Efficiency Optimization in a Simple Mathematical Model of Underground Thermal Energy Storage

Underground thermal energy storage (UTES) using boreholes is a promising technology for handling the intermittency of renewable energy sources. Because efficiency requires large installations, simulation is crucial in system design. The goal of this study is to develop general explicit relationships for the system efficiency in terms of basic systems parameters, within a highly idealized model of UTES. The boreholes carrying the heat-transporting fluid are described as a density C_s(r) of sources or sinks. The system is charged for a time t_c, and then discharged for an equal time. Heat diffuses with coefficient D_th in the soil during both phases. By numerically solving the corresponding reaction-diffusion partial differential equations, we obtain the total energy extracted E_out given C_s(r). We then use functional differentiation to find the function C_s(r) that maximizes E_out given a total amount C_tot of C_s. A step-function form for C_s(r), with width R_s, is very close to optimal. We obtain explicit formulas for E_out and R_s. For D_th=0, E_out is proportional to (C_tot t_c ) and R_s is proportional to (C_tot t_c )^1/3. The loss in E_out from thermal diffusion is proportional to -(D_th)^1/2, while R_s is essentially independent of D_th.