John D. Garrison Physics Department, San Diego State University, San Diego, ca 92182-1233, U. S. A




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A PROGRAM FOR CALCULATION OF SOLAR ENERGY COLLECTION BY FIXED AND TRACKING COLLECTORS

John D. Garrison

Physics Department, San Diego State University, San Diego, CA 92182-1233, U.S.A.,

email: jgarriso@mail.sdsu.edu, fax: 619-594-5485, ISES member



Abstract- SOCOL, a realistic and versatile FORTRAN program, has been developed to estimate net solar energy collected by a solar collector per unit collection area. This program was developed to study the properties of various solar collectors. It is made useful to a wide spectrum of users by allowing them to choose any or all of 15 possible solar collector types for calculation and comparison. Additional collectors can be included without undo labor. Either or both of two selective absorbers can be selected for energy collection calculations. SOCOL allows input for a third selective absorber. SOCOL is programmed to use solar radiation and surface meteorological data taken from The National Solar Radiation Data Base (NSRDB) for 239 stations over the USA. It can be adjusted to read other data sets. It takes 20 seconds on a Compaq Presario 2700 1.13 GHz computer to calculate net solar energy collection per unit area for one solar collector design using each of two selective absorbers at 5 fixed absorber temperatures for all the daylight hours of one year at one location. The program output includes sums of solar energy collection for each day, month and year along with averages and distributions. Averages and distributions for the solar radiation and surface meteorological data are also obtained so solar energy collection can be related to these data. SOCOL can be down-loaded from web site: www.sci.sdsu.edu/SOCOL/.

  1. INTRODUCTION



The FORTRAN program SOCOL calculates the net solar thermal energy collected per unit area by any of a variety of solar thermal collectors and a planar PV collector for solar electricity for a particular site and year. It allows comparison of different collectors. It is useful for estimating energy collection by a particular collector at a particular location for various fixed operating temperatures and orientations of the collector, or comparing energy collection at different locations.

The net amount of solar thermal energy collected per unit area by a collector is the amount of energy absorbed by the absorbing surface minus the energy lost by the absorbing surface to the environment per unit area. The thermal conduction losses by supports for the absorber can be made small and are neglected. Energy collection and energy losses by a complete energy system are not considered here.

Many methods already exist for analysis and design of a complete solar energy system. They are very useful and well tested. These include the simpler f-chart method (Klein, et al, 1977; Beckman, et al,1977), the Utilizability method (Whillier, 1953; Liu and Jordan, 1963; Klein, 1978; Collares-Pereira and Rabl, 1979), and the more thorough and involved, but quite flexible, mathematical simulation methods, such as TRNSYS (Klein and Beckman, 1976; Klein, et al, 1990; Duffie and Beckman, 1991), for example.

This work is a long overdue continuation and much improved version of an earlier study (Garrison, et al, 1978). Rabl has done an excellent, somewhat similar study, which is discussed further below (Rabl,1981). Rabl’s work has been used by Gordon and Rabl (1982) for an analysis of process heat plants without storage. Brunold, et al (1994), compare energy collection by two evacuated collectors and one air flat plate collector with glass capillary transparent insulation.

SOCOL contains parts of a program SOLRAD, used by (Gueymard and Garrison, 1998) for example, so that solar energy collection can be related to the properties of the solar radiation and surface meteorological data. SOCOL goes one step beyond the work of Marion and Wilcox (1994), who use solar radiation data from the National Solar Radiation Data Base (NSRDB, 1992; NSRDB, 1995) to estimate the direct and diffuse solar radiation incident on flat plate, concentrating and tracking collectors with varying orientations and locations. Examples of calculations by SOCOL have been discussed earlier (Garrison, 2000,2002).


  1. THE FORTRAN PROGRAM SOCOL

2.1 The data

When SOCOL is started it requests: The station; year of the data; tilt c and azimuthal angle c of the collector array; angle limits on the sky and collector view horizons; range of numbers of the types of collectors to be calculated; surface albedo; choice of output sent to the output file; a reduced radiation loss (low loss ) number; and today’s date. Two absorbers are used as standards for calculation of solar energy collection. One is more suitable for low temperature operation of a collector. The other is more suitable for higher temperature operation of the collector. If energy collection by a collector using another absorber is desired then the additional input required for this absorber is: the normal absorptance; 5 hemispherical emittance values for five absorber operating temperatures; and a weighting factor (Discussed in Subsection 2.3 below). Input for the Planar PV array is discussed below.

The program then reads the solar radiation and surface meteorological data for one year from a file. The input solar radiation and surface meteorological data currently used are the National Solar Radiation Data Base for 239 US stations available from the National Climatic Data Center, NOAA, U.S. Department of Commerce, Washington, D.C. Solar radiation and surface meteorological data for Canadian stations obtained from Atmospheric Environment Service, Downsview, Ontario, Canada have also been used (Garrison,2000). SOCOL contains information concerning: the selective absorbers used as standards in the program; loss properties and angular response of 15 solar collectors; station data; corrections for deviation of the orbital motion of the earth from circular motion; and data needed to estimate the distribution of diffuse radiation over the sky (Perez, et al, 1993).

2.2 Collector designs

Fig. 1 shows simplified transverse cross sections of eight fixed collector designs and their identifying numbers whose energy collection properties have been included in this program. One single glazed air flat plate collector (Number 1, with the absorbing surface in air) and seven evacuated collectors (Numbers 3,5-9,12, with the absorber surfaces in vacuum) are shown. The vacuum envelope for the evacuated collectors is a glass tube. Solar energy collection is by a plane parallel array of identical collector tubes, with the plane of the collection area for each tube in the plane of the array. SOCOL calculates energy collection for these eight types of collectors and seven others not shown. These 15 collectors are discussed below. The collector concentration C, shown with each collector cross section in Fig. 1, is taken to be the ratio of normal incidence energy collection area to the absorber surface area. The individual collectors will now be discussed.

• [1,2] Air Flat Plate - In the top upper left of Fig. 1 is shown a simplified partial cross section of a single glazed air flat plate collector (Number1). A double glazed air flat plate collector (Number 2), whose energy collection is also calculated, is not shown. These two collectors are discussed in (Duffie and Beckman,1991, Chap. 6) and (Rabl, 1985, Chap. 1).

• [3] Vacuum Tubular (dewar) - Just below the single glazed air flat plate collector in Fig. 1 is shown a simplified transverse cross section of a fixed evacuated glass tubular (dewar) collector. Tubes of this type have been discussed by (Beekley and Mather,1975; Schmidt, et al, 1990). Nippon Electric Glass in Japan and others have used larger diameter tubes of this type for their ICS collector. For this study, the inner absorber tube is taken to have a diameter which is 92% of the diameter of the outer glass tube. When tube axes are oriented in approximately a polar axis direction, they act much like a tracking collector, since the collecting area viewed from any direction perpendicular to the tube axis does not change, except for shielding by neighboring tubes. Because of this feature, this collector collects more solar energy per unit collection area at low operating temperatures than any other collector considered here. Its energy collection per unit absorber area is the lowest of any of the collectors considered here, since its concentration is only C = 1/=0.32. Thus, its energy loss by radiation per unit collection area is large relative to the other evacuated collectors with higher concentration. This loss can be reduced by the order of 20% by the use of a silver mirror on the inner surface of the outer glass tube on the lower non-collecting portion of the tube, and by the use of a very low emissive coating on the corresponding outer surface of the inner tube. A low emittance, thermally floating shield can be placed between the inner and outer tubes in this region to reduce further this region’s loss by about a factor of two.

• [4] Vacuum U-Trough - The simplified cross section of this collector tube has the absorber surface consist of a semicircular trough in the lower half of the outer glass tube with absorber on both inside and outside surfaces. Its semicircular cross section is identical to the lower half of the dewar collector [Number 3]. The energy collection by this collector is intermediate between that of the dewar collector and the vacuum cusp collector [Number 5] discussed next. The loss of this collector can also be reduced by the use of a silver mirror on the inner surface of the glass vacuum envelope tube and by the use of a very low emissive coating on the corresponding outer surface of lower part of the semicircular trough. A low emittance, thermally floating shield can be placed between the inner trough and outer glass tube in this region to reduce further this region’s loss by about a factor of two.

• [5] Vacuum Cusp –The simplified cross section of this collector tube is shown just below the vacuum tubular collector cross section. The surface of the cusp is coated with selective absorber. For this study, the cusp is assumed to have a width which is 92% of the diameter of the outer vacuum envelope. The cusps in an array of these tubes act as a trap for solar radiation, since the reflected part of rays incident on the absorber surface are often again incident on the absorber and mostly absorbed. The properties of this collector place it intermediate between the vacuum U-trough collector (Number 4) and horizontal fin collector (Number 6) in collection and loss properties. The radiation loss by this collector can be reduced by the order of 30% by silvering the inner surface of the lower half of the glass tube and placing a very low emissive coating on the bottom of the cusp. Placing a thermally floating low emittance fin just below the cusp bottom will reduce the bottom loss further by a factor of about two. With this reduction, this collector can collect more energy per unit collection area than any of the other fixed collectors discussed here at an operating temperature near 200 C, and more than all other collectors except the dewar and U-trough collectors with loss reduction at lower temperatures.

• [6] Vacuum Horizontal Fin - A simplified transverse cross section of an evacuated, horizontal fin collector tube is shown just below the vacuum cusp collector in Fig. 1. Collectors of this type are shown in (Duffie and Beckman,1991, Chap. 6; Rabl, 1985, Chap 1). The internal fin “flat plate” is coated with a selective absorber. For this study, the internal fin is assumed to have a width which is 92% of the diameter of the outer vacuum envelope. The concentration is taken to be 0.49, reduced from 0.50 by the effect of an energy collection tube thermally in contact with the internal fin (not shown). Commercial production of this type of collector tube has been by Philips in the Netherlands (Bloem, et al, 1982); Fournelle Energie Technologies, Canada; Thermomax Technologies, England; Corning of France; Philco Italiana of Italy and Nippon Electric Glass of Japan, and others. The energy loss by this collector can be reduced by the order of 40% by coating the inner surface of the lower half of the glass tube with silver and by placing a very low emissive coating on the lower surface of the fin. Placing a thermally floating shield just below the bottom of the fin will reduce further the bottom loss by about a factor of two. With this loss reduction this collector can collect more energy than any other of the fixed collectors near 300 C.

The subsequent collector designs (Numbers 7 – 12) indicated in Fig. 1 and discussed below have a mirror on the inside surface of the lower half of the glass tube to concentrate solar radiation onto the inner absorber surface. The internal silver mirrors are assumed to have a reflectance  0.95, independent of angle of incidence of the solar radiation on the mirror. They do not lend themselves to further loss reduction, as can be done for collector numbers 3 - 6. The loss reductions possible with collectors 3 - 6, called low loss, are calculated in SOCOL by input of the number one for the number requested by SOCOL.

• [7] Vacuum Vertical Fin – At the top right of Fig. 1 is shown a simplified transverse cross section of an evacuated vertical fin collector. For N-S or polar axis orientation of these collector tubes, this collector design can increase energy collection in the early morning and late afternoon, relative to the evacuated horizontal fin collector.

• [8] Vertical Half Fin - At the top of Fig. 1 on the right, just below the vertical fin collector is shown a simplified view of the transverse cross section of an evacuated vertical half-fin collector tube. Tubes of this type are discussed by (Winston, et al, 1997, 1998; Duff, et al 1997) and references found therein. This has ideal CPC concentration (Welford and Winston, 1989) with acceptance half angle of 90o with C = 0.89.

• [9] Horizontal Half Fin - In Fig. 1 on the right, just below the cross section of the evacuated vertical half-fin collector tube, is a simplified view of the transverse cross section of an evacuated horizontal half-fin collector tube. CPC tubes of this type are also discussed by (Winston, et al,1997, 1998; Duff, et al 1997) and references found therein.

• [10 -12] CPC Shaped Glass - On the right side of Fig. 1 below the evacuated horizontal half-fin collector is shown a cross section of a fixed evacuated CPC shaped glass solar collector tube with acceptance half-angle of 35o (Number 12). This collector with small acceptance angle has the highest energy collection per unit absorber area of all the fixed collectors discussed here. It has the lowest heat loss per unit collection area of any of the fixed collectors considered here. Representative references for this type of collector are: (Snail, et al 1984; Garrison and Fischer-Cripps,1997) and references found therein. Energy collection by this type of collector tube with acceptance half-angles of 60o (Number 10) and 45o (Number 11) is also included in SOCOL

. • [13 -14] Parabolic Tracking – Evacuated parabolic tracking collectors are not shown in Fig. 1. SOCOL can calculate energy collection by a single axis parabolic tracking collector (Number13). This tracking collector is modeled to be similar to the Luz Corporation SEGS arrays LS-2 and LS-3 (Cohen, et al, 1993) who quote a concentration of C=71 for the LS-2 design. This concentration is the ratio of parabolic mirror width to absorber tube diameter, rather than circumference (Gordon, 2001). Here, C = 71/ 22.6. The LS-3 has a normal incidence optical efficiency of 0.80, about the same as the vacuum tubular collector used here with no neighboring tubes. SOCOL can calculate energy collection by a two axis parabolic tracking collector (Number 14). This is assumed to have a concentration of 500 and a normal incidence optical efficiency of 0.80.

. • [15] Planar PV – This is the most common form of solar electric collector, not shown in Fig. 1. It consists of a plane array of solar cells. The input for this collector consists of the normal efficiency o of this collector at 20 C, a number for the variation of the relative efficiency o with incident angle, and a number for a linear (assumed) variation of efficiency with temperature relative to 20 C (in percent change per degree Celsius). The variation of relative efficiency with incident angle is expected to have approximately the same form as the variation of relative absorptance of the selective absorbers of the thermal collectors and can be specified in the same manner (discussed in Section 2.3 and shown in Fig. 2).

_______________

If one wishes to design a best collector for a given temperature, one might wish to try other designs besides the 15 discussed above. For example, the U-trough collector does not need to be semicircular in cross section, but can be an arc of a circle of larger radius of curvature, placing the design intermediate between the U-trough and the horizontal fin of infinite radius of curvature. The U-trough or arc can also be inverted into the upper half of the vacuum envelope tube. Also a “V” trough can be tried. Such trials are time consuming since they require ray tracing to determine the angular response of each design.

__________________

The air flat plate collectors (Numbers 1,2) have a rectangular energy collection area. The absorbing surface and its bottom insulation are contained in a sealed rectangular box. Both orthogonal transverse dimensions of the absorber surface are assumed to be large compared to the height of the upper part of the side walls of the sealed box which are above the absorber surface, so edge effects will be small.

There exist a number of air flat plate solar collectors with modifications to the basic flat plate design. See for example, (Oliva, et al, 2000) and (Goetzberger, et al, 1991). Oliva, et al describe an air flat plate solar collector with a honeycomb-type transparent insulation cover. Goetzberger, et al describe a bifacial collector with concentration and absorber surface insulation. Energy collection by collectors of this type can be calculated using SOCOL by including their angular response and loss characteristics in the program.

The evacuated collector tubes are assumed to be long relative to the width across the tube in the transverse direction so that end effects are small. The tubes in an array are assumed to have a spacing that is 20% of the transverse tube width. Knowing this spacing permits calculation of the effect of scattering and attenuation by neighboring tubes on the energy collection of a tube. The effect of the spacing on solar energy collection is small, of the order of 1%. Exceptions are: The vacuum tubular (dewar) collector (Number 3), which collects about 15 to 20% more energy when the tubes of an array have a wide separation and the tubes are oriented parallel to the polar axis. The other exceptions are the U-trough, cusp and vertical fin collectors (Number 4,5,7) which also collect somewhat more energy when widely separated and with the polar axis orientation.

2.3 The absorber surface

The selective absorber literature has been searched rather thoroughly in an attempt to find all absorbers with measurements of the variation of absorptance  with the angle of incidence on the absorber. Although the number of selective absorbers discussed in the literature is of the order of 1000 or more, only 25 measurements of the absorptance as a function of angle of incidence have been found. Two of these 25 selective absorbers have been selected for use in these studies: The black chrome on Watts nickel absorber of Pettit and Sowell (1976) with normal incidence absorptance of  = 0.95 and the highly selective cermet absorber of Zhang and Mills (1992), sample R517CuB, with = 0.92. The mathematical form of the variation of / with incident angle used to fit the data here is

/ = 1 - exp[- c(90 – A)d], (1)

where A is the angle of incidence on the absorber surface in degrees. The adjustable parameters c and d are varied to yield least square fits to the measured values of absorptance for each absorber. In fitting the measured values, it is important that the unmeasured value: /= 0 at A = 90o is included. Fig. 2 shows the variation of the Pettit-Sowell and Zhang-Mills absorbers with angle of incidence. Smooth selective absorbers with a high selectivity ratio  (is the hemispherical emittance) apparently have a variation of / with incidence angle close to that of the Zhang-Mills absorber. See (Reed, 1977).

Energy collection for another absorber requires as input for SOCOL the normal absorptance  of the new absorber; and the position of the curve for / for the new absorber relative to those for the Pettit-Sowell and Zhang-Mills curves in Fig.2. In specifying this position, the position of the Pettit-Sowell absorber curve is taken to be 1.0 and the position of the Zhang-Mills absorber curve 0.0 for linear interpolation or extrapolation. The Planar PV is treated in the same manner with , replaced by o.

2.4 Window transmission and reflection

The window glass for all of these collector designs is assumed to be soda lime glass. The optical properties of soda lime glass are presented in detail by (Rubin, 1985). For this study, the transmission of soda lime glass as a function of incident angle has been approximated by

 = 2.782 cos G (1-1.011 cos G+0.342 cos2G), (2)

where G is the angle of incidence on the glass surface. Attenuation and bending of radiation in the glass is small, and has been neglected: reflection  = 1 - 

2.5The angular response

The collector angular response is defined equal to the optical efficiency times the cosine of the angle of incidence of the solar radiation on the collection area. This replaces the incidence angle modifier used in most work. The angle of incidence on the collector is defined in this study in terms of two angles: X the angle the sun’s rays make with the direction of the axes of the collector tubes unit vector in each array, and X, the angle the projection of the direction of the incident solar radiation onto the plane transverse to the collector axis direction makes with the unit vector normal to the array area n. Angles X and X are indicated in Fig.3. The cosine of the angle of incidence on the collector area is the product: cosX sinX. The optical efficiency  at angles Xand X is determined analytically and/or by ray tracing of a group of equally spaced rays incident upon the collection area at angles Xand X. Eqs. (1) and (2) are used. The product of the radiation energy intensity incident with angles Xand X times the angular response for these two angles is equal to the amount of incident radiation energy intensity which is collected per unit collection area per unit time. Because of the longitudinal symmetry along the collector axis direction of all the collector designs, it is sufficient to do analytic calculation or ray tracing only in the transverse plane where X = 90o. The collectors have longitudinal symmetry and left-right symmetry of the transverse cross section of the collectors (other than the horizontal half-fin collector). These symmetries make the angular response values obtained for angles Xand X between 0o and 90o determine the angular response at other values of  X and X. In SOCOL, the angular response of the horizontal half-fin collector is the average of the angular response of this collector with the half-fin on the left and on the right side of the tube. An array is assumed to be made up of an equal number of “left” and “right” tubes.

Fig. 4 shows the angular response as a function of Xand X for the vertical fin collector of Fig. 1 using the Pettit-Sowell absorber. Angular responses of other collectors are shown in (Garrison, 2000). The angular response of the collector designs is somewhat greater for collectors using the Pettit-Sowell absorber, because of the larger value of o and also the slower drop-off of / with increasing angle of incidence. The gain in solar energy collection by this increase in angular response using the Pettit and Sowell absorber is largely cancelled at the lower collector operating temperatures by the greater losses associated with the much larger emittance of this absorber. At higher operating temperatures, solar energy collection is much larger using the Zhang-Mills absorber.

In SOCOL, the angular response for each collector is represented by a 19x19 element bivariate histogram of angular response values for equally spaced intervals from 0-90o in both X and X. SOCOL does a table look-up operation for the angular response using values it calculates for X and X.

2.6 Collector Losses

In SOCOL, energy loss is calculated for five different values of the absorber operating temperature: T = 40, 70, 120, 200, and 300 C. These have been assumed constant over the collection time and area. These temperatures are data in SOCOL and can be changed easily. The loss coefficient of the top surface of the absorber in the air flat plate collectors is obtained by the method of Klein (1975) as given in (Duffie and Beckman, 1991), Eq. 6.4.9. The heat loss from the lower side of the absorber is determined by a loss coefficient taken to be hp = 0.6 W/m2- C (about 5 cm of polyurethane foam).

The evacuated collector designs lose thermal energy mainly by radiation from the absorber surface. A first approximation is to assume that the glass window is at ambient temperature. Going one step further, this loss is treated as a two step process: Radiation from the absorbing surface to the glass window and convection and radiation from the window to ambient. The temperature of the window is needed for this calculation. It is estimated by iteration until the two steps of the process transfer energy at the same rate. Radiation from the absorbing surface to the glass window is estimated by the equation

q = (T4 – TG4)/[(1 - )/ A +1/ AFAG + (1 - G)/GAG ] (3)

for a two surface enclosure (Incropera and deWitt, 1990), Eq. 13.23, p. 771. The view factor FAG is set equal to one for all the collectors treated in SOCOL except the U-trough [Number 4], cusp (Number 5) and vertical fin (Number 7) where it is set equal to 0.50 (inside), 0.75 (upper part) and 0.72, respectively. The other symbols in Eq. (3) are:  the Stefan-Boltzmann constant; T, the absorber operating temperature; TG, the glass window temperature; A, the absorber area; G, the glass emittance taken to be 0.88; and AG, the area of the glass window. The absorber hemispherical emittances  for the Pettit-Sowell and Zhang-Mills absorbers are: 0.115, 0.12, 0.14, 0.17, 0.20 and 0.0275, 0.028, 0.030, 0.033, 0.039, respectively, at the 5 operating temperatures. Eq. (3) takes the following form when the known values are inserted

q =5.67x10-8 ( T4 – TG4)/[(C/) + b) (W/m2). (4)

C is the collector concentration. Values of C and b are tabulated as data in SOCOL. The second step of the heat transfer from window to ambient is calculated by the equation

q = 5.0x10-8 (TG4 – TSKY4) + 15 (TG –TA) (W/m2) (5)

The first term on the right is an estimation of the radiation loss, while the second term is an estimation of the convection loss. TSKY is the sky temperature. TA is the ambient temperature. TSKY is calculated using Berdahl and Martin (1984), if the dew point temperature is in the input surface meteorological data. Otherwise Swinbank (1963) is used. By symmetry there should be no net radiation transfer between the neighboring tubes in an array. Generally, the temperature drop from the window glass to ambient temperature is small relative to the drop from the absorber to the glass. The surface meteorological data on wind for each hour or day has not been used to vary the coefficient of the convection loss. The approximation using Eq. 5 calculates this loss in the same manner for all evacuated collectors. Any person desiring to improve this calculation can modify SOCOL. As a help, there are numerous comments throughout SOCOL to identify the different calculations.

2.7 Surface albedo

SOCOL calculates the contribution of solar radiation scattered by the ground in front of the collector to solar energy collection. It assumes that the scattering by the ground is diffuse. Often, this scattering has a forward component. To account for this effect, the albedo used as input to SOCOL can be increased. The contribution from ground scattering is generally quite small. Ground scattering has a larger effect on diffuse radiation collection.

    1. Calculation of solar energy collection

The contribution of each part of the sky to the diffuse radiation is determined using the prescription of Perez, Seals and Michalsky (1993) with sky luminance replaced by sky irradiance. The total contribution of the diffuse radiation to solar thermal energy collection is obtained by numerical integration, summing the contributions of 400 elements equally spaced over the sky. To this is added the contribution of an additional number of elements below the horizontal for ground reflection. For both direct and diffuse radiation, the thermal energy collected per unit collection area for each sky element for each hour is taken to be the product of the mean incident radiation energy intensity from the direction of an element of the sky Xand X the collector angular response and the time duration. The net total energy collection is given by

ET = Eb + Edloss (6)

where Eb, is the direct or beam energy collected, Ed is the diffuse energy collected, loss is the energy loss by convection and radiation, and ET is the net total energy collected, all per unit collection area. Whenever the loss exceeds the sum Eb + Ed for any hour, the net total energy collection ET is set equal to zero. The calculation of the energy collection from diffuse radiation is time consuming. The time to run each hour of data is greatly reduced by reducing the number of points in the sky from 400 to 100, for example, with some reduced precision of the calculation.

The equations for the solar time and direction of the sun as a function of time, along with other needed equations are in the Appendix. The orientation of the collector axes in the plane of the collector array is assumed to have only two possible conditions, either horizontal, or lying in the vertical plane containing the normal to the collector array.



  1. TESTS OF THE PROGRAM


SOCOL has been tested in many different ways. For each of a few hours selected at random during the collection year, all the results of the equations in the energy collection part of the program have been hand calculated and sometimes visualized with figures, and then compared with the results obtained by SOCOL. The collection of solar energy for particular hours has been tested for proper behavior. For example, when the normal to the array is horizontal and the plane of the array is vertical, energy collection at different azimuthal angles of the array normal are compared to see if the behavior is as expected. Thus, there should be no direct radiation collection when the normal points north and the hour is in the winter half of the year. Also, there is less diffuse radiation collection when the normal points north. When the plane of the collector array is horizontal and the angular response is set equal to the cosine of the zenith angle (optical efficiency =1), the diffuse energy collection and the direct energy collection for each hour are the same as the measured diffuse and direct radiation on a horizontal surface.

When the angular response is set equal to the cosine of the incidence angle on the collector array and the collection area is tilted at the latitude angle and faces south at Albuquerque, NM, USA, the calculated annual energy collections per unit area for the years 1976-1979 inclusive are: 8641, 8275, 7832, and 8009 MJ/m2. The mean is 8189 ± 352 (176) MJ/ m2 where the 352 is the standard deviation of a single year and 176 is the standard deviation of the mean. Marion and Wilcox (1994) give a corresponding value of 8400 MJ/m2 and Rabl (1981) gives a corresponding value of 8000 MJ/m2. It is not known what years Marion and Wilcox, and Rabl have used for their values.

Finally, the solar energy collection of a double glazed air flat plate collector has been calculated using both the Pettit-Sowell and Zhang-Mills selective absorbers and compared with results that are obtained using the method of Rabl (1981). The results of the calculation by SOCOL and comparison with Rabl are presented in Table 1. In the table, T is the absorber operating temperature, TA, is the ambient temperature (mean for the year) used by Rabl, , is the normal optical efficiency used here in Rabl, and Q, is the annual energy collection in GJ/m2. The other symbols are as in Rabl. The results by these two methods are in good agreement.

The difference between the calculated energy collection by one collector and another arises only from differences in the angular response and differences in heat loss. The heat losses have been checked carefully by hand calculation. The angular responses have also been checked carefully. The angular responses of the CPC shaped glass evacuated collector are probably the most prone to error. The first two of these with half-angles of acceptance of 60o and 45 o were repeated. The average of the angular responses over the 19x19 bivariate histogram for the two determinations differ by about 1% for both the 60o and 45 o collectors. All ray tracings for these use a density of 10 rays per collection width of one tube. This ray density extends across the tube and neighboring tubes in the plane transverse to the tube axis.


4. SAMPLE RESULTS BY SOCOL

Table 2 shows annual energy collection per unit collection area as a function of absorber temperature at Albuquerque, New Mexico, USA and Seattle, Washington, USA. This is for nine collector designs: 1, air flat plate; 3, vacuum tubular; 4, U-trough; 5, cusp; 6, horizontal fin; 8, vertical half-fin; 9, horizontal half-fin; 12, 35o CPC shaped glass tube; and 13, single axis parabolic tracking. The values in Table 2 for each collector use the axes orientations: E–W, N-S, and polar. The values are for the selective absorber (Pettit-Sowell or Zhang-Mills) which yields the highest energy collection at each temperature. For the few cases at lowest temperature where the Pettit-Sowell absorber collects the most energy, the number is put in italics. The array normal is tilted at the latitude angle for the E-W (and polar) orientation. The low loss energies in the table are for collectors 3,4,5 and 6 when the lower part of the collectors have low emissive coatings and thermally floating shields, as discussed earlier. The bordered energy collection numbers in bold type in Table 2 are the highest values at each temperature. Energy collection by all collectors is highest for the polar axis orientation, except for the CPC shaped glass collectors. At these latitudes, the E-W axis orientation collects more energy than the N-S orientation. The evacuated collectors outperform the air flat plate collectors significantly. Energy collection per unit absorber area is obtained by multiplying by the concentration. This is of interest since the selective absorber is generally a more expensive part of the collector. The concentration must be suitably changed for the low loss cases.

Fig. 5 shows the net annual energy collection ET for 35 US stations for the year 1979 ordered by increasing mean annual clearness index KT for a single glazed air flat plate collector using the Pettit-Sowell absorber at a temperature of T = 40 C and T = 70 C. ET also varies to a lesser degree with latitude, annual mean daylight ambient temperature and surface albedo. This accounts largely for the fluctuation in points in Fig. 5. Also shown is a least square fit to the net energy collection using the relation: ET = 1.282[1-0.0079(T - Ta)] [(0.95 (Kb+Kd(-0.149+0.92cos(thetaL)) 104 - 6(T - Ta)] where Kb is the annual mean direct beam index, Kd is the annual mean diffuse index, TA is the annual mean daylight ambient temperature and L is the latitude. The correlation between the fluctuations in the lines connecting the calculated energy and the RMS fit indicates validity in the choice of the four variables selected (eg. surface albedo would not be a useful variable).

Table 2 and Fig. 5 are representative of the types of information which can be obtained using SOCOL. Additional examples may be found in (Garrison, 2000, 2002). SOCOL calculates mean values of KT , Kd , Kb, cloud amount and opacity, ambient temperature, in addition to ET ,Ed ,Eb, and energy loss for each hour, day, month and the entire year.


  1. SUMMARY


The FORTRAN program SOCOL is a program of rather general utility which realistically predicts net hourly solar energy collection for one year or any part thereof at a particular site for which one has data. This net energy collection can be for any of 15 collector types contained in SOCOL. Any selective absorber can be used for the absorber surface. The orientation of the collector array can be with the collector axis horizontal or with the axis lying in a vertical plane containing the array normal. The normal to the collector array can be tilted at any angle with respect to the vertical and with any azimuthal angle about vertical. Net solar energy collection for a collector not included in SOCOL can be calculated by inserting the angular response table and loss charactistics of this collector in SOCOL. The uncertainty of the calculation of net solar energy collection is believed to be about ±5%. This is indicated by comparison with results of Rabl (1981) and Marion and Wilcox (1994).


ACKNOWLEDGEMENTS- Jeff Gordon has made a number of very helpful suggestions which have improved this paper. Carl Lampert provided advice concerning selective absorbers and provided an additional reference for information. The reviewers suggested placing SOCOL on a Web site and the U-Trough design. Herb Shore provided his time to install the FORTRAN compiler on the Compaq laptop computer. Jim Varnell, Bill Morris, Denis Poon, and Susan Langsford of the College of Sciences Computer Group continue to provide the able and friendly help needed in computer operations. Denis Poon provided assistance in placing SOCOL and supporting material on its web site.


NOMENCLATURE


A absorber area (m2)

AG glass window area (m2)

B angle constant in equation of time

C collector concentration

E time correction (hours)

Eb direct radiation energy collection (KJ/m2) and (MJ/m2)

Ed diffuse radiation energy collection (KJ/m2) and (MJ/m2)

ET net total radiation collection (KJ/m2) and (MJ/m2)

H standard time (hours)

I hourly global radiation (J/m2 - hr)

Id hourly diffuse radiation (J/m2 - hr)

Ib hourly direct normal radiation (J/m2 - hr)

Io hourly normal extraterrestrial radiation (J/m2 - hr)

Ioh hourly extraterrestrial radiation on horizontal surface (W/m2)

KT clearness index

Kd diffuse index

Kb direct (beam)index

T absorber temperature (K)

TA ambient temperature (K)

TD dew point temperature (C)

TG window glass temperature (K)

TSKY sky temperature (K)

b two surface enclosure constant (for loss calculations)

hp air flat plate collector bottom loss coefficient (W/m2-K)

no number of days since beginning of year

q energy intensity (W/m2)

t solar time, t = 0 at solar noon (hours)

io unit vector in direction of sun

jo unit vector normal to io and ko (= -io x ko )

ko unit vector normal to earth’s orbital plane

k unit vector parallel to earth’s axis (north)

i unit vector normal to k in plane of io and k

j unit vector east at solar noon = k x i

i unit vector normal to earth’s surface at equator at collector longitude

j’ unit vector, east at collector longitude

k’ unit vector parallel to earth’s axis, equals k

n normal to the plane of the collector array

s south at the latitude and longitude of the collector array

v vertical at collector latitude and longitude

H direction of tube axis when horizontal

P direction of tube axis when in plane of n and v

 selective absorber absorptance

o selective absorber normal absorptance

 the angle between the sun direction and the polar axis (See Fig. 1A)

 selective absorber hemispherical emittance

G glass window hemispherical emittance, G = 0.88

 efficiency at incident angle  (optical or PV)

 earth rotation angle,  =0 at solar noon

C angle that the projection of n onto the horizontal plane makes with south [east of south is positive]

L longitude, L = 0o at Greenwich, England

LO time zone longitude (multiple of 15o)

o earth’s orbital angle , o = 0 June 21

R phase correction to o for circular orbit approximation

See (Goldstein, 1983) Sec. 3.8, pp. 98-102.

S the angle the projection of the direction io of sun onto horizontal plane

makes with south

X the angle the projection of io onto the plane transverse to tube

axis makes with normal to array plane n

 reflectance

G ground reflectance, albedo (assumes diffuse reflection)

 Stefan-Boltzmann constant,  = 5.67x10-8 (W/m2K4)

 window transmission

A incidence angle on selective absorber

G incidence angle on glass window

L latitude

N angle earth’s axis makes with the normal to the earth’s orbital plane, N = 23.452o

S angle sun direction makes with south

X angle sun direction makes with tube axis

Z zenith angle

 angle of incidence on collector array plane

C angle normal to collector array makes with vertical


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