91Ó°ÊÓ

One scheme for extending the operation of gas turbine blades to higher temperatures involves applying a ceramic coating to the surfaces of blades fabricated from a superalloy such as inconel. To assess the reliability of such coatings, an apparatus has been developed for testing samples under laboratory conditions. The sample is placed at the bottom of a large vacuum chamber whose walls are cryogenically cooled and which is equipped with a radiation detector at the top surface. The detector has a surface area of \(A_{d}=10^{-5} \mathrm{~m}^{2}\), is located at a distance of \(L_{\text {sl }}=1 \mathrm{~m}\) from the sample, and views radiation originating from a portion of the ceramic surface having an area of \(\Delta A_{c}=10^{-4} \mathrm{~m}^{2}\). An electric heater attached to the bottom of the sample dissipates a uniform heat flux, \(q_{b}^{\prime \prime}\), which is transferred upward through the sample. The bottom of the heater and sides of the sample are well insulated. Consider conditions for which a ceramic coating of thickness \(L_{c}=0.5 \mathrm{~mm}\) and thermal conductivity \(k_{c}=\) \(6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) has been sprayed on a metal substrate of thickness \(L_{s}=8 \mathrm{~mm}\) and thermal conductivity \(k_{s}=\) \(25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The opaque surface of the ceramic may be approximated as diffuse and gray, with a total, hemispherical emissivity of \(\varepsilon_{c}=0.8\). (a) Consider steady-state conditions for which the bottom surface of the substrate is maintained at \(T_{1}=1500 \mathrm{~K}\), while the chamber walls (including the surface of the radiation detector) are maintained at \(T_{w}=90 \mathrm{~K}\). Assuming negligible thermal contact resistance at the ceramic- substrate interface, determine the ceramic top surface temperature \(T_{2}\) and the heat flux \(q_{b}^{\prime \prime}\). (b) For the prescribed conditions, what is the rate at which radiation emitted by the ceramic is intercepted by the detector?

Step by step solution

01

Heat Transfer through Ceramic Coating

To find the heat transfer through the ceramic coating, we can use Fourier's Law applied to a one-dimensional, steady conduction as follows: \(q_c = k_c\frac{T_1-T_2}{L_c}\) We will use this equation later to link the ceramic coating and the metal substrate.

02

Heat Transfer through Metal Substrate

Similarly, we will find the heat transfer through the metal substrate using Fourier's Law: \(q_s = k_s\frac{T_2-T_w}{L_s}\)

03

Equate the Heat Transfers

Since the heat transfer through the ceramic coating must equal the heat transfer through the metal substrate, we can equate the two equations we derived earlier: \(k_c\frac{T_1-T_2}{L_c} = k_s\frac{T_2-T_w}{L_s}\) We can now solve this equation for T2: \(T_2 = \frac{k_c L_s T_1 + k_s L_c T_w}{k_c L_s + k_s L_c}\) Substitute the given values: - \(k_c = 6 \text{ W/mK}\) - \(L_c = 0.5 \times 10^{-3} \text{ m}\) - \(k_s = 25 \text{ W/mK}\) - \(L_s = 8 \times 10^{-3} \text{ m}\) - \(T_1 = 1500 \text{ K}\) - \(T_w = 90 \text{ K}\) to get the value for \(T_2\): \(T_2 = 663.87 \text{ K}\)

04

Calculate the Heat Flux\(q_{b}''\)

Now to find the heat flux, we can substitute T2 in any of the two heat transfer equations from step 1 or step 2. Using the ceramic heat transfer equation: \(q_{b}'' = q_c = k_c\frac{T_1-T_2}{L_c}\) \(q_{b}'' = 6\frac{1500-663.87}{0.5 \times 10^{-3}}\) \(q_{b}'' = 10,054.4 \text{ W/m}^2\) So the heat flux is \(10,054.4 \text{ W/m}^2\).

05

Radiation Emitted by the Ceramic

For part (b), we will use the following relationship for the rate at which radiation is emitted: \(q_{r} = \varepsilon_{c} \sigma (T_2^4 - T_w^4)\) where: - σ is the Stefan–Boltzmann constant (\(5.67 \times 10^{-8} \text{W/m}^2\text{K}^4\)) - ε is the emissivity of the ceramic (\(0.8\)) Substitute the values to find the radiation emitted by the ceramic: \(q_{r} = 0.8 \times 5.67 \times 10^{-8} (663.87^4 - 90^4)\) \(q_{r} = 975.52 \text{ W/m}^2\)

06

Radiation Intercepted by Detector

Now we can calculate the radiation emitted by the ceramic intercepted by the detector using the following relationship: \(q_{d} = q_r \frac{A_d}{\Delta A_c}\) Substitute the given values and calculated radiation emitted: - \(A_d = 10^{-5}\text{ m}^2\) - \(\Delta A_c = 10^{-4}\text{ m}^2\) - \(q_r = 975.52 \text{ W/m}^2\) to get the value for \(q_d\): \(q_{d} = 975.52 \frac{10^{-5}}{10^{-4}}\) \(q_{d} = 97.55 \text{ W}\) So the rate at which radiation is intercepted by the detector is \(97.55 \text{ W}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ceramic Coating

A ceramic coating is a thin layer of ceramic material applied to a substrate, typically metal. In the context of gas turbine blades, this coating serves as a protective layer to handle high temperatures. Ceramic materials are chosen for their ability to withstand extreme heat while also providing thermal insulation.

The ceramic coating discussed here is applied to turbine blades made from a superalloy such as Inconel. This helps the blade endure high operational temperatures by reducing the thermal load on the metallic substrate beneath the ceramic layer.

Ceramic coatings have several benefits:

• Heat Resistance: Ceramics can tolerate high temperatures much better than metals, helping protect the underlying alloy.
• Thermal Barrier: By acting as a thermal barrier, ceramics reduce the temperature experienced by the blade, enhancing its lifespan.
• Weight Efficiency: Since they are thin, ceramic coatings add minimal weight to the blade.

Overall, the efficacy of a ceramic coating is heavily influenced by its thermal conductivity, which determines how effectively it can insulate the underlying metal substrate from heat.

Radiation Detector

In the setup for this exercise, a radiation detector plays a crucial role in measuring the effectiveness of the ceramic coating. The detector is placed in a vacuum chamber and is responsible for capturing the radiation emitted by the ceramic surface.

The placement and surface area of the radiation detector are critical for accurate readings. It captures the radiation originating from the surface area of the ceramic, allowing for the determination of heat transfer efficiency and coating performance.

• Location: Positioned at the top of the vacuum chamber, the radiation detector measures the radiation intercepted within its area.
• Purpose: Its function is to measure the radiant energy that emits from the ceramic surface and reaches the detector.
• Calculation Aid: Helps in calculating the rate of radiation, which involves applying concepts of emissivity and the Stefan-Boltzmann law.

The radiation detector's role in assessing radiative heat transfer provides insights into the thermal performance of the coating and can help improve thermal efficiency strategies.

Thermal Conductivity

Thermal conductivity is a measure of a material's ability to conduct heat. This property is vital when considering the application of ceramic coatings in high-temperature environments like turbines.

In our exercise, two different materials are examined for their thermal conductivity: ceramic and metal substrates. The ceramic has a thermal conductivity of 6 W/m·K, while the metal substrate has a higher conductivity of 25 W/m·K. This difference impacts how heat flows through the multilayered structure of coating and substrate.

• Importance: High thermal conductivity in metals means heat flows quickly, whereas lower values in ceramics slow the heat transfer, providing insulation.
• Calculations: By applying Fourier's Law, we can calculate the heat transfer through both materials, understanding how each layer affects the thermal flow.
• Balancing Act: In applications like turbine blades, it's crucial to balance the thermal conductivity to ensure that the metal remains protected by the ceramic layer without excessive heat transfer.

Understanding thermal conductivity is fundamental to designing efficient ceramic coatings, ensuring that they can provide the desired thermal insulation while maintaining structural integrity under operational stresses.