IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i9p1608-d415379.html
   My bibliography  Save this article

On the Existence of Solutions of a Two-Layer Green Roof Mathematical Model

Author

Listed:
  • J. Ignacio Tello

    (Department of Fundamental Mathematics, School of Sciences, UNED, 28040 Madrid, Spain)

  • Lourdes Tello

    (Department of Applied Mathematics, ETS Arquitectura, Universidad Politécnica de Madrid, Av. Juan de Herrera 4, 28040 Madrid, Spain)

  • María Luisa Vilar

    (Department of Applied Mathematics, ETS Arquitectura, Universidad Politécnica de Madrid, Av. Juan de Herrera 4, 28040 Madrid, Spain)

Abstract

The aim of this article is to fill part of the existing gap between the mathematical modeling of a green roof and its computational treatment, focusing on the mathematical analysis. We first introduce a two-dimensional mathematical model of the thermal behavior of an extensive green roof based on previous models and secondly we analyze such a system of partial differential equations. The model is based on an energy balance for buildings with vegetation cover and it is presented for general shapes of roofs. The model considers a vegetable layer and the substratum and the energy exchange between them. The unknowns of the problem are the temperature of each layer described by a coupled system of two partial differential equations of parabolic type. The equation modeling the evolution of the temperature of the substratum also considers the change of phase of water described by a maximal monotone graph. The main result of the article is the proof of the existence of solutions of the system which is given in detail by using a regularization of the maximal monotone graph. Appropriate estimates are obtained to pass to the limit in a weak formulation of the problem. The result goes one step further from modeling to validate future numerical results.

Suggested Citation

  • J. Ignacio Tello & Lourdes Tello & María Luisa Vilar, 2020. "On the Existence of Solutions of a Two-Layer Green Roof Mathematical Model," Mathematics, MDPI, vol. 8(9), pages 1-17, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1608-:d:415379
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/9/1608/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/9/1608/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Jaffal, Issa & Ouldboukhitine, Salah-Eddine & Belarbi, Rafik, 2012. "A comprehensive study of the impact of green roofs on building energy performance," Renewable Energy, Elsevier, vol. 43(C), pages 157-164.
    2. Mansoureh Gholami & Alberto Barbaresi & Patrizia Tassinari & Marco Bovo & Daniele Torreggiani, 2020. "A Comparison of Energy and Thermal Performance of Rooftop Greenhouses and Green Roofs in Mediterranean Climate: A Hygrothermal Assessment in WUFI," Energies, MDPI, vol. 13(8), pages 1-15, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Mihalakakou, Giouli & Souliotis, Manolis & Papadaki, Maria & Menounou, Penelope & Dimopoulos, Panayotis & Kolokotsa, Dionysia & Paravantis, John A. & Tsangrassoulis, Aris & Panaras, Giorgos & Giannako, 2023. "Green roofs as a nature-based solution for improving urban sustainability: Progress and perspectives," Renewable and Sustainable Energy Reviews, Elsevier, vol. 180(C).
    2. Taleghani, Mohammad, 2018. "Outdoor thermal comfort by different heat mitigation strategies- A review," Renewable and Sustainable Energy Reviews, Elsevier, vol. 81(P2), pages 2011-2018.
    3. Moreno, Álex & Chemisana, Daniel & Lamnatou, Chrysovalantou & Maestro, Santiago, 2023. "Energy and photosynthetic performance investigation of a semitransparent photovoltaic rooftop greenhouse for building integration," Renewable Energy, Elsevier, vol. 215(C).
    4. Daniel Mora-Melià & Carlos S. López-Aburto & Pablo Ballesteros-Pérez & Pedro Muñoz-Velasco, 2018. "Viability of Green Roofs as a Flood Mitigation Element in the Central Region of Chile," Sustainability, MDPI, vol. 10(4), pages 1-19, April.
    5. Vera, Sergio & Pinto, Camilo & Tabares-Velasco, Paulo Cesar & Bustamante, Waldo, 2018. "A critical review of heat and mass transfer in vegetative roof models used in building energy and urban enviroment simulation tools," Applied Energy, Elsevier, vol. 232(C), pages 752-764.
    6. Carlo Alberto Campiotti & Carlo Bibbiani & Alberto Campiotti & Evelia Schettini & Corinna Viola & Giuliano Vox, 2016. "Innovative sustainable strategies in agro-food systems and in buildings for energy efficiency," RIVISTA DI STUDI SULLA SOSTENIBILITA', FrancoAngeli Editore, vol. 2016(2), pages 79-96.
    7. De Valck, Jeremy & Beames, Alistair & Liekens, Inge & Bettens, Maarten & Seuntjens, Piet & Broekx, Steven, 2019. "Valuing urban ecosystem services in sustainable brownfield redevelopment," Ecosystem Services, Elsevier, vol. 35(C), pages 139-149.
    8. Jim, C.Y., 2014. "Air-conditioning energy consumption due to green roofs with different building thermal insulation," Applied Energy, Elsevier, vol. 128(C), pages 49-59.
    9. Arunima Sarkar Basu & Francesco Pilla & Srikanta Sannigrahi & Rémi Gengembre & Antoine Guilland & Bidroha Basu, 2021. "Theoretical Framework to Assess Green Roof Performance in Mitigating Urban Flooding as a Potential Nature-Based Solution," Sustainability, MDPI, vol. 13(23), pages 1-34, November.
    10. Berardi, Umberto & GhaffarianHoseini, AmirHosein & GhaffarianHoseini, Ali, 2014. "State-of-the-art analysis of the environmental benefits of green roofs," Applied Energy, Elsevier, vol. 115(C), pages 411-428.
    11. Xiaobin Yang & Zhilong Chen & Hao Cai & Linjian Ma, 2014. "A Framework for Assessment of the Influence of China’s Urban Underground Space Developments on the Urban Microclimate," Sustainability, MDPI, vol. 6(12), pages 1-31, November.
    12. Yi-Yu Huang & Tien-Jih Ma, 2019. "Using Edible Plant and Lightweight Expanded Clay Aggregate (LECA) to Strengthen the Thermal Performance of Extensive Green Roofs in Subtropical Urban Areas," Energies, MDPI, vol. 12(3), pages 1-27, January.
    13. Irene Zluwa & Ulrike Pitha, 2021. "The Combination of Building Greenery and Photovoltaic Energy Production—A Discussion of Challenges and Opportunities in Design," Sustainability, MDPI, vol. 13(3), pages 1-29, February.
    14. Refahi, Amir Hossein & Talkhabi, Hossein, 2015. "Investigating the effective factors on the reduction of energy consumption in residential buildings with green roofs," Renewable Energy, Elsevier, vol. 80(C), pages 595-603.
    15. Yara Nasr & Henri El Zakhem & Ameur El Amine Hamami & Makram El Bachawati & Rafik Belarbi, 2024. "Comprehensive Assessment of the Impact of Green Roofs and Walls on Building Energy Performance: A Scientific Review," Energies, MDPI, vol. 17(20), pages 1-52, October.
    16. Manso, Maria & Teotónio, Inês & Silva, Cristina Matos & Cruz, Carlos Oliveira, 2021. "Green roof and green wall benefits and costs: A review of the quantitative evidence," Renewable and Sustainable Energy Reviews, Elsevier, vol. 135(C).
    17. Mansoureh Gholami & Alberto Barbaresi & Patrizia Tassinari & Marco Bovo & Daniele Torreggiani, 2020. "A Comparison of Energy and Thermal Performance of Rooftop Greenhouses and Green Roofs in Mediterranean Climate: A Hygrothermal Assessment in WUFI," Energies, MDPI, vol. 13(8), pages 1-15, April.
    18. Bruno, Roberto & Bevilacqua, Piero, 2022. "Heat and mass transfer for the U-value assessment of opaque walls in the Mediterranean climate: Energy implications," Energy, Elsevier, vol. 261(PA).
    19. Coma, Julià & Pérez, Gabriel & Solé, Cristian & Castell, Albert & Cabeza, Luisa F., 2016. "Thermal assessment of extensive green roofs as passive tool for energy savings in buildings," Renewable Energy, Elsevier, vol. 85(C), pages 1106-1115.
    20. Stevanović, Sanja, 2013. "Optimization of passive solar design strategies: A review," Renewable and Sustainable Energy Reviews, Elsevier, vol. 25(C), pages 177-196.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1608-:d:415379. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.