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An EarthAir Heat Exchanger (EAHE) is a device that consists of one or more buried ducts through which air is forced to flow. The surrounding soil is responsible for enabling thermal exchanges along with the installation, making the temperature at the outlet milder than the inlet. The objective of this work is to ally a numericalanalytical approach with the Constructal Design method and Exhaustive Search technique to minimize the soil volume occupation (V), minimize the air flow pressure drop (PD), and maximize the thermal potential (TP) of a Tshaped EAHE. Starting from a conventional EAHE composed of a straight duct, called Reference Installation (RI), two degrees of freedom (DOF) were considered: the ratio between the length of the bifurcated branch and the length of the main branch (L_{1}/L_{0}) and the ratio between the diameter of the bifurcated branch and the diameter of the main branch (D_{1}/D_{0}). Comparing with RI, different Tshaped EAHE geometries were identified to reduce V by 23% and PD by 62% and to increase TP by 21%; and when these three performance parameters were concomitantly considered another Tshaped EAHE geometric configuration allowed to reach an improvement of around 27% when compared with the RI.
constructal design, computational fluid dynamics (CFD), earthair heat exchanger (EAHE), finite volume method (FVM), numerical simulation, thermal comfort
Energy consumption is an important factor to be taken into account in engineering projects. Regarding the Civil Construction industry, one of the main concerns of the market players is to develop products with low electrical energy usage to provide comfort to its users. Studies show that 10% of global energy expenditure results from traditional air conditioning systems, cooling, or heating devices [1], and it can reach levels of 20 to 40 % in developed countries [2, 3].
In Brazil, it is possible to notice the rise in electricity consumption due to air conditioners within the last years. A local company was able to estimate this growth as being three times the levels of consumption of 2006 due to the expanding rate of these equipment sales, taken a 9% increase each year for the period of 2005 to 2017 [4].
As a consequence of this demand, the development of technologies to reach acceptable comfort levels with less electricity consumption has increasingly become a focus of various studies. Another course of action for solving this problem is investigating ways to use renewable energies instead of an electrical one. It is known that the radiation emitted by the Sun is the main source of clean energy [5] and, in Vaz et al. [6, 7], one can note that the soil can be used as a reservoir of thermal energy with the adoption of EarthAir Heat Exchangers (EAHE) to reduce the usage of traditional air conditioning systems. These devices are relatively simple to build and install, mainly in new buildings, since their materials can be easily found on the market and no specialized workforce is needed to put them into service. In addition to this, its operating principle is also simple: the air is forced to flow into buried ducts exchanging heat with the surrounding soil, resulting in a milder air temperature at the EAHE outlet when compared to its inlet temperature.
Many studies were carried out aiming to investigate the EAHE thermofluiddynamic behavior using different approaches, such as experimental [710], analytical [1113], and numerical [1418]. Since experimental research is timeconsuming and financially onerous, in reason of the period needed to be monitored to achieve a good level of knowledge of its behavior – taken as one entire year to embrace all seasons, other approaches appear as the best way to fulfill this necessity of studies. Hence, the numerical method, usually solved with the help of computers, is the faster way to develop new research concerning its geometry optimization and operational parametric analysis.
Regarding the numerical researches, two main streams can be identified in the literature. On one side, it stands up for the implementation of onedimensional analytical studies, such as given by Papakostas et al. [19], Benkert et al. [20], and De Paepe and Janssens [21]. This methodology is considered adequate for a preliminary project and designing aspects since it is quickly solved and generates satisfactory results. However, it is only appropriate for simpler installations, such as straight duct buried on nonstratified soil. For more complex geometric configurations and soil stratifications, the two and threedimensional models are usually adopted, being this the other stream of studies. In this particular type of work, the full soil thermophysical properties, such as density, specific heat, and thermal conductivity, are taken into account, and the air flow inside ducts are considered unsteady, turbulent, and incompressible, in contrast to the laminar modeling adopted by some 1D studies.
Vaz et al. [6, 7] carried out a pioneer experimental study for an EAHE installed in the city of Viamão, in the south of Brazil. It was developed a complex geometry with multiple ducts, and the air and soil temperatures were monitored every 30 min throughout the year 2007. The study also included experiments in determining the thermophysical properties of the soil and the air, aiming to reproduce computationally the thermofluiddynamic behavior observed in loco. Since that, other works took place to improve the computational model developed in Vaz et al. [6], like in Brum et al. [15], where it was validated using a reduced model, making it possible to simulate the EAHE operation with less computational effort. After that, in Rodrigues et al. [16], it was proposed to enhance this model with the employment of a simpler turbulence model, allowing a reduction in processing time with no accuracy loss.
Another widely used approach for the analysis of Heat Transfer systems consists in the application of the Constructal Design method in association with the computational modeling, as in Adewumi et al. [22] and Gulotta et al. [23]; being this approach also used for the study of EAHE, as in Rodrigues et al. [16] and Brum et al. [24].
The Constructal Design is based on the Constructal Law theory developed by Bejan [25], which states that "for a finitesize flow system to persist in time (to live) it must evolve such that it provides greater and greater access to the currents that flow through it."
In this context, the present article aims to study a Tshaped EAHE for the city of Viamão, in the central region of the state of Rio Grande do Sul, in southern Brazil. The Constructal Design was applied to propose different EAHE Tshaped geometries that can be properly compared to each other. In other words, the Constructal Design was adopted to define the search space. In sequence, the thermal behavior of each EAHE configuration was numerically simulated, while its soil volume occupation and air flow pressure drop were analytically determined. Then, using the Exhaustive Search technique, it was possible to identify the best Tshaped EAHE geometric configurations that maximize the thermal potential (TP), minimize the soil volume occupation (V), and minimize the air flow pressure drop (PD). Finally, considering these three performance parameters concomitantly and performing a vector analysis, it was possible to indicate the optimized geometry for the Tshaped EAHE.
It is important to highlight that the contribution of the present work if compared with other studies, as [6, 7, 15, 16, 24], relies on the analysis of the Tshaped geometry for an EAHE (having one inlet and two outlets); allowing to investigate the influence of two degrees of freedom (DOF) over the EAHE performance: The ratio between the length of the bifurcated branches and the length of the main branch (L_{1}/L_{0}); and the ratio between the diameter of the bifurcated branches and the diameter of the main branch (D_{1}/D_{0}).
The Constructal Design application starts based on an EAHE adopted as a reference and composed of a straight duct 30 m long and 110 mm in diameter, having one inlet and one outlet for the air flow. This case, called Reference Installation (RI), is depicted in Figure 1, being buried at a depth of 3 m on a 15 m high portion of the soil, as recommended in Brum et al. [15].
From RI, the Tshaped EAHE geometric configurations were defined, having the aspect indicated in Figure 2.
Figure 1. Reference Installation (RI)
Figure 2. Tshaped EAHE: (a) perspective; and (b) superior view
The Tshaped geometries were generated respecting the following constraints (see Figure 2b): The total duct length (L_{Total}) given by:
$L_{T o t a l}=L_{R I}=L_{0}+2 L_{1}=30 \mathrm{~m}$ (1)
where, L_{RI} is the length of the RI duct (in m, see Figure 1), L_{0} is the length of the main branch (m), and L_{1} is the length of the bifurcated branch (m); so the total duct volume (V_{Duct}) is defined as:
$V_{\text {Duct }}=\frac{\pi}{4} L_{R I} D_{R I}^{2}=\frac{\pi}{4} L_{0} D_{0}^{2}+\frac{\pi}{4} L_{1} D_{1}^{2}=0.285 \mathrm{~m}^{3}$ (2)
being D_{RI} the RI diameter (in m, see Figure 1), D_{0} the diameter of the main branch (m), and D_{1} the diameter of the bifurcated branch (m). As in RI (see Figure 1), for the Tshaped geometries, the installation depth is h = 3 m, and the soil portion height is H = 15 m (see Figure 2a).
In addition to these constraints, the two degrees of freedom (DOF) were varied. The first DOF was the ratio L_{1}/L_{0}, varying in a range from 0.1 to 7.0 and generating the installation I1 to I15, as illustrated in Figure 3.
Figure 3. Tshaped EAHE obtained due to the L_{1}/L0 variation
The ducts length for the main and bifurcated branches of the EAHEs depicted in Figure 3 are in Table A of the Appendix.
The second varied DOF was the ratio D_{1}/D_{0}. Initially, its value was assumed to be equal to 1.00, being D_{1} = D_{0} = 110 mm (i.e., the same diameter of RI). Thereafter, variations in this DOF were stipulated, considering contractions when D_{0} > D_{1} (D_{1}/D_{0} = 0.50 and D_{1}/D_{0} = 0.75  see Table B of Appendix) and expansions when D_{0} < D_{1} (D_{1}/D_{0} = 1.25 and D_{1}/D_{0} = 1.50  see Table C of Appendix).
Therefore, considering the variations of the ratios L_{1}/L_{0} and D_{1}/D_{0}, a total of 75 Tshaped EAHE installations were proposed to form the search space of this analysis.
In sequence, the RI and the 75 Tshaped installations were analyzed. To do so, the thermal behavior of these EAHE was numerically simulated through a computational model in the Fluent software, which is based on the Finite Volume Method (FVM), allowing to define the thermal potential (TP) of each installation. In its turn, the soil volume (V) occupied for each installation and its air pressure drop (PD) were analytically defined.
2.1 Numerical approach
The employed computational model was previously verified and validated by Brum et al. [15] and Rodrigues et al. [16] by means the numerical and experimental results of Vaz et al. [6, 7]. So, for brevity, the computational model verification and validation were not presented here in detail.
It is important to mention that regardless of the L_{1}/L_{0} and D_{1}/D_{0} values, the distance d between the ducts and the domain wall (see Figure 2b) was equal to 2 m. The reason behind it is to avoid jeopardized results due to boundary conditions set in these walls [16].
From this, and considering the Constructal Design method application, the geometry of the computational domains of the RI (see Figure 1) and the Tshaped EAHEs (see Figures 2 and 3) were defined. The spatial discretization of these computational domains was generated according to Rodrigues et al. [16], being the mesh composed of tetrahedral computational cells with a size equivalent to 3D for the soil, while in the duct the size of computational cells was D/3. Remembering that D is the EAHE duct diameter which due to the variation of the DOF D_{1}/D_{0} can assume different values, but always being the smallest diameter adopted to promote the domain discretization. Moreover, to avoid mesh generation issues, the ducts' thickness and material properties were disregarded. Consequently, the air is considered to pass through the boreholes in the soil portion, being this simplification widely adopted in EAHE numerical simulations, as in Vaz et al. [6], Brum et al. [15, 24], and Rodrigues et al. [16].
The numerical simulations were performed by solving the energy equation for the soil [26]:
$\frac{\partial T}{\partial t}=\frac{\partial}{\partial x_{j}}\left(\alpha_{s} \frac{\partial T}{\partial x_{j}}\right)$ (3)
where, T is the soil temperature field (K), t is the time (s), $x_{j}$ represents the spatial coordinates (m), a_{s} is the soil thermal diffusivity (m²/s), and j = 1, 2, and 3; in addition to the conservation equations of mass, momentum, and energy for the transient, incompressible, and turbulent forced convective air flow inside the EAHE duct, respectively, given by [26, 27]:
$\frac{\partial \overline{v_{i}}}{\partial x_{i}}=0$ (4)
$\frac{\partial \overline{v_{i}}}{\partial t}+\frac{\partial\left(\overline{v_{i} v_{j}}\right)}{\partial x_{j}}=\frac{1}{\bar{\rho}} \frac{\partial \bar{p}}{\partial x_{j}} \delta_{i j}+\frac{\partial}{\partial x_{j}}\left[v\left(\frac{\partial \overline{v_{i}}}{\partial x_{j}}+\frac{\partial \overline{v_{j}}}{\partial x_{i}}\right)\tau_{i j}\right]$ (5)
$\frac{\partial \bar{T}}{\partial t}+\frac{\partial}{\partial x_{j}}\left(\overline{v_{j} T}\right)=\frac{\partial}{\partial x_{j}}\left(\alpha \frac{\partial \bar{T}}{\partial x_{j}}q_{j}\right)$ (6)
where, the over line represents the timeaveraged terms, x_{i} are the spatial coordinates (m), v_{i} are the velocity in Cartesian directions (m/s), d_{ij} is the Kronecker delta, p is the pressure (Pa), u is the air kinematic viscosity (m²/s), a is the air thermal diffusivity (m²/s), and i, j = 1, 2, 3. The terms t_{ij} and q_{j} that arise in the filtering process of the momentum and energy conservation equation, respectively, need to be modeled and can be written as [28]:
$\tau_{i j}=\overline{v_{i}^{\prime}} \overline{v_{j}^{\prime}}$ (7)
$q_{j}=\overline{v_{i}^{\prime}} \overline{T^{\prime}}$ (8)
where, the (') indicates the time varying fluctuating component.
To deal with the closure problem, the RANS kε model is employed, which is based on the solution of two additional transport equations. For incompressible flows, the closure terms of Eqns. (7) and (8) are given by [28]:
$\tau_{i j}=v_{t}\left(\frac{\partial \bar{v}_{i}}{\partial x_{j}}+\frac{\partial \bar{v}_{j}}{\partial x_{i}}\right)\frac{2}{3} k \delta_{i j}$ (9)
$q_{j}=\alpha_{t} \frac{\partial \bar{T}}{\partial x_{j}}$ (10)
where, u_{t} is the kinematic eddy viscosity (m^{2}/s), k is the turbulent kinetic energy (m^{2}/s^{2}) and a_{t} is the thermal eddy diffusivity (m^{2}/s). The values of u_{t} and a_{t} can be defined as:
$v_{t}=C_{\mu} \frac{k^{2}}{\varepsilon}$ (11)
$\alpha_{t}=\frac{v_{t}}{\operatorname{Pr}_{t}}$ (12)
where, C_{μ} = 0.09 and Pr_{t} = 1.00. The turbulent kinetic energy (k) and turbulent dissipation (ε) are, respectively [28]:
$\frac{\partial k}{\partial t}+\bar{v}_{j} \frac{\partial k}{\partial x_{j}}=\tau_{i j} \frac{\partial \bar{v}_{i}}{\partial x_{j}}+\frac{\partial}{\partial x_{j}}\left[\left(v+\frac{v_{t}}{\sigma_{k}}\right) \frac{\partial k}{\partial x_{j}}\right]\varepsilon$ (13)
$\frac{\partial \varepsilon}{\partial t}+\bar{v}_{j} \frac{\partial \varepsilon}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\left[\left(v+\frac{v_{t}}{\sigma_{\varepsilon}}\right) \frac{\partial \varepsilon}{\partial x_{j}}\right]+C_{\varepsilon 1} \frac{\varepsilon}{k} \tau_{i j} \frac{\partial \bar{v}_{i}}{\partial x_{j}}C_{\varepsilon 2} \frac{\varepsilon^{2}}{k}$ (14)
being C_{ε}_{1} = 1.44 and C_{ε}_{2 }= 1.92.
In the current study, the mathematical model was numerically solved through the FVM with the aid of the Fluent software, where a transient and pressurebased solution was performed. The firstorder upwind scheme was set to deal with the arising instabilities of the advective terms. In relation to the pressurevelocity coupling, the SIMPLE algorithm was used. Concerning the convergence criteria, the residues for the mass, momentum, and energy were considered converged when the residues between two consecutive iterations are lower than 1×10^{3}, 1×10^{3}, and 1×10^{}^{6}, respectively. The processing time was equivalent to two simulated years for all the simulations, totaling 17520 time steps of 3600 s (1 h) each. However, the first simulated year is neglected to avoid interference of the initial conditions in the results.
As in Vaz et al. [6] and Brum et al. [15], for each numerical simulation, the initial temperature condition of the entire domain is assumed equal to the average soil temperature of 291.70 K (18.70℃).
Regarding the boundary conditions, also based on Vaz et al. [6] and Brum et al. [15], the inlet air prescribed velocity is 3.3 m/s; and the prescribed annual temperature variation at the air inlet, T_{in}(t), and at the superior soil surface, T_{ss}(t), are, respectively, given by:
$T_{i n}(t)=296.18+6.92 \cdot \sin \left(200 \times 10^{9} \cdot t+26.42\right)$ (15)
$T_{s s}(t)=291.70+6.28 \cdot \sin \left(200 \times 10^{9} \cdot t+26.24\right)$ (16)
The transient functions presented in Eqns. (15) and (16) were generated from the adjustment of the experimental data monitored during 2007 in the city of Viamão [7]. The other soil surfaces of the computational domain were considered thermally insulated (qʺ = 0 W/m²), while a manometric pressure was assumed at the EAHE outlets (p_{out} = 0 Pa) and nonslip and impermeability conditions were imposed at the duct walls.
Moreover, the soil and air thermophysical properties were considered the same way as in Vaz et al. [7]. The geotechnical profile of Viamão soil is homogeneous and with clayey characteristics, being its density (r), thermal conductivity (l), and specific heat (c_{p}) assumed as isotropic and constant [29]. Concerning the air, these properties, together with its absolute viscosity (m), were assumed as constants in agreement with Brum et al. [15] and Lee et al. [30]. The values of the thermophysical properties for the soil and the air are presented in Table 1.
Table 1. Thermophysical properties for the air and the soil
Material 
r (kg/m^{3}) 
l (W/m·K) 
c_{p} (J/kg·K) 
m (kg/m·s) 
Soil 
1,800 
2.1 
1,780 
 
Air 
1.16 
0.0242 
1,010 
1.798×10^{5} 
2.2 Performance parameters
As previously stated, three different performance parameters were evaluated for this study: the occupied soil volume (V), the air pressure drop (PD), and the EAHE thermal potential (PT). The performance parameters were normalized through the Reference Installation (RI) results to perform a geometric optimization using the Exhaustive Search technique to the Tshaped EAHE geometric configurations proposed by the Constructal Design method.
Therefore, the normalized soil volume occupied by the Tshaped EAHEs is obtained as:
$V_{N}=\frac{V_{T}}{V_{R I}}=\frac{W \cdot L \cdot H}{V_{R I}}$ (17)
being V_{T}, W, L, and H, respectively, the volume, width, length, and high of the Tshaped soil portion (see Figure 2b), and V_{RI} = 2,250 m^{3} is the RI soil volume (see Figure 1). Wherefore, as H is kept constant and equal to 15 m, V_{T} depends directly on the L_{1}/L_{0} value, since this ratio changes the values of L and W of the soil portion.
In its turn, the pressure drop (PD) in the forced turbulent air flow inside the EAHE duct comprises distributed losses in its straight stretches (PD_{d}) and localized losses related to the flow discontinuities (PD_{l}), being defined as [31]:
$P D=\sum P D_{d}+\sum P D_{l}$ (18)
where:
$P D_{d}=f \cdot \frac{L_{d}}{D} \cdot \frac{v^{2}}{2 g}$ (19)
and
$P D_{l}=K_{l} \cdot \frac{v^{2}}{2 g}$ (20)
being ƒ the friction factor; L_{d} the length of the straight duct (m); v the air velocity (m/s); D the duct diameter (m); g the gravitational acceleration (9.81 m/s^{2}); and K_{l} the localized loss coefficient. It is possible to obtain the friction factor f with a correlation developed by Petukhov [32]:
$f=\left(0.79 \ln \operatorname{Re}_{D}1.64\right)^{2}$ (21)
where, Re_{D} is the air flow Reynolds number, given by [27]:
$\operatorname{Re}_{D}=\frac{\rho \cdot v \cdot D}{\mu}$ (22)
Concerning Eq. (20), for the Tshaped EAHEs, the Tjoint was considered having a K_{l} = 1.8; while the contractions or expansions due to the D_{1}/D_{0} variations were defined as indicated in [31] for the loss coefficients for flow through sudden area changes, being K_{l} = 0.40; 0.16; 0.12; and 0.30, respectively, for D_{1}/D_{0} = 0.50; 0.75; 1.25; and 1.50.
From this, the normalized pressure drop (PD_{N}) for the Tshaped EAHE installations was obtained as:
$P D_{N}=\frac{P D_{T}}{P D_{R I}}$ (23)
where, PD_{T} is the pressure drop of the Tshaped EAHE and PD_{RI} = 3.81 m is the pressure drop of the RI.
Finally, the Thermal Potential (TP) was adopted to the thermal performance of the EAHEs, being determined from an annual averaged air temperature. The TP of the EAHE installation can be expressed by [15, 16]:
$T P=\sqrt{\frac{\sum_{i=1}^{1460}\left(T(t)_{i}^{o u t}T(t)_{i}^{i n}\right)^{2}}{1460}}$ (24)
where, T(t)^{out} is the transient air temperature (in K or ℃) at the duct outlet; T(t)^{in} is the transient air temperature (in K or ℃) at the duct inlet (prescribed temperature condition), and i varies from 1 to 1460 representing the outlet temperature measurements performed every 21,600 s during the second year of the numerical simulation. Thereby, the normalized TP (TP_{N}) is given by:
$T P_{N}=\frac{T P_{T}}{T P_{R I}}$ (25)
being TP_{T}the thermal potential of the Tshaped EAHE and TP_{RI} = 3.35℃ the thermal potential of the RI.
It is important to highlight that to reach the superior EAHE performance, while V_{N} and PD_{N} should be minimized, TP_{N} must be maximized. When these parameters are analyzed individually, there is no problem; however, for a global analysis taking into account concomitantly the three performance parameters, it was necessary to consider the minimization of $T P_{N}^{1}$ Therefore, for the global evaluation it was adopted a vector approach, given by:
$\mid$ Performance $\mid=\sqrt{\left(V N_{N}\right)^{2}+\left(P D_{N}\right)^{2}+\left(T P_{N}^{1}\right)^{2}}$ (26)
From Eq. (26), an ideal hypothetical EAHE would have a null value for the $\mid$ Performance $\mid$ achieved with a V_{N} = 0, PD_{N} = 0, and $T P_{N}^{1}=0$. Hence, the Tshaped EAHE having the lowest value of $\mid$ Performance$\mid$ will be the one with the best overall performance.
2.3 Geometric optimization
Figure 4. Constructal design associated with Exhaustive Search procedure
As earlier mentioned, the Constructal Design application allows defining the geometric configurations of the Tshaped EAHE, forming the search space. In addition to that, these cases were compared among each other, promoting a geometric optimization using the Exhaustive Search technique.
Figure 4 illustrates the steps of the optimization procedure. For each of the five values of D_{1}/D_{0}, fifteen values for the ratio L_{1}/L_{0} were adopted (see Figure 3). From this, it was possible to identify the optimal value of L_{1}/L_{0}, (L_{1}/L_{0})_{o}, being the once optimized geometric configuration that leads to the superior performance of the Tshaped EAHE. After that, it was defined the twice optimized L_{1}/L_{0}, (L_{1}/L_{0})_{2o}, and the once optimized D_{1}/D_{0}, (D_{1}/D_{0})_{o}, reaching the superior performance among all cases of the search space.
This optimization procedure was employed for each performance parameter, i.e., by means Eqns. (17), (23), and (25); as well as concomitantly for the tree performance parameters through Eq. (26).
As previously indicated, a Reference Installation (RI) was adopted to normalize the Tshaped EAHEs performance. The thermal behavior of RI was presented in Fig. 5, showing the consistency of the numerical results generated with the computational model.
From Figure 5, the annual air variation of the RI outlet air is in perfect agreement with the results of Brum et al. [15] and Rodrigues et al. [16]. Since the computational model was verified and validated in these references, it is possible to prove the coherence of the numerical results generated in the present work. One can note in Figure 5 that the EAHE RI can provide milder temperatures almost every year. It can reach up to approximately 8℃ during the summer (months of January and December) and around +2℃ during the winter (months of June and July).
Besides, the annual averaged thermal potential for the RI was TP_{RI} = 3.35℃, while the soil volume occupation and the pressure drop for The RI were, respectively, V_{RI} = 2,250 m^{3} and PD_{RI} = 3.81 m. Hence, these values were used as a normalizing factor in the Tshaped EAHE cases.
So, with the Constructal Design and Exhaustive Search association (see Figure 4), it is possible to perform a geometric evaluation and a geometric optimization, i.e., in addition to defining the optimized geometry that leads to the superior performance of the Tshaped EAHE, it was also possible to evaluate the influence of the degrees of freedom over the performance parameters.
Taking that into account, the results obtained individually and concomitantly for the three performance parameters are presented in sequence.
3.1 Occupied soil volume
From Eq. (17), the normalized soil volume occupied for each Tshaped EAHE of Figure 3 was defined and shown in Figure 6 for the five values of D_{1}/D_{0}.
Figure 6 shows the influence of the ratio L_{1}/L_{0} between the length of the main and bifurcated branches on the normalized soil volume occupation, with the curves for the different values of D_{1}/D_{0} practically superimposed. This no relevant influence of the D_{1}/D_{0} DOF was already expected because the soil portion does not suffer significant change due to D_{1}/D_{0} variation. On the other hand, regarding the L_{1}/L_{0} DOF effect, one can assume that the minimization of the occupied soil occurs when L_{1} >> L_{0}, since in all studied cases resulted in installation 15 as the best option. Therefore, the best ratio for soil occupation parameter was taken as (L_{1}/L_{0})_{o} = 7.0, reaching reductions around 23 % and 94 % if compared with RI and the worst Tshaped geometry (with L_{1}/L_{0} = 0.5), respectively.
Despite the small influence of the D_{1}/D_{0}, following the optimization procedure (see Figure 4), Figure 7 presents the values of (L_{1}/L_{0})_{o} as a function of the D_{1}/D_{0} variation.
As earlier observed, the effect of D_{1}/D_{0} is no significant. A reduction of less than 0.4 % is achieved between the twice optimized configuration, with (L_{1}/L_{0})_{2o} = 7.0 and (D_{1}/D_{0})_{o} = 0.50, and the worst geometry of Fig. 7, with (L_{1}/L_{0})_{o} = 7.0 and D_{1}/D_{0} = 1.50.
Figure 5. Annual air temperature variation for the RI
Figure 6. Influence of L_{1}/L_{0} over V_{N}, for each D_{1}/D_{0}
Figure 7. Influence of D_{1}/D_{0} over (V_{N})_{min}, from (L_{1}/L_{0})_{o} values
In summary, it is possible to reach out that the minimum soil occupation was achieved by the installation (L_{1}/L_{0})_{oo} = 7.0 and (D_{1}/D_{0})_{o} = 0.50, with a twice minimized volume of (V_{N})_{2min} = 0.765 (representing a soil volume of 1,721.25 m^{3}).
3.2 Pressure drop
Regarding the pressure drop imposed on the EAHE air flow, it was analytically calculated in its normalized version, according to Eq. (23). The effect of the ratio L_{1}/L_{0} over the normalized pressure drop (PD_{N}) is presented in Figure 8 for the five ratios of D_{1}/D_{0}.
One can infer in Figure 8 that the L_{1}/L_{0 }DOF has a low influence over the pressure drop, being the main difference a consequence of the D_{1}/D_{0 }DOF. However, it is possible to highlight L_{1}/L_{0} = 7.0 as the once optimized geometric configuration to minimize the pressure drop among these configurations. If compared the performances of the best and worst Tshaped EAHE geometries for each D_{1}/D_{0}, improvements of 14.90 %, 19.25 %, 58.16 %, 67.66 %, and 66.67 % were, respectively, achieved. Moreover, in a general way, only the cases with D_{1}/D_{0} ≥ 1.00 and L_{1}/L_{0} ≥ 1.0 lead to a reduction in pressure drop compared to RI.
Figure 8. Influence of L_{1}/L_{0} over PD_{N}, for each D_{1}/D_{0}
Figure 9. Influence of D_{1}/D_{0} over (PD_{N})_{min}, from (L_{1}/L_{0})_{o} values
Another aspect indicated by Figure 8 is that from D_{1}/D_{0} = 1.00 and L_{1}/L_{0} = 2.0 is possible to observe a practically linear behavior for the pressure drop; while for D_{1}/D_{0} values less than 1.00 fluctuations occur in the pressure drop magnitude due to the L_{1}/L_{0} variation.
In sequence, the optimized values of L_{1}/L_{0} that conduct to the once minimized PD_{N} (identified in Figure 8) were plotted as a function of D_{1}/D_{0} ratio in Figure 9.
The results of Figure 9 indicate that for D_{1}/D_{0} = 1.00, 1.25, and 1.50, the values of (PD_{N})_{min} are similar, being 0.47, 0.38, and 0.44, respectively, allowing a decrease of pressure drop in relation to RI; whereas for D_{1}/D_{0} = 0.50 and 0.75, a pressure drop augmentation occurred. From this, the optimized Tshaped EAHE considering the both DOF is defined by (L_{1}/L_{0})_{oo} = 7.0 and (D_{1}/D_{0})_{o} = 1.25, which reaches a (PD_{N})_{2min} = 0.38 (representing a pressure drop of 1.45 m) being a performance 62% superior to RI.
3.3 Thermal potential
The thermal potential is taken as the difference between the air temperature at the outlet and the inlet of the EAHE, measuring its capability to provide or remove heat from the air flow. So, as defined by Eq. (25), Figure 10 presents the normalized thermal potential of the Tshaped EAHEs according to the L_{1}/L_{0 }variation for all studied D_{1}/D_{0}.
Analyzing Figure 10 and remembering that the thermal potential should be maximized, it is possible to infer that the superior performances for the Tshaped EAHEs were achieved for small values of L_{1}/L_{0}. When D_{1}/D_{0} = 0.50, 0.75, and 1.00, (L_{1}/L_{0})_{o} = 0.1; and when D_{1}/D_{0} = 1.25 and 1.50, (L_{1}/L_{0})_{o} = 0.5. Besides, in a general way, the effect of L_{1}/L_{0} variation over TP_{N} is more pronounced as D_{1}/D_{0} decreases. This fact is proved if the best and worst geometries for D_{1}/D_{0} = 0.50 and D_{1}/D_{0} = 1.50 are compared. In the last situation a difference of around only 15% occurs, while in the first one a difference of almost 50% is reached.
After the first stage of the optimization procedure (see Figure 4) regarding the thermal potential performance, the influence of D_{1}/D_{0} DOF over the (L_{1}/L_{0})_{o} was evaluated in Figure 11.
Figure 10. Influence of L_{1}/L_{0} over TP_{N}, for each D_{1}/D_{0}
Figure 11. Influence of D_{1}/D_{0} over (TP_{N})_{min}, from (L_{1}/L_{0})_{o} values
One can note in Figure 11, as well as in Figure 10, that only the geometries with D_{1}/D_{0} = 1.25 and 1.50 reached superior performances than the RI. The Tshaped EAHE geometry with (L_{1}/L_{0})_{oo} = 0.5 and (D_{1}/D_{0})_{o} = 1.50 achieves a (TP_{N})_{2max} = 1.21 (representing a thermal potential of 4.05℃), being 21% better than the RI.
3.4 Global performance
Finally, the three performance parameters already individually analyzed were concomitantly taken into account. The goal here was to identify the optimized Tshaped EAHE that at the same time reduces the soil volume and pressure drop and increases the thermal potential. Eq. (26) was employed to the 75 Tshaped EAHE, and its results can be viewed in Figure 12 as a function of L_{1}/L_{0} variation.
The results of Figure 12 indicate that the best geometric configuration was always obtained with L_{1}/L_{0} = 7.0, regardless of the value of D_{1}/D_{0}.
Besides, one can observe in Fig. 12 that the Tshaped EAHE installations with D_{1}/D_{0} = 0.75 and mainly D_{1}/D_{0} = 0.50 presented inferior performances, since the smallest values in the vector analysis indicate the superior global performance. This result was already expected, once these cases have the worst performance regarding the pressure drop (see Figure 8) and thermal potential (see Figure 10) evaluations. Because of that, these geometries were removed from Figure 12, allowing in Figure 13 a better visualization. In addition, the result of Eq. (26) to the RI was also included in Figure 13, making possible a more comprehensive discussion.
Figure 12. Influence of L_{1}/L_{0} in the vector analysis, for each D_{1}/D_{0}
Figure 13. Influence of L_{1}/L_{0} in the vector analysis, for the RI and the Tshaped EAHE with D_{1}/D_{0} ≥ 1.00
Figure 14. Influence of D_{1}/D_{0} over Performance_{min}, from (L_{1}/L_{0})_{o} values
It is possible to infer in Figure 13 that thirtytwo installations of Tshaped EAHE achieved an overall performance better than RI, being the ones with L_{1}/L_{0} ≥ 2.5 for D_{1}/D_{0} = 1.00 and with L_{1}/L_{0} ≥ 2.0 for D_{1}/D_{0} ≥ 1.25. This is an important finding since several Tshaped configurations can be used with superior performance than the traditional EAHE with a straight duct.
After that, the effect of D_{1}/D_{0} variation over the (L_{1}/L_{0})_{o} of Figure 12 concerning the vector analysis is depicted in Figure 14, together with the RI.
Figure 14 shows that as D_{1}/D_{0} increases, the overall performance becomes better. Hence, the global optimized Tshaped EAHE geometric configuration is obtained, having (L_{1}/L_{0})_{oo} = 7.0 and (D_{1}/D_{0})_{o} = 1.50. This geometry was able to improve the overall performance by 27.26 % when compared with the RI. However, the geometries reached relevant global improvements having D_{1}/D_{0} = 1.0 and D_{1}/D_{0} = 1.25, with (L_{1}/L_{0})_{o} = 7.0 in both cases, being 16.29% and 24.95%, respectively, also in comparison with RI.
In this work, it was presented an investigation about a Tshaped EAHE combining the analytical approach for the evaluation of the soil volume occupation (V) and air flow pressure drop (PD), while the thermal potential (TP) was numerically analyzed. Employing the Constructal Design and the Exhaustive Search, it was possible to compare 75 different configurations under the same operational parameters, distinguishing themselves by the ratio of the lengths of bifurcated and main branches (L_{1}/L_{0}) together with the variation of the ratio of the diameters between both (D_{1}/D_{0}). These Tshaped geometric configurations were proposed based on an EAHE with a straight duct, named Reference Installation (RI). Also, the results for the Tshaped EAHEs were normalized based on the RI results.
Firstly, the three performance parameters were individually considered. After that, they were taken into account in a concomitant way. So, it was possible to identify a specific optimized Tshaped geometric configuration for each performance parameter, and the optimized geometry, indicated to improve all performance parameters at the same time.
Concerning the soil volume occupation, as expected, the effect of D_{1}/D_{0} variation was insignificant, unlike the L_{1}/L_{0} ratio that has a significant influence on the minimization of V. The optimized geometry was obtained with (L_{1}/L_{0})_{o} = 7.0, regardless of D_{1}/D_{0}, reaching a reduction in soil volume around 23% if compared with RI. For the air flow pressure drop analysis, a significant reduction in relation to RI of 62% was achieved by the Tshaped EAHE with (L_{1}/L_{0})_{oo} = 7.0 and (D_{1}/D_{0})_{o} = 1.25, having the L_{1}/L_{0} ratio a lower influence than D_{1}/D_{0} over the PD minimization. In its turn, the thermal performance was maximized when (L_{1}/L_{0})_{oo} = 0.5 and (D_{1}/D_{0})_{o} = 1.50, reaching an improvement of 21 % in comparison with RI, being relevant to the influence of L_{1}/L_{0} and D_{1}/D_{0}. Finally, in the global performance parameters analysis, the Tshaped EAHE geometry defined by (L_{1}/L_{0})_{oo} = 7.0 and (D_{1}/D_{0})_{o} = 1.50 conduct to an overall performance about 27% superior to RI.
These findings clearly show the importance of performing geometric evaluations to identify the configurations that conduct to the superior performance regarding the EAHE devices. In addition, the Constructal Design method proved to be a suitable tool to understand the influence of the degrees of freedom over the EAHE performance.
The Tshaped EAHE can be indicated to be used instead of the traditional EAHE composed by a straight duct due to its better performance in all performed investigations. Highlighting that in urban areas, the use o Tshaped EAHE brings additional advantages due to its minor soil volume occupation and capability of attending two build environments simultaneously.
G.C. Rodrigues thanks to CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brazil) for master's scholarship (Finance Code 001). L.C. Victoria thanks to CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil) and I.S. Vaz thanks to FAPERGS (Fundação de Amparo à Pesquisa do Rio Grande do Sul, Brazil) for academic scolarships. E.D. Dos Santos, L.A. Isoldi and L.A.O. Rocha thank CNPq for research grants (306024/20179, 306012/20170, 307791/20190). The authors also thank to FAPERGS by the financial support (Edital 02/2017  PqG  17/255100011112).
c_{p} 
specific heat, J/kg·K 
D 
diameter, m 
d 
distance, m 
f 
friction factor 
g 
gravitational acceleration, m/s^{2} 
H 
height, m 
h 
depth, m 
K 
loss coefficient 
k 
turbulent kinetic energy, m^{2}/s^{2} 
L 
length, m 
PD 
pressure drop, m 
Re 
Reynolds number 
T 
temperature, K (or °C) 
TP 
thermal Potential, K (or °C) 
t 
time, s 
V 
volume, m^{3} 
v 
velocity, m/s 
x 
spatial coordinates, m 
Greek symbols 

$\alpha$ 
thermal diffusivity, m^{2}/s 
$\varepsilon$ 
turbulent dissipation rate, J/kg·K 
$\lambda$ 
thermal conductivity, W/m·K 
$\rho$ 
density, kg/m^{3} 
$\mu$ 
absolute viscosity, kg/m·s 
Subscripts and Superscripts 

0 
main branch 
1 
bifurcated branch 
Duct 
duct 
d 
straight stretch 
i 
indicial notation 
in 
inlet 
j 
indicial notation 
l 
localized loss 
N 
normalized 
RI 
reference installation 
s 
soil 
T 
Tshaped 
t 
thermal 
Total 
total 
out 
outlet 
Table A. Ducts length of the Tshaped EAHE
L_{1}/L_{0} 
L_{0} (m) 
L_{1} (m) 
0.1 
25.00 
2.50 
0.5 
15.00 
7.50 
1.0 
10.00 
10.00 
1.5 
7.50 
11.25 
2.0 
6.00 
12.00 
2.5 
5.00 
12.50 
3.0 
4.28 
12.84 
3.5 
3.75 
13.12 
4.0 
3.33 
13.32 
4.5 
3.00 
13.50 
5.0 
2.73 
13.63 
5.5 
2.50 
13.75 
6.0 
2.31 
13.84 
6.5 
2.14 
13.92 
7.0 
2.00 
14.00 
Table B. Ducts diameter of the Tshaped EAHE (D_{0} > D_{1})

D_{1}/D_{0} = 0.5 
D_{1}/D_{0} = 0.75 

L_{1}/L_{0} 
D_{0} (m) 
D_{1} (m) 
D_{0} (m) 
D_{1} (m) 
0.1 
0.1180 
0.0590 
0.1140 
0.0860 
0.5 
0.1400 
0.0700 
0.1244 
0.0933 
1.0 
0.1560 
0.0780 
0.1307 
0.0980 
1.5 
0.1660 
0.0830 
0.1342 
0.1006 
2.0 
0.1740 
0.0870 
0.1360 
0.1020 
2.5 
0.1800 
0.0900 
0.1380 
0.1035 
3.0 
0.1840 
0.0920 
0.1392 
0.1044 
3.5 
0.1880 
0.0940 
0.1400 
0.1050 
4.0 
0.1910 
0.0955 
0.1408 
0.1056 
4.5 
0.1930 
0.0965 
0.1413 
0.1059 
5.0 
0.1950 
0.0975 
0.1416 
0.1062 
5.5 
0.1970 
0.0985 
0.1421 
0.1066 
6.0 
0.1980 
0.0990 
0.1424 
0.1068 
6.5 
0.2000 
0.1000 
0.1428 
0.1071 
7.0 
0.2010 
0.1005 
0.1430 
0.1072 
Table C. Ducts diameter of the Tshaped EAHE (D_{0} < D_{1})

D_{1}/D_{0} = 1.25 
D_{1}/D_{0} = 1.50 

L_{1}/L_{0} 
D_{0} (m) 
D_{1} (m) 
D_{0} (m) 
D_{1} (m) 
0.1 
0.1052 
0.1315 
0.1000 
0.1500 
0.5 
0.0880 
0.1100 
0.0733 
0.1100 
1.0 
0.0938 
0.1172 
0.0812 
0.1218 
1.5 
0.0920 
0.1150 
0.0790 
0.1185 
2.0 
0.0913 
0.1142 
0.0778 
0.1167 
2.5 
0.0907 
0.1134 
0.0770 
0.1155 
3.0 
0.0904 
0.1130 
0.0765 
0.1147 
3.5 
0.0900 
0.1125 
0.0760 
0.1140 
4.0 
0.0898 
0.1123 
0.0757 
0.1136 
4.5 
0.0896 
0.1120 
0.0754 
0.1132 
5.0 
0.0894 
0.1118 
0.0752 
0.1128 
5.5 
0.0893 
0.1117 
0.0751 
0.1126 
6.0 
0.0892 
0.1115 
0.0749 
0.1124 
6.5 
0.0892 
0.0114 
0.0748 
0.1123 
7.0 
0.0891 
0.1113 
0.0747 
0.1121 
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