## 2 Methods
### Graph-Based Design Method  
In a development similar to that of [PERSON] and [PERSON] [35], the following nondimensional quantities are defined: \(\overline{T}\), a measure of the EAHX thermal efficiency,  
\[\overline{T}=\frac{T_{\text{a0}}-T_{\text{a}}(x)}{T_{\text{a0}}-T_{\text{so} }}[=\varepsilon]\, \tag{6}\]  
and \(\overline{x}\), a fraction of the EAHX characteristic length,  
\[\overline{x}=\frac{x}{l^{\ast}}\, \tag{7}\]with the characteristic length,  
\[L^{*}=\frac{\rho_{a}(\pi r_{0}^{2}v)c_{a}}{2\pi r_{0}U}\;, \tag{8}\]  
defining the distance from the EAHX inlet for which the reduction in temperature difference \(T_{a0}-T_{a}(x)\) equals \(63\%(\approx 1-1/e)\) of the maximum reduction possible, i.e., \(T_{a0}-T_{sa0}\). Redefining Equation (3) in terms of \(\overline{T}\) and \(\overline{x}\), the following simplified expression is obtained:  
\[\overline{T}=1-\exp(-\overline{x})\;. \tag{9}\]  
Taking advantage of the concept of EAHX characteristic length, using Equation (8) together with Equations (2) and (9), and the expression for volumetric flow rate,  
\[\dot{V}=\pi r_{o}^{2}v\;; \tag{10}\]  
a set of graphs to assist large-diameter EAHX design is obtained--see Figures 3-5. In deriving these graphs, air and soil properties and temperatures from the case study described in Section 3 were considered. Using Equation (2), the top graph in Figure 3 relates soil depth \(z\) and outdoor air temperature \(T_{a0}\) to EAHX maximum temperature gradient \(T_{a0}-\hat{T}_{sa0}(z)\). The bottom graph in Figure 3 is derived from Equation (9) and presents lines of constant fraction \(x/L^{*}\) as a function of temperature differences \(T_{a0}-T_{aL}\) and \(T_{sa0}-\hat{T}_{sa0}(z)\). Figure 4, obtained from Equation (10), presents lines of constant airflow rate (per pipe), \(\dot{V}\), for different diameters and airflow velocities. Figure 3: Graphs to assist EAHX design: maximum EAHX temperature gradient \(T_{a0}-\hat{T}_{sa0}(z)\) and fraction of characteristic length \(x/L^{*}\)[36]. The graphs in Figure 5 were derived from Equation (8) and depict lines of constant pipe length \(L\) for specific fractions of characteristic length \(x/L^{*}\), different diameters, and different airflow velocities. To understand how Figures 3-5 assist in the _design_ of an EAHX, consider the following specifications, which will be referred to as \"Case 1\":  
* Pipe burial depth \(z=5.5\) m;
* EAHX inlet air temperature: \(T_{\text{a0}}=37\)\({}^{\circ}\)C;
* EAHX airflow rate: \(\dot{V}=8000\) m\({}^{3}\)/h. Moreover, assume that the pipe length \(L\) should not exceed 70 m (due to property limits) and the desired EAHX exit air temperature is \(T_{\text{aL}}=28\)\({}^{\circ}\)C.  
Using the pipe burial depth \(z=5.5\) m and \(T_{\text{a0}}=37\)\({}^{\circ}\)C, from Figure 3, point (ii), the maximum temperature gradient \(T_{\text{a0}}-T_{\text{s0}}(z=5.5)\) is 17 K. Since the desired EAHX inlet to outlet temperature difference \(T_{\text{a0}}-T_{\text{aL}}\) is 9 K, the minimum fraction \(x/L^{*}\) meeting specifications is 0.75, point (i) in Figure 3.  
Figure 4: Graph to assist EAHX design: airflow rate \(\dot{V}\) per pipe [36]. Figure 5: Graph to assist EAHX design: pipe length \(L\) for different fractions \(x/L^{*}\)[36]. From Figure 4, it can be seen that the specified airflow rate could be met with different pipe layouts. A single pipe with 1.0 m diameter and airflow velocity \(\sim\)2.8 m/s (point 10) would meet the requirement. Other options are represented in points 40 and 10 for two and four parallel pipes with 1.0 and \(\sim\)0.7 m diameter, respectively. For these layouts, pipe airflow velocity is equal to \(\sim\)1.4 m/s. Switching focus to the \(x/L^{*}=0.75\) (bottom) graph in Figure 5, the shaded area depicts the domain meeting the pipe length specification, \(L\leq 70\) m. Points 22 and 22 represent two (1 m diameter) 70 m and four (0.7 m diameter) 50 m long parallel pipes, respectively. These are the two possible design solutions.