A Novel Method for Estimating the Thermal Performance of Multi-Block Wall Systems Using Thermal Impedance Z-Value under Transient Uncontrolled Heat Transfer Conditions

Abstract

Climate change is one of the biggest challenges today. An increasing population accelerates the construction of concrete houses and the use of air conditioners, thereby leading to an increase in energy consumption. When the walls of buildings are well-designed and insulated, energy consumption can be reduced. Therefore, it is important to measure the thermal performance of wall systems accurately. The existing traditional methods of measuring R- and U-values provide acceptable solutions for steady-state controlled, uncontrolled or transient state-controlled conditions. However, a need to develop a novel approach for transient state-uncontrolled realistic conditions has been identified. The present study involves both experimental and numerical investigations. An in situ model room with dimensions of 1.60 m × 1.73 m × 1.50 m was built for the experimental work, and a series of experiments were conducted. For numerical work, two models using Ansys Fluent 2021/2022 and MATLAB Simulink 2021/2022 were developed. The real-time experimental data were fed into numerical models to predict the thermal behavior of the wall system. The results include the evaluation of a concept called ‘Time-Lag’ for all three models. ‘Time-Lag’ is the time taken for the heat energy to flow across the wall system. The Time-Lag for the experimental model was 8 h 45 min, while for MATLAB and Ansys models, it was 8 h 22 min. (average) and 7 h 30 min, respectively. Minor variations validate the accuracy of the numerical models. Further, a novel method using a new parameter in building systems called ‘thermal impedance Z-value’ was developed to estimate the real-time thermal performance of walls using MATLAB Simulink. The Z-value measures the ability of a wall system to resist the flow of heat (thermal resistance, R-value) combined with its ability to store heat energy (thermal capacitance, C_th-value). It is evaluated for steady-state and dynamic (transient) systems. For the steady-state system, the Z-values on the outer and inner walls were 18.2683 K/W and 18.6761 K/W, respectively with a minor difference of 0.4078 K/W at the end of 72 h. For the dynamic system, the Z-value did not reach a constant value and fluctuated in a particular pattern during 24 h of the solar cycle with average values of 3.2969 K/W on the outer and 1.2886 K/W on the inner walls at the end of 72 h, thus presenting more accurate and realistic thermal performance results of a wall system.

1. Introduction

Due to climate change, several parts of the world experience extremely hot or extremely cold climatic conditions. Accordingly, to maintain human comfort, the demand for cooling and heating systems in buildings is increasing, thereby raising overall energy consumption and greenhouse gas emissions. On a global scale, nearly 40% of energy is used for cooling and heating systems of buildings [1]. The Kingdom of Bahrain being a Middle Eastern country, experiences a hot and humid climate for most of the year. According to the Ministry of Electricity and Water (EWA), the average annual population growth rate from 2014 to 2018 is 5% [2]. As a result, primary energy consumption in Bahrain increased from 1.11 × 10⁸ MWh in 2000 to 1.96 × 10⁸ MWh in 2019 [3]. The per-capita consumption of energy rose to 112.81 MWh in 2019, which is five times the global average and three times the Middle Eastern average. Meanwhile, the per-capita electricity consumption was 21 MWh, seven times the global average and five times the Middle Eastern [4]. During the baseline period, 49.5% of total grid electricity production was consumed by the residential sector, while 36.6% was consumed by the commercial sector. Approximately 60% and 55% of the electricity in these sectors was consumed by the residential and commercial sectors, respectively, annually [5].

In general, elements of the building such as roofs, doors, windows, concrete blocks, and plaster have great potential to reduce energy consumption and contribute to energy efficiency initiatives. Nearly 10% to 45% of energy is lost due to leakage from the solid boundaries of the buildings, such as doors, windows, and roofs [6]. Therefore, the design of each building element is important, particularly the concrete block, as it constitutes the largest area of a building system. Typically, the lifespan of buildings is around 40 to 50 years or sometimes longer. If these buildings are constructed in such a way that minimizes energy losses through their elements, then by 2050, the overall energy demand could decrease by one-third [7]. The roof of a building is connected to its walls. Research has shown that constructing ‘cool roofs’, which reflect more sunlight, can reduce the energy consumption of a building by 30% [8]. Therefore, roofs can reduce energy consumption significantly, followed by walls and columns [9].

Additionally, argon gas can be used as insulation within windowpanes to reduce heat transfer. A test result showed that heat transfer across the windows was reduced by 10.9% when the quantity of argon gas increased from 0 to 95% [10]. Furthermore, 25 mm thick plaster layers on the inner and outer surfaces of a building block improved the thermal performance by 79.34% [11]. The optimization of the cavity of a building block enhanced the thermal resistance by 10.67% [11]. Consequently, appropriate thermal testing of the building blocks and wall systems is essential to determine their thermal performance. Based on these findings, researchers can provide valuable suggestions and recommendations to improve the performance of the wall systems.

The thermal assessment of building block systems is either conducted on individual blocks or on multiple blocks (in the form of a wall). For each system, mainly two methods are employed: experimental and numerical. Apart from the various experimental methods, some include the use of infrared thermography [12] or hot box calorimeter [13] to measure the thermal performance of building systems. Similarly, among numerical methods, Computational Fluid Dynamics (CFD) and Finite Element Analysis (FEA) are used for the thermal assessment of building elements [14]. The thermal and physical properties of the building elements also play a significant role in the performance of wall systems [15], which is investigated in detail in [16] using CFD programming.

It is crucial to validate the numerical results using experimental results or vice versa. Research that combines both analytical and experimental methods tend to be more valuable. The experimental analysis of building block systems can be conducted in the lab under controlled conditions or in situ under uncontrolled conditions. In 2020, a combination of experimental and numerical studies was conducted on an individual block system to estimate the thermal conductivity of the block using in-lab controlled conditions, the temperature difference across the block surfaces maintained between 30 °C and 40 °C. Measurements were taken every hour until the system reached a steady state, and validation was performed using the CFD model [17].

In [18], experimental measurements were obtained for individual masonry blocks under controlled environments and compared with a mathematical model to evaluate their thermal performance. The experimental results of a multi-layered wall system were compared with numerical ones in [19,20] with modeled data. However, since these studies were conducted in the lab under controlled conditions, it was still not clear how the performance of a block or a wall would be affected under realistic uncontrolled conditions. In [21], a small model room measuring 600 cm × 730 cm × 220 cm was built in an open environment using in situ method to study the performance of various building wall systems. Similarly, in [22], different models of wall systems using the in situ method were built. However, in [21,22], the measurements were noted after 24 h, and the analysis was based on the assumption that the system reaches a steady state within that time, which eventually reflects the real-time transient conditions. This clearly showcases a gap in this field.

For the experimental in situ method, a common standard procedure was to be followed. A standard was well implemented by investigators in their studies [21,22,23,24]. This standard was developed in the form of International Standard-ISO 9869 [25] and American Standard-ASTM 1046 [26], which guide the evaluation of the ‘thermal resistance R-value’ of a building block system. These include all the details required for procedures to be followed, measurement methods, equipment to be used, and conditions to be maintained for the in situ experimental method. The accuracy and validity of the two standards were well compared in [27].

According to the ISO 9869 standard [25], two methods are used to evaluate the R-value of the building block system. The first method is called the Heat Flux Method (HFM), which requires a heat flux at the wall surface and the difference in surface temperatures on either side of the wall to estimate the R-value of the block system under steady-state conditions. The second method is called the temperature-based method (TB), which only requires the difference in surface temperatures on either side of the wall to estimate the R-value of the block system under steady/transient conditions. Out of these two methods, the most popular method is HFM, which uses two thermocouples to measure temperature and one sensor to measure heat flux. However, two major issues arise when using the HFM:

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Longer experimental durations.

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Possibility of inaccuracy in estimating the R-value of a block system.

ISO 9869 [25] further provides guidelines for the following conditions to be satisfied before stopping the experimentation and measurements:

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The experimentation should include the collection of measurements at fixed time intervals, and the total experimentation period should be longer than 72 h.

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The difference between the R-value at the end of the experiment should be less than 5% as compared with the R-values noted 24 h before.

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The R-value analyzed from the first time period with an integer part of (2 × no. of test days/3) should be less than 5% of the value obtained from the last time period of the same duration.

While following the ISO standards, Arash Rasooli in 2018 [24] suggested an additional change to the standard average in situ method that could reduce the experimentation time while simultaneously maintaining the accuracy of the results. The change involved the use of two heat flux sensors instead of one on a wall system. The new heat flux sensor was placed opposite to the first one on the other side of the wall. The thermal resistance R-value was obtained from both sensors which eventually converged with time, and the average R-value of the wall system was obtained for a comparatively shorter time period. In this way, the entire experimental process was simplified; however, the heating and cooling processes of the wall surfaces were completely controlled using a heater and cooler to vary the surface temperatures with respect to time. In other words, although the system was considered to be transient, it was maintained under controlled conditions. In 2018, John Smith, also conducted in situ transient experimental analysis under uncontrolled conditions; however, the R-value was calculated using the Fourier equation for steady-state conditions [28] and not transient conditions.

In summary,

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Concrete blocks are an important element and have a high potential to reduce energy losses in building systems.

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Most experimental and numerical studies have been conducted on individual blocks, and few have been conducted on multi-block systems.

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Experimental methods for individual block systems must be conducted in the lab in a controlled environment under steady-state conditions [17].

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Experimental methods for a multi-block system can be conducted in the lab under controlled and steady-state conditions, or in situ under controlled/uncontrolled and steady /transient state conditions.

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In [21,22,23], tests were conducted using in situ methods by following the standards [25,26]; however, in the analysis, the system was considered to reach steady-state conditions after a certain time period.

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In [24], again, the in situ method was adopted by incorporating an extra heat flux sensor to reduce the data collection time; however, while doing so, one of the wall surfaces was controlled using a heater and cooler, which finally led to a controlled condition.

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In [28], an in situ transient uncontrolled condition was considered; however, the R-value was calculated using the steady-state conduction equation.

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The widely used R- and U-values measure the thermal performance of wall systems under either steady-state-controlled or transient-controlled conditions.

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There are limited approaches addressed in the Kingdom of Bahrain; most focus either on the improvement of roofs of buildings [29] or on the thermal assessment of individual block systems under steady-state-controlled conditions using the mini-scale hot box method [30].

1.1. Research Gap

There exists a clear gap in the evaluation of the thermal performance of a multi-block wall system under uncontrolled transient conditions, thus representing real-world environmental conditions.

1.2. Need for a New Approach to Estimate the Thermal Performance of a Multi-Block Wall System

Thermal resistance (R-value) or its reciprocal thermal transmittance (U-value) is a key parameter for assessing the thermal performance of buildings. Each country has its own set of building rules and regulations, for example, [31] applies to Bahrain. The construction and refurbishment of buildings are permitted based on the U-value or R-value established according to the minimum energy performance requirement and the local climate [32,33]. The higher the R-value, the better the wall insulates against heat. Architects and builders use R-values to select materials that provide the necessary insulation to meet energy-efficiency standards and maintain indoor comfort.

The U-value (U = 1/R) [34], or thermal transmittance, measures the rate of heat transfer through a building element (e.g., a wall, window, or roof). The lower the U-value, the better the wall insulates against heat. The U-value is used in building regulations to ensure that buildings meet the minimum energy efficiency requirements. It is also useful for calculating heating and cooling loads in buildings, which impact energy consumption and HVAC system sizing. The U-value is calculated as the reciprocal of the sum of the thermal resistances of all the layers in the building element (U = 1/(R1 + R2 + … + Rn)) [34].

In this way, it is clear that the R-value provides detailed information about the individual materials of a building wall system, while the U-value provides comprehensive information about the building wall system. Thus, R- and U-values are preferred for measuring thermal performance in building applications due to their simplicity, practicality, relevance to steady-state conditions, ease of measurement, and established use in building codes and standards. However, the major drawback of these two parameters (R-value and U-value) is that they provide results by considering steady-state conditions, thereby ignoring the changes in the wall’s performance with respect to time. Indirectly, they ignore the effects of solar radiation intensity and wind velocities during a solar cycle of 24 h, and therefore, they do not provide real-time results for the thermal performance of a building wall system. Consequently, there is a need to develop a novel approach for building wall systems to estimate thermal performance under transient, realistic conditions.

1.3. Scope of Study

This study focuses on the experimental and numerical analysis of an IMSI-based multi-block wall system under real-time transient conditions. Various building blocks are manufactured in Bahrain. However, in this paper, the investigation is conducted on a widely used block in Bahraini buildings named the IMSI block. IMSI stands for ‘Insulated Reinforced Masonry System International, Ltd. Bahrain’. This is a concrete block with inserts of polystyrene insulation thereby having a unique design and structural integrity that is not found in other insulation blocks in Bahrain. The development process of the experimental and numerical models is described in Section 2.

For experimental analysis, a model room measuring 1.60 m × 1.73 m × 1.50 m was built with IMSI blocks in accordance with the appropriate set of rules set by the Bahraini government [31], including two layers of plaster on the internal and external sides of the walls. A set of experiments was conducted from June 25 to June 29, 2024, in the model room, and the measurements were extracted for the east-facing wall, which was open and subjected to real environmental conditions including wind and solar radiation. Moreover, the design of the IMSI block was developed using SolidWorks 2021/2022 software and two numerical models were developed using Ansys Fluent 2021/2022 software and MATLAB Simulink 2021/2022 software.

The two major theoretical concepts of this study: the concepts of Time-Lag and thermal impedance (Z-value), are presented in Section 3. In Section 4, the heat transfer across the wall system in the experimental and numerical models is studied, compared, and validated to eventually develop a new method to investigate how a multi-block system would perform thermally under uncontrolled transient realistic conditions using the new concepts of ‘Time-Lag’ and ‘thermal impedance’. Finally, Section 5 draws conclusions and highlights future developments that will eventually reduce energy consumption and benefit the whole country.

2. Methodology

A combination of experiments and simulations are used to achieve the objectives of the present research.

2.1. Development of Experimental Model: In Situ

In Figure 1, a detailed description of the development of the model room in an open environment with in situ conditions and the required measuring equipment is presented. The construction of the model room was conducted according to the regulations of the government [31] which included layers of 20 mm of plaster on the inner and outer sides of the wall. The east-facing wall of the model room was chosen for analysis. The remaining walls and roof were insulated from the inside while the door was insulated from the outside using polystyrene foam sheets. The measuring equipment and data collection methods used are:

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Pyranometer SP-420 (pre-calibrated) [Figure A10] to measure the solar radiation incident on the east wall.

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K-type Thermocouples (pre-calibrated) to measure the temperatures of the inner and outer air and the wall surfaces.

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Data Logger Pico Log TC-08 (pre-calibrated) [Figure A11] to measure and record the temperatures. Its specifications are mentioned in [35].

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Software program named Pico Log-6 for Pico Log data logger to extract the data of temperatures.

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Software program named Apogee for pyranometer SP-420 to extract the data of solar radiation.

The experiments were conducted and required measurements on the outer and inner sides of the wall were obtained. Because the temperatures of the wall surfaces vary with respect to time in accordance with the ambient conditions and because the walls do not have any heater or cooler; this experimental room represents a transient uncontrolled condition.

2.2. Development of Numerical Simulation Model: Ansys Fluent

In this section, the development of a numerical model using Ansys Fluent 2021/2022 is presented. The assembly of the IMSI block with concrete and insulation parts is shown in Figure 2. To reduce complexity, the geometry of the IMSI block was simplified by neglecting some complex features (circled in red), such as the surface of the PU foam and non-perfect rectangular geometry as shown in Figure 2. The detailed SolidWorks (2021/2022) design is presented in Figure A1, Figure A2 and Figure A3.

A simplified geometry of the assembly is shown in Figure 3. A 20 mm layer of plaster was applied on all six sides of the block. The breakdown of the design to convert into two-dimensional simplified geometry and its breakdown into smaller sections and the direction of heat flowing across it (represented by blue arrows) is shown in Figure 4.

The plaster at the top, bottom, left and right is the medium for heat transfer. To capture the thermal resistance of the plaster between the bricks on the left and right sides, the thickness of these plaster pieces was added to the top and bottom sides of the 2D model. Therefore, 10 mm plaster layers were placed on top–bottom and left–right sides, while 20 mm plaster layers were placed on front–back sides of the block. In this way, because the plaster layers were considered on all six sides of the block, the model was approximated as a multi-block wall system model. Further, since both the geometry and boundary conditions were symmetric, only half of the geometry was modeled (Figure 5) and simulated to reduce the simulation time. The details of the mesh and the material properties entered for the simulation are presented in Figure A4 and Figure A5, respectively.

Boundary conditions were applied on the inner and outer wall surfaces of the geometry.

At the inner wall surface, the boundary conditions considered were as follows:

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Temperature of the inner air, $T_{innerair}$ (°C) from the experimental data.

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The radiative heat flux in the inner wall was found using the internal radiation from the surrounding wall assuming that they are at thermal equilibrium with the internal air (due to the insulation).

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The convective heat flux on the inner wall surface q_in (W/m²) was obtained from the temperature difference between the inner wall and the inner ambient air from the experimental data using the following formula:

$$q_{in}=h_{in}\times (T_{innerwallsurface}-T_{innerair})$$

(1)

where h_in = 5 W/m² °C [Figure A12].

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The total heat flux in the inner surface of the wall was calculated by adding the convective and the radiative heat fluxes.

$$q_{in,total}=h_{in}\times \left(T_{innerwallsurface}-T_{innerair}\right)+q_{radfrominnersurroundings}$$

(2)

At the outer wall surface, the boundary conditions considered are as follows:

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Temperature of the outer air, $T_{outerair}$ (°C) from the experimental data.

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The radiative heat flux in the outer wall was found using the surrounding building radiation on the outer wall surface, and approximating the surrounding building temperature and the emmisivity from the existing literature for buildings in Bahrain.

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The convective heat flux on the outer wall surface q_out (W/m²) was obtained from the temperature difference in the outer wall and the external ambient air from the experimental data using the following formula:

$$q_{out}=h_{out}\times (T_{outerwallsurface}-T_{outerair})$$

(3)

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The total convective heat flux on the outer wall surface q_out (W/m²) was obtained from the temperature difference in the outer wall and the outer ambient air from the experimental data using the following formula:

$$q_{out,total}=q_{solarirradiance}{+h}_{out}\times \left(T_{outerair}-T_{outerwallsurface}\right)-q_{reflected}+q_{radfromsurroundings}$$

(4)

where h_out = 22.32 W/m² °C [Figure A12]; emissivity of the outer wall surface, $\in$ = 0.95 [36]; Stephen Blotzman’s constant $\sigma =5.67\times {10}^{-8}$; solar irradiation from the experimental data, $q_{solarirradiance}$(W/m²); radiation reflected from outer wall surface, $q_{reflected}$(W/m²), is obtained using the following formula:

$$q_{reflected}=\in \times \sigma \times ({T_{outerwallsurface}}^4-{T_{outerair}}^4)$$

(5)

The inputs were entered into the model and the outputs were extracted in terms of the inner and outer wall surface temperatures. These values were then compared with the experimental values. Graphical representations are prepared and presented. The remaining settings of Ansys Fluent are as per Figure A6.

2.3. Development of Numerical Simulation Model: MATLAB Simulink

In this section, the development of a transient numerical wall system model for simulation using MATLAB 2021/2022 software is presented. Again, a simplified geometry similar to the Ansys Fluent simulation was used. An exploded view of the block and the nomenclature of each layer are presented in Figure 6.

The Simulink model of MATLAB for the inner and outer sides of the wall was prepared. The inputs and outputs for each side are as below.

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For the inner side of the wall:

• Input data from the in situ experimental data:

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  Inner air temperature (K).

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  Inner convective heat transfer coefficient h_in (W/m²K) [Figure A12].

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For the outer side of the wall:

• Input data from the in situ experimental data:

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  Outer air temperature (K).

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  Outer convective heat transfer coefficient h_out (W/m²K) [Figure A12].

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  Solar radiation (W/m²).

• Output data:

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  Heat transferred by convection on the inner and outer surfaces of the wall (W/m²).

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  Heat transferred by radiation on the outer surface of the wall (W/m²).

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  Inner wall surface temperature.

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  Outer wall surface temperature.

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  Overall heat flux.

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  Overall thermal resistance and thermal impedance.

The Simulink model for the wall is presented in Figure 7. The thermal capacitance consisting of the thermal resistance (R) and thermal mass (TM) for each layer of the wall is clearly presented in this figure. The thermal resistance of each layer [34] is calculated using the Formula (6)

$$R={\displaystyle \frac{∆x}{kA}}$$

(6)

where R = thermal resistance (K/W) across the thickness of the material; Δx = thickness (m) of the material; k = thermal conductivity (W/m K) of the material; A = cross-section area (m²) perpendicular to the path of heat flow.

An example of the calculation of thermal resistance R and thermal mass TM is shown in Figure A7. Additionally, Figure A8 and Figure A9 show the Simulink models for the inner and outer sides of the wall, respectively.

3. Background Theory

3.1. Concept of Time-Lag

To understand heat transfer across a wall system, the foremost requirement is to observe changes in the behavior of the in situ wall system with respect to time. In realistic conditions, as the intensity of solar radiation and the wind velocity vary with respect to the hours of the day (time), it is obvious that the behavior of the wall system also varies with time. A certain amount of heat may enter and exit the wall system depending on the properties of its constituent materials. Therefore, to evaluate thermal performance, it is important to observe the time required for heat to travel across the wall surface under realistic conditions. This concept is called ‘Time-Lag’. The Time-Lag is the delay between the peak temperatures on the outer and inner surfaces of a building wall. It measures how long it takes for the heat to travel through the wall material, indicating the wall’s ability to moderate temperature fluctuations. This characteristic is crucial in evaluating the thermal performance of building materials, as a higher time lag generally indicates better thermal insulation and improved comfort for the building occupants. It helps reduce the heating and cooling loads, thus enhancing energy efficiency [37]. In June 2023, numerical analysis was conducted using a one-dimensional model of transient heat conduction through building enclosure walls. This research examined the effects of thermophysical building envelope parameters on transient heat exchange, peak cooling, and heating load in northern part of India. Results demonstrate that when the thermal conductivity of the wall increased, the time lag reduced, whereas it increased as the wall thickness increased [38]. In March 2024, again, numerical investigations were conducted on building walls in Morocco’s hot and humid climate to analyse key thermal parameters including Time-Lag, to show that the Time-Lag was extended by 5 h in a double wall compared to a single wall system [39]. Another numerical study discussed the effect of various thermal parameters, including thermal mass and Time-Lag, on the demand of energy and the thermal comfort of buildings with rammed earth walls. This underscores the discrepancies between nominal design data and in situ measured values, particularly in traditional constructions [40]. From these studies, it is clear that the concept of Time-Lag is mainly used till date for numerical analysis and that it has not yet been implemented to evaluate the thermal performance of realistic in situ transient conditions. Eventually, the estimation of the thermal performance of a multi-block system under uncontrolled transient conditions using the concept of Time-Lag will help to fill the gap in this research field.

3.2. Introduction of a Novel Concept ‘Thermal Impedance Z-Value’ in a Building Wall System

Thermal impedance is a measure of the resistance of a material to transient heat transfer conditions. It is represented by ‘Z’ and is defined as the ratio of the temperature difference across the material and the power dissipation causing that difference [41]. Unlike steady-state measures like thermal resistance (R-value), which always has a constant value; thermal impedance (Z-value) considers the time-dependent aspects of heat transfer that fluctuate with respect to time.

$$Z\left(t\right)={\displaystyle \frac{∆T\left(t\right)}{P}}$$

(7)

where Z = thermal impedance (K/W), $∆T$ = change in temperature across the material (K), P = power dissipation causing the temperature difference (W).

The work in [41] is about the application of impedance to measure the ideal volumetric specific heat of an external wall system. Thus, the scope of the implementation of the concept of impedance to measure thermal resistance under multi-dimension transient heat transfer conditions was extended. The work [42] describes the temperature distribution in the IGBT (Insulated-Gate Bipolar Transistor) modules using the concept of impedance. The concept of impedance is used primarily in electrical engineering. An electric circuit opposes the passage of an alternating current (AC) through it and, the measure of this opposition is called impedance. Impedance is a complex quantity, composed of ‘resistance R (real part)’ and ‘reactance (imaginary part) X [43]’:

$$Z=R+jX$$

(8)

where Z is the impedance, R is the resistance, j is the imaginary unit, and X is the reactance.

In thermodynamics, an analogy to electrical impedance can be drawn when examining thermal systems, specifically through the concept of ‘thermal resistance R’ and ‘thermal reactance ωC_th’ in the context of heat transfer. In thermal systems, impedance can be written as

$$Z_{th}=R_{th}+j\omega C_{th}$$

(9)

where Z_th = thermal impedance (K/W), R_th = thermal resistance (K/W), ω = angular frequency of temperature oscillations (rad/s), C_th = thermal capacitance (J/K)

In a thermal system, thermal resistance (R_th) describes how much a material resists heat flow, whereas thermal capacitance (C_th) describes how much heat a material can store. A building wall system under realistic conditions is subjected to solar radiation and wind whose intensity varies with respect to time. The wall system, based on its materials’ characteristics, tends to store and resist heat flow. Hence, thermal impedance (Z-value) can offer valuable advantages when a detailed understanding of the dynamic thermal performance of a wall system is required. It can complement existing metrics like R- and U-values thereby providing more precise and realistic information about the thermal behavior of building wall systems.

4. Results and Analysis

The in situ experimental setup and numerical models of Ansys and MATLAB were used to simulate and investigate the thermal performance of a multi-block IMSI-based wall system. It is worth noting that the experimental data include the data collected on one typical day, which are repeated for four days, while this is not the same for the simulation parts. The following studies were conducted to understand and validate the results obtained:

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Variation in inner and outer wall surface temperatures.

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Time taken for the heat to travel across the wall system in terms of ‘Time-Lag’.

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Variation in heat flux on the inner and outer surfaces of the wall system.

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Thermal impedance Z-value for steady state and dynamic (transient) systems.

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The following assumptions were considered in these studies:

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One-dimensional heat transfer condition which is across the thickness of the wall.

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No heat transfer takes place along the length and width of the wall.

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Roof and three walls other than the east wall are perfectly insulated.

4.1. Variation in Surface Temperatures of Wall System

The variation in inner wall and outer wall surface temperatures with respect to time for all three modes is shown in Figure 8 and Figure 9, respectively. It is observed that a similar pattern appears on all four days, and both Ansys and MATLAB were able to predict the trend accurately when compared to the experimental data. While analyzing Figure 8 in detail, it can be observed that MATLAB slightly overpredicted, while Ansys slightly underpredicted. For each day of a 24 h cycle, it is clear that the inner wall surface temperature increases from 10:15 h to around 21:00 h and then drops until around 11:00 h, repeating the same pattern the following day. A significant difference in the values is observed at their peaks nearing midnight, while a minor difference is observed for the rest of the day. Additionally, the ambient inner side air temperature is always higher than the inner wall surface temperature. This is because, from the experimental point of view, the roof, the remaining three walls, and the door of the room were insulated for the entire time. The air does not find any space to escape, and the heat is trapped in the room, thereby increasing the temperature of the ambient air in the room.

On the other hand, while analyzing Figure 9 in detail, it is observed that for each day of a 24 h cycle, the outer wall surface temperature increases from 05:00 h to around 12:00 h (noon time) and then drops until around 05:00 h, repeating the same pattern for the following day. A significant difference in the values is observed when their values are at a minimum, almost nearing early morning, while minor differences are observed for the rest of the day. Also, the ambient outer side air temperature is always lower than the outer wall surface temperature. This is because, from an experimental point of view, the ambient outer air is under the effect of wind, solar radiation, and convection currents, while the outer wall, being a solid, is directly subjected to the sun’s radiation, which acts as an absorber of heat and therefore, its temperature is always higher than the ambient air temperature.

4.2. Evaluation and Comparison of Time-Lag

For Time-Lag, it is important to know the peak temperature values of the outer and inner surfaces of the wall system. Figure 10 and Figure 11 represent the variations in inner and outer wall surface temperatures, respectively, for Ansys and MATLAB simulations to predict the Time-Lag for each model under consideration.

The Time-Lag parameter represents the time taken for the heat to be transferred from the outer surface to the inner surface of the wall. The value of Time-Lag is therefore estimated from the time difference between the peak temperatures of the outer and inner walls for each day of analysis.

Table 1. Time-Lag for simulation and experimental methods.

Method	Number of Days	Peak Temperature of Outer Wall (°C)	Peak Temperature of Inner Wall (°C)	Time of Peak Temperature of Outer Wall (h:min:s)	Time of Peak Temperature of Inner Wall (h:min:s)	Time-Lag
Ansys Fluent Simulation	Day 1	40.91698	37.48118	13:00:00	20:30:00	7 h.30 min.
	Day 2	42.03129	37.74437	12:45:00	20:15:00
	Day 3	42.10404	37.76667	12:30:00	20:00:00
	Day 4	42.11011	37.76854	12:15:00	19:45:00
MATLAB Simulink Simulation	Day 1	42.7034	38.5367	12:03:00	20:34:00	8 h.30 min.
	Day 2	42.7882	38.5834	12:03:00	20:34:00	8 h.30 min.
	Day 3	42.985	38.5769	11:59:00	20:15:00	8 h.15 min.
	Day 4	42.9559	38.5798	12:00:00	20:13:00	8 h.13 min.
Experimental	Days 1 to 4	43.242	38.141	11:30:00	20:15:00	8 h.45 min.

presents the Time-Lag for all three models. For the Ansys simulation, the peak temperatures and their timings for the four days vary; however, the Time-Lag remains 7 h 30 min. For MATLAB simulation, the Time-Lag is almost the same, with a minor difference ranging from 8 h 13 min to 8 h 30 min. For, the experimental model in which a typical day was mimicked for four days, the Time-Lag value is 8 h 45 min. In this way, it is observed that the Time-Lag was very well predicted by the Ansys and MATLAB simulations, as their results are very close to the experimental results.

4.3. Variation in Solar Radiations on the Wall System

The intensity of solar radiation is captured every 15 min by a Pyranometer SP-420. The variation in solar radiation with respect to time is shown in Figure 12 for 24 June 2024. The radiations begin with 0.1 W/m² at 4:30 h (sunrise), gradually increase from morning till afternoon, and gradually drop in the evening, eventually ending with 0.4 W/m² at 18:45 h (sunset). It remains 0 W/m² for the rest of the time (no sunrays). The variations in solar radiation are more or less the same as expected. However, the peak values are observed between 8:00 to 9:00 h. This may be due to the fact that June is the peak summer season in Bahrain, and the intensity of solar radiation is high in the early hours of the day. If the experiment had been conducted in winter, then the same graph would have slightly shifted towards the right. Also, the model room was built in an open environment, and therefore, the wall under study also experiences shadows of the surrounding buildings, trees, and other structures.

4.4. Variation in Heat Flux on the Wall System

The variation in heat flux at the inner and outer surfaces of the wall for Ansys Fluent and MATLAB simulations with respect to time is shown in Figure 13. It is clear that the predictions by these two simulations are very similar, with minor variations. The minor variations are due to the fact that Ansys Fluent considers the wall as a continuous system, while MATLAB divides the wall into separate parts.

4.5. Thermal Impedance for a Multi-Block Wall System

As discussed earlier, the ‘thermal impedance Z-value’ is a crucial parameter that can offer a more detailed and comprehensive understanding of a material’s thermal behavior under varying conditions. Therefore, in this section, the thermal impedance of the wall system was evaluated using the MATLAB Simulink Simulation model. Further, to observe the difference in the thermal performance of the wall for a steady-state system and for a dynamic (transient) system, the ‘thermal impedance’ was evaluated for each of these systems separately.

4.5.1. Thermal Impedance Z-Value for Steady State Wall System

For steady-state conditions, the average temperature on the outer surface and the inner surface of the wall was fixed at 42 °C and 30 °C, respectively, and entered into the model as inputs. The outputs of the model were the heat flux on the outer and inner wall surfaces. These heat flux variations were plotted against time in Figure 14. It is clear that initially, a huge variation in the heat flux occurred, particularly in the heat flux on the outer wall on the first day (24 h). However, at the end of the third day (72 h), the flux values on both surfaces became almost constant. At the end of the fourth day (96 h), neglecting the negative sign, which merely represents the direction, it was observed that the thermal impedance on inner (6.0843 W/m²) and outer (6.2206 W/m²) wall surfaces was almost similar, with a minor difference of 0.1363 W/m².

From the thermal impedance equation, $Z=∆T/Q$, the model was amended to read the thermal impedance at both the inner and outer sides of the wall. Again, since it is a steady-state condition, the average temperatures on the outer and inner surfaces of the wall were fixed at 42 °C and 30 °C, respectively. The variation in thermal impedance with respect to time is presented in Figure 15. Initially, a huge variation in the thermal impedance occurred, particularly in the outer wall, on the first day (24 h). However, at the end of the third day (72 h), the impedance values on both surfaces became almost constant. The reason for this is that after 3 days, the wall became saturated and the wall could no longer store any more heat. In the beginning, the heat flux increased and then became constant. At the end of the fourth day (96 h), neglecting the negative sign, it was observed that the thermal impedance on the outer (18.2683 K/W) and inner (18.6761 K/W) wall surfaces was almost similar with a minor difference of 0.4078 K/W. The constant value of thermal impedance Z signifies that in the long run, whatever heat enters the wall system exits the system under steady-state conditions. This also validates why experiments to date have been required to be conducted for 72 h.

4.5.2. Thermal Impedance for Dynamic State Wall System

A dynamic wall system is one in which the heat flux and the temperature of the inner and outer wall surfaces vary with respect to time. A dynamic system thereby represents a realistic in situ condition in which the thermal performance of the wall varies with respect to time. Therefore, experiments were conducted for four days on the east-facing wall of an in situ test room. The parameters extracted from the setup to evaluate the thermal impedance of the dynamic state wall system were the heat flux and temperature of the inner and outer wall surfaces at regular time intervals. The impedance at the inner and outer wall surfaces was evaluated using equation $Z=∆T/Q$ and plotted against time in Figure 16 for all four days of experimentation. Almost similar patterns of thermal impedance on the inner and outer walls were observed for all four days. For the inner wall, huge fluctuations were observed between 11:00 to 12:00 h (noon time) followed by slightly lower fluctuations between 23:00 to 03:00 h (nighttime). These are the times when, due to Time-Lag, the heat either penetrates through the wall and reaches the inner surface or the heat moves out of the wall system. On the other hand, for the outer wall, huge fluctuations were observed between 05:00 to 07:00 h (morning time) and 16:00 to 18:00 h (evening time) when the intensity of solar radiation was either too high or too low, respectively. The average value of thermal impedance on the outer surface was 3.2969 K/W, while on the inner surface was 1.2886 K/W. Eventually, in a dynamic (transient) wall system, the Z value will never reach a stabilized value. On the contrary, it fluctuates in a particular pattern with respect to time. This is because, during the daytime, conduction of heat through the wall and the storage of heat in the wall will simultaneously take place, while at night time, continuous release of heat from the wall towards the inside and outside of the room (two directions) will take place.

Finally, the Z-value in a steady state system reaches a constant value as the wall is assumed to be saturated with heat after a certain time period, and the impedance merely depicts that whatever heat enters, exits the system. However, in the real world, the wall continuously stores and releases heat depending on the ambient conditions. Therefore, the Z-value in a dynamic system never reaches a constant value. On the contrary, it fluctuates in some regular pattern depending on the 24 h ambient cycle. An interesting point to be noted is that each wall type will have its own pattern of thermal impedance depending on the design and material characteristics of the wall system.

5. Conclusions

In the present study, a thorough review of the literature dealing with the measurement of thermal performance of building wall systems was conducted. It was observed that the widely used traditional methods of evaluation of the ‘thermal resistance R-value’ and ‘thermal transmittance U-value’ provide acceptable solutions for wall systems under steady-state-controlled and uncontrolled conditions or for walls under transient state-controlled conditions. However, these methods are not suitable for transient state uncontrolled conditions, which was identified as the main research gap. Consequently, the need for the development of a novel method to evaluate the thermal performance of a multi-block wall system under transient, uncontrolled realistic conditions was demonstrated in detail.

To simulate realistic transient conditions, an in situ experimental model is developed in the form of a model room subjected to ambient conditions. Appropriate measuring equipment and acquisition systems were used to extract real-time data. Further, two numerical models using Ansys Fluent and MATLAB Simulink were developed and validated using the experimental results. Additionally, these numerical models contributed to the estimation of the thermal performance of wall systems under transient, uncontrolled conditions. The results of the three models were analyzed in detail using appropriate graphical representations.

To estimate the thermal performance, it is important to study the time taken for the heat to be transferred across the wall system. Therefore, the concept of ‘Time-Lag’ is implemented in all three models. Minor differences in the values of Time-Lag were observed in the numerical models as compared to the experimental model. In this way, the numerical models were validated.

Furthermore, a novel method was introduced by considering the concept of ‘thermal impedance Z-value’ in a building wall system. The major concluding points for the thermal impedance in a wall system are:

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The estimation of the thermal performance of a wall system involves its ability to resist the flow of heat (thermal resistance, R-value) combined with its ability to store the heat energy (thermal capacitance, C_th-value).

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Thermal impedance Z-value is a novel parameter for a building wall system that includes both the R-value and the C_th-value. Therefore, the thermal impedance Z-value, of a wall system provides precise results in the estimation of the overall thermal performance of a building wall system.

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In a steady-state system, the thermal impedance on the inner and outer wall surfaces gradually converges to a constant value at the end of four days. The reason for this is that after around 72 h, the wall becomes saturated and stops storing additional heat, thereby representing a controlled steady-state non-realistic condition.

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In a dynamic-state system, a fluctuating pattern is observed on a daily basis, and therefore, the convergence phenomenon does not occur. The value of thermal impedance fluctuates in a particular pattern with respect to time. This occurs because during the daytime, the wall system continuously tends to store or transfer heat, while during nighttime, it tends to release the heat inside and out in two directions, thereby representing an uncontrolled transient dynamic-state realistic condition.

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As a result, there exists a significant variation in the values of thermal impedance for the steady-state and dynamic (transient) state systems, with the dynamic system providing more realistic results than the steady-state system.

Limitations of the experimental in situ model of the present research include:

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Setting up and maintaining an in situ experimental system proved costly and logistically challenging.

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In situ experiments require long-term monitoring. Ensuring continuous data collection and managing data gaps was challenging.

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The performance of the wall system can be influenced by adjacent buildings or structures, such as HVAC systems, neighboring walls, or shading elements. These factors were not considered in the present study.

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For analysis, 1D (along wall thickness) heat transfer conditions were considered. However, in actual practice, heat transfer also takes place along the length and width of the wall system.

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For the estimation of ‘thermal impedance Z-value’, the roof and walls other than the east wall (under study) were insulated to estimate an accurate value. However, in actual practice, the Z-value will be slightly different as heat transfer does take place through the roof and all the walls of a house.

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The velocity of wind in contact with the wall system was not measured separately using any instrument. However, the effect of variations in the wind velocity was indirectly considered in the convective heat transfer coefficient.

In the future, this novel approach to the estimation of the thermal performance of building wall systems by applying the concept of ‘thermal impedance Z-value’ can be implemented for wall systems:

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With blocks of varied design configurations.

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With blocks of varied concentrations of aggregate and insulation materials.

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Under varied climatic conditions across different seasons.

The ‘thermal impedance Z-value’ of different wall system types can be evaluated, and their patterns can be studied in detail. These patterns of Z-value may help the researchers to compare, estimate, and improve the thermal performance of building walls.