Application of the Heat Flow Meter Method and Extended Average Method to Improve the Accuracy of In Situ U-Value Estimations of Highly Insulated Building Walls

Abstract

In the context of remodeling old buildings, enhancing insulation performance in the exterior skin necessitates an accurate assessment of a wall’s thermal performance. The conventional method for determining the thermal transmittance (U-value) of a wall is the heat flow meter (HFM) as outlined in the ISO 9869-1. However, this measurement is susceptible to errors influenced by indoor and outdoor environmental conditions and the wall’s material composition. This study evaluates the U-value of an internally insulated wall, specifically constructed for this purpose, utilizing both the average and dynamic methodologies of an HFM. Furthermore, it introduces a novel estimation method: the extended average method (EXAM). The effectiveness of this proposed method is ascertained by comparing the accuracy and convergence of the U-value estimations with those derived from existing methodologies. Additionally, the study explores the limitations of the HFM by analyzing the heat flow traversing the interior of a wall. The findings revealed that the EXAM method enhanced the precision of U-value estimation in all scenarios. Particularly, in walls with superior insulation, the HFM tended to underestimate the heat flow observed indoors, leading to negative errors. The EXAM method, incorporating considerations of both insulation and structural materials, offers an accurate in situ measurement of the U-value relative to the HFM.

1. Introduction

Aligned with global objectives for carbon neutrality, the building sector is increasingly recognizing the need for energy conservation [1]. A fundamental aspect in minimizing a building’s energy demands involves ensuring that the thermal performance of the building envelope is optimized [2]. The effectiveness of insulation is typically gauged using parameters such as thermal transmittance (U-value) or thermal resistance (R-value).

According to the ISO 6946 (2017), the thermal transmittance of building elements can be theoretically calculated based on material thermal conductivity and thickness [3]. This calculation method is widely utilized owing to its simplicity and reliance on accurate wall layer information. However, the actual thermal transmittance measured in the field may significantly deviate from the designed value due to factors such as insulation material aging, moisture presence, construction errors, and material damage [4,5,6,7,8,9,10]. This discrepancy between designed and actual thermal transmittance values can impact the strategies for remodeling old buildings and enhancing skin insulation performance, highlighting the need for in-field diagnostic measures of building skin insulation.

Several non-destructive methods exist for field measurements of wall U-values and R-values, such as the heat flow meter (HFM), hot box, infrared thermography, and temperature-based method (TBM). The selection of a method typically depends on factors such as complexity, accuracy, cost, and time requirements. Among these, the HFM method of the ISO 9869-1, known for its relatively simple equipment and estimation process, is commonly adopted. This standard delineates two data analysis methods: the average method (AM) and dynamic method (DM) that involve analyzing temperature and heat flow data across building walls to derive an in situ U-value or R-value [11]. The AM requires a prolonged measurement period for steady-state values, whereas the DM adjusts for heat flow changes through modifications in the heat equation and various parameters [12].

Numerous researchers have employed this methodology to estimate the U-value or R-value of walls, assess discrepancies from the design values, and identify optimal measurement conditions. Adhikari et al. found that the difference between calculated and measured thermal transmittance values ranged from 2% to 58%, depending on the wall material used in the building [13]. Evangelisti’s study also experimentally confirmed a substantial error margin, spanning from 14% to 112%, between the theoretical and measured heat transmittance [14]. Ficco et al. conducted field measurements of the U-value for buildings designed with the U-value ranging from 0.37 W/m²·K to 3.3 W/m²·K, and they reported an error of 2–233% between calculated and measured values [15]. Gaspar et al. conducted a field measurement test of the U-value on a building facade, confirming that the convergence period of the measurement was determined depending on the indoor and outdoor temperature difference conditions of the test environment [16].

Highly insulated walls, with substantial thermal resistance, further challenge the accuracy of thermophysical property evaluations [17,18].

Asdrubali et al. measured the in situ U-value of insulated green buildings, with theoretical U-values ranging from 0.23 W/m²·K to 0.33 W/m²·K, employing the average method [19]. They found a discrepancy of 4% to 75% between the theoretically calculated U-values and the measured ones. Similarly, Bros-Williamson investigated an insulation structure designed with a U-value of 0.10 W/m²·K; the in situ measurement yielded a U-value of 0.23 W/m²·K, indicating a divergence of 10% to 65% from the theoretical value [20]. In a related study by O’Hegarty, field U-values for five insulated walls were measured, with theoretical U-values ranging from 0.118 W/m²·K to 0.191 W/m²·K. They also identified a significant deviation between the design and the measured values, with an error margin of 9–300% [21].

These studies collectively demonstrate that on-site measurements of U-values, particularly when utilizing the HFM method on highly insulated exterior walls, can yield results that significantly deviate from theoretically calculated values. To address the various limitations inherent in field measurements, several studies have proposed methodologies for improvement from diverse perspectives.

One critical factor influencing U-value estimation is the measurement and sensing process, which includes considerations such as the location of sensor attachment and the presence of surrounding heat source devices [22,23,24]. Studies have also explored optimal conditions for U-value field measurements by analyzing various environmental conditions [25,26,27]. Additionally, approaches involving mathematical modeling have been attempted, such as regression modeling, advanced data analysis methods, probabilistic gray box modeling, and inverse modeling [28,29,30]. Furthermore, an extended concept of the average method has been proposed, entailing modifications to the data interpretation method [31]. However, it remains uncertain whether these methods necessitate new techniques, such as mathematical modeling, or constitute a comprehensive improvement over the HFM method.

To enhance the limitations of the HFM method identified in highly insulated exterior walls, the concept of the extended average method (EXAM) is introduced in this study. This concept utilizes the thermal conductivity measurement method for insulation materials. To assess the efficacy and degree of improvement of this proposed method, the field U-value of a highly insulated exterior wall was estimated through experimentation. The improvement effect was subsequently verified through a comparative analysis with the existing method.

2. Methods

2.1. Estimation Methods of the U-Value

We reference the ISO 9869-1 for the on-site estimation of a wall’s U-value. The ISO 9869-1 delineates the protocol for measuring the thermal resistance and thermal transmittance of building elements in situ. Central to this standard is the utilization of the HFM method. This approach entails the measurement of temperatures both inside and outside the building element, concurrently with the measurement of heat flow. These measurements collectively facilitate the assessment of the thermal performance of the building element. The HFM method operates under the assumption of steady-state conditions and necessitates an extended measurement duration. It is during this period that temperature and heat flux data, essential for the U-value estimation, are collected. The ISO 9869-1 prescribes two distinct methods for this estimation: the average method and dynamic method. In this study, both estimation techniques were used to estimate the U-value of a wall.

2.1.1. Average Method

The AM posits that the cumulative average of observed internal and external temperatures along with heat flux values over an extended period can yield a steady-state thermal resistance value (R-value) [12]. This method in the field involves estimating the R-value by dividing the average heat flux density by the average temperature difference. The formula for this calculation is the following:

$$R_{AM}={\sum }_{j=1}^n\left(T_{i,j}-T_{e,j}\right)/{\sum }_{j=1}^n{q}_{si,j},$$

(1)

where $R_{AM}$ represents the thermal resistance value estimated by the average method. $n$ is the number of measurement data, $q_{si,j}$ denotes the density of heat flow rate, and $T_{i,j}$ and $T_{e,j}$ are the internal and external air temperatures, respectively. The estimated value of $R_{AM}$ converges gradually with each calculation. For a measurement to be considered complete via the average method, it must satisfy three conditions. Firstly, the measurement duration must be at least three days. Secondly, the estimated R_AM value at the end of the test should not deviate by more than ±5% from the value obtained one day before the test’s conclusion. Thirdly, the $R_{AM}$ value estimated from the data acquired during the initial INT (2 × D_T/3) days of the measurement period should not deviate by more than ±5% from the $R_{AM}$ values estimated from the data of the last period of the same duration. In this context, D_T represents the total duration of the measurement in days, and INT denotes the integer part of a number. The inverse of $R_{AM}$, as determined by the aforementioned method, is defined as the thermal transmittance (U-value).

2.1.2. Dynamic Method

The DM is a sophisticated analytical technique compared with the AM, designed to deduce the steady-state properties of building elements from HFM measurements. This method is particularly applicable in environments exhibiting significant temperature and heat flow variations [11]. Despite its sophistication, which offers the advantage of potentially reducing the testing duration for heavy elements, the complexity of the DM has led to its less frequent utilization in the U-value estimation of building elements, as noted by Ficco et al. [15] and Mandilaras et al. [32]. The DM conceptualizes the heat equation through a series of parameters and a select number of time constants [33]. This method articulates the heat flow rate at a specific time, $t_i$, as a function not only of the temperatures at that precise moment but also of all preceding times. This relationship is encapsulated in the following equation [11]:

$$\begin{array}{ll}{q}_i=\mathsf{\Lambda }\left(T_{si,i}-T_{se,i}\right)& +K_1{\dot{T}}_{si,i}-K_2{\dot{T}}_{se,i}+{\displaystyle \sum _n}{P}_n{\displaystyle \sum _{j=i-p}^{i-1}}{T}_{si,j}(1-{\beta }_n){\beta }_n(i-j)\\ & +\sum _n{Q}_n\sum _{j=i-p}^{i-1}{T}_{se,j}(1-{\beta }_n){\beta }_n(i-j),\end{array}$$

(2)

where ${\dot{T}}_{si,i}$ and ${\dot{T}}_{se,i}$ represent the time derivatives of indoor and outdoor temperatures, respectively. $K_1,K_2,P_n,Q_n$, and $\mathsf{\Lambda }$ are variables of the wall that change according to the time constants ${\tau }_n$. Although these variables capture the dynamic characteristics of the wall, they lack specific meanings or defined definitions [6]. ${\beta }_n$ is an exponential function of the time constant and is defined as the following:

$${\beta }_n=exp\left(-\frac{\mathsf{\Delta }t}{{\tau }_n}\right)$$

(3)

To appropriately represent the interrelation among $q_i$, $T_{si,i}$, and $T_{se,i}$ in Equation (2), the time constants need to be adjusted for estimating the optimal parameters. Typically, assuming three time constants ${(\tau }_1,{\tau }_2,{\cdots \tau }_m)$, the introduction of time constants results in 2m + 3 unknown parameters, such as ${\mathsf{\Lambda },K}_1{,K}_2,P_n,Q_n$, owing to the influence of these time constants.

$$\overrightarrow{q}=\left(x\right)\overrightarrow{Z},$$

(4)

where $\overrightarrow{q}$ is a vector with heat flow density data $\left(q_i\right)$ as a component, and (X) is a rectangular matrix with 2m + 3 columns. $\overrightarrow{Z}$ is a vector composed of 2 m+ 3 unknown parameters (such as ${\mathsf{\Lambda },K}_1{,K}_2,P_n,Q_n$). ${\left(X\right)}^{\prime }$ is the transposed matrix of $\left(X\right)$.

From this set of equations, $\overrightarrow{Z}\ast$ can be deduced as follows, and $\overrightarrow{q}\ast$ can be estimated from each value of $\overrightarrow{Z}\ast$.

$$\begin{array}{c}\overrightarrow{Z}\ast ={\left[{\left(X\right)}^{\prime }·\left(X\right)\right]}^{-1}·{\left(X\right)}^{\prime }·\overrightarrow{q}\\ \overrightarrow{q}\ast =\left(X\right)·\overrightarrow{Z}\ast \end{array}$$

(5)

Furthermore, nine unknown parameters were determined, derived from three distinct time constants. The equations were established based on measurements taken at intervals of 10 min. To resolve these linear equations, the least squares fitting method was utilized. For the optimal solution of these equations, the Solver tool was employed. Additionally, the programming language R was used for the programming aspects of this study.

2.1.3. Extended Average Method

The HFM method, as outlined in the ISO 9869-1, is frequently employed for the on-site assessment of a wall’s thermal performance due to its relatively straightforward experimental procedures and analysis methods, particularly the average method. However, this method may encounter limitations in accurately evaluating the thermophysical properties of composite walls with high-performance insulation. Such walls exhibit high thermal resistance in the thickness direction, impeding vertical heat transfer, which can potentially reduce the precision of assessments [17,18]. To address this, the current study seeks to enhance the existing HFM average method by incorporating concepts from the ISO 8301 [34] aiming to measure the thermal transmittance of highly insulated walls on-site with greater accuracy.

The ISO 8301 prescribes a technique for determining the thermal resistance of insulation materials under standardized conditions, utilizing a heat flow meter. As depicted in Figure 1, under steady-state conditions, this method involves measuring the heat flow density passing through one or two heat flow meters. This is achieved by establishing a specific temperature differential using heating and cooling plates placed on either side of the insulation specimen. According to the ISO 8301, the measured data are employed to calculate thermal resistance, as illustrated in the following formula:

$$R_s=\frac{∆T}{fe},$$

(6)

$$R_d=\frac{∆T}{\frac{1}{2}(f_1{e}_1+f_2{e}_2)},$$

(7)

where $R_s$ is the thermal resistance when a single heat flow meter is arranged, and $R_d$ is the thermal resistance when two heat flow meters are configured. $f$ is the calibration factor of the heat flow meter (W/mV/m²), $e$ is the heat flow meter output (mV), and $∆T$ is the temperature difference between the heating plate and the cooling plate.

The concept of utilizing two heat flow meters, as outlined in the ISO 8301, was adapted and extended to the analysis of walls, as illustrated in Figure 2. Recognizing that a highly insulated wall typically comprises various materials, including both insulation and structural components, this study employed two heat flow meters for each material type. This approach was implemented to precisely determine the thermophysical properties of each distinct material component within the wall.

The methodology encapsulated in Equation (7) was adopted for estimating the thermal resistance of each component of a wall, namely the insulation material and the structural element, utilizing the dual heat flow meter arrangement approach. Additionally, to approximate a steady-state condition in real-world environments, the averaging method from the ISO 9869-1 was employed. This approach led to the definition of an extended average method, which was formulated as the following:

$$R_{EXAM\_in}={\sum }_{j=1}^n\left(T_{i,j}-T_{m,j}\right)/{\sum }_{j=1}^n\left[\frac{1}{2}\left(q_{si,j}+q_{sm,j}\right)\right],$$

(8)

$$R_{EXAM\_out}={\sum }_{j=1}^n\left(T_{m,j}-T_{e,j}\right)/{\sum }_{j=1}^n\left[\frac{1}{2}(q_{sm,j}+q_{se,j})\right],$$

(9)

$$R_{EXAM}=R_{EXAM\_in}+R_{EXAM\_out},$$

(10)

where $R_{EXAM\_in}$ is the thermal resistance of the inner material estimated by the extended average method, and $–R_{EXAM\_out}$ is the thermal resistance of the outer material (m²·K/W) estimated by the expanded average method. $T_i$ and $T_e$ are the indoor and outdoor air temperatures, respectively, and $T_m$ is the temperature (K) at the insulation–structure interface. $q_{si}$ and $q_{se}$ are the heat flow amounts indoors and outdoors, respectively, and $q_{sm}$ is the heat flow amount passing between the insulation and the structural area (W/m²). $R_{EXAM}$ is the total thermal resistance (m²·K/W), i.e., the sum of the thermal resistances of each part, essentially reflecting the thermal resistance of the wall as determined by the expanded average method. Therefore, the thermal transmittance $U_{EXAM}$ (W/m²·K) determined via the expanded average method is defined as the following:

$$U_{EXAM}=\frac{1}{R_{EXAM}}$$

(11)

2.2. Test Wall

The experimental wall under study formed part of the exterior of an experimental building constructed in May 2021, to investigate the thermal performance of highly insulated structures (as depicted in Figure 3). This building, essentially a single room, encompassed a floor area of 31.8 m² and a ceiling height of 3.2 m. It was strategically located to ensure unobstructed solar radiation, free from shading by adjacent structures, and was oriented to face directly south and north.

The building’s exterior wall was distinguished by three different types based on the location of insulation and heat capacity. Each wall type measured 2.1 m in width and 3.2 m in height. In designing these walls, special attention was given to minimizing the impact of thermal bridges, particularly in the central area where measurements were conducted. Additionally, these walls were specifically manufactured for experimental purposes, with temperature and heat flow sensors embedded within, as illustrated in Figure 4. These sensors were crucial for determining the thermophysical properties inside the wall.

We primarily focused on concrete walls with internal insulation, which were the north-facing walls less influenced by solar radiation. The composition of the test wall, along with the physical characteristics of each layer, are detailed in

Table 1. Layers and thermophysical properties of the test walls.

Layer No.	Material Layer (Inside–Outside)	Thickness (m)	Thermal Conductivity (W/m·K)	Thermal Resistance (m2·K/W)	Theoretical U-Value (W/m2·K)
1	Internal surface			0.110	0.145
2	Wall paper	0.005	0.170	0.029
3	Gypsum board	0.020	0.180	0.111
4	Polyisocyanurate (PIR) insulation	0.130	0.020	6.500
5	Reinforced concrete	0.200	2.300	0.087
6	Cement mortar	0.010	1.400	0.007
5	External surface			0.043

. The theoretical thermal transmittance (U-Value) of these walls, calculated in accordance with the ISO 6946, was 0.145 (W/m²·K).

2.3. In Situ Measurements

To accurately estimate the thermal transmittance of the insulated walls, we conducted extensive continuous monitoring from June 2022 to May 2023. Key environmental and operational factors were observed throughout this duration. As depicted in Figure 4, an array of instruments, including heat flow meters, temperature sensors, and surface temperature sensors, were strategically installed at various locations—indoors, outdoors, and within the structure of the target wall. These instruments were tasked with the measurement of heat flow and temperature. The data collected from these measurements were then systematically compiled using a data logger. The main technical specifications of the measuring equipment utilized are concisely summarized in

Table 2. Technical specifications of the major measuring equipment.

Type of Equipment	Model and Manufacturer	Range	Accuracy
Temperature sensor	TC-PVC-T, Omega EngineeringInc., Norwalk, CT, USA	−250 °C to 350 °C	±0.5 °C
Heat flux sensor	HFP01, Hukseflux, Delft, The Netherlands	±2000 W/m²	±5%
Data logger	GL-820, Graphtec AmericaInc., Brooklyn Heights, OH, USA	20–50,000 mV	±0.1% of reading

.

This study’s approach to estimating the thermal transmittance of a wall hinged on the analysis of datasets, each spanning 7 days, which were continuously measured. For this experiment, the data were segmented into five distinct sections. This segmentation was based on the average air temperature difference ($∆T_{air}$) between the indoor and outdoor environments over the 7 days.

Table 3. Selection of a thermal resistance measurement case based on the indoor and outdoor temperature differences.

Case	Dataset Selection Conditions (7-Day Average $\mathsf{\Delta }{\mathit{T}}_{\mathit{a}\mathit{i}\mathit{r}}$)	Number of Datasets that Met the Criteria (Number of 7-Day Datasets)	7-Day Average $\mathsf{\Delta }{\mathit{T}}_{\mathit{a}\mathit{i}\mathit{r}}$ of Datasets
Case 1	10 °C (±1 °C)	20	9.92 °C
Case 2	15 °C (±1 °C)	31	14.97 °C
Case 3	20 °C (±1 °C)	7	19.84 °C
Case 4	25 °C (±1 °C)	21	24.89 °C
Case 5	30 °C (±1 °C)	7	29.90 °C

presents a detailed overview of the specific cases, categorized according to the standards of indoor and outdoor temperature differences. In total, this study utilized 86 weeks’ worth of datasets. To increase the number of data cases, we created each 7-day case by shifting one day at a time instead of using a fixed 7-day period from Monday to Sunday as a case. Consequently, we were able to generate 86 sets of 7-day cases from a typical year’s 54 weeks. This comprehensive collection of data was instrumental in facilitating a thorough analysis of the thermal and physical properties of the insulated wall under study.

3. Results and Discussion

3.1. Estimation of the U-Value

To estimate the thermal transmittance of the insulated wall amidst variably changing indoor and outdoor temperatures and heat flows, we employed the HFM method as specified in the ISO 9869-1, utilizing both the average and dynamic methods. Concurrently, the thermal transmittance for the same period was also estimated using the expanded average method (EXAM), as defined in this study.

The accuracy of the thermal transmittance estimates obtained through the HFM method and EXAM was evaluated against the thermal transmittance calculated by the ISO 6946’s method. This method bases the thermal transmittance calculation on the thermal conductivity of each material comprising a wall. Ideally, the in situ measured thermal transmittance should align closely with the value obtained from the calculation method. However, in real-world settings, the thermal transmittance value may deviate from the calculated one due to various factors, including the aging of insulation material, construction issues, and abnormal environmental conditions at the site [0]. Therefore, the accuracy of the field measurement method’s thermal transmittance estimation was assessed through a comparative analysis with the calculation method.

Furthermore, as outlined in Section 2.1.1, the convergence of the average method was verified against the criteria set forth in the ISO 9869-1. Similarly, the convergence of the extended average method was also reviewed, employing the same set of indicators.

3.1.1. Average Method

The in situ U-value for each subcase was estimated using the average method as per Equation (1). The estimates were then evaluated against the three convergence conditions outlined in the ISO 9869-1. Typically, if all three conditions are satisfied, the estimated U-value is considered reflective of the in situ value for a specific wall. However, the thermal transmittance values for each subcase were initially analyzed before verifying their adherence to the convergence conditions.

The estimation of heat transmittance using the average method, as depicted in Figure 5, showed variations influenced by indoor and outdoor temperature conditions. Each line represents a subcase within each main case, and it can be observed that the U-value was estimated as data accumulated over time within each subcase. A common observation was the significant variability in thermal transmittance estimates during the initial stages of measurement (approximately 2–3 days), primarily due to non-steady-state environmental conditions. As the measurement period progressed, a trend of convergence to a single value was noted, attributed to the cumulative averaging of data. Observation revealed that the heat transmittance estimate curve tended to converge more rapidly as the indoor and outdoor temperature difference increased.

On the 7th day of each case, the estimated thermal transmittance rates were analyzed, as shown in Figure 6. The average thermal transmittance values for the cases were the following: 0.135 W/m²K for Case 1, 0.137 W/m²K for Case 2, 0.135 W/m²K for Case 3, 0.135 W/m²K for Case 4, and 0.135 W/m²K for Case 5. When considered solely as average values, the thermal transmittance estimates across all cases appeared quite similar. However, in scenarios with relatively small temperature differences between indoors and outdoors, particularly in Case 1, the range of estimated thermal transmittance values was approximately 25%. The standard deviation of these values was 0.012 for Case 1, 0.005 for Case 2, 0.003 for both Cases 3 and 4, and 0.004 for Case 5. Hence, in Case 1, a relatively large variance in estimated thermal transmittance values was observed, rendering it challenging to determine a reliable thermal transmittance value for the wall.

In accordance with the ISO 9869-1, the evaluation of the convergence condition for the thermal transmittance value estimated by the average method is a crucial step in determining its validity. In this study, all cases were examined to ascertain whether they satisfied the three prescribed convergence conditions. Given that the first condition, which stipulates a minimum measurement period of 3 days, was fulfilled in all cases, the focus was on quantitatively assessing the fulfillment of Conditions 2 and 3. Figure 7 presents the resultant values for these conditions across each case. Here, a thermal transmittance value was considered to comply with the ISO standard if both Conditions 2 and 3 fell within a 5% margin.

An analysis of the percentages of subcases meeting both conditions in each case revealed that Cases 4 and 5 exhibited the highest compliance, with approximately 71.4% of subcases satisfying the criteria. This was followed by 42.8% in Case 3, 35.5% in Case 2, and only 5.0% in Case 1, where merely one subcase met all three conditions.

These outcomes appear to be consistent with the observations from the thermal transmittance rate estimation curve shown in Figure 5. Notably, Case 1 exhibited significant fluctuations over time in the heat transmittance estimate curve. This suggests that for Case 1, where the indoor/outdoor temperature difference was approximately 10 °C, a measurement period extending beyond 7 days might have been necessary for a more precise estimate. Consequently, in such a scenario (Case 1), the reliability of the thermal transmittance rate estimated over a 7-day period could be questionable.

3.1.2. Dynamic Method

The in situ U-value for each analysis case, as estimated using the dynamic method outlined in the ISO 9869-1, was analyzed, as presented in Figure 8. The average U-values determined for each case were the following: 0.183 for Case 1, 0.136 for Case 2, 0.136 for Case 3, 0.135 for Case 4, and 0.134 for Case 5. Notably, except for Case 1, the U-values for the remaining cases were highly similar, ranging between 0.134 and 0.136. This similarity indicates a consistency in the U-value estimates across these cases.

Moreover, the standard deviation of the estimated U-values for the subcases was found to be minimal, with values of 0.004 for Case 2, 0.002 for both Cases 3 and 4, and 0.005 for Case 5. A small standard deviation suggests limited variability in the estimated U-values within these subcases. Thus, for Cases 2 to 5, which exhibited specific indoor and outdoor temperature difference conditions, the reliability of the U-value estimation can be considered relatively high.

Conversely, Case 1 stands out due to its higher average U-value estimate compared with the other cases, with certain subcases featuring U-values exceeding 0.18. Additionally, the standard deviation for the estimated values in these subcases was calculated to be 0.11. This higher standard deviation implies a greater degree of variability in the U-values across the subcases of Case 1, indicating increased uncertainty in the U-value estimation for this particular scenario.

Case 1 comprises subcases where the average indoor–outdoor temperature difference over the analysis period of 7 days was approximately 10 °C. This period corresponds to the spring and fall seasons in Korea, which exhibit high physical variability in the surrounding environment owing to seasonal changes. Therefore, it is estimated that the error in Case 1 was larger than that in other cases owing to this variability. Additionally, these results suggest that if the heat flux and indoor–outdoor temperature differences are not consistent and vary significantly, the dynamic method may produce larger errors than the average method. Consequently, further studies are required to investigate the impact of changes in the physical elements of the surrounding environment using the in situ U-value measurement methods (dynamic and average methods).

3.1.3. Extended Average Method

Furthermore, Equations (8)–(11), embodying the concept of the extended average method proposed, were applied to estimate the U-value. Additionally, the compliance of this method with the average method convergence conditions set by the ISO 9869-1 was evaluated. The in situ thermal transmittance rate estimated for each analysis case is depicted in Figure 9. The average U-value for each case was determined to be 0.141 for Case 1, 0.151 for Case 2, 0.150 for Case 3, 0.149 for Case 4, and 0.150 for Case 5.

Similar to the results obtained using other HFM methods, in Case 1, where the average indoor and outdoor temperature difference was 10 °C, the estimated U-values of the subcases ranged from a minimum of 0.067 to a maximum of 0.159. This variation is reflected in the standard deviation, which was analyzed to be 0.018 for Case 1. In contrast, the standard deviations for the remaining cases were markedly lower, with 0.006 for Case 2, 0.002 for Case 3, 0.001 for Case 4, and 0.005 for Case 5, indicating a disparity of up to 18 times relative to Case 1. Thus, the methodology presented in this study also exhibited lower reliability in estimating the U-value under the specific indoor and outdoor environmental conditions of Case 1, as opposed to the other cases.

This study also involved an evaluation of the second and third conditions of the average method convergence criteria as stipulated by the ISO 9869-1. The values for Condition 2 and Condition 3 across each case were analyzed, as presented in Figure 10. According to the ISO standard, a compliance with both Condition 2 and Condition 3 was indicated if their values were within a 5% margin.

The analysis revealed that for each case, the percentage of subcases meeting both conditions varied by the following: Case 4 exhibited the highest compliance at 95.2%, followed by Case 5 at 85.7%, Case 3 at 71.4%, Case 2 at 58.1%, and Case 1 at 25%. Generally, as the difference between indoor and outdoor temperatures increased, the likelihood of meeting the convergence conditions also tended to increase. Notably, in Cases 4 and 5, nearly all the subcases satisfied all the required conditions, with only one exception in each case.

3.2. Comparison of Methods

Section 3.1 presented the in situ U-value estimation results, based on which the effectiveness and limitations of the extended average method proposed in this research were evaluated in comparison to the existing HFM method. This evaluation focused primarily on two aspects: the accuracy of U-value estimation and the satisfaction of convergence criteria.

Accuracy in this context refers to the discrepancy between the theoretical U-value as per the ISO 6946 (outlined in Table 1) and the estimated in situ U-value.

Table 4. Average U-values and absolute error rates compared with the theoretical U-values.

ID	U-Value (W/m2K)			Error with Uth * (%)
	AM	DM	EXAM	AM	DM	EXAM
Case 1	0.135	0.183	0.141	9.97	37.83	6.86
Case 2	0.137	0.136	0.151	5.63	5.85	4.15
Case 3	0.137	0.136	0.150	5.64	6.25	3.14
Case 4	0.135	0.135	0.149	6.76	6.83	2.57
Case 5	0.135	0.134	0.150	7.06	7.27	3.37
Average	0.136	0.145	0.148	7.01	12.81	4.02

* U_th refers to the theoretical U-value specified in the ISO 6946.

in this study details the results of the in situ U-value estimation and the corresponding accuracy analysis for both the AM and DM of the HFM, as well as for the extended average method. This table highlights the average U-value and the absolute average error rate for the subcases in each method. Furthermore, Figure 11 provides a visual representation of the error rate for the in situ U-value of each case in the form of a boxplot, allowing for a clear comparison of the error rates across the subcases.

A key observation from this analysis is that the extended average method demonstrated an improvement in the accuracy of in situ U-value estimation when compared with the traditional HFM approach. The average error rates for the average method and dynamic method across all cases were calculated to be 7.01% and 12.81%, respectively. While these rates indicate reasonable accuracy (below 15%), the extended average method further enhanced this accuracy, presenting an overall average error rate of 4.02% for all cases. This represents a reduction of 2.99% compared with the average method and 8.79% compared with the dynamic method.

A detailed examination of each case revealed the most significant improvement in accuracy under the conditions of Case 1 when compared with the dynamic method. While the dynamic method exhibited an average error rate of 37.83% in Case 1, indicating a high level of inaccuracy, the extended average method drastically improved this to 6.86%, signifying a substantial increase in accuracy. Moreover, for Cases 2, 3, 4, and 5, the extended average method consistently demonstrated a reduction in the error rate, ranging from a minimum of 1.70% to a maximum of 4.26%, when compared with the dynamic method.

The field-measured thermal transmittance via the average method of the HFM exhibited commendable accuracy, with an error rate of less than 10% across all cases. Despite this high level of precision, the extended average method further demonstrated its efficacy by reducing the error rate by a minimum of 1.48% and a maximum of 4.19% compared with the average method. While the extended average method did not achieve absolute superiority in accuracy over the HFM in every subcase, its overall performance was notably better across all experimental cases, when categorized by indoor and outdoor temperature differences. This indicates that, on average, the extended average method enhances the precision of thermal transmittance estimation under various environmental conditions, thereby demonstrating a significant improvement in the field of thermal performance measurement.

3.3. Heat Flow Characteristics of Highly Insulated Wall

As shown in Figure 11, the error rate distributions for both the HFM method and the extended average method were bifurcated into negative and positive directions. The in situ U-value estimations via the average and dynamic methods generally exhibited errors skewed toward the negative direction, with only a few subcases being exceptions. This trend indicates that the in situ U-values estimated by these methods were typically lower than the theoretical U-values derived from standard calculations.

The elucidation of this phenomenon’s cause was derived using the U-value inference formula from the average method. This formula articulates the U-value as the quotient of heat flow and the temperature differential between indoor and outdoor environments. Within this framework, the indoor and outdoor temperature values are classified as primary natural variables, while the heat flow emerges as a secondary variable, instigated by the aforementioned temperature difference. In our study, the primary variable—namely, the indoor/outdoor temperature differential—was confined to increments of 10, 15, 20, 25, and 30 °C. Concurrently, the denominator remained constant, leading to the inference that the U-value predominantly hinged on the heat flow, i.e., the numerator. Consequently, this suggests that the measured heat flow might have been insufficient to accurately estimate the theoretical heat transmittance. To corroborate this hypothesis, an analysis of the heat flow behavior within the wall was conducted.

To decipher the wall’s heat flow characteristics, we examined the correlation between each case’s cumulative indoor and outdoor temperature differential and the corresponding cumulative heat flow, as depicted in Figure 12. The slope of the line in the figure, representing the ratio of cumulative indoor/outdoor temperature difference to cumulative heat flow, is indicative of the heat transmittance as estimated by Equation (1). Employing this equation, we calculated the theoretical U-value’s heat flow (illustrated by a black dashed line) inversely under the conditions of the measured indoor/outdoor temperature difference, thereby facilitating an analysis of the heat flow characteristics based on this line.

As observed in Figure 12, the heat flows $q_{in}$ (measured on the indoor side of the inner insulation wall) and $q_{middle}$ (measured inside the wall, between the insulation and structural areas) exhibit marked disparities. Predominantly, $q_{in}$ was situated below the theoretical heat flow $q_{\mathrm{t}\mathrm{h}}$, indicating that under identical indoor and outdoor temperature differences, $q_{in}$ was smaller than $q_{\mathrm{t}\mathrm{h}}$. This resulted in a lesser slope for the $q_{in}$ line, representing the U-value, compared with the $q_{\mathrm{t}\mathrm{h}}$ line, thus signifying that the U-value estimated by $q_{in}$ was generally lower than its theoretical counterpart. This observation aligns with the findings presented in Figure 11. Conversely, the heat flow $q_{middle}$ measured within the wall was positioned above the theoretical $q_{\mathrm{t}\mathrm{h}}$. This suggests that $q_{middle}$ exceeded $q_{\mathrm{t}\mathrm{h}}$, leading to the conclusion that the thermal transmittance value estimated by $q_{middle}$ surpassed the theoretical thermal transmittance.

The utilization of high-performance insulation materials was anticipated to accentuate this phenomenon. Hence, in walls employing such materials, the disparate heat flow characteristics of the insulation and structural materials must be duly considered. However, as the HFM method predominantly reflects indoor heat flow, it is likely to yield underestimations in the U-value of walls with high-performance insulation.

4. Conclusions

In this study, the in situ thermal transmittance of internal insulation exterior walls under variably changing real-world conditions was estimated via the HFM method as outlined in the ISO 9869-1, encompassing both average and dynamic methodologies. The precision of the estimated U-values was assessed through comparison with the theoretical U-values specified in the ISO 6946, alongside an evaluation of convergence characteristics in accordance with the criteria set by the ISO 9869-1. Further, this study identified the characteristics of heat flow through the wall and considered the limitations inherent in the HFM method, particularly in the context of highly insulated walls. To enhance the HFM method, an extended average method was proposed, with its efficacy verified through comparative analyses with existing methods.

A specially designed experimental wall was constructed for the accurate estimation of the in situ U-value, with various parameters being measured over an extensive period from June 2022 to May 2023. The U-value of the wall was determined using data from 7 consecutive days, focusing on cases where the average 7-day indoor and outdoor temperature differential was 10, 15, 20, 25, and 30 °C.

This study revealed that the average thermal transmittance values, as per the different methods, were 0.136 W/m²K for the average method, 0.145 W/m²K for the dynamic method, and 0.148 W/m²K for the expanded average method. Notably, Case 1, characterized by a 10 °C indoor and outdoor temperature difference, exhibited the lowest accuracy. Furthermore, the significant standard deviation in the estimated thermal transmittance values of Case 1’s subcase indicated challenges in ascertaining a highly reliable on-site thermal transmittance value under such conditions.

When comparing the in situ U-value estimation accuracy by methodology, the extended average method proposed in this study proved superior to the standard HFM in all scenarios. The average error rates for all cases were 7.01% for the average method and 12.81% for the dynamic method, while the extended average method demonstrated a significantly lower average error rate of 4.02%, thereby indicating a marked improvement in accuracy.

The enhanced efficacy of the extended average method compared with the standard HFM is attributed to its consideration of heat flow characteristics within highly insulated walls. The U-value determined by the HFM was generally lower than the theoretical U-value, a discrepancy arising from the minimal heat flow measured on the indoor surface due to the high-performance insulation material. Therefore, to accurately estimate the in situ U-value of a highly insulated wall, the heat flow through both the insulation and structural areas must be separately considered.

However, the challenge lies in the general impossibility of non-destructively measuring the heat flow within a wall, i.e., between the insulation material and the structural material. Thus, the EXAM cannot be regarded as a non-destructive in situ U-value estimation method. This study’s significance lies in its presentation of a concept to improve the accuracy of in situ U-value measurements, along with the verification of this concept on a specially manufactured wall. Future research efforts will focus on experiments designed to predict and estimate the heat flow inside a wall non-destructively.