 and 3.48 cm of precipitable water vapor. Remaining data from the MSA model atmosphere is used along with each of the generated profiles to compute e ^  values using the computer program. The six sets of e ^ values obtained for six water vapor profiles are then analyzed simultaneously. For any wavelength band centered at X, the six values of are examined. Each corresponds to a different value of w. Figure 3.12 illustrates the typical variation of with preciptable water vapor amount w for a few wavelength bands. The figure suggests that it is possible to determine correlation coefficients — though different for each wavelength band —in order 46 to express spectral emissivity as a function of water vapor amount Such a correlation must be developed independentiy for each wavelength band due to the complex absorption spectra of water vapor. These results are obtained using variations of the basic water vapor profile in the midlatitude summer atmosphere (MSA). Within each wavelength band a curve fitting was performed using the DOLSF routine available on the UBC computing system. DOLSF uses a least squares technique to fit a polynomial of one independent variable and can statistically determine the degree of that polynomial which gives the best fit Rather than fit a polynomial between e ^  and w such as seen in Fig. 3.12, it is more common to plot the spectral transmittance written as: versus absorber amount w. Further, it is advantageous to fit the polynomial between In(r^) and w since taking the logarithm has an effect of linearizing the curve; and consequently fewer coefficients are needed to describe the polynomial. Summarizing, a polynomial is obtained to describe the variation of ln(r^) with w yielding unique coefficients aQ^  a^ for each wavelength band in the range 5 to 43 Mm. Then the spectral emissivity is obtained as: = l - e x p ( a o X + a n M , + a 2 X H , + a 3X (3.23a) It is found from the computations that the maximum degree of the best-fit polynomial is three, consequently the general form of Eq. (3.23a) with four coefficients is used. 47 A similar procedure is employed to determine the spectral coefficients bj^ for ozone in the ten wavelength bands between 9.17 and 10.09 Mm. Ozone amount was varied from 0.20 to 0.48 cm(NTP) which covers all the values listed in Table 2.1. The spectral emissivity of the atmosphere in the above range is correlated to ozone amount to yield e . = l-exp(b , + b.. 2 ) (3.23b) a A OA -LA where b^, b^ are the spectral coefficients for ozone absorption obtained from the curve-fitting routine. It is seen that for ozone a first degree polynomial gives the best fit between ln(r^) and z\ consequently only two spectral coefficients are necessary in the range 9.17<X< 10.09 urn. Table 3.1(a) presents the spectral coefficients a^ to be used in Eq. (3.23a) along with the value of w. This equation expresses e ^ as a function of w at any desired wavelength outside the ozone range 9.17 to 10.09 um. Table 3.1(b) presents the spectral coefficients to compute e ^  from Eq. (3.23b) along with ozone amount(z) read from data such as Table 2.1. This gives values of solely as a function of z in the range 9.17<X< 10.09 /im. Summarizing, the detailed algorithm to compute e ^ presented in Sec. 3.3 requires a large amount of data and is complex to use. A simple correlation has been obtained which computes e ^  directly from the amount of absorbing gas present in the atmosphere. The simple formula uses coefficients which are derived using the midlatitude summer atmospheric model but it is demonstrated in the next section that the same coefficients are valid for all the other model atmospheres corresponding to different latitudes and seasons. The 48 only input required for the simple formula is the absorbing-gas content of the atmosphere. The above correlations apply under clear sky conditions. This completes the presentation of the simplified technique to compute spectral atmospheric emissivity. Using these values the spectral atmospheric radiation (R^ ). the total longwave radiation (R) or the total atmospheric emissivity (e ) can be computed. In the following pages a comparison is made 3. between results obtained from the detailed layer-by-layer computations and those obtained from the proposed simple correlation developed in this section. 3.5 COMPARISON OF RESULTS FROM THE DETAILED METHOD AND THE SIMPLIFIED TECHNIQUE In this section a comparison is made between results obtained from the two methods. The detailed method when applied to the five Model Atmospheres, gave results which have been presented earlier (Figs. 3.7 to 3.11). To obtain those results the layer-by-layer technique was employed, together with absorption and detailed atmospheric data. Each plot of e ^  shown there corresponds to one of the five Model Atmospheres. The precipitable water vapor and ozone can be computed from the concentration profile in each case and is also noted on the figures. The simple correlation is developed using the MSA. It relates e ^  to the values of w and z through a set of coefficients that depend on the wavelength being considered. The range of values of w and z considered in order to obtain the coefficients are such as to cover all naturally occurring atmospheres. Values of for each of the model atmospheres are now obtained using their values of w and z as the only input data for the correlation. The spectral emissivity values obtained from the correlation are then 49 compared to those obtained from the detailed method of Sec. 3.3. Figures 3.13 to 3.17 illustrate the excellent agreement between values of e ^  and obtained from the two methods for all the five standard atmospheres. It confirms the hypothesis that spectral emissivity is mainly dependent on water vapor and ozone amount in the atmosphere. Even though the MSA has been used to develop the spectral coefficients for the correlation, the same coefficients give good agreement for the other four standard atmospheres as well. The five standard atmospheres cover all latitudes and seasons that occur on the earth; it can be concluded, therefore, that the coefficients are location independent The maximum difference between e ^  computed from the two methods is 4%. Also noted in Fig. 3.13(b) to 3.17(b) are the total emissivity e& and the total atmospheric radiation R obtained from the two methods. The simplified method yields values of e& and R which are within 1% of those from the detailed layer-by-layer technique. Ramsey et al.[20] have measured the atmospheric state in one instance to use the layer-by-layer method and obtained the result shown in Table 3.2. The detailed method yields values which are in good agreement with measurements. Thus it has been demonstrated that the simplified technique, based on a correlation developed in this chapter, gives results in very good agreement with those from a rigorous layer-by-layer method. The correlation has been tested for the five Model Atmospheres; these cover all geographic latitudes and seasons. It has also been verified by comparing its results with one case of actual measurements which are shown in Table 3.2 . In all cases the agreement has been very good. 50 3.6 CONCLUSIONS The simple correlation developed in this chapter (Eqs. (3.23a) and (3.23b)) requires, as input, only the amount of precipitable water vapor(>v) and ozone(z) contained in the atmosphere at any given moment, in order to compute the spectral atmospheric emissivity. Both w and z can be easily estimated: the former from simple measurements of temperature and relative humidity, and the latter from tabulated seasonal values such as in Table 2.1. Values of spectral emissivity e ^  from the correlation agree well with those from a detailed technique which requires a large amount of atmospheric data and computational effort The correlation gives excellent agreement with the presently used rigorous technique and has been tested for model atmospheres that include all seasons and latitudes. Hence the presently used method to estimate e ^  and can be simplified into a simple correlation equation. This simplified technique can be used to estimate the net infrared radiative loss from the surface of a solar pond under clear skies. Chapter 4 MONTHLY AVERAGE TRANSMTTTANCE- ABSORPTANCE PRODUCT FOR A SOLAR POND 4.1 INTRODUCTORY REMARKS The solar radiation incident on a solar pond at any instant can be computed using the method presented in Chapter 2. However, in order to estimate the useful energy available from a pond one needs to know the solar energy absorbed by the pond bottom and the thermal losses from the pond. Figure 4.1 shows how the solar pond works as a heat trap. If the pond were homogenous, the solar energy absorbed by the pond bottom would be convected upward by the fluid and lost to ambient air. By providing an increasing salt concentration (density) with depth, the tendency for natural convection is supressed. The pond then becomes stratified. Due to the action of wind-induced waves a thin homogenous layer called an upper convection zone (UCZ) develops. The middle non-convecting zone (NCZ) provides the required insulation by supressing convection. A typical density and temperature profile is also shown in the figure. The lower converting zone (LCZ) is a mixed zone and acts as the thermal storage layer. Hot brine from this layer is circulated through heat exchangers to recover the useful energy. Under steady state conditions the energy absorbed by the pond bottom depends upon the extinction coefficient of the brine solution, pond depth, absorptance of the pond bottom surface and the angle of incidence of the solar beam at the pond surface. The extinction coefficient represents the absorption of solar radiation by the brine solution. Measured values of the extinction coefficient for various salt solutions are presented by Lund and Keinonen[40]. 51 52 The extinction coefficients are used to compute the transmission of solar radiation through an exponential relation(Appendix D, Eq. 6). It is useful to develop a parameter called the 'transmittance-absorptance product' which directly relates the absorbed energy to the solar energy incident on the top surface of the pond. The objective of this chapter is to obtain the average daily values for such a parameter. The daily value of the parameter is an energy-weighted quantity calculated using hourly values of total incident radiation and hourly values of the parameter. Hence, it depends on atmospheric conditions, latitude and pond characteristics. The above computation is carried out for a 'characteristic' day of each month. The characteristic day of the month has solar irradiation equal to the average value of solar radiation received outside the earth's atmosphere during that month. The dependence of the daily average value of the parameter on other variables is investigated. Once the monthly variation of the transmittance-absorptance product is known and the monthly average losses are computed, one can express the monthly average pond efficiency in the form of the standard flat plate solar collector equation. The useful energy delivered annually by the pond can then be easily estimated and the pond performance simulated for various conditions. The development of the monthly average daily transmittance-absorptance product, denoted by (TO) is presented in this Chapter and an application is shown in the Appendix. 4.2 DEVELOPMENT OF. THE TRANSMITTANCE-ABSORPTANCE PRODUCT The well known Hottel-Whillier-Bliss (HWB) equation for the useful energy from a flat plate solar collector is[39]: 53 where (ra) is the instantaneous value of the transmittance-absorptance product for the solar collector, (usually assumed constant over an hour), i is the instantaneous solar radiation (direct plus diffuse) assumed constant during that hour, U L is the collector heat loss coefficient based on inlet fluid temperature (T.) and T is the ambient temperature during that hour. The quantity (ra) represents the fraction of the incident solar energy which is absorbed by the absorber plate in a flat plate solar collector. An expression to evaluate (ra) for a flat plate solar collector is given in Ref[39]. The collector heat loss coefficient includes the effect of conduction, convection and radiation from the absorber plate. Analogous to Eq. (4.1) one can envisage a similar equation for a solar pond. However, the thermal mass of the pond is so large that the time scale of temperature changes is of the order of several weeks. It is appropriate, then, to replace the hourly average values in Eq. (4.1) by daily average values and the resulting equation becomes: } U = (TCX)H - U (f - T ) L i a (4.2) The above equation gives the monthly-average daily useful energy (Qu) from the pond as the difference of the daily energy absorbed by the pond and the daily thermal losses from the pond. The absorbed energy is computed from the daily solar radiation incident on the pond (H) and the daily transmittance-absorptance product (7a) and these values can be taken to be monthly average daily values. The heat loss coefficient from a solar pond includes the conductive and convective losses from the pond. Radiative losses are usually ignored for simplicity. In this study it is proposed that an additional loss term representing longwave radiative loss from the surface of the solar pond be included in Eq. 4.2 . The water surface behaves as a perfect 54 absorber/radiator of longwave radiation. The instantaneous (ra) of Eq. (4.1) is an optical parameter that represents the fraction of incident radiation that is absorbed by the solar collector absorbing plate. Its magnitude depends on the angle of incidence of the solar beam and the optical properties of its glass cover system. In Eq. (4.2) the hourly value (ra) is replaced by a daily average value (7a) for the solar pond. This parameter is developed from the hourly values (TO.) by a suitable simulation technique which considers the angle of incidence, absorption of solar radiation in the brine solution, absorption in the pond bottom and multiple reflections between the pond bottom and the water-air interface. The (7a) value for any month is computed for a 'characteristic' day of that month and can be regarded as an average daily value for the entire month. The following subsections show how the hourly and then the daily average values of the transmittance-absorptance product are obtained. 4.2.1 DEVELOPMENT OF THE HOURLY TRANSMITTANCE-ABSORPTANCE PRODUCT (ra) FOR A SOLAR POND Consider the solar pond shown in Fig. 4.2. The pond has a length L, depth D and width W. It contains a brine solution whose concentration varies 0 with depth and has an extinction coefficient of kp , see Appendix D for extinction coefficient The refractive index of the brine solution is N . The absorptance of the pond bottom for solar wavelengths is a p and it is assumed to be a diffuse reflector. The walls are assumed to be perfect absorbers of radiation (black). The following analysis pertains to direct solar radiation. Solar ponds are usually located in places that have abundant sunshine and only a few" cloudy periods each year. Under clear skies the diffuse radiation is a small fraction of 55 the total solar radiation and comes from a small region around the solar disc. The global incident energy can then be treated as beam radiation alone and the hourly values (ra) in the following analysis refer to this situation. However, if the diffuse component is large it must be treated separately; and for this purpose values of the transmittance-absorptance for diffuse radiation are presented later. The ray-tracing method is employed to determine the energy absorbed by the bottom smface. A ray of light is followed through each reflection between the pond bottom and the water-air interface by accounting for all the attenuation that occurs in its path. A portion of the incident solar beam is reflected from the top surface of the pond, the remainder enters the brine and strikes the bottom as direct radiation after being attenuated due to absorption in the brine solution. As the bottom surface is not a perfect absorber, a portion of the incident beam is reflected and the remainder absorbed. The portion that is not absorbed is reflected as diffuse radiation since the bottom surface is painted or lined with HDPE i.e. diffuse reflectors. This diffused reflection is assumed isotropic and can be approximated as beam energy reflected at 60°, an approximation that is frequently made in solar energy studies. This equivalent beam-radiation undergoes successive attenuations by absorption (r ^) and reflection at the water-air interface both of which remain constant since the incidence angle is 60° each time. The fraction absorbed at the pond bottom is given by a finite series containing (Nr£j) terms, where (^Tef) depends on 6 z, x, D and L. A simple algebra yields the following expression for the fraction of incident energy that 56 is absorbed: (TO) = ( l - r . ) T . a i P (4.3) where r^  is the fraction reflected off the air-water interface upon first incidence, is the transmittance due to absorption in the first pass through the pond. The formulae required to calculate the various terms in Eq. (4.3) are summarized in Appendix D. The parallel and perpendicular components of the unpolarized incident beam have different values of r^  and r^ , so the above equation must be applied to each of them separately. It must be emphasized, first, that (ra) is an instantaneous value since the incidence angle (#z) changes throughout the day; at best 6 and hence (ra) can be assumed constant during an hour. Secondly, the above explanation is made for the case in which the ray propagates along the length of the pond (L). The general case is slightly more complex. The movement of the sun from east to west and its zenith movement causes 8 to change at every moment The position of the sun is determined by 8 and the azimuth angle (i//) as illustrated in Fig. 4.3. These two angles can be computed for any moment, on any day, at any given location (i.e. latitude) using the sun-earth astronomical relations presented in Appendix D. The value of 6z determines the value of r^  but the azimuth angle xp determines the general direction in which the ray propagates, shown by the vertical plane indicated by dotted lines in Fig. 4.3. Hence the azimuth angle is used to calculate the number of reflections that occur (N f) before the ray strikes the side wall. It is evident that N j, also depends on the position of the element on the pond surface with respect to the perimeter of the pond. 57 It would be impractical to determine (ra) for every hour for each day of the month. An alternative which is used frequently in solar energy work, is to perform the computations for a single day in each month: this day is called a characteristic day for the month. Such characteristic days are predetermined and are listed in Ref[13], for example. This reduces the task of computing (ra) to only twelve days in a year — one characteristic day for each month. From the above discussion it is clear that the hourly values of (ra) depend on the following variable parameters: (i) pond dimensions D, L and W (ii) brine absorption coefficient kp (iii) absorptance of pond bottom a p (iv) month of year and hour of day (v) geographic latitude <j> In addition, the case of a solar pond in which the walls are reflective in order to boost the solar radiation reaching the pond bottom has been considered but no improvement in (ra) is observed for ponds larger than a few square meters in area. In order to compute hourly values of (ro) for the twelve characteristic days of the year, a computer program is developed. As inputs, the program requires the latitude, pond area, pond depth, extinction coefficient of the brine solution and absorptance of the pond bottom. It then computes (ra) during each hour from sunrise to sunset for each of the twelve characteristic days (months) for three different pond depths D=l, 2 and 3 m. The program is . used to generate a bank of data for latitudes of 0, 15, 30, 45 and 60 degrees; extinction coefficients of 0.4, 0.6, 0.8 nr1 and bottom absorptance of 0.3, 0.6, 0.9 besides the three values of depth. Table 4.1 from Ref[40] presents the 58 measured extinction coefficients of different salt solutions. For a given hour, day and month the program first computes 8^ and ii. It then divides the entire pond surface into a grid and selects the first element of that grid. Using values of 8^ and \p and the pond dimensions it computes the number of reflections (N f) that can occur for a ray originating from that element It then computes (TO.) for that element using Eq. (4.3). After obtaining (ra) for each element in this manner it computes the average value for the entire pond surface for that particular hour. The procedure is repeated for each hour from sunrise to sunset and then for each of the twelve characteristic days. The size of each element of the grid is important, since a fine mesh will increase computation time. A coarse grid reduces computation time as well as accuracy. Hence a suitable size of mesh is obtained as follows: for constant values of all other parameters, the mesh size was gradually decreased and (ra) evaluated in each case until no further changes in (ra) were observed. This occurred at an element size of 0.3*0.3 m and a convenient size of 0.25*0.25 m was then selected. Figure 4.4 shows a typical variation of (ja) with solar time in the month of January at three different latitudes. The conditions are listed in the figure; it is observed that pond area greater than a few square meters has no influence on (ra) so it is not explicitly reported. The graph is symmetric about solar noon, hence only the half day variation is shown. Early morning and late afternoon hours correspond to large angles of incidence, resulting in large reflective losses (r^ ) and hence low (ra) values. In any given month, the angle of incidence at high latitudes is greater than that at lower latitudes at any given instant; hence reflective losses are large even at midday. This is the main reason for smaller (ra) values at high latitudes as observed in Fig. 4.4. 59 The next section describes how the large number of (ra) values obtained for various conditions are reduced into a more usable form. 4.2.2 DEVELOPMENT OF THE DAILY TRANSMrTTANCE- ABSORPTANCE PRODUCT (7a) FOR A SOLAR POND The hourly values of (ra) such as those illustrated in Fig. 4.4 are derived from the detailed computations of the last Section. Those calculations form a basis from which the daily average values are developed. A parameter like (7a) enables one to determine the energy absorbed by a solar pond knowing the value of the daily global solar radiation incident on the pond (H). Monthly-average values of H are routinely measured at meteorological stations around the world and are also available in the form of global radiation maps. As seen earlier, the energy absorbed by the pond bottom throughout a day is the sum of the hourly absorbed energy from sunrise to sunset During the course of a day, the energy incident on the pond changes each hour. The fraction of this incident energy which is absorbed also changes from hour to hour. Hence the value of the incident energy and the absorbed fraction must be considered simultaneously to determine the absorbed energy each hour. Results from the previous Section are used together with the estimated hourly radiation from Chapter 2, to derive the hourly absorbed values. Summation of the hourly values over the entire day yield the daily absorbed radiation which can then be expressed as a daily fraction called (7a). Both the hourly values of incident radiation and the absorbed fraction depend on latitude, month, hour of the day, atmospheric parameters such as precipitable water vapor, ozone and aerosols and pond parameters such as depth, brine extinction coefficient and absorptance of the bottom surface. 60 As an example, consider a location at latitude 0 = 30 N say, in the month of June. From the material presented in Chapter 2 one can compute hourly values of solar radiation reaching the earth's surface on the characteristic day of June. Such a calculation will need the amount of ozone present in the atmosphere in June at 30° N latitude (available from Table 2.1). The water vapor amount is assumed constant at >v=1.5 cm since its effect on the global solar radiation reaching the earth's surface is negligible. The amount of aerosol in the atmosphere is fixed at 0=0.2 and a = 1.3 . For the case being considered, the hourly variation of the beam radiation under these conditions is shown in Fig. 4.5(a) and the corresponding hourly variation of (ra) is shown in Fig. 4.5(b). The fraction of the daily radiation which is absorbed is given as: ss I ( T a ) I At sr ( T a ) = ss Z I At sr where I (W nr2) is the instantaneous value of incident radiation assumed constant during each hour and At is the number of seconds in an hour. The denominator in Eq. (4.4) is simply the monthly-average total radiation incident during the day (H) . The above analysis applies to the situation when the diffuse component is negligible. If this is not the case then the beam and diffuse components must be considered separately; using (7a) values with the beam component and (Ta)^ values (presented later) with the diffuse component of incident radiation. A computer program is developed to perform the computations of (7a) for each month, for latitudes upto 60° N, for various pond parameters such as depth, extinction coefficient and bottom absorptance. This program takes the 61 (TO.) values computed by the previous program as one input, the other being atmospheric parameters and the formulation of Chapter 2 to estimate hourly beam and diffuse solar radiation; knowing which the daily energy absorbed by the pond represented by the numerator in Eq. (4.4) can be calculated. 4.3 DISCUSSION OF THE DAILY TRANSMITTANCE-ABSORPTANCE RESULTS The technique used to develop values of ( 7a ) has been described in the previous section. In this section the values of ( 7 a ) obtained for various pond parameters at different latitudes are presented. These results depend on the hourly radiation and hourly transmittance absorptance values computed for a given month at a given location. The hourly solar radiation was simulated for the characteristic day of each month using monthly-average values of w and z corresponding to the geographic latitude of the location. The aerosol amount (0) does affect the total solar radiation. However, a constant value lying within the range of values for natural atmospheres is considered adequate since we are interested only in the long term, average performance of a solar pond. All results, except Figure 4.12, refer to beam radiation. The diffuse component is expected to be small in most solar pond applications and Figures 4.7 to 4.11 may be used. However, if the diffuse fraction of solar radiation is appreciably large, then the two components must be treated separately: Figures 4.7 to 4.11 for the beam component and Figure 4.12 for the diffuse component of solar radiation. Although the hourly computation of (ra) requires the pond dimensions(L,W) explicitly, the effect of L and W on (7a) is negligible for ponds bigger than a few square meters in area. Hence pond dimensions L, W 62 do not appear in the final results. The pond depth D is, however, a critical parameter as will be seen shortly. The brine extinction coefficient kp is another influential parameter. The results are presented for bottom absorptance a ^ =0.9 . It is observed that the value of (ra) varies linearly with a p as shown in Fig. 4.6. For this reason, the value of ( 7 a ) for 0^*0.9 can be obtained by scaling the value of ( 7 a ) read from the graphs. The results are grouped according to the location (latitude) of the solar pond. Depending on the latitude either one set of Figs. 4.7 to 4.11 is selected. For other latitudes an interpolation may be performed. Each set of figures has three graphs (a), (b), and (c) corresponding to brine extinction coefficient kp = 0.4, 0.6 and 0.8 nr1 respectively. Then, each of the graphs has a set of curves corresponding to pond depth D=l, 2 and 3 m. Figures 4.7 (a), (b), (c) show that at the equator the values of ( 7 a ) are larger than elsewhere and the annual variation is small. Also, the daily radiation recieved during any month in equatorial regions does not vary much for at least nine months of dry weather. This allows a solar pond to operate at constant load under sunny conditions for long periods of time. The influence of pond depth is clear from either Figure 4.7 (a), (b) or (c). The effect of kp can be seen if the figures (a), (b) and (c) are examined simultaneously. Both D and kp decrease ( 7a ) . However, larger values of D imply a thicker insulating zone (NCZ) which reduces the upward conductive heat loss from the pond. Since this study is restricted to the study of radiative parameters only, the influence of D on heat loss coefficient has not been investigated. An examination of Figs. 4.7 to 4.11 reveals that at higher latitudes in the northern hemisphere, the ( 7 a ) for the summer months is the same as that for the equator for similar values of D and k . The annual variation is large, 63 however. This occurs because during the winter months in high latitudes the sun remains low throughout the day, causing large reflective losses due to the larger angles of incidence. This permits only a small portion of the incident radiation to penetrate the pond. The case of high extinction coefficient values shown in the figures serves a dual purpose. It is used for a brine solution which absorbs solar radiation strongly[40]. It can also be used to evaluate the ( 7a ) after a period of operation of the pond. During operation the pond clarity (transmission) is reduced owing to debris, algae growth, formation of iron-oxide suspension in brine, etc. A better estimate of absorbed energy can be made by considering the increase of k with time. P Knowing the average daily radiation in each month (H) and the average daily value of (7a) for the corresponding month, the average energy absorbed during each day of that month is written as: \ b s = <4-5a> However, if diffuse radiation is large then the above equation can be rewritten as: Q , = (TO)H, + (TO), H, (4.5b) xabs b d d where (7a) for beam radiation is read from one of Figs. 4.7 to 4.11, (7a ) d is the corresponding parameter for diffused radiation to be discussed shortly and H^ , H^  are measured values (or estimated from methods presented in Reft 13]) of the daily beam and diffuse radiation incident on the pond surface. 64 Figure 4.12 shows the variation of the daily transmittance-absorptance product for diffuse radiation (Ta)^ with kp and D. The figure was derived assuming a constant incident angle of 60° and is therefore independent of the month, latitude and hour of the day. The other parameters are w=1.5 cm, z=0.30 cm, a = 1.3, *=0.2 at 30°N latitude. The graph is for ap=0.9 and the results can be scaled for other values of a in the same manner as P explained regarding Fig. 4.6 . Usually, under clear skies the diffuse radiation is small and Figs. 4.7 to 4.11 should suffice. For locations having a large fraction of diffuse solar energy Fig. 4.12 can be employed. 4.4 CONCLUSIONS This chapter presents the development of a daily transmittance-absorptance product ( 7a ) for a solar pond. Such a parameter considers the various optical losses and estimates the daily solar energy absorbed by the pond bottom under steady state conditions. Such a parameter varies monthly and depends on latitude, pond depth, extinction coefficient of the brine solution and the absorptance of the pond bottom. The results are presented graphically in Figs. 4.7 to 4.11 for cases when the diffuse radiation is small; Fig. 4.12 can be used for the diffuse component when necessary. Hence this chapter presents a simple technique to estimate the energy absorbed in a solar pond knowing only the daily average solar radiation received at that location. The longwave radiative loss from the pond surface can be computed from the simplified method presented earlier. Together with the other thermal losses usually considered, pond temperatures can be better estimated. Since this study deals only with the radiative parameters for a solar pond, the calculation of pond temperatures is recommended for further 65 investigations. CONLUSIONS The simple algorithm to compute diffuse solar spectral radiation developed by Brine and Iqbal[9] can be extended up to solar zenith angles of 80 degrees through a set of correction factors developed in this study. The results obtained by the use of correction factors agree with those from a rigorous method (BRITE[21]), which in turn, agrees with observed data for a variety of atmospheric conditions to within 15% . Longwave radiative thermal loss from a pond surface can be estimated using a simple correlation developed in this study. The atmospheric radiation computed through this correlation agrees to within 1% of that obtained from a detailed method for a variety of atmospheric conditions. Unlike the presently used detailed methods, the proposed correlation requires only two atmospheric parameters which are readily available. The fraction of the daily incident solar radiation which is absorbed at the bottom of a solar pond can be computed in a simple manner using a proposed parameter called the transmittance-absorptance product. This parameter has been developed for different latitudes for a variety of pond characteristics. 6 6 RECOMMENDATIONS FOR FURTHER STUDY The results obtained using the simplified algorithm for longwave atmospheric radiation need to be compared with observed data for a variety of geographic locations. Good agreement has been observed for the cases shown in Table 3.2 . The correlation has been based on Standard atmospheres. However, the presence of temperature inversions will change the value of R. The results obtained from the correlation under such localized conditions needs to be investigated. The proposed transmittance-absorptance product for a solar pond should be used to predict the performance of existing ponds so that the results can be compared with available data. The next step in a further investigation should be to incorporate the infrared thermal loss in addition to the other losses from a pond and to compute the useful energy available from a pond after the temperatures have been determined. Such an investigation needs to be extended to various latitudes and different pond parameters. 67 REFERENCES H. Weinberger, The physics of the solar pond. Sol. Energy 8(2), 45- 56 (1964). H. Tabor, Large-area solar collectors for power production. Sol. Energy 7(4), 189-194 (1963). A. Sonn and R. Letan, Thermal analysis of a solar pond power plant operated with a direct contact boiler. AS ME Transactions 104, 262-269 (1982) J.D. Wright, Selection of a working fluid for an organic Rankine cycle coupled to a salt-gradient solar pond by direct-contact heat exchanger. ASME Transactions 104, 286-291 (1982) A. Rabl and C. Nielsen, Solar ponds for space heating. Sol. Energy 17, 1-12 (1975). T.A. Newell and R.F. Boehm, Gradient zone constraints in a salt-stratified solar pond. ASME Transactions 104, 280-285 (1982). Y.S. Cha, W.T. Sha and W.W. Schertz, Modelling of the surface convective layer of salt-gradient solar ponds. Journal of Solar Energy Engineering 104, 293-298 (1982). B. Leckner, The spectral distribution of solar radiation at the earth's surface - elements of a model. Sol. Energy 20(2), 143-150 (1978). D.T. Brine and M. Iqbal, Diffuse and global solar spectral irradiance under clear skies. Sol. Energy 30(5), 447-453 (1983). R.A. McClatchey, R.W. Fenn, J.E.A. Selby, F.E. Volz and J.S. Goring, Optical Properties of the Atmosphere (3rd ed.), Air Force Cambridge Research Lab., AFCRL-72-0497 (1972). J.E.A. Selby and R.A. McClatchey, Atmospheric transmittance from 0.25 to 28.5 microns. Computer Code Lowtran 3. Air Force Cambridge Research Lab., AFCRL-TR-75-0255, AD-A017734 (1975). J.EA. Selby, F.Y. Kneizys, J.H. Chetwind Jr. and R.A McClatchey, Atmospheric Transmittance/Radiance: Computer Code Lowtran 4, Air 68 69 Force Geophys. Lab., A F G L - T R - 7 8 - 0 0 5 3 (1978). M . Iqbal, " A n Introduction to Solar Radiation". Academic Press Canada, Toronto, 1983. F. Kasten, A new table and approximate formula for relative optical airmass. Arch. Meteor. Geophys. Bioklimatol, B 14, 206-223 (1966). N . Robinson, "Solar Radiation". American Elsevier, New York, 1966. L. Elterman, Ultraviolet, Visible and Infrared Attenuation for Altitudes to 50 km. A i r Force Cambridge Research Lab., A F C R L - 6 8 - 0 1 5 3 , Environment Research Paper No. 285 (1968). D. Deirmendjian, "Electromagnetic Scattering on Spherical Polydispersions". American Elsevier, New York, 1966. F .Y . Kneizys, E P . Shettle, W.O. Gallery, J . H . Chetwynd, L .W. Abrea Jr., J . E A . Selby, R.W. Fenn and R.A. McClatchey, Atmospheric Transmittance/Radiance: Computer Code Lowtran 5. A i r Force Geophys. Lab., A F G L - T R - 8 0 - 0 0 6 7 (1980). R.E. Bird, A simple spectral model for direct normal and diffuse horizontal irradiance. Sol. Energy 32, 461-471 (1984). E. Vigroux, Contribution a l'etude experimentale de l'ozone. Ann. Phys. 8, 709-762 (1953). R . E Bird, Terrestrial solar spectral modelling. Solar Cells 7 (1983). R.E. Bird and C. Riordan, Simple Solar Spectral Model for Direct and Diffuse Irradiance on Horizontal and Tilted Planes at the Earth's Surface for Cloudless Atmospheres. Solar Energy Research Institute, SERI/TR-215-2436 (1984). C . G . Justus and M . V . Paris, A model for solar spectral irradiance at the bottom and top of a cloudless atmosphere. Submitted to /. of Climate and Applied Meteor., 1984. P. Berdahl and R. Fromberg, The thermal radiance of clear skies.5o/. Energy 29(4), 299-314 (1982). 70 K. Kondratyev," Radiation in the Atmosphere", Academic Press, New York, 1969. P. Varanasi, S. Chou and S.S. Penner, Absorption coeficients of water vapor in the 600 - 1000 c n r 1 region. J. Quant. Spectrosc. Radial. Transfer 8, 1537-1541 (1968). K. Bignell, F . Saiedy and P.A. Sheppard, O n the atmospheric infrared continuum. /. Opt. Soc. of Am. 53(4), 466-479 (1963). H.R. Carlon and C.S. Harden, Mass spectroscopy of ion-induced water clusters: an explanation of the infrared continuum absorption. App. Optics 19(11), 1776-1786 (1980). A . Angstrom, A study of the radiation of the atmosphere. Smithson. Inst. Misc. Coll. 65(3), 1-159 (1918). D. Brunt, Notes on radiation in the atmosphere. Quart. J. Roy. Meteor. Soc. 58, 389-418 (1932). W . H . Dines and L . H . G . Dines, Monthly mean values of radiation from various parts of the sky at Benson, Oxfordshire. Memoirs Roy. Meteor. Soc. 2(11), (1927). W . C . Swinbank, Longwave radiation from clear skies. Quart. J. Roy. Meteor. Soc. 89, 339-348 (1963). S.B. Idso and R . D . Jackson, Thermal radiation from the atmosphere. /. of Geophys. Res. 74(23), 313-319 (1970). G . Clark and C P . Al len, The estimation of atmospheric radiation from clear and cloudy skies. Proc. Second Nat. Pass. Solar ConfNol. 2, Philadelphia (1978). K. Kondratyev, "Radiative Heat Exchange in the Atmosphere". First English ed., translated by O.Teddar. Pergamon Press, 1965. J.W. Ramsey, H . D . Chiang and R.J . Goldstein, A study of incoming longwave radiation from a clear sky. /. of App. Meteor. 21, 566-578 (1982). W . L Wolfe and G . J . Zissis (editors), "The Infrared Handbook", 71 published by The Infrared Information and Analysis Center, Env. Res. Ins l of Michigan, 1978. 38. E M . Sparrow and R . D . Cess, "Radiation Heat Transfer", M c G r a w - H i l l , 1978. 39. J.A. Duffie and W . A . Beckman, "Solar Engineering of Thermal Processes", John Wiley, 1980. 40. P.D. Lund and R.S. Keinonen, Radiation transmission measurements for solar ponds. Sol. Energy 33(3), 237-240 (1984). -l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r mQ = 1 -w : 2 cm z = 0.35 cm oc = 1.3 p = 0.2 -C J 0 = 0-95 -F =0.92 rc -S = 0.2 3 X -(L25 0.55 (L8S LIS 1.45 1.75 2J5 2.35 2.65 235 3J5 3.55 335 Wavelength (Urn) Figure 2.1 Solar spectral radiation outside the earth's atmosphere and at ground level under typical conditions -i 1 1 1 1 1 1 r - i 1 ' I 1 r i i i i i i Mesophere S OJD S t rat o sphere Troposphere _i i i i i L j i i i i u o.o 576.0 Density (gm m 3 ) 1152.0 Figure 2.2 Variation of air density with altitude Figure 2.4 Variation of ozone concentration with altitude Adapted from [6] 74 i r r n — i — | — i — i — | — i — r 10"' 2 S ID"' 2 5 1 2 5 10 2 5 1 0 J 2 Panicle concentration ( c m - 3 ) Particle radius iitm) (a) (b) Figure 2.5 (a)Vertical distribution of particulate concentration From Elterman [9] (b)Particle size distribution for three haze models. H = Stratospheric, L = Continental, M = Coastal. From Deirmendjian [17] 0.25 1.3"7 Wavelength (Mm) 2.5 Figure 2.6 Rayleigh spectral transmittance as a function of airmass I I I I I I I I I U I I I I A I A. I I 1 1 1 1 L 0.4 1.26 2.12 2.99 3.85 Wavelength (lim) Figure 2.8 Spectral transmittance due to water vapor absorption ( w=l cm) • ' II' ' ' ^ W 'V 7 ' ' 'I 1 1 M 0.4 - I 1 1 1 1 1 I I I I I • • « • • • » • • • 1.26 2.12 2.99 Wavelength (Lrm) 3.85 Figure 2.9 Spectral transmittance of uniformly mixed gases (m =1) 2.414 Wavelength (Urn) Figure 2.11 Diffuse spectral irradiance as a Angstrom's turbidity coefficient, 8 function of 78 S || 1 1 1 I 1 I I I 1 1 1 1 1 1 1 1 1 1 1 1 P 1 1 , 1 2.414 Wavelength (Urn) Figure 2.12 The various components of the spectral diffuse irradiance on a horizontal surface Table 2.1 Seasonal variation of atmospheric o z o n e [H Latitude Month Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec 90° N 0.33 0.39 0.46 0.42 0.39 0.34 0.32 0.30 0.27 0.26 0.28 0.30 80° N 0.34 0.40 0.46 0.43 0.40 0.36 0.33 0.30 0.28 0.27 0.29 0.31 70° N 0.34 0.40 0.45 0.42 0.40 0.36 0.34 0.31 0.29 0.28 0.29 0.31 60° N 0.33 0.39 0.42 0.40 0.39 0.36 0.34 0.32 0.30 0.38 0.30 0.31 50° N 0.32 0.36 0.38 0.38 0.37 0.35 0.33 0.31 0.30 0.28 0.29 0.30 40° N 0.30 0.32 0.33 0.34 0.34 0.33 0.31 0.30 0.28 0.27 0.28 0.29 30° N 0.27 0.28 0.29 0.30 0.30 0.30 0.29 0.28 0.27 0.26 0.26 0.27 20° N 0.24 0.26 0.26 0.27 0.28 0.27 0.26 0.26 0.26 0.25 0.25 0.25 10° N 0.23 0.24 0.24 0.25 0.26 0.25 0.25 0.24 0.24 0.23 0.23 0.23 0° 0.22 0.22 0.23 0.23 0.24 0.24 0.24 0.23 0.23 0.22 0.22 0.22 10° S 0.23 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.23 20° S 0.24 0.25 0.24 0.25 0.25 0.25 0.25 0.26 0.26 0.26 0.26 0.25 30° S 0.27 0.28 0.26 0.27 0.28 0.28 0.29 0.31 0.32 0.32 0.29 0.29 40° S 0.30 0.29 0.28 0.29 0.31 0.33 0.35 0.37 0.38 0.37 0.34 0.32 50° S 0.31 0.30 0.29 0.30 0.32 0.36 0.39 0.40 0.40 0.39 0.37 0.35 60° S 0.32 0.31 0.30 0.30 0.33 0.38 0.41 0.42 0.42 0.40 0.39 0.35 70° S 0.32 0.31 0.31 0.29 0.34 0.39 0.43 0.45 0.43 0.40 0.38 0.34 80° S 0.31 0.31 0.31 0.28 0.35 0.40 0.44 0.46 0.42 0.38 0.36 0.32 90° S 0.31 0.30 0.30 0.27 0.34 0.38 0.43 0.45 0.41 0.37 0.34 0.31 Table 2.2 Rayleigh spectral attenuation coefficient Wavelength Eq(2-7) Eq(2-8) Eltermanfc#s] 0.270 1 831 1 933 1 928 0.280 1 573 i 650 i 645 O. 30O 1 187 1 225 1 222 0.320 O 912 0 929 0 927 0.340 O 713 0 719 0 717 0.360 0 564 0 565 0 564 0.380 0 453 0 451 0 450 0.400 0 367 0 364 0 364 0.450 0 227 0 224 0 223 0.500 0 148 0 145 0 145 0.550 0 100 0 098 0 098 0.60O 0 070 0 069 0 069 0.650 0 051 0 050 0 050 0.700 0 037 0 037 0 037 O.BOO 0 022 0 021 0 021 0.900 0 013 0 013 0 013 1 .026 0 008 0 0O8 0 007 1 .060 0 007 0 007 0 003 1 .670 0 001 0 001 0 001 2 . 170 0 0 0 0 0 0 3.500 0 0 0 0 0 0 4 .OOO 0 0 0 0 0 0 Table 2.3 Spectral absorption coefficients for water vapor X k w X X k w X X k w X 0.69 O.I60£ - 01 0.93 0 270£ + 02 1.85 0.22O£ + 04 0.70 0.240£ - 01 0.94 0.380£ + 02 1.90 0 140£ + 04 0.71 0.125£ - 01 0.95 0 4 I 0 £ + 02 1.95 0 160£ + 03 0.72 0 I00£ + 01 0.96 0.260£ + 02 2.00 0.290£ + 01 0.73 0 870£ + 00 0.97 0 310£ + 01 2.10 0.220£ + 00 0.74 0.610£ - 01 0.98 0 I48£ + 01 2.20 0.330£ + 00 0.75 0 I00£ - 02 0.99 0.125£ + 00 2.30 0.590£ + 00 0.76 O.IOO£ - 04 1.00 0.25O£ - 02 2.40 0.2O3£ + 02 0.77 0 I00£ - 04 1.05 0 I00£ - 04 2.50 0.310£ + 03 0.78 0 600£ - 03 1.10 0 320£ + 01 2.60 0.150£ + 05 0.79 0.175£ - 01 1.15 O.230£ + 02 2.70 0.220£ + 05 0.80 0 360£ - 01 1.20 0 I60£ - 01 2.80 0.8OO£ + 04 081 0.330£ + 00 1.25 0.180£ - 03 2.90 0.650£ + 03 0.82 0.153£ + 01 1.30 0.290£ + 01 3 00 0.240£ + 03 0.83 0.660£ + 00 1.35 0.200£ + 03 3.10 0.230£ + 03 0.84 0.155£ + 00 1.40 0 I I 0 £ + 04 3.20 0.100£ + 03 0.85 0 300£ - 02 1.45 0 I50£ + 03 3.30 0.120£ + 03 086 0 I00£ - 04 1.50 0 150£ + 02 3.40 0 195£ + 02 0.87 0.100£ - 04 1.55 0 170£ - 02 3.50 0.36O£ + 01 0.88 0.260£ - 02 1.60 0.100£ - 04 3.60 0.310£ + 01 0.89 0 630£ - 01 1.65 0 100£ - 01 3.70 0.250£ + 01 090 0.210£ + 01 1.70 0.510£ + 00 3.80 0.140£ + 01 0 91 0 I60£ + 01 1.75 0 400£ + 01 3.90 0.170£ + 00 0.92 0.125£ + 01 1.80 0 I30£ + 03 4.00 0 450£ - 02 Table 2.4 Spectral absorption coefficients for mixed gases X k X X V X * g X 0.76 0.300£ + 01 1.75 0 I 0 0 £ - 04 2.80 0.150£ + 03 0.77 0.210£ + 00 1.80 0.100£ - 04 2.90 0.13O£ + 00 1.85 0 I 4 5 £ - 03 3.00 0.950£ - 02 1.25 0.730£ - 02 1.90 0.710£ - 02 3.10 0.100£ - 02 1.30 0.400£ - 03 1.95 0.200£ + 01 3.20 0.800£ + 00 1.35 0.110£ - 03 2.00 0.300£ + 01 3.30 0.190£ + 01 1.40 0.100£ - 04 2.10 0.240£ + 00 3.40 0 130£ + 01 1.45 0t>40£ - 01 2.20 0.38O£ - 03 3.50 O.750£ - 01 1.50 0.630£ - 03 2.30 0.1 lOf - 02 3.60 0.100£ - 01 1 55 0 100£ - 01 2.40 0.170£ - 03 3.70 0 195£ - 02 1.60 0.640£ - 01 2.50 O.I40£ - 03 3.80 0.400£ - 02 1.65 0.145£ - 02 2.60 0 660£ - 03 3.90 0.290£ + 00 1.70 0.100£ - 04 2.70 0.100£ + 03 4.00 0.250£ - 01 Table 2.5 Spectral absorption coefficients for ozone X k o X X k o X X oX 0.290 38.000 0.485 0.017 0.595 0.120 0.295 20.000 0.490 0.021 0.600 0.125 0300 10.000 0.495 0.025 0.605 0.130 0.305 4.800 0.500 0.030 0.610 0.120 0.310 2.700 0.505 0.035 0.620 0.105 0.315 1.350 0.510 0.040 0.630 0.090 0 320 0.800 0.515 0.045 0.640 0.079 0.325 0.380 0.520 0.048 0.650 0.067 0.330 0.160 0.525 0.057 0.660 0.057 0.335 0.075 0.530 0.063 0670 0.048 0.340 0.040 0.535 0.070 0.680 0036 0.345 0.019 0.540 0.075 0.690 0.028 0.350 0.007 0.545 0.080 0.700 0.023 0 355 0.000 0.550 0.085 0.710 0.018 0.445 0.003 0.555 0.095 0.720 0.014 0.450 0.003 0.560 0.103 0.730 0.011 0.455 0.004 0.565 0.110 0.740 0.010 0.460 0.006 0.570 0.120 0.750 0.009 0465 0.008 0.575 0.122 0.760 0.007 0.470 0.009 0.580 0.120 0.770 0.004 0.475 0.012 0.585 0.118 0.780 OOOO 0480 0.014 0.590 0.115 0.790 0.000 Table 2.6 F ratio c for an aerosol fl, 0 10 20 30 40 50 60 70 80 85 f c 0.92 092 0.90 0.90 0.90 0.85 0.78 0.68 060 050 Table 2.7 Diffuse correction factors for various amounts of aerosol AIRMASS Wavelength 1 00 1 .25 1 .50 2.00 3 00 4 00 6 00 (urn) DIFFUSE CORRECTION FACTORS FOR BETA = 0 05 0.35 0 97 0.97 1 .00 1 .06 1 22 1 40 1 93 0.40 1 03 1 .06 1 .09 1 . 15 1 32 1 45 1 86 0.45 0 97 1 OO 1 .02 1 .08 1 22 1 31 1 58 0.50 1 01 0.99 1 .01 1 .05 1 20 1 25 1 44 0.55 0 97 0.94 0.95 1 .01 1 14 1 18 1 32 0.71 1 20 1 . 12 1 . 14 1 . 14 1 23 1 18 1 17 0.78 1 13 1 .06 1 .07 1 .09 1 20 1 15 1 21 0.99 2 87 2.68 2 .69 2.74 2 96 2 83 2 90 2 . 10 0 74 0.79 0.85 0.85 1 06 1 OO 1 12 4 . 10 0 74 0. 79 0.85 0.85 1 06 1 OO 1 12 DIFFUSE CORRECTION FACTORS FOR BETA =0 10 0.35 1 01 1 .01 1 .07 1.21 1 63 2 19 4 30 0.40 1 02 1 .05 1 . 12 1 .25 1 59 1 91 3 03 0.45 0 95 0.97 1 .04 1 . 13 1 38 1 55 2 09 0.50 1 04 0.94 0.97 1 .04 1 22 1 30 1 56 0.55 1 01 0.90 0.93 0.98 1 15 1 17 1 33 0.71 1 22 1 .07 1 .06 1 . 10 1 31 1 15 1 10 0. 78 1 15 1 .04 1 .04 1.14 1 22 1 1 1 1 09 0.99 2 73 2 .45 2 .44 2 .65 2 77 2 46 2 33 2 . 10 0 74 0.76 0.83 0.83 1 02 0 94 0 91 4 . 10 0 74 0.76 0.83 0.83 1 02 0 94 0 91 DIFFUSE CORRECTION FACTORS FOR BETA =0 15 0. 35 1 07 0.94 1 .07 1 . 29 1 93 2 98 7 41 0.40 1 02 1 . 10 1 .35 1 .34 1 74 2 48 3 79 0.45 0 97 1 .01 1 .24 1 .20 1 46 1 65 2 28 0.50 1 07 0.96 0.99 1 .05 1 33 1 39 1 69 0. 55 1 04 0.93 0.95 1 .00 1 24 1 26 1 40 0.71 1 23 1 .05 1 .04 1 .09 1 21 1 13 1 09 0. 78 0 98 0.98 0.97 0.96 1 13 1 02 1 03 0.99 2 26 2.23 2 . 19 2. 15 2 44 2 16 2 07 2 . 10 0 82 0.85 0.96 0.89 1 12 1 07 0 96 4 . 10 0 82 0.85 0.96 0.89 1 12 1 07 0 96 DIFFUSE CORRECTION FACTORS FOR BETA =0 20 0.35 0 99 1 .03 1 .22 1 . 33 2 12 3 46 10 04 0.40 0 96 1 .01 1 .25 1 . 18 1 63 2 17 3 82 0.45 0 91 0.95 1 . 15 1 .06 1 33 1 54 2 17 0.50 1 09 0.98 0.97 1 .04 1 25 1 30 1 60 0.55 1 07 0.94 0.93 0.99 1 16 1 16 1 31 0.71 1 13 1 .02 1 .02 1 .03 1 16 1 07 0 98 0.78 1 05 0.93 0.95 0.95 1 13 1 00 0 95 0.99 2 37 2.07 2.09 2 .06 2 36 2 03 1 82 2. 10 0 75 0.71 0.84 0.92 1 16 1 12 1 25 4. 10 0 75 0.71 0.84 0.92 1 16 1 12 1 25 83 1 *J > o Q. 1 U o 0 1 f °3 \ A \ \ \ . y v 4 > Atmosphere 10 L 15 20 A t m o s p h e r i c w i n d o w 25 W a v e l e n g t h [ y n ] Figure 3.1 Absorption spectra for atmospheric gases and absorption spectrum of the atmosphere[17] 0 0 H H A. u V = M 0 U 1 0 Hz V = 0-502*10 Hz v=1-137»10 Hz Figure 3.2 (a) Fundamental vibrations of the water vapor molecule Figure 3.2 (b) Structure of the water vapor molecule 0^ 14 0 1 t 0 - 4 0 9 x 1 0 Hz 14 V = 0 - 2 0 2 x 1 0 Hz 0 v =0-714 Figure 3.3 Fundamental vibrations of the C O , molecule Wavenumber 1.00 „ 0.80 v e 2 0.60 6 3 0.40 c H 0.20 0.00, /n iii i r e Wavenumber Figure 3.4 Typical atmospheric transmittance due to molecular absorption in a bandwidth of 0.012 microns N V l N M 077777EoTt Figure 3.5 '3 ' 2 Schematic layered atmosphere for transmission calculations 85 130.0 202.0 214.0 226.0 23S.0 250.0 262.0 274.0 286.0 298.0 310.0 Tenperature (K) Figure 3.6 (a) Temperature variations for standard atmospheres R ¥ i i i i i 1 1 1 1 1 1 1 1 1 i 1 1 1 r Figure 3.6(b) Ozone distribution for standard atmospheres 86 a IQ a a 'o a o 'o a enrage-C in " 3 0/ 1 1 1 1 ^ 5 ^ 5 WATER VAPOR = 2-23 cm OZONE = 0-34 cm LAYER-BY-LAYER METHOD I I ' i i — i — i — i — i — i — i — 1 — 1 — 1 — u 5 e 11 14 1 ? 20 » • * 29 32 35 M 41 Wavelength (Urn) Figure 3.7(a) Spectral emissivity of the mid-latitude summer atmosphere 8 | 1 1 1 1 1 1 1- - i 1 1 1 1 1 1 1 r i 1 1 1 r B( T n . \ ) T Q = 294 K LAYER-BY-LAYER METHOD Wavelength (Urn) Figure 3.7(b) Spectral atmospheric radiation for the mid-latitude summer atmosphere 87 0 5 e 11 1 4 17 20 23 26 29 32 35 38 41 Wavelength (Um) Figure 3.8(a) Spectral emissivity of the mid-winter atmosphere -latitude Si in 1 1 1 1 1 1 1 1 * 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 'E t C -T Q = 2 7 2 K c -c o . 0 «N B ( T Q , A ) -• 0 0 R LAYER-BY-LAYER METHOD Ik. O - -tn _ V) 0 tn 1 1 1 1 1 1 1 1 1 1 i t 1 1 1 1 1 1 1 1 1 1 ' i " * M 1 0 5 e 11 1 4 n 20 23 26 29 32 35 39 41 Wavelength (Um) Figure 3.8(b) Spectral atmospheric radiation for the mid-latitude winter atmosphere — o in 0/ CL V) ~i 1 1 r WATER VAPOR = 3 25 cm OZONE = 0-21 cm LAYER-BY-LAYER METHOD 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I I 5 « » 14 n 20 23 26 29 32 35 38 41 Wavelength (Um) Figure 3.9(a) Spectral emissivity of the tropical atmosphere R | "T I I I I I I I I I I I I I 1 1 1 1 1 1 1 1 1 1 -'E N '£ c o '•6 a a. T Q = 3 00 K LAYER-BY-LAYER METHOD Figure 3.9(b) Spectral atmospheric radiation for the tropical atmosphere 89 1 f1 " i i T i i i i i . r~i'i _ -o bcgriDO 'o d <s'e> A B'O (b odwatei da» (to WATER VAPOR = 1-75 cm OZONE = 0- 39 cm LAYER-BY-LAYER METHOD j i i i ' i - i — i — i i i i i i i ' ' << 14 17 20 23 26 29 32 35 38 41 Wavelength (bm) R I — i — r Figure 3.10(a) Spectral emissivity of the sub-arctic summer atmosphere n 1 1 1 1 1 r E B ' E _ BIT-.A) TQ = 2 8 7 K LAYER-BY-LAYER METHOD Wavelength (bm) Figure 3.10(b) Spectral atmospheric radiation for the sub-arctic summer atmosphere 'mo 'o ci o '<D d o 'o o -g>4 WATER VAPOR = 0-31 cm OZONE = 0- 53 c m L A Y E R - B Y - L A Y E R METHOD _ l I I I I I I I I L ' i I I I I I I I 1 L_ 5 8 II 14 17 20 23 26 23 32 3S 38 41 Wavelength (Um) Figure 3.11(a) Spectral emissivity of the sub-arctic winter atmosphere - i — i — i — i — i — i — i — i — i — i — i — i — i i i i r B( T„.\ ) T Q = 2 5 7 K I A Y F R - B Y - L A Y E R METHOD Figure 3.11(b) Spectral atmospheric radiation for the sub-arctic winter atmosphere 9 1 1-0 1-5 20 2-5 Water Yapor (cm) 30 3-5 Figure 3.12 Typical variation of spectral emissivity with water vapor content (w) at various wavelengths T 1 1 1—-i—Tr: w 3 E a. CO 6 i*iA»rW(l» WATER VAPOR = 2-23 cm OZONE = 0-34 cm . j i i i i i i J i i_ -1 I 1 I 1 I I I L _ _ L _ 5 8 11 14 17 20 23 26 29 32 35 39 41 Wavelength (Lim) Figure 3.13(a) Comparison of e ^ vaiues from the detailed technique and the simplified procedure for the M S A ' E 5 c o X) a u C L - i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r -i 1 1 r-e Q R H -h 0.72 332 W m O O 0.73 334 W m - 2 -f- Layer-by-layer method <t> Simplified technique J I ' I 1 I I 1 I L_ 14 17 20 23 26 29 32 35 38 41 Wavelength (Urn) Figure 3.13(b) Comparison of R~ values from the detailed technique and the simplified procedure for the M S A VI tn 1 - i O y a. to 3IB 'm Q) WATER VAPOR =0-78 cm OZONE = OA 2 cm -I I I I L_ _ l I L . -I I I I I I I I I l l 5 8 II 14 17 20 23 26 29 32 35 38 4 Wavelength (bm) Figure 3.14(a) Comparison of e ^ values from the detailed technique and the simplified procedure for the M W A ~i i I I I I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 4 - h 0 . 6 5 2 0 3 W m " 2 0 . 6 6 2 0 4 W m -2 'E 5 c o o "5 o Layer-by-layer method <J> Simplified technique Q. Wavelength (bm) Figure 3.14(b) Comparison of values from the detailed technique and the simplified procedure for the M W A WATER VAPOR = 3-25 cm OZONE = 0-21 cm _i i i i_ i i i i i i i i i i i i i _ 17 20 23 26 29 Wavelength (Um) 32 35 38 41 Figure 3.15(a) Comparison of e ^ values from the detailed technique and the simplified procedure for the Tropical atmosphere ~i 1 1 1 1 1 r n 1 1 r ~i 1 1 1 1 1 1 r R -I 1_ o.8A O O 0-83 355 W m 3 5 1 W m - 2 . -f- Layer-by-layer method <J> Simplified technique 17 20 23 26 29 32 Wavelength (Um) 38 41 Figure 3.15(b) Comparison of R , values from the detailed technique and the simplified procedure for the Tropical atmosphere WATER VAPOR = 1-75 cm OZONE -- 0- 39 cm -I 1 1 1 1 1 1 1 1 1 1 I I I 1 I I I I 1 i ' l L_ 5 8 11 14 17 20 23 26 29 32 35 38 41 Wavelength (Um) Figure 3.16(a) Comparison of e . values from the detailed technique and the simplified procedure for the S A S atmosphere - i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 r R 0 . 6 9 2 6 7 W m - a O O 0 . 7 0 2 6 8 W m -1 -f- Layer-by-layer method <^ > Simplified technique 17 20 23 26 29 32 Wavelength (Um) Figure 3.16(b) Comparison of values from the detailed technique and the simplified procedure for the S A S atmosphere 96 > <9 — o C L U l 3, T 1 1 r WATER VAPOR = 0-31 cm OZONE = 0- 53 cm i i i i i i i i i t t i i i i i i i > t i 14 17 20 23 26 29 32 35 36 41 Wavelength ( Um) Figure 3.17(a) Comparison of the e values from the detailed technique and the simplified procedure for the SAW atmosphere i 1 1 1 1 1 1 1 1 1 r ~i 1 1 r 0.60 - i 1 r R K 9 Wm ' E o O O 0.60 H 9 Wm -2 P S a. U) -f- Layer-by-layer method <J> Simplified technique 38 41 Wavelength (Um) Figure 3.17(b) Comparison of values from the detailed technique and the simplified procedure for the SAW atmosphere Table 3.1(a) Coefficients (a.^) to compute spectral atmospheric emissivity: in the range 5.25<X<8.92 and 10.24<X<42.54 Mm X aox a l X a2X a3X 5 25 -o 2783E+01 0 0 0 0 0 0 6 00 -o 2912E+01 0 0 0 0 0 0 G .75 -o 3027E+01 0 0 0 0 0 0 7 .06 - 0 28G1E+01 - 0 2633E+00 0 9713E-01 -0 '1083E-01 7 14 -0 3032E+01 - 0 G330E-01 0 2358E-01 -0 2S44E-02 7 22 -0 2432E+01 -0 8079E+00 0 2929E+00 -o 3228E-0 t 7 36 - 0 8956E+00 - 0 2395E+01 0 8017E+00 - 0 838GE-01 7 56 - 0 2246E+01 - 0 1077E+01 0 3873E+00 -0 4248E-01 7 80 -0 1670E+01 - 0 1742E+01 0 6115E+00 -0 6602E-01 7 95 0 9471E-01 -0 1407E+01 0 2606E+00 -0 2137E-01 8 .08 -o 3757E-01 - 0 1970E+00 0 0 0 0 8 22 - 0 2107E-01 - 0 1392E+00 0 0 0 0 8 46 - 0 3747E-01 - 0 1981E+00 0 0 0 0 8 70 -o 1523E-01 -o 1151E+00 0 0 0 0 8 92 -o 1252E-01 -0 1024E+00 0 0 0 0 10 24 -o 9938E-02 - 0 9025E-01 0 0 0 0 10 49 - 0 9937E-02 - 0 9043E-01 0 0 0 0 10 74 -0 7549E-02 - 0 7680E-01 0 0 0 0 11 05 -0 7549E-02 -0 7696E-01 0 0 0 0 11 .44 - 0 9934E-02 - 0 9103E-01 0 0 0 0 11 95 - 0 3677E-01 - 0 2049E+00 0 0 0 0 12 37 - 0 9664E-01 - 0 3645E+00 0 0 0 0 12 55 0 1096E+00 - 0 1258E+01 0 231GE+00 -0 2307E-01 12 84 0 1125E+00 - 0 1044E+01 0 2057E+00 -o 2334E-01 13 13 0 100GE+00 -0 1458E+01 0 2528E+00 -o 2 117E-01 13 31 o 1392E+00 - 0 2596E+01 0 606 1E+00 -o 4798E-01 13 47 -0 1G38E+01 - 0 2362E+01 0 8261E+00 -o 889GE-01 13 67 - 0 3448E+01 - 0 3108E+00 0 1151E+00 -0 1287E-01 13 90 - 0 3570E+01 -0 1752E+00 0 651BE-01 -0 7304E-02 14 50 - 0 3745E+01 0 0 0 0 0 0 15 50 - 0 3802E+01 0 0 0 0 0 0 16 17 - 0 3565E+01 - 0 3466E+00 0 1284E+00 -0 1436E-01 16 50 -o 3702E+01 - 0 1956E+00 0 7277E-01 - 0 8155E-02 17 17 - 0 1589E+01 - 0 2G49E+01 0 9243E+00 -o 9936E-01 17 71 - 0 6018E+00 - 0 3487E+01 0 1145E+01 -0 1182E+00 17 80 o 1G17E+00 -0 2657E+01 0 5902E+00 -0 4437E-01 17 93 0 9641E-01 - 0 175GE+01 0 2888E+00 -o 1930E-01 18 06 0 1625E+00 -0 2659E+01 0 5891E+00 - 0 4417E-01 18 29 -0 5892E+00 -0 3523E+01 0 1156E+01 - 0 1 192E+00 18 56 0 1640E+00 -0 2663E+01 o 5872E+00 -o 4380E-01 18 79 0 9618E-01 -0 1757E+01 0 2867E+00 -o 1910E-01 18 97 0 1651E+00 - 0 2666E+01 0 585GE+00 - 0 4349E-01 19 0G 0 1316E+00 - 0 3429E+01 0 9470E+00 - 0 8586E-01 19 47 - 0 5649E+00 - 0 3592E+01 0 1177E+01 -0 12 13E+00 19 90 -o 1554E+01 -0 2818E*01 0 9819E+00 -o 1054E+00 20 02 -0 5540E+00 - 0 3623E+01 0 118GE+01 -0 1221E+00 20 07 -o 1218E+00 -0 3743E+01 0 1153E+01 - 0 1136E+00 20 32 0 1682E+00 - 0 2674E+01 0 5803E+00 -0 4253E-01 20 42 0 1684E+00 - 0 2674E+01 0 5799E+00 -o 4247E-01 20 62 -0 1124E+00 -0 3766E+01 0 1159E+01 -0 114 1E+00 20 92 0 1693E+00 - 0 2677E+01 0 5779E+00 - 0 4212E-01 21 44 -o 3299E+01 -0 9502E+00 0 3485E+00 - 0 3871E-01 22 00 - 0 4049E+01 - 0 3881E-01 0 1454E-01 -0 1G37E-02 22 30 -0 5128E+00 -0 3736E+01 0 1220E+01 -0 1254E+00 22 70 -0 3317E+01 - 0 9792E+00 0 3592E+00 -0 3989E-01 23 30 -0 49G3E+00 -0 3780E+01 0 1233E+01 - 0 1266E+00 23 83 - 0 4136E+01 - 0 1521E-03 0 5754E-04 - 0 G510E-05 24 1 1 - 0 3799E+01 - 0 4383E<-00 0 1625E+00 -0 1817E-01 Table 3.1(a) continued 98 X aox a l X a 2 X a 3 X 24 . 23 0 18 12E+00 -o 3541E+01 0 9590E+00 -o 8549E-01 24 . 35 -o 7116E-01 -o 1459E+01 0 1350E+00 0 0 24 . 47 0 1175E+00 - 0 1075E+01 0 2107E+00 -o 2487E-01 24 . 59 -o 7101E-01 -o 1460E+01 0 1348E+00 0 0 24 . 7 1 0 1849E+00 -0 3549E+01 0 9597E+00 -0 8544E-01 24 . 83 -o 4730E+00 - 0 3842E+01 0 1251E+01 -0 1283E+00 25 .45 - 0 4181E+01 0 0 0 0 0 0 26 . 28 -o 2410E+01 - 0 2155E+01 0 7722E+00 - 0 8449E-01 26 . 79 - 0 3850E+01 - 0 4629E+00 0 1716E+00 - 0 1919E-01 27 . 18 - 0 4223E+01 -0 1666E-03 0 6302E-04 -0 7140E-05 27 .52 - 0 4361E+00 -0 3937E+01 0 1279E+01 -0 1309E+00 27 . 86 - 0 4239E+01 -0 1691E-03 0 6393E-04 -0 7240E-05 28 . 50 - 0 4254E+01 0 0 0 0 0 0 29 . 50 - 0 4275E+01 0 0 0 0 0 0 30 . 50 - 0 4295E+01 0 0 0 0 o 0 3 1 .50 -o 4314E+01 0 0 0 0 0 0 32 .50 - 0 4331E+01 0 0 0 0 0 0 33 .50 - 0 4349E+01 0 0 0 0 0 0 34 . 50 - 0 4365E+01 0 0 0 0 0 0 35 .50 - 0 4380E+01 0 0 0 0 0 0 36 .50 - 0 4395E+01 0 0 0 0 0 0 37 . 50 - 0 4409E+01 0 0 0 0 0 0 38 .02 - 0 4180E+01 -0 3026E+00 0 1127E+00 -0 1264E-01 38 . 20 -0 2376E+01 -0 2449E+01 0 8761E+00 -0 9575E-01 38 . 48 - 0 3293E+00 -0 4202E+01 0 1355E+01 -0 1379E+00 38 . 86 - 0 2374E+01 - 0 2461E+01 0 8803E+00 - 0 9621E-01 39 . 15 -o 4299E+01 -o 1688E+00 0 6303E-01 -0 7082E-02 39 . 38 - 0 4433E+01 0 0 0 0 0 0 40 .00 -0 4441E+01 0 0 0 0 0 0 40 85 - 0 445 1E+01 0 0 0 0 0 0 4 1 .31 -0 4383E+01 -0 9456E-01 0 3538E-01 - 0 3980E-02 4 1 . 5 1 -0 3148E+01 - 0 1619E+01 0 5898E-»-00 -0 6522E-01 41 . 72 - 0 8150E+00 -0 3975E+01 0 1336E+01 -0 1400E+00 42 ,oo -0 6870E-01 -0 1475E+01 0 1261E+00 0 0 42 . 34 0 1763E+00 -o 2S94E+01 0 5190E+00 -o 3289E-01 42 .83 -0 4226E+01 -0 3168E+00 0 1179E+00 -0 1323E-01 42 . 54 -o 2364E+01 -o 2524E+01 0 9024E+00 -o 9859E-01 Table 3.1(b) Coefficients ( b ^ ) to compute spectral atmospheric emissivity : e = i - e x p ( b + b z) in the range 8.92<X<10.24 Lim X box b l X 9 .17 - 0 2103E+00 -o 1053E+00 9 . 29 - 0 2222E+00 -o 3140E+00 9 . 36 -o 2511E+00 -o 5707E+00 9 . 44 -0 2887E+00 - 0 7959E+00 9 .54 - 0 2852E+00 - 0 7757E+00 9 .68 - 0 2700E+00 -0 6888E+00 9 . 8 0 -o 2405E+00 -o 4872E+00 9 .90 - 0 2225E+00 - 0 3075E+00 10 .00 - 0 2125E+00 -o 1282E+00 10.09 - 0 2103E+00 - 0 2439E-01 99 Table 3.2 Comparison of measured values of atmospheric radiation (R) with values computed from the detailed technique Run Dumber Air tem-perature C C ) Experi-mental C a l c u -lation Differ-ence % 400 22 . 2 332 0 336 0 1 . 2 401 29 .0 388 3 385 4 0 .7 402 16 . 3 307 0 305 4 0 .7 403 20 .4 339 4 328 5 3 . 2 404 15 .6 278 7 291 4 4 . 5 405 15 .3 300 6 296 3 1 .2 406 19 .4 318 3 309 5 2 .8 408 1 1 . 5 278 5 281 8 1 . 1 410 1 1 . 3 265 3 264 8 0 .4 41 1 6 .9 253 7 253 9 0 . 0 412 12 .6 285 8 284 2 0 . 5 413 8 .6 260 7 261 3 0 .2 414 7 . 2 253 1 260 1 2 .7 415 - 1 0 . 7 151 4 146 0 3 .5 419 - 17 .4 148 1 152 6 3 . 0 100 brine return hot brine tempN 60 C /salt ' ' cone. 2 0 % Figure 4.1 Schematic of a typical salt-gradient solar pond Figure 4.2 Schematic to estimate the hourly solar radiation absorbed in a solar pond Figure 4.3 Schematic to determine the hourly value of ( r a ) for any solar position 8 , i/> ("COO 0° Hour angle cO k =0-4 m" P 105 Figure 4.4 Typical hourly variation of ( r a ) for various latitudes J U N E 0 = 30° N H o u r a n g l e , CO ( d e g r e e s ) Figure 4.5(a) Hourly variation of direct beam radiation for a typical case at 0 = 3 O ° N in June Figure 4.5(b) Hourly variation of the ( r a ) value for a typical case at 0 = 3 O ° N in June 1 0 4 i i i i i 1 1 1 1 1—i 1 i 1 1 1 1 1 1 1 r A A Pond depth=1m o © Pond depth=2m « o Pond depth=3m QJ in -e k = 0-i, m P o c y 0- 9 -1 -© e--i 1 1 1 1 1 1 1 1 1 i i i i i i i i i i i 2 3 * 5 6 7 6 9 10 11 12 Month (Jan=1....Dec=12) Figure 4.7(a) Monthly variation of ( r a ) at 0 ° N latitude for k =0.4 m " P 105 _ *- — « Pond depth=3m CD o. cn T 1 1 r n 1 1 1 1 1 1 1 1 i i i i i i r -A Pond depth=1m - o Pond depth=2m kp =0-6 m o c =0-9 P -1 •o •o- -e - © o- e - - -n <p < o -e> -o ^ _1 \ I I I L 1 2 3 4 I I I I I I 5 6 7 8 9 10 11 12 Month (Jan=1....Dec=12) Figure 4.7(b) Monthly variation of ( r a ) at 0 ° N latitude for k =0.6 nf 1 P > T r o = ~ 1 1 1 ' ' 1 1 i 1 r—i 1 1 1 1 1 1 1 r—, r (- A A Pond depth=ln o o Pond depth=2ii • * Pond depth=3a k p = 0-8 m -1 oCp-.0-9 A A- _ A -- - 0 - - - 0 - - - 0 - -o- - - o - - e > - - e - - © - - - o - -o- - - i J 1 1 1 1 ' 1 1—1 L _ J 1 1 | 3 4 5 6 7 8 9 Month (Jan=1....Dec=12) 10 II 12 Figure 4.7(c) Monthly variation of ( r a ) at 0 ° N latitude for k =0.8 m P 107 Figure 4.8(a) Monthly variation of ( f a ) at 15° N latitude for k =0.4 m" 1 P 108 "i 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 r A — -A Pond depth=1m o o Pond depth=2m *- — o Pond depth=3m k =0-6 m P o c p = 0,9 QJ \2 •o -©— - - e — - o o —o-<v -« — - © — — - * — _ J I I I I I I I I I I I ' i ' ' ' I L 2 3 4 5 6 7 8 9 Month (Jan=1....Dec=12) 10 11 12 Figure 4.8(b) Monthly variation of (7a) at 15° N latitude for k =0.6 m" 1 P i i i i i i i i n i i i i i i i i i i i r (_ A A Pond depth=1» o o Pond depth=2n « » Pond depth=3n k =0 -8 m"1 P o C =0-9 P a 9 I ro I 3 . . a - - « - ' -° ° °- - - ° " ' e - ' o - - <D - . . 0 . J I I I I I I I 1 I I I I I I 1 I I I I L 2 3 4 5 6 7 8 9 10 II 12 Month (Jan=l....Dec=12) Figure 4.8(c) Monthly variation of ( f a ) at 1 5 ° N latitude for k =0.8 m"' 110 i i i i i i i i i 1 1 1 1 1 1 1 1 1 1 1 r A A Pond depth=lin o o Pond depth=2m -1 « » Pond depth=3m -e —e_ k = 0-4 m P oc p= 0-9 J 1 i I i I I I i i i i i i i 2 3 4 5 6 7 8 9 10 II 12 Month U3n=1....Dec=12) Figure 4.9(a) Monthly variation of ( r a ) at 30°N latitude for k =0.4 m'x P I l l -i 1 1 r ~ o o Pond depth=2m _ «- —o Pond depth=3m cx m «=•' r o I 5 * i 1 1 1 1 1 1 1 1 1 - — i 1 1 i i r s— -A Pond depth=1* k =0-6 m P o C p = 0-9 -1 •o • o-_ -o -©— - -e— - o o — -o e>— _ ' I I I I I \ I L 1 I I 1 I I I 1 L 2 3 4 5 6 7 8 9 10 11 Month (Jan=1....Dec=12) Figure 4.9(b) Monthly variation of (TO ) at 30 N latitude for k =0.6 m " P 1 1 2 ~ i i i i i i i i 1 1 1 1 1 1 1 1 1 1 1 1 r _ A A Pond depth=1n ro I 3 o o Pond depth=2n « o Pond depth=3n k =0-8 m P o C =0-9 P -1 - -o o o- - - o - - a - . e . -i 1 1 1 1 1 1 1 1 i i i i i i i i i i i i 5 6 7 8 9 10 II Month (Jan=1....Dec=12) Figure 4.9(c) Monthly variation of (ra) at 30 N latitude for k =0.8 m" P 113 Figure 4.10(a) Monthly variation of ( r a ) at 4 5 ° N latitude for k =0.4 m" P 114 o-L «- — o Pond depth=3m 2 ^ CL O n 1 1 1 1 1 1 1 1 i r i 1 1 1 1 1 1 1 r •A Pond depth=1m - o Pond depth=2m k p =0-6 m =0-9 P -1 •o . cr — o -o I I I I I ' I I I I 1 I I I I I 1 I I I L_ 2 3 4 5 6 7 B 9 10 11 12 Month (Jan=1....Dec=12) Figure 4.10(b) Monthly variation of ( r c O at 45 N latitude for k p = 0 . 6 m " 1 1 5 n 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 r - A Pond depth=1n -o Pond depth=2» - * Pond depth=3a k = 0-8 m P o C p : 0 . 9 QJ £ ci n •o - -o--O- - -O- - - Q <D - -O-I I I I | J I I I I I I L J I L 5 6 7 8 Month (Jan=1....Dec=12) 11 12 Figure 4.10(c) Monthly variation of ( r a ) at 45°N latitude for k =0.8 m " P 116 "i r ~i 1 1 1 r T 1 1 1 1 1 1 r -i 1 r A A Pond depth=ln) o © Pond depth=2m « * Pond depth=3m k = 0-4 m P oc p= 0-9 -1 o I I I I I I I I I ° l 2 3 4 5 6 7 8 9 10 II 12 Month (Jan=1....0ec=12) Figure 4.11(a) Monthly variation of ( r a ) at 60 N latitude for k =0.4 m"' P 117 A — -A Pond depth=1m © o Pond depth=2m L <s- — < > Pond depth=3m L -| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r / V Q-- 0 - — k =0-6 m P o c =0-9 P -1 \ V J 1 1 i_ J I I L J I I I I I I I I I L 5 6 7 8 Month (Jan=1....Dec=12) Figure 4.11(b) Monthly variation of ( r a ) at 60°N latitude for k =0.6 m " P 118 — i 1 1 1 1 1 1 1 1 1 1 F ~ — i 1 1 i i i i i r A A Pond depth=1n s o Pond depth=2n » t> Pond depth=3n k p = 0-8 m o C =0-9 P -1 V -o- . O- - - o - - o . O --o -o -o-_ l 1 I l _ I 2 3 I I I I I I I I I I 1 1 1 L 5 6 7 8 Month (Jan=1....Dec=12) 11 12 Figure 4.11(c) Monthly variation of ( r a ) at 60 N latitude for k =0.8 m~" P 119 Figure 4.12 Variation of ( r a ) . for diffuse radiation with k and D 1 120 APPENDIX A Extraterrestrial Solar Spectral Irradiance at Mean Sun-Earth Distance X J onX X *onX X T onX X ! onX 0 250 64 56 0 440 1755 00 0 650 1597 50 1 150 551.04 0 255 91 25 0 445 1929 49 0 660 1555 00 1 200 4 9 7 . 9 9 0 260 122 50 0 450 2099 99 0 670 1505 00 1 250 4 6 9 . 9 9 0 265 253 75 0 455 2017 51 0 680 1472 50 1 300 4 3 6 . 9 9 0 270 275 00 0 460 2032 49 0 690 1415 02 1 350 3 8 9 . 0 3 0 275 212 50 0 465 2000 00 0 700 1427 50 1 400 3 5 4 . 0 3 0 280 162 50 0 470 1979 99 0 710 1402 50 1 450 318.99 0 285 286 25 0 475 2016 25 0 720 1355 00 1 500 2 9 6 . 9 9 0 290 535 00 0 480 2055 OO 0 730 1355 00 1 550 2 7 3 . 9 9 0 295 560 0 0 0 485 1901 26 0 740 1300 00 1 600 247.02 0 300 527 50 0 490 1920 0 0 0 750 1272 52 1 650 2 3 4 . 0 2 0 305 557 50 0 495 1965 0 0 O 760 1222 50 1 700 2 1 5 . 0 0 0 310 602 51 0 500 1862 52 0 770 1 187 50 1 750 187.00 0 315 705 OO O 505 1943 75 0 780 1 195 00 1 800 170.00 0 320 747 50 0 510 1952 50 0 790 1 142 50 1 850 149.01 0 325 782 50 0 515 1835 01 O 800 1 144 70 1 900 136.01 0 330 997 50 0 520 1802 49 0 810 1113 0 0 1 950 1 2 6 . 0 0 0 335 906 25 0 525 1894 99 0 820 1070 00 2 000 118.50 0 340 960 0 0 0 530 1947 49 0 830 104 1 00 2 100 93 . 0 0 o 345 877 50 O 535 1926 24 o 840 1019 99 2 200 74 . 75 0 350 955 0 0 0 540 1857 50 o 850 994 00 2 300 63 . 25 0 355 1044 99 0 545 1895 01 0 860 1002 00 2 400 5 6 . 5 0 0 360 940 OO 0 550 1902 50 0 870 972 OO 2 500 48 .25 0 365 1 125 01 0 555 1885 00 0 880 966 00 2 600 42 . 0 0 0 370 1165 0 0 0 560 1840 02 0 890 945 00 2 700 36 .50 0 375 1081 25 0 565 1850 0 0 0 900 913 0 0 2 800 32 . 0 0 0 380 1210 00 0 570 1817 50 0 910 876 00 2 900 28 . 0 0 0 385 931 25 0 575 184B 76 0 920 841 0 0 3 000 24 . 75 0 390 1200 OO 0 580 1840 0 0 0 930 830 OO 3 100 21 .75 0 395 1033 74 0 585 1817 50 0 940 801 oo 3 200 19.75 0 400 1702 49 0 590 1742 90 0 950 778 00 3 300 17 . 25 0 405 1643 75 0 595 1785 OO 0 960 77 1 00 3 400 15 . 75 0 4 10 17 10 OO 0 600 1720 OO o 970 764 00 3 500 14 . 0 0 0 415 1747 50 0 605 1751 25 0 980 769 00 3 .600 12 .75 0 420 1747 50 0 610 1715 00 0 990 762 00 3 . 7 0 0 1 1 . 50 0 425 1692 51 0 620 1715 00 1 000 743 99 3 . 8 0 0 10.50 0 4 30 1492 50 0 630 1637 50 1 050 665 98 3 . 9 0 0 9 . 5 0 0 435 1761 25 0 640 1622 50 1 100 606 04 4 . 0 0 0 8 . 50 a) C.Fr'c-hlich and C W e h r l i , Spectral distribution of solar irradiance from 250 nm to 25000 nm. World Radiation Center, Davos, Switzerland. Presented in Ref[6] b) X in ym and i . in (W n r 2 unr1) APPENDIX B 1 2 1 Spectral Absorption Coefficients of Water Vapor and Ozone in the Range 5.25 to 42.83 um Table B 1 Effective mass absorption coefficient for water vapor X AX k wX X AX k wX X AX kwX 5 . 25 o 500 4 ooo 13 31 0 160 0 1PC 2-1 47 0 120 0 .050 6 . 0 0 1 000 20 000 13 47 0 160 0 400 24 . 59 0 120 0. I00 6 .75 0 500 15 000 13 67 0 250 0 800 24 . 71 0 120 0 .200 7 .06 0 130 0 750 13 90 0 200 0 900 24 . 83 0 120 0. 300 7 .14 0 025 1 ooo 14 50 1 000 20 000 25 45 1 120 2 . 500 7 .22 0 130 0 550 15 50 1 OOO 20 000 26 28 0 520 0. 500 7 .36 0 150 0 300 16 17 0 340 0 800 26 79 0 520 0 .8C0 7 .56 0 225 0 500 16 50 0 330 0 900 27 18 0 270 2 . 500 7 .80 0 210 0 400 17 17 1 000 0 400 27 52 0 400 0 .300 7 .95 0 100 0 080 17 71 0 090 0 300 27 86 0 280 2 . 500 8 .08 0 160 0 015 17 80 0 090 0 150 28. 50 1 000 11.000 8 . 22 0 130 0 010 17 93 0 170 0 100 29 50 1 000 12.000 8 .46 0 350 0 015 18 06 o 090 0 150 30 50 1 000 15.000 8 . 70 0 130 0 008 18 29 0 360 0 300 31 50 1 000 15.000 8 .92 0 300 0 007 18 56 0 180 0 150 32 50 1 000 1 5 .000 9 . 17 0 190 0 006 18 79 0 280 0 100 33 50 1 000 16.000 9 . 2 9 0 060 0 006 18 97 0 080 0 150 34 50 1 000 9 .000 9 .36 0 080 0 0O6 19 06 0 100 0 200 35 50 1 000 20 .000 9 .44 0 080 0 006 19 47 0 720 0 300 36 50 1 000 20 .000 9 .54 0 130 0 006 19 90 0 140 0 400 37 50 1 000 20 .000 9 . 6 8 0 130 0 006 20 02 0 100 0 300 38 02 0 100 0 .900 9 . 8 0 0 120 0 006 20 07 0 200 0 250 38 20 0 190 0 . 500 9 . 9 0 0 090 0 006 20 32 0 100 0 150 38 48 0 380 0 . 300 10.OO o 090 0 006 20 42 0 100 0 150 38 86 0 380 0 . 500 10.09 0 090 0 006 20 62 0 300 0 250 39 15 0 200 1 .000 10.24 o 220 0 006 20 92 0 300 0 150 39 38 0 250 20 .000 10.49 0 280 0 006 21 44 0 730 0 650 40 00 1 000 20 .000 10.74 0 220 0 005 22 00 0 400 1 200 40 85 0 710 20 .000 1 1 .05 0 400 0 005 22 30 0 200 0 300 4 1 31 0 200 1 . 100 1 1 .44 0 380 0 006 22 70 0 600 0 650 4 1 51 0 200 0 .600 1 1 .95 0 650 0 015 23 30 o 600 0 300 41 72 0 210 0. 350 12.37 0 180 0 030 23 83 0 460 2 500 42 00 0 4 10 0. 100 12 .55 0 180 0 065 24 1 1 0 1 10 0 800 42 34 0 210 0. 150 12.84 0 390 0 050 24 23 0 120 0 200 42 54 0 210 0. 500 13.13 0 200 0 080 24 35 0 120 0 100 42 83 c 350 0 . 900 Table B 2 Effective mass absorption coefficient for ozone  X AX k o X 9 . 17 O. 190 35 . 9 . 29 0 . 0 6 0 120. 9 . 36 0 . 0 8 0 260. 9 .44 0 . 0 8 0 420 . 9 .54 0 . 130 400 . 9 . 6 8 0 . 130 330. 9 . 8 0 0 . 120 200. 9 . 9 0 0 . 0 9 0 1 10. 10.OO 0 . 0 9 0 40 . 10 .09 0 . 0 3 0 7 . APPENDIX C 122 M o d e l Atmospheres Used as a Basis for the Computation of Atmospheric Radiation .Ht Pressure Temp Density Water vapor Ozone (km) (mbar) (K) (kg m 3) -3 (kg m ) (kg m 3) Tropical 60E-04 0 . 0 101E+04 300. 0 . 117E+04 0 20E+02 0 1. 0 904E+03 294 . 0 . 106E+04 0 10E+02 0 60E-04 2 . 0 805E+03 288 . 0 . 969E+03 0 90E+01 0 50E-04 3 . 0 715E+03 284 . 0 . 876E+03 0 50E+01 0 50E-04 4 . 0 633E+03 277 . 0 . 795E+03 0 20E+01 0 50E-04 5 . 0 559E+03 270. 0 . 720E+03 0 20E+01 0 40E-04 e. 0 492E+03 264 . 0 . 650E+03 0 90E+00 0 40E-04 7 . 0 432E+03 257 . 0 . 586E+03 0 50E+O0 0 40E-04 8 . o 378E+03 250. 0 . 529E+03 0 30E+00 0 40E-04 9 . 0 329E+03 244 . 0 471E+03 0 10E+00 0 40E-04 10. 0 286E+03 237 . 0 420E+03 0 50E-01 0 40E-04 1 1 . 0 247E+03 230. 0 374E+03 0 20E-01 0 4CE-04 12 . 0 213E+03 224 . 0 332E+03 0 60E-02 0 40E-04 13. 0 182E+03 217 . 0 293E+03 0 20E-02 0 40E-04 14 . 0 156E+03 210. 0 258E+03 0 10E-02 0 40E-04 15 . 0 132E+03 204 . 0 226E+03 0 80E-03 0 50E-04 1S . 0 111E+03 197 . 0 197E+03 0 60E-03 0 50E-04 17 . 0 937E+02 195 . 0 168E+03 0 60E-03 0 70E-04 18 . 0 789E+02 199 . 0 138E+03 0 50E-03 0 90E-04 19 . 0 666E+02 203 . 0 1 15E+03 o 50E-03 0 10E-03 20 . 0 565E+02 207 . o 951E+02 0 40E-03 0 20E-03 21 . 0 480E+02 211. 0 794E+02 0 50E-03 0 20E-03 22 . 0 409E+02 215 . 0 664E+02 0 50E-03 0 30E-03 2 3 . 0 350E+02 217 . 0 562E+02 0 50E-03 0 30E-03 24 . 0 300E+02 219. 0 476E+02 0 60E-03 0 30E-03 2 5 . 0 257E+02 221 . 0 404E+02 0 70E-03 0 30E-03 30 . 0 122E+02 232 . 0 183E+02 0 40E-03 0 20E-03 35 . o 600E+01 243 . 0 860E+01 0 10E-03 0 90E-04 40 . 0 305E+01 254 . 0 418E+01 0 40E-04 0 40E-04 45 . 0 159E+01 265 . 0 210E+01 0 20E-04 0 10E-04 50 . 0 854E+0O 270. 0 110E+01 0 G0E-05 0 40E-05 70 . 0 579E+00 219. 0 921E-01 0 10E-06 0 90E-07 100. o 3QOE-03 210. 0 500E-03 0 10E-08 0 40E-10 Ht Pressure Temp Density Water vapor Ozone (km) (mbar) (K) (kg m~3) (kg m 3 ) (kg m 3) Midlatitude Summer 0 . 0 101E+04 294 . 0 119E+04 0 14E+02 0 60E-04 1. 0 902E+03 290. 0 108E+04 0 93E+01 0 60E-04 2 . 0 802E+03 285 . 0 976E+03 0 59E+01 0 60E-04 3 . 0 710E+03 279 . 0 885E+03 0 33E+01 0 62E-04 4 . 0 628E+03 273 . 0 799E+03 0 19E+01 0 64E-04 5. 0 554E+03 267 . 0 721E+03 0 10E+01 0 66E-04 6 . 0 487E+03 261 . 0 648E+03 0 61E+00 0 69E-04 7 . 0 426E+03 255 . 0 583E+03 0 37E+00 0 75E-04 8 . 0 372E+03 248 . 0 523E+03 0 21E+00 0 79E-04 9 . 0 324E+03 242 . 0 567E+03 0 12E+00 0 86E-04 10. 0 281E+03 235 . 0 416E+03 0 64E-01 0 90E-04 1 1 . 0 243E+03 229 . 0 369E+03 0 22E-01 0 11E-03 12 . 0 209E+03 222 . 0 327E+03 0 60E-02 0 12E-03 13 . 0 179E+03 216. 0 288E+03 0 18E-02 0 15E-03 14 . 0 153E+03 216. 0 246E+03 0 10E-02 0 18E-03 15 . 0 130E+03 216. 0 210E+03 0 76E-03 0 19E-03 16 . 0 111E+03 216. 0 179E+03 0 64E-03 0 21E-03 17 . 0 950E+02 216. 0 154E+03 0 56E-03 0 24E-03 18 . 0 812E+02 216 . 0 131E+03 0 50E-03 0 28E-03 19 . 0 695E+02 217 . 0 111E+03 0 49E-03 0 32E-03 20 . 0 595E+02 218. 0 945E+02 0 45E-03 0 34E-03 21 . 0 510E+02 2 19. 0 806E+02 0 51E-03 0 36E-03 22 . 0 437E+02 220. 0 687E+02 0 51E-03 0 36E-03 23 . 0 376E+02 222 . 0 587E+02 0 54E-03 0 34E-03 24 . 0 322E+02 223 . 0 501E+02 0 60E-03 0 32E-03 25 . 0 277E+02 224 . 0 429E+02 0 67E-03 0 30E-03 30. 0 132E+02 234 . 0 197E+02 0 36E-03 0 20E-03 35 . 0 652E+01 245 . 0 926E+01 0 11E-03 0 92E-04 40 . o 333E+01 258 . 0 451E+01 0 43E-04 0 41E-04 45 . 0 176E+01 270. 0 227E+01 0 19E-04 0 13E-04 50 . 0 951E+00 276 . 0 120E+01 0 63E-05 0 43E-05 70 . 0 671E-01 2 18. 0 107E+00 0 14E-06 0 86E-07 100. 0 300E-03 2 10. 0 500E-03 0 10E-08 0 4 3 E - 10 Midlatitude Winter 0. 0 102E+04 272 . 0 130E+04 0 40E+01 0 60E-04 1 . 0 897E+03 269 . 0 116E+04 0 30E+01 0 50E-04 2 . 0 790E+03 265 . 0 104E+04 0 20E+01 0 50E-04 3. 0 694E+03 262 . 0 923E+03 0 10E+01 0 50E-04 4 . 0 608E+03 256 . 0 828E+03 0 70E+00 0 50E-04 5 . 0 531E+03 250. 0 74-1 E+03 0 40E+00 0 60E-04 6 . 0 463E+03 244 . 0 661E+03 0 20E+00 0 60E-04 7 . 0 402E+03 238 . 0 589E+03 0 80E-01 0 80E-04 8 . 0 347E+03 232 . 0 522E+03 0 40E-01 0 90E-04 9 . 0 299E+03 226 . 0 462E+03 0 20E-01 0 10E-03 10. 0 257E+03 220. 0 407E+03 0 80E-02 0 20E-C3 1 1 . 0 220E+03 219. 0 350E+03 0 70E-02 0 20E-03 12 . o 188E+03 219 . 0 300E+03 o 60E-02 o 30E-03 13 . 0 161E+03 218. 0 257E+03 0 20E-02 0 30E-03 14 . 0 138E+03 218 . 0 221E+03 0 10E-02 0 30E-03 15 . 0 118E+03 2 17. 0 189E+03 0 80E-03 0 30E-03 16 . 0 101E+03 217 . 0 162E+03 0 60E-03 0 40E-03 17 . 0 861E+02 216. 0 139E+03 0 60E-03 0 40E-03 18 . 0 735E+02 216. 0 119E+03 0 50E-03 0 40E-03 19 . 0 628E+02 • 215 . 0 102E+03 0 5OE-03 0 40E-03 20. 0 537E+02 215 . 0 869E+02 0 40E-03 0 40E-03 21 . 0 458E+02 215 . 0 742E+02 0 50E-03 0 40E-03 22. . 0 391E+02 215. 0 634E+02 0 50E-03 0 40E-03 23 . 0 334E+02 215. 0 541E+02 0 50E-03 0 40E-03 24 . 0 286E+02 215. 0 462E+02 0 60E-03 0 40E-03 25 . 0 243E+02 215 . r 0 . 395E+02 0 70E-03 0 30E-03 30 . 0 111E+02 2 17. 0. 178E+02 0 40E-03 0 20E-03 35 . 0 518E+01 228 . 0 . 792E+01 0 10E-03 0 90E-04 40 . 0 253E+01 243. 0 363E+01 0 40E-04 0 40E-04 45 . 0 129E+01 259 . 0 174E+01 0 20E-04 0 10E-04 50 . 0 682E+00 266 . 0 895E+00 0 60E-05 0 40E-05 70 . 0 467E-01 231 . 0 705E-01 0 10E-06 0 SOE-07 100. 0 300E-03 210. 0 500E-03 0 10E-08 0 40E-10 Ht Pressure Temp Density Water vapor Ozone (km) (mbar) (K) (kg m3) (kg m3) (kg m^ ) Subarctic Summer 0 . 0 101E+04 287 . 0 122E+04 0 90E+01 0 50E-04 1. 0 896E+03 282 . 0 111E+04 0 60E+01 0 50E-04 2 . 0 793E+03 276. 0 997E+03 0 40E+01 0 60E-04 3 . 0 700E+03 271 . 0 899E+03 0 30E+01 0 60E-04 4 . 0 616E+03 266 . 0 808E+03 0 20E+01 0 60E-04 5 . 0 541E+03 260. 0 722E+03 0 10E+01 0 60E-04 6 . 0 473E+03 253 . 0 652E+03 0 50E+00 0 70E-04 7 . 0 413E+03 246 . 0 585E+03 0 30E+00 0 80E-04 8 . 0 359E+03 239 . 0 523E+03 0 10E-01 0 80E-04 9 . 0 311E+03 232 . 0 466E+03 0 40E-01 0 10E-03 10. 0 268E+03 225 . 0 412E+03 0 20E-01 0 10E-03 1 1 . 0 230E+03 225. 0 356E+03 0 90E-02 0 20E-03 12 . 0 198E+03 225 . 0 30GE+03 0 60E-02 0 20E-03 13 . 0 170E+03 225. 0 263E+03 0 20E-01 0 30E-03 14 . 0 146E+03 225 . 0 226E+03 0 10E-02 0 30E-03 15 . 0 125E+03 225. 0 194E+03 0 80E-03 0 30E-03 16 . 0 108E+03 225 . 0 167E+03 0 60E-03 0 30E-03 17 . 0 928E+02 225 . 0 144E+03 0 60E-03 0 40E-03 18 . 0 798E+02 225. 0 124E+03 0 50E--03 0 40E-03 19 . 0 686E+02 225 . 0 106E+03 0 50E-03 0 40E-03 20 . 0 589E+02 225. 0 913E+02 0 40E-03 0 40E-03 2 1 . 0 507E+02 225 . 0 785E+02 0 50E-03 o 40E-03 22 . 0 436E+02 225 . 0 675E+02 0 50E-03 0 30E-03 23 . 0 375E+02 225 . 0 581E+02 0 50E-03 0 30E-03 24 . 0 323E+02 226. 0 496E+02 0 60E-03 0 30E-03 25 . 0 278E+02 228 . 0 425E+02 0 70E-03 0 30E-03 30 . 0 134E+02 235. 0 198E+02 0 40E-03 0 10E-03 35 . 0 661E+01 247 . o 932E+01 0 10E-03 0 90E-04 40 . 0 340E+01 262 . 0 453E+01 0 40E-04 0 40E-04 45 . 0 181E+01 274 . 0 231E+01 0 20E-04 0 10E-04 50. 0 987E+00 277 . 0 124E+01 o 60E-05 0 . 40E-05 70 . 0 707E-01 216. 0 114E+00 0 10E-06 0 . 90E-07 100. 0 300E-03 210. 0 500E-03 0 10E-08 0 4 0 E - 1 0 Subarctic Winter 0. 0 101E+04 257 . 0 137E+04 0 10E+01 0 40E-04 1 . 0 887E+03 259 . 0 119E+04 o 10E+01 0 40E-04 2 . 0 778E+03 256 . 0 106E+04 0 90E+00 0 40E-04 3 . 0 680E+03 253 . 0 937E+03 0 70E+00 0 40E-04 4 . o 593E+03 248 . 0 834E+03 0 40E+00 0 40E-04 5 . o 516E+03 241 . 0 746E+03 0 20E+00 0 50E-04 6 . 0 447E+03 234 . 0 665E+03 0 106+00 0 50E-04 7 . 0 385E+03 227. 0 590E+03 0 50E-01 0 70E-04 8 ; 0 331E+03 221 . 0 523E+03 0 10E-01 0 90E-04 9. 0 283E+03 217 . 0 454E+03 0 80E-02 0 20E-03 10. 0 242E+03 217 . 0 388E+03 0 50E-02 0 20E-03 1 1 . 0 207E+03 217 . 0 332E+03 0 40E-02 0 30E-03 12 . 0 177E+03 217 . 0 283E+03 0 30E-02 0 40E-03 13 . 0 151E+03 217 . 0 242E+03 0 20E-02 0 50E-03 14 . 0 129E+03 217 . 0 207E+03 0 10E-02 0 50E-03 15 . 0 110E+03 217 . 0 177E+03 0 80E-03 0 60E-03 16 . 0 943E+02 217 . 0 152E+03 0 60E-03 0 60E-03 17 . 0 806E+02 216. 0 130E+03 0 60E-03 0 60E-03 18 . 0 688E+02 215 . 0 111E+03 0 50E-03 0 60E-03 19 . 0 588E+02 215. 0 953E+02 0 50E-03 0 60E-03 20 . 0 501E+02 214. o 816E+02 0 40E-03 0 60E-03 21 . 0 428E+02 214 . 0 698E+02 0 50E-03 0 50E-03 22 . 0 365E+02 213. 0 597E+02 0 50E-03 0 50E-03 23 . 0 311E+02 212. 0 510E+02 0 50E-03 0 40E-03 24 . 0 265E+02 212 . 0 436E+02 o 60E-03 0 40E-03 25 . 0 226E+02 211. 0 372E+02 0 70E-03 0 30E-03 30 . 0 102E+02 216 . 0 164E+02 0 40E-03 0 20E-03 35 . 0 470E+01 222. 0 737E+01 0 10E-03 0 90E-04 40 . 0 224E+01 235 . 0 333E+01 0 40E-04 0 40E-04 4 5 . 0 111E+01 247 . 0 157E+01 0 20E-04 0 10E-04 50. 0 572E+00 259. 0 768E+00 0 60E-05 0 40E-05 70. 0 402E-01 246 . 0 569E-01 0 10E-06 0 90E-07 100. 0 300E-03 210. 0 500E-03 0 10E-08 0 40E-10 125 A P P E N D I X D Summary of Formulae for Solar-Geometry and Reflection/Transmission in a Partially Transparent Medium (1) Angle of incidence for beam radiation (8 ) cos 6 = sl n 6 sin<j> + cosS cos<j> cosw (2) Solar azimuth angle (\p) cos = (sinct sina) - s i n 6 ) / c o s ( 9 0 - 0 )costj) (3)Solar sunrise/sunset hour angle (co ) (4) Angle of refraction (Snell's law) n , s i n 9, = n„ s i n 0 (5)Reflection of unpolarized radiation (Fresnel's equations) S' cosco s r = -tantj) tan<5 s i n 2 ( 9 2 + e L ) t a n 2 ( 9 2 - ej_) r r 126 (6) Absorption of radiation by a partially transparent medium (Bouger's law) x = e x p ( - k i l ) Nomenclature specific to this Appendix k. Extinction coefficient of the medium (m ^") Z path length of ray in the absorbing medium (m) n^,n^ Refractive indicies of mediums on either side of the interface r Average reflectance of the unpolarized radiation r _ l _ » r H Reflectance of the perpendicular and parallel components of unpolarized radiation, respectively 6 Solar declination angle (degrees) assumed constant over a day. Ref. [6] presents tabulated values for each day of the year to Hour angle of the sun (degrees); morning (+)ve values, evening (-)ve values, noon zero <JU , 0 ) Surise/sunet hour angle (degrees); that hour sr ss » w angle which corresponds to sunrise or sunset 9^ ,02 Angle of incidence,angle of refraction for a ray 

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