Simplified Solar Collector Model: Hourly Simulation of Solar Boundary Condition for Multi‐energy Optimization

Simplified Solar Collector Model: Hourly Simulation of Solar 
Boundary Condition for Multi‐energy Optimization 
Grahovac, M.1, Liedl, P.2, Frisch, J.3, Tzscheutschler, P.4
International Graduate School of Science and Engineering (IGSSE), Project Team 2.08
Technische Universität München, 80290 München, Germany,
email: mgrahovac@tum.de
Abstract
The tool we are developing provides guidelines concerning building envelope and
heating/cooling energy sources in order to increase energy efficiency of buildings by
including the subject of energy consumption in the very early stage of building design.
Building size and climate can be varied. To calculate the right mix of energy sources and
conversion equipment that minimizes costs, energy or CO2 emissions numerous parameters
concerning various space heating or cooling technologies need to be assessed.
To perform multi-energy plant optimization we need to determine the upper boundary of
thermal energy gained by solar collectors. Based on weather data and maximal available solar
collector area, solar collector gains for different outlet temperatures can be simulated. For
design purposes collector tilt angle has been optimized. The developed collector model has a
very short calculation time and can easily be integrated into plant system model and its
optimization. Due to its hourly resolution the simulated profile can be used for thermal
storage optimization. We compared this simplified model with a more detailed one developed
in Trnsys and obtained satisfying results.
Keywords: solar thermal collector, model, tilt angle, optimization, solar gains
1. Introduction 
The goal of our project is to create an easy-to-use tool for architects, whose application
insures implementation of energy efficient solutions into design process from the earliest
stages of commercial buildings design. The in-development tool provides
recommendations for the construction of envelope and energy supply system, taking the
climate and location into account. This means that the heating and cooling energy generation
system – the plant has to be optimized with respect to investment and running costs,
utilisation of renewables and/or emissions.
The major difficulty for optimizing the space heating and cooling energy source (hereafter the
plant) is the fact that during conceptual building design there is little information about the
distribution system, flow rates and temperatures. On the other hand, if the architect has the
rough idea about which energy source and conversion devices would be appropriate for
certain climate and building size and shape, he or she can bare it in mind during further
design. Lots of systems impose restrictions on building design and can later not be introduced.

1 Lehrstuhl für Energiewirtschaft und Anwendungstechnik, mgrahovac@tum.de
2 Lehrstuhl für Bauklimatik und Haustechnik, petra.liedl@lrz.tum.de
3 Lehrstuhl für Computation in Engineering, frisch@bv.tum.de
4 Project Team Leader, Lehrstuhl für Energiewirtschaft und Anwendungstechnik, ptzscheu@tum.de
1

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The way to overcome this lack of information is to simulate the plant using its thermal
capacities. Nowadays it is rather easy to simulate the approximate building heating and/or
cooling loads in hourly resolution. The plant needs to cover these loads and the losses by the
transport and distribution systems, which can be estimated. The energy necessary is observed
in hourly steps to enable taking the part load ratio, hourly weather data and thermal storage
into consideration. Due to high temporal resolution we can also provide information on the
number of hours per year when the plant can hardly satisfy the load, which helps not to over
dimension the system components.
In this paper we present a simplified thermal collector model and its verification compared to
more detailed Trnsys model. This model simulates the yearly solar gains profile based on
weather and radiation data for the location, available collector area, collector type and
performance parameters, and average collector temperature. It has been tailored to serve as
solar collector capacity estimation for energy mix optimization.
Since we are implementing this model in plant simulation and optimization during the early
design phase, the model accuracy tolerances are quite broad. For this reason some
assumptions were introduced in order to accelerate the simulation. The simplified model has
been compared to the equivalent model in Trnsys (Transient system simulation software),
yielding satisfyingly similar solar gain profiles. To perform plant optimization, advantages
such as calculation speed and simplicity of utilisation are essential. The results of this
comparison are presented in further chapters. There we compare results for three cities,
Belgrade, Munich and Shanghai, which is one of our project’s objective locations.
2. Solar thermal collector performance 
There are two kinds of solar thermal collectors available on the market, flat plate and
evacuated tube. The major difference in their performance is the fact that evacuated tube
collectors have smaller thermal losses and thus can achieve higher outlet temperatures.
Nevertheless, the ratio of absorber to gross area is higher for flat plate collectors. The prices
of evacuated tube collectors are significantly higher. Thus the choice of optimal collector
depends on its utilisation purpose and climate conditions.
Solar collector models are usually based on quadratic efficiency model originating from
theoretical equations developed by Duffie and Beckmann [1]. Collector efficiency can be
formulated as:
a T
Δ
a
T 2
Δ
(1)
1
2
η =η −
−

0
I
I
Where η represents optical collector efficiency (conversion factor), a and a are linear and
0
1
2
quadratic loss coefficients, T
Δ is temperature difference between collector fluid and
ambience, and I is solar radiation on collector surface.
American ASHRAE Standard 93-77 [2] and European EN 12975 [3] prescribe the test
procedures for obtaining collector parameters. They have a slightly different definition of
temperature difference – ASHRAE uses outlet and EN average collector fluid temperature. It
is important to notice to which area collector parameters have been referenced to: gross,
aperture or absorber.
Additional important parameter for collector performance simulation is IAM (Incidence
Angle Modifier), which corrects the measured optical efficiency in cases when solar radiation
is not perpendicular to collector surface. More details can be found in [1].
2

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A large amount of collector performance data can be found in [4]. We averaged the
parameters for several collectors and used these values in both simulations. The data are
presented in Table 1.
1

Flat Plate Collector
0.9
Evacuated Tube Collector
0.8
0.7
y
c
n
i
e 0.6
c
ffi 0.5
r
E
to
c 0.4
l
l
e
o
C 0.3
0.2
0.1
0
0
10
20
30
40
50
60
70
80
90
100
Temperature difference (T
- T
), °C
ave
amb

Figure 1. Collector efficiency drops with increased temperature difference between collector fluid and
surroundings. Evacuated tube collectors are well isolated and perform better than flat plate. Data from
Table 1 used.
Table 1. Test rated collector properties (With respect to the absorber area), [4]

Test Flow/
Optical
Loss coef. a1
Loss coef. a2
IAM at 50°
Absorber/Gross Gross area,
Eff., η0
W/(m2K)
W/(m2K2)
incidence
area, %
kg/(m2 h)
Flat
plate
79.46
4.0363 0.0078 0.8522 89.82
71.36
Evacuated
tube
78.79
2.0933 0.0072 1.1417 59.65
50.69
3. Solar radiation on tilted surface 
According to eq. (1) the thermal loss is occasionally greater than the product of optical
efficiency and current solar radiation. Obviously we consider only the positive energy gains
useful. To calculate these gains we need hourly solar radiation intensities on tilted surface.
The review of methods to calculate the radiation on tilted surface is given in [5].
Complete weather data for lots of locations are provided by global meteorological database
for engineers, planners and education – Meteonorm. The global, diffuse and beam radiation
on horizontal and ambience temperature are the data necessary to compare the models.
In this work we used Trnsys radiation processor, see [9], to obtain the radiation on tilted
surface and introduced this data into our simplified model.
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3.1. Tilt angle 
Solar collector performance is largely influenced by its correct positioning. The collectors
should face south (azimuth angle is 0°) if located on northern hemisphere and vice versa.
There are also recommendations about collector tilt – the angle between collector surface and
horizontal. The basic one is that collector tilt should be equal to latitude for good
performance. To reach the seasonal maximum, the tilt should be increased by 15° (max +20°)
during the winter and decreased by 10° during the summer, since the zenith angle changes
during the year.
Since our goal is to maximize the energy output, and not the power, as is the case with PV, we
can optimize the tilt with the objective to maximize the integral of hourly solar gains. This
will maximize the annual solar fraction of the plant. This angle is not necessarily equal to the
one which maximizes the solar radiation at the top of the collector. [6]
The advantage in planning beforehand is the fact we can optimize building design to
accommodate the optimal collector tilt.
3.2. Tilt angle optimization 
We performed optimizations of collectors tilt angles on yearly and seasonal bases to
maximize the annual solar energy gain. Simulations were performed in Trnsys, the
optimizations in GenOpt, using Hooke-Jeeves optimization algorithm. This optimization
algorithm starts with a choice of a base point. This is followed by exploratory moves,
changing the values of one variable at a time until the new base point has been found.
Exploratory moves are performed again, followed by finding the next base point. This is
repeated until convergence. Moghadam et al. [7] offer a Matlab based tool to optimize tilt,
witch could increase the optimization speed.
Table 2 shows results of tilt angle optimizations for Shanghai, Belgrade and Munich. It is
interesting that winter and summer optima for flat plate collectors agree with the
recommendations mentioned at the beginning of the chapter 3.1. As expected, due to the
geometry of evacuated tube collectors, seasonal tilt deviations from latitude are not as high.
Table 2. Collector tilt angles: latitude, yearly and seasonal optima
Flat
Plate Evacuated
Tube
Latitude
Yearly
Seasonal Opt, °
Yearly
Seasonal Opt, °

°N
Opt, °
Winter Summer
Opt, °
Winter Summer
Shanghai 31 26 46
11.5 26
43
13
Belgrade
45
36
60
24.5
35
56
22.5
Munich 48 39 63
28 37 60
25
4. Simplified solar collector performance model 
The goal was to create a collector model which can simulate solar collector hourly energy
gains. We demanded from the model to be quick, simple and flexible in terms of further
incorporation with other energy converters in the plant simulation. The designer (the user) has
to enter the solar radiation data for the location and specify the area available for solar
collectors. At the moment the model assumes this area to be the collector gross area and
corrects the area used within the calculation if collector parameters refer to absorber area.
User can specify the type (flat plate or evacuated tube) and purpose of collectors, such as floor
or radiation heating, absorption cooling etc. This defines the collector fluid output
temperature levels. The purpose of the model is plant system optimization, so collector area
4

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and output temperature can become optimization variables. If not provided by the user,
averaged collector parameters are used.
The collector efficiency is modelled as quadratic function of temperature difference:
a T
(
− T ) a T
(
− T
2
)
(2)
η = η ⋅ IAM
1
m
amb
2
m
amb
−
−

0
I
I
where T is average collector fluid temperature and the rest of symbols have been defined in
m
eq. (1).
According to previous equation, optical collector gains are:
(3)
Q = η ⋅ IAM ⋅ I ⋅ A
o
0
and thermal losses due to temperature difference are:
Q =
(
−
) −
(
−
2
) ⋅

tl
(a T T
a T
T
1
m
amb
2
m
amb
)
(4)
A
A is the user provided gross collector area. Useful solar gains are the positive difference
between optical gains and thermal losses:
⎧Q − Q Q

,
> Q
o
tl
o
tl
(5)
Q =

u
⎨
⎩ Q
,
0
≤ Q
o
tl
We assume constant average fluid temperature inside the collector, as well as constant
incidence angle modifier, IAM.
To verify the performance of the model, it has been compared to more detailed model
simulated in Trnsys. There we assumed constant collector inlet temperature and disregarded
the performance changes that are a result of using flow rate other than the test flow used to
obtain collector parameters, see Figure 2. Since we are using the models for planning and
design purposes, the impacts of these parameters can be disregarded.
2
Shanghai, Flat Plate collector, Tilt = Latitude
Shanghai, Evacuated Tube collector, Tilt = Latitude
1.5
on, %
i
ati
1
v
e
0.5
r Gain D
a
Sol
0
Annual -0.5
-1
40
50
60
70
80
90
100
Collector Flow, kg/(m2h)

Figure 2. Collector flow rates other than test flow rate and their influence on annual solar gain. The
deviation can be neglected.
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5. Comparison Results 
Results of both Trnsys and simplified model one year (8760 hours) simulation for all three
locations are presented in following chapters.
5.1. Annual solar gains 
Integral of solar gains over one year yields the annual gain. Annual gains per square meter flat
plate collector are presented in Figure 3. Both simulation models yield similar results.
650
623
Collector tilt: Latitude
Trnsys
Simplified 617
a
model
model
Collector tilt: Yearly optimum
2
600
Collector tilt: Seasonal optimum
589
588
578
577
kWh/m
s,
n
550
Gai
r
Trnsys
Simplified 529
o
522
model
model
ll
ect
501
o
498
500
489 491
r C
Trnsys
Simplified
482
la
476
o
model
model
453
451
ual S
446
450
443
Ann
400
Shanghai
Belgrade
Munich

Figure 3. Flat plate collector annual gains comparison for three locations and three collector tilt
configurations. Trnsys and simplified model yield similar result. Referred to gross collector area.
650
634
Trnsys
628
Simplified
Collector tilt: Latitude
model
model
607
609
Collector tilt: Yearly optimum
a
2
598
599
600
Collector tilt: Seasonal optimum
h/m
W
,
k
Trnsys
Simplified 559
550
ins
model
model
550
Ga
537
531 533
534
r
t
o
Trnsys
Simplified 521
517
c
l
l
e
model
model
o
498
498
500
490
490
r C
a
Sol
nual 450
An
400
Shanghai
Belgrade
Munich

Figure 4. Evacuated tube collector annual gains comparison for three locations and three collector tilt
configurations. Trnsys and simplified model yield similar result. Referred to gross collector area.
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Differences in annual gain obtained in case the collector tilt equals the latitude and ones with
the yearly optimized tilt are not significant. In case we change the collector tilt two times per
year the annual gain does increase. The decision on the suitable tilt angle has to be made
based on the utilisation. If we utilise solar collectors for solar cooling, it is reasonable to use
the tilt angle similar to summer optimum, see chapter 3.2.
The same comparison is given in Figure 4 for evacuated tube collectors. Annual gains are
somewhat higher than those obtained with the flat plate collector. Although the thermal losses
of evacuated tubes collector are lower, the absorber to gross area ratio for these collectors is
smaller, see Table 1. The data refer to 1 m2 gross collector area.
5.2. Monthly solar gains 
Figure 5 shows flat plate collector monthly solar gains calculated by the simplified model. In
this case collector tilt is equal to the latitude. Due to its mild winters and the smallest latitude
compared to other cities, Shanghai has the largest gains during winter months, but is being
overrun by Belgrade during the summer. Ambience temperatures during the summer are even
higher than those in Belgrade, but the high air humidity disturbs the beam radiation and thus
reduces the solar gain.
100
Shanghai
Belgrade
2
83
m
Munich
78
h/
80
W
k
68
s,
65
65
65
64
61
60
57
r Gain
53
51
t
o
49
c
48
47
45
46
45
l
l
e
44
42
42
o
38
40
36
r C
36
35
l
a
o
30
29
28
S
25
23
24
hly
21
20
15
15
14
14
Mont
10
0
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec

Figure 5. Flat plate collector monthly solar gains simulated for all three locations using the simplified
model. Latitude tilt.
5.3. Hourly solar gains 
Hourly solar gain profile for Shanghai using flat plate collector with the latitude tilt is shown
in Figure 6. Solar gains reach its peak during the late summer, after the air has dried out
during the warm summer months.
Such hourly profiles are a part of plant simulation, where they are, together with other energy
converter and storage component profiles, optimized to satisfy the hourly building loads.
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700
50
600
40
2 m

°
C
500
,
W/
30
r
e,
s
u
in
r
at
a 400
e
p
r
G
20
m
la
e
300
o
T
S
10
ent
r
ly 200
u
bi
m
Ho
A
0
100
0
-10
Jan
Feb
Mar
Apr
Mai
Jun
Jul
Aug
Sep
Oct
Nov
Dec

Figure 6. Hourly flat plate collector solar gain and ambience temperature profile for Shanghai. Latitude
tilt.
5.4. Average Collector Fluid Temperature 
800

Flat Plate, Simplified
Evacuated Tube, Simplified
700
Flat Plate, Trnsys
2
/
m
h 600
W
,
k
s
in
a 500
r
G
la
o
l
S
a 400
u
n
n
A
300
200
30
35
40
45
50
55
60
65
70
75
80
Average Collector Fluid Temperature, °C

Figure 7. Annual solar gains decrease with the increase of average collector fluid temperature.
Thermal losses have a huge influence on collector efficiency. The average fluid temperature
varies depending on the application of the collector generated heat. For floor heating
utilisation temperatures of 40°C are high enough, but if we want to run an absorption chiller,
collector outlet temperature should be at least two times higher. How this influences the
annual solar gains is illustrated in Figure 7. In the case of flat plate collectors, the increase of
fluid temperature by 10°C decreases the gains by over 100 kWh/m2. The dependence is less
emphasized for evacuated tube collectors, but it is still significant. For this reason are the
evacuated tubes collectors more suitable for colder climates.
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6. Conclusion and Outlook 
There is a good agreement between the results obtained with simplified and Trnsys model.
Trnsys model is more detailed, which causes slower performance. Detailed model is to be
implemented in case of detailed system simulation, where all design parameters are already
known. Trnsys model calculates the radiation on tilted surface, whereas the simplified model
uses these as input. We can overcome this by using weather data provided radiation on
horizontal and calculating radiation on tilted by the suitable method [5], [8].
Our simplified model neglects average collector temperature variations and uses constant test
flow for its calculations. Incidence angle modifier also remains constant. Despite these
approximations the model accuracy is satisfying for early design purposes. Quick calculation
yields an hourly solar gain profile. This profile, together with the available collector area
provided by the user, is then used as solar capacity upper boundary within a multi-energy
space heating and/or cooling system. Collector area and inlet temperature (if not set to be
fixed) are optimization variables within the system. Hourly resolution contributes to better
thermal storage simulation.

References:
1. Duffie, J.A. and Beckman, W.A., Solar Engineering of Thermal Processes, John Wiley &
Sons, Inc., New York (1991). ISBN 0-471-51056-4.
2. ASHRAE Standard 93-77, 1977.
3. DIN EN 12975 Thermische Solaranlagen und ihre Bauteile – Kollektoren – Teil 2:
Prüfverfahren; Deutsche Fassung EN 12975-2:20066. Kapitel 1.4.8.4.
4. http://www.solarenergy.ch/
5. Evseev, E.G., und Kudish, A.I., The assessment of different models to predict the global
solar radiation on a surface tilted to the south, Solar Energy 83, Nr. 3, 377-388.
6. Shariah, A., Al-Akhras, M., und Al-Omari, I. A., Optimizing the tilt angle of solar
collectors, Renewable Energy 26, no. 4 (August 2002): 587-598.
7. Moghadam, H., Tabrizi, F.F., und Sharak, A.Z., Optimization of solar flat collector
inclination, Desalination In Press, Corrected Proof (o. J.).
8. Hartley, L. E. u. a., The optimisation of the angle of inclination of a solar collector to
maximise the incident solar radiation, Renewable Energy 17, Nr. 3, 1999: 291-309.
9. TRNSYS Program Manual. Solar Energy Laboratory, University of Wisconsin, Madison,
USA; 1996.
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