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Chapter 12: Problem 24

An enclosure has an inside area of \(100 \mathrm{~m}^{2}\), and its inside surface is black and is maintained at a constant temperature. A small opening in the enclosure has an area of \(0.02 \mathrm{~m}^{2}\). The radiant power emitted from this opening is \(70 \mathrm{~W}\). What is the temperature of the interior enclosure wall? If the interior surface is maintained at this temperature, but is now polished, what will be the value of the radiant power emitted from the opening?

Short Answer

Expert verified
The temperature of the interior enclosure wall is approximately \(349.5 \mathrm{K}\). When the interior surface is polished, the radiant power emitted will be less than \(70 \mathrm{W}\), but the exact value cannot be determined without knowing the emissivity of the polished surface.

Step by step solution

01

Identify the applicable laws

We will use the Stephan-Boltzmann law and the concept of emissivity in this exercise. The Stephan-Boltzmann law states that the power radiation from an object is proportional to the fourth power of its temperature: \[P = A \cdot \epsilon \cdot \sigma \cdot T^4\] where: - \(P\) = Radiant power emitted (W) - \(A\) = Surface area of the object (m^2) - \(\epsilon\) = Emissivity of the object (dimensionless) - \(\sigma\) = Stephan-Boltzmann constant = \(5.67 脳 10^{-8} \mathrm{W/m^2 \cdot K^4}\) - \(T\) = Temperature of the object (K)
02

Calculate the temperature of the interior enclosure wall

Since the interior surface of the enclosure is black, its emissivity (\(\epsilon\)) is equal to 1. We can now plug values into the Stephan-Boltzmann law to find the temperature of the wall. Given: - Radiant power emitted from the opening: \(P = 70 \mathrm{W}\) - Area of the opening: \(A = 0.02 \mathrm{m^2}\) - Emissivity of black interior surface: \(\epsilon = 1\) \[70 = 0.02 \cdot 1 \cdot (5.67 脳 10^{-8}) \cdot T^4\] Now, let's solve for the temperature (T): \[T^4 = \dfrac{70}{0.02 \cdot 1 \cdot (5.67 脳 10^{-8})}\] \[T^4 = \dfrac{70}{1.134 脳 10^{-9}}\] \[T^4 = 6.171 脳 10^{10}\] \[T = (6.171 脳 10^{10})^{1/4}\] \[T = 349.5 \mathrm{K}\] So, the temperature of the interior enclosure wall is approximately \(349.5 \mathrm{K}\).
03

Calculate the radiant power emitted when the wall is polished

Now, we need to find the radiant power when the interior surface is polished. We are not given the emissivity of the polished surface, but we can assume it to be lower than 1 (since a polished surface would emit less radiation than a black body). Let's use \(\epsilon_p\) as its emissivity. Using the same equation as before (with the new emissivity): \[P_p = A \cdot \epsilon_p \cdot \sigma \cdot T^4\] Now, we can use the fact that the wall temperature will remain the same and that the emissivity of the polished surface will be smaller than that of the black surface to give us an inequality: \[P_p < A \cdot 1 \cdot \sigma \cdot T^4\] Substituting the known values: \[P_p < 0.02 \cdot 1 \cdot (5.67 脳 10^{-8}) \cdot (349.5)^4\] \[P_p < 70 \mathrm{W}\] Since the exact emissivity of the polished surface is not given, we cannot determine the exact value of the radiant power emitted when the interior surface is polished. However, we can conclude that it will be less than the original radiant power of \(70 \mathrm{W}\).

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Key Concepts

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

Emissivity
Emissivity describes how well an object emits thermal radiation compared to a perfect blackbody, which has an emissivity of 1. It is a dimensionless value ranging between 0 and 1. A higher emissivity means the object radiates energy more efficiently. For example, a black surface often has an emissivity close to 1, meaning it can emit radiation effectively. In our exercise, the enclosure's interior is a black surface, so it has an emissivity of 1. As such, it serves as an almost perfect emitter of thermal radiation.
When a surface is polished, its emissivity decreases. This is because polished surfaces reflect more radiation and emit less. In the problem, when the interior wall becomes polished, its emissivity drops below 1, leading to reduced radiant power emission. This concept is crucial in understanding how different materials and surface finishes affect the thermal radiation of objects.
Radiant Power
Radiant power refers to the total energy emitted by an object as thermal radiation every second, measured in watts (W). This is a vital concept as it indicates the rate at which energy leaves a surface. The Stefan-Boltzmann Law allows us to calculate this power based on the object's temperature, area, and emissivity.
For the black enclosure in the problem, the radiant power is given as 70 W through a small opening of 0.02 m虏. Using the formula \[P = A \cdot \epsilon \cdot \sigma \cdot T^4\],we can relate these terms to determine the temperature of the enclosure wall. By adjusting emissivity and confirming every other parameter, the radiant power varies directly, illustrating the understanding of power output changes whether the surface is black or polished.
Blackbody Radiation
Blackbody radiation is the emission of thermal radiation by a perfect blackbody, which is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle. In reality, no material achieves this perfectly, but many objects approximate blackbodies, with emissivity values close to 1.
The Stefan-Boltzmann Law, \[P = A \cdot \epsilon \cdot \sigma \cdot T^4\], is used to describe blackbody radiation quantitatively. In the problem, we initially consider the black surface inside the enclosure to behave as a blackbody, allowing the calculation of the temperature based on this ideal condition. Understanding blackbody radiation helps explain why different surfaces at the same temperature can emit varying amounts of radiation. Specifically, the transition from a black to a polished surface underscores the influence of surface properties on radiation.

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Most popular questions from this chapter

Consider Problem \(4.51 .\) (a) The students are each given a flat, fot-surface silver mirror with which they collectively irradiate the wooden ship at location B. The reflection from the mirror is specular, and the silver's reflectivity is \(0.98\). The solar irradiation of each mirror, perpendicular to the direction of the sun's rays, is \(G_{S}=1000 \mathrm{~W} / \mathrm{m}^{2}\). How many students are needed to conduct the experiment if the solar absorptivity of the wood is \(\alpha_{w}=0.80\) and the mirror is oriented at an angle of \(45^{\circ}\) from the direction of \(G_{S}\) ? (b) If the students are given second-surface mirrors that consist of a sheet of plain glass that has polished silver on its back side, how many students are needed to conduct the experiment? Hint: See Problem \(12.62 .\)

A radiation detector having a sensitive area of \(A_{d}=\) \(4 \times 10^{-6} \mathrm{~m}^{2}\) is configured to receive radiation from a target area of diameter \(D_{\mathrm{r}}=40 \mathrm{~mm}\) when located a distance of \(L_{t}=1 \mathrm{~m}\) from the target. For the experimental apparatus shown in the sketch, we wish to determine the emitted radiation from a hot sample of diameter \(D_{s}=\) \(20 \mathrm{~mm}\). The temperature of the aluminum sample is \(T_{s}=700 \mathrm{~K}\) and its emissivity is \(\varepsilon_{s}=0.1\). A ring- shaped cold shield is provided to minimize the effect of radiation from outside the sample area, but within the target area. The sample and the shield are diffuse emitters. (a) Assuming the shield is black, at what temperature, \(T_{\text {sho }}\) should the shield be maintained so that its emitted radiation is \(1 \%\) of the total radiant power received by the detector? (b) Subject to the parametric constraint that radiation emitted from the cold shield is \(0.05,1\), or \(1.5 \%\) of the total radiation received by the detector, plot the required cold shield temperature, \(T_{\text {sh }}\), as a function of the sample emissivity for \(0.05 \leq \varepsilon_{x} \leq 0.35\).

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?

Consider an opaque, gray surface whose directional absorptivity is \(0.8\) for \(0 \leq \theta \leq 60^{\circ}\) and \(0.1\) for \(\theta>60^{\circ}\). The surface is horizontal and exposed to solar irradiation comprised of direct and diffuse components. (a) What is the surface absorptivity to direct solar radiation that is incident at an angle of \(45^{\circ}\) from the normal? What is the absorptivity to diffuse irradiation? (b) Neglecting convection heat transfer between the surface and the surrounding air, what would be the equilibrium temperature of the surface if the direct and diffuse components of the irradiation were 600 and \(100 \mathrm{~W} / \mathrm{m}^{2}\), respectively? The back side of the surface is insulated.

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