the canyon effect
TRANSCRIPT
BSc Thesis Biosystems Engineering
Chair group Biomass Refinery and Process Dynamics
The canyon effect:
Decay of diffuse light between vertical plates
Marco Saglibene
June, 6th 2013
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The canyon effect:
Decay of diffuse light between vertical plates
The relationship between the decay of diffuse light
and the height and distance of vertical plates for the purpose of more efficient microalgae cultivation
Course name : BSc thesis Biosystems Engineering
Course code : YEI-80324
Credits : 24
Date : June 6th 2013
Student : Marco Saglibene
Registration number : 81-05-31-724-210
Study programme : Agrotechnologie (BAT)
Supervisors : Dr. Rachel van Ooteghem en Ing. Kees van Asselt
Examinator : Dr. Ton van Boxtel
Chair group : Leerstoelgroep Biomass Refinery and Process Dynamics
Bornse Weilanden 9
6708 WG Wageningen
Tel: +31 (317) 48 21 24
Fax: +31 (317) 48 49 57
e-mail: [email protected]
Abstract In order to make the production of algae for biofuels feasible, it must take place on a larger scale and at lower
production costs compared to how it is today. Vertical flat plate reactors, which are known to be best used for
algae growth, are subject to the canyon effect. This is the extinction of light between plates towards the ground
surface. Robinson and Stone (2004) presented a model to describe the canyon effect for diffuse light for an urban
street canyon. This research evaluated the applicability of this model on a simulated flat plate reactor.
The test setup was constructed out of painted MDF plates. The albedos of the used materials were determined
and are from high to low 0.658 for white paint, 0.076 for grey paint, 0.044 for black plastic, 0.042 for black paint
and 0.018 for the black cloth. The diffusivity of the reflectance was qualitatively examined and was for a ray at a
tilt angle of 45° better than for 10°, where it was mostly specular. Due to the limited width of the test setup, the
measurement values were extrapolated and the sensors were corrected for their specific and cosine course errors.
The result showed that the diffusivity of the incoming light in the test setup was reasonable, but not ideal
because light entering the test setup parallel to the ground plane was too low. Because of the diffusivity
level of the incoming light, the adapted measurement values were too low when measured close to the light
source and also due to the specular reflectance increasingly too high in the direction of the ground. This effect
was most visible for components with a high albedo value.
The specular reflectance caused the adapted measurement values to not coincide with the prerequisites of the
Robinson and Stone model. However, reflectance of a flat plate reactor in practice is probably also not diffuse.
Therefore, the Robinson and Stone model can be used only as a prior estimation for irradiance for any distance
or height of flat plate reactors within of the adapted measurement values compared to the irradiance at the
horizontal plane.
Keywords: canyon effect, flat plate reactor, algae, diffuse, lambertian, extinction, light, albedo,
Robinson and Stone
Preface Nine month ago I started this four months thesis research project. When I started I knew the subject was going to
be tough and for four months I worked fulltime on it. But after these four months, the courses started again and
time became a critical factor while I was still struggling with data processing. Using all available time,
sometimes at the expense of other courses and even of friends and family, the process steadily continued until
now. Now the time has finally come that this thesis is finished. Although the results made it quite hard to
evaluate the applicability of the Robinson and Stone model, I still hope the contents of this report will come to
good use and that somebody benefits from it. The subject is challenging and provides enough possibilities for
further research!
This thesis research would have lasted at least for another nine months if some people wouldn’t have helped. I
would like to thank Joost van Opheusden for his clear explanations on some mathematical issues that pushed me
in the right direction when I needed it, Bert Heusinkveld for lending the sensors and providing support to get
started with the subject, Jeroen de Vree who provided freely essential materials, like the plastic and the diffuse
cloth and Vida Mohammadkhani who saved the diffusivity by providing the prismatic glass.
Further I would like to thank my supervisors, Kees van Asselt and Rachel van Ooteghem. They stood always
ready to discuss problems that were arisen and provided me with any form of support needed. I would also like
to thank all other members of the chair group for their small and big contributions in any form. Lastly I would
like to thank Ton van Boxtel for being the examiner. I hope I can also thank him after he has graded me…
Last of all, but for me the most important one, is my girlfriend Janna Poortinga. She provided me with the
mental support to bring this thesis to a good end. She kept me going on every time I faced a setback while
struggling with the data.
Nomenclature Symbol Unit Description
Albedo, fraction of reflected divided by incoming electromagnetic energy
Wavelength of the light
Spectral reflectivity
Spectral irradiance
Solid angle
(or ) Angle between the surface normal and the ground plane
(or ) Angle within the ground plane
Diffuse irradiance incident on a plane of slope
Diffuse horizontal irradiance
A plane of slope where
Angular height of an equivalent continuous skyline
Height opposing building starting from measurement height
Width of the urban street canyon
Irradiance received on due to reflectance of upper opposing buildings
Irradiance received on due to reflectance of lower opposing buildings
Global irradiance incident on
Diffuse reflectance factor (=albedo)
Same as , but now the angle with the lower obstructing surface
Irradiance received on due to ground reflectance
Irradiance on the ground
Angle between the centre of the urban street canyon and the walls
Height of the flat plate reactor (MDF plates)
Height of the sensors
Distance between the plates
Width of the diffuse cloth
Irradiance fraction coming from a component
Average irradiance fraction onto a component
Extrapolated irradiance value for the horizontal plane
Overall corrected, extrapolated, normalized, and averaged measured values
Maximum number of lights that were used for a measurement
By Robinson and Stone model calculated values after iterations
Relative difference of calculated irradiance to CENA value compared to the
irradiance at the horizontal plane just above the flat plate reactor row
Relative difference of calculated irradiance compared to the corresponding
Overall corrected and extrapolated measurement values
Measured irradiance value
Sensor specific correction factor
Cosine correction factor
Theoretical measurement value
cc Average relative deviation to theoretical cosine course
Average fraction of sensor output in relation to the starting angle
Calculated average fraction of sensor output in relation to starting angle at Calculated average fraction of sensor output in relation to a starting angle
wcc As , but weighted with
Function that approximates
Superscripts
Concerning irradiance onto the ground
Concerning a measured value
Average starting angle for the view with the ground or the sky
Average stopping angle for the view with the plate
Subscripts
White paint
Grey paint
Black paint
Black cloth
Black plastic
Boundary between sky and plate
Boundary between sky and black plastic
Intersection between sky, plate and black plastic
Boundary between ground and plate
Boundary between ground and black plastic
Intersection between ground, plate and black plastic
Boundary between plate and black plastic
Concerning the sky
Concerning the ground
Concerning the plate
Concerning the endless row
Concerning the test setup
Starting angle, not concerning the test setup
Table of contents
1. Introduction ..................................................................................................................................................... 1
1.1 Problem description ............................................................................................................................... 1
1.2 Goal ....................................................................................................................................................... 1
1.3 Research questions ................................................................................................................................ 2
1.4 Delimitations ......................................................................................................................................... 2
2. Theory ............................................................................................................................................................. 5
2.1 Albedo and reflectivity .......................................................................................................................... 5
2.2 Reflection of light .................................................................................................................................. 5
2.3 Incoming and reflected radiance ............................................................................................................ 5
2.3.1 Cosine course of incoming radiance ................................................................................................. 5
2.3.2 Distribution of the quantity of reflected radiance .............................................................................. 6
2.4 Ray tracing and radiosity methods ........................................................................................................ 6
2.5 Integrating over the surface of a sphere ................................................................................................. 6
2.6 The Robinson and Stone model ............................................................................................................. 7
3. Materials and methods .................................................................................................................................... 9
3.1 Test setup ............................................................................................................................................... 9
3.2 Data collecting and processing ............................................................................................................ 10
3.3 Sensors ................................................................................................................................................. 10
3.3.1 Correcting the sensor specific error ................................................................................................. 10
3.4 Creating a diffuse light source ............................................................................................................. 10
3.5 Measuring albedo ................................................................................................................................ 12
3.6 Determining Lambertian reflectivity ................................................................................................... 12
3.7 Extrapolating: from test setup to endless row ...................................................................................... 13
3.7.1 Irradiance from the sky ................................................................................................................... 13
3.7.2 Irradiance from the ground .............................................................................................................. 15
3.7.3 Irradiance from the opposing plate .................................................................................................. 15
3.8 Correcting the sensors cosine course error .......................................................................................... 17
3.9 Irradiance onto the ground and plate ................................................................................................... 17
3.9.1 Irradiance onto the ground .............................................................................................................. 17
3.9.2 Irradiance onto the plate .................................................................................................................. 19
3.10 Correcting, extrapolating, normalizing and averaging the measurement data ..................................... 19
3.10.1 Correcting and extrapolating the measurement values ............................................................... 19
3.10.2 Irradiance at the horizontal plane ................................................................................................ 19
3.10.3 Normalizing and averaging the corrected and extrapolated measurement values ....................... 20
3.11 Using the Robinson and Stone model .................................................................................................. 20
3.12 Comparing calculated with measured values ....................................................................................... 22
4. Results ........................................................................................................................................................... 23
4.1 Sensor specific error correction ........................................................................................................... 23
4.2 Diffusivity of light source .................................................................................................................... 23
4.3 Albedos ................................................................................................................................................ 23
4.4 Lambertian reflectivity ........................................................................................................................ 23
4.5 Calculated and measured values and their relative differences ........................................................... 24
5. Discussion ..................................................................................................................................................... 31
5.1 Test setup ............................................................................................................................................. 31
5.2 Sensors ................................................................................................................................................. 31
5.3 Albedos ................................................................................................................................................ 32
5.4 Lambertian reflectance ........................................................................................................................ 32
5.5 Extrapolations ...................................................................................................................................... 32
5.6 Robinson and Stone model .................................................................................................................. 33
5.7 Applicability of the model ................................................................................................................... 33
5.8 Accuracy of the values ........................................................................................................................ 33
6. Conclusions ................................................................................................................................................... 35
6.1 Albedo of the components ................................................................................................................... 35
6.2 Diffusivity of the reflectance ............................................................................................................... 35
6.3 Extrapolation of the measurement values ............................................................................................ 35
6.4 Applicability of the Robinson and Stone model .................................................................................. 35
7. Recommendations ......................................................................................................................................... 37
7.1 Test setup ............................................................................................................................................. 37
7.2 Sensors ................................................................................................................................................. 37
7.3 Improvement of the model .................................................................................................................. 37
References ............................................................................................................................................................. 39
Appendices ............................................................................................................................................................ 41
Appendix A: Calculation of integrals ................................................................................................................ 41
Appendix B: Determining sensor characteristics .............................................................................................. 42
Appendix C: Determining the sensor specific error .......................................................................................... 45
Appendix D: Long equations ............................................................................................................................ 46
Appendix E: Determining the cosine course error ............................................................................................ 47
Appendix F: Corrections for the cosine course error ........................................................................................ 51
Appendix G: Extrapolations .............................................................................................................................. 53
Appendix H: Irradiance onto ground and plate ................................................................................................. 56
Appendix I: Iterated calculated values and similarity of Robinson and Stone model and extrapolation method
.......................................................................................................................................................................... 57
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1. Introduction The growing world population and emerging economies provide an increasing pressure on the natural resources
of the earth. The oil reserves are not inexhaustible and if we do not want to marginalize the area for nature for
the purpose of food production, more sustainable production methods are needed. The cultivation of microalgae
can play an important role in this according to Wolkers et al. (2011). Microalgae can be grown in areas where
other forms of food production are not possible and they can grow well on waste such as diluted manure or
carbon dioxide from flue gases. The microalgae convert this waste into raw materials such as proteins, starch,
pigments and oils. These materials can be used in many ways. Some possibilities include the production of
biodiesel, bioplastics, dyes and meat substitutes.
Currently, the cultivation of microalgae takes primarily place in
Asia and North America. Global production in 2010 was
approximately 5 000 tonnes of dried microalgae per year (Wijffels
and Barbosa, 2010). Mainly high value products such as carotenes
and omega-3 fatty acids are produced with an average price of €
250/kg dry biomass. The production takes place in open ponds,
single-layer horizontal tube reactors, vertical three-dimensional
tube reactors or , as shown in Figure 1, vertical flat plate reactors.
Production of algae for biofuels must take place on a much larger
scale at lower production costs to make it feasible. According to
Wijffels and Barbosa (2010) this means on a practical level that
the production must increase by at least a factor of three while at the same time the production costs must
decrease by a factor of ten. In order to achieve this, a multi-disciplinary leap must be made with the development
of microalgae technology. With genetic engineering can, for example, be examined whether microalgae strains
can be produced that yield more lipids. Also design can be optimized so that production can increase and
production costs can decrease.
The motivation for this work is to improve the design, concerning height and distance of vertical flat plate
reactors, to optimize algae production.
1.1 Problem description Microalgae are photosynthesizing organisms (Slegers et al., 2013)
and the research of Wijffels and Barbosa (2010) and Cuaresma et al.
(2011) showed that vertical reactors are more effective than
horizontal ones. Vertical reactors are, however, subject to the
canyon effect as can be seen in Figure 2 for a real canyon. The
canyon effect means that the light gradually extinguishes between
the vertical reactors in the direction of the earth. There is a model
that describes the canyon effect for an urban street canyon
(Robinson and Stone, 2004). However, the problem is that the
practical applicability of this model, that represents the quantity of
light as a function of the height and distance in angles, is unknown
for vertical flat plate reactors.
1.2 Goal The goal of this research is to test the applicability of the Robinson and Stone urban street canyon model for
diffuse light (Robinson and Stone model) on vertical flat plate reactors. If the Robinson and Stone model is not
applicable enough, the aim is to indicate how the Robinson and Stone model can be improved. When the canyon
effect for diffuse light can be modeled, the appropriate distance and height of the vertical flat plate reactors can
be calculated for each species of microalgae. As a result, the design of vertical reactors can be improved and
microalgae production can be made more efficient.
Figure 2. The canyon effect. The light
extinguishes while looking down the sides of
the canyon. Source: cdn2.vtourist.com
Figure 1. Vertical flat plate photobioreactors
with algae. Source: asulightworks.com
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1.3 Research questions To get a better understanding of the canyon effect so that microalgae can be cultivated more efficiently, the
following research question was drawn:
Can the urban street canyon model of Robinson and Stone (2004) be used to describe the extinction of
diffuse light between vertical plates with a certain height and a certain distance or can the model be made
more accurate?
To answer these questions a very important expected variable is reflection. To be able to identify this variable
two subquestions were formulated:
1. What is the reflectivity, or the albedo, of the components?
2. How diffuse, or how Lambertian, is the reflectance?
To answer these questions a vertical algae reactor row was simulated by means of white and grey painted MDF-
plates on which diffuse light fell. The ground was also simulated by an MDF-plate which was painted black and
white. This gave different combinations for difference in reflectance.
A real flat plate reactor is considerably longer in the horizontal direction than the length of the test setup.
Because the Robinson and Stone model assumes an endless street canyon, it had to be tried to extrapolate the
measured values to match the values of an endless row. This led to the third subquestion:
3. Can the measured values be extrapolated to match accurately the values of an endless row?
1.4 Delimitations Direct light is light coming from the sun and falls directly on an object. This means the object can ‘see’ the solar
disc. Diffuse light is coming from all directions from the sky due to scattering. This diffuse light can be isotropic
or anisotropic. Anisotropic diffuse light means that the quantity of light coming from the sky is not equal from
all directions, while isotropic means that the quantity of light coming from any direction of the sky is equal to
that coming from any other direction (Nicodemus et al., 1977). In this research direct light was not taken into
account and the sky was considered isotropic because this forms the basis for the Robinson and Stone model.
Due to the conservation of energy all light falling on a surface will be absorbed, reflected or transmitted (Cohen
and Wallace, 1993). When light is transmitted it passes through the reactor and can fall on the next reactor. It
should be noted that this transmitted light can be used by algae for photosynthesis, while absorbed light cannot.
If the reflected light is falling on the opposing reactor it can again be absorbed, reflected or transmitted and the
transmitted fraction of this reflected light can again be used by algae for photosynthesis. This research only
concerns absorbed and reflected light. Since MDF-plates were used to simulate a flat plate reactor, transmittance
did not take place because MDF is opaque for visible light. The Robinson and Stone model also does not take
transmittance into account.
Algae are phototrophic, which means they use sunlight as their energy source. Of the full electromagnetic
spectrum of the sun only a part is used for photosynthesis. This part concerns the light with wavelengths ranging
from 400 to 700 (Cuaresma et al., 2011) which is referred to as photosynthetically active radiation or
PAR which coincides largely with the visible light range of the spectrum (Mahmoud et al., 2007). The unit of
PAR is based on the number of photons that fall during a second on a square meter and is expressed in SI units
as . One mole is equivalent to photons. This description of the electromagnetic
radiance in purely physical terms is called radiometry
(Zwinkels et al., 2010). In this research the quantity of PAR
was measured because PAR represents the part of the
electromagnetic spectrum being used by algae.
In practice the amount of light falling on a flat plate reactor
will depend on its position relative to the other reactors as
can be seen in Figure 3. The first and the last reactor will
have no opposing reactor at one side which will result in a
different amount of light falling on it (blue arrows). Also
within a row differences occur. In the middle of a row
(yellow arrow) a reactor ‘sees’ its opposing reactor both to
the left and to the right. When going to the end of a row it
Figure 3. Representation of a field with vertical flat
plate reactors. Blue arrows depict outer reactors.
Red arrows depict outer sides of the reactors, the
yellow arrow depicts the center of a row.
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only ‘sees’ the opposing reactor either to the left or to the right (red arrows). If no other obstacles block the view
to the sky the reactor receives more light at the side of a row than in the centre. This research is based on the
‘worst case scenario’, so the amount of light in the centre of an endless row (the yellow arrow) was determined.
In practice vertical flat plate reactors are placed outside. Due to different weather conditions the irradiance on an
arbitrary point on the flat plate reactor can differ. When there is for example haze or rain in between or onto the
flat plate reactors bioreactors, water particles can act as absorbing and reflecting elements. Refraction of the light
can occur as well. This research was done in a laboratory to eliminate weather conditions.
Most flat plate reactors used for algae cultivation are made of some glasslike transparent material that might
have a more specular than diffuse reflection. The Robinson and Stone model assumes ideal diffuse reflecting
surfaces. To come as close as possible to diffuse reflectance the MDF-plates were painted with matte paint. Paint
giving a more specular reflection was not examined.
The sensors used in this research have certain dimensions but they were treated as sensors with an endless small
sensitive surface area. Such a small area is called a point from now on in this report. It was necessary to treat the
sensors as point sensors to be able to correct for certain measurement errors caused by the sensors.
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5
Equation 1
2. Theory 2.1 Albedo and reflectivity Albedo and reflectivity (or reflectance) are terms that are widely used in the field of remote sensing. Albedo is
defined as reflectivity of some surface to irradiance from the sun, which can be described as in equation 1 (Sailor
et al., 2006):
∫
∫
Albedo
Wavelength
Spectral reflectivity
Spectral irradiance
By this definition reflectivity is a component of albedo. Figure
4 shows the reflectance as a function of wavelength for a
tobacco leaf (Knipling, 1970). As can be seen from the Figure
the reflectance of the leaf differs per wavelength whereas the
formula for albedo integrates over the wavelength. Therefore
reflectance is the amount of electromagnetic energy reflected
by an object for a certain wavelength while albedo is the
amount of reflected electromagnetic energy of a spectrum. In
this research PAR was used, which consists of a range of
wavelengths, so every time reflectance or reflectivity is
mentioned, the definition of albedo is meant.
2.2 Reflection of light When light falls on a surface, part of it is reflected while the direction of a ray changes (scattering of light). The
way this reflection takes place depends on microscopic interactions between the light and the surface (Whitted,
1980). The reflection of a ray of light can, according
to Cohen and Wallace (1993), occur on three ways:
mirror reflection, diffuse reflection and glossy
reflection, as depicted in Figure 5. When a ray falls
on an object with some arbitrary angle relative to the
surface normal, the mirror reflection, or specular
reflection, reflects this ray with that same arbitrary
angle but now in the opposite direction of the
surface normal. The diffuse reflection, often called Lambertian reflection, reflects the ray in any direction with
the same radiance. The glossy reflection reflects most light specularly, but also scatters a part of it. Most surfaces
are non-ideal and reflect light in all three manners at the same time, depicted in Figure 5 as “BRDF”. Some
surfaces reflect more specularly while other surfaces reflect more diffuse. The amount of the contribution of a
reflection type can also depend on the angle of the incoming ray of light (Cogley, 1979 and Woolley, 1971). For
most surfaces the reflection will gradually be more specular when the incoming ray of light is more horizontal,
so when the angle of the incoming ray gets closer to 90° relative to the surface normal.
The Robinson and Stone model assumes all reflectance to be diffuse.
2.3 Incoming and reflected radiance 2.3.1 Cosine course of incoming radiance The quantity of irradiance on a surface depends on the angle between these two. Imagine a uniform beam of
light, e.g. a laser beam, which gives a completely round reflectance area when shining perpendicular on a
surface. When the surface is tilted the reflectance area becomes an ellipse. This ellipse will increase in size while
the angle between the surface and the beam increases until the angle reaches , then the beam and the surface
are parallel and no light falls on the surface anymore. This means that when the angle increases the total
irradiance is divided over a larger surface area until . If measuring the irradiance on some small area of the
surface that was completely irradiated when the angle was between until , the amount of irradiance for this
small area would decrease from the maximum value at perpendicular till zero at parallel. The decrease between
Figure 5. Possibilities of reflection. Source: Cohen and Wallace (1993)
Figure 4. Reflectance spectrum of a tobacco leaf. Source: Knipling (1970)
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this maximum and zero value follows a course corresponding to the cosine of the angle between the surface
normal and the incoming beam (Cohen and Wallace, 1993).
2.3.2 Distribution of the quantity of reflected radiance
Light reflecting from a Lambertian surface can
graphically be expressed as in Figure 5 “diffuse” and
Figure 7 left, but can also be expressed as in Figure 7
right. Both visualizations are true but they depict two
different descriptions of Lambertian reflectance
(Mobley, 2013). The right representation depicts the
intensity of the reflected light from one point. The
intensity of reflected light decreases following a cosine
course where the intensity is at maximum for the ray
that is reflected perpendicular to the surface. The left representation depicts the radiance, not the intensity. As
Figure 6 shows both angles are equal in size, but the area on the surface for blue, , is smaller than for red,
. When the area is resized to match the area of the angle (the red in Figure 6) becomes smaller.
This decrease follows a cosine course, just as it did for the intensity (Chelle, 2006). So the intensity decreases
when the angle becomes larger relative to the surface normal, but the view angle ( ) decreases by the same
factor, resulting in an angle-independent quantity of radiance (Figure 7, left).
2.4 Ray tracing and radiosity methods Tsangrassoulis and Bourdakis (2003) presented two ways of solving light equations. One is the ray tracing
method and the other is the radiosity method. With ray tracing the paths of rays are followed of which each ray
has a weight that corresponds to its level of intensity. When a ray encounters a surface, it is absorbed if its
intensity level is below a certain value, or it is reflected. When it is reflected new rays are formed of which each
has its own intensity level. This method excels when dealing with specular reflections. With the radiosity method
the scene is split up into patches where each patch receives and exhibits Lambertian reflectance. This is an
iterative process until, due to the absorption fraction, the amount of irradiance falls below a certain value. When
using this method the entire scene is described with angles between the patches. This method is very effective for
diffuse reflections. In this research matte paint was used to come as close as possible to diffuse reflectance,
which coincides with the prerequisite of the Robinson and Stone model.
2.5 Integrating over the
surface of a sphere When looking from some small area, a point, on a
surface, the sky can be represented as half of a unit
sphere as depicted in Figure 8. The point on the surface
lies on the origin whereas the surface is the xy-plane.
From the surface of the half sphere light radiates towards
the origin. The point on the surface can only ‘see’ sky. If
the sky is obstructed or if the point is not facing straight
up, the point ‘sees’ less sky (since part of the view will
now be either the obstruction or the surrounding
surface). To know how much radiance goes to the point,
the surface area of the half sphere must be known. To
solve this, spherical coordinates are best used. Spherical
coordinates are more practical than Cartesian coordinates (De Gee, 2005). Now two angles are needed, as
described by Cohen and Wallace (1993), the first is [ ] and describes the angle between the z-axis (the
Figure 7. Two graphical expressions of Lambertian
surfaces. Source: oceanopticsbook.info
Figure 6. Increase of surface area when view angle
remains constant but moves from perpendicular to
parallel. Source: Chelle, 2006
Figure 8. Representation of the sky dome as a half
sphere and the two angles and used to create it.
z
7
Equation 2
Equation 3
Equation 4
surface normal) and the xy-plane (the ground plane). The other is [ ] and describes the angle within the xy-
plane. The last parameter is radius . If runs from to with the result is a quarter of the
unit circle. When this quarter is rotated with running from to the result is a half sphere. However
when is smaller than the resulting horizontal circle after the rotation, parallel to the ground plane,
becomes smaller. This is also depicted in Figure 8 by the red area. Close to the z-axis the width is small, but the
width gradually enlarges when moving towards the ground plane. This enlargement of the width when
enlarges follows a sine course. The total width is dependent on the radius as well. If the radius becomes smaller,
the width also becomes smaller. This leads to the Jacobian: ( ) which is the result of the usage of spherical
coordinates (De Gee, 2005). But the unit sphere is used, so , and the Jacobian is simplified to: ( ). The
surface area, , can be calculated by the usage of a double integral, equation 2 (Nicodemus et al., 1977):
∫ ∫ ( )
The surface area of a full sphere is , so the surface area of a half sphere is . This means the
surface area in solely dependents on radius . This is well known for a
circle, as depicted in Figure 9. The circumference of a circle is [m] which
holds for any , so the circumference and the radius are related. The
circumference can therefore just as well be given in as in . Between a
sphere and its radius a similar relation exists, but with an extra dimension. The
squared radius is equal to a certain area of the sphere. This area is a fraction of
the total surface area. When a different radius is taken, the area differs as well,
but the fraction compared to the total area stays the same. This represents the
solid angle in steradians [ ]. The surface area of a sphere can of course be
calculated in , but can also be calculated in , because the surface area equals
for any . So a steradian is related to . The steradian is an SI unit
and is the two dimensional equivalent of the radian and is, just as a radian, in
fact dimensionless. This means the solution of equation 2 is correct if it is .
The calculation is described in Appendix A: Calculation of integrals and the
solution is indeed .
However, as said, there is a cosine course between the angle of an incoming light
ray, which depends on angle , and the amount of irradiance that the point in the
origin receives. To know to what extent the point in the origin receives
irradiance for the full sky, equation 2 has to be altered. This results in equation 3
for a half sphere (Nicodemus et al., 1977):
∫ ∫ ( ) ( )
The unit of both equation 2 and equation 3 is [ ] (Nicodemus et al., 1977). The calculation of equation 3 is
described in Appendix A: Calculation of integrals and the solution is .
2.6 The Robinson and Stone model The Robinson and Stone model consists of five equations that were used for this research. The first (Equation 4):
( )
Diffuse irradiance incident on a plane of slope
Diffuse horizontal irradiance [ ]
A plane of slope , (where 0 is hor. facing up, is vertical and hor. facing down) [ ]
Angular height of an equivalent continuous skyline; the obstruction angle of the view [ ]
Equation 4 calculates the irradiance on a certain height in the urban street canyon. It depends on the amount of
diffuse horizontal irradiance, the elevation angle of the plane itself and the height of the opposing buildings. This
equation gives the amount of irradiance on the plates that directly originated from the sky and does not take
reflectance in consideration. Because the reflectance is absent, equation 4 does not have to be iteratively solved.
The derivation of is somewhat complex, but has been simplified by Robinson and Stone (2006). The can be
Figure 9. Relation between
radius and circumference for
a circle (top) and between
radius and surface area for a
sphere (bottom). Source: mathisfun.com
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Equation 6
Equation 7
Equation 8
Equation 9
Equation 5
derived from the tangential relation between the width of the urban street canyon and the height from the point of
measurement to the top of the opposing building, see Figure 10. This relation can be written as equation 5:
(
)
Height opposing building starting from measurement height [ ]
Width of the urban street canyon [ ]
The second and third equation (resp. equation 6 and equation 7) describe the amount of irradiance caused by
reflectance of the opposing buildings. Equation 6 is valid for irradiance coming from above horizontal and
equation 7 is valid for irradiance coming from below the horizontal plane, see Figure 10.
( ) ( )
( ) ( )
Irradiance on due to reflectance of upper (above hor. plane) opposing buildings [ ]
Irradiance on due to reflectance of lower (below hor. plane) opposing buildings [ ]
Global irradiance incident on [ ]
Diffuse reflectance factor [ ]
Same as , but now the angle with the lower obstructing surface [ ]
The is the outcome from equation 4 for the opposing wall. The gives the factor of reflectance where one
means that all incoming radiance is reflected by the opposing wall and zero means all radiance is absorbed by the
opposing wall.
The fourth equation (equation 8) describes the amount of irradiance resulting from ground reflectance:
( )
Irradiance received on due to ground reflectance [ ]
The is the irradiance on where the height is zero. This is the result of equation 9. The reflected radiance
from the ground is build up of the reflectance of both sides of the urban street canyon and the radiance directly
from the sky, equation 9:
( ( ) ( ))
( ( ))
( ( ))
Irradiance on ground [ ]
Angle between the centre of the urban
street canyon and the walls [ ]
The first wall
The second (opposed to the first) wall
The first part of equation 9 yields the irradiance as a
direct result from the sky dome. The second and third part
describe the irradiance as a result of reflectance for both
sides of the urban street canyon. So when iterating this
equation the first part must be omitted.
The total irradiance for can be calculated by iteratively
solving equation 6, equation 7, equation 8 and equation 9.
Figure 10 is a graphical representation of an urban street
canyon and visualizes the , , and .
Figure 10. Urban
street canyon
geometry. Source:
Robinson and
Stone, 2004
9
3. Materials and methods 3.1 Test setup In order to simulate a flat plate reactor row three MDF plates were used. The dimensions of two of these plates
that were used as flat plate reactor walls were 244.6 by 122.2 . To simulate the ground the third MDF
plate was used, the dimensions were 40.0 by 122.2 . The thickness of all MDF plates was 1.8 . MDF
was the smoothest woody material available in Gamma, the local construction market. The smoother a material
is, the less light energy will be absorbed by internal shadowing (Goldschleger et al., 2001). The MDF plates
were painted with matte paint from the Histor Matte Lak series, in order to get reflectance as diffuse as possible.
The side plates were painted white (color code 6400) on one side and grey (code 6907) on the other. The ground
plate was also white on one side but the other side was painted black (code 6372), see Figure 11. To get an as
smooth as possible result, the paint was applied with a foam roller. For each paint a new roller was used to
prevent possible mixing of paints. Five holes with a diameter of 2.4 were drilled over the length of one side
plate to fit five light measuring sensors. The center of the top and bottom hole were 1.2 from the end of the
plate, the other three holes were evenly distributed over the plate. The sensors are described in section 3.3
Sensors. This sensor containing plate is referred to as <self>, the opposing plate is referred to as <plate> and the
ground plate as <ground>. The plates were examined in the combinations presented in Table 1.
Table 1. The examined combinations.
Combination: 1 2 3 4 5 6
‘Self’ Grey paint Grey paint White paint White paint Grey paint Grey paint
‘Plate’ Black cloth Black cloth White paint White paint Grey paint Grey paint
‘Ground’ Black paint White paint Black paint White paint Black paint White paint
A black cloth is introduced in Table 1 for combinations 1 and 2. This was done in an attempt to get (almost) no
reflectance from <plate>. For combination 1 this meant that all
measured light comes directly from the light source if there is no
reflectance by the black <plate> and the black <ground>.
Combination 2 adds only reflectance from <ground>. For
combinations 3 and 5 there is no reflectance from ‘ground’, but
there is reflectance between <self> and <plate>. It was expected
that white paint reflects more light than grey paint. For
combinations 4 and 6 reflectance occurs for all three parts.
Whether or not the black cloth and the black painted <ground>
indeed have negligible reflectance levels was tested by measuring
their albedo. This is described in section 3.5 Measuring albedo.
With metal parts (branded Syboka) a frame was built that
positioned the MDF plates, see Figure 11. <Self> was at a fixed
position. <Plate> could move in vertical direction towards <self>
resulting in difference in distance between the plates. All
examined distances are presented in Table 2. When <plate> was
positioned at 40 distance to <self>, <ground> could move
towards the light source resulting in difference in height. The
examined heights are presented in Table 3.
Table 2. Examined distances between the plates while <ground> was at a distance of 244.6 from the light source.
Examined distances (at height = 244.6 cm) [cm] 40 35 30 25 20 15 10 5
Table 3. Examined heights (distance between <ground> and light source) while the plates were at a distance of 40.0
.
Examined heights (at distance = 40.0 cm) [cm] 244.6 184.1 123.5 63.0 2.4
The light source is on the left side of Figure 11 and is described in section 3.4 Creating a diffuse light source.
The test setup was positioned in a horizontal direction. In practice flat plate reactors are positioned vertically,
however the horizontal positioning was used because of its practical benefits since all parts of the test setup were
easily accessible without the hassle of using a ladder to make adjustments.
Figure 11. The frame that positioned the MDF
plates. The upper plate (<self>) contains
sensors. The bottom plate (<plate>) can move
as depicted by the red arrow. The ground plate
(<ground>) can move according to the green
arrow. The white (on <self>), grey (on <plate>)
and black (on <ground>) paint is visible. The
light source is on the left side.
10
To refrain daylight from disturbing the measurements the complete test setup was covered with agricultural
plastic, obtained from AlgaePARC Wageningen. This plastic is black on one side and white on the other. The
black side was used to cover the test setup. Also parts of the frame that were in a position to disturb the
measurements by means of reflectance were covered with black plastic. The black plastic was used under the
assumption that it does not reflect light.
3.2 Data collecting and processing In this research a number of experiments were done where PAR measuring sensors were mounted on a test setup.
These sensors were connected to a 22 bit Agilent 34970A data acquisition system. Via an RS232 connection this
data logger was connected to a 64 bit PC running Microsoft Windows 7 Entreprise. This PC was running a
standalone National Instruments LabVIEW 2010, version 10.0f2 32 bit, script written by Ing. Kees van Asselt
(Wageningen University). This program, named hotrod_ProcEng_Agilent, controlled the data logger. The
measurement data collected with hotrod_ProcEng_Agilent was processed to a text file which was then read in by
Microsoft Office Excel 2010 for further processing. A 64 bit PC running openSUSE 12.2 was used to perform
calculations by means of MathWorks Matlab 7.9.0.529 (R2009B).
3.3 Sensors The sensors used in this research were six QSO-S PAR Photon Flux Sensors from Decagon Devices. Figure 12
was downloaded from the Decagon Devices website and raises the suspicion
that the sensors are manufactured by Apogee Instruments. The diameter is
, the height is and the length of the cable is . The top of the
sensor consists of a domed white lens with a field of view of . This sensor
gives a voltage as output which must be multiplied by five to transform the
measurement value in to , the unit of PAR. A tool was
made to fit all sensors equally in <plate>. The beginning of the domed lenses
were aligned with <plate>. In order to get acquainted with the characteristics
of the sensors to perform measurements reliably on the simulated flat plate
reactor, some prior experiments were done. First the linearity was examined. Second the sensors were check for
an off-set. Third the possible differences between ports of the data logger were tested and last the output of the
sensors during a longer period of time was tested for drift. A description of these experiments is presented in
Appendix B: Determining sensor characteristics. The sensors were found linear for the range of 20 to 1 000
. This means that all measurements on the simulated flat plate reactor row had to yield output values within
this range. The sensors did not have an off-set, so no correction had to be done. The measurement values were
not influenced by the used ports of the data logger. One sensor was found to be prone of drift and was replaced.
3.3.1 Correcting the sensor specific error An experiment to test a possible gain of the sensors was not performed due to a lack of a calibrated sensor.
Therefore the average output values of the sensors were taken as the value that each sensor should have been
given during measurements. In order to find the deviations from this mean value to be able to calibrate the
sensors, an experiment was done. This experiment is described in Appendix C: Determining the sensor specific
error. Due to the sensor’s linearity the sensor specific error correction factors, , are constant for the range of
to . The output of the sensors during measurements at the test setup was always corrected for
the sensor specific error by means of multiplying the measured values with these correction factors.
The irradiance levels closer to the light source are higher than further away. Therefore the sensor with the
smallest correction factor was placed closest to the bottom of the test setup. When the measurement value is
the sensor specific error corrected value is
(a bit) lower. This extents the linearity range a
little. For the same reason the sensor with the
largest correction factor is placed towards the
light source to measure the irradiance on the
horizontal plane. A schematic view of the test
setup and the positioning of the sensors is
depicted in Figure 13.
3.4 Creating a diffuse light source Two Variolux 2500 lamps were used, see Figure 15. Each lamp consisted out of two independently operable
halogen lights. All examined distances and heights (Table 2 and Table 3) were examined with four,
Figure 12. QSO-S PAR Photon
Flux Sensor from Decagon
Devices. Source: Decagon
Figure 13. Schematic view of the test setup. The grey plane on
the left is the light source. The six black ovals represent the
positioning of the sensors.
11
Figure 14. Hotspots with acrylic fiber cloth. The green
line and the sewing seam are visible at the right side.
Figure 15. Lamps shining on the prismatic glass and
behind that the acrylic fiber cloth.
three, two and one light turned on as long as no sensor would give a measurement value outside the range of
linearity. An exception was made for the lower boundary of linearity. If a measurement with all four lights
turned on resulted in a measurement value below 20 , this value was still kept because it was not possible to
increase the amount of light since no more than four lights were available. All measurements on the test setup
were done during one minute with a sampling time of two seconds. Then these results were averaged. The light
entering the test setup had to be made diffuse first. In order to achieve this, an acrylic fiber cloth (diffuse cloth)
was folded three times, resulting in eight layers. The cloth was obtained from AlgaePARC, Wageningen. The
length was 60 and the width was 122.2 after folding which was identical to the width of the plates of the
test setup. In the middle of the diffuse cloth a green stripe and a sewing seam were present (Figure 14). This
piece of diffuse cloth was the only obtainable piece and therefore the stripe and seam had to be ignored. The
diffuse cloth consists out of a meshed network of fibers. This causes the light to get diffracted. However the sole
usage of the acrylic fiber cloth led to hotspots, as can be seen in Figure 14. Also the diffusivity was inadequate
since much higher irradiance levels were observed near the cloth’s perpendicular then towards its parallel. To
overcome these problems a prismatic glass plate, length and width , was introduced. This is a
glass plate consisting of numerous tiny cones. These cones ensure lights get refracted. In order to check if this
refraction lessens the intensity of the hotspots and at the same time improves the diffusivity, a focused beam of
light from a laser was shone perpendicular to the glass plate. The resulting pattern was photographed, see Figure
17. The focused beam was diverged to over an distance of . The angle of divergence is then:
(
) . The width of the prismatic glass is 10.2 smaller than the diffuse cloth. Therefore
the distance between the prismatic glass and the diffuse cloth had to be at least: (
) (
) .
As can be seen in Figure 17 the resulting pattern is not a circle with equally distributed intensities, therefore the
distance was, for certainty, set to during measurements. The result of the combination of prismatic glass
and diffuse cloth is depicted in Figure 16. Figure 15 shows the used combination itself. There are no visible
hotspots for the eye anymore. In order to check for both hotspots and the level of diffusivity the radiance was
measured at four arbitrary points. At each point at the diffuse cloth three measurements were done, one at angle
(perpendicular to the diffuse cloth), one at angle and one at angle (parallel to the
diffuse cloth). If the diffuse cloth was endlessly long and wide the output of the sensors at must be
Figure 16. No visible hotspots. Behind the acrylic fiber cloth
the prismatic glass is placed. The two arrows depict the
placement of the sensors in ‘self’ during measurements.
Figure 17. Schematic view of laser and prismatic
glass. On the bottom the pattern as a result of
refraction by the prismatic glass is visible.
12
of the output at . This is because at the sensors ‘sees’ the diffuse cloth for of its total field of
view of whereas at it ‘sees’ the diffuse cloth completely within its field of view. For the same
reason the output of the sensor must be at . Each measurement was done for one minute with a
sampling time of two seconds and the outputs were corrected for the sensor specific error. Just as with the
determination of the sensor specific error, see Appendix C: Determining the sensor specific error, the
measurements at the different angles were not done simultaneously because the measurements had to be done
at the same place. A control sensor was used which had a fixed position. By dividing the measurement value by
the value of the control sensor the deviations caused by the fluctuations of the mains over time were eliminated.
3.5 Measuring albedo Figure 5 depicts reflectance patterns. For the combined pattern, ‘BRDF’, the radiance is not equal in all
directions. However equation 1, that calculates the albedo by dividing reflected radiance by the incoming
radiance while integrating over wavelengths, does not take direction into account. Albedo is therefore a
simplification since it only looks at quantities of reflected electromagnetic radiance without taking into account
where that radiance is going to.
When measuring albedo the distance between the light source and the to be measured surface must be as small as
possible (Reifsnyder, 1967). Otherwise the radiance at an angle close to will not be able to hit the to be
measured surface. This also counts for the reflected radiance. When taking this distance into account the
influence of the reflectance pattern also diminishes when a diffuse light source is used. The radiance coming to
the surface causes an ‘BRDF’ pattern, but since that radiance was diffuse the protrusion of the pattern goes in all
directions, yielding diffuse reflectance.
To measure the albedo the painted plates were placed in front of the diffuse cloth with a distance of , see
Figure 18. During one minute every two seconds a
measurement was done with one, two, three and
four lights turned on. One sensor was attached to
the MDF plate and measured the incoming
radiance while the other was attached to the
diffuse cloth and measured the reflected radiance.
Both sensors were positioned in such a way that
each other’s influence on the measurements was
as small as possible. This was done by placing the
sensors in the middle of the half of the opposing
object they were receiving radiance from. The
distance between the diffuse cloth and the plate
was not smaller than because the sensor
measuring the reflectance would otherwise block
too much of the incoming light to the plate because of its own dimensions. This way the white, grey and black
painted MDF plates, the black cloth and the black plastic were examined, resulting in respectively , , ,
and . The measurement values of both sensors were corrected for their sensor specific error. The albedo was
yielded by dividing the reflected values by the incoming values.
3.6 Determining Lambertian reflectivity Matte paint was used to paint the MDF plates in order to get
the reflectance as Lambertian (diffuse) as possible. In order to
gain some insight in the level of success a qualitative
experiment was done. A laser beam was shone onto the white
and grey painted MDF plates under an angle of with a
white A4 paper that was placed perpendicular to the MDF
plate, see Figure 19. The laser was tilted from the MDF
plate causing reflectance onto the A4 paper. Then a picture
was taken of the A4 paper. Another picture was taken with the
laser at a tilted angle of with the MDF plate. The laser had
a focused beam of light but if the reflectance was lambertian
there are no bright spots visible on the A4 paper, since these
are caused by specular reflectance. Also the shape of the
reflectance, a half sphere as depicted in Figure 5 “diffuse”,
must be equal for both angles and paints.
Figure 18. Albedo measurement setup. Left sensor measures
incoming radiation, right sensor the reflected radiation.
Figure 19. Setup to qualitatively examine the level
of Lambertian reflectance. A photograph was
taken of the white paper.
13
3.7 Extrapolating: from test setup to endless row The used test setup is wide. In practice a flat plate reactor row will be much longer. The Robinson and
Stone model assumes even an endless long row. Therefore the measured values resulting from the test setup had
to be extrapolated to the values that would have been measured at an endless row. This was done by using
modified versions of equation 3. All incoming fractions were determined: the diffuse light that fell directly on a
(sensor in the) plate, reflected light from the ground and reflected light from the opposing plate. For all examined
distances and heights for every sensor height these fractions were calculated.
In essence these fractions are solid angles. They represent the angular fraction a sensor ‘sees’ of the components
of the test setup and of a flat plate reactor, imagined as an endless long test setup.
3.7.1 Irradiance from the sky Endless row The irradiance decreases when angle increases, this means that equation 3 can be used, but that the boundaries
have to be altered. See Figure 20 A and B. The sky is represented as the blue plane, which can be imagined
endless long in Figure 20 A. When a line is drawn from the origin, which represents a to be measured point in
the flat plate reactor, to the upper boundary of the depicted sky plane, the line intersects with the half sphere. The
angle of this intersection depends on the angle that was used to draw the line. The blue line, ( ), in Figure
20 A represents all these intersections. From this ( ) until the origin receives radiance from the
sky. The lowest value for (as said, at the z-axis and goes to when reaching the xy-plane) is
found when because then the shortest distance between the origin and the sky plane is found.
Because the sky plane runs parallel to the y-axis, when ( equals ). So the function of ( ) has to be known since this function forms the lower boundary of the inner integral
of equation 3. If the sky plane of Figure 20 A was wider (a larger z-axis value), ( ) would be smaller. In
practical sense this means that the distance between the flat plate reactors would be bigger. If the sky plane was
closer to the origin, ( ) would also be smaller. In practice this means that the height of the flat plate reactor
would be smaller. A smaller ( ) means that the fraction of the sky projected on the half sphere enlarges, so the
origin receives more radiance from the sky. In the test setup the distances between the plates was according to
Table 2 and the heights according to Table 3. However the center of the assumed point sensors lies
higher than the examined heights what follows from the sensor’s diameter of . This means that the height
of the plates for the calculations was adapted to the sensor heights presented in Table 4.
Table 4. Height of the five sensors in relation to <ground>.
Height of the sensors [cm] 243.4 182.9 122.3 61.8 1.2
Figure 20 A (left) and B (right). A represents an endless row, B represents the test setup. The blue planes represent
the sky, the blue (in A) and red (in B) lines represent the starting angle θ for all ϕ values and the yellow plane in B
represents all ϕ values when the row is not endless long. The red arrow in B shows the yellow plane beneath the red
line that represents the surface area where radiance is received from for the test setup.
Now ( ) had to be determined, this is the ( ) that describes the boundary between the sky and
<plate>. It also depended on the width of the plates, which is . See Figure 21 A and B for a graphical
representation. In this figure lies between the horizontal xy-plane and the blue line that limits the width of the
14
Equation 11
Equation 12
Equation 10
diffuse cloth, while has to lie between the z-axis and this line. This means that equals . For
( ), Figure 21 A, this leads to
equation 10:
( )
(
)
( ) Boundary between sky and plate [ ]
Height of simulated flat plate reactor [ ]
Height of the sensors [ ]
Distance between the plates [ ]
Then was introduced, see Figure 21 B. In this figure a
black line has been drawn that indicates a shift caused by
on the yellow plane that represents the diffuse cloth. The
distance between the sensor and the diffuse cloth at is
known: , this is represented by the thin red line. The
length of the shift in the diffuse cloth was then found by the
tangential relation between and the distance of the sensor
to the diffuse cloth at . Now the distance between the
sensor and the diffuse cloth at can be found by applying
the Pythagorean Theorem. This resulted in equation 11:
( )
(
√( ( ) ( )) ( ) )
Now ( ) is known, the irradiance fraction for the endless row was calculated by means of altering
equation 3. The outcome of this altered equation was divided by , which was the outcome of equation 3, to
give the fraction (between 0 and 1) of the total unobstructed sky (the full half sphere), see equation 12:
( ) ∫ ∫ ( ) ( )
( )
( ) Irradiance fraction for the sky for the endless row [ ]
( ) Equation 11 [ ]
Test setup The test setup had a limited width which
means it was limited in the direction as can
be seen in Figure 20 B. If enlarges, also
has to enlarge which means ( ) had
to be determined. This function describes the
boundary between the sky and the black
plastic (the black plastic started where the
plates ended). Using the same methodology as
was used to determine ( ) for
equation 11, ( ) was determined,
resulting in equation13.
( )
( ( )
)
( ) Boundary between sky and
black plastic [ ]
Width of diffuse cloth [ ]
Since both ( ) and ( ) are functions, the irradiance fraction could not be calculated with one
double integral. Figure 22 shows the two dimensional course for a ( ) and a ( ). As can be
Equation 13
Figure 22. Two dimensional representation of ( ) and
( ) for , , and
. The shaded and the by green arrows indicated
area represents ( ).
Figure 21. A (top) and B (bottom). Graphical aid to
determine θ(ϕ). For A: ϕ=0 and for B: ϕ=ϕ. The
yellow plane represents the diffuse cloth. The green
plane represents <self>. The black point represents
a sensor. Variable A, B and C are first used in
equation 10, D is first used in equation 13.
( )
( )
15
Equation 15
Equation 16
Equation 14
seen in the figure there are two intersections. These intersections had to be found first in order to set up the
integral. The intersections were found after rewriting equation 13 (positive) to equation 14:
( ) ( ( ) ( )
)
Then ( ) ( ) was used to find the ( ) values of the intersection. This
yielded a very long equation which is therefore presented in Appendix D: Long equations, equation 35. After the
( ) values of the intersection were known, they were used as input values of equation 11 to
calculate the associated ( ) value.
After these intersection values were known, the integral was determined with the aid of ( ) of equation
11, resulting in equation 15:
( )
(∫ ∫ ( ) ( )
( )
( )
( )
∫ ∫ ( ) ( )
( )
( )
( )
)
( ) Irradiance fraction for the sky for the test setup [ ]
( ) value of intersection [ ]
( ) Equation 11 [ ]
( ) value of intersection [ ]
( ) Equation 13 [ ]
The first integral of equation 15 calculates the shaded part of Figure 22 and the second integral calculates the
two parts depicted by the green arrows.
3.7.2 Irradiance from the ground Endless row and test setup The dimensions of the diffuse cloth (sky) and <ground> were during all measurements identical. This means the
irradiance fractions calculated with equation 12, the ( ), and equation 15, the
( ), also applied for <ground>. However, an inversion had to be done. The
upper sensor ‘saw’ much more sky than it ‘saw’ <ground>. For the lower sensor the opposite was true. Since the
dimensions of sky and <ground> were identical the upper sensor ‘saw’ just as much sky as the lower sensor
‘saw’ <ground>. This means that ( ) and ( ) are
equal to ( ) and ( ) but in reversed order concerning height
of the sensor. The subscript stands for ground. The same height conversion was also done for the intersections
( ( ) ( )).
3.7.3 Irradiance from the opposing plate Endless row Seen from a sensor in the flat plate reactor the opposing plate is also in its zenith,
, as depicted two dimensionally in Figure 23 with the horizontal blue line. Therefore the
irradiance fractions caused by the parts of the opposing plate above and below zenith had to
be calculated separately. Practically this means determining the radiance from above and
below the horizontal plane. The ( ) that describes the boundary for the sky is known:
( ) of equation 11. The ( ) that describes the boundary for <ground> with
<plate> can easily be determined since the course of the function is similar to ( ) of
equation 11, but now the distance between the sensor and the ground is needed, not the
distance between the sensor and the diffuse cloth. This means that had to be rewritten
to , see equation 16.
( )
(
√( ( ) ) )
Figure 23. The
blue line
represents the
zenith of the
green point that
divides the
opposing plate in
two fractions,
one above and
one below the
zenith line.
16
Equation 17
Equation 18
Equation 19
( ) Boundary between ground and plate [ ]
By using ( ) and ( ) of equation 11 and equation 16 the integral was set up, resulting in
equation 17:
( )
∫ ∫ ( ) ( ) ( )
∫ ∫ ( ) ( )
( )
( ) Irradiance fraction for the plate for the endless row [ ]
( ) Equation 11 [ ]
( ) Equation 16 [ ]
The first integral of equation 17 calculates for the above horizontal and the second integral for the below
horizontal portion.
After the calculations for all examined distances and heights for the sky, the ground and the opposing plate were
performed, a check was done. Because ( ), ( ) and ( ) are
fractions, the sum for any respective sensor height had to be one in order for these equations to be correct.
Test setup By using the same methodology as was used to determine ( ) of equation 11, ( ) was
determined, resulting in equation 18.
( ) (
√( ( ) ) )
( ) Boundary between plate and black plastic [ ]
Equation 18 holds for both above and below the horizontal plane since the width, , of the opposing plate is
above and below the horizontal plane the same.
The intersections for above and below horizontal plane, ( ( ) ( )) and
( ( ) ( )), had to be known to be able to set up the integral. However, the
( ( ) ( )) values were already known since they were already calculated for
( ) , equation 15. The ( ( ) ( )) values were also already known,
since they are identical to ( ( ) ( )) but in reversed order concerning sensor height, as
was just the same for ( ).
Now all intersection values were known the integral to calculate the irradiance fraction was set up, see equation
19:
( )
(∫ ∫ ( ) ( )
( )
( )
( )
∫ ∫ ( ) ( )
( )
( )
( )
∫ ∫ ( ) ( )
( )
( )
( )
∫ ∫ ( ) ( )
( )
( )
( )
)
17
( )
Irradiance fraction for the plate for the test setup [ ]
( ) value of intersection with the sky [ ]
( ) Equation 11 [ ]
( ) value of intersection with the sky [ ]
( ) Equation 18 [ ]
( ) value of intersection with the ground [ ]
( ) Equation 16 [ ]
( ) value of intersection with the ground [ ]
The first two integrals of equation 19 calculate the above horizontal and the second two the below horizontal
portion.
3.8 Correcting the sensors cosine course error When light radiates to the sensor at an angle of the output of the sensors must be zero, however the
sensors have domed lenses. This dome caused a measurement value when light came in at an angle of because the light did not go over the dome as would be the case with a flat lens. An experiment was done to
evaluate the actual cosine course of the sensors and compared that with the theoretical cosine course that the
sensors should have exhibited. See Appendix E: Determining the cosine course error for a detailed description
of the experiment and the yielded equations. Then the percentage difference of the actual cosine course to the
theoretical cosine course was calculated. See equation 37 and equation 38.
However, light coming in at a larger angle contributes less to the total measurement value than when it comes
in at a smaller angle . The contribution to the total measurement value at this larger angle also depends on the
starting angle where irradiance is received from. When this starting angle enlarges, the contribution to the total
measurement value of the irradiance received from the larger angle increases. This is described by equation 39,
equation 40 and equation 41.
Then by means of equation 42 and equation 43 the average starting angles for respectively <ground> and the sky
were determined. For <plate> the average stopping angle had to be determined since <plate> is ‘visible’ for the
sensor from zenith till the end of the plate where the black plastic begun, see equation 44.
The percentage difference of the actual cosine course to the theoretical cosine course and the weight of the
irradiance coming in under a certain angle corrected for its starting angle were combined to form equation 45.
Now, by means of equation 46, equation 47 and equation 48 the respective cosine course correction factors
(
) for the sky, (
) for <ground> and (
) for <plate> were calculated.
3.9 Irradiance onto the ground and plate In section 3.7 Extrapolating: from test setup to endless row the irradiance fractions received by the sky, plate and
ground were calculated. However, these irradiance fractions describe the amount of radiance that is received if
the amount of radiance send out by sky, plate and ground are all equal to each other. This is because the solid
angles were calculated and each point of the half sphere radiates as much as any other point on the half sphere. If
the sky produced one unit of light then the ground and plate produced in reality less than one unit since a portion
of the light was absorbed. This means the amount of irradiance received by <ground> and <plate> from the sky
had to be calculated. Multiple reflections between the components were omitted.
3.9.1 Irradiance onto the ground The irradiance onto the ground can be imagined by a two
dimensional view of an endless row, see Figure 24 A and B. In
both figures there is one point in the sky that radiates diffuse light
onto the ground. The irradiance on the ground is highest straight
under the point. This is depicted by the grey lines. The further
away from the grey line the lower the irradiance is due to the
enlarging angle between the ground and the point. In Figure 24 A
the highest irradiance is in the middle and the lowest at the sides.
For Figure 24 B the highest is at the left side while the lowest is
at the right side. It is lower than on the right side of Figure 24 A.
If it is now imagined that the sky radiates from every point in the
two dimensional sky, the highest irradiance is still in the middle
Figure 24 A (left) and B (right). Schematic
view of a flat plate reactor row. In A and B
one point of the sky radiates diffuse light onto
the ground. The grey line shows where the
irradiance on the ground is highest.
18
Equation 20
Equation 21
Equation 22
Equation 23
because there the average angle is smallest between the ground and all points of the sky. At the sides is the
largest average angle between the ground and all points of the sky. Therefore the irradiance is lower at the sides
than in the middle. This means the canyon effect not only resides on the side plates but also on the ground. In
practical sense for a flat plate reactor row that has a distance of and a height of this means that
the irradiance fraction at the sides is ( (
)) compared to the center. The test setup also had a
limited width and therefore the center point received most irradiance, both to the sides with the black plastic and
to the plates it is lower. However, the difference in irradiance fraction is small and therefore one average value
was calculated. This was again done by integrating over a half sphere. The sphere was ‘placed’ on the diffuse
cloth, facing <ground> and the functions that describe the borders between the components were determined.
First a function was setup that describes the border between the ground and the plates by using the same
methodology as was used to determine ( ) for equation 11, see equation 20:
( ) (
√( ( ) ) )
( ) Boundary between the ground and the plates for irradiance onto the ground [ ]
For the test setup a function had to be set up that described the boundary between the ground and the black
plastic, see equation 21:
( ) (
√( ( ) ) )
( ) Function that describes the border between the ground and the black plastic for irradiance onto
the ground [ ]
To find the intersections between
( ) and
( ) of equation 20 and equation 21,
( )
was rewritten, see equation 22:
(
)
(
√
( )
( ( ))
)
Then the
( ) value of the intersections was found by applying
( ) (
) .
The ( ) intersection values are described by equation 23:
( )
(
√ √ )
( ) values of intersection between ground, plate and black plastic for irradiance onto the ground
[ ]
The associated
( ) values were found by entering ( ) as in equation 20.
Then the integrals were set up. Just as in Figure 22, the to be integrated area was split up into pieces in order to
be able to solve the integrals. When looking straight down the half sphere the centre point of the ground lies in
the zenith, but the projection of the ground is equal for both sides. Each to be integrated part had therefore to be
multiplied by two. This led to equation 24 for the irradiance fraction onto the ground:
19
Equation 24
Equation 25
(
)
( ∫ ∫ (
) (
)
( )
( )
( )
∫ ∫ (
) (
)
( )
( )
( )
)
(
)
Average irradiance fraction onto the ground for the test setup [ ]
( ) value of intersection for irradiance onto the ground [ ]
( ) Equation 20 [ ]
( ) value of intersection for irradiance onto the ground [ ]
( ) Equation 21 [ ]
For the endless row the irradiance fraction onto the ground was then easily calculated since there is no black
plastic limiting the width meaning that only ( ) of equation 20 remained. This led to equation 25:
(
) ∫ ∫ (
) (
)
( )( )
(
) Average irradiance fraction onto the ground for the endless row [ ]
( ) Equation 20 [ ]
3.9.2 Irradiance onto the plate The irradiance onto the plate differs per height since it is subject to the canyon effect. The ( ) and
( ) of equation 12 and equation 15 formed the irradiance fractions onto <self> for every sensor
height for the test setup and the endless row as a result of the irradiance of the sky. Exactly the same amount of
irradiance was received by the opposing plate. Therefore the ( ) and ( ) were
used as the irradiance fractions onto the plate for every sensor height. When looking at the irradiance that was
received by <self> by using this method a simplification was applied. The irradiance <self> received from
<plate> was assumed homogeneous, meaning that when calculating the reflected radiance by <plate> the entire
plate was assumed to reflect as much as it reflects at the sensor height where was calculated for.
3.10 Correcting, extrapolating, normalizing and averaging the
measurement data 3.10.1 Correcting and extrapolating the measurement values All measurement values were corrected for their sensor specific and cosine course errors and were extrapolated
to values for an endless row. These corrected and extrapolated values, (
),
from now on abbreviated to , were calculated by means of equation 36, which is, due to its length,
presented in Appendix D: Long equations.
3.10.2 Irradiance at the horizontal plane In practice reflected light in a flat plate reactor row can disappear into the sky. At the test setup this light hits the
diffuse cloth and could have been reflected back into the test setup. This reflected light was not measured by the
sensor measuring the irradiance at the horizontal plane because it was placed outside the test setup in order not to
disturb the measurements, see Figure 13. Therefore the irradiance values for the horizontal plane, (
20
Equation 26
Equation 27
), from now on abbreviated to , were extrapolated from the
of the top sensor, see equation 26:
(
)
( ( ( ) (
)
( ) ( ) )) ( )
(
)
Extrapolated irradiance values for the horizontal plane [ ]
The share of the irradiance values of the ground and the plate contributed to the of the top sensor and
were therefore subtracted from the , so only the share caused by the sky remained. That value was
divided by the fraction irradiance caused by the sky to reconstruct the value that would have been yielded if the
sky was in the sensor’s total field of view of .
3.10.3 Normalizing and averaging the corrected and extrapolated
measurement values As long as the lowest measurement value with four lights turned on was at least the measurement was
redone with three, two and one light(s) turned on. This yielded lower irradiance values for all sensors. The
fraction of irradiance of a sensor in <plate> in relation to the irradiance at the horizontal plane remained constant
regardless of the quantity of incoming light to the test setup. This means the fraction of irradiance in relation to
the irradiance at the horizontal plane had to be calculated. Because the fraction remained equal regardless of the
number of lights that were used, the average fraction values were calculated, see equation 27:
(
)
∑
(
)
Overall corrected, extrapolated, normalized, and averaged values [ ]
Maximum number of lights that were used for a measurement [ ]
(
) is from now on abbreviated to .
The calculated values by Robinson and Stone were compared with the values.
3.11 Using the Robinson and Stone model The Robinson and Stone model was presented in section 2.6 The Robinson and Stone model.
If the experiment to measure the albedo, as described in section 3.5 Measuring albedo, yielded that black cloth
and black paint actually did not reflect any light, the applicability of the individual equations of the Robinson
and Stone model could be evaluated. For combination 5, see Table 1, only radiance by the sky was received,
which means equation 4 alone described the . For combination 6 equation 8 and equation 9 are
added to this and for combination 1 to 4 equation 6 and equation 7. If the albedo values of black cloth and black
paint make it impossible to evaluate the individual equations, then it is only possible to evaluate the entire
Robinson and Stone model.
For the Robinson and stone model first the angles (for equation 4 and equation 5), (for equation 7 and
equation 8) and (for equation 9), all are depicted in Figure 10, were determined. Only had to be determined
21
Equation 28
Equation 29
Equation 30
Equation 31
because in all examined situations the height of <self> was equal to that of <plate>. This means that .
The angles , and were determined by means of respectively equation 28, equation 29 and equation 30.
Note: equation 28 is similar to equation 5, but equation 28 is adapted for the test setup.
( ) (
)
( ) (
)
( ) ( ) (
)
In all examined situations both plates of the test setup were perpendicular to the ground. This means that (used
in equation 4, equation 6, equation 7 and equation 8) equals to .
The were normalized to the horizontal irradiance entering the test setup. This means that , used
in equation 4 and equation 9, also had to have a value of one.
Step 1 The irradiance fractions on the plates caused by the sky, , were calculated by means of equation 4 per plate
for every sensor height, see Figure 25.
Step 2 The irradiance fractions due to reflected radiance by both plates onto each other was calculated by means of
equation 6 and equation 7, see Figure 26 and Figure 27, for the reflected radiance from <plate> onto <self> both
for above and below the horizontal plane. The that is used as input equals that was calculated at step 1.
The equals the albedo, , or , of the respective opposite plate. This way the reflected radiance for
above and below the horizontal plane, and , was jointly calculated. When no black cloth was used the
irradiance for both plates is equal but if <plate> was covered with black cloth these values differ because <self>
was then grey.
Step 3 By means of equation 9 the irradiance fraction onto the ground, , caused by radiance directly from the sky
and reflected radiance from both plates, was calculated, see Figure 28. The variable equals either or
since <self> was limited to white or grey. Variable equals , or because <plate> was either white,
grey or covered with the black cloth. The equals the summed outcomes of equation 6 and equation 7
( ) for <self> and for that of <plate>, calculated in step 2.
Step 4 Irradiance fractions on the plates, , caused by the ground were calculated by means of equation 8, see Figure
29 for a visualization for one of the plates. The equals that was calculated in step 3. The equals
or depending on whether the white or black side of the ground plate was used.
Step 5 After the calculations described in step 1 to 4 were done the irradiance fractions on <self> for every sensor
height, , were calculated for all examined distances and heights by applying equation 31:
So follow by the summation of step 1, step 2 and step 4 for all outcomes concerning <self>.
Figure 25. Equation
4. Irradiance from
the sky onto the
plates.
Figure 26. Equation
7. Irradiance from
the opposing plate
below horizontal.
Figure 27. Equation
6. Irradiance from
the opposing plate
above horizontal.
Figure 28. Equation
9. Irradiance from
the sky and the
plates onto the
ground.
Figure 29. Equation
8. Irradiance onto
the plate from the
ground.
22
Equation 33
Equation 34
Equation 32
Step 6 The reflected radiance can again be reflected, therefore some steps were repeated. Step 2 was first repeated. In
this step (step 6) the equals the summation of (from step 2) for the opposing plate, so the
calculated in step 2 for <self> forms in this step (step 6) the for <plate> and vice versa.
Step 7 Step 3 had to be repeated, but since the sky does not reflect light (that was also the reason step 1 was not
repeated) now equals zero. The and now equal the outcomes of step 6 for respectively <self> and
<plate>.
Step 8 Then step 4 was repeated where now equals the calculated in step 7.
Step 9 After this second iteration, the irradiance fractions, , were calculated by applying equation 32:
This means the irradiance on <self> now equals the summation of the irradiance calculated in step 5 but with the
reflected energy calculated in steps 6 and 8 added.
Step 10 Steps 6 to 9 were repeated until in total five iterations were done. This yielded the calculated values of the
Robinson and Stone model, the .
3.12 Comparing calculated with measured values Both the and the (by the Robinson and Stone calculated values) represent a fraction of
irradiance compared to the irradiance received at the horizontal plane just above the flat plate reactor row. The
were assumed to be correct. The were compared with the by subtracting
the from the and multiplying the outcome by 100 to get the percentage difference of
irradiance for all measured heights on <self> for all examined distances and heights compared to the irradiance
at the horizontal plane just above the flat plate reactor row, see equation 33.
(
)
( )
(
)
Relative differences of irradiance to compared to the irradiance at the horizontal
plane just above the flat plate reactor row [ ], from now on abbreviated to .
The by the Robinson and Stone model calculated irradiance fractions after five iterations [ ]
Also the percentage difference of the calculated values to the were determined, equation 34.
(
)
(
)
(
)
Relative differences of irradiance compared to the corresponding [%], from now
on abbreviated to .
For example, the is and the is , then the is ( )
and the is (
) .
23
4. Results The results of the sensors cosine course error correction, the extrapolations for irradiance caused by the sky,
ground and opposing plate and the irradiances onto the ground and plate are intermediate results in order to
calculate the and are, due to their vast numbers, presented in respectively Appendix F:
Corrections for the cosine course error, Appendix G: Extrapolations and Appendix H: Irradiance onto ground
and plate.
4.1 Sensor specific error correction The sensor specific error correction factors, , to correct each sensor to the average value of all sensors is
presented in Table 5.
Table 5. The correction factors for all sensors to correct for the sensor specific error, .
Sensor nr.: 9003 8755 8585 8998 8754 8620
Sensor position: Bottom Second Middle Fourth Top Diffuse cloth
Sensor specific correction factor [ ] 0.979 0.986 0.990 1.020 1.022 1.027
Sensor 8620 has with the largest sensor specific error. Each measured value had to be multiplied with
and therefore sensor 8620 was placed towards the diffuse cloth. Sensor 9003 was placed nearest to <ground>
because it extended the lower boundary of linearity of 20 a bit.
4.2 Diffusivity of light source The results of the experiment where the diffuse cloth was examined at 4 arbitrary points is presented in Table 6.
Table 6. Results of the measurements at four arbitrary points at the diffuse cloth at three angles. Relative deviation to
mean must be if there are no hotspots. Mean relative irradiance must be at , at and at for the diffusivity to be ideal.
Arbitrary position: 1 2 3 4 1 2 3 4 1 2 3 4
Angle with diffuse cloth [ ] 0 45 90
Relative deviation to mean [ ] -1.1 8.7 -11.5 3.9 -6.3 5.8 -7.0 7.5 -9.9 -0.1 -2.9 12.8
Mean relative irradiance [ ] 100 79.7 37.8
The relative deviation to the mean of the measurements was measured to be up to , so although hotspots
are not visible for the eye, see Figure 16, the results show that hotspots were present.
The mean relative irradiance at was higher and at was lower than it should have been for
the diffusivity level to be ideal.
4.3 Albedos The results of the albedo measurements are presented in Table 7.
Table 7. Albedos of examined materials, .
White paint ( ) Grey paint ( ) Black paint ( ) Black cloth ( ) Black plastic ( )
Albedo [ ] 0.658 0.076 0.042 0.018 0.044
As expected white paint has the highest albedo, it reflected almost two thirds of the incoming radiance. Grey
paint reflected almost ten times less than white paint. Black paint reflected about half as much as grey paint. The
albedo value of black cloth is almost of the albedo of grey paint. The black plastic, that was assumed to not
reflect any light, reflected almost the same as black paint.
4.4 Lambertian reflectivity For the white paint the resulting photograph of the reflection pattern at a tilt angle of is shown as Figure 30
and for as Figure 31. For the gray paint at see Figure 32 and at see Figure 33. The blue arrows in the
figures indicate the boundary of the white A4 paper, above the arrow, and the examined plate, below the arrow.
24
Figure 30. Reflection pattern of the laser beam at a tilt
angle of 45° on white paint.
Figure 31. Reflection pattern of the laser beam at a tilt
angle of 10° on white paint.
Figure 32. Reflection pattern of the laser beam at a tilt
angle of 45° on grey paint.
Figure 33. Reflection pattern of the laser beam at a tilt
angle of 10° on grey paint.
The upper two figures, they are the reflection patterns of the white paint, show more reflected red light than the
lower two that were the reflections of the grey paint. This is in line with the measured albedos for white and grey
paint (see Table 7). The left two figures, where the laser was under an angle of , exhibit reasonable
lambertian reflectance since the pattern resembles somewhat the silhouette of a half sphere. However, there
appears to be more red light left of the point caused by the laser on the plate, meaning that the reflectance is
partly specular. The right two figures, where the laser was under an angle of , show more specular
reflectance. This is because a bright spot is clearly visible on the white A4 paper. This bright spot is in a straight
line with the incoming laser beam. This means the angle of incidence is equal to the angle of reflection. Due to
this bright spot it is possible that the amount of reflectance was higher than it was at .
4.5 Calculated and measured values and their relative
differences Table 8 to Table 17 show the results of the (upper left corner), the (upper right corner),
the (lower left corner) and the (lower right corner) for every sensor height for all examined
distances and heights. The are presented in a red to green to red color gradient from to to
and the are presented in the same color gradient but from to to .
The and and their relative differences, also in relation to the irradiance at the horizontal
plane, for combination 4 (all components were white) and an example of the similarity of the Robinson and
Stone model and the extrapolation method presented in this paper, are presented in Appendix I: Iterated
calculated values and similarity of Robinson and Stone model and extrapolation method.
25
Table 8. Results for combination 1 with variable distance (<self>=grey, <plate>=black cloth and <ground>=black).
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 8.34E-03 1.20E-02 2.53E-02 8.17E-02 4.90E-01 1.42E-02 2.10E-02 3.96E-02 1.02E-01 4.87E-01
35 6.58E-03 9.26E-03 1.97E-02 6.61E-02 4.87E-01 1.16E-02 1.69E-02 3.17E-02 8.32E-02 4.85E-01
30 5.02E-03 6.84E-03 1.47E-02 5.11E-02 4.85E-01 9.10E-03 1.29E-02 2.44E-02 6.53E-02 4.82E-01
25 3.66E-03 4.77E-03 1.03E-02 3.72E-02 4.81E-01 6.99E-03 9.76E-03 1.81E-02 4.81E-02 4.78E-01
20 2.50E-03 3.06E-03 6.69E-03 2.48E-02 4.75E-01 4.47E-03 6.38E-03 1.17E-02 3.12E-02 4.72E-01
15 1.55E-03 1.72E-03 3.79E-03 1.44E-02 4.65E-01 2.82E-03 3.99E-03 7.17E-03 1.84E-02 4.62E-01
10 8.03E-04 7.65E-04 1.70E-03 6.54E-03 4.45E-01 1.26E-03 1.80E-03 3.22E-03 7.98E-03 4.42E-01
5 2.71E-04 1.91E-04 4.26E-04 1.66E-03 3.88E-01 3.52E-04 5.43E-04 9.39E-04 2.03E-03 3.85E-01
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 -0.59 -0.90 -1.43 -2.00 0.25 -41.46 -42.76 -36.08 -19.67 0.50
35 -0.51 -0.76 -1.20 -1.71 0.25 -43.42 -45.22 -37.83 -20.52 0.51
30 -0.41 -0.61 -0.97 -1.41 0.25 -44.81 -47.08 -39.66 -21.66 0.52
25 -0.33 -0.50 -0.77 -1.10 0.25 -47.62 -51.15 -42.73 -22.76 0.52
20 -0.20 -0.33 -0.50 -0.64 0.25 -44.04 -52.01 -42.59 -20.67 0.54
15 -0.13 -0.23 -0.34 -0.40 0.26 -45.18 -56.85 -47.11 -21.68 0.56
10 -0.05 -0.10 -0.15 -0.14 0.26 -36.27 -57.53 -47.33 -17.98 0.60
5 -0.01 -0.04 -0.05 -0.04 0.28 -23.00 -64.83 -54.70 -18.15 0.73
Table 9. Results for combination 2 with variable distance (<self>=grey, <plate>=black cloth and <ground>=white).
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 3.27E-02 1.60E-02 2.66E-02 8.23E-02 4.90E-01 7.16E-02 2.60E-02 3.79E-02 9.63E-02 4.87E-01
35 2.78E-02 1.21E-02 2.06E-02 6.65E-02 4.88E-01 6.13E-02 2.02E-02 3.03E-02 7.92E-02 4.85E-01
30 2.31E-02 8.73E-03 1.52E-02 5.14E-02 4.85E-01 5.12E-02 1.50E-02 2.30E-02 6.18E-02 4.82E-01
25 1.86E-02 5.92E-03 1.07E-02 3.73E-02 4.81E-01 3.80E-02 9.92E-03 1.58E-02 4.34E-02 4.78E-01
20 1.43E-02 3.67E-03 6.86E-03 2.48E-02 4.75E-01 2.90E-02 6.63E-03 1.08E-02 2.92E-02 4.72E-01
15 1.02E-02 1.99E-03 3.86E-03 1.44E-02 4.65E-01 2.00E-02 3.98E-03 6.57E-03 1.72E-02 4.62E-01
10 6.35E-03 8.46E-04 1.72E-03 6.55E-03 4.45E-01 1.16E-02 2.02E-03 3.39E-03 8.20E-03 4.42E-01
5 2.68E-03 2.01E-04 4.28E-04 1.66E-03 3.88E-01 3.45E-03 5.71E-04 9.68E-04 2.07E-03 3.85E-01
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 -3.89 -0.99 -1.13 -1.40 0.28 -54.34 -38.20 -29.86 -14.58 0.57
35 -3.35 -0.81 -0.97 -1.27 0.27 -54.62 -40.00 -32.08 -16.07 0.55
30 -2.81 -0.63 -0.78 -1.04 0.26 -54.86 -41.82 -33.76 -16.83 0.54
25 -1.94 -0.40 -0.51 -0.61 0.26 -50.96 -40.37 -32.54 -14.01 0.54
20 -1.47 -0.30 -0.39 -0.44 0.26 -50.64 -44.60 -36.36 -15.06 0.54
15 -0.98 -0.20 -0.27 -0.28 0.26 -48.89 -49.96 -41.14 -16.06 0.56
10 -0.52 -0.12 -0.17 -0.16 0.27 -45.15 -58.01 -49.37 -20.07 0.60
5 -0.08 -0.04 -0.05 -0.04 0.28 -22.30 -64.76 -55.76 -19.72 0.73
26
For both Table 8 and Table 9 the show an increasing negative percentage gradient from the top to the bottom sensor
height for every distance, but for combination 2 at bottom sensor height this percentage is more negative than for combination
1. A gradient per sensor height can also be seen when looking at variable distance. If the distance between the plates was
smaller, the became more negative for the middle and second sensor heights. The fourth and top sensor heights
remained almost constant. The bottom sensor height became less negative.
When looking at the Table 8 shows all percentages are within while Table 9 shows negative percentages up to
almost for a distance of 40 . When the distance became smaller the percentages became less negative, except at top
sensor heights the percentage remained constant.
The of Table 10 are more positive for the top sensor height while they are more negative for the other sensors heights
compared to those of Table 8 and Table 9. The courses of the second and middle sensor height are similar, but the percentages
for the bottom sensor height did not decrease when the distance became smaller, they remained more or less constant.
The are positively highest for the top sensor height and negatively highest for the fourth sensor height. From fourth
to bottom sensor height the values become less negative. Also from fourth to bottom sensor heights the percentages went
closer to zero when the distance was decreased, while they remained almost constant for the top sensor height.
Table 10. Results for combination 3 with variable distance (<self>=white, <plate>=white and <ground>=black).
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 1.16E-02 2.76E-02 6.22E-02 1.90E-01 7.28E-01 3.84E-02 7.86E-02 1.42E-01 2.74E-01 5.79E-01
35 9.14E-03 2.18E-02 4.92E-02 1.58E-01 7.27E-01 3.12E-02 6.63E-02 1.23E-01 2.48E-01 5.77E-01
30 6.93E-03 1.65E-02 3.72E-02 1.25E-01 7.26E-01 2.49E-02 5.48E-02 1.03E-01 2.19E-01 5.73E-01
25 5.01E-03 1.18E-02 2.65E-02 9.26E-02 7.23E-01 1.89E-02 4.31E-02 8.31E-02 1.87E-01 5.69E-01
20 3.39E-03 7.70E-03 1.73E-02 6.28E-02 7.19E-01 1.33E-02 3.14E-02 6.14E-02 1.48E-01 5.62E-01
15 2.06E-03 4.41E-03 9.89E-03 3.71E-02 7.11E-01 8.41E-03 2.01E-02 4.02E-02 1.06E-01 5.50E-01
10 1.05E-03 1.99E-03 4.45E-03 1.71E-02 6.95E-01 4.30E-03 1.00E-02 1.98E-02 5.80E-02 5.26E-01
5 3.42E-04 5.01E-04 1.12E-03 4.36E-03 6.42E-01 1.22E-03 2.32E-03 4.08E-03 1.19E-02 4.54E-01
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 -2.68 -5.09 -7.95 -8.33 14.88 -69.75 -64.81 -56.09 -30.45 25.70
35 -2.20 -4.44 -7.35 -9.03 15.04 -70.69 -67.05 -59.90 -36.43 26.08
30 -1.79 -3.83 -6.61 -9.42 15.22 -72.11 -69.90 -63.98 -43.03 26.54
25 -1.39 -3.14 -5.66 -9.40 15.44 -73.49 -72.72 -68.13 -50.36 27.14
20 -0.99 -2.37 -4.42 -8.56 15.73 -74.48 -75.44 -71.86 -57.67 27.99
15 -0.64 -1.57 -3.03 -6.92 16.16 -75.50 -78.09 -75.41 -65.11 29.39
10 -0.33 -0.80 -1.53 -4.10 16.93 -75.67 -80.15 -77.52 -70.59 32.22
5 -0.09 -0.18 -0.30 -0.76 18.81 -71.90 -78.44 -72.58 -63.41 41.44
Table 11 shows the same pattern as Table 10. The lowest sensor height is closest to for the . When looking at the
top sensor height it stands out that for a height of the percentage is only while for the other heights this
percentage was much larger. At this height the white plates were hardly visible for the sensor while the black <ground> was
taking almost of the field of view of the sensor.
The show again positive values for the top sensor height. For the other sensors heights they are all negative and
become more negative to the bottom sensor height at increasing height of the test setup.
27
Table 11. Results for combination 3 with variable height (<self>=white, <plate>=white and <ground>=black).
Height [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
243.4 1.16E-02 2.76E-02 6.22E-02 1.90E-01 7.28E-01 3.84E-02 7.86E-02 1.42E-01 2.74E-01 5.79E-01
182.9
1.95E-02 5.81E-02 1.87E-01 7.24E-01
5.82E-02 1.38E-01 2.67E-01 5.75E-01
122.3
4.00E-02 1.75E-01 7.15E-01
1.04E-01 2.60E-01 5.72E-01
61.8
1.19E-01 6.80E-01
2.08E-01 5.60E-01
1.2
5.15E-01
4.97E-01
Height [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
243.4 -2.68 -5.09 -7.95 -8.33 14.88 -69.75 -64.81 -56.09 -30.45 25.70
182.9
-3.87 -8.04 -7.98 14.94
-66.43 -58.04 -29.90 25.99
122.3
-6.43 -8.49 14.35
-61.66 -32.69 25.10
61.8
-8.91 11.96
-42.82 21.35
1.2
1.82
3.66
The pattern in Table 12 for both the and the is similar to that of Table 10 for the top sensor height to the
middle sensor heights. The bottom and second sensor heights show more negatively and especially more negative
than they did in Table 10 when <ground> was black instead of white.
Table 12. Results for combination 4 with variable distance (<self>=white, <plate>=white and <ground>=white).
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 3.66E-02 3.18E-02 6.35E-02 1.91E-01 7.28E-01 1.66E-01 1.18E-01 1.52E-01 2.76E-01 5.79E-01
35 3.08E-02 2.47E-02 5.01E-02 1.58E-01 7.27E-01 1.46E-01 9.89E-02 1.30E-01 2.48E-01 5.77E-01
30 2.54E-02 1.84E-02 3.77E-02 1.25E-01 7.26E-01 1.27E-01 8.12E-02 1.09E-01 2.19E-01 5.73E-01
25 2.02E-02 1.29E-02 2.68E-02 9.27E-02 7.23E-01 1.05E-01 6.20E-02 8.55E-02 1.84E-01 5.69E-01
20 1.54E-02 8.32E-03 1.75E-02 6.29E-02 7.19E-01 8.39E-02 4.43E-02 6.31E-02 1.47E-01 5.62E-01
15 1.08E-02 4.68E-03 9.96E-03 3.71E-02 7.11E-01 6.20E-02 2.76E-02 4.05E-02 1.04E-01 5.50E-01
10 6.63E-03 2.07E-03 4.47E-03 1.71E-02 6.95E-01 3.85E-02 1.28E-02 1.93E-02 5.57E-02 5.26E-01
5 2.76E-03 5.11E-04 1.12E-03 4.36E-03 6.42E-01 1.49E-02 2.81E-03 4.13E-03 1.19E-02 4.54E-01
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 -12.94 -8.61 -8.83 -8.49 14.91 -77.97 -73.03 -58.17 -30.77 25.75
35 -11.51 -7.42 -8.00 -9.01 15.06 -78.87 -74.98 -61.50 -36.30 26.12
30 -10.20 -6.28 -7.11 -9.40 15.23 -80.08 -77.30 -65.33 -42.94 26.56
25 -8.47 -4.90 -5.87 -9.15 15.45 -80.73 -79.13 -68.66 -49.66 27.15
20 -6.85 -3.60 -4.56 -8.45 15.73 -81.68 -81.21 -72.32 -57.32 28.00
15 -5.12 -2.29 -3.06 -6.72 16.16 -82.53 -83.01 -75.42 -64.43 29.39
10 -3.19 -1.07 -1.49 -3.86 16.93 -82.79 -83.86 -76.90 -69.32 32.22
5 -1.21 -0.23 -0.30 -0.75 18.81 -81.41 -81.80 -72.83 -63.31 41.44
28
All in Table 13 are larger than . The positive values are found for the top sensor height, the other sensors
heights have negative . The show the same pattern, positive for the top sensor height and a gradient towards
the bottom sensor heights where the increase negatively. There is also a gradient when looking at the height of the
test setup. When the height decreased the of a sensor height became more negative or less positive.
Table 13. Results for combination 4 with variable height (<self>=white, <plate>=white and <ground>=white).
Height [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
243.4 3.66E-02 3.18E-02 6.35E-02 1.91E-01 7.28E-01 1.66E-01 1.18E-01 1.52E-01 2.76E-01 5.79E-01
182.9
5.33E-02 6.37E-02 1.89E-01 7.25E-01
2.14E-01 1.82E-01 2.82E-01 5.75E-01
122.3
9.07E-02 1.83E-01 7.18E-01
3.08E-01 3.14E-01 5.72E-01
61.8
2.17E-01 6.96E-01
4.43E-01 5.62E-01
1.2
8.13E-01
6.98E-01
Height [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
243.4 -12.94 -8.61 -8.83 -8.49 14.91 -77.97 -73.03 -58.17 -30.77 25.75
182.9
-16.11 -11.79 -9.33 15.02
-75.14 -64.93 -33.08 26.13
122.3
-21.71 -13.11 14.59
-70.53 -41.72 25.52
61.8
-22.65 13.42
-51.13 23.89
1.2
11.57
16.58
The and of Table 14 are very similar to those of Table 8 but the values for the top sensor height are slightly
more positive. For fourth to bottom sensor heights the values are more negative.
Table 14. Results for combination 5 with variable distance (<self>=grey, <plate>=grey and <ground>=black).
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 8.54E-03 1.27E-02 2.68E-02 8.62E-02 5.04E-01 1.77E-02 2.56E-02 4.50E-02 1.07E-01 4.94E-01
35 6.74E-03 9.78E-03 2.09E-02 6.98E-02 5.02E-01 1.47E-02 2.13E-02 3.73E-02 9.01E-02 4.92E-01
30 5.14E-03 7.23E-03 1.56E-02 5.41E-02 4.99E-01 1.17E-02 1.70E-02 2.98E-02 7.20E-02 4.89E-01
25 3.74E-03 5.04E-03 1.10E-02 3.93E-02 4.96E-01 9.01E-03 1.30E-02 2.27E-02 5.45E-02 4.85E-01
20 2.55E-03 3.24E-03 7.09E-03 2.62E-02 4.90E-01 6.61E-03 9.39E-03 1.64E-02 3.92E-02 4.79E-01
15 1.58E-03 1.83E-03 4.02E-03 1.52E-02 4.80E-01 4.26E-03 6.04E-03 1.05E-02 2.47E-02 4.69E-01
10 8.18E-04 8.12E-04 1.80E-03 6.94E-03 4.60E-01 2.32E-03 3.29E-03 5.73E-03 1.31E-02 4.49E-01
5 2.75E-04 2.03E-04 4.52E-04 1.76E-03 4.02E-01 5.38E-04 8.00E-04 1.43E-03 3.22E-03 3.90E-01
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 -0.92 -1.30 -1.83 -2.10 1.02 -51.73 -50.56 -40.54 -19.56 2.06
35 -0.79 -1.15 -1.64 -2.03 1.03 -54.10 -54.02 -44.07 -22.51 2.08
30 -0.66 -0.98 -1.42 -1.79 1.03 -56.20 -57.48 -47.71 -24.91 2.11
25 -0.53 -0.80 -1.17 -1.51 1.04 -58.45 -61.19 -51.70 -27.78 2.15
20 -0.41 -0.62 -0.93 -1.29 1.06 -61.35 -65.52 -56.81 -33.05 2.20
15 -0.27 -0.42 -0.65 -0.95 1.07 -62.95 -69.78 -61.75 -38.28 2.29
10 -0.15 -0.25 -0.39 -0.62 1.11 -64.72 -75.35 -68.59 -46.99 2.47
5 -0.03 -0.06 -0.10 -0.15 1.18 -48.88 -74.68 -68.36 -45.22 3.01
29
Table 15 shows all values within , just as it was for variable distance (Table 14). The are slightly
positive for the top sensor height and increase negatively towards the bottom sensor height. The most negative value was found
for the bottom sensor height at a height of .
Table 15. Results for combination 5 with variable height (<self>=grey, <plate>=grey and <ground>=black).
Height [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
243.4 8.54E-03 1.27E-02 2.68E-02 8.62E-02 5.04E-01 1.77E-02 2.56E-02 4.50E-02 1.07E-01 4.94E-01
182.9
1.42E-02 2.69E-02 8.62E-02 5.04E-01
2.46E-02 4.00E-02 1.09E-01 4.94E-01
122.3
2.90E-02 8.62E-02 5.04E-01
4.21E-02 1.11E-01 4.94E-01
61.8
8.92E-02 5.03E-01
1.19E-01 4.93E-01
1.2
5.06E-01
4.95E-01
Height [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
243.4 -0.92 -1.30 -1.83 -2.10 1.02 -51.73 -50.56 -40.54 -19.56 2.06
182.9
-1.04 -1.31 -2.32 1.04
-42.26 -32.74 -21.22 2.10
122.3
-1.31 -2.52 1.02
-31.11 -22.60 2.07
61.8
-2.97 0.98
-25.00 1.99
1.2
1.17
2.36
Table 16 shows the same pattern for the and as in Table 9 but the positive values are more positive and the
negative values are more negative.
Table 16. Results for combination 6 with variable distance (<self>=grey, <plate>=grey and <ground>=white).
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 3.29E-02 1.67E-02 2.80E-02 8.68E-02 5.05E-01 8.72E-02 3.56E-02 4.78E-02 1.09E-01 4.94E-01
35 2.80E-02 1.26E-02 2.17E-02 7.02E-02 5.03E-01 7.67E-02 2.88E-02 3.96E-02 9.15E-02 4.92E-01
30 2.32E-02 9.12E-03 1.61E-02 5.43E-02 5.00E-01 6.54E-02 2.22E-02 3.14E-02 7.35E-02 4.89E-01
25 1.87E-02 6.19E-03 1.13E-02 3.95E-02 4.96E-01 5.36E-02 1.63E-02 2.37E-02 5.57E-02 4.85E-01
20 1.44E-02 3.85E-03 7.26E-03 2.63E-02 4.90E-01 4.14E-02 1.11E-02 1.67E-02 3.91E-02 4.79E-01
15 1.03E-02 2.09E-03 4.10E-03 1.53E-02 4.80E-01 2.95E-02 6.86E-03 1.07E-02 2.47E-02 4.69E-01
10 6.36E-03 8.92E-04 1.82E-03 6.95E-03 4.60E-01 1.78E-02 3.61E-03 5.86E-03 1.32E-02 4.49E-01
5 2.69E-03 2.13E-04 4.54E-04 1.76E-03 4.02E-01 5.03E-03 9.42E-04 1.61E-03 3.55E-03 3.90E-01
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 -5.43 -1.89 -1.98 -2.20 1.05 -62.27 -53.11 -41.38 -20.22 2.13
35 -4.87 -1.61 -1.79 -2.13 1.05 -63.54 -56.11 -45.15 -23.25 2.13
30 -4.22 -1.31 -1.53 -1.92 1.05 -64.48 -58.87 -48.63 -26.12 2.14
25 -3.49 -1.01 -1.24 -1.62 1.05 -65.12 -61.92 -52.37 -29.04 2.17
20 -2.70 -0.72 -0.95 -1.28 1.06 -65.24 -65.26 -56.65 -32.67 2.21
15 -1.92 -0.48 -0.67 -0.94 1.08 -65.19 -69.50 -61.89 -38.18 2.29
10 -1.15 -0.27 -0.40 -0.62 1.11 -64.34 -75.28 -68.95 -47.23 2.47
5 -0.23 -0.07 -0.12 -0.18 1.18 -46.55 -77.39 -71.77 -50.37 3.01
30
In Table 16 the were most negative for the bottom sensor height. This pattern is also visible in Table 17, the lowest
sensor height for every height except is most negative. The value for stands out because it is the largest positive
value. The grey plates were hardly visible for the sensor while it ‘saw’ the white <ground> for almost of its total field of
view.
The of Table 17 are positive for the top sensor height. The other sensor heights show negative values. The
measurement at 1.2 also stands out with its because it is again the largest positive value.
Table 17. Results for combination 6 with variable height (<self>=grey, <plate>=grey and <ground>=white).
Height [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
243.4 3.29E-02 1.67E-02 2.80E-02 8.68E-02 5.05E-01 8.72E-02 3.56E-02 4.78E-02 1.09E-01 4.94E-01
182.9
4.67E-02 3.23E-02 8.78E-02 5.05E-01
1.06E-01 5.09E-02 1.12E-01 4.94E-01
122.3
7.72E-02 9.42E-02 5.06E-01
1.54E-01 1.25E-01 4.94E-01
61.8
1.81E-01 5.18E-01
2.64E-01 4.94E-01
1.2
8.05E-01
6.93E-01
Height [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
243.4 -5.43 -1.89 -1.98 -2.20 1.05 -62.27 -53.11 -41.38 -20.22 2.13
182.9
-5.93 -1.86 -2.40 1.11
-55.94 -36.59 -21.46 2.25
122.3
-7.64 -3.05 1.26
-49.73 -24.44 2.55
61.8
-8.28 2.38
-31.36 4.83
1.2
11.12
16.04
31
5. Discussion 5.1 Test setup In order to be sure the canyon effect was measurable, the MDF plates were used in lengthwise direction to
simulate the flat plate reactor row. The Robinson and Stone model assumes an endless long row. The canyon
effect was clearly measured as can be seen by the in Table 8 to Table 17 when looking from top
to bottom sensor height. However to go from measured values to extrapolations had to be done. If
the MDF plates were used in widthwise direction the extrapolation factors would have been smaller which would
have resulted in that were for a larger part made out of the measured values. However, the
influence of the sensors on the measurements would have increased if the setup was in widthwise direction
because the total surface area of the five sensors was which for this research was divided
over the length of and would have been divided over if the plates were used in widthwise
direction. But the surface area of the MDF plate where light will be received from is ,
which is much more than the surface area of the sensors and therefore the influence of the sensors on the
measurements would have been small.
In practice light reflecting towards the sky leaves the flat plate reactor row. For the test setup the sky was
simulated by means of a diffuse cloth. Although no albedo measurement could be done for practical reasons, the
diffuse cloth potentially had a high albedo since it was white. The sensor measuring the irradiance at the
horizontal plane was placed outside the test setup in order not to disturb the measurements, as can be seen in
Figure 13. However, reflected light from the test setup that potentially reflected onto the diffuse cloth was not
measured by this sensor. Therefore equation 26 was used to calculate the irradiance at the horizontal plane. Since
this calculation was based on the top sensor, which resided within the test setup, possible reflectance by the
diffuse cloth was accounted for.
Table 6 shows hotspots were present on the diffuse cloth. When looking at individual places on the diffuse cloth
the irradiance was anisotropic, but when looking at the diffuse cloth as a whole it was isotropic, because the
results showed both higher and lower irradiance values were present and the diffuse cloth was therefore an
acceptable isotropic light source for this research.
Also shown in Table 6 is that the mean relative irradiance measured at an angle of was too high and at
was too low compared to ideally diffuse. So, radiance levels almost parallel to the diffuse cloth were
too low and perpendicular to the diffuse cloth were too high. This means for the upper part of the test setup that
the irradiance was too low and for the lower part that the irradiance was too high compared to an ideally diffuse
light source. This coincides with the results presented in Table 8 to Table 17. The top sensor height
and were for all combinations positive, meaning that the were larger than the .
For all other sensor height values the and were negative, so the were larger
than the .
5.2 Sensors The sensor specific error correction factors, see Table 5, are based on the average value of the six sensors.
Without a calibrated sensor it cannot be certain that the corrected value is the true value. However, since the
sensors were linear there is only the possibility of a gain error. These possible errors were removed however. By
means of calculating the the fractions compared to the irradiance at the horizontal plane were
taken. Through the process of dividing by the horizontal plane irradiance these possible errors were eliminated.
The cosine course error correction factors, Table 20 to Table 25, were found by an experiment with a laser beam.
However, in the test setup the incoming light was diffuse. This means that for the determination of the correction
factors a small ray of light, the laser beam, was shone on the dome of the sensor. This small ray led to a small dot
on the dome at the to be measured angle while in reality, because the incoming light was diffuse, always half
of the dome was lid, even if angle was close to . This means the cosine course error of the sensors can
exhibit in reality a smaller cosine course error when diffuse light shines on it instead of a laser beam.
The sensors measured the quantity of light in the wavelength range of 400 to 700 (PAR) but it was not
known whether the sensitivity of the sensors was equal for all wavelengths. Also the amount of light radiated by
the lamps or reflected by the paint within this range was unknown. It is possible that, for example, the amount
for 700 was much larger than for 400 and because reflectance can vary with wavelength (Knipling,
1970) the measured values can either be too high or too low.
The linearity of the sensors was determined for the range to . Therefore all measured values had
to be within this range. All values were lower than and all but one value were higher than .
Only the bottom sensor for combination 1 (where <self> was grey, <plate> was black cloth and <ground> was
black paint) at the distance of a value outside this range was measured, namely .
32
5.3 Albedos A distance between diffuse cloth and the to be measured material of 18 was used. Ideally this distance should
have been , but this was impossible due to the sensors dimensions. But a smaller distance also caused a
visible shadow of the sensor measuring the reflected light onto the material it was measuring. This means that
the measured albedo values, see Table 7, were likely to be a bit lower than the true albedo values.
The albedo values of black paint and black cloth were with respectively 55.3% and 23.7% compared to grey
paint too high to neglect. The of Table 8 showed in the direction from top to bottom sensor height an
increasing negative percentage. This increase was probably due to the black painted <ground>. At top sensor
height most irradiance is caused by the sky and because the diffuse cloth radiated too little at large angles, the
are positive. For all other sensor heights the were negative, meaning the were
larger than the , which could have been the result of the reflectance of the black cloth.
Because the albedo values of black paint and black cloth were high compared to grey paint, it was not possible to
evaluate the equations of the Robinson and Stone model individually.
The black plastic was assumed not to reflect any light, however the albedo, Table 7, shows that 4.4% of the
incoming light was reflected. Therefore this caused an error on the . When the distance between
the plates increased, more plastic had to be used to cover the test setup. This was also necessary when the height
increased. This means the highest errors were present when the distance, or the height, was large. When the
reflectance of <ground> and <plate> were smallest, this effect should best be visible. The combination with least
reflectance is again Table 8. However, this least reflectance combination was only examined with difference in
distance. When looking from 5 distance to 40 distance for all sensor heights a trend is visible where the
become closer to 0%. As discussed the opposite effect was expected. Therefore another source was
present causing a much larger opposite effect. That source is discussed in section 5.4 Lambertian reflectance.
5.4 Lambertian reflectance Figure 30 to Figure 33 show the qualitative results of the reflectance pattern of the white and grey paint. When
the tilt angle of the laser was large ( ) the result was reasonable diffuse, but some specular reflectance was
visible because more of the red light is visible on the left side of the bright spot on the plate caused by the laser
than on the right side. When the tilt angle was the reflectance pattern was mostly specular because a bright
spot was visible. Almost all reflected light is visible to the left of where the laser light reflected on the plate. In
the test setup the fourth sensor had the largest angle with the diffuse cloth at a distance of . The angle was
( ) For the bottom sensor this largest angle was even much smaller: ( ) . This means the reflectance in the test setup was mostly specular and not diffuse. This
had a large effect on the measurement values. If the reflectance at a tilt angle of , Figure 30 and Figure 32, is
estimated to be between and left of the reflection spot on the plate, then it is estimated to be
left of the reflection spot on the plate for a tilt angle of . If the reflectance would have been diffuse,
would have been reflected towards the diffuse cloth and would have been reflected towards the ground.
Since the diffuse cloth was the component of the test setup that radiated most, the estimated percentages led to
irradiance values that were increasingly too large towards the ground. The results presented in Table 8 to Table
17 support this. Towards the bottom sensor height the are negatively increasing, meaning the
were larger than the . The effect caused by the specular reflectance surpassed the
opposite effect caused by the black plastic as discussed in section 5.3 Albedos.
5.5 Extrapolations All extrapolations, both for incoming and reflected radiance see Table 26 to Table 41, were based on isotropic
diffuse light. As discussed in the previous sections, the diffuse cloth was not fully diffuse and the reflectance
caused by the paint was more specular than diffuse. This resulted in measurement values that were higher than
should have been if all light was diffuse. When these values were extrapolated, this error was also extrapolated
resulting in that were increasingly too high in the direction of the bottom sensor height. When
looking at the combinations with a white <ground> (Table 9, Table 12, Table 13, Table 16 and Table 17) this
extrapolation of the error is clearly visible due to the high albedo of white paint. Both the and
are much higher in absolute sense, than for the associated combinations with a black <ground> (Table
8, Table 10, Table 11, Table 14 and Table 15). The extrapolation error is also present for the combinations where
<plate> was white (Table 10, Table 11, Table 12 and Table 13). Again both the and are
higher in absolute sense than their associated combinations with a grey <plate> (Table 14, Table 15, Table 16
and Table 17). This means the comparison between the calculated values by the Robinson and Stone model and
the became increasingly impossible for heights in the test setup further away from the diffuse
cloth as well as for increasing albedo values.
33
To calculate the irradiance on the opposing plate the irradiance caused by the sky was used. The irradiance on
the plate lowers when moving away from the diffuse cloth towards the <ground>. This is the canyon effect.
However, to calculate the irradiance onto the opposing plate the irradiance value corresponding to the sensor
height was used. By this method the opposing plate is assumed isotropic with this single value, where in reality
the part of the opposing plate in the direction of the diffuse cloth had a continuous higher irradiance value and in
the direction of the <ground> it had a continuous lower irradiance value. As a result of this method the irradiance
caused by the opposing plate is simplified. The Robinson and Stone model, equation 6 and equation 7 uses this
same simplification. Therefore, due to the use of this simplification method no deviations between the
extrapolated part of the and the calculated was introduced. But the part of the
caused by the measurements did, of course, not follow this simplification. This means a deviation
between the Robinson and Stone model and the reality is to be expected, even when all light is truly diffuse.
While calculating the only the first reflection was calculated for while for calculating the
five iterations were done. This was done because the measurement values were already build up of the
total of reflections, therefore iterations were omitted when calculating the . Five iterations for the
calculated values were considered to be enough, see Table 42, because the largest relative increase from
to was below 5%. The largest increase from to as fraction of the
horizontal plane irradiance was less than for all values.
For all extrapolations a half sphere was used. Mathematically this is only correct for a sensor at ground level.
The top height sensor was above ground level meaning the horizon was just below its horizontal plane
and the half sphere should have been little more than half a sphere. However, the amount of additional irradiance
would have been very small and was therefore neglected.
5.6 Robinson and Stone model The Robinson and Stone model is in essence also based on equation 3 but it can only be used for an endless row
situation because the double integrals are absent. Therefore, mathematically the Robinson and Stone model is
equal to the extrapolation method presented in this report, see the example presented in Table 43. The Robinson
and Stone model also uses the irradiance value at some plane (sky, plate or ground) and uses that in combination
with the albedo to calculate the irradiance value of the plane of interest while taking the cosine course at the
plane of interest into consideration. It does, however, use one value per plane. Since the sky is assumed isotropic
and the canyon effect on the ground is small, this is reasonable. But for the opposing plate the Robinson and
Stone model uses the irradiance value that is present on the opposing plate at the same height. However, in
practice above this height the irradiance is higher and below this height it is lower due to the canyon effect itself.
Equations 6, equation 7 and equation 9 use this single value for the plates. For equation 6 and equation 9 this
leads to an underestimation and for equation 7 to an overestimation of the irradiance.
Because the Robinson and Stone model and the extrapolations presented in this report coincide mathematically,
the differences between the and the must be explained by the specular reflectance of
the used paints.
5.7 Applicability of the model In reality a flat plate reactor is constructed out of some glasslike material where the water and algae reside in.
This means transmittance of light can take place, meaning the Lambert-Beer law has to be incorporated into the
model (Slegers et al., 2013). Light radiating on the flat plate reactor partly reflects, but on glass it reflects
specular (Cohen and Wallace, 1993). Also refraction takes place, meaning Schnell’s law and the Fresnell
equations also have to be incorporated (Slegers et al., 2011). Due to this specular reflectance a ray tracing
method could be more appropriate to describe the light for the glasslike material than the Robinson and Stone
model. However, light falling on the algae can also be reflected. If algae can be imagined as small circular
bodies, their reflectance is diffuse if the algae concentration in the whole flat plate reactor is constant (Slegers et
al., 2013). To describe this light reflected by the algae, the Robinson and Stone model could be appropriate.
Further, Equation 4 can always be used to calculate the irradiance caused directly by the sky, be it under the
assumption of sky isotropy.
5.8 Accuracy of the values The are presented with three digits. A higher accuracy was found meaningless because of the
uncertainty of the sensor specific error correction, the cosine course error correction and the albedo values. Also
the distances and heights were measured with a meter stick. The folding and unfolding of it could have caused
distances to be measured with some degree of error. MathWorks Matlab was used to solve the integrals.
However, most integrals could not be solved analytically and had to be solved numerically.
34
35
6. Conclusions 6.1 Albedo of the components During this research measurements at an simulated flat plate reactor were done in order to investigate how much
light fell on certain points. This was done to check the applicability of the Robinson and Stone model for flat
plate reactors instead of for urban street canyons where this model originally was set up for. A very important
variable was the albedo of the materials the simulated flat plate reactor was made up of. The albedo gives the
quantity of light energy reflected from a surface as fraction of the quantity of light energy that radiated onto that
surface. The test setup was made up of white, grey and black painted MDF plates, black cloth and black plastic.
White paint reflected most light. It reflected of the light, the grey paint reflected , black paint and
black plastic reflected almost the same with and respectively. The black cloth reflected least, only
of the light that fell on it was reflected. The black paint and black cloth were assumed to reflect no light,
however their albedos in relation to the grey paint were too large to neglect.
6.2 Diffusivity of the reflectance The Robinson and Stone model is valid when all components exhibit diffuse and isotropic radiance. This begins
with the sky. The sky was simulated by means of two lamps, each consisting out of two lights, which shone onto
a prismatic glass that refracted the light that then entered eight layers of acrylic fiber cloth that diffracted the
light. The light entering the test setup was not fully diffuse because too much radiance left the cloth at an
angle of and too little radiance left at an angle of . The diffusivity of the white and grey paint was qualitatively determined by taking photographs of the reflectance
patterns. It was found that both the white and the grey paint reflected the light mostly diffuse, but also partly
specular, for an incoming ray at an angle of from the surface. When the incoming ray was lowered to the reflectance was mostly specular. The test setup had a length of 244.6 and a maximum distance of .
This means that at the bottom of the plate the angle with the diffuse cloth was at maximum . The distance
between the plates was at minimum , meaning that the angle with the sky for a large part of the plates was
smaller than resulting in mostly specular, instead of diffuse, reflectance.
6.3 Extrapolation of the measurement values In order to extrapolate the measured values yielded at the test setup, which had a limited width, to match the
values for an endless row, the test setup’s components were projected on a sphere. By means of integrating over
angles and irradiance fractions were calculated for incoming and reflected radiance for all components of
the test setup for all examined distances and heights. The integrals assume diffuse and isotropic planes, which
was mostly true for the sky. The ground received little more irradiance in the middle than at the sides, for a flat
plate reactor row with a height of and a distance of the irradiance fraction is at the sides
compared to the irradiance at the center. The ground was therefore almost isotropic and using one irradiance
fraction value for the ground was justified. The plates, subject to the canyon effect, received most irradiance
closest to the diffuse cloth. This continuous irradiance gradient was made isotropic by taking the irradiance
fraction at the height of interest and assume that fraction for the entire plate. This is a simplification that
estimates the irradiances fraction close to the diffuse cloth too high and further away too low. The exact over-
and underestimation caused by this simplification is unknown. However, the Robinson and Stone model uses
this same simplification method for the plates, meaning the calculated values could be compared to the
extrapolated measurement values. As said, the integrals assume the planes not only to be isotropic, but also
diffuse. The paint used in the test setup was found to reflect mostly specular when the incoming radiance was
close to the surface parallel meaning that more than 50% of the reflected light followed the direction of the
original ray whereas this would be exactly 50% for diffuse reflectance. The result of this specular instead of
diffuse reflectance was that irradiances measured in the test setup were too high further away from the diffuse
cloth. The measurement values were extrapolated on a mathematically correct way, but the error caused by
specular reflectance was extrapolated as well.
6.4 Applicability of the Robinson and Stone model The results showed that all for grey and black components were within difference with the
in relation to the irradiance at the horizontal plane just above the flat plate reactor. When a
component was white, this difference was up to . This higher difference was the result of the specular
reflectance in combination with the higher albedo value of white paint.
The relative differences of the in relation to the were higher. They were up to if
only grey and black components were used and up to if white components were used.
36
As discussed the non diffuse (but specular) reflectance of the paint caused the to be non
compliant with the diffusivity prerequisite of the Robinson and Stone model. Therefore the
cannot be used to evaluate the applicability of the Robinson and Stone model for a flat plate reactor row.
However, mathematically the extrapolating method presented in this report and the Robinson and Stone model
coincide and therefore the Robinson and Stone model is assumed to be theoretically correct for diffuse
reflectance except for the irradiance fractions caused by the opposing plate because of the isotropic
simplification, as discussed.
In practice the components of a flat plate reactor row also do not reflect all light diffuse. It was discussed that
algae can reflect light diffusely, but a flat plate reactor wall, which is made out of some glasslike material,
reflects light specular. The Robinson and Stone model assumes diffuse light and can therefore be used for the
fraction of light reflected by the algae, but, as was discussed, equation 6, equation 7 and equation 9 needs to be
corrected for the canyon effect on the plates and the transmittance of light has to be incorporated into the model.
The Robinson and Stone model, however, can be used as it is to give an estimation of the extinction of diffuse
light within compared to the irradiance at the horizontal plane for any height of interest and for any
distance and height of the flat plate reactor when a combined albedo value for all components of is used.
37
7. Recommendations 7.1 Test setup The canyon effect was well measurable with the used test setup. For combination 1 it was even impossible to
yield a measurement value of 20 (the minimum to stay within the linear range of the sensors) for the bottom
sensor height while the distance was 5 . The used paint was found to be mostly specular, meaning the amount
of irradiance at the bottom sensor height was higher than would be the case for diffuse reflectance. When
measuring real diffuse reflectance the height of the test setup should be decreased in order to be able to measure
within the linear range of the sensors. A height of half the used height during this research is expected to be more
than sufficient.
The width of the test setup should be larger. This decreases the need for extrapolating the measurement values.
Although the extrapolation methods presented in this research are mathematically sound, they follow the same
methodology as the Robinson and Stone model. Relying heavily on these extrapolation methods to evaluate the
applicability of the Robinson and Stone model should therefore be avoided.
In order to fulfill the prerequisites of the Robinson and Stone model, better diffuse reflecting materials should be
used. Maybe acrylic fiber cloth can be used for the plates and the ground since its diffusivity was better than that
of the paint. However, the diffuse cloth was used as a light source. It is possible that when used as a reflector, the
diffusivity is lower. As a light source this acrylic fiber cloth was reasonable, but could be better. For further
research it is advised to use a better diffuse light source.
During this research transmittance of light was absent. To simulate a flat plate reactor more accurately,
transmittance should be included. This can be done by the usage of a (small) flat plate reactor filled with algae.
7.2 Sensors The QSO-S PAR Photon Flux Sensors from Decagon Devices were found to have a faulty cosine course. This
error enlarged while angle enlarged. Because the ground and the sky were mostly ‘seen’ by a sensor from a
large angle till the measurements were taken in the sensors measurable range where this error was
largest. To avoid this, sensors without an erroneous cosine course should be used. Sensors without a domed lens
are probably less prone to this. If sensors with a flat lens are not available, the determination of the cosine course
error corrections should be done with a diffuse light source instead of with a laser.
7.3 Improvement of the model It is expected that the walls of a flat plate reactor reflect light mostly specular and that the algae within the flat
plate reactor reflect light mostly diffuse. Ray tracing methods excel when reflectance is specular, therefore a ray
tracing method can be used to evaluate the reflectance of the flat plate reactor walls. However, radiosity methods
are best used for diffuse light and therefore the Robinson and Stone model could be used for the diffuse part. The
Robinson and Stone model can be improved by adding transmittance and by using the light extinction gradient
on the walls instead of using the isotropic assumption. The transmittance can be added by alteration of the in
the Robinson and Stone equations. This can be done by replacing the by ( ) where is the
fraction of light that is lost due to transmittance to the next flat plate reactor row and is the fraction of light
that is gained from transmittance by the next flat plate reactor row. The difference between the quantity of
and can be calculated by means of the Lambert-Beer law. However, research is needed to evaluate the
necessity of this, since the amount of transmitted light can in fact be very small.
The use of the actual light extinction gradient on the walls needs to be incorporated in equation 6, equation 7 and
equation 9 and can possibly be done by integrating over angle for the values for all heights while using the
cosine of to incorporate the cosine course of the irradiance.
38
39
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40
41
Appendices
Appendix A: Calculation of integrals The solution of equation 2 is derived as follows:
∫ ∫ ( )
∫ ( )
∫ (
) ( ( ))
∫
The solid angle .
The solution of equation 3 is derived as follows:
∫ ∫ ( ) ( )
∫ [ ( )
]
∫ ( )
( )
∫
[
]
The solid angle .
42
Appendix B: Determining sensor characteristics Linearity of the sensors Before measuring the light in the simulated flat plate reactor, the characteristics of the sensors had to be known.
According to Decagon Devices the sensor is linear for the range – . This linearity was tested by
dimming gradually a light source, consisting of seven light bulbs of each, hanging approximately five
meters above the sensor. At regular light intensity intervals the output of the sensors was measured and
compared to the mean value of the six sensors. Then this experiment was redone, but now with a distance
between light source and sensors of approximately three meters in order to get higher output values from the
sensors. The setup was surrounded by black plastic, so daylight could not disturb the measurements.
The results are depicted in Figure 34. The sensors were at a distance of approximately resulting in the range
in Figure 34 from to while
the lights were dimmed from almost
full to off. From to the
resulting values are more or less
constant since all values stay within a
range of . From to about
the difference starts to increase,
however the values are still within a
range of compared to the values
from to . Below
the difference sharply increases or
decreases for each sensor.
Then the distance between the lights
and the sensors was decreased to
approximately . Measurements
were again taken but now resulting in
the range from to
resulting in almost linear deviations
compared to the mean of the sensors. Therefore the sensors were assumed to be linear until the claimed
by Decagon Devices. Striking are the results for sensor 8969, depicted in orange. This sensor has the
largest difference to the mean for the entire measured range and has an almost identical course for the two
measured ranges which was, together with susceptibility to drift, the reason to replace it.
Off-set of the sensors To be sure the sensors did not have an off-set, the lights were turned off and a measurement was taken every
minute during fourteen minutes. One
port of the data logger was wired short
while the other five ports were
connected to a sensor. If the tested
sensors have an off-set, they are non-
zero, while the short wired port has a
zero value. If all ports of the sensors,
including the short wired port have an
off-set, than the off-set is the result of
the data logger. The results of the
experiment are depicted in Figure 35.
The short wired port shows the same
course as the other ports that had
sensors connected to it while the light
source was turned off. The difference
to decreases as time increases for
all ports. Because this course is the
same for all six ports, the data logger
caused the off-set. Although the off-set was small, less than directly after the measurement was started,
the choice was made to wait for at least fifteen minutes before starting any measurement.
Figure 34. Measured output compared as percentage to the mean output
of all sensors.
Figure 35. Measured voltage in darkness to find the off-set of the sensors
or of the data logger.
43
Influence of the ports of the data logger Two identical measurements were done to identify possible differences in output of the data logger when the
sensors were connected via different ports of the data logger. The sensors were positioned under seven
light bulbs and their output was measured during one minute per measurement.
Figure 36. Two measurements with the output voltage as a function of time. The only difference between the two
measurements is the data logger ports where the respective sensors were connected to.
Figure 36 represents the results of the two one-minute measurements with a two second sample time where
seven light bulbs were at a distance of approximately five meters from the sensors. The difference
between the two measurements is that the sensors were connected to the data logger via different ports. The
differences between the lines in both measurements are equal, resulting in no change in measured output per port
of the data logger.
Stability of the output of the sensors in time Another experiment was done to check if the output of the sensors was stable during a longer period of time. The
sensors were placed under a light source and every two seconds a measurement was taken during ten minutes. If
there would be drift, one or more sensors would show an increase or decrease in time relative to the other
sensors. If all sensors would have the same amount of drift, their output voltage would change during the
measurement resulting in an overall increase or decrease. However, if peaks or dips are measured, the light
source is the cause. This holds if the output signal of the sensors is (almost) the same after the peak or dip as it
was before the anomaly.
Figure 37 shows the results. The sensors did not exhibit drift because the distance between the resulting lines
remains constant. The general course
shows two dips approximately at 180
seconds and at 560 seconds. After 570
seconds a sharp increase is visible for
all sensors. Since everything remained
constant during this measurement,
these dips must have been caused by
the light source. The light bulbs were
connected directly to the mains power
supply. The mains voltage probably
fluctuates causing fluctuations of the
light intensity levels send out by the
bulbs. This way the mains voltage
fluctuations resulted directly in the
fluctuations that can be seen in Figure
37.
Discussion sensor characteristics After the first three experiments, described in this section, and some other tests to get acquainted with the
sensors and data logger were done, the choice was made to replace sensor 8969 by another sensor. This choice
was made because sensor 8969 systematically gave an output value which was significantly lower than the
average of the sensors. Also, as can be seen in Figure 34, for the results below the line representing sensor
Figure 37. Results of a ten minute during measurement to check whether
the sensors remain stable in time or that they exhibit drift.
44
8969 increases when going from to . Almost exactly the same course of increase is visible for the values
above which were measured when the distance of the sensors to the light source was three instead of five
meters. Since the course is almost identical, this seems a systematic error of the sensor (drift). Some of the other
sensors also increased or decreased slightly, but then their course of the output values smaller than was not
identical to those larger than . After the replacement of sensor 8968 by sensor 8755 the experiments on the
simulated flat plate reactor were done while the sensors and ports were configured according to Table 18. The
sensor numbers were already present on the cables of the sensors, which indicate that these number are probably
assigned to the sensors by the manufacturer.
Table 18. Configuration of sensors and ports of the data logger for all non-calibrating experiments.
Sensor number: 9003 8998 8585 8755 8754 8620
Data logger port number: 1 4 3 2 5 6
Conclusions sensor characteristics The experiments described in this section, which were done to examine the characteristics of the sensors, yielded
that the sensors did not have an off-set but that the data logger always had to be turned on for at least fifteen
minutes before starting measurements in order to minimize the effects of a small off-set caused by the data
logger. The sensors are considered to be linear for the range from to and, by dropping sensor
8969, do not exhibit drift. The measured output does not depend on the port of the data logger it is connected to.
However, after the above mentioned experiments to check the characteristics of the sensors a calibrating
experiment had to be done in order to find a possible gain of the sensors. This experiment could not be done
because of a lack of a calibrated sensor. Therefore the average output value of the sensors was considered to be
the correct value for any given amount of light.
45
Appendix C: Determining the sensor specific error The average output of the sensors was considered to be the correct output value that every sensor should have
been given during measurements. In order to know the correction factor that had to be applied to correct the
measured output of the sensor, the following experiment was done.
During testing it was noticed that the output value of a sensor changes significantly if its position even slightly
changes. So, to determine the sensor specific error it was necessary that
each sensor was tested at exactly the same position. However the mains
can fluctuate (Figure 37) making it necessary to test each sensor at the
same time. This conflict between time and place has been resolved by
using two sensors as can be seen in Figure 38. One sensor was a control
sensor which was at the same position during the time that all other
sensors were tested. By dividing the average measurement value with the
average value of the control sensor the fluctuations caused by the mains
were eliminated and the outcome of the tested sensors as fraction of the
control sensor was known. In order to keep the position as stable as
possible two metal discs with three holes were placed on a foam mat. One
disc for the control sensor, the other for the to be tested sensor. The
sensors were also mounted on similar metal discs with three holes. When
changing the to be tested sensor the bottom and the top discs were aligned carefully. This was checked by gently
pushing three bolts through the holes. After the five sensors were examined another sensor was used as control
sensor in order to be able to determine the sensor specific error of the first control sensor. The fraction values
that were yielded this way were averaged and the relative deviation for each sensor compared to this average was
determined. To yield the sensor specific error correction factor, , this relative deviation was subtracted from
one. The sensor specific error correction factors was valid for every measurement value between and
since the sensors were assumed to be linear for this range.
During this experiment the light source consisted out of seven light bulbs which were at full power.
Every two seconds during one minute a measurement was taken. To make sure daylight could not interfere with
the measurements, the test setup was wrapped in black plastic and two MDF plates.
Figure 38. Test setup to determine
the sensor specific error.
46
Equation 35
Equation 36
Appendix D: Long equations Equation 35:
( )
(
√ √
√ √
( ) values of intersection between sky, plate and black plastic [ ]
Equation 36:
(
)
( ( ) ( (
) ( ) (
) ( ) (
) (
) ( ) ( ) ))
( ( ) ( ) (
)
( ) ( ) )
(
)
Overall corrected and extrapolated values [ ]
Sensor specific error correction factor of the respective sensor [ ]
( ) Measured values [ ]
Albedo of the respective component, either , , or [ ]
47
Equation 37
Equation 38
Appendix E: Determining the cosine course error The irradiance from a ray of light decreases if the angle of that ray compared to the surface perpendicular
increases. This follows a cosine course. The sensors have a domed lens which led to the suspicion that the cosine
course is faulty since light under an angle of (the surface parallel) could still hit the dome. The internal
functioning of the sensors is unpublished by Decagon Devices and can only be known by destructive
examination. Since this was not an option the following experiment was done.
Test setup A red laser beam was used to simulate a ray of light. The laser was securely fastened to a slat which was bolted
to a wooden plate. This way the slat could rotate around
the bolt. Straight above the centre of the bolt the sensor
was mounted. On the wooden plate lines were drawn to
mark the angle of the laser beam with the sensor. Figure
39 is a picture of the used test setup. From
perpendicular to parallel of the sensor the angles were
marked according to Table 19. At every angle the laser
was turned on and kept on during ten seconds in which
every two seconds a measurement was taken. This was
done for every to be measured angle until was
reached. Then the measurements were redone but now
from to resulting in total in ten
measurements per angle per sensor. The width of the laser beam was too large for the larger angles since some of
the light fell over the sensor. The tip of an injection needle was used to create a small hole in aluminum foil. This
was placed in front of the laser and reduced the width of the beam. At the angles larger than a very small
portion still went over the sensor since a bit of a red glow was visible on the black plastic that lay over the test
setup in order to refrain daylight from affecting the measurements. It is possible that this light reflected on the
dome of the sensor. It can also be that the light was transmitted through the dome and then radiated onto the
black plastic. Since the cause can only be known by destructive examination the choice was made to neglect the
radiance that fell over the sensor.
Table 19. Angles at which a measurement was taken to identify the cosine course of the sensors.
Measured angles [°] 0 11 23 34 45 51 56 62 68 73 79 84 87 90
Determining the sensor’s actual cosine course Because the dimensions of the six sensors were the same and at larger angles some of the light of the laser beam
went over the sensors, causing a (small) measurement error, an average actual cosine course for all sensors was
calculated. The output value of the sensors obtained at was used as the value to determine the theoretical
cosine course by means of multiplying this value by the cosine of the angle, equation 37:
( ) ( ) ( )
( ) Theoretical output value at angle [ ]
( ) Measured output value at measured angle [ ]
Then the relative difference between the
theoretical cosine course and the measured
values was determined, see Figure 40. A
trendline was drawn through these points and
its function was determined which gave the
average relative error for the sensors cosine
course per angle, equation 38:
( )
( ) Average relative deviation to the
theoretical cosine course at angle [%]
Figure 39. Test setup to determine the cosine error.
Figure 40. The average relative deviation from the sensors actual cosine course
with the theoretical cosine course. A trendline and its function is included.
48
Equation 41
Equation 40
Equation 39
Correcting the sensor’s actual cosine course for the angle of irradiance The relative deviation from the cosine course is not what had to be corrected for since the portion of the
irradiance at a certain angle also had to
be accounted for. This is because the
weight to the total output of the sensor
from a ray at a larger angle is smaller
than for that at a smaller angle. The
average weight of the sensors output
compared to the output at starting angle
was calculated for every angle
based on the measured values (equation
39). This means that for every angle the
average output of all measurements for
that angle was taken and divided by the
average output measured at an angle of
. The weight of the sensors output
compared to the output at starting angle
was then plotted as function of
the angle, see Figure 41, the blue line
(partly behind the red line) that
represents the starting angle at 0°. A
trendline was drawn and a polynomial
was determined for it, the complete function is given as equation 40. When light comes in under a starting angle
greater than then the outcome of equation 40 for every angle must be divided by the outcome of equation 40
at starting angle ( ) to find the fraction of the sensors output compared to the output at the
starting angle while the course remains equal. This forms equation 41 and is graphically represented in Figure
41, the lines that represent calculated weight values to the sensors output for rays starting at an angle of 0°, 45°
and at 84°.
( ) ( ) ( )
( ) Average fraction of the sensor output at measured angle compared to the output at measured starting
angle [ ]
( )
( ) Calculated average fraction of the sensor output as function of angle when starting angle [ ]
( ) ( ) ( )
( ) Calculated average fraction of the sensor output at angle compared to the output at starting angle [ ]
Starting and stopping angle for measurements in the test setup The starting angles ( ) and ( ) for measurements in the test setup is identical to
respectively ( ) (equation 11) for the sky and ( ) (equation 16) for the ground, but both are
bounded in direction because of the limited width of the test setup. These limits were already found. The
intersections for the sky were ( ) and for the ground ( ). The starting angles where had
to be corrected for ran from ( ) for the sky or ( )for the ground till . The
starting angles therefore describe the smallest angle where radiance is received from.
For the opposing plate the starting angle was in fact a stopping angle because the opposing plate was visible
from till ( ). ( ) is identical to ( ) of equation 18,. In direction it
is bounded by ( ) and ( ). The stopping angle therefore describes the largest angle
where radiance is received from.
Now ( ), ( ) and ( ) were known the average angles were calculated,
see respectively equation 42, equation 43 and equation 44:
Figure 41. Weight to the sensors output for rays with starting angles , and compared to the output at the respective starting angle. A
trendline and its function is included for starting angle . (calc) stands
for calculated.
49
Equation 44
Equation 45
Equation 42
Equation 43
( )
( )∫ ( )
( )
( )
( ) Average starting angle for the sky [ ]
( )
( )∫ ( )
( )
( )
( ) Average starting angle for the ground [ ]
( )
( ) ( )(∫ ( )
( )
∫ ( ) ( )
)
( ) Average stopping angle for the plate [ ]
Calculating the correction factor When the average relative deviation to the theoretical cosine course, ( ) of equation 38, is multiplied by the
calculated average fraction of the sensor’s output at angle compared to the output at starting angle , ( ) of
equation 41, the relative deviation, weighted for the sensors actual cosine course, to the theoretical cosine course
for every angle could be calculated. This is equation 45:
( ) ( ) ( )
( ) Relative deviation, weighted for the sensors actual cosine course to the theoretical cosine course at
angle [ ]
The results for starting angles at , and are depicted in Figure 42. The ( ) always starts at the
corresponding ( ) value because the ( ) represents the unweighted deviation of the actual cosine course to
the theoretical cosine course. The weighted deviation starts therefore at the unweighted deviation percentage and
while angle increases starts to add the weight factor.
For the sky and ground the ( )
always ran from the average starting
angle
( ) for the sky or
( ) for the ground to
. Therefore a formula was made with
Microsoft Excel that needed the average
starting angle as input and would then
draw the figure of ( ) while, by
means of a trendline, giving its function
as polynomial ( ) (and ( ) ( ). By integrating this polynomial
from the average starting angle to 90°
and dividing it by 90 minus the average
starting angle, the result is the
percentage deviation from the true
cosine course. With this deviation known the correction factor that
had to be applied to correct the output values of the sensors was
calculated. The calculation is presented as equation 46 and
equation 47.
Figure 42. Average relative deviation to the
theoretical cosine course for cc (unweighted
deviation) and three starting angles weighted
for the starting angle till 90° (wcc).
50
Equation 46
Equation 47
Equation 48
(
)
(
( )∫ ( )
( )
)
(
) Correction factor to correct for the faulty actual cosine course of the sensors for the
sky [ ]
(
)
(
( )∫ ( )
( )
)
(
) Correction factor to correct for the faulty actual cosine course of the sensors for the
ground [ ]
For the plate the ( ) always ran from to the average stopping angle
( ).
Therefore the ( ) could be used directly, but because the stopping angle is in direction must be used as
( ). By integrating over ( ) from to the average stopping angle and then divide that by the average
starting angle, the percentage deviation from the true cosine course was found. With this deviation known the
correction factor that had to be applied to correct the output values of the sensors was calculated. The calculation
is presented as equation 48.
(
)
(
( )∫ ( )
( )
)
(
) Correction factor to correct for the faulty actual cosine course of the sensors for the
plate [ ]
51
Appendix F: Corrections for the cosine course error The correction factors to correct for the sensors cosine course error for every sensor for all examined distances
and heights, calculated by means of equation 22 and equation 23, is presented in Table 20 to Table 25. When the
sensor was closer to <ground> or the height of the test setup was increased, the correction factors became
smaller in absolute sense and when the distance became smaller, the correction factors also became smaller in
absolute sense.
The sensor that measured the irradiance at the horizontal plane ‘saw’ the diffuse cloth in its total field of view of
for every measurement. Therefore ran from to which resulted in a correction factor of .
Table 20. Cosine course error correction factors (
) [ ] for the sky with variable distance.
Distance [cm] Bottom Second Middle Fourth Top Diffuse cloth
40 0.626 0.670 0.733 0.823 0.946 0.949
35 0.606 0.649 0.711 0.806 0.945 0.949
30 0.584 0.625 0.686 0.784 0.945 0.949
25 0.559 0.597 0.656 0.756 0.944 0.949
20 0.532 0.566 0.621 0.720 0.943 0.949
15 0.501 0.531 0.579 0.673 0.941 0.949
10 0.468 0.490 0.528 0.610 0.937 0.949
5 0.431 0.443 0.466 0.520 0.922 0.949
Table 21. Cosine course error correction factors (
) [ ] for the sky with variable height.
Height [cm] Bottom Second Middle Fourth Top Diffuse cloth
243.4 0.626 0.670 0.733 0.823 0.946 0.949
182.9
0.670 0.733 0.823 0.946 0.949
122.3
0.733 0.823 0.946 0.949
61.8
0.823 0.946 0.949
1.2
0.946 0.949
Table 22. Cosine course error correction factors (
) [ ] for the ground with variable distance.
Distance [cm] Bottom Second Middle Fourth Top Diffuse cloth
40 0.946 0.823 0.733 0.670 0.626 0.949
35 0.945 0.806 0.711 0.649 0.606 0.949
30 0.945 0.784 0.686 0.625 0.584 0.949
25 0.944 0.756 0.656 0.597 0.559 0.949
20 0.943 0.720 0.621 0.566 0.532 0.949
15 0.941 0.673 0.579 0.531 0.501 0.949
10 0.937 0.610 0.528 0.490 0.468 0.949
5 0.922 0.520 0.466 0.443 0.431 0.949
Table 23. Cosine course error correction factors (
) [ ] for the ground with variable height.
Height [cm] Bottom Second Middle Fourth Top Diffuse cloth
243.4 0.946 0.823 0.733 0.670 0.626 0.949
182.9
0.946 0.823 0.733 0.670 0.949
122.3
0.946 0.823 0.733 0.949
61.8
0.946 0.823 0.949
1.2
0.946 0.949
52
Table 24. Cosine course error correction factors (
) [ ] for the opposing plate with variable distance.
Distance [cm] Bottom Second Middle Fourth Top Diffuse cloth
40 0.993 0.992 0.992 0.992 0.993 0.949
35 0.992 0.991 0.991 0.991 0.992 0.949
30 0.990 0.989 0.989 0.989 0.990 0.949
25 0.988 0.986 0.987 0.986 0.988 0.949
20 0.984 0.983 0.983 0.983 0.984 0.949
15 0.977 0.977 0.978 0.977 0.977 0.949
10 0.968 0.970 0.970 0.970 0.968 0.949
5 0.958 0.960 0.960 0.960 0.958 0.949
Table 25. Cosine course error correction factors (
) [ ] for the opposing plate with variable height.
Height [cm] Bottom Second Middle Fourth Top Diffuse cloth
243.4 0.993 0.992 0.992 0.992 0.993 0.949
182.9
0.993 0.992 0.992 0.993 0.949
122.3
0.992 0.991 0.992 0.949
61.8
0.991 0.991 0.949
1.2
0.987 0.949
53
Appendix G: Extrapolations The results for the irradiance fractions from the sky, ground and plate for all examined distances and heights are
presented in Table 26 to Table 37.
Irradiance from the sky Table 26 and Table 27 are the result of equation 12 and Table 28 and Table 29 are the result of equation 15.
Further away from the diffuse cloth and when the distance between the plates became smaller, the fractions
became smaller.
Table 26. Irradiance fractions for the endless row with variable distance, ( ) [ ].
Distance [cm] Bottom Second Middle Fourth Top
40 6.62E-03 1.15E-02 2.48E-02 8.03E-02 4.85E-01
35 5.09E-03 8.91E-03 1.93E-02 6.49E-02 4.83E-01
30 3.76E-03 6.59E-03 1.44E-02 5.02E-02 4.80E-01
25 2.62E-03 4.61E-03 1.01E-02 3.65E-02 4.76E-01
20 1.68E-03 2.96E-03 6.55E-03 2.43E-02 4.70E-01
15 9.47E-04 1.67E-03 3.72E-03 1.41E-02 4.60E-01
10 4.21E-04 7.46E-04 1.66E-03 6.42E-03 4.40E-01
5 1.05E-04 1.87E-04 4.17E-04 1.63E-03 3.83E-01
Table 27. Irradiance fractions for the endless row with variable height, ( ) [ ].
Height [cm] Bottom Second Middle Fourth Top
243.4 6.62E-03 1.15E-02 2.48E-02 8.03E-02 4.85E-01
182.9
1.15E-02 2.48E-02 8.03E-02 4.85E-01
122.3
2.48E-02 8.03E-02 4.85E-01
61.8
8.03E-02 4.85E-01
1.2
4.85E-01
Table 28. Irradiance fractions for the test setup with variable distance, ( ) [ ].
Distance [cm] Bottom Second Middle Fourth Top
40 2.01E-03 4.49E-03 1.32E-02 6.17E-02 4.84E-01
35 1.55E-03 3.48E-03 1.03E-02 5.05E-02 4.82E-01
30 1.14E-03 2.59E-03 7.76E-03 3.95E-02 4.79E-01
25 7.99E-04 1.81E-03 5.49E-03 2.90E-02 4.76E-01
20 5.14E-04 1.17E-03 3.57E-03 1.95E-02 4.70E-01
15 2.90E-04 6.62E-04 2.03E-03 1.14E-02 4.60E-01
10 1.29E-04 2.95E-04 9.12E-04 5.21E-03 4.40E-01
5 3.24E-05 7.40E-05 2.29E-04 1.33E-03 3.83E-01
Table 29. Irradiance fractions for the test setup with variable height, ( ) [ ].
Height [cm] Bottom Second Middle Fourth Top
243.4 2.01E-03 4.49E-03 1.32E-02 6.17E-02 4.84E-01
182.9
4.49E-03 1.32E-02 6.17E-02 4.84E-01
122.3
1.32E-02 6.17E-02 4.84E-01
61.8
6.17E-02 4.84E-01
1.2
4.84E-01
54
Irradiance from the ground The irradiance fractions from the ground are equal to the irradiance fractions from the sky, but the order is
reversed, meaning the fractions for the top sensor are now the fractions for the bottom sensor, etc.
Table 30. Irradiance fractions for the endless row with variable distance, ( ) [ ].
Distance [cm] Bottom Second Middle Fourth Top
40 4.85E-01 8.03E-02 2.48E-02 1.15E-02 6.62E-03
35 4.83E-01 6.49E-02 1.93E-02 8.91E-03 5.09E-03
30 4.80E-01 5.02E-02 1.44E-02 6.59E-03 3.76E-03
25 4.76E-01 3.65E-02 1.01E-02 4.61E-03 2.62E-03
20 4.70E-01 2.43E-02 6.55E-03 2.96E-03 1.68E-03
15 4.60E-01 1.41E-02 3.72E-03 1.67E-03 9.47E-04
10 4.40E-01 6.42E-03 1.66E-03 7.46E-04 4.21E-04
5 3.83E-01 1.63E-03 4.17E-04 1.87E-04 1.05E-04
Table 31. Irradiance fractions for the endless row with variable height, ( ) [ ].
Height [cm] Bottom Second Middle Fourth Top
243.4 4.85E-01 8.03E-02 2.48E-02 1.15E-02 6.62E-03
182.9
4.85E-01 8.03E-02 2.48E-02 1.15E-02
122.3
4.85E-01 8.03E-02 2.48E-02
61.8
4.85E-01 8.03E-02
1.2
4.85E-01
Table 32.Irradiance fractions for the test setup with variable distance, ( ) [-].
Distance [cm] Bottom Second Middle Fourth Top
40 4.84E-01 6.17E-02 1.32E-02 4.49E-03 2.01E-03
35 4.82E-01 5.05E-02 1.03E-02 3.48E-03 1.55E-03
30 4.79E-01 3.95E-02 7.76E-03 2.59E-03 1.14E-03
25 4.76E-01 2.90E-02 5.49E-03 1.81E-03 7.99E-04
20 4.70E-01 1.95E-02 3.57E-03 1.17E-03 5.14E-04
15 4.60E-01 1.14E-02 2.03E-03 6.62E-04 2.90E-04
10 4.40E-01 5.21E-03 9.12E-04 2.95E-04 1.29E-04
5 3.83E-01 1.33E-03 2.29E-04 7.40E-05 3.24E-05
Table 33. Irradiance fractions for the test setup with variable height, ( ) [ ].
Height [cm] Bottom Second Middle Fourth Top
243.4 4.84E-01 6.17E-02 1.32E-02 4.49E-03 2.01E-03
182.9
4.84E-01 6.17E-02 1.32E-02 4.49E-03
122.3
4.84E-01 6.17E-02 1.32E-02
61.8
4.84E-01 6.17E-02
1.2
4.84E-01
55
Irradiance from the opposing plate Table 34 and Table 35 are the results of equation 17 and Table 36 and Table 37 are the results of equation 19.
The bottom sensor ‘saw’ just as much of the opposing plate as the top sensor. Therefore all results were mirrored
over the middle sensor, which ‘saw’ most of the opposing plate, where distance was variable. When height was
variable the results are mirrored over the center value. The small deviations were the result of rounding errors for
the height.
Table 34. Irradiance fractions for the endless row with variable distance, ( ) [ ].
Distance [cm] Bottom Second Middle Fourth Top
40 5.08E-01 9.08E-01 9.50E-01 9.08E-01 5.08E-01
35 5.12E-01 9.26E-01 9.61E-01 9.26E-01 5.12E-01
30 5.16E-01 9.43E-01 9.71E-01 9.43E-01 5.16E-01
25 5.21E-01 9.59E-01 9.80E-01 9.59E-01 5.21E-01
20 5.28E-01 9.73E-01 9.87E-01 9.73E-01 5.28E-01
15 5.39E-01 9.84E-01 9.93E-01 9.84E-01 5.39E-01
10 5.59E-01 9.93E-01 9.97E-01 9.93E-01 5.59E-01
5 6.17E-01 9.98E-01 9.99E-01 9.98E-01 6.17E-01
Table 35. Irradiance fractions for the endless row with variable height, ( ) [ ].
Height [cm] Bottom Second Middle Fourth Top
243.4 5.08E-01 9.08E-01 9.50E-01 9.08E-01 5.08E-01
182.9
5.03E-01 8.94E-01 8.93E-01 4.90E-01
122.3
4.90E-01 8.37E-01 4.75E-01
61.8
4.32E-01 4.21E-01
1.2
1.50E-02
Table 36. Irradiance fractions for the test setup with variable distance,
( ) [ ].
Distance [cm] Bottom Second Middle Fourth Top
40 2.41E-01 4.41E-01 4.67E-01 4.41E-01 2.41E-01
35 2.61E-01 4.88E-01 5.10E-01 4.88E-01 2.61E-01
30 2.84E-01 5.39E-01 5.58E-01 5.39E-01 2.84E-01
25 3.10E-01 5.97E-01 6.11E-01 5.97E-01 3.10E-01
20 3.41E-01 6.61E-01 6.71E-01 6.62E-01 3.41E-01
15 3.78E-01 7.33E-01 7.39E-01 7.33E-01 3.78E-01
10 4.31E-01 8.13E-01 8.15E-01 8.13E-01 4.31E-01
5 5.35E-01 9.01E-01 9.02E-01 9.01E-01 5.35E-01
Table 37. Irradiance fractions for the test setup with variable height,
( ) [ ].
When adding all the corresponding
for the endless row for sky,
ground and opposing plate, they sum up to
one, meaning the integrals are correct.
When adding these values for the test setup
they are less than one. The remaining
fraction is caused by the black plastic that
was assumed to not reflect any light.
Height [cm] Bottom Second Middle Fourth Top
243.4 2.41E-01 4.41E-01 4.67E-01 4.41E-01 2.41E-01
182.9
2.40E-01 4.35E-01 4.34E-01 2.39E-01
122.3
2.34E-01 4.02E-01 2.34E-01
61.8
2.01E-01 2.02E-01
1.2
4.37E-04
56
Appendix H: Irradiance onto ground and plate The results for the irradiance fractions onto the ground for all examined combinations are presented in Table 38
to Table 41. The endless row values were calculated with equation 25, for the test setup with equation 24. The
irradiance fractions onto the opposing plate are equal to those on <self> as presented in Table 26 to Table 29.
When the distance became smaller, less radiance fell on the ground but more radiance fell on the opposing plate.
When the height became smaller, more radiance fell on the ground and less radiance fell on the opposing plate.
Table 38. Irradiance fractions onto the ground for the endless row with variable distance, (
) [ ].
Distance [cm] Ground
40 3.33E-03
35 2.55E-03
30 1.88E-03
25 1.30E-03
20 8.35E-04
15 4.70E-04
10 2.09E-04
5 5.22E-05
Table 39. Irradiance fractions onto the ground for the endless row with variable height, (
) [ ].
Height [cm] Ground
243.4 5.93E-03
182.9 1.31E-02
122.3 4.86E-02
61.8 9.40E-01
1.2 5.93E-03
Table 40. Irradiance fractions onto the ground for the test setup with variable distance,
(
) [ ].
Distance [cm] Ground
40 1.01E-03
35 7.77E-04
30 5.72E-04
25 3.98E-04
20 2.55E-04
15 1.43E-04
10 6.38E-05
5 1.60E-05
Table 41. Irradiance fractions onto the ground for the test setup with variable height,
(
) [ ].
Height [cm] Ground
243.4 2.34E-03
182.9 7.14E-03
122.3 3.90E-02
61.8 9.40E-01
1.2 2.34E-03
57
Appendix I: Iterated calculated values and similarity of Robinson and
Stone model and extrapolation method Table 42 shows the calculated values after four and after five iterations for combination 4, where most reflectance was present
due to the highest albedo value (white paint ). The relative differences from to , calculated by
( ) ( ) are presented at the bottom right. The difference at the top and bottom sensor
height was less than . The largest difference was found for the middle sensor height, but was still below .
The relative increase from to as fraction of the irradiance at the horizontal plane, calculated by
( ) , shows the largest increase was . They are presented at the bottom right.
Table 42. The calculated values after four ( ) and five ( ) iterations for combination 4 with variable distance (<self>=white,
<plate>=white and <ground>=white) and the relative difference from to and from to as fraction of the horizontal plane.
values [ ]
values [ ]
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 3.65E-02 3.09E-02 6.11E-02 1.85E-01 7.26E-01 3.66E-02 3.18E-02 6.35E-02 1.91E-01 7.28E-01
35 3.08E-02 2.40E-02 4.81E-02 1.53E-01 7.25E-01 3.08E-02 2.47E-02 5.01E-02 1.58E-01 7.27E-01
30 2.54E-02 1.78E-02 3.62E-02 1.20E-01 7.24E-01 2.54E-02 1.84E-02 3.77E-02 1.25E-01 7.26E-01
25 2.02E-02 1.25E-02 2.57E-02 8.91E-02 7.21E-01 2.02E-02 1.29E-02 2.68E-02 9.27E-02 7.23E-01
20 1.54E-02 8.00E-03 1.67E-02 6.03E-02 7.17E-01 1.54E-02 8.32E-03 1.75E-02 6.29E-02 7.19E-01
15 1.08E-02 4.49E-03 9.52E-03 3.55E-02 7.09E-01 1.08E-02 4.68E-03 9.96E-03 3.71E-02 7.11E-01
10 6.63E-03 1.98E-03 4.27E-03 1.63E-02 6.92E-01 6.63E-03 2.07E-03 4.47E-03 1.71E-02 6.95E-01
5 2.76E-03 4.88E-04 1.07E-03 4.16E-03 6.38E-01 2.76E-03 5.11E-04 1.12E-03 4.36E-03 6.42E-01
Relative differences from to [ ]
to as fraction of horizontal plane irradiance [ ]
Distance [cm] Bottom Second Middle Fourth Top
Bottom Second Middle Fourth Top
40 0.09 2.84 3.87 3.30 0.28 0.00 0.09 0.24 0.61 0.20
35 0.09 3.12 4.06 3.57 0.29 0.00 0.07 0.20 0.54 0.21
30 0.08 3.40 4.23 3.83 0.30 0.00 0.06 0.15 0.46 0.22
25 0.07 3.69 4.39 4.09 0.31 0.00 0.05 0.11 0.36 0.23
20 0.07 3.97 4.52 4.32 0.33 0.00 0.03 0.08 0.26 0.24
15 0.06 4.23 4.63 4.52 0.36 0.00 0.02 0.04 0.16 0.26
10 0.05 4.47 4.72 4.67 0.43 0.00 0.01 0.02 0.08 0.30
5 0.05 4.66 4.78 4.77 0.66 0.00 0.00 0.01 0.02 0.42
Table 43 shows an example of the similarity of the Robinson and Stone model and the extrapolation method presented in this
report. The shown fractions are of ( ), the right side is equal to Table 26 (calculated with equation 12) and the
left side is the outcome of equation 4 of the Robinson and Stone model. All values are identical.
Table 43. Irradiance from the sky on <self> as fraction of the horizontal irradiance [ ]
Calculated by the Robinson and Stone model
Calculated by the extrapolation method
Distance [cm] Bottom Second Middle Fourth Top Bottom Second Middle Fourth Top
40 0.006618 0.011545 0.024772 0.080251 0.485007 0.006618 0.011545 0.024772 0.080251 0.485007
35 0.005091 0.008911 0.019297 0.064929 0.482867 0.005091 0.008911 0.019297 0.064929 0.482867
30 0.003755 0.006593 0.014396 0.050197 0.480016 0.003755 0.006593 0.014396 0.050197 0.480016
25 0.002617 0.004606 0.010130 0.036489 0.476028 0.002617 0.004606 0.010130 0.036489 0.476028
20 0.001679 0.002963 0.006555 0.024291 0.470054 0.001679 0.002963 0.006555 0.024291 0.470054
15 0.000947 0.001673 0.003719 0.014108 0.460127 0.000947 0.001673 0.003719 0.014108 0.460127
10 0.000421 0.000746 0.001663 0.006420 0.440427 0.000421 0.000746 0.001663 0.006420 0.440427
5 0.000105 0.000187 0.000417 0.001628 0.383314 0.000105 0.000187 0.000417 0.001628 0.383314