Table of Contents
Why calculate soil temperature?
Calculating soil temperature is helpful for electrical power engineers, especially for performing power cable ampacity calculations or earthing system designs. Here are some reasons why it is important:
Power cable designs: Soil temperature affects the current-carrying capacity of power cables. Soil ambient temperature is typically assumed, but the assumed values are usually too conservative, especially for deeply buried cables (greater than 2-3 metres). If you calculate soil temperature rather than assume it, then a more accurate current rating of the power cables may be achievable.
Earthing/grounding designs: Soil electrical resistivity affects performance and varies with temperature. It is especially important to perform soil temperature calculations for soils subject to freezing during winter, as their electrical resistivity dramatically increases when the soil freezes. Note that several additional steps are required, and the process is referred to as seasonal analysis.
How weather affects soil temperature
Weather conditions directly affect ground soil temperature. The ground’s surface has a complex and balanced heat transfer system consisting of conduction, convection, thermal radiation, and moisture evaporation. Soil temperature is closely linked to ambient air temperature.
Air temperature varies sinusoidally over a 24-hour cycle. These daily fluctuations are primarily driven by the sun’s position and heating effect during the day, followed by cooling during the night. Soil temperature variation from daily weather cycles occurs at depths much below 1 m. Therefore, daily (diurnal) air temperature variations can be ignored when calculating soil temperature.
The annual air temperature varies sinusoidally and is related to the soil temperature, varying with the same angular frequency. Parameters such as the annual average air temperature, the annual fluctuation in air temperature, and phase angles define this relationship (refer to the equations).
Due to the ground’s high thermal inertia, the maximum ground temperature at a few meters depth occurs several months later than the maximum air temperature, and the minimum ground temperature occurs several later than the minimum air temperature.
Equations for soil temperature calculations
Soil temperature at a particular depth can be readily calculated from air temperature for a day of the year.
The equation for daily average soil temperature is as follows:
(1) \(\begin{aligned}
T(z,t)=T_{mag}+Ae^{-\lambda z}\sin\left[\omega(t-t_s)-\lambda z\right]\; , \; \omega = \frac{2\pi}{Y}\; , \; \lambda=\sqrt{\frac{\pi}{aY}}\;
\end{aligned}\)
Where \(T(z,t)\) is the soil temperature (°C) at a depth \(z\) (m) for a particular time \(t\) (days), \(T_{mag}\) is the average annual ground temperature (equal to the average annual air temperature plus an additional allowance for surface cover (1-5 °C)), \(A\) is the amplitude of yearly fluctuation of air temperature, \(Y=365.2422\) is one year in days, \(t_s\) is the time of early spring when the average daily air temperature (close to the ground) becomes equal to the average annual air temperature (days), \(a\) is the soil thermal diffusivity (\(\frac{\text{m}^2}{\text{d}}\)).
Yearly variations of the average daily soil temperature decrease exponentially with depth. The equation for calculating the maximum depth where the yearly variation of average daily soil temperature falls below 1% of the value in the air is as follows:
(2) \(\begin{aligned}
z_{\max}=49.6548\sqrt{a}\
\end{aligned}\)
Thermal diffusivity influences the depth to which climatic effects penetrate the soil. Soils with higher diffusivity will show temperature changes at greater depths than soils with lower diffusivity.
Thermal diffusivity values for typical soil types include clay = 0.055, sandy soil = 0.063, silt = 0.115, loam = 0.069, and peat = 0.0088. The equation is as follows:
(3) \(a=\frac{k}{pc} \;\)
Where \(k\) is the soil thermal conductivity (W/m/°C), \(p\) is soil mass density (kg/m3), \(c\) is the specific heat capacity of soil (J/kg/°C).
Example soil temperature calculations
\(T_{mag}\) = 3 °C, \(A\) = 14 °C, \(t_s\) = 120 days, \(a\) and clay soil type where = 0.025 \(\frac{\text{m}^2}{\text{d}}\).
The first plot below uses Equation (1) above, showing air and soil temperature variations over a year. The critical depth \(z_{max}\) at which soil temperature variation becomes negligible is 7.851 m.
The second plot also uses Equation (1) above, showing the soil temperature variation with depth for various days in the year. The soil temperature distributions are very different from peak summer to peak winter.
The first day of Spring (day 120) shows the soil is still colder than the air temperature due to remnants of the winter that just ended.
Conclusions
Calculating soil temperature is essential for accurate power cable ampacity calculations, as it directly influences the current-carrying capacity of buried cables.
Soil electrical resistivity varies with temperature, making soil temperature calculations critical for effective earthing system designs, particularly in regions prone to freezing.
Weather conditions significantly impact soil temperature, which is closely linked to ambient air temperature and vary seasonally.
The equations provided allow for precise calculation of soil temperature at specific depths based on air temperature, enabling engineers to predict thermal behaviour accurately.
The analysis highlights the importance of considering seasonal variations in soil resistivity to ensure the reliability and safety of grounding systems.
Ongoing monitoring and adjustments of grounding systems are necessary to maintain their effectiveness and safety throughout different seasons.
References
[1] Larwa, B. (2019). Heat Transfer Model to Predict Temperature Distribution in the Ground. Energies, 12(1), 25.
[2] OYEWOLE, J.A., OLASUPO, T., AKINPELU, J.A., FABORO, E.O. (2018). Prediction of Soil Temperature at Various Depths Using a Mathematical Model. Journal of Applied Sciences and Environmental Management, 22(9), 1417-1420.
[3] Nofziger, D.L. (2003). Soil Temperature Changes with Time and Depth: Theory.
[4] Hanks, R.J., & Rasmussen, V.P. (2005). Simulating Soil Temperatures Using Numerical Methods. Journal of Natural Resources and Life Sciences Education.
[5] Ahmad, M.F., & Rasul, G. (2008). Prediction of Soil Temperature by Air Temperature; A Case Study for Faisalabad. Pakistan Meteorological Department.
