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Chapter 19: Problem 21

We might think of a porous material as being a composite in which one of the phases is a pore phase. Estimate upper and lower limits for the room- temperature thermal conductivity of an aluminum oxide material having a volume fraction of \(0.25\) of pores that are filled with still air.

Short Answer

Expert verified
The upper limit of the room-temperature thermal conductivity of the given composite material is approximately \(22.506 \ W/mK\), and the lower limit is approximately \(0.0998 \ W/mK\).

Step by step solution

01

Understand the given information

We are given an aluminum oxide material with a volume fraction of 0.25 pores filled with still air. The room temperature thermal conductivities of aluminum oxide (also known as alumina) and air are approximately \(30 \ W/mK\) and \(0.025 \ W/mK\), respectively.
02

Upper limit of thermal conductivity

To find the upper limit of thermal conductivity (\(k_{max}\)) for the composite material, we apply the Rule of Mixtures (also known as a series model) as follows: \(k_{max} = (1 - V_{air})k_{alumina} + V_{air}k_{air}\), where \(V_{air}\) is the volume fraction of pores filled with air. Substituting the given volume fraction and the thermal conductivities of alumina and air, we get: \(k_{max} = (1 - 0.25) \times 30 + 0.25 \times 0.025 \\ k_{max} = 0.75 \times 30 + 0.25 \times 0.025 \\ k_{max} = 22.5 + 0.00625 \\ k_{max} ≈ 22.506 \ W/mK \)
03

Lower limit of thermal conductivity

To find the lower limit of thermal conductivity (\(k_{min}\)) for the composite material, we apply the inverse Rule of Mixtures (also known as a parallel model) as follows: \(\frac{1}{k_{min}} = \frac{(1 - V_{air})}{k_{alumina}} + \frac{V_{air}}{k_{air}}\) Substituting the given volume fraction and the thermal conductivities of alumina and air, we get: \(\frac{1}{k_{min}} = \frac{0.75}{30} + \frac{0.25}{0.025} \\ \frac{1}{k_{min}} = 0.025 + 10 \\ \frac{1}{k_{min}} = 10.025\) Finally, we solve for \(k_{min}\): \(k_{min} = \frac{1}{10.025} ≈ 0.0998 \ W/mK\)
04

Provide the final answer

We have found the upper and lower limits for the room-temperature thermal conductivity of an aluminum oxide material having a volume fraction of 0.25 of pores filled with still air. The upper limit is around \(22.506 \ W/mK\) and the lower limit is around \(0.0998 \ W/mK\).

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