Question

Enhanced geothermal systems (EGS) consist of two or more boreholes that extend several kilometers below ground level into the hot bedrock. Since drilling these holes can cost millions, one concern is that the heat provided by the bedrock cannot pay back the initial investment. How long can \(0.669 \mathrm{~km}^{3}\) of granite deliver an average of \(13.9 \mathrm{MW}\) of power, if its initial temperature is \(168.3^{\circ} \mathrm{C}\) and its final temperature is \(103.5^{\circ} \mathrm{C}\) ? [The density of granite is 2.75 times that of water, and its specific heat is \(0.790 \mathrm{~kJ} /\left(\mathrm{kg}^{\circ} \mathrm{C}\right)\).

Short answer

Answer: The granite can provide an average power of 13.9 MW for approximately 215.7 years.

Step-by-step solution

Find the volume of granite in cubic meters

To do this, we will need to convert the volume of granite from cubic kilometers to cubic meters. Use the conversion factor 1km³ = 1x10^9 m³: \(0.669 \mathrm{~km}^3 = 0.669 \times 10^9 \mathrm{~m}^3\)

Calculate the mass of the granite

Since the density of granite is given as 2.75 times that of water, and the density of water is 1000 kg/m³, we find the density of granite as follows: Density of granite = 2.75 * (Density of water) = 2.75 * (1000 kg/m³) = 2750 kg/m³ Now we can find the mass of the granite by multiplying its volume by its density: Mass = Volume * Density Mass = 0.669 * 10^9 m³ * 2750 kg/m³ = 1.84025 * 10^12 kg #Step 2: Calculate the energy extracted from the granite#

Calculate the temperature difference

Find the difference between the initial and final temperatures of the granite: \(\Delta T = T_{initial} - T_{final} = 168.3^{\circ} \mathrm{C} - 103.5^{\circ} \mathrm{C} = 64.8^{\circ} \mathrm{C}\)

Calculate the energy extracted

Use the mass of the granite, the granite's specific heat, and the difference in temperature to calculate the amount of energy extracted from the granite using the formula Q = mcΔT: Energy extracted(Q) = Mass(m) * Specific heat(c) * Temperature difference(ΔT) Q = 1.84025 * 10^12 kg * 0.790 kJ/(kg·°C) * 64.8°C = 9.45287 * 10^13 kJ #Step 3: Determine how long the granite can provide the power#

Convert power from MW to kJ/s

The given power is in MW (Megawatts). To be able to use it in our calculations, we need to convert it to kJ/s. We can use the conversion factor: 1 MW = 1x10^6 W = 1x10^6 J/s = 10^3 kJ/s: Power = 13.9 MW * 10^3 kJ/MW = 13.9 * 10^3 kJ/s

Calculate the time it can provide the power

To find how long the granite can deliver the given power (in seconds), we need to divide the total energy extracted from the granite by the power: Time = Energy Extracted / Power Time = (9.45287 * 10^13 kJ) / (13.9 * 10^3 kJ/s) = 6.79918 * 10^9 s #Step 4: Convert time to a more convenient unit#

Convert time to years

To get a more convenient unit, let's convert the time from seconds to years: 1 year = 365.25 days * 24 hours/day * 3600 s/hour ≈ 3.1536 * 10^7 s Time = 6.79918 * 10^9 s / (3.1536 * 10^7 s/year) ≈ 215.7 years The granite can deliver an average of 13.9 MW of power for approximately 215.7 years.