Multi-Objective Numerical Analysis of Horizontal Rectilinear Earth–Air Heat Exchangers with Elliptical Cross Section Using Constructal Design and TOPSIS

Abstract

This study presents a numerical evaluation of a Horizontal Rectilinear Earth–air Heat Exchanger (EAHE), considering the climatic and soil conditions of Viamão, Brazil, a subtropical region. The Constructal Design method, combined with the Exhaustive Search, was utilized to define the system constraints, degree of freedom, and performance indicators. The degree of freedom was characterized by the aspect ratio between the vertical and horizontal lengths of the elliptical cross-section duct (H/L). The performance indicators for the EAHE configurations were assessed based on thermal potential (TP) and pressure drop (PD). The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) was applied for multi-objective evaluation, and a methodology for EAHE is proposed. The problem was solved using FLUENT software (version 2024 R2), which employs the Finite Volume Method to solve the conservation equations for mass, momentum, and energy. The (H/L)_T,o = 6.0 configuration showed a 16.4% increase in thermal performance for heating and 15.9% for cooling compared to the conventional circular duct. Conversely, the (H/L)_F,o = 1.0 configuration reduced pressure loss by 65.33%. The integration of Constructal Design with TOPSIS facilitated the identification of optimized geometries that achieve a balance between performance indicators and those that specifically prioritize thermal or fluid dynamic aspects, being this approach an original scientific contribution of the present work.

1. Introduction

In recent years, there has been a significant increase in the search for sustainable air conditioning solutions. Currently, about one-third of the energy consumed in residential and commercial buildings is used for heating and cooling [1,2]. This high energy consumption can be mitigated using a device known as earth–air heat exchanger (EAHE). The EAHEs have the ability to harness the superficial geothermal energy available in the soil, presenting themselves as a promising and sustainable technology capable of cooling or heating built spaces with minimal environmental impact [3,4,5,6,7].

Solar radiation is an abundant source of energy in the form of light and heat. This source can significantly contribute to reducing the use of non-renewable resources [8]. The EAHE utilizes this energy reserve by employing one or more buried ducts through which the air flows, facilitating the heat exchange with the soil [9]. During warmer seasons, heat is transferred from the air to the soil, aiding in cooling. Conversely, in cooler seasons, the soil transfers heat to the air, aiding in warming [10]. The advantages of the EAHEs include: the ability to adapt the system to various types of soil and climatic conditions; the use of air as the working fluid; low maintenance costs; and their environmentally friendly nature, with no combustion or pollutant emissions [11,12].

The thermal performance of an EAHE is influenced by a variety of parameters, including air flow conditions (velocity and mass flow), geometric and dimensional characteristics of the ducts, and installation-related factors (thermophysical properties of the soil and air, installation depth, and climatic conditions) [10]. There are several studies dedicated to investigate these influences over the EAHE thermal performance [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. Despite that, according to Wei et al. [28] and Benhammou et al. [29], there is a significant gap in the current literature regarding the analysis of the influence of duct cross-sectional geometry on the thermal performance of EAHEs. Recent studies show that the thermal performance of EAHEs is directly influenced by the surface area available for thermal exchange in the ducts. Wei et al. [28] investigated the performance of ducts with a rectangular cross-section in a straight EAHE and found that, in addition to increasing the heating and cooling capacity of the air, this configuration provided a more stable temperature in the duct walls and reduced thermal disturbance in the soil, compared to EAHEs equipped with ducts of circular cross-section. Jahanbin [30] investigated the application of ducts with an elliptical cross-section in vertical U-shaped EAHEs and observed that this configuration resulted in up to a 17% improvement in the reduction of the system’s thermal resistance compared to U-shaped ducts with a circular cross-section. Benhammou et al. [29] conducted a case study on EAHEs for air cooling and found that the use of ducts with rectangular and elliptical cross-sections was more effective than conventional circular cross-section ducts. Specifically, rectangular and elliptical ducts can provide a heat transfer area between the air and soil that was approximately 88% and 93% larger, respectively, compared to systems with circular geometry. Consequently, these results highlight the importance of expanding studies on duct cross-sections in EAHEs, and this work will seek to address this gap by exploring elliptical cross-section ducts in rectilinear EAHEs.

In this context, the Constructal Design method is an appropriate technique for evaluating the geometric configuration of finite-dimensional flow systems. This field originated with the definition of Constructal Theory, which posits that the design of flow systems, as well as their behavior, is a universal physical phenomenon that can be interpreted as the tendency for the evolution of flow system designs [31,32]. In engineering, the Constructal Design method has been adopted to define performance indicators, constraints, and degrees of freedom of systems, allowing for the modification of their geometries with the goal of achieving a geometric configuration that improves efficiency and reduces imperfections within these parameters [33,34]. Recently, Constructal Design has been successfully applied in studies on geometric variations in EAHEs [10,35,36,37,38]. Therefore, the application of the Constructal Design enables defining the several different geometric configurations to be investigated in such a way that allows a fair comparison between them, i.e., forming the search space. If the Exhaustive Search technique, which is an optimization method, is associated to the Constructal Design, it is possible to determine the optimized geometry for each performance indicator and, in addition, to understand how the degrees of freedom influence the performance indicator [39,40,41,42,43,44,45].

In the geometric investigation of EAHEs, performance analysis is of utmost importance. The thermal capacity of these systems can be quantitatively measured by the thermal potential (TP), which expresses the temperature variation of the air between the inlet and outlet sections of the duct. In terms of fluid dynamic analysis, the evaluation is carried out through the pressure drop (PD) along the EAHE, representing the resistance encountered by the air flow as it passes through the system [10]. This approach provides a comprehensive understanding of the system’s capabilities and limitations across different geometric configurations. Additionally, the combined evaluation of these indicators characterizes a multi-objective problem.

The solution to multi-objective problems can be achieved using the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). This method is based on the idea that the ideal solution to a problem should have the smallest Euclidean distance from the positive ideal solution and, simultaneously, be as far as possible from the negative ideal solution [46]. It is noteworthy that TOPSIS is appropriate for problems where performance indicators are addressed quantitatively and has been successfully employed in systems that combine thermal and fluid dynamic analyses, as demonstrated in studies such as Cui et al. [47]; Li et al. [48]; Razera et al. [49]; Sun et al. [50] and Keykhah et al. [51]; Trilok and Gnanasekaran [52].

Given this context, the present work aims to numerically evaluate the horizontal rectilinear EAHE with an elliptical cross-section through Constructal Design method associated with the Exhaustive Search technique, considering the climatic and soil characteristics of the city of Viamão in southern Brazil. The general objective of the study is to maximize the heat transfer between the external air and the soil, as well as to minimize the system’s head loss, through geometric variation of the elliptical cross-section of the EAHE. Regarding the multi-objective evaluation of the problem, a general methodology for applying the TOPSIS method to EAHEs is developed and presented for the first time, in which each studied geometry will be evaluated concerning the performance indicators TP and PD, with different weights of importance.

2. Mathematical Modeling

The mathematical model associated with the functioning of the EAHE is based on the fundamental principles of conservation of mass, momentum, and energy. For the soil layer, the effects of heat diffusion are mathematically modeled as defined by [53]:

$${\displaystyle \frac{\partial T}{\partial t}}={\alpha }_s{\displaystyle \frac{{\partial }^{2}T}{\partial x_i^2}}i = 1, 2, 3 \mathrm{in} t \times \Omega ,$$

(1)

in which, Einstein’s index notation is used. The spatial domain, Ω, is represented by the variable x_i, along direction i. The variable t denotes the temporal domain, α_s is the thermal diffusivity of the soil, assumed to be constant, and T represents the temperature.

For the flow inside the duct, the system is treated as transient, incompressible, turbulent, and undergoing forced convection heat transfer, as in Rodrigues et al. [10]. Considering the simplifying assumptions, the conservation equations for mass, momentum, and energy are defined as follows [54,55]:

$${\displaystyle \frac{\partial ({\overline{v}}_i)}{\partial x_i}}=0 i = 1, 2, 3 \mathrm{in} t \times \Omega ,$$

(2)

$${\displaystyle \frac{\partial \overline{v_i}}{\partial t}}+{\displaystyle \frac{\partial \left(\overline{v_i}\overline{v_j}\right)}{\partial x_j}}=-{\displaystyle \frac{1}{\overline{\rho }}}{\displaystyle \frac{\partial \overline{p}}{\partial x_j}}{\delta }_{ij}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\upsilon \left({\displaystyle \frac{\partial \overline{v_i}}{\partial x_j}}+{\displaystyle \frac{\partial \overline{v_j}}{\partial x_i}}\right)-{\tau }_{ij}\right] i,j = 1, 2, 3 \mathrm{in} t \times \Omega ,$$

(3)

$${\displaystyle \frac{\partial \overline{T}}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\overline{v_j}\overline{T}\right)={\displaystyle \frac{\partial }{\partial x_j}}\left({\alpha }_a{\displaystyle \frac{\partial \overline{T}}{\partial x_j}}-q_j\right) i,j = 1, 2, 3 \mathrm{in} t \times \Omega ,$$

(4)

the overline (—) denotes the temporal average. The variable v represents the flow velocity, p is the static pressure of the fluid, δ_i,j is the Kronecker delta, υ is the dynamic viscosity of the air, and α_a is the thermal diffusivity of the air, assumed to be constant.

The turbulence model used to address the closure problem was the Reynolds-Averaged Navier-Stokes (RANS) k-ε model [10]. This approach involves solving two partial differential equations for transport: one for turbulent kinetic energy (k) and one for the rate of dissipation of turbulent kinetic energy (ε), given by [54,55]:

$${\displaystyle \frac{\partial k}{\partial t}}+{\overline{v}}_j{\displaystyle \frac{\partial k}{\partial x_j}}={\tau }_{ij}{\displaystyle \frac{\partial {\overline{v}}_i}{\partial x_j}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\upsilon {\displaystyle \frac{{\upsilon }_t}{{\sigma }_k}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]-\epsilon i,j,k = 1, 2, 3 \mathrm{in} t \times \Omega ,$$

(5)

$${\displaystyle \frac{\partial \epsilon }{\partial t}}+{\overline{v}}_j{\displaystyle \frac{\partial \epsilon }{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\upsilon +{\displaystyle \frac{{\upsilon }_t}{{\sigma }_{\epsilon }}}\right){\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right]+C_{\epsilon 1}{\displaystyle \frac{\epsilon }{k}}{\tau }_{ij}{\displaystyle \frac{\partial {\overline{v}}_i}{\partial x_j}}-C_{\epsilon 2}{\displaystyle \frac{{\epsilon }^2}{k}} i,j = 1, 2, 3 \mathrm{in} t \times \Omega .$$

(6)

The constants $C_u$ = 0.09, $C_{\epsilon 1}$ = 1.44, $C_{\epsilon 2}$ = 1.92, and $P{r}_t$ = 1.0 are found in Wilcox [54] and Launder and Spalding [55], and they are applicable to a wide range of turbulent flows [10].

The problem under analysis is illustrated in Figure 1, which shows an idealized EAHE consisting of a rectilinear duct buried in the soil.

In the computational model, the soil is represented with a depth (H_S) of 15 m, a width (W_S) of 5 m, and a length (L_S) of 25.77 m (Figure 1a). The distance from the duct to the sidewalls of the soil (w_l) is 2.5 m. The installation depth of the duct relative to the soil surface is h = 1.6 m, characterizing a shallow installation depth EAHE. It is interesting to inform that these dimensions are based on the experimental and numerical study performed in Vaz et al. [9]. For the boundary conditions of the computational domain, zero heat flux conditions are applied to the lateral faces and the bottom face of the soil $(\partial T/\partial x=\partial T/\partial y=\partial T/\partial z=0)$. Prescribed temperature conditions that vary with time are applied at the soil surface and the inlet region, represented by T_s(s) and T_in(s), respectively. These boundary conditions, T_s(s) and T_in(s), were established based on experimental data provided by Vaz et al. [9], reflecting the climatic conditions of Viamão, Rio Grande do Sul, Brazil, for the year 2007. For the flow, a constant prescribed velocity condition of v = 3.3 m/s is applied at the duct inlet, and zero-gauge pressure is applied at the outlet region, P_o = 0 Pa, as specified by Vaz et al. [9]. Impermeability and no-slip conditions are applied to the walls of the duct. The thermo-physical properties of the air and soil are detailed in

Table 1. Properties of the fluid and soil.

Properties	Symbol	Air	Soil
Density (kg/m3)	ρ	1.16	1800
Thermal Conductivity (W/m·K)	k	0.0242	2.1
Specific Heat (J/kg·K)	Cp	1010	1780
Dynamic Viscosity (kg/m·s)	μ	1.789 × 10−5	-

, which describe the specific conditions of the installation site [9].

As already mentioned, the Constructal Design method, combined with the Exhaustive Search technique, was employed to determine the constraints, degrees of freedom, and objectives in the geometric evaluation of the elliptical cross-sectional duct of the EAHE. The system has two geometric constraints: the cross-sectional area of the duct (A_D), which is maintained constant at 0.0095 m², and the volume of the duct (V_D), also constant at 0.2449 m³. This implies that the length of the duct (L_D = L_S) is fixed at 25.77 m. Figure 1b illustrates the geometric definition of the elliptical cross-section of the duct. The degree of freedom is characterized by the ratio between the vertical axis (H) and the horizontal axis (L) of the elliptical duct (H/L). The equations describing the system’s geometry for any value of H/L are defined as follows:

$$A_D={\displaystyle \frac{\pi \cdot H\cdot L}{4}},$$

(7)

$$L=\sqrt{{\displaystyle \frac{A_D\cdot 4}{\pi \cdot \left(\frac{H}{L}\right)}}},$$

(8)

$$H={\displaystyle \frac{A_D\cdot 4}{L\cdot \pi }}.$$

(9)

Figure 1b illustrates the various geometries resulting from changes in the H/L ratio, which is examined within the range 2.0 ≤ H/L ≤ 6.0. As the ratio H/L approaches its lower limit (H/L = 0.2), the elliptical duct’s cross-section becomes increasingly elongated horizontally relative to the ground surface. Conversely, as H/L approaches its upper limit (H/L = 6.0), the cross-section of the duct elongates vertically relative to the ground surface. Additionally, when H/L = 1.0, indicating a circular cross-section, the diameter is given by d = H = L = 0.11 m, which is the same diameter of the EAHE’s duct studied by Vaz et al. [9].

Regarding the thermal performance of the system, the EAHE is evaluated by calculating the thermal potential (TP). TP is defined as the temperature difference between the air at the outlet and inlet of the device and can be calculated at each time interval. It is typically expressed as an average over a specified period (daily, monthly, or annually) [10]:

$$TP={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\left[T_i^o(t)-T_i^{in}(t)\right]},$$

(10)

where $T_i^o(t)$ and $T_i^{in}(t)$ represent the temperatures of the air at the outlet and inlet of the EAHE system, respectively, evaluated at time interval t for a number N of monitored data points.

For the fluid dynamic evaluation of the EAHE, the performance indicator used is the pressure drop within the duct, defined as [10]:

$$PD=P_{in}-P_o,$$

(11)

where P_in and P_o represent the average pressure values at the inlet and outlet faces of the EAHE duct, respectively.

In this study, there is a need to evaluate EAHEs through the maximization of TP and the simultaneous minimization of PD, thus defining a multi-objective problem. The TOPSIS method emerges as a robust and suitable approach for such analysis, recognized for its ability to model quantitative criteria [49]. This method is based on the principle that the optimal solution in a multi-objective system should present the shortest Euclidean distance to the positive ideal solution, while maximizing the Euclidean distance to the negative ideal solution. The positive ideal solution is the one that provides the best results for the desired criteria, whereas the negative ideal solution provides the worst results for the same criteria [46]. In this context, this work aims to introduce a detailed methodological procedure for applying the TOPSIS method to EAHEs, focusing on the maximization of TP and the minimization of PD, with the goal of jointly optimizing thermal performance and energy efficiency of the systems. Thus, the TOPSIS method is developed according to the following procedure:

• Create a matrix (x_i,j)_m×n with m alternatives (sets of geometries) and n objectives (performance indicators of the system), as illustrated in Figure 2.

For the evaluation of the multi-objective problem, it is recommended to use the absolute values of the TP, since TP assumes negative values in the context of air cooling. In this way, the maximization of TP is considered a benefit criterion for both heating and cooling. This consideration is applied in the subsequent steps of the procedure.

2.

Construct the normalized decision matrix (r_i,j)_m×n (Figure 3) transforming the performance indicators TP and PD into dimensionless attributes, as follows:

$$r_{i,j}={\displaystyle \frac{x_{i,j}}{\sqrt{{\displaystyle \sum _{i=1}^m{x}_{i,j}^2}}}}.$$

(12)

3.

Construct the weighted normalized decision matrix (V_i,j)_m×n (Figure 4) by assigning importance weights to each of the performance indicators evaluated, as follows:

$$V_{i,j}=r_{i,j}\cdot w_j,$$

(13)

$$w=(w_1,w_2,\dots ,w_j,\dots ,w_n),{\displaystyle \sum _{j=1}^n{w}_j=1}.$$

(14)

In this study, w₁ represents the weight associated with the thermal relevance (TP), and w₂ represents the weight associated with the fluid dynamic relevance (PD).

4.

Classification of benefit and non-benefit parameters: TP is the benefit parameter, which should be maximized, meaning higher values are preferable. On the other hand, PD within the duct is the non-benefit parameter, which should be minimized, meaning lower values are desirable.

5.

Definition of the artificial positive ideal solution (V⁺) and negative ideal solution (V⁻): the artificial positive ideal solution (V⁺) represents the optimal condition of the system for both benefit and non-benefit criteria. In this study, it corresponds to the highest weighted normalized value of TP and the lowest weighted normalized value of PD. Conversely, the artificial negative ideal solution (V⁻) reflects the worst conditions for these criteria, meaning the lowest weighted normalized value of TP and the highest weighted normalized value of PD. The V⁺ and V⁻ are respectively, defined as:

$$V^{+}=\left.\left.\left\{\left(\left.\underset{i}{\max }{V}_{i,j}\right|j\epsilon J\right)\right.,\left(\left.\underset{i}{\min }{V}_{i,j}\right|j\epsilon J^{\prime }\right)\right|i=1,2,\dots ,m\right\},$$

(15)

$$V^{-}=\left.\left.\left\{\left(\left.\underset{i}{\min }{V}_{i,j}\right|j\epsilon J\right)\right.,\left(\left.\underset{i}{\max }{V}_{i,j}\right|j\epsilon J^{\prime }\right)\right|i=1,2,\dots ,m\right\}.$$

(16)

6.

Calculation of the Euclidean distance for each alternative m (each evaluated EAHE geometry) relative to the V⁺ and V⁻: the distance between each alternative is measured using the dimensionless Euclidean distance, which represents the relative proximity of each weighted normalized value to the V⁺ (as illustrated in Figure 5a) and to the V⁻ (as shown in Figure 5b). These distances are calculated using, respectively, the following equations:

$$S_i^{+}=\sqrt{{\displaystyle \sum _{j=1}^m{(V_{i,j}-V_j^{+})}^2},}$$

(17)

$$S_i^{-}=\sqrt{{\displaystyle \sum _{j=1}^m{(V_{i,j}-V_j^{-})}^2}}.$$

(18)

7.

Calculate the multi-objective performance indicator: the scores of the evaluated geometries are calculated based on the Euclidean distance that each geometry has from the positive and negative ideal solutions, as defined by:

$$C_i={\displaystyle \frac{S_i^{-}}{(S_i^{+}+S_i^{-})}},0<C_i<1.$$

(19)

An illustration of the C_i matrix is shown in Figure 6. Solutions with higher values of C_i are closer to the positive ideal solution and simultaneously farther from the negative ideal solution, within the range 0 < C_i < 1.

A comparison of the results is conducted using the Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), and Mean Percentage Error (MPE) indicators. According to Wilks [56], RMSE is widely used to assess the difference between estimated values and reference values. Thus, when it equals zero, a perfect agreement between the evaluated data is indicated. In the present study, the RMSE value is obtained in Celsius (°C) degrees, and its determination is given by:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}{{\displaystyle {\sum }_{i=1}^n\left(X_i^{Est}-X_i^{Ref}\right)}}^2}.$$

(20)

The MAPE indicator consists of a measurement of the absolute percentage error concerning a reference, indicating the relative difference between the estimated values and those adopted as reference [56]. In this work, the MAPE is determined from:

$$\mathrm{MAPE}={\displaystyle \frac{100}{n}}{\displaystyle {\sum }_{i=1}^n\left|\frac{X_i^{Ref}-X_i^{Est}}{X_i^{Ref}}\right|}.$$

(21)

The MPE is a statistical metric used to measure the accuracy of a forecasting model compared to reference values. An important characteristic of MPE is that it can be positive or negative, indicating whether the data is, on average, overestimated or underestimated compared to actual [56]. In this work, the MPE is determined from:

$$\mathrm{MPE}={\displaystyle \frac{100}{n}}{\displaystyle {\sum }_{i=1}^n\left(\frac{X_i^{Ref}-X_i^{Est}}{X_i^{Ref}}\right)}.$$

(22)

For all statistical metrics, Equations (20)–(22), $X_i^{Est}$ represents any parameter under analysis (in the present work TP and PD), and $X_i^{Ref}$ the corresponding value, but this time from the reference. Here, i denotes the count of values, ranging from 1 to 31 for obtaining monthly average values.

3. Numerical Modeling

The computational geometric domain was generated using Design Modeler, while its spatial discretization was performed with Meshing, both tools from the ANSYS^® package (version 2024 R2). Numerical solutions were obtained through the commercial computational fluid dynamics software FLUENT^® (version 2024 R2), which is based on the Finite Volume Method (FVM) [57,58,59]. It is important to note that, in this study, a decoupled numerical solution approach for thermal and fluid dynamic problems was adopted, as proposed by Rodrigues et al. [10]. This approach aims to reduce processing costs in geometry discretization and numerical simulation. This solution methodology is well-suited for airflow problems with forced convection heat transfer, without compromising the accuracy of the results [10,60].

For the thermal problem, a non-uniform three-dimensional mesh with tetrahedral finite volumes was used. The dimensions of the finite volumes were defined as three times the duct diameter (3d) in the region corresponding to the soil, and d/3 for the flow domain inside the duct, where d represents the duct diameter when H/L = 1.0 (d = 0.11 m, based on Vaz et al. [9]). This spatial discretization for the computational domain (see Figure 1a) was adopted in accordance with the studies by Rodrigues et al. [10] and Rodrigues et al. [35]. The problem was solved in unsteady state with a time step of Δt = 1800 s, totaling 35,040 time steps, which corresponds to a simulation time equivalent to 2 years. Only the last year of simulation was considered in the analysis due to the time required for soil temperature stabilization [61]. For the fluid dynamics problem, the computational domain was only composed by the duct and an isothermal numerical simulation was carried out for each cross-section (see Figure 1b). An air flow with a prescribed velocity of 3.3 m/s at the duct inlet was considered. The geometries were discretized using hexahedral finite volumes, based on the ratio d/14, also in agreement with Rodrigues et al. [10]. The system was solved in unsteady state with a time step of Δt = 0.1 s. A total of 240 time steps were simulated, ensuring that the fluid passed through the duct three times. For the results, the average pressure difference between the inlet and outlet of the duct over the last 160 time steps were considered.

The k–ε turbulence model was adopted to address the effects of turbulent flow in the system. The treatment of advective terms was conducted using the Second-Order Upwind interpolation scheme, applied to the solution of the momentum, energy (related to the thermal problem), and pressure discretization equations. Pressure-velocity coupling was performed using the Semi-Implicit Method for Pressure-Linked Equations Consistent (SIMPLE) algorithm. Convergence of the calculations was considered achieved when the residuals of the mass, momentum, and energy conservation equations, between consecutive iterations, fell below 10⁻³, 10⁻³, and 10⁻⁶, respectively.

For validating the computational thermal model, the experimental study by Vaz et al. [9] was used. For its verification, the results of the current study were compared with the numerical studies by Brum et al. [61] and Vivaz et al. [62]. The results are presented in Figure 7. Qualitatively, the current study successfully replicated the trend in the system’s thermal behavior over the 365 days evaluated, capturing the peaks and valleys of the experimental curve quite accurately. However, a magnitude difference was noted, with this study’s results showing slightly lower values compared to the experimental data. Quantitatively, the difference between the numerical results revealed a MAPE of 11.50% and an RMSE of 2.67 °C. This discrepancy may be attributed to the experimental study using three buried ducts at slightly different velocities, whereas the current model considers only one duct, potentially affecting the thermal behavior of the system. In terms of verification, there was satisfactory agreement with the numerical models by Brum et al. [61] and Vivaz et al. [62], showing similar curve behavior and compatible magnitudes. The quantitative discrepancies with these models indicated a MAPE of 3.14% and 2.77%, and an RMSE of 0.72 °C and 0.62 °C, respectively, when compared to the studies by Brum et al. [61] and Vivaz et al. [62].

In turn, the verification of the computational fluid dynamic model aimed at evaluating the pressure drop in the system. The rectilinear duct presented in this study (H/L = 1.0) was verified using the analytical solution proposed by Pritchard and Mitchell [63]:

$$\mathrm{\Delta }{P}_A=P{D}_A=f\cdot {\displaystyle \frac{L_d}{d}}\cdot {\displaystyle \frac{\rho v^2}{2}},$$

(23)

where, L_d represents the duct length (L_d = 25.77 m) and f is the friction factor, given by [63]:

$$f={\left\{-1.8\log \left[{\left({\displaystyle \frac{R/d}{3.7}}\right)}^{1.11}+{\displaystyle \frac{6.9}{R{e}_d}}\right]\right\}}^{-2},$$

(24)

in this expression, R is the relative roughness of the material and Re_d is the Reynolds number [63,64]:

$$R{e}_d={\displaystyle \frac{\rho \cdot v\cdot d}{\mu }},$$

(25)

Thus, for the analytical model, considering the assumption of a smooth duct (R/d = 0), a PD_A = 36.63 Pa was obtained. For the numerical model used in this study, the same geometry resulted in a PD = 36.43 Pa. The results demonstrate a high level of agreement between the numerical model and the analytical model, with a relative difference of 0.55%. Thus, the fluid dynamic model is considered appropriate for the study.

4. Results

The case study results were obtained through numerical simulations that analyzed different geometries of the Rectilinear EAHE with elliptical cross section. As earlier informed, the geometric degree of freedom of the EAHE is defined by the aspect ratio between the vertical and horizontal axes of the elliptical duct cross section (H/L), which was evaluated in a range from 0.20 to 6.0 (see Figure 1b). The simulations considered temperature variations and the geotechnical profile specific to the city of Viamão. The evaluation period covered 365 days, corresponding to one climatic year.

In the thermal analysis of EAHE, the main objective is to maximize the heat transfer between the ground and the air. This maximization process is evidenced by achieving the highest values of TP for air heating (cold periods) and the lowest TP values for air cooling (hot periods). Figure 8 illustrates the daily average thermal potential (${\overline{TP}}^D$) of the EAHEs for: (a) month with the highest ${\overline{T_{in}}}^M$ (February) and (b) month with the lowest ${\overline{T_{in}}}^M$ (July). The selection of these two months, representing the annual thermal extremes, provides an overview of the system’s behavior. The analysis also considers the extreme and intermediate values of the aspect ratio H/L (0.2; 0.6; 1.0; 2.5; 6.0), providing a general overview of the results for the evaluated geometric range. The curves indicate that the system’s behavior with different aspect ratios H/L is similar, as they follow similar trends. However, the values of ${\overline{TP}}^D$ show distinct magnitudes in each case. It is observed that the geometries corresponding to the geometric extremes (H/L = 0.2 and 6.0) exhibited more favorable magnitudes in terms of heat transfer throughout the days compared to the intermediate geometries, thereby achieving the best performance related to ${\overline{TP}}^D$ for both cooling (Figure 8a) and heating (Figure 8b). On the other hand, the H/L = 1.0 ratio, corresponding to the circular cross section duct, showed the worst performance in terms of ${\overline{TP}}^D$, standing out as the least favorable configuration. This thermal behavior was also observed in the other months of the year.

Figure 9 presents a daily analysis considering the TP and the T_in of the EAHE, focusing on the days with the most pronounced thermal conditions from the 2007 data series—the hottest day (day with highest ${\overline{T_{in}}}^D$, Figure 9a) and the coldest day (day with lowest ${\overline{T_{in}}}^D$, Figure 9b). In the cooling scenario (Figure 9a), there is a trend of improvement in the EAHE’s performance concerning TP as T_in increases. During the daily maximum temperature period (14.5–16 h), all EAHEs recorded TP values better than −10 °C, demonstrating the system’s capability to reduce the air outlet temperature to levels consistent with thermal comfort (22.5 °C to 25.5 °C in summer, with a relative humidity of 65%, as specified by ABNT NBR 16401-2 [65] and ASHRAE [66]), without the need for additional cooling devices. The configuration with the aspect ratio of H/L = 6.0 stood out for its superior performance, achieving a cooling TP close to −12 °C during the period of highest T_in, outperforming the worst-performing geometry, characterized by the aspect ratio of H/L = 1.0, by approximately 13% (1.4 °C). Although the hottest day in the data series was 30 March 2007, the best cooling TP was obtained on 5 September 2007, at 14 h, with −16.18 °C for H/L = 6.0, showing a difference of −1.95 °C compared to the lowest-performing geometry, H/L = 1.0 (TP = 14.23 °C). In the heating condition (Figure 9b), a similar behavior is observed, although the system shows its best performance during the period of lowest T_in (22–23.5 h). In this interval, all geometries evaluated with H/L provided a significant increase in the ambient temperature, exceeding 11 °C. Despite this good performance, the use of auxiliary heating methods is still necessary for the system’s outlet temperature to reach the recommended thermal comfort levels for winter (21.5 °C to 23.5 °C, with a relative humidity of 65%, as per the ABNT NBR 16401-2 [65] and ASHRAE [66] guidelines). Once again, the configuration with H/L = 6.0 demonstrated a greater heat transfer capacity, achieving a maximum heating TP of 13.4 °C during the period of lowest T_in (23.5 h), surpassing the worst-performing geometry, characterized by H/L = 1.0, by approximately 15% (1.7 °C). Although the coldest day in the data series was 11 July 2007, the highest heating TP was recorded on 12 July 2007, at 6.5 h, with 16.09 °C for H/L = 6.0, showing a difference of 2.06 °C compared to the H/L = 1.0 geometry (TP = 14.03 °C).

In a more comprehensive assessment, Figure 10 presents the annual average thermal potential (${\overline{TP}}^A$) and the heat transfer surface area (A_SHT) between the duct and the soil as a function of the H/L ratio, considering: (a) air cooling and (b) air heating. In the context of air cooling (Figure 10a), where lower ${\overline{TP}}^A$ values indicate superior system performance, it is observed that the duct’s cross sectional geometry has a direct influence on heat transfer in the EAHE, as designs with larger heat transfer areas demonstrate better results. Thus, the ducts with the extreme H/L values within the evaluated range showed the best performance. Conversely, as the H/L ratio approaches 1.0, a decrease in system performance is observed, reaching its most critical point at H/L = 1.0. In this regard, as previously observed in earlier analyses, the optimal thermal geometry was found to be at (H/L)_T,o = 6.0. For the air heating case (Figure 10b), where higher ${\overline{TP}}^A$ values are now desirable, a similar behavior was identified, with the optimal thermal geometry also obtained at (H/L)_T,o = 6.0 and the worst performance at H/L = 1.0. Another point to note is that the geometries with the same A_SHT value (H/L = 0.5 and 2.0; H/L = 0.2 and 5.0) exhibited almost identical thermal performance. This suggests that, for the duct diameter evaluated in this study, variations in duct shape, whether elongated longitudinally or transversely to the ground, do not result in significant changes in system performance in terms of TP. This means that the orientation of the duct shape—whether longitudinal or transverse—can be adapted based on practical considerations (such as construction or spatial constraints) without compromising the system’s thermal capacity.

Table 2. Monthly statistical analysis through the MPE.

MPE (%)
	Cooling Potential					Heating Potential
Month	Number of Days	H/L				Number of Days	H/L
		0.2	0.6	3.0	6.0		0.2	0.6	3.0	6.0
January	27	−13.04	−1.47	−6.93	−15.23	4	−15.68	−2.34	−9.79	−17.27
February	26	−10.93	−0.83	−5.17	−13.21	2	14.01	−4.97	−2.15	31.80
March	25	−12.09	−1.44	−6.47	−13.99	6	−16.68	−2.16	−9.36	−19.09
April	16	−15.86	−0.92	−7.32	−21.45	14	−24.53	−2.98	−12.98	−28.54
May	2	−10.46	−1.06	−5.55	−11.86	29	−15.01	−1.63	−7.88	−17.57
June	11	−13.76	−1.46	−7.60	−15.23	19	−13.25	−1.54	−7.15	−15.47
July	3	−13.12	−1.66	−7.47	−14.72	28	−13.30	−1.57	−7.15	−15.48
August	8	−12.36	−1.68	−7.14	−13.74	23	−10.79	−2.07	−6.67	−18.06
September	23	−13.55	−1.45	−7.16	−15.95	7	−9.34	−0.55	−3.98	−11.43
October	23	−14.44	−1.52	−7.78	−16.68	8	−12.47	−1.83	−7.29	−13.73
November	23	−16.92	−1.59	−8.74	−19.73	7	−7.87	−1.74	−5.67	−7.60
December	30	−13.46	−1.49	−7.16	−15.65	1	−17.42	−4.59	−11.63	−17.85
Weighted Average		−13.55	−1.37	−7.07	−15.98		−13.67	−1.88	−7.68	−16.44

presents a monthly quantitative assessment of the results through the calculation of the MPE statistical indicator. In this assessment, the geometry given by H/L = 1.0, corresponding to the conventional circular cross sectional duct, was used as the reference for the analysis. A negative sign in the MPE (%) values indicates that the system’s performance is superior compared to the reference, while a positive sign indicates lower performance. Additionally, the calculations were based on the ${\overline{TP}}^D$. Regarding the MPE indicator, the geometry with H/L = 6.0 showed the greatest performance difference compared to the reference across all months for the air-cooling condition. A similar behavior was observed for heating days, except in February, when the reference (H/L = 1.0) demonstrated better performance, and in November, when the geometry with H/L = 0.2 achieved the best response in heat transfer between the soil and the air. On average, the configuration with (H/L)_T,o = 6.0 resulted in a 16.4% improvement in thermal response for heating and a 15.9% improvement for cooling compared to the conventional circular cross sectional duct. It is important to note that the number of heating or cooling days in each month was used as a weighting factor for calculating the averages. Still, the better MPE (%) values were observed for the air heating condition, suggesting that the geometric variation of the system through H/L ratio has a slightly more significant impact during these periods. These results are significant, and the adoption of elliptical geometry, by enhancing the system’s thermal performance, enables the exploration of more compact installations. This translates, for example, into the possibility of reducing the length of EAHEs, facilitating their integration into various architectural and spatial urbans contexts.

In terms of the fluid dynamic evaluation of the EAHE, the primary goal is to minimize the pressure difference between the air inlet and outlet of the duct, thereby reducing the system’s power demand on the fluid blower or fan. In this context, Figure 11 illustrates the system’s fluid dynamic performance through the PD performance indicator and the A_SHT. It was observed that as the geometric ratio H/L of the duct approaches the extremes (H/L = 0.2 and H/L = 6.0), there is a significant increase in the system’s pressure drop. This behavior can be attributed to the increase in the heat transfer surface area (A_SHT). Consequently, the curve describing the PD behavior resembles the TP curve for air heating (Figure 10b). This indicates that, in both cases, the geometric extremes produce significant variations in system performance. However, while an increase in TP is beneficial (for air-heating condition), resulting in greater heating capacity, an increase in PD has the opposite effect; the elevated pressure drop requires greater effort from the ventilation system, leading to higher energy consumption. Thus, it underscores the importance of finding a balance between maximizing the thermal capacity of the EAHE and minimizing its energy consumption to combine performance with sustainability. Hence, the best performing case in terms of PD, (H/L)_F,o = 1.0, showed a reduction in pressure drop of approximately 65.33% compared to the worst performing case, H/L = 6.0.

Figure 12 presents the multi-objective analysis of the system using the performance indicator C_i (TOPSIS) for: (a) air cooling and (b) air heating. In this analysis, different importance weights were assigned to thermal and fluid dynamic aspects. Additionally, the annual average thermal potential was considered as the thermal performance indicator for heating and cooling. Thus, when the fluid dynamic problem assumes greater importance relative to the thermal problem (w₁ < w₂, lower region of the graphs in Figure 12a,b), the geometries with the best performance (represented by the highest C_i values), are concentrated in the range of 0.5 < H/L < 2.0. As the relevance of the fluid dynamic criterion decreases and the importance of the thermal criterion increases (shifting from the base to the center of the graph), this same geometric range continues to show the best multi-objective results, highlighting the robustness of these configurations, even when the weights assigned to the thermal and fluid dynamic aspects are equal (w₁ = w₂ = 0.5). This behavior indicates that the system’s geometric variation, through the H/L ratio, results in greater sensitivity to fluid dynamic considerations. However, as the thermal criterion becomes more important (w₁ > w₂, region extending from the center to the top of the graph), there is a shift in optimal geometries toward the extremes of the H/L ratio. In these conditions, geometries with H/L < 0.3 and H/L > 3.0 become more favorable for solving the multi-objective problem. It is worth noting that this behavior is almost identical for both heating and cooling scenarios (Figure 12a,b).

Figure 13 illustrates the optimal multi-objective indicator (C_i,opt) and the corresponding optimal geometries ((H/L)_M,o) for different weight combinations assigned to the thermal and fluid dynamic problems (w₁ and w₂). The C_i,opt curves show that the highest C_i,opt values are obtained when w₁ < w₂, meaning that the fluid dynamic criterion is given more emphasis than the thermal one. This behavior reinforces that the PD has greater sensitivity to the H/L degree of freedom variation, highlighting that the optimal geometries obtained for w₁ < w₂ are closer to the positive ideal solution compared to the geometries obtained for w₁ > w₂. The (H/L)_M,o curves indicate that the optimal geometries differ between the heating and cooling cases, depending on the different combinations of w₁ and w₂. This underscores the need to adjust the system based on its primary objective, whether it is to maximize the performance for heating or cooling. For the air-heating condition, the geometries corresponding to (H/L)_M,o = 0.9 and (H/L)_M,o = 0.7 showed the best multi-objective performance, covering a wider range of weight combinations between w₁ and w₂. On the other hand, for air-cooling, the geometry (H/L)_M,o = 0.8 proved more suitable in most of the analyzed situations.

5. Conclusions

This work presented a numerical analysis of the thermal and fluid dynamic behavior of rectilinear EAHEs with elliptical cross section duct. The evaluation was conducted using the Constructal Design method, combined with Exhaustive Search and TOPSIS. The results were simulated to determine the aspect ratio H/L values that would lead to the best outcomes in terms of the system’s thermal potential for heating and cooling, as well as minimizing the pressure drop along the EAHE duct.

Regarding the thermal problem, a similar behavior trend was observed for the different H/L values evaluated, although with varying magnitudes. The geometries corresponding to the geometric extremes evaluated (H/L = 0.2 and 6.0) exhibited more favorable magnitudes in terms of heat transfer over the days compared to the intermediate geometries, achieving the best performance related to TP for both cooling and heating. The H/L = 1.0 ratio, corresponding to the circular cross sectional duct, reached the worst performance in terms of TP, presenting itself as the least favorable configuration and highlighting the importance of the study. It is noteworthy that the increase in the duct’s cross sectional surface area has a direct influence on the heat transfer in the EAHE, resulting in better outcomes. Quantitatively (through the statistical indicator MPE), the configuration given by (H/L)_T,o = 6.0 provided a 16.4% improvement in the system’s thermal response for heating and 15.9% for cooling compared to the conventional circular cross section duct (H/L = 1.0). Concerning the fluid dynamic analysis, it was observed that while more elongated cross-sectional ducts increase TP, they also result in a substantial increase in pressure drop (PD), leading to higher energy consumption by the system. In contrast, the configuration with (H/L)_F,o = 1.0 demonstrated a significant reduction in pressure drop, approximately 65% lower compared to the worst-performing configuration, given by H/L = 6.0. By integrating Constructal Design with the TOPSIS method, it was possible to identify the geometric regions with the best performance within the analyzed spectrum, highlighting both the geometries that offer a balance between performance indicators and those that prioritize specific thermal or fluid dynamic aspects.

In future works, the approach associating Constructal Design and TOPSIS will be used to investigate different types of cross section for the duct, as well as to study different types of EAHE (such as: horizontal T-shaped, Y-shaped, and serpentine and horizontal and vertical helical).