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Chapter 12: Problem 46

For a prescribed wavelength \(\lambda\), measurement of the spectral intensity \(I_{\lambda,}(\lambda, T)=\varepsilon_{\lambda} I_{\lambda, b}\) of radiation emitted by a diffuse surface may be used to determine the surface temperature, if the spectral emissivity \(\varepsilon_{\mathrm{A}}\) is known, or the spectral emissivity, if the temperature is known. (a) Defining the uncertainty of the temperature determination as \(d T / T\), obtain an expression relating this uncertainty to that associated with the intensity measurement, \(d I_{\lambda} / I_{\lambda}\). For a \(10 \%\) uncertainty in the intensity measurement at \(\lambda=10 \mu \mathrm{m}\), what is the uncertainty in the temperature for \(T=500 \mathrm{~K}\) ? For \(T=1000 \mathrm{~K}\) ? (b) Defining the uncertainty of the emissivity determination as \(d \varepsilon_{\lambda} / s_{\lambda}\), obtain an expression relating this uncertainty to that associated with the intensity measurement, \(d I_{\lambda} / I_{\lambda}\). For a \(10 \%\) uncertainty in the intensity measurement, what is the uncertainty in the emissivity?

Short Answer

Expert verified
In summary, for a given spectral intensity uncertainty of 10%, the uncertainty in temperature is 50 K for T = 500 K and 100 K for T = 1000 K. Additionally, the uncertainty in the spectral emissivity is 10%.

Step by step solution

01

Understanding the given relations

We are given the relation between the spectral intensity I位(位,T), spectral emissivity 蔚位, and the temperature T : \( I_{\lambda} (\lambda, T)= \varepsilon_{\lambda} I_{\lambda, b} \) Our goal is to find expressions for the uncertainties in temperature (dT/T) and spectral emissivity (d蔚位/蔚位) in terms of the uncertainty in the intensity measurement (dI位/I位).
02

Differentiating the equation with respect to T to find dT/T

First, we will differentiate I位(位,T) with respect to the temperature, T, while keeping 位 constant: \( \frac{dI_{\lambda}(\lambda, T)}{dT} = \varepsilon_{\lambda} \frac{dI_{\lambda, b}}{dT} \) Now, we will divide both sides of the equation by the product of I位(位,T) and T: \( \frac{1}{I_{\lambda}(\lambda, T) T} \frac{dI_{\lambda}(\lambda, T)}{dT} = \frac{\varepsilon_{\lambda}}{I_{\lambda}(\lambda, T) T} \frac{dI_{\lambda, b}}{dT} \) Next, we will multiply both sides by dT to obtain the uncertainty in temperature, dT, in terms of the uncertainties in intensity: \( \frac{dI_{\lambda}}{I_{\lambda}} \frac{dT}{T} = \frac{dI_{\lambda, b}}{I_{\lambda, b}} \frac{dT}{T} \) Thus, we get: \( \frac{dT}{T} = \frac{dI_{\lambda}}{I_{\lambda}} \)
03

Finding the uncertainty in temperature for the given values of 位, I位, and T

We know the intensity uncertainty, dI位/I位 = 0.1 (10%), and we have the following given values: 位 = 10 渭m T = 500 K and T = 1000 K Using the obtained relation in the previous step, we can find the uncertainty in temperature for the given values of 位, I位, and T: For T = 500 K: \( \frac{dT}{T} = 0.1 \) \( dT = 0.1 * 500 = 50 K \) For T = 1000 K: \( \frac{dT}{T} = 0.1 \) \( dT = 0.1 * 1000 = 100 K \)
04

Obtaining the expression for d蔚位/蔚位

We have the given expression: \( I_{\lambda} (\lambda, T)= \varepsilon_{\lambda} I_{\lambda, b} \) Divide both sides by 蔚位: \( \frac{I_{\lambda} (\lambda, T)}{\varepsilon_{\lambda}} = I_{\lambda, b} \) Now, differentiate both sides with respect to 蔚位, keeping 位 and T constant: \( \frac{d(I_{\lambda} (\lambda, T))}{d\varepsilon_{\lambda}} = \frac{dI_{\lambda, b}}{d\varepsilon_{\lambda}} \) Divide both sides by the product of I位(位,T) and 蔚位: \( \frac{1}{I_{\lambda}(\lambda, T) \varepsilon_{\lambda}} \frac{dI_{\lambda}(\lambda, T)}{d\varepsilon_{\lambda}} = \frac{1}{\varepsilon_{\lambda} I_{\lambda, b}} \frac{dI_{\lambda, b}}{d\varepsilon_{\lambda}} \) Therefore, we obtain the relation: \( \frac{d\varepsilon_{\lambda}}{\varepsilon_{\lambda}} = \frac{dI_{\lambda}}{I_{\lambda}} \)
05

Finding the uncertainty in emissivity for the given intensity uncertainty

We know that the intensity uncertainty, dI位/I位, is 0.1 (10%). Using the obtained relation in the previous step, we can find the uncertainty in the emissivity: \( \frac{d\varepsilon_{\lambda}}{\varepsilon_{\lambda}} = 0.1 \) This expression tells us that the uncertainty in the emissivity is 10%.

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Key Concepts

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

Uncertainty in Temperature Determination
The determination of temperature using spectral intensity measurement hinges on the precision of the measured values. When dealing with the uncertainty in temperature determination, denoted as \(d T / T\), it is essentially a reflection of how accurate the temperature measurement is compared to the true value.

According to the step by step solution, this uncertainty can be directly linked to the uncertainty in spectral intensity measurement \(d I_{\lambda} / I_{\lambda}\). The exercise shows, through differentiation and manipulation of the heat transfer equations, that an uncertainty in spectral intensity directly translates to an uncertainty in the temperature reading.

This concept is crucial because accurate temperature readings are vital in numerous scientific and engineering applications, such as material testing and thermal system monitoring. An error in temperature reading can lead to significant deviations from expected performance or behavior in systems. Therefore, understanding and minimizing uncertainties is key to obtaining reliable results. For instance, at two different temperatures (\(500 K\) and \(1000 K\)), a \(10\%\) intensity uncertainty leads to temperature uncertainties of \(50 K\) and \(100 K\) respectively, which are large enough to affect experimental outcomes.
Spectral Intensity Measurement
Spectral intensity measurement is the process of quantifying the amount of electromagnetic radiation at a specific wavelength emitted by an object. The spectral intensity, \(I_{\lambda}(\lambda, T)\), relates to the object's temperature \(T\) and its spectral emissivity \(\varepsilon_{\lambda}\).

The step by step solution provided highlights that the accuracy of measuring spectral intensity is pivotal for determining both temperature and emissivity of a material. An uncertainty in the intensity measurement can stem from several sources such as detector sensitivity, calibration errors, or environmental factors. As in the example, a \(10\%\) uncertainty in spectral intensity directly translates to the same percentage of uncertainty in the calculated emissivity, indicating a strong dependence of material property determination on the measurement techniques used.

Professionals must carefully calibrate their instruments and account for potential errors in their experimental design to ensure the precise determination of these properties. Awareness and control over measurement uncertainties enable better material behavior predictions under various conditions.
Differentiating Heat Transfer Equations
Differentiating heat transfer equations plays a fundamental role in understanding and solving thermal problems, as illustrated in the exercise's approach to finding relationships between uncertainties. It involves taking the derivative of heat transfer expressions with respect to a certain variable to investigate how changes in one parameter affect another.

In the given problem, the differentiation with respect to temperature and emissivity, while keeping wavelength constant, provided a way to connect the uncertainty in spectral intensity to the uncertainties in temperature and emissivity. These mathematical manipulations provide a practical approach to estimating potential errors in temperature and emissivity determinations, giving students and professionals alike a deeper insight into the sensitivity of their systems to measurement errors.

This understanding is important for developing accurate thermal models and simulations that are used in a wide array of industries, including aerospace, automotive, and electronics, where heat transfer plays a crucial role in the design and performance of different components and systems.

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Most popular questions from this chapter

Two small surfaces, \(A\) and \(B\), are placed inside an isothermal enclosure at a uniform temperature. The enclosure provides an irradiation of \(6300 \mathrm{~W} / \mathrm{m}^{2}\) to each of the surfaces, and surfaces A and B absorb incident radiation at rates of 5600 and \(630 \mathrm{~W} / \mathrm{m}^{2}\), respectively. Consider conditions after a long time has elapsed. (a) What are the net heat fluxes for each surface? What are their temperatures? (b) Determine the absorptivity of each surface. (c) What are the emissive powers of each surface? (d) Determine the emissivity of each surface.

The 50 -mm peephole of a large furnace operating at \(450^{\circ} \mathrm{C}\) is covered with a material having \(\tau=0.8\) and \(\rho=0\) for irradiation originating from the furnace. The material has an emissivity of \(0.8\) and is opaque to irradiation from a source at room temperature. The outer surface of the cover is exposed to surroundings and ambient air at \(27^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of \(50 \mathrm{~W} / \mathrm{m}^{2}=\mathrm{K}\). Assuming that convection effects on the inner surface of the cover are negligible, calculate the heat loss by the furnace and the temperature of the cover.

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 \(10 \mu \mathrm{m}\) ?

A roof-cooling system, which operates by maintaining a thin film of water on the roof surface, may be used to reduce air-conditioning costs or to maintain a cooler environment in nonconditioned buildings. To determine the effectiveness of such a system, consider a sheet metal roof for which the solar absorptivity \(\alpha_{5}\) is \(0.50\) and the hemispherical emissivity \(\varepsilon\) is \(0.3\). Representative conditions correspond to a surface convection coefficient \(h\) of \(20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), a solar irradiation \(G_{S}\) of \(700 \mathrm{~W} / \mathrm{m}^{2}\), a sky temperature of \(-10^{\circ} \mathrm{C}\), an atmospheric temperature of \(30^{\circ} \mathrm{C}\), and a relative humidity of \(65 \%\). The roof may be assumed to be well insulated from below. Determine the roof surface temperature without the water film. Assuming the film and roof surface temperatures to be equal, determine the surface temperature with the film. The solar absorptivity and the hemispherical emissivity of the film-surface combination are \(\alpha_{S}=\) \(0.8\) and \(s=0.9\), respectively.

A thermocouple whose surface is diffuse and gray with an emissivity of \(0.6\) indicates a temperature of \(180^{\circ} \mathrm{C}\) when used to measure the temperature of a gas flowing through a large duct whose walls have an emissivity of \(0.85\) and a uniform temperature of \(450^{\circ} \mathrm{C}\). (a) If the convection heat transfer coefficient between the thermocouple and the gas stream is \(h=125 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and there are negligible conduction losses from the thermocouple, determine the temperature of the gas. (b) Consider a gas temperature of \(125^{\circ} \mathrm{C}\). Compute and plot the thermocouple measurement error as a function of the convection coefficient for \(10 \leq 5\) \(h \leq 1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). What are the implications of your results?
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