Question

A 70-cm-diameter flat black disk is placed at the center of the ceiling of a $1-\mathrm{m}\times 1-\mathrm{m}\times 1-\mathrm{m}$ black box. If the temperature of the box is ${620}^{\circ }\mathrm{C}$ and the temperature of the disk is ${27}^{\circ }\mathrm{C}$, the rate of heat transfer by radiation between the interior of the box and the disk is (a) $2\mathrm{kW}$ (b) $5\mathrm{kW}$ (c) $8\mathrm{kW}$ (d) $11\mathrm{kW}$ (e) $14\mathrm{kW}$

Short answer

Answer: (b) $5\mathrm{kW}$

Step-by-step solution

Convert temperatures to Kelvin

Before we start our calculations, we need to convert the temperatures from Celsius to Kelvin for both the box and the disk. To convert temperatures from Celsius to Kelvin, add 273.15 to the Celsius temperature: Box temperature: 620°C + 273.15 = 893.15 K Disk temperature: 27°C + 273.15 = 300.15 K

Calculate the area of the disk

We need to find the area of the flat black disk. Since it's a disk, its area is given by the formula A = πr² where r is the radius and A is the area. First, find the radius of the disk: The diameter is 70cm, so the radius is half of that which is 35 cm. To convert it to meters, divide by 100: 35cm / 100 = 0.35 m. Now calculate the area of the disk using the formula A = πr²: A = π(0.35)^2 = 0.3849 m²

Calculate the rate of heat transfer by radiation

To find the rate of heat transfer between the box and the disk, use the Stefan-Boltzmann Law formula: Q = Aσ(Tb^4 - Td^4) where Q is the rate of heat transfer, A is the area of the disk, σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/(m²K^4)), Tb is the temperature of the box, and Td is the temperature of the disk. Now plug in the values that we have: Q = (0.3849 m²)(5.67 x 10^-8 W/(m²K^4))((893.15 K)^4 - (300.15 K)^4) ≈ 5070 W Now convert the rate of heat transfer to kW by dividing by 1000: Q = 5070 / 1000 ≈ 5.07 kW

Choose the correct answer

Now look at the given options and choose the closest value to our calculated heat transfer rate (5.07 kW): (a) $2\mathrm{kW}$ (b) $5\mathrm{kW}$ (c) $8\mathrm{kW}$ (d) $11\mathrm{kW}$ (e) $14\mathrm{kW}$ Thus, the correct answer is (b) $5\mathrm{kW}$.

Key concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle in thermodynamics relating to heat transfer by radiation. It specifies how the thermal radiation emitted by a black body is proportional to the fourth power of its absolute temperature. The law is mathematically expressed by the equation:
$$Q=A\sigma (T_b^4-T_d^4)$$
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The equation tells us that the rate at which a body radiates energy increases rapidly as the temperature rises. For instance, doubling the temperature of an object increases its thermal radiation by a factor of 16, as this relationship is to the fourth power ($2^4=16$). Due to this, temperature differences play a significant role in the transfer of heat by radiation, which is crucial when calculating the energy exchange between objects at different temperatures, such as in our exercise with the disk and the box.

Temperature Conversion

In physics problems, it's essential to use the correct units. For temperature-related calculations, we often need to convert Celsius to Kelvin - the standard unit in the International System (SI) for thermodynamic temperature. The conversion is straightforward but crucially important for ensuring accurate calculations in thermodynamics. To convert from Celsius to Kelvin, we use the following relationship:
$$K=\text{degree}C+273.15$$
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Adding 273.15 to the Celsius temperature gives us the temperature in Kelvin necessary for computations involving the Stefan-Boltzmann Law, as seen in our example exercise. Precision is key, and overlooking this step can lead to incorrect answers. This simple yet essential conversion allows us to accurately assess how heat is transferred by radiation between objects at different temperatures, like our flat black disk and the interior of the box in the given problem.

Area Calculation of a Disk

The determination of an object's surface area is often required when dealing with heat transfer calculations. For objects like our flat black disk, we use area calculation formulas unique to their geometrical shape.
The area ($A$) of a disk is calculated by the formula: $$A=\pi r^2$$
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To calculate the area in our example, we first need to determine the radius, which is half of the diameter. After finding the radius, we apply the formula. Although the calculation itself is straightforward, precision is necessary—it's about using the correct units and calculating to adequate significance. Here, ensuring the radius is in meters is key, as the Stefan-Boltzmann Law uses SI units for area. Therefore, if the radius is initially given in centimeters, it must be converted to meters before applying the area formula. Understanding how to calculate the area correctly enables us to quantify the heat transfer across the disk in our sample problem.