Explicit multipole formula for the local thermal resistance in an energy pile-the line-source approximation

Research output: Contribution to journalArticle


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.


Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Building Technologies
  • Energy Systems


  • Energy piles, Ground source heat pump (GSHP), Local thermal resistance, Minimum thermal resistance, Multipole method, Pile foundations, Thermal piles
Original languageEnglish
Article number5445
Issue number20
Publication statusPublished - 2020
Publication categoryResearch