# Nanotechnology Applications in Ground Heat Exchanger Pipes: A Review
## 2 Literature Review
### Nanofluids  
#### 2.1.1 Theoretical and Numerical Simulations  
Several theoretical models based on differential equations have been reported in the literature to simulate the thermodynamic behaviour of GSHPs [12]. The main goal of these mathematical models is to determine the heat temperature carried by the fluid along the pipe from the borehole under certain operating conditions. For example, based on the transient heat conduction in the borehole, the temperature distribution of the ground might be obtained by solving the following partial differential equation:  
\[\frac{\partial^{2}T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T}{\partial r} =\frac{1}{\alpha}\frac{\partial T}{\partial r}\quad r_{b}<r<\infty\]  
\[-2\pi r_{b}\lambda\frac{\partial T}{\partial r}=q\quad\ r=r_{b}\ \ t>0\]  
\[T-T_{0}=0\quad\ r>r_{b}\ \ t>0\]  
where \(T\) stands for the ground temperature distribution at a distance \(r\), \(r_{b}\) represents the borehole radius, \(q\) is the heating rate per length of the source, \(t\) is the time, \(T_{0}\) is the initial temperature of the ground, and \(\lambda\) and \(\alpha\) are the thermal conductivity and diffusivity of the ground. Some numerical methods are commonly used to solve the previous equations. However, additional models are required to improve the approximations of the heat temperature carried by the fluid along the pipe, as in [12]. Moreover, such models must consider the GSHP fluid properties to obtain accurate results to optimize the GHSP performance. Therefore, nanofluids are attractive in the effort to achieve higher efficiencies and reduce the system size. The volumetric concentrations of nanofluids in the range from 0.1% to 1% can result in a reduction in the borehole thermal resistance. Additionally, they can reduce the borehole length depending on the kind of nanofluid. For example, with a 1% concentration of Cu, graphite, Ag, and CuO, the reductions in the length of boreholes were 2.3, 0.52, 0.33 and 0.14, respectively [13]. The authors who found this discussed and compared different papers focused on nanofluids used in geothermal systems to improve heat transfer. They concluded that a heat transfer enhancement is dependent on several factors, including the type of nanofluid, concentration, and system specification. Thus, the reduction in the heat exchanger sizes and the borehole sizes used in the geothermal-based system affect the efficiency of the system. The application of nanofluids in heat pipes, due to their superior thermophysical properties, was addressed in [14]. This work summarizes and provides the outcomes of experimental and theoretical studies of some nanofluids as working fluids in heat pipes, such as metals (Cu, Ag, and Gold, etc.) and metal-oxides (Al\({}_{2}\)O\({}_{3}\), CuO, MgO, ZnO, TiO\({}_{2}\), Fe\({}_{2}\)O\({}_{3}\), and SiO\({}_{2}\), etc.) to obtain an enhanced thermal performance. The results include the calculations of thermal efficiency, thermal resistance, the effective thermal conductivity, surface temperature gradient and the convective heat transfer coefficient in the evaporator and condenser section. Different configurations of heat pipes were reported according to different sizes and shapes related to the application requirements, integrating summaries and tables. In conclusion, the heat transfer mechanisms depend on the type of heat pipe, the characteristics of the nanofluids, the design and operating parameters of the heat pipes, etc. An important drawback in the vertical GHE is the existence of possible nanoparticle sedimentation when the system remains static. To solve this, the paper [15] carried out numerical simulations with nanoparticles of Al\({}_{2}\)O\({}_{3}\)-water and Fe\({}_{3}\)O\({}_{4}\)-water at constant temperature (26 \({}^{\circ}\)C) to visualize the sedimentation inside the exchanger. When the fluid is static, the nanoparticles accumulate after many hours of sedimentation. This can be averted by a high-speed flow, a pulsed flow, or by optimizing the geometry of the bottom borehole.