The average rate at which energy is conducted outward through the ground surface in a certain region is 30.9 mW/m2, and the average thermal conductivity of the near-surface rocks is 3.11 W/m·K. Assuming a surface temperature of 9.48°C, find the temperature (in Celsius) at a depth of 35.0 km (near the base of the crust). Ignore the heat generated by the presence of radioactive elements.
heatflow= conductivity*Area/depth*deltaTemp
heatflow/area=conductivity/depth*(Surfacetemp-lowertemp)
solve for lower temp
To solve this problem, we will use Fourier's law of heat conduction. This law states that the rate at which heat is conducted through a material is proportional to the temperature gradient and the thermal conductivity of the material.
First, let's convert the given thermal conductivity of the near-surface rocks from W/m·K to mW/m·K by multiplying it by 1000.
Thermal conductivity = 3.11 W/m·K * 1000 = 3110 mW/m·K
Next, we need to find the temperature gradient. In this case, we are given the rate at which energy is conducted outward through the ground surface, which is 30.9 mW/m². The temperature gradient can be calculated using the formula:
Temperature gradient = Energy conducted / Thermal conductivity
Temperature gradient = 30.9 mW/m² / 3110 mW/m·K
Temperature gradient = 0.00992 K/m
Now, we can find the change in temperature with depth. We are given the depth as 35.0 km, which we need to convert to meters by multiplying it by 1000.
Depth = 35.0 km * 1000 = 35000 m
The change in temperature with depth can be calculated using the formula:
Change in temperature = Temperature gradient * Depth
Change in temperature = 0.00992 K/m * 35000 m
Change in temperature = 347.2 K
Finally, we need to find the temperature at a depth of 35.0 km. The surface temperature is given as 9.48°C, which we need to convert to Kelvin by adding 273.15.
Surface temperature = 9.48°C + 273.15 = 282.63 K
Temperature at depth = Surface temperature + Change in temperature
Temperature at depth = 282.63 K + 347.2 K
Temperature at depth = 629.83 K
Now, to convert the temperature back to Celsius, subtract 273.15:
Temperature at depth = 629.83 K - 273.15 = 356.68°C
Therefore, the temperature at a depth of 35.0 km (near the base of the crust) is approximately 356.68°C.