Heat transfer in a geothermal heat-pump system – an analytical assessment Y. YANG*, M. DATCHEVA{ , D. KO ¨ NIG* and T. SCHANZ* A heat transfer model of a vertical borehole heat exchange system (open-loop) is introduced. Outside the borehole, a linearly increasing initial temperature for the ground is employed. Inside the borehole, the heat transfer procedure is divided into passive and active steps. An analytical solution of the heat transfer problem for a system of semi-infinite ground with a finite line source is derived and the open- loop case is discussed. The temperature distribution both inside and outside the borehole is obtained for a long operation time. A model parameter study is performed and the results are discussed. The analytical solutions and parameter study results are also compared with the respective U-tube model (closed-loop system). It is shown that, compared with the U-tube model, the open-loop system can improve the heat exchange system by increasing system effectiveness. KEYWORDS: environmental engineering; rocks/rock mechanics; temperature effects; theoretical analysis; time dependence ICE Publishing: all rights reserved NOTATION a ground thermal diffusivity a f fluid thermal diffusivity c f fluid specific heat H depth of borehole k ground thermal conductivity k f fluid thermal conductivity q b heating rate per unit length q f heating rate q geo geothermal heat flux R thermal resistance between pumped fluid and the borehole wall R 11 resistance between pipes R 12 resistance between pipe and borehole wall r, z spatial coordinates r, – Z dimensionless spatial coordinates r b radius of borehole r p radius of pipe T dimensionless time factor t time factor t* retention time V p pumping rate H dimensionless ground temperature H f dimensionless fluid temperature h ground temperature h b temperature at midpoint of the line source h f fluid temperature h H fluid temperature at bottom of borehole h 0 initial temperature h* ambient temperature r f fluid density INTRODUCTION Geothermal heat is a type of energy produced mainly through radioactive decay in the core of the earth, about 6000 km below the surface. This energy is renewable and often sufficiently available even in cold areas such as Canada and northern Europe. Geothermal heat-pump systems utilise the ground as a heat source. The system, which works by supplying heat to the ground during the summer and extracting heat during the winter, is more efficient than electric or gas/oil heating systems. A geothermal heat-pump system can reduce energy consump- tion by up to 44% compared with an air-source heat-pump system and up to 72% compared with conventional electrical heating and air conditioning (Omer, 2008). Another important advantage of geothermal heat-pump systems is that they are less damaging to the environment. Emissions of harmful gases, such as CO 2 , SO 2 and NO x , are obviously less than the use of electricity, natural gas, oil or coal for heating and/or cooling (Bose et al., 2002). After the first oil crisis in the 1970s and due to their advantages, investigations and applications of geothermal heat-pump systems increased and the systems were seen as more and more attractive. The number of geothermal heat-pump system installations is growing continuously, at an annual rate of 10–30% in recent years (Sanner et al., 2003). The cost of facility installation, repair and maintenance is also relatively low. A typical geothermal heat-pump system consists of a heat pump and a vertical heat exchanger system installed into the ground at depths ranging from 40 to 200 m. A heat-pump system model that involves two working modes, passive and active, is considered in this paper. In the heating mode, a certain amount of fluid is injected into the borehole and is then pumped out through a pipe after being heated or cooled to the required temperature. Many different factors influence geothermal heat-pump system performance, including the ground temperature distribu- tion, possible freezing and thawing of the soil, thermal resistance of the pipe and of the grouting material, etc. Suitable modelling and an analytical solution of the problem describing geothermal heat-pump system response Manuscript received 5 August 2013; first decision 29 November 2013; accepted 10 March 2014. Published online at www.geotechniqueletters.com on 30 June 2014. *Chair of Foundation Engineering, Soil and Rock Mechanics, Ruhr-Universita ¨ t Bochum, Bochum, Germany {Institute of Mechanics, Bulgarian Academy of Sciences, Sofia, Bulgaria Yang, Y. et al. (2014) Ge ´ otechnique Letters 4, 139–144, http://dx.doi.org/10.1680/geolett.13.00055 139