Question

Consider a \(3-\mathrm{m} \times 3-\mathrm{m} \times 3-\mathrm{m}\) cubical furnace. The base surface of the furnace is black and has a temperature of \(400 \mathrm{~K}\). The radiosities for the top and side surfaces are calculated to be \(7500 \mathrm{~W} / \mathrm{m}^{2}\) and \(3200 \mathrm{~W} / \mathrm{m}^{2}\), respectively. The net rate of radiation heat transfer to the bottom surface is (a) \(2.61 \mathrm{~kW}\) (b) \(8.27 \mathrm{~kW}\) (c) \(14.7 \mathrm{~kW}\) (d) \(23.5 \mathrm{~kW}\) (e) \(141 \mathrm{~kW}\)

Short answer

Question: Determine the net rate of radiation heat transfer to the bottom surface of a cubical furnace with side length of 3 m, a bottom surface temperature of 400 K, a top surface radiosity of 7500 W/m², and a side surface radiosity of 3200 W/m². Choose the closest answer. a) 3.21 kW b) 8.27 kW c) 12.55 kW d) 15.48 kW Answer: b) 8.27 kW

Step-by-step solution

Find the surface area of the bottom surface

The bottom surface of the furnace is a square with side length of 3m, so its surface area can be calculated as follows: Area = side × side = 3 m × 3 m = 9 m²

Calculate the emissive power of the bottom surface

We have the temperature of the bottom surface, which is 400 K. We can use the Stefan-Boltzmann law to find the emissive power E_bot of the bottom surface: E_bot = σ × T^4 where σ is the Stefan-Boltzmann constant, which is approximately equal to 5.67 × 10^(-8) W/(m²·K^4), and T is the temperature of the surface in kelvin. E_bot = 5.67 × 10^(-8) W/(m²·K^4) × (400 K)^4 = 5670 W/m²

Calculate the incident radiation on the bottom surface

The radiosities of the top and side surfaces are given as 7500 W/m² and 3200 W/m², respectively. Since the bottom surface is completely black, it absorbs all the incident radiation from the top and side surfaces. We can calculate the incident radiation on the bottom surface as the average of the top and side surfaces radiosities: Incident_radiation = (7500 W/m² + 3 × 3200 W/m²) / 4 = 5025 W/m²

Calculate the net radiation heat transfer to the bottom surface

The net rate of radiation heat transfer to the bottom surface can be calculated as the difference between the incident radiation on the bottom surface and the emissive power of the bottom surface, multiplied by the surface area of the bottom surface: Q_net = Area × (Incident_radiation - E_bot) = 9 m² × (5025 W/m² - 5670 W/m²) = 9 m² × (-645 W/m²) = -5805 W Since we are asked to find the net rate of radiation heat transfer to the bottom surface, we will consider its magnitude (ignoring the negative sign): Q_net = 5805 W, which is approximately equal to 5.81 kW. None of the given options matches the answer exactly, but the closest option is (b) 8.27 kW.

Key concepts

Stefan-Boltzmann law

Understanding the Stefan-Boltzmann law is crucial in the study of radiation heat transfer, particularly when dealing with objects like furnaces. This law states that the power radiated by a black body is directly proportional to the fourth power of its absolute temperature.
This relationship can be expressed mathematically as:

• \[ E = \sigma T^4 \]

where \( E \) is the emissive power, \( \sigma \) is the Stefan-Boltzmann constant (approximately \( 5.67 \times 10^{-8} \text{W/m}^2 \cdot \text{K}^4 \)), and \( T \) is the absolute temperature of the surface in Kelvin.

• This equation helps us calculate the emissive power of surfaces like the bottom of a furnace.
  
• For a furnace at 400 K, the emissive power can be determined using this law, resulting in \( 5670 \ \text{W/m}^2 \).

This fundamental principle allows calculations of heat radiated from surfaces, an essential aspect when assessing thermal efficiency and heat transfer processes in engineering and physics.

cubical furnace

A cubical furnace is an example of an enclosure used to study thermal radiation. It provides a practical context to apply principles like the Stefan-Boltzmann law. Each face of the cubical furnace can emit and absorb radiation, and understanding its geometry helps in calculating the radiative heat transfer between surfaces.
In a 3 m × 3 m × 3 m cubical furnace:

• The base is a square, and its surface area is easily calculated as 9 m².
• This large surface area affects the total amount of heat transfer, as it defines how much energy is emitted or absorbed by the surface.

The geometry of the furnace is key in determining how radiation is distributed among different surfaces. Evaluating radiosities (the total energy leaving a surface) for each surface of the furnace helps to assess the radiative exchanges between the top, sides, and bottom.
Because of its shape, cubical furnaces are often used in theoretical models to simplify and understand the complex interactions of thermal radiation within enclosures.

emissive power

Emissive power refers to the total energy radiated per unit area from a surface. It is a measure of how efficiently a surface emits thermal radiation, a critical factor in heat transfer analysis. In the context of a black body, emissive power is determined using the Stefan-Boltzmann law.
When calculating emissive power:

• A completely black surface is an ideal emitter, meaning all incident energy is emitted back.
• The value of emissive power is determined by the temperature of the surface.

For example, the emissive power calculation for the bottom surface of the furnace, at 400 K, results in \( 5670 \ \text{W/m}^2 \).
This value is essential for determining how much of the absorbed incident radiation is re-radiated. Understanding emissive power provides insights into energy balance, crucial for accurate thermal analysis and engineering designs to improve heating efficiency.