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Chapter 14: Problem 92

Bare, dark-colored basalt has a thermal conductivity of 3.1 $\mathrm{W} /(\mathrm{m} \cdot \mathrm{K}),$ whereas light-colored sandstone's thermal conductivity is only \(2.4 \mathrm{W} /(\mathrm{m} \cdot \mathrm{K}) .\) Even though the same amount of radiation is incident on both and their surface temperatures are the same, the temperature gradient within the two materials will differ. For the same patch of area, what is the ratio of the depth in basalt as compared with the depth in sandstone that gives the same temperature difference?

Short Answer

Expert verified
Answer: The ratio of the depth in basalt to the depth in sandstone is 31/24.

Step by step solution

01

Write the thermal conduction equation for both materials

We set up two equations for the thermal conduction of basalt and sandstone as follows: For basalt: \(Q_{1} = k_{1}A\frac{dT}{dx_{1}}\) For sandstone: \(Q_{2} = k_{2}A\frac{dT}{dx_{2}}\) where \(k_{1}\) and \(k_{2}\) are the thermal conductivities of basalt and sandstone, \(A\) is the same area for both materials, and \(dx_{1}\) and \(dx_{2}\) are the depths in basalt and sandstone, respectively.
02

Equate the heat flux for both materials

Since the heat flux (rate of heat transfer) is the same for both materials, we can equate \(Q_{1}\) and \(Q_{2}\): \(k_{1}A\frac{dT}{dx_{1}} = k_{2}A\frac{dT}{dx_{2}}\)
03

Cancel the common terms and rearrange the equation to find the ratio of depths

The area (A) and the temperature difference (\(dT\)) are the same for both materials, so we can cancel them from both sides of the equation: \(k_{1}\frac{1}{dx_{1}} = k_{2}\frac{1}{dx_{2}}\) Now, we can rearrange the equation to get the ratio of depths: \(\frac{dx_{1}}{dx_{2}} = \frac{k_{1}}{k_{2}}\)
04

Calculate the ratio of depths in basalt and sandstone with given thermal conductivities

Insert the given thermal conductivities, \(k_{1} = 3.1 W/(m\cdot K)\) for basalt and \(k_{2} = 2.4 W/(m\cdot K)\) for sandstone into the equation: \(\frac{dx_{1}}{dx_{2}} = \frac{3.1}{2.4}\)
05

Simplify the fraction and calculate the ratio

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor: \(\frac{dx_{1}}{dx_{2}} = \frac{31}{24}\) So, the ratio of the depth in basalt as compared with the depth in sandstone that gives the same temperature difference is \(\frac{31}{24}.\)

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