A city occupies a circular area 15km in diameter and uses energy at the rate of about 10GW. Make the approximation that the top 4 kilometers of Earth's crust consists of granite (density 2700kg/m3, specific heat 840J/kg.K, and thermal conductivity 3.37 W/m.K), and that the average temperature in this layer is 50oC higher than the surface temperature.
a) Estimate the total energy that could be extracted in cooling the 4-km-thick layer of crust beneath the city to the temperature of the surface.
b) If the efficiency in converting this low-quality thermal energy to useful energy is 5%, how long would the 4-km geothermal resource below the city last?
c) Show that the initial rate at which the geothermal energy is replenished by conduction from below is far less than the rate of extraction, implying that this use of geothermal energy is probably not renewable

Assume that granite (which has a thermal conductivity of 3.37, a density of 2700 kg/m3, and a specific heat of 840 J/kg-K) makes up the top 4 kilometers of the Earth's crust.

The effectiveness of utilizing shallow geothermal energy is significantly influenced by the thermo-physical characteristics of the rock and soil. The term "Fourier's Law" for heat conduction refers to this. Despite being frequently stated as a scalar, Conduction-based thermal energy transfer is more likely to take place. Instead of moving for convection, the warm water would remain at the top. There are three methods that thermal energy can be transferred: conduction, convection, and radiation. Another method of transferring thermal energy is convection. Heat is transferred through convection when particles move through a fluid. Energy travels through space in electromagnetic waves. The Sun heats the Earth by thermal radiation.

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