Question

Consider two infinitely long concentric cylinders with diameters 20 and \(25 \mathrm{~cm}\). The inner surface is maintained at \(700 \mathrm{~K}\) and has an emissivity of \(0.40\), while the outer surface is black. If the rate of radiation heat transfer from the inner surface to the outer surface is \(2400 \mathrm{~W}\) per unit area of the inner surface, the temperature of the outer surface is (a) \(605 \mathrm{~K}\) (b) \(538 \mathrm{~K}\) (c) \(517 \mathrm{~K}\) (d) \(451 \mathrm{~K}\) (e) \(415 \mathrm{~K}\)

Short answer

Answer: (b) 538 K

Step-by-step solution

Identify the given information

We have two concentric cylinders. The inner cylinder has diameter 20 cm, temperature 700 K, and emissivity 0.40. The outer cylinder has an unknown temperature and is a blackbody (emissivity=1). The rate of radiation heat transfer from the inner surface to the outer surface is 2400 W per unit area.

Apply the Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the heat transfer rate per unit area Q/A is proportional to the temperature difference between the two surfaces raised to the fourth power. So, we have: Q/A = ε_inner * σ * (T1^4 - T2^4) where ε_inner is the emissivity of the inner surface, σ is the Stefan-Boltzmann constant (5.67 * 10^-8 W/m².K⁴), T1 is the temperature of the inner surface, and T2 is the temperature of the outer surface.

Insert given values and solve for T2

We know the Q/A, ε_inner, and T1 values, so we can plug those values in and solve for T2: 2400 = 0.40 * (5.67 * 10^-8) * (700^4 - T2^4) Now we need to isolate T2^4. First, divide both sides by (0.40 * (5.67 * 10^-8)): 2400 / (0.40 * (5.67 * 10^-8)) = 700^4 - T2^4 Next, subtract 700^4 from both sides: 700^4 - 2400 / (0.40 * (5.67 * 10^-8)) = T2^4 Now, we need to find the fourth root of the value on the right side of the equation to determine the temperature of the outer surface: T2 = (700^4 - 2400 / (0.40 * (5.67 * 10^-8)))^(1/4) T2 ≈ 538 K

Select the correct answer

We have calculated the temperature of the outer surface to be 538 K, so the correct answer is: (b) 538 K

Key concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle in radiation heat transfer. It helps in calculating the heat exchange between surfaces, particularly when the surfaces are at different temperatures. The law states that the power radiated from a black body is proportional to the fourth power of its absolute temperature.
In mathematical form, the power per unit area (\( Q/A \)) radiated by a surface is given by:
\[ Q/A = σT^4 \] where \( σ \)is the Stefan-Boltzmann constant, approximately \(5.67 \times 10^{-8} \, \mathrm{W/m^2K^4} \). When dealing with surfaces that are not perfect black bodies, the equation incorporates emissivity:\[ Q/A = \varepsilon σ(T_{inner}^4 - T_{outer}^4) \] where \( \varepsilon \)is the emissivity of the surface. This equation is critical for analyzing heat transfer in systems like our concentric cylinders.

Emissivity

Emissivity (\(\varepsilon \)) is a measure of a material's ability to emit thermal radiation relative to a perfect black body.
The value of emissivity ranges from 0 to 1, where:

• A black body has an emissivity of 1, meaning it is a perfect emitter of radiation.
• An emissivity of 0 means the surface does not emit thermal radiation.

This property significantly affects the amount of heat radiating from a surface. For example, in our exercise, the inner surface of the cylinder has an emissivity of 0.40, indicating that it emits 40% of the thermal radiation that a perfect black body would emit at the same temperature.
Understanding emissivity is crucial because it allows us to determine how surfaces will interact in terms of heat transfer, especially when calculating the radiative heat transfer rates using the Stefan-Boltzmann equation.

Temperature calculation

Calculating temperature in radiation heat transfer problems requires applying the Stefan-Boltzmann Law with known variables of emissivity, initial temperature, and heat transfer rate.
From our exercise, we know the equation:\[ 2400 = 0.40 \times 5.67 \times 10^{-8} (T_{inner}^4 - T_{outer}^4) \] This equation forms the basis for solving for the unknown temperature (\(T_{outer}\)) of the outer cylinder.
First, rearrange the equation to isolate \(T_{outer}^4\):\[ T_{outer}^4 = T_{inner}^4 - \frac{2400}{0.40 \times 5.67 \times 10^{-8}} \] Then calculate \(T_{outer}\) by taking the fourth root of the result.
This process demonstrates how small changes in variables, such as emissivity and temperature, can significantly impact the results, making precision vital in these calculations.

Concentric cylinders

Concentric cylinders are a common configuration in thermal engineering problems. This arrangement, where one cylinder is inside the other, provides a model to study the thermal radiation transfer between the two surfaces.
In this exercise, the concentric cylinders have known diameters and temperatures.The inner cylinder's surface radiates energy to the outer cylinder's surface.
Concentric cylinders are helpful for understanding complex heat transfer situations. They allow us to apply fundamental heat transfer concepts, such as radiation and emissivity, in a controlled environment.
Such configurations are used in devices like heat exchangers, which rely on efficient thermal energy transfer. Understanding how energy moves from one surface to another is essential to design systems for thermal management.