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Chapter 3: Problem 211

A cylindrical pin fin of diameter \(0.6 \mathrm{~cm}\) and length of \(3 \mathrm{~cm}\) with negligible heat loss from the tip has an efficiency of 0.7. The effectiveness of this fin is (a) \(0.3\) (b) \(0.7\) (c) 2 (d) 8 (e) 14

Short Answer

Expert verified
Answer: The fin effectiveness of the given cylindrical pin fin is 0.7.

Step by step solution

01

Understand Given Information and Required Formula

We are given the following information: - Diameter \((D) = 0.6\,\text{cm} = 0.006\,\text{m}\) - Length \((L) = 3\,\text{cm} = 0.03\,\text{m}\) - Fin Efficiency \((\eta_f)= 0.7\) We're tasked to find the fin effectiveness. The formula for fin effectiveness in terms of fin efficiency, heat transfer rate from an unfinned area of the same base dimensions \((q_s)\), and actual heat transfer from the fin \((q_f)\) is as follows: Fin Effectiveness \((\varepsilon) = \frac{q_f}{q_s} = \frac{\eta_f \cdot q_s}{q_s}\)
02

Substitute Given Values into the Formula

Now, let's substitute the given values of the fin efficiency into the formula: \(\varepsilon = \frac{0.7 \cdot q_s}{q_s}\)
03

Simplify and Calculate the Fin Effectiveness

We can simplify the equation for fin effectiveness by canceling out the \(q_s\) term: \(\varepsilon = 0.7\) Fin Effectiveness \((\varepsilon) = 0.7\) The correct answer is (b) 0.7.

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Key Concepts

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

Heat Transfer
Heat transfer is a fundamental concept in thermal engineering that involves the movement of thermal energy from one place to another. This process occurs due to a temperature difference within a body or between different bodies, and it can occur through three main mechanisms: conduction, convection, and radiation.

Conduction is the transfer of heat through a solid material when there is a temperature gradient. In our example with the cylindrical pin fin, heat conducts from the base where the fin is attached to the cooler surroundings at the tip, even though in this exercise the tip is considered to have negligible heat loss.

Convection occurs when a fluid (either liquid or gas) is heated and the less dense portion rises, while the denser and cooler part sinks. Fins increase the surface area exposed to the fluid, which enhances heat transfer by convection.

Radiation is the transfer of heat in the form of electromagnetic waves without the need for a physical medium. All objects with a temperature above absolute zero (-273.15°C) emit thermal radiation.

Understanding the principles of heat transfer is crucial when analyzing the functionality and efficiency of a fin since fins are designed to maximize heat dissipation through conduction and convection.
Fin Efficiency
Fin efficiency, denoted as \(\eta_f\), is a measure of how well a fin conducts heat relative to its ideal capacity. The ideal, or maximum, heat transfer would occur if the entire fin were at the base temperature, which is highly unlikely due to the drop in temperature along the fin's length. The efficiency is thus defined as the actual heat transfer rate from the fin to the air divided by the heat transfer rate if the entire fin were at the base temperature.

An efficiency of 0.7, as in the given exercise, indicates that the fin is transferring 70% of the heat compared to the ideal case. It's also important to note that the efficiency of a fin is affected by its geometric characteristics, material properties, and the surrounding thermal environment. For instance, a fin's performance can change based on factors like its shape and size, the conductivity of the material it's made from, and the convective heat transfer coefficient of the fluid in contact with the fin.
Cylindrical Pin Fin
A cylindrical pin fin is a particular type of heat transfer enhancement device commonly used to increase the heat dissipation from a surface. These fins are shaped like rods or pins and extend out from the surface to be cooled. They work on the principle of increasing the surface area exposed to cooling air or fluid, thus facilitating greater heat transfer from the surface to the cooling medium through conduction and convection.

In the given exercise, we have a cylindrical pin fin with a diameter of 0.006 m and a length of 0.03 m. The simplicity of a cylindrical pin fin's geometry often makes it easy to manufacture and analyze, but its performance can vary depending on the fin's aspect ratio and the conditions of the cooling medium. This type of fin is particularly effective in forced convection scenarios where air or fluid is moved over the fin by mechanical means.

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Most popular questions from this chapter

Using cylindrical samples of the same material, devise an experiment to determine the thermal contact resistance. Cylindrical samples are available at any length, and the thermal conductivity of the material is known.

In a combined heat and power (CHP) generation process, by-product heat is used for domestic or industrial heating purposes. Hot steam is carried from a CHP generation plant by a tube with diameter of \(127 \mathrm{~mm}\) centered at a square crosssection solid bar made of concrete with thermal conductivity of \(1.7 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The surface temperature of the tube is constant at \(120^{\circ} \mathrm{C}\), while the square concrete bar is exposed to air with temperature of \(-5^{\circ} \mathrm{C}\) and convection heat transfer coefficient of \(20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the temperature difference between the outer surface of the square concrete bar and the ambient air is to be maintained at \(5^{\circ} \mathrm{C}\), determine the width of the square concrete bar and the rate of heat loss per meter length.

A 50 -m-long section of a steam pipe whose outer (鈧�) diameter is \(10 \mathrm{~cm}\) passes through an open space at \(15^{\circ} \mathrm{C}\). The average temperature of the outer surface of the pipe is measured to be \(150^{\circ} \mathrm{C}\). If the combined heat transfer coefficient on the outer surface of the pipe is \(20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine (a) the rate of heat loss from the steam pipe; \((b)\) the annual cost of this energy lost if steam is generated in a natural gas furnace that has an efficiency of 75 percent and the price of natural gas is $$\$ 0.52 /$$ therm ( 1 therm \(=105,500 \mathrm{~kJ})\); and \((c)\) the thickness of fiberglass insulation \((k=0.035 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) needed in order to save 90 percent of the heat lost. Assume the pipe temperature to remain constant at \(150^{\circ} \mathrm{C}\).

A hot surface at \(80^{\circ} \mathrm{C}\) in air at \(20^{\circ} \mathrm{C}\) is to be cooled by attaching 10 -cm-long and 1 -cm-diameter cylindrical fins. The combined heat transfer coefficient is \(30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and heat transfer from the fin tip is negligible. If the fin efficiency is \(0.75\), the rate of heat loss from 100 fins is (a) \(325 \mathrm{~W}\) (b) \(707 \mathrm{~W}\) (c) \(566 \mathrm{~W}\) (d) \(424 \mathrm{~W}\) (e) \(754 \mathrm{~W}\)

Hot water at an average temperature of \(70^{\circ} \mathrm{C}\) is flowing through a \(15-\mathrm{m}\) section of a cast iron pipe \((k=52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) whose inner and outer diameters are \(4 \mathrm{~cm}\) and \(4.6 \mathrm{~cm}\), respectively. The outer surface of the pipe, whose emissivity is \(0.7\), is exposed to the cold air at \(10^{\circ} \mathrm{C}\) in the basement, with a heat transfer coefficient of \(15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The heat transfer coefficient at the inner surface of the pipe is \(120 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Taking the walls of the basement to be at \(10^{\circ} \mathrm{C}\) also, determine the rate of heat loss from the hot water. Also, determine the average velocity of the water in the pipe if the temperature of the water drops by \(3^{\circ} \mathrm{C}\) as it passes through the basement.
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