# Potential building resiliency indicators to heat waves suitable for housing renovations

Marie Tanchou, Olivier Greslou, Jean-Marie Alessandrini, Moncef Krarti,
Pascal Stabat

► To cite this version:

Marie Tanchou, Olivier Greslou, Jean-Marie Alessandrini, Moncef Krarti, Pascal Stabat. Potential building resiliency indicators to heat waves suitable for housing renovations. Building Simulation 2023, 18th international Conference of the International Building Performance Simulation Association (IBPSA), Sep 2023, Shanghai, China. hal-05004570

HAL Id: hal-05004570

https://cstb.hal.science/hal-05004570v1

Submitted on 25 Mar 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Potential building resiliency indicators to heat waves suitable for housing renovations

Marie Tanchou¹,², Olivier Greslou¹, Jean-Marie Alessandrini¹, Moncef Krarti³, Pascal Stabat²

¹CSTB, France

²Mines Paris, PSL University, France

³University of Colorado, USA

## Abstract

Thermal comfort and resiliency to heat waves are becoming increasingly important issues for housing renovations. As part of renovating existing building stock, it is becoming necessary to assess and enhance the abilities of buildings to face heat waves through good utilization of their envelope thermal inertia as an alternative of installing air conditioning systems.

This paper proposes metrics to evaluate the potential of a building's thermal inertia to different heat wave sequences. These metrics depend on the building's constructions and the physical properties of the materials used. The proposed metrics for building thermal resiliency are developed and assessed using a series of simulation analyses to account for various thermal stresses associated with heat waves using simplified housing units. Then, the robustness of these metrics is tested using models for a typical housing in France subject to heat wave sequences. Throughout the study, the analysis is conducted using a state-of-art whole building energy simulation tool, EnergyPlus. A resiliency assessment method is proposed based on the calculation of time constant and final temperature.

## Highlights

* An assessment method of the resiliency of buildings to heat waves.

* A comprehensive calculation of the building time constant, based on design features.

* A sensitivity analysis on building parameters to validate the robustness of the resiliency indicators.

## Introduction

In the context of global warming, the Earth will have to face, in the next decades, both an increase in the mean surface temperature, and in the risk of extreme heat wave events. Between 1850 and 2021, the temperature on the Earth's surface has already increased by 1.1°C and this increase should reach 1.5 °C by 2040 (IPCC, 2022). In France, between 1970 and 2016, a mean of 12 heat waves by department was recorded, and, for most departements, more than the half of these heat waves have occured after 2004 (Santé Publique France, 2019). As consequence, the discomfort hours in dwellings have increased and the health risks are an increasing cause of concern. Indeed, the heat wave in Europe in 2003 caused the death of almost 15 000 inhabitants (Pirard et al., 2005).

Furthermore, the renovation of buildings is a major stake in the context of greenhouse gas emission reduction. The retrofit of dwellings is today focused on the reduction of heating consumption in France. It seems essential to consider also the resilience of housings to heat waves.

In order to help building contractors in providing solutions for summer comfort in renovated dwellings, it seems useful to develop an easy-to-calculate indicator of resiliency to heat waves based on design data.

This paper aims to build such an indicator using numerical experimentations with a state-of-the-art building simulation tool.

## Literature review

The question of resiliency of buildings to heat waves has been widely studied in the literature. Some studies try to determine the resiliency of a building stock, or of representative buildings, in terms of summer comfort (Encinas and De Herde, 2013) or in the particular case of a heat wave (Kuczyński and Staszczuk, 2020; Zinzi et al., 2020) or of a urban heat island (Zinzi et al., 2020). They give either indications on comfort, with maximal temperatures or hours of discomfort (Encinas and De Herde, 2013; Kuczyński and Staszczuk, 2020; Zinzi et al., 2020), or on cooling consumption (Kuczyński and Staszczuk, 2020; Zinzi et al., 2020). Such studies inform qualitatively on the most relevant building features that can assist in reducing the dangerous effect of heat waves but the quantitative data are relevant only on the particular case study, under particular climatic sequences. Moreover, most of these data are obtained through simulations or experimentations. Thus, the transposition of the method to other buildings would be time-consuming. That is why, the use of easy-to-calculate resiliency indicators, based on buildings' design data and on heat waves' characteristics, could help in enhancing the global knowledge on the thermal response of buildings face to heat waves, whatever their constructive features or the climatic sequence considered.

A literature review of the existing indicators used to evaluate the thermal response of a building highlights three main indicators: the time lag, the decrement factor and the time constant. Time lag and decrement factor are often assessed together (Asan, 1998; Kontoleon and Bikas, 2007; Quagraine et al., 2020; Sun et al., 2013) and reflect the behaviour of the internal conditions of a building submitted to periodic external sollicitations. There is not any analytical expression to determine them, thus, a simulation or an experimentation is needed. These
indicators are therefore not suitable for a wide range of different building configurations under different climatic sequences.

The third indicator highlighted is the time constant. It represents the thermal response of the internal conditions of a building submitted to a step outside temperature change. This time can be assessed on the temperature evolutions from simulations (Orosa and Oliveira, 2012) or experimentations. However, there are also analytical expressions that allow to calculate this time, either through the resolution of the heat equation (Janssen, 1982; Tsilingiris, 2004), or by parameter identification of a resistance and a capacity (Antonopoulos and Tzivanidis, 1995; Hoffman and Feldman, 1981; Orosa and Oliveira, 2012). The parameter identification of a RC equivalent system is easier than the resolution of the heat equation, but more approximative (Antonopoulos and Tzivanidis, 1995). These approximative results are mainly due to a difficulty to parameter the equivalent capacity. (Antonopoulos and Tzivanidis, 1995; Orosa and Oliveira, 2012; Tsilingiris, 2004). Thus, few articles present an easy-to-calculate time constant indicator and, among them, the proper consideration of the equivalent capacity remains a source of uncertainty in this time constant. Moreover, these indicators are used to evaluate the thermal response of buildings in general but have not been used to assess thermal resiliency of buildings face to heat waves.

If the time constant is the most interesting indicator for our purpose, it appears necessary to adapt it in particular by proposing a new way to assess easily the capacity.

## Methodology

The methodology considered in this study to develop heat wave resiliency indicators that are easy to calculate is based on temperature steps applied to a basic building energy model. Then, new design and operating features are added gradually to the basic building model. At each stage, the indicators are adapted, in order to take into account the effects of all the design and operating features. Finally, the developed indicators are validated using simulation results from a detailed building energy model.

### Proposed metrics

For this study, the resiliency is defined as the time that indoor temperatures exceed a threshold level for the case of an infinitely long heat wave. This time is related to the dynamic response (i.e., time constant) of the building, but also to the internal temperature (i.e., final temperature). These two indicators are considered in this study for different building features.

The time constant describes the dynamic thermal response of a building to a sudden external stimulus. This indicator is based on the thermal-electrical analogy and the assumption that a building can be represented by one resistance and one capacitance as shown in Figure 1. A unique capacitance is a simplified representation of the building dynamics but is not sufficient to evaluate all

oscillation modes. In this case, only the main mode over long term variation is sought.

<figure><img src="image_2.png" /><figcaption>Figure 1: Simplified RC model building representation</figcaption></figure>

The indoor temperature using such an RC system is subject to the following heat balance equation:

$$RC \cdot \frac{dT_{int}}{dt} + (T_{int} - T_0) = T_{exteq} - T_0 \quad (1)$$

The solution of Eq. (1) can be expressed as:

$$T_{int}(t) - T_0 = (T_{exteq} - T_0) \cdot (1 - e^{-\frac{t}{\tau}}) \quad (2)$$

The time constant of the RC system is defined as: $\tau = RC$. The values of R and C are determined based on the building design features. First, a simple building energy model is developed using the whole building energy simulation tool, EnergyPlus. The simple building energy model is exposed to a temperature step representing the average temperature variation during a heat wave. The time constant is the time required by the indoor air temperature to reach 66% of the final temperature amplitude (i.e., the magnitude of the internal temperature step magnitude).

In addition to the time constant, the final temperature is the second indicator that needs to be estimated to evaluate the resiliency of a building subject to heat waves. The final temperature is obtained using the heat balance equation when the building reaches steady-state conditions. Thus, the final temperature differs from the external temperature but is an equivalent external temperature $T_{exteq}$ that accounts for any external solicitations.

With the knowledge of both $T_{exteq}$ and $\tau$, and the initial temperature $T_0$, it is possible to determine the time needed from the indoor temperature to exceed a predefined threshold, and thus, determine the resiliency time to a heat wave.

### External solicitation

The building energy models considered in this study are exposed to time step temperature variations to determine the time constant and the final temperature. Each heat wave is characterized by a sudden change in minimum and maximum outdoor air temperatures during several days (Pascal et al., 2013), as well as in a decrease of wind speed. For the preliminary analysis, the heat wave is represented by a positive step in mean outdoor air temperature with the wind speed set to zero. This setting allows to evaluate the effects of a high increase in the mean outdoor air temperature over a long period without considering the daily temperature variations.

Moreover, all the internal gains are set to zero for the simplified energy model since these are not related to a heat wave. The solar gains through the windows are neglected since perfect external solar shading devices are considered for the preliminary analysis. Thus, the effects
of solar radiation are only considered on opaque walls using an equivalent outside sol-air temperature (Bouchié et al., 2014):

$$T_{ext} = T_{air} + \frac{\alpha_{I}}{h_{ce}} \quad (3)$$

And $T_{air}$ is the air temperature, $\alpha$ is the solar absorptance of the walls, $I$ is the solar radiation on the walls, and $h_{ce}$ is the external convection coefficient.

All weather features are adjusted to minimize potential of the building to thermally discharge to model the worst case conditions for resiliency performance. First, using an equivalent mean outdoor temperature prevents night-time thermal discharge from the building through natural ventilation. Second, the cloud cover is maximized to prevent long wave radiation discharge during nighttime.

The heat wave is represented by the outdoor air temperature variation shown in Figure 2.

<figure><img src="image_3.png" /><figcaption>Figure 2: Outdoor temperature scenario</figcaption></figure>

The temperature $T_2$ in Figure 2 consists in the average equivalent sol-air temperature of the heat wave. For this study, the sequence used is the heat wave that occurred in Lyon during July 2015 and lasted 8 days, based on the IBM thresholds defined by the French government (Pascal et al., 2006). Table 1 lists the values for both T1 and T2 and associated setting periods considered in the simulation analysis. The complete meteorological data for the heat wave were extracted from the Meteo France database, through MEREEN tool. The length of simulation analysis period is extended for several months in order to let the building reach steady-state conditions.

<table><caption>Table 1: Outdoor air temperature settings</caption><thead><tr><td>Temperature</td><td>Value</td><td>Duration</td></tr></thead><tbody><tr><td>T<sub>1</sub></td><td>25 °C</td><td>01/04 → 08/07</td></tr><tr><td>T<sub>2</sub></td><td>48.5 °C</td><td>09/07 → 30/11</td></tr></tbody></table>

### Case-study

These metrics are first developed using a basic simplified building energy model representing a test cell with various additional features. The developed metrics are then verified using results from a more realistic building energy model for a single-family house.

### Test cell

The simplified building energy model is a parallelepipedal test cell with characteristics listed in Table 2. The test cell is attached to the ground, and all the walls have the same composition and thickness. In the preliminary analysis, the heat transfer through ground floor is neglected and is not modelled.

<table><caption>Table 2: Test cell model characteristics</caption><tr><td>Height</td><td>3.5 m</td></tr><tr><td>Length</td><td>5 m</td></tr></table>

<table><tr><td>Width</td><td>5 m</td></tr><tr><td>Openings</td><td>None</td></tr><tr><td>Air leakage</td><td>None</td></tr><tr><td>HVAC systems</td><td>None</td></tr></table>

Many features, as reported in Table 3, of this test-cell model are then gradually modified as part of a series of sensitivity analyses to assess the correlations between both indicators (i.e., time constant and the final temperature) and these modified features.

<table><caption>Table 3: Different configurations of the test-cell model</caption><thead><tr><td>Case-study's element</td><td>Value range</td></tr></thead><tbody><tr><td>Wall's specific capacity</td><td>250 – 1000 – 2000 J/kg.K</td></tr><tr><td>Wall's thickness</td><td>5 – 10 – 15 – 20 cm</td></tr><tr><td>Wall's insulation</td><td>Internal – external – without</td></tr><tr><td>Wall's emissivity</td><td>0.005 – 0.5 – 0.9</td></tr><tr><td>Internal mass (thickness, exchange surface,</td><td>None – Concrete floor (15 cm, 25 m²) – Concrete floor (15 cm, 50 m²) – Concrete floor (20 cm, 25 m²)</td></tr><tr><td>Windows</td><td>None – 1% – 2% – 20% glazing</td></tr><tr><td>Air infiltration</td><td>None – 0.014 m³/s</td></tr><tr><td>Wall's heterogeneity</td><td>All walls with same composition – one wall in wood</td></tr></tbody></table>

### Single-family house

The developed indicators are then verified and refined using analyses using the simulation results obtained for a single-family house. The simulated house is a 100m² single story house, representative of the average sized houses constructed after 1989 in France. The house consists of a dwelling, an attached garage, an attic, and a crawl space. All internal gains due to people, light, and equipment are set to zero for the preliminary analyses conducted for this study.

<figure><img src="image_4.png" /><figcaption>Figure 3: Single family house energy model</figcaption></figure>

Four configurations of the house are tested, to cover a wide range of RC values: with internal walls' insulation and thin external floor insulation (configuration 1), with internal walls' insulation and thick external floor insulation (configuration 2), with external walls' insulation and thin external floor insulation (configuration 3), with external walls' insulation and thick external floor insulation (configuration 4).

The simulation analysis using the energy model of this house aims at the evaluation of new effects on resiliency indicators, not considered by the test-cell energy model
including the impacts of the ground, the ceiling, and any adjoining spaces.

## Discussion of sensitivity analysis results

A series of sensitivity analyses is carried out to evaluate the impact of various building features on the time constant and the final temperature. The time constant and the final temperature are determined from the indoor temperature variations obtained from the simulation results of the test cell model.

### Impact of the walls' thermal mass

The first sensitivity analysis considers walls' thermal mass. This thermal mass varies in terms of specific heat capacity (Cp) and thickness (d). Other properties of the structural material remain the same: $λ = 1.13$ W/m.K and $ρ = 2000$ kg/m³. Figure 4 shows the indoor temperature variations for two thicknesses and three specific heat capacities. The estimated values for both the time constant and the final temperature are listed in the table associated with Figure 4.

<figure><img src="image_5.png" /></figure>

<table><caption>Figure 4: Impact of walls' composition</caption><thead><tr><td>d (cm)</td><td>Cp (J/kg.K)</td><td>τ (h)</td><td>T<sub>final</sub> (°C)</td></tr></thead><tbody><tr><td>10</td><td>250</td><td>4</td><td>45.21</td></tr><tr><td>10</td><td>1000</td><td>11</td><td>45.21</td></tr><tr><td>10</td><td>2000</td><td>20</td><td>45.21</td></tr><tr><td>20</td><td>250</td><td>8</td><td>45.23</td></tr><tr><td>20</td><td>1000</td><td>25</td><td>45.23</td></tr><tr><td>20</td><td>2000</td><td>49</td><td>45.23</td></tr></tbody></table>

As indicated by the results shown in Figure 4, the time constant depends substantially on both Cp and d. On the contrary, the final temperature is only affected by walls' thickness, and thus, walls' thermal resistance even though this effect is rather negligible.

### Impact of the insulation and of its position

The effects of walls' thermal insulation thickness and placement on the time constant and on the final temperature are evaluated in this section. Other properties of the insulation material remain the same: $λ = 0.042$ W/m.K and $ρ = 10$ kg/m³. The non-insulated case is compared to an externally insulated and internally insulated wall configurations. The three configurations and their associated values for time constant and final temperature are shown in Figure 5.

The final temperature depends only slightly on the insulation level and is not affected by the insulation position. The time constant is affected by both position and level of the insulation, but the effect of the position is much more significant than that of the level of the insulation. Indeed, the internal insulation blocks the exchanges between indoor air and the wall thermal mass. On the contrary, the external insulation enhances these exchanges as any exchange with outdoors and indoor air is mostly blocked by the insulation layer.

<figure><img src="image_6.png" /></figure>

<table><caption>Figure 5: Impact of walls' insulation</caption><thead><tr><td>Insulation</td><td>τ (h)</td><td>T<sub>final</sub> (°C)</td></tr></thead><tbody><tr><td>None</td><td>11</td><td>45.21</td></tr><tr><td>Internal</td><td>12</td><td>45.28</td></tr><tr><td>External</td><td>155</td><td>45.28</td></tr></tbody></table>

### Impact of the walls' thermal emissivity

The effects of walls' long wave radiative properties are determined using four walls' emissivity ($ε$) values. The results of the wall emissivity sensitivity analysis are summarized in Figure 6.

<figure><img src="image_7.png" /></figure>

<table><caption>Figure 6: Impact of walls' long wave emissivity</caption><thead><tr><td>ε</td><td>τ (h)</td><td>T<sub>final</sub> (°C)</td></tr></thead><tbody><tr><td>0.9</td><td>11</td><td>45.21</td></tr><tr><td>0.5</td><td>14</td><td>45.60</td></tr><tr><td>0.1</td><td>21</td><td>46.94</td></tr><tr><td>0.005</td><td>27</td><td>48.27</td></tr></tbody></table>

Both the time constant and the final temperature are affected by walls' thermal emissivity. Indeed, exchanges
of the building envelope system with the sky are enhanced when the walls' emissivity increases. Since the walls become rapidly warmer than the sky, they emit heat towards the sky and wall-sky thermal exchanges tend to diminish the heat effectively transferred to the indoor air.

## Impact of air infiltration

The performance of configurations with various air infiltration rates is evaluated. These air infiltrations are implemented both on non-insulated, on externally and on internally insulated configurations, since the results are quite different, depending on the insulation placement. The impacts of air infiltration rate are illustrated in the Figure 7. Both the time constant and the final temperature are affected by the presence of air infiltration. Air infiltration reduces the time constant but increases the final temperature for both insulated and non-insulated configurations. However, the effects of air infiltration are more important for insulated wall configurations. Indeed, for insulated walls, the decrease in the time constant is to 33% against only 9% for non-insulated cases due to air infiltration.

<figure><img src="image_8.png" /><figcaption>Figure 7: Impact of air infiltration</figcaption></figure>

## Impact of glazed windows

The addition of glazed windows in a wall, even in the absence of solar radiation, modifies the wall's performance since it degrades the R and the C values. This impact is presented in Figure 8, for both insulated and non-insulated configurations. The results of Figure 8 are similar to those obtained for the effects of air infiltration shown in Figure 7. The impact of glazed windows on the final temperature is less significant than that on the time constant. The presence of glazed windows results in a decrease in the time constant by 60% for the insulated walls and by 9% for the non-insulated walls. Indeed, the

time constant is reduced when the glazed windows are added to the walls due to both a reduced capacity, and a lower thermal resistance of the building envelope.

<figure><img src="image_9.png" /><figcaption>Figure 8: Impact of glazed windows</figcaption></figure>

## Impact of internal mass

Various configurations of internal mass are considered including both insulated and non-insulated walls. The results of the internal mass sensitivity analysis are depicted in Figure 9.

<figure><img src="image_10.png" /><figcaption>Figure 9: Impact of internal mass</figcaption></figure>

As indicated in Figure 9, the final temperature is not affected by the internal mass, regardless of the insulation

<table><thead><tr><td>Insulation</td><td>Glazing</td><td>τ (h)</td><td>T<sub>final</sub> (°C)</td></tr></thead><tbody><tr><td>None</td><td>None</td><td>11</td><td>45.21</td></tr><tr><td>None</td><td>20%</td><td>10</td><td>45.14</td></tr><tr><td>Internal</td><td>None</td><td>12</td><td>45.28</td></tr><tr><td>Internal</td><td>20%</td><td>5</td><td>45.86</td></tr><tr><td>External</td><td>None</td><td>155</td><td>45.28</td></tr><tr><td>External</td><td>20%</td><td>63</td><td>45.86</td></tr></tbody></table>

<table><thead><tr><td>Insulation</td><td>Int. mass</td><td>τ (h)</td><td>T<sub>final</sub> (°C)</td></tr></thead><tbody><tr><td>None</td><td>No</td><td>11</td><td>45.21</td></tr><tr><td>None</td><td>Yes</td><td>14</td><td>45.21</td></tr><tr><td>Internal</td><td>No</td><td>12</td><td>45.28</td></tr><tr><td>Internal</td><td>Yes</td><td>45</td><td>45.28</td></tr><tr><td>External</td><td>No</td><td>155</td><td>45.28</td></tr><tr><td>External</td><td>Yes</td><td>187</td><td>45.28</td></tr></tbody></table>

<table><thead><tr><td>Insulation</td><td>Air Infiltration Rate</td><td>τ (h)</td><td>T<sub>final</sub> (°C)</td></tr></thead><tbody><tr><td>None</td><td>0 vol/h</td><td>11</td><td>45.21</td></tr><tr><td>None</td><td>0.7 vol/h</td><td>10</td><td>45.63</td></tr><tr><td>Internal</td><td>0 vol/h</td><td>12</td><td>45.28</td></tr><tr><td>Internal</td><td>0.7 vol/h</td><td>8</td><td>46.45</td></tr><tr><td>External</td><td>0 vol/h</td><td>155</td><td>45.28</td></tr><tr><td>External</td><td>0.7 vol/h</td><td>104</td><td>46.45</td></tr></tbody></table>
level and position. On the contrary, the time constant is highly affected by the addition of internal mass, particularly for the case of internal insulation. Indeed, the overall thermal resistance of the building envelope remains the same, but the thermal capacity of the building is highly enhanced when the internal mass is added.

### Impact of boundary conditions (ground temperature)

In this section, the energy model for single-family house is used to investigate the impacts of stationary boundary conditions including the influence of the ground temperature. Specifically, two configurations are modeled for the single-family house, with two different ground temperatures. In both cases, the ground temperature is considered stationary. The results of the impacts of ground temperature on both time constant and final temperature are displayed in Figure 10. Only the final temperature is affected by the ground temperature as the time constant remains unchanged. Indeed, only the indoor temperature is affected by the heat transfer between the building and the ground medium.

<figure><img src="image_11.png" /></figure>

<table><caption>Figure 10: Impact of ground temperature</caption><thead><tr><td>T<sub>gnd</sub> (°C)</td><td>τ (h)</td><td>T<sub>final</sub> (°C)</td></tr></thead><tbody><tr><td>18</td><td>105</td><td>37.53</td></tr><tr><td>10</td><td>106</td><td>34.99</td></tr></tbody></table>

### Discussion of sensitivity analysis results

As outlined by the series of sensitivity analyses, both the final temperature and the time constant are affected by several design features of the building. Specifically, the final temperature depends on all characteristics that can enhance or limit the thermal exchanges between indoor and outdoor conditions. As these exchanges are enhanced, the closer is the final temperature to the outdoor temperature. For the simplified energy model, the indoor-outdoor thermal exchanges are dominated by heat exchanges with the outdoor air and the sky. Thus, the parameters that have noticeable impacts on the final temperature include the level of thermal insulation, the walls' thickness and thermal properties, the walls' radiative properties, the presence of air infiltration, and window to wall ratio. All these parameters affect the overall thermal resistance of the building envelope.

The time constant is affected by the same parameters that impact the final temperature as well as by those that can enhance or limit heat storage within the building, and thus

its thermal capacity. The parameters that affect the internal capacity of buildings include the specific heat capacity and the thickness of walls, the accessibility of thermal mass through the position of thermal insulation, the level of internal mass, and the reduction of thermal mass caused by the presence of glazed windows. The higher are the thermal resistance and capacity of a building, the longer is the period that the indoor temperature takes to reach steady-state conditions, that is the time constant of the building. Finally, the parameter that has the most significant impact on the time constant is the presence of thermal insulation and its position. The effects on the time constant of glazing and air infiltration can also be significant particularly for the case of external insulation.

### Development of resiliency indicators

#### Time constant

As explained in the methodology, the time constant is calculated as the product of equivalent thermal resistance, R, and thermal capacitance, C. The calculated time constant is compared to the time constant estimated based on the results obtained from EnergyPlus simulation.

#### The equivalent thermal resistance

The overall building thermal resistance takes into account the convective, radiative and conductive resistances of all building envelope components including walls, roofs, floors and windows. The air infiltration is also considered by the overall building thermal resistance as expressed by the following equation:

$$R = \frac{1}{h_{inf} + \frac{1}{R_{tr}}} \quad (4)$$

With: $h_{inf} = \frac{q_{air}*Cp_{air}}{S_{tot}}$, which expresses the thermal load for the infiltration, with the air flow rate $q_{air}$, the specific heat capacity of the air $Cp_{air}$, and the total external surface $S_{tot}$. $R_{tr}$ is the global resistance of the walls and $R_{tr} = \frac{1}{h_{re}+h_{ce}} + R_{cond} + \frac{1}{h_{ci}}$. The internal heat radiation between walls is neglected. For the case of a building with n different walls and windows, the conduction equivalent resistance can be written as:

$$R_{cond} = \frac{S_{tot}}{\sum_{i=1}^{n} \left( \frac{1}{R_{cond_i}} * S_i \right) + U_{gl} * S_{gl}} \quad (5)$$

With $U_{gl}$ and $S_{gl}$ the glazing transmission coefficient and surface, and $S_i$ and $R_{cond_i}$ the surface and the conduction resistance related to the wall i.

For each wall i, composed of k layers with a thickness $d_k$ and a thermal conductivity $\lambda_k$, the conduction resistance can be expressed as:

$$R_{cond_i} = \sum_{j=1}^{k} \frac{d_j}{\lambda_j} \quad (6)$$

Throughout this study, the external and internal convective coefficients $h_{ce}$ and $h_{ci}$ are fixed to a mean observed value, equal to 2.

Moreover, the external long wave radiative coefficient is expressed by:

$$h_{re} = \sum_j F_j \cdot h_{rj} \quad (7)$$
With $j = (sky, ground, air)$, $F_j$ are the associated form factors, and $h_{rj}$ are the associated radiative coefficients which are calculated using the following expression:

$$h_{rj} = 4 \cdot \sigma \cdot \varepsilon \cdot T_j^3 \qquad (8)$$

With $\sigma$ the Stefan-Boltzmann constant and $\varepsilon$ the long wave emissivity of the walls.

### The equivalent thermal capacitance

As shown by (Antonopoulos and Tzivanidis, 1996), the equivalent thermal capacitance for a building cannot be estimated simply as the product of the mass and specific heat capacity of various components. Indeed, such an approach does not consider the position of thermal mass relative to the interior or exterior surfaces. However, the calculation method described in (NF EN ISO 13786, 2017) takes into account the relative positions of various layers with high thermal inertia. In this calculation methodology, each wall separating two zones m and n is represented by a quadrupole. Then, using the superposition theorem, a sinusoidal temperature variation is applied to m and n sides of the walls, one after the other, and the thermal admittance $Y_{mm}$ (or $Y_{nn}$) and the thermal periodical transmission coefficient $Y_{mn}$ (or $Y_{nm}$) between the zones are calculated for each variation. Finally, the expression of the thermal capacitance per surface unit is given for both sides m and n. The following equation expresses this capacitance for side m:

$$\kappa_m = \frac{T}{2\pi} * |Y_{mm} - Y_{mn}| \qquad (9)$$

The choice of T, the period of the sinusoidal variation, determines the charging and discharging period of the thermal capacitance. Thus, the longer the period is, the closer is the heat stored to its full capacity. In this study, the heat wave is represented by a step variation of outdoor air temperature. Thus, the calculation period considered for the thermal capacitance of a building should represent a full use of its heat storage capacity. In case of a step temperature variation, the capacity is fully used when steady state conditions are reached. Thus, the calculation period should be at least $T = 3\tau$. Throughout this study, the period of one year is select to meet this criterion for all the configurations considered in the analysis.

In case of more than one wall's type, including the floor and the ceiling, the equivalent capacitance is the area-weighted capacitances for all the walls.

### Verification Analysis

The calculated time constants using the equivalent thermal resistance and capacitance are compared to those estimated using the simulation results. This comparison analysis results are depicted in Figure 11 considering all the design configurations for the test cell. A clear linear correlation can be observed, with a slope close to 1 and a R² value of 0.99 indicating an excellent match between the calculated and simulated values for the time constant. The four design configurations of the house are simulated using EnergyPlus, with the same heat wave sequence, in order to validate the time constant indicator obtained using the test cell for residential buildings with attached spaces (i.e., attic, crawl space and garage). The simulated

and calculated time constants for the house are found to be close to those obtained for the test-cell as indicated in Figure 12.

<figure><img src="image_12.png" /><figcaption>Figure 11: Simulated versus calculated values of time constant for the case of energy model for the test-cell</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 12: Simulated versus calculated values of time constant for the case of energy model for the house</figcaption></figure>

### Final temperature

The final temperature $T_{exteq}$ reached by the indoor air can be calculated using heat balance equation under steady-state conditions. For the case when the building is subject to two different boundary conditions, such as the exterior ambient temperature ($T_{ext}$) and the ground temperature ($T_{gnd}$), the final temperature can be determined as:

$$T_{exteq} = \left( \frac{T_{ext}}{R_{ext}} + \frac{T_{sol}}{R_{sol}} \right) \cdot \frac{R_{ext} \cdot R_{sol}}{R_{ext} + R_{sol}} \qquad (10)$$

Where $R_{ext}$ and $R_{gnd}$ are the equivalent resistances, as explained in the *Time constant* section, of respectively the surfaces facing outdoors and the ground. $T_{ext}$ represents the equivalent temperature as described in (Bouchié et al., 2014):

$$T_{ext} = T_{air} - \frac{F_{sky} \cdot h_{rsky} \cdot (T_{air} - T_{sky})}{h_{ce} + h_{re}} \qquad (11)$$

Figure 13 shows the time variations of internal temperatures and calculated final temperatures specific to the case of the house. The calculated final temperatures are close to those obtained from the simulation results. Thus, the final temperature can be easily determined using only building design features and weather characteristics.
<figure><img src="image_14.png" /><figcaption>Figure 13: Comparison between calculated and simulated final temperatures</figcaption></figure>

## Summary and conclusions

In this paper, the calculation approaches of two resiliency indicators are presented. The first indicator is the time constant of a building when subjected to a step temperature increase and is calculated using the product of equivalent thermal resistance, R, and capacitance, C (i.e., $τ = RC$). The equivalent resistance accounts for all resistive properties of the building envelope systems through conductive, radiative, and convective heat transfer mechanisms. The equivalent capacitance is calculated using the method of ISO 13786, which allows to consider the position of the most capacitive layers for each wall. The second indicator is the final temperature which represents the indoor temperature that building experiences at the end of the step signal. Thus, both indicators can be fully determined using building design features and external characteristics without the need of any whole-building simulation analysis. To conclude, the use of these indicators allows to determine the time needed by a building to reach a threshold temperature, when this building, in initially steady-state conditions, is subject to a sudden outdoor temperature change. It gives a first assessment on the overall resiliency of a building to heat waves. These indicators could be complemented by other indicators to evaluate the daily outdoor temperature variation impacts on resiliency. For instance, night ventilation could improve the time constant and thus the resiliency of the building. Likewise, the internal load's daily profiles impact the resiliency of the building. All these aspects will be considered in a further work. Such a set of indicators will make possible to characterize the dynamics of a building under real heat wave sequences.

## References

* Antonopoulos, K.A., Tzivanidis, C., 1996. Finite-difference prediction of transient indoor temperature and related correlation based on the building time constant. International Journal of Energy Research.

* Antonopoulos, K.A., Tzivanidis, C., 1995. Time constant of greek buildings. Energy 20, 785–802.

* Asan, H., 1998. Effects of wall's insulation thickness and position on time lag and decrement factor. Energy and Buildings 28, 299–305.

* Bouchié, R., Derouineau, S., Abele, C., Millet, J.-R., 2014. Conception et validation d'un capteur de mesurage de la température extérieure équivalente d'une paroi opaque d'un bâtiment. Presented at the Conférence IBPSA France.

* Encinas, F., De Herde, A., 2013. Sensitivity analysis in building performance simulation for summer comfort assessment of apartments from the real estate market. Energy and Buildings 65, 55–65.

* Hoffman, M.E., Feldman, M., 1981. Calculation of the thermal response of buildings by the total thermal time constant method. Building and Environment 16, 71–85.

* IPCC, 2022. Sixième rapport du GIEC.

* Janssen, J.E., 1982. APPLICATION OF BUILDING THERMAL-RESISTANCE MEASUREMENT TECHNIQUES. ASHRAE Transactions.

* Kontoleon, K.J., Bikas, D.K., 2007. The effect of south wall's outdoor absorption coefficient on time lag, decrement factor and temperature variations. Energy and Buildings 39, 1011–1018.

* Kuczyński, T., Staszczuk, A., 2020. Experimental study of the influence of thermal mass on thermal comfort and cooling energy demand in residential buildings. Energy 195, 116984.

* NF EN ISO 13786, 2017. Performance thermique des composants de bâtiment - Caractéristiques thermiques dynamiques - Méthodes de calcul.

* Orosa, J.A., Oliveira, A.C., 2012. A field study on building inertia and its effects on indoor thermal environment. Renewable Energy 37, 89–96.

* Pascal, M., Wagner, V., Le Tertre, A., Laaidi, K., Honoré, C., Bénichou, F., Beaudeau, P., 2013. Definition of temperature thresholds: the example of the French heat wave warning system. International Journal of Biometeorology 57, 21–29.

* Pirard, P., Vandentorren, S., Pascal, M., Laaidi, K., Le Tertre, A., Cassadou, S., Ledrans, M., 2005. Summary of the mortality impact assessment of the 2003 heat wave in France. Eurosurveillance 10, 7–8.

* Quagraine, K.A., Ramde, E.W., Fiagbe, Y.A.K., Quansah, D.A., 2020. Evaluation of time lag and decrement factor of walls in a hot humid tropical climate. Thermal Science and Engineering Progress 20, 100758.

* Santé Publique France, 2019. Evolutions de l'exposition aux vagues de chaleur et de la mortalité associée en France métropolitaine entre 1970 et 2013.

* Sun, C., Shu, S., Ding, G., Zhang, X., Hu, X., 2013. Investigation of time lags and decrement factors for different building outside temperatures. Energy and Buildings 61, 1–7.

* Tsilingiris, P.T., 2004. On the thermal time constant of structural walls. Applied Thermal Engineering 24, 743–757.

* Zinzi, M., Agnoli, S., Burattini, C., Mattoni, B., 2020. On the thermal response of buildings under the synergic effect of heat waves and urban heat island. Solar Energy 211, 1270–1282.