CHEMICAL ENGINEERING TRANSACTIONS
VOL. 76, 2019
A publication of
The Italian Association
of Chemical Engineering
Online at www.aidic.it/cet
Guest Editors: Petar S. Varbanov, Timothy G. Walmsley, Jiří J. Klemeš, Panos Seferlis
Copyright © 2019, AIDIC Servizi S.r.l.
ISBN 978-88-95608-73-0; ISSN 2283-9216
Multi-Objective Optimisation of Flat Plate Solar Collector-
Networks
Juan R. Lizárraga-Morazán, Guillermo Martínez-Rodríguez, Amanda L. Fuentes-
Silva, Martín Picón-Núñez*
Department of Chemical Engineering, University of Guanajuato, Noria Alta s/n, Guanajuato, Gto., Mexico
picon@ugto.mx
The use of solar heat for industrial applications is of growing interest in the academic and governmental areas.
The development of thermal integration techniques, design approaches for minimum cost of solar collector fields
and its operation are fundamental to motivate the interest of the process industries. About the design of a solar
field, the investment and pay-back period are fundamental for decision making, therefore, cost minimisation is
of paramount importance. The present work deals with the minimisation of the total cost of a network of solar
collectors using a multi-objective optimisation approach. The optimisation approach uses a new cost equation
for flat plate solar collectors and a validated thermal model. The optimisation model is solved in a Mat-Lab
GAMS platform. It is found that tube diameter, the tube length and collector width, are the variables that most
impact the total cost, and in the case of solar networks, the number of collectors in series to achieve a fixed
temperature. With respect to commercial geometries, the cost of the optimised designs is 41 % cheaper.
1. Introduction
The transition to the large-scale use of solar energy into processing plants is still an outstanding issue to attend.
Among the challenges to overcome are the costs associated with the integration of the thermo-solar plants into
the productive processes and the payback period. Karagiorgias et al. (2001) reported that the cost of the network
of solar collectors represents 54 % of the total installation costs. The system consisted of the network of solar
collectors, a heat storage system, heat exchangers and the piping. Atkins et al. (2010) stated that one of the
major challenges in the integration of solar energy in industrial processes are related to the intermittent nature
of the solar radiation and the high capital cost associated. In a low temperature solar collector network, the total
number of collectors are determined by the number of collectors in series and number of collectors in parallel.
The network structure depends on the heat load, the delivery temperature, the pressure drop, and the operating
and environmental conditions. Picón-Núñez et al. (2016) found that the structure of a solar field is a combination
of series-parallel arrays, where the number of lines in parallel are determined as a function of the mass flow rate
to meet the required heat load and proposed a methodology to size the collector surface area and determined
that inlet temperature is a design variable that largely impacts the outlet temperature. To date, only a few
publications focus on the minimisation of the number of collectors in a network. In their work, Martínez-
Rodríguez et al. (2018) quantified how the change on the operating conditions can reduce the size of the
network. According to the International Renewable Energy Agency (IRENA), the payback time of a thermo-solar
installation must be lower than 3 years for the investment to be reasonable.
Walmsley et al. (2018) carried out an analysis of various energy indices that measure the profitability and
sustainability of renewable energy electricity plants. Out of the five indices studied, the most relevant are the
Energy Return on Investment (EROI) and the Energy Payback Time (EPT). The major contribution of this paper
is the introduction of the time-worth for the calculation of energy. An analogy is established with the time-worth
of money and provides the means to understand the temporary characteristic of the investment on and the
generation of energy.
Even though the installation of thermo-solar plants has increased in the world, its number is still not significant.
In this regard, the quest for the reduction of the solar-collector-network size is a subject that still needs to be
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DOI: 10.3303/CET1976161
Paper Received: 18/03/2019; Revised: 05/05/2019; Accepted: 07/05/2019
Please cite this article as: Lizarraga-Morazan J.R., Martinez-Rodriguez G., Fuentes-Silva A.L., Picon-Nunez M., 2019, Multi-Objective
Optimisation of Flat Plate Solar Collector-Networks, Chemical Engineering Transactions, 76, 961-966 DOI:10.3303/CET1976161
further developed. IRENA reported that in 2014, the number of industrial thermo-solar plants in the world came
up to 140. Most thermo-solar plants in Europe are installed to supply district heating and Denmark takes the
lead in this regard (Bava et al., 2015). In a solar plant located in Taars, Denmark, Tian et al. (2018) designed
and simulated the performance of a network with 5,960 m2 of flat plate collectors and 4,039 m2 of parabolic
concentrators. They concluded that the cost of district heating can be reduced from 5 % to 9 % using renewable
energy.
The reported works in the open literature on the optimisation of the variables associated to solar collector
networks is scarce. In this regard, Hajabdollahi and Hajabdollahi (2017) developed a thermo-economic model
of a flat-plate collector which was later optimised. This optimisation searched for the maximum thermal efficiency
and the minimum annual cost. The main limitation of a thermo-economic optimisation is however, that they do
not consider the actual size of the components of the system.
The present work introduces a multi-objective NLP optimisation methodology to determine the size of a network
of flat plate solar collectors (FPSCN) arranged in series. The approach uses a new cost equation to minimise
the total cost and the maximisation of the thermal efficiency based on temperature increment, which is the
difference between the fluid inlet and outlet temperatures. The total cost equation involves the surface area, the
pumping systems and the operating costs. The system was solved using GAMS (General Algebraic Modeling
System) and the solver CONOPT (Continuous Nonlinear Optimisation). This work focuses on the design and
optimisation of the collector field and does not consider the storage system for the operation of the plant at this
stage.
2. Sizing solar collector networks
The methodology for the design of a network of solar collectors is taken from Martínez-Rodríguez et al. (2019).
The approach determines the network with the minimum area that meets the heat load and the target
temperature throughout the year. The application of the methodology to two case studies are reproduced in
Table 1. In case study 1 (dairy products), is possible to supply the total process heat duty at a temperature of
95 °C (Quijera et al., 2011); while in case study 2, the solar network supplies a small fraction of the total load
(0.067) at 99 °C (Oseguera-Villaseñor, 2016).
Table 1: Solar network design and operating data for two case studies (Martínez-Rodríguez et al., 2019).
Case study 1 Case study 2
Production process Dairy products: yogurt, cheese
and milk drinks
Bioetanol: fuel
Plant feed 20. 6 t/day 22.66 t/day
Plant throughput 20.5 t/day 4.96 t/day
Boiler heat load 4,401.01 kWh 97,914.4 kWh
Boiler operating time (360 day/year)
5 h/day
(350 day/year)
24 h/day
Process hot temperature ºC 95 °C 204 °C
Heat load from solar network 4,401.01 kWh 5,662.00 kWh
Solar fraction, f 1 0.067
Solar target temperature 95 °C 99 °C
Solar network structure (series-parallel) 28x34 29x74
3. Optimisation model
The thermal model reported in Martínez-Rodríguez et al. (2019) was implemented in the Mat-Lab platform, and
then exported to the GAMS environment. The optimisation of the flat plate solar collector (FPSC) is based on
the minimisation of the total cost as a function of the minimum temperature increment between the inlet and
outlet collector temperatures. The temperature increment of a commercial collector whose total costs are known
is taken as a reference. Figure 1 shows the optimised designs for different values of temperature increment
fixing the inlet temperature to 60 °C. For a temperature rise of 1.2 °C, the total costs are 936 USD for the
commercial collector and 632 USD for the optimized design. This indicates a cost reduction of 32.48 %.
The multi-objective optimisation methodology based on restrictions was used to solve the model. A sensitivity
analysis was performed to identify the design variables that have a largest impact upon the size of the network.
From the results, these variables are: the tube length, tube diameter, collector width, number of collectors in
series and the specified network target temperature. These variables define the network of solar collectors that
minimises the total cost for a fixed target temperature.
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The Mat-Lab model provides the thermal data for a specific network geometry and the GAMS model optimises
the geometry required to meet the heat load. The Mat-Lab and GAMS models can operate independently.
Therefore, for a specific case, the GAMS model optimises a single collector or a network of collectors by means
of a multi-objective function.
Figure 1: Pareto front of optimal configurations of a FPSC as a function of temperature increment (T).
To carry out the multi-objective optimisation, a new cost model for plat plate collectors was developed. The
model was validated by comparing it against the models published by Herrera-Alcázar and Andrade-Vallejo
(2010) and Bava et al. (2015). These models differ in the way they are structured, however the results they
produce are very similar between each other. The proposed multi-objective optimisation model is as follows:
𝑚𝑖𝑛 𝑍 = [𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡𝑠, 𝑁𝑒𝑡𝑤𝑜𝑟𝑘 𝑎𝑟𝑒𝑎] s.t. ℎ(𝑥) = 0; 𝑔(𝑥) ≤ 0 (1)
Where ℎ(�̅�) is given by the mathematical expressions that make up the model, and 𝑔(�̅�) represents the group
of restrictions of the system:
𝑇0 ≤ 𝑇𝑝𝑚; 𝑇𝑐𝑜𝑣𝑒𝑟 ≤ 𝑇𝑝𝑚 (2)
The expression for the total cost is:
𝑇𝑜𝑡𝑎𝑙 𝐶𝑜𝑠𝑡 = 𝛾0 +
𝐴𝐶 𝑁𝐶
𝜋
(𝛾1𝑑 + 𝛾2 +
𝛾3
𝑑
) + 𝑊𝐿𝛾4 + 𝛾5 (
�̇�𝐻𝑏
𝑒𝑓𝑓
) + 𝛾10
�̇�𝐿𝜇
𝜋𝜌𝑑4
(3)
Where 𝛾𝑖 are adjustment parameters, 𝑁𝑐 is the number of collectors in series, �̇� is the mass flow rate, 𝐻𝑏 is the
pump head, 𝑑 and L are the diameter and length of the tubes, w is the collector width, 𝑒𝑓𝑓 is the efficiency of
the pump, 𝜌 and 𝜇 are the density and viscosity of water. Eq(3) includes the costs of manufacture, installation
and accessories as well as the operating costs (due to pumping).
The expression to determine the network surface area is represented by:
C
LwNareaNetwork = (4)
The nonlinear optimisation problem was solved using the GAMS platform; Eq(1) was discretised using a fifth
order Runge-Kutta method and the CONOPT solver. Table 2 shows the geometry of a commercial flat plate
collector and the range within which some variables are to be optimised.
Table 2: Commercial FPSC geometry.
Features Geometry Optimisation range
No. Tubes 8 8
Length, m 1.97 0.5≥ L ≤ 3
Width, m 0.9 0.5≥ w ≤ 3
Diameter, m 0.051 0.008≥ di ≤ 0.051
963
4. Results
The operating conditions of the network of solar collectors for the two case studies are shown in Table 3. Figure
2 shows the variation of total cost against the surface area for the optimised designs. The design to achieve the
target temperature using commercial collectors for each case study is also shown for the purposes of
comparison.
Table 3: Operating data.
Operating conditions Case study 1 Case study 2
Mass flow rate per collector line, kg/s 4.5 4.5
Inlet temperature, °C 60 60
Target temperature (outlet temperature), °C 95 99
Irradiance, W/m2 725.32 761.13
Ambient temperature, °C 19.73 20.14
Wind velocity, m/s 1.6 1.0
Figure 2: Pareto front of optimal configurations of FPSC Networks. Comparison between optimised networks
and networks designed using commercial geometries for: (a) Case study 1, and (b) Case study 2.
From Figure 2 it can be observed that the total cost grows rapidly as the surface area reaches the minimum
value that can thermodynamically achieve the specified target temperature. As the surface area is reduced, the
number of collectors in series required is increased thus affecting the total cost. For instance, for case 1, the
minimum surface area that reaches 95 °C is 46.5 m2. This design condition requires 186 collectors (0.5 m long
and 0.5 m wide) in series. The results for the optimised solution indicate that the target temperature can be
achieved with a surface area of 50 m2 that requires only 8 collectors in series (3 m long and 3 m wide).
Comparatively, if commercial collectors are used, 49.6 m2 are needed. The cost of the design using commercial
units is 24,570 USD while cost of the optimised design is 12,192 USD (Figure 2a). For case study 2 (Figure 2b),
the minimum possible thermodynamic design requires a surface area of 47.5 m2 with a total of 190 collectors
(0.5 m long and 0.5 m wide) in series. The optimised design achieves the target temperature with 50.3 m2 and
a total cost of 14,96.76 USD while the design using commercial collectors requires a surface area of 51.4 m2
with a total cost of 25,445 USD.
Figure 2 provides more information to aid the designer in making the right decision based on the objective. For
instance, if the objective is to go for the minimum cost, there are other options that with an increment in surface
area will meet the objective. The different optimal point in the Pareto front, indicates different collector
geometries and different number of collectors in series. In the case land for the installation of the solar plant is
limited, then the choice must go in the direction of reduced surface area.
The designer may also be interested in evaluating the effect upon surface area of the inlet temperature to the
network. Further analysis is shown in Figure 3 using the information of case study 2. Figure 3a shows the Pareto
front for an inlet temperature of 45 °C and Figure 3b for 60 °C. The network design using the commercial
collectors is included in both diagrams for comparison. Taking as a reference the surface area from commercial
collectors, it can be observed that reduction of the inlet temperature from 60 °C to 45 °C results in an increment
of approximately 9 m2 of surface area in the case of the optimised designs. This represents a cost increment of
almost 20 %.
(a) (b)
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Figure 3: Effect of inlet temperature upon the network surface area for case study 2: (a) Inlet temperature 45 °C,
(b) inlet temperature 60 °C.
For a feed temperature of 60 °C, Table 4 shows the detailed results of the optimisation for each of the case
studies and the geometries of the commercial and optimised designs. It can be observed that for the same
surface area between the optimised network and the commercial one, the optimised network fulfils the target
temperature with a 50 % reduced cost for case 1, and a cost reduction of 41 % for case 2. The substantial
reduction in the total costs justifies the optimisation and more importantly, the determination of the optimal
geometry of the collectors.
Table 4: Comparison between commercial and optimised network designs.
Payback time is another important parameter to consider in any thermo-solar installation. The analysis in this
work did not include maintenance costs. Table 5 shows the payback time for the different case studies. What is
important to highlight is that the main reason that hinders the widespread use of solar energy in industrial
processes apart from the intermittent nature of the solar resource is the high cost associated. The former can
be dealt with by means of proper design and energy storage (Walmsley et al., 2015) while the second, as it is
shown in this work, can be reduced substantially by optimising the collector geometry. Since it is estimated that
the solar filed takes almost 54 % (Karagiorgias et al., 2001) of all the associated costs, and since it is
demonstrated that the costs can be reduced up to 50 %, then the total cost of the installation, can in principle,
be reduce approximately 25 %. This cost reduction can make solar plants more competitive.
Table 5: Optimised costs and payback time.
(a) (b)
Case 1 Case 2
FPSC Network Features Commercial Optimised Commercial Optimised
Length, m 1.97 2.80 1.97 2.61
Width, m 0.90 0.81 0.90 0.67
Tube diameter, m 0.051 0.01 0.051 0.008
Number of collectors in series, Nc 28 22 29 29
Surface area of series arrangement, m2 49.6 50.0 51.4 50.3
Installed costs, USD 24,357.73 12,008.38 25,227.65 14,609.97
Operating costs, USD 60.58 108.82 60.58 205.01
Total costs, USD 24,418.31 12,117.20 25,288.25 14,814.98
Savings 50 % 41 %
Case study 1 Case study 2
Commercial Optimised Commercial Optimised
Network arrangement, series-parallel 28x34 22x34 29x74 29x74
Total number of collectors 952 748 2,146 2,146
Total surface area, m2 1,687.89 1,696.46 3,804.85 3,752.71
Total cost of network, USD 830,223 411,985 1,871,329 1,096,309
Operating days/year 350 350 350 350
Payback time, years 1.5 0.75 2.63 1.54
965
5. Conclusions
This work introduces an optimisation approach for flat plate solar collector plants. It uses a new cost equation
and focuses only on the solar field. The main conclusions of this work are:
• The geometrical variables that most affect the cost of a solar collector are: length, inner tube diameter and
collector width.
• In the case of solar collector networks, the variable to optimise is the number of collectors in series for a
fixed target temperature.
• For a single solar collector, the cost reduces as the surface area decreases. However, for a network of
collectors in series, cost does not necessarily reduce as surface area reduces. The reason for this being
that for smaller surface areas, smaller collectors are needed but in increased number which increases the
cost.
• The multi-objective optimisation approach using Mat-Lab GAMS is an effective tool for solar collector
networks.
• The payback time keeps a relation to the solar fraction (fraction of the process heat load that is substituted
by solar energy). Case study 1 exhibits a solar fraction of 1 and a payback period of 9 months, and case
study 2, with a solar fraction of 0.067, has a payback time of 19 and a half months. In all cases, the payback
time is below the recommended by the International Renewable Energy Agency.
• Further analysis to consider the storage system into the overall optimisation approach is underway.
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