Explicit Multipole Formula for the Local Thermal Resistance in an Energy Pile—The Line-Source Approximation
This paper presents a closed-form quite handy formula for the local thermal resistance <i>R</i><sub>b</sub> 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 th...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-10-01
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| Series: | Energies |
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| Online Access: | https://www.mdpi.com/1996-1073/13/20/5445 |
| _version_ | 1797550507169939456 |
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| author | Johan Claesson Saqib Javed |
| author_facet | Johan Claesson Saqib Javed |
| author_sort | Johan Claesson |
| collection | DOAJ |
| description | This paper presents a closed-form quite handy formula for the local thermal resistance <i>R</i><sub>b</sub> 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 <i>R</i><sub>b0</sub> for the zero-order approximation (<i>J</i> = 0), where only line heat sources at the pipes are used, is presented. The errors using zeroth-order approximation (<i>J</i> = 0) are shown to be quite small by comparisons with eight-order approximation (<i>J</i> = 8) with its accuracy of more than eight digits. The relative error for the local thermal resistance <i>R</i><sub>b0</sub> for the zero-order approximation (<i>J</i> = 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 <i>R</i><sub>bmin</sub> is, with some exceptions, obtained when the pipes lie directly in contact with the pile periphery. A neat formula for this minimum is presented. |
| first_indexed | 2024-03-10T15:31:18Z |
| format | Article |
| id | doaj.art-2ba4a54397974b29a9f60e18ff3a9f96 |
| institution | Directory Open Access Journal |
| issn | 1996-1073 |
| language | English |
| last_indexed | 2024-03-10T15:31:18Z |
| publishDate | 2020-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Energies |
| spelling | doaj.art-2ba4a54397974b29a9f60e18ff3a9f962023-11-20T17:37:49ZengMDPI AGEnergies1996-10732020-10-011320544510.3390/en13205445Explicit Multipole Formula for the Local Thermal Resistance in an Energy Pile—The Line-Source ApproximationJohan Claesson0Saqib Javed1Building Physics, Lund University, 221 00 Lund, SwedenBuilding Services, Lund University, 221 00 Lund, SwedenThis paper presents a closed-form quite handy formula for the local thermal resistance <i>R</i><sub>b</sub> 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 <i>R</i><sub>b0</sub> for the zero-order approximation (<i>J</i> = 0), where only line heat sources at the pipes are used, is presented. The errors using zeroth-order approximation (<i>J</i> = 0) are shown to be quite small by comparisons with eight-order approximation (<i>J</i> = 8) with its accuracy of more than eight digits. The relative error for the local thermal resistance <i>R</i><sub>b0</sub> for the zero-order approximation (<i>J</i> = 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 <i>R</i><sub>bmin</sub> is, with some exceptions, obtained when the pipes lie directly in contact with the pile periphery. A neat formula for this minimum is presented.https://www.mdpi.com/1996-1073/13/20/5445energy pilesthermal pilespile foundationslocal thermal resistanceminimum thermal resistancemultipole method |
| spellingShingle | Johan Claesson Saqib Javed Explicit Multipole Formula for the Local Thermal Resistance in an Energy Pile—The Line-Source Approximation Energies energy piles thermal piles pile foundations local thermal resistance minimum thermal resistance multipole method |
| title | Explicit Multipole Formula for the Local Thermal Resistance in an Energy Pile—The Line-Source Approximation |
| title_full | Explicit Multipole Formula for the Local Thermal Resistance in an Energy Pile—The Line-Source Approximation |
| title_fullStr | Explicit Multipole Formula for the Local Thermal Resistance in an Energy Pile—The Line-Source Approximation |
| title_full_unstemmed | Explicit Multipole Formula for the Local Thermal Resistance in an Energy Pile—The Line-Source Approximation |
| title_short | Explicit Multipole Formula for the Local Thermal Resistance in an Energy Pile—The Line-Source Approximation |
| title_sort | explicit multipole formula for the local thermal resistance in an energy pile the line source approximation |
| topic | energy piles thermal piles pile foundations local thermal resistance minimum thermal resistance multipole method |
| url | https://www.mdpi.com/1996-1073/13/20/5445 |
| work_keys_str_mv | AT johanclaesson explicitmultipoleformulaforthelocalthermalresistanceinanenergypilethelinesourceapproximation AT saqibjaved explicitmultipoleformulaforthelocalthermalresistanceinanenergypilethelinesourceapproximation |