# Parametric Study of Thermodynamics in the Mediterranean Courtyard as a Tool for the Design of Eco-Efficient Buildings

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## Abstract

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## 1. Introduction

“Courtyards can be defined as unbuilt spaces which are delineated by the interior facades of the buildings, or those spaces which are situated within the interior alignments of a plot.”[1]

“A space that is open to the sky, typically centralized and with a geometrical shape, frequently surrounded by porticoes, which constitutes the organizing nucleus of the spaces that lay around it, and symbolically represents the community associated to it.”[2]

**Figure 1.**Courtyard’s vertical section. In grey colour: building section. In others colours: air at different temperatures. (

**a**) Stratification; (

**b**) Convection; (

**c**) Flow patterns (Images from the CFD model).

## 2. Numerical Model

_{0}is the relative density. Using the perfect gas law we can approximate the relative density by the relative temperature T/T

_{0}(for more details see [7]). By ν we denote the kinematic viscosity and p is the pressure.

_{w}, the temperature of the wall, by assuming a constant temperature at each hour at each partition of the wall. We compute this temperature as the stationary solution of the partial differential equation, used to solve the following equation:

_{w}is the length of the wall, σ is the Stephan-Boltzman constant, ε is the emissivity of the wall, α is the absorbance of the wall, T

_{ref}is a reference temperature and I is the radiation reaching the wall. By using this form to compute the temperature of the wall we assume that the mean temperature of the wall is almost independent of the temperature in the interior of the building. Although we consider a constant value for the transmittance, the absorbance and the emissivity, it can be considered different values of these constant at each of the walls of the courtyard, depending on the material. Let us remark that this boundary condition can be improved by taking into account the reflections of the solar radiation on the walls of the courtyard. The reference temperature is a parameter of the model which can be defined in terms of the air temperature and measure data.

## 3. Streamlines in Simplified Shape Courtyards

## 4. Temperature Evolution in Simplified Shape Courtyards

**Figure 6.**(

**a**) Non dimensional temperature in the courtyard base; (

**b**) Non dimensional temperature formula from Sánchez [6].

**Figure 7.**(

**a**) Graphic of the non dimensional temperature related to the depth ratio of the courtyard as a heat source at three different heights. Three different flow patterns are shown for D = 0.1, D = 1, D = 5 which correspond with changes of non dimensional temperature. The four lines represent θ at three different heights in the courtyard and the average temperature of the whole courtyard; (

**b**) Non dimensional temperature formula when the courtyard acts as a thermal source [6].

_{max}is θ

_{ref}= 25. At the base of the courtyard fresh air without pollutants is provided (as a heat drain). Re-circulation and mixtures should respond to the previously proposed equation, exchanging a stream of pollutant by a heat stream. This expression presumes a reasonable symmetric behavioural pattern for these natural phenomena mechanics, on the scale studied.

**Figure 8.**(

**a**) Graphic of the non dimensional temperature depending on the depth ratio of the courtyard being a heat drain at three different heights. The figure shows three different flow patterns for D = 0.1, D = 1, D = 5 that corresponds to the non dimensional temperature changes. The four lines represent θ at three different heights in the courtyard and the average temperature of the whole courtyard; (

**b**) Formula for the non dimensional temperature for courtyards acting as a heat drain.

#### 4.1. Depth Ratio D < 1. Non-Mediterranean Courtyards. Example of Courtyard for Winter Conditions

#### 4.2. Depth Ratio D = 1. The Noble Mediterranean Courtyards. Example of a Thermodynamically Balanced Courtyard

**Figure 9.**(

**a**) Plan of the Cathedral of Santiago de Compostela, Galicia, north of Spain; (

**b**) Image of the cloister of the Cathedral of Santiago de Compostela with indicated proportions. Depth ratio (D) = 0.3.

**Figure 10.**Courtyard of Palazzo Farnese. Rome (16 th century). Architects: Antonio da Sangallo the Younger and Michelangelo. Depth ratio of the courtyard D = 1 (

**a**) section (

**b**) plan (

**c**) inside image.

#### 4.3. Courtyards with Depth Ratio D > 1. Mediterranean Popular Patios. Example of a Courtyard with High Air Confinement

#### 4.4. Depth Ratio D >1 Courtyard Connected to the Outside. The Advantages of Unenclosed Deep Courtyards. One Typical Typology of the Mediterranean Deep Courtyards

**Figure 13.**(

**a**) House on Lepanto Street, Seville (Spain). Section; (

**b**) Entrance hall in a house in Medina Sidonia, Cádiz, Spain.

_{2}, etc). Ventilating these spaces improves the living conditions. In vernacular houses this is often achieved by the strategies discussed above, such as cross ventilation from a courtyard to the street. In larger and more complex buildings, where manual interaction is less likely, and in those buildings that demand high conditions of security and comfort, it is necessary to renew the air in these spaces through technological and mechanical means. Better and safer air conditions can be achieved, as the following investigation will show.

## 5. Temperature Evolution in the Courtyard of the Monte Málaga Hotel

_{2}= 2.9. This courtyard is connected to another larger one, D

_{1}= 0.77 (see Figure 14).

**Figure 14.**(

**a**) Image of the Monte Málaga Hotel; (

**b**) the deep courtyard, and (

**c**) section with air circulation diagram.

^{2}per year of electricity (year 2010). This is below the average consumed by hotels with similar characteristics, 165–200 kW h/m

^{2}per year according to the IDAE (Institute for Energy Diversification and Saving of Energy, Ministry of Science and Technology, Government of Spain) [12].

**Figure 15.**Red: Outside monitored temperature. Blue: Monitored temperature in the courtyard (

**a**) Monitored temperatures on the 27th April 2010; (

**b**) Monitored temperatures on the 5th May 2010. The courtyard acts as a heat drain on the morning and on the 27th as a heat source on the night. Monte Málaga Hotel, Málaga, Spain (Design work was completed in 2002 and the construction completed in 2005 by J. M. Rojas and J. R. Montoya, architects).

_{ref}is set taking into account the measured temperature at 7:00 a.m.

#### 5.1. Test 1—Stratification Effect: The Courtyard as a Heat Drain

**Figure 16.**(

**a**) The geometry of the courtyard’s section [Figure 14(c)] has been divided into segments of lines to find the equations that describe them and introduce them into the numerical model; (

**b**) Definition of the courtyard domain with coordinates of the segments in meters (code for Freefem++).

**Figure 17.**Test 1. Stratification of the air in the courtyard by buoyancy effects. Temperature: °C; T: hours.

#### 5.2. Test 2—Linking Solar Radiation, Forced Extraction and Wind

**Figure 18.**Test 2. Air temperatures. Forced extraction and wind. Solar radiation is imposed as boundary conditions. Temperature: °C; T: hours.

**Figure 19.**Test 2. Air velocities. Forced extraction and wind. Solar radiation is imposed as boundary conditions. Velocities: m/s. T: hours.

**Figure 20.**Red: Outside monitored temperature on the 27th April 2010. Blue: Monitored temperature in the courtyard on the 27th April 2010. Orange: numerical results.

## 6. Conclusions

## Acknowledgements

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**MDPI and ACS Style**

Rojas, J.M.; Galán-Marín, C.; Fernández-Nieto, E.D.
Parametric Study of Thermodynamics in the Mediterranean Courtyard as a Tool for the Design of Eco-Efficient Buildings. *Energies* **2012**, *5*, 2381-2403.
https://doi.org/10.3390/en5072381

**AMA Style**

Rojas JM, Galán-Marín C, Fernández-Nieto ED.
Parametric Study of Thermodynamics in the Mediterranean Courtyard as a Tool for the Design of Eco-Efficient Buildings. *Energies*. 2012; 5(7):2381-2403.
https://doi.org/10.3390/en5072381

**Chicago/Turabian Style**

Rojas, Juan M., Carmen Galán-Marín, and Enrique D. Fernández-Nieto.
2012. "Parametric Study of Thermodynamics in the Mediterranean Courtyard as a Tool for the Design of Eco-Efficient Buildings" *Energies* 5, no. 7: 2381-2403.
https://doi.org/10.3390/en5072381