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Chapter 12: Problem 94

A manufacturing process involves heating long copper rods, which are coated with a thin film, in a large furnace whose walls are maintained at an elevated temperature \(T_{w}\). The furnace contains quiescent nitrogen gas at 1-atm pressure and a temperature of \(T_{x}=T_{w}\). The film is a diffuse surface with a spectral emissivity of \(\varepsilon_{\lambda}=0.9\) for \(\lambda \leq 2 \mu \mathrm{m}\) and \(\varepsilon_{\lambda}=0.4\) for \(\lambda>2 \mu \mathrm{m}\). (a) Consider conditions for which a rod of diameter \(D\) and initial temperature \(T_{i}\) is inserted in the furnace, such that its axis is horizontal. Assuming validity of the lumped capacitance approximation, derive an equation that could be used to determine the rate of change of the rod temperature at the time of insertion. Express your result in terms of appropriate variables. (b) If \(T_{w}=T_{\mathrm{o}}=1500 \mathrm{~K}, T_{i}=300 \mathrm{~K}\), and \(D=10 \mathrm{~mm}\), what is the initial rate of change of the rod temperature? Confirm the validity of the lumped capacitance approximation. (c) Compute and plot the variation of the rod temperature with time during the heating process.

Short Answer

Expert verified
In this problem, we analyze the heating process of a copper rod coated with a thin film in a furnace. Using the lumped capacitance approximation, we derived the equation for the rate of change of the rod's temperature: \( \frac{dT}{dt} = \frac{A_{r} \sigma \varepsilon_{r}}{mc} ( \int_{0}^{\infty} E_{w\lambda} d\lambda - \int_{0}^{\infty} E_{r\lambda} d\lambda ) \) For given values of \(T_w = 1500K, T_i = 300K, D = 10mm\), we found the initial rate of change of the rod temperature and confirmed the validity of the lumped capacitance approximation using the Biot number. Finally, we computed and plotted the variation of the rod temperature with time during the heating process by solving the derived equation, giving us insight into the behavior of the heating process for the copper rod.

Step by step solution

01

Problem Setup: Governing conservation of energy equation

We need to find the rate of change of the rod's temperature as it is inserted into the furnace. This involves conservation of energy applied to the rod and its interaction with the surrounding environment. The governing equation for this process is given by the heat energy balance on the rod: \( \frac{dQ}{dt} =mc\frac{dT}{dt}= Q_{in} - Q_{out} \) where \(m\) is the mass of the rod, \(c\) is the specific heat of the rod, and \(T\) is the temperature of the rod at a given time. \(Q_{in}\) and \(Q_{out}\) represents the heat energy input and output of the system, respectively.
02

Find the heat energy input and output associated with radiation

The heat energy input comes from the surrounding walls at temperature Tw and the heat energy output comes from the rod. We can use the Stefan-Boltzmann law to find the radiative heat exchange between the rod and the walls: \( Q_{in} = A_{r} \sigma \varepsilon_{r} \int_{0}^{\infty} E_{w\lambda} d\lambda \) \( Q_{out} = A_{r} \sigma \varepsilon_{r} \int_{0}^{\infty} E_{r\lambda} d\lambda \) where \(A_r\) is the surface area of the rod, \(\sigma\) is the Stefan-Boltzmann constant, \(\varepsilon_r\) is the emissivity of the rod, \(E_{w\lambda}\) and \(E_{r\lambda}\) are the spectral blackbody emissive power of the walls and the rod, respectively.
03

Derive the equation for the lumped capacitance approximation

Since we are assuming the lumped capacitance approximation, we can simplify the equation as: \( mc\frac{dT}{dt} = A_{r} \sigma \varepsilon_{r} ( \int_{0}^{\infty} E_{w\lambda} d\lambda - \int_{0}^{\infty} E_{r\lambda} d\lambda ) \) Note that we need to find \(\int_{0}^{\infty} E_{w\lambda} d\lambda\) and \(\int_{0}^{\infty} E_{r\lambda} d\lambda\) using the spectral emissivity given.
04

Evaluate (a)

Now we have the equation to find the rate of change of the rod's temperature at the time of insertion using the lumped capacitance approximation: \( \frac{dT}{dt} = \frac{A_{r} \sigma \varepsilon_{r}}{mc} ( \int_{0}^{\infty} E_{w\lambda} d\lambda - \int_{0}^{\infty} E_{r\lambda} d\lambda ) \)
05

Evaluate (b)

Given \(T_w = T_o = 1500K, T_i = 300K, D = 10mm\), first find the initial rate of change of the rod temperature by plugging in these values to the equation derived in Step 3. Next, check the validity of the lumped capacitance approximation using the Biot number. The Biot number is defined as: \(Bi = \frac{hD}{k}\) where \(h\) is the convective heat transfer coefficient and \(k\) is the thermal conductivity of the material. If \(Bi << 1\), then the lumped capacitance approximation is valid.
06

Evaluate (c)

Compute and plot the variation of the rod temperature with time during the heating process by solving the equation derived in Step 3. To do this, it's necessary to evaluate the integral terms and solve the resulting ordinary differential equation. This will give the variation of temperature with time. Finally, plot the rod temperature as a function of time to visualize the heating process. Upon following these steps, you should now have a better understanding of the heating process for a copper rod coated with a thin film in a furnace, using the lumped capacitance approximation.

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Key Concepts

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

Lumped Capacitance Approximation
Understanding the lumped capacitance approximation is key for students grappling with problems in heat transfer, particularly when dealing with temperature changes of objects in response to their environments. This method simplifies complex heat transfer situations by assuming that the temperature within a solid varies little and can be considered uniform at any given point in time. In essence, it portrays the object as a 'lump' with a capacitance equivalent to the thermal capacity of the entire object.

The utility of the lumped capacitance method hinges on certain conditions, primarily being that the Biot number, which is a dimensionless quantity denoting the ratio of internal conduction resistance to external convection resistance, must be much smaller than one (\(Bi << 1\)). In the case of the copper rod heated in the furnace in the given problem, ensuring that the Biot number is sufficiently small confirms the validity of using this approximation. Once this criterion is met, the problem becomes much easier to solve as it circumnavigates the complexities of detailed internal temperature distributions.
Stefan-Boltzmann Law
Radiative heat transfer is a fundamental concept in thermal physics, and the Stefan-Boltzmann law provides a mathematical basis to calculate the heat emitted by blackbody radiation. According to this law, the power radiated per unit area of a blackbody is directly proportional to the fourth power of the blackbody's absolute temperature. Mathematically, this is expressed as:

\( P = \text{e} \times \text{A} \times \text{蟽} \times T^4 \) where:
  • \text{P} is the total power radiated per unit area,
  • \text{e} is the emissivity of the material,
  • \text{A} is the surface area,
  • \text{蟽} is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} W/m^2K^4\)), and
  • \text{T} is the absolute temperature in kelvins.

In the context of the exercise, the Stefan-Boltzmann law is used to derive an equation that quantifies the heat exchange due to radiation between the furnace walls and the copper rod. These concepts are indispensable for students aiming to manage real-world problems related to heat transfer through radiation.
Radiative Heat Exchange
Radiative heat exchange is a mode of heat transfer that occurs through electromagnetic waves and does not require any medium to take place. It is the dominant form of heat transfer in space, especially between the sun and the earth. In engineering contexts, understanding and calculating radiative heat exchange is crucial for designing systems that have to manage thermal energy efficiently.

In our exercise, radiative heat exchange must consider the spectral emissivity of the copper rod's film coating, given for two separate wavelengths. The net heat exchange is the difference between the incoming and outgoing radiative heat energy. To tackle these types of problems, students must employ the concept of spectral blackbody emissive power which informs us about the distribution of energy emitted at different wavelengths for a given temperature. Importantly, upon insertion of the copper rod into the furnace, the heat exchange formula derived from the Stefan-Boltzmann law can predict the initial rate at which the rod's temperature will begin to rise. Integrating these calculations over time can then provide a curve that represents the temperature variation of the rod, an essential piece of the manufacturing process analysis in the given scenario.

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Most popular questions from this chapter

Consider an opaque horizontal plate that is well insulated on its back side. The irradiation on the plate is \(2500 \mathrm{~W} / \mathrm{m}^{2}\), of which \(500 \mathrm{~W} / \mathrm{m}^{2}\) is reflected. The plate is at \(227^{\circ} \mathrm{C}\) and has an emissive power of \(1200 \mathrm{~W} / \mathrm{m}^{2}\). Air at \(127^{\circ} \mathrm{C}\) flows over the plate with a heat transfer convection coefficient of \(15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine the emissivity, absorptivity, and radiosity of the plate. What is the net heat transfer rate per unit area?

A thermograph is a device responding to the radiant power from the scene, which reaches its radiation detector within the spectral region 9-12 \(\mu \mathrm{m}\). The thermograph provides an image of the scene, such as the side of a furnace, from which the surface temperature can be determined. (a) For a black surface at \(60^{\circ} \mathrm{C}\), determine the emissive power for the spectral region \(9-12 \mu \mathrm{m}\). (b) Calculate the radiant power (W) received by the thermograph in the same range \((9-12 \mu \mathrm{m})\) when viewing, in a normal direction, a small black wall area, \(200 \mathrm{~mm}^{2}\), at \(T_{s}=60^{\circ} \mathrm{C}\). The solid angle \(\omega\) subtended by the aperture of the thermograph when viewed from the target is \(0.001 \mathrm{sr}\). (c) Determine the radiant power \((\mathrm{W})\) received by the thermograph for the same wall area \(\left(200 \mathrm{~mm}^{2}\right)\) and solid angle \((0.001 \mathrm{sr})\) when the wall is a gray, opaque, diffuse material at \(T_{x}=60^{\circ} \mathrm{C}\) with emissivity \(0.7\) and the surroundings are black at \(T_{\text {sur }}=23^{\circ} \mathrm{C}\).

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?

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?

The oxidized-aluminum wing of an aircraft has a chord length of \(L_{c}=4 \mathrm{~m}\) and a spectral, hemispherical emissivity characterized by the following distribution. (a) Consider conditions for which the plane is on the ground where the air temperature is \(27^{\circ} \mathrm{C}\), the solar irradiation is \(800 \mathrm{~W} / \mathrm{m}^{2}\), and the effective sky temperature is \(270 \mathrm{~K}\). If the air is quiescent, what is the temperature of the top surface of the wing? The wing may be approximated as a horizontal, flat plate. (b) When the aircraft is flying at an elevation of approximately \(9000 \mathrm{~m}\) and a speed of \(200 \mathrm{~m} / \mathrm{s}\), the air temperature, solar irradiation, and effective sky temperature are \(-40^{\circ} \mathrm{C}, 1100 \mathrm{~W} / \mathrm{m}^{2}\), and \(235 \mathrm{~K}\), respectively. What is the temperature of the wing's top surface? The properties of the air may be approximated as \(\rho=0.470 \mathrm{~kg} / \mathrm{m}^{3}, \mu=1.50 \times\) \(10^{-5} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}, k=0.021 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(P r=0.72\).
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