1 Introduction

The consumption of energy in buildings accounts for nearly a third of energy utilization in the world in which half of this energy is consumed in heating and cooling [1]. Buildings are claimed to be responsible for 30% of CO2 emission which substantially contribute to climate change [2]. To mitigate the effect of CO2 emission on climate change, many policies and international agreements were implemented [3,4,5]. Thus, there is an increasing attempt for energy saving in buildings involves the accurate estimation of the thermal performance of these buildings to meet national codes and standards for certification requirements. In order to accomplish this purpose, it is essential to analyze the transient thermal behavior of the building envelope to minimize heat losses or gains [6,7,8,9].

The thermophysical properties of the building envelope are considered key parameters in studying the energy performance of buildings and without prior knowledge of these properties, the building thermal performance cannot be predicted correctly. Particularly, when analyzing thermally existing and old buildings, material and structure of the envelope components may not be documented. Moreover, the knowledge of thermal properties in the diagnosis of the wall construction is important to select the appropriate renovation method. To overcome this difficulty, the thermo-physical parameters are estimated based on data obtained by in situ measurements of surface temperature and heat flux [10,11,12,13,14,15,16,17]. The analysis and measurements of in situ helps to assess the thermal behavior of the building when the material properties are not known precisely [13, 18]. Different methods and models were proposed in the literature to analyze the thermal performance of buildings. The selection of the approach depends on the required information and the available input data [19, 20]. For example, Sassine [9] proposed a method based on Fourier analysis to estimate both the thermal capacitance and thermal conductivity for a building wall by monitoring the inner and outer surface temperatures and outer heat flux (Fig. 1). Figure 2 shows a good comparison between the measured and calculated heat flux using the optimized values of equivalent thermal resistance (R) and thermal capacitance (C).

Fig. 1.
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Schematic diagram of the experimental setup (Sassine [9])

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Fig. 2.
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Comparison of the heat flux between measured and calculated results for the optimized values of R and C (Sassine [9])

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Biddulph et al. [13] used both lumped thermal mass model and Bayesian statistical analysis to predict the U-value and effective thermal mass. Figure 3a Illustrates the electrical equivalent circuit for heat transfer through the wall for the single thermal mass (STM) model and the arrangement of the thermometers and heat flux plate.

The method presented by the authors required few days of measurements to provide an estimate of the effective thermal mass (Fig. 3b). The authors claimed that in-situ observing using thermometers and heat flux plates allows precise description of the thermo-physical properties of walls.

Fig. 3.
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(a) Schematic diagram of the model, (b) Measured internal (solid line) and external (dashed line) air temperatures. Estimated thermal mass temperature predicted by the STM model is shown by the dotted line.

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Tadeu et al. [21] used an iterative method coupled with Newton Raphson method to estimate the thermal resistance of multilayer walls in a dynamic condition (Fig. 4). A nonlinear system was achieved by applying temperatures and fluxes on external surfaces. Various experimental measurements were shown in that study and the anticipated thermal resistance was compared with the numerical results. Further, the thermal resistance of an outer wall subjected to actual outdoor conditions was predicted using the suggested method, and the equivalent thermal conductivity was estimated correctly. The relative error between both results were found in that study below 8%.

Fig. 4.
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Geometry of the problem under investigation (Tadeu et al. [21])

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Faye et al. [22] presented an approach to measure experimentally the effective heat capacity of a heterogenous and symmetrical wall element as depicted in Fig. 5. The authors in that investigation developed an analytical model based on the thermal quadrupole method. The wall element was exposed to sinusoidal boundary conditions (Fig. 5) to obtain the required data (temperature and heat fluxes) for the analytical model. The authors claimed in that study that their developed model was a useful tool to estimate the effective heat capacity of asymmetrical heterogeneous material.

Fig. 5.
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Wall element with boundary conditions under investigation (Faye et al. [22])

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Cucumo et al. [23] developed a method based on experimental estimation of the in-situ building’s wall conductance in various period of time using wall inside heat flux and both surface temperature on the inner and outer surfaces. A finite difference code was also used in this analysis. Their results agreed with results obtained a mean progressive method recommended by EN 12494 pre-regulation. The authors indicated that later method is limited since the thermal energy stored in the wall must be considered negligible compared with the energy passing through the wall during testing. Chaffar et al. [10] developed a method to characterize thermally a wall designed for in situ applications (Fig. 6a). The wall was exposed to a heat flux and temperature measurements on the other side were recorded using infrared thermography. The thermophysical properties of the wall were then estimated by an inverse method (thermal conductivity and volumetric heat) based on Levenberg–Marquardt algorithm. The authors illustrated that their developed method agreed well with the measured results (Fig. 6b).

Fig. 6.
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(a) Schematic of the proposed model, (b) Normalized curves measured and simulated for the gypsum tile (Chaffar et al. [10])

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Commercial software such as DOE-2, BLAST, TRNSYS, and ENERGYPLUS were also used to predict the energy performance of entire buildings based on the knowledge of building’s dimensions, material properties, occupants, etc. [24,25,26,27,28,29]. Most of these software assumed one-dimensional analysis to describe the thermal behavior of the building envelop. However, several structural and material configurations of building envelope components contain two and three-dimensional bridging effects, which cannot be modelled by one-dimensional analysis  [30]. Magnier and Haghighat [25] presented multi-objective study on building optimization to reduce energy consumption using TRNSYS simulations, genetic algorithm, and artificial neural network. The optimization process was illustrated in Fig. 7. The authors indicated that their proposed optimization process exhibited substantial reduction in energy consumption and improvement in thermal comfort. Fumo et al. [27] developed a simplified model to predict the energy consumption per hour using utility bills by employing a series of predetermined coefficients to the monthly data of energy consumption. The predetermined coefficients, which were obtained by running EnrgyPlus software, has the advantage to provide information without the need of executing a comprehensive dynamic simulation of the building.

Fig. 7.
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Optimization process (Magnier and Haghighat [25])

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Buildings with complex geometries, which lead to thermal bridging effects, need more realistic, two or three-dimensional analysis to account for these effects. One can note from above that most of the models utilized in analyzing thermal performance of buildings were simple and assumed one-dimensional situation to determine the thermophysical properties of the wall structure. The objective of this work is to develop a numerical model to predict the thermodynamics properties of building envelop components. As such, the optimization process is incorporated into the finite element method for solving one and two-dimensional coefficient inverse problems of heat conduction. The model is used to obtain the thermo-physical material properties of an equivalent wall, which provides a matching thermal flux behavior of a multilayer wall under the same transient thermal boundary conditions and accounts for thermal bridges. These properties can be used in commercial software such as ENERGYPLUS and DOE-2 to study in details the thermal performance of buildings. Another aim of this study is to incorporate the influence of thermal bridging on the equivalent wall thermo-physical properties using two-dimensional model. Generally, thermal bridging takes place when there is a break in, or penetration of the building envelope (e.g., insulation) and it produced by many circumstances such as the junctions between the wall and floor, windows and doors.

2 Physical Description of the Model

The finite element simulation is considered in this investigation to estimate the thermo-physical properties of a wall structure using two cases. In the first case, a simple 4-layer wall structure is analyzed, and in the second case, thermal bridging effects are introduced in the wall structure. The cross-sections of both cases are shown in Fig. 8 with their structure and materials. The thickness and corresponding thermal material properties of each layer, in these cases, are presented in Fig. 9. The entire wall thickness is 36 cm. The insulation material used in this study is PolyFoam, which is considered to have a very low thermal conductivity.

Fig. 8.
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Thermo-physical properties for different cases considered in thus investigation (a) without a thermal bridge (b) with a thermal bridge.

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Fig. 9.
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Thermo-physical properties of different layers of the wall structure

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The thermo-physical properties of the equivalent wall are estimated based on the coupling between the experimental data of the ambient temperature and the finite element model-based optimization. In this analysis, the heat flux of the equivalent wall will be matched to the heat flux calculated using the actual wall structure. Furthermore, the constitutive characterization of a wall thermal behavior under transient thermal conditions is considered as an inverse heat conduction problem for thermo-physical parameters estimation. These parameters are the density of different layers of the wall structure (ρ), thermal conductivity (k), and the specific heat capacity (c). The boundary conditions used in this study were based on the actual measurements of the outside temperature.

To estimate these parameters, the response of the model; q(t), is required to be fitted to the actual response by adjusting the set of constitutive parameters. In this work, the measured transient heat flux q(xm,ti) at some specific locations xm and time instants ti (Fig. 10) is employed for the estimation of the unknown thermo-physical parameters. Therefore, the optimization approach is applied to minimize the difference between the measured and the computed heat flux functions, while the thermo-physical parameters are considered as constraints. The main idea of the inverse method is to fit a mathematical model to a set of experimentally available data. This process is clearly summarized in Fig. 10. The objective function is designed using a least-squares formulation such that best fit design parameters are achieved by its minimization. Consequently, the numerical technique of the optimization theory is employed based on the finite element method. In the optimization part of the analysis, the measured heat flux q(xm,ti) is compared with the computed value obtained from the function qeq(xm, ti, ρ, k, c).

Fig. 10.
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Optimization process under investigation. Actual wall structure vs. equivalent wall

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3 Results and Discussion

Two cases were considered in this study. The first case consists of four layers without thermal bridge while the second case includes the effect of thermal bridge.

First Case Study: Multilayer Wall without Thermal Bridge

For the transient thermal analysis, convective boundary conditions are applied at both sides of the wall. The inside air temperature is assumed to be constant at 22 °C while the ambient air temperature is a cyclic function of time with a period of 24 h and temperatures between 35.5 °C and 48 °C as shown in Fig. 11. The temperature function corresponds to a summer day in Kuwait. Furthermore, the initial wall temperature is set to 22 °C, and the analysis is performed for a time period of 48 h.

Fig. 11.
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Ambient air temperature corresponds to a summer day in Kuwait

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Figure 12 compares the heat flux behavior at the internal faces of the actual wall structure and two equivalent walls as a function of time. A very good correlation between the presented function can be depicted from Fig. 12, which makes the applicability of the applied modeling approach practical. The maximum heat flux value appears at the end of the considered time period and reaches a value of 10.17 W/m2 in the case of the original wall and a value of 9.55 W/m2 in the case of the equivalent walls (6% relative error).

Fig. 12.
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Comparison of the temporal heat flux values between the original wall structure and the equivalent wall model Heat flux vs. time

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Table 1 illustrates a comparison of thermophysical properties between the original wall structure and equivalent wall in the absence of thermal bridge. The optimization results shown in Table 1 also highlighted the fact that the equivalent wall results are not unique but lead to the nearly same actual response of the heat flux (two solutions: equivalent wall 1 and equivalent wall 2). Though, the difference in the resulting thermophysical properties is still insignificant. The obtained two sets of parameters leading to the similar behavior of the heat flux curves is depicted in Fig. 12.

Table 1. Comparison of the thermophysical Properties between the original wall structure and equivalent walls in the absence of thermal bridge effect
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Second Case Study: Multilayer Wall with Thermal Bridge Effect

The effect of thermal bridge on estimating the thermophysical properties of the equivalent wall is considered in the second case. Similar initial and boundary conditions to the first case were applied in the second case. The schematic diagram of the second case with thermal bridge is depicted in Fig. 8b. The thermal bridge layer was introduced along the center of the wall structure which makes the model two dimensional. The thermo-physical properties of the thermal bridge are identical to the thermal properties of the cement block. An optimization process is carried out like the one in the first case. Table 2 exhibits a comparison of the thermophysical properties of the original wall compared with the equivalent wall situation. Since optimization process does not lead to a unique solution, Table 2 illustrates three optimized solutions where are close to each other. Comparison of the temporal variation of the heat flux between the original wall and the optimized equivalent walls are depicted in Fig. 13. Equivalent wall 1 was found to exhibit the smallest relative error (5.1%) compared with original wall heat flux. The maximum relative errors of other equivalent walls (2 & 3) compared with the original wall were 8.4% and 12.3%, respectively. Therefore, equivalent wall 1 provides more accurate results than other equivalent walls.

Table 2. Comparison of the thermophysical Properties between the original wall structure and equivalent walls with thermal bridge effect
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Fig. 13.
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Comparison of the temporal heat flux values between the original wall structure and the equivalent wall model Heat flux vs. time

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4 Conclusion

A finite element method coupled with optimization process is used in this investigation to estimate the thermo-physical properties of an equivalent wall from a measured heat flux. The heat flux measurements were based on an actual temperature data. Two cased were considered in this study. The first study included multilayer wall in the absence of thermal bridging while the second case included the later effect. Due to the existence of thermal bridging, two-dimensional analysis is assumed for the second case. The results presented in this investigation illustrated a very good correlation between the original multilayer wall structure and equivalent wall for both studied cases. The results also suggested that the effect of thermal bridging should be reduced as they increase the design loads and consequently increasing the size of equipment and power consumption. The method proposed in this investigation may allow to simulate the equivalent wall structure with the optimized thermophysical properties in commercial software. It also has the capability for solving two-dimensional problems in order to capture thermal bridging effects.