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Transient heat conduction from a vertical rod buried in a semi-infinite medium with variable heating strength

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Abstract

In this paper, a variable heating strength model (VHS model) is developed to predict transient heat conduction from a vertical rod buried in a semi-infinite medium. Unlike past studies, the current VHS model permits a VHS along the rod. Both axial heat conduction through the rod and lateral heat conduction to the surrounding ground are modeled. A derived distribution of axial heating strength is then applied to a finite line heat source model to predict transient temperature changes in the surrounding medium. The predicted results show how the rod’s radius and ground’s thermal conductivity affect the vertical variation of heating strength and temperature response. Additional simulations predict the long-term temperature increase in the ground, due to a power transmission tower installed in a region of initially frozen ground.

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Abbreviations

A c :

cross-sectional area (m2)

D :

diameter (m)

H :

height (m)

h :

heat transfer coefficient (W/m2 K)

k :

thermal conductivity (W/m K)

L :

length (m)

q′′:

heat flow rate per unit area (W/m2)

R :

radius (m)

R′:

thermal resistance (Km/W)

T :

temperature (K)

z :

vertical coordinate (m)

Q :

total heat flow rate (W)

q :

heating strength (W/m)

m :

coefficient defined in Eq. 9 (1/m2)

α:

thermal diffusivity (m2/s)

ρ:

radial coordinate (m)

θ:

temperature response or difference (K)

b:

rod/tower

b0:

ground surface level of the rod

c:

fluid

g:

heat conduction medium (ground)

gs:

surface of medium

pf:

pin fin

References

  1. Breger DS, Hubbell JE, El Hasnaoui H, Sunderland JE (1996) Thermal energy storage in the ground: comparative analysis of heat transfer modeling using U-tubes and boreholes. Solar Energy 56(6):493–503

    Article  Google Scholar 

  2. Lund PD, Ostman MB (1985) A numerical model for seasonal storage of solar heat in the ground by vertical pipes. Solar Energy 34(4):351–366

    Article  Google Scholar 

  3. Rabin EK (1996) Thermal analysis of a helical heat exchanger for ground thermal energy storage in arid zones. Int J Heat Mass Transf 39(5):1051–1065

    Article  MathSciNet  Google Scholar 

  4. Zeng HY, Diao NR, Fang ZH (2002) A finite line-source model for boreholes in geothermal heat exchangers. Heat Transf-Asian Res 31(7):558–567

    Article  Google Scholar 

  5. Liu X, Wang D, Fang Z (2001) Modeling on heat transfer of a vertical bore in geothermal heat exchangers. Build Energy Environ 20:1–3

    Google Scholar 

  6. Bi Y, Chen L, Wu C (2002) Ground heat exchanger temperature distribution analysis and experimental verification. Appl Therm Eng 22(2):183–189

    Article  Google Scholar 

  7. Deng Y, Fedler CB (1992) Multi-layered soil effects on vertical ground-coupled heat pump design. Trans ASAE 35(2):687–694

    Google Scholar 

  8. Pearson SW (1979) Thermal performance verification of thermal vertical support members for the trans-alaska pipeline. J Energy Resour Technol 101:225–231

    Google Scholar 

  9. Eckert ERG, Drake RM Jr (1972) Analysis of heat and mass transfer. McGraw-Hill, New York, pp. 103–106

    MATH  Google Scholar 

  10. Yang S (1993) Heat transfer, 2nd edn. Higher Education Press, pp. 43–46

  11. Incropera FP, DeWitt DP (1996) Fundamentals of heat and mass transfer, 4th edn. Wiley, New York

    Google Scholar 

  12. Naterer GF (1996) Conduction shape factors of long polygonal fibres in a matrix. Numer Heat Transf A 30(7):721–738

    Google Scholar 

  13. Anteby I, Shai I (1993) Modified conduction shape factors for isothermal bodies embedded in a semi-infinite medium. Numer Heat Transf A 23(2):233–245

    Google Scholar 

  14. Chung M, Jung P, Rangel RH (1999) Semi-analytical solution for heat transfer from a buried pipe with convection on the exposed surface. Int J Heat Mass Transf 42(20):3771–3786

    Article  MATH  Google Scholar 

  15. Sodha MS, Sawhney RL, Sengupta A (1994) Shape factor for an underground vertical infinite cylindrical structure. Int J Energy Res 18(4):431–436

    Google Scholar 

  16. Fang T (2004) Similarity solutions for heat conduction in a semi-infinite medium with power law thermal conductivity. Int Commun Heat Mass Transf 31(4):477–485

    Article  Google Scholar 

  17. Naterer GF (2001) Applying heat-entropy analogies with experimental studies of interface tracking in phase change heat transfer. Int J Heat Mass Transf 44(15):2917–2932

    Article  MATH  Google Scholar 

  18. Gau C, Viskanta R (1984) Melting and solidification of a metal system in a rectangular cavity. Int J Heat Mass Transf 27:113–123

    Article  Google Scholar 

  19. Ho CJ, Viskanta R (1984) Heat transfer during melting from an isothermal vertical wall. ASME J Heat Transf 106:12–19

    Article  Google Scholar 

  20. Kowalewski TA, Cybulski A, Rebow M (1998) Particle image velocimetry and thermometry in freezing water. In: 8th international symposium on flow visualization

  21. Wisniewski TS, Kowalewski TA, Rebow M (1998) Infrared and liquid crystal thermography in natural convection. In: 8th international symposium on flow visualization

  22. Yavuzturk C, Spitler JD (1999) A short time step response factor model for vertical ground loop heat exchangers. ASHRAE Trans 105(2):475–485

    Google Scholar 

  23. Carslaw HS, Jeager JC (1993) Conduction of heat in solids, 2nd edn. Oxford University Press, Oxford

    Google Scholar 

  24. Ingersoll LR, Zobe OJ, Ingersoll AC (1954) Heat conduction with engineering, geological and other applications. Revised Edition, The University of Wisconsin Press

  25. Hart DP, Couvillion R (1986) Earth-coupled heat transfer, Dublin, OH, National Water Well Association, pp 11 – 16

  26. Beier RA, Smith MD (2002) Borehole thermal resistance from line-source model of in-situ tests. ASHRAE Trans 108:2112–2119

    Google Scholar 

  27. Naterer GF (2003) Heat Transfer in single and multiphase systems. CRC, Boca Raton

    Google Scholar 

  28. A solar design manual for Alaska, University of Alaska, Fairbanks Cooperative Extension Service Publications EEM-01255, http://www.uaf.edu/coop-ext/publications/eehpubs.html#eem

  29. Lunardini VJ (1981) Heat transfer in cold climates. Van Nostrand Reinhold Co., New York

    Google Scholar 

  30. National climate data and information archive, http://www.climate.weatheroffice.ec.gc.ca/climateData/canada_e.html

  31. Gosselin C, Mareschal J (2003) Variations in ground surface temperature histories in the Thompson Belt, Manitoba, Canada: environment and climate changes. Glob Planet Change 39:271–284

    Article  Google Scholar 

  32. Hinkel KM (1997) Estimating seasonal values of thermal diffusivity in thawed and frozen soils using temperature time series. Cold Regions Sci Technol 26:1–15

    Article  Google Scholar 

  33. Andersland OB, Ladanyi B (1994) An introduction to frozen ground engineering. Chapman & Hall, New York

    Google Scholar 

  34. Hammer TA, Ryan WL, Zirjacks WL (1985) Ground temperature observations. In: Thomas G. Krzewinski, Rupert G. Tart Jr (eds) Thermal design considerations in frozen ground engineering

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Acknowledgements

Support of this research from Manitoba Hydro and NSERC (Natural Sciences and Engineering Research Council of Canada) is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mechanical and Manufacturing Engineering, University of Manitoba, 15 Gillson Street, Winnipeg, MB, Canada, R3T 2N2

    X. Duan

  2. Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, ON, Canada, L1H 7K4

    G. F. Naterer

  3. Line Maintenance Services, Transmission Construction and Line Maintenance Division, Manitoba Hydro, 1100 Waverly Street, Winnipeg, MB, Canada, R3C 2P4

    M. Lu & W. Mueller

Authors
  1. X. Duan
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  2. G. F. Naterer
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  3. M. Lu
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  4. W. Mueller
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Correspondence to G. F. Naterer.

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Duan, X., Naterer, G.F., Lu, M. et al. Transient heat conduction from a vertical rod buried in a semi-infinite medium with variable heating strength. Heat Mass Transfer 43, 547–557 (2007). https://doi.org/10.1007/s00231-006-0124-8

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  • DOI: https://doi.org/10.1007/s00231-006-0124-8

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