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).
2.
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
2
(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),
3
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.
4
• [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.
5
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 90 o
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.
6
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
this collector at 20 C, a number for the variation of the relative efficiency
o of
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.
__________________
7
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
8
/ = 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
new absorber; and the position of the curve for
 of the
/ 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
9
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
and
in each array,
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 Xand 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
other values of

X
and
X.
Xand X between 0o and 90o determine the angular response at
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
10
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 F AG 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; T G,
the glass window temperature; A, the absorber area; G, the glass emittance taken to be 0.88; and AG, the
11
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
( T – TG )/[(C/) + b) (W/m2).
4
4
(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. T SKY 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.
2.8 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,
12
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
Xand X the collector angular response and the time duration. The net total energy collection is given by
ET = Eb + Ed – loss
(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 E T 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.
3.
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
13
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, T A, 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 60 o 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.
14
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 K T for a single glazed air flat plate collector using the Pettit-Sowell
absorber at a temperature of T = 40 C and T = 70 C. E T 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: E T = 1.282[10.0079(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, T A is the annual mean daylight ambient temperature and
15
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.
5.
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 UTrough 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.
16
NOMENCLATURE
A
AG
B
C
E
Eb
Ed
ET
H
I
Id
Ib
Io
Ioh
KT
Kd
Kb
T
TA
TD
TG
TSKY
b
absorber area (m2)
glass window area (m2)
angle constant in equation of time
collector concentration
time correction (hours)
direct radiation energy collection (KJ/m2) and (MJ/m2)
diffuse radiation energy collection (KJ/m2) and (MJ/m2)
net total radiation collection (KJ/m2) and (MJ/m2)
standard time (hours)
hourly global radiation (J/m2 - hr)
hourly diffuse radiation (J/m2 - hr)
hourly direct normal radiation (J/m2 - hr)
hourly normal extraterrestrial radiation (J/m2 - hr)
hourly extraterrestrial radiation on horizontal surface (W/m2)
clearness index
diffuse index
direct (beam)index
absorber temperature (K)
ambient temperature (K)
dew point temperature (C)
window glass temperature (K)
sky temperature (K)
two surface enclosure constant (for loss calculations)
hp
no
q
t
air flat plate collector bottom loss coefficient (W/m2-K)
number of days since beginning of year
energy intensity (W/m2)
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
k
unit vector normal to earth’s orbital plane
unit vector parallel to earth’s axis (north)
unit vector normal to k in plane of io and k
unit vector east at solar noon = k x i
unit vector normal to earth’s surface at equator at collector longitude
unit vector, east at collector longitude
unit vector parallel to earth’s axis, equals k
normal to the plane of the collector array
south at the latitude and longitude of the collector array
vertical at collector latitude and longitude
i
j
i’
j’
k’
n
s
v
H
direction of tube axis when horizontal
17
P

o
direction of tube axis when in plane of n and v



 

selective absorber normal absorptance

G




C







selective absorber hemispherical emittance
efficiency at incident angle  (optical or PV)
earth rotation angle,
 =0 at solar noon
angle that the projection of n onto the horizontal plane makes with
south [east of south is positive]
longitude,
L = 0o at Greenwich, England
time zone longitude (multiple of 15o)
earth’s orbital angle ,
o = 0 June 21
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 between the sun direction and the polar axis (See Fig. 1A)
glass window hemispherical emittance, G = 0.88
L
LO
o
R

G

selective absorber absorptance









the angle the projection of io onto the plane transverse to tube
axis makes with normal to array plane n
reflectance
ground reflectance, albedo (assumes diffuse reflection)
8
Stefan-Boltzmann constant,  = 5.67x10-
(W/m2K4)
window transmission
A
G
L
N
incidence angle on selective absorber
incidence angle on glass window
latitude
angle earth’s axis makes with the normal to the earth’s orbital plane,
N = 23.452o
S
X
Z

C
angle sun direction makes with south
angle sun direction makes with tube axis
zenith angle
angle of incidence on collector array plane
angle normal to collector array makes with vertical
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19
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21
APPENDIX
SUN POSITION AND SOLAR ENERGY COLLECTION EQUATIONS
A1. INPUT DATA FROM NSRDB:
Each datum has symbol, name, units and FORTRAN program array name (in italics):
loh, extraterrestrial radiation on horizontal surface (KJ/m2-hr) NETRH; Io, hourly direct normal
extraterrestrial radiation (KJ/m2-hr) NETR; In , hourly direct normal radiation (KJ/m2-hr) NBRAD; Id ,
hourly diffuse radiation (KJ/m2-hr) NDRAD; cloud amount (in fraction of one) CLOUD; cloud opacity
amount (in fraction of one) OPAC; TA, ambient temperature (K) ATEMP; TD, dew point temperature (C)
DTEMP; (Hourly radiation has been converted from Wh/m2; cloud amounts converted from tenths of sky
covered; TA converted from C.)
A2. SOLAR RADIATION INDICES
Clearness index: KT = IG / Ioh, RAD; Diffuse index: Kd = Id / Ioh, DRAD;
Direct index:
Kb = Ib / Io, BRAD
A3. EARTH’S ORBITAL MOTION
The orbit of the earth is treated as circular with orbital angular correction
R to o, the orbital angle. The
time has correction E called Equation of Time. The distance from the sun varies over the year. This is
included in the NSRDB input data as a variation of NETR.
Fig. 1A shows the relation of two rectangular coordinate systems relating the earth’s orbital
motion to the direction of the sun. Unit vectors of Fig. 1A are: io, sun direction; ko, normal to orbital plane
and parallel to orbital angular momentum vector; jo = ko x io, the direction at right angles to plane of io and
ko; k, parallel to earth’s axis and rotational angular momentum; j, east at solar noon; i = j x k, normal to k
in plane of io and k. For a more accurate determination of sun direction which is suitable for high
concentration tracking collectors see (M. Blanco-Muriel, et al, 2001).
Orbital angle:
o = 2 (no-172)/365 + R,
where no is number of days since beginning of the year.
22
Relations: cos= cos o sin N; k = cos  io + sin o sin N jo + cos N ko; j = k x io / sin 
A4. SOLAR TIME AND ROTATION OF EARTH
B = 2 (no – 1)/365
Equation of time (Iqbal, 1983), p. 11:
E = 0.0002865 + 0.0071358 cos B –0.12253 sin B – 0.055829 cos 2B – 0.1562 sin 2B (hours)
Solar time: t = H – 12 + (L -
LO)/15
+ E (hours) L is the longitude at the collector site, and
LO is the
time zone longitude.
A5. DEVELOPMENT OF ENERGY COLLECTION EQUATIONS
The new unit vectors in Fig. 2A are associated with rectangular coordinates rotating with the
earth: i’, unit vector perpendicular to the earth at the equator at the collector longitude; j’, east at collector
longitude; k’, same as k. Also shown in Fig. 2A are s, south at collector latitude and longitude; v, vertical
at the collector latitude and longitude. Fig. 3A shows the rectangular coordinate system used at the
collector latitude and longitude, with orthogonal unit vectors s, j’ (east), and v. The normal to the collector
array
n and the direction of the sun io are also shown. Two other unit vectors not in the figure are used:
H, direction of collector tube axes when horizontal; P, direction of collector tube axes when in the
plane of n and v .
Earth rotation angle:  =  t/12
Relations: io = cos k + sin
j’ = - sin
cos L sin
sin
 i + cos j,
(radians,
< 0 before solar noon)
.
 i; io  i = sin  io  j = 0; io  k = cos  i’ = cos  i + sin  j;
v = cos L i’+ sin L k’; s = sin L i’ - cos L k’; v = cos L cos i +
 j + sin L k; s = sin L cos i
+ sin L sin
 j - cos L k; n = sin c cos c s +
c sin c j’ + cos c v; H = - sin c s + cos c j’; P = sin c v – cos c cos c s –
23
cos c sin
c j’
Cosine of angle of incidence of radiation on array plane:
cos  = io 
n = sin [ sin c cos c sin L cos  sin c sin c sin 
cos c cos L cos  cos cos c sin L - sin c cos c cos L ]
Cosine of zenith angle Z: cos Z v io = cos L cos
 sin  + sin L cos 
Cosine of angle sun direction makes with s:
cos S = io s = sin  cos sin L – cos  cos L
Cosine of angle that projection of sun direction onto horizontal plane makes with south:
cos S = cos S / sin Z
Cosine of axial angle X with H: cos XH = sin Z [cos c sin
S – sin c cos S ]
Cosine of axial angle X with P: cos XP = cosZ sin c - sin Z cosc[cos S cos c + sin S sin
c]
Cosine of projected angle
X in plane transverse to H:
Cosine of projected angle
X in plane transverse to P:
cos  X H = cos sin XH
cos XP = cos sin  XP
Hourly direct radiation collection: Eb = Kb Io x (angular response) x time EB
Hourly total radiation collection: ET = Eb + Ed – loss
24
(ET > 0) ET
TABLE 1- DOUBLE GLAZED FLAT PLATE COLLECTOR ANNUAL ENERGY COLLECTION
-A COMPARISON OF RABL MODEL WITH SOCOL SIMULATIONAlbuquerque
Icoll = 8.0 GJ/m2-y, I = 0.60 kW/m2, TA =13 C (Rabl Data)
Zhang-Mills Absorber
SOCOL Simulation (NSRDB data)
Rabl Model ( o = 0.75)
T
U
X
Q
Q
(annual)
Q (ave)
Q/o
C
W/m2-C kW/m2 GJ/m2 GJ/m2
GJ/m2 GJ/m2 GJ/m2
GJ/m2
GJ/m2
1976
1977
1978
1979
Average
13
0.00
0.000
8.00
6.00
6.10
5.88
5.55
5.68
5.80
40
2.09
0.075
6.98
5.24
5.32
5.12
4.82
4.94
5.05
70
2.32
0.176
5.70
4.28
4.29
4.11
3.87
3.96
4.06
120
2.58
0.368
2.80
2.10
2.63
2.52
2.37
2.42
2.49
Pettit-Sowell Absorber
SOCOL Simulation (NSRDB data)
Rabl Model ( o = 0.77)
T
U
X
Q
Q
(annual)
Q (ave)
Q/o
C
W/m2-C kW/m2 GJ/m2 GJ/m2
GJ/m2 GJ/m2 GJ/m2
GJ/m2
GJ/m2
1976
1977
1978
1979
Average
13
0.00
0.000
8.00
6.16
6.58
6.34
5.98
6.12
6.26
40
2.23
0.080
6.91
5.32
5.55
5.34
5.03
5.15
5.27
70
2.50
0.190
5.53
4.26
4.41
4.23
3.98
4.07
4.17
120
2.85
0.410
3.32
2.56
2.52
2.42
2.27
2.32
2.38
25
TABLE 2 – ANNUAL SOLAR ENERGY COLLECTION (MJ/M2)
Year = 1979
COLLECTOR
ARRAY
Axis
1 AIR FLAT PL 1G E-W
3
DEWAR
Polar
N-S
E-W
3
LOW LOSS
Polar
DEWAR
N-S
E-W
4
U TROUGH
Polar
N-S
E-W
4
LOW LOSS
Polar
U TROUGH
N-S
E-W
5
CUSP
Polar
N-S
E-W
5
LOW LOSS
Polar
CUSP
N-S
E-W
6 HORIZONTAL Polar
FIN
N-S
E-W
6
LOW LOSS
Polar
HORIZONTAL
N-S
FIN
E-W
8
VERTICAL
Polar
HALF FIN
N-S
E-W
9 HORIZONTAL Polar
HALF FIN
N-S
E-W
12 35 DEG CPC
Polar
N-S
E-W
13
ONE AXIS
Polar
PARABOLIC
N-S
TRACK
E-W
T40
=
5583
7396
6344
6697
7475
6429
6837
7231
6199
6802
7279
6247
6849
7031
6060
6637
7093
6121
6725
6331
5275
6203
6554
5513
6425
6005
5105
5625
6206
5245
6091
4469
3518
5541
6231
5640
5016
Albuquerque
70
120
4087
7079
6027
6383
7245
6192
6546
6919
5886
6492
7041
6009
6613
6770
5798
6377
6931
5960
6537
6121
5057
5995
6290
5222
6162
5887
4988
5508
5963
4978
5829
4286
3346
5252
6214
5623
4998
2058
6317
5274
5644
6714
5664
6026
6170
5146
5762
6461
5434
6044
6136
5171
5758
6529
5556
6140
5618
4560
5493
6026
4962
5899
5592
4693
5216
5669
4688
5537
4104
3166
5066
6166
5575
4946
26
KT =0.64
200
300
87
4445
3406
3858
5354
4304
4720
4338
3326
3992
5002
3970
4629
4557
3585
4235
5474
4507
5120
4367
3327
4252
5331
4274
5210
4811
3916
4455
4910
3932
4780
3660
2739
4587
6015
5423
4787
0
850
447
674
2433
1580
2021
866
454
796
1953
1209
1771
1354
779
1304
3120
2218
2909
1800
1080
1739
3746
2739
3639
3055
2206
2774
3191
2285
3088
2762
1928
3486
5637
5045
4403
40
2961
4372
3871
3980
4444
3942
4050
4277
3778
4037
4330
3830
4089
4174
3706
3947
4243
3775
4015
3768
3195
3686
3842
3302
3760
3598
3146
3371
3648
3129
3569
2648
2037
3226
2945
2686
2354
Seattle
70
120
1953
4080
3576
3691
4239
3736
3848
3989
3487
3752
4106
3605
3867
3930
3460
3705
4088
3620
3862
3571
2999
3490
3739
3166
3657
3485
3032
3258
3534
3014
3456
2574
1966
3089
2931
2672
2340
810
3441
2918
3071
3770
3257
3389
3361
2842
3138
3601
3088
3371
3388
2902
3174
3723
3248
3501
3130
2552
3052
3492
2919
3411
3213
2758
2989
3263
2742
3186
2401
1795
2913
2895
2636
2303
KT = 0.43
200
300
3
2119
1615
1824
2725
2197
2394
2069
1568
1899
2506
1982
2316
2236
1745
2070
2877
2382
2682
2193
1627
2127
2901
2321
2826
2571
2099
2365
2631
2103
2559
2011
1416
2494
2788
2528
2197
0
193
79
153
939
598
763
214
86
198
699
412
632
432
220
412
1350
942
1249
662
360
635
1789
1261
1729
1389
981
1241
1467
1014
1415
1352
840
1702
2537
2272
1951
FIGURE CAPTIONS
Fig. 1. A schematic view of the transverse cross sections of 8 of the 15 solar collectors used in this
study. The number of each collector is placed just to the left of each transverse cross sectional view. The
collector concentration C is also shown.
Fig. 2.
The variation of absorptance over normal absorptance with incidence angle on the
selective absorbers of Pettit and Sowell (1976) and Zhang and Mills (1992).
Fig. 3. A projected view of the circular cross section of evacuated collector glass tube in the plane
transverse to the tube axis.
n is the normal to the plane of the collector array. X is the angle a sun ray
makes with tube axis direction
. x is the angle a projection of a sun ray in the plane transverse to
makes with respect to n. io is
the direction of the sun.
Fig.4. The angular response of the vertical fin collector (Number 7) using the Pettit and Sowell
(1992) selective absorber.
Fig. 5. The variation of net annual solar energy collection by a single glazed air flat plate collector
shown with increasing mean annual clearness index KT using values calculated by SOCOL for 33 stations.
Points are connected by a solid line. The collector is tilted towards the equator at the latitude angle. A least
square fit of a function of the five variables Kd, K d, L, T and Ta, to the net annual energy collection is
shown by the points connected by the dashed line.
Fig. 1A. Two rectangular coordinate systems giving the orientation of the earth with respect to the
sun and the earth’s orbital plane. Unit vector io is the direction to the sun. Unit vector ko is the normal to
the earth’s orbital plane and parallel to the orbital angular momentum. Unit vector jo = ko x io. Unit vector
k is parallel to the earth’s axis and rotational angular momentum. Unit vector i lies in the plane of k and io
and is perpendicular to k. Unit vector j = k x i. The angle
 is the earth’s orbital angle about the sun and
N is the fixed angle the earth’s axis makes with respect to the normal to the orbital plane ko.
27
Fig. 2A. The relation of the unit vectors i’, j’ and k’ of the rectangular coordinate system rotating
with the earth to unit vectors i, j and k of Fig. 1A. The unit vectors k and k’ coincide. i’ is normal to the
earth’s surface at the equator at the longitude of the solar collector site. At solar noon at this longitude, the
earth’s rotational angle
is zero and i’ is parallel to i. The unit vector
j’ is east at the solar collector
longitude. The unit vector v is vertical at the latitude and longitude of the solar collector site. The unit
vector s is horizontal and directed south at the collector latitude and longitude. s and v lie in the plane of i’
and k’. v makes an angle with i’ equal to the latitude, L.
Fig. 3A. The orientation of the direction of the sun io and the normal to the solar collector array n
with respect to the unit vectors s, j’ and v of Fig. 2A. c is the angle of n with respect to v.
the projection of
c is the angle
n on the horizontal plane makes with s. The angle io makes with v is z (zenith angle).
The angle io makes with s is s. The angle the projection of io onto the horizontal plane makes with s is
s.
28
C = 0.51
C = 1.0
7
1
C = 0.89
C = 0.32
3
8
C = 0.39
C = 0.89
5
9
C = 0.49
C = 1.4
12
6
29
30
n
(Normal to Array Plane)
x
io
(Sun Direction)
x

(Tube Axis)
31
VACUUM VERTICAL FIN
o = 0.95
1.0
0.8
0.6
Angular
Response
0.4
90
0.2
70
50
0.0
30
0
20
40
10
60
x (deg)
32
80
x (deg)
33
ko
k
N = 23.452
o
N

jo
o
o
io
i
j
o is zero June 21
34
v
k,k'
(north)
L
j'
j
i 
i'
s
35
(east)
v
n
c
z
io
j' (east)
s
s
c
s
36