Energy Sustainability of Urban Areas by Green Systems: Applied Thermodynamic Entropy and Strategic Modeling Means

Abstract

Global warming, anthropogenic pressure, and urban expansion at the expense of green spaces are leading to an increase in the incidence of urban heat islands, creating discomfort and health issue for citizens. This present research aimed at quantifying the impact of nature-based solutions to support decision-making processes in sustainable energy action plans. A simple method is provided, linking applied thermodynamics to physics-informed modeling of urban built-up and green areas, high-resolution climate models at urban scale, greenery modeling, spatial georeferencing techniques for energy, and entropy exchanges evaluation in urban built-up areas, with and without vegetation. This allows the outdoor climate conditions and thermo-hygrometric well-being to improve, reducing the workload of cooling plant-systems in buildings and entropy flux to the environment. The finalization and post-processing of obtained results allows the definition of entropy footprints. The main findings show a decrease in greenery’s contribution for different scenarios, referring to a different climatological dataset, but an increase in entropy that becomes higher for the scenario with higher emissions. The comparison between the entropy footprint values for different urban zones can be a useful support to public administrations, stakeholders, and local governments for planning proactive resilient cities and anthropogenic impact reduction and climate change mitigation.

1. Introduction

Sustainability is a very broad and complex concept in the current climate change and global warming situation [1,2,3,4,5,6,7,8].

In the urban context, the concept of sustainability implies that urban areas should be considered open thermodynamic systems, thus requiring physical definition of entropy, which some of the literature has interpreted based on the connections between entropy and irreversibility [9].

From literature studies comparison, it is deduced that different applications of entropy to urban systems can generate misunderstanding, especially if applied to bioenergetics, ecology, and evolutionary biology [10,11,12,13]. Nevertheless, with the Second Law of Thermodynamics, strong physical approaches can be formulated, at large and small scales, and crucial insights into the sustainability and efficiency of urban processes can be drawn. In detail, statistical entropy, including Shannon’s diversity, and informational entropy have been applied to generally describe diversity, heterogeneity, and spatial patterning of biological systems [14].

While some research studies are oriented towards the study of entropy flows as indicators of urban ecosystems’ integrity [15], others aim to apply thermodynamics entropy to the environment [16], often combining GIS and remote sensing techniques [17]. For instance, there are several literature studies on entropy assessment at large and small scales [18,19,20,21,22,23,24,25], using spatial quantitative–qualitative analyses, implementing GIS, Spearman correlation models, and fractal models. The “low-entropy city” has been explored by considering the role of urban green infrastructure [26,27,28,29], also referring to Prigogine and Stengers [30] and, for the Italian context, to [31]. Recent research has used information entropy, highlighting that the higher the value of information entropy, the more diverse the land use functions [32].

Similarly, a recent study has highlighted that urban microclimate models can play an important role for sustainability, management of energy sources, urban design, and urban heat island (UHIs) control [33]. Here, the Shannon entropy coefficient has been used to study spatial and temporal entropy connected to the effective temperature index (ET) [34]. Furthermore, the urban entropy patterns [35] and Entropy Life Cycle Assessment have been applied as a screening tool for ecological footprint [36]. In this context, accurate urban-scale modeling, for energy and entropy analyses, requires high-resolution spatial data (on the order of kilometers) to incorporate orographic effects and reliably represent anthropogenic emission sources [37,38,39].

Moreover, comparisons of climatic historical data with high-resolution climate models (i.e., the so-called Convection Permitting regional Models, CPMs) to obtain results applicable to future scenarios and quantify impacts across different projections are necessary [40,41,42,43,44,45] to study entropy dynamics. In fact, CPMs, derived from downscaling regional climate models [37,38], allow more accurate simulations of energy balance, providing detailed information on the statistics and return periods of extreme events.

In this literature background, our method fills the literature’s lack of physics-thermodynamics-informed modeling for the entropy assessment of urban areas. Starting from the physics of the built urban system, through the application of the Laws of Thermodynamics, develops a simple, easy-to-implement method useful for quantifying, via georeferenced modeling, entropy flows at transient conditions and entropic impacts of built-up urban areas, evaluating the effects of greenery to counteract climate change. This approach aligns with the IPCC (2025) [46] issues.

The method is based on a simplified and easy-to-implement model (without the need for large amounts of measured/detected data or obtained from the literature evidence and/or specific data sources). Entropy is the physical quantity that can be used to define the anthropic weight within the evolution of physical phenomena (i.e., heat exchanges at urban scale). The city is one of the contexts in which the environment is most forced by human action from morphology (relationship between built-up and green spaces) and energy use point of view. Therefore, the entropy variation (with and without greenery) can provide a measure of sustainable scenarios, considering the future trends of global warming.

The method provides fundamental indicators, i.e., the entropy footprints, and enables spatial assessment of energy use and entropy fluxes and environmental stress, distinguishing high-impact zones where an increase in greenery and climate adaptation strategies are most urgent. By applying entropy footprints analysis, it is possible to find out effective solutions for urban design, orienting policies and social action towards resilience and sustainability transitions and energy planning for mitigation of UHIs.

2. Materials and Methods

Energy and entropy analyses of the urban system are implemented by means of the First and Second Law of Thermodynamics application for two conditions, i.e., with and without greenery areas.

The proposed approach integrates simplified model of greenery and a building system to compute leaf surface temperatures, thermal energy exchanges, and entropy balance evaluations of any urban area. We use the by now common term “sustainability” to mean low environmental impact, energy quality, and durability [39,47]. So, the production of entropy is related to energy flows, thermal exchanges, and not to material waste or pollution. Therefore, the entropy “discharged” into the surrounding environment is read as waste heat only. Referring to [39,47], we start from the concept that real sustainability can only be obtained if total irreversible entropy flux, due to anthropogenic weight, is lower than the entropy flux from the Sun.

After an introduction to the general theoretical framework (Section 2.1), Section 2.2 briefly shows and discuss how to organize the climate database and the classification of urban green spaces, which are necessary inputs to identify and estimate all the model parameters, applied at the case study of a Florentine area in Italy (Section 2.3).

The process and method implementation are provided in the flowchart of Figure 1 and cited as a reference.

2.1. Theoretical Framework

This study introduces a thermodynamics-informed entropy footprint method to quantify entropy exchanges in urban systems. The urban environment, conceptualized as an open thermodynamic system, is analyzed under two conditions—with and without greenery—through the application of the First and Second Laws of Thermodynamics. The energy balance is applied initially to buildings, and subsequently to the vegetation coverage, treating the leaf surface as continuous, homogeneous, and isotropic. This simplification enables the estimation of the leaf surface temperature and the resulting heat exchange with the environment—via latent and sensible heat—while neglecting physiological and soil interactions, i.e., disregarding the morphology, the physiological and water processes, of plant and leaves, with the soil and the atmosphere (point 4 of the flowchart in Figure 1).

The energy balance at steady-state conditions is described by governing equations applied to the composed system vegetation-air-building. The Second Law of Thermodynamics allows us to compute entropy variations resulting from the building’s heat exchange and solar energy gains, as shown in the flowchart (branches 5–6 of Figure 1). Energy balance at the leaf surface is based on the method proposed in [48], which combine contributions of solar radiation, infrared emission, convection and evapotranspiration:

$$GHI·{\alpha }_l= {\epsilon }_l·\sigma ·{\left(T_l + 273\right)}^4 + k_1{\left({\displaystyle \frac{v_w}{D_l}}\right)}^{0.5}\left(T_l - T_a\right) + L_l\left({\displaystyle \frac{{\rho }_{vl} - {RH}_a{\rho }_{va}}{r_l + k_2\frac{D_l^{0.3}{W}_l^{0.2}}{{v_w}^{0.5}}}}\right)$$

(1)

GHI is the global horizontal solar radiation on a specific site [W/m²], T_l and T_a are, respectively, the surface temperature of the leaves and the ambient temperature [°C]; v_w is the wind velocity [m/s] and RH_a is the relative humidity of the air [-]. σ is the Stefan–Boltzmann constant [5.67 × 10⁻⁸ W/m² K⁴]. Parameters such as absorption coefficient α_l [-], emissivity ε_l [-] and evaporation resistance of leaves r_l [s/m], latent heat of vaporization L_l [kg/m³], vapor density at the surface leaves’ temperature (ρ_vl) and at the ambient temperature (ρ_va) [kg/m³], the characteristic dimension of leaves, in the direction transverse (W_l) and parallel (D_l) [m] to the wind, and empirical coefficients k₁ [J m⁻² s^−1/2 °C⁻¹)] and k₂ [s^1/2 m⁻¹] are taken from the literature experimental evidence [48].

Starting from the knowledge of the leaves’ surface temperature, the green coverage is considered as an infinitely extended flat surface neglecting its thickness.

Therefore, the simplified energy balance in absence and presence of green coverage is expressed by the following equations (from Equations (2)–(5)). In particular, in Equations (2) and (3), the quantity of heat released into the air under the green coverage is considered. In Equations (4) and (5), the same quantity is described without greenery.

In the absence of green cover:

$${\alpha }_{a}GHI+h_g{(T}_g-T_{a-g})+{\epsilon }_{effective}\sigma \left(T_g^4-T_{a-g}^4\right)-m{c}_p(T_{a-g}-T_a)=0$$

(2)

$$(1-{\alpha }_a){\alpha }_{g}GHI-h_g{(T}_g-T_{a-g})-{\epsilon }_{effective}\sigma \left(T_g^4-T_{a-g}^4\right)=0$$

(3)

In the presence of the green coverage:

$${\epsilon }_{effective}\sigma \left(T_l^4-T_{a-g}^4\right)-h_g{(T}_{a-g}-T_g)+h_l{(T}_l-T_{a-g})-m{c}_p(T_{a-g}-T_a)=0$$

(4)

$${\epsilon }_{effective}\sigma \left(T_l^4-T_g^4\right)+h_s{(T}_{a-g}-T_g)+{\epsilon }_{effective}\sigma \left(T_{a-g}-T_g\right)=0$$

(5)

with

$${m=v}_w{\rho }_a$$

(6)

The meaning of the physical parameters used is ${\alpha }_a$ air adsorption coefficient, assumed 0.2 [48] in the visible range, while in the infrared wavelength range is taken equal to the emissivity ${\epsilon }_a$ 0.9; $h_s$ is the heat transfer coefficient by convection between the ground and the air above [W/m² °C] (calculated through dimensionless numbers and their connected empirical correlations); $T_g$ is the ground surface temperature [°C]; $T_{a - g}$ is the air temperature above the ground [°C]; ${\epsilon }_{effective}$ is the effective emissivity between two bodies/surfaces as a function of the emissivity of each: ${\epsilon }_{effective}={\left({\displaystyle \frac{1}{{\epsilon }_1}} + {\displaystyle \frac{1}{{\epsilon }_2}} - 1\right)}^{-1}$ (e.g., for radiative heat exchange between the ground and the air above it, ${\epsilon }_1$ and ${\epsilon }_2$ are referred to them; ${\epsilon }_{effective}$ is different for each term); $m$ is the air flow rate on the considered surface as a function of wind velocity [kg/s m²]; ${\rho }_a$ is the air density at ambient temperature [kg/m³]; $T_a$ is the air temperature according to climate data [°C]; ${\alpha }_g$ is the ground adsorption coefficient [-] (e.g., 0.85 for asphalt, and the same value is used for emissivity ${\epsilon }_g$ [48]); and $T_l$ is the leaves’ surface temperature [°C] from Equation (1); $h_l$ is the heat transfer coefficient by convection between the leaves and air below [W/m²K] (for natural convection as a function of dimensionless numbers and connected correlations [49]).

The balances iteratively solved allow for determining the air temperature, affected by the presence of green cover, and the sunny ground without greenery, replacing, respectively, T_e and T_e−G of Equations (7) and (8). For this computation the overall average over summer hours of all the parameters must be considered. Due to the boundary conditions of central hours of some summer days, the leaf temperature can be lower than that one of the surrounding environment. Taking into account this phenomena, the outside air temperature values tend to decrease. The heat exchange of the building (branch 5 of the flowchart in Figure 1) in the absence (Q_b) and presence of greenery (Q_b−G), can be calculated as follows:

$$Q_b=U A_b\left(T_e - T_i\right)$$

(7)

$$Q_{b-G}=U A_b\left(T_{e-G}-T_i\right)$$

(8)

where T_e is the external air temperature without the green covering, T_e−G is the external air temperature with greenery and T_i the internal design temperature of the building (28 °C) in compliance with [50,51], A_b is the total dispersing surface and U the overall heat exchange coefficient that depends on the type of construction and average age of each building provided in [52]. In particular, the overall heat transfer coefficient U is related to the type of construction and materials used, which depend on the building’s age. It significantly affects the amount of heat exchanged between the building and environment, mutual radiative heat exchange, increasing the overall entropy.

Considering the most significant climatic condition, i.e., summer, the total heat transfer occurs from the external environment to the internal one of building. Assuming a seasonal coefficient of performance of 2.7 (SCOP) the heat released to the external environment via a cooling system can be calculated in the absence and presence of greenery (Equations (9) and (10)), from the First Law of Thermodynamics:

$$Q_{b\_rel}=Q_b{\displaystyle \frac{\left(1+SCOP\right)}{SCOP}}$$

(9)

$$Q_{b-G\_rel}=Q_{b-G}{\displaystyle \frac{\left(1+SCOP\right)}{SCOP}}$$

(10)

Referring to the Second Law of Thermodynamics (branch 6 of the flowchart in Figure 1), the entropy variation [J/K] of a building at real environmental conditions, in the absence and presence of greenery, can be calculated with the following equations:

$${∆S}_b=Q_{b\_rel}\left({\displaystyle \frac{1}{T_e}}\right)$$

(11)

$${∆S}_{b-G}=Q_{b-G\_rel}\left({\displaystyle \frac{1}{T_{e-G}}}\right)$$

(12)

with T_e and T_e−G in Kelvin.

Using temperatures T_e, T_e−G, and Q_Sun (i.e., horizontal global solar radiation), A_urban is the total surface and α_m the mean absorption coefficient of the urban area (the latter is usually 0.6); the Sun entropy that naturally exists can be assessed by Equations (13) and (14):

$${∆S}_{Sun, e}={\alpha }_m{A}_{urban}\left({\displaystyle \frac{Q_{Sun}}{T_e}}\right)$$

(13)

$${∆S}_{Sun, e-G}={\alpha }_m{A}_{urban}\left({\displaystyle \frac{Q_{Sun}}{T_{e-G}}}\right)$$

(14)

Applying the Second Law of Thermodynamics, two entropy footprints (hereinafter referred to ef and ef*) can be calculated. ef and ef* are, respectively, the entropy weight of the urban built-up area in the absence and presence of greenery. They are expressed by the ratio between the entropy variation in buildings and the entropy variation due to the solar energy gain of the area. They provide a measure of the irreversibility produced in the environment and are calculated as follows:

$$ef={\displaystyle \frac{{∆S}_b}{{∆S}_{Sun, e}}}$$

(15)

$$e{f}^{*}={\displaystyle \frac{{∆S}_{b-G}}{{∆S}_{Sun, e-G}}}$$

(16)

Finally, a new irreversibility indicator can be defined by the ratio between ef and ef*. This ratio expresses the building’s entropy contribution when greenery is present compared to that generated in the absence of greenery.

2.2. Modeling Approach, Data Sources, and Pre-Processing

The proposed method, applied to a real urban area (Florence, Italy), is structured and follows the flowchart provided in Figure 1. The thermo-hygrometric modeling incorporates microclimatic influences and vegetation contributions inside the Urban Canopy Layer (UCL)—defined as the volume between the ground and the top of buildings and vegetation [53].

This region, where urban climate dynamics directly affect human comfort and energy demands [54], is characterized through high-resolution climate modeling and vegetation assessment. The modeling process includes remote sensing, meteorological data, and GIS-based building classification. Our research focuses on UCL-UHI because it is commonly characterized by field measurements and is the zone where thermal–hygrometric comfort, environmental quality, and people well-being are assessed [54]. Climatic inputs are derived from historical records and future projections using regional climate models. The climate change scenarios are based on the Very High-Resolution PROjections for Italy (VHR_PRO_IT) under RCP4.5 and RCP8.5 pathways [55,56]. These models provide hourly data at 2.2 km resolution for five datasets: historical (1981–2005), near-term, and long-term projections under both RCP4.5 and RCP8.5 forcing scenarios (Appendix A.1,). Starting from the modeling of past and future climate scenarios, five datasets are obtained: (i) historical years 1981–2005, (ii) RCP4.5 years 2006–2038, (iii) RCP4.5 years 2039–2070, (iv) RCP8.5 years 2006–2038, and (v) RCP8.5 years 2039–2070.

Each dataset includes air temperature, humidity, wind speed, and solar radiation that are collected with hourly frequency, for the entire time span over which they are defined. To apply entropy balances, a typical weather year (TWY) is generated for each dataset, condensing 30-year data into an annual sequence while maintaining representativeness (Appendix A.2). Details on climatological modeling and climate change scenarios construction are provided in Appendix A.1.

The characterization of greenery can be carried out by a dedicated modeling (branch 1 of the flowchart in Figure 1). Green cover surfaces and structural characteristics of urban vegetation (e.g., different species, total height, and diameter at breast height) can be collected either through the public tree census (which provides information on trees located in public areas) or by using optical and LiDAR remote sensing data, when census information is not available. This is the case of private green areas (i.e., private gardens), which still play a pivotal role in the total share of urban green surfaces. Moreover, each plant, with all its properties for specific species, can easily be referenced and connected to the urban area with GIS techniques.

In particular, to characterize green surface and tree species, the individual tree crown segmentation algorithm (ITC), based on the Canopy Height Model (CHM) have to be applied, using a variable moving window size [57]: treetops of trees in the upper canopy strata are identified through the local maxima algorithm [58] with a variable window filter [59]. The moving window [60] is used to scan the CHM so that any cell found to be the highest within the window can be classified as a treetop. The Marker-Controlled Watershed Segmentation algorithm [61] is then applied to delineate urban tree canopies.

In particular, the ITC algorithm allows for the quantification of the (i) number of urban trees (also including those not covered by the public tree census), and (ii) their canopy extent. Using available in situ data, or from Green-Trees Census and Municipal Green Plans, it is possible to classify each ITC-based canopy into broadleaved/conifer with very high-resolution imagery, by means of the application of machine learning models (e.g., Random Forest [62]).

The model’s accuracy have to be assessed using out-of-bag (OOB) samples [63,64], which consist of data points that are not included in the bootstrap sample used to train a given tree [65]. These samples are used as a validation set for each decision tree in the Random Forest [66]. The OOB error is computed by aggregating the predictions made by each tree on its respective OOB samples. This procedure is usually applied for model’s performance evaluation, without requiring a separate validation set, and provides the accuracy assessment [62].

Practical application to any area, as that one studied, allows the characterization of the existing vegetation and identification of basic species-specific parameters (e.g., the emissivity coefficient in the infrared and the evaporation resistance and/or vapor permeability of leaves). Usually, these parameters are obtained through direct measurement [67] or derived from dedicated studies carried out under local conditions [68]. For example, Tiralla et al. [69] collected information on ε for many different trees, shrubs, and herbaceous species, along with their standard deviation. Similarly, Mashabatu et al. [68] provided some useful information about the mean specific leaf resistance of five fruit tree orchard species. In Section 2.3, the above modeling method implementation is explained and adapted for the specific greenery of the urban area studied.

2.3. The Case Study: Typical Built-Up Urban Area of Florence

An urban area of the city of Florence (Figure 2), of Tuscany region (Italy), is the case study. The studied urban area belongs to the Municipal Environmental Energy Plan (PEAC) [70] (branch 3 of the flowchart in Figure 1). This built-up urban area is a representative of the city’s urban development and includes buildings of different uses and construction periods (age classes of buildings) (Figure 2, in blue). In this area asphalt and green surfaces are present (Figure 2, in green).

The limits of this area were identified using the spatial boundary provided by the Italian National Institute of Statistics for the year 2011 [71], which includes the densely inhabited areas near recognized aggregation centers, i.e., squares and public complexes (Figure 2, in red). For this reason, the peripheral areas and those with low interaction between green system and built-up zones were excluded. All the necessary environmental parameters are derived from available databases/open-sources and census data from 2011 [71]. Georeferenced building data (i.e., three-dimensional geometry of buildings and their house number and construction age class, thermo-physical features, volume, height, base area, type of plant system, number of occupants, intended use, etc.) were extracted from PEAC [70,72]. By means of database interfacing and necessary information extraction, many useful data were derived from Tuscany Region GEOscopio (https://www.regione.toscana.it/-/geoscopio (accessed on 12 August 2025)) with QGIS software (version 3.40.1-Bratislava) post-processing [73].

The Florence Municipality covers 102.4 km², with a population of 352,622 in the urban area; the urban area case study is 6464 hectares (A_urban), with 115,000 buildings corresponding to 192 million m² total dispersing surface (A_b). The global heat exchange coefficient U [W/m² K] for all these buildings has an average value of 2.1 W/m²K. The intended use of most buildings is “residential”; then follows “services” and “commercial”.

The total dispersing surface of each building is connected with the global heat exchange coefficient as a function of the construction age class of each building, which takes into account the thermo-physical characteristics of the building envelope, as reported in the PEAC database [70,72].

Table 1. Global heat exchange coefficient by building construction age class [W/m²K].

Age Class	<1940	1940–1970	1971–1980	>1980
U [W/m2K]	2.4	1.9	1.8	1.2

shows the value of the global heat exchange coefficient deduced from PEAC.

For green areas computation (branch 1 of the flowchart in Figure 1), the ITC algorithm was applied on the specific CHM, provided by the Florence Municipality for the year 2023, with 0.5 m spatial resolution. Two thresholds were applied at resulting segmentation for filtering out the non-tree features. Selecting trees, with heights between 3 and 35 m, the mean Normalized Difference Vegetation Index (NDVI)) was calculated for each identified tree crown (based on Sentinel-2 data composite for May–July 2023) using a minimum NDVI threshold of 0.5 [74]. Therefore, 273,528 trees, of public and private zones, were identified, for 9,858,840 m² total canopy cover, that is 15% of the total selected urban area.

This resulting total surface is used as the whole leaf-surface coverage (in Equation (1)) that is a square, with 3140 m side length, replacing the values for both the parameters D and W in Equation (1).

The specific trees classification was carried out: segmented tree crowns intersection with tree census points from the city of Florence, which provided a subset of 43,469 tree crowns for coniferous or broadleaf; five percentiles computation (10th, 30th, 50th, 70th, and 90th) based on four predictors and for each identified crowns [75] (i.e., the CHM and three bands from a high-resolution Florentine orthophoto; 0.2 m, for year 2023). Then, 20 predictors available for each tree crown were computed. Random Forest (RF) model [62] allowed for predicting tree species class (coniferous or broadleaf) using the above 20 predictors as input features. The Google Earth Engine RF classifier was exploited, setting the number of decision trees to 300. Modeling performance was evaluated by the OOB aggregating predictions and comparing reference data, and then constructing a confusion matrix. Due to the unbalance inside the dataset (i.e., 6269 coniferous trees vs. 37,200 broadleaves), the RF classification model performance was assessed by the Matthews Correlation Coefficient (MCC) [76,77] based on the confusion matrix. In detail, the MCC is a correlation measure between categorical predictions and observations: it equals 1 for perfect predictions, 0 for random agreement, and −1 when predictions and observations are entirely mismatched. The trained model was applied to the 273,528 tree crowns remaining, using the 20 predictors to classify all of them into coniferous and/or broadleaves, with a 0.83 MCC value. This classification provided a subset of 43,469 tree crowns (coniferous or broadleaf) according to the municipality’s tree census.

Therefore, for the present study, the overall canopy coverage turned out 760,191 m² conifers and 9,098,648 m² broadleaves. Their mean emissivity values (inside the 8–14 μm interval) were derived from [78]: the emissivity was 0.982 ± 0.009 for conifers and 0.985 ± 0.010 for broadleaves; 0.985 emissivity mean value, weighted by the covering surface. Fundamental literature evidence [79] was used for the evaporation resistance of the oak leaves (i.e., 2000 s/m) that was applied to conifers and broadleaves trees. The used values of vapor diffusivity, despite not directly available for conifer species, and absorption coefficient of solar radiation 0.8, refer to [48].

Using to the aforementioned greenery parameters, and to the TWYs for each dataset, all the balances of Equations (1)–(5) were carried out for summer periods, i.e., from 01/04 to 31/10.

From the available hourly data, the hot hours (HHs) were selected, matching the following boundary conditions:

$$GHI > 200{\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2}} 0.1{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}< v_w < 10.7{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}} T_a > 28 °\mathrm{C}$$

Those constraints identify the set of hot hour measurements, such that all the balances have physical meaning. The conditions on the solar radiation GHI and the exclusion of small values of v_w ensure thermodynamic balance validity [56]. Extreme wind conditions are avoided, taking into account the Beaufort wind force scale, i.e., an empirical scale that relates wind velocity to observed conditions at sea or on land: for the upper bound chosen, the scale indicates a condition of fresh breeze, characterized by the presence of wind that begins to move the large branches, but not the entire structure of the tree [79].

Filtering the summer data with the above boundary conditions, all the balances are solved for the two configurations (i.e., green presence and absence) in order to determine the hourly values of leaf temperature T_l, air volume temperature T_a−g, and ground temperature T_g. The average values of the external temperature with green coverage are

$${{T_{e-G}=<T}_{a-g}}^{withG}{>}_{HH}$$

and without green coverage

$${{T_e=<T}_{a-g}}^{withoutG}{>}_{HH}$$

where the < >_HH represents the average over the selected hours (HH).

The mean values are used to solve Equations (7)–(14) and calculate the entropy footprint of Equations (15) and (16). The average value of GHI over the HH subset was used for the solar radiation Q_Sun in Equations (13) and (14).

3. Results

All the balances of Section 2 are solved under the hypothesis and boundary conditions extensively described above, obtaining the main following results: hourly summer temperature for the leaf coverage T_l; bottom mean air temperature T_e−G, i.e., between the leaf coverage and ground; mean air temperature above the ground without leaf coverage, T_e; cooling energy need of buildings in the presence and absence of greenery, respectively (Q_b−G and Q_b); heat released to the external environment, in the presence and absence of greenery (Q_b−G_rel and Q_{b_rel}); entropy increase, in the presence and absence of greenery, i.e.,${ ∆S}_{b - G}$ and ${∆S}_b$; entropy footprints ef and ef*, due to the aforementioned thermodynamic process; and entropy increases due to the Sun radiation hitting the ground ${∆S}_{Sun, e - G}$ and ${∆S}_{Sun, e}$.

In particular, a preliminary global sensitivity analysis was carried out, in order to evaluate how much greenery properties (i.e., emissivity, vapor resistance, and absorption coefficients) impact the balances and affect results. This sensitivity analysis showed a variation in leaf temperature of no more than one hundredth of a degree, and therefore, an absolute error lower than 2%, also showing the method robustness and its applicability to different contexts. Therefore, in the ranges of all the possible values for parameters, the leaf temperature variation obtained is negligible compared to the contribution of the parameters related to the boundary conditions (solar radiation, wind velocity, relative humidity).

Table 2. Studied area results: main temperatures and entropy footprints with and without greenery.

	Standard TWY	Historical TWY	RCP4.5 2006–2038 TWY	RCP4.5 2039–2070 TWY	RCP8.5 2006–2038 TWY	RCP8.5 2039–2070 TWY
$T_i$ [°C]	28.0
$A_{urban}$ [m2]	64,645,324
$A_b$ [m2]	192,582,468
U [W/m2K]	2.1
$T_e$ [°C]	31.8	32.5	33.0	33.1	32.7	33.8
$T_{e - G}$ [°C]	31.3	32.1	32.5	32.7	32.3	33.4
$GHI [$ W/m2]	641.3	610.6	606.4	589.6	597.0	581.3
$Q_b$ [MWh]	1540	1831	2016	2085	1897	2337
$Q_{b - G}$ [MWh]	1347	1670	1844	1921	1731	2174
$Q_{b\_rel}$ [MWh]	2110	2509	2763	2857	2600	3202
$Q_{b - G\_rel}$ [MWh]	1846	2289	2527	2633	2372	2979
${∆S}_{Sun, e}$ [MJ/K]	293,658	278,941	276,589	268,791	272,575	264,473
${∆S}_{Sun, e - G}$ [MJ/K]	294,117	279,303	276,973	269,146	272,942	264,819
${∆S}_b$ [MJ/K]	24,925	29,566	32,510	33,597	30,620	35,578
${∆S}_{b - G}$ [MJ/K]	21,836	27,008	29,777	31,001	27,967	35,006
$ef$ [%]	8.5%	8.5%	9.7%	10.4%	9.1%	12.1%
$e{f}^{*}$ [%]	7.4%	7.6%	8.7%	9.4%	8.1%	11.1%
$ef/e{f}^{*}$ [%]	87.5%	89.1%	89.6%	90.6%	89.2%	91.8%

shows the main parameters for the five investigated scenarios, referring to [80]: Standard TWY, Historical TWY, “RCP4.5 years 2006–2038” TWY, “RCP4.5 years 2039–2070” TWY, “RCP8.5 years 2006–2038” TWY, and “RCP8.5 years 2039–2070” TWY.

It must be noticed that the entropy footprints are already updated according to the weighted correction b found in Equation (A2), that is −0.021. The calibration effect is provided by Figure 3 (top graph): entropy footprints for the different datasets, highlighting the distance between the entropy footprints value calculated in respect to the same period (1981–2005) but with different sources (Standard TWY vs. Historical TWY).

After calibration, the entropy footprints value is shown in Figure 3 (center graph) that also provides results for the dataset of years 1981–2005, varied by 0.2% at maximum.

Comparing ef* with ef, and considering their ratio, a drop in the entropy production can be found, due to the green coverage presence. All this acts by modifying the environmental constraints of temperature (only the humidity changes are considered in Equation (1) for leaf temperature calculation) creating cooler boundary conditions for the building plants, which can operate with a reduced external thermal load during summer.

For all the scenarios, the entropy decreases by almost 1% in respect to the contribution of the Sun radiation hitting the ground. The low value is due to that reference at the denominator, which is relatively high.

Using the percentage between the values of each entropy footprint (i.e., ef and ef*), due to the green coverage, the entropy footprint is 9.2–12.5% depending on the considered scenario, but with lower changes resulting in the projections at different scales (Table 2). These trends are provided in Figure 3 and in the bar graph in Figure 4, with a more direct and easier way to understand and compare.

It is important to note that differences between the values of the entropy footprints are small, but statistically significant and important. As a matter of fact, the entropy footprints are close to 10% because they refer to thermal energy coming from the Sun, which, precisely because it exists naturally, can be taken as a basic reference. This is relatively greater than the contribution that buildings and their plant-systems can provide. For this reason, a slight variation (at most 4.3%) in the entropy footprints can be noted.

On the other hand, Table 2 shows the absolute values of the entropy produced, which, for the cases examined alone, reaches the order of magnitude of 10⁵ MJ/K. The difference between the scenarios with and without greenery is 3000 MJ/K for a standard TWY. This means that, if the study is extended to the whole urban area, the impact becomes physically and statistically appreciable, confirming the importance of the presence of greenery, especially for those scenarios with more critical conditions.

The effectiveness of the greenery presence is more evident if the absolute value of the difference in entropy released in the environment by the built-up urban area with or without is considered.

In

Table 3. Maximum temperature [°C] for each dataset.

Standard	Historical	RCP4.5 2006–2038	RCP4.5 2039–2070	RCP8.5 2006–2038	RCP8.5 2039–2070
36.8	40.0	42.9	41.9	43.3	43.7

the ambient temperature T_a that corresponds to the maximum temperature of each dataset is reported: these values were used to evaluate the entropy difference.

∆S, shown in Figure 5.

As a matter of fact, for RCP4.5 there is a smaller increase in the maximum temperature compared to the RCP8.5 scenario, with a lower overheating effect of the buildings, a greater cooling effect by the plants and therefore a lower energy consumption of the heat pump system (HP) for cooling.

4. Discussion

This study proposes a useful tool, based on the physics of real systems, matching thermodynamics (of First and Second Law) with advanced modeling techniques for evaluating green systems effectiveness to counteract climate change and reduce the entropy of the urban built-up system, ensuring responsive resilience cities. The method can be applied at different urban scales.

The study offers crucial indicators based on thermodynamics, i.e., the entropy footprints, which can be used for quantifying sustainability in urban areas, supporting the decision-making policies of local governments and public administrations for urban and energy planning oriented to sustainability and green energy transition.

Comparing the entropy footprint values of different zones of any urban built-up area, effective interventions can be directly oriented to entropy flows reduction in building–plant systems, in the presence and absence of greenery, referring to dynamic climate change scenarios. The entropy analysis of the urban area allows for energy–environmental sustainability evaluation, in terms of the low environmental impact, energy quality, and durability.

By the method application, quantification of energy–environmental benefits due to entropy flows reduction that effectively counteracts of climate change effects can be assessed: it allows a general but significant evaluation of the lower impact due to air conditioning systems when buildings are immersed in greenery (see Figure 4).

From this perspective, the methodological approach, based on entropy footprint, is innovative because, using the Second Law of Thermodynamics, allows the quantification of greenery presence effects on any urban system by a simplified and easy-to-implement approach without the need for large amounts of measured/detected data or obtained from the literature evidence and/or specific data sources. It is robust–rigorous, as it is validated and confirmed by the important relevant literature evidence that by means of energy analyses application to urban systems (i.e., First Law of Thermodynamics) show the real cooling demand reduction in buildings due to greenery presence (e.g., NBS integration, green roofs and green walls), remote sensing data on surface temperature drops, and measured air temperature differences between green and non-green areas in urban systems [54,81,82,83,84,85,86,87,88,89].

The innovative contribution of the method also lies in the fact that, through the quantification of the entropy footprints (ef and ef*) and the indicator obtained from their ratio, a homogeneous scale, of physical significance, can be built, identifying the most critical and vulnerable areas on which to intervene as a priority. This allows us to use and apply a physical-entropy-informed criterion for controlling and planning interventions over time. As a matter of fact, results comparison (Table 2) shows that the contribution due to the greenery decreases for different scenarios referred to the different climatological dataset, but at the same time the increase in entropy becomes higher for the scenario with higher emissions (RCP8.5).

However, it must be considered that the proposed method is simplified because all the thermal properties of building materials are unified in the global heat exchange coefficient that is linked to the age class. Moreover, the present study aims at urban-level analyses rather than at the performance/design of building–plant systems and indoor scenarios assessment. In line with the two Laws of Thermodynamics, building energy balances for summer period, consider the temperature levels at which heat exchanges occur. Furthermore, any increase in greenery or energy efficiency interventions and NBS integration on buildings would certainly lead to a lowering of the ambient temperature together with a reduced energy consumption by the cooling systems. The latter, thanks to the energy efficiency of buildings, would entail significant improvements in environmental impacts decrease, i.e., the entropy production reduction (Table 3).

Another limitation/simplification of the method is due to a basic general evaluation of the overall heat exchange of buildings with the external environment as a function of their overall heat exchange coefficient and volume (disregarding their mass and thermal inertia and heat capacity of specific building materials, linked to the accumulation of heat due to solar radiation). This latter choice is motivated by the fact that most available databases and the literature do not provide specific information on the construction and thermo-physical characteristics of buildings.

However, the method proposed can be directly applied by public administrations, all stakeholders, and local governments in the perspective of planning the energy green and sustainability transition for livable and comfortable urban areas and resilient cities.

These aspects are also supported by the recent literature results [90] that show how at least three quarters of the current carbon storage of urban trees in Central and Southern Europe will be seriously threatened by 2070.

With adequate and available data, the entropy footprint evaluation can provide useful quantification of sustainability due to the current conditions of any urban area and effective analyses of different strategies to combat climate change, UHIs, and anthropogenic effects in cities, in the short and long term, considering that conversion of impervious surfaces into green areas has high costs and requires time [91].

In the sustainability, resilience, and transition perspectives, the method implementation could be used as predictive tool. Indeed, it provides a physical-entropy-informed support for practical applications at large scales, focused on selection of suitable species for NBS and integrated green systems, maintenance of urban functional diversity, and permeability [92,93].

More in general, results comparison, obtained at small scale, show that further applications of the method can be for the management, design, and implementation of energy and environmentally sustainable projects, considering the entropy footprints based on the necessary operating times in relation to the developments of climate change effects over the years.

Future research directions will concern the method’s application to different urban realities, both in terms of complexity and infrastructures and the presence of different green areas and water surfaces (e.g., fountains, rivers, lakes, etc.) for different climatic contexts. A further interesting development of the research will concern the study of entropy footprint at regional level, in the local context, but also at a wider national level.

Starting from the entropy footprints connected to the different development of green areas, intensive vegetation and strong integration of NBS, and the choice of effective strategies in the medium to short term, the method foresees future developments, including its implementation at a large scale, and of an objective function to be minimized/maximized in relation to the decision-making policies and strategic environmental, energy, and proactive resilience choices of specific managerial groups and stakeholders (taking into account economic aspects).

In this perspective, the research will use AI and implement finalized machine learning techniques, especially oriented towards a detailed study of criticalities and vulnerability of the urban areas.

5. Conclusions

This study proposes an integrated method, simple and easy to implement, to quantify the energy–environmental sustainability of urban areas. The research links applied thermodynamics of Second Law to strategic modeling of biophysical thermo-hygrometric balances of leaf surfaces, high-resolution climate at urban scale, identification and characterization of plants for specific species and vegetation, and spatial georeferencing applied techniques for entropy exchanges evaluation within the urban context.

The focus, i.e., the entropy flow reduction and entropy impact of urban built-up areas assessment, with and without greenery, constitutes the innovative thermodynamic-entropy framework to quantify the benefits of urban greening.

The simple method proposed can thus be easily implemented for energy and entropy analyses on a small and large urban scale.

Given that most databases and the literature lack detailed thermo-physical data on buildings, but provide georeferenced and general attributes (e.g., age-based heat exchange coefficients, volume, surface area, height, system type, and use), the proposed simplified method relies on readily available information. It evaluates the energy and entropy balance of the built environment in relation to green space distribution, without accounting for detailed building-system thermo-physics. Thus, this method can be a crucial tool for energy planning and thermo-hygrometric comfort improvement in urban areas. It allows also programming activities of authorities, local municipal governments, disaster management authorities and the same citizens, to better understand the relationship between the potential for reducing anthropogenic impacts, the UHI mitigation, climate change adaptation, housing factors and urban areas development, urban forestry, and NBS and/or green systems integration. The method can be easily applied for quantifying proactive resilience and sustainability transition, as it provides a homogeneous scale of entropy footprints, i.e., irreversibility, as a measure of sustainability, and a rational criterion for comparing and integrating these evaluations. While not suited for deterministic modeling, this approach might serve as a decision-support tool for evaluating ex-ante and ex-post urban energy strategies, including energy savings, renewable integration, and nature-based solutions, through entropy-based assessment (with entropy footprints) of adaptive sustainability and resilience.

Nevertheless, basic limitations of the method concern its effectiveness and applicability, especially when moving from small scale to large scale (i.e., from single building to group of buildings, block and whole urban area, from city to regional scale). Its reliability depends on the availability, quality, and quantity of information and usable data (e.g., climate and local weather conditions, buildings’ features and their air conditioning systems, greenery).

Despite the method, by means of the entropy footprints assessment, which can provide a rational criterion to compare complex innovative and sustainable technology solutions, capable of leading to important irreversibility reduction and energy efficiency increase, its application to different urban areas and then to larger scale of analysis (e.g., regional and national scale) can show possible implementation downsides mainly due to the reliability and typology of necessary data.