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Chapter 4: Problem 3

Give examples of radiative heat transfer processes dominated by surface- tosurface transfer and of processes for which the effects of absorption and emission in the space between surfaces (participating medium) are essential.

Short Answer

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Question: Identify and describe examples of two types of radiative heat transfer processes: one dominated by surface-to-surface transfer and another where absorption and emission in the space between surfaces play a significant role. Answer: An example of surface-to-surface radiative heat transfer is heat transfer between two parallel plates with a vacuum between them, such as in a vacuum thermos. In this case, heat transfer is solely due to the exchange of radiation between the surfaces of the plates. On the other hand, an example where absorption and emission between surfaces play a crucial role is the Earth's atmosphere. Here, the participating medium consists of various gases, water vapor, and aerosols. The atmosphere absorbs and emits solar radiation and longwave radiation originating from the Earth's surface, which is essential in understanding the greenhouse effect.

Step by step solution

01

Example of Surface-to-Surface Transfer

An example of surface-to-surface radiative heat transfer would be heat transfer between two parallel plates with a vacuum between them. In this case, the particles in the vacuum do not participate in the heat transfer process, and the heat transfer is solely due to the exchange of radiation between the surfaces of the plates. This type of heat transfer is typically seen in a vacuum thermos, where the space between the double walls is evacuated to minimize heat transfer by conduction and convection. The shiny inner surfaces of the thermos also minimize the heat transfer by reflecting the radiations.
02

Example of Participating Medium Transfer

An example of radiative heat transfer where absorption and emission in the space between surfaces play a significant role is in the Earth's atmosphere. In this case, the participating medium consists of various gases, water vapor, and aerosols. The incoming solar radiation is partly absorbed and partly scattered by the atmosphere. The atmosphere also absorbs and emits longwave radiation that originates from the Earth's surface. This participating medium heat transfer is essential in understanding the greenhouse effect, where certain gases (like carbon dioxide and water vapor) absorb outgoing longwave radiation, causing the Earth's atmosphere to warm up.

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Most popular questions from this chapter

Equation (4.8) is valid when the heat conductivity is independent on the direction. How would a generalization look like for the case of heat conductivity being different in the three coordinate directions? Which physical properties, e.g., of biomass material, could cause such difference?

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Consider a layer of biomass of thickness \(2 \mathrm{~L}\), with moisture removal at both sides. In a simple modeling approach, the drying process is assumed to be mass transfer limited and to proceed at constant or slowly changing temperature. Assuming that the sample does not shrink during drying, moisture removal can be expressed using Fick's law for unsteady diffusion of moisture in the direction orthogonal to the layer. Experimentally, it was found that the effective moisture diffusivity \(D_{e f f}\) is temperature dependent and has the form of an Arrhenius relationship: \(D_{e f f}=D_{0} \exp \left(-\frac{\mathrm{E}_{\mathrm{a}}}{\mathrm{R}_{\mathrm{u}} \mathrm{T}}\right)\), depending on pre-exponential factor \(D_{0}\) and the activation energy \(\mathrm{E}_{\mathrm{a}}\) (Gebreegziabher et al., 2013). Let \(X(x, \mathrm{t})\) denote the sample moisture concentration as a function of position (distance from the symmetry plane) and time. Assume that the initial value inside the layer is \(X(x, 0)=X_{0}\) and outside the layer is \(X(x, 0)=X_{e}\). The normalized moisture concentration is defined as \(9=\frac{X-X_{e}}{X_{0}-X_{e}}\) a. Write an evolution equation for \(\theta(x, \mathrm{t})\) containing a transient term and a diffusion term according to Fick's law. b. Assuming that the temperature remains constant during drying, solve the equation by the method of separation of variables to obtain the solution $$ \vartheta(x, \mathrm{t})=\sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2 n+1) \pi} \exp \left(-\frac{(2 n+1)^{2} \pi^{2} D_{e f f}}{4 \mathrm{~L}^{2}} \mathrm{t}\right) \cos \left(\left(n+\frac{1}{2}\right) \pi \frac{x}{\mathrm{~L}}\right) $$ Hint: The problem is mathematically similar to the problem of diffusion of heat from a slab considered, e.g., in Mills (1999). The factor \(\left(D_{e f f} t / L^{2}\right)\) appearing in the exponential function is the analogue of the Fourier number Fo appearing in Table \(4.1\) and is the dimensionless time of the process. c. Determine the surface moisture flux at \(x=L\). d. Identify accurate approximate solutions for the case of very long drying time and very short drying time. e. Describe a procedure to determine the pre-exponential factor and the activation energy from experiments.
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