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Chapter 2: Problem 19

The resistance wire of an 800 -W iron is 15 in long and has a diameter of \(D=0.08 \mathrm{in}\). Determine the rate of heat generation in the wire per unit volume, in \(\mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{3}\), and the heat flux on the outer surface of the wire, in \(\mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}\), as a result of this heat generation.

Short Answer

Expert verified
Question: Determine the rate of heat generation in an 800-watt iron with a 15-inch long resistance wire and a diameter of 0.08 inches per unit volume and the heat flux on the outer surface of the wire. Answer: To determine the rate of heat generation per unit volume and the heat flux on the outer surface of the wire, follow these steps: 1. Convert the given quantities to consistent units: Length = 15/12 ft, Diameter = 0.08/12 ft, and Power = 800/3.412 Btu/h. 2. Calculate the wire's volume, cross-sectional area, and surface area using the cylinder formulas. 3. Calculate the rate of heat generation in the wire per unit volume: q_gen = Power / V. 4. Calculate the heat flux on the outer surface of the wire: q_flux = Power / A_s.

Step by step solution

01

Convert given quantities to consistent units

Our given dimensions are in inches, and we need to convert them to feet for consistency with the other given units. There are 12 inches in a foot, so the length and diameter of the wire in feet are: Length: \(15 \, in = \dfrac{15}{12} \, ft \) Diameter: \(0.08 \, in = \dfrac{0.08}{12} \, ft\) Next, convert the power, currently in watts, to Btu per hour: Power: \(800 \, W = \dfrac{800}{3.412} \, Btu/h\)
02

Calculate wire's volume, cross-sectional area, and surface area

We need to find the volume, cross-sectional area, and surface area in order to calculate the heat generation and heat flux. Use the following formulas for a cylinder's geometry: Volume: \(V = \pi \cdot r^{2} \cdot L\) Cross-sectional area: \(A_{c} = \pi \cdot r^{2}\) Surface area: \(A_{s} = 2\pi \cdot r \cdot L\) Where \(L\) and \(r\) are the length and radius of the wire in feet, respectively.
03

Calculate the rate of heat generation in the wire per unit volume

The rate of heat generation in the wire per unit volume (\(q_{gen}\)) can be found using the total power and wire's volume: \(q_{gen} = \dfrac{Power}{V} \, = \dfrac{800(Btu/h)}{3.412 \cdot \pi \cdot (\dfrac{0.08}{12})^{2}\cdot(\dfrac{15}{12})} Btu/h\cdot ft^3\)
04

Calculate the heat flux on the outer surface of the wire

The heat flux on the outer surface of the wire (\(q_{flux}\)) can be found using the total power and wire's surface area: \(q_{flux} = \dfrac{Power}{A_{s}} \, = \dfrac{800(Btu/h)}{3.412 \cdot 2\pi \cdot (\dfrac{0.08}{12})\cdot (\dfrac{15}{12})} Btu/h\cdot ft^2\) Follow these steps to calculate the rate of heat generation in the wire per unit volume and the heat flux on the outer surface of the wire.

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