Thermal Characteristics of a Greenhouse for
Aquaculture
G.R. Branfield
Thesis presented in partial fulfilment of the requirements for the degree of
Master of Science in Mechanical Engineering at the University of
Stellenbosch.
Thesis supervisor: Prof. D.G. Kröger
April 2006
Declaration
I, the undersigned, hereby declare that the work contained in this thesis is my own
original work and that I have not previously in its entirety or part submitted it at any
university for a degree.
Signature: …………………………
Date: …………………………
Abstract
Successful housing and breeding of exotic animals or plants often requires an
environment that is quite different to the ambient conditions present. The current study
approached the problem of sub-optimal water temperatures experienced by Central
African Bream (Tilapia) housed within a South African greenhouse during winter
months. A theoretical and experimental study of fundamental heat and mass transfer
processes relevant to an aquacultural greenhouse was conducted. Experimental results
were generally in agreement with those of previous researchers; while evaporation tests
were found to concur particularly well with an analytical equation developed. The
experimental results were used to develop a simple glass greenhouse model to evaluate
the expected thermal behaviour during the coldest time of the year. Manipulation of the
model revealed that water has the ability to absorb large quantities of solar radiation and
regulate temperature fluctuations within such a system, and that the appropriate use of
thermal insulation during both the night and day can maintain acceptable water
temperatures for extended periods of time. With the conclusions drawn from the
experimentation and modelling done, an optimised conceptual greenhouse design was
presented, along with associated guidelines and principles for attaining the required water
temperatures, and consequently providing the exotic fish specie with a healthy
environment.
i
Opsomming
Daar bestaan dikwels omgewings toestande wat verskil van die wat ideaal is vir die teel
en aanhou van eksotiese plante en diere, daarom is dit partykeer noodsaaklik om deur
middel van tegniese innovasie die regte toestande na te streef vir suksevolle resultate. Die
huidige studie benader die probleem ondervind deur Kurpers van Sentrale Afrika wat
bloodgestel is aan sub-optimale water temperature binne ‘n Suid Afrikaanse kweekhuis
gedurende winter maande. ‘n Teoretiese en eksperimentele studie van fundamentele hitte
en massa oordrag prosesse relevant tot aquakultuur kweekhuise is onderneem.
Eksperimentele resultate het in die algemeen met die van vorige navorsers saamgestem
en verdampingstoetse het veral goed gepas by ‘n nuut ontwikkelde vergelyking. Die
eksperimentele resultate is gebruik om ‘n eenvoudige glas kweekhuis te ontwikkel wat
die verwagte termiese gedrag kan evalueer gedurende die koudste maande van die jaar.
Manipulasie van die model het gewys dat water die vermoe het om groot hoeveelhede
sonstraling te absorbeer en om temperatuur veranderings te reguleer in so ‘n stelsel. Dit is
verder gewys dat die regte gebruik van termiese insulasie gedurende dae en nagte ‘n
aanvaarbare water temperature oor lang periodes kan waarborg. ‘n Optimeerde
kweekhuis konsep ontwerp is voorsgestel deur gebruik te maak van experimentele
resultate en van analietiese modelle. Riglyne is ook voorgestel vir die behaling van
korrekte water temperatuur wat ‘n gesonde omgewing vir eksoties plante en diere
waarborg.
ii
Dedication
This thesis is dedicated to those who make it all worthwhile, my loving and supportive
family and my insanely crazy friends, with an insatiable zest for life.
iii
Acknowledgements
The financial assistance of the National Research Foundation (NRF) towards this
research is hereby acknowledged. Opinions expressed and presented conclusions are
those of the author and are not necessarily to be attributed to the National Research
Foundation.
The author wishes to acknowledge the contributions of the following people:
Prof. D.G. Kröger, for supervising this study and providing invaluable guidance during
both undergraduate and postgraduate projects, for promoting great aspirations and the
long discussions about the “important things in life.”
Dr. D. Brink, for organising financial assistance towards the current research; also for
introducing the author to the field of Aquaculture and for help and friendship throughout
the duration of this study.
Mr. C. Zietsman, for his enthusiasm towards this project and assistance with the
associated experimental work.
iv
Table of Contents
Declaration
Abstract i
Opsomming (Afrikaans) ii
Dedication iii
Acknowledgements iv
Table of Contents v
List of Tables viii
List of Figures ix
Nomenclature xii
Chapter 1: Introduction
1.1 Background 1
1.2 Problem statement and objectives 2
1.3 Engineering approach 4
1.4 Literature study 5
1.5 Overview of chapters 8
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
2.1 An analytical approach 9
2.2 Experimental convective heat transfer coefficient from a horizontal 9
surface exposed to the natural environment
2.2.1 Experimental apparatus and procedure 10
2.2.2 Results obtained 12
2.3 Evaporation from a water surface exposed to the natural 13
environment
2.3.1 Analysis 14
2.3.2 Experimental apparatus and procedure 16
2.3.3 Results and discussion 17
2.4 Conclusion 23
v
Chapter 3: Thermal behaviour and energy storage of water
3.1 Storage of solar energy in water 25
3.2 Open body of water exposed to the natural environment 26
3.2.1 Analysis 26
3.2.2 Experimental setup and procedure 28
3.2.3 Results and discussion 29
3.3 Solar collector with plastic covered water tank 33
3.3.1 Analysis 33
3.3.2 Experimental apparatus and procedure 34
3.3.3. Results and discussion 36
3.4 Solar collector with condensation 40
3.5 Conclusion 46
Chapter 4: Thermal behaviour and energy storage of water
4.1 Modelling objectives 48
4.2 Fundamental system 49
4.2.1 Analysis 49
4.2.1.1 Flat glass roof 52
4.2.1.2 Vertical glass walls 54
4.2.1.3 Concrete floor 56
4.2.1.4 Enclosed air 57
4.2.2 Simulation results 57
4.3 Night-time insulation usage 60
4.4 Day and night-time insulation usage 62
4.5 Conclusion 63
Chapter 5: Aspects of greenhouse design
5.1 System considerations 65
5.2 Proposed improved design 65
5.3 Aspects of optimal design analysis 68
5.4 Welgevallen system analysis 73
Chapter 6: Conclusion
6.1 Research findings 78
vi
6.2 Greenhouse conclusions 80
6.3 Further investigation 82
References 83
Appendix A: Solar Radiation
A.1 Direction of beam radiation A.1
A.2 Ratio of beam radiation on tilted surface to that on horizontal A.4
surface
Appendix B: Evaporation
B.1 Analysis – Evaporation from a water surface B.1
B.2 Analysis – Temperature gradient within water layer B.9
B.3 Evaporation pan photograph B.10
B.4 Numerical example (section 2.3) B.11
Appendix C: Water tank experiments
C.1 Open water tank photograph and spindle drawing C.1
C.2 Numerical example (section 3.2) C.2
C.3 Solar collector with plastic-covered water tank: Photograph and C.9
analysis
C.4 Numerical example (section 3.3) C.15
Appendix D: Water tank solar characteristics
D.1 Cover solar characteristics D.1
D.2 Absorber plate effective absorptivity D.4
Appendix E: Greenhouse sunlight properties
E.1 Area of transmitted sunlight E.1
vii
List of Tables
Table 2.1: Comparison between cumulative evaporation rates. 22
Table B.1: Evaporation pan experimental data on April 14
th
2005 at 12.494 h. B.11
Table C.1: Experimental data and physical parameters of water tank. C.2
Table C.2: Experimental data and physical parameters of plastic covered water C.15
tank.
viii
List of Figures
Figure 1.1: Author at Welgevallen tilapia system. 3
Figure 2.1: Experimental apparatus used for determining convection heat transfer 11
coefficient.
Figure 2.2: Comparison between predicted and measured dimensionless 13
convective heat transfer coefficients.
Figure 2.3: Heat fluxes present on water surface exposed to natural environment. 14
Figure 2.4: Absorptivity of water. 15
Figure 2.5: Front and top views of evaporation pan. 16
Figure 2.6: Solar irradiation and wind speed readings on 13/14 April 2005 20
Figure 2.7: Measured water-, ambient air and dew point temperatures. 21
Figure 2.8: Comparative evaporation rates during daytime operation. 22
Figure 2.9: Predicted evaporation rates compared over period of 24 hours. 23
Figure 3.1: Drawing of water tank with associated heat fluxes. 27
Figure 3.2: Front and top views of experimental setup. 29
Figure 3.3: Solar irradiation and wind speed readings 30
Figure 3.4: Ambient air-, water surface- and dew-point temperatures. 31
Figure 3.5: Water temperature distribution. 31
Figure 3.6: Comparative evaporation rates. 32
Figure 3.7: Schematic drawing of solar collector with plastic-covered water tank. 33
Figure 3.8: Experimental apparatus. 35
Figure 3.9: Total solar irradiation and wind speed readings. 36
Figure 3.10: Water temperature profile. 37
Figure 3.11: Ambient air temperature and predicted- and measured glass cover and 38
mean water temperatures.
Figure 3.12: Results from same analysis as above, including effect of glass cover 38
stored energy.
Figure 3.13: Ambient air temperature and predicted- and measured glass cover and 39
mean water temperatures during the night.
Figure 3.14: Ambient air temperature and predicted- and measured glass cover 40
and mean water temperatures with Sisalation usage.
ix
Figure 3.15: Condensation of fine droplets. 41
Figure 3.16: Size of droplets attained once system became stabilised. 42
Figure 3.17: Water temperature distribution. 44
Figure 3.18: Measured- and predicted glass and mean water temperatures. 44
Figure 3.19: Results attained for night-time operation. 45
Figure 3.20: Measured- predicted temperatures with Sisalation usage. 45
Figure 4.1: Sketch of fundamental greenhouse. 48
Figure 4.2: Path of the sun from above on the 1
st
of January and the 1
st
51
of July.
Figure 4.3: Energy balance on the roof of the greenhouse. 52
Figure 4.4: Heat fluxes imposed on vertical glass window. 54
Figure 4.5: Heat fluxes associated with concrete floor. 56
Figure 4.6: Beam- and diffuse solar irradiation profiles. 58
Figure 4.7: Relative humidity- and wind speed readings. 58
Figure 4.8: Predicted temperatures within greenhouse with concrete floor. 59
Figure 4.9: Predicted temperatures in greenhouse with 100 mm water floor. 60
Figure 4.10: Predicted temperatures in greenhouse with concrete floor and thermal 61
insulation usage at night.
Figure 4.11: Predicted temperatures in greenhouse with water floor and thermal 62
insulation usage at night.
Figure 4.12: Predicted greenhouse temperatures. 63
Figure 5.1: Side and plan views of the proposed greenhouse structure. 66
Figure 5.2: Simplified greenhouse model. 69
Figure 5.3: Mean water tank temperature on June 21
st
. 70
Figure 5.4: Comparative glass cover temperatures. 71
Figure 5.5: Predicted glass greenhouse temperatures on December 21. 72
Figure 5.6: Side view of greenhouse structure. 73
Figure 5.7: Plan view of Welgevallen greenhouse layout. 74
Figure 5.8: Profile of corrugated polycarbonate 75
Figure A.1: Randomly orientated surface in direct sunlight with associated angles. A.1
Figure B.1: Concentration or temperature distribution. B.1
Figure B.2: Flow development on surface. B.4
Figure B.3: Photograph of evaporation pan. B.10
x
Figure C.1: Photograph of open water tank C.1
Figure C.2: Thermocouple spindle used to determine water temperature profile C.2
Figure C.3: Solar collector with plastic-covered water tank. C.14
Figure D.1: Solar radiation striking a cover of thickness t
c
and being either D.2
transmitted, absorbed or reflected.
Figure D.2: Effective transmittance-absorptance product. D.5
Figure E.1: Relative orientation of glass greenhouse. E.1
Figure E.2: Sketch of condition 1. E.2
Figure E.3: Sketch of condition 2. E.3
Figure E.4: Sketch of condition 3. E.4
xi
Nomenclature
A Area, m
2
a Absorption coefficient, m
-1
C Coefficient
C
e
Extinction coefficient, m
-1
C
f
Friction factor
c
p
Specific heat capacity, J.kg
-1
.K
-1
D Diffusion coefficient, m
2
.s
-1
DOY Day of the year
g Gravitational acceleration, 9.81 m.s
-2
h Convective heat transfer coefficient, W.m
-2
.K
-1
I Incident solar irradiation, W.m
-2
i
fg
Latent heat of vaporization, J.kg
-1
k Thermal conductivity, W.m
-1
.K
-1
m Mass or mass flowrate, kg or kg.s
-1
n Refraction index
P Annual phase angle
p Pressure, Pa
q Heat flux, W.m
-2
R Gas constant, kJ.kg
-1
.K
-1
T Temperature, °C or K
t Thickness or time, m or s
v Velocity, m.s
-1
x Distance, m
Y Year
YADJ Leap year adjustment
z Depth, m
Dimensionless numbers
Gr Grashof number, β ΔT g L
3
/ ν
2
xii
Le Lewis number, k/(D μ c
p
)
Nu Nusselt number, h x / k
Pr Prandtl number, μ c
p
/ k
Sc Schmidt number, μ / (ρ D)
Greek symbols
α Solar absorptivity or solar altitude angle, °
Thermal diffusivity, )/(
p
ck ρ , m
2
/s
α’ Effective absorptivity
β Slope angle, ° or fraction of solar energy absorbed at open water surface
γ Psychrometric constant or surface azimuth angle, °
γ
s
Solar azimuth angle, °
γ
s
’ Pseudo solar azimuth angle, °
γ* Adjusted psychrometric constant
∆ Slope of saturation pressure line, dTdP , Pa.K
-1
δ Declination angle, ˚
ε Emissivity
θ Incidence angle, ˚
μ Dynamic viscosity, N.s.m
-2
ρ Reflectivity or density, kg.m
-3
ρ’ Effective reflectivity
σ Stefan-Boltzman’s constant, 5.67 · 10
-8
W.m
-2
.K
-4
τ
α
Transmissivity due to absorption
τ
’
Effective transmissivity
υ Kinematic viscosity, m
2
.s
-1
φ Latitude angle, ˚
φ
l
Longitude angle, ˚
φ
m
Local meridian, ˚
Ψ Local time, h
ω Hour angle or specific, ˚ or kg H
2
O / kg dry air
xiii
Subscripts
a Ambient air
af Air-film
av Air-vapour mixture
ave Average
b Beam
c Cover
ca Convection to ambient air
ce Convection from enclosure
col Colburn
cw Convection from water surface
D Mass diffusion
d Diffuse
dp Dew-point
e Enclosure
ea Enclosed air
en Energy
ev Evaporation
exp Experimental
f Film or floor
fw Film-water
g Glass
ge Glass surface facing east
gn Glass surface facing north
gr Glass surface facing roof
gs Glass surface facing south
gw Glass surface facing west
h Horizontal
horiz Horizontal
i Ambient condition or iteration
m Mean
mon Monteith
xiv
o Saturated condition
p Plate
ra Radiation to environment
re Radiation from enclosure
rw Radiation from water surface
s Surface
sat Saturated
sky Sky
v Vapour
vert Vertical
w Wind or water
wb Wet-bulb
z Zenith
xv
CHAPTER 1
Introduction
1.1 Background
1.2 Problem statement and objectives
1.3 Engineering approach
1.4 Literature study
1.5 Overview of chapters
1.1 Background
Aquaculture is defined as the controlled production of fish, shellfish and aquatic plants
for human consumption, industrial use and recreational purposes.
The rapid expansion of the global population has put tremendous pressure on the food
resources of the world; this high demand presents both a dire need to feed those without
food, and a market for a cheap protein-rich food that is easily attainable. Aquaculture
has not only got the ability to boost worldwide food resources and -security, but can
also assist in an economic and socio-economic manner by increasing export income and
reducing imports, while also creating a demand for technology and training, and
generating employment.
Aquaculture is the fastest growing food industry in the world; since it is accepted that
the quantity of fish supplied by traditional methods is unlikely to increase substantially
in the future due to factors such as over-fishing, pollution and habitat destruction,
aquaculture production will need to increase in an attempt to meet the growing demand
for fisheries products.
According to the United Nations’ Food and Agriculture Organization (FAO), world
aquaculture production increased from 3.5 million tonnes in 1970 to a quantity in
excess of 40 million tonnes in 2000, which relates to a compounded annual growth rate
1
Chapter 1: Introduction
of approximately 9.2%. This compares favourably to an increase of 1.4% and 2.8% for
capture fisheries and terrestrial meat farming respectively (FAO 2003). Between 1995
and 2000, Africa was the continent that showed the largest increase in aquaculture
production (30.8%), with South Africa being responsible for less than 1% of African
production. The South African aquaculture industry amassed a total of 4383 tonnes in
produce in 2003, valued at R237 million at farm gate level (FAO 2003).
1.2 Problem statement and objectives
The current study is concerned with the tilapia group of fish, in particular the species
Oreochromis mossambicus. Tilapias are endemic to Africa, but their potential as an
aquaculture species has led to a widespread distribution in the past 50 years. O.
mossambicus was the first tilapia species to be exported from Africa to the Indonesian
island of Java in 1939 (Popma and Lovshin, 1994), the success has been such that the
current worldwide annual production has reached a value of 1,26 million tonnes (FAO,
2003). This is valued at US$1 706 million and has lead to tilapia being considered the
third most important fish species group worldwide.
The South African tilapia industry is only in the introductory phase with approximately
15 producers nationwide, the majority of which are located in the warmer northern and
eastern regions of the country.
Welgevallen Research Centre is situated in Stellenbosch in the Western Cape and
houses a smaller quantity of tilapia (O. mossambicus) within an intensive recirculation
system in a greenhouse environment. Figure 1.1 shows a photograph taken within
greenhouse-covered aquaculture system.
The fact that tilapias are tolerant toward lower water quality and have the ability to
utilize a wide variety of natural food sources render it a suitable species for use in
aquaculture. They are also prolific breeders and generally disease resistant (Popma and
Masser, 1999; Roberts and Sommerville, 1982).
2
Chapter 1: Introduction
The optimal temperature range for tilapia is between 25 – 36°C; but the main biological
constraint to the development of commercial tilapia farming is their inability to
withstand temperatures below 10°C (Chervinski, 1982) and early sexual maturation that
results in fish spawning before reaching market size (Philapaert and Ruvet, 1982). The
problem of precocious breeding and excessive population growth due to the early onset
of sexual maturation is a well-known phenomenon in tilapia aquaculture (Hepher and
Pruginin, 1982; Mair et al., 1997; Mair and Abella, 1997). A tilapia population that
reaches sexual maturity at an earlier age will have an increased spawning frequency and
stunted growth rates. Fish exposed to an optimal water temperature range may reach
sexual maturity at sizes of 150 – 200g, while those kept at colder temperatures may
breed at sizes as small as 10g (Jalabert and Zohar, 1982).
Figure 1.1: Author at Welgevallen tilapia system.
The cold winter climate experienced in the south-western region of South Africa results
in sub-optimal water temperatures, which adversely influence the general health of the
fish, breeding success, production and ultimately economic success of the species. The
3
Chapter 1: Introduction
primary objective of this study is to determine a way in which the environment enclosed
within the greenhouse can be thermally controlled to meet the needs of the species of
interest. This should preferably be done in an environmentally friendly manner,
considering the current system and the construction of other such systems in the future.
If possible, the most likely manner in which this can be achieved during winter without
external input, such as gas or electricity, is to maximise the solar input entering the
system, while minimising the losses experienced from the system at night. The method
in which the recirculating water is heated should be done in the most environmentally
friendly manner, with little or no running costs and should in no way hinder the
performance of any feature necessary for the well-being of the fish.
1.3 Engineering approach
Attempts have been made in the past to regulate temperatures within similar greenhouse
systems on a purely superficial basis, without complete knowledge of all the interacting
components present. Optimal thermal control of such a system requires that the problem
be divided into the appropriate fundamental heat and mass transfer modes for a
thorough understanding of the behaviour of the system to be designed.
Tests were performed on numerous experimental set-ups that replicated some of the
physical situations found within the greenhouse system. These systems were then
analysed theoretically and the results compared to those obtained through testing. Once
satisfied that the heat and mass transfer modes present in the greenhouse were fully
understood, mathematical modelling of the system could begin. Dynamic simulation is
regarded as one of the most powerful approaches to understanding the interactions of a
complex system (Cuenco, 1989).
Modelling began with a very fundamental glass system, having a high solar input
coupled with large losses. Modifications could be made to the model with respect to
orientation, size, construction materials, thermal insulation and others, from which
conclusions could be drawn as to whether or not the required temperature range is
4
Chapter 1: Introduction
obtainable during periods of adverse weather conditions and recommendations
formulated for the construction and maintenance of future systems of this nature.
Simulations associated with the modelling of greenhouses can be classified either as
passive- or active systems. A passive simulation would be used to determine what
temperatures could be expected within such a system during operation. If the predicted
temperatures do not meet the design specifications, then an active system needs to be
employed to determine the appropriate heating or cooling load required.
1.4 Literature study
The most useful paper found in the literature with respect to the current study is by Zhu
et al. (1998) and deals with the thermal modeling of greenhouse pond systems (GPS).
As stated in section 1.3, this source confirms that greenhouse pond systems can indeed
provide a good alternative for maintaining water temperature in aquacultural facilities.
It also verifies the fact that the thermal characteristics of such systems are not properly
understood and that many pond greenhouses are simply copied from those used in
horticulture.
Numerous authors have modelled pond temperature, some of which are Klemetson and
Rogers (1985), Carthcart and Wheaton (1987), Losordo and Piedrahita (1991) and Zhu
et al. (1998). The model developed by Klemetson and Rogers (1985) assumes saturated
air conditions and no wind across the water surface; the results indicate an increase in
water temperature of between 2.8 – 4.4°C for any month in the year when compared to
an open air pond. Conflicting results were obtained by Little and Wheaton (1987), who
predict an increase of between 9 - 10°C in the pond water temperature with the use of a
greenhouse cover.
This paper states, as expected, that the primary losses from such a system are due to
radiation from within the enclosure to the cover, convection from the cover to the
environment and radiation from the cover to the sky; also that reducing these 3 flux
densities is the principle measure for maintaining water temperature. The model used
by Zhu et al. (1998) is similar to that used in this study with the air and water being
5
Chapter 1: Introduction
considered as homogenous, with the exception that the former considers only one-
dimensional heat flow in a vertical direction and ignores the influence of the sides of
the greenhouse.
The manner in which the condensation on the inner surfaces of the greenhouse affects
the transmission of light entering the system is not stated in the publication. However,
as far as thermal radiation exchange between the inner surfaces is concerned, the area
covered by condensation is determined as a fraction of total area and treated as having
radiation parameters equivalent to that of water. It is also important to note that the
internal relative humidity was always assumed to be near 100%.
The performance of polyethylene, PVC, standard glass and low-emissivity glass were
compared as cover materials for the GPS. Polyethylene achieves a 5.2°C increase in
temperature when compared to the outside air temperature, PVC and low-emissivity
glass produce equally good results while that of standard glass is somewhat less.
Another publication of interest is by Pieters and Deltour (1999) and deals with the
modelling of solar input in greenhouses. Its purpose was to determine the relative
importance of constructional parameters that influence the solar collecting efficiency of
greenhouses. Even though the greenhouse considered contained a tomato crop,
interesting results of application to the current study were obtained. While the study by
Zhu et al. (1998) is fairly comprehensive and of a very similar nature to this thesis, the
fact that it neglected to mention how the light transmission was determined is of
concern to the author and is the reason for the former publication being considered.
Pieters and Deltour (1999) state that increasing the solar transmittance of greenhouses
has always been an important research topic in horticultural engineering, and thus is of
importance to the current study. Condensation on glass products is theoretically
expected to develop as a film, while dropwise condensation is expected on most plastic
cladding materials. The drops in the latter case generally grow until a certain size, after
which run-off occurs; this is however only applicable to non-horizontal glazing.
Numerous simulations and experimental studies have been performed by Pieters
(1995), Pieters et al. (1997) and Pollet and Pieters (1999) to mention just a few. They
confirm that film condensation or flat droplets result in a small decrease or even an
6
Chapter 1: Introduction
increase in glazing transmissivity, while drops with high contact angles can give rise
transmittance reductions of up to 40%.
It is interesting to note that when condensation forms on polyethylene its radiative
properties change significantly; it becomes practically opaque to thermal radiation and
behaves like low-emissivity glass. As found by Zhu et al. (1998), polyethylene
performs better than glass does; thus even with the significant decrease in light
transmission, the gain in thermal radiation is greater. Note that the glass and
polyethylene compared were not the same thickness, and were measured to be 0.004 m
and 0.0001 m respectively.
Pieters and Deltour (1999) find that condensation is found only during low- or zero
solar radiation levels, this is however for a tomato crop with far less available moisture
than in an aquaculture facility and thus continual condensation is expected occur on the
cladding of a GPS. It was found that the presence of condensation has little or no effect
on the absorptance of the cladding. Thus it could be concluded that a greenhouse of
glass construction without any horizontal glass surfaces is unlikely to perform any
worse due to internal condensation.
Critten and Bailey (2002) published a review of greenhouse engineering developments
during the 1990’s, which covers much of the progress made in this field in recent years.
A point of interest is the use of a reflecting surface on the wall/s receiving the least
solar radiation to enhance the light entering the system, and the use of movable screens
to provide thermal insulation and minimise night-time radiative losses.
Nijskens et al. (1984) consider heat transfer through covering materials in greenhouses.
This paper deals with the associated heat fluxes and considers the use of double walls to
minimise convective heat losses. This is found to have definite benefits and is used
extensively throughout much of Europe and North America in residential structures
with great success. The application thereof is however quite expensive, and thus this
form of convection suppression will be retained unless urgently required.
7
Chapter 1: Introduction
1.5 Overview of chapters
The content of this study is summarised in the paragraphs that follow.
Chapter 2 contains all experimental work not associated with water storage. Theoretical
models are developed for convective heat transfer coefficients and evaporation from
horizontal surfaces, which are then compared to results obtained through
experimentation.
Further experimental work was considered in Chapter 3. This dealt with the theoretical
modelling and experimentation with regard to the open water tank, the solar collector
with plastic covered water tank and the solar collector with condensation. The influence
of thermal insulation on the above systems is also investigated. As before, the results
obtained through experimentation are then compared to those predicted by the
theoretical models developed.
The modelling of a fundamental greenhouse is covered in chapter 4. A simple glass
model is developed, and the thermal behaviour is then analysed on the day of the winter
solstice with either a water or concrete layer serving as the floor. The effect of thermal
insulation usage on the predicted system temperatures is then investigated during both
the day and night.
Chapter 5 contains a conceptual greenhouse design that is expected to absorb the
greatest quantity of solar radiation available, while simultaneously experiencing
minimal losses to the environment. Recommended modifications are also supplied to
enable the greenhouse system at Welgevallen to meet the necessary design
specifications.
Chapter 6 concludes this study and briefly discusses the results obtained in each section,
with suggestions on how further studies in this field could be directed.
8
CHAPTER 2
Convective- and evaporative heat transfer
from a horizontal surface
2.1 An analytical approach
2.2 Experimental convective heat transfer coefficient from a horizontal surface
exposed to the natural environment
2.3 Evaporation from a water surface exposed to the natural environment
2.4 Conclusion
2.1 An analytical approach
Any system not fully insulated and exposed to environmental conditions will
experience a net loss or gain in energy, comprised of heat fluxes entering and/or leaving
the system of interest. While significant effort has been put into understanding the
possible modes of heat transfer, the calculation of convective and evaporative heat
transfer still relies largely on the use of empirical correlations. These correlations are
the result of experimental work performed by various researchers, with the result that
opinions are often divided as to which specific correlation is in fact the most accurate.
The lack of an analytical approach to the above issues presents a potential margin of
error, and while inevitable in some cases, is highly undesirable.
2.2 Experimental convective heat transfer coefficient from a horizontal
surface to the natural environment
Reviewing the content available in the literature on convective heat transfer coefficients
between a horizontal surface and the environment revealed that a large degree of
uncertainty exists in this field. Lombaard (2002) quotes numerous sources that verify
the above statement, one of which is Duffie and Beckman (1991) who state that, “from
9
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
the preceding discussion it is apparent that the calculation of the wind induced heat
transfer coefficient is not well established.”
A potential source of these discrepancies may be that the experimental apparatus and
procedure is often poorly defined. Also it was found that the definition of the
convective heat transfer coefficient is often inconsistent, with many researchers making
use of correlations not in dimensionless form. This presents the problem that the
variation in thermophysical properties of the air are not taken into account, and thus it is
unlikely that a simple equation will accurately model the complex convective heat
transfer process.
Kröger (2002) theoretically analysed the problem of convective heat transfer between a
horizontal surface and the natural environment; his findings were later confirmed
experimentally and refined by Burger and Kröger (2004).
Testing was performed on an apparatus similar to that employed by Burger and Kröger
(2004) in an attempt to verify their results. The reader is referred to Burger and Kröger
(2004) for a theoretical analysis of this system.
2.2.1 Experimental apparatus and procedure
Experiments were conducted on a polystyrene plate, with sides measuring 1.020 x
1.020 m and a thickness of 0.05 m. A simple drawing of the experimental apparatus
used is shown in figure 2.1. The surface was covered in a matt black paint (α
s
≈ 0.93, ε
s
≈ 0.90) and was positioned such that the height of the upper surface was flush with the
surrounding ground. The lower surface of the polystyrene plate was reinforced to resist
warping while exposed to solar radiation.
Five type-T thermocouples were used to measure the temperature across the surface of
the horizontal plate, while another four were used to measure the vertical ambient air
temperature distribution until a height of 1 meter above the surface. Figure 2.1 shows a
zoomed in view of a thermocouple at the polystyrene surface, which measures the
10
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
temperature as close to the surface as possible. A light layer a black paint covers the
exposed wires and helps prevent inaccurate temperature readings.
A weather station (Davis Weather Monitor II) was used to measure the ambient air- and
dew-point temperatures, as well as the wind speed at a height 1 m above the ground and
the barometric pressure. A Kipp and Zonen pyranometer was used to measure the total
solar radiation. Diffuse radiation readings were measured by shielding the pyranometer
from direct sunlight for a period long enough for stable readings to be taken.
All data was collected in one-minute intervals and averaged over a period of ten
minutes. The location of the University of Stellenbosch Solar Energy Laboratory is 100
m above sea level, at a latitude of 33.98˚ S and a longitude of 18.85˚ E.
Figure 2.1: Experimental apparatus used for determining convection heat transfer
coefficient.
11
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
2.2.2 Results obtained
Burger and Kröger (2004) present the two equations that follow to determine the
experimental convective heat transfer coefficient between a horizontal surface and the
natural environment under day- and night-time conditions respectively.
3
1
22
3
1
)(
)(
0026.02106.0
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⋅⋅⋅−⋅
⋅
⎥
⎦
⎤
⎢
⎣
⎡
−⋅⋅
⋅
⋅+
=
ρ
μ
μ
ρ
kcTTg
T
TTg
T
v
h
pas
m
as
m
w
a
(2.1)
32
)/(
0022.087.3
kc
cv
h
p
pw
a
⋅
⋅⋅
+=
μ
ρ
(2.2)
The values of the thermophysical properties in the above equations are evaluated at the
arithmetic mean temperature, T
m
between the measured surface and ambient air
temperatures, T
s
and T
a
. Equation (2.1) is applicable when the surface temperature T
s
is
greater than the ambient temperature T
a
. While the use of equation (2.2) is
recommended when the surface temperature T
s
is less than the ambient temperature T
a
,
or when T
s
is slightly larger than T
a
when equation (2.2) is larger than equation (2.1).
Experimental data is not available to check the validity of equation (2.2), since the
formation of dew occurs rapidly after sunset during the time of the year when testing
was done at the given location, and condensation on the system changes the analysis
entirely.
Burger and Kröger (2004) consider an energy balance on a horizontal surface exposed
to the natural environment, and give equation (2.3) for the experimental convective heat
transfer coefficient during daytime operation.
()[ ][ ]
)(
0060.0727.0
44
as
adpsshs
e
TT
TTTI
h
−
⋅+−⋅−⋅
=
σεα
(2.3)
12
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
Experimentally determined results are compared to values predicted by equations (2.1)
in figure 2.2.
Figure 2.2: Comparison between predicted and measured dimensionless convective heat
transfer coefficients.
2.3 Evaporation from a water surface exposed to the natural
environment
Solar applications such as solar ponds, swimming pools or roof ponds with stringent
energy or economic requirements demand that the associated heat fluxes be accurately
determined. If it is considered that evaporation is responsible for more than half the
energy loss from a free water surface, an inability to accurately determine the loss of
energy from such a system could result in a significant error.
Sartori (1999), Yilmaz (1999) and Shah (2002) all made comparisons between the
theoretical and experimental expressions available for the prediction of water
evaporation rates from free water surfaces. They all indicated that significant
discrepancies exist between the available expressions.
13
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
Many authors in the literature refer to work published by Dalton (1802); he states that
the rate of evaporation from a water surface is proportional to the difference in vapour
pressure between the water surface and the surrounding air, he also goes on to say that
this proportionality is influenced by wind speed.
Based on the analogy between heat- and mass transfer, the majority of the expressions
available have a format similar to
fgawwev
ippvBAm /)()( φ−⋅⋅+= , where A and B are
coefficients that are simply determined through repetitive experimental work. The
analysis, experimental procedure and results obtained in the present study will be
discussed in the section that follows, with a full numerical example available in
Appendix B.4.
2.3.1 Analysis
By applying a steady state energy balance to the surface of a film of water on an
insulated base that is exposed to the natural environment as shown schematically in
figure 2.3, it is possible to obtain an expression for the rate of evaporation per unit
surface area i.e.
fgawaskywwhwen
iTThTTIm /)]()([
44
−−−−= σεα (2.4)
Figure 2.3: Heat fluxes present on water surface exposed to natural environment.
14
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
The term α
w
I
h
represents the net solar radiation absorbed by the surface, h
a
(T
w
-T
a
) is the
convective heat transfer, m
en
i
fg
the evaporative heat transfer, with the subscript en
referring to the above energy balance, while ε
w
σ(T
w
4
-T
sky
4
) is the radiative heat transfer
from the water surface.
The absorptivity of the water surface α
w
is given by Holman (1986) as a function of the
zenith angle θ
z
. This relationship is shown graphically in figure 2.4.
Figure 2.4: Absorptivity of water.
One could argue that the above relationship is only valid for a large expanse of water
and that a very thin film of water is essentially transparent, absorbing practically no
solar radiation. The equation above has been rewritten for the current application and
was originally a relationship between solar altitude and albedo. All solar radiation
incident on a large body of water and not reflected at the surface will be absorbed.
Similarly if a thin film of water covers an insulated black surface, it is assumed that all
solar radiation not reflected at the water surface will be absorbed upon striking the
highly absorptive surface.
15
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
The convective heat transfer coefficient h
a
is represented by equation (2.1), while the
sky temperature T
sky
is determined according to Berdahl and Fromberg (1982) for
daytime- and night-time operation in equations (2.5) and (2.6) respectively.
[]
4
1
4
)0060.0727.0(
adpsky
TTT ⋅+= (2.5)
4
1
4
)0062.0741.0(
adpsky
TTT ⋅+= (2.6)
Where T
dp
is the dew-point temperature measured in degrees Celsius.
2.3.2 Experimental apparatus and procedure
The evaporation pan used to experimentally determine the rate of evaporation from a
water surface is shown in figure 2.5.
Figure 2.5: Front and top views of evaporation pan.
It consisted of a 50 mm thick horizontal polystyrene plate having an effective upper
surface area of approximately 0.97 m
2
, which was painted with a waterproof matt black
paint. A 3mm high bead of silicon sealant was run along the perimeter of the pan, with
16
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
the purpose of containing a 1- 2 mm deep layer of water. Five type-T thermocouples
were embedded in the surface of the plate with the purpose of measuring the water
temperature. Four of the five were positioned in the corners of the pan, 150mm from
adjacent sides, while the fifth was placed in the centre.
The wind speed, ambient air- and dew-point temperatures were measured with the aid
of a weather station at a height of 1 m above the ground. A Kipp and Zonen
pyranometer was used measure the total incident solar radiation on the surface, while
diffuse solar radiation readings were measured by shielding the pyranometer from
direct sunlight for a period long enough for stable measurements to be taken. All tests
were conducted on clear sunny days.
2.3.3 Results and discussion
The evaporation rate from the water surface was measured by adding consecutive
quantities of water (500 ml) to the evaporation pan at a temperature similar to that of
the remaining water, and recording the period of time taken for each to evaporate. With
the average temperature of the water known during the particular period, the mass
flowrate or evaporation rate could be determined. This evaporation rate will be denoted
by m
exp.
The author presents the following equation for the prediction of the approximate
evaporation rate for cases where the water surface temperature is measurably higher
than that of the ambient air. The derivation of equation (2.7) is given in Appendix B.1,
equation (B.31).
()( )
mvivowavpavoavivom
Tppvkcgm /)(0104.0)/()(1061.7
31
224
−⋅+−×=
−
ρρρμ (2.7)
In the above expression ρ
avi
represents the density of the moist air at the humidity and
temperature of the ambient air, while ρ
avo
is the density of saturated air at the
temperature of the water. The symbol ρ
av
is the average between the two
aforementioned densities, while (p
vo
- p
vi
) is the difference in vapour pressure between
the water and air respectively. The symbol m
vom
represents the mean mass flowrate of
17
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
the vapour at the saturated condition. All thermophysical properties are calculated at the
mean temperature T
m
between the ambient air T
a
and water temperature T
w
. Note that
this expression is only applicable when the water temperature T
w
exceeds the ambient
air temperature T
a
.
The use of the expression below is recommended when the ambient air temperature T
a
exceeds the water surface temperature T
w
or when density differences are very small
(see equation B.34).
mv
vivo
p
w
p
vom
TR
pp
k
Dc
v
c
m
)(
Pr
0022.087.3
3
2
32
−
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ ⋅⋅
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⋅
=
ρ
ρ
(2.8)
Equation (2.8) is found to be in good agreement with equations recommended by Tang
et al. (2004) for the evaporation of water from both open water- and wetted surfaces.
Monteith and Unsworth (1990) provide a relationship that is used extensively in the
evapotranspiration field. The predicted evaporation rate is given by equation (2.9).
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+Δ
⋅−+⋅⋅Δ
=
fg
wvivsihw
mon
i
AhppI
m
*
)(
γ
α
(2.9)
Such an expression deals with an analysis that is quite different to that of a water
surface exposed to the natural environment. However, if the results provided by
equation (2.7) are of a similar magnitude to those predicted by an accepted relationship,
such as that by Monteith and Unsworth (1990), this would further suggest that the
expression developed (equation 2.7) may well be of reasonable accuracy.
The derivation of this expression is tedious and unlikely to be of any benefit to the
current study, for further information on this equation the reader is referred to Westdyk
and Kröger (2003).
The adjusted psychrometric constant in equation (2.9) is represented by γ* which is
determined by
18
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅⋅
=⋅=
)(
3
2
3
2
622.0
*
wb
T
fg
atmpma
p
i
pc
cD
k
Le
μ
γγ (2.10)
where Le is the Lewis number, defined as ( )
p
cDk ⋅⋅μ . The symbol c
pma
is the specific
heat capacity of the moist air and i
fg(Twb)
is the latent heat of vaporization of water,
evaluated at the wet-bulb temperature T
wb
. The ∆ symbol is the average slope of the
saturation pressure line between the ambient wet-bulb temperature T
wb
and water
surface temperature T
w
and is given by equation (2.11). The difference in vapour
pressure between the air and the saturation pressure at the same dry-bulb temperature is
represented by (p
vsi
– p
vi
). The water surface area is given by the symbol A, while i
fg
is
the latent heat of vaporisation. The temperature T is the average between the surface T
s
and wet-bulb temperatures T
wb
.
v
satfg
RT
Tpi
⋅
⋅
=Δ
2
)(
(2.11)
The rates of evaporation that will be compared in the following section are the
experimentally measured evaporation rate m
exp
and the given by equations (2.4), (2.7)
and (2.9).
Since the driving force behind the evaporation process is the difference in vapour
pressure, which is a function of ambient air- and water temperature, care needs to be
taken with respect to the measurement of these two values.
Ambient air temperature readings were taken at a height of 1 m above the ground with
the aid of the weather station and with auxiliary thermocouples used to measure the
temperature gradient in the air above the water surface. Only a slight deviation was
found between the two sets of data; the auxiliary thermocouples were preferred since
they were the same type of thermocouple used in the measuring of the water surface
temperature.
An analysis was performed on equation (2.7) to determine the sensitivity of the
expression to an increase or decrease in measured water temperature by 1°C. It was
19
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
found that equation (2.7) is very sensitive to change early in the morning and towards
evening, while a difference in the predicted evaporation rate of approximately 10% was
found during the day. These findings suggest that extreme care should be taken when
measuring the water surface temperature T
w
.
With the above taken into account, the heat fluxes associated with a thin film of water
on a black surface exposed to the natural environment were analysed theoretically. The
results of this analysis can be summarised by equation (2.12), the details of which can
be found in Appendix B (equation B.37).
zITT
hmeasuredwsurfacew
⋅−= (2.12)
where z is the depth of the water film, measured in meters.
Evaporative testing was performed from 17.164 h on April the 13
th
for a period of 24
hours, during which no cloud cover was present. The solar irradiation and wind speed
present during this time are shown in figure 2.6.
Figure 2.6: Solar irradiation and wind speed readings on 13/14 April 2005.
20
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
Figure 2.7 shows a comparative plot between the water-, ambient air-, dew-point
temperatures. Note that T
w
is the surface water temperature, as corrected according to
equation (2.12). The figure indicates that the water surface temperature only exceeds
the ambient air temperature between 8.09 h and 14.75 h; thus equation (2.7) is only
applicable during this period. It is also interesting to note that as soon as solar
irradiation readings diminish, the water surface temperature drops below the dew-point
temperature and thus condensation can be expected to occur.
Figure 2.7: Measured water-, ambient air and dew point temperatures.
Table 2.1 compares the quantity of water evaporated between 8.319 h and 14.752 h,
with the amount of water that is expected to evaporate during this period according to
equations (2.4), (2.7) and (2.9). The error margin is calculated with respect to the
experimentally measured quantity.
It is clearly shows that the best results obtained throughout the course of the day are
attained with the use of equation (2.7).
21
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
Table 2.1: Comparison between cumulative evaporation rates.
Equation Measured [kg] Error [%]
m
exp
3.979
m
en
(2.4) 4.270 7.29
m
vom
(2.7) 4.066 2.19
m
mon
(2.9) 4.739 19.10
The above evaporation rates are compared graphically in figure 2.8 and 2.9. Figure 2.8
displays results between approximately 8.00 h and 15.00 h, while figure 2.9 shows the
predicted evaporation rates over a period of 24 hours. Note that in figure 2.8 the
experimentally measured values m
exp
are assumed to be constant during each particular
period of analysis. Observation of figure 2.9 shows that when the water temperature T
w
drops below the dew-point temperature T
dp
, equations (2.4) and (2.8) predict
condensation on the water surface, as expected.
Figure 2.8: Comparative evaporation rates during daytime operation.
22
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
Figure 2.9: Predicted evaporation rates compared over period of 24 hours.
2.4 Conclusion
The results regarding the convective heat transfer coefficient in figure 2.2 shows that
the trend of the experimental results is similar to that predicted by equation (2.1). The
results obtained are considered acceptable and equations (2.1) and (2.2) will be used
where applicable.
The results of the evaporation tests were far more interesting than those considered
above. In the introduction to section 2.3 it was stated that this was the first time that an
expression predicting evaporation rates was supported by good experimental results. A
summary of the predicted evaporation rates with respect to the experimentally measured
rate of evaporation was given in table 2.1. The results show that equation (2.7) predicts
the rate of evaporation the most accurately of the given expressions, with an error
margin of just more than 2%.
23
Chapter 2: Convective- and evaporative heat transfer from a horizontal surface
It is found that all of the evaporation rates over-predict the experimentally measured
value early in the morning, and under-predict towards afternoon. This however has the
net result of cancelling each other out, to provide better than expected cumulative
results. It is expected that the lack of wind on this particular morning (see figure 2.6)
lead to an increase of moist air directly above the experimental apparatus. This resulted
in an increased vapour pressure of the air above the water, and thus less of a potential
for evaporation. The excessive evaporation found to occur in the afternoon could
possibly be attributed to an increase in the radiation reflected from the buildings
surrounding the apparatus, and thus an increase in the energy absorbed by the water
surface.
If the testing on the 13
th
/14
th
April is considered, it is interesting to note that portion of
the total heat flux for which convection is responsible is less than 10%. This suggests
that even if the convective heat transfer coefficient used is not very accurate,
evaporative results within acceptable accuracy are still possible.
24
CHAPTER 3
Thermal behaviour and energy storage of
water
3.1 Storage of solar energy in water
3.2 Open body of water exposed to the natural environment
3.3 Solar collector with plastic covered water tank
3.4 Solar collector with condensation
3.5 Conclusion
3.1 Storage of solar energy in water
It has been shown by many researchers (see Duffie and Beckman (1991)) that although
thermal energy can be stored in numerous forms, water is the medium that is used most
extensively, and is often both inexpensive and readily available. Holman (1986) states
that solar radiation is absorbed very rapidly in the top layer of a water body, followed
by an approximately exponential decay with depth in the water. This relationship is
given by equation (B.35)
Thus according to Kreider et al. (1989), water has the ability to both act as a collector
of solar energy and serve as an excellent storage medium. This characteristic is
exploited with the use of solar ponds; which are classified either as salt gradient ponds
or freshwater solar ponds. In salt gradient solar ponds the use of salt leads to an increase
in water density and thus suppressed convection, while freshwater solar ponds utilise
transparent membranes to achieve the same goal.
Neither of the above systems are applicable to an aquaculture system, since not all
species are tolerant of salt water, especially at high concentrations and the use of a
membrane would retain gasses that would otherwise need to be expelled from the body
of water. An algal build-up on the membrane would also lead to decreased levels of
25
Chapter 3: Thermal behaviour and energy storage of water
both available oxygen and transmitted sunlight. However, the thermal behaviour of
such a system without any evaporative losses could provide important insight into
possible greenhouse improvements at a later stage, and since a rapid increase in water
temperature is attained with the latter of the two, it will be considered in more detail in
section 3.3.
Sartori (1996) compares the thermal behaviour of a solar still and a solar evaporator. A
solar evaporator is simply a volume of water that is exposed to environmental
conditions, while a solar still makes use of a cover. This is a very interesting study,
since a comparison between the two is essentially a performance evaluation of the use
of a greenhouse to heat water, which is aligned with the objective of this study.
3.2 Open body of water exposed to the natural environment
As mentioned above, the thermal behaviour of a body of water that is exposed to the
natural environment is of interest since it essentially represents an aquaculture pond
system, which experiences high thermal losses. In such an experimental setup these
losses can be improved upon with the use of a cover, or a greenhouse in practise, and
thus provides a basis for comparison. It is a simple system upon which improvements
can be made. Such a system could be considered an extension of the evaporation
concept considered in section 2.3, the only difference between the two systems is that
since the depth of the water film is substantially more in this case, the thermal inertia of
the water body needs to be taken into account.
3.2.1 Analysis
Figure 3.1 shows a horizontal water surface of unit area, contained within a tank of
depth, t
w
that has a well-insulated base and sides, and is exposed to the environment
during the day.
26
Chapter 3: Thermal behaviour and energy storage of water
Figure 3.1: Drawing of water tank with associated heat fluxes.
If an energy balance is applied to the volume of water in the figure, the heat fluxes can
be summed as follows:
0)()(
44
=−−−−−−
dt
dT
ctimTThTTI
wm
pwwfgevawaskywwhw
ρσεα
(3.1)
The terms in the above equation represent the absorbed solar radiation, the radiative-,
convective- and evaporative heat fluxes, and the quantity of heat responsible for the
change in stored energy of the water tank respectively. All but the last term have been
described in detail in Chapter 2.
The last term consists of the water density ρ
w
, tank depth t
w
, water specific heat capacity
c
pw
and the rate of change in mean water temperature dT
wm
/dt. Note that since
convection and radiation are surface phenomena the water surface temperature T
w
has
been used in those terms, while the stored energy term is concerned with a change in
the mean water temperature T
wm
.
If the ambient conditions and water temperatures are known, equation (3.1) can be
rewritten and the evaporation rate m
ev
from the body of water can be determined.
27
Chapter 3: Thermal behaviour and energy storage of water
3.2.2 Experimental setup and procedure
An apparatus that could be classified as a solar evaporator was constructed and utilised
for experimental testing. Figure 3.2 shows the apparatus used, while a photograph of the
system can be found in Appendix C (figure C.1).
The water tank is filled to the surface and the water is heated by solar radiation,
resulting in radiative and convective gains or losses, changes in temperature of the
water body and evaporative losses, which are monitored regularly. Temperatures are
measured at specific heights with the use of the thermocouple spindle, from which the
water temperature distribution can be determined.
The base of the water tank is black, so the assumption is made that all solar radiation
not reflected at the surface is absorbed by the water body. The sides of the water tank
are covered in aluminium foil to prevent heat absorption by the side walls, while the
water surface is maintained as close as practically possible to the top of the tank in an
effort to minimize detrimental edge effects.
A “thermocouple spindle” was made to measure the temperature of the water at various
depths. It was comprised of a thin PVC frame that held seven type T thermocouples that
rotated slowly in a horizontal plane through the water at the appropriate depths required
to determine a sufficiently accurate temperature profile in the water tank. Measurements
were taken at 190, 180, 170, 100, 30, 20 and 10 mm from the base, as well as at the
base of the water tank. A drawing of the thermocouple frame is given in Appendix C
(figure C.2).
The wind speed and ambient- and dew-point temperatures were measured with the aid
of a weather station at a height of 1 m above the ground. A Kipp and Zonen
pyranometer was used measure the total incident solar radiation present at the water
surface. Diffuse solar radiation was measured by shielding the pyranometer from direct
sunlight for a period long enough for stable readings to be taken.
28
Chapter 3: Thermal behaviour and energy storage of water
Figure 3.2: Front and top views of experimental setup.
3.2.3 Results and discussion
Data was recorded from late afternoon on the 28
th
April 2005 for a period of 24 hours.
This data consisted of temperatures at depths throughout the water body, solar radiation
readings and various ambient conditions. Figures 3.3 displays the total solar irradiation
and wind speed readings, while the ambient air, water and dew-point temperatures are
shown in figure 3.4. All given values are averaged over a period of ten minutes.
Figure 3.3 shows that although not particularly windy, wind was still present throughout
the entire period. This would generally suggest that significant evaporation should have
occurred, however due to the nature of the experimental setup the temperature of the
water surface is far less than that encountered in section 2.3.
29
Chapter 3: Thermal behaviour and energy storage of water
Figure 3.3: Solar irradiation and wind speed readings.
Observation of figure 3.4 shows that the value of the water surface temperature never
drops below that of the dew-point temperature, and thus evaporation is continuously
taking place.
Figure 3.5 displays the water temperature profile found to occur during the period of
analysis, it shows substantial cooling of the upper layer of water whenever solar
radiation is absent. In general however, the change in temperature remains
approximately constant from 20 mm below the surface until about 40 mm from the
base. From the figure it is evident that the base is always at a greater temperature than
that midway up when solar radiation is present.
30
Chapter 3: Thermal behaviour and energy storage of water
Figure 3.4: Ambient air-, water surface- and dew-point temperatures.
Figure 3.5: Water temperature distribution.
31
Chapter 3: Thermal behaviour and energy storage of water
The use of some of the equations for evaporation rates employed in section 2.3 are
reused in this analysis. The measured evaporation rate, as determined before is
designated by m
exp
.
Rewriting equation (3.1) in terms of the evaporation rate gives
fg
wm
pwwfgevawaskywwhwen
i
dt
dT
ctimTThTTIm
⎟
⎠
⎞
⎜
⎝
⎛
−−−−−−= ρσεα )()(
44
(3.2)
This is the evaporation rate as determined with the use of an energy balance, m
en
and is
used for comparison in conjunction with the evaporation rates as predicted by equations
(2.8) and (2.9) in figure 3.6 below.
Figure 3.6: Comparative evaporation rates.
32
Chapter 3: Thermal behaviour and energy storage of water
3.3 Solar collector with plastic covered water tank
As mentioned in section 3.1, freshwater solar ponds utilise transparent membranes in an
attempt to hinder convective losses and eliminate evaporative losses from a body of
water.
An experimental setup was utilised that consisted of a body of water that was covered
by a thin plastic film and contained within an insulated tank with a glass cover. The
glass cover and water temperatures were measured while the system was exposed to
ambient conditions during the day and night. The system was then modelled
theoretically, and the predicted results compared to those measured; Lombaard (2002)
conducted similar tests with satisfying results on similar apparatus. This use of thermal
insulation across the upper surface of the glass cover is also investigated.
3.3.1 Analysis
Figure 3.7 shows a schematic drawing of the solar collector with plastic-covered water
tank. By applying the conservation of energy law to the glass cover and the water tank,
equations (3.3) and (3.4) are obtained respectively (see Appendix C).
Figure 3.7: Schematic drawing of solar collector with plastic-covered water tank.
33
Chapter 3: Thermal behaviour and energy storage of water
)(1
11
1
)1)(1(
1
)1)(1(
44
1
cw
fc
hd
dcdc
dcdc
hb
bcbc
bcbc
TTII −
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−++⋅
−
−−
+⋅
−
−−
−
εε
σ
τρ
τρ
τρ
τρ
α
α
α
α
)(1
5830
Pr
Pr
1708
144.11
3
1
cw
a
cwcwcw
cwcw
TT
t
kGr
Gr
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−⎟
⎠
⎞
⎜
⎝
⎛
+
⎥
⎦
⎤
⎢
⎣
⎡
−++
()
44
3
1
22
3
1
)006.0727.0()(
)(
)(
0026.02106.0
adpccac
pac
m
ac
m
w
TTTTT
kcTTg
T
TTg
T
v
+−+−⋅
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
⎥
⎦
⎤
⎢
⎣
⎡
−
+
= σε
ρ
μ
μ
ρ
(3.3)
1
2
2
2
2
22
2
1
)21(
1
1
)21(
1
)1(
)1(
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−+
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−+
−
⎥
⎦
⎤
⎢
⎣
⎡
−
−
bf
bfwbaf
bf
bfwbafbaf
dc
bf
bfwbaf
bf
bfwbafbaf
bcbc
bcbc
α
α
α
α
α
α
τρρ
τρρρ
ρ
τρρ
τρρρ
τρ
τρ
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−+
−
⎥
⎦
⎤
⎢
⎣
⎡
−
−
+−⋅
df
dfwdaf
df
dfwdafdaf
dcdc
dcdc
hbbw
I
α
α
α
α
τρρ
τρρρ
τρ
τρ
ρ
2
2
22
2
1
)21(
1
)1(
)1(
)1(
hddw
df
dfwdaf
df
dfwdafdaf
dc
I)1(
1
)21(
1
1
2
2
ρ
τρρ
τρρρ
ρ
α
α
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−+
−⋅
−
)(1
5830
Pr
Pr
1708
144.11
31
cw
a
cwcwcw
cwcw
TT
t
kGr
Gr
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−⎟
⎠
⎞
⎜
⎝
⎛
+
⎥
⎦
⎤
⎢
⎣
⎡
−+=
dt
dT
ctTT
w
pwwwcw
fc
ρ
εε
σ +−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−++
−
)(1
11
44
1
(3.4)
3.3.2 Experimental apparatus and procedure
The solar collector of concern consists of a 1 m x 1 m x 0.195 m black fibreglass water
tank. A clear 0.2 mm thick Luminal Anti-Fog plastic covers the surface of the water
with the purpose of supressing evaporation. A 1 m x1 m x 4 mm glass sheet covers the
solar collector and is separated from the plastic film by a 0.1 m air space.
The water tank is insulated by 50 mm thick polystyrene, such that negligible conduction
losses occur. The inner sides of the water tank are covered in highly reflective
34
Chapter 3: Thermal behaviour and energy storage of water
aluminum foil to minimize any heat transfer between opposing sides, the water and the
enclosed air gap. The solar collector was supported on a square pedestal with reflective
sides that housed much of the data logging equipment. The apparatus is displayed figure
3.8.
The thermocouple frame mentioned in section 3.2 was employed to measure the water
temperature at various depths, with the only difference being that the tank was only
filled to a height of 170 mm from the base. Two thermocouples were placed beneath the
base of the fiberglass water tank and three on the glass lid of the solar collector. Those
on the glass lid were covered by highly reflective foil to minimize unwanted radiative
contributions.
Figure 3.8: Experimental apparatus.
The total incident solar radiation was measured with aid of a Kipp and Zonen
pyranometer. Diffuse radiation values were determined by casting a shadow over the
instrument until such time that it stabilized. The beam radiation component was then
35
Chapter 3: Thermal behaviour and energy storage of water
simply the difference between the measured total and diffuse radiation values. The
weather station mentioned in figure 3.8 was used to measure the ambient temperature,
wind speed and dew point temperature at one-minute intervals. A photograph of the
solar collector is shown in Figure C.3.
3.3.3 Results and discussion
Tests were conducted on a continuous basis during extended periods of clear weather.
The purpose of the tests was to predict the glass cover- and mean water temperatures
with the theoretical models developed and to compare these temperatures with those
measured through experimentation. The apparatus was given sufficient time to
acclimatize with the environment and data taken on the 12
th
of January 2005 was
considered suitable for analysis.
Observation of figure 3.9 shows that the wind speed was approximately constant
throughout the period of analysis until the latter portion of the afternoon, at which point
it was found to increase. The continuous solar radiation profile shows that no clouds
were present during the period in which experimental readings were being taken.
Figure 3.9: Total solar irradiation and wind speed readings.
36
Chapter 3: Thermal behaviour and energy storage of water
Figure 3.10 shows the water temperature profile in the solar collector. The profile is
seen to be approximately exponential, while relative cooling at the water surface (170
mm) becomes evident as the day progresses and the ambient temperature starts to
decrease.
Figure 3.11 displays the measured and predicted glass cover- and mean water
temperatures. The predicted temperatures are determined by solving equations (3.3) and
(3.4) simultaneously, a mediocre agreement is found to exist between these values and
those measured.
It was stated earlier that the decision was made not to include the change in stored
energy of the glass cover in the current analysis. This choice was made since this
particular inclusion was not expected to have any significant effect on the results;
Lombaard (2002) attained acceptable results on an identical apparatus without the
inclusion of this term.
Figure 3.10: Water temperature profile.
37
Chapter 3: Thermal behaviour and energy storage of water
Figure 3.11: Ambient air temperature and predicted- and measured glass cover and
mean water temperatures.
Figure 3.12: Results from same analysis as above, including effect of glass cover
stored energy.
38
Chapter 3: Thermal behaviour and energy storage of water
The results attained after having taken the stored energy term of the glass into account
is displayed in figure 3.12; as expected the two figures differ very little from one
another.
Displayed below are the results obtained with the use of equations (3.3) and (3.4)
during night-time operation. The sky temperature in equation (3.3) needs to be
represented by equation (2.6) for night-time operation.
Similarly, the results obtained with placing thermal insulation across the upper surface
of the glass cover correspond reasonably well with one another. The thermal insulation
used is sold under the name Sisalation, and is comprised of a thin layer (1 mm thick) of
laminated aluminium. It was placed over the upper surface of the solar collector and
sealed such that it was airtight. This form of insulation offers little resistance to
conductive losses, but with a very low emissivity (ε ≈ 0.1) it substantially reduces
radiative losses from the particular system.
Figure 3.13: Ambient air temperature and predicted- and measured glass cover and
mean water temperatures during the night.
39
Chapter 3: Thermal behaviour and energy storage of water
Figure 3.14: Ambient air temperature and predicted- and measured glass cover and
mean water temperatures with Sisalation usage.
If the emissivity of the upper glass surface in equation (3.3) is set to that of the
Sisalation, solving equations (3.3) and (3.4) simultaneously gives the results shown in
figure 3.14 above. As with the modelling of the night-time operation, relatively good
results were obtained.
3.4 Solar collector with condensation
If the same experimental setup as that in section 3.3 is considered, the thermal
behaviour of the system is changed to a large degree by the removal of the plastic film
covering the water surface.
Initially evaporation occurs from the water surface into the air until the air is fully
saturated. Following this, condensation starts to occur on the lower surface of the glass
plate and on the inner sides of the enclosure. As time progresses the condensed droplets
within the enclosure increase in size, until such time that the gravitational force on the
40
Chapter 3: Thermal behaviour and energy storage of water
droplets exceeds the surface tension and other microscopic forces retaining the droplets
on the surface; this results in a condition referred to as runoff. This process then repeats
itself. While this phenomenon occurs on the vertical sides of the enclosure, droplets on
the horizontal surface reach sizes that far exceed those elsewhere in the system.
A literature study (see Chapter 1) revealed that horizontal roofs are very rarely used in
greenhouses since pitched roofs assist in natural ventilation within the system,
coincidentally this would also avoid the above problem within a saturated system. This
may suggest why the author was unable to find any publications regarding condensation
on horizontal glass plates. Technical papers found often dealt either with polyethylene,
PVC or glass structures, however correlations associated with the influence of
condensation on light transmission and reflection are only concerned with droplets with
a maximum diameter of 1.5 – 2 mm on an inclined or vertical surface. This presented a
problem since initially the droplets were very small, but once stabilised, the droplets
reached diameters of approximately 10 – 12 mm (see figures 3.15 and 3.16).
Figure 3.15: Condensation of fine droplets.
41
Chapter 3: Thermal behaviour and energy storage of water
Figure 3.16: Size of droplets attained once system became stabilised.
Another problem encountered with modelling the above system is explained below.
When the system appears to have stabilised and the rate of evaporation equals the rate
of condensation, then the influence of evaporation and condensation can be ignored,
and the system can be modelled in a similar manner to that in section 3.3. However,
when an increase or decrease in the glass cover temperature occurs, further evaporation
or condensation will occur within the enclosure. This must be taken into account if the
system is to be accurately modelled, but adds complexities to an already uncertain
model.
With this in mind, it was attempted to model the system with constant
glass/condensation solar properties, and under the assumption that no
evaporation/condensation occurs within the system. Adapting the governing equations
developed in section 3.3, the following glass/condensation properties were found to
predict glass- and water temperatures that tracked the measured temperatures most
accurately: ρ’ = 0.1, τ’ = 0.55, α’ = 0.35.
42
Chapter 3: Thermal behaviour and energy storage of water
It is interesting to view the temperature distribution in the water tank when compared to
that found with the use of a plastic film on the water surface. Since the absorptivity of
the cover is greater than before, the cover temperature exceeds the mean water
temperature, which results in heating of the upper water layer and a greater variation in
temperature (see figure 3.17).
Figure 3.18 displays the measured and predicted results obtained for the current system
during daytime operation. The predicted results are obtained by solving equations (3.5)
and (3.6) simultaneously. Note that the mean water temperature has been used in the
calculations that follow.
() (
cw
a
cw
ecw
ww
hdhb
TT
t
k
hTTII −+−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+++⋅
−
)(1
11
35.0
44
1
εε
σ ) (3.5)
()
44
3
1
22
3
1
)006.0727.0()(
)(
)(
0026.02106.0
adpccac
pac
m
ac
m
w
TTTTT
kcTTg
T
TTg
T
v
+−+−⋅
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
⎥
⎦
⎤
⎢
⎣
⎡
−
+
= σε
ρ
μ
μ
ρ
()()
dt
dT
ctTTTT
t
k
hII
w
pwwwcw
ww
cw
a
cw
ehdhb
ρ
εε
σ +−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−++−=+⋅
−
)(1
11
55.0
44
1
(3.6)
Note that the emissivities within the enclosure are replaced with those of water, while
the convective heat transfer coefficient within the system is represented by h
e
. If the
water temperature is greater than the cover temperature then the correlation offered by
Holland et al. (1975) in equation (C.9) is applicable, while if the cover temperature
exceeds the water temperature, Holman (1986) recommends the use of equation (3.7).
()
a
cw
cwcwe
t
k
Grh ⋅⋅⋅=
2.0
Pr58.0 (3.7)
The results are also displayed for the system during night-time operation, and night-
time operation with the use of Sisalation in figures 3.19 and 3.20 respectively. Note that
43
Chapter 3: Thermal behaviour and energy storage of water
the sky temperature in equation (3.5) needs to be represented by equation (2.6) and not
(2.5) as during daytime operation.
Figure 3.17: Water temperature distribution.
Figure 3.18: Measured- and predicted glass and mean water temperatures.
44
Chapter 3: Thermal behaviour and energy storage of water
Figure 3.19: Results attained for night-time operation.
Figure 3.20: Measured- predicted temperatures with Sisalation usage.
45
Chapter 3: Thermal behaviour and energy storage of water
3.5 Conclusion
The results of section 3.2 are summarised in figure 3.6. It shows a reasonable
correlation between the measured evaporation rates from the water tank m
exp
, and those
predicted by equation (2.8). The use of an energy balance with equation (3.2) shows
very erratic results, which if averaged out over a longer period of time, may too provide
reasonable results. This erratic nature may be attributed to the term in equation (3.2)
associated with the change in internal energy of the volume of water. This change is a
function of the change in average temperature of the water and is highly sensitive to
small changes in this value.
Equation (2.9) provides a very poor prediction of the evaporation rate. During the day it
overestimates the mass flow from the water surface by a very large margin, but
provided a reasonable estimate when applied to the wetted surface in Chapter 2. This
expression is used extensively in evapotranspiration calculations that involve the
evaporation of moisture from leaves or other botanical surfaces. Clearly applications
such as this are quite different to the current analysis since they do not consider a
change in internal energy of the system, this may explain the inaccuracy obtained in
figure 3.6. A full numerical example is available is Appendix C.
The solar collector with plastic-covered water tank was considered in section 3.3, the
following can be concluded from the associated experimental work.
The use of a transparent membrane on the surface of a body of water to suppress
evaporation is very effective and mean water temperatures of approximately 35 ˚C
greater than the ambient air temperature were attained. Figure 3.11 compares the
measured- and predicted glass cover and mean water temperatures, with a maximum
temperature difference between the measured and predicted values of approximately 5-
and 3 ˚C respectively. This error is found with both an initial temperature difference
and a cumulative error.
The fact that the predicted water temperature was too high and the cover temperature
suggests that the heat transferred between the water surface and cover is less than
46
Chapter 3: Thermal behaviour and energy storage of water
expected. This is unlikely however, since Lombaard (2002) followed the same
procedure with better results, even though he only compared the measured and
predicted temperatures over a period of an hour.
As shown by figure 3.10, very little difference exists between the mean- and surface
water temperatures; and the comparison between figure 3.11 and 3.12 show that
neglecting the stored energy in the glass cover was an acceptable approximation. The
apparatus was well sealed and infiltration of ambient air into the enclosure was not
permitted, thus the reason for the difference in measured- and predicted temperatures is
still unknown.
Good results were obtained on the same apparatus during night-time operation, with
and without the use of Sisalation. Figures 3.13 and 3.14 indicate a maximum difference
of approximately 2 ˚C for the glass cover temperature and less for the mean water
temperature.
Section 3.4 was included since a saturated system is of great interest to the current
study, most points of interest have however already been considered. Theoretical
modelling of the daytime operation was simply included as an approximation, while the
night-time models are of more importance.
The results obtained were relatively good, with figures 3.18, 3.19 and 3.20 showing
better correlations than those in section 3.3.
47
CHAPTER 4
Greenhouse modelling
4.1 Modelling objectives
4.2 Fundamental system
4.3 Night-time insulation usage
4.4 Day- and night-time insulation usage
4.5 Conclusion
4.1 Modelling objectives
In this chapter a basic greenhouse model is developed, with the intention of quantifying
the heat fluxes to and from the system when exposed to ambient conditions. The
horizontal roof and walls are of glass construction, while a concrete slab serves as a
floor; since glass has a high transmissivity, this implies that the quantity of solar energy
incident on the system is very high, but also that the losses incurred are extensive. Once
complete, this model will determine the temperatures that can be attained within such a
greenhouse. A sketch of the greenhouse is shown in figure 4.1.
Figure 4.1: Sketch of fundamental greenhouse.
48
Chapter 4: Greenhouse modelling
This model will also be used to determine the influence of insulation usage on certain
surfaces of the greenhouse, to determine the maximum temperatures that are
theoretically possible during winter operation.
4.2 Fundamental system
The basic model was of rectangular construction, with a 100 mm concrete floor, and a 4
mm standard glass flat roof and walls. Dry air conditions were assumed to exist within
the structure.
The weather profile used in the model is from Sishen, South Africa (23.00° E, 27.67°
S). Ambient air temperature, humidity, total- and diffuse solar radiation and wind
speed-readings are given on an hourly basis for this particular location. Skies are
assumed to be clear and the wind is assumed to blow across the roof and three of the
four walls.
Since the winter period is the focus of this study, the thermal behaviour of the
greenhouse on June the 21
st
will be determined. This date is selected since it is the
winter solstice in the Southern Hemisphere, and the day on which the available solar
radiation is at it’s least.
4.2.1 Analysis
The given system is analysed in the following manner: the structure is divided into
seven parts, namely the roof, the floor, the four walls and the enclosed air within the
greenhouse. An energy balance is applied to each of these components while exposed to
the selected ambient conditions, with the result that each of these temperatures are
predicted for any time and day throughout the year. Note that the floor, roof, walls and
enclosed air are each assumed to have a constant temperature throughout, and
conduction between adjacent surfaces is assumed negligible.
The first step is to determine the nature of the solar radiation on all of the greenhouse
surfaces.
49
Chapter 4: Greenhouse modelling
A measure of the position of the sun in the sky is known as the zenith angle θ
z
, and the
calculation thereof is critical to the current analysis. The angle formed between the sun
and the normal to a particular surface is known as the incidence angle, θ. As stated in
Appendix A, the incidence- and zenith angles can be calculated according Duffie and
Beckman (1991) with the aid of equations (A.1) and (A.2) for any randomly orientated
surface.
Note that at any time of the day it is only possible for two of the four vertical walls to
experience direct sunlight, and thus two of the surfaces will have incidence angles
greater than 90˚.
The geometric factor R
b
is a ratio between the intensity of the solar radiation on a
particular surface and that on a horizontal surface; it is defined by equation (A.9) and is
has to be considered if non-horizontal surfaces are to be analysed.
Determining the portion of sunlight that is transmitted, absorbed and reflected from
each of the greenhouse surfaces is crucial. From Appendix D, equations (D.9) and
(D.13) express the reflectivity and transmissivity for any non-opaque surface; and can
then be used in conjunction equations (D.4), (D.5) and (D.6) to calculate the effective
reflectivity, -transmissivity and –absorptivity of the particular surface.
The above relations hold for a wall of the greenhouse which is in direct sunlight,
however if one or more of the walls does not receive direct solar radiation then the
situation becomes somewhat more complex. It becomes necessary to determine through
which particular wall the light has been transmitted before striking the wall of interest,
and the associated area for which that portion of light is responsible. The reader is
referred to Appendix E for the appropriate equations.
The areas in Appendix E are calculated with use of the horizontal- and vertical
components of the incident sunlight, which have yet to be defined. Determining these
values requires that it be known in which quadrant the sun is present. Figure 4.2 below
shows a theoretical plot of the sun’s path on January the 1
st
and July the 1
st
.
50
Chapter 4: Greenhouse modelling
Figure 4.2: Path of the sun from above on the 1
st
of January and the 1
st
of July.
The horizontal component of the incident sunlight is simply calculated by taking the
difference between the solar- and surface azimuth angles
shoriz
'γγθ −= (4.1)
The vertical component is represented by the expression below.
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
−
horiz
s
vert
θ
α
θ
cos
tan
tan
1
(4.2)
With the above taken into consideration, if a glass window is not in direct sunlight it
can be determined through which window the transmitted light that strikes the particular
window has passed and the area for which that portion of light is responsible.
Now that the solar calculations have been taken into account, energy conservation
equations can be developed for the respective greenhouse surfaces.
51
Chapter 4: Greenhouse modelling
4.2.1.1 Flat glass roof
Consider the heat fluxes associated with the roof of the greenhouse.
Figure 4.3: Energy balance on the roof of the greenhouse.
If a control volume is drawn around the roof, the energy entering and leaving the
control volume can be summed as shown below.
racahghgraceh
qqIIqqI ++⋅+⋅=++ τρ
()( ) 01 =−−++−−⋅+
racaracegghdhb
qqqqII ρτ
0=−−++⋅+⋅
racaracedghdbghb
qqqqII αα (4.3)
The subscripts ce and re represent the convective and radiative heat transfer within the
greenhouse, while ca and ra represent the convective and radiative heat transfer to the
environment.
Note that with respect to figure 4.3 the convective heat fluxes can be expressed as:
)(
greaece
TThq −⋅= (4.4)
)(
agraca
TThq −⋅= (4.5)
52
Chapter 4: Greenhouse modelling
The internal convective heat transfer coefficient for a horizontal surface is found
according to Holman (1986) in equation (4.6) below:
Nu
L
k
h
e
⋅= (4.6)
where Nu is the Nusselt number with a value defined by one of the following applicable
relations.
()
31
Pr13.0 GrNu = cooled surface facing downward, Gr Pr < 2.10
8
()
31
Pr16.0 GrNu = cooled surface facing downward, 2.10
8
< Gr Pr < 10
11
()
51
Pr58.0 GrNu = heated surface facing downward
The convective heat transfer coefficient on the upper surface is determined according to
Burger and Kröger (2004) as shown in equations (2.1) and (2.2).
The radiative heat transfer within the greenhouse is likely to become quite a complex
analysis, the benefits of which are doubtful. While the difference in temperature
between the floor and the other surfaces within the greenhouse may be substantial, the
temperature of the glass surfaces are expected to be similar and thus the only radiative
transfer considered within the enclosure is that between the concrete floor and the glass
surfaces.
The radiative heat flux heat flux reaching the roof from the concrete floor is then
rfgr
grf
ra
RA
TT
q
⋅
−⋅
=
)(
44
σ
(4.7)
Where the resistance to transfer R
rf
is:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
−
+
⋅
+
⋅
−
=
ff
f
rfrrr
r
rf
AFAA
R
ε
ε
ε
ε
1
11
. (4.8)
53
Chapter 4: Greenhouse modelling
The radiative heat flux leaving the upper surface of the roof is emitted to a sky
temperature T
sky
defined by equations (2.5) and (2.6), and where
)(
44
skygRgra
TTq −⋅⋅= σε (4.9)
Then if equation (4.3) is considered, it can be rewritten as:
()() ( ) ( )
()
( )
rfr
grf
greae
dd
dd
hd
bb
bb
hb
RA
TT
TThII
⋅
−⋅
+−⋅+
⋅−
−⋅−
⋅+
⋅−
−⋅−
⋅
44
1
11
1
11
σ
τρ
τρ
τρ
τρ
α
α
α
α
() ( ) 0
44
=−⋅⋅−−⋅−
skygrgagra
TTTTh σε (4.10)
4.2.1.2 Vertical glass walls
In a similar manner to which the glass roof was analysed, consider the vertical glass
walls. Assume for the sake of notation that the window in the analysis faces north.
54
Chapter 4: Greenhouse modelling
Figure 4.4: Heat fluxes imposed on vertical glass window.
Equation (4.3) is applicable to this window as well; the only difference in this case is
the change in convective heat transfer coefficients.
Empirical correlations for combined forced- and free convection are not readily
available for vertical plates due to the numerous complexities present, which are not
found on horizontal plates. Thus in an attempt to solve this problem, it was decided that
heat transfer coefficients for both free- and forced convection would be determined and
the larger of the two would be retained for use in the model.
For free convection Churchill and Chu (1975) supply the following relation:
()()
278
169
61
Pr492.01
387.0
825.0
+
+=
Ra
Nu (4.11)
While for flat plate flow equation (4.12) is applicable for both laminar and turbulent
conditions, Holman (1986):
( )871Re037.0Pr
8.031
−⋅⋅=Nu (4.12)
Note that the applicable characteristic length is only half the length of the window
across which the wind blows, since it is the average heat transfer coefficient across the
surface that is desired.
Therefore the conservation of energy equation applicable to the vertical walls is very
similar to equation (4.10) for the horizontal surface.
()() ( ) ( )
()
( )
nfn
gnf
gneae
dd
dd
hd
bb
bb
hb
RA
TT
TThII
⋅
−⋅
+−⋅+
⋅−
−⋅−
⋅+
⋅−
−⋅−
⋅
44
1
11
1
11
σ
τρ
τρ
τρ
τρ
α
α
α
α
() ( ) 0
44
=−⋅⋅−−⋅−
skygngagna
TTTTh σε (4.13)
55
Chapter 4: Greenhouse modelling
4.2.1.3 Concrete floor
The analysis of the concrete floor differs from that of the vertical and horizontal glass
sheets, in that the concrete slab is opaque, the thermal inertia is taken into consideration
and while exposed to the enclosed air on one surface, it is assumed to be fully insulated
on the other.
The conservation of energy equation then reads as follows:
()
dt
dT
cmqqII
f
precehch
⋅⋅+++⋅−= α1 (4.14)
If equations (2.1) or (2.2) are used to determine the natural convective heat transfer
coefficient for the upper surface, then it remains only for the radiative heat transfer
between the floor and the glass surfaces to be taken into account as shown in equation
(4.7) and (4.8).
Rewriting equation (4.14) as follows:
() ()
( ) ( ) ( )
fwf
gwf
fnf
gnf
frf
grf
eafehc
RA
TT
RA
TT
RA
TT
TThI
⋅
−⋅
−
⋅
−⋅
−
⋅
−⋅
−−⋅−⋅−=
444444
10
σσσ
α
( ) ( ) ( )
dt
TT
c
A
m
RA
TT
RA
TT
initialff
p
ffsf
gsf
fef
gef
−
⋅⋅−
⋅
−⋅
−
⋅
−⋅
−
4444
σσ
(4.15)
Figure 4.5: Heat fluxes associated with concrete floor.
56
Chapter 4: Greenhouse modelling
4.2.1.4 Enclosed air
Since the temperature of the enclosed air, T
ea
is highly unlikely to reach very high
temperatures, and since short-wave radiation has no influence on air, the radiation
exchange within the enclosure and direct solar radiation have no influence on the
enclosed body of air.
This means that the air will only experience convective heat gains / losses from the
greenhouse surfaces and a change in internal energy as shown below.
( ) ( ) ( ) ( )
eagsseeagnneeagrreeaffe
TTAhTTAhTTAhTTAh −⋅⋅+−⋅⋅+−⋅⋅+−⋅⋅
() ()
( )
0=
−
⋅⋅+−⋅⋅+−⋅⋅
dt
TT
cmTTAhTTAh
initialeaea
paeageeeeagwwe
(4.16)
Note that the internal convective heat transfer coefficients h
e
in the above expression,
are those applicable to each of the appropriate surfaces.
It is now possible for equations (4.10), (4.13), (4.15) and (4.16) to be solved
simultaneously with the aid of the Gauss-Seidel method to determine the predicted
temperatures. This is possible if an estimate of the initial floor- and enclosed air
temperatures is made, the above-mentioned equations can be solved to determine the
applicable temperatures at each time step.
4.2.2 Simulation results
The simulation was run on June the 21
st
for a glass greenhouse with a 100 mm concrete
floor, and for the same greenhouse with a 100 mm layer of concrete (ρ
c
= 2300 kg.m
-3
,
c
pc
= 880 J.kg
-1
.K
-1
, α
c
= 0.61, ε
c
= 0.9) replaced by a layer of water of the same
thickness. The greenhouse was given arbitrary dimensions of 15 m in length, 5 m in
width and a height of 2.8 m. The building faces directly north, with the 15 m length
running from east to west. Figures 4.6 and 4.7 show the solar irradiation, and relative
humidity- and wind speed readings expected on June the 21
st
. Take note that the sun
rises and sets at 07:00 and 17:00 respectively. The wind speed measured at a height of
57
Chapter 4: Greenhouse modelling
10 m above the ground, ranges approximately between 2.8 and 3.9 m.s
-1
, while the least
humid time of the day can be seen to be at about 15:00.
Figure 4.6: Beam- and diffuse solar irradiation profiles.
Figure 4.7: Relative humidity- and wind speed readings.
58
Chapter 4: Greenhouse modelling
The floor-, enclosed air and mean glass temperatures predicted by the model for a 100
mm concrete floor are displayed in figure 4.8, the figure includes the ambient air
temperature.
Figure 4.8: Predicted temperatures within greenhouse with concrete floor.
Figure 4.8 shows that the mean glass temperature T
g ave
reaches temperatures in the
region of 5˚C while the ambient air temperature T
a
is always well above this value.
Similarly, the enclosed air temperature T
ea
only exceeds the ambient air temperature
between 10:00 and 18:00 and thus the purpose of the greenhouse is almost defeated,
since for the greater part of the day the air outside is warmer than that inside. The
concrete floor can be seen to reach a temperature of approximately 28˚C.
If a water layer of equal thickness replaces the concrete slab, then the temperatures
displayed in figure 4.9 are predicted. As with the concrete floor, the water layer is
assumed to be fully insulated on the lower surface, also the air within the structure is
assumed to be 100% saturated. The thermophysical properties of saturated air can be
determined according to Kröger (1998).
59
Chapter 4: Greenhouse modelling
Figure 4.9: Predicted temperatures in greenhouse with 100 mm water floor.
The figure displays a similar mean glass temperature to that in figure 4.8; the enclosed
air- and water temperatures are however slightly higher than those achieved with the
concrete floor. The rate of change in temperature of these values is less than in figure
4.8; this is attributed to the fact that the product of the density and specific heat capacity
of water is greater than for concrete, and thus more energy is required to change the
temperature of a set quantity of water than concrete.
4.3 Night-time insulation usage
This section incorporates the use of insulation into the model developed in section 4.2.
It was decided that the insulation used would be placed on all the outer surfaces of the
structure at sunset and removed at sunrise. Many types of thermal insulation are
commercially available, and focus on reducing either convective or radiative losses; the
assumption is made that the insulation used in the model is 100% effective and permits
no convective or radiative losses from the outer surfaces of the structure.
60
Chapter 4: Greenhouse modelling
Figure 4.10 shows the predicted temperatures associated with the concrete floored
greenhouse with full insulation during the night.
Figure 4.10: Predicted temperatures in greenhouse with concrete floor and thermal
insulation usage at night.
Figure 4.10 shows that when insulation is used, the enclosed temperatures tend towards
just more than 32˚C, which is substantially higher than that achieved without the use of
insulation as depicted by figure 4.8. It also shows that the temperatures achieved by the
concrete floor and enclosed air are greater than without the use of insulation. A sharp
drop is the predicted temperatures is visible when the insulation is removed at daybreak,
this is particularly prevalent if the mean glass temperature is considered which falls to
temperatures only slightly higher than those achieved without insulation usage.
Conversely, a sharp increase is found to occur with the addition of insulation at sunset.
Predicted results for the greenhouse with 100 mm water floor and night-time insulation
usage are displayed graphically in figure 4.11.
61
Chapter 4: Greenhouse modelling
Figure 4.11: Predicted temperatures in greenhouse with water floor and thermal
insulation usage at night.
The figure indicates that the system temperatures tend towards approximately 42˚C
when insulation is used. An increase in the mean daytime temperature of approximately
20 and 10˚C was experienced by the floor (water) and enclosed air respectively,
compared to when insulation was present.
The mean glass temperature is also found to increase slightly by replacing the concrete
floor with a water layer.
4.4 Day- and night-time insulation
This section extends the analysis considered in the latter part of section 4.3. A glass
greenhouse with a 100 mm water layer floor is fully insulated during the night; since
the day of interest is June the 21
st
, when the sun is only in the first and second quadrants
(see figure 4.2), the southern wall is permanently insulated since it receives no direct
sunlight. The decision was also made to retain the insulation on the western wall until
midday, at which point the insulation was added to the eastern wall and retained until
62
Chapter 4: Greenhouse modelling
dawn the following day. These changes would ensure that no sunlight would pass
through the entire system without influencing the heat gain of the greenhouse. The
predicted temperatures of the greenhouse are shown in figure 4.12.
Figure 4.12: Predicted greenhouse temperatures.
The figure shows that the system temperatures now tend to 63˚C, approximately 21˚C
more than without the daytime insulation. Since the some of the walls are now non-
opaque the solar absorptivity is now far higher and the eastern-, western and southern
walls now reach very high temperatures, while the glass roof and northern wall don’t
exceed 27˚C. The solar absorptivity of the insulated surfaces is taken as 0.6, which is
an average value similar to that of brick or concrete.
4.5 Conclusion
This chapter modelled the thermal behaviour of a flat roofed glass greenhouse which
was very poor at retaining energy; this occurred to the extent that the air within the
greenhouse became cooler than the ambient air at times, see figure 4.8. At first glance
this may suggest the presence of an error, but if it is considered that all the glass
63
Chapter 4: Greenhouse modelling
surfaces have a high emissivity (ε
g
= 0.9) and absorb very little of the incident solar
energy, low temperatures are probable during winter weather conditions.
An all glass structure has very poor heat retention, since the absorptivity of glass is so
low the solar energy that doesn’t strike the floor, which has a higher solar absorptivity,
essentially “passes through the system unnoticed” and is wasted. Thus, the performance
of such a system can be increased either by increasing the heat retained in the floor,
reducing the solar energy passing through the system or minimising any night-time
losses experienced.
Sections 4.2 and 4.3 clearly show that a volume of water is more efficient in receiving
and providing energy than an equal volume of concrete. This potential for receiving
energy is due to the greater solar absorptivity of water than concrete, and is based on
the assumption that all solar radiation not reflected at the water surface is absorbed
(compare the α-value of 0.61 for concrete to figure 2.4 for water). As mentioned above,
the ability of a material to store energy is dependant on the density / specific heat
capacity product, which in this case is greater for water than for concrete.
Figures 4.10 and 4.11 show that substantial gains can be made by insulating a glass
greenhouse during the night, while figure 4.12 shows that significant gains can be made
using insulation both during the day and night. Using darker insulation and thus
increasing the solar absorptance of the walls could increase this gain in energy further.
The systems described are idealised and the results are approximate; the model does
however give an indication of the potential gains possible with insulation usage.
64
CHAPTER 5
Aspects of greenhouse design
5.1 System considerations
5.2 Proposed improved design
5.3 Aspects of optimal design analysis
5.4 Welgevallen system analysis
5.1 System considerations
Chapter 4 dealt with theoretical modelling of a glass greenhouse, and illustrated how
the temperatures within such a system can be elevated substantially if the associated
thermal behaviour is well understood.
In this chapter an optimal design is proposed for the housing of species such as tilapia
that require elevated temperatures. This does not build upon another design, but is that
system which in the opinion of the author will receive the greatest quantity of solar
radiation and simultaneously experience minimal losses.
Thereafter, the tilapia system at the Welgevallen Research Centre is analysed, and
recommendations presented as to how the system can be improved upon without major
structural changes, with the result that the temperature of the water is increased to a
sufficient level.
5.2 Proposed optimal design
The primary objective of this study is to increase the temperature of a body of water
contained within a greenhouse during the cold winter season; the greenhouse
configuration that has been calculated to most efficiently achieve this objective is
shown in figure 5.1.
65
Chapter 5: Aspects of greenhouse design
Figure 5.1: Side and plan views of the proposed greenhouse structure.
The side view in the figure shows a cross-sectional view of the greenhouse from the
east. The south wall, floor, rear roof section and lower north wall are opaque and fully
66
Chapter 5: Aspects of greenhouse design
insulated, while the upper north wall and front roof section are of glass or plastic
construction, with a high transmissivity. Other than these two transparent surfaces, all
surfaces within the enclosure have a high solar absorptivity. Water is assumed to cover
the entire surface area of the floor and is filled up to a set height, this is fully insulated
and losses only occur from the upper surface. Note that the roof is pitched; this prevents
the build-up of condensation on the roof panels and assists with natural ventilation
during summer months.
The plan view of the structure shows a quadrilateral with north and south walls parallel
to each other, while the eastern and western walls are slanted in opposing directions by
an angle θ
wall
. The walls are angled such that the net energy into the greenhouse is
greater than the loss to the environment. It is also required that the inner surfaces of the
eastern and western walls have a high solar absorptance.
The available solar radiation is expected to be the lowest on the 21
st
of June, the winter
solstice in the southern hemisphere. South Africa lies approximately between the
latitudes of 23˚ and 34˚S and according to equation (A.3), at midday on the 21
st
of June
2005 the zenith angle at the respective latitudes is 46.44˚ and 57.44˚. This indicates that
if θ
roof
is set to approximately 45˚ at northern latitudes and approximately 30˚ further
south, then no sunlight will be wasted on the external surface of the rear roof section,
regardless of where the structure is positioned in the country.
Equation (A.3) can be solved for a zenith angle of 90˚ to determine the time of sunrise
and sunset at a particular location. On June the 21
st
at 23˚ and 34˚S, sunrise occurs at
06.71 and 07.13, while sunset happens at 17.29 and 16.87 respectively (all times are
measured in solar time). Thus, is could be assumed that on average sunrise and sunset
occurs at 07:00 and 17:00 on this particular day, as was indicated by the Sishen solar
profile at 27.67˚S (see figure 4.6).
Observation of figures 4.9, 4.10 and 4.11 show a distinct decrease in the predicted
temperatures after the removal of the thermal insulation at dawn. These temperatures
then increase as the day progresses, they reach a maximum value and then decrease
slightly, after which the insulation is replaced on the system for the night. This form of
operation is clearly sub-optimal.
67
Chapter 5: Aspects of greenhouse design
The figures indicate that the gain in energy of the system exceeds the losses incurred at
approximately 09:00, two hours after sunrise. Similarly, the predicted temperatures
reach their peak at roughly 15:00, two hours before sunset. Thus, it recommended that
for the current greenhouse the entire system should be fully insulated until two hours
after sunrise, at which point the insulation on the front roof section and north wall
should be removed, and should be replaced two hours before sunset.
At 09:00 and 15:00, the horizontal component of the sunlight incident on the system is
calculated with equation (4.1) to be 46.31˚ and 43.13˚ at 23˚ and 34˚S respectively.
Then if the wall angle θ
wall
is set to appropriately 45˚, the thermal insulation will be
removed once θ
horiz
equals θ
wall
in the morning and replaced again when that occurs in
the afternoon.
Thus for any location within the given latitudes, the appropriate roof θ
roof
and wall
angle θ
wall
can be determined for the maximum gain in solar energy. Take note that this
particular layout has been designed with South Africa as the proposed location.
Applying the above in another country will require alterations to the given design.
5.3 Optimal design analysis
When the northern wall and front roof section of the system in figure 5.1 are covered
with thermal insulation, the system is fully insulated and ideally no energy can either
enter or leave the greenhouse. Once removed however, it is necessary to determine
whether or not a net loss or gain is experienced by the system when exposed to low
solar radiation, and cold but clear weather. If the result is such that the greenhouse
experiences an increase in energy, then the primary objective of this study has been
attained. If the greenhouse is able to maintain the temperature of the enclosed water at
the upper limit of the temperature range when skies are clear, then if the system is
sealed during periods of cloudy cold weather, the storage potential of the water is such
that enough energy should be retained to keep the water temperature above the lower
limit until the clear weather returns.
68
Chapter 5: Aspects of greenhouse design
Figure 5.2: Simplified greenhouse model.
Observation of the system reveals that it bears a striking resemblance to that of a solar
collector. Modelling of the exact system presents numerous complications, and thus it is
highly simplified in the following model in an attempt to evaluate the storage potential
of the proposed greenhouse. The model above is essentially identical to the plastic-
covered water tank considered in section 3.3. The system is displayed in figure 5.2.
The ambient temperature, dew-point temperature and solar radiation readings applicable
to June the 21
st
in Sishen will be used in conjunction with the above model. The initial
temperature of the water will be set to the upper temperature limit applicable to tilapia,
namely 35˚C, while the depth of the water t
w
will be taken as 500mm. Since the model
is highly simplified the convective heat transfer coefficients, h
a
and h
e
will be taken as
10 W.m
-2
.K
-1
. The absorptivity and transmissivity of the glass cover are taken as 0.05
and 0.85 respectively.
The governing equations for the glass cover and water tank are given by equations (5.1)
and (5.2):
69
Chapter 5: Aspects of greenhouse design
Figure 5.3: Mean water tank temperature on June 21
st
.
() ()() (
4444
1
11
skyccacacw
cw
cwehc
TTTThTTTThI −⋅+−=−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−++−+⋅ σε
εε
σα ) (5.1)
() ()
( )
3600
1
11
44 initialww
pwwcw
cw
cwehc
TT
ctTTTThI
−
⋅⋅+−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−++−=⋅ ρ
εε
στ (5.2)
Solving equations (5.1) and (5.2) simultaneously produces the results displayed in
figures 5.3 and 5.4.
The above figure displays the comparative water temperatures attained without
insulation, insulation usage while dark and the use of insulation from 00:00 to 09:00
and from 15:00 to 24:00 as recommended. The mean water temperature is found to
decrease without insulation usage, indicating a net loss from the system, while the
proposed usage can be found to achieve the best results. The influence of removing the
insulation 2 hours after sunrise and adding it 2 hours before sunset, as suggested in
section 5.2 is evident if the two sets of insulated results are compared. The glass cover
temperatures in the above analysis are displayed in figure 5.4.
70
Chapter 5: Aspects of greenhouse design
Figure 5.4: Comparative glass cover temperatures.
It has been shown that the proposed system will definitely attain the required water
temperature during mid-winter, no mention has however been made about the operation
of the system during warmer summer months.
In section 4.2 a glass greenhouse with a 100 mm concrete floor was considered and the
predicted temperatures displayed in figure 4.7 for operation on June the 21
st
. The
temperatures were generally lower than the range acceptable to tilapia, and thus
improvements on the system needed to be considered. Figure 5.5 shows the predicted
results for the same system modelled on December the 21
st
, the summer solstice.
The figure shows a vast improvement to that considered during winter months, and
even though large losses are incurred high temperatures are still possible during the day.
Such a system is obviously useless from an aquaculture perspective, since any species
housed in such a system would perish quickly if temperatures of this magnitude were
produced. Thus, it needs to be ensured that the above cannot happen to the proposed
system.
71
Chapter 5: Aspects of greenhouse design
Figure 5.5: Predicted glass greenhouse temperatures on December 21.
The two most important concepts to consider when keeping a structure cool are shading
and ventilation. During summer it is suggested that the entire roof and eastern and
western walls be fully insulated, and that insulation not be used on the transparent north
window during the night. Thus, the only way in which sunlight can enter the system is
through the north window. However figure 4.2 indicates that the zenith angle is very
small during summer months and therefore the quantity of solar radiation passing
through the northern window is expected to be very little. Furthermore, the use of an
overhang could be considered on the front section of the roof to provide shading of the
northern window if necessary.
If it is found that the system still becomes too warm, ventilation could be considered.
Since the proposed structure has a pitched roof, air could be extracted just below the
apex of the roof through the eastern and western walls. Due to the fact that warm air is
less dense than cooler air, ventilation will occur naturally if openings are provided in
the lower half of the walls. If this is still not satisfactory, forced ventilation could be
employed through the use of fans.
72
Chapter 5: Aspects of greenhouse design
5.4 Welgevallen system analysis
It was stated in Chapter 1 that one of the goals of this study was to increase the water
temperature in the aquaculture system housing tilapia at the Welgevallen Research
Centre. It was required that guidelines be provided on how the heat gain of the system
could be improved by making non-structural alterations to the existing greenhouse. A
side view of the structure is given in figure 5.6 as viewed from the west; the structure
faces slightly west of north.
Figure 5.6: Side view of greenhouse structure.
The walls, door and roof of the structure are primarily constructed from corrugated
polycarbonate (Modek Greca, clear 1 mm) with a steel frame. The northern wall has 7
polycarbonate windows that stretch the full length of the wall, whereas the roof has a
relatively flat pitch, elevated at 65˚ to the vertical. The lower section of the northern and
western walls is facebrick, while a slab of concrete serves as the floor. The southern
wall of the enclosure is cement, while the area further south (further right in the figure)
is shaded by an overhang that shelters a walkway.
A plan view of the greenhouse is shown in figure 5.7, it displays 14 PVC tanks that
house the tilapia, 12 of which are in direct sunlight. Four large cylindrical biofilters are
positioned opposite the entrance, while behind the biofilters; a polycarbonate wall
separates the current structure from a cooler adjacent greenhouse.
73
Chapter 5: Aspects of greenhouse design
Figure 5.7: Plan view of Welgevallen greenhouse layout.
74
Chapter 5: Aspects of greenhouse design
The following observations were made with regard to the energy flows to and from the
system:
Since the polycarbonate sheeting is corrugated (see figure 5.8), it rarely fits flush
against the steel structure supporting the greenhouse. Attempts have been made to
minimise the infiltration of unconditioned outside air entering the system by injecting a
foam sealant into many of the cavities and joints.
Figure 5.8: Profile of corrugated polycarbonate.
The greenhouse is approximately north facing, and thus has the potential to receive
large quantities of solar radiation. The roof is slightly pitched, with both halves of the
roof transmitting sunlight. The polycarbonate panels are hazy in appearance, due to the
accumulation of dirt and as a result of ageing. The presence of condensation within the
system is negligible.
The eastern section of the greenhouse is separated from an adjacent cooler greenhouse
by the same polycarbonate sheeting used in the walls and roof. This wall is likely to be
a source of substantial heat loss. An air-conditioning unit is mounted on this wall and
simultaneously cools the adjacent greenhouse, while heating the current system.
The tanks are blue in colour and are of PVC construction; they are supported by
galvanised steel stands and thus do not experience conductive losses through the base
into the floor. Black nets cover the upper surface of all the tanks, and thus the absorbed
solar radiation should be sufficient. The water is cycled from the tanks, passing through
a belt filter (solids removal) and biofilters before returning to the source. A marginal
75
Chapter 5: Aspects of greenhouse design
quantity of water is lost through the operation of the belt filter, and thus the water cycle
is essentially a closed system.
The southern wall is painted white and can reach a thickness of 0.75 m in places. This
wall has four archways that lead into a room that is fully shaded at all times; it contains
two tanks and a control panel at the entrance. This room, although open, is far cooler
than anywhere else in the greenhouse and definitely a source of heat loss from the main
enclosure.
There is very little that can add to the heat load within the system, three large lights
hang from the roof to provide artificial lighting when required. A pump drives the water
cycle, while the belt filter motor and two small fans are also found within the enclosure.
The influence of these components is expected to be negligible.
The nearest surrounding structures to the greenhouse are sufficiently far away, such that
the influence of shading should be negligible during times of the day when the quantity
of available solar radiation is of significance.
Application of the following recommendations will assist in increasing the enclosed
water temperature:
The entry of outside air into the greenhouse is detrimental to the current objective.
Thus, it should be made absolutely sure that the entire building is airtight. All doors
should have airtight seals, or airlocks as used in the tunnel systems at the same location.
The air and pump used for providing aeration to the aquaculture system should be
located within the greenhouse structure.
The polycarbonate panels allow solar radiation to enter the system, but also have a high
emissivity leading to large radiative losses, they offer marginal conductive resistance to
heat leaving the system and the corrugations may well lead to turbulent forced
convective losses from the outer surface. It is recommended that the panels be
maintained as clean as practically possible to permit maximum transmission of solar
radiation.
76
Chapter 5: Aspects of greenhouse design
The gain in useful solar energy through the eastern and western walls is unlikely to be
substantial during winter months, while the loss of energy through the eastern wall to
the cooler greenhouse is unacceptable. Both walls should be fully insulated at all times,
thus eliminating any light transmission and radiative and convective losses from the
system. Although not ideal because the pitch of the roof is a bit low, the southern
portion of the roof could also be fully insulated since the losses incurred may well
exceed the gain in solar radiation.
It was stated above that the shaded southern room adjoining the greenhouse is expected
to be a large source of heat loss. This can be solved in one of two manners: If the two
shaded tanks are ignored, insulating the whole south wall could seal off the entire
shaded portion of the greenhouse and thus eliminate the heat loss. Otherwise the outer
surfaces of the shaded room could be insulated, this would retain heat within the system
and the large separating walls could provide significant energy storage. In both cases,
while often not aesthetically pleasing, darkening the walls would result in increased
solar absorptance.
As recommended in Chapter 4, full thermal insulation should be added to the exposed
northern wall and front roof section approximately two hours before sunset and retained
until two hours after sunrise. This proposal may however pose a labour issue since
covering both panels could become quite labour intensive, thus the use of radiative
insulation such as Sisalation (see Chapter 3) might need to be considered. Substantial
gains can still be achieved with Sisalation and may prove a better option since it is
easier to handle, and lends itself to mechanised application. The environmental integrity
of the insulation needs to considered in conjunction with the capital- and installation
cost involved, as well as the above-mentioned factors, to determine which form of
insulation is best suited to the specific application.
77
CHAPTER 6
Conclusion
6.1 Research findings
6.2 Greenhouse conclusions
6.3 Further investigation
6.1 Research findings
The first tests performed were those to determine the convective heat transfer
coefficient between a horizontal surface and the natural environment. The literature
reveals a wide range of correlations attempting to predict the convective heat transfer
coefficient, with varying results. The process was analysed theoretically by Kröger
(2002) and later refined with experimental work by Burger and Kröger (2004). The
results obtained through testing in the present study were sufficiently accurate when
compared to those predicted by Burger and Kröger (2004).
Experiments were conducted to determine the evaporation rate from a water surface
exposed to the natural environment. As with the convective heat transfer coefficient,
many empirical correlations are available for predicting the rate of evaporation from a
water surface, with a large variation found to occur in the rates predicted. The results
obtained in this study were very pleasing and the author was fortunate to obtain a
difference of less than three percent (see table 2.1) between the quantity of water
actually evaporated throughout the day and the amount predicted by the relevant
expression (equation 2.7). This is the first time that experimental results have been
found to correlate well with a theoretically determined evaporation rate. Note that this
correlation requires that the water surface temperature exceed the ambient air
temperature.
A body of water within an insulated tank was then considered, with the upper surface
exposed to the environment. This experiment was conducted since the apparatus
78
Chapter 6: Conclusion
mimicked the fundamentals of a solar pond, the results of which are of interest to this
study. The ambient air temperature exceeded the measured water surface temperature
throughout the duration of the experiment and thus the correlation applicable to the
evaporation pan (equation 2.11) was unfortunately not of application. The results
obtained by alternate expressions were not as good, but displayed reasonable trends.
Monteith and Unsworth (1990) published an equation for evaporation rate prediction,
primarily for horticultural applications. Good results were obtained with the use of this
expression in earlier experiments, but was found to be clearly limited to applications
without a stored energy component, since unsatisfactory results were obtained in the
current experiment.
Experimental work was done to confirm the results obtained by Lombaard (2002)
regarding the coefficient of extinction for glass, as used in the solar collector. The
findings matched those obtained previously precisely, with a value of 13 m
-1
.
The performance of a solar collector with a plastic covered water tank was investigated
in section 3.3. The apparatus serves as both an excellent storage medium and collector
of solar energy, with high water temperatures being attained during the day. The
daytime operation of the system was modelled, as well as the operation of the system
during the night, when the solar collector releases the energy stored during the day. The
influence of thermal insulation on the reduction of night-time losses from the system
was also investigated.
The temperatures predicted by the model representing daytime operation did not agree
particularly well with those measured experimentally. This particular model was the
source of much frustration and was really thoroughly investigated, with the source of
the error still eluding the author. Predicted night-time temperatures, with and without
the use of thermal insulation did however correlate much better with those measured.
Experiments were later performed on the same solar collector incorporating water
storage without the use of the plastic film. The behaviour of the system was found to
change entirely. Due to evaporation at the water surface, the horizontal glass cover soon
became burdened with heavy condensation. Condensation of this nature is rarely found
in practical applications and thus the influence of such condensation on sunlight is not
79
Chapter 6: Conclusion
well documented. Fixed solar properties of the glass/condensation were assumed with
surprisingly good results. The lack of plastic film on the water / air interface provided a
water temperature profile that was substantially warmer at the surface during daytime
operation. This is attributed to the greater solar absorptance of the glass/condensate,
which indirectly heats the upper portion of the water tank.
Modelling the same system during night-time operation with and without insulation
usage provided very good results, with a better correlation between predicted- and
measured temperatures found than with the use of a plastic film.
6.2 Greenhouse conclusions
In Chapter 4 a fundamental greenhouse model was developed, it was of glass
construction with an insulated 100 mm concrete floor. This served as a reference, since
it was exposed to large quantities of solar radiation and thus could possibly absorb large
quantities of solar energy, but was simultaneously very basic in design and experienced
large night-time losses.
When compared to an identical structure with a 100 mm deep water layer as a floor, the
model was found to predict warmer temperatures within the system, while the rate of
change of the temperatures was also found to be less. The former characteristic is
attributed to the fact that water has a greater solar absorptivity than concrete, while the
latter occurs because the specific heat capacity of water is greater than that of concrete.
The influence of thermal insulation was investigated during the night. With the absence
of sunlight, the entire structure was encapsulated in insulation, ideally permitting no
radiative or convective losses from the outer surfaces of the structure. Similar results
were found as before, with the system containing water stabilising at a higher
temperature during the night than that with a concrete floor. It was observed in both
scenarios that the floor- and enclosed air temperatures were found to decrease once the
insulation was removed at daybreak, and started to increase only after a period of about
two hours. A similar occurrence was observed before sunset. This was clearly indicative
of sub-optimal insulation usage and required further consideration.
80
Chapter 6: Conclusion
Lastly, insulation was used permanently on the south wall and on the west wall until
midday, and on the east wall from midday onwards; as before complete insulation was
used during the night. This simulation was run only on the greenhouse containing water
and as expected, a further increase in temperature was achieved. It was found that
increasing the inner solar absorptance of the walls could increase this even further.
An optimal greenhouse design was presented in section 5.2 that in the opinion of the
author acts as an excellent solar collector while simultaneously minimising the loss of
energy incurred during the night. This section proposes concepts to those interested in
building structures of this nature on how to optimise the solar input to the system,
without attaining unreasonable temperatures during warmer summer months.
It was further proven with the aid of a very simple model that the conceptual system
would indeed experience a net gain in energy, even on the day of the year with the least
available solar radiation, provided that clear skies were present. Thus, if the water
within the system is kept at the upper temperature limit (35ºC), the proposed system
would be able to maintain an acceptable water temperature during extended periods of
rain or cloud.
Section 5.3 analyses the greenhouse covered aquaculture system at Welgevallen
Research Centre in Stellenbosch. The layout is described briefly and all possible flows
of energy, entering or leaving the system are analysed. Recommendations were then
presented as to how the potential problems could be solved with little structural
changes. The author believes that if the given suggestions are applied to this particular
system, that there should be no reason why the necessary temperatures are not attained.
The supplied recommendations are conceptual in nature and thus the practical
implications thereof may still need to be considered. It is important to note that
environmental conditions such as sunlight and rain rapidly degrade certain materials,
and may make certain types of insulation redundant.
81
Chapter 6: Conclusion
6.3 Further investigation
As mentioned, the optimised greenhouse design developed in Chapter 5 presented
educated guidelines for those involved in the construction of such buildings to benefit
from. Extending this study with an exact theoretical analysis of this system could prove
to be very interesting; firstly in winter to see what temperatures can in fact be attained,
and during summer to determine if the precautionary measures put in place prevent the
system from overheating. An analysis of the most applicable form of thermal insulation
to be used in the current system would also prove to be very useful.
Since many aquaculture species are also faced with the problem that the ambient
conditions regularly exceed their upper temperature limit; a study of a similar nature
focussed on cooling the enclosed water cycle, coupled with the current study, would
comprehensively cover this topic.
Lastly, current trends in intensive aquaculture have been focussed on the development
of hybrid systems. Such facilities incorporate previously closed system aquaculture
farms with horticultural applications. In the opinion of the author linking two or more
systems, each with a different focus but the same temperature requirements, such as
aquaculture and hydroponics, should be the direction in which further studies in this
field are directed.
82
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87
APPENDIX A
Solar radiation
A.1 Direction of beam radiation
A.2 Ratio of beam radiation on tilted surface to that on horizontal surface
A.1 Direction of beam radiation
The relationship between any particular surface orientated relative to the earth, and the
position of the sun at any time can be described in terms of several angles. This is
apparent if figure A.1 is considered.
Figure A.1: Randomly orientated surface in direct sunlight with associated angles.
The angles listed in the figure are described below:
β Slope, this is the angle between the surface of interest and the horizontal plane,
0˚ < β < 180˚.
A.1
Appendix A: Solar radiation
θ
z
Zenith angle, the angle between the vertical and the line to the sun.
α
s
Solar altitude angle, the angle between the horizontal and the line to the sun.
γ
s
Solar azimuth angle, the angular displacement from south of the projection of
beam radiation on the horizontal plane. Note that displacements east of south
are considered negative.
γ Surface azimuth angle, the deviation of the projection on a horizontal plane of
the normal to the surface from the local meridian. Due south equals zero, with
east negative.
It is often necessary to calculate the angle formed by the beam radiation on a surface
and the normal to that surface, this is known as the angle of incidence, θ. This angle is
described by the following equations:
ωγβδωγβφδ
ωβφδγβφδβφδθ
sinsinsincoscoscossinsincos
coscoscoscoscossincossincossinsincos
++
+−=
(A.1)
)cos(sinsincoscoscos γγβθβθθ −+=
szz
(A.2)
However, by definition the incidence angle for a horizontal surface is the zenith angle.
Thus equation (A.1) can be rewritten as shown in equation (A.3) if the surface is
horizontal.
ωφδφδθ coscoscossinsincos +=
z
(A.3)
The undefined terms in the above equations are the latitude angle φ, hour angle ω and
the declination angle δ. The latitude angle is a measure of the angular position of the
specific location north or south of the equator, while the hour angle is the angular
displacement of the sun east or west of the local meridian. The hour angle is measured
in degrees and can be determined according to equation (A.4).
()noonSolartimeSolar −=15ω (A.4)
A.2
Appendix A: Solar radiation
The declination angle is defined as the angular position of the sun at solar noon with
respect to the equator. The South African Weather Bureau suggests that the following
equation be used to determine the declination angle (measured in radians).
)4789.12sin(00665.0)4075.1sin(40602.000661.0 −+−+= PPδ (A.5)
)0996.13sin(00298.0 −+ P
The P symbol is the annual phase angle (measured in radians) and is a function of the
day of the year and the number of the specific year.
)417.0(0172028.0 ++= YADJDOYP (A.6)
DOY represents the day of the year and YADJ is a leap year adjustment given by the
South African Weather Bureau, and can be determined as follows:
()()[4/1int45.225.0 ]−−−= YYYADJ (A.7)
where Y indicates the number of the specific year.
Thus with the use of equations (A.4) through (A.7), the inclination angle in either
equation (A.1) or (A.3) can be evaluated. The utilisation of equation (A.2) however still
requires that the solar- and surface azimuth angles be correctly defined.
The solar azimuth angle (measured in degrees) is represented by the following
expression, where γ
s
’ is introduced as a pseudo solar azimuth angle:
180
2
1
'
21
321
⋅
⎟
⎠
⎞
⎜
⎝
⎛ ⋅−
⋅+⋅⋅=
CC
CCC
ss
γγ (A.8)
where
z
s
θ
δω
γ
sin
cossin
'sin
⋅
= .
The values of C
1
, C
2
and C
3
are determined according to the criteria below.
A.3
Appendix A: Solar radiation
C
1
= 1 if |ω| < ω
ew
φ
δ
C
1
= -1 otherwise where
ωcos
tan
tan
=
ew
C
2
= 1 if φ (φ - δ) ≥ 0
C
2
= -1 otherwise
C
3
= 1 ω ≥ 0
C
3
= -1 otherwise
A.2 Ratio of beam radiation on tilted surface to that on horizontal
surface
Solar performance calculations often require that the quantity of solar radiation on an
oblique surface be calculated. This is given in terms of a ratio known as the geometric
factor R
b
, which can be expressed as shown in equation (A.9).
zh
b
I
I
R
θ
θ
θ
cos
cos
== (A.9)
All equations supplied in section (A.1) and (A.2) were obtained from Duffie and
Beckman (1991).
A.4
APPENDIX B
Evaporation
B.1 Analysis – Evaporation from a water surface
B.2 Analysis – Temperature gradient within water layer
B.3 Evaporation pan photograph
B.4 Numerical example (section 2.3)
B.1 Analysis - Evaporation from a water surface
The problem of evaporation from a horizontal water surface that is exposed to the
natural environment is analysed for cases where the water surface temperature is
measurably higher than that of the ambient air. An empirical equation is recommended
in the case where the temperature difference is relatively small and for application at
night.
Consider a stationary fluid (binary mixture consisting of air and water vapour) in which
the concentration c
vi
of
the species of interest (water vapour) is initially uniform.
Beginning with the time t = 0, the concentration at the z = 0 boundary or surface is
maintained at a greater level c
vo
as shown in figure B.1 (a).
Figure B.1: Concentration or temperature distribution.
Water vapour will diffuse into the medium to form a concentration boundary layer, the
thickness of which increases with time. The mathematical equation of time dependent
B.1
Appendix B: Evaporation
diffusion in a binary mixture, expressed in terms of the molar concentration c is as
follows:
t
c
x
c
D
∂
∂
=
∂
∂
2
2
(B.1)
The diffusion flux is driven solely by the concentration gradient strictly in an isothermal
and isobaric medium. Equation (B.1) is however a good approximation in many non-
isothermal systems, where temperature differences are relatively small. If changes in
temperature are small the diffusion coefficient D can be assumed to be constant.
Equation (B.1) is analogous to the time-dependent equation for heat conduction into a
semi-infinite body i.e.
t
T
z
T
∂
∂
=
∂
∂
2
2
α
(B.2)
If the temperature of a semi-infinite solid is initially uniform at T
i
and a sudden increase
in temperature to T
o
occurs at z = 0 as shown in figure B.1 (b), Schneider
(1955) shows
that the temperature gradient at z = 0 is given by
()()
2/1
/ tTT
z
T
oi
πα−=
∂
∂
(B.3)
The corresponding heat flux is
( )
()
2/1
t
TTk
z
T
kq
io
T
πα
−
=
∂
∂
−=
(B.4)
An effective heat transfer coefficient can be expressed in terms of this heat flux i.e.
()()
2/1
// tkTTqh
ioTTt
πα=−= (B.5)
B. 2
Appendix B: Evaporation
By solving equation (B.2) for the case where the solid at an initial uniform temperature
T
i
is suddenly exposed to a constant surface heat flux q
q
, the latter can, according to
Holman
(1986), be expressed in terms of an effective surface temperature T
oq
as
[ ]
2/1
)/(2/)( παtTTkq
ioqq
−=
(B.6)
The corresponding effective heat transfer coefficient is defined as
[ ]
2/1
)/(2/)/( παtkTTqh
ioqqqt
=−=
(B.7)
It follows from equations (B.5) and (B.7) that for the same temperature difference i.e.
for
)()(
ioioq
TTTT −=−
TqTtqt
hhhh /2// ==π (B.8)
Although equation (B.2) is applicable to a solid, it is a good approximation when
applied to a thin layer of gas or vapour near a solid surface.
Due to the analogy between mass and heat transfer the solution of equation (B.1) gives
the following relations corresponding to equations (B.3) to (B.8) respectively:
If the initial concentration at z = 0 is suddenly increased to c
vo
()(
2/1
/ tDcc
z
c
vovi
v
π−=
∂
∂
)
(B.9)
The corresponding vapour mass flux is
()()[]
2/1
/ tDcc
z
c
Dm
vivov
π−=
∂
∂
−=
(B.10)
An effective mass transfer coefficient can be expressed in terms of this mass flux i.e.
B. 3
Appendix B: Evaporation
[]
2/1
)/()/( tDccmh
vivovDt
π=−=
(B.11)
If vapour is generated uniformly at a rate m
vm
at z = 0, this mass flux can be expressed
in terms of an effective concentration c
vom
to give analogous to equation (B.6)
[ ]
2/1
)/(2/)( πDtccDm
vivomvm
−= (B.12) 2/)/)((
2/1
tDcc
vivom
π−=
The corresponding effective mass transfer coefficient is defined as
2/)/()/(
2/1
tDccmh
vivomvmDmt
π=−=
(B.13)
It follows from equations (B.11) and (B.13) that for the same effective difference in
concentration i.e. for
)()(
vivovivom
cccc −=−
DDmDtDmt
hhhh /2// ==π
(B.14)
These latter equations are applicable in the region of early developing concentration
distribution in a semi-infinite region of air exposed to a water or wet surface.
According to Merker
(1987) for a Rayleigh number Ra ≥ 1101, unstable conditions
prevail with the result that water vapour is transported upwards away from the wetted
surface by means of “thermals” as shown in figure B.2.
Figure B.2: Flow development on surface.
B. 4
Appendix B: Evaporation
The generation of such thermals is periodic in time, and both spatial frequency and rate
of production are found to increase with an increase in heating rate.
For an analysis of the initial developing vapour concentration distribution near the
suddenly wetted surface at z = 0, consider figure B.1 (a).
The approximate magnitude of the curvature of the concentration profile is the same as
the change in slope across the relatively small concentration layer thickness or
height δ
zdc
v
∂/
D
i.e.
()()
0
//
0
2
2
−
∂∂−∂∂
≈
∂
∂
==
D
zvzv
v
zczc
z
c
D
δ
δ
(B.15)
Figure B.1 (a) suggests the following concentration gradient scales:
(),0/ =∂∂
=
D
zv
zc
δ
()()
Dvoviozv
cczc δ// −≈∂∂
=
Substitute these gradients into equation (B.15) and find
()
222
Dvoviv
cczc δ// −−≈∂∂ (B.16)
The approximate magnitude of the term on the right-hand side of equation (B.1) can be
deduced by arguing that the average concentration of the δ
D
-thick region increases from
the initial value c
vi
by a value of ( ) 2/
vivo
cc − during the time interval of length t.
()(tcctc
vivov
2// −=∂∂ )
(B.17)
According to equations (B.1), (B.16) and (B.17) find
()( )( )Dtcccc
vivoDvovi
2
2
// −≈−− δ
or
B. 5
Appendix B: Evaporation
()
2/1
2Dt
D
=δ
(B.18)
The concentration layer becomes unstable when
() 1101Ra
3
=−= )/( μρρρδ kcg
pavavoaviDu
(B.19)
where
()2/
aviavoav
ρρρ +=
At this condition
{ }[ ]
3/1
)(/33.10
pavavoaviDu
cgk ρρρμδ −=
(B.20)
From equations (B.18) and (B.20) find
{ }[ ] Dcgkt
pavavoaviu
/)(/31.53
3/2
ρρρμ −=
(B.21)
Substitute equation (B.21) into equation (B.11) to find
[ ] 0773.0/})({/
3/1
=− Dcgkh
pavavoaviDt
ρρρμ (B.22)
The average mass transfer coefficient during the period t is found by integrating
equation (B.11) i.e.
()[]
DtuD
htDh 22
21
==
/
/ π
or upon substitution of equation (B.22)
{}[]1550)(
31
.// =− Dcgkh
/
pavavoaviDt
ρρρμ (B.23)
If the surface generates vapour at a uniform rate it follows from equation (B.14) that
2/
DDm
hh π=
B. 6
Appendix B: Evaporation
or
{ }[ ] 243.0/)(/
3/1
=− Dcgkh
pavavomaviDm
ρρρμ (B.24)
It is stressed that these equations are only applicable to the first phase of the heat or
mass transfer process (growth of concentration layer) and do not include the second
phase during which thermals exist (breakdown of concentration layer). No simple
analytical approach is possible during this latter phase, although the mean mass transfer
coefficient during the breakdown of the concentration layer will probably not differ
much from the first phase. This would mean that the mean mass transfer coefficient
over the cycle of conduction or concentration layer growth and breakdown is of
approximately the same value as that obtained during the first phase of the cycle.
By following a procedure similar to the above, the analogous problem of heat transfer
during natural convection above a heated horizontal surface for a constant surface
temperature of T
o
can be analysed to find according to Kröger
(2002)
{}[]
{ }[ ] 155.0/)(/)(
/
3/1
3/1
22
=−=
−
kcgkhkcTT
gTh poiTpio
T
ρρρμρ
μ
(B.25)
and for the case of uniform heat flux q
(){}[]( ){ }[ ] 243.0///
3/13/1
22
=−=− k
c
gk
hkcTT
gT
h
poqiqpioqq
ρρρ
μ
ρ
μ
(B.26)
where
()2/
oi
ρρρ +=
Note the similarity between equations (B.23), (B.24) and (B.25), (B.26) respectively.
These equations are applicable to natural convection mass and heat transfer
respectively.
In the absence of winds, effective values of
avi
ρ in equations (B.23) and (B.24) and
i
ρ
in equation (B.25) and (B.26) respectively, change with time . )(
u
tt >
B. 7
Appendix B: Evaporation
During windy periods (forced convection) evaporation rates generally increase with
increasing wind speed. According to the Reynolds-Colburn analogy and the analogy
between mass and heat transfer, the following relations exist (Holman 1986)
w
Dw
f
wp
w
v
Sh
C
vc
Prh
3/23/2
c
2
==
ρ
or
)2(
32
ScvCh
wfDw
= (B.27)
In general the rate of mass transfer or evaporation from a horizontal wetted surface at a
uniform concentration c
vo
is thus
)(][
vivoDwDvo
cchhm −+=
(B.28)
If the vapour is generated uniformly at z = 0 find the rate of evaporation according to
equations (B.24), (B.27) and (B.28). For relatively small temperature differences, the
concentrations can be replaced by the partial vapour pressures i.e. where
and Furthermore, for air-water vapour mixtures
oivvv
TRpc/=
)/2(
iooi
TTT += K.kgJ52461 /.=
v
R
..60Sc ≈
()( ){}[ ]
oivivowfavpavomavivom
TppvCkcgm /)(735.1/1078.8
31
224
−+−×=
−
ρρρμ (B.29)
Since the thermal conductivity of water is not negligible, it is not possible to achieve a
truly uniform heat flux situation. The value of the dimensionless mass transfer
coefficient as given by equation (B.24) may thus be less than 0.243, i.e. it will be some
value between 0.243 and 0.155 as given by equation (B.23). Burger and Kröger (2004)
report the results of experiments conducted during analogous heat transfer tests between
a low thermal conductivity horizontal surface and the environment. They obtain a value
of 0.2106 instead of the theoretical value of 0.243 given in equation (B.26) and the
analogous equation (B.24). They furthermore obtain a value of C
f
= 0.0052 based on a
wind speed measured 1 m above the test surface. With these values equation (B.26)
applied over a wetted surface can be extended to become
B. 8
Appendix B: Evaporation
(){}[] ({})[ ]
31
2
31
2
/0026.02106.0/
avomaviavwavomaviavpq
gvkgch ρρμρρρρμ −+=− (B.30)
If the above values (0.2106 and C
f
= 0.0052) are substituted into equation (B.29) find
()( ){}[ ]
oivivowavpavomavivom
Tppvkcgm /)(0104.0/1061.7
31
224
−+−×=
−
ρρρμ (B.31)
When density differences are very small and conditions relatively stable or at night
when T
oq
< T
i
and the heat flux is uniform Burger and Kröger (2004) recommend
32
)/(
0022.087.3
kc
cv
h
p
pavw
q
μ
ρ
+= (B.32)
A mass transfer coefficient that is analogous to equation (B.32) is given by
32
3/23/2
0022.087.3
Sc
v
Sc
Pr
SccSc
Pr
c
h
h
w
pavpav
q
Dm
+
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
=
ρρ
(B.33)
The corresponding uniform mass transfer rate is
mv
vivo
p
w
p
vom
TR
pp
k
Dc
v
c
m
)(
Pr
0022.087.3
3
2
32
−
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ ⋅⋅
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⋅
=
ρ
ρ
(B.34)
This expression is applicable at night and during the day when the value for m
vom
is
found to be larger that that given by equation (B.32).
B.2 Analysis - Temperature gradient within water layer
According to Holman (1986) the intensity of solar radiation in clear water at a distance
z from the surface is given by
z
hz
eII
⋅−
=
16.0
6.0 (B.35)
B. 9
Appendix B: Evaporation
For a thin film of thickness z ≈ 0.0015 m the heat flux near the surface of the film is
thus approximately
hz
II
dz
dT
k 6.0≈≈
(B.36)
or the surface temperature can be described by
hmeasuredo
h
measuredooq
IT
k
zI
TT 0015.06.0 −≈−≈ (B.37)
B.3 Evaporation pan
A photograph of the evaporation pan employed in section 2.3 is shown below. Note the
thick polystyrene plate painted black and maintained as level as practically possible,
with the associated weather station is just out of picture on the right.
Figure B.3: Photograph of evaporation pan.
B. 10
Appendix B: Evaporation
B.4 Numerical example
The numerical example that follows makes use of experimental data between the times
of 12.419 and 12.585 solar time on April 14
th
2005. Take note that all numerical values
listed from this point onwards would have been averaged over a period of ten minutes.
All related experimental data and physical parameters are listed in Table B.1 below.
Table B.1: Evaporation pan experimental data on April 14
th
2005 at 12.494 h.
Water surface temperature T
w
31.12
o
C (304.27 K)
Ambient air temperature T
a
27.83
o
C (300.98 K)
Dew point temperature T
dp
14.70
o
C
Wind speed v
w
1.13 m.s
-1
Atmospheric pressure p
a
100990 Pa
Total incident solar radiation I
h
742.10 W.m
-2
Evaporation pan surface area A
0.97 m
2
Long-wave emissivity of water ε
w
0.9
Evaporation rate - Measured m
exp
0.000 218 kg.s
-1
Evaporation rate – Equation (2.4) m
en
0.000 231 kg.s
-1
Evaporation rate – Equation (2.7) m
vom
0.000 179 kg.s
-1
Evaporation rate – Equation (2.9) m
mon
0.000 239 kg.s
-1
Time interval - local 13h10 – 13h19
Average time - local Ψ 13.242
Average time - solar 12.494 h
Location latitude angle Ф
33.98
o
S
Location longitude angle Ф
l
18.85
o
E
South African standard meridian Ф
m
30.00
o
E
As described in section 2.3, 500 ml quantities of water were added in successive periods
to the evaporation pan, where the time taken to evaporate the water was recorded and in
turn converted into an evaporation rate. The period applicable to the current numerical
example is between 12.035 h and 12.668 h, 38 minutes in length. The density of the
water was considered constant for each period; with the temperature at 12.035 h a
B. 11
Appendix B: Evaporation
density of 994.6 kg.m
-3
was used. Therefore the measured evaporation rate for this
period of time could be calculated to be
1
exp
.218000.0
1000
6.994
6038
5.0
−
=⋅
⋅
= skgm (B.38)
The next evaporation rate to be evaluated is that predicted by the energy balance m
en
.
This equation however requires that the incidence angle be determined to evaluate the
absorptivity of the water surface with the aid of figure 2.4.
Utilising equations (A.3) to (A.7), the incidence angle at 12.494 h on the 14
th
of April
2005 can be determined.
)1292.0cos()593.0cos()16584.0cos()593.0sin()16584.0sin(cos −−+−=
z
θ
rad
z
7689.0=θ
o
053.44=
For an incidence angle of 44.053°, figure 2.4 gives an approximate absorptivity for
beam radiation of 0.9467, while the absorptivity for diffuse radiation is 0.9207. The
solar radiation absorbed by the horizontal water surface can now be determined under
the assumption that all radiation not reflected from the water surface is absorbed.
Diffuse solar radiation was measured to be approximately 8% of the total incident
radiation.
WI
hw
99.70008.010.7429207.092.010.7429467.0 =⋅⋅+⋅⋅=⋅α
The average temperature between the ambient air, T
a
and the saturated air just above the
water, T
w
is
K
TT
T
aw
m
63.302
2
)98.30027.304(
2
)(
=
+
=
+
=
B. 12
Appendix B: Evaporation
The following thermophysical properties of the air are evaluated at the mean
temperature, T
m
according to Kröger (1998).
Density, ρ 1.16243 kg.m
-3
Specific heat capacity, c
p
1007.05 J.kg
-1
.K
-1
Thermal conductivity, k 0.026424 W.m
-1
.K
-1
Dynamic viscosity, μ 1.85894 ·10
-5
kg.m
-1
.s
-1
From these values it is then possible to determine the Prandtl number
708449.0
026424.0
05.10071085894.1
Pr
5
=
⋅⋅
=
⋅
=
−
k
c
p
μ
Now the convective component from equation (2.4) can be evaluated.
)(
)(
)(
0026.02106.0
)(
3
1
22
3
1
aw
paw
m
aw
m
w
awa
TT
kcTTg
T
TTg
T
v
TTh −⋅
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
⎥
⎦
⎤
⎢
⎣
⎡
−
+
=−⋅
ρ
μ
μ
ρ
)98.30027.304(
16243.1026424.005.1007)98.30027.304(81.9
63.3021085894.1
)98.30027.304(81.91085894.1
63.30216243.1
13.10026.02106.0
3
1
22
5
3
1
5
−⋅
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
⋅⋅⋅−⋅
⋅⋅
⎥
⎦
⎤
⎢
⎣
⎡
−⋅⋅
⋅
⋅+
=
−
−
W323.26=
Similarly, the radiative component in equation (2.4) can be evaluated
( )
)98.300)7.14006.0
727.0(27.304(1067.59.0)006.0727.0(
4
48
44
⋅⋅+
−⋅⋅⋅=+−
−
adpww
TTTσε
W021.96=
B. 13
Appendix B: Evaporation
If the latent heat of vaporization of water at 304.27 K is determined as shown below
32
31904020014.0139568.12703.86754.1814833
sssfg
TTTi ⋅−⋅+⋅−=
32
27.30410140209043.027.304139568.1227.304703.58674.3483181 ⋅−⋅+⋅−=
(B.39)
1
.2426499
−
= kgJ
Now with all the terms necessary to complete equation (B.42), the predicted
evaporation rate m
en
can be determined.
( )[ ][ ]
fg
adpwsawhw
en
i
TTTTThI
m
44
0060.0727.0)( ⋅+−⋅−−⋅−⋅
=
σεα
2426499
021.96323.2699.700 −−
=
(B.40)
1
.238000.0
−
= skg
But this is applicable to an area of 1 m
2
, for an area of 0.97 m
2
the predicted
evaporation rate m
en
becomes 0.000 231 kg.s
-1
.
To determine the evaporation rate according to equation (2.7) it is first necessary to
calculate the following values.
The vapour pressure, p
vi
in the air can be determined with the aid of the following
equation.
)
85.287
1915.5406
(
11
)
1915.5406
(
11
10368745.210368745.2
−
−
⋅⋅=⋅⋅= eep
dp
T
vi
(B.41)
2
.63.1651
−
= mN
The humidity ratio, w
i
of the ambient air is given by
63.1651100990
63.1651622.0622.0
−
⋅
=
−
⋅
=
via
vi
i
pp
p
w (B.42)
B. 14
Appendix B: Evaporation
airdrykgOHkg /010342.0
2
=
With the aid of these two values, the density of the ambient air can be determined.
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⋅
⎥
⎦
⎤
⎢
⎣
⎡
+
−⋅+=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅
⎥
⎦
⎤
⎢
⎣
⎡
+
−⋅+=
98.30008.287
100990
)622.0010342.0(
010342.0
1)010342.01(
08.287)622.0(
1)1(
a
a
i
i
iavi
T
p
w
w
wρ
(B.43)
3
.161561.1
−
= mkg
The partial vapour density of the ambient air, ρ
vi
is determined below.
98.3005.461
63.1651
⋅
=
⋅
=
aw
vi
vi
TR
p
ρ (B.44)
3
.01189.0
−
= mkg
Similarly, these equations can be applied to the saturated air at the surface of the water.
The vapour pressure, p
vo
is
)
27.304
1915.5406
(
11
)
1915.5406
(
11
10368745.210368745.2
−
−
⋅⋅=⋅⋅= eep
w
T
vo
(B.45)
2
.045.4552
−
= mN
The humidity ratio of the saturated air at T
w
is
045.4552100990
045.4552622.0622.0
−
⋅
=
−
⋅
=
voa
vo
o
pp
p
w
airdrykgOHkg /029360.0
2
=
Then the density of the air at the surface of the water can be calculated as follows:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅
⎥
⎦
⎤
⎢
⎣
⎡
+
−⋅+=
w
a
o
o
oavo
T
p
w
w
w
08.287)622.0(
1)1(ρ
B. 15
Appendix B: Evaporation
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⋅
⎥
⎦
⎤
⎢
⎣
⎡
+
−⋅+=
27.30408.287
100990
)622.0029360.0(
029360.0
1)029360.01
3
.136439.1
−
= mkg
The partial vapour density of the saturated air, ρ
vo
can be found according to the
equation that follows:
27.3045.461
045.4552
⋅
=
⋅
=
ww
vo
vo
TR
p
ρ
3
.032417.0
−
= mkg
The average density of the moist air between the water surface and the ambient is
3
.149.1
2
)136439.1161561.1(
2
)(
−
=
+
=
+
= mkg
avoavi
av
ρρ
ρ
Equation (2.7) can now be evaluated
()( )
63.302
)63.1651045.4552(
)13.10052.0002.2))149.1026424.0/(05.1007
136439.1161561.11085894.181.9((106053.7
3
1
2
2
54
−
×⋅⋅+⋅×
−⋅⋅⋅⋅⋅=
−−
vom
m
(B.46)
1
.179000.0
−
= skg
Lastly, if the evaporation rate predicted by equation (2.9) is to be evaluated, it is first
necessary to determine the values that follow.
6135103
6135103
98.3001091332.598.3001046784.298.30031334.2103605.1
1091332.51046784.231334.2103605.1
⋅⋅+⋅⋅−⋅+⋅=
⋅⋅+⋅⋅−⋅+⋅=
−−
−−
aaapv
TTTc
11
..83.1886
−−
= KkgJ
83.1886010342.005.1007 ⋅+=
⋅+=
pvipapma
cwcc
B. 16
Appendix B: Evaporation
11
..50.1026
−−
= KkgJ
Now if the latent heat of vaporization at the wet-bulb temperature (T
wb
= 293.31 K) is to
be determined equation (B.39) can be used to find
32
31.29310140209043.031.293139568.1231.293703.58674.3483181 ⋅−⋅+⋅−=
fg
i
1
.4884522
−
= kgJ
then the adjusted psychrometric constant can be expressed as shown below
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⋅
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅⋅⋅
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅⋅
=⋅=
−
2452488622.0
10099050.1026
05.100716878.11026.0
026424.0
622.0
*
3
2
4
)(
3
2
3
2
wb
T
fg
atmpma
p
i
pc
cD
k
Le
ρ
γγ
(B.47)
433.61=
The slope of the saturated vapour pressure line Δ, can be calculated with the aid of the
following equation
v
satfg
RT
Tpi
⋅
⋅
=Δ
2
)(
(B.48)
where all properties are evaluated at the temperature T, which is the average between
the surface- T
s
and wet-bulb temperatures T
wb
.
K
TT
T
wbs
79.298
2
31.29327.304
2
=
+
=
+
=
With equation (B.39) the latent heat of vaporization at this temperature is
32
79.29810140209043.079.298139568.1279.298703.58674.3483181 ⋅−⋅+⋅−=
fg
i
1
.5064392
−
= kgJ
B. 17
Appendix B: Evaporation
The saturation vapour pressure p
sat
(T) at this temperature can be determined according
to equation (B.41)
)
79.298
1915.5406
(
11
)
1915.5406
(
11
10368745.210368745.2
−−
⋅⋅=⋅⋅= eep
T
sat
2
.54.3292
−
= mN
1
2
.95.194
5.46179.298
54.32922439506
−
=
⋅
⋅
=Δ KPa
Lastly the difference between the vapour pressure of the air at the ambient condition p
vi
and the saturated vapour pressure at that same dry-bulb temperature p
vsi
is given by
63.165184.3747 −=−
vivsi
pp
Pa21.2096=
Now it is possible for the evaporation rate according to equation (2.9) to be evaluated.
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⋅+⋅
=
2426499
97.0
43.6195.194
821.209699.70095.194
mon
m
1
.239000.0
−
= skg (B.49)
B. 18
APPENDIX C
Water tank experiments
C.1 Open water tank photograph and spindle drawing
C.2 Numerical example (section 3.2)
C.3 Solar collector with plastic-covered water tank: Photograph and analysis
C.4 Numerical example (section (3.3)
C.1 Open water tank
A photograph of the open water tank is shown below. The tank is in the centre of the
picture with the weather station to the left and the pyranometer to the right.
Figure C.1: Photograph of open water tank.
C.1
Appendix C: Water tank experiments
Observation of the above figure shows that the “thermocouple spindle” used to
determine the water temperature profile is located in the centre of the tank, with the
upper portion visible above the water level. It is shown in detail in the figure C.2.
Figure C.2: Thermocouple spindle used to determine water temperature profile.
C.2 Numerical example
Table C.1 supplies the measured experimental data and apparatus parameters applicable
to the water tank between 12.301 and 12.451 solar time on the 29
th
of April 2005. As
before, all numerical data listed has been averaged over a period of ten minutes.
Table C.1: Experimental data and physical parameters of water tank.
Water surface temperature T
w
23.62
o
C (296.77 K)
Ambient air temperature T
a
33.20
o
C (306.35 K)
C. 2
Appendix C: Water tank experiments
Dew-point temperature T
dp
11.66
o
C
Wind speed v
w
0.28 m.s
-1
Atmospheric pressure p
a
100830 Pa
Total incident solar radiation I
h
729.77 W.m
-2
Water tank surface area A
1 m
2
Depth of water tank t
w
0.190 m
Long-wave emissivity of water ε
w
0.9
Evaporation rate – Measured m
exp
0.000 096 kg.s
-1
Evaporation rate – Equation (3.2) m
en
0.000 093 kg.s
-1
Evaporation rate – Equation (2.8) m
vom
0.000 052 kg.s
-1
Evaporation rate – Equation (2.9) m
mon
0.000 231 kg.s
-1
Time interval - local 13h00 – 13h09
Average time - local Ψ 13.075 h
Average time - solar 12.376 h
Mean water temperature (12.209 h) T
wm
i
23.423
o
C (296.573 K)
Mean water temperature (12.376 h) T
wm
i-1
23.071
o
C (296.221 K)
Location latitude angle Ф
33.98
o
S
Location longitude angle Ф
l
18.85
o
E
South African standard meridian Ф
m
30.00
o
E
As in section 2.3, a quantity of water was added to the water tank and the period of time
taken for that particular quantity of water to evaporate was recorded and converted into
the measured evaporation rate m
exp
. In this case 750 ml of water was evaporated within
a period of 130 minutes at an average density of 997.4 kg.m
-3
. The measured
evaporation rate is then
1
exp
.096000.0
1000
4.997
60130
75.0
−
=⋅
⋅
= skgm (C.1)
The solar absorptivity of water is a function of the position of the sun in the sky and
thus it is important that the incidence angle be determined. The same procedure is
followed as in Appendix A and the result is that the incidence angle is found to be
48.823˚.
C. 3
Appendix C: Water tank experiments
This means that the solar altitude at this time of the day was 41.177°, which with the aid
of figure 2.4 gives a solar absorptivity for water under clear skies to be 0.9392.
WI
hw
333.68408.077.7299207.092.077.7299392.0 =⋅⋅+⋅⋅=⋅α
W333.684=
The average temperature between the ambient air, T
a
and the saturated air just above the
water, T
w
is
K
TT
T
aw
m
56.301
2
)35.30677.296(
2
)(
=
+
=
+
=
The following thermophysical properties of the air are evaluated at the mean
temperature T
m
according to Kröger (1998).
Density, ρ 1.16469 kg.m
-3
Specific heat capacity, c
p
1007.01 J.kg
-1
.K
-1
Thermal conductivity, k 0.026342 W.m
-1
.K
-1
Dynamic viscosity, μ 1.85404 ·10
-5
kg.m
-1
.s
-1
Then with the use of these three values, the Prandtl number can be determined.
708765.0
026342.0
01.10071085404.1
Pr
5
=
⋅⋅
=
⋅
=
−
k
c
p
μ
Now the convective component from equation (3.2) can be evaluated with the use of
equation (2.2)
()
()
()77.29635.306
026342.0/01.10071085404.1
01.100716469.128.0
0022.087.3
)/(
0022.087.3)(
3
2
5
32
−⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⋅⋅
⋅⋅
+=
−⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⋅
⋅⋅
+=−
−
wa
p
pw
waa
TT
kc
cv
TTh
μ
ρ
C. 4
Appendix C: Water tank experiments
W772.45=
Similarly, the radiative component in equation (3.3) can be evaluated as shown below.
( ) ( )
)35.306)66.11006.0727.0(77.296(1067.59.0
)006.0727.0(
448
4444
⋅⋅+−⋅⋅⋅=
+−=−
−
adpwwskyww
TTTTT σεσε
W628.37=
Evaluating the stored energy within the water tank requires that the following properties
be determined at the mean temperature of the water tank (296.573 K). The density of
water at this temperature according to Kröger (1998) is
1
6
20
2
963
)10090321.1
1009782.7107164.31049343.1(
−−
−−−
⋅⋅−
⋅⋅+⋅⋅−⋅=
wm
wmwmwm
T
TTρ
1620
2963
)573.29610090321.1
573.2961009782.7573.296107164.31049343.1(
−−
−−−
⋅⋅−
⋅⋅+⋅⋅−⋅=
3
.42.997
−
= mkg
The specific heat capacity of water at 296.573 K according to Kröger (1998) is
6
13
2
23
1017582.2
1011283.51080627.21015599.8
wm
wmwmpwm
T
TTc
⋅⋅−
⋅⋅+⋅⋅−⋅=
−
−
11
..32.4182
−−
= KkgJ
and the change in mean water temperature over a period of 10 minutes is
1
1
.000587.0
600
221.296573.296
1060
−
−
=
−
=
⋅
−
=
sK
TT
dt
dT
i
wm
i
wmwm
C. 5
Appendix C: Water tank experiments
The change in internal energy can then be evaluated as shown below
587000.032.4182190.042.997 ⋅⋅⋅=⋅⋅⋅
dt
dT
ct
wm
mpwwm
ρ
W054.465=
The evaporative heat flux leaving the water surface and is represented by
fgevev
imq ⋅=
If the latent heat of vaporization of water at 296.77 K is determined by equation (B.39)
to be 2 444 300 J.kg
-1
, then the evaporation rate as predicted by the energy balance m
en
is represented by
()[ ][ ]
fg
wm
pmwwmadpssawhw
en
i
dt
dT
ctTTTTThI
m
⋅⋅⋅−⋅+−⋅−−⋅−⋅
=
ρσεα
44
0060.0727.0)(
3004442
054.465628.37772.45333.684 −−+
=
(C.2)
1
.093000.0
−
= skg
If the evaporation rate according to equation (2.8) is to be evaluated
()00956.0021209.0
026342.0
00026.016469.101.1007
01.100716469.1
1
)026342.0/01.10071085404.1(
01.100716469.128.0
0022.087.3
3
2
3
2
5
−⋅
⎟
⎠
⎞
⎜
⎝
⎛ ⋅⋅
×
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⋅⋅
⋅⋅
⋅+=
−
vom
m
(C.3)
1
.052000.0
−
= skg
The evaporation rate predicted by equation (2.9) can be evaluated with the calculation
of the following values.
The specific heat capacity of the vapour is
C. 6
Appendix C: Water tank experiments
613
5103
98.3001091332.5
98.3001046784.298.30031334.2103605.1
⋅⋅+
⋅⋅−⋅+⋅=
−
−
pv
c
11
..10.1892
−−
= KkgJ
The vapour pressure of the air p
vi
can be determined with equation (B.41)
)
81.284
1915.5406
(
11
10368745.245.0
−
⋅⋅⋅= ep
vi
Pa61.1351=
Then with the use of the above result it is possible to find the humidity ratio ω
i
of the
air with equation (B.42).
61.1351100830
61.1351622.0
−
⋅
=
i
w
airdrykgOHkg /008451.0
2
=
The following relationship is used to calculate the specific heat capacity of the moist air
10.1892008451.001.1007 ⋅+=
pma
c
11
..19.1023
−−
= KkgJ
Now if the latent heat of vaporization at the wet-bulb temperature (T
wb
= 289.30 K) is to
be determined with equation (B.39).
32
30.28910140209043.030.289139568.1230.289703.58674.3483181 ⋅−⋅+⋅−=
fg
i
1
.9844612
−
= kgJ
then the adjusted psychrometric constant can be expressed as in equation (B.47)
325.62
2461984622.0
10083019.1023
01.100716469.11026.0
026342.0
*
3
2
4
=
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⋅
⋅
⎟
⎠
⎞
⎜
⎝
⎛
⋅⋅⋅
=
−
γ
C. 7
Appendix C: Water tank experiments
The slope of the saturated vapour pressure line Δ, can be calculated with the aid of the
equation (B.48), where T is defined by
K
TT
T
wbs
04.293
2
30.28977.296
2
=
+
=
+
=
With equation (B.39) the latent heat of vaporization at this temperature is
32
04.29310140209043.004.293139568.1204.293703.58674.3483181 ⋅−⋅+⋅−=
fg
i
1
.1454532
−
= kgJ
The saturation vapour pressure p
sat
(T) at this temperature can be determined according
to equation (B.41)
)
04.293
1915.5406
(
11
)
1915.5406
(
11
10368745.210368745.2
−−
⋅⋅=⋅⋅= eep
T
sat
2
.16.2322
−
= mN
1
2
.75.143
5.46104.293
16.23222453145
−
=
⋅
⋅
=Δ KPa
Lastly the difference between the vapour pressure of the air at the ambient condition p
vi
and the saturated vapour pressure at that same dry-bulb temperature p
vsi
is given by
Papp
vivsi
64.378261.135125.5134 =−=−
It is now possible for the evaporation rate according equation (2.9) to be evaluated.
1
.231000.0
2444300
1
33.6275.143
779.464.378233.68475.143
−
=
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⋅+⋅
=
skg
m
mon
(C.4)
C. 8
Appendix C: Water tank experiments
C.3 Solar collector with plastic-covered water tank: Photograph and
analysis
The solar collector is analysed with the intention of obtaining a set of equations that can
be solved simultaneously to accurately predict the measured glass cover- and mean
water temperatures.
As shown schematically in figure 3.7, the solar collector is simply comprised of a
volume of water covered by a plastic film that is exposed to solar radiation through a
glass cover. The purpose of the plastic film is to prevent the condensation of water on
the lower face of the glass lid and thus the effects of evaporation, condensation and the
influence of condensation on light transmission can be omitted in this analysis.
Applying an energy balance to the glass cover shown in figure 3.7 and find
dt
dT
ctqqIhIhIhIhqqIhIh
c
pcccracaddcbbcddcbbcrwcwdb
ρττρρ ++++++=+++ '''' (C.5)
The terms with b and d subscripts represent the beam and diffuse components as
discussed previously, while ρ’ and τ’ are the effective reflectivity and transmissivity of
the glass cover as defined in Appendix D. The terms q
cw
and q
rw
represent the
convective and radiative heat transfer respectively between the cover and the water
tank. Similarly, q
ca
and q
ra
are the convective and radiative heat transfer between the
cover and the environment.
It is important to note that the solar collector is well insulated and thus losses through
conduction are considered negligible, while highly reflective sides used in the solar
collector ensure very little radiation heat exchange between the glass cover and the
sides, or the water surface and the sides.
Equation (C.5) can be rewritten in the following format
C. 9
Appendix C: Water tank experiments
dt
dT
ctqqqqIhIh
c
pcccracarwcwddcdcbbcbc
ρτρτρ ++=++−−+−− )''1()''1( (C.6)
As mentioned in Appendix D, Mills (1992) states that ρ’+τ’+α’ = 1, which means that
equation (C.6) can be rewritten as
dt
dT
ctqqqqIhIh
c
pcccracarwcwddcbbc
ραα ++=+++ '' (C.7)
The beam and diffuse cover absorptivities can now be substituted into equation (C.7)
from equation (D.6). This results in the cover energy balance as given by equation (C.8)
racarwcwd
dcdc
dcdc
b
bcbc
bcbc
qqqqIhIh +=++⋅
−
−−
+⋅
−
−−
α
α
α
α
τρ
τρ
τρ
τρ
1
)1)(1(
1
)1)(1(
(C.8)
Note that the last term representing the change in stored energy of the glass cover
dtdTct
cpccc
ρ , has been omitted since it will be shown that the magnitude of this term
is negligible under all conditions. Also note the addition of the cover subscripts c in the
absorptivity terms from equation (D.6).
Now it remains for the convective and radiative terms associated with equation (C.8) to
be evaluated.
The convective losses to the environment q
ca
will be determined according to equation
(2.1) and (2.2) as given by Burger and Kröger (2004). Note that the surface
temperatures T
s
from equation (2.1) have simply been replaced by the cover
temperatures T
c
. This equation is only applicable when the cover temperature exceeds
that of the ambient air, otherwise equation (2.2) would have been used instead.
The radiative heat transfer to the environment q
ra
is also evaluated in same manner as in
Appendix B, with the use of either equation (2.5) or (2.6) as recommended by Berdahl
and Fromberg (1982).
C. 10
Appendix C: Water tank experiments
The appropriate convective heat transfer coefficient between the water surface and the
glass cover, when the water surface is at a greater temperature than the glass, is
supplied by Holland et al. (1975) as shown below
a
cwcwcw
cwcw
e
t
kGr
Gr
h
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
⎟
⎠
⎞
⎜
⎝
⎛
+
⎥
⎦
⎤
⎢
⎣
⎡
−+= 1
5830
Pr
Pr
1708
144.11
3
1
(C.9)
Note that the Prandtl and Grashof numbers are determined at the mean temperature
between the water and glass cover.
Lastly, if the water surface and the glass cover are considered as two infinite parallel
plates then the radiative heat exchange between the two can be calculated according to
Mills (1992).
)(1
11
44
1
cw
wc
rw
TTq −
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+=
−
εε
σ (C.10)
where the cover and water emissivities are given by ε
c
and ε
w
.
If the appropriate equations are substituted into equation (C.8), it is possible for the
energy balance on the cover to be fully expanded for steady-state conditions as shown
below
)(1
11
1
)1)(1(
1
)1)(1(
44
1
cw
fc
hd
dcdc
dcdc
hb
bcbc
bcbc
TTII −
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−++⋅
−
−−
+⋅
−
−−
−
εε
σ
τρ
τρ
τρ
τρ
α
α
α
α
)(1
5830
Pr
Pr
1708
144.11
3
1
cw
a
cwcwcw
cwcw
TT
t
kGr
Gr
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−⎟
⎠
⎞
⎜
⎝
⎛
+
⎥
⎦
⎤
⎢
⎣
⎡
−++ (C.11)
()
44
3
1
22
3
1
)006.0727.0()(
)(
)(
0026.02106.0
adpccac
pac
m
ac
m
w
TTTTT
kcTTg
T
TTg
T
v
+−+−⋅
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
⎥
⎦
⎤
⎢
⎣
⎡
−
+
= σε
ρ
μ
μ
ρ
C. 11
Appendix C: Water tank experiments
Note that the emissivity of the water ε
w
has simply been replaced with that of the plastic
film ε
f
. It is assumed that good contact exists between the plastic film and the water
surface, and since the film is very thin (t
f
≈ 0.0002 m) it is assumed to be at the same
temperature as that of the water.
Now if an energy balance is applied to the water tank as shown in figure 3.7 then the
following equation is produced
dt
dT
ctqqII
w
pwwwrwcwhddwdchbbwbc
ρατατ ++=+ )'()'( (C.12)
The transmittance-absorptance product can be determined from equation (D.16), but
since the transmittance of the water tank is zero then the sum of the effective
reflectivity and absorptivity must add up to unity.
Equation (D.16) then becomes
()
⎥
⎦
⎤
⎢
⎣
⎡
−
−
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅−
−
=
22
2
1
)1(
'1
)'1(
'
bcbc
bcbc
dcbf
bf
w
α
α
τρ
τρ
ρρ
ρ
ατ (C.13)
Then if equation (D.4) is substituted into equation (C.13) for the effective reflectivity,
which is in turn substituted into equation (C.12), then the energy balance on the water
tank becomes
hb
bf
bfwbaf
bf
bfwbafbaf
dc
bf
bfwbaf
bf
bfwbafbaf
bcbc
bcbc
I⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−+
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−+
−
⎥
⎦
⎤
⎢
⎣
⎡
−
−
−1
2
2
2
2
22
2
1
)21(
1
1
)21(
1
)1(
)1(
α
α
α
α
α
α
τρρ
τρρρ
ρ
τρρ
τρρρ
τρ
τρ
()
()
hd
df
dfwdaf
df
dfwdafdaf
dc
df
dfwdaf
df
dfwdafdaf
dcdc
dcdc
I
1
2
2
2
2
22
2
1
)21(
1
1
21
1
1
1
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−+
−⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−+
−
⎥
⎦
⎤
⎢
⎣
⎡
−
−
+
α
α
α
α
α
α
τρρ
τρρρ
ρ
τρρ
τρρρ
τρ
τρ
()
cw
a
cwcwcw
cwcw
TT
t
kGr
Gr
−⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
⎟
⎠
⎞
⎜
⎝
⎛
+
⎥
⎦
⎤
⎢
⎣
⎡
−+= 1
5830
Pr
Pr
1708
144.11
31
dt
dT
ctTT
w
pwwwcw
fc
ρ
εε
σ +−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−++
−
)(1
11
44
1
(C.14)
C. 12
Appendix C: Water tank experiments
Equation (C.14) presented above is the water tank’s steady-state energy equation and is
based on the assumption that good contact exists between the plastic film and the water
surface. However it was found in practice that while exposed to the sun, a small layer of
air developed between the plastic film and the water. This occurred because the
solubility of oxygen in water is decreased with an increase in temperature and thus the
cumulative result of the oxygen being released from solution would be the air gap that
formed.
The presence of this air gap changes the above analysis slightly, such that the effective
transmissivity of the plastic film needs to incorporate the air layer that has formed. The
result is equation (C.15) below.
1
2
2
2
2
22
2
1
)21(
1
1
)21(
1
)1(
)1(
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−+
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−+
−
⎥
⎦
⎤
⎢
⎣
⎡
−
−
bf
bfwbaf
bf
bfwbafbaf
dc
bf
bfwbaf
bf
bfwbafbaf
bcbc
bcbc
α
α
α
α
α
α
τρρ
τρρρ
ρ
τρρ
τρρρ
τρ
τρ
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−+
−
⎥
⎦
⎤
⎢
⎣
⎡
−
−
+−⋅
df
dfwdaf
df
dfwdafdaf
dcdc
dcdc
hbbw
I
α
α
α
α
τρρ
τρρρ
τρ
τρ
ρ
2
2
22
2
1
)21(
1
)1(
)1(
)1(
hddw
df
dfwdaf
df
dfwdafdaf
dc
I)1(
1
)21(
1
1
2
2
ρ
τρρ
τρρρ
ρ
α
α
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−+
−⋅
−
)(1
5830
Pr
Pr
1708
144.11
31
cw
a
cwcwcw
cwcw
TT
t
kGr
Gr
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−⎟
⎠
⎞
⎜
⎝
⎛
+
⎥
⎦
⎤
⎢
⎣
⎡
−+=
dt
dT
ctTT
w
pwwwcw
fc
ρ
εε
σ +−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−++
−
)(1
11
44
1
(C.15)
The change in plastic film transmissivity is employed by simply multiplying the solar
terms by a factor of (1 - ρ
w
).
Note that in the above equations the af and fw subscripts refer to the air-film and film-
water interfaces respectively.
Lastly it needs to stated that a number of assumptions were made regarding the
derivation of equations (C.11) and (C.15). It was mentioned previously that the thermal
C. 13
Appendix C: Water tank experiments
capacitance of the plastic film was considered negligible and that conductive losses
through the sides of the well-insulated water tank were also ignored, however it also
needs to be mentioned that the changes in thermal capacity of the enclosed air is
ignored. This assumption is based on the fact that under steady-state conditions, the
natural convective heat transfer between the water surface and the air is the same as that
between the enclosed air and the glass cover.
The other assumption made is that the water temperature predicted will be compared to
the measured mean water temperature. Stored energy is a function of mean water
temperature and convection and radiation a function of water surface temperature; it
will be shown at a later stage that the stored energy term in equation (C.15) has a much
larger influence on the predicted temperatures than the radiative or convective terms
and thus more accurate results are expected with this approach.
Figure C.3 shows a photograph of the solar collector with plastic-covered water tank
during daytime operation.
Figure C.3: Solar collector with plastic-covered water tank.
C. 14
Appendix C: Water tank experiments
C.4 Numerical example (section 3.3)
Table C.2 supplies the measured experimental data and apparatus parameters applicable
to the water tank between 11.500 and 11.650 solar time on the 12
th
of January 2005.
The predicted mean water temperature and glass cover temperature were found to be
329.3403 K and 310.8348 K respectively. As before, all numerical data listed has been
averaged over a period of ten minutes.
Table C.2: Experimental data and physical parameters of plastic covered water tank.
Predicted glass cover temperature T
c
37.6848
o
C (310.8348 K)
Predicted mean water temperature T
w
i
56.1903
o
C (329.3403 K)
Predicted previous mean water temperature T
w
i-1
55.5806
o
C (328.7306 K)
Ambient air temperature T
a
30.81
o
C (303.96 K)
Dew-point temperature T
dp
16.03
o
C
Wind speed v
w
2.09 m.s
-1
Atmospheric pressure p
a
100550 Pa
Total incident solar radiation I
h
1057.757 W.m
-2
Water tank surface area A
1 m
2
Air gap height t
a
0.100 m
Depth of water tank t
w
0.170 m
Long-wave emissivity of plastic film ε
f
0.88
Long-wave emissivity of glass cover ε
c
0.9
Time interval - local 11h23 – 11h32
Average time - local Ψ 12.458 h
Average time - solar 11.575 h
Measured glass cover temperature T
cm
42.5869
o
C (315.7369 K)
Measured mean water temperature T
wm
54.8409
o
C (327.9909 K)
Location latitude angle Ф
33.98
o
S
Location longitude angle Ф
l
18.85
o
E
South African standard meridian Ф
m
30.00
o
E
C. 15
Appendix C: Water tank experiments
Employing the method described in section 3.3.2, the measured diffuse radiation was
found to be roughly 6.5% of the total solar radiation, ie.
2
.754.68757.1057065.0065.0
−
=⋅== mWIhI
hd
This means that the rest of the total solar radiation was represented by the beam
component.
2
.003.989754.68757.1057
−
=−=−= mWIII
dhhbh
However, since the surface of the glass cover is 1m
2
the above radiation components
could simply be written as 68.754 W and 989.003 W respectively.
Determining the angle at which beam radiation strikes a surface is critical, particularly
when calculations regarding surface reflectivities are considered. This angle, measured
with respect to the normal of the surface is known as the incidence angle, which for a
horizontal surface is equivalent to the zenith angle. Appendix A contains all relevant
information regarding the directional characteristics of sunlight. The angle of incidence
in this particular case is found to be
°= 605.13
z
θ
It is now possible the numerical value of the incidence angle to be used in conjunction
with equation (D.9) from Appendix D to determine the appropriate interface
reflectivities.
The beam reflectivity for the air-glass interface is
⎥
⎦
⎤
⎢
⎣
⎡
+
−
+
+
−
=
−
−
−
−
)526.1605.13sin(sin605.13(sin
)526.1605.13sin(sin605.13(sin
))526.1605.13sin(sin605.13(tan
))526.1605.13sin(sin605.13(tan
2
1
12
12
12
12
bc
ρ
043418.0=
while the beam reflectivity of the air-plastic film interface can be found to be
C. 16
Appendix C: Water tank experiments
⎥
⎦
⎤
⎢
⎣
⎡
+
−
+
+
−
=
−
−
−
−
)6.1605.13sin(sin605.13(sin
)6.1605.13sin(sin605.13(sin
))6.1605.13sin(sin605.13(tan
))6.1605.13sin(sin605.13(tan
2
1
12
12
12
12
baf
ρ
053316.0=
The beam reflectivity for the air-water interface is determined below for evaluating the
influence of the air layer beneath the plastic film.
⎥
⎦
⎤
⎢
⎣
⎡
+
−
+
+
−
=
−
−
−
−
))333.1/605.13(sinsin605.13(sin
))333.1/605.13(sinsin605.13(sin
))333.1605.13sin(sin605.13(tan
))333.1605.13sin(sin605.13(tan
2
1
12
12
12
12
bw
ρ
020410.0=
Similarly, the reflectivity of the film-water interface can be calculated with the aid of
equation (D.11) and the incidence angle.
⎥
⎦
⎤
⎢
⎣
⎡
+
−
=
−−
−−
))333.1605.13sin(sin)6.1605.13sin((sintan
))333.1605.13sin(sin)6.1605.13sin((sintan
2
1
112
112
bfw
ρ
⎥
⎦
⎤
⎢
⎣
⎡
+
−
+
−−
−−
))333.1605.13sin(sin)6.1605.13sin((sinsin
))333.1605.13sin(sin)6.1605.13sin((sinsin
2
1
112
112
008293.0=
Now to determine the diffuse reflectivities, the same equations are used with an
effective incidence angle of 60˚.
⎥
⎦
⎤
⎢
⎣
⎡
+
−
+
+
−
=
−
−
−
−
)526.160sin(sin60(sin
)526.160sin(sin60(sin
))526.160sin(sin60(tan
))526.160sin(sin60(tan
2
1
12
12
12
12
dc
ρ
093468.0=
⎥
⎦
⎤
⎢
⎣
⎡
+
−
+
+
−
=
−
−
−
−
)6.160sin(sin60(sin
)6.160sin(sin60(sin
))6.160sin(sin60(tan
))6.160sin(sin60(tan
2
1
12
12
12
12
daf
ρ
105240.0=
⎥
⎦
⎤
⎢
⎣
⎡
+
−
+
+
−
=
−
−
−
−
))333.160sin(sin60(sin
))333.1/60(sinsin60(sin
))333.160sin(sin60(tan
))333.160sin(sin60(tan
2
1
12
12
12
12
dw
ρ
C. 17
Appendix C: Water tank experiments
059690.0=
⎥
⎦
⎤
⎢
⎣
⎡
+
−
=
−−
−−
))333.160sin(sin)6.160sin((sintan
))333.160sin(sin)6.160sin((sintan
2
1
112
112
dfw
ρ
⎥
⎦
⎤
⎢
⎣
⎡
+
−
+
−−
−−
))333.160sin(sin)6.160sin((sinsin
))333.160sin(sin)6.160sin((sinsin
2
1
112
112
010742.0=
The transmissivity due to absorptance of the cover and the plastic film, due to beam
radiation can be calculated as follows according to equation (D.13).
948732.0
))526.1/605.13(sincos(sin/004.013))/sin(cos(sin/
11
===
−−
⋅−−
ee
cbacec
nntC
bc
θ
α
τ
960367.0
))6.1/252.21(sincos(sin/0002.0200
))/sin(cos(sin/
1
1
===
−
−
⋅−
−
ee
fbafef
nntC
bf
θ
α
τ
As in the case of the diffuse reflectivities, the transmissivity due to absorptance for
diffuse solar radiation is calculated in the same manner as the beam transmissivities, but
with an incidence angle of 60˚.
938798.0
))526.1/60(sincos(sin/004.013))/60sin(cos(sin/
11
===
−−
⋅−−
ee
cacec
nntC
dcα
τ
953543.0
))6.1/60(sincos(sin/0002.0200
))/60sin(cos(sin/
1
1
===
−
−
⋅−
−
ee
fafef
nntC
df
θ
α
τ
The extinction coefficients used in the above calculations were determined by
Lombaard (2002) and were confirmed by the author with experimental tests.
To evaluate the convective heat transfer from the glass cover to the environment, the
thermophysical properties of the air at a mean cover / ambient temperature are required.
K
TT
T
ac
am
3974.307
2
)96.3038348.310(
2
)(
=
+
=
+
=
Density, ρ 1.139407 kg.m
-3
Specific heat capacity, c
p
1007.243 J.kg
-1
.K
-1
C. 18
Appendix C: Water tank experiments
Thermal conductivity, k 0.026792 W.m
-1
.K
-1
Dynamic viscosity, μ 1.880767 ·10
-5
kg.m
-1
.s
-1
Similarly it is necessary that the same thermophysical properties be determined to
calculate the natural convective heat transfer between the water surface and the glass
plate. These properties are determined at the mean temperature between the two
surfaces as shown below.
K
TT
T
wc
cm
0876.320
2
)3403.3298348.310(
2
)(
=
+
=
+
=
Density, ρ 1.094234 kg.m
-3
Specific heat capacity, c
p
1007.858 J.kg
-1
.K
-1
Thermal conductivity, k 0.027762 W.m
-1
.K
-1
Dynamic viscosity, μ 1.938320 ·10
-5
kg.m
-1
.s
-1
Then with the use of these three values, the Prandtl number can be determined.
703667.0
027762.0
858.100710938320.1
Pr
5
=
⋅⋅
=
⋅
=
−
k
c
p
μ
The natural convective heat transfer within the enclosed air space requires that the
Grashof number be calculated. According to Mills (1992), the effective length used in
the calculation of the Grashof number in the case of a thin air enclosure can be taken as
the distance between the two opposing surfaces.
2
23
)(
)(2
cwcw
cwacw
cw
TT
tgTT
Gr
μ
ρ
+
−
= (C.16)
057.1807468
)10938320.1()8348.3103403.329(
094234.11.081.9)8348.3103403.329(2
25
23
=
⋅⋅+
⋅⋅⋅−⋅
=
−
The density and specific heat capacity of the water at the mean temperature, T
w
are
provided below, as determined according to Kröger (1998).
C. 19
Appendix C: Water tank experiments
Water density
3
.174.985
−
= mkg
w
ρ
Water specific heat capacity
11
..798.4181
−−
= KkgJc
pw
It is now possible to substitute all the necessary values into the energy balance on the
glass cover given by equation (3.3).
)(1
88.0
1
9.0
1
1067.5
754.68
938798.0093468.01
)938798.01)(093468.01(
003.989
948732.0043418.01
)948732.01)(043418.01(
44
1
8
cw
TT −
⎟
⎠
⎞
⎜
⎝
⎛
−+⋅+
⋅
⋅−
−−
+⋅
⋅−
−−
−
−
1.0
027762.0
1
5830
703667.0057.1807468
703667.0057.1807468
1708
144.11
3
1
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
⎟
⎠
⎞
⎜
⎝
⎛
⋅
+
⎥
⎦
⎤
⎢
⎣
⎡
⋅
−++
()
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
⋅⋅−⋅
⋅⋅
⎥
⎦
⎤
⎢
⎣
⎡
−⋅⋅⋅
⋅
⋅+
=−
−
−
3
1
22
5
3
1
5
139407.1026792.0243.1007)96.3038348.310(81.9
3974.30710880767.1
)96.3038348.310(81.910880767.1
3974.307139407.1
09.20026.02106.0
cw
TT
() ( )
448
96.303)03.16006.0727.0(1067.59.096.303 ⋅⋅+−⋅⋅+−⋅
−
cc
TT
This equation can then be further simplified to
)96.303(4627.12)(0675.2)(105452.41814.45884.50
448
−⋅=−⋅+−⋅++
−
ccwcw
TTTTT
( )
448
5277.289101030.5 −⋅+
−
c
T (C.17)
Similarly, equation (3.4) representing the energy balance on the water tank can be
completed as shown below.
C. 20
Appendix C: Water tank experiments
⎥
⎦
⎤
⎢
⎣
⎡
⋅⋅−
⋅⋅⋅−+
−
⎥
⎦
⎤
⎢
⎣
⎡
⋅−
⋅−
2
2
22
2
960367.0008293.0053316.01
960367.0008293.0)053316.021(053316.0
1
)948732.0043418.01(
948732.0)043418.01(
003.989)020410.01(
938798.0008293.0053316.01
938798.0008293.0)053316.021(053316.0
093468.01
1
2
2
⋅−⋅
⎥
⎦
⎤
⎢
⎣
⎡
⋅⋅−
⋅⋅⋅−+
−⋅
−
⎥
⎦
⎤
⎢
⎣
⎡
⋅⋅−
⋅⋅⋅−+
−
⎥
⎦
⎤
⎢
⎣
⎡
⋅−
⋅−
+
2
2
22
2
953543.0010742.0105240.01
953543.0010742.0)105240.021(105240.0
1
)938798.0093468.01(
938798.0)093468.01(
754.68)059690.01(
953543.0010742.01
953543.0010742.0)105240.021(105240.0
093468.01
1
2
2
⋅−
⎥
⎦
⎤
⎢
⎣
⎡
⋅−
⋅⋅−+
⋅−⋅
−
)(
1.0
027762.0
1
5830
703667.0057.1807468
703667.0057.1807468
1708
144.11
31
cw
TT −
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
⎟
⎠
⎞
⎜
⎝
⎛ ⋅
+
⎥
⎦
⎤
⎢
⎣
⎡
⋅
−+=
1060
)7306.328(
798.4181170.0174.985)(1
88.0
1
9.0
1
1067.5
44
1
8
⋅
−
⋅⋅+−
⎟
⎠
⎞
⎜
⎝
⎛
−+⋅+
−
− w
cw
T
TT
Which can be simplified into equation (C.18).
)(105452.4)(0675.29190.478616.812
448
cwcw
TTTT −⋅⋅+−⋅=+
−
(C.18)
)7306.328(2765.1167 −⋅+
w
T
Solving equations (C.17) and (C.18) simultaneously produces predicted glass cover-
and mean water temperatures of 310.8348 K and 329.3403 K. Unfortunately these
temperatures are found to differ by quite a substantial margin from the measured glass
and water temperatures of 315.7369 K and 327.9909 K.
C. 21
APPENDIX D
Water tank solar characteristics
D.1 Cover solar characteristics
D.2 Absorber plate effective absorptivity
If the water tank is to be accurately modelled, it is necessary that the behaviour of the
solar radiation be fully understood. The components into which the solar radiation that
strikes the solar collector is divided will be explained in detail in the section that
follows.
D.1 Cover solar characteristics.
The literature provides two alternate sets of equations for determining the solar
characteristics of a non-opaque cover. Duffie and Beckman (1991) provide equations
that are only applicable to a cover that has the same media at the upper and lower
interfaces, while Modest (1993) provides similar equations without the afore-mentioned
limitation.
Mills (1992) states that that the sum of the effective reflectivity, transmissivity and
absorptivity of a transparent medium is equal to one. Summing the two sets of
equations given in the literature (Duffie and Beckman (1991) and Modest (1993)), it was
found that those provided by the latter added up to one. Thus, the equations provided by
Modest (1993) will be used to determine the solar characteristics of the glass cover.
Figure D.1 shows the behaviour of solar radiation that strikes a cover of thickness t
c
.
Note that ρ represents the interface reflectivity between the cover and the air, and τ
α
the
transmittance due to absorptance. The symbols n
1
, n
2
and n
3
represent the medium at the
upper interface, the material from which the cover is constructed and the medium at the
lower interface. Equations (D.1), (D.2) and (D.3) are recommended by Modest (1993)
D.1
Appendix D: Water tank solar characteristics
for determining the effective reflectivity ρ’, effective transmissivity τ’ and effective
absorptivity α’ of a transparent cover.
()
2
2312
22
1223
12
1
1
'
α
α
τρρ
τρρ
ρρ
⋅⋅−
⋅−⋅
+= (D.1)
()()
2
2312
2312
1
11
'
α
α
τρρ
τρρ
τ
⋅⋅−
⋅−⋅−
= (D.2)
()( )()
2
2312
2312
1
111
'
α
αα
τρρ
ττρρ
α
⋅⋅−
−⋅⋅+⋅−
= (D.3)
Figure D.1: Solar radiation striking a cover of thickness t
c
and being either
transmitted, absorbed or reflected.
However, if the media above and below the cover is the same then these equations
simplify to those given below in equations (D.4), (D.5) and (D.6).
()
⎥
⎦
⎤
⎢
⎣
⎡
⋅−
⋅−
+=
22
22
1
1
1'
α
α
τρ
τρ
ρρ (D.4)
()
2
2
2
1
1
'
α
α
τρ
τρ
τ
⋅−
⋅−
= (D.5)
D. 2
Appendix D: Water tank solar characteristics
()()
α
α
τρ
τρ
α
⋅−
−⋅−
=
1
11
' (D.6)
Now if these values are to be determined it is first necessary for the reflectivities and
transmissivity due to absorption to be determined. Both are functions of incidence angle
θ
z
and the refraction indices.
Modest (1993) recommends the use of Fresnel’s equation to determine the interface
reflectivities as given by equation (D.7).
()
()
()
()
⎥
⎦
⎤
⎢
⎣
⎡
+
−
+
+
−
=
21
2
21
2
21
2
21
2
12
sin
sin
tan
tan
2
1
θθ
θθ
θθ
θθ
ρ (D.7)
The incidence is given by θ
1
in this case, while θ
2
represents the refractive angle of the
interface.
Observation of equation (D.7) indicates that if the refractive angle is determined, then
the reflectivity can be calculated, since the angle of incidence has already been defined
in Appendix A.
With the aid of Snell’s law and the incidence angle and refractive indices, the angle of
refraction can be determined.
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
2
11
2
sin
arcsin
n
n θ
θ (D.8)
Equation (D.8) can be substituted into (D.7) with the result below
()()( )( )
()
⎥
⎦
⎤
⎢
⎣
⎡
+
−
+
+
−
=
2111
2
2111
2
2111
2
2111
2
12
/sinarcsinsin
/sinarcsinsin
/sinarcsintan
/sinarcsintan
2
1
nn
nn
nn
nn
θθ
θθ
θθ
θθ
ρ (D.9)
Similarly at the lower surface interface, the refractive angle θ
3
is
D. 3
Appendix D: Water tank solar characteristics
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
3
11
3
22
2
sin
arcsin
sin
arcsin
n
n
n
n θθ
θ (D.10)
which results in a lower interface reflectivity of
()( )
⎥
⎦
⎤
⎢
⎣
⎡
+
−
=
311211
2
311211
2
12
/sinarcsin/sinarcsintan
/sinarcsin/sinarcsintan
2
1
nnnn
nnnn
θθ
θθ
ρ
()( )
⎥
⎦
⎤
⎢
⎣
⎡
+
−
+
311211
2
311211
2
/sinarcsin/sinarcsinsin
/sinarcsin/sinarcsinsin
2
1
nnnn
nnnn
θθ
θθ
(D.11)
Now if equations (D.4), (D.5) and (D.6) are to be evaluated it remains for the
transmissivity due to absorptance to be determined. This can be found according to
Bouguer’s law for a partially transparent medium with an extinction coefficient C
e
and
thickness t
c
.
)cos(/
2
θ
α
τ
ce
tC
e
⋅−
= (D.12)
This can be rewritten with the use of Snell’s law as in equation (D.8) to give the
following result.
))/sin(cos(arcsin/
211
nntC
ce
e
θ
α
τ
⋅−
= (D.13)
It has been shown in Appendix A how to determine the incidence angle, then with the
use of the equations listed above it is possible to calculate the effective reflectivity,
transmissivity and absorptivity of any non-opaque cover of thickness t
c
.
D.2 Absorber plate effective solar absorptivity.
Since the solar properties of a solar collector cover have been discussed in section D.2,
it only remains for the solar absorptivity of the absorber plate to be evaluated.
D. 4
Appendix D: Water tank solar characteristics
Figure D.2 shown below presents the incident solar radiation penetrating the cover and
being absorbed and reflected numerous times from both the absorber plate and solar
collector cover.
If the assumption is made that the reflected energy is diffuse and if multiple reflections
are taken into consideration, then according to Duffie and Beckman (1991) the
transmittance-absorptance product for the absorber plate shown in figure D.2 is
()
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅−−
=
dp
pp
ρα
τ
αατ
)1(1
'
' (D.14)
However, if the definition of the effective transmissivity as given by Modest in equation
(D.5) is substituted into equation (D.14) then the transmittance-absorptance product can
be rewritten as
()
⎥
⎦
⎤
⎢
⎣
⎡
−
−
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅−−
=
22
2
1
)1(
)1(1
'
α
α
τρ
τρ
ρα
α
ατ
dp
p
p
(D.15)
Note that α
p
is the absorptance of the absorber plate and that is ρ
d
the diffuse reflectivity
of the underside of the cover.
Figure D.2: Effective transmittance-absorptance product.
D. 5
APPENDIX E
Greenhouse sunlight properties
E.1 Area of transmitted sunlight
In Chapter 4 it is stated that for the total solar input to a panel of an all-glass structure
to be calculated, it becomes necessary to determine what portion of the sunlight
indirectly incident on a particular window has been transmitted through which
window in direct sunlight. Consider figure E.1.
Figure E.1: Relative orientation of glass greenhouse.
The above statement is explained in the following manner: All four walls, the roof and
floor receive sunlight in the above structure. However, if the sun is shining in the line
of sight then the eastern- and northern walls and roof will receive direct sunlight,
while the floor, southern- and western walls receive light that has been transmitted
E.1
Appendix E: Greenhouse sunlight properties
through the other surfaces. The total sunlight incident on a surface in indirect sunlight
can be evaluated by equation (E.1). It means that the total effective transmissivity is
the sum of the transmissivity of each window multiplied by the associated area, the
sum is then divided by the total area of the window in indirect sunlight.
tot
n
nn
A
A
∑
⋅
=
1
'
τ
τ
(E.1)
E.1 Area of transmitted sunlight.
Condition 1:
Figure E.2: Sketch of condition 1.
Light passes through the eastern wall and strikes the floor and south wall of the
greenhouse. If the incident light on the eastern wall is broken up into horizontal and
vertical components, as shown in the sketch, then case 1 is applicable between the
following limits.
• tan
-1
(width/length) < θ
HORIZ
< 90˚
E. 2
Appendix E: Greenhouse sunlight properties
• 0˚ < θ
VERT
< tan
-1
((height/width) tan θ
HORIZ
) < θ
HORIZ
< 90˚
While the appropriate areas are:
A1 = (height width) / tan θ
HORIZ
– 0.5 width
2
tan θ
VERT
/ (tan θ
HORIZ
)
2
A2 = 0.5 width
2
/ tan θ
HORIZ
Condition 2:
Light strikes the eastern wall as in condition 1, however different limits are now
applicable.
• 0˚ < θ
HORIZ
< tan
-1
((width/height) tan θ
VERT
)
• tan
-1
(height/length) < θ
VERT
< 90˚
The appropriate areas are:
A1 = 0.5 height
2
/ tan θ
VERT
A2 = (height width) / tan θ
VERT
– 0.5 height
2
tan θ
HORIZ
/ (tan θ
VERT
)
2
Figure E.3: Sketch of condition 2.
E. 3
Appendix E: Greenhouse sunlight properties
Condition 3:
Light passing through the eastern window now strikes the floor, south window and
western window. This condition occurs under the following limitations.
• 0˚ < θ
HORIZ
< tan
-1
(width/length)
• 0˚ < θ
VERT
< tan
-1
(height/length)
The appropriate areas are:
A1 = length (height - 0.5 length tan θ
VERT
)
A2 = length (width - 0.5 length tan θ
HORIZ
)
A3 = (width - length tan θ
HORIZ
) (height - length tan θ
VERT
)
Figure E.4: Sketch of condition 3.
E. 4