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Chapter 9: Problem 95

Two concentric spheres with diameters of \(5 \mathrm{~cm}\) and \(10 \mathrm{~cm}\) are having the surface temperatures maintained at \(100^{\circ} \mathrm{C}\) and \(200^{\circ} \mathrm{C}\), respectively. The enclosure between the two concentric spherical surfaces is filled with nitrogen gas at \(1 \mathrm{~atm}\). Determine the rate of heat transfer through the enclosure.

Short Answer

Expert verified
Answer: The rate of heat transfer through the enclosure is approximately 65 Watts.

Step by step solution

01

Determine the radii of the spheres and their temperature difference

We are given the diameters of the spheres as 5 cm and 10 cm. We will find their radii by dividing the diameters in half: \(r_1 = \dfrac{5}{2} \mathrm{~cm} = 2.5 \mathrm{~cm}\) \(r_2 = \dfrac{10}{2} \mathrm{~cm} = 5 \mathrm{~cm}\) The surface temperatures of the two spheres are given as \(T_1 = 100^{\circ}\mathrm{C}\) and \(T_2 = 200^{\circ}\mathrm{C}\). We will find the temperature difference as: \(\Delta T = T_2 - T_1 = 200^{\circ}\mathrm{C} - 100^{\circ}\mathrm{C} = 100^{\circ}\mathrm{C}\)
02

Find the thermal conductivity of nitrogen gas

Next, we need to find the thermal conductivity of nitrogen gas. At 1 atm and room temperature, the thermal conductivity of nitrogen gas is approximately \(k = 0.026 \mathrm{~W/m \cdot K}\).
03

Calculate the rate of heat transfer through the enclosure

Now we will use the formula for the rate of heat transfer through a spherical shell: \(q = 4 \pi k \dfrac{r_1 r_2}{r_2 - r_1} \Delta T\) We will convert the radii to meters (\(1 \mathrm{~cm} = 0.01 \mathrm{~m}\)) and plug in the values to find the heat transfer rate: \(q = 4 \pi (0.026 \mathrm{~W/m \cdot K}) \dfrac{(2.5 \cdot 0.01)(5 \cdot 0.01)}{(5 \cdot 0.01) - (2.5 \cdot 0.01)} (100^{\circ}\mathrm{C})\) \(q = 4 \pi (0.026) \dfrac{0.0125}{0.025} (100)\) \(q = 4 \pi (0.026) (0.5) (100)\) \(q \approx 65 \mathrm{~W}\) The rate of heat transfer through the enclosure is approximately 65 Watts.

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Most popular questions from this chapter

A \(50-\mathrm{cm} \times 50-\mathrm{cm}\) circuit board that contains 121 square chips on one side is to be cooled by combined natural convection and radiation by mounting it on a vertical surface in a room at \(25^{\circ} \mathrm{C}\). Each chip dissipates \(0.18 \mathrm{~W}\) of power, and the emissivity of the chip surfaces is 0.7. Assuming the heat transfer from the back side of the circuit board to be negligible, and the temperature of the surrounding surfaces to be the same as the air temperature of the room, determine the surface temperature of the chips. Evaluate air properties at a film temperature of \(30^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption?

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Consider a fluid whose volume does not change with temperature at constant pressure. What can you say about natural convection heat transfer in this medium?

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