# Constructal Evaluation of Polynomial Meta-Models for Dynamic Thermal Absorptivity Forecasting for Mixed-Mode nZEB Heritage Building Applications

## Abstract

**:**

^{−1}/2 m

^{−2}K and will be developed further to improve its absolute location accuracy for scenarios wherein the thermal absorptivity value is lower than 50 Ws

^{−1}/2 m

^{−2}K.

## 1. General Context, Findings, and Outcomes

- Thermal comfort models and international standards;
- Thermal characteristics of fabrics and thermodynamic parameters;
- MM ventilation strategies and building operation;
- Design of Experience (DOE) methods and Response Surface Methodology (RSM);
- Constructal law and entropy analysis.

#### Case Study: Heritage Building and Nearly Zero-Energy Building (nZEB) Categorization, or Why Indoor Thermal Comfort Is a Crucial Factor for Overall Energy Performance

## 2. Introduction, Thematic Subjects, and Interdisciplinary Aspects

_{2}concentration, and odors, are now quantified; hence, in order to be built, a well-conceived building must satisfy strict ad hoc quantitative dynamic standards that evolve over time [1,6]. This explains why, during the late twentieth and, in particular, twenty-first century, many quantitative studies focused on sunlight and thermal energy consumption in the building sector.

_{2}concentration and odors, hygienic issues highlighting the necessity of introducing clean and open air into indoor environments date back to the late eighteenth century, when the industrial revolution was globalized. These hygienic parameters linked to architectural spatialization appeared for the first time in the form of an architectural manifest in the Athens Charter [4,5,6,7,8,9].

## 3. State of the Art

#### 3.1. Thermal Comfort Theories and Related International Standards

- The local air velocity, which influences the heat exchange between the body and the surrounding environment via convection due to changes in the evaporation on the skin’s surface. Convection is heat transfer via the bulk movement of molecules within fluids such as gases and liquids, including molten rock (rheid). Convection includes sub-mechanisms of advection (the directional bulk-flow transfer of heat) and diffusion (the non-directional transfer of energy or mass particles along a concentration gradient). Air velocity affects heat transfer due to the temperature difference between the air and body surfaces and also promotes evaporation from the skin. In MM buildings, this parameter is very important, especially when the building operates in natural ventilation mode in order to avoid local discomfort issues. In order to ensure thermally comfortable conditions, we should generally limit the indoor air velocity to around 0.2–0.3 m/s.
- The ambient air temperature and the surficial wall temperature, which directly influence thermal comfort. The “cold wall” effect refers to the cold feeling experienced when an occupant stands near the elements of the building envelope (walls or windows). This occurs because the temperature of these elements is significantly lower than that of ambient air.
- Metabolism, i.e., the biological thermoregulatory process (thermal body homeostasis) that regulates the internal heat of the human body; for healthy organisms, the internal temperature at the skin level is maintained around 36.6 °C.
- The air’s relative humidity, i.e., the ratio of the partial pressure of water vapor to the equilibrium vapor pressure of water at the same temperature. Studies have revealed that humid air transmits sensations of cold or heat much more effectively than dry air because it contains water vapor (see also [45,46,47]).

#### 3.1.1. The Concept of Koestel and Tuve [14]

#### 3.1.2. Fanger’s PMV and PPD Concepts [12,15]

^{TM}, introducing two original indexes, Predicted Mean Vote (PMV) and Predicted Percentage of Dissatisfied (PPD). The combination of these two indexes was expected to be capable of predicting the optimal indoor thermally comfortable conditions. Fanger’s indexes represented an update on the Effective Draft Temperature (EDT) concept by taking into consideration the three modes of heat transfer (radiation, conduction, and convection), as well as metabolic mechanisms such as respiration and perspiration. The output of Fanger’s calculations was the Predicted Mean Vote (PMV) index, which, according to his theory, could forecast the potential thermal vote of a number of occupants located within the studied indoor conditions [12,15]. To improve the predictive capacity of the model, he introduced a complementary index named Predicted Percent of Dissatisfaction (PPD) to complement the PMV [12,15]. The PPD index predicted the percentage of occupants in discomfort. Fanger’s PMV index expressed the average thermal sensation experienced by a large group of individuals based on the wide thermal sensation scale proposed by ASHRAE, and a value of 10% for the PPD index corresponded to a value between −0.5 and +0.5 for the PMV index [11,12,15,20].

#### 3.1.3. Gagge’s Two-Node Model and the Updated Effective Temperature (ET) Concept [16,17,20]

#### 3.1.4. The Berkeley Model [19]

#### 3.1.5. Adaptive Thermal Comfort Concept

#### 3.1.6. International Standards and Databases

#### 3.2. Thermodynamic Properties of Clothing (ISO 9920)

#### 3.3. MM Building Operation

#### 3.4. Design of Experience (DOE) Methods and Response Surface Methodology

_{i}, referring to the main parameters that, according to the updated ASHRAE 55 international standard [13], influence thermal comfort.

_{i}), where y is considered the response (thermal absorptivity) and x

_{i}represents the variables investigated in each set of simulations (fabric density, thickness, and thermal capacity and minimum outside temperature). According to the RSM approach, the values of x

_{i}were the coordinates of a simulation point inside our investigated domain, and y was the calculated value of the response at that specific point. To calculate y, we used the ASHRAE 55 [13] adaptive thermal comfort model updated by Kim et al. [28]. In the past, we successfully employed DOE methods and RSM to:

- Identify and propose an empirical polynomial equation for calculating the overall thermal resistance of a complex sophisticated dynamic composite building envelope element departing from the in-wall air-gap thickness values [59];
- Develop accurate regression models for daylight factor prediction within buildings at an early design stage, when the relevant data have not been precisely determined (dimension of glazing area, materials, opacities) [61];
- Link wide-ranging geometrical and non-geometrical glazing options for daylight effectiveness estimation at an early design stage [62].

## 4. Materials and Methods

#### Mathematical Formulation of the Problem

_{cl}, in clo) predictive model in relation to the outdoor temperature at 6 a.m. according to the research findings of Schiavon and Lee [27]. The mathematical formulation of this approach was as follows ([13,27] cited in [28]):

_{cl}(in clo) is dynamic clothing insulation, and T

_{out6a.m.}is the measured outside temperature at 6 a.m. The R-squared value of this predictive model was 0.33. According to their observed values, Kim et al. [28] found that the ASHRAE model underestimated the clo value when T

_{out6a.m.}< 18 °C and overestimated the clo value when T

_{out6a.m.}> 18 °C. Nevertheless, Kim et al. [28] detected up to a few degrees (°C) of inconsistency between T

_{outMIN}and T

_{out6a.m.}and proposed an improved version of the ASHRAE Standard 55 model. Their predictive model was built in relation to the outside daily minimum air temperature, T

_{outMIN}, instead of T

_{out6a.m.}. This model had a very high R-squared value of 0.37 [28] and was formulated mathematically as follows:

_{clothing}is the overall thermal resistance of the fabric. The thermal resistance of a fabric (R

_{clothing}) is connected mathematically to the fabric thickness σ (in m) and thermal conductivity k (in Wm

^{−1}K

^{−1}) through the following equation [46]:

^{1/2}m

^{−2}K) is a surface property that changes based on the use of the fabric (laundry, wear, ageing) and the finishing processes. This parameter was of interest in our study since it allows the assessment of the fabric characteristics in terms of “cool-warm” sensations, according to Frydrych et al. [45] and Hes et al. [47,48]. Specifically, fabrics with a low value of thermal absorptivity (b) provide a “warm” feeling, according to Hes et al. [47,48]. The finishing of a fabric also significantly influences the thermal absorptivity parameter. It has been observed and reported that elastomeric finishing produces lower thermal absorptivity values and, consequently, warmer sensations than starch finishing [45,46,47,48,49,50,51]. As a rule of thumb, according to the literature [45,46,47,48,49,50,51], fabrics with a regular, flat, and smooth surface have a high thermal absorptivity value, offering a cooler sensation, while fabrics of lower regularity and smoothness and higher surface roughness have a lower b value, offering a warmer feeling. The parameter of thermal absorptivity (b) was calculated according to the following relation [45,47,48]:

^{−1}K

^{−1}, ρ is the volumetric fabric density in kg/m

^{3}, and c

_{p}is the specific heat capacity of the fabric in Jkg

^{−1}K

^{−1}. Thus, developing Equation (6), we obtain:

## 5. Results and Discussion

_{i}is the value of the ith factor (i = 1, 2, 3, …), and x

_{j}is the value of the jth factor (j = 1, 2, 3, …). The parameters were carefully selected to produce a composite factorial design and a Hoke D6 design, whereby, as in our past research [58,59,60,61,62], the effect of each factor was evaluated at three different levels (two for the investigation and one for the validation of the statistical model) according to codified values of −1, 0, +1. In order to cover an extremely large domain of potential scenarios and extend the applicability of our polynomial function, the thickness of the fabric ranged between 0.00002 m and 0.007 m, while the median value was equal to 0.0350 m.

^{3}, 750 kg/m

^{3}, and 1000 kg/m

^{3}, while the fabric thermal capacity values were 1000 J/kg K, 1500 J/kg K, and 2000 J/kg K.

- Provided us a good in-sample fit, associated with low error measurements and normalized residuals (NMRSE for training data, −0.406478699; NMRSE for test data, −1.417656707).
- Avoided systematic random overfitting and slightly underfit the data by providing us a satisfactory out-of-sample forecast accuracy.

- When the residual is less than −2, the model’s output is less than the expected output.
- When the residual is greater than 2, the model’s output is greater than the expected output.

^{2}values: 0.99603 for the Hoke D6 model and 0.9829 for the composite factorial regression model.

^{2}= 0.9901. On the other hand, for the test scenarios, the composite factorial regression model was revealed to have a very poor forecast capacity.

^{2}value of 0.58705 for the testing scenarios. We observed a decrease of around 40% in the accuracy of the model. Thus, the Hoke D6 design performed better for our case study, and Equation (11) took the following mathematical form after the incorporation of the calculated coefficients:

#### 5.1. Analysis of Standardized Residuals’ Entropy

_{b}[57], which is obtained by the general equation:

^{2}values are shown in Figure 6. We observe in Figure 6 that the residuals were generally distributed around a 6th-order polynomial trend. This 6th-order polynomial equation represented the potential number of microstates.

#### 5.2. Constructal Evaluation and Evolution of the Hoke D6 Regression Model

^{−1/2}m

^{−2}K. Thus, we had to find a way to create scenarios that could train our model to freely achieve maximum fitness and adaptability, even though these two objectives are contradictory, local, and disunited [78].

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Test dataset comparison. Both DOE methods provided models with satisfactory data trends and good fit to absolute location for the test datasets. (

**a**) Outputs of Hoke D6 design model compared to those of Kim et al.’s model [28]; (

**b**) outputs of composite factorial regression model compared to those of Kim et al.’s model [28]. y axis—b in Ws

^{1/2}m

^{−2}K; x axis—the number of the tested scenario.

**Figure 2.**Training dataset comparison. Least-square line equation and the square of the linear correlation coefficient are also shown for (

**a**) Hoke D6 design and (

**b**) composite factorial design. Both axes represent b in Ws

^{1/2}m

^{−2}K.

**Figure 3.**Test dataset comparison. Hoke D6 design model’s satisfactory fit to Kim et al.’s model [28]: data trends followed the pattern of Kim et al.’s model. We can also observe from this graph the poor forecast capacity of the composite factorial plan regression model, since it presented a very bad fit to the absolute location and did not follow the data trends and the pattern outlined by Kim et al.’s model. y axis—b in Ws

^{1/2}m

^{−2}K; x axis—the number of the tested scenario.

**Figure 4.**Test dataset comparison. Least-square line equation and the square of the linear correlation coefficient are also shown for (

**a**) Hoke D6 design and (

**b**) composite factorial design. Both axes represent b in Ws

^{1/2}m

^{−2}K.

**Figure 5.**Codification equations for each parameter. Each parameter had to be codified before using the regression equation that resulted from the Hoke D6 simulation scenarios. y axis—codified values; x axis—real values for (

**a**) fabric thickness σ, (

**b**) fabric density ρ, (

**c**) fabric thermal capacity c

_{p}, and (

**d**) minimum outside temperature T

_{out_min}.

**Figure 6.**Graphical distribution of the overall standardized residuals, trendline, and R

^{2}values. x axis—b in Ws

^{1/2}m

^{−2}K; y axis—entropy in J/K.

**Figure 7.**Graphical distribution of the overall entropy and trendline. x axis—b in Ws

^{1/2}m

^{−2}K; y axis—entropy in J/K.

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**MDPI and ACS Style**

Mavromatidis, L.
Constructal Evaluation of Polynomial Meta-Models for Dynamic Thermal Absorptivity Forecasting for Mixed-Mode nZEB Heritage Building Applications. *Energies* **2023**, *16*, 429.
https://doi.org/10.3390/en16010429

**AMA Style**

Mavromatidis L.
Constructal Evaluation of Polynomial Meta-Models for Dynamic Thermal Absorptivity Forecasting for Mixed-Mode nZEB Heritage Building Applications. *Energies*. 2023; 16(1):429.
https://doi.org/10.3390/en16010429

**Chicago/Turabian Style**

Mavromatidis, Lazaros.
2023. "Constructal Evaluation of Polynomial Meta-Models for Dynamic Thermal Absorptivity Forecasting for Mixed-Mode nZEB Heritage Building Applications" *Energies* 16, no. 1: 429.
https://doi.org/10.3390/en16010429