Abstract
Borehole thermal resistance is both an important design parameter and a key performance characteristic of a ground heat exchanger. Another quantity that is particularly important for ground heat exchangers is the internal thermal resistance between the heat exchanger pipes. Both these resistances can be calculated to a high degree of accuracy by means of the well-known multipole method. However, the multipole method has a fairly intricate mathematical algorithm and is thus not trivial to implement. Consequently, there is considerable interest in developing explicit formulas for calculating borehole resistances. This paper presents derivation and solutions of newly derived second-order and higher-order multipole formulas for calculating borehole thermal resistance and total internal thermal resistance of single U-tube ground heat exchangers. A new and simple form of the first-order multipole formula is also presented. The accuracy of the presented formulas is established by comparing them to the original multipole method. The superiority of the new higher-order multipole formulas over the existing formulas is also demonstrated.
Highlights
The use of ground source heat pump (GSHP) systems to provide heating and cooling in buildings has increased at a rapid rate in the last two decades or so
This paper has presented new second-order and third-order multipole formulas for calculating
This paper has presented new second-order and third-order multipole formulas for calculating borehole thermal resistance and total internal thermal resistance
Summary
The use of ground source heat pump (GSHP) systems to provide heating and cooling in buildings has increased at a rapid rate in the last two decades or so. The theoretical methods to estimate borehole thermal resistance include analytical [5,6,7] or empirical [8,9,10,11] formulas based on one-dimensional or two-dimensional steady-state conductive heat transfer in the borehole. Lamarche et al [12] calculated a root-mean-square difference of 0.003 between the first-order multipole method and a finite element numerical model for 72 studied cases with varying grout and soil conductivities, and pipe sizes and positing. It is possible to simplify the multipole method to explicit closed-form formulas assuming that heat exchanger pipes are placed symmetrically about the center of the borehole. We present derivation and solutions of new second-order and higher-order multipole formulas for calculating borehole thermal resistance of single U-tube ground heat exchangers with symmetric pipes. The presented formulas are tested against the original multipole method (i.e., the tenth-order multipole calculation) as well as previously-derived zeroth-order and first-order multipole formulas
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