Chapter 3

Clothing made of several thin layers of fabric with trapped air in between, often called ski clothing, is commonly used in cold climates because it is light, fashionable, and a very effective thermal insulator. So it is no surprise that such clothing has largely replaced thick and heavy old-fashioned coats. Consider a jacket made of five layers of \(0.1-\mathrm{mm}\)-thick synthetic fabric \((k=0.13 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) with \(1.5\)-mm- thick air space \((k=0.026 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) between the layers. Assuming the inner surface temperature of the jacket to be \(28^{\circ} \mathrm{C}\) and the surface area to be \(1.25 \mathrm{~m}^{2}\), determine the rate of heat loss through the jacket when the temperature of the outdoors is \(0^{\circ} \mathrm{C}\) and the heat transfer coefficient at the outer surface is \(25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). What would your response be if the jacket is made of a single layer of \(0.5-\mathrm{mm}\)-thick synthetic fabric? What should be the thickness of a wool fabric ( \(k=0.035 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) if the person is to achieve the same level of thermal comfort wearing a thick wool coat instead of a five-layer ski jacket?

A 5-m-wide, 4-m-high, and 40-m-long kiln used to cure concrete pipes is made of 20 -cm-thick concrete walls and ceiling \((k=0.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). The kiln is maintained at \(40^{\circ} \mathrm{C}\) by injecting hot steam into it. The two ends of the kiln, \(4 \mathrm{~m} \times 5 \mathrm{~m}\) in size, are made of a 3 -mm-thick sheet metal covered with 2 -cm-thick Styrofoam \((k=0.033 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). The convection heat transfer coefficients on the inner and the outer surfaces of the kiln are \(3000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. Disregarding any heat loss through the floor, determine the rate of heat loss from the kiln when the ambient air is at \(-4^{\circ} \mathrm{C}\).

A 4-m-high and 6-m-wide wall consists of a long \(18-\mathrm{cm} \times\) 30 -cm cross section of horizontal bricks \((k=0.72 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) separated by \(3-\mathrm{cm}\)-thick plaster layers \((k=0.22 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). There are also 2 -cm-thick plaster layers on each side of the wall, and a 2-cmthick rigid foam \((k=0.026 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) on the inner side of the wall. The indoor and the outdoor temperatures are \(22^{\circ} \mathrm{C}\) and \(-4^{\circ} \mathrm{C}\), and the convection heat transfer coefficients on the inner and the outer sides are \(h_{1}=10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(h_{2}=20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. Assuming one-dimensional heat transfer and disregarding radiation, determine the rate of heat transfer through the wall.

A 12-m-long and 5-m-high wall is constructed of two layers of \(1-\mathrm{cm}\)-thick sheetrock \((k=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) spaced \(16 \mathrm{~cm}\) by wood studs \((k=0.11 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) whose cross section is \(16 \mathrm{~cm} \times 5 \mathrm{~cm}\). The studs are placed vertically \(60 \mathrm{~cm}\) apart, and the space between them is filled with fiberglass insulation \((k=0.034 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). The house is maintained at \(20^{\circ} \mathrm{C}\) and the ambient temperature outside is \(-9^{\circ} \mathrm{C}\). Taking the heat transfer coefficients at the inner and outer surfaces of the house to be \(8.3\) and \(34 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively, determine \((a)\) the thermal resistance of the wall considering a representative section of it and (b) the rate of heat transfer through the wall.

A 10-in-thick, 30-ft-long, and 10-ft-high wall is to be constructed using 9 -in-long solid bricks \(\left(k=0.40 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right)\) of cross section 7 in \(\times 7\) in, or identical size bricks with nine square air holes \(\left(k=0.015 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right)\) that are 9 in long and have a cross section of \(1.5\) in \(\times 1.5 \mathrm{in}\). There is a \(0.5\)-in-thick plaster layer \(\left(k=0.10 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}{ }^{\circ} \mathrm{F}\right)\) between two adjacent bricks on all four sides and on both sides of the wall. The house is maintained at \(80^{\circ} \mathrm{F}\) and the ambient temperature outside is \(30^{\circ} \mathrm{F}\). Taking the heat transfer coefficients at the inner and outer surfaces of the wall to be \(1.5\) and \(4 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\), respectively, determine the rate of heat transfer through the wall constructed of \((a)\) solid bricks and (b) bricks with air holes.

In an experiment to measure convection heat transfer coefficients, a very thin metal foil of very low emissivity (e.g., highly polished copper) is attached on the surface of a slab of material with very low thermal conductivity. The other surface of the metal foil is exposed to convection heat transfer by flowing fluid over the foil surface. This setup diminishes heat conduction through the slab and radiation on the metal foil surface, while heat convection plays the prominent role. The slab on which the metal foil is attached to has a thickness of \(25 \mathrm{~mm}\) and a thermal conductivity of \(0.023 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). In a condition where the surrounding room temperature is \(20^{\circ} \mathrm{C}\), the metal foil is heated electrically with a uniform heat flux of \(5000 \mathrm{~W} / \mathrm{m}^{2}\). If the bottom surface of the slab is \(20^{\circ} \mathrm{C}\) and the metal foil has an emissivity of \(0.02\), determine \((a)\) the convection heat transfer coefficient if air is flowing over the metal foil and the surface temperature of the foil is \(150^{\circ} \mathrm{C}\); and \((b)\) the convection heat transfer coefficient if water is flowing over the metal foil and the surface temperature of the foil is \(30^{\circ} \mathrm{C}\).

What is an infinitely long cylinder? When is it proper to treat an actual cylinder as being infinitely long, and when is it not?

Can the thermal resistance concept be used for a solid cylinder or sphere in steady operation? Explain.

Consider a short cylinder whose top and bottom surfaces are insulated. The cylinder is initially at a uniform temperature \(T_{i}\) and is subjected to convection from its side surface to a medium at temperature \(T_{\infty}\), with a heat transfer coefficient of \(h\). Is the heat transfer in this short cylinder one- or twodimensional? Explain.

Steam at \(320^{\circ} \mathrm{C}\) flows in a stainless steel pipe \((k=\) \(15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) ) whose inner and outer diameters are \(5 \mathrm{~cm}\) and \(5.5 \mathrm{~cm}\), respectively. The pipe is covered with \(3-\mathrm{cm}\)-thick glass wool insulation \((k=0.038 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). Heat is lost to the surroundings at \(5^{\circ} \mathrm{C}\) by natural convection and radiation, with a combined natural convection and radiation heat transfer coefficient of \(15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Taking the heat transfer coefficient inside the pipe to be \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the rate of heat loss from the steam per unit length of the pipe. Also determine the temperature drops across the pipe shell and the insulation.