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Chapter 7: Problem 84

Air at \(20^{\circ} \mathrm{C}(1 \mathrm{~atm})\) is flowing over a \(5-\mathrm{cm}\) diameter sphere with a velocity of \(3.5 \mathrm{~m} / \mathrm{s}\). If the surface temperature of the sphere is constant at \(80^{\circ} \mathrm{C}\), determine \((a)\) the average drag coefficient on the sphere and \((b)\) the heat transfer rate from the sphere.

Short Answer

Expert verified
Answer: The average drag coefficient on the sphere is 0.001287, and the heat transfer rate from the sphere is 2.470 W.

Step by step solution

01

Calculate the Reynolds number for the flow

The Reynolds number is a dimensionless quantity that compares the inertial forces to the viscous forces of a flow, and can be calculated using the formula: Re = (ρ * V * D) / μ where ρ is the air density, V is the air velocity, D is the sphere's diameter, and μ is the air dynamic viscosity. For air at 20°C (1 atm), we have ρ = 1.204 kg/m³ and μ = 1.815e-5 kg/(m*s). The velocity and diameter are given as V = 3.5 m/s and D = 0.05 m. Calculating the Reynolds number: Re = (1.204 * 3.5 * 0.05) / 1.815e-5 = 18,641
02

Obtain the drag coefficient from the Reynolds number

The drag coefficient (Cd) for a sphere can be obtained from the Reynolds number using empirical correlations. One such correlation is the Schlichting formula: Cd = 24/Re Using the calculated Reynolds number, we can now find the drag coefficient: Cd = 24 / 18,641 = 0.001287 Thus, the average drag coefficient on the sphere is 0.001287.
03

Calculate the Nusselt number for the flow

The Nusselt number is a dimensionless number that represents the ratio of convective heat transfer to conductive heat transfer. For a sphere, an empirical correlation can be used to determine the Nusselt number based on the Reynolds and Prandtl numbers: Nu = 2 + 0.4 * Re^0.5 * Pr^(1/3) For air at 20°C, the Prandtl number (Pr) can be approximated as 0.7. Using the previously calculated Reynolds number, we can now calculate the Nusselt number: Nu = 2 + 0.4 * 18,641^0.5 * 0.7^(1/3) = 9.982
04

Calculate the heat transfer coefficient from the Nusselt number

The heat transfer coefficient (h) can be obtained from the Nusselt number using the formula: h = Nu * k / D where k is the thermal conductivity of the air and D is the sphere's diameter. For air at 20°C, k = 0.0262 W/(m*K). Calculating the heat transfer coefficient: h = 9.982 * 0.0262 / 0.05 = 5.228 W/(m²*K)
05

Calculate the heat transfer rate from the sphere

Finally, the heat transfer rate (Q) can be calculated using the formula: Q = h * A * ΔT where A is the surface area of the sphere and ΔT is the temperature difference between the surface and the air. The surface area of a sphere can be calculated using the formula: A = 4 * π * (D/2)^2 Substituting the given values: A = 4 * π * (0.05/2)^2 = 0.007853 m² ΔT = 80°C - 20°C = 60 K Calculating the heat transfer rate: Q = 5.228 * 0.007853 * 60 = 2.470 W In conclusion, the average drag coefficient on the sphere is 0.001287, and the heat transfer rate from the sphere is 2.470 W.

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Most popular questions from this chapter

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