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A STUDY OF  THE  RADIATIVE  PARAMETERS FOR  DESIGN OF A  SOLAR POND  By AUROBINDO K E N N E T H 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 F U L F I L M E N T O F  T H E REQUIREMENTS F O R T H E D E G R E E O F MASTER O F APPLIED SCIENCE in THE FACULTY  O F G R A D U A T E STUDIES  Department of Mechanical Engineering  We accept this thesis as conforming to the required standard  T H E UNIVERSITY  O F BRITISH C O L U M B I A  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 forfinancialgain 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 c  81  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  iii  99  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 (k ^) using the methods of fl  Angstrom and Bird  75 N  2.7(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 3.2(b) Structure of the water vapor molecule  :  3.3 Fundamental vibrations of the CO2 molecule 3.4  Typical atmospheric transmittance due  83  to molecular  bandwidth of 0.012 microns  83 84 absorption in a 84  3.5 Schematic layered atmosphere for transmission calculations  iv  84  3.6(a) Temperature variations for standard atmospheres 3.6(b) Ozone distribution for standard atmospheres 3.7(a) Spectral emissivity of the mid-latitude summer atmosphere  85 85 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 3.13(b) Comparison of  92  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 3.14(b) Comparison of R^  values from the detailed technique and the  simplified procedure for the MWA 3.15(a) Comparison of  93  values from the detailed technique and the  simplified procedure for the Tropical atmosphere 3.15(b) Comparison of R^  93  94  values from the detailed technique and the  simplified procedure for the Tropical atmosphere  v  94  3.16(a) Comparison  of e ^ values from the detailed technique and  simplified procedure for the SAS atmosphere 3.16(b) Comparison  of  95  values from the detailed technique and the  simplified procedure for the SAS atmosphere 3.17(a) Comparison  of e ^ &  95  values from the detailed technique and  simplified procedure for the SAW atmosphere 3.17(b) Comparison  the  the 96  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 e  z  .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 (a )  103  p  4.7(a) Monthly variation of (7a) at 0 N latitude for k=0.4 nr  104  1  p  4.7(b) Monthly variation of (7a) at 0 N latitude for k = 0.6 nr  105  1  p  4.7(c) Monthly variation of (7a) at 0 N latitude for k=0.8 nr  106  1  p  4.8(a) Monthly variation of (7a) at 15 N latitude for k = 0.4 nr  1  p  4.8(b) Monthly variation of (7a) at 15 N latitude for k=0.6 nr  1  p  4.8(c) Monthly variation of (7a) at 15 N latitude for k=0.8 nr  4.9(a) Monthly variation of (7a) at 30 N latitude for k=0.4 nr  1  p  4.9(b) Monthly variation of (7a) at 30 N latitude for k=0.6 nr  1  vi  108 109  1  p  p  107  110 Ill  4.9(c) Monthly variation of (TO) at 30 N latitude for k=0.8 nr  112  1  p  4.10(a) Monthly variauon of (7a) at 45 N latitude for k = 0.4 nr  1  p  4.10(b) Monthly variation of (7a) at 45 N latitude for k=0.6 nr  1  p  4.10(c) Monthly variation of (7a) at 45 N latitude for k = 0.8 nr  113 114 115  1  p  4.11(a) Monthly variauon of (7a) at 60 N latitude for k=0.4 nr  1  p  116  4.11(b) Monthly variation of (7a) at 60 N latitude for k=0.6 nr  1  117  4.11(c) Monthly variation of (7a) at 60 N latitude for k = 0.8 nr  1  118  p  p  4.12 Variation of (7a), for diffuse radiation with k and D  vii  119  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 nr unr ] 1  2  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  F  Ratio of energy scattered in the forward direction to total  c  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 nr day ] Monthly-average daily beam solar radiation incident on 2  H, b  1  the solar pond [kJ nr day"] J  Hj  1  Monthly-average daily diffuse solar radiation incident on the solar pond [kJ nr day ] 2  1  i  Running index  I  Instantaneous global solar radiation incident on a flat plate solar collector [W nr ] 2  Aerosol-scattered spectral diffuse irradiance arriving on a horizontal surface [W nr nmr ] J  viii  1  Multiply reflected spectral diffuse irradiance arriving on a horizontal surface [W m~ Mm" ] 2  I^  1  Direct beam spectral irradiance on a surface normal to the solar rays [W nr Mm ] 2  i ^ Q n  -1  Extraterrestrial spectral  irradiance  at mean  sun-earth  distance on a surface normal to the solar rays [W nr  2  um ] 1  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  k^ Q  Coefficient of attenuation due to absorption by ozone [cm ] -1  kp  Extinction coefficient of the brine solution [nr ]  k^  Coefficient of attenuation due to Rayleigh scattering by  1  air molecules k^ w  Coefficient of attenuation due to absorption by water vapor [cm ] -1  k^  Absorption coefficient for an atmospheric absorber in the infrared wavelengths [m kg"] 2  k^  Effective  infrared  1  absorption  coefficient for a given  wavelength band of width AX [m kg"] J  L  Length of solar pond [m]  m  Airmass at actual pressure  m_  Ozone mass  a  ix  1  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 nr ] 2  Monthly-average daily energy absorbed in a solar pond [kJ nr day ] 2  1  Monthly-average daily useful energy available from a solar pond [kJ nr day ] 2  1  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 nr ] 2  Relative humidity Spectral longwave radiation received  at ground level  [W nr /im-] 2  1  Spectral longwave radiation received at ground level from layer (i) of the N-layered atmosphere. [W m  J  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 nr /inr ] 2  1  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 nr K"] 2  Heat  1  loss coefficient for a solar  pond  based on  monthly-average values [W nr K"] 2  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 nr ] 3  Spectral albedo of the cloudless atmosphere Spectral albedo of the ground Stefan-Boltzmann constant [W nr K"] 2  Instantaneous  value  4  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  Table  nf  Contents  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  3.  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 2.5 Conclusions A SEMPLJTEED TECHNIQUE TO COMPUTE SPECTRAL ATMOSPHERIC RADIATION  21 23 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 radiation  35  3.4 Simplified Radiation  Technique  to compute  to Compute  spectral atmospheric Spectral  Atmospheric  43  3.5 Comparison of Results from the Detailed Method and the Simplified Technique 48 3.6 Conclusions 4.  MONTHLY  50 AVERAGE  TRANSMITTANCE-ABSORPTANCE  PRODUCT FOR A SOLAR POND  51  4.1 Introductory Remarks  51  4.2 Development of the Transmittance-Absorptance Product 4.2.1 Development of the hourly transmittance-absorptance product (ra) for a solar pond  52  4.2.2 Development of the daily product (ra) for a solar pond  transmittance-absorptance  54 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  121  APPENDIX C  1  APPENDIX D  1 2  xvi  2  2  5  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 nr  2  and  $200 nr for a solar pond and a flat plate solar collector, respectively. A solar 2  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 m) in the U.S. - have also 2  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:  Useful energy  Absorbed solar energy  Conduction, convection, evaporation losses  Atmospheric radiation + inc ident on pond  Inf rared radiation f r o m pond to sky  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 nr and compared to the first term (approx. 500 W 2  nr ) 2  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 processes, namely, scattering and  types of attenuation  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 and  extent to which solar radiation  number  of  molecules  in  the  path  is attenuated  of  a  solar  depends on the  ray.  The  type  mass of  an  atmospheric constituent in a column of unit cross sectional area is called optical mass given by:  m  = act  / p ds s  (2.1)  where (ds) is the differential  path length and p is the density at that point in  the  shows the  atmosphere.  altitude  Figure  2.2  typical  variation  air  density  with  in an atmosphere. The continuously varying density causes the ray to  follow a curved path and the integration in Eq. (2.1) the  of  actual  slant  path  of  the  ray.  More  must be performed along  useful  for  non-dimensional quantity called relative optical mass (m ) r  calculations  is  the  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  r  =  where 6  [cos 6  z  + 0.15(93.885 - 6 ) z  - 1  -  2 5 3  ]-  1 (  2  <  3  )  is the solar zenith angle. For stations not at sea level the air mass  must be pressure corrected:  9  P m  m  a  s  (2.4)  r 1013.25  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 m  r  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 m  o  assuming that  all ozone was concentrated at height h and h = 22 km : 3  m  o  =  3  (1 + h /r )[ cos 6 + 2(h_/r )]-0.5 2  3  e  (2.5)  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]: *" = 0.493  T  (2.6a)  10 where p is the partial pressure of water vapor in saturated air, and T y  ambient temperature. The value of p  y  &  is the  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 (k ^). Leckner[8] uses the following expression for the r  same coefficient:  k  rA  =  °-  0087  35 A ~  4,08  Yet another expression has been proposed by Kneizys et a/.[18]:  (2.7)  12  k  -  [A  (115.6406  A  -  1.335T-1  (2.8)  rX  In  Table  Eq.  2.2  (2.7)  a  comparison  and Eq. (2.8)  of  the  attenuation  is given. Values of  coefficients  k^  obtained  computed by  from  Elterman  and  considered quite accurate are also given for comparison. In the ultraviolet region some  difference  strongly  is  observed  in this region  but  this  compared to  algorithm of Iqbal[13], equation (2.7) The  complete  expression for  can  be  Rayleigh  ignored  since  attenuation.  ozone absorbs  Consistent with  the  will be employed in this study. transmittance  due  to  Rayleigh  attenuation  ( T ^ ) is then written as:  T  exp(-0.008735 A ~ '  =  r A  where  4  m  fl  is the  0 8  m )  (2.9)  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 A E R O S O L A T T E N U A T I O N 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  absorption  and  molecules,  Mie  general, is  called  scattering  scattering Mie  by  aerosols  scattering.  produces  more  is  Unlike forward  much  Rayleigh than  greater  than  scattering backward  by  their air  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 , aX  =  M"  (2.10)  a  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 k ^ in which a  the coefficients are wavelength dependent:  -a k  W  aX h  e  =  r  3  n  (2.H)  X  e  and /3j and  a  ±  = 1.0274  for  X < 0.5 urn  a  2  = 1.2060  for  X > 0.5 urn  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 (?"^) is obtained from: a  T  aX  =  « P < - a X »a> k  (  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  Leckner[8] the  work  Leckner's  of  vapor  developed an expression for absorption by water vapor from  McClatchey et  study  presented  al.[ 10,11,12]  the  absorption  who  employed detailed methods.  coefficients  (k ^)  at  w  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  wA  T  exp[-0.2385 k  =  w A  w  1^/(1 + 20.07 ^ * ,  n^) ' ] 0  (2.13)  4 5  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 w=l  2.8  shows a  cm. As expected,  the  typical  variation  transmittance  of  T  w  ^ with  wavelength for  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 ( g^) T  T . gX  where mixed  =  exp[-1.41 k , m /(1+118.93 k gA a gA L  m )°* ] a 45  ;  (2.14)  is the effective absorption coefficient presented in Table 2.4. Since 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 (k ^) at wavelength intervals which did not correspond to Q  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:  T  oA  =  e x  P  ( _ k  oX  z  V  (2-15)  where z is the amount of ozone in the vertical direction and m  r  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 (J y) is then given by[13]: n  nX  1  =  ^nX^rX  where i ^ Q n  T a  X  T  X gX T  W  T D  X^  (2-16)  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 arigorouscalculation. 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 (m ) and in the case of water vapor absorption, unfortunately, a Q  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 ^ ( ^ ) where i ^ is the extraterrestrial normal cos  on  z  solar spectral intensity. The  Q n  scattered diffuse radiation reaching the earth's  surface after scattering by air molecules, aerosols and absorption by gases is given by[13]:  f  drA  =  \  c o s  T  T w A  g  A  T  AI ' 0  D  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  characteristics are  poorly specified.  Approximate values of a and Q are used to first obtain T ^  from Eq. (2.12).  To calculate  since  aerosol  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  haX  *  KnX  c  o  can be written in the simple form[13]:  s  9  Z  \  T X  gA o X T  K V^aX^rJ  < - > 2  18  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  ***  '  t dr> f  * i  +  n X  cos 6 J y  where p'^ and  is:  fr,"^, gX a X P  are the ground, and atmospheric spectral reflectances  respectively. The atmospheric reflectance for diffuse radiation is calculated from[13]: p  ax  "  Kx  T  gx  T  ox  [O.5(I-T; )X; X  a+  (i- .) « - ( I - T ; ) T ; J F  O  X  (2.20)  21  I  I  drA  + I + I daA 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 =  C  A< drA f  +  !  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 therigorouscode 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, daX. and I,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 nr respectively. J  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 ± 1 5 % 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 earth's surface under  integrated) direct and  a given  difuse radiation  reaching the  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  = 0.958*10" Mm 4  H  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  X = 2.72 Mm 2  The water vapor spectrum above 13 Mm  X= 3  2.64 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  X=4.2 Lim 2  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~ Mm. In the infrared, ozone 4  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 Tsky 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,  R  =  e  where T  a  a T  4 a  (3.2)  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 ):  e  a  = A  -  exp(-  B  c  P„,) v  (3.3)  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 p is y  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: 0.5 e  a  "  D  +  E  P  v  (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 p  y  results basically from a correlation  between temperature and humidity. The nature of the correlation between e and p  v  &  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:  a  =  0.92  (3.6)  x 10  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  e  fl  =  1 - 0.261  exp[-7.77 x 10~ (273-T ) ] 4  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  a  =  0.787 + 0.764 Zn (T /273) <ip  Berdhal  and  Fromberg[24]  (3.8)  reported  results  of  2945 clear-sky  measurements made at Tucson(AZ), Gaithersburg(MD), and StLouis(MO): e a  = 0.741 + 0.0062 T. dp  (3.9)  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 p , T or T^ one y  a  can compute e . Using Eq. (3.2) and the measured value of T 2.  one can  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 (*^) can be obtained as: a  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  1, 1 etc. Each layer is considered isothermal, homogenous in 2  3  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  =  (3.11)  / R dX 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>  R  -  q  2/ / o o  H  (3-12;  B(T,X) E ( t ) dt dX 2  where q^ is total optical depth of the atmosphere and E is given by 2  E (t) 2  =  1 / e x p (~tx — ) dy o  (3.13)  where n is cost^. In general, the optical depth for any absorbing species from ground level to an  38 elevation (h) is defined by h q  h  =  / o  k  where  x  p(h') dh'  ( 3  .  1 4 )  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 I 2B(T.,X) i=l  C  q  i+l /  p=l /  exp(—)dy dt  1  (3.15)  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: R  =  x  N Z 2B(T ,X )[E (q.) - E ( q . ) ] i=l i  3  3  + 1  where E (q ) 3  As  i  =  i -q, / u exp(-^)dp o  (3.16)  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 j  N R  =  x  j  I 1  =  2B(T ,X ) * 1  /  j  [E ( ) - E (q 3  q i  3  1 + 1  )]dX  (3.17)  X -AX/2  1  Rather than evaluate qj at each spectral line an effective optical thickness is computed for each band from:  «i,AX  hx  =  ^ o  PO»')dh'  s  (3.18)  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:  R  N I  x  j  i  =  2B(T.,X.)[E (q 3  i)AX  ) - E (q. 3  + 1 > A X  )]AX  (3.19)  1  The effect of all absorbing gases are given by the sum of individual effective optical thicknesses, and Ramsey et al. wrote thefinalform of Eq. (3.19) as : \  =  j  x  2B(T ,X.)[E (Z q i  3  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 thefivemodel 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 thesefiguresshows 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  SIMPLIFIED  TECHNIQUE  TO  COMPUTE  SPECTRAL  ATMOSPHERIC  RADIATION  In  Sec.  3.3  it has  been  shown that the  detailed method, though  preferred in  principle, requires a large amount of data as i n p u t In reality it is impractical to specify with  the variation o f temperature, pressure and absorbing gas concentration  altitude  at  precipitable  water  parameters  such  which  easily  are  any  given  vapor  as  moment can  surface  be  air  measured.  at  a  location.  determined  temperature  Using E q . (2.6)  However,  from  (T )  and  the  value  the  'surface' relative of  w  amount  of  meteorological humidity  can  be  (R^)  obtained  as[8]:  w  =  0  .493 ^  < ' > 2  6a  a  Thus, the amount of water vapor can be quite easily determined from standard measurements. The amount of ozone(z) in the atmosphere changes  like  water  vapor.  Monthly  average  values  been monitored and are presented in Table 2.1 slow  seasonal  variations and  so  average  does not undergo short term at  different  latitudes  have  . Ozone amount undergoes only  monthly  values  used  from  Table  2.1  should be sufficiently accurate. The carbon consider  dioxide it  appropriate total  values has  of  w a  explicitly;  and z can be fixed  and  variables for  easily  concentration the  amounts  in of  obtained as shown above. the water  atmosphere, vapor  and  one  Since  need  ozone  not  become  a suitable correlation. It is logical to expect that the  (and spectral) atmospheric radiation depends  o f radiating gases present in the atmosphere.  i n some  way  on the  amount  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)  The  appropriate units are: R^ (dimensionless). Knowing R^  (3.10)  (W  nr  2  Mm ), B (W -1  from the detailed  nr  2  Mm )  and  1  method presented in  Sec. 3.3, the total atmospheric emissivity can be witten as: 00  / £  a  =  ~  R^dX  ~ a T a R  (3.21a) (3.21b)  a  that is, as a ratio of the total downward atmospheric radiation R(W  nr ) to 2  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 amount (w) outside this range of wavelengths.  and with water vapor  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 tofita polynomial of one independent variable and can statistically determine the degree of that polynomial which gives the best fit Rather thanfita 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 tofitthe 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-exp(a  o X  + a  n M  ,+  a H, 2 X  + 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 )  aA  OA  where b^,  (3.23b)  -LA  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 arigorouslayer-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 thefigure.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  is the collector heat  L  temperature (T.) and T  loss coefficient based  on  inlet  fluid  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  L i  -  T  )  a  (4.2)  The above equation gives the monthly-average daily useful energy (Q ) from u  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 PRODUCT (ra) FOR  HOURLY TRANSMITTANCE-ABSORPTANCE  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 k  p  , 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 (N j) terms, where r£  (^ f) Te  depends on 6, x, D and L. A simple z  algebra yields the following expression for the fraction of incident energy that  56  is absorbed:  (TO)  =  (l-r.)T.a  i  (4.3)  P  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 (# ) changes throughout the day; at best 6 z  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 by 8 can  to change at every moment The position of the sun is determined  and the azimuth angle (i//) as illustrated in Fig. 4.3. These two angles 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 6  z  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 k  p  (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 nr and bottom absorptance of 0.3, 0.6, 1  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 )  =  sr ss Z  ( T a ) I At  sr  I At  where I (W nr ) is the instantaneous value of incident radiation assumed 2  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 ( 7 a ) 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  dimensions(L,W) explicitly, the effect of L and W  (ra)  requires the pond  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 k is another p  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 offigureshas three graphs (a), (b), and (c) corresponding to brine extinction coefficient kp = 0.4, 0.6 and 0.8 nr respectively. Then, each of the graphs has a set of 1  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 k can be seen if thefigures(a), (b) and (c) are examined p  simultaneously. Both D and k  p  decrease ( 7 a ) . 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 ( 7 a ) 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:  \bs  <- >  =  4  5a  However, if diffuse radiation is large then the above equation can be rewritten as:  Q , x  abs  =  (TO)H, + ( T O ) , H ,  b  d  d  (4.5b)  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 k  p  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 a=0.9 and p  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 ( 7 a ) 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 arigorousmethod (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.  66  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. 45- 56 (1964).  Sol.  Energy  8(2),  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. 1-12 (1975).  Sol.  Energy  17,  T.A. Newell and R.F. 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"Solar  Center,  Heat  Engineering  Env.  Res.  Transfer",  of  Thermal  measurements  for  -l  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  m w  z oc p 0  F c r  S 3 X  (L25  0.55  (L8S  1.45  LIS  1.75  2.35  2J5  2.65  1  r  1  = 1 : 2 cm = 0.35 cm = 1.3 = 0.2 = 0-95 =0.92  Q  CJ  1  -  -  -  = 0.2  235  -  3.55  3J5  335  Wavelength (Urn) Figure 2.1  -i  1  1  Solar spectral atmosphere and conditions 1  1  1  1  r  radiation outside at ground level  i  i  i  i  i  i  the under  -i  earth's typical  1  '  I  1  r  Mesophere  S  OJD  S t rat o sphere  Troposphere _i  i  i  i  i  L  j  i  o.o  i  i  576.0  Density Figure 2.2  i  u  1152.0  (gm m ) 3  Variation of air density  with altitude  Figure 2.4  Variation of ozone Adapted from [6]  concentration  with  altitude  74  i  10"' 2  S  rrn—i—|—i—i—|—i—r  ID"' 2  5  1  2  5  Panicle concentration ( c m  10 - 3  2  5  1 0  J  2  )  Particle radius  (b)  (a)  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  Figure 2.6  iitm)  Rayleigh airmass  1.3"7 Wavelength  (Mm)  spectral  transmittance  2.5  as  a  function  of  U  I I I I I 1.26I I I I  0.4  A  I2.12I I I  A.2.99I  I  I  1  1  1 3.85 L  1  Wavelength (lim) Figure 2.8  • ' II'  Spectral transmittance absorption ( w = l cm)  '  ' ^  'V  W  7 '  due  ' 'I  1  to  water  vapor  1  M 0.4  -I  1  1  1  1  1  1.26  I  I  I  I  I  •  •  2.12  «  •  •  •  »  •  •  •  3.85  2.99  Wavelength (Lrm) Figure 2.9  Spectral (m =1)  transmittance  of  uniformly  mixed  gases  2.414  Wavelength  Figure  2.11  (Urn)  Diffuse spectral irradiance as Angstrom's turbidity coefficient, 8  a  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 Figure 2.12  (Urn)  The various components of the irradiance on a horizontal surface  spectral  diffuse  Table 2.1 Seasonal variation of atmospheric  ozone[H  Month Latitude 90° N 80° N 70° N 60° N 50° N 40° N 30° N 20° N 10° N 0° 10° S 20° S 30° S 40° S 50° S 60° S 70° S 80° S 90° S  Feb  Mar  Apr  May  June July  Aug Sept  Oct  Nov  Dec  0.33 0.39 0.34 0.40 0.34 0.40 0.33 0.39 0.32 0.36 0.30 0.32 0.27 0.28 0.24 0.26 0.23 0.24 0.22 0.22 0.23 0.24 0.24 0.25 0.27 0.28 0.30 0.29 0.31 0.30 0.32 0.31 0.32 0.31 0.31 0.31 0.31 0.30  0.46 0.46 0.45 0.42 0.38 0.33 0.29 0.26 0.24 0.23 0.24 0.24 0.26 0.28 0.29 0.30 0.31 0.31 0.30  0.42 0.43 0.42 0.40 0.38 0.34 0.30 0.27 0.25 0.23 0.24 0.25 0.27 0.29 0.30 0.30 0.29 0.28 0.27  0.39 0.40 0.40 0.39 0.37 0.34 0.30 0.28 0.26 0.24 0.24 0.25 0.28 0.31 0.32 0.33 0.34 0.35 0.34  0.34 0.36 0.36 0.36 0.35 0.33 0.30 0.27 0.25 0.24 0.24 0.25 0.28 0.33 0.36 0.38 0.39 0.40 0.38  0.30 0.30 0.31 0.32 0.31 0.30 0.28 0.26 0.24 0.23 0.24 0.26 0.31 0.37 0.40 0.42 0.45 0.46 0.45  0.26 0.27 0.28 0.38 0.28 0.27 0.26 0.25 0.23 0.22 0.24 0.26 0.32 0.37 0.39 0.40 0.40 0.38 0.37  0.28 0.29 0.29 0.30 0.29 0.28 0.26 0.25 0.23 0.22 0.24 0.26 0.29 0.34 0.37 0.39 0.38 0.36 0.34  0.30 0.31 0.31 0.31 0.30 0.29 0.27 0.25 0.23 0.22 0.23 0.25 0.29 0.32 0.35 0.35 0.34 0.32 0.31  Jan  0.32 0.33 0.34 0.34 0.33 0.31 0.29 0.26 0.25 0.24 0.24 0.25 0.29 0.35 0.39 0.41 0.43 0.44 0.43  0.27 0.28 0.29 0.30 0.30 0.28 0.27 0.26 0.24 0.23 0.24 0.26 0.32 0.38 0.40 0.42 0.43 0.42 0.41  Table 2.2 Rayleigh spectral attenuation  Wavelength 0.270 0.280 O. 30O 0.320 0.340 0.360 0.380 0.400 0.450 0.500 0.550 0.60O 0.650 0.700 O.BOO 0.900 1 .026 1 .060 1 .670 2 . 170 3.500 4 .OOO  Eq(2-7) 1 1 1 O O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  831 573 187 912 713 564 453 367 227 148 100 070 051 037 022 013 008 007 001 0 0 0  Eq(2-8) 1 933 i 650 1 225  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  929 719 565 451 364 224 145 098 069 050 037 021 013 0O8 007 001 0 0 0  coefficient  Eltermanfc#s] 1 928 i 645 1 222  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  927 717 564 450 364 223 145 098 069 050 037 021 013 007 003 001 0 0 0  Table 2.3  X 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 081 0.82 0.83 0.84 0.85 086 0.87 0.88 0.89 090 0 91 0.92  k  Spectral absorption coefficients for water vapor  X  wX  O.I60£ 0.240£ 0.125£ 0 I00£ 0 870£ 0.610£ 0 I00£ O.IOO£ 0 I00£ 0 600£ 0.175£ 0 360£ 0.330£ 0.153£ 0.660£ 0.155£ 0 300£ 0 I00£ 0.100£ 0.260£ 0 630£ 0.210£ 0 I60£ 0.125£  + + + + + + + + +  Table 2.4  X  k  01 01 01 01 00 01 02 04 04 03 01 01 00 01 00 00 02 04 04 02 01 01 01 01  0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80  k  X  wX  0 270£ 0.380£ 0 4I0£ 0.260£ 0 310£ 0 I48£ 0.125£ 0.25O£ 0 I00£ 0 320£ O.230£ 0 I60£ 0.180£ 0.290£ 0.200£ 0 II0£ 0 I50£ 0 150£ 0 170£ 0.100£ 0 100£ 0.510£ 0 400£ 0 I30£  + + + + + + + + + + + + + + + + +  02 02 02 02 01 01 00 02 04 01 02 01 03 01 03 04 03 02 02 04 01 00 01 03  1.85 1.90 1.95 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3 00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00  k  wX  0.22O£ 0 140£ 0 160£ 0.290£ 0.220£ 0.330£ 0.590£ 0.2O3£ 0.310£ 0.150£ 0.220£ 0.8OO£ 0.650£ 0.240£ 0.230£ 0.100£ 0.120£ 0 195£ 0.36O£ 0.310£ 0.250£ 0.140£ 0.170£ 0 450£  + + + + + + + + + + + + + + + + + + + + + + + -  04 04 03 01 00 00 00 02 03 05 05 04 03 03 03 03 03 02 01 01 01 01 00 02  Spectral absorption coefficients for mixed gases  X  X  0.76 0.77  0.300£ + 01 0.210£ + 00  1.25 1.30 1.35 1.40 1.45 1.50 1 55 1.60 1.65 1.70  0.730£ 0.400£ 0.110£ 0.100£ 0t>40£  -  02 03 03 04 01  0.630£ 0 100£ 0.640£ 0.145£ 0.100£  -  03 01 01 02 04  1.75 1.80 1.85 1.90 1.95 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70  V 0I00£ 0.100£ 0I45£ 0.710£ 0.200£ 0.300£ 0.240£ 0.38O£ 0.1 l O f 0.170£ O.I40£ 0 660£ 0.100£  + + + +  X 04 04 03 02 01 01 00 03 02 03 03 03 03  2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00  *gX 0.150£ 0.13O£ 0.950£ 0.100£ 0.800£ 0.190£ 0 130£ O.750£ 0.100£ 0 195£ 0.400£ 0.290£ 0.250£  + + + + + + -  03 00 02 02 00 01 01 01 01 02 02 00 01  Table 2.5  X  k  Spectral absorption coefficients  X  oX  0.485 0.490 0.495 0.500 0.505 0.510 0.515 0.520 0.525 0.530 0.535 0.540 0.545 0.550 0.555 0.560 0.565 0.570 0.575 0.580 0.585 0.590  38.000 20.000 10.000 4.800 2.700 1.350 0.800 0.380 0.160 0.075 0.040 0.019 0.007 0.000 0.003 0.003 0.004 0.006 0.008 0.009 0.012 0.014  0.290 0.295 0300 0.305 0.310 0.315 0 320 0.325 0.330 0.335 0.340 0.345 0.350 0 355 0.445 0.450 0.455 0.460 0465 0.470 0.475 0480  Table 2.6  fl, 0 f  c  0.92  k  F  c  for ozone  X  oX  0.017 0.021 0.025 0.030 0.035 0.040 0.045 0.048 0.057 0.063 0.070 0.075 0.080 0.085 0.095 0.103 0.110 0.120 0.122 0.120 0.118 0.115  oX 0.120 0.125 0.130 0.120 0.105 0.090 0.079 0.067 0.057 0.048 0036 0.028 0.023 0.018 0.014 0.011 0.010 0.009 0.007 0.004 OOOO 0.000  0.595 0.600 0.605 0.610 0.620 0.630 0.640 0.650 0.660 0670 0.680 0.690 0.700 0.710 0.720 0.730 0.740 0.750 0.760 0.770 0.780 0.790  ratio for an aerosol  10  20  30  40  50  60  70  80  85  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 2.00  1 00  1 .25  0.35 0.40 0.45 0.50 0.55 0.71 0.78 0.99 2 . 10 4 . 10  0 1 0 1 0 1 1 2 0 0  97 03 97 01 97 20 13 87 74 74  DIFFUSE CORRECTION FACTORS 0.97 1 .00 1 .06 1 .06 1 .09 1 . 15 1 OO 1 .02 1 .08 1 .05 0.99 1 .01 0.94 0.95 1 .01 1 . 12 1 . 14 1 . 14 1 .06 1 .07 1 .09 2.68 2 .69 2.74 0.79 0.85 0.85 0. 79 0.85 0.85  FOR BETA = 0 1 22 1 1 32 1 1 22 1 1 20 1 1 14 1 1 23 1 1 20 1 2 96 2 1 06 1 1 06 1  05 40 45 31 25 18 18 15 83 OO OO  1 1 1 1 1 1 1 2 1 1  93 86 58 44 32 17 21 90 12 12  0.35 0.40 0.45 0.50 0.55 0.71 0. 78 0.99 2 . 10 4 . 10  1 1 0 1 1 1 1 2 0 0  01 02 95 04 01 22 15 73 74 74  DIFFUSE CORRECTION FACTORS 1 .07 1 .01 1.21 1 .05 1 . 12 1 .25 0.97 1 .04 1 . 13 0.94 0.97 1 .04 0.90 0.93 0.98 1 .07 1 .06 1 . 10 1 .04 1 .04 1.14 2 .45 2 .44 2 .65 0.76 0.83 0.83 0.76 0.83 0.83  FOR BETA =0 1 63 2 1 59 1 1 38 1 1 22 1 1 15 1 1 31 1 1 22 1 2 77 2 1 02 0 1 02 0  10 19 91 55 30 17 15 11 46 94 94  4 3 2 1 1 1 1 2 0 0  30 03 09 56 33 10 09 33 91 91  0. 35 0.40 0.45 0.50 0. 55 0.71 0. 78 0.99 2 . 10 4 . 10  1 1 0 1 1 1 0 2 0 0  07 02 97 07 04 23 98 26 82 82  DIFFUSE CORRECTION FACTORS 0.94 1 .07 1 . 29 1 . 10 1 .35 1 .34 1 .01 1 .24 1 .20 0.96 1 .05 0.99 0.93 0.95 1 .00 1 .05 1 .04 1 .09 0.98 0.97 0.96 2.23 2 . 19 2. 15 0.85 0.96 0.89 0.85 0.96 0.89  FOR BETA =0 1 93 2 1 74 2 1 46 1 1 33 1 1 24 1 1 21 1 1 13 1 2 44 2 1 12 1 1 12 1  15 98 48 65 39 26 13 02 16 07 07  7 3 2 1 1 1 1 2 0 0  41 79 28 69 40 09 03 07 96 96  0.35 0.40 0.45 0.50 0.55 0.71 0.78 0.99 2. 10 4. 10  0 0 0 1 1 1 1 2 0 0  99 96 91 09 07 13 05 37 75 75  DIFFUSE CORRECTION FACTORS 1 .03 1 .22 1 . 33 1 .01 1 .25 1 . 18 0.95 1 . 15 1 .06 0.97 1 .04 0.98 0.94 0.93 0.99 1 .02 1 .02 1 .03 0.95 0.95 0.93 2.07 2 .06 2.09 0.71 0.84 0.92 0.71 0.84 0.92  FOR BETA =0 2 12 3 1 63 2 1 33 1 1 1 25 1 16 1 1 16 1 1 13 1 2 2 36 1 16 1 1 16 1  20 46 17 54 30 16 07 00 03 12 12  10 3 2 1 1 0 0 1 1 1  04 82 17 60 31 98 95 82 25 25  Wavelength (urn)  1 .50  3 00  4 00  6 00  83 f  °3  1  *J  \  >o Q.  1  U  o  0  1  A\ \ \ . y 4  v  >  10  Atmosphere  15  20  L  Atmospheric window  Wavelength  Figure 3.1  [yn]  Absorption spectra for atmospheric gases and absorption spectrum of the atmosphere[17]  A. 0  0  H  u V = M0U10  25  Hz  H  V = 0-502*10  Hz  v=1-137»10  Figure 3.2 (a) Fundamental vibrations of the vapor molecule  Figure 3.2 (b) Structure of the water vapor  water  molecule  Hz  0^  0  Hz  V =0-202x10  14 0-409x10  Figure 3.3  0  t  1  14  v =0-714  Hz  Fundamental vibrations of the C O , molecule  Wavenumber  1.00  /n  „ 0.80  ire  v e  2  0.60  6  3 c H  0.40  iii  0.20 0.00,  Wavenumber  Figure 3.4  Typical atmospheric transmittance due to molecular absorption in a bandwidth of 0.012 microns  N  N  Vl  '3 '2  M  077777EoTt  Figure 3.5  Schematic layered atmosphere for transmission calculations  85  130.0  202.0  214.0  226.0  23S.0  250.0  262.0  274.0  Tenperature (K)  286.0  298.0  310.0  Figure 3.6 (a) Temperature variations for standard atmospheres  R ¥  i  i  i  i  Figure 3.6(b)  i  1  1  1  1  1  1  1  1  1  i  1  1  1  Ozone distribution for standard atmospheres  r  86 1  C  in  0/ "  3  1  1  a IQ a a 'o a o 'o a enrage-  ^5^5  1  WATER VAPOR = 2-23  cm  OZONE = 0-34  cm  LAYER-BY-LAYER  '  I I  5  11  e  14  Figure 3.7(a) 8 |  1  1  1  1  1  1  20  1?  i — i — i — i — i — i — i —  » • * 29 Wavelength (Urn)  32  -i  B( T . \  1  1  1  1  1  1  —  1  —  1  M  1  r  i  1  1  )  T  Q  =  294 K  LAYER-BY-LAYER  Wavelength  Figure 3.7(b)  1  35  METHOD  (Urn)  Spectral atmospheric radiation for the mid-latitude summer atmosphere  —  41  Spectral emissivity of the mid-latitude summer atmosphere  1-  n  i  METHOD  1  r  u  87  0  e  5  11  17  14  Figure 3.8(a)  Si  1  1  1  1  1  1  20 23 26 Wavelength (Um)  29  Spectral emissivity winter atmosphere  1  1  *  1  1  1  1  1  of the  1  35  32  1  1  38  41  mid--latitude  1  1  1  1  1  1  1  in  'E  T  =  Q  272  K  tC  c  co  «N  -  B(T ,A) Q  . 0  LAYER-BY-LAYER  • 0 0  Ik.  R  METHOD  -  O  -  tn  V)  _  0  tn  0  1  5  e  1  1 11  1  1 14  Figure 3.8(b)  1 1  n  1  1  1  i  t  1  1  1  20 23 26 29 Wavelength (Um)  1  1 1  32  Spectral atmospheric radiation for mid-latitude winter atmosphere  1  35  1  the  1  39  1  'i"*  41  M  1  ~i  1  r  1  WATER  VAPOR = 3 25 cm OZONE = 0-21  — in  cm  o  0/ CL  LAYER-BY-LAYER  V)  5  1  1  1  1  1  «  5  1  1  1  1  1  1  1  1  1  20  n  14  »  1  1  23  Wavelength Figure 3.9(a) R |  "T  I  I  I  I  I  I  1  26  I  29  I 32  35  I  I  I  I  I  I  38  I  1  1  1  1  1  atmosphere 1  1  1  'E N  T  '£  c o a  '•6  Q  =  3 00  LAYER-BY-LAYER  K  METHOD  a.  Figure 3.9(b)  41  (Um)  Spectral emissivity of the tropical I  METHOD  Spectral atmospheric tropical atmosphere  radiation for the  1  1  -  89 1  f " i 1  T i  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  j  i  <<  i  i  14  '  -i—i—i  i  17  20  23  26  i  i  29  i 32  i  METHOD  i  i  35  '  '  38  Wavelength (bm)  Figure 3.10(a) Spectral emissivity of the sub-arctic summer atmosphere R I—i—r  n  1  1  1  1  1  r  E B 'E _  T  Q  = 2 87 K  BIT-.A) LAYER-BY-LAYER  METHOD  Wavelength (bm)  Figure 3.10(b) Spectral atmospheric radiation for the sub-arctic summer atmosphere  41  'mo 'o ci o '<D d o 'o o -g>4  WATER VAPOR = 0-31  cm  OZONE  cm  = 0- 53  LAYER-BY-LAYER  _l 5  I 8  I  I II  I  I 14  I  I  I  17  '  L 20  23  i  I  26  Wavelength  I  I  23  32  I  I  METHOD  I  3S  I 38  (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  T  i  Q  r  = 2 57 K  B( T„.\ )  IAYFR-BY-LAYER  METHOD  Figure 3.11(b) Spectral atmospheric radiation for the sub-arctic winter atmosphere  1  L_ 41  91  1-0  Figure 3.12  1-5 Water  20 Yapor  2-5 (cm)  30  Typical variation of spectral emissivity with water vapor content (w) at various wavelengths  3-5  T  1  1  6 i*iA»rW(l»  1—-i—Tr:  WATER VAPOR = 2-23 cm OZONE = 0-34  cm  w 3  E  a.  CO  -1 5  I  1  8  I  1  11  I  I  14  I  L__L_  17  .j  20  i  23  i  i  i  26  i  29  i  J  i  32  i_  35  39  41  Wavelength (Lim)  Figure 3.13(a) Comparison of e ^ vaiues technique and the simplified MSA -i  1  1  1  1  1  1  1  1  1  1  1  1  1  1  e  1  from the detailed procedure for the 1  r  -i  1  1  r-  R  Q  H  -h  0.72  332  Wm  O  O  0.73  334  W m  -2  'E 5  c o  -f- Layer-by-layer method  X)  a  <t> Simplified  technique  u CL  J 14  17  20  23  26  I 29  '  I  32  1  I 35  I  1 38  I L_ 41  Wavelength (Urn)  Figure 3.13(b) Comparison of R~ values technique and the simplified MSA  from the procedure  detailed for the  3IB  'm  Q)  WATER  VAPOR = 0 - 7 8 cm OZONE = O A 2 cm  VI tn  1  -i  Oy  a.  to  -I 5  8  I  I  I  II  L_  _l  14  17  I  L.  20  -I  I  23  I  26  Wavelength  I  i  I  I  I  I  4-  1  1  1  h  1  I  I  32  I  I  l  35  l  38  4  (bm)  Figure 3.14(a) Comparison of e ^ values technique and the simplified MWA ~i  I  29  1  1  from the detailed procedure for the 1  1  1  1  1  1  1  0.65  203W m "  0.66  204 W m  1  2  -2  'E 5 c o  o "5 o  Layer-by-layer method <J> Simplified technique  Q.  Wavelength (bm)  Figure 3.14(b) Comparison of technique and the MWA  values simplified  from the procedure  detailed for the  r  WATER  VAPOR = 3-25 cm OZONE = 0-21  _i  i  i  i_  i  17  i  i  i  i  i  Figure 3.15(a) Comparison of e ^ values technique and the simplified Tropical atmosphere ~i  1  1  1  1  1  i  20 23 26 29 Wavelength (Um)  r  n  1  1  r  i  i  i  32  cm  i  i  35  i_  38  41  from the detailed procedure for the ~i  1  1  1  1  1  r  1  R -I  O  -2.  1_ o.8A  355 W m  0-83  351 W m  O  -f- Layer-by-layer method <J> Simplified technique  17  20  23 26 29 Wavelength (Um)  Figure 3.15(b) Comparison of R, values technique and the simplified Tropical atmosphere  32  from the procedure  38  41  detailed for the  WATER VAPOR = 1-75  cm  OZONE -- 0- 39 cm  -I 5  1  1  8  1  1  11  1  1  14  1  1  17  1  1  20  I  I  23  I  I  1  26  I  29  I  I  1  32  i  35  '  l  L_  38  41  Wavelength (Um)  Figure 3.16(a) Comparison of e . values from technique and the simplified procedure atmosphere -i  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  i  the detailed for the S A S 1  1  1  1  r  R  O  O  - a  0.69  2 67 W m  0.70  -1 2 68 W m  -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 3, T  1  1  r  WATER VAPOR = 0-31 OZONE >  —  cm  = 0- 53 cm  <9  o  CL  Ul  i  i  i  i  i  14  i  i  17  i  i  20  t  t  23  i  1  1  1  1  1  1  1  O  'E  1  1  r  O  i  29  Wavelength  Figure 3.17(a) Comparison of the technique and the SAW atmosphere  i  i  26  i  i  i  >  35  36  t  i 41  ( Um)  e  values  simplified ~i  i 32  1  1  r  from the detailed procedure for the -i  1  R  r  0.60  K 9 Wm  0.60  H 9 Wm  -2  o  P  -f- Layer-by-layer method  S  a.  <J> Simplified technique  U)  38  Wavelength  Figure 3.17(b) Comparison of technique and the SAW atmosphere  41  (Um)  values simplified  from the procedure  detailed for the  Table 3.1(a)  Coefficients emissivity:  in  X 5 25  6 00 G .75 7 .06 7 14 7 22 7 36 7 56 7 80 7 95 8 .08 8 22 8 46 8 70 8 92 10 24 10 49 10 74 11 05 11 .44 11 95 12 37 12 55 12 84 13 13 13 31 13 47 13 67  13 90  14 50 15 50 16 17 16 50 17 17 17 71 17 80 17 93 18 06 18 29 18 56 18 79 18 97 19 0G 19 47 19 90 20 02 20 07 20 32 20 42 20 62 20 92 21 44 22 0 0 22 30 22 70 23 30 23 83 24 1 1  a  the  (a.^) to  range  ox  -o 2783E+01 -o 2912E+01 -o 3027E+01  - 0 28G1E+01 -0 3032E+01 - 0 2432E+01 - 0 8956E+00 - 0 2246E+01 -0 1670E+01 0 9471E-01 -o 3757E-01 - 0 2107E-01 - 0 3747E-01 -o 1523E-01 -o 1252E-01 -o 9 9 3 8 E - 0 2 -0 9937E-02 -0 7549E-02 -0 7549E-02 -0 9934E-02 - 0 3677E-01 - 0 9664E-01 0 1096E+00 0 1125E+00 0 100GE+00 o 1392E+00 - 0 1G38E+01 - 0 3448E+01 - 0 3570E+01 - 0 3745E+01 - 0 3802E+01 - 0 3565E+01 -o 3702E+01 - 0 1589E+01 - 0 6018E+00 o 1G17E+00 0 9641E-01 0 1625E+00 - 0 5892E+00 0 1640E+00 0 9618E-01 0 1651E+00 0 1316E+00 - 0 5649E+00 -o 1554E+01 -0 5540E+00 -o 1218E+00 0 1682E+00 0 1684E+00 -0 1124E+00 0 1693E+00 -o 3299E+01 - 0 4049E+01 -0 5128E+00 -0 3317E+01 -0 49G3E+00 - 0 4136E+01 - 0 3799E+01  5.25<X<8.92  a  0 0 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0  -o  -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0  -0  -0  -0 -0 -0 -0 -0 -0 -0  compute  and  lX  0 0 0 2633E+00 G330E-01 8079E+00 2395E+01 1077E+01 1742E+01 1407E+01 1970E+00 1392E+00 1981E+00 1151E+00 1024E+00 9025E-01 9043E-01 7680E-01 7696E-01 9103E-01 2049E+00 3645E+00 1258E+01 1044E+01 1458E+01 2596E+01 2362E+01 3108E+00 1752E+00 0 0 3466E+00 1956E+00 2G49E+01 3487E+01 2657E+01 175GE+01 2659E+01 3523E+01 2663E+01 1757E+01 2666E+01 3429E+01 3592E+01 2818E*01 3623E+01 3743E+01 2674E+01 2674E+01 3766E+01 2677E+01 9502E+00 3881E-01 3736E+01 9792E+00 3780E+01 1521E-03 4383E<-00  spectral  a  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  atmospheric  10.24<X<42.54  2X  0 0 0 9713E-01 2358E-01 2929E+00 8017E+00 3873E+00 6115E+00 2606E+00 0 0 0 0 0 0 0 0 0 0 0 0 231GE+00 2057E+00 2528E+00 606 1E+00 8261E+00 1151E+00 651BE-01 0 0 1284E+00 7277E-01 9243E+00 1145E+01 5902E+00 2888E+00 5891E+00 1156E+01 5872E+00 2867E+00 585GE+00 9470E+00 1177E+01 9819E+00 118GE+01 1153E+01 5803E+00 5799E+00 1159E+01 5779E+00 3485E+00 1454E-01 1220E+01 3592E+00 1233E+01 5754E-04 1625E+00  a  0 0 0 -0 -0  -o  -0 -0 -0 -0 0 0 0 0 0 0 0 0 0 0 0 0 -0  -o -o -o -o -0 -0 0 0 -0 -0  -o  -0 -0  -o  -0 -0  -o -o -0 -0 -0  -o -0 -0 -0  -o  -0 -0 -0 -0 -0 -0 -0 -0 -0  Mm  3X  0 0 0 '1083E-01 2S44E-02 3228E-0 t 838GE-01 4248E-01 6602E-01 2137E-01 0 0 0 0 0 0 0 0 0 0 0 0 2307E-01 2334E-01 2 117E-01 4798E-01 889GE-01 1287E-01 7304E-02 0 0 1436E-01 8155E-02 9936E-01 1182E+00 4437E-01 1930E-01 4417E-01 1 192E+00 4380E-01 1910E-01 4349E-01 8586E-01 12 13E+00 1054E+00 1221E+00 1136E+00 4253E-01 4247E-01 114 1E+00 4212E-01 3871E-01 1G37E-02 1254E+00 3989E-01 1266E+00 G510E-05 1817E-01  98 Table 3.1(a)  a  X 24 . 23 24 . 35 24 . 47 24 . 59 24 . 7 1 24 . 83 25 . 4 5 26 . 28 26 . 79 27 . 18 27 .52 27 . 86 28 . 50 29 . 50 30 . 50 3 1. 5 0 32 . 5 0 33 . 5 0 34 . 50 35 . 5 0 36 . 5 0 37 . 50 38 .02 38 . 20 38 . 48 38 . 86 39 . 15 39 . 38 40 . 0 0 40 85 4 1 .31 4 1. 5 1 41 . 72 42 42 . 34 42 . 8 3 42 . 54  ,oo  ox  a  continued  lX  -o 7 1 1 6 E - 0 1  -o 3541E+01 -o 1459E+01  -o  -o  0 0  0  -o -0 -o -0 -0 -0 -0 -0 -0 -0  -o  -0 -0 -0 -0 -0 -0 -0 -0 -0 -0  -o -0 -0 -0 -0 -0 -0 -0 0 -0  -o  18 12E+00 1175E+00 7101E-01 1849E+00 4730E+00 4181E+01 2410E+01 3850E+01 4223E+01 4361E+00 4239E+01 4254E+01 4275E+01 4295E+01 4314E+01 4331E+01 4349E+01 4365E+01 4380E+01 4395E+01 4409E+01 4180E+01 2376E+01 3293E+00 2374E+01 4299E+01 4433E+01 4441E+01 445 1E+01 4383E+01 3148E+01 8150E+00 6870E-01 1763E+00 4226E+01 2364E+01  Table 3.1(b)  -0  -0 -0 0 -0 -0 -0 -0 -0 0 0 0 0 0 0 0 0 0 0 -0 -0 -0 -0  -o 0 0 0 -0 -0 -0 -0  -o -0 -o  a  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  1075E+01 1460E+01 3549E+01 3842E+01 0 2155E+01 4629E+00 1666E-03 3937E+01 1691E-03 0 0 0 0 0 0 0 0 0 0 3026E+00 2449E+01 4202E+01 2461E+01 1688E+00 0 0 0 9456E-01 1619E+01 3975E+01 1475E+01 2S94E+01 3168E+00 2524E+01  Coefficients ( b ^ ) to emissivity :  9590E+00 1350E+00 2107E+00 1348E+00 9597E+00 1251E+01 0 7722E+00 1716E+00 6302E-04 1279E+01 6393E-04 0 0 0 0 0 0 0 0 0 0 1127E+00 8761E+00 1355E+01 8803E+00 6303E-01 0 0 0 3538E-01 5898E-»-00 1336E+01 1261E+00 5190E+00 1179E+00 9024E+00  compute  = i - exp (b  e  2X  spectral  + b z)  in the range 8.92<X<10.24 Lim  X 9.17 9 . 29 9 . 36 9 . 44 9.54 9.68 9.80 9 .90 10.00 10.09  b  -0 -0  -o -0 -0 -0  -o -0 -0 -0  ox  2103E+00 2222E+00 2511E+00 2887E+00 2852E+00 2700E+00 2405E+00 2225E+00 2125E+00 2103E+00  b  lX  -o 1053E+00 -o 3140E+00 -o 5707E+00  - 0 7959E+00 - 0 7757E+00 - 0 6888E+00 4872E+00 - 0 3075E+00 1282E+00 - 0 2439E-01  -o -o  a  3X  -o 8 5 4 9 E - 0 1 0 0  -o 2 4 8 7 E - 0 1  0 -0 -0 0 -0 -0 -0 -0 -0 0 0  o 0 0 0 0 0 0 0 -0 -0 -0 -0 -0 0 0 0 -0 -0 -0 0  -o -0 -o  0 8544E-01 1283E+00 0 8449E-01 1919E-01 7140E-05 1309E+00 7240E-05 0 0 0 0 0 0 0 0 0 0 1264E-01 9575E-01 1379E+00 9621E-01 7082E-02 0 0 0 3980E-02 6522E-01 1400E+00 0 3289E-01 1323E-01 9859E-01  atmospheric  99  Table 3.2  Run Dumber  400 401 402 403 404 405 406 408 410 41 1 412 413 414 415 419  Comparison of measured values of radiation (R) with values computed detailed technique  Air temperature CC)  Experimental  22 . 2 29 .0 16 . 3 20 .4 15 .6 15 .3 19 .4 1 1. 5 1 1. 3 6 .9 12 .6 8 .6 7 .2 -10 . 7 - 17 .4  332 388 307 339 278 300 318 278 265 253 285 260 253 151 148  atmospheric from the  Calculation  Difference  % 0 3 0 4 7 6 3 5 3 7 8 7 1 4 1  336 0 385 4 305 4 328 5 291 4 296 3 309 5 281 8 264 8 253 9 284 2 261 3 260 1 146 0 152 6  1.2 0.7 0.7 3 .2 4.5 1 .2 2.8 1.1 0.4 0.0 0.5 0.2 2.7 3.5 3.0  100  /salt  tempN  brine return  60 C  hot brine Figure 4.1  Figure 4.2  Schematic  ''cone.  of a typical salt-gradient  Schematic to estimate the absorbed in a solar pond  hourly  20%  solar pond  solar  radiation  Figure 4.3  Schematic to determine the hourly for any solar position 8 , i/>  value  of  (ra)  k =0-4 m" P  ("COO  0°  Figure 4.4  Hour angle Typical latitudes  hourly  105  cO  variation  of  (ra)  for  various  JUNE 0  Hour Figure 4.5(a)  Figure 4.5(b)  angle ,  CO  = 30° N  (degrees)  Hourly variation of direct beam typical case at 0 = 3 O ° N in June  Hourly variation of the ( r a ) case at 0 = 3 O ° N in June  radiation  value  for  for  a  a  typical  104  i i i 1 A Pond depth=1m © Pond depth=2m o Pond depth=3m  i  i  A o «  1  1  1  1—i  1  1  i  1  1 k  1  1  1  r  = 0-i, -1 m  P  ocy  QJ  1  0- 9  in  -©  -e  -i  1 2  1  1  1  3  1 *  1  1  1  5  Month  Figure 4.7(a)  1  i  6  Monthly k =0.4 P  i 7  i  i  i  6  i  e-  i  9  i  i  10  i  i  11  12  (Jan=1....Dec=12)  variation m"  of  (ra)  at  0°N  latitude  for  105  T  1  1  n  r  1  1  1  1  1  1  1  1  -A Pond depth=1m - o Pond depth=2m _ *- — « Pond depth=3m  i  i  i  k  i  i  i  r  -1  =0-6 m  p  oc  P  =0-9  CD  o. cn  •o-  •o  <p  <  _1  1  -e -  \  2  I  I 3  I  L  4.7(b)  Monthly k =0.6  P  o-  e-  o  -e>  -o  8  9  I  4  I  5  6  Month  Figure  ©  variation nf 1  7  I  10  -  ^  I  I  11  (Jan=1....Dec=12)  of  (ra)  at  0°N  latitude  -n  for  I  12  ~ (- A o • 1  1  ' ' A Pond depth=ln o Pond depth=2ii * Pond depth=3a 1  1  1  i  1  r—i  1  1  1  1  1  1  k  1  = 0-8  p  r—,  r  m-1  oC -. -9 p  >  ro  0  T =  A  A- _  --0---0---0-  J  1  1  1  3  Figure 4.7(c)  1  '  A-  -o- -  1  4  Monthly k =0.8 P  -o--e>--e--©---o  1—1  L_J  1  5  6  7  Month  1  |  8  9  (Jan=1....Dec=12)  variation m  of  (ra)  - -o- - - i  at  0°N  10  latitude  II  for  12  107  Figure 4.8(a)  Monthly k =0.4 P  variation m" 1  of  (fa)  at  15° N  latitude  for  108  "i A—  1  1  1  1  1  1  1  1  1  1  1  1  1  r  I  L  -A Pond depth=1m k  o o Pond depth=2m *- — o Pond depth=3m  QJ  1  1  i  P  oc  p  =0-6  m  = 0,9  \2  -©— - - e —  •o  I  2  I  I  3  I  I  4  I  I  I  5  I  6  4.8(b)  Monthly k =0.6 P  I  7  Month  Figure  o  —  -«  <v  J  - o  —o-  - © —  I  '  8  i  _  - * —  —  '  9  '  10  '  11  12  (Jan=1....Dec=12)  variation m" 1  of  (7a)  at  15° N  latitude  for  i  i  (_ A  i  i  i  i  i  i  n  i  i  i  i  i  i  i  i  i  i  i  r  A Pond depth=1»  o  o Pond depth=2n  «  » Pond depth=3n  k  =0-8  m"  1  P o C  =0-9  P  a9 I  ro I  3  . . a - - « - ' -°  J  I 2  I  I  I  3  I 4  °  I  I  °-  1  5  Figure 4.8(c)  I 6  Month  - -  I  ° "' -'oe  I  I  I  I  8 (Jan=l....Dec=12)  Monthly variation k =0.8 m"'  7  of ( f a ) at  1 9  - <D - . . . 0  I  I 10  I  I II  1 5 ° N latitude for  L 12  110  A o «  i  i A o »  i Pond Pond Pond  i i i depth=lin depth=2m depth=3m  i  i  i  1  1  1  1  1  1  1  1  1  k  -1 = 0-4 m  P  1  1  r  o c = 0-9 p  -e  J  1 2  i  I 3  i  Figure 4.9(a)  I 4  I  —e_  I i i i i i i 5 6 7 8 Month U3n=1....Dec=12)  Monthly  variation  k =0.4 P  m'  x  of  (ra)  i 9  at  30°N  10  latitude  12  II  for  I l l  i  1  1  1  1  1  1  1  1  1 —i -  1  1  i  i  s— -A Pond depth=1* ~ o o Pond depth=2m _ «- —o Pond depth=3m  -i  r k  P  oC  p  1  =0-6  1  -1 m  = 0-9  cx m «=•' ro I  -©— - - e — - o  • o-  •o  —  _ -o  o  e>— _  -o  5 *  '  I  2  I  I  3  Figure 4.9(b)  I  I  4  Monthly k =0.6 P  \  I  5  6 Month  L  variation m"  1  I  I  7 8 9 (Jan=1....Dec=12)  of  (TO)  at  1  30 N  I  I  I  10  latitude  1  L  11  for  r  112  ~i _ A o «  i  i i i i A Pond depth=1n o Pond depth=2n o Pond depth=3n  i  1  1  i  1  1  1  1  1  1  1  1  k  P  r  -1  =0-8  P oC  1  1  m  =0-9  ro I 3  - -o  -i  1  1  1  1  1  o  1  1 5  Figure 4.9(c)  Monthly  k =0.8 P  o- - - o - - a - .  1  Month  i 6  i 7  i  i  8  (Jan=1....Dec=12)  variation  m"  i  of  (ra)  at  i  .  e  i  9  30 N  i  i  i  10  latitude  i  II  for  i  113  Figure 4.10(a) Monthly k =0.4  P  variation  m"  of  (ra)  at  45°N  latitude  for  114  1  n  1  1  1  1  1  1  1  i  i  r  1  1  •A Pond depth=1m - o Pond depth=2m L «- — o Pond depth=3m  1  k  o-  1  1  1  1  r  -1  =0-6 m  p  =0-9  P  2 CL  ^ O  . cr  •o  —o  -o I  I  2  I  I 3  I  '  I  4  I 5  I  Month  Figure 4.10(b) Monthly k =0.6 p  I 6  1  I  7  I  I B  I  I 9  1  I 10  I  I 11  (Jan=1....Dec=12)  variation m"  of  ( r c O at  45 N  latitude  for  L_ 12  115  n  1 1 1 1 1 - A Pond depth=1n -o Pond depth=2» - * Pond depth=3a  1  1  i  1  1  1  1  1  1  1 k  1  1  1  1  r  = 0-8 m  P  oC : .9 p  0  QJ £  ci n  I  I  I  I  -O- - -O- - - Q  -o-  •o -  |  J  I  I  5  Month  Figure 4.10(c) Monthly k =0.8 P  I  I  6  <D -  I  I  7  L  J  -O-  I  L  8  11  (Jan=1....Dec=12)  variation m"  of  (ra)  at  45°N  latitude  for  12  116  "i A o  r ~i 1 1 A Pond depth=ln) © Pond depth=2m  «  1  r  T  1  1  1  1  1  1 k  * Pond depth=3m  -i  r  P  = 0-4 m  1  r  I  I  -1  o c = 0-9 p  o ° l  I  I  I  I  2  3  4  5 6 7 8 Month (Jan=1....0ec=12)  Figure 4.11(a) Monthly variation k =0.4 m"'  P  of ( r a ) at  9  I  I  10  I  II  60 N latitude for  12  117  -|  1  1  1  1  1  1  1  1  1  1  1  1  1  1  A— -A Pond depth=1m © o Pond depth=2m L <s- — < > Pond depth=3m  1  1  k  1  1  1  r  I  L  -1 =0-6 m  P  o c =0-9 P  L  \  /  V  V Q-  J  1  1  i_  J  I  -0-  —  I 5  L  J  6  I 7  I  I 8  I  I  I  I  I  Month (Jan=1....Dec=12)  Figure  4.11(b) Monthly variation k =0.6 m " P  of  ( r a ) at  60°N  latitude  for  118  —i A s »  1 1 1 1 1 A Pond depth=1n o Pond depth=2n t> Pond depth=3n  1  1  1  1  1  1  1  F~—i  k  i  p  oC  i  i  i  i  r  -1 = 0-8 m P  =0-9  V -o  . O-  -o-  -  - o - -o .  O -  -o o-  _ l  I  1 2  I  l_  I  I  I  I  5  3  Month  Figure 4.11(c) Monthly k =0.8  P  I  I  6  I  7  I  I  I  1  1  1  L  11  8  (Jan=1....Dec=12)  variation m~"  of  (ra)  at  60  N  latitude  for  12  119  Figure 4.12  Variation and D  of  (ra).  for  diffuse  radiation  with  k 1  120  APPENDIX A  Extraterrestrial  X 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 4 10 415 420 425 4 30 435  J  X  onX  64 91 122 253 275 212 162 286 535 560 527 557 602 705 747 782 997 906 960 877 955 1044 940 1 125 1165 1081 1210 931 1200 1033 1702 1643 17 10 1747 1747 1692 1492 1761  56 25 50 75 00 50 50 25 00 00 50 50 51 OO 50 50 50 25 00 50 00 99 OO 01 00 25 00 25 OO 74 49 75 OO 50 50 51 50 25  Solar Spectral Irradiance Sun-Earth Distance  0 0 0 0 0 0 0 0 0 0 0 0 0 O 0 0 0 0 0 O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  X  *onX 1755 1929 2099 2017 2032 2000 1979 2016 2055 1901 1920 1965 1862 1943 1952 1835 1802 1894 1947 1926 1857 1895 1902 1885 1840 1850 1817 184B 1840 1817 1742 1785 1720 1751 1715 1715 1637 1622  440 445 450 455 460 465 470 475 480 485 490 495 500 505 510 515 520 525 530 535 540 545 550 555 560 565 570 575 580 585 590 595 600 605 610 620 630 640  0 0 0 0 0 0 0 0 0 0 0 O 0 0 0 O 0 0 0 o o 0 0 0 0 0 0 0 0 0 0 0 o 0 0 1 1 1  00 49 99 51 49 00 99 25 OO 26 00 00 52 75 50 01 49 99 49 24 50 01 50 00 02 00 50 76 00 50 90 OO OO 25 00 00 50 50  at  T  650 660 670 680 690 700 710 720 730 740 750 760 770 780 790 800 810 820 830 840 850 860 870 880 890 900 910 920 930 940 950 960 970 980 990 000 050 100  Mean  X  onX  1597 1555 1505 1472 1415 1427 1402 1355 1355 1300 1272 1222 1 187 1 195 1 142 1 144 1113 1070 104 1 1019 994 1002 972 966 945 913 876 841 830 801 778 77 1 764 769 762 743 665 606  50 00 00 50 02 50 50 00 00 00 52 50 50 00 50 70 00 00 00 99 00 00 OO 00 00 00 00 00 OO oo 00 00 00 00 00 99 98 04  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4  150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 000 100 200 300 400 500 600 700 800 900 000 100 200 300 400 500 .600 .700 .800 .900 .000  a) C.Fr'c-hlich and CWehrli, 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  unr ) 1  !  onX  551.04 497.99 469.99 436.99 389.03 354.03 318.99 296.99 273.99 247.02 234.02 215.00 187.00 170.00 149.01 136.01 126.00 118.50 93 . 0 0 74 . 75 63 . 25 56.50 48 .25 42 . 0 0 36 . 5 0 32 . 0 0 28 . 0 0 24 . 75 21 .75 19.75 17 . 25 15 . 75 14 . 0 0 12 .75 1 1 . 50 10.50 9.50 8 . 50  121  APPENDIX B  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 AX  X 5 . 25 6.00 6 .75 7 .06 7.14 7.22 7.36 7.56 7 .80 7 .95 8 .08 8 . 22 8.46 8 . 70 8.92 9 . 17 9.29 9.36 9.44 9.54 9.68 9.80 9.90 10.OO 10.09 10.24 10.49 10.74 1 1 .05 1 1 .44 1 1 .95 12.37 12 .55 12.84 13.13  o 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  o 0  o 0 0 0 0 0 0 0 0 0  500 000 500 130 025 130 150 225 210 100 160 130 350 130 300 190 060 080 080 130 130 120 090 090 090 220 280 220 400 380 650 180 180 390 200  k 4 20 15 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  wX ooo 000 000 750  ooo 550 300 500 400 080 015 010 015 008 007 006 006 0O6 006 006 006 006 006 006 006 006 006 005 005 006 015 030 065 050 080  13 13 13 13 14 15 16 16 17 17 17 17 18 18 18 18 18 19 19 19 20 20 20 20 20 20 21 22 22 22 23 23 24 24 24  31 47 67 90 50 50 17 50 17 71 80 93 06 29 56 79 97 06 47 90 02 07 32 42 62 92 44 00 30 70 30 83 1 1 23 35  k wX  AX  X  0 0 0 0 1 1 0 0 1 0 0 0  o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  o 0 0 0 0  160 160 250 200 000 OOO 340 330 000 090 090 170 090 360 180 280 080 100 720 140 100 200 100 100 300 300 730 400 200 600 600 460 1 10 120 120  0 0 0 0 20 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 2 0 0 0  1PC 400 800 900 000 000 800 900 400 300 150 100 150 300 150 100 150 200 300 400 300 250 150 150 250 150 650 200 300 650 300 500 800 200 100  X 2-1 47 24 . 59 24 . 71 24 . 83 25 45 26 28 26 79 27 18 27 52 27 86 2 8 . 50 29 50 30 50 31 50 32 50 33 50 34 50 35 50 36 50 37 50 38 02 38 20 38 48 38 86 39 15 39 38 40 00 40 85 4 1 31 4 1 51 41 72 42 00 42 34 42 54 42 83  Table B 2 Effective mass absorption coefficient for ozone X  AX  9 . 17 9 . 29 9 . 36 9.44 9.54 9.68 9.80 9.90 10.OO 10.09  O . 190 0.060 0.080 0.080 0 . 130 0 . 130 0 . 120 0.090 0.090 0.030  k  oX  35. 120. 260. 420. 400. 330. 200. 1 10. 40. 7.  AX 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0  c  120 120 120 120 120 520 520 270 400 280 000 000 000 000 000 000 000 000 000 000 100 190 380 380 200 250 000 710 200 200 210 4 10 210 210 350  k  wX  0.050 0 . I00 0.200 0 . 300 2 . 500 0 . 500 0.8C0 2 . 500 0.300 2 . 500 11.000 12.000 15.000 15.000 1 5.000 16.000 9.000 20.000 20.000 20.000 0.900 0 . 500 0 . 300 0 . 500 1 .000 20.000 20.000 20.000 1 . 100 0.600 0 . 350 0 . 100 0 . 150 0 . 500 0 . 900  122 APPENDIX C  Model  .Ht  Pressure (mbar)  (km) 0.  1.  2. 3. 4. 5.  e.  7 . 8. 9. 10. 11 . 12 . 13. 14 . 15 . 1S . 17 . 18 . 19 . 20. 21 . 22 . 23. 24 . 25. 30. 35 . 40. 45 . 50. 70. 100.  Atmospheres Used as a Basis for the Computation of Atmospheric Radiation  0 0 0 0 0 0 0 0  o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  o 0 0 0 0  o  101E+04 904E+03 805E+03 715E+03 633E+03 559E+03 492E+03 432E+03 378E+03 329E+03 286E+03 247E+03 213E+03 182E+03 156E+03 132E+03 111E+03 937E+02 789E+02 666E+02 565E+02 480E+02 409E+02 350E+02 300E+02 257E+02 122E+02 600E+01 305E+01 159E+01 854E+0O 579E+00 3QOE-03  Temp  Density  (kg m ) (K) Tropical 3  300. 294 . 288 . 284 . 277 . 270. 264 . 257 . 250. 244 . 237 . 230. 224 . 217 . 210. 204 . 197 . 195 . 199 . 203 . 207 . 211. 215 . 217 . 219. 221 . 232 . 243 . 254 . 265 . 270. 219. 210.  0. 0. 0. 0. 0. 0. 0. 0. 0. 0 0 0 0 0 0 0 0 0 0 0  o 0 0 0 0 0 0 0 0 0 0 0 0  117E+04 106E+04 969E+03 876E+03 795E+03 720E+03 650E+03 586E+03 529E+03 471E+03 420E+03 374E+03 332E+03 293E+03 258E+03 226E+03 197E+03 168E+03 138E+03 1 15E+03 951E+02 794E+02 664E+02 562E+02 476E+02 404E+02 183E+02 860E+01 418E+01 210E+01 110E+01 921E-01 500E-03  Water vapor  Ozone  (kg m )  (kg m )  -3  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  o 0 0 0 0 0 0 0 0 0 0 0 0 0  20E+02 10E+02 90E+01 50E+01 20E+01 20E+01 90E+00 50E+O0 30E+00 10E+00 50E-01 20E-01 60E-02 20E-02 10E-02 80E-03 60E-03 60E-03 50E-03 50E-03 40E-03 50E-03 50E-03 50E-03 60E-03 70E-03 40E-03 10E-03 40E-04 20E-04 G0E-05 10E-06 10E-08  3  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  60E-04 60E-04 50E-04 50E-04 50E-04 40E-04 40E-04 40E-04 40E-04 40E-04 40E-04 4CE-04 40E-04 40E-04 40E-04 50E-04 50E-04 70E-04 90E-04 10E-03 20E-03 20E-03 30E-03 30E-03 30E-03 30E-03 20E-03 90E-04 40E-04 10E-04 40E-05 90E-07 40E-10  Ht (km) 0.  1.  2 . 3 . 4 . 5. 6. 7. 8 . 9. 10. 11 . 12 . 13 . 14 . 15 . 16 . 17 . 18 . 19 . 20. 21 . 22 . 23 . 24 . 25 . 30. 35 . 40. 45 . 50. 70. 100.  Pressure (mbar) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0  101E+04 902E+03 802E+03 710E+03 628E+03 554E+03 487E+03 426E+03 372E+03 324E+03 281E+03 243E+03 209E+03 179E+03 153E+03 130E+03 111E+03 950E+02 812E+02 695E+02 595E+02 510E+02 437E+02 376E+02 322E+02 277E+02 132E+02 652E+01 333E+01 176E+01 951E+00 671E-01 300E-03  0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  102E+04 897E+03 790E+03 694E+03 608E+03 531E+03 463E+03 402E+03 347E+03 299E+03 257E+03 220E+03 188E+03 161E+03 138E+03 118E+03 101E+03 861E+02 735E+02 628E+02 • 537E+02 458E+02 391E+02 334E+02 286E+02 243E+02 111E+02 518E+01 253E+01 129E+01 682E+00 467E-01 300E-03  Temp  Density  (kg m~ ) (K) Midlatitude Summer 3  294 . 290. 285 . 279 . 273 . 267 . 261 . 255 . 248 . 242 . 235 . 229 . 222 . 216. 216. 216. 216. 216. 216 . 217 . 218. 2 19. 220. 222 . 223 . 224 . 234 . 245 . 258 . 270. 276 . 2 18. 2 10.  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  119E+04 108E+04 976E+03 885E+03 799E+03 721E+03 648E+03 583E+03 523E+03 567E+03 416E+03 369E+03 327E+03 288E+03 246E+03 210E+03 179E+03 154E+03 131E+03 111E+03 945E+02 806E+02 687E+02 587E+02 501E+02 429E+02 197E+02 926E+01 451E+01 227E+01 120E+01 107E+00 500E-03  Water vapor  Ozone  (kg m )  (kg m )  3  3  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  14E+02 93E+01 59E+01 33E+01 19E+01 10E+01 61E+00 37E+00 21E+00 12E+00 64E-01 22E-01 60E-02 18E-02 10E-02 76E-03 64E-03 56E-03 50E-03 49E-03 45E-03 51E-03 51E-03 54E-03 60E-03 67E-03 36E-03 11E-03 43E-04 19E-04 63E-05 14E-06 10E-08  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  60E-04 60E-04 60E-04 62E-04 64E-04 66E-04 69E-04 75E-04 79E-04 86E-04 90E-04 11E-03 12E-03 15E-03 18E-03 19E-03 21E-03 24E-03 28E-03 32E-03 34E-03 36E-03 36E-03 34E-03 32E-03 30E-03 20E-03 92E-04 41E-04 13E-04 43E-05 86E-07 4 3 E - 10  0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  40E+01 30E+01 20E+01 10E+01 70E+00 40E+00 20E+00 80E-01 40E-01 20E-01 80E-02 70E-02 60E-02 20E-02 10E-02 80E-03 60E-03 60E-03 50E-03 5OE-03 40E-03 50E-03 50E-03 50E-03 60E-03 70E-03 40E-03 10E-03 40E-04 20E-04 60E-05 10E-06 10E-08  0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  60E-04 50E-04 50E-04 50E-04 50E-04 60E-04 60E-04 80E-04 90E-04 10E-03 20E-C3 20E-03 30E-03 30E-03 30E-03 30E-03 40E-03 40E-03 40E-03 40E-03 40E-03 40E-03 40E-03 40E-03 40E-03 30E-03 20E-03 90E-04 40E-04 10E-04 40E-05 SOE-07 40E-10  Midlatitude Winter 0. 1 . 2 . 3. 4 . 5. 6. 7 . 8 . 9. 10. 11 . 12 . 13 . 14 . 15 . 16 . 17 . 18 . 19 . 20. 21 .  22. . 23. 24 . 25 . 30. 35 . 40. 45 . 50. 70 . 100.  272 . 269 . 265 . 262 . 256 . 250. 244 . 238 . 232 . 226 . 220. 219. 219 . 218. 218 . 2 17. 217 . 216. 216. 215 . 215 . 215 . 215. 215. 215. 215 . 2 17. 228 . 243. 259 . 266 . 231 . 210.  r  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0. 0. 0. 0 0 0 0 0  130E+04 116E+04 104E+04 923E+03 828E+03 74-1 E+03 661E+03 589E+03 522E+03 462E+03 407E+03 350E+03 300E+03 257E+03 221E+03 189E+03 162E+03 139E+03 119E+03 102E+03 869E+02 742E+02 634E+02 541E+02 462E+02 395E+02 178E+02 792E+01 363E+01 174E+01 895E+00 705E-01 500E-03  Ht (km)  Pressure (mbar)  2 . 3 . 4 . 5. 6. 7 . 8 . 9. 10. 11 . 12 . 13 . 14 . 15 . 16 . 17 . 18 . 19 . 20. 21 . 22 . 23 . 24 . 25 . 30. 35 . 40. 45 . 50. 70. 100.  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  101E+04 896E+03 793E+03 700E+03 616E+03 541E+03 473E+03 413E+03 359E+03 311E+03 268E+03 230E+03 198E+03 170E+03 146E+03 125E+03 108E+03 928E+02 798E+02 686E+02 589E+02 507E+02 436E+02 375E+02 323E+02 278E+02 134E+02 661E+01 340E+01 181E+01 987E+00 707E-01 300E-03  0. 1 . 2 . 3 . 4 . 5. 6. 7 . 8 ; 9. 10. 11 . 12 . 13 . 14 . 15 . 16 . 17 . 18 . 19 . 20. 21 . 22 . 23 . 24 . 25 . 30. 35 . 40. 45. 50. 70. 100.  0 0 0 0 o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  101E+04 887E+03 778E+03 680E+03 593E+03 516E+03 447E+03 385E+03 331E+03 283E+03 242E+03 207E+03 177E+03 151E+03 129E+03 110E+03 943E+02 806E+02 688E+02 588E+02 501E+02 428E+02 365E+02 311E+02 265E+02 226E+02 102E+02 470E+01 224E+01 111E+01 572E+00 402E-01 300E-03  0.  1.  Temp  Density  (kg m ) (K) Subarctic Summer 3  287 . 282 . 276. 271 . 266 . 260. 253 . 246 . 239 . 232 . 225 . 225. 225 . 225. 225 . 225. 225 . 225 . 225. 225 . 225. 225 . 225 . 225 . 226. 228 . 235. 247 . 262 . 274 . 277 . 216. 210.  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0  122E+04 111E+04 997E+03 899E+03 808E+03 722E+03 652E+03 585E+03 523E+03 466E+03 412E+03 356E+03 30GE+03 263E+03 226E+03 194E+03 167E+03 144E+03 124E+03 106E+03 913E+02 785E+02 675E+02 581E+02 496E+02 425E+02 198E+02 932E+01 453E+01 231E+01 124E+01 114E+00 500E-03  Water vapor  Ozone  (kg m )  (kg m^)  3  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0  90E+01 60E+01 40E+01 30E+01 20E+01 10E+01 50E+00 30E+00 10E-01 40E-01 20E-01 90E-02 60E-02 20E-01 10E-02 80E-03 60E-03 60E-03 50E--03 50E-03 40E-03 50E-03 50E-03 50E-03 60E-03 70E-03 40E-03 10E-03 40E-04 20E-04 60E-05 10E-06 10E-08  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0. 0. 0  50E-04 50E-04 60E-04 60E-04 60E-04 60E-04 70E-04 80E-04 80E-04 10E-03 10E-03 20E-03 20E-03 30E-03 30E-03 30E-03 30E-03 40E-03 40E-03 40E-03 40E-03 40E-03 30E-03 30E-03 30E-03 30E-03 10E-03 90E-04 40E-04 10E-04 40E-05 90E-07 40E-10  0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0  10E+01 10E+01 90E+00 70E+00 40E+00 20E+00  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  40E-04 40E-04 40E-04 40E-04 40E-04 50E-04 50E-04 70E-04 90E-04 20E-03 20E-03 30E-03 40E-03 50E-03 50E-03 60E-03 60E-03 60E-03 60E-03 60E-03 60E-03 50E-03 50E-03 40E-03 40E-03 30E-03 20E-03 90E-04 40E-04 10E-04 40E-05 90E-07 40E-10  Subarctic Winter 257 . 259 . 256 . 253 . 248 . 241 . 234 . 227. 221 . 217 . 217 . 217 . 217 . 217 . 217 . 217 . 217 . 216. 215 . 215. 214. 214 . 213. 212. 212. 211. 216 . 222. 235 . 247 . 259. 246 . 210.  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0  137E+04 119E+04 106E+04 937E+03 834E+03 746E+03 665E+03 590E+03 523E+03 454E+03 388E+03 332E+03 283E+03 242E+03 207E+03 177E+03 152E+03 130E+03 111E+03 953E+02 816E+02 698E+02 597E+02 510E+02 436E+02 372E+02 164E+02 737E+01 333E+01 157E+01 768E+00 569E-01 500E-03  106+00  50E-01 10E-01 80E-02 50E-02 40E-02 30E-02 20E-02 10E-02 80E-03 60E-03 60E-03 50E-03 50E-03 40E-03 50E-03 50E-03 50E-03 60E-03 70E-03 40E-03 10E-03 40E-04 20E-04 60E-05 10E-06 10E-08  125 APPENDIX  Summary  of  (1) Angle  cos  of incidence  6  for  Formulae  Reflection/Transmission  =  Solar-Geometry  in a Partially Transparent  for beam  sln6  D  Medium  radiation (8 )  sin<j> + c o s S cos<j> cosw  (2) Solar azimuth angle (\p)  cos  =  (sinct  (3)Solar sunrise/sunset  cosco s r  =  sina) - s i n 6 ) / c o s ( 9 0 - 0  hour angle (co ) S'  -tantj) tan<5  (4) Angle of refraction  n,sin  9,  =  (Snell's  n„  law)  sin 0  (5)Reflection of unpolarized radiation (Fresnel's  sin (9  2  +  tan (9  2  - ej_)  2  2  r  r  )costj)  e ) L  equations)  and  126 (6) Absorption of radiation by a partially transparent medium (Bouger's law)  x  =  exp(-kil)  Nomenclature s p e c i f i c to this Appendix  k.  Extinction c o e f f i c i e n t of the medium (m ^")  Z  path length of ray i n the absorbing medium (m)  n^,n^  Refractive i n d i c i e s of mediums on either side of the interface  r r  Average reflectance of the unpolarized radiation  _l_» H r  6  Reflectance of the perpendicular and p a r a l l e l components of unpolarized radiation, respectively Solar declination angle (degrees) assumed constant over a day. Ref. [6] presents tabulated values f o r each day of the year  to  Hour angle of the sun (degrees); morning (+)ve values, evening (-)ve values, noon zero  <JUsr , 0ss )  Surise/sunet hour angle (degrees); that hour » w angle which corresponds  9^,02  to sunrise or sunset  Angle of incidence,angle of r e f r a c t i o n f o r a ray