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Chapter 3: Problem 225

Two finned surfaces with long fins are identical, except that the convection heat transfer coefficient for the first finned surface is twice that of the second one. Which of the following conditions is accurate for the efficiency and effectiveness of the first finned surface relative to the second one? (a) Higher efficiency and higher effectiveness (b) Higher efficiency but lower effectiveness (c) Lower efficiency but higher effectiveness (d) Lower efficiency and lower effectiveness (e) Equal efficiency and equal effectiveness

Short Answer

Expert verified
Answer: (a) Higher efficiency and higher effectiveness.

Step by step solution

01

Define the Efficiency and Effectiveness of a fin

Efficiency refers to the ratio of the actual heat transfer rate from the fin to the maximum possible heat transfer rate in the absence of fin. Effectiveness refers to the ratio of the actual heat transfer rate from the fin to the heat transfer rate from the base without fin. Using these definitions, we'll analyze which condition holds true for the given problem.
02

Determine the comparison criteria

Let's consider the following variables: - Efficiency of the first finned surface: η1 - Efficiency of the second finned surface: η2 - Effectiveness of the first finned surface: ε1 - Effectiveness of the second finned surface: ε2 - Convection heat transfer coefficient for first finned surface: h1 (h1 = 2 * h2) Latex formulas (please use the code mentioned below in Latex editors for correct formatting): Efficiency (η) = \\frac{tan (mL) - mL}{tan (mL)} Effectiveness (ε)= η (\\frac{P}{A_b}) Where: - mL = ml (characteristic length, non-dimensional parameter) - P = fin perimeter - A_b = unfinned base area h is included in m. However, we don't need to use these formulas directly as our focus is on the comparison of the efficiency and effectiveness.
03

Compare Efficiency

Since the fins are long and identical except h: η1 = η(h1, k, geometry) η2 = η(h2, k, geometry) η1 > η2 High convection heat transfer coefficient promotes heat transfer, thus first finned surface will have higher efficiency than the second one.
04

Compare Effectiveness

Based on the formula for effectiveness: ε1 > ε2 Increasing h, results in increased effectiveness. Therefore, the first finned surface will have higher effectiveness than the second one.
05

Find the Correct Option

Based on the analysis, the first finned surface has higher efficiency and higher effectiveness compared to the second finned surface. This corresponds to the option (a) Higher efficiency and higher effectiveness.

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Most popular questions from this chapter

A \(0.4-\mathrm{cm}\)-thick, 12 -cm-high, and \(18-\mathrm{cm}\)-long circuit board houses 80 closely spaced logic chips on one side, each dissipating $0.04 \mathrm{~W}$. The board is impregnated with copper fillings and has an effective thermal conductivity of $30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. All the heat generated in the chips is conducted across the circuit board and is dissipated from the back side of the board to a medium at \(40^{\circ} \mathrm{C}\), with a heat transfer coefficient of $52 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. (a) Determine the temperatures on the two sides of the circuit board. (b) Now a \(0.2-\mathrm{cm}\)-thick, \(12-\mathrm{cm}\)-high, and \(18-\mathrm{cm}\)-long aluminum plate $(k=237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\( with \)8642 \cdot \mathrm{cm}-$ long aluminum pin fins of diameter \(0.25 \mathrm{~cm}\) is attached to the back side of the circuit board with a \(0.02-\mathrm{cm}\)-thick epoxy adhesive \((k=1.8 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). Determine the new temperatures on the two sides of the circuit board.

A 10-m-long, 8-cm-outer-radius cylindrical steam pipe is covered with 3 -cm- thick cylindrical insulation with a thermal conductivity of $0.05 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. If the rate of heat loss from the pipe is \(1000 \mathrm{~W}\), the temperature drop across the insulation is (a) \(58^{\circ} \mathrm{C}\) (b) \(101^{\circ} \mathrm{C}\) (c) \(143^{\circ} \mathrm{C}\) (d) \(282^{\circ} \mathrm{C}\) (e) \(600^{\circ} \mathrm{C}\)

Circular fins of uniform cross section, with diameter of \(10 \mathrm{~mm}\) and length of \(50 \mathrm{~mm}\), are attached to a wall with surface temperature of \(350^{\circ} \mathrm{C}\). The fins are made of material with thermal conductivity of \(240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), they are exposed to an ambient air condition of \(25^{\circ} \mathrm{C}\), and the convection heat transfer coefficient is $250 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine the heat transfer rate and plot the temperature variation of a single fin for the following boundary conditions: (a) Infinitely long fin (b) Adiabatic fin tip (c) Fin with tip temperature of \(250^{\circ} \mathrm{C}\) (d) Convection from the fin tip

A 2.2-m-diameter spherical steel tank filled with iced water at $0^{\circ} \mathrm{C}$ is buried underground at a location where the thermal conductivity of the soil is \(k=0.55 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The distance between the tank center and the ground surface is \(2.4 \mathrm{~m}\). For a ground surface temperature of \(18^{\circ} \mathrm{C}\), determine the rate of heat transfer to the iced water in the tank. What would your answer be if the soil temperature were \(18^{\circ} \mathrm{C}\) and the ground surface were insulated?

A total of 10 rectangular aluminum fins $(k=203 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ are placed on the outside flat surface of an electronic device. Each fin is \(100 \mathrm{~mm}\) wide, \(20 \mathrm{~mm}\) high, and $4 \mathrm{~mm}$ thick. The fins are located parallel to each other at a center- to-center distance of \(8 \mathrm{~mm}\). The temperature at the outside surface of the electronic device is \(72^{\circ} \mathrm{C}\). The air is at $20^{\circ} \mathrm{C}\(, and the heat transfer coefficient is \)80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. Determine \)(a)$ the rate of heat loss from the electronic device to the surrounding air and \((b)\) the fin effectiveness.
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