Fundamentals of the Thermal Analysis of Complex Arrangements of Underground Heat Sources
Abstract
:1. Introduction
2. Basic Principles
2.1. Steady-State Temperatures
- r—distance between a point source (with losses W) and the considered point,
- λ—thermal conductivity of the soil, and
- c—constant.
2.2. Transient Temperatures
3. Some Basic Examples
3.1. Line Source with Locally Varying Losses
3.2. Line Source with a 90° Bend
4. Temperature and Rating Analysis of a Complicated Cable Route
4.1. Steady State, without Longitudinal Heat Fluxes
- = temperature rise of conductor section is because of the dielectric losses of cable, (K)
- T1, T2, T3 = thermal resistance of the insulation, armor bedding and the jacket, respectively, (K.m/W).
- λ1, λ2 = sheath and armor loss factors, respectively
- = total losses of section js in cable j, (W).
- δi,j is the Kronecker-symbol with δi,j = 1 for i = j and δi,j = 0 otherwise.
- is the thermal resistance between section js of cable j and section is of cable, (K·m/W). Following (3), with Ns,j sectors (i.e., point sources) of the influencing cable j, we have:
4.2. Steady State, with Consideration of the Longitudinal Heat Fluxes
4.3. Transient Behaviour (without Longitudinal Heat Fluxes)
4.4. Transient Behaviour with Consideration of the Longitudinal Heat Fluxes
5. Conclusions
- The most simple geometric definition is achieved by using equidistant points along the cable axis.
- This method is much easier to handle compared with the finite line sources with their special systems of coordinates.
- It provides a very simple formulae for expressing temperature rises, relating only to the distance r between the considered point and the point source.
- It enables the consideration of varying temperatures along the cable conductor.
- It enables the analysis of transient conductor temperatures even for complicated, nonlinear cable runs.
- It enables the analysis of transient cable temperatures, taking into account the longitudinal heat fluxes, even for very complicated, nonlinear cable runs.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Point | x | y | z | z′ | Explanation |
---|---|---|---|---|---|
m | m | m | m | ||
cable 2 | |||||
2/1 | 1.00 | 2.00 | 18.00 | 18.00 | beginning of bend 1 |
2/2 | 1.80 | 2.00 | 20.20 | 20.40 | crossing point with cable 4 |
2/3 | 2.70 | 2.00 | 20.70 | 21.35 | crossing point with cable 5 |
2/4 | 3.70 | 2.00 | 21.00 | 22.50 | crossing point with cable 6 |
2/5 | 6.02 | 2.00 | 21.00 | 24.72 | crossing point with cable 7 |
2/6 | 7.41 | 2.00 | 21.00 | 26.14 | crossing point with cable 8 |
2/7 | 8.02 | 2.00 | 21.00 | 26.70 | beginning of bend 2 |
2/8 | 8.70 | 2.00 | 21.10 | 26.72 | crossing point with cable 9 |
2/9 | 11.00 | 2.00 | 24.00 | 31.40 | end of bend 2 |
cable 5 | |||||
5/1 | 5.59 | 2.00 | 13.00 | 13.00 | beginning of cable rising (L = 2.0 m) |
5/2 | 5.59 | 1.40 | 15.00 | 15.00 | end of cable rising (L = 1.4 m) |
5/3 | 5.59 | 1.40 | 16.59 | 16.70 | beginning of the bend |
5/4 | 2.70 | 1.40 | 20.70 | 21.90 | crossing point with cable 2 |
cable 8 | |||||
8/1 | 10.59 | 2.00 | 13.00 | 13.00 | beginning of cable rising (L = 2.0 m) |
8/2 | 10.59 | 1.40 | 15.00 | 15.00 | end of cable rising (L = 1.4 m) |
8/3 | 10.59 | 1.40 | 16.59 | 16.70 | beginning of the bend |
8/4 | 7.41 | 1.40 | 21.00 | 22.30 | crossing point with cable 2 |
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Brakelmann, H.; Anders, G.J.; Zajac, P. Fundamentals of the Thermal Analysis of Complex Arrangements of Underground Heat Sources. Energies 2021, 14, 6813. https://doi.org/10.3390/en14206813
Brakelmann H, Anders GJ, Zajac P. Fundamentals of the Thermal Analysis of Complex Arrangements of Underground Heat Sources. Energies. 2021; 14(20):6813. https://doi.org/10.3390/en14206813
Chicago/Turabian StyleBrakelmann, Heiner, George J. Anders, and Piotr Zajac. 2021. "Fundamentals of the Thermal Analysis of Complex Arrangements of Underground Heat Sources" Energies 14, no. 20: 6813. https://doi.org/10.3390/en14206813