How Do You Calculate Temperature at a Depth Based on Surface Conduction Rates?

[mm2424]

Homework Statement

The average rate at which energy is conducted outward through the ground surface in North America is 54 mW/m^2, and the average thermal conductivity of the near-surface rocks is 2.5 W/mK. Assuming a surface temperature of 10 C, find the temperature at a depth of 35 km. Ignore the heat generated by the presence of radioactive elements.

Homework Equations

P = kA(Th-Tc)/L

where P is the conduction rate
k = thermal conductivity
A = face area
L = thickness
Th = temperature of hot reservoir
Tc = temperature of cold reservoir

The Attempt at a Solution

I can plug all values into the equation for conduction rate except for A. Is there something I'm missing in terms of the surface area for this problem? In the answers I've seen online, people seem to assume A is 1m^2. However, I can't figure out why that assumption is made. (The answer seems to be 766 C, which is found using A=1m^2).

Thanks for your help!

 

[Delphi51]

You could apply the formula to any area you like.
But if you are going to use the 54 mW/m^2 you must express A in m^2 as well.
In the old days we used to put in the units for all quantities and check to make sure they worked out the same on both sides of a formula. That took too much time, but when in doubt, it is still worth doing occasionally.

 

[mm2424]

But if we had arbitrarily chosen an A of 2 m^2, wouldn't the answer to this question be different? The only quantity that would change in the equation would be A, no? I'm confused because the problem doesn't seem to tell you how large of an area we're considering, whereas the solution seems to automatically use a value of 1 m^2.

 

[Delphi51]

Oops, I gave you a dumb answer!
When you work out the units on the right side, taking k to be in W/mK, you get Watts.
That formula is giving you the power, not the power per square meter as the wording of the question implies. It is only natural that if you double the area, you get double the power flowing through.

Mix up in formulas? The formula is given as ΔQ/ΔT = kA*ΔT here:
http://en.wikipedia.org/wiki/Thermal_conductivity#Equations

 

[asteriatic]

I would like to point out that the equation for conduction rate, P = kA(Th-Tc)/L, assumes a uniform and constant surface temperature and thermal conductivity. However, in reality, the surface temperature may vary and the thermal conductivity of rocks can also vary depending on their composition. Therefore, the assumption of A=1m^2 may not always be accurate and could lead to errors in the calculated temperature at a depth of 35 km. To accurately solve this problem, it would be necessary to obtain more specific and accurate data on the surface temperature and thermal conductivity of the rocks in the area. Additionally, the presence of radioactive elements can also affect the temperature at a depth of 35 km, so it may not be appropriate to ignore their heat generation in this calculation. More research and data collection would be needed to accurately solve this conduction rate problem.