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Isothermal furnaces with small apertures approximating a blackbody are frequently used to calibrate heat flux gages, radiation thermometers, and other radiometric devices. In such applications, it is necessary to control power to the furnace such that the variation of temperature and the spectral intensity of the aperture are within desired limits. (a) By considering the Planck spectral distribution, Equation \(12.30\), show that the ratio of the fractional change in the spectral intensity to the fractional change in the temperature of the furnace has the form $$ \frac{d I_{\lambda} / I_{\lambda}}{d T / T}=\frac{C_{2}}{\lambda T} \frac{1}{1-\exp \left(-C_{2} / \lambda T\right)} $$ (b) Using this relation, determine the allowable variation in temperature of the furnace operating at \(2000 \mathrm{~K}\) to ensure that the spectral intensity at \(0.65 \mu \mathrm{m}\) will not vary by more than \(0.5 \%\). What is the allowable variation at \(10 \mu \mathrm{m}\) ?

Step by step solution

01

Understand Planck's Spectral Distribution equation

Planck's Spectral Distribution equation is given by $$ I_{\lambda} = \frac{2 \pi h c^2 }{\lambda^5}\frac{1}{\exp \left(\frac{hc}{\lambda k_B T}\right) - 1} $$ Here, \(I_{\lambda}\) denotes the spectral intensity, and h, c, kB, λ, and T represent Planck's constant, the speed of light, Boltzmann constant, wavelength, and temperature, respectively.

02

Differentiate the equation with respect to λ and T

Differentiate Planck's Spectral Distribution equation with respect to λ and T. $$ \frac{d I_{\lambda}}{d T} =\frac{2 \pi h c^2 }{\lambda^5}\frac{d}{dT}\left(\frac{1}{\exp \left(\frac{hc}{\lambda k_B T}\right) - 1}\right) $$ Let \(C_2=\frac{hc}{k_B}\), then the above expression becomes $$ \frac{d I_{\lambda}}{d T} = \frac{2 \pi h c^2 }{\lambda^5}\frac{d}{dT}\left(\frac{1}{\exp \left(\frac{C_2}{\lambda T}\right) - 1}\right) $$

03

Calculate the derivative

Calculate the derivative of the above equation. $$ \frac{d I_{\lambda}}{d T} = \frac{2 \pi h c^2 }{\lambda^5}\frac{\frac{C_2}{\lambda T^2}}{\left(\exp \left(\frac{C_2}{\lambda T}\right) - 1\right)^2}\cdot\exp \left(\frac{-C_2}{\lambda T}\right) $$ Now, we can re-write the equation as: $$ \frac{d I_{\lambda}}{d T} = \frac{C_2}{\lambda T^2}\frac{I_{\lambda}^2}{1-\exp \left(-\frac{C_2}{\lambda T}\right)} $$

04

Formulate the given ratio

Formulate the ratio equation as mentioned in the problem. $$ \frac{d I_{\lambda}/I_{\lambda}}{d T/T} = \frac{C_2}{\lambda T}\frac{1}{1-\exp \left(-C_2 / \lambda T\right)} $$

05

Calculate the allowable variation in temperature for given conditions

For \(0.65 \mu m (6.5\times10^{-7}m)\) and \(2000 K\), we have: $$\frac{d I_{\lambda} / I_{\lambda}}{d T / T} = 0.005 \text{(for } 0.5 \% \text{ change)}$$ Plugging values into the ratio equation, $$0.005 =\frac{C_2}{(6.5\times10^{-7}) (2000)}\frac{1}{1-\exp \left(-\frac{C_2}{(6.5\times10^{-7})(2000)}\right)}$$ From this expression, we can find the value of \(\frac{dT}{T}\): $$\frac{dT}{T} = \left(\frac{d I_{\lambda} / I_{\lambda}}{0.005}\right)$$ Then, we can find the allowable variation in temperature, dT, by multiplying the result by T (2000 K). Now, for \(10 \mu m (10^{-5}m)\), the steps to calculate the allowable variation in temperature will be similar. Firstly, plug the given values into the ratio equation, and then compute the value of \(\frac{dT}{T}\). Finally, to find the allowable variation in temperature (dT), multiply the result by the temperature (2000 K).

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Key Concepts

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

Planck's Law

Planck's Law is a fundamental principle in the field of quantum mechanics and thermal radiation. It describes how electromagnetic radiation is emitted by a blackbody in thermal equilibrium. A blackbody is an idealized object that perfectly absorbs and emits all frequencies of radiation. Planck's Law provides the mathematical framework to understand this phenomenon.

At the heart of Planck's Law is the formula for the spectral intensity of radiation, which can be written as:

• \[ I_{\lambda} = \frac{2 \pi h c^2 }{\lambda^5}\frac{1}{\exp \left(\frac{hc}{\lambda k_B T}\right) - 1} \]

Here, each variable represents a fundamental component:

• \(I_{\lambda}\) is the spectral intensity at a wavelength \(\lambda\).
• \(h\) is Planck's constant.
• \(c\) is the speed of light.
• \(k_B\) is the Boltzmann constant.
• \(T\) is the absolute temperature of the blackbody.

This equation highlights how the intensity of radiation depends on the wavelength and the temperature of the blackbody. Understanding Planck's Law is crucial for dealing with topics in radiation heat transfer and calibrating radiometric devices.

Spectral Intensity

Spectral intensity is a measure of how much energy is emitted from a surface at a particular wavelength per unit time, area, and solid angle. It offers insights into how energy is distributed across different wavelengths.

In the context of blackbody radiation, spectral intensity is a key quantity described by Planck's Law. The function is complex but essential, allowing you to predict how a blackbody emits radiation depending on its temperature and the specific wavelength you are interested in.

• Spectral intensity can increase dramatically with temperature, indicating more energy being emitted.
• At longer wavelengths, intensity tends to decrease, resulting in fewer energy emissions.
• Changes in intensity are influenced by temperature and wavelength, demonstrating their interrelatedness.
• Engineers and scientists use spectral intensity to design systems like thermal cameras and radiometers, which require accurate thermal readings.

These properties underline the importance of mastering spectral intensity, as it impacts everything from environmental monitoring to the development of heating systems.

Temperature Variation

Temperature variation is a crucial concept in understanding and controlling blackbody radiation. It directly affects the spectral intensity of emitted radiation. When dealing with devices such as isothermal furnaces, precise temperature control is required to ensure accurate measurements and consistency.

To grasp the impact of temperature variation, consider how the spectral intensity changes when temperature shifts. Using part of the solution equation,

• \[ \frac{d I_{\lambda}/I_{\lambda}}{d T/T} = \frac{C_{2}}{\lambda T}\frac{1}{1-\exp \left(-C_{2}/\lambda T\right)} \]

You can see the need to carefully calibrate temperature. This ratio tells you the sensitivity of intensity to small changes in temperature for a given wavelength. It is pivotal in ensuring that radiometric devices function accurately.

• Small temperature variances can lead to significant spectral intensity changes.
• Managing temperature variation helps in maintaining the integrity of measurements.
• In calibration processes, keeping variations in check leads to reliable device outputs.

Understanding temperature variation is essential for applications needing high precision, such as optical measurements and scientific experiments.

Heat Flux Calibration

Heat flux calibration involves the accurate measurement and adjustment of heat transfer rates, often using blackbody sources. This process ensures that radiometric devices provide correct readings, critical in various industrial and scientific applications.

Calibration relies heavily on Planck's Law and the principles of spectral intensity and temperature control. Accurate heat flux measurements are essential when using instruments like heat flux gages and radiation thermometers.
Here are the steps and considerations in the calibration process:

• The isothermal furnace's temperature must remain stable, as fluctuations can skew results.
• Small apertures that mimic a blackbody are often used, as they provide a consistent radiation source.
• Heat flux needs to be adjusted to match the desired levels, usually requiring iterative modifications.
• Devices are calibrated over specific wavelengths, considering atmospheric conditions and material properties.

Such meticulous calibration practices ensure that the readings reflect true physical conditions, allowing for precision in subsequent analyses and applications.