## 3 Results  
The numerical simulations are performed for different values of (a) the dimensionless diameter of the insulation, \(W\), in the range between 2 and 6, (b) the dimensionless center-to-center pipe spacing, \(L\), in the range between 1.2 and 3.6, (c) the dimensionless burial depth, \(H\), in the range between 1 and 10, (d) the ratio between the thermal conductivities of the insulating material and the ground, \(K\), in the range between 0.01 and 0.5 and (e) the dimensionless temperature of the return pipe, \(T_{\rm r}\), in the range between 0 and 1. The selected local results are displayed in Figures 3 and 4, in which equispaced isotherm contours are plotted for \(H=6\), \(W=4\) and \(L=1.3,1.75\) and 2.5. As expected, the temperature field in the ground is featured by an eccentric distribution with a higher isotherm concentration in the region between the ground surface and the pre-insulated double pipe. On the other hand, the highest temperature gradients are localized within the insulation, in which the distance between the pipes plays the most important role in determining the heat transfer rate from the supply pipe to the return pipe, and from each pipe to the ground surface. In fact, for small center-to-center spacings, most of the heat loss from the supply pipe is \"gained\" by the return pipe; conversely, when the pipe spacing is large enough, the higher insulation thickness between the pipes results in a growth of the amount of heat transferred to the ground surface. Therefore, the thermal resistance existing between each pipe and the ground surface is affected by the heat transfer rate occurring from the supply pipe to the return pipe, which means that \(\rho_{\rm pg}\) increases as \(L\) is decreased. This is clearly illustrated in Figures 5 and 6, in which the heat flux at the supply and return pipe surfaces are plotted for \(L=1.3,1.75\) and 2.5. Additionally, \(\rho_{\rm pg}\) is expected to increase as the burial depth \(H\) is increased, as reported in Figure 7. In parallel, the thermal resistance \(\rho_{\rm pp}\) existing between the supply and return pipes increases as \(L\) is increased and \(H\) is decreased, as displayed in Figure 8. Hence, an optimal dimensionless pipe spacing of \(L_{\rm opt}\) for the minimum heat loss from the supply pipe for a prescribed temperature imbalance assigned between the pipes has to be expected, as displayed  
Figure 4: Close-up of the isotherm contour plots for \(H=6\), \(W=4\), \(T_{\rm r}=0.5\), \(K=0.1\) and \(L=1.3\), to \(2.5\). Figure 5: Distributions of the dimensionless local heat loss from the supply pipe \(q_{\rm s}\) versus angular coordinate \(\theta\) using \(L\) as a parameter. Figure 6: Distributions of the dimensionless local heat loss from the return pipe \(q_{\rm r}\) versus angular coordinate \(\theta\) using \(L\) as a parameter. The effects of \(W\), \(H\) and \(K\) on the heat loss from the supply pipe are then pointed out in Figures 10-12 in which, for any independent variable, a number of distributions of \(q_{\rm s}\) are plotted versus \(L\) using the variable itself as a parameter. It is apparent that the minimum heat loss increases as the insulation diameter decreases, whereas all of the other controlling variables are plotted in Figures 10-12. Figure 8: Distributions of the dimensionless thermal resistance \(\rho_{\rm PP}\) versus \(H\) using \(L\) as parameter. Figure 7: Distributions of the dimensionless thermal resistance \(\rho_{\rm PB}\) versus \(H\) using \(L\) as parameter. Figure 9: Distributions of the dimensionless heat loss from the supply pipe \(q_{\rm s}\) versus \(L\) using \(T_{\rm r}\) as a parameter. parameters have more limited effects, at least within their investigated ranges of variability. Furthermore, it can be noticed that the optimal pipe spacing increases as the insulation diameter and the burial depth are increased.