# Parametric Methodology to Optimize the Sizing of Solar Collector Fields in Series-Parallel Arrays

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model

_{mu}is a constant mass flow rate of water with constant temperature. The heat exchanger stands for the operation with different fluids and pressures, as in applications for space heating or other applications where the climate conditions reach freezing temperatures.

#### 2.1. Storage Tank

_{pc}, C

_{pL}and C

_{ps}, are constant in all ranges of operation temperature.

#### 2.2. Solar Collector Field

#### Solar Radiation

#### 2.3. Techno-Economic Analysis

## 3. Methodology

- Determine the optimal number of collectors from a thermal analysis (all collectors connected in parallel), where installation cost is approximated as a percentage of the total investment cost equivalent to the collectors and the thermal storage tank.
- Analyze different serial–parallel arrays keeping the number of collectors determined in the previous step, with the objective of optimizing the payback time without affecting the solar collection area. Iterations are performed varying the number of collectors in series and their corresponding number of rows. In each iteration (corresponding to a specific array) the following steps are carried out:
- The installation costs are calculated according to the different pipe diameters and the pumping cost corresponding to the required power of the pump.
- The solar fraction and payback time are calculated and compared with the previous iteration.
- The optimal array is established when the payback time is less than the value of the previous iteration.

## 4. Results and Discussions

#### 4.1. Model Validation

#### 4.2. TRNSYS Comparison

^{2}. That area corresponds to 160 collectors connected in parallel and results in a payback time of 7.43 years and a solar fraction of 0.772.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${A}_{a}$ | Aperture area [m^{2}] |

${A}_{c}$ | Gross area [m^{2}] |

${A}_{cost}$ | Auxiliary cost [$] |

${A}_{r}$ | Receiver area [m^{2}] |

${A}_{s}$ | Storage tank area [m^{2}] Storage tank area [m^{2}] |

${b}_{0}$ | Constant for calculation of incidence angle modifier [-] |

${b}_{1}$ | Constant for calculation of incidence angle modifier [-] |

${b}_{2}$ | Constant for calculation of incidence angle modifier [-] |

$B$ | Previous unpaid balance [$] |

${B}^{+}$ | Current unpaid balance [$] |

${C}_{pc}$ | Specific heat of collector fluid [J/kg K] |

${C}_{pL}$ | Specific heat of load fluid [J/kg K] |

${C}_{ps}$ | Specific heat of storage tank fluid [J/kg K] |

${D}_{rate}$ | Discount rate [$] |

${E}_{r}$ | Relative error [-] |

${\overline{E}}_{r}$ | Mean relative error [-] |

$f$ | Solar fraction [-] |

${F}_{R1}$ | Collector removal factor [-] |

${F}_{R\epsilon}$ | Collector removal factor with heat exchanger [-] |

${F}_{sav}$ | Fuel savings [$] |

${G}_{T}$ | Irradiance on a tilted surface [W/m^{2}] |

${I}_{b}$ | Beam irradiation on a tilted surface [W/m^{2}] |

${I}_{d}$ | Diffuse irradiation on a tilted surface [W/m^{2}] |

${I}_{T}$ | Total irradiation on a tilted surface [W/m^{2}] |

$K$ | Constant for calculation of useful heat of N identical collectors connected in series [-] |

${K}_{\theta}$ | Incidence angle modifier [-] |

${L}_{loc}$ | Longitude of the location [°] |

${M}_{cost}$ | Maintenance cost [$] |

${\dot{m}}_{c}$ | Collector flow mass [kg/s] |

${\dot{m}}_{L}$ | Load flow mass [kg/s] |

${\dot{m}}_{s}$ | Storage tank flow mass [kg/s] |

${M}_{s}$ | Fluid mass in storage tank [kg] |

$N$ | Number of collectors in series [-] |

NTU | Number of transfer units [-] |

${P}_{cost}$ | Pumping power cost [$] |

$PT$ | Payback time [years] |

$P{T}_{ent}$ | Payback time previous integer values [years] |

$P{T}_{ent}{}^{+}$ | Payback time current integer values [years] |

$P{T}_{frac}$ | Payback time current fractional values [years] |

${Q}_{u,N}$ | Useful heat of N series collectors [J] |

${Q}_{pl}$ | Thermal losses in interconnection pipe [J] |

${S}_{sav}$ | Solar Savings [$] |

$t$ | Time (s) |

${T}_{a}$ | Ambient temperature [°C] |

${T}_{co}$ | Collector array outlet temperature [°C] |

${T}_{in1}$ | Inlet fluid temperature of first collector [°C] |

${T}_{in2}$ | Inlet fluid temperature of second collector [°C] |

${T}_{L}$ | Temperature required in the load [°C] |

${T}_{mu}$ | Make-up water temperature [°C] |

${T}_{out1}$ | Outlet fluid temperature of first collector [°C] |

${T}_{out2}$ | Outlet fluid temperature of second collector [°C] |

${T}_{s}$ | Fluid temperature inside the storage tank [°C] |

${T}_{s-n}$ | Instantaneous fluid temperature inside the tank [°C] |

${T}_{s,herr}$ | Fluid temperature inside storage tank calculated by the tool [°C] |

${T}_{s,TRNSYS}$ | Fluid temperature inside storage tank calculated by TRNSYS [°C] |

${U}_{L}$ | Collector overall loss coefficient [W/m^{2} K] |

${U}_{L}^{\prime}$ | Modified collector overall loss coefficient [W/m^{2} K] |

${U}_{s}$ | Storage tank overall loss coefficient [W/m^{2} K] |

$V$ | Volume of hot water storage tank [L] |

${V}_{L}$ | Daily volume required in the load [L] |

Greek symbols | |

$\beta $ | Collector slope [°] |

$\Delta t$ | Time difference [s] |

$\Delta T$ | Temperature difference [°C] |

${\epsilon}_{L}$ | Heat exchange effectiveness [-] |

${\eta}_{L}$ | Optical efficiency [-] |

${\eta}_{o}^{\prime}$ | Modified optical efficiency [-] |

$\theta $ | Incidence angle of beam radiation [°] |

${\sigma}_{r}$ | Standard deviation of relative error [-] |

$\varphi $ | Latitude of the location [°] |

## References

- Kalogirou, S. The potential of solar industrial process heat applications. Appl. Energy
**2003**, 76, 337–361. [Google Scholar] [CrossRef] - Krummenacher, P.; Muster, B. Solar Process Heat for Production and Advanced Applications: Methodologies and Software Tools for Integrating Solar Heat Into Industrial Processes; IEA SHC Task 49HC Task 49—Deliverable B1; Solar Heating and Cooling Programe: Paris, France, 2015. [Google Scholar]
- Muster, B.; Ben Hassine, I.; Helmke, A.; Heß, S.; Krummenacher, P.; Muster, B.; Schmitt, B.; Schnitzer, H. Solar Process Heat for Production and Advanced Applications: Integration Guideline; IEA SHC Task 49HC Task 49—Deliverable B2; Solar Heating and Cooling Programe: Paris, France, 2015. [Google Scholar]
- Farjana, S.H.; Huda, N.; Mahmud, M.A.P.; Saidur, R. Solar process heat in industrial systems—A global review. Renew. Sustain. Energy Rev.
**2018**, 82, 2270–2286. [Google Scholar] [CrossRef] - Shrivastava, R.L.; Kumar, V.; Untawale, S.P. Modeling and simulation of solar water heater: A TRNSYS perspective. Renew. Sustain. Energy Rev.
**2017**, 67, 126–143. [Google Scholar] [CrossRef] - Sornek, K. The comparison of solar water heating system operation parameters calculated using traditional method and dynamic simulations. In E3S Web of Conferences; EDP Sciences: Les Ulis, France, 2016; Volume 10, p. 4. [Google Scholar]
- Liang, H.; You, S.; Zhang, H. Comparison of different heat transfer models for parabolic trough solar collectors. Appl. Energy
**2015**, 148, 105–114. [Google Scholar] [CrossRef] - Xu, L.; Wang, Z.F.; Yuan, G.F.; Sun, F.H.; Zhang, X.L. Thermal performance of parabolic trough solar collectors under the condition of dramatically varying DNI. Energy Procedia
**2015**, 69, 218–225. [Google Scholar] [CrossRef] - Wojcicki, D.J. The application of the typical day concept in flat plate solar collector models. Renew. Sustain. Energy Rev.
**2015**, 49, 968–974. [Google Scholar] [CrossRef] - Kicsiny, R. Multiple linear regression based models for solar collectors. Sol. Energy
**2014**, 110, 496–506. [Google Scholar] [CrossRef] - Lauterbach, C.; Schmitt, B.; Jordan, U.; Vajen, K. The potential of solar heat for industrial processes in Germany. Renew. Sustain. Energy Rev.
**2012**, 16, 5121–5130. [Google Scholar] [CrossRef] - Pietruschka, D.; Fedrizzi, R.; Orioli, F.; Söll, R.; Stauss, R. Demonstration of three large scale solar process heat applications with different solar thermal collector technologies. Energy Procedia
**2012**, 30, 755–764. [Google Scholar] [CrossRef] - Mauthner, F.; Hubmann, M.; Brunner, C.; Fink, C. Manufacture of malt and beer with low temperature solar process heat. Energy Procedia
**2014**, 48, 1188–1193. [Google Scholar] [CrossRef] - Zahler, C.; Iglauer, O. Solar process heat for sustainable automobile manufacturing. Energy Procedia
**2012**, 30, 775–782. [Google Scholar] [CrossRef] - Frey, P.; Fischer, S.; Drück, H.; Jakob, K. Monitoring results of a solar process heat system installed at a textile company in Southern Germany. Energy Procedia
**2015**, 70, 615–620. [Google Scholar] [CrossRef] - Schramm, S.; Adam, M. Storage in solar process heat applications. Energy Procedia
**2014**, 48, 1202–1209. [Google Scholar] [CrossRef] - Bava, F.; Dragsted, J.; Furbo, S. A numerical model to evaluate the flow distribution in a large solar collector field. Sol. Energy
**2017**, 143, 31–42. [Google Scholar] [CrossRef] - Lauterbach, C.; Schmitt, B.; Vajen, K. System analysis of a low-temperature solar process heat system. Sol. Energy
**2014**, 101, 117–130. [Google Scholar] [CrossRef] - Silva, R.; Pérez, M.; Fernández-Garcia, A. Modeling and co-simulation of a parabolic trough solar plant for industrial process heat. Appl. Energy
**2013**, 106, 287–300. [Google Scholar] [CrossRef] - Karagiorgas, M.; Galatis, K.; Tsagouri, M.; Tsoutsos, T.; Botzios-Valaskakis, A. Solar assisted heat pump on air collectors: A simulation tool. Sol. Energy
**2010**, 84, 66–78. [Google Scholar] [CrossRef] - Bunea, M.; Duret, A.; Franck, E.; Péclat, L.; Citherlet, S. Medium temperature solar thermal installation with heat storage for industrial applications. In Eurosun 2014: International Conference on Solar Energy and Buildings; International Solar Energy Society: Aix-les-Bains, France, 2014; p. 10. [Google Scholar]
- Kulkarni, G.N.; Kedare, S.B.; Bandyopadhyay, S. Determination of design space and optimization of solar water heating systems. Sol. Energy
**2007**, 81, 958–968. [Google Scholar] [CrossRef] [Green Version] - Picón-Núñez, M.; Martínez-Rodríguez, G.; Fuentes-Silva, A.L. Design of solar collector networks for industrial applications. Appl. Therm. Eng.
**2014**, 70, 1238–1245. [Google Scholar] [CrossRef] - Bava, F.; Furbo, S.; Perers, B. Simulation of a solar collector array consisting of two types of solar collectors, with and without convection barrier. Energy Procedia
**2015**, 70, 4–12. [Google Scholar] [CrossRef] - Almeida, P.; Carvalho, M.J.; Amorim, R.; Mendes, J.F.; Lopes, V. Dynamic testing of systems—Use of TRNSYS as an approach for parameter identification. Sol. Energy
**2014**, 104, 60–70. [Google Scholar] [CrossRef] - Silva, R.; Berenguel, M.; Pérez, M.; Fernández-Garcia, A. Thermo-economic design optimization of parabolic trough solar plants for industrial process heat applications with memetic algorithms. Appl. Energy
**2014**, 113, 603–614. [Google Scholar] [CrossRef] - Kalogirou, S.A. Solar Energy Engineering: Processes and Systems, 1st ed.; Elsevier Inc.: London, England, 2009; ISBN 9780123745019. [Google Scholar]
- Duffie, J.; Beckman, W.A. Solar Engineering of Thermal Processes, 4th ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2013; ISBN 9780470873663. [Google Scholar]
- Rabl, A. Active Solar Collectors and Their Applications, 1st ed.; Oxford University Press: New York, NY, USA, 1985; ISBN 0-19-503546-1. [Google Scholar]
- Perez, R.; Seals, R.; Ineichen, P.; Stewart, R.; Menicucci, D. A new simplified version of the Perez diffuse irradiance model for tilted surfaces. Sol. Energy
**1987**, 39, 221–231. [Google Scholar] [CrossRef] - Perez, R.; Stewart, R.; Seals, R.; Guertin, T. The Development and Verification of the Perez Diffuse Radiation Model; Atmospheric Sciences Research Center: Albany, NY, USA, 1988; p. 176. [Google Scholar]
- Thulukkanam, K. Heat Exchanger Design Handbook, 2nd ed.; CRC Press, Taylor & Francis Group: Boca Raton, FL, USA, 2013; ISBN 978-1-4398-4212-6. [Google Scholar]
- Venegas-Reyes, E. Diseño, Construcción y Evaluación de un Arreglo de Concentradores de Canal Parabólico para Calor de Proceso, Dirección General de Bibliotecas; UNAM: Ciudad de México, México, 2013. [Google Scholar]
- Solar Rating & Certification Corporation. ICC-SRCC OG-100 ICC-SRCC Certified Solar Collector # 2007033A; Solar Rating & Certification Corporation: Brea, CA, USA, 2018; pp. 1–5. [Google Scholar]
- Comisión Federal de Electricidad Public Service Tariff. Available online: https://app.cfe.mx/Aplicaciones/CCFE/Tarifas/Tarifas/Tarifas_industria.asp?Tarifa=CMAS&Anio=2016# (accessed on 7 May 2019).
- SAGARPA. Especificaciones Técnicas para Sistemas de Calentamiento de Agua con Energía Térmica Solar; Fideicomiso de Riesgo Compartido: Ciudad de México, México, 2011.

**Figure 7.**The standard deviation of relative error between TRNSYS and the proposed tool using the temperature profile of the water inside the storage tank.

Parameter | Value |
---|---|

Site | Delicias, Chihuahua, Mexico |

Latitude, $\varphi $ | 28°11′36″ |

Longitude, ${L}_{loc}$ | 105°28′16″ |

Make-up water temperature, ${T}_{mu}$ | 20 °C |

Temperature required in the load, ${T}_{L}$ | 90 °C |

Daily volume required, ${V}_{L}$ | 25,000 L |

Mass flow rate required in the load, ${\dot{m}}_{L}$ | 0.6314 kg/s |

Mass flow rate in the storage tank, ${\dot{m}}_{s}$ | 0.6314 kg/s |

Period of operation (standard time) | 7:00–17:00 h |

Volume of hot water (storage tank), $V$ | 25,000 L |

Overall heat loss coefficient of the hot water tank, ${U}_{s}$ | 1 W/m^{2} K |

Heat exchanger effectiveness, ${\epsilon}_{L}$ | 0.85 |

Heat capacity ^{1} ($C{p}_{c}=C{p}_{L}$) | 4186 J/kg K |

^{1}Constant.

**Table 2.**Parameters of collector used in simulations [34].

Data | Value |
---|---|

Gross area, ${A}_{c}$ | 4.158 m^{2} |

Slope, $\beta $ | 28.18° |

Optical efficiency, ${\eta}_{L}$ | 0.458 |

Overall heat loss coefficient, ${U}_{L}$ | 1.579 W/m^{2} °C |

Mass flow rate, ${\dot{m}}_{c}$ | 0.1 kg/s |

Constant for calculation of incidence angle modifier, ${b}_{0}$ | 0.0074 |

Constant for calculation of incidence angle modifier, ${b}_{1}$ | −7.0 × 10^{−4} |

Constant for calculation of incidence angle modifier, ${b}_{2}$ | 9.0 × 10^{−6} |

Data | Value |
---|---|

Single collector cost | US $628.57 |

Hot water storage tank cost | US $14 285.71 |

Installation cost | US $35 200.00 |

Annual inflation rate | 5% |

Annual interest rate | 8% |

Discount rate | 8% |

Taxes | 16% |

Cost of gas fuel | US $16.9/GJ |

Nominal Diameter [Inches] | Pipe Cost [USD] |
---|---|

0.75 | $ 15.61 |

1.00 | $ 23.52 |

1.25 | $ 31.95 |

1.50 | $ 40.92 |

2.00 | $ 60.42 |

2.50 | $ 82.04 |

3.00 | $ 105.78 |

4.00 | $ 159.58 |

6.00 | $ 292.52 |

8.00 | $ 459.25 |

Series N | Rows | Pumping Power Cost [USD] | Pipe Cost [USD] |
---|---|---|---|

1 | 160 | $ 2 287.95 | $ 19 878.51 |

2 | 80 | $ 1 720.94 | $ 5 678.39 |

3 | 53 | $ 1 264.46 | $ 2 917.33 |

4 | 40 | $ 1 075.36 | $ 1 850.75 |

5 | 32 | $ 634.87 | $ 1 302.49 |

6 | 27 | $ 636.29 | $ 1 000.37 |

7 | 23 | $ 637.51 | $ 758.67 |

8 | 20 | $ 637.51 | $ 577.40 |

9 | 18 | $ 541.86 | $ 495.57 |

10 | 16 | $ 434.55 | $ 413.74 |

11 | 15 | $ 326.72 | $ 372.82 |

12 | 13 | $ 327.65 | $ 331.91 |

13 | 12 | $ 216.89 | $ 299.95 |

14 | 11 | $ 217.37 | $ 236.05 |

15 | 11 | $ 217.64 | $ 236.05 |

Solar Fraction | Payback Time (Years) | |
---|---|---|

Optimal—thermal analysis (160 rows) | 0.772 | 7.43 |

Optimal—technical-economic analysis (5 series, 32 rows) | 0.763 | 6.13 |

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**MDPI and ACS Style**

Venegas-Reyes, E.; Ortega-Avila, N.; Rodríguez-Muñoz, N.A.; Nájera-Trejo, M.; Martín-Domínguez, I.R.; Ibarra-Bahena, J.
Parametric Methodology to Optimize the Sizing of Solar Collector Fields in Series-Parallel Arrays. *Processes* **2019**, *7*, 294.
https://doi.org/10.3390/pr7050294

**AMA Style**

Venegas-Reyes E, Ortega-Avila N, Rodríguez-Muñoz NA, Nájera-Trejo M, Martín-Domínguez IR, Ibarra-Bahena J.
Parametric Methodology to Optimize the Sizing of Solar Collector Fields in Series-Parallel Arrays. *Processes*. 2019; 7(5):294.
https://doi.org/10.3390/pr7050294

**Chicago/Turabian Style**

Venegas-Reyes, Eduardo, Naghelli Ortega-Avila, Norma A. Rodríguez-Muñoz, Mario Nájera-Trejo, Ignacio R. Martín-Domínguez, and Jonathan Ibarra-Bahena.
2019. "Parametric Methodology to Optimize the Sizing of Solar Collector Fields in Series-Parallel Arrays" *Processes* 7, no. 5: 294.
https://doi.org/10.3390/pr7050294