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Chapter 13: Problem 176

Consider an enclosure consisting of \(N\) diffuse and gray surfaces. The emissivity and temperature of each surface as well as the view factors between the surfaces are specified. Write a program to determine the net rate of radiation heat transfer for each surface.

Short Answer

Expert verified
Based on the step by step solution provided, here is a short answer question: Question: Briefly explain the process of calculating the net rate of radiation heat transfer for each surface in an N-surface enclosure. Answer: To calculate the net rate of radiation heat transfer for each surface, follow these steps: 1) Understand radiosity and irradiation concepts. 2) Use the Radiation Heat Transfer Equation for gray diffuse surfaces to link radiosity and irradiation. 3) Define the view factors between surfaces and create a linear system of equations to determine the radiosity for each surface. 4) Implement an algorithm to solve the linear system of equations for radiosity values. 5) Calculate the net rate of radiation heat transfer using the difference between radiosity and irradiation multiplied by the surface area.

Step by step solution

01

Understand radiosity and irradiation.

Radiosity, J, represents the total rate of radiation energy leaving a surface per unit area. It is composed of the emitted radiation and the reflected radiation. Similarly, irradiation, G, is the total rate of radiation energy incident on a surface per unit area. It describes how much radiation energy is received by the surface from all the other surfaces.
02

Radiation heat transfer equation for gray diffuse surfaces.

For gray diffuse surfaces, we can use the following formula linking radiosity (J) and irradiation (G): \(J_i = \epsilon_i \sigma T_i^4 + (1 - \epsilon_i)G_i\), where \(i\) is the surface index, \(\epsilon_i\) is the emissivity of surface \(i\), \(\sigma\) is the Stefan-Boltzmann constant, and \(T_i\) is the surface temperature.
03

Define the view factors and create the linear system of equations.

For an N-surface enclosure, the irradiation on each surface depends on the view factors and radiosity of all the other surfaces. The view factor, \(F_{ij}\), represents the fraction of the energy leaving surface \(i\) that strikes surface \(j\). The irradiation on surface \(i\) can be expressed as: \(G_i = \sum_{j=1}^N F_{ij} J_j\) Using this equation, we can create a system of linear equations to determine the radiosity for each surface.
04

Implement an algorithm to solve the linear system of equations.

To solve the system of linear equations, we use the iterative method or any scientific programming language to perform the matrix inversion and multiplication, resolving the values \(J_i\) for each surface.
05

Calculate the net rate of radiation heat transfer for each surface.

Once we have obtained the radiosity for each surface, we can calculate the net rate of radiation heat transfer (q) for each surface with the following equation: \(q_i = A_i (J_i - G_i)\) where \(A_i\) is the area of surface \(i\). The net rate of radiation heat transfer for each surface \(q_i\) represents the heat exchanged by the surface due to radiative mechanisms.

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Most popular questions from this chapter

The base surface of a cubical furnace with a side length of \(3 \mathrm{~m}\) has an emissivity of \(0.80\) and is maintained at \(500 \mathrm{~K}\). If the top and side surfaces also have an emissivity of \(0.80\) and are maintained at $900 \mathrm{~K}$, the net rate of radiation heat transfer from the top and side surfaces to the bottom surface is (a) \(194 \mathrm{~kW}\) (b) \(233 \mathrm{~kW}\) (c) \(288 \mathrm{~kW}\) (d) \(312 \mathrm{~kW}\) (e) \(242 \mathrm{~kW}\)

Consider an infinitely long three-sided triangular enclosure with side lengths \(5 \mathrm{~cm}, 3 \mathrm{~cm}\), and \(4 \mathrm{~cm}\). The view factor from the \(5-\mathrm{cm}\) side to the \(4-\mathrm{cm}\) side is (a) \(0.3\) (b) \(0.4\) (c) \(0.5\) (d) \(0.6\) (e) \(0.7\)

A large number of long tubes, each of diameter \(D\), are placed parallel to each other and at a center-to-center distance of s. Since all of the tubes are geometrically similar and at the same temperature, these could be treated collectively as one surface \(\left(A_{j}\right)\) for radiation heat transfer calculations. As shown in Fig. P13-140, the tube bank \(\left(A_{j}\right)\) is placed opposite a large flat wall \(\left(A_{j}\right)\) such that the tube bank is parallel to the wall. (a) Calculate the view factors \(F_{i j}\) and \(F_{j i}\) for $s=3.0 \mathrm{~cm}\( and \)D=1.5 \mathrm{~cm}$. (b) Calculate the net rate of radiation heat transfer between the wall and the tube bank per unit area of the wall when $T_{i}=900^{\circ} \mathrm{C}, T_{j}=60^{\circ} \mathrm{C}, \varepsilon_{i}=0.8\(, and \)\varepsilon_{j}=0.9$. (c) A fluid flows through the tubes at an average temperature of $40^{\circ} \mathrm{C}\(, resulting in a heat transfer coefficient of \)2.0 \mathrm{~kW} / \mathrm{m}^{2} \cdot \mathrm{K}\(. Assuming \)T_{i}=900^{\circ} \mathrm{C}, \varepsilon_{i}=0.8\( and \)\varepsilon_{j}=0.9$ (as above) and neglecting the tube wall thickness and convection from the outer surface, calculate the temperature of the tube surface in steady operation.

Consider a \(10-\mathrm{ft} \times 10-\mathrm{ft} \times 10-\mathrm{ft}\) cubical furnace whose top and side surfaces closely approximate black surfaces and whose base surface has an emissivity \(\varepsilon=0.4\). The base, top, and side surfaces of the furnace are maintained at uniform temperatures of $800 \mathrm{R}, 1600 \mathrm{R}\(, and \)2400 \mathrm{R}$, respectively. Determine the net rate of radiation heat transfer between \((a)\) the base and the side surfaces and \((b)\) the base and the top surfaces. Also, determine the net rate of radiation heat transfer to the base surface.

A 5-m-diameter spherical furnace contains a mixture of \(\mathrm{CO}_{2}\) and \(\mathrm{N}_{2}\) gases at \(1200 \mathrm{~K}\) and \(1 \mathrm{~atm}\). The mole fraction of \(\mathrm{CO}_{2}\) in the mixture is \(0.15\). If the furnace wall is black and its temperature is to be maintained at \(600 \mathrm{~K}\), determine the net rate of radiation heat transfer between the gas mixture and the furnace walls.
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