Question

A \(20-\mathrm{cm}\)-diameter hot sphere at \(120^{\circ} \mathrm{C}\) is buried in the ground with a thermal conductivity of $1.2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The distance between the center of the sphere and the ground surface is \(0.8 \mathrm{~m}\), and the ground surface temperature is \(15^{\circ} \mathrm{C}\). The rate of heat loss from the sphere is (a) \(169 \mathrm{~W}\) (b) \(20 \mathrm{~W}\) (c) \(217 \mathrm{~W}\) (d) \(312 \mathrm{~W}\) (e) \(1.8 \mathrm{~W}\)

Short answer

Based on the given information, after performing the calculations and applying Fourier's law for heat transfer through conduction, the rate of heat loss from the hot sphere to the ground is approximately 20 W.

Step-by-step solution

Understand the given information and convert to standard units

The given information is: - Sphere diameter: 20 cm - Sphere temperature: 120°C - Ground surface temperature: 15°C - Thermal conductivity of ground: 1.2 W/m·K - Distance between sphere center and ground surface: 0.8 m First, convert the sphere diameter to meters: 20 cm = 0.2 m The sphere radius (r) is half of the diameter: r = 0.1 m

Apply the formula for heat transfer through conduction

The formula for the rate of heat transfer (Q) through conduction is given by Fourier's law: Q = k × A × (T₁ - T₂) / d Where: - Q is the rate of heat transfer (in Watts), - k is the thermal conductivity (1.2 W/m·K), - A is the surface area of the sphere (in square meters), - T₁ is the sphere temperature (120°C), - T₂ is the ground surface temperature (15°C), - d is the distance between the sphere center and the ground surface (0.8 m).

Calculate the surface area of the sphere

The surface area of a sphere can be calculated using the formula: A = 4πr^2 Where r is the radius of the sphere. Plugging in the values: A = 4π(0.1)^2 A ≈ 0.1256 m²

Substitute the values into the heat transfer formula

Now, we can plug all the values into Fourier's law formula to find the rate of heat loss: Q = (1.2) × (0.1256) × (120 - 15) / 0.8

Calculate the rate of heat loss

Now, simply do the arithmetic: Q ≈ (1.2) × (0.1256) × (105) / 0.8 Q ≈ 20.0464 W

Choose the correct answer

The calculated rate of heat loss is approximately 20 W, which corresponds to answer (b) in the given options. So, the correct answer is: (b) 20 W