SOLAR CELLS ON A DUTCH MULTI-SPAN
VENLO-TYPE GREENHOUSE
A STUDY ON THE EFFECT OF A RANGE OF ROOFTOP TILT- AND ORIENTATION
ANGLES ON THE TOTAL ENERGY, INCIDENT ON A TILTED SOLAR CELL AND
TRANSMITTED THROUGH THE GREENHOUSE COVER
Written by Floris Max van der Staaij
20 August 2020
MSc Thesis Project. Physics & Astronomy; Advanced Matter and Energy Physics (AMEP).
Student number: 10907963 (UvA)
Project size: 60 EC
Supervisor/ Examiner: Asoc. Prof. Elizabeth von Hauff Supervisor 2/ Examiner 2: Prof. Dr. Davide Iannuzzi Faculty of research: VU Amsterdam
TABLE OF CONTENTS
SUMMARY ... 5
1
INTRODUCTION ... 9
2
BACKGROUND ... 11
2.1
MULTI-SPAN VENLO-TYPE GREENHOUSE ... 11
2.2
ORGANIC PHOTOVOLTAICS ... 12
2.2.1
OPERATING PRINCIPLE ... 12
2.2.2
HISTORICAL DEVELOPMENT AND CURRENT STATUS ... 12
2.3
PHOTOVOLTAIC GREENHOUSE ... 13
2.3.1
CROP YIELD BY LIGHT QUANTITY ... 13
2.3.2
CURRENT STATUS AND FUTURE OPPORTUNITIES ... 14
2.4
RESEARCH OBJECTIVE ... 14
2.5
THESIS OUTLINE ... 15
3
DATASET ... 16
4
LIGHT INCIDENT ON A PHOTOVOLTAIC MODULE ... 17
4.1
INTRODUCTION ... 17
4.2
SOLAR LOCATION ... 17
4.2.1
SOLAR TIME ... 17
4.2.2
SOLAR ANGLES ... 18
4.3
IRRADIANCE ON A TILTED PLANE ... 19
4.3.1
TOTAL IRRADIANCE ... 20
4.3.2
DIRECT IRRADIANCE ... 21
4.3.3
DIFFUSE IRRADIANCE ... 21
4.4
TOTAL ANNUAL INCIDENT ENERGY ... 23
5
LIGHT TRANSMISSION OF A GREENHOUSE COVER ... 25
5.1
INTRODUCTION ... 25
5.2
GLASS SHEET REFLECTION AND TRANSMISSION ... 26
5.3
TRANSMITTANCE BY GREENHOUSE COVER ... 28
5.3.1
TOTAL IRRADIANCE TRANSMITTED ... 29
5.3.2
TRANSMISSION OF DIRECT IRRADIANCE ... 29
5.3.3
TRANSMISSION OF DIFFUSE IRRADIANCE ... 38
5.4
TOTAL ANNUAL TRANSMITTED ENERGY ... 38
6
DISCUSSION ... 40
6.1
DISCUSSION OF RESULTS ... 40
6.2
LIMITATIONS ... 41
6.3
OUTLOOK ... 42
7
CONCLUSION ... 43
8
APPENDIX ... 44
8.1
DATASET ... 44
8.1.1
ANNUAL INSOLATION PER TIME OF THE DAY ... 44
8.1.2
SOLAR ELEVATION ANGLE ... 45
8.2
MODEL VALIDATION ... 45
8.2.1
SOLAR LOCATION ... 45
8.2.2
PV MODULE ... 46
8.2.3
GREENHOUSE ... 52
8.3
SUPPLEMENTARY FIGURES ... 57
REFERENCES ... 59
“He who lives by the crystal ball soon
learns to eat ground glass”
– Edgar Russel Fiedler
ACKNOWLEDGEMENT
I wish to thank various people for their contribution to this project and my personal development, during the two years of my master’s degree.
First and foremost, I would like to express my very great appreciation to my research
supervisor Asoc. Prof. Elizabeth von Hauff, for all the lessons I’ve learned while doing this project. All the conversations we had on a broad spectrum of topics, throughout the years, were both very enjoyable and of great value to my skills as a researcher. Also, the freedom I received to operate independently during this project has taught me a lot. I will never forget the humorous discussions we had together with Isabelle. Thank you, Elizabeth, for always believing in Isabelle and me, and especially for continuously being critical on our work. As a result, I have become far more critical on myself too.
Second, deep gratitude is expressed towards Prof. Dr. Davide Iannuzzi. In some ways, Davide
has walked many paths I would once like to discover. His openness, sincerity and vision, has motivated me to keep on working at the highest pace possible, knowing that if I do so, I will once reach my goals in life too. And if I do so, there will come a moment in life when I will be able to return the favor. Thanks Davide, for giving Isabelle and me the chance to learn from you. As a small first return, hereby, I will promise to read your book after my thesis is finished (no fingers crossed this time).
Third, I would like to thank Tom Veeken, for the couple of mentoring sessions I received.
Tom’s guidance was particularly helpful during moments when I got stuck and was in need for both a positive attitude and critical thoughts on how to best move forward. By discussing my progression on the project, Tom gave me much insight into what pathways to take and which to never enter. Thanks, Tom, you showed to be a wise teacher with a lot of patience.
Forth, to all members of the Hybrid Solar Energy Group, located at the VU. Alba
Fonseca-Topp, Achidi Frick, Micha Hillenius, Ivo Jak, Rhea Lambregts, Pan Li, Sudeshna Maity, Lotte Schaap, David Schreuder, Martin Slaman and Luuk Wagenaar, thanks all for the discussions we had during our weekly meetings, I’ve learned a lot from you. Especially to Ivo, thank you for correcting my thesis.
Last but not least, I am very grateful to have collaborated together with Isabelle van Keulen,
during my Master’s. I experienced our teamwork as both very productive and a lot of fun. In my opinion, Isabelle can best be described as a work horse, equipped with maximum horsepower. She can be considered a talented person, which possesses both strong social skills and a very high intelligence. Therefore, knowing that she is still unsure about her future, I insist that she uses her skills to positively impact the world, since not many people in the world will be capable of leaving the same amount of impact as she can. It was always a great pleasure to work together with her, and if I keep on dreaming, maybe one day we will run a huge company together. And if not, I am still looking forward to seeing her for the rest of my life, knowing our friendship will last forever. Only one remark remains, which cannot be left unnoted. Some years before 30-10-2018, she made me a promise she did not yet fulfil. And, as we all know; a promise made, must be a promise kept. So, Isabelle, please tell me… what are you waiting for?
SUMMARY
Worldwide, the use of energy is expanding at an enormous rate. Consequently, climates are changing, air is becoming more polluted and global warming is turning into a serious threat to humanity. Therefore, societal changes are demanding a lowering of 𝐶𝑂! emissions, although energy consumption by the Dutch horticulture sector has been rising over the last decades. Renewable, clean energy sources are required to keep the Dutch horticulture sector up and running, while meeting the global energy commitments. Agrivoltaic greenhouse systems offer an interesting solution to the current issue, by partly supplying the greenhouse with locally harvested solar energy. Organic solar cells are an emerging third- generation photovoltaic (PV) technology, which have the potential to be made low-cost, flexible and semi-transparent. As a result, electricity and agricultural crop production can be combined on the same area of land.
The Netherlands is known for its innovative horticulture sector, providing high-quality crops. Light quantity is frequently one of the limiting factors to crop development. Hence, to achieve high light transmittance throughout the year, one of the most commonly build structures is the multi-span Venlo-type greenhouse. A theoretical model is constructed to examine to what extend the ideal rooftop tilt and orientation of a Dutch multi-span Venlo-type greenhouse matches with the ideal tilt and orientation for OPV modules fixed on top of the greenhouse roof structure.
The effect of a range of specific rooftop tilt- and orientation angle on the total energy, incident on a tilted PV module and transmitted through the greenhouse rooftop, is graphically illustrated. The ideal tilt angle of the PV module was found to match only slightly with the ideal rooftop angle of the greenhouse, if both directed in their ideal orientation. In addition, the ideal orientation of the PV module shows to be highly unfavorable, relative to the ideal orientation of the greenhouse, and vice versa. Furthermore, the range in maximum and minimum total energy was found to be much larger for the PV module. In sum, further research is required, before economically the ideal balance between the angles can be identified.
NOMENCLATURE
Symbol Value Dimension Description SOLAR LOCATION
𝐿𝑇 hour Time on location
𝑇"#$ hour Coordinated Universal Time
𝑇%&'()*+( hour Time zone correction
𝐿𝑆𝑇𝑀 hour Local standard time meridian
𝐸𝑂𝑇 hour Equation of time
𝐵 rad Equation of time correction factor
𝑛 Number of the day (January 1st corresponds to 𝑛 = 1 and
December 31st corresponds to 𝑛 = 365)
𝐿𝑆𝑇 hour Local solar time
𝑇𝐶 hour Net time correction factor
𝜆 deg Longitude
∅ deg Latitude
𝜔 deg Solar hour angle
𝛿 deg Declination angle
𝛾, deg Solar azimuth angle
𝜎) deg Solar zenith angle
PV MODULE
𝜉 deg Angle of incidence
𝛽 deg PV module tilt angle
𝛾 deg PV module azimuth angle (oriented in the South corresponds to 𝛾 = 0°)
𝐺 𝑊𝑚-! Total irradiance on a horizontal plane
𝐺. 𝑊𝑚-! Beam irradiance on a horizontal plane
𝐺/ 𝑊𝑚-! Diffuse irradiance on a horizontal plane
𝐺# 𝑊𝑚-! Total irradiance on a tilted plane
𝐺.# 𝑊𝑚-! Total beam irradiance on a tilted plane
𝐺/# 𝑊𝑚-! Total diffuse irradiance on a tilted plane
𝑟. Beam conversion factor
𝐷𝑁𝐼 𝑊𝑚-! Direct normal irradiance
𝐺0+ 𝑊𝑚-! Extraterrestrial irradiance on a surface normal to the sun
𝐺,1 1367 𝑊𝑚-! Solar constant
𝐸0 Eccentricity correction factor
𝐺0 𝑊𝑚-! Extraterrestrial global solar irradiance on a horizontal
plane
𝐴& Anisotropy index
𝑏 Muneer-family correction factor, varies per sky condition
𝑎2 0.712 Muneer-1 constant 2
𝑎! 0.6883 Muneer-1 constant 3
REFLECTION AND TRANSMISSION COEFFICIENTS
𝑑 3.2 𝑚 Gutter-to-gutter distance
𝜓 deg Rooftop tilt angle
𝜉3 deg Greenhouse azimuth angle (gutter oriented in the South corresponds to 𝜉
3 = 0°)
𝑅45 Single surface reflectivity of a parallel polarized beam of light
𝑅4( Single surface reflectivity of a perpendicular polarized beam of light
𝜃& deg Angle of incidence with respect to the normal of the surface
𝜃% deg Angle of transmission with respect to the normal of the surface
𝑛2 1 Refractive index of air
𝑛! 1.5 Refractive index of glass
𝑛1*6(7 1.5 Refractive index of the greenhouse cover material
𝐶 Variable introduced to redefine the definitions of the single surface reflectivity 𝑅
45 and 𝑅4(
𝑄 Single surface reflectivity correction factor
𝐶5., 𝑚-2 Power absorption coefficient
𝐷 4 ∙ 10-8 𝑚 Optical pathlength
𝐷′ 𝑚 Redefined optical pathlength
𝑅′45 Double surface reflectivity of a parallel polarized beam of light
𝑅′4( Double surface reflectivity of a perpendicular polarized beam of light
𝑇45 Double surface transmission of a parallel polarized beam
of light
𝑇4( Double surface transmission of a perpendicular polarized
beam of light
𝑅 Reflection coefficient of glass cover material
𝑇 Transmission coefficient of glass cover material
GREENHOUSE TRANSMITTANCE
𝐺9: 𝑊𝑚-! Transmission coefficient of greenhouse cover
𝑇1*6(7,/&7 Transmission coefficient of direct irradiance incident on the greenhouse cover
𝑇1*6(7,/&<< Transmission coefficient of diffuse irradiance incident on the greenhouse cover
ℎ Height, on the z-axis, of the ridge relative to the gutter
𝕀 Three-dimensional vector of a beam of light incident on
a roof pane
ℝ2 Three-dimensional vector of roof pane 1
ℝ! Three-dimensional vector of roof pane 2
ℕ2 Three-dimensional vector normal to roof pane 1
ℕ! Three-dimensional vector normal to roof pane 2
𝜃2& Angle of incidence of a light beam reaching roof pane 1 𝜃!& Angle of incidence of a light beam reaching roof pane 1
ℛ2 Two-dimensional vector of roof pane 1
ℛ! Two-dimensional vector of roof pane 2
𝐼 Two-dimensional vector of a beam of light incident on a roof pane N= Two-dimensional vector normal to the beam of light incident on a roof pane
𝒟 Two-dimensional vector of the gutter’s orientation
ℛ !? Two-dimensional vector projection of vector ℛ! on 𝒟
𝜆! Multiplication factor between 𝒟 and ℛ !?
𝑓2 Fraction the light beam
𝑓! Fraction the light beam
𝑇2 Transmission coefficient of roof pane 1
𝑇! Transmission coefficient of roof pane 2
𝓃 Number of roof spans passed by the beam of light
𝑇<2 Transmission coefficient of the fraction of light 𝑓2
𝑇<! Transmission coefficient of the fraction of light 𝑓!
𝑅! Reflection coefficient of roof pane 2
R@ABCDE Reflection on inner sides of pane 2
𝜗 deg Angular correction factor used as step one to mirror back to the third quadrant 𝜅 deg Angular correction factor used as step two to mirror back
to the third quadrant
𝐼E@FGD@HIG@JA Factor describing the distribution of diffuse irradiance across the sky
APPENDIX
𝐴𝑀 Air mass index
𝑘% Clearness index
𝑘%′ Modified clearness index
𝑓 Reindl irradiance estimation model correction factor
∈ Perez irradiance estimation model brightness coefficient
𝐹2, 𝐹!, 𝐹22, 𝐹2!,
𝐹28, 𝐹!2, 𝐹!!, 𝐹!8
Perez irradiance estimation model correction factors
∆ Perez irradiance estimation model correction factor
𝓆2, 𝓆! Perez irradiance estimation model correction factor
𝑅𝑀𝑆𝐸 Root-mean-square-error
1 INTRODUCTION
Over the last decades, worldwide energy demand has been rising rapidly due to population growth and the industrial evolution [1]–[3]. As a consequence, mostly traditional fossil fuels have been burned at a tremendous rate, leading to a series of environmental problems such as climate change, air pollution and acid rain [3]–[5]. In 2017, approximately more than 80% of the estimated total energy consumption still originated from fossil fuels, while primarily hydroelectric-, nuclear- and renewable energy sources provided the remainder [1], [6]. Therefore, in spite of past growth in the use of alternative energy sources, the world requires even less dependence on fossil fuels to meet the international Paris Climate Agreement [6].
In a single hour, the amount of power from the sun that strikes earth is enough to supply the worldwide electricity demand for one year [7], [8]. Energy harvested from the sun can be converted into electricity, heat or fuel [9]. In 2018, the share of electrical energy in total final energy consumption was around 19% [10]. Solar photovoltaic technologies (PV) generate electric power by the use of solar cells, which convert solar energy into a flow of electrons by the photovoltaic effect. In 2018, the amount of dollars invested in new renewable power capacity far exceeded investments in fossil and nuclear power capacity. Moreover, the largest share of capital was invested in solar PV technologies [6]. Nowadays, due to rapidly reducing costs [11], solar PV is the fastest growing power generating technology among all other renewable energy technologies. Therefore, PV technologies offer great potential to partly replace fossil fuels [8].
The Netherlands is known for its strong and competitive horticulture sector [12], selected as one of the key innovations and growth industries by the Dutch Government [13]. Accordingly, Dutch horticulture businesses have increased rapidly in recent decades [14], [15]. In 2017, the Dutch glass-made greenhouses covered a total area of 9079 ha, of which 4992 ha is utilized for the cultivation of vegetables [15]. Over the last decennaries, greenhouses primarily burned natural gas, to supply their energy demands [16]. However, since the 1990s, societal demands started to require a lowering of 𝐶𝑂!
emissions in the greenhouse horticulture sector [12], [16]. Accordingly, a transition was induced towards the use of more renewable and clean energy sources [17]. As a result, in the last decade, the use of electricity has risen relative to thermal energy. In 2018, the energy usage consisted around 74% of heat and 26% of electricity [17]. For all Dutch greenhouse farmers, energy cost form a major part of the total production cost and are expected to keep on growing in the upcoming decade [12].
Agrivoltaics is referred to a system which consists of PV modules installed several meters above ground where soil crops are grown [18]–[20]. In doing so, agricultural and electricity production are combined, leading to both effective use of land and a higher community acceptance towards solar PV technologies [21]–[23]. In higher latitude locations such as the Netherlands, the amount of available sunlight for crop production shows to be limited [24]–[26]. Since additional shading can be detrimental for both the plant’s productivity and growth [28]–[32], crystalline silicon (c-Si) or other opaque PV technologies constructed on the greenhouse’s rooftop will most likely negatively affect annual crop yield inside a greenhouse located in the Netherlands [33].
Organic photovoltaics (OPV) is an emerging thin-film PV technology that possesses unique properties such as a tunable absorption spectrum, flexibility and light-weightiness [34]–[39]. Record lab efficiency for OPV has exceeded 18%, with reported device lifetimes ranging from months to several years [40]. Moreover, the ultra-thin, molecular layers of an OPV solar cell absorb only a small amount of the available photons in the visible part of the spectrum. As a result, organic solar cells are able to transmit part of the visible light spectrum, while still being able to absorb energy from the other parts of the light spectrum [38], [41], [42]. Thus, OPV greenhouses show to be an interesting future
niche application, offering a local solution to partly provide greenhouses with sustainable energy [12], [16], [27], [43].
In the Netherlands, an ideal greenhouse rooftop tilt and orientation is considered crucial, to reach high transmission of light through the greenhouse’s cover material. Consequently, in the Netherlands, a commonly build greenhouse design is the Venlo-type greenhouse, which is known for its high transmissivity of light [27]. Since the greenhouse’s rooftop tilt- and orientation angle is fixed after the greenhouse is built, PV modules put on top of a greenhouse are dependent on the geometrics of the building design. However, due to the relatively low efficiency of large active-area OPV modules [44], correct placement of a fixed rooftop module can be considered essential too, to harvest a significant amount of solar energy over the solar cell’s lifespan.
This thesis examines to what extend the rooftop tilt- and orientation angle of a Dutch multi-span Venlo-type greenhouse matches with the tilt- and orientation angle for PV modules fixed on top of the greenhouse roof structure. By the aid of a theoretical model, the impact of a range of rooftop tilt- and orientation angles, on the amount of energy transmitted through the greenhouse cover, is compared with the effect of a range of PV module tilt- and orientation angles, on the amount of energy incident on the tilted module surface.
2 BACKGROUND
2.1 Multi-span Venlo-type greenhouse
In the Netherlands, high greenhouse cover transmittance is considered a key priority due to the limiting amounts of solar irradiance, especially throughout the winter period [24]–[26]. Therefore, mostly thin, single-layer, glass-made cover materials are being used. On the contrary, poor insulation by the cover material leads to high energy losses, particularly during the winter period [45], [46]. The greenhouse cover’s light transmissivity relies on a lot of factors such as the greenhouse’s orientation, roof tilt angle, geometrical shape of the roof, dimensions of the structure, shading by construction elements, position and transmissivity of the cover material, ageing of the cover material, dust accumulation and the effect of wetting by condensation and rain on both the inner and outer side of the covering material [24]–[26], [45], [47]–[52].
One of the most common greenhouse building structures, in the Netherlands, is the symmetric Venlo-type greenhouse, which is known for its high transmissivity of light [27]. The Venlo-type greenhouse is considered a high technology greenhouse, usually consisting of roof ventilation, automated controls, and offering superior crop and environmental performance. In 2016, the majority of Dutch greenhouse owners with plans of building a new greenhouse in the future prefer to build a Venlo-type greenhouse. In their opinion, new greenhouse designs have too long payback periods [53]. The Venlo-type greenhouse can be constructed as a single-span or multi-span greenhouse. Both types of construction are illustrated in the figure below. A glass multi-span greenhouse is characterized by modular structures on top of the greenhouse’s roof, which typically leads to a more robust design [45].
Fig. 2-1 Left: Single-span Venlo-type greenhouse. Right: Multi-span Venlo-type greenhouse.
In 2011, more than 90% of the Dutch greenhouse area was covered by single glass [46]. The cost of a glass structure is higher than a plastic structure. However, shading by the greenhouse’s building structure is minimized, leading to both high light transmissivity and a longer lifetime of the cover material, relative to the use of plastic materials. Moreover, plastic sheet covers are used mostly in countries where the amount of light incident on the greenhouse is not considered a limiting factor to crop growth [27]. Also, an anti-reflection coating can be deposited onto the glass cover material, or some greenhouse owners even install special diffuse glass, to increase the amount of diffuse light in the greenhouse [26], [46], [53], [54].
2.2 Organic photovoltaics
2.2.1
Operating principle
The active layer of an OPV module consists of carbon-based molecular materials [37], [55]. The operating principle of OPV is illustrated in the figure below [56].
Fig. 2-2 Schematic illustration of the working principle of OPV. The red block represents a donor material, while the blue block is representing a acceptor material [56].
Absorbed photons with energies greater than the bandgap result in the creation of a coulombically bound electron-hole pair, also called an exciton. However, before any free charge carriers are created, the exciton must first be dissociated. A two-component system, consisting of an electron donor (D) and electron acceptor (A), is utilized to achieve exciton dissociation [37], [55]–[58].
Firstly, photoabsorption leads to the excitation of an electron, from the donor’s highest occupied molecular orbital (HOMO) to its lowest unoccupied molecular orbital (LUMO), creating an electron-hole pair. Secondly, the exciton migrates to the donor:acceptor (D:A) interface, via a chemical potential gradient. In order for exciton dissociation to occur, the photogenerated electron-hole pair has to reach the D:A interface, before relaxation to the ground state occurs via non-radiative and intrinsic radiative processes. The charge transfer (CT) complex will be favorable to occur when the difference in energy between the D- and A LUMO is greater than the binding energy of the electron-hole pair. Thirdly, free charge carriers are created by charge-transfer to the LUMO of the A material. Finally, the free charge carriers are diffused into the bulk and collected at the anode and cathode [37], [55]–[58].
2.2.2
Historical development and current status
Over the past 20 years, the photovoltaic (PV) market has expanded tremendously, rising from just 252 MW installed per year in 2000 to 109 GW installed per year in 2018 [6]. Silicon wafer-based technology consistently has been the most dominant commercial PV technology, representing over 95% of the entire PV market in 2018 [11]. The remaining PV market share is distributed over different thin film technologies, such as Cadmium Telluride (CdTe), amorphous silicon (a-Si) and Copper Indium Gallium Selenide (CIGS) and other emerging PV technologies, such as OPV.
It was long expected that these thin-film PV technologies would gain increasing market share and eventually compete with silicon [59]–[61]. However, silicon’s dominant market position can be attributed to a variety of factors such as its abundance of raw materials, low-cost manufacturing, high power conversion efficiencies (PCE) and substantial lifetimes [11], [62]–[65]. As a result, thin film PV
technologies have not managed to gain the expected competitive traction against the global silicon PV market [11], [59], [66], [67].
During the first decades after the invention of thin-film OPV modules, the active layer was generally composed of single-component organic materials which yielded low PCEs [68]. In 1986, Tang [69] first showed that, by bringing together the donor and acceptor material in one bilayer cell, a better driving force was provided to overcome the exciton binding energy. Both Yu et al. [70] and Halls et al. [71] independently reported in 1995 on a bulk-heterojunction configuration, which offered enhanced PCEs via an extended donor-acceptor heterojunction. Until 2006, the potential of semi-transparent organic PV modules was not realized [38], [72], [73].
Extensive research has led to significant increases in the record efficiencies of OPV [74] over the past decades, as is illustrated in the figure below. The timeline regarding organic cells can found in the bottom right part of the figure, indicated by the orange filled circles.
Fig. 2-3 Certified best research-cell efficiencies over time for various PV technologies, provided by NREL [74]. Both past and future progress in OPV is built on the emergence of new materials and device manufacturing technologies, beside fundamental research on the active layer’s morphology and the physics of the device [75], [76]. OPV, at present, still experiences commercialisation challenges due to its shorter lifetime and lower efficiency, compared to commercial PV technologies. Moreover, despite the interesting advantages and impressive learning curve, OPV has not yet succeeded as a commercial technology on the large scale [77]–[80].
2.3 Photovoltaic greenhouse
2.3.1
Crop yield by light quantity
Crop yield is defined as the total amount of agricultural production harvested per unit of land area. The effect of opaque PV modules put on top of a greenhouse construction, has been studied a lot in the past
[81]. The development of a crop is dependent on the type of irradiance, spectrum specific wavelengths and the light quantity [82]–[84].
In the Netherlands, the effect of the quantity of light incident on the crop shows to be of great importance to its growth response. Some plants show to adapt better to lower than optimal levels of light, while other show more detrimental effects on crop growth [33]. For most crops produced in a Dutch greenhouse, on average a 1% increase of light induces a 0.5 to 1% increment in crop yield, depending on the crop type [85]. Furthermore, Dutch Greenhouses are known to produce high-quality crops, therefore leading to relatively high market prices [86]. Thus, for Dutch greenhouses which produce crops with a high economic importance, the use of opaque solar cells shows to be a highly improbable match.
2.3.2
Current status and future opportunities
Studies with regard to the economic feasibility of photovoltaic greenhouses suggest immense future potential, if production costs are low enough, the negative effect on crop yield is not significant and substantial energy can be generated [23], [87]. Nevertheless, at present, most semi-transparent OPV cells exhibit suboptimal photovoltaic properties such as a low power conversion efficiency, low average transmittance in the PAR region, or both [38], [41], [87]. Most likely, therefore, no commercial OPV module seems to exist nowadays for the greenhouse sector. Some studies suggest that commercial feasibility will be achieved if OPV modules can be fabricated at significantly lower costs [41], [87], [88]. If so, power conversion efficiency only has to reach a couple of percentages [88].
The photosynthetically active radiation (PAR) is the portion of the light spectrum utilised by plants for photosynthesis, which designates the same spectral range as visible (VIS) light from 400 to 700nm [89]. Since the cover of a Venlo-type greenhouses is made with glass materials, the greater part of the UV spectrum is already absorbed by the cover itself and therefore not available to generate electricity [90]. Hence, in the most ideal case, light in the PAR region would be transmitted through the module to be used for the growth of the crops, while most of the near-infrared (NIR) light is absorbed by the OPV module [38], [91], [92].
Depending on the type of crop, the absorption spectra of wavelengths in the solar light spectrum differs. As green light is only weakly absorbed by most plants [33], [84], it is likely to have the least effect on crop yield if eliminated from the light spectrum [33], [93]. Hence, a way to solve the current low efficiency and transmittance issue of OPV, is to synthesize active layer materials which primarily absorb green light and other parts of the VIS spectrum which are not utilized by the crop grown beneath the OPV module. As a result, agrivoltaic greenhouse systems based on OPV technology could offer great advantages to the horticulture sector [12], [16], [23], [27], [43], if spectral matching between the absorption spectra of the greenhouse crop and the OPV module is done properly [94].
2.4 Research objective
The tilt- and orientation angle of a multi-span Venlo-type greenhouse defines the tilt- and orientation angle of a fixed PV module placed on top of the greenhouse. In this study a theoretical model is constructed in the language Wolfram Mathematica, which analyses historical data of measured light irradiance in the Netherlands. The theoretical model offers the possibility to compute the effect of a range of greenhouse rooftop tilt- and orientation angles on the total annual energy, per square meter, transmitted through the greenhouse cover material. In addition, the model computes the effect of a range of PV module tilt- and orientation angles on the total annual energy, per square meter, to be utilized for
the generation of electricity. The effect of the range of varying angles on the total annual energy available, for crop growth and electricity production, is graphically illustrated for both cases and subsequently compared to each other.
By comparing the results computed by the theoretical model, the objective of this study is to achieve better understanding, to what extent, a range of rooftop tilt- and orientation angles of a multi-span Venlo-type greenhouse located in the Netherlands matches with the corresponding tilt- and orientation angles of an PV module.
2.5 Thesis outline
The subsequent chapters are as follows:
Chapter 3 presents the dataset utilized for this study.
Chapter 4 explains the methodology of modelling irradiance on a PV module. Firstly, it
describes how the solar time and its location can be calculated. Second, it explains the effect of a tilted plane, with a certain orientation, on the irradiance incident on the PV module. Lastly, it presents the impact of a range of tilt- and orientation angles on the annual energy, per square meter, incident on a tilted PV module.
Chapter 5 describes the methodology of modelling how roof top geometry impacts the light
transmission of a greenhouse cover. Based on vector analysis, the effect of a certain rooftop tilt and orientation, the amount of light transmitted through the greenhouse structure, can be computed by the use of this model. In the end, it presents the impact of a range of tilt- and orientation angles on the annual energy, per square meter, transmitted through the greenhouse cover of a multi-span Venlo-type rooftop structure.
Chapter 6 compares and discusses the final results of the thesis. Furthermore, the theoretical
model’s limitations are addressed and, lastly, an outlook on the study is presented.
Chapter 7 provides a conclusion on the most important results of this study.
Chapter 8 further investigates the dataset’s characteristics. Moreover, the chapter discusses
3 DATASET
In this study, data is taken from the online solar radiation tool provided by the ‘Photovoltaic Geographical Information System’ (PVGIS), made available by the European (EU) Commission’s science and knowledge service [95]. The EU tool provides datasets of hourly measured direct, diffuse, and reflected irradiance on both horizontal and tilted surfaces, of which the tilt and orientation of the PV module can be specified manually. All data manipulation is executed by the aid of Mathematica, version 12.1 [96].
A full-time series of hourly values of the solar irradiance, by its corresponding coordinated universal time (UTC) and solar altitude angle, is retrieved by the use of the PVGIS tool. Data from the complete year of 2016 is used in this paper, which is the last available year for which data can be obtained via the online tool. The dataset is set to the latitude 52.007 and longitude 4.419, which describes a location in the Netherlands, where currently a greenhouse is situated. Moreover, the greenhouse is located in the countryside of the Dutch province ‘South-Holland’, which consists of multiple regions where greenhouses are located [97]. A top view of the dataset’s location can be found in the appendix’s supplementary figure S1.
To validate part of the model created in this paper, the PVGIS tool is also used to directly manipulate the data of irradiance on a horizontal plane to irradiance on a tilted plane, by specifying a certain tilt and azimuth and let the tool do the calculations by itself. By doing so, the tool provides datasets based on its own model estimations, which then can be used to validate the model its output. The dataset’s characteristics are analyzed and discussed in section 8.1. of the appendix. Furthermore, validation of all parts of the model utilized in this paper, are discussed in the appendix.
4 LIGHT INCIDENT ON A PHOTOVOLTAIC MODULE
4.1 Introduction
Most weather stations measure the global irradiance on a horizontal surface. A horizontal surface can be described by a zero-tilt angle and a directionless surface. To calculate the irradiance on an inclined surface, the measurement of its corresponding local solar time, solar position, and many other factors need to be taken into consideration, which is further discussed in section 4.2. In addition, numerous angle calculations play a role in translation from the horizontal plane to the inclined plane.
Since a different installation angle and orientation strongly affects the amount of solar radiation incent on the PV module, many studies have been done on this subject. In the past, multiple models were created for the estimation of the hourly global irradiance on inclined surfaces [98]–[104]. In this paper, the estimation models by Muneer [105], Reindl [10], and Perez [107] are utilized for this purpose.
In the end of this chapter, the final result of the theoretical model its computation is presented based on the methodology explained in this chapter, resulting in the graphical illustration of the dependency of the PV module’s tilt- and orientation angle on the total annual energy incident on the PV module. Additionally, the result is validated with existing literature.
4.2 Solar location
4.2.1
Solar time
In the dataset used in this paper, each hourly device measurement was done in Coordinated Universal Time (UTC). The time on a specific location, defined as the local time, LT, is described by the UTC, 𝑇"#$ in hours, and the specific time zone of the location. Depending on the day of the year and the analyzed location, the time zone, 𝑇%&'()*+(, can shift according to the daylight-saving time (DST). Many but not all countries in the Northern hemisphere use DST in the summertime, which usually starts in March-April and ends in September-November when the countries return to wintertime. The local time is derived below.
𝐿𝑇 = 𝑇"#$+ 𝑇%&'()*+( (4.1)
The local standard time meridian, LSTM, is used as a reference meridian for a particular time zone, also comparable to the prime meridian used for the Greenwich mean time (GMT) [108], [109]. The LSTM is expressed in the following equation.
𝐿𝑆𝑇𝑀 = 15 ∙ (𝐿𝑇 − 𝑇"#$) (4.2)
The equation of time, EOT in minutes, corrects for the eccentricity of the earth’s axial tilt and orbit [109], [110]. B is expressed in radians and n stands for the number of days since the beginning of the year, for example, the 1st of January corresponds to n = 1, while the 1st of February corresponds to n =
32 [109]–[111]. The following mathematical description of the EOT is accurate to within 0.5 minutes [110].
𝐸𝑂𝑇 = 229.2 ∗ (0.000075 + 0.001868 c𝑜𝑠[𝐵] − 0.032077 s𝑖𝑛[𝐵] − 0.014615 c𝑜𝑠[2 ∙ 𝐵] − 0.04089 s𝑖𝑛[2 ∙ 𝐵])
(4.3)
with B formulated as,
𝐵 = 2 𝜋 (𝑛 − 1) 365
(4.4)
The local solar time, LST, is set as the moment for which the LST is noon and the sun is at its highest point in the sky. Generally, the LT varies from the LST due to the eccentricity of the Earth’s orbit and the human definition of time zones and DST.
As the earth rotates 1 degree every 4 minutes, within a certain time zone, a variation of the LST by longitude variations within the time zone needs to be taken into consideration. The net time correction factor, TC, is therefore introduced, which also takes into account the EOT [109]. The longitude of the location under study, in degrees, is described by the symbol 𝜆 and ranges from, 𝜆 = 0 ° at the Prime Meridian, to 𝜆 = +180 ° eastward, and 𝜆 = −180 ° westward [99], [109].
𝑇𝐶 = 𝐸𝑂𝑇 − 4 (𝐿𝑆𝑇𝑀 − 𝜆) (4.5)
𝐿𝑆𝑇 = 𝐿𝑇 +𝑇𝐶 60
(4.6)
Finally, the solar hour angle (SHA), 𝜔 in degrees, can be calculated, which at noon should be equal to zero. The SHA is considered positive in the afternoon and is calculated and completes a 360-degrees cycle in exactly 24 hours, hence rotating with 15 degrees per hour [109], [112].
𝜔 = 15 ∙ (𝐿𝑆𝑇 − 12) (4.7)
4.2.2
Solar angles
Inclined plane irradiance models take into account the analyzed location by its longitude, 𝜆 in degrees, and latitude, ∅ in degrees, in combination with the time and day of the year. In this section, the mathematical relationships are discussed which correctly describe the required solar components which need to be taken into consideration for each time-related measurement in the dataset.
The declination angle, 𝛿 in degrees, is used to describe the angular position of the sun at solar noon, with respect to the plane of the equator. Values of the declination angle range from 𝛿 = −23.45° to 𝛿 = 23.45°. Around the 21st of December, the Northern hemisphere is inclined far away from the
sun, leading to the minimum value of the declination angle. Similarly, but opposite, around the 21st of
June, the southern hemisphere is inclined far away from the sun, leading to the maximum value of the declination angle in the Northern hemisphere. During both spring and fall equinoxes, which begin around the 21st of March and the 21st of September respectively, the declination angle is 𝛿 = 0 °, as the
sun passes directly over the equator [109], [111], [113], [114].
The declination angle can be described by the use of the number of the day, 𝑛, as stated in the following equation.
𝛿 = 23.45 s𝑖𝑛 w2𝜋284 + 𝑛365 x (4.8) At the equator, the latitude is equal to ∅ = 0 °, while at the Northern and Southern hemisphere is reaches ∅ = 90 ° and ∅ = −90 °, respectively [99], [101], [104], [109], [113]. The sun’s angular displacement is defined as the solar azimuth angle, 𝛾, in degrees, which ranges from 𝛾,= 0 ° at the south to 𝛾,=
±180 ° at the north, changing either positive towards the west or negative towards the east [99], [104], [109]. The range of azimuthal displacement of the sun, during each day, is maximal during the summer period and minimal during the winter period. Based on the sign of the solar hour angle, the solar azimuth angle can be calculated as by the following equations [115].
If 𝜔 > 0, 𝛾,= {𝑐𝑜𝑠 }𝑠𝑖𝑛(𝜎)) s𝑖𝑛(∅) − s𝑖𝑛(𝛿) 𝑠𝑖𝑛(𝜎)) c𝑜𝑠(∅) ~ -2 { (4.9) If 𝜔 < 0, 𝛾, = −1 ∙ {𝑐𝑜𝑠 }𝑠𝑖𝑛(𝜎)) s𝑖𝑛(∅) − s𝑖𝑛(𝛿) 𝑠𝑖𝑛(𝜎)) c𝑜𝑠(∅) ~ -2 { (4.10)
The solar zenith angle, 𝜎) in degrees, describes the angle between the zenith and the center of the sun [99], [109], [113]. 𝛼, in degrees, represents the solar altitude angle, which describes the elevation of the sun by the angle between the horizontal plane and the sun’s center. A schematic illustration of the solar azimuth and solar altitude angle can be found in the appendix’s supplementary figure S2. The solar altitude angle and solar zenith angle are complementary angles and the cosine of either one equals the sine of the other. The solar zenith angle can be expressed by the following two equations, stated below.
𝜎) = 90 − 𝛼, (4.11)
𝜎) = c𝑜𝑠[𝑐𝑜𝑠(𝛿) c𝑜𝑠(∅) c𝑜𝑠(𝜔) + s𝑖𝑛(𝛿) s𝑖𝑛(∅) ]-2 (4.12)
The suns location throughout a day of the year is described by the solar altitude angle and solar azimuth angle, as is illustrated in the figure below. Dependency of the solar angles on the day of the year, and validation of the model used in this paper, is further discussed in section 8.2.1 of the appendix.
4.3 Irradiance on a tilted plane
The angle of incidence (AOI), 𝜉 in degrees, describes the angle between the normal to the PV’s plane and the incident beam radiation [99], [109], [113]. These angles are mostly used to calculate the estimated irradiance on an inclined surface, as is further discussed in the next sections of this chapter.
𝜉 = c𝑜𝑠[𝑐𝑜𝑠(𝜎)) c𝑜𝑠(𝛽) + s𝑖𝑛(𝜎)) s𝑖𝑛(𝛽) c𝑜𝑠(𝛾,− 𝛾) ]-2 (4.13)
where 𝛽, in degrees, is the surface of the PV module’s inclined angle to the horizontal plane. Moreover, 𝛾, in degrees, explains the PV module’s azimuth angle, which is the angle between the normal to the surface and the south, similarly, as the solar azimuth angle (𝛾,) ranging from 𝛾 = [−180 °, 180 °]. All solar and module related angles discussed in this section are illustrated in the following figure.
Fig 4-1 Illustration of the solar and PV module related angles [114].
4.3.1
Total irradiance
Total irradiance on a horizontal surface, G, is described by the sum of the beam irradiance on the horizontal plane, 𝐺., and the diffuse irradiance on the horizontal plane, 𝐺/. In some literature, diffuse
irradiance on the horizontal plane is also described as the diffuse horizontal irradiance (DHI).
𝐺 = 𝐺.+ 𝐺/ (4.14)
For an inclined plane, irradiance received from ground or water reflections also need to be taken into consideration. Furthermore, if the plane is tilted, the global irradiance, 𝐺#, consists of the sum of the
direct beam solar radiation, 𝐺.#, the diffuse solar radiation, 𝐺/# and the reflective solar radiation [116]– [119]. The different types of irradiance are schematically illustrated in the appendix’s supplementary figure S3. The fraction of incident radiation reflected by ground or water is not relevant in the case of rooftop constructed solar cells. This relation is expressed in the following equation form.
𝐺# = 𝐺.#+ 𝐺/# (4.15)
The total annual energy incident on a tilted module, 𝐸K in 𝑘𝑊ℎ ∙ 𝑚-!, is calculated by a summation of
4.3.2
Direct irradiance
The direct beam solar radiation can be estimated based purely on the beam irradiance on the horizontal plane 𝐺. and the geometrical relation between the surface’s tilt and the basic solar angles described in
the section above [99], [100], [109]. The ratio between these two angles, $*,(M)
O&+(P!), is known as the beam
conversion factor 𝑟.. Some papers also relate the direct beam solar radiation to the direct normal irradiance (DNI) and the cosine of its AOI, 𝜉, further derived below [109], [113], [120].
𝐺.# = 𝐺.∙ 𝑟. = 𝐺.∙ }𝑐𝑜𝑠(𝜉) 𝑠𝑖𝑛(𝛼,)~ = 𝐷𝑁𝐼 ∙ c𝑜𝑠(𝜉) (4.16) where, 𝐷𝑁𝐼 = 𝐺. 𝑠𝑖𝑛(𝛼,) (4.17)
4.3.3
Diffuse irradiance
Since the spatial distribution is generally unknown and time-dependent, correctly estimating the diffuse irradiance on an inclined plane remains complex [106]. In general, diffuse irradiance models for inclined surfaces can be separated into two different categories; isotropic and anisotropic models. The first model on the estimation of the irradiance on an inclined surface was published in the 1960s [121], [122] was an isotropic diffuse sky model, which assumed that diffuse irradiance is uniformly distributed over the sky hemisphere. Nevertheless, before the first model was created, Kondratyev and Manolova in 1960 [122] and others had already proven that assumption was incorrect. Therefore, more realistic models were created afterwards, based on the assumption of an anisotropic diffuse sky model [99], [100], [102], [104], [106], [107], [109], [120].
4.3.3.a Muneer
In 1990, Muneer [105] published his method on how to estimate the irradiance on an inclined surface. The model is known to treat the shaded and sunlit surfaces separately and further distinguish between overcast and non-overcast conditions of sunlit surfaces. Over the last decades, this method is used for many research purposes. Also, it is still used for many research nowadays [120], [123], [124]. Moreover, the European Commission’s photovoltaic geographical information system’s online tool [95], from which the dataset used in this thesis is extracted, is also based on this model [100]. Therefore, it is assumed to give lifelike enough outcomes to be used in this paper.
On a given day, the rate of energy 𝐺0+ describes the extraterrestrial irradiance on a surface normal to the sun [99], [109]. The eccentricity correction factor, 𝐸0, is calculated according to the method of Spencer [125]. This parameter is based on the position of the earth on the elliptic path around the sun, to calculate the effect of the distance of the earth to the sun on the incoming radiation, which is depending on the day of the year.
𝐺0+= 𝐺,1∙ 𝐸0 = 𝐺,1∙ }1 + 0.033 c𝑜𝑠 w2 𝜋 𝑛
365x~ (4.18)
where 𝐺,1 is known as the solar constant, assumed to be equal to 1367 W/𝑚!. The extraterrestrial global
solar irradiance on the horizontal plane, 𝐺0, can then be calculated with the use of [99], [109].
𝐺0= 𝐺0+∙ c𝑜𝑠(𝜎)) (4.19)
Together with the beam irradiance on the horizontal surface, now a datapoint’s corresponding anisotropy index, 𝐴&, can be calculated according to [99], [109].
𝐴& = w𝐺. 𝐺0x
(4.20)
The Muneer family of models contains multiple versions, however, the output of these models is not very different from each other [120]. In this paper, two methods of this family are analyzed, which referred to as Muneer-1 and Muneer-2 in the following parts of this paper. In the section below only the methodology of the diffuse irradiance estimation model Muneer-1 is discussed.
o Muneer-1
Muneer-1 is a diffuse irradiance model estimating the irradiance incident on a tilted plane. The methodology of the model follows a version of Muneer’s model [99], which is also implemented by others in existing literature [98], [102], [120]. In section 8.2.2.a and 8.2.2.b of the appendix, four different estimation models; Muneer-1, Muneer-2, Reindl, and Perez, are validated relative to the dataset of PVGIS. The validation is done by analyzing the normalized-root-mean-square-errors for each method, taking into account different scenarios. Muneer-1 showed to be most promising, relative to the other estimation models, and is therefore used in this paper.
The classification of the sky conditions in this model is implemented by both the solar radiation’s AOI, 𝜉, and if any direct beam irradiance on the horizontal surface was measured. The term,
! .
Q(8R!.), is modeled empirically for non-overcast and sunlit (e.g. unshaded) conditions, which implies
the condition 𝐺 > 𝐺/, by the following relation.
2𝑏
𝜋(3 + 2𝑏)= (𝑎0− 𝑎2 𝐴&− 𝑎! 𝐴& !) (4.21)
The values 𝑎0, 𝑎2, 𝑎! are location-dependent parameters [99], [120]. This paper has used the values for locations in Europe, 𝑎0= 0.00263, 𝑎2= 0.712, 𝑎!= 0.6883, as specified by Muneer in 2004 [99]. Moreover, these values were also implemented by the model of Muneer used in the tool of PVGIS [100].
Now, the diffuse irradiance on the inclined surface can be calculated by implementing the following formulas.
𝐺/# = 𝐺/∙ „}1 + c𝑜𝑠(𝛽) 2 ~ + 2𝑏 𝜋(3 + 2𝑏)∙ …(s𝑖𝑛(𝛽) − 𝛽 c𝑜𝑠(𝛽) − 𝜋 †𝑠𝑖𝑛 w 𝛽 2x‡ ! ˆ ∙ (1 − 𝐴&) + (𝐴&∙ 𝑟.)‰ (4.22) If 𝐺 = 𝐺/ and 𝜉 < ‚Q !ƒ, 𝐺/#= 𝐺/∙ „}1 + c𝑜𝑠(𝛽) 2 ~ + 2𝑏 𝜋(3 + 2𝑏)∙ …(s𝑖𝑛(𝛽) − 𝛽 c𝑜𝑠(𝛽) − 𝜋 †𝑠𝑖𝑛 w 𝛽 2x‡ ! ˆ‰ With 𝑏 = 1.68 (4.23) If 𝜉 ≥ ‚Q !ƒ, 𝐺/# = 𝐺/∙ „} 1 + c𝑜𝑠(𝛽) 2 ~ + 2𝑏 𝜋(3 + 2𝑏)∙ …(s𝑖𝑛(𝛽) − 𝛽 c𝑜𝑠(𝛽) − 𝜋 †𝑠𝑖𝑛 w 𝛽 2x‡ ! ˆ‰ With 𝑏 = 5.73 (4.24)
4.4 Total annual incident energy
By running all calculations of the model, the dependency of the PV module’s tilt- and orientation angle on the total annual energy incident on the PV module is computed and graphically illustrated in the figure below.
Fig. 4-2 Illustration of the annual total energy incident on a tilted PV module, in the year 2016.
Based on the output of the model and as illustrated in the figure above, it can be concluded that the ideal tilt of the PV module is centered around a tilt of 40 degrees, with its surface oriented towards the south around an angle of 4 degrees. The optimal tilt and orientation show to match exactly with the calculations of the European PVGIS tool [95].
Furthermore, it can be noticed that the optimum, red egg-shaped, area is relatively large, leading to conclude that there is some flexibility in the tilt and orientation angle close to the optimum. When the tilt and orientation angle are set at more unideal angles away from the optimum area, the total amount of energy incident on the PV module is decreasing at a high rate.
A symmetrical pattern around the optimal module tilt angle can be identified on the x-axis. The symmetrical pattern is as expected, since the sun describes a parabola shaped path across the sky, which is discussed in more detail in section 8.2.1 of the appendix.
All separate parts of the model that constitute to the final result presented in this chapter are individually validated and discussed in section 8.2.2 of the appendix.
- ° - ° - ° ° ° ° ° ° ° ° ° ° ° ° ° - ° - ° - ° ° ° ° ° ° ° ° ° ° ° ° ° [ / ] Total energy [𝑘𝑊ℎ ∙ 𝑚!"]
5 LIGHT TRANSMISSION OF A GREENHOUSE COVER
5.1 Introduction
In this chapter, the effect of the greenhouse’s orientation and roof tilt angle on the amount of irradiance transmitted through the greenhouse cover is further analyzed. The shading by frame construction elements of the greenhouse is neglected in this paper since it is considered beyond the scope of this study [126]. In the case of a multi-span Venlo-type greenhouse, shading causes around 5 to 6% less light to be transmitted [127]. Also, wetting of the cover material is neglected, as a result of its complicated effect on transmissivity and unpredictability throughout the year.
In general, the characteristic triangular roof structure of the multi-span Venlo-type greenhouse is repeated over the full rooftop structure. The basic parameters describing the geometry and orientation of a Venlo type greenhouse roof structure are illustrated in the figure below. For a multi-span Venlo-type greenhouse, the gutter describes the lowest point of the rooftop structure. The ridge of a greenhouse can be considered the highest point of the roof structure.
Fig. 5-1 Greenhouse geometry of a multi-span Venlo-type greenhouse. The angle 𝝃𝒈 describes the azimuth angle of the
greenhouse between the gutter and the south, being positive in value towards the west. The distance d describes the length between the gutters and 𝝍 the roof’s angle of tilt [128].
The tilt of the Venlo-type roof is described by the angle 𝜓, in degrees, and the gutter-to-gutter distance of the greenhouse construction is described by d, in meters. In the model used in this paper, the distance between two gutters is set at 3.2 meters, which is a well-known dimensional size for Dutch greenhouses built according to a multi-span Venlo-type greenhouse design [129].
Due to symmetry, the azimuth angle of the greenhouse between the gutter and the south, 𝜉3 in degrees, ranges between 𝜉3= −90° and 𝜉3 = 90°, for which in both scenarios the gutter is oriented towards the E-W direction, while if 𝜉3= 0°, the greenhouse’s gutter is oriented in the N-S direction. Therefore, symmetry of the roof structure can be used to reduce computing time, for the calculation of the transmission of light incoming from all solar coordination angles. Mirroring by symmetry is further discussed in section 5.3.2.b. of this chapter.
In the end of this chapter, the final result of the theoretical model its computation is shown based on the methodology discussed in this chapter. The dependency of the greenhouse’s rooftop tilt and
gutter orientation on the total annual energy incident transmitted through the greenhouse cover is graphical illustrated. Additionally, the result is validated with existing literature.
5.2 Glass sheet reflection and transmission
In this section, the methodology is discussed to calculate the reflection and transmission coefficient of a single clear glass sheet with a thickness of 4 mm, which is a common thickness for clear glass used as the covering material of a Dutch greenhouse [26], [48], [54], [127], [130]. Clear glass is not compromised of additives other than those meant to affect the manufacturing process [131]. Ultimately, the fraction of light being absorbed, reflected, and transmitted will depend on the angle of incidence relative to the material. As illustrated in figure 8-3, discussed in the appendix, each moment of the year the angle of the sun will differ, and consequently, the corresponding angle of incidence will alter too.
The Fresnel equations are utilized to calculate the absorption, reflectance, and transmittance of a single transparent sheet of glass, with a certain material thickness. Relative to the plane of incidence, the beam incident to the sheet of material can be considered as composed of both a parallel and perpendicular polarized beam of light. The single surface reflectivity’s, 𝑅45 and 𝑅4(, are calculated by the Fresnel equations [30], [40], and described by the following relations:
𝑅45 = }𝑠𝑖𝑛(𝜃&− 𝜃%) 𝑠𝑖𝑛(𝜃&+ 𝜃%)~ ! (5.1) 𝑅4( = }𝑡𝑎𝑛(𝜃&− 𝜃%) 𝑡𝑎𝑛(𝜃&+ 𝜃%)~ ! (5.2)
where 𝜃&, in degrees, is the angle of incidence with respect to the normal of the surface and 𝜃%, in degrees, is the angle of transmission in the glass, with respect to the normal. The angle 𝜃% is related by
Snell’s equation [48], [132] to the angle 𝜃& and the refractive indexes of the two media at the point of
incidence. The refractive indexes, 𝑛2 and 𝑛!, correspond to air and glass, respectively.
𝑛2∙ s𝑖𝑛(𝜃&) = 𝑛!∙ s𝑖𝑛(𝜃%) (5.3)
By assuming that the refractive index of air is equal to one [48], [132], the following relation can be stated for the refractive index, 𝑛1*6(7, of the glass cover material.
𝑛1*6(7= 𝑠𝑖𝑛(𝜃&) 𝑠𝑖𝑛(𝜃%)
(5.4)
The variable C is introduced to rewrite the equations for both 𝑅45 and 𝑅4(:
𝑅45= }𝑐𝑜𝑠(𝜃&) − 𝐶) 𝑐𝑜𝑠(𝜃&) + 𝐶)~ ! (5.5) 𝑅4( = }𝑛1*6(7! ∙ c𝑜𝑠(𝜃&) − 𝐶) 𝑛1*6(7! ∙ c𝑜𝑠(𝜃 &) + 𝐶) ~ ! (5.6)
where,
𝐶 = 𝑛1*6(7∙ 𝐶𝑜𝑠(𝜃%) = Œ𝑛1*6(7 !− s𝑖𝑛(𝜃&)! (5.7)
Clearly, reflection and transmission of the greenhouse’s cover depends on the refractive index of the cover material. In addition, the refractive index depends again on multiple factors, such as for instance the temperature of the glass material and the photon’s wavelength incident on the glass material. Consequently, the refractive index of glass increases as the wavelength of incident light gets shorter [133].
In this study, the refractive index of the glass cover material its dependency on temperature and photon energy is neglected. Furler [134] studied the optical properties of homogeneous silica glasses, which states that the dependence of the reflection and transmission coefficients on the refractive index and wavelengths between 300 and 4600 nm is small. Moreover, measurements of Matsuoka shows that the refractive index differed with less than 1%, when silica glass ranged in temperature from -43.4 to 45.2 degrees, for varying photon energies between 2.27 eV (546 nm) and 5.21 eV (238 nm) [133]. Additionally, Bot [48] claims that spectral differences are not significant in the interaction with the cover material, and multiple studies use a constant value for the refractive index of the cover material. Therefore, the refractive index of the cover material is set equal to 1.5, which is a more commonly used value for clear glass, in the field of study [48], [135]–[137].
For the glass sheet, both surfaces of the sheet have to be taken into account to calculate the corresponding transmittance of the overall glass cover. The effect of having two surfaces on the incoming beam of light, leading to multiple reflections between the two surfaces, can be corrected for by using the method described below [48], [138].
Absorption by glass sheet is taken into consideration by both the power absorption coefficient, 𝐶5., in 𝑚-2, and the optical pathlength, 𝐷 in 𝑚, in the glass. In this paper, the power absorption
coefficient was set equal to 2.5 𝑚-2 and an optical pathlength of 4 𝑚𝑚 was used, corresponding to a
thin single clear glass greenhouse cover material. The equation is expressed in the following relation.
𝑄 = 𝐸𝑥𝑝(−2 𝐶5.,∙ 𝐷′) (5.8) where, 𝐷′ = 𝐷 •1 − w𝑠𝑖𝑛(𝜃&) 𝑛 x ! (5.9)
The single surface reflectivities can be corrected by the use of the factor Q. This factor takes into account the absorption by the glass, leading to the correct surface relativities.
𝑅′45= 𝑅45+𝑅45∙ •1 − 𝑅45‘
!
∙ 𝑄 (1 − 𝑅45! ∙ 𝑄)
𝑅′4(= 𝑅4(+𝑅4(∙ •1 − 𝑅4(‘
!
∙ 𝑄 (1 − 𝑅4(! ∙ 𝑄)
(5.11)
Light which reaches earth directly from the sun is known to be unpolarized. However, when natural light reaches the atmosphere, part of the sunlight is scattered by molecules in the air and will hence become polarized before reaching earth’s surface. Since only a small fraction of light becomes polarized, in this thesis it is assumed that the incoming beam of light on purely unpolarized [48], [139], [140]. Hence, the total reflection and transmission, R, of the glass cover can be calculated by the following relation:
𝑅 =𝑅′45+ 𝑅′4( 2
(5.12)
As mentioned above, the beam incident to the sheet of material is considered as composed of both a parallel and perpendicular polarized beam of light. Hence, the total transmission is again composed of both the transmittance of the parallel and perpendicular polarized beam of light, described as 𝑇45 and
𝑇4(, respectively. By doing so, the total transmission of the glass cover, T, is derived.
𝑇 =𝑇45+ 𝑇4( 2 (5.13) where, 𝑇45=•1 − 𝑅′45‘ ! ∙ 𝑄2/! (1 − 𝑅45T ∙ 𝑄) (5.14) 𝑇4(=•1 − 𝑅′4(‘ ! ∙ 𝑄2/! (1 − 𝑅4(T ∙ 𝑄) (5.15)
The validation of the model’s output on the reflection and transmission coefficient, with the corresponding angle of incidence, is further discussed in section 8.2.3.a of the appendix.
5.3 Transmittance by greenhouse cover
In the Northern Hemisphere, the sun rises in the southeast and sets in the southwest, which leads to a different transmissivity of the greenhouse’s roof structure at each moment of the day. As concluded in section 8.2.3.a. of the appendix, ultimately, the angle of incidence between the cover and the sunlight’s beam of light will greatly determine the amount of direct irradiance transmitted by the greenhouse cover. Hence, the geometry of the greenhouse’s roof structure and orientation are of impact on the total amount of light beneath the glass cover that is able to reach the crops.
In the past, many theoretical models have been constructed to analyze the transmissivity of the greenhouse cover [48], [137], [141]–[143]. In the sections below, the methodology is described which is used to model the impact of the greenhouse’s direction and roof angle on the amount of light
transmitted through the cover of the greenhouse. The theoretical model used in this paper is described by De Zwart [128], and is based on the method of Bot [48].
5.3.1
Total irradiance transmitted
Both the angular distribution of the radiant emittance of the sky and the daytime change in the angular position of the sun, play a key role in the computation of the energy receipts. However, sky conditions are time-dependent and generally unknown. Consequently, as already discussed in section 4.3.3., the precise spatial distribution of diffuse irradiance during each measurement is difficult to reconstruct. Especially locations that are known to have complex, varying cloud patterns throughout the year are difficult to take into account. Therefore, calculating the correct diffuse transmission of the greenhouse’s cover remains difficult. For this reason, an assumption must be done on the distribution of the radiant emittance of the sky before any data can be used for computations, regarding the diffuse irradiance.
Thus, the transmittance of the direct irradiance, T1*6(7,/&7, and diffuse irradiance, T1*6(7,/&<<,
need to be computed separately. The total irradiance transmitted through the greenhouse cover, 𝐺UV, is calculated by the following equation:
𝐺9: = 𝐺.∙ 𝑇1*6(7,/&7+ 𝐺/∙ 𝑇1*6(7,/&<< (5.16)
Now, the total annual energy transmitted through the greenhouse cover, 𝐸9: in 𝑘𝑊ℎ ∙ 𝑚-!, is
calculated by a summation of the total irradiance transmitted, over all rows in the dataset.
5.3.2
Transmission of direct irradiance
5.3.2.a Vector notations
The greenhouse cover’s transmission of direct light, is analyzed by a vector analysis of De Zwart [128].
Fig. 5-2 Vector coordinates of the greenhouse and incoming light beam vector I, with the azimuth angles, 𝝃𝒈 and 𝜸𝒔, of
the greenhouse and the sun respectively. The solar altitude is represented by the angle 𝜶𝒔 [128].
The vectors from the origin, point (0, 0, 0) in the figure above, to the ridges of the planes to the right and left of the gutter are described by the gutter-to-gutter distance, 𝑑, and the height, h, of the ridge relative to the origin, both described in meters.
γs
αs
h =1
2∙ 𝑑 ∙ t𝑎𝑛(𝜓)
(5.17)
As discussed before, the beam of solar light can be described by its elevation angle, 𝛼,, and azimuth angle 𝛾,. The light beam, 𝕀, is derived as a three-dimensional vector by the following equation.
𝕀 = … 𝑠𝑖𝑛•𝛾,− 𝜉3‘ ∙ c𝑜𝑠(𝛼,) 𝑐𝑜𝑠•𝛾,− 𝜉3‘ ∙ c𝑜𝑠(𝛼,) − s𝑖𝑛(𝛼,) ˆ (5.18)
Since the greenhouse’s rooftop structure has a symmetrical shape, merely light incoming from one quadrant of the xy-plane has to be taken into consideration to be able to calculate the effect of light incoming from all four quadrants. For this method, only light reaching the greenhouse from the third quadrant of the xy-plane is analyzed. Mirroring of light incoming from the other quadrants to the third quadrant is further discussed at the end of this section.
One roof triangle consists of two panes, tilted with the same angle and directed in exactly the opposite side of the x-axis. Thus, if the gutter is oriented in the N-S direction, the panes are oriented in the E-W direction, and vice versa. The panes of the roof, ℝ2 and ℝ!, can be described by the following set of vectors: ℝ2,! = •± 1 2∙ 𝑑 0 ℎ – †01 0 ‡ (5.19)
where the second vector of the above equation describes the gutter’s orientation, hence being the gutter’s direction vector 𝔾. To calculate the corresponding transmittance and reflectance coefficient of a light beam 𝕀 reaching a glass pane, the angle of incidence relative to the pane needs to be taken into account. For this purpose, the normal vectors, ℕ2 and ℕ!, of the roof panes ℝ2 and ℝ!, respectively,
are expressed below. ℕ2,! = • ℎ 0 ∓ 1 2∙ 𝑑 – (5.20)
Now, the angles of incidence, 𝜃2& and 𝜃!&, corresponding to the light beam reaching the roof panes 𝑅2 and 𝑅!, respectively, can be calculated by the use of an adapted cosine rule, providing an angle of incidence ranging between 0 and 2
!𝜋.
θ2@,!@= arccos }|œ𝕀 ∗ ℕ2,!•| |𝕀| ∙ |ℕ2,!|~
(5.21)
The ‘∗’ operator, stated between the square brackets, describes a scalar product between the two vectors. In the denominator of the equation, the product between the mean lengths of the vectors is described.
