# Topology Optimization Method of a Cavity Receiver and Built-In Net-Based Flow Channels for a Solar Parabolic Dish Collector

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Physical Model

#### 2.1. Net-Based Thermal–Fluid Model

**p**is the pressure vector of each node and

**K**is the friction factor matrix.

#### 2.2. Heat Loss Model

_{s}is the interior cavity surface temperature, ${T}_{cav-surface}$ is the exterior cavity surface temperature, and ${T}_{s-ave}$ is the average cavity surface temperature. The temperature difference between the interior and exterior surfaces of the cavity is quite small since an embedded channel structure is adopted (see the verification example in Section 4.2). In the heat loss model, the temperature inside and outside the cavity is regarded as approximately equal, as shown in the following equation:

#### 2.2.1. Conduction Heat Loss

_{s}is the mean surface area, ${T}_{\infty}$ is the ambient temperature, ${\sigma}_{insu}$ is the thickness of the insulation, and ${k}_{insu}$ is the thermal conductivity of the insulation. ${h}_{out}$ is the heat transfer coefficient at the exterior of the insulated receiver, which can be obtained using the following:

#### 2.2.2. Convection Heat Loss

#### 2.2.3. Radiation Heat Loss

^{2}K

^{4}) is the Stefan Boltzmann Constanta, and ${\epsilon}_{eff}$ is the effective emissivity of the receiver surface, estimated using the following equation:

#### 2.2.4. Thermal Efficiency

## 3. Formulation of the Optimization Problem

#### 3.1. Objective Function

#### 3.2. Design Variables

#### 3.3. Implementation of Optimization by GA

## 4. Results and Discussion

#### 4.1. Analysis of Thermal Efficiency

#### 4.1.1. Relationship between Heat Loss and Cavity Temperature

#### 4.1.2. Relationship between Thermal Efficiency and Receiver Size

#### 4.2. Numerical Examples for Validation under Uniform Heat Flux

#### 4.3. Optimization Results of the GA under Inhomogeneous Heat Flux

_{2}as an example. The heated surface is divided into 11 regions in these optimization examples. Although the heat flux distribution on the entire heating surface is nonuniform, the heat flux is assumed to be uniformly distributed in each sub-region, as shown in Equation (37):

^{2}. The coefficient vector of heat flux (W/m

^{2}) is assumed to be $C=\left[1,1,2,4,9,8,6,5,4,3,1\right]$.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | Temperature matrix |

${A}_{s}$ | Mean surface area (m^{2}) |

${A}_{0}$ | Area of receiver aperture (m^{2}) |

${c}_{p}$ | Specific heat (J kg^{−1} K^{−1}) |

C | Distribution vector of heat flux (W/m^{2}) |

$d$ | Diameter of branch channel (m) |

${d}_{cav}$ | Cavity diameter (m) |

${d}_{ap}$ | Receiver aperture diameter (m) |

$dept{h}_{s}$ | Cavity depth (m) |

${D}_{out}$ | Exterior surface diameter of the receiver cavity (m) |

${D}_{ground}$ | Diameter of the ground structure for the channel model (m) |

${f}_{\mathrm{t}}$ | Load vector for temperature field (W) |

$F$ | View factor |

$g$ | Gravitational acceleration (m s^{−2}) |

$Gr$ | Grashhof number |

${h}_{out}$ | Heat transfer coefficient for convection at the exterior of an insulated receiver (W m^{−2} K^{−1}) |

$J$ | Objective function |

${k}_{\infty}$ | Thermal conductivity of atmospheric air (W m^{−1} K^{−1}) |

${k}_{insu}$ | Thermal conductivity of insulation (W m^{−1} K^{−1}) |

$K$ | Friction factor matrix |

$m$ | Node number |

$Nu$ | Nusselt number |

${N}_{e}$ | Number of solid elements |

${N}_{RC}$ | Radiation conduction number |

$Pr$ | Prandtl number |

$p$ | Pressure (Pa) |

${p}_{in}$ | Inlet pressure (Pa) |

${p}_{0}$ | Inlet pressure of the initial structure (Pa) |

$p$ | Pressure vector (Pa) |

$Q$ | Heat transfer capacity (W) |

$q$ | Volume flow (m^{3} s^{−1}) |

${q}_{in}$ | Inlet volume flow (m^{3} s^{−1}) |

$r$ | Friction factor (Pa m^{−6} s^{−2}) |

$Ra$ | Rayleigh number |

$Re$ | Reynolds number |

${R}_{th}$ | Thermal resistance |

$St{d}_{0}$ | Standard deviation of the solid domain temperature for the initial structure (°C) |

$T$ | Temperature variable (°C) |

$T$ | Temperature vector (°C) |

${T}_{s}$ | Interior cavity surface temperature (°C) |

${T}_{cav-surface}$ | Exterior cavity surface temperature (°C) |

${T}_{s-ave}$ | Average cavity surface temperature (°C) |

${T}_{\infty}$ | Ambient temperature (°C) |

${V}_{0}$ | Volume constraint |

${V}_{wind}$ | Wind velocity (m s^{−1}) |

Greek symbols | |

$\mu $ | Dynamic viscosity (Pa s) |

$\rho $ | Density (kg m^{−3}) |

$\xi $ | Design variable in optimization problem |

$\xi $ | Vector of topology design variables |

${\xi}_{m}$ | Threshold for deleting a branch |

$\theta $ | Receiver inclination angle (°) |

$\epsilon $ | Emissivity of receiver surface |

$\alpha $ | Absorptivity of receiver surface |

$\beta $ | Volume expansion coefficient (K^{−1}) |

${\sigma}_{insu}$ | Thickness of insulation (m) |

${\tau}_{i}$ | Intercept factor |

${\tau}_{shad}$ | Shading factor |

${\rho}_{mirror}$ | Mirror reflectivity |

${\eta}_{th}$ | Thermal efficiency |

${\eta}_{0}$ | Thermal efficiency of the initial structure |

Subscripts | |

$amb$ | Ambient |

$avg$ | Average |

$ap$ | Aperture |

$cav$ | Cavity |

$cond$ | Conduction heat transfer |

$conv$ | Convection heat transfer |

$eff$ | Effective |

$em$ | Emission |

$forced$ | Forced condition |

$insu$ | Insulation |

$ini$ | Initial |

$mirror$ | Mirror reflective surface |

$max$ | Maximum |

$min$ | Minimum |

$nat$ | Natural condition |

$out$ | Outer surface |

$rad$ | Radiative heat transfer |

$ref$ | Reflection |

## References

- Islam, M.T.; Huda, N.; Abdullah, A.B.; Saidur, R. A comprehensive review of state-of-the-art concentrating solar power (CSP) technologies: Current status and research trends. Renew. Sustain. Energy Rev.
**2018**, 91, 987–1018. [Google Scholar] [CrossRef] - Kumar, A.; Prakash, O.; Dube, A. A review on progress of concentrated solar power in India. Renew. Sustain. Energy Rev.
**2017**, 79, 304–307. [Google Scholar] [CrossRef] - Wang, J.; Yang, S.; Jiang, C.; Zhang, Y.; Lund, P.D. Status and future strategies for Concentrating Solar Power in China. Energy Sci. Eng.
**2017**, 5, 100–109. [Google Scholar] [CrossRef] - Liu, D.; Xin-Feng, L.; Bo, L.; Si-Quan, Z.; Yan, X. Progress in thermochemical energy storage for concentrated solar power: A review. Int. J. Energy Res.
**2018**, 42, 4546–4561. [Google Scholar] [CrossRef] - Prasad, D.R.; Senthilkumar, R.; Lakshmanarao, G.; Krishnan, S.; Prasad, B.N. A critical review on thermal energy storage materials and systems for solar applications. AIMS Energy
**2019**, 7, 507–526. [Google Scholar] [CrossRef] - Tian, Y.; Zhao, C. A review of solar collectors and thermal energy storage in solar thermal applications. Appl. Energy
**2013**, 104, 538–553. [Google Scholar] [CrossRef][Green Version] - Chen, Y.; Zhang, Y.; Wang, D.; Hu, S.; Huang, X. Effects of design parameters on fatigue–creep damage of tubular supercritical carbon dioxide power tower receivers. Renew. Energy
**2021**, 176, 520–532. [Google Scholar] [CrossRef] - Conroy, T.; Collins, M.N.; Grimes, R. A review of steady-state thermal and mechanical modelling on tubular solar receivers. Renew. Sustain. Energy Rev.
**2020**, 119, 109591. [Google Scholar] [CrossRef] - Maytorena, V.; Hinojosa, J. Effect of non-uniform concentrated solar flux on direct steam generation in vertical pipes of solar tower receivers. Sol. Energy
**2019**, 183, 665–676. [Google Scholar] [CrossRef] - He, Y.-L.; Wang, K.; Qiu, Y.; Du, B.-C.; Liang, Q.; Du, S. Review of the solar flux distribution in concentrated solar power: Non-uniform features, challenges, and solutions. Appl. Therm. Eng.
**2019**, 149, 448–474. [Google Scholar] [CrossRef] - Daabo, A.M.; Mahmoud, S.; Al-Dadah, R.K. The optical efficiency of three different geometries of a small scale cavity receiver for concentrated solar applications. Appl. Energy
**2016**, 179, 1081–1096. [Google Scholar] [CrossRef][Green Version] - Loni, R.; Asli-Areh, E.A.; Ghobadian, B.; Kasaeian, A.B.; Gorjian, S.; Najafi, G.; Bellos, E. Research and review study of solar dish concentrators with different nanofluids and different shapes of cavity receiver: Experimental tests. Renew. Energy.
**2020**, 145, 783–804. [Google Scholar] [CrossRef] - Bellos, E.; Bousi, E.; Tzivanidis, C.; Pavlovic, S. Optical and thermal analysis of different cavity receiver designs for solar dish concentrators. Energy Convers. Manag. X
**2019**, 2, 100013. [Google Scholar] [CrossRef] - Karwa, N.; Jiang, L.; Winston, R.; Rosengarten, G. Receiver shape optimization for maximizing medium temperature CPC collector efficiency. Sol. Energy
**2015**, 122, 529–546. [Google Scholar] [CrossRef][Green Version] - Zou, C.; Zhang, Y.; Feng, H.; Falcoz, Q.; Neveu, P.; Gao, W.; Zhang, C. Effects of geometric parameters on thermal performance for a cylindrical solar receiver using a 3D numerical model. Energy Convers. Manag.
**2017**, 149, 293–302. [Google Scholar] [CrossRef] - Venkatachalam, T.; Cheralathan, M. Effect of aspect ratio on thermal performance of cavity receiver for solar parabolic dish concentrator: An experimental study. Renew. Energy
**2019**, 139, 573–581. [Google Scholar] [CrossRef] - Xiao, L.; He, S.; Wu, S.-Y.; Chen, Z.-L. Optical-thermal conversion characteristics of cylindrical receiver with built-in helically coiled tubes. Sustain. Energy Technol. Assess.
**2020**, 37, 100626. [Google Scholar] [CrossRef] - Karimi, R.; Gheinani, T.T.; Avargani, V.M. A detailed mathematical model for thermal performance analysis of a cylindrical cavity receiver in a solar parabolic dish collector system. Renew. Energy
**2018**, 125, 768–782. [Google Scholar] [CrossRef] - Albarbar, A.; Arar, A. Performance Assessment and Improvement of Central Receivers Used for Solar Thermal Plants. Energies
**2019**, 12, 3079. [Google Scholar] [CrossRef][Green Version] - Wang, G.; Niu, S.; Yu, S.; Lin, J.; Chen, Z.; Hu, P. Parametric Study on Integrated Thermal and Mechanical Performance of Molten Salt Receiver for Solar Tower Power Plant. Int. J. Thermophys.
**2020**, 41, 23. [Google Scholar] [CrossRef] - Fuqiang, W.; Zhexiang, T.; Xiangtao, G.; Jianyu, T.; Huaizhi, H.; Bingxi, L. Heat transfer performance enhancement and thermal strain restrain of tube receiver for parabolic trough solar collector by using asymmetric outward convex corrugated tube. Energy
**2016**, 114, 275–292. [Google Scholar] [CrossRef] - Li, Z.; Tang, D.; Du, J.; Li, T. Study on the radiation flux and temperature distributions of the concentrator–receiver system in a solar dish/Stirling power facility. Appl. Therm. Eng.
**2011**, 31, 1780–1789. [Google Scholar] [CrossRef] - Tao, Y.; He, Y.; Cui, F.; Lin, C. Numerical study on coupling phase change heat transfer performance of solar dish collector. Sol. Energy
**2013**, 90, 84–93. [Google Scholar] [CrossRef] - Adkins, D.R.; Andraka, C.E.; Moss, T.A. Development of a 75-kW Heat-Pipe Receiver for Solar Heat-Engines; Sandia National Labs.: Albuquerque, NM, USA, 1995. [Google Scholar]
- Wang, P.; Liu, D.; Xu, C. Numerical study of heat transfer enhancement in the receiver tube of direct steam generation with parabolic trough by inserting metal foams. Appl. Energy
**2013**, 102, 449–460. [Google Scholar] [CrossRef] - Zheng, Z.-J.; Li, M.-J.; He, Y.-L. Optimization of porous insert configurations for heat transfer enhancement in tubes based on genetic algorithm and CFD. Int. J. Heat Mass Transf.
**2015**, 87, 376–379. [Google Scholar] [CrossRef] - Zheng, Z.; Xu, Y.; He, Y. Thermal analysis of a solar parabolic trough receiver tube with porous insert optimized by coupling genetic algorithm and CFD. Sci. China Technol. Sci.
**2016**, 59, 1475–1485. [Google Scholar] [CrossRef] - Du, S.; Li, Z.; He, Y.; Li, D.; Xie, X.; Gao, Y. Experimental and numerical analysis of the hydraulic and thermal performances of the gradually-varied porous volumetric solar receiver. Sci. China Technol. Sci.
**2020**, 63, 1224–1234. [Google Scholar] [CrossRef] - Guo, J.; Huai, X. Multi-parameter optimization design of parabolic trough solar receiver. Appl. Therm. Eng.
**2016**, 98, 73–79. [Google Scholar] [CrossRef] - Du, S.; He, Y.-L.; Yang, W.-W.; Liu, Z.-B. Optimization method for the porous volumetric solar receiver coupling genetic algorithm and heat transfer analysis. Int. J. Heat Mass Transf.
**2018**, 122, 383–390. [Google Scholar] [CrossRef] - de Risi, A.; Milanese, M.; Laforgia, D. Modelling and optimization of transparent parabolic trough collector based on gas-phase nanofluids. Renew. Energy
**2013**, 58, 134–139. [Google Scholar] [CrossRef] - Moloodpoor, M.; Mortazavi, A.; Ozbalta, N. Thermal analysis of parabolic trough collectors via a swarm intelligence optimizer. Sol. Energy
**2019**, 181, 264–275. [Google Scholar] [CrossRef] - Zadeh, P.M.; Sokhansefat, T.; Kasaeian, A.; Kowsary, F.; Akbarzadeh, A. Hybrid optimization algorithm for thermal analysis in a solar parabolic trough collector based on nanofluid. Energy
**2015**, 82, 857–864. [Google Scholar] [CrossRef] - Montes, M.; Rovira, A.; Martínez-Val, J.; Ramos, A. Proposal of a fluid flow layout to improve the heat transfer in the active absorber surface of solar central cavity receivers. Appl. Therm. Eng.
**2012**, 35, 220–232. [Google Scholar] [CrossRef] - Bopche, S.; Rana, K.; Kumar, V. Performance improvement of a modified cavity receiver for parabolic dish concentrator at medium and high heat concentration. Sol. Energy
**2020**, 209, 57–78. [Google Scholar] [CrossRef] - Zou, C.; Zhang, Y.; Falcoz, Q.; Neveu, P.; Zhang, C.; Shu, W.; Huang, S. Design and optimization of a high-temperature cavity receiver for a solar energy cascade utilization system. Renew. Energy
**2017**, 103, 478–489. [Google Scholar] [CrossRef] - Dorn, W.C.; Gomory, R.E.; Grenberg, H. Automatic design of optimal structures. J. de Mec.
**1964**, 3, 25–52. [Google Scholar] - Liu, J.; Li, R.; Wang, K. Net-based topology optimization approach for cooling channels. Int. J. Therm. Sci.
**2020**, 156, 106494. [Google Scholar] [CrossRef] - Zhang, C.; Yang, T. Optimal maintenance planning and resource allocation for wind farms based on non-dominated sorting genetic algorithm-II. Renew. Energy
**2020**, 164, 1540–1549. [Google Scholar] [CrossRef] - Xing, Y.; Chen, K. Solution of Mine Ventilation Network; China University of Mining and Technology Press: Xuzhou, China, 2015. [Google Scholar]
- Loni, R.; Asli-Ardeh, E.A.; Ghobadian, B.; Bellos, E.; Le Roux, W.G. Numerical comparison of a solar dish concentrator with different cavity receivers and working fluids. J. Clean. Prod.
**2018**, 198, 1013–1030. [Google Scholar] [CrossRef] - Gholamalizadeh, E.; Chung, J.D. Thermal Analysis of the Receiver of a Standalone Pilot Solar Dish–Stirling System. Entropy
**2018**, 20, 429. [Google Scholar] [CrossRef][Green Version] - Cui, F.-Q.; He, Y.-L.; Cheng, Z.-D.; Li, Y. Modeling of the Dish Receiver with the Effect of Inhomogeneous Radiation Flux Distribution. Heat Transf. Eng.
**2013**, 35, 780–790. [Google Scholar] [CrossRef]

**Figure 1.**Three-dimensional schematic for the hemispherical-shaped receiver and the dish collector under it.

**Figure 2.**Network-type flow channel and heat transfer model, where ${Q}_{ss}$ represents the heat exchange between solid subdomains and Q

_{sf}represents the heat exchange between solid subdomains and HTF.

**Figure 12.**Comparisons between the temperature field (°C) and pressure field (Pa) predicted by net-based thermal–fluid model and those predicted by numerical simulation.

**Figure 14.**The distribution of inhomogeneous heat flux (W/m

^{2}) of the optimization result for J

_{2}.

**Figure 15.**Topological structures of reference design (helical channel) and those of optimization results.

**Figure 17.**Comparisons between the physical fields of optimization results and those of the reference design (helical channel).

Design Variables | Definition | Lower Bound (mm) | Upper Bound (mm) |
---|---|---|---|

$d$ | diameter of channels | 1 | 5 |

${d}_{cav}$ | diameter of cavity | 250 | 350 |

${d}_{ap}$ | diameter of aperture | 160 | 200 |

${\delta}_{insu}$ | thickness of insulation | 25 | 75 |

Parameters | Definition | Values |
---|---|---|

${T}_{\infty}$ | Ambient temperature | 20 °C |

$\theta $ | Receiver inclination angle | 45° |

${\epsilon}_{s}$ | Emissivity of receiver surface | 0.83 |

${\alpha}_{s}$ | Absorptivity of radiative surface | 0.75 |

${Q}_{b}$ | Solar beam radiation energy | 300 W |

${\tau}_{i}$ | Intercept factor | 0.94 |

${\tau}_{shad.}$ | Shading factor | 0.95 |

${\rho}_{mirror}$ | Mirror reflectivity | 0.85 |

${V}_{wind}$ | Wind velocity | 1 m/s |

F | View factor | 1 |

Material | ρ [kg/m^{3}] | υ [m^{2}/s] | k [W/m K] | c_{p} [J/(kg K)] |
---|---|---|---|---|

Water | 997.56 | 8.91 × 10^{−7} | 0.62 | 4181.72 |

Copper | 8940 | - | 386 | 386 |

Grid Number | ${\mathit{T}}_{\mathit{s}}$ [°C] | Deviation | ${\mathit{P}}_{\mathit{i}\mathit{n}}$ [Pa] | Deviation |
---|---|---|---|---|

3,464,861 | 76.66 | 0.39% | 93.21 | −0.40% |

4,932,816 | 76.58 | 0.29% | 95.72 | 2.29% |

7,571,626 | 76.36 | - | 93.58 | - |

Objective | Reference Case | J_{1} | ${\mathit{J}}_{2}$ | MOO |
---|---|---|---|---|

Inlet pressure (Pa) | 14,980 | 15,093 | 1817 | 1832 |

Standard deviation of solid temperature (°C) | 9.95 | 1.67 | 11.17 | 2.82 |

Thermal efficiency (%) | 79.10% | 72.26% | 81.64% | 79.25% |

Total number of branches | 202 | 359 | 667 | 405 |

Average channel diameter (mm) | 3.23 | 2.56 | 1.90 | 1.68 |

${d}_{cav}$(m) | 0.30 | 0.25 | 0.33 | 0.31 |

${d}_{ap}$(m) | 0.18 | 0.20 | 0.16 | 0.17 |

${\delta}_{insu}$(m) | 0.05 | 0.03 | 0.07 | 0.05 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, J.; Li, R.; Chen, Y.; Zheng, J.; Wang, K.
Topology Optimization Method of a Cavity Receiver and Built-In Net-Based Flow Channels for a Solar Parabolic Dish Collector. *Entropy* **2023**, *25*, 398.
https://doi.org/10.3390/e25030398

**AMA Style**

Liu J, Li R, Chen Y, Zheng J, Wang K.
Topology Optimization Method of a Cavity Receiver and Built-In Net-Based Flow Channels for a Solar Parabolic Dish Collector. *Entropy*. 2023; 25(3):398.
https://doi.org/10.3390/e25030398

**Chicago/Turabian Style**

Liu, Jun, Renfu Li, Yuxuan Chen, Jianguo Zheng, and Kun Wang.
2023. "Topology Optimization Method of a Cavity Receiver and Built-In Net-Based Flow Channels for a Solar Parabolic Dish Collector" *Entropy* 25, no. 3: 398.
https://doi.org/10.3390/e25030398