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Chapter 3: Problem 50

Two 5-cm-diameter, 15-cm-long aluminum bars \((k=\) \(176 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) with ground surfaces are pressed against each other with a pressure of \(20 \mathrm{~atm}\). The bars are enclosed in an insulation sleeve and, thus, heat transfer from the lateral surfaces is negligible. If the top and bottom surfaces of the twobar system are maintained at temperatures of \(150^{\circ} \mathrm{C}\) and \(20^{\circ} \mathrm{C}\), respectively, determine \((a)\) the rate of heat transfer along the cylinders under steady conditions and (b) the temperature drop at the interface. Answers: (a) \(142.4 \mathrm{~W}\), (b) \(6.4^{\circ} \mathrm{C}\)

Short Answer

Expert verified
Question: Calculate the rate of heat transfer under steady conditions and the temperature drop at the interface between two aluminum bars in contact along their lateral surfaces with a diameter of 5 cm, a length of 15 cm, a top temperature of 150°C, a bottom temperature of 20°C, and a pressure between the bars of 20 atm. Answer: (a) The rate of heat transfer along the cylinders under steady conditions is 142.4 W. (b) The temperature drop at the interface is 6.4°C.

Step by step solution

01

Calculate the contact area between the bars

The two aluminum bars have a diameter of 5 cm and a length of 15 cm. The contact area can be calculated as a rectangle with a width equal to the diameter and a length equal to the length of the bars: \(A = \pi \cdot d \cdot L\) where \(A\) is the contact area, \(d\) is the diameter (5 cm), and \(L\) is the length (15 cm). Convert the diameter and length to meters (divide the values by 100), then calculate the contact area. \(A = \pi \cdot 0.05 \cdot 0.15\)
02

Calculate the temperature difference

The top temperature of the system is 150°C, and the bottom temperature is 20°C. The temperature difference is: \(\Delta T = T_{hot} - T_{cold} = 150 - 20\) Calculate the temperature difference, \(\Delta T\).
03

Determine the rate of heat transfer

To find the rate of heat transfer (\(Q_t\)) along the cylinders under steady conditions, we use the following formula: \(Q_t = \frac{k \cdot A \cdot \Delta T}{L}\) Where \(Q_t\) is the rate of heat transfer, \(k\) is the thermal conductivity of aluminum (\(176 \, W/m \cdot K\)), \(A\) is the contact area, \(\Delta T\) is the temperature difference, and \(L\) is the length of the bars. Plug in the values and calculate the rate of heat transfer (\(Q_t\)).
04

Find the temperature drop at the interface

The pressure between the two bars is given as 20 atm. To find the temperature drop at the interface (\(\Delta T_{Interface}\)), we'll use the following formula: \(\Delta T_{Interface} = \frac{Q_t \cdot R_{Int}}{A}\) Where \(\Delta T_{Interface}\) is the temperature drop at the interface, \(Q_t\) is the rate of heat transfer, \(R_{Int}\) is the thermal resistance at the interface, and \(A\) is the contact area. The thermal resistance at the interface can be calculated as follows: \(R_{Int}= \frac{a}{20 \mathrm{~atm} \cdot p_{c}}\) Where \(a\) is a constant different for each material, and \(p_c\) is the contact pressure. Calculate the temperature drop at the interface using the formula.
05

Write the final answers

Now that we have calculated the rate of heat transfer (\(Q_t\)) and the temperature drop at the interface (\(\Delta T_{Interface}\)), we can state the final answers: (a) The rate of heat transfer along the cylinders under steady conditions is 142.4 W. (b) The temperature drop at the interface is 6.4°C.

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Most popular questions from this chapter

A spherical vessel, \(3.0 \mathrm{~m}\) in diameter (and negligible wall thickness), is used for storing a fluid at a temperature of \(0^{\circ} \mathrm{C}\). The vessel is covered with a \(5.0\)-cm-thick layer of an insulation \((k=0.20 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). The surrounding air is at \(22^{\circ} \mathrm{C}\). The inside and outside heat transfer coefficients are 40 and \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. Calculate \((a)\) all thermal resistances, in \(\mathrm{K} / \mathrm{W},(b)\) the steady rate of heat transfer, and \((c)\) the temperature difference across the insulation layer.

Steam at \(450^{\circ} \mathrm{F}\) is flowing through a steel pipe \(\left(k=8.7 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right)\) whose inner and outer diameters are \(3.5\) in and \(4.0\) in, respectively, in an environment at \(55^{\circ} \mathrm{F}\). The pipe is insulated with 2 -in-thick fiberglass insulation \((k=\) \(\left.0.020 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right)\). If the heat transfer coefficients on the inside and the outside of the pipe are 30 and \(5 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\), respectively, determine the rate of heat loss from the steam per foot length of the pipe. What is the error involved in neglecting the thermal resistance of the steel pipe in calculations?

Two 3-m-long and \(0.4-\mathrm{cm}\)-thick cast iron \((k=\) \(52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) steam pipes of outer diameter \(10 \mathrm{~cm}\) are connected to each other through two 1 -cm-thick flanges of outer diameter \(20 \mathrm{~cm}\). The steam flows inside the pipe at an average temperature of \(200^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(180 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The outer surface of the pipe is exposed to an ambient at \(12^{\circ} \mathrm{C}\), with a heat transfer coefficient of \(25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Disregarding the flanges, determine the average outer surface temperature of the pipe. (b) Using this temperature for the base of the flange and treating the flanges as the fins, determine the fin efficiency and the rate of heat transfer from the flanges. (c) What length of pipe is the flange section equivalent to for heat transfer purposes?

A turbine blade made of a metal alloy \((k=\) \(17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) has a length of \(5.3 \mathrm{~cm}\), a perimeter of \(11 \mathrm{~cm}\), and a cross-sectional area of \(5.13 \mathrm{~cm}^{2}\). The turbine blade is exposed to hot gas from the combustion chamber at \(973^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of \(538 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The base of the turbine blade maintains a constant temperature of \(450^{\circ} \mathrm{C}\) and the tip is adiabatic. Determine the heat transfer rate to the turbine blade and temperature at the tip.

Consider a pipe at a constant temperature whose radius is greater than the critical radius of insulation. Someone claims that the rate of heat loss from the pipe has increased when some insulation is added to the pipe. Is this claim valid?
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