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Chapter 1: Problem 141

A soldering iron has a cylindrical tip of \(2.5 \mathrm{~mm}\) in diameter and \(20 \mathrm{~mm}\) in length. With age and usage, the tip has oxidized and has an emissivity of \(0.80\). Assuming that the average convection heat transfer coefficient over the soldering iron tip is \(25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the surrounding air temperature is \(20^{\circ} \mathrm{C}\), determine the power required to maintain the tip at \(400^{\circ} \mathrm{C}\).

Short Answer

Expert verified
Answer: 1. Calculate the surface area of the tip: \(A = 2\pi(0.00125)(0.02) \approx 0.000157 \mathrm{~m}^{2}\) 2. Calculate the heat transfer by convection: \(Q_{conv} = (25)(0.000157)(400-20) \approx 1.491 \mathrm{~W}\) 3. Calculate the heat transfer by radiation: \(Q_{rad} = (0.80)(5.67\times 10^{-8})(0.000157)((400+273)^{4}-(20+273)^{4}) \approx 1.164 \mathrm{~W}\) 4. Calculate the total heat transfer and power required: \(P_{required} = Q_{total} = Q_{conv} + Q_{rad} = 1.491 + 1.164 \approx 2.655 \mathrm{~W}\) The power required to maintain the temperature of the soldering iron tip at \(400^{\circ} \mathrm{C}\) is approximately \(2.655 \mathrm{~W}\).

Step by step solution

01

Calculate the surface area of the tip

The first step is to find the surface area of the cylindrical tip. The surface area of a cylinder can be calculated using the formula \(A=2 \pi r h\) where \(r\) is the radius of the base and \(h\) is the height. The diameter is given as \(2.5 \mathrm{~mm}\), so the radius is \(1.25 \mathrm{~mm}\). Convert the radius and height to meters: - radius: \(1.25 \mathrm{~mm} = 0.00125 \mathrm{~m}\) - length: \(20 \mathrm{~mm} = 0.02 \mathrm{~m}\) Now, calculate the surface area of the tip: \(A = 2\pi(0.00125)(0.02)\)
02

Calculate the heat transfer by convection

The heat transfer by convection can be calculated using the formula \(Q_{conv} = hA(T_{tip} - T_{air})\), where \(h\) is the convection heat transfer coefficient, \(A\) is the surface area, \(T_{tip}\) is the temperature of the tip, and \(T_{air}\) is the air temperature. Plug in the values given: \(Q_{conv} = (25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K})(2\pi(0.00125)(0.02))(400-20)\)
03

Calculate the heat transfer by radiation

The heat transfer by radiation can be calculated using the formula \(Q_{rad} = \epsilon \sigma A(T_{tip}^{4} - T_{air}^{4})\), where \(\epsilon\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant \((5.67\times 10^{-8} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4})\), and \(A\) is the surface area. Plug in the values given: \(Q_{rad} = (0.80)(5.67\times 10^{-8})(2\pi(0.00125)(0.02))((400+273)^{4}-(20+273)^{4})\)
04

Calculate the total heat transfer and power required

To determine the power required to maintain the temperature, we need to add the heat transfer by convection and radiation: \(Q_{total} = Q_{conv} + Q_{rad}\) Finally, the power required is equal to the total heat transfer, so: \(P_{required} = Q_{total}\) Calculate the values using the above equations and obtain the power required to maintain the tip at \(400^{\circ} \mathrm{C}\).

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Most popular questions from this chapter

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