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SOLAR SYSTEMS IN THE ECO‐VILLAGE
AT THE UNIVERSITY OF MANITOBA
Heather King
SOLAR SYSTEMS IN THE ECO‐VILLAGE
AT THE UNIVERSITY OF MANITOBA
Heather King
A thesis submitted in conformity with
the requirements for the degree of
BACHELOR OF SCIENCE (MECH. ENG)
at the University of Manitoba
Supervisor: Dr. E. Bibeau
Department of Mechanical and Manufacturing Engineering University of Manitoba
2007
ii
ABSTRACT Solar technologies are vastly emerging across the globe as a renewable, environmentally
acceptable alternative to fossil fuels and other non‐renewable energy resources. The
“eco‐village”, an old straw bale barn located on the University of Manitoba’s Ft. Garry
campus, is currently being used to study solar technologies; more specifically, to
examine the use and efficiency of solar air and water collectors. Once installed, data
must be logged from these solar collectors to enable the energy output and
instantaneous efficiency for each collector to be determined. This report discusses, in
detail, the various types of solar collectors that are to be installed at the “eco‐village”.
Calculations are carried out to determine the optimum location and position the solar
collectors should be installed at in order to maximize the amount of solar radiation
received by the collectors throughout the year. Research into the transfer of heat
throughout each collector allowed for instantaneous efficiency models to be created for
each collector, in which a user simply has to enter the required variable input data and
the efficiency of the collector is calculated. The report concludes by examining the
economical and environmental benefits of installing a system of solar collectors on the
University of Manitoba’s Ft. Garry campus and the capabilities this proposed system of
solar collectors would have in supplementing the university’s current district heating
system.
iii
ACKNOWLEDGEMENTS I would like to thank the following people for their assistance during the course of this
undergraduate research project:
Dr. Eric Bibeau Assistant Professor, Department of Mechanical
Engineering, University of Manitoba G. Paul Zanetel, P.Eng Associate Professor, Engineer‐in‐Residence, Department of
Mechanical Engineering, University of Manitoba Daniel Friesen, P. Eng Consultant, Daniel Lepp Friesen Consulting Mike Ferley, P.Eng Energy Advocate, Physical Plant, University of Manitoba Stephanie Zubriski Graduate Student, Faculty of Engineering, University of
Manitoba Dr. Kris Dick, P.Eng Adjunct Professor, Engineer‐in‐Residence, Design
Engineering, University of Manitoba
iv
NOMENCLATURE
sh = specific enthalpy of evaporation of steam at working pressure
bI = the direct radiation received by the collector
effI = the effective solar radiation on the collector
NI = the normal terrestrial solar radiation received at ground level
sm
= steam mass flow rate
aT = ambient air temperature
Q = heat transferred from the steam
LQ.
= thermal energy lost to the environment
LU = the overall heat loss coefficient
= Stefan‐Boltzmann constant = 5.67 x 10‐8 W/m²∙K4
= density of the heat transfer fluid
= viscosity of the heat transfer fluid
Σ = tilt angle of the flat plane of the collector Φ = direction of the angle of tilt Evacuated Tube Collector nomenclature:
icA , = inner tube diameter times collector length
ocA , = outer tube diameter times collector length
ihA , = heat pipe inner diameter times collector length
ohA , = heat pipe inner diameter times collector length
icD , = inner diameter of the collector tube
ocD , = outer diameter of the collector tube
ocondD , = outer diameter of the heat transfer condenser
ihD , = inner diameter of the heat transfer pipe
ohD , = outer diameter of the heat transfer pipe
evacF = the shape factor between the selectively coated outer surface of the
heat pipe and the inner surface of the collector tube
rF = flow rate factor
ch = heat transfer coefficient of the outer surface of the collector tube
eh = heat transfer coefficient of the inner surface of the collector tube
hmh = heat transfer coefficient between the heat pipe fluid and the manifold
fluid
v
phh = heat transfer coefficient between the fin plate and the heat transfer pipe
fluid
ck = thermal conductivity of the glass collector tube
hk = thermal conductivity of the copper heat transfer tube
fink = thermal conductivity of the fin plate
L = the length of the heat transfer pipe
fm
= flow rate of the fluid through the manifold
mPr = Prandtl number of the heat transfer fluid in the manifold
pPr = Prandtl number of the fluid within the heat transfer pipe
mRe = Reynolds number of the heat transfer fluid in the manifold
pRe = Reynolds number of the fluid within the heat transfer pipe
cT = mean temperature of the collector plate
hT = mean temperature of the heat pipe
ifT , = temperature of the fluid in the manifold before heating occurs
hmQ.
= thermal energy transferred from the heat pipe fluid to the manifold fluid
mU = mean velocity of the fluid within the heat pipe
V = mean velocity of the air flowing over the evacuated tube collector = kinematic viscosity of the liquid in the heat transfer pipe
c = emissivity of the collector tube
h = emissivity of the heat transfer pipe
Flat‐Plate Collector nomenclature:
cA = flat plate collector area
D = outer diameter of the fluid tube 'F = collector efficiency factor
RF = collector heat removal factor
h = convective heat transfer coefficient from the inner fluid tube to the fluid k = thermal conductivity of the fluid tube
m = flow rate of the heat transfer fluid through the heat transfer tubes
fT = mean temperature of the heat transfer fluid
W = distance between centre of two fluid tubes δ = thickness of the fluid tube = heat transfer tube absorptance ε = heat transfer tube emittance
vi
TABLE OF CONTENTS
ABSTRACT .............................................................................................................................ii ACKNOWLEDGEMENTS ....................................................................................................... iii NOMENCLATURE ................................................................................................................. iv LIST OF FIGURES ................................................................................................................ viii LIST OF TABLES .................................................................................................................... ix 1 INTRODUCTION ....................................................................................................... 1 1.1 Purpose and Scope.............................................................................................. 1 1.2 Layout of Thesis .................................................................................................. 2
2 REVIEW OF LITERATURE .......................................................................................... 3 2.1 Solar Energy ........................................................................................................ 3 2.2 Solar Collectors ................................................................................................... 5 2.2.1 Solar Evacuated Tube Collector .................................................................... 6 2.2.2 Solar Water Flat‐Plate Collector ................................................................... 9 2.2.3 Solar Air Flat‐Plate Collector ....................................................................... 10 2.2.4 Solar Wall Air Heating Solar Collector ........................................................ 11
3 SOLAR COLLECTOR LOCATION .............................................................................. 13 3.1 Geographical Location ...................................................................................... 14 3.2 Incident Angle ................................................................................................... 14 3.2.1 Methodology ............................................................................................... 15 3.2.2 Calculations ................................................................................................. 17
3.3 Solar Irradiance ................................................................................................. 24 3.3.1 Methodology ............................................................................................... 24 3.3.2 Calculations ................................................................................................. 25
3.4 Chapter Summary ............................................................................................. 29 4 HEAT TRANSFER ANALYSIS .................................................................................... 30 4.1 Evacuated Tube Collector Efficiency ................................................................... 30 4.1.1 Methodology ............................................................................................... 30 4.1.2 Calculations ................................................................................................. 38
4.2 Flat‐Plate Collector Efficiency ........................................................................... 40 4.2.1 Methodology ............................................................................................... 40 4.2.2 Calculations ................................................................................................. 43
vii
5 ECONOMIC ANALYSIS ............................................................................................ 47 5.1 Background ....................................................................................................... 47 5.2 Boiler Energy Analysis ....................................................................................... 49 5.3 Collector Energy Analysis .................................................................................. 49 5.4 Cost Analysis ..................................................................................................... 53 5.5 Chapter Summary ............................................................................................. 57
6 DISCUSSION OF RESULTS .............................................................................................. 58 6.1 Benefits of Installing Solar Collectors ............................................................... 58 6.2 Possible Collector Locations ............................................................................. 59 6.3 Evacuated Tube or Flat‐Plate? .......................................................................... 63
7 CONCLUSIONS AND RECOMMENDATIONS ........................................................... 65 7.1 Solar Collector Location .................................................................................... 65 7.3 Instantaneous Efficiency Calculators ................................................................ 66 7.4 Economic Analysis ............................................................................................. 66 7.5 Final Recommendations ................................................................................... 67
8 REFERENCES .......................................................................................................... 68 APPENDIX A – COLLECTOR SPECIFICATIONS ..................................................................... 70 APPENDIX B ‐ RETSCREEN DATA FOR WINNIPEG INT. AIRPORT ....................................... 83 APPENDIX C – OPTIMUM TILT ANGLES AND DIRECTIONS: JAN –DEC 2007 ..................... 85
viii
LIST OF FIGURES Figure 1: Earth's Energy Budget .......................................................................................... 3
Figure 2: Flat‐Plate Collector............................................................................................... 5
Figure 3: "Eco‐Village" Evacuated Tube Collector .............................................................. 6
Figure 4: Single Evacuated Tube Model .............................................................................. 7
Figure 5: Heat Pipe Schematic ........................................................................................... 8
Figure 6: Heat Transfer Schematic ...................................................................................... 8
Figure 7: EnerWorks Flat‐Plate Collector ............................................................................ 9
Figure 8: Flat‐Plate Collector Schematic ........................................................................... 10
Figure 9: Sunsiaray Northern Comfort Flat‐Plate Collector .............................................. 11
Figure 10: Solar Wall Schematic ....................................................................................... 12
Figure 11: Solar Geometry ................................................................................................ 13
Figure 12: Angle of Elevation of the Sun ‐ Sept 15/07 ...................................................... 19
Figure 13: Azimuth Angle of the Sun ‐ Sept 15/07 ........................................................... 21
Figure 14: Direct Radiation Received with a Tilt Angle of 45° ‐ Sept 15/07 ..................... 27
Figure 15: Varying Degrees of Tilt Angle ........................................................................... 28
Figure 16: Comparison of 40° ‐ 50° Tilt Angles ................................................................. 28
Figure 17: "Eco‐Village" Collector Schematic ................................................................... 30
Figure 18: Equivalent Thermal Circuit ............................................................................... 31
Figure 19: Evacuated Tube Efficiency Calculator .............................................................. 39
Figure 20: Flat‐Plate Collector Efficiency Calculator ......................................................... 45
Figure 21: Distribution of Powerhouse Boiler Usage........................................................ 47
Figure 22: University of Manitoba Campus Map .............................................................. 60
Figure 23: Hybrid Map of the University .......................................................................... 61
Figure 24: Rooftop Solar Collector Locations ................................................................... 62
ix
LIST OF TABLES
Table 1: Hour Angle ‐ Sept 15/07 ...................................................................................... 17
Table 2: Angle of Elevation of the Sun above the Eco‐Village ‐ Sept 15/07 ..................... 18
Table 3: Azimuth Angle of the Sun ‐ Sept 15/07 ............................................................... 20
Table 4: Direct Radiation Received by the Collector ‐ Sept 15/07 ................................... 26
Table 5: Summary of Location Results .............................................................................. 29
Table 6: Days of Operation in 2006 ‐ Boilers 5 and 6 ....................................................... 48
Table 7: Theoretical Collector Efficiencies ........................................................................ 50
Table 8: SRCC Heating Applications .................................................................................. 50
Table 9: Solar Radiation Values......................................................................................... 51
Table 10: Quantity of Collectors Needed.......................................................................... 52
Table 11: Collector Area Needed ...................................................................................... 52
Table 12: Cost to Substitute Boiler 5 with Solar Energy ................................................... 54
Table 13: Economic Analysis of a System of 10 Collectors ............................................... 56
Table 14: Rooftop Availability ........................................................................................... 62
Table 15: Optimum Tilt Angle ‐ January 2007 .................................................................. 86
Table 16: Optimum Tilt Angle ‐ February 2007 ................................................................ 86
Table 17: Optimum Tilt Angle ‐ March 2007..................................................................... 87
Table 18: Optimum Tilt Angle ‐ April 2007 ....................................................................... 87
Table 19: Optimum Tilt Angle ‐ May 2007 ........................................................................ 88
Table 20: Optimum Tilt Angle ‐ June 2007 ........................................................................ 88
Table 21: Optimum Tilt Angle ‐ July 2007 ......................................................................... 89
Table 22: Optimum Tilt Angle ‐ August 2007 .................................................................... 89
Table 23: Optimum Tilt Angle ‐ September 2007 ............................................................. 90
Table 24: Optimum Tilt Angle ‐ October 2007 .................................................................. 90
Table 25: Optimum Tilt Angle ‐ November 2007 .............................................................. 91
Table 26: Optimum Tilt Angle ‐ December 2007 .............................................................. 91
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 1 Heather King
1 INTRODUCTION
The following chapter introduces the reader to this report, with the aim of clarifying the
purpose, scope, and layout the following undergraduate thesis presents.
1.1 Purpose and Scope The purpose of this thesis is to examine and discuss the energy potential of installing solar
collectors at the University of Manitoba. The primary intention of this report is to
conclude on whether it would be beneficial for the University of Manitoba to install solar
collectors on campus. If installed, these collectors would either 1) provide hot water that
would tie into the university’s district heating system or 2) heat air that could be
circulated as a heating source through campus buildings.
The “eco‐village”, a straw‐bale barn located on the University of Manitoba’s Fort Garry
campus, currently has a single evacuated tube collector installed onto its south facing
wall. Plans to install two flat‐plate collectors, both a solar water collector and a solar air
collector, as well as a solar wall collector, are in the works. Using manufacturer data for
the evacuated tube collector and the solar water flat‐plate collector, theoretical
efficiencies have been calculated. The results from these analyses were then applied to a
hypothetically larger quantity of collectors in order to study the economic potential of
installing a system of solar collectors on campus.
The scope of this project is limited to evacuated tube collectors and solar water flat plate
collectors. Solar air collectors, such as the second flat plate collector and the solar wall
collector, and photo‐voltaic (PV) systems could also be analyzed in the same manner;
however time and equipment constraints limited this report to the two solar technologies
mentioned.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 2 Heather King
1.2 Layout of Thesis Following the introduction to this report, a basic overview of solar energy and solar
energy technologies is provided. The schematics of the evacuated tube collector that is
currently mounted on the “eco‐village” and the proposed solar water flat‐plate collector
are provided and discussed in detail. Brief overviews of the solar air collector and solar
wall collector that the university plans to install along with the solar water collectors are
also given, however the scope of this thesis limits the discussion of these solar air
collectors, and therefore no analysis will be carried out on these collectors at this time.
The first calculation to be discussed is that of the optimal collector location on campus in
regards to the sun. Methodology and calculations related to this objective are stated and
explained. A heat transfer analysis is then applied to both the evacuated tube system and
the flat‐plate collector system in order to determine the theoretical efficiencies of these
collectors.
Finally, an economic analysis is conducted on the proposed systems of collectors.
Discussion in this section includes the number of collectors needed, the proposed location
for these collectors on campus, and the potential energy they would provide in relation to
the amount of energy currently used to heat water on campus. A recommendation is
then presented on whether or not installing a system of evacuated tube collectors at the
University of Manitoba is economically and environmentally advantages.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 3 Heather King
2 REVIEW OF LITERATURE In order to provide the reader with an introductory lesson on solar collector technologies,
the following background section and literature review has been provided. This section
will discuss current solar technologies, with specific focus on the collectors that are to be
installed at the University of Manitoba’s “eco‐village”.
2.1 Solar Energy Solar energy is, in the simplest of descriptions, energy from the sun. This energy travels to
the earth in the form of electromagnetic radiation and is used to support virtually all life
on earth. Approximately 1367 W/m² of energy is available from the sun outside the
Earth’s atmosphere; however some of this energy is absorbed as it passes through the
Earth’s atmosphere. For instance, on a clear day, the amount of solar energy available
from the sun is in the region of 1000 W/m² [1].
The schematic below shows how the incoming solar radiation from the sun is divided as it
enters the Earth’s atmosphere [2].
Figure 1: Earth's Energy Budget
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 4 Heather King
As the diagram shows, the majority of the incoming solar energy is absorbed by land and
oceans; this is a large amount of free, renewable energy which could be converted to heat
energy or stored for later use.
Heat and light, direct forms of energy from the sun, and solar based resources account for
more than 99.9 percent of the available flow of renewable energy [3]. Solar based
resources include wind power, hydroelectricity, biomass and solar collectors. This thesis
report will further discuss solar energy collection by the use of solar collectors, such as
evacuated tube collectors and flat‐plate collectors. Further discussion on these different
types of collectors follows in the proceeding sections.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 5 Heather King
2.2 Solar Collectors A solar collector is a device that extracts energy from the incident radiation of the sun and
converts it into a more useful form of energy, such as heat. Solar collectors can be used
to heat air or water and are often seen atop of homes and industrial buildings. The image
below shows a flat plate collector mounted to the roof top of a house in Australia [4].
Figure 2: Flat‐Plate Collector
The two most commonly found solar collectors are flat plate collectors and evacuated
tube collectors. A flat‐plate collector generally consists of a weatherproofed and durable
insulated box, containing an absorber sheet and built in heat transfer pipes. The collector
is placed in the path of sunlight, which allows the radiation incident on the plate to heat
up the water within the heat transfer pipes, causing it to circulate through the system by
natural convection. This heated water is usually then accumulated in a storage tank
above the collector.
Evacuated tube collectors operate on the same general principle as the flat plate
collector; however these collectors use evacuated tubes to absorb the sun’s incident rays.
More detail is provided on evacuated tube collectors and flat‐plate collectors in the
following sections.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 6 Heather King
Evacuated tube collectors and flat‐plate collectors are referred to as solar water heating
devices. However, solar collector technology is not limited to water heating; solar air
heating technology is also commonly seen. Solar air heating collectors operate with the
same general principles as water heating collectors, the only difference between the two
technologies being that air is the medium to be heated, not water or a heat transfer fluid.
2.2.1 Solar Evacuated Tube Collector The image below shows the evacuated tube collector that is currently installed on the
“eco‐village”. This collector was manufactured by Apricus Solar Co. and is installed at a tilt
angle of 45° on the buildings south facing wall. Collector specifications for the Apricus
solar collector are given in Appendix A.
Figure 3: "Eco‐Village" Evacuated Tube Collector
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 7 Heather King
The evacuated tube collector works by absorbing radiation from the sun through a system
of 30 evacuated tubes. Each evacuated tube consists of two glass tubes made from
extremely strong borosilicate glass. The outer tube is transparent, which allows light rays
to pass through with minimal reflection and maximum absorption. The inner tube is
coated with a special selective coating, in this case Al‐N/Al, which gives the collector
excellent solar radiation absorption and minimal reflection properties. A vacuum is
created by fusing the top of these two tubes together and pumping out the air contained
in the space between the two layers of glass. A schematic of a single evacuate tube is
given below.
Figure 4: Single Evacuated Tube Model
As you can see from the above figure, a copper heat pipe is inserted into the inner
borosilicate tube. This heat pipe works as a condenser, based on this principle that liquids
boil at a lower temperature when the surrounding air pressure is decreased. The liquid
used in the heat pipe is purified water.
The heat pipes used in Apricus solar collectors have a boiling point of only 30°C, allowing
any liquid in the heat pipe to vaporize if the heat pipe is heated above 30°C. This water
vapor rapidly rises to the top of the heat pipe, transferring heat along the heat pipe and
exchanging it into the propylene glycol/water heat transfer fluid flowing in the manifold.
As the heat is lost at the top of the condenser, the water vapor condenses to form a liquid
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 8 Heather King
and returns to the bottom of the heat pipe to once again repeat the process. The heat
transfer fluid used in our application is a 50:50 propylene glycol water solution, in order to
achieve the lower freezing point necessary for northern winters. This process is
diagramed in Figure 5.
Figure 5: Heat Pipe Schematic
Figure 6: Heat Transfer Schematic
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 9 Heather King
The figure shown on the previous page illustrates the flow of heat through the evacuated
tube collector, from the incident solar energy from the sun collected by the evacuated
tubes, to the transfer of heat from the condenser into the water running through the
copper header. Once heated, the water in the copper header will be collected in a
storage tank for future distribution into the university’s district heating system.
2.2.2 Solar Water Flat‐Plate Collector
At the time of completion of this thesis, an EnerWorks solar water flat‐plate collector had
been ordered by the university, for installation in the “Eco‐Village”. The EnerWorks flat‐
plate collector is a high performance collector that boasts an absorbance rate of 94% of
the sun’s energy [14]. The figure below shows two EnerWorks flat‐plate collectors
mounted to a home in North America.
Figure 7: EnerWorks Flat‐Plate Collector
Solar hot water flat‐plate solar collectors follow the same general principle as an
evacuated tube collector; however, the heat transfer process within the collector is much
simpler than that of the evacuated tube collector. The diagram on the following page
illustrates how the flat‐plate collector works.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 10 Heather King
Figure 8: Flat‐Plate Collector Schematic
Energy from the sun strikes the flat, glazed surface of the collector, travelling through the
glass surface to be absorbed by the absorber plate. The heat obtained from this energy
absorption warms up the heat transfer fluid that is flowing through the collector. This
fluid flow enters the collector through an inlet pipe on one corner of the collector, travels
through the fluid flow tubes, and exits the collector through an outlet tube located at the
opposite corner of the collector. From the outlet tube, heat is transferred through a heat
exchanger into a water flow. This heated water is collected in a storage tank for future
use.
Detailed product specifications for the EnerWorks solar hot water flat‐plate collector to be
installed on at the “Eco‐Village” can be found in Appendix A.
2.2.3 Solar Air Flat‐Plate Collector
The solar air flat‐plate collector works with exactly the same principles are the solar water
flat‐plate collector, the one and only difference being that the fluid that circulates through
the collector is air, not a heat transfer fluid such as water or a propylene glycol solution.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 11 Heather King
A Sunsiaray Inc. Northern Comfort flat plate collector is to be installed at the “eco‐village”.
An image of this collector can be seen in Figure 9 below.
Figure 9: Sunsiaray Northern Comfort Flat‐Plate Collector
2.2.4 Solar Wall Air Heating Solar Collector
The solar wall is a solar system for heating or pre‐heating ventilation air for buildings. If
you were to look at a building which contained a solar wall, it would simply look like the
building was composed of conventional metal cladding. This metal cladding, however,
contains thousands of tiny perforations which allow fresh air to pass through on the way
to the building’s heating and ventilation (HVAC) system. As the air passes through these
holes into the building, it accumulates free heat from the cladding, which has been
warmed by the sun during the day. Ventilation fans installed inside the building create
negative pressure to draw air in through the perforations. Once the warmed up air has
passed through the solar wall it rises up to be collected into a distribution ducting system.
One of the biggest benefits of solar walls is that they even work at night, as the heat that
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 12 Heather King
would otherwise escape freely from the building is captured within the face of the metal
cladding and reused for ventilation pre‐heat.
The image below shows a schematic of the principle of solar wall operation [16].
Figure 10: Solar Wall Schematic
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 13 Heather King
3 SOLAR COLLECTOR LOCATION
The absolute value of solar radiation available for utilization in a particular site on Earth is
dependent on the relation between the location of the site and the location of the sun. In
order to maximize the amount of solar radiation available we must optimize the variables
in this relation.
The diagram below shows the variables that must be optimized [6]:
Figure 11: Solar Geometry
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 14 Heather King
Where:
A = the angle of elevation of the sun
Z = the azimuth angle of the sun
Φ = the azimuth angle from due south
θ = the incident angle of the collector
Ψ = the elevation angle
Σ = the tilt angle of the plane
In order to show the variation in optimal solar collector location throughout the year,
each variable has been calculated for the 1st of each month for the year 2007. Appendix C
shows the results of these calculations. Step by step calculations for September 15th,
2007 are shown in the proceeding sections.
3.1 Geographical Location The first variable that must be specified is the position coordinates of the solar collector.
For calculation purposes we will use the coordinates of the eco‐village.
The eco‐village has the following coordinates [5]:
Location Straw Bale BuildingService Street 1 SW University of Manitoba Winnipeg, MB Canada
Latitude 49.81° N
Longitude 97.13° W
Elevation Above Sea Level
238 m (781 ft)
Table 3.1 ‐ Location of the Eco‐Village
3.2 Incident Angle
The following section discusses the methodology for calculating the incident angle upon
the evacuated tube collector installed on the “eco‐village”. Sample calculations for
September 15th, 2007 are also included. The incident angle calculated in the following
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 15 Heather King
sections will apply to both evacuated tube and flat‐plate collectors mounted on the
University of Manitoba campus.
3.2.1 Methodology
The incident angle of the collector, θ, is the angle between the sun’s rays and the normal
to the plane surface of the collector. This angle is given by the following relation:
)])[cos()(sin(cos))(cos(sincos ZAA [3‐1]
By calculating the incident angle we can determine the optimum solar collector position
relative to the sun.
The following are the steps that must be followed to determine the incident angle of the
sun in relation to a solar collector located at the University of Manitoba. For example
purposes, calculations will be done for each step of the process for the date of September
15th, 2007.
Step 1: Calculate D, the angle of declination.
The angle of declination, D, represents the amount by which the Earth’s north polar axis is
tilted towards the sun. This value varies on a daily basis and can be approximated by the
following equation:
)284(
365
360sin45.23 nD [3‐2]
Where:
n = the nth day of the year
Step 2: Calculate H, the hour angle.
The hour angle, H, expresses the time of day in which one 24‐hour day is represented as
360 degrees of angle.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 16 Heather King
tH 15 [3‐3]
Where:
t = the time in hours (decimally) from solar noon
Step 3: Calculate A, the angle of elevation of the sun.
The angle of elevation of the sun above the eco‐village can be calculated, on an hourly
basis, by the following equation:
))(sin(sin))(cos)(cos(cossin LDLHDA [3‐4]
Where:
D = the declination angle, calculated above.
H = the hour angle, calculated above.
L = the latitude of the collector site.
The angle of elevation will be positive above the horizon and negative below the horizon.
Step 4: Calculate Z, the azimuth angle of the sun.
The azimuth angle of the sun, Z, can be calculated as follows:
A
HDZ
cos
sincossin
[3‐5]
Where:
D = the declination angle, calculated above.
H = the hour angle, calculated above.
A = the angle of elevation of the sun, calculated above.
The azimuth angle will be positive to the east and negative to the west, if measured from
the south.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 17 Heather King
Step 5: Determine the optimum tilt angle of the plane, Σ, and the optimum
direction of the angle of tilt, Φ, to maximize the incident angle, θ.
In order to maximize the amount of solar irradiance we want to get the incident angle
between the sun's rays and the normal to the plane surface of the collector to as close to
zero as possible.
Using equation [3‐1], where )])[cos()(sin(cos))(cos(sincos ZAA , we can see
that, in order to make equal to zero, we must set cos equal to 1. By setting cos
equal to 1, the solver function in Excel can be used to optimize Σ, the tilt angle of the
plane, and Φ, the direction of the angle of tilt.
3.2.2 Calculations
The following computations use the methodology
outlined in section 3.2.1 to optimize Σ, the tilt
angle of the plane, and Φ, the direction of the
angle of tilt, for September 15th, 2007.
Complete, tabulated data is available in
Appendix C for the 1st of every month from
January 2007 to December 2007.
Step 1: Calculate D, the angle of declination.
September 15th, 2007 is the 258th day of the year,
therefore 258n .
Using equation [3‐2],
)]25.365
258284(2[45.23
D
3616.23D [degrees]
Table 1: Hour Angle ‐ Sept 15/07
Hour t H
12:00 AM 10.60 159.00
1:00 AM 11.60 174.00
2:00 AM 12.60 189.00
3:00 AM 13.60 204.00
4:00 AM 14.60 219.00
5:00 AM 15.60 234.00
6:00 AM 16.60 249.00
7:00 AM 17.60 264.00
8:00 AM 18.60 279.00
9:00 AM 19.60 294.00
10:00AM 20.60 309.00
11:00AM 21.60 324.00
12:00 PM 22.60 339.00
1:00 PM 23.60 354.00
2:00 PM 0.60 9.00
3:00 PM 1.60 24.00
4:00 PM 2.60 39.00
5:00 PM 3.60 54.00
6:00 PM 4.60 69.00
7:00 PM 5.60 84.00
8:00 PM 6.60 99.00
9:00 PM 7.60 114.00
10:00 PM 8.60 129.00
11:00 PM 9.60 144.00
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 18 Heather King
Step 2: Calculate H, the hour angle.
On September 15th, 2007, solar noon occurred at 1:24 pm [7]. Using equation [3‐3], the
hour angle can be calculated. The table to the on the previous page shows the values of
the hour angle, in degrees, for each hour of the 24‐hour clock for September 15th, 2007.
Step 3: Calculate A, the angle of elevation of the sun.
From Table 1, the latitude of the eco‐village is 49.81° N.
By substituting values for D, H, and L into equation [3‐4] we can obtain a numerical result
for sinA, and thus are able to calculate the angle of elevation of the sun above the eco‐
village, A for September 15th, 2007. The following table shows these results:
Hour sinA A
12:00 AM ‐0.57 ‐34.92
1:00 AM ‐0.61 ‐37.72
2:00 AM ‐0.61 ‐37.40
3:00 AM ‐0.56 ‐34.02
4:00 AM ‐0.47 ‐28.14
5:00 AM ‐0.35 ‐20.46
6:00 AM ‐0.20 ‐11.63
7:00 AM ‐0.04 ‐2.17
8:00 AM 0.13 7.49
9:00 AM 0.29 16.97
10:00 AM 0.44 25.81
11:00 AM 0.55 33.45
12:00 PM 0.63 39.17
1:00 PM 0.67 42.13
2:00 PM 0.67 41.79
3:00 PM 0.62 38.22
4:00 PM 0.53 32.05
5:00 PM 0.41 24.12
6:00 PM 0.26 15.11
7:00 PM 0.10 5.56
8:00 PM ‐0.07 ‐4.09
9:00 PM ‐0.23 ‐13.46
10:00 PM ‐0.38 ‐22.10
11:00 PM ‐0.49 ‐29.48
12:00 AM ‐0.57 ‐34.92
Table 2: Angle of Elevation of the Sun above the Eco‐Village ‐ Sept 15/07
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 19 Heather King
The graph shown on the following page shows how the sun rises and falls during the 24‐
hour period of September 15th, 2007. Sunrise occurs when the angle of elevation of the
sun crosses the zero axes from a negative value to a positive one. Likewise, sunset occurs
when the angle of elevation of the sun crosses the zero axes from a positive value to a
negative value.
Figure 12: Angle of Elevation of the Sun ‐ Sept 15/07
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 20 Heather King
Step 4: Calculate Z, the azimuth angle of the sun.
By substituting values for D, H, and A into equation [3‐5] we can obtain a numerical result
for sinZ, and thus are able to calculate the azimuth angle of the sun, Z, for September 15th,
2007.
The table below shows these results.
Table 3: Azimuth Angle of the Sun ‐ Sept 15/07
Hour sinZ Z
12:00 AM ‐0.44 ‐25.90
1:00 AM ‐0.13 ‐7.59
2:00 AM 0.20 11.35
3:00 AM 0.49 29.37
4:00 AM 0.71 45.49
5:00 AM 0.86 59.63
6:00 AM 0.95 72.26
7:00 AM 0.99 83.98
8:00 AM 1.00 84.53
9:00 AM 0.99 81.89
10:00 AM 0.86 59.61
11:00 AM 0.70 44.75
12:00 PM 0.46 27.51
1:00 PM 0.14 8.10
2:00 PM ‐0.21 ‐12.10
3:00 PM ‐0.52 ‐31.15
4:00 PM ‐0.74 ‐47.90
5:00 PM ‐0.89 ‐62.34
6:00 PM ‐0.99 ‐81.89
7:00 PM ‐1.00 ‐86.84
8:00 PM ‐0.99 ‐81.68
9:00 PM ‐0.94 ‐69.82
10:00 PM ‐0.84 ‐56.95
11:00 PM ‐0.67 ‐42.43
12:00 AM ‐0.44 ‐25.90
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________________________________________________________________________ December 4th, 2007 21 Heather King
Figure 13: Azimuth Angle of the Sun ‐ Sept 15/07
The graph shown above shows the oscillation of the azimuth angle of the sun for a 24‐
hour period on September 15th, 2007. It can be seen that the azimuth angle is at a
maximum to the east at sunrise, as the sun rises from the east. Likewise, the azimuth
angle is at a maximum to the west at sunset, as the sun sets to the west.
Step 5: Determine the optimum tilt angle of the plane, Σ, and the optimum
direction of the angle of tilt, Φ, to maximize the incident angle, θ.
Using equation [3‐1], the values for A, the angle of elevation of the sun, and Z, the
azimuth angle of the sun, which were calculated above, and the solver function in
Microsoft Excel, it was possible to determine optimum values for the tilt angle of the
plane, Σ, and the direction of the angle of tilt, Φ. The optimum values for Σ and Φ for
September 15th, 2007 are shown in the table on the following page.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 22 Heather King
Hour Opt Σ Opt φ Max cosθ Max θ
12:00 AM 124.92 ‐25.90 1.00 0.00
1:00 AM 127.72 ‐7.59 1.00 0.00
2:00 AM 127.40 11.35 1.00 0.00
3:00 AM 124.02 29.37 1.00 0.00
4:00 AM 118.14 45.49 1.00 0.00
5:00 AM 110.46 59.63 1.00 0.00
6:00 AM 101.63 72.26 1.00 0.00
7:00 AM 92.17 83.98 1.00 0.00
8:00 AM 82.51 84.53 1.00 0.00
9:00 AM 73.03 81.89 1.00 0.00
10:00 AM 64.19 59.61 1.00 0.00
11:00 AM 56.55 44.75 1.00 0.00
12:00 PM 50.83 27.51 1.00 0.00
1:00 PM 47.87 8.10 1.00 0.00
2:00 PM 48.21 ‐12.10 1.00 0.00
3:00 PM 51.78 ‐31.15 1.00 0.00
4:00 PM 57.95 ‐47.90 1.00 0.00
5:00 PM 65.88 ‐62.34 1.00 0.00
6:00 PM 74.89 ‐81.89 1.00 0.00
7:00 PM 84.44 ‐86.84 1.00 0.00
8:00 PM 94.09 ‐81.68 1.00 0.00
9:00 PM 103.46 ‐69.82 1.00 0.00
10:00 PM 112.10 ‐56.95 1.00 0.00
11:00 PM 119.48 ‐42.43 1.00 0.00
Table 3.4 – Optimizing the Tilt Angle and Direction of the Tilt Angle of the Evacuated Tube Collector on the “Eco‐Village”
If we were to plot the curves of the optimum tilt angle and the direction of the tilt angle
as the day progressed, we would see curves similar to the ones plotted on the following
page for September 15th, 2007.
The first plot shows the variation in optimum tilt angle. Please note that this graph only
shows the values of the tilt angle that are equal to or less than 90°. An angle greater than
90° occurs when the sun is below the horizon; a time when irradiance levels are zero.
The second plot shown is that of the change in the optimum direction of the angle of tilt
as the day progresses. It can be seen from the plot that the optimum tilt direction is 0°;
the time at which the optimum direction of the collector is to face directly south.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 23 Heather King
Figure 3.4 – Optimum Tilt Angle: Sept 15/07
Figure 3.5 – Optimum Direction of the Tilt Angle: Sept 15/07
Determining the most beneficial, constant tilt angle for the evacuated tube collector
however is not as simple. The optimization of this constant angle will be discussed to
further depth in the following section.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 24 Heather King
3.3 Solar Irradiance
The following section discusses the methodology for calculating the solar irradiance
directed upon a collector installed at the “eco‐village”. Sample calculations for September
15th, 2007 are also included.
3.3.1 Methodology
The amount of solar irradiance that reaches either the installed evacuated collector or the
proposed flat plate collector throughout the day is related to the incident angle of the
collector, θ. The incident angle of the collector is the angle that a direct beam of light
from the sun makes with the line normal to the plane of the collector, as seen in Figure
3.1.
As discussed in Section 3.2, in order to receive optimum solar radiation on the plane of
the collector, this angle should be set as close to zero as possible. However, due to the
fact that the sun is not a stationary object and rises and falls throughout the day, it is not
possible to keep this angle constantly at zero unless a pivoting solar collector was
probable, which is not the case. Thus, in order to optimize the amount of radiation
received by the collector, we must determine the tilt angle that accumulates the most
incident rays onto the plate throughout the day.
The following two formulas are utilized in the proceeding section to determine the best
tilt angle for the evacuated tube collectors. The first equation that we will use is Equation
[3‐1],
)])[cos()(sin(cos))(cos(sincos ZAA [3‐1]
In this equation A and Z are known values. The direction of the tilt angle, Φ, will be set to
0°, as discussed in Section 3.2.2. This leaves us with two variables, Σ, the tilt angle of the
plane, and θ, the incidence angle of the collector.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 25 Heather King
The second formula utilized relates the daily solar radiation values for Winnipeg to the
incidence angle of the collector [6]. This equation allows us to determine the irradiance
upon the collector as it varies throughout the day.
cosNb II [3‐6]
Where:
bI = the direct radiation received by the collector
NI = the normal terrestrial solar radiation received at ground level
Values for NI can be obtained from RETScreen. The RETScreen data for Winnipeg
International Airport is included in Appendix B.
3.3.2 Calculations The evacuated tube collector that is currently installed on the “eco‐village” is mounted so
that it faces directly south and has a tilt angle of 45°. Therefore, Σ = 45° and Φ = 0°. From
the RETScreen data in Appendix B we can see that, for the month of September, the daily
solar radiation, NI , is 3.61 kWh/m²/d or 0.300833 kWh/m²/h. Using Equations [3‐1] and
[3‐6] with the values of Σ, Φ, and NI noted above and the values of A and Z as calculated
in the previous sections, a curve of the direct radiation received by the collector, bI ,
throughout the day can be plotted for September 15th, 2007. This plot can be seen on
page 27.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 26 Heather King
Hour A Z Σ φ cosθ θ In
[kWh/m²/h] Ib
[kWh/m²/h]
12:00 AM ‐34.92 ‐25.90 45.00 0 0.116774 83.29406 0 0
1:00 AM ‐37.72 ‐7.59 45.00 0 0.12188 82.99939 0 0
2:00 AM ‐37.40 11.35 45.00 0 0.121307 83.03244 0 0
3:00 AM ‐34.02 29.37 45.00 0 0.115095 83.39089 0 0
4:00 AM ‐28.14 45.49 45.00 0 0.103666 84.04967 0 0
5:00 AM ‐20.46 59.63 45.00 0 0.0878 84.96296 0 0
6:00 AM ‐11.63 72.26 45.00 0 0.068577 86.06774 0 0
7:00 AM ‐2.17 83.98 45.00 0 0.047307 87.28847 0 0
8:00 AM 7.49 84.53 45.00 0 0.15901 80.85058 0.300833 0.05
9:00 AM 16.97 81.89 45.00 0 0.301765 72.43636 0.300833 0.09
10:00 AM 25.81 59.61 45.00 0 0.629878 50.95885 0.300833 0.19
11:00 AM 33.45 44.75 45.00 0 0.808808 36.02035 0.300833 0.24
12:00 PM 39.17 27.51 45.00 0 0.93284 21.11805 0.300833 0.28
1:00 PM 42.13 8.10 45.00 0 0.993521 6.525449 0.300833 0.30
2:00 PM 41.79 ‐12.10 45.00 0 0.986717 9.349024 0.300833 0.30
3:00 PM 38.22 ‐31.15 45.00 0 0.912891 24.09208 0.300833 0.27
4:00 PM 32.05 ‐47.90 45.00 0 0.777073 39.00663 0.300833 0.23
5:00 PM 24.12 ‐62.34 45.00 0 0.58852 53.94791 0.300833 0.18
6:00 PM 15.11 ‐81.89 45.00 0 0.280605 73.70371 0.300833 0.08
7:00 PM 5.56 ‐86.84 45.00 0 0.107324 83.8389 0.300833 0.03
8:00 PM ‐4.09 ‐81.68 45.00 0 0.051657 87.03896 0 0
9:00 PM ‐13.46 ‐69.82 45.00 0 0.07263 85.83495 0 0
10:00 PM ‐22.10 ‐56.95 45.00 0 0.09128 84.76275 0 0
11:00 PM ‐29.48 ‐42.43 45.00 0 0.106337 83.89582 0 0
Table 4: Direct Radiation Received by the Collector ‐ Sept 15/07
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 27 Heather King
The plot below shows how the incident radiation upon the evacuated tube collector
increase as the day moves towards mid‐day, then falls again as the afternoon progresses.
Figure 14: Direct Radiation Received with a Tilt Angle of 45° ‐ Sept 15/07
One of the aims of this thesis was to determine the optimum tilt angle; the angle in which
the most irradiance would be collected by the evacuated tube collector throughout the
period of one day. In order to compare the potentials of different tilt angles, the above
calculations were done while varying the tilt angle, Σ. As with the example above, the
direction of tilt, Φ, was set to 0°.
The plot on the following page shows the direct radiation received by the evacuated tube
collector, bI , as the day of September 15th, 2007 progresses for tilt angles of 5°, 15°, 25°,
35°, 45°, 55° and 65°. From this it can be concluded that the optimum tilt angle is 50°,
however any tilt value in the range of 45° to 55° would produce optimal results.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 28 Heather King
Figure 15: Varying Degrees of Tilt Angle
Figure 16: Comparison of 40° ‐ 50° Tilt Angles
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 29 Heather King
3.4 Chapter Summary Analysis on the location of the sun with regards to the evacuated tube collector currently
installed on the “eco‐village” has allowed me to determine the optimum mounting
location for an evacuated tube or flat plate collector.
The following table summarizes these results:
Location Flat, horizontal surface, such as a roof top or an empty field.
Tilt Angle, Σ 50°
Direction of Tilt Angle, Φ 0° (South Facing)
Table 5: Summary of Location Results
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________________________________________________________________________ December 4th, 2007 30 Heather King
4 HEAT TRANSFER ANALYSIS
The following section examines the transfer of heat through the Apricus evacuated tube
collector and the EnerWorks flat‐plate collector. A heat transfer model is created for each
collector and the overall efficiencies of the individual collectors are determined.
4.1 Evacuated Tube Collector Efficiency
The proceeding section discusses and outlines the methodology used to calculate the
working efficiency of the Apricus evacuated tube collector that is installed at the “eco‐
village” at the University of Manitoba. In order for these calculations to be accurately
applied, the necessary data must be acquired from the solar collector system, through the
use of data acquisition equipment.
4.1.1 Methodology
Figure 17 below shows a schematic of a single evacuated tube, comparable to one of the
tubes used in the collector currently mounted at the “eco‐village”. This tube consists of
an outer glass cover which acts as an envelope around a fin plate, which is selectively
coated and attached to a heat pipe absorber. The energy obtained from the solar
radiation incident on the collector travels as heat and is transferred from the heat pipe
evaporator fluid to the heat pipe condenser, and finally to the fluid flowing through the
manifold.
Figure 17: "Eco‐Village" Collector Schematic
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________________________________________________________________________ December 4th, 2007 31 Heather King
The heat flow through the evacuated tube to the manifold can be modeled as a thermal
circuit. This model allows us to simplify the system for use in the following calculations.
Shown below is the thermal circuit equivalent of an evacuated tube [8], as seen in the
“eco‐village” solar collector.
Figure 18: Equivalent Thermal Circuit
In the above diagram the nomenclature is as follows:
icA , = collector tube inner diameter times collector length
ocA , = collector tube outer diameter times collector length
ihA , = heat pipe inner diameter times collector length
ohA , = heat pipe outer diameter times collector length
och , = heat transfer coefficient of the outer surface of the collector tube
hmh = heat transfer coefficient between the heat pipe fluid and the manifold
fluid
aT = ambient air temperature
cT = mean temperature of the collector tube
hT = mean temperature of the heat pipe
ifT , = temperature of the fluid in the manifold before heating occurs
effI = the effective solar radiation on the collector
LQ.
= thermal energy lost to the environment
hmQ.
= thermal energy transferred from the heat pipe fluid to the manifold fluid
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________________________________________________________________________ December 4th, 2007 32 Heather King
LU = the overall heat loss coefficient
= the product of the absorptivity and transmitivity of the glass tube
Assuming steady‐state conditions, Norton (1992) and Tiwari (2002) describe the rate of
useful energy produced by the evacuated tube collector, in Watts, as:
)]()[(,
.
afiLeffocru TTUIAFQ [4‐1]
Where:
rF , the flow rate factor, is equal to:
)1()(
1
,
,
ph
L
ohhm
ocLr
h
U
Ah
AUF [4‐2]
And,
phh = heat transfer coefficient between the fin plate and the heat transfer
pipe Once the rate of useful thermal energy is determined, the efficiency of the evacuated
tube collector can be calculated as:
oceff
u
AI
Q
,
.
[4‐3]
The following outline the steps taken to calculate the efficiency of the evacuated tube
collector installed at the “eco‐village”:
Step 1: Calculate hc,o, the heat transfer coefficient of the outer surface of the
collector tube.
The heat transfer coefficient of the collector tube can be calculated as:
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 33 Heather King
ihoc
acaccoc AA
TTTTVh
,,
22
,
))((8.37.5
[4‐4]
Where:
c = emissivity of the glass collector tube
= Stefan‐Boltzmann constant = 5.67 x 10‐8 W/m²∙K4
V = mean velocity of the air flowing over the evacuated tube collector
Step 2: Calculate hc,i, the heat transfer coefficient of the evacuated envelope
(inner surface of the collector tube).
The heat transfer coefficient of the inner surface of the collector tube can be calculated
as:
)ln(2 ,
,,,
ic
ocoh
cic
DDDk
h [4‐5]
Where:
ocD , = outer diameter of the collector tube
icD , = inner diameter of the collector tube
ohD , = outer diameter of the heat transfer pipe
ck = thermal conductivity of the glass collector tube
Step 3: Calculate hh,o, the heat transfer coefficient of the outer surface of the heat
transfer pipe.
The heat transfer coefficient of the outer surface of the heat transfer pipe can be
calculated as:
)()1(1)1(
))((
,
,
22
,
ic
ih
c
c
evach
h
chchoh
AA
F
TTTTh
[4‐6]
Where:
c = emissivity of the collector tube
h = emissivity of the heat transfer pipe
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________________________________________________________________________ December 4th, 2007 34 Heather King
evacF = the shape factor between the selectively coated outer surface of the
heat pipe and the inner surface of the collector tube
Step 4: Determine the overall heat loss coefficient of the collector, LU .
The overall heat loss coefficient of the collector can be calculated as:
ocicoh
L
hhh
U
,,,
1111
[4‐7]
Where the heat transfer coefficients ohh , , ich , and och , were calculated in the previous
three steps.
Step 4: Calculate hhm, the heat transfer coefficient between the heat pipe
condenser and the fluid flowing through the manifold.
The heat transfer coefficient between the heat pipe condenser and the fluid flowing
through the manifold can be calculated as external flow over a cylinder in cross flow. This
heat transfer can be expressed as:
h
ocondD
hm k
DNuh , [4‐8]
Where:
ocondD , = outer diameter of the heat transfer condenser
hk = thermal conductivity of the copper condenser
DNu = Nusselt number of the fluid flowing through the manifold
In order to calculate DNu , the Nusselt number of the fluid flowing through the manifold,
we first need to calculate the Reynolds number of that fluid.
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________________________________________________________________________ December 4th, 2007 35 Heather King
The Reynolds number, mRe , can be calculated as follows:
focond
fm D
m
,
4Re
[4‐9]
Where:
fm
= the flow rate of the fluid through the manifold
f = dynamic viscosity of the fluid within the heat pipe
Once the Reynolds number of the heat transfer fluid within the manifold has been
determined, the value can be substituted into the following Churchill and Bernstein [15]
correlation for a cylinder in cross flow:
5/48/5
4/13/2
3/12/1
282000
Re1
]Pr)/4.0(1[
PrRe62.03.0
DmD
DNu [4‐10]
Where:
mPr = Prandtl number of the heat transfer fluid in the manifold
Substituting the value for the average Nusselt number obtained by the above correlation
into equation [4‐8], the heat transfer coefficient between the heat pipe condenser and
the heat transfer fluid flowing through the manifold can be calculated.
Step 5: Calculate phh , the heat transfer coefficient between the fin plate and the
heat transfer pipe fluid. Flow between the fin plate and the heat transfer pipe fluid can be considered as flow over
a flat plate. This flow will be assumed to be laminar.
The first step to calculating this value is to determine the Reynolds number of the liquid
within the heat transfer pipe.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 36 Heather King
This can be done by using the following relations:
LU
pRe [4‐11]
Where:
U = the velocity of the liquid within the heat transfer pipe
L = the length of the heat transfer pipe
= the kinematic viscosity of the liquid in the heat transfer pipe.
Please note that since the velocity of the liquid within the heat transfer pipe, in this case
purified water, is not known, we will assume it to be very small (~0.0001 m/s).
Once the Reynolds number has been determined, the following relation can be used to
find the heat transfer coefficient between the fin and the heat transfer pipe.
L
kh finpp
ph
3/12/1 PrRe664.0 [4‐12]
Where:
pPr = the Prandtl number of the fluid within the heat transfer pipe
fink = the thermal conductivity of the fin plate
Step 6: Calculate rF , the flow rate factor of the evacuated tube collector.
Using equation [4‐2], as shown below, and the values calculated above for LU , hmh , and
phh , the flow rate factor of the evacuated tube collector can be calculated.
)1()(
1
,
,
ph
L
ohhm
ocLr
h
U
Ah
AUF [4‐2]
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________________________________________________________________________ December 4th, 2007 37 Heather King
Step 7: Calculate
Q , the rate of useful energy produced by the collector.
Using equation [4‐1], as shown below, and the values calculated above for rF and LU ,
the rate of useful energy produced by the evacuated tube collector can be calculated.
)]()[(,
.
afiLeffocru TTUIAFQ [4‐1]
Step 8: Calculate , the instantaneous efficiency of the evacuated tube collector.
Using equation [4‐3], as shown below, and the value of uQ
calculated in step 7, the
instantaneous efficiency of the evacuated tube can be calculated.
oceff
u
AI
Q
,
.
[4‐3]
The eight steps described in the preceding section can be used to calculate the
instantaneous efficiency of the Apricus evacuated tube solar water collector that is
currently installed at the University of Manitoba’s “eco‐village”. As an alternative to
manually calculating each step of the sequence, a Microsoft Excel spreadsheet has been
derived which calculates the instantaneous efficiency of the Apricus evacuated tube
collector based on the variable input parameters of the solar collector system (mass flow
rate, wind velocity, collector temperature, ambient air temperature and manifold fluid
temperature). This program can be found in the disk attached to the appendix of this
thesis report.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 38 Heather King
4.1.2 Calculations As previously mentioned, a Microsoft Excel spreadsheet was created to calculate the
instantaneous efficiency of the Apricus evacuated tube collector. This spreadsheet takes
seven variable parameter inputs and, along with the known collector and heat transfer
fluid parameters, calculates the efficiency of the Apricus evacuated tube collector.
The seven variable inputs necessary for the spreadsheet to calculate the efficiency are:
1. The temperature of the ambient air surrounding the collector, in degrees K.
2. The temperature of the heat transfer fluid entering the manifold, in degrees K.
3. The mean temperature of the collector tube, in degrees K.
4. The mean temperature of the heat pipe, in degrees K.
5. The mass flow rate, in kg/s, of the liquid within the heat pipe.
6. The velocity of the air flow over the collector, in m/s.
7. The month of the year. A scroll down menu is available for the user to choose the
month of the year that the preceding three parameters were collected. Once the
month of the year is known, the spreadsheet can calculate the appropriate
incident radiation value received by the collector.
Unfortunately, since the data acquisition equipment necessary to log the required inputs
from the Apricus evacuated tube collector have not been received at the “eco‐village” at
this time, no data readings were available for the input values in order to run a test of the
program. In lieu of entering obtained data as the input values, a theoretical calculation
will be applied to test to efficiency calculator program.
For this theoretical calculation, the following input parameters will be used:
1. aT = 293 K = 20 °C 5.
m = 0.0001 kg/s
2. fT = 288 K = 15 °C 6. V = 0. 1 m/s
3. tubeT = 298 K = 25 °C 7. Month of the Year = September
4. pipeT = 303 K = 30 °C
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 39 Heather King
The image below shows the results of this calculation using the excel spreadsheet:
Apricus Evacuated Tube Collector Efficiency Calculator
Required Inputs: Ambient Air Temperature: 293 [K]
Temperature of Fluid flowing into the Manifold: 288 [K]
Mean Temperature of the Collector Tube: 298 [K]
Mean Temperature of the Heat Transfer Pipe: 303 [K]
Air Flow Velocity over the Collector: 0.1 [m/s]
Flow Rate of Heat Transfer Fluid in the Heat Pipe: 0.0001 [kg/s]
Month of the Year: September
Known Collector Parameters:
Collector Tube Length: 1.8000 [m]
Heat Pipe Length: 1.8000 [m]
Collector Tube Outer Diameter: 0.0580 [m]
Collector Tube Inner Diameter: 0.0480 [m]
Heat Pipe Outer Diameter: 0.0080 [m]
Heat Pipe Inner Diameter: 0.0073 [m]
Condenser Outer Diameter: 0.0200 [m]
Shape Factor between Heat Pipe and Collector Surfaces: 0.5000
Emissivity of the Glass Collector Tube: 0.9250
Emissivity of the Copper Heat Transfer Pipe: 0.0500
Stefan‐Boltzmann Constant: 5.67E‐08 [W/m²∙K^4]
Thermal Conductivity of the Copper Fin Plate: 4.01E+02 [W/m‐K]
Thermal Conductivity of the Condenser: 4.01E+02 [W/m‐K]
Thermal Conductivity of the Glass Collector Tube: 1.4 [W/m‐K]
Heat Transfer Fluid Kinematic Viscosity: 0.00813 [N∙s/m²]
Prandtl Number of Manifold Heat Transfer Fluid: 1.39615
Prandtl Number of Heat Pipe Fluid: 5.5
Effective Radiation Incident on Collector: 285.9 [W/m²]
Velocity of Liquid in the Heat Transfer Pipe: 0.001 [m/s]
Kinematic Viscosity of Liquid in the Heat Transfer Pipe: 8.33E‐07 [m²/s] Collector Absorptance: 0.94
Calculated Collector Parameters:
Collector Tube Outer Surface Area: 0.1044 [m²]
Collector Tube Inner Surface Area: 0.0864 [m²]
Heat Transfer Pipe Inner Surface Area: 0.0131 [m²]
Condenser Outer Surface Area: 0.0006 [m²]
Collector Tube Outer Surface Heat Transfer Coefficient: 1.4466 [W/m²·K] Collector Tube Inner Surface Heat Transfer Coefficient: 1849.4837 [W/m²·K]
Heat Transfer Pipe Outer Surface Heat Transfer Coefficient: 13222.1196 [W/m²·K] Collector Overall Heat Loss Coefficient: 1.4453 [W/m²∙°C]
Manifold Heat Transfer Fluid Reynolds Number: 0.7831
Manifold Heat Transfer Fluid Nusselt Number: 0.8604
Heat Transfer Coefficient Between Heat Pipe Condenser and Manifold Fluid: 4.2915E‐05 [W/m²·K] Reynolds Number of Fluid in the Heat Transfer Pipe: 2160.0864
Heat Transfer Coefficient Between Fin Plate and Heat Pipe Fluid: 12135.6508 [W/m²·K] Flow Rate Factor: 0.7899
Rate of Useful Energy from the Collector: 22.7572 [W]
Efficiency: 76.24 %
Figure 19: Evacuated Tube Efficiency Calculator
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 40 Heather King
4.2 Flat‐Plate Collector Efficiency
The proceeding section follows the same general principles in describing the methodology
and calculations used to determine the working efficiency of the EnerWorks flat‐plate
collector, as were used to determine the working efficiency of the Apricus evacuated tube
collector. Variation, however, can be seen in the formulae utilized for the flat‐plate
collector from those used in analyzing the efficiency of the evacuated tube collector.
As mentioned in the previous section, the necessary data must be acquired for the flat‐
plate system before any calculations can be carried out.
4.2.1 Methodology
Assuming steady‐state conditions and pump forced flow, Norton (1992) and Tiwari (2002)
describe the rate of useful energy from the collector, in Watts, as:
)]()[(.
afLeffcRu TTUIAFQ [4‐10]
Where:
cA = flat plate collector area
RF , the collector heat removal factor, is equal to:
f
LC
Lc
fR
mC
FUA
UA
CmF
'
exp1 [4‐11]
'F , the collector efficiency factor, is equal to:
))(
()(
1'
FDWD
W
hD
WUF
L
[4‐12]
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 41 Heather King
F, the fin efficiency, is equal to:
2/)]([
2/)](tanh[
DWm
DWmF
[4‐13]
k
Um L [4‐14]
In which,
W = distance between centre of two tubes
D = outer diameter of the tube
k = thermal conductivity of the tube
δ = thickness of the tube
h = convective heat transfer coefficient from the inner tube to the fluid
Once the rate of useful thermal energy is determined, the efficiency of the flat‐plate
collector can be calculated as:
ceff
u
AI
Q.
[4‐15]
Unlike with evacuated tube collectors, the overall heat loss coefficient of the flat‐plate
collector, UL, is a known value. Tiwari and Ghosal (2005) state that the value of the overall
heat transfer coefficient for a flat plate collector with a single glass cover at an inclination
angle, in an operating range of ambient to 70°C, can be considered as UL = 7.5 W/m² °C.
Therefore, the only variable left to calculate is h, the convective heat transfer coefficient
from the inner tube to the fluid. The steps needed to calculate this value are shown on
the next page.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 42 Heather King
Step 1: Determine DRe , the Reynolds number of the heat transfer fluid within
the tube.
The Reynolds number within the inner heat transfer tubes of the flat‐plate collector can
be determined using the following formula:
D
mD
4
Re [4‐16]
Where:
m = flow rate of the heat transfer fluid through the heat transfer tubes
= viscosity of the heat transfer fluid
Step 2: Determine the Nusselt number of the heat transfer fluid within the tube.
Fluid flow within the inner tubes of the flat‐plate collector can be described as laminar
internal flow. In a circular tube that is characterized by a uniform surface heat flux and
laminar, fully developed flow, the Nusselt number is a constant; it is independent of DRe ,
Pr, and axial location. Therefore,
36.4Nu [4‐17]
Step 3: Determine h, the convective heat transfer coefficient from the inner tube
to the heat transfer fluid.
The convective heat transfer coefficient from the inner tube to the heat transfer fluid can
be determined by combining equation [4‐17] and the following relation.
36.4k
hDNuD [4‐18]
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 43 Heather King
Once the heat transfer coefficient between the inner tube and the heat transfer fluid has
been determine, the fin efficiency, collector efficiency factor, and collector heat removal
factor can be determined. This then allows for the rate of useful energy from the
collector to be calculated, by utilizing equation [4‐10], as shown.
)]()[(.
afLeffcRu TTUIAFQ [4‐10]
The instantaneous efficiency of the EnerWorks flat‐plate solar water collector can thus be
determined by applying equation [4‐15]:
ceff
u
AI
Q.
[4‐15]
As an alternative to manually calculating each step of the sequence, a Microsoft Excel
spreadsheet has been derived which calculates the instantaneous efficiency of the
EnerWorks flat‐plate collector based on the variable input parameters of the solar
collector system (mass flow rate, ambient air temperature, and heat transfer fluid
temperature). This program can be found in the disk attached to the appendix of this
thesis report.
4.2.2 Calculations As mentioned above, a Microsoft Excel spreadsheet was created to calculate the
instantaneous efficiency of the EnerWorks flat‐plate collector. This spread sheet takes
four variable parameter inputs and, along with the known collector and heat transfer fluid
parameters, calculates the instantaneous efficiency of the EnerWorks flat‐plate collector.
The four variable inputs necessary for the spreadsheet to calculate the efficiency are:
1. The temperature of the ambient air surrounding the collector, in degrees K.
2. The temperature of the heat transfer fluid entering the collector, in degrees K.
3. The mass flow rate, in kg/s, of the heat transfer fluid entering the collector.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 44 Heather King
4. The month of the year. A scroll down menu is available for the user to choose the
month of the year that the preceding three parameters were collected. Once the
month of the year is known, the spreadsheet can calculate the appropriate
incident radiation value received by the collector.
Unfortunately, since the EnerWorks flat‐plate collector and the necessary data acquisition
equipment, have not been received and installed at the “eco‐village” at this time, no data
readings were available for the input values in order to run a test of the program. In lieu
of entering obtained data as the input values, a theoretical calculation will be applied to
test to efficiency calculator program.
For this theoretical calculation, the following input parameters will be used:
1. aT = 293 K = 20 °C 3.
m = 0.0001 kg/s
2. fT = 288 K = 15 °C 4. Month of the Year = September
The image on the next page shows the results of this calculation using the excel
spreadsheet.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 45 Heather King
EnerWorks Flat-Plate Collector Efficiency Calculator
Required Inputs:
Mass Flow Rate of Heat Transfer Fluid: 0.0001 [kg/s]
Ambient Air Temperature: 293 [K]
Mean Heat Transfer Fluid Inlet Temperature: 288 [K]
Month of the Year: September
Known Collector Parameters:
Outer Diameter of Heat Transfer Tube: 0.01 [m]
Heat Transfer Tube Length: 2.445 [m]
Viscosity of Heat Transfer Fluid: 0.00813 [N∙s/m²]
Nusselt Number of the Heat Transfer Fluid: 4.36
Thermal Conductivity of the Heat Transfer Tube: 401 [W/m·K] Overall Collector Heat Loss Coefficient: 7.5 [W/m²∙°C]
Distance Between Centre of 2 Heat Transfer Tubes: 0.02 [m]
Thickness of Heat Transfer Tube: 0.001 [m]
Specific Heat of the Heat Transfer Fluid: 795 [J/kg·K] Effective Radiation Incident on the Collector: 285.9 [W/m²]
Collector Absorptance: 0.94
Calculated Collector Parameters:
Flat‐Plate Collector Area: 0.02445 [m²] Reynolds Number of the Heat Transfer Fluid: 1.5661
Heat Transfer Coefficient from the Inner Tube to the Fluid: 174836 [W/m²·K] Fin Efficiency: 0.9994
Collector Fin Efficiency: 0.9997
Collector Heat Removal Factor: 0.3903
Rate of Useful Energy from the Collector: 2.9226 [W]
Collector Efficiency: 41.8 %
Figure 20: Flat‐Plate Collector Efficiency Calculator
From the efficiency calculator output shown in Figure 20, it can be seen that, during the
month of September, if the ambient air surrounding the collector is at 25°C and the heat
transfer fluid flowing through the collector at a mass flow rate of 0.0001 kg/s is measured
to have a mean temperature of 45°C, the efficiency of the collector at that instant is
41.8%.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 46 Heather King
It should be noted that the heat transfer fluid used in the EnerWorks flat‐plate collector is
a 50:50 propylene glycol water solution. All heat transfer fluid properties used in the
efficiency calculator spreadsheet are for a 50:50 propylene glycol water solution.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 47 Heather King
5 ECONOMIC ANALYSIS
In order to determine the energy potential of installing solar collectors on campus, an
economic analysis must be undertaken. This analysis will look at the cost of purchasing
and installing a system of solar collectors, the number of solar collectors needed to
substantially affect the current district heating system at the University of Manitoba, and
finally, the pay‐back period that installing a system of collectors would produce.
5.1 Background
The University of Manitoba’s powerhouse currently has 6 steam boilers in operation.
These boilers are mainly used for domestic hot water and reheat, with the distribution
being approximately as follows:
Figure 21: Distribution of Powerhouse Boiler Usage
Of the six boilers in operation at the powerhouse, boiler 5 and boiler 6 are summer
boilers, each with an operating capacity of 15,000 lbs of steam per hour. In order to look
at the potential of installing a system of solar collectors on campus, an analysis will be
undertaken in which the cost of replacing one of these boilers with solar energy will be
determined.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 48 Heather King
The following table summarizes the number of days boilers 5 and 6 were in operation in
2006, along with the quantity of steam produced each month [10].
Operating Days Boiler 5
Steam Produced [klbs]
Operating Days Boiler 6
Steam Produced [klbs]
JAN 0 0 0 0
FEB 0 0 0 0
MAR 0 0 0 0
APR 4 49 0 0
MAY 3 86 20 3129
JUN 2 246 2 246
JUL 21 4336 21 4336
AUG 31 7350 31 7350
SEP 9 2270 8 1954
OCT 7 511 8 574
NOV 0 0 0 0
DEC 0 0 0 0
TOTAL 77 14848 90 17589
Table 6: Days of Operation in 2006 ‐ Boilers 5 and 6
Both boilers 5 and 6 have an average operating efficiency of 81% and operate for roughly
the same amount of days per month [10]. For analysis sake, the following discussion will
determine the potential of replacing the heat obtained from boiler 5 with solar energy. By
taking the total pounds of steam produced by boiler 5 in August, as is the dominant
month for use of the boiler, it can be seen that the boiler produces, at most, 9879 lbs of
steam per hour during August.
The following shows the conversion used to obtain this value:
Convert to lbs/hour: hours
day
days
month
klbs
lbsklbs
24
1
31
110007350 = 9879 lbs/hour
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 49 Heather King
5.2 Boiler Energy Analysis
In order to eliminate the use of boiler 5, the equivalent to 9879 lbs/hour of steam must be
able to be produced by a system of solar collectors. The following relation is used to
determine the amount of power, in KW, that this flow rate of steam produces:
3600es hm
Q
[5‐1]
Where:
sm
= steam flow rate (kg/h)
sh = specific enthalpy of evaporation of steam at working pressure (kJ/kg)
Q = heat transferred from the steam (kW)
Noting that 9879 lbs/hour is equivalent to 4481 kg/h, and that the specific enthalpy of
evaporation of steam at 100 psi (the working pressure) is 267.1 kJ/kg [11], we can see
that:
5.3323600
)1.267)(4481(Q kW
Therefore, a system of collectors must be able to produce ~332.5 kW in order for boiler 5
to be eliminated from the district heating system.
5.3 Collector Energy Analysis
As the solar collectors have not yet been tested to determine their working efficiencies, all
energy analysis calculations will be done using the theoretical efficiency of the collectors,
as given on their respective product specification sheets.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
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These efficiencies are as follows:
Product Efficiency
Apricus Evacuated Tube Collector 0.717
EnerWorks Flat‐Plate Collector I
TT
I
TT aiai2)(0187.0)(014.4
7166.0
Table 7: Theoretical Collector Efficiencies
The Solar Rating & Certification Corporation (SRCC) breaks the different heating
applications into the following categories:
(T inlet – T ambient) Heating Application
A ‐5 °C Pool heating in warm climate.
B 5 °C Pool heating in cool climate.
C 20 °C Water heating in warm climate.
D 50 °C Water heating in cool climate.
E 90 °C Industrial process water heating.
Table 8: SRCC Heating Applications
During the summer months, when boiler 5 is operational, the climate in Winnipeg can be
described as warm. Therefore, in order to determine the efficiency of the EnerWorks Flat‐
Plate collector, )( ai TT will be set to 20 °C.
From the RETScreen data for Winnipeg International Airport, provided in Appendix B, the
daily solar radiation, in W/m² can be calculated based on the average monthly hours of
sunlight. These values can be seen in Table 9.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
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Month Daily Solar Radiation
[kWh/m²/d] Average Monthly Hours of Sunlight
[W/m²]
JAN 1.39 8.75 158.9
FEB 2.39 10.15 235.5
MAR 3.75 12 312.5
APR 4.97 13.75 361.5
MAY 5.78 15.375 375.9
JUN 6.17 16.125 382.6
JUL 6.33 15.75 401.9
AUG 5.28 14.375 367.3
SEP 3.61 12.625 285.9
OCT 2.19 10.75 203.7
NOV 1.31 9.125 143.6
DEC 1.03 8.251 124.8
ANNUAL 3.69 12.25 301.2
Table 9: Solar Radiation Values
For the month of August, in which boiler 5 is most prominently used, it can be seen that
the daily solar radiation value is 367.3 W/m².
In order to estimate the theoretical efficiency of the EnerWorks solar collector during
August, I, the solar irradiance incident upon the collector, will be set as 367.3 W/m². The
efficiency can be calculated as:
I
TT
I
TT aiai2)(0187.0)(014.4
7166.0
3.367
)20(0187.0
3.367
)20(014.47166.0
2
4776.0
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 52 Heather King
The following table summarizes the number of solar collectors needed to obtain enough
heat energy from solar radiation to eliminate the use of boiler 5.
Collector Solar
Radiation for August [W/m²]
Efficiency
Solar Energy Converted to Heat Energy [W/m²]
Collector Absorber Area [m²]
Solar Energy Converted to Heat Energy
[W]
Number of Collectors Needed to Produce 332.5
kW of Heat
Apricus Evacuated Tube Collector
367.3 0.7166 263.207 2.400 631.697 418
EnerWorks Flat‐Plate Collector
367.3 0.4776 175.422 2.691 472.062 560
Table 10: Quantity of Collectors Needed
As Table 10 shows, 418 evacuated tube collectors, or 560 flat‐plate collectors, would need
to be installed by the University of Manitoba in order for the powerhouse to eliminate the
use of boiler 5.
To put this value into perspective, the area required to house these collectors can be
calculated. Table 11 shows a brief calculation of the area needed to house these
theoretical solar collector systems. Please note that the area of a standard Canadian
football field is approximately equal to 1 acre.
Collector Number of Collectors Needed
Area of Land Needed per Collector [m²]
Total Area of Land Needed for
Collector System [m²]
Total Area of Land Needed for Collector
System [acres]
Equivalent Number of Football Fields
Apricus Evacuated Tube Collector
418 4.348 1817.464 0.449 ~0.45
EnerWorks Flat‐Plate Collector
560 2.873 1608.880 0.398 ~0.4
Table 11: Collector Area Needed
From this calculation it can be seen that, if you were to take an area of land approximately
the size of approximately half of a football field and fill it with side by side evacuated tube
solar collectors, enough energy would be accumulated to eliminate the use of boiler 5.
This land may seem like a slightly large area to put aside solely for use by solar collectors,
however it is not unreasonable.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 53 Heather King
The University of Manitoba recently purchased the land on which the Southwood Golf and
Country Club resides. This facility is located adjacent to the University of Manitoba and
consists of roughly 120 acres of land, 240 times the area needed to house the solar
collectors; land that could easily be home to a system of solar collectors.
5.4 Cost Analysis
A brief cost analysis will be carried out to determine if there is any financial benefit for the
university to install a system of solar collectors to replace boiler 5. This cost analysis will
compare the cost of purchasing the solar collectors to the cost of running boiler 5. As this
is a simplified cost analysis estimate, factors such as installation and maintenance costs
are not included in the calculation.
Carbon Finance Intel, a Canadian company which trades greenhouse gasses financial
incentives for both Canadian and international customers, states the current trading
prices of C02 reduction as ~ €22 per tonne of CO2 (stated 28 Nov 2007 [17]). This value
converts to approximately $32.40 CAD per tonne of CO2.
In order to determine the financial incentive the University of Manitoba would obtain by
reducing their CO2 emissions, it must be determined how much CO2 boiler 5 releases
during the month of august. Boiler 5 was in use for 31 days during august 2006 and
produced a total of 247,380 kWh’s of energy (332.5 kW x 31 days x 24 hours/day). As the
boiler is only 81% efficient, 305,407.4 kWh’s of energy from natural gas were used by the
boiler to produce this steam output. For every kWh of energy from a greenhouse gas that
is consumed, 0.21 kg’s of CO2 are released into the atmosphere. This means that boiler 5
released approximately 64,135.6 kg’s (64.14 tonnes) of CO2 into the surroundings during
its operating period. Based on the current trading values of CO2 ($32.40 CAD per tonne
CO2), the university would receive an incentive of approximately $2,078.14 during the
month of august if boiler 5 was replaced with solar energy. Over the course of one year
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 54 Heather King
this reimbursement would equal $24,937.68. This financial incentive will be taken into
consideration in the following cost analysis.
Apricus Evacuated Tube Collector
EnerWorks Flat‐Plate Collector
Cost Per Collector [$] 2425.00 1750.00
Number of Collectors Needed
418 560
Amount of Energy Produced [kW]
332.5 332.5
Total Cost of Solar Collector System [$]
1,013,650.00 980,000.00
Cost of Boiler 5 Operation [$]
87,957 87,957
Carbon Dioxide Reduction Incentive
[S] 24,938 24,938
Annual Energy Savings
112,895 112,895
Simple Payback 9.0 years 8.7 years
Table 12: Cost to Substitute Boiler 5 with Solar Energy
The values in Table 12 were determined as follows:
1. Cost per collector: Estimate quoted by Apricus and EnerWorks reference collector
specifications.
2. Number of collectors needed: Calculated in Table 10.
3. Total cost of solar collector systems: Cost per collector x number of collectors
needed.
4. Cost of Boiler 5 Operation: Calculated based on an average natural gas price in
Winnipeg of $0.024/kWh. Boiler 5 used 305,407.4 kWh’s during august 2006, at a
cost of $7329.78. Over the course of one year this would equal $87,957.36.
5. Carbon dioxide reduction incentive: As discussed above.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 55 Heather King
6. Annual Energy Savings: Equivalent to the cost of operating boiler 5 plus the CO2
reduction incentive received by the university.
7. Simple payback: The cost to purchase the system divided by the annual energy
savings, in years.
From the results shown in Table 12, it can be seen that installing a system of solar
collectors on campus to replace the use of boiler 5 is financially feasible. This conclusion
is made for two reasons:
1) The average life span of a solar collectors averages around 20 – 30 years, making
the simple payback period of purchasing the solar collectors shorter than the life
of the collectors themselves. This would mean that the university would finish
paying off the purchasing cost of the collectors before their life span had ended,
allowing the university to actually make a profit out of the use of solar collectors
over natural gas steam boilers.
2) Other financial benefits for installing a system of solar collectors exist, beyond that
of the CO2 reduction incentive. Federal government incentives, such as the
Renewable Energy Deployment Initiative (REDI). REDI provides funds for up to
25% of the purchase and installation cost of a system of collectors by a business or
institution, up to a maximum value of $80,000. If the full spectrum of available
government incentives is utilized in the installation of solar collectors on campus,
the simple payback period will drop even further, creating more revenue for the
university.
Installing 418 evacuated tube collectors or 560 flat‐plate collectors on campus is a large
risk for the university as no solar collector system is currently set up to test the results
with. It is good, in this case, to look into the advantages of installing a smaller system of
collectors on campus. Would the results, both financial and environmental, be the same?
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 56 Heather King
The same analysis was carried out for a system of 100 solar collectors; both the Apricus
evacuated tube collector and the EnerWorks flat‐plate collector. The results of this
analysis are contained in Table 13.
Apricus Evacuated Tube Collector
EnerWorks Flat‐Plate Collector
Cost Per Collector [$] 2425.00 1750.00
Number of Collectors 100 100
Total Cost of Solar Collector System [$]
242,500.00 175,000.00
Average Annual Solar Radiation [kWh/m²/d]
3.690 3.690
Average Annual Solar Radiation [kW/m²]
0.301 0.301
Collector Absorber Area [m²]
2.400 2.691
Average Annual Incident Radiation [kW]
0.723 0.811
Collector Efficiency 0.717 0.478
Amount of Energy Produced By One Collector
[kW] 0.518 0.387
Amount of Energy Produced By 100 Collectors [kW]
51.802 38.711
% of District Heating System Supplemented
4.296 3.210
Cost to Produce Equivalent Heat by District
Heating System [$] 10622.31 7937.92
Carbon Dioxide Prevented from Entering the
Atmosphere [tonnes] 92.95 69.46
Carbon Dioxide Reduction Incentive [S]
3011.58 2250.50
Annual Energy Savings $13,633.89 $10,188.42
Simple Payback 17.8 years 17.2 years
Table 13: Economic Analysis of a System of 10 Collectors
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 57 Heather King
The results in Table 13 show the same type of results for the system of 100 solar
collectors to that of the larger quantity of collectors; however the simple payback period
is about double to length. It should be noted, however, that the results from Table 13
analyze a system of collectors over the course of one year and use the average annual
daily solar radiation, whereas the larger system of collectors used to replace boiler 5
looked specifically at the summer months, where more solar radiation is available.
5.5 Chapter Summary
This chapter examined the economic potential that installing a system of solar collectors
at the University of Manitoba would produce. An analysis was carried out to determine
how many solar collectors, either evacuated tube or flat‐plate, would be needed to
eliminate the use of one of the smaller, summer load boilers, and what the economic
value of installing this theoretical system would be. As the previous sections determined,
it is both environmentally and financially advantages to replace one of the boilers from
the university’s powerhouse with solar energy. The payback period that installing a
system of collectors induced would be much less than the lifespan of the collectors
themselves, creating both a large reduction in CO2 emissions and a long‐term profit for
the university.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 58 Heather King
6 DISCUSSION OF RESULTS
After analyzing the potential of installing solar collectors at the University of Manitoba’s
“eco‐village”, many distinctive results and debates arose. These results are discussed in
the following sections.
6.1 Benefits of Installing Solar Collectors
There are two ways to view the theoretical addition of a system of solar collectors to the
University of Manitoba campus. The first view and most obvious view would be to look at
the cost that installing a system of collectors ensues and compare it to the payback period
installing the system of collectors holds. From the results seen in Section 5.4, it is
apparent that the installation of a system of collectors on campus in economically
advantages for the university in the long run, and should be highly regarded as an energy
conservation method.
The second view in which to regard the installation of solar collectors on campus is from
an environmental standpoint. The use of steam boilers emits CO2 into the atmosphere,
adding to the Earth’s already highly increasing level of greenhouse gasses and global
warming. Solar energy, on the other hand, is a completely harm free, renewable energy
source. Yes, the price we pay for these collectors now may be significantly higher than
that of a steam boiler or natural gas, but what about 50 years from now? Somewhere
down the line, whether our generation or the next, someone will have to pay the price for
global warming and other easily preventable man‐made disasters. Many government
incentives now exist in which funding for some, if not most, of the cost of purchasing and
installing the collectors is provided. Incentives also exist for the reduction of CO2 from the
environment, with some emission trading markets stating exchange rates of up to $32.40
CAD per tonne of CO2 reduced from release into the atmosphere.
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 59 Heather King
It may be completely unreasonable to convert the entire University of Manitoba’s heating
system to solar energy, but how much would it really take to convert 5, or even 10% of
the university’s annual heating use? We can see from the example in Section 5.4 that
substituting even 4% of the university’s district heating system from boilers to solar
energy would prevent around 93 tonnes of CO2 from being released into the atmosphere
per year, not to mention benefitting the University’s appeal and environmental
conformity from a green stance.
6.2 Possible Collector Locations
The following page shows a map of the University of Manitoba. Also shown in the image
is the Southwood Golf and Country Club. As mention previously, these 120 acres of land
were recently purchased by the University of Manitoba and will be acquired for use by the
University in 2010. In order to supplement around 4% of the district heating system,
approximately 400m² (~0.1 acres) of land would be needed. Of the 120 acres about to be
acquired by the university, it must certainly be possible to set aside a tenth of an acre for
solar collectors.
One could argue that that by designating a certain area of land for solar collectors, the
University would be losing potential land for agriculture. This is not necessarily the case.
The land underneath the collectors would still be cultivatable; it would just not receive as
much sunlight as a fully exposed area of land. However, there are many plants and crops
that do not need the sun to thrive. Fungi, such as mushrooms, do not need chlorophyll to
grow, and potatoes, once their dormancy is broken, can be fully harvested in a shaded
area. In addition, plants, such as hostas, astillbies, ferns, and impatients, will bloom to
fully potential in a shaded environment.
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Figure 22: University of Manitoba Campus Map
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In order to show just how little land is needed to significantly contribute to the University
of Manitoba’s district heating system with solar energy, the results from the previous
chapters have been used to estimate how much solar energy could be obtained from
installing solar collectors on the engineering and agriculture building’s rooftops.
The image below shows a hybrid map of the agriculture building and the engineering
building, on the left and the right of the view respectively.
Figure 23: Hybrid Map of the University
From the image above it can be seen that a majority of the rooftops of these buildings are
flat and unused. The hatched red areas on the following image mark potential solar
collector system locations on the Architecture and Engineering building’s rooftops.
Architecture Building
EngineeringBuilding
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Figure 24: Rooftop Solar Collector Locations
By estimating the area, in m², of rooftop contained in the hatched areas, the following
analysis was completed to determine the approximate percentage of the district heating
system that could be supplemented by installing solar collectors in the available rooftop
space.
Engineering Building Roof Architecture Building Roof
Evacuated Tube
Collectors Flat‐Plate Collectors
Evacuated Tube Collectors
Flat‐Plate Collectors
Roof‐top Area Available [m²] 1000 1000 1500 1500
Area Needed Per Collector [m²]
4.348 2.873 4.348 2.873
Max # of Collectors Possible 230 348 345 522
Energy Produced by 1 Collector [kW]
0.518 0.387 0.518 0.387
Energy Produced by Collector System [kW]
119.135 134.702 178.703 202.054
% of District Heating System Supplemented
9.9 11.2 14.8 16.8
Table 14: Rooftop Availability
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This breakdown shows that, by installing a system of solar collectors in the un‐utilized
space on top of one or more campus buildings, the university could easily obtain enough
heat energy from the sun to significantly supplement the district heating system and
drastically reduce CO2 emissions.
6.3 Evacuated Tube or Flat‐Plate?
The two types of solar collectors that were studied in detail throughout this report were
evacuated tube collectors and flat‐plate collectors. Both operate on the same general
principle – utilizing energy from the sun to heat water. However, there are a few distinct
differences between the collectors that must be discussed.
1. Evacuated tubes, in general, have a higher efficiency than flat‐plate
collectors. This result was seen when the efficiency calculators were used
for both the evacuated tube collector and the flat‐plate collector under the
same conditions. The evacuated tube collector gave an efficiency of
76.24% while the flat‐plate collector gave an efficiency of 41.8%. These
results match up to the efficiencies stated on both collectors specification
sheets. The Apricus evacuated tube collector suggested a product
efficiency of 71.77%, while the EnerWorks flat‐plate collector was
estimated to have an efficiency of 47.78% in the summer months.
2. Of the two types of collectors, flat‐plate collectors tend to sell at a price
reasonably less than evacuated tube collectors. The cost of an Apricus
evacuated tube collectors is approximately $2425 per collector, while an
EnerWorks flat‐plate collector would cost around $1750 per collector.
3. As for weathering the elements and surviving a winter in Winnipeg, both
collectors should fair the same. However, if a problem were to occur, an
evacuated tube collector may be easier to fix. For instance, if a hailstorm
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was to damage a flat‐plate collector, it is most likely that the whole
collector would have to be replaced. With an evacuated tube collector,
however, only one or two of the tubes may have been damaged, allowing
the individual tubes to be easily replaced.
If the University of Manitoba were to install a system of solar collectors on campus, it is a
matter of quantity over quality as to whether evacuated tube collectors or flat plate
collectors should be installed. The calculations and analysis completed during this study
confirmed that the payback periods for each system were equivalent; however
maintenance costs were not included in those calculations. Therefore, it is suggested that
evacuated tube collectors be installed over flat‐plate collectors, as they would save the
university money in the long run.
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7 CONCLUSIONS AND RECOMMENDATIONS The aim of this thesis report was to study and analyze the potential, both economic and
environmental, that installing a system of solar collectors at the University of Manitoba
could provide. This study began by proposing the installation of four solar collectors at
the “eco‐village”; an evacuated tube collector, two flat‐plate collectors (water and air),
and a solar wall collector.
The scope of this thesis project, unfortunately, was limited as “eco‐village” solar collector
research project as a whole is in its beginning phase. The solar collectors have been
purchased by the university but these collectors, along with the necessary data acquisition
equipment, were not accessible for installation at the completion of this thesis project
time period.
Despite the lack of available equipment, this project studied and analyzed many aspects
of a proposed installation of solar collectors on campus. The following conclusion and
recommendations can be made upon the completion of this thesis project.
7.1 Solar Collector Location
By utilizing the theories of Tiwari & Ghosal, and the work done by Ametek Inc., the
optimum angular location of a solar collector at the University of Manitoba was
determined. It was found that, in order to receive the maximum amount of solar
radiation incident on the collector plane over the course of one year, the tilt angle of the
plane should be set to 50° with a tilt angle direction of 0° (facing directly south). It is
recommended to install the “eco‐village” collectors at these tilt angle coordinates.
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7.3 Instantaneous Efficiency Calculators Analyzing the heat transfer flows through the Apricus evacuated tube collector and the
EnerWorks flat‐plate collectors allowed for the creation of a Microsoft Excel spreadsheet
for each collector. This collector requires the user to input the variable collector
parameters, such as temperatures and fluid flow, and calculates the instantaneous
efficiency of the collector. As there is currently no installed, operational solar collector, it
was not possible to verify the accuracy of these calculators through experimentation. It is
highly recommended that these formulae be verified once the applicable equipment is
installed.
From the efficiency calculator models created, it was determined that, during the summer
months, an evacuated tube collector located at the “eco‐village” would have an efficiency
of 76.2% while a flat‐plate collector would have an efficiency of 41.8%. These results are
accurate when compared to the solar collector efficiencies stated by the respective
manufacturers.
7.4 Economic Analysis A brief economic analysis was performed on several theoretical solar collector systems.
This analysis showed both pros and cons to the installation of a system of solar collectors
at the University of Manitoba. The following gives an overview of the conclusions
obtained.
By placing a system of evacuated tube collectors in 1000m² of empty space upon the
rooftop of the Engineering Building, approximately 10% of the university’s district heating
load could be supplemented. This would decrease the amount of CO2 released into the
atmosphere by roughly 220 tonnes a year.
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By placing a system of flat‐plate collectors in 1500m² of empty space upon the rooftop of
the Architecture Building, approximately 16.8% of the university’s district heating load
could be supplemented. This would decrease the amount of CO2 released into the
atmosphere by roughly 372 tonnes a year.
The cost for installing enough evacuated tube solar collectors on campus to eliminate the
use of one of the U of M powerhouse’s smaller, summertime boilers is approx. $1.013
million; installing enough flat‐plate collectors would cost approx. $0.98 million. If we take
these costs, minus the financial incentives available for reducing CO2 emissions, and
compare them to the cost of running a boiler with natural gas, lead to payback periods of
9.0 and 8.7 years respectively. It was concluded that the switch from steam boiler heating
to that of solar energy is beneficial to the university in the long run.
Due to time constraints on this thesis project, different effects on solar radiation levels
were not examined. Factors such as the amount of cloudy days per year, snow coverage
on collectors, and collector maintenance should be studied before a system of solar
collectors is installed at the University of Manitoba.
7.5 Final Recommendations
This thesis report concluded that installing a system of solar collectors at the University of
Manitoba is both an economically and environmentally viable project. Further research
and analysis of the solar collectors to be installed at the “eco‐village” should be carried
out to reinforce this claim, and a full, economic and financial cost evaluation should be
initiated.
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8 REFERENCES
[1] “About Solar Energy”, Canadian Renewable Energy Network, [Online Document], 2000 Jul 10, [cited 2007 Oct 27], Available HTTP: http://www.canren.gc.ca/tech_appl/index.asp?CaId=5&PgId=121 [2] “Cheaper and Efficient Energy”, Solar Benefits, [Online Document], no date given, [cited 2007 Oct 27], Available HTTP: http://www.solar‐benefits.com/ [3] “Solar Energy”, Wikipedia, [Online Document], no date given, [cited 2007 Oct 27], Available HTTP: http://en.wikipedia.org/wiki/Solar_energy [4] “The Facts about Solar Hot Water”, World of Energy, [Online Document], 2007 Feb 27, [cited 2007 Oct 27], Available HTTP: http://www.worldofenergy.com.au/factsheet_solarhotwater/07_fact_solarhotwater_how.html [5] “Winnipeg”, Wikipedia, [Online Document], no date given, [cited 2007 Oct 28], Available HTTP: http://en.wikipedia.org/wiki/Winnipeg [6] Ametek, Inc., Solar Energy Handbook: Theory and Applications, 2nd Edition, Chilton Book Company, 1984. [7] “Sunrise and Sunset in Winnipeg”, Time and Date AS, [Online Document], no date given, [cited 2007 Sept 29], Available HTTP: http://www.timeanddate.com/worldclock/astronomy.html?n=265&month=9&year=2007&obj=sun&afl=‐11&day=1 [8] Tiwari, G. N., Solar Energy Technology Advances, 1st Edition, Nova Science Publishers, 2006. [9] “Technical Information”, Apricus Solar Co., [Online Document], 2007, [cited 2007 Nov 15], Available HTTP: http://www.apricus.com/html/solar_collector_technical_info.htm [10] “University of Manitoba Powerhouse Annual Report 2006”, University of Manitoba, 2006. [11] Çengel, Y.A. & Boles, M.A., Thermodynamics: An Engineering Approach, 4th Edition, McGraw‐Hill Higher Education, 2002. [12] “Solar Collector Efficiency”, Energistic Systems, [Online Document], no date given, [cited 2007 Nov 21], Available HTTP: http://www.energisticsystems.us/pdfs/ce_brochure.pdf
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
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[13] “Profitability Index Calculator”, MONEY‐zine.com, [Online Document], 2006, [cited 2007 Nov 22], Available HTTP: http://www.money‐zine.com/Calculators/Investment‐Calculators/Profitability‐Index‐Calculator/ [14] “Residential Products”, EnerWorks Inc., [Online Document], 2007, [cited 2007 Nov 24], Available HTTP: http://www.enerworks.com/Res_Products_Collectors.asp [15] Incropera et al, Fundamentals of Heat and Mass Transfer, 6th Edition, John Wiley & Sons, 2007. [16] “World's Largest Solar Wall at Canadair Facility”, Natural Resources Canada, [Online Document], 2005 Sep 26, [cited 2007 Nov 27], Available HTTP: http://www.oee.nrcan.gc.ca/publications/infosource/pub/ici/caddet/english/r336.cfm?attr=20 [17] “Carbon Finance Intel”, Carbon Finance Intel, [Online Document], 2006, [cited 2007 Nov 28], Available HTTP: www.carbonfinance.ca
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APPENDIX A – COLLECTOR SPECIFICATIONS This appendix lists the technical specifications of the two collectors studied throughout
this report.
The Apricus Evacuated Tube Collector specifications can be found at the following site: http://www.trendsetterindustries.com/pdf/FTD‐801‐APCollectorSpecificationsRev.1.6.pdf The EnerWorks Flat‐Plate Collector specifications can be found at the following site: http://www.enerworks.com/Res_Products.asp
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Apricus Solar Collector Specifications:
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EnerWorks Flat‐Plate Collector Specifications:
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APPENDIX B ‐ RETSCREEN DATA FOR WINNIPEG INT. AIRPORT This appendix gives the RETScreen data for Winnipeg International Airport that was
utilized throughout this report.
RETScreen International can be accessed at www.retscreen.net.
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APPENDIX C – OPTIMUM TILT ANGLES AND DIRECTIONS: JAN –DEC 2007
The following appendix calculates the optimum tilt angles and tilt directions for a solar
collector installed at the University of Manitoba’s Ft. Garry campus for the months of
January to December 2007.
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Table 15: Optimum Tilt Angle ‐ January 2007
01‐Feb‐07
n = 32
Solar Noon: 12:42 PM
Hour L D t H sinA A sinZ Z Opt Σ Opt φ
6:00 AM 49.81 ‐17.52 17.30 259.50 ‐0.34 ‐20.00 1.00 86.25 110.00 86.25
7:00 AM 49.81 ‐17.52 18.30 274.50 ‐0.18 ‐10.47 0.97 75.19 100.47 75.19
8:00 AM 49.81 ‐17.52 19.30 289.50 ‐0.02 ‐1.40 0.90 64.05 91.40 64.05
9:00 AM 49.81 ‐17.52 20.30 304.50 0.12 6.81 0.79 52.33 83.19 52.33
10:00 AM 49.81 ‐17.52 21.30 319.50 0.24 13.77 0.64 39.62 76.23 39.62
11:00 AM 49.81 ‐17.52 22.30 334.50 0.33 19.00 0.43 25.73 71.00 25.73
12:00 PM 49.81 ‐17.52 23.30 349.50 0.38 22.04 0.19 10.81 67.96 10.81
1:00 PM 49.81 ‐17.52 0.30 4.50 0.38 22.56 ‐0.08 ‐4.65 67.44 ‐4.65
2:00 PM 49.81 ‐17.52 1.30 19.50 0.35 20.50 ‐0.34 ‐19.87 69.50 ‐19.87
3:00 PM 49.81 ‐17.52 2.30 34.50 0.28 16.10 ‐0.56 ‐34.21 73.90 ‐34.21
4:00 PM 49.81 ‐17.52 3.30 49.50 0.17 9.77 ‐0.74 ‐47.38 80.23 ‐47.38
5:00 PM 49.81 ‐17.52 4.30 64.50 0.04 2.01 ‐0.86 ‐59.46 87.99 ‐59.46
6:00 PM 49.81 ‐17.52 5.30 79.50 ‐0.12 ‐6.76 ‐0.94 ‐70.78 96.76 ‐70.78
7:00 PM 49.81 ‐17.52 6.30 94.50 ‐0.28 ‐16.15 ‐0.99 ‐81.80 106.15 ‐81.80
8:00 PM 49.81 ‐17.52 7.30 109.50 ‐0.44 ‐25.81 ‐1.00 ‐86.88 115.81 ‐86.88
Table 16: Optimum Tilt Angle ‐ February 2007
01‐Jan‐07
n = 1
Solar Noon: 12:32 PM
Hour L D t H sinA A sinZ Z Opt Σ Opt φ
6:00 AM 49.81 ‐23.01 17.47 262.05 ‐0.38 ‐22.38 0.99 80.35 112.38 80.35
7:00 AM 49.81 ‐23.01 18.47 277.05 ‐0.23 ‐13.05 0.94 69.66 103.05 69.66
8:00 AM 49.81 ‐23.01 19.47 292.05 ‐0.08 ‐4.34 0.86 58.82 94.34 58.82
9:00 AM 49.81 ‐23.01 20.47 307.05 0.06 3.40 0.74 47.38 86.60 47.38
10:00 AM 49.81 ‐23.01 21.47 322.05 0.17 9.77 0.57 35.06 80.23 35.06
11:00 AM 49.81 ‐23.01 22.47 337.05 0.25 14.38 0.37 21.75 75.62 21.75
12:00 PM 49.81 ‐23.01 23.47 352.05 0.29 16.84 0.13 7.64 73.16 7.64
1:00 PM 49.81 ‐23.01 0.47 7.05 0.29 16.91 ‐0.12 ‐6.78 73.09 ‐6.78
2:00 PM 49.81 ‐23.01 1.47 22.05 0.25 14.59 ‐0.36 ‐20.92 75.41 ‐20.92
3:00 PM 49.81 ‐23.01 2.47 37.05 0.18 10.10 ‐0.56 ‐34.28 79.90 ‐34.28
4:00 PM 49.81 ‐23.01 3.47 52.05 0.07 3.82 ‐0.73 ‐46.67 86.18 ‐46.67
5:00 PM 49.81 ‐23.01 4.47 67.05 ‐0.07 ‐3.84 ‐0.85 ‐58.16 93.84 ‐58.16
6:00 PM 49.81 ‐23.01 5.47 82.05 ‐0.22 ‐12.50 ‐0.93 ‐69.02 102.50 ‐69.02
7:00 PM 49.81 ‐23.01 6.47 97.05 ‐0.37 ‐21.81 ‐0.98 ‐79.70 111.81 ‐79.70
8:00 PM 49.81 ‐23.01 7.47 112.05 ‐0.52 ‐31.44 ‐1.00 ‐89.21 121.44 ‐89.21
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Table 17: Optimum Tilt Angle ‐ March 2007
01‐Apr‐07
n = 91
Solar Noon: 1:33 PM
Hour L D t H sinA A sinZ Z Opt Σ Opt φ
6:00 AM 49.81 4.02 16.45 246.75 ‐0.20 ‐11.57 0.94 69.32 101.57 69.32
7:00 AM 49.81 4.02 17.45 261.75 ‐0.04 ‐2.23 0.99 81.10 92.23 81.10
8:00 AM 49.81 4.02 18.45 276.75 0.13 7.42 1.00 87.44 82.58 87.44
9:00 AM 49.81 4.02 19.45 291.75 0.29 16.98 0.97 75.64 73.02 75.64
10:00 AM 49.81 4.02 20.45 306.75 0.44 26.02 0.89 62.80 63.98 62.80
11:00 AM 49.81 4.02 21.45 321.75 0.56 33.99 0.74 48.15 56.01 48.15
12:00 PM 49.81 4.02 22.45 336.75 0.64 40.16 0.52 31.02 49.84 31.02
1:00 PM 49.81 4.02 23.45 351.75 0.69 43.68 0.20 11.41 46.32 11.41
2:00 PM 49.81 4.02 0.45 6.75 0.69 43.85 ‐0.16 ‐9.36 46.15 ‐9.36
3:00 PM 49.81 4.02 1.45 21.75 0.65 40.65 ‐0.49 ‐29.16 49.35 ‐29.16
4:00 PM 49.81 4.02 2.45 36.75 0.57 34.70 ‐0.73 ‐46.55 55.30 ‐46.55
5:00 PM 49.81 4.02 3.45 51.75 0.45 26.88 ‐0.88 ‐61.43 63.12 ‐61.43
6:00 PM 49.81 4.02 4.45 66.75 0.31 17.92 ‐0.96 ‐74.42 72.08 ‐74.42
7:00 PM 49.81 4.02 5.45 81.75 0.15 8.39 ‐1.00 ‐86.28 81.61 ‐86.28
8:00 PM 49.81 4.02 6.45 96.75 ‐0.02 ‐1.27 ‐0.99 ‐82.25 91.27 ‐82.25
Table 18: Optimum Tilt Angle ‐ April 2007
01‐Mar‐07
n = 60
Solar Noon: 12:41 PM
Hour L D t H sinA A sinZ Z Opt Σ Opt φ
6:00 AM 49.81 ‐8.29 17.32 259.75 ‐0.22 ‐12.93 1.00 87.56 102.93 87.56
7:00 AM 49.81 ‐8.29 18.32 274.75 ‐0.06 ‐3.29 0.99 81.03 93.29 81.03
8:00 AM 49.81 ‐8.29 19.32 289.75 0.11 6.06 0.94 69.48 83.94 69.48
9:00 AM 49.81 ‐8.29 20.32 304.75 0.25 14.70 0.84 57.20 75.30 57.20
10:00 AM 49.81 ‐8.29 21.32 319.75 0.38 22.16 0.69 43.66 67.84 43.66
11:00 AM 49.81 ‐8.29 22.32 334.75 0.47 27.86 0.48 28.52 62.14 28.52
12:00 PM 49.81 ‐8.29 23.32 349.75 0.52 31.21 0.21 11.88 58.79 11.88
1:00 PM 49.81 ‐8.29 0.32 4.75 0.53 31.75 ‐0.10 ‐5.53 58.25 ‐5.53
2:00 PM 49.81 ‐8.29 1.32 19.75 0.49 29.39 ‐0.38 ‐22.57 60.61 ‐22.57
3:00 PM 49.81 ‐8.29 2.32 34.75 0.41 24.49 ‐0.62 ‐38.30 65.51 ‐38.30
4:00 PM 49.81 ‐8.29 3.32 49.75 0.30 17.60 ‐0.79 ‐52.41 72.40 ‐52.41
5:00 PM 49.81 ‐8.29 4.32 64.75 0.16 9.33 ‐0.91 ‐65.10 80.67 ‐65.10
6:00 PM 49.81 ‐8.29 5.32 79.75 0.00 0.20 ‐0.97 ‐76.84 89.80 ‐76.84
7:00 PM 49.81 ‐8.29 6.32 94.75 ‐0.16 ‐9.39 ‐1.00 ‐88.23 99.39 ‐88.23
8:00 PM 49.81 ‐8.29 7.32 109.75 ‐0.33 ‐19.02 ‐0.99 ‐80.11 109.02 ‐80.11
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01‐May‐07
n = 121
Solar Noon: 1:26 PM
Hour L D t H sinA A sinZ Z Opt Σ Opt φ
6:00 AM 49.81 14.90 16.57 248.50 ‐0.03 ‐1.84 0.90 64.11 91.84 64.11
7:00 AM 49.81 14.90 17.57 263.50 0.13 7.23 0.97 75.43 82.77 75.43
8:00 AM 49.81 14.90 18.57 278.50 0.29 16.78 1.00 86.60 73.22 86.60
9:00 AM 49.81 14.90 19.57 293.50 0.45 26.43 0.99 81.75 63.57 81.75
10:00 AM 49.81 14.90 20.57 308.50 0.58 35.78 0.93 68.78 54.22 68.78
11:00 AM 49.81 14.90 21.57 323.50 0.70 44.25 0.80 53.36 45.75 53.36
12:00 PM 49.81 14.90 22.57 338.50 0.78 50.96 0.56 34.21 39.04 34.21
1:00 PM 49.81 14.90 23.57 353.50 0.82 54.69 0.19 10.91 35.31 10.91
2:00 PM 49.81 14.90 0.57 8.50 0.81 54.41 ‐0.25 ‐14.21 35.59 ‐14.21
3:00 PM 49.81 14.90 1.57 23.50 0.77 50.20 ‐0.60 ‐37.02 39.80 ‐37.02
4:00 PM 49.81 14.90 2.57 38.50 0.68 43.20 ‐0.83 ‐55.61 46.80 ‐55.61
5:00 PM 49.81 14.90 3.57 53.50 0.57 34.57 ‐0.94 ‐70.63 55.43 ‐70.63
6:00 PM 49.81 14.90 4.57 68.50 0.42 25.15 ‐0.99 ‐83.36 64.85 ‐83.36
7:00 PM 49.81 14.90 5.57 83.50 0.27 15.49 ‐1.00 ‐85.10 74.51 ‐85.10
8:00 PM 49.81 14.90 6.57 98.50 0.10 5.98 ‐0.96 ‐73.94 84.02 ‐73.94
Table 19: Optimum Tilt Angle ‐ May 2007
01‐Jun‐07
n = 152
Solar Noon: 1:26 PM
Hour L D t H sinA A sinZ Z Opt Σ Opt φ
6:00 AM 49.81 22.04 16.57 248.50 0.07 3.87 0.86 59.81 86.13 59.81
7:00 AM 49.81 22.04 17.57 263.50 0.22 12.65 0.94 70.71 77.35 70.71
8:00 AM 49.81 22.04 18.57 278.50 0.38 22.03 0.99 81.47 67.97 81.47
9:00 AM 49.81 22.04 19.57 293.50 0.53 31.68 1.00 87.29 58.32 87.29
10:00 AM 49.81 22.04 20.57 308.50 0.66 41.23 0.96 74.69 48.77 74.69
11:00 AM 49.81 22.04 21.57 323.50 0.77 50.13 0.86 59.33 39.87 59.33
12:00 PM 49.81 22.04 22.57 338.50 0.84 57.48 0.63 39.19 32.52 39.19
1:00 PM 49.81 22.04 23.57 353.50 0.88 61.76 0.22 12.81 28.24 12.81
2:00 PM 49.81 22.04 0.57 8.50 0.88 61.43 ‐0.29 ‐16.65 28.57 ‐16.65
3:00 PM 49.81 22.04 1.57 23.50 0.84 56.64 ‐0.67 ‐42.23 33.36 ‐42.23
4:00 PM 49.81 22.04 2.57 38.50 0.75 49.01 ‐0.88 ‐61.60 40.99 ‐61.60
5:00 PM 49.81 22.04 3.57 53.50 0.64 39.98 ‐0.97 ‐76.49 50.02 ‐76.49
6:00 PM 49.81 22.04 4.57 68.50 0.51 30.39 ‐1.00 ‐88.85 59.61 ‐88.85
7:00 PM 49.81 22.04 5.57 83.50 0.35 20.75 ‐0.98 ‐80.02 69.25 ‐80.02
8:00 PM 49.81 22.04 6.57 98.50 0.20 11.43 ‐0.94 ‐69.28 78.57 ‐69.28
Table 20: Optimum Tilt Angle ‐ June 2007
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 89 Heather King
Table 21: Optimum Tilt Angle ‐ July 2007
01‐Aug‐07
n = 213
Solar Noon: 1:35 PM
Hour L D t H sinA A sinZ Z Opt Σ Opt φ
6:00 AM 49.81 17.91 16.42 246.25 ‐0.01 ‐0.71 0.87 60.58 90.71 60.58
7:00 AM 49.81 17.91 17.42 261.25 0.14 8.14 0.95 71.81 81.86 71.81
8:00 AM 49.81 17.91 18.42 276.25 0.30 17.57 0.99 82.81 72.43 82.81
9:00 AM 49.81 17.91 19.42 291.25 0.46 27.23 1.00 85.81 62.77 85.81
10:00 AM 49.81 17.91 20.42 306.25 0.60 36.73 0.96 73.22 53.27 73.22
11:00 AM 49.81 17.91 21.42 321.25 0.71 45.55 0.85 58.26 44.45 58.26
12:00 PM 49.81 17.91 22.42 336.25 0.80 52.84 0.63 39.38 37.16 39.38
1:00 PM 49.81 17.91 23.42 351.25 0.84 57.34 0.27 15.56 32.66 15.56
2:00 PM 49.81 17.91 0.42 6.25 0.85 57.71 ‐0.19 ‐11.18 32.29 ‐11.18
3:00 PM 49.81 17.91 1.42 21.25 0.81 53.83 ‐0.58 ‐35.75 36.17 ‐35.75
4:00 PM 49.81 17.91 2.42 36.25 0.73 46.90 ‐0.82 ‐55.43 43.10 ‐55.43
5:00 PM 49.81 17.91 3.42 51.25 0.62 38.27 ‐0.95 ‐70.93 51.73 ‐70.93
6:00 PM 49.81 17.91 4.42 66.25 0.48 28.83 ‐0.99 ‐83.82 61.17 ‐83.82
7:00 PM 49.81 17.91 5.42 81.25 0.33 19.17 ‐1.00 ‐84.66 70.83 ‐84.66
8:00 PM 49.81 17.91 6.42 96.25 0.17 9.68 ‐0.96 ‐73.64 80.32 ‐73.64
Table 22: Optimum Tilt Angle ‐ August 2007
01‐Jul‐07
n = 182
Solar Noon: 1:33 PM
Hour L D t H sinA A sinZ Z Opt Σ Opt φ
6:00 AM 49.81 23.12 16.45 246.75 0.07 3.77 0.85 57.87 86.23 57.87
7:00 AM 49.81 23.12 17.45 261.75 0.21 12.40 0.93 68.74 77.60 68.74
8:00 AM 49.81 23.12 18.45 276.75 0.37 21.70 0.98 79.41 68.30 79.41
9:00 AM 49.81 23.12 19.45 291.75 0.52 31.32 1.00 89.53 58.68 89.53
10:00 AM 49.81 23.12 20.45 306.75 0.66 40.92 0.98 77.23 49.08 77.23
11:00 AM 49.81 23.12 21.45 321.75 0.77 50.00 0.89 62.35 40.00 62.35
12:00 PM 49.81 23.12 22.45 336.75 0.85 57.70 0.68 42.80 32.30 42.80
1:00 PM 49.81 23.12 23.45 351.75 0.89 62.54 0.29 16.63 27.46 16.63
2:00 PM 49.81 23.12 0.45 6.75 0.89 62.79 ‐0.24 ‐13.67 27.21 ‐13.67
3:00 PM 49.81 23.12 1.45 21.75 0.85 58.34 ‐0.65 ‐40.49 31.66 ‐40.49
4:00 PM 49.81 23.12 2.45 36.75 0.78 50.85 ‐0.87 ‐60.64 39.15 ‐60.64
5:00 PM 49.81 23.12 3.45 51.75 0.67 41.87 ‐0.97 ‐75.89 48.13 ‐75.89
6:00 PM 49.81 23.12 4.45 66.75 0.53 32.29 ‐1.00 ‐88.38 57.71 ‐88.38
7:00 PM 49.81 23.12 5.45 81.75 0.39 22.65 ‐0.99 ‐80.48 67.35 ‐80.48
8:00 PM 49.81 23.12 6.45 96.75 0.23 13.31 ‐0.94 ‐69.80 76.69 ‐69.80
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 90 Heather King
01‐Sep‐07
n = 244
Solar Noon: 1:29 PM
Hour L D t H sinA A sinZ Z Opt Σ Opt φ
6:00 AM 49.81 7.72 16.52 247.75 ‐0.14 ‐8.02 0.93 67.85 98.02 67.85
7:00 AM 49.81 7.72 17.52 262.75 0.02 1.26 0.98 79.50 88.74 79.50
8:00 AM 49.81 7.72 18.52 277.75 0.19 10.89 1.00 89.10 79.11 89.10
9:00 AM 49.81 7.72 19.52 292.75 0.35 20.49 0.98 77.30 69.51 77.30
10:00 AM 49.81 7.72 20.52 307.75 0.49 29.62 0.90 64.32 60.38 64.32
11:00 AM 49.81 7.72 21.52 322.75 0.61 37.71 0.76 49.31 52.29 49.31
12:00 PM 49.81 7.72 22.52 337.75 0.69 43.99 0.52 31.43 46.01 31.43
1:00 PM 49.81 7.72 23.52 352.75 0.74 47.48 0.19 10.66 42.52 10.66
2:00 PM 49.81 7.72 0.52 7.75 0.74 47.42 ‐0.20 ‐11.39 42.58 ‐11.39
3:00 PM 49.81 7.72 1.52 22.75 0.69 43.82 ‐0.53 ‐32.08 46.18 ‐32.08
4:00 PM 49.81 7.72 2.52 37.75 0.61 37.47 ‐0.76 ‐49.85 52.53 ‐49.85
5:00 PM 49.81 7.72 3.52 52.75 0.49 29.32 ‐0.90 ‐64.78 60.68 ‐64.78
6:00 PM 49.81 7.72 4.52 67.75 0.34 20.17 ‐0.98 ‐77.70 69.83 ‐77.70
7:00 PM 49.81 7.72 5.52 82.75 0.18 10.57 ‐1.00 ‐89.49 79.43 ‐89.49
8:00 PM 49.81 7.72 6.52 97.75 0.02 0.94 ‐0.98 ‐79.11 89.06 ‐79.11
Table 23: Optimum Tilt Angle ‐ September 2007
01‐Oct‐07
n = 274
Solar Noon: 1:18 PM
Hour L D t H sinA A sinZ Z Opt Σ Opt φ
6:00 AM 49.81 ‐4.22 16.70 250.50 ‐0.27 ‐15.72 0.98 77.59 105.72 77.59
7:00 AM 49.81 ‐4.22 17.70 265.50 ‐0.11 ‐6.12 1.00 89.29 96.12 89.29
8:00 AM 49.81 ‐4.22 18.70 280.50 0.06 3.50 0.98 79.24 86.50 79.24
9:00 AM 49.81 ‐4.22 19.70 295.50 0.22 12.76 0.92 67.36 77.24 67.36
10:00 AM 49.81 ‐4.22 20.70 310.50 0.36 21.21 0.81 54.44 68.79 54.44
11:00 AM 49.81 ‐4.22 21.70 325.50 0.47 28.31 0.64 39.91 61.69 39.91
12:00 PM 49.81 ‐4.22 22.70 340.50 0.55 33.40 0.40 23.50 56.60 23.50
1:00 PM 49.81 ‐4.22 23.70 355.50 0.59 35.83 0.10 5.54 54.17 5.54
2:00 PM 49.81 ‐4.22 0.70 10.50 0.58 35.22 ‐0.22 ‐12.85 54.78 ‐12.85
3:00 PM 49.81 ‐4.22 1.70 25.50 0.52 31.65 ‐0.50 ‐30.29 58.35 ‐30.29
4:00 PM 49.81 ‐4.22 2.70 40.50 0.43 25.67 ‐0.72 ‐45.94 64.33 ‐45.94
5:00 PM 49.81 ‐4.22 3.70 55.50 0.31 17.96 ‐0.86 ‐59.77 72.04 ‐59.77
6:00 PM 49.81 ‐4.22 4.70 70.50 0.16 9.13 ‐0.95 ‐72.20 80.87 ‐72.20
7:00 PM 49.81 ‐4.22 5.70 85.50 ‐0.01 ‐0.32 ‐0.99 ‐83.85 90.32 ‐83.85
8:00 PM 49.81 ‐4.22 6.70 100.50 ‐0.17 ‐9.99 ‐1.00 ‐84.68 99.99 ‐84.68
Table 24: Optimum Tilt Angle ‐ October 2007
Solar Energy Systems in the Eco-Village at the University of Manitoba _________________________________________________________________________________________________
________________________________________________________________________ December 4th, 2007 91 Heather King
01‐Nov‐07
n = 305
Solar Noon: 1:12 PM
Hour L D t H sinA A sinZ Z Opt Σ Opt φ
6:00 AM 49.81 ‐15.36 16.80 252.00 ‐0.39 ‐23.25 1.00 86.47 113.25 86.47
7:00 AM 49.81 ‐15.36 17.80 267.00 ‐0.23 ‐13.59 0.99 82.17 103.59 82.17
8:00 AM 49.81 ‐15.36 18.80 282.00 ‐0.07 ‐4.19 0.95 71.04 94.19 71.04
9:00 AM 49.81 ‐15.36 19.80 297.00 0.08 4.59 0.86 59.53 85.41 59.53
10:00 AM 49.81 ‐15.36 20.80 312.00 0.21 12.36 0.73 47.19 77.64 47.19
11:00 AM 49.81 ‐15.36 21.80 327.00 0.32 18.63 0.55 33.66 71.37 33.66
12:00 PM 49.81 ‐15.36 22.80 342.00 0.39 22.92 0.32 18.88 67.08 18.88
1:00 PM 49.81 ‐15.36 23.80 357.00 0.42 24.77 0.06 3.19 65.23 3.19
2:00 PM 49.81 ‐15.36 0.80 12.00 0.41 23.97 ‐0.22 ‐12.67 66.03 ‐12.67
3:00 PM 49.81 ‐15.36 1.80 27.00 0.35 20.61 ‐0.47 ‐27.89 69.39 ‐27.89
4:00 PM 49.81 ‐15.36 2.80 42.00 0.26 15.07 ‐0.67 ‐41.93 74.93 ‐41.93
5:00 PM 49.81 ‐15.36 3.80 57.00 0.14 7.85 ‐0.82 ‐54.72 82.15 ‐54.72
6:00 PM 49.81 ‐15.36 4.80 72.00 ‐0.01 ‐0.58 ‐0.92 ‐66.51 90.58 ‐66.51
7:00 PM 49.81 ‐15.36 5.80 87.00 ‐0.17 ‐9.78 ‐0.98 ‐77.72 99.78 ‐77.72
8:00 PM 49.81 ‐15.36 6.80 102.00 ‐0.33 ‐19.38 ‐1.00 ‐88.92 109.38 ‐88.92
Table 25: Optimum Tilt Angle ‐ November 2007
01‐Dec‐07
n = 335
Solar Noon: 12:18 PM
Hour L D t H sinA A sinZ Z Opt Σ Opt φ
6:00 AM 49.81 ‐22.11 17.70 265.50 ‐0.33 ‐19.54 0.98 78.53 109.54 78.53
7:00 AM 49.81 ‐22.11 18.70 280.50 ‐0.18 ‐10.28 0.93 67.80 100.28 67.80
8:00 AM 49.81 ‐22.11 19.70 295.50 ‐0.03 ‐1.73 0.84 56.78 91.73 56.78
9:00 AM 49.81 ‐22.11 20.70 310.50 0.10 5.78 0.71 45.08 84.22 45.08
10:00 AM 49.81 ‐22.11 21.70 325.50 0.21 11.84 0.54 32.42 78.16 32.42
11:00 AM 49.81 ‐22.11 22.70 340.50 0.28 16.03 0.32 18.77 73.97 18.77
12:00 PM 49.81 ‐22.11 23.70 355.50 0.31 17.97 0.08 4.38 72.03 4.38
1:00 PM 49.81 ‐22.11 0.70 10.50 0.30 17.48 ‐0.18 ‐10.20 72.52 ‐10.20
2:00 PM 49.81 ‐22.11 1.70 25.50 0.25 14.60 ‐0.41 ‐24.34 75.40 ‐24.34
3:00 PM 49.81 ‐22.11 2.70 40.50 0.17 9.62 ‐0.61 ‐37.61 80.38 ‐37.61
4:00 PM 49.81 ‐22.11 3.70 55.50 0.05 2.93 ‐0.76 ‐49.87 87.07 ‐49.87
5:00 PM 49.81 ‐22.11 4.70 70.50 ‐0.09 ‐5.04 ‐0.88 ‐61.25 95.04 ‐61.25
6:00 PM 49.81 ‐22.11 5.70 85.50 ‐0.24 ‐13.92 ‐0.95 ‐72.10 103.92 ‐72.10
7:00 PM 49.81 ‐22.11 6.70 100.50 ‐0.40 ‐23.36 ‐0.99 ‐82.87 113.36 ‐82.87
8:00 PM 49.81 ‐22.11 7.70 115.50 ‐0.54 ‐33.02 ‐1.00 ‐85.77 123.02 ‐85.77
Table 26: Optimum Tilt Angle ‐ December 2007