Abstract
This paper presents a closed-form quite handy formula for the local thermal resistance Rb between the temperature of the bulk heat-carrier fluid in the pipes, equally spaced on a concentric circle inside a circular energy pile, and the mean temperature at the periphery of the pile. The so-called multipole method is used to calculate the temperature field. An important improvement of the multipole method is presented, where Cauchy’s mean value theorem of analytical functions is used. The formula for thermal resistance Rb0 for the zero-order approximation (J = 0), where only line heat sources at the pipes are used, is presented. The errors using zeroth-order approximation (J = 0) are shown to be quite small by comparisons with eight-order approximation (J = 8) with its accuracy of more than eight digits. The relative error for the local thermal resistance Rb0 for the zero-order approximation (J = 0) lies below 5% for a wide range of input parameter values. These ranges are judged to cover most practical cases of application. The smallest local thermal resistance Rbmin is, with some exceptions, obtained when the pipes lie directly in contact with the pile periphery. A neat formula for this minimum is presented.
Highlights
Ground-source heat pump (GSHP) systems are among the most energy-efficient and environmentally clean heating and cooling technologies available today [1]
The most common application of GSHP systems is with vertical borehole heat exchangers [3]
The high capital cost of installing borehole heat exchangers is one of the underlying reasons that hamper the widespread deployment of GSHP systems in many places around the world
Summary
Ground-source heat pump (GSHP) systems are among the most energy-efficient and environmentally clean heating and cooling technologies available today [1]. The larger number of pipes and their many possible arrangements in energy piles the calculation of thermal resistance calculation more complex than for borehole heat exchangers. The original model with 10th-order multipole was provided as a FORTRAN code of approximately 600 lines It has since been incorporated into several building energy simulation software and ground heat exchanger design programs for calculating the thermal resistance of borehole heat exchangers with single and double U-pipes [17,18]. Its implementation in design and simulation programs is intricate and requires considerable programming expertise and effort This has led to the development of closed-form multipole formulas that, under certain conditions, can be as accurate as the original multipole method in calculating the thermal resistance of ground heat exchangers. The accuracy of the presented formula is established by a detailed parametric study that brackets all or almost all real-world energy piles with a circular cross-section
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