Chapter 4

A long 18-cm-diameter bar made of hardwood \((k=\) $\left.0.159 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=1.75 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)\( is exposed to air at \)30^{\circ} \mathrm{C}$ with a heat transfer coefficient of \(8.83 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the center temperature of the bar is measured to be \(15^{\circ} \mathrm{C}\) after \(3 \mathrm{~h}\), the initial temperature of the bar is (a) \(11.9^{\circ} \mathrm{C}\) (b) \(4.9^{\circ} \mathrm{C}\) (c) \(1.7^{\circ} \mathrm{C}\) (d) \(0^{\circ} \mathrm{C}\) (e) \(-9.2^{\circ} \mathrm{C}\)

Consider a 7.6-cm-long and 3-cm-diameter cylindrical lamb meat chunk $\left(\rho=1030 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=3.49 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right.\(, \)k=0.456 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=1.3 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$ ). Fifteen such meat chunks initially at \(2^{\circ} \mathrm{C}\) are dropped into boiling water at \(95^{\circ} \mathrm{C}\) with a heat transfer coefficient of $1200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The amount of heat transfer during the first \(8 \mathrm{~min}\) of cooking is (a) \(71 \mathrm{~kJ}\) (b) \(227 \mathrm{~kJ}\) (c) \(238 \mathrm{~kJ}\) (d) \(269 \mathrm{~kJ}\) (e) \(307 \mathrm{~kJ}\)

Consider a 7.6-cm-diameter cylindrical lamb meat chunk $\left(\rho=1030 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=3.49 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.456 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$, \(\alpha=1.3 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\) ). Such a meat chunk intially at \(2^{\circ} \mathrm{C}\) is dropped into boiling water at \(95^{\circ} \mathrm{C}\) with a heat transfer coefficient of $1200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The time it takes for the center temperature of the meat chunk to rise to \(75^{\circ} \mathrm{C}\) is (a) \(136 \mathrm{~min}\) (b) \(21.2 \mathrm{~min}\) (c) \(13.6 \mathrm{~min}\) (d) \(11.0 \mathrm{~min}\) (e) \(8.5 \mathrm{~min}\)

A potato may be approximated as a \(5.7-\mathrm{cm}\) solid sphere with the properties $\rho=910 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4.25 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\(, \)k=0.68 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, and \)\alpha=1.76 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$. Twelve such potatoes initially at \(25^{\circ} \mathrm{C}\) are to be cooked by placing them in an oven maintained at \(250^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(95 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The amount of heat transfer to the potatoes by the time the center temperature reaches \(90^{\circ} \mathrm{C}\) is (a) \(1012 \mathrm{~kJ}\) (b) \(1366 \mathrm{~kJ}\) (c) \(1788 \mathrm{~kJ}\) (d) \(2046 \mathrm{~kJ}\) (e) \(3270 \mathrm{~kJ}\)

A small chicken $(k=0.45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \quad \alpha=0.15 \times\( \)10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) can be approximated as an \(11.25-\mathrm{cm}\)-diameter solid sphere. The chicken is initially at a uniform temperature of \(8^{\circ} \mathrm{C}\) and is to be cooked in an oven maintained at \(220^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). With this idealization, the temperature at the center of the chicken after a 90 -min period is (a) \(25^{\circ} \mathrm{C}\) (b) \(61^{\circ} \mathrm{C}\) (c) \(89^{\circ} \mathrm{C}\) (d) \(122^{\circ} \mathrm{C}\) (e) \(168^{\circ} \mathrm{C}\)

A potato may be approximated as a \(5.7-\mathrm{cm}\)-diameter solid sphere with the properties $\rho=910 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4.25 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\(, \)k=0.68 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, and \)\alpha=1.76 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\(. Twelve such potatoes initially at \)25^{\circ} \mathrm{C}$ are to be cooked by placing them in an oven maintained at \(250^{\circ} \mathrm{C}\) with a heat transfer coefficient of $95 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The amount of heat transfer to the potatoes during a 30 -min period is (a) \(77 \mathrm{~kJ}\) (b) \(483 \mathrm{~kJ}\) (c) \(927 \mathrm{~kJ}\) (d) \(970 \mathrm{~kJ}\) (e) \(1012 \mathrm{~kJ}\)

When water, as in a pond or lake, is heated by warm air above it, it remains stable, does not move, and forms a warm layer of water on top of a cold layer. Consider a deep lake $\left(k=0.6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, c_{p}=4.179 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)$ that is initially at a uniform temperature of \(2^{\circ} \mathrm{C}\) and has its surface temperature suddenly increased to \(20^{\circ} \mathrm{C}\) by a spring weather front. The temperature of the water \(1 \mathrm{~m}\) below the surface \(400 \mathrm{~h}\) after this change is (a) \(2.1^{\circ} \mathrm{C}\) (b) \(4.2^{\circ} \mathrm{C}\) (c) \(6.3^{\circ} \mathrm{C}\) (d) \(8.4^{\circ} \mathrm{C}\) (e) \(10.2^{\circ} \mathrm{C}\)

A large chunk of tissue at \(35^{\circ} \mathrm{C}\) with a thermal diffusivity of \(1 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\) is dropped into iced water. The water is well-stirred so that the surface temperature of the tissue drops to \(0^{\circ} \mathrm{C}\) at time zero and remains at \(0^{\circ} \mathrm{C}\) at all times. The temperature of the tissue after 4 min at a depth of $1 \mathrm{~cm}$ is (a) \(5^{\circ} \mathrm{C}\) (b) \(30^{\circ} \mathrm{C}\) (c) \(25^{\circ} \mathrm{C}\) (d) \(20^{\circ} \mathrm{C}\) (e) \(10^{\circ} \mathrm{C}\)

The 35-cm-thick roof of a large room made of concrete $\left(k=0.79 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=5.88 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)$ is initially at a uniform temperature of \(15^{\circ} \mathrm{C}\). After a heavy snowstorm, the outer surface of the roof remains covered with snow at \(-5^{\circ} \mathrm{C}\). The roof temperature at \(12 \mathrm{~cm}\) distance from the outer surface after $2 \mathrm{~h}$ is (a) \(13^{\circ} \mathrm{C}\) (b) \(11^{\circ} \mathrm{C}\) (c) \(7^{\circ} \mathrm{C}\) (d) \(3^{\circ} \mathrm{C}\) (e) \(-5^{\circ} \mathrm{C}\)

Conduct the following experiment at home to determine the combined convection and radiation heat transfer coefficient at the surface of an apple exposed to the room air. You will need two thermometers and a clock. First, weigh the apple and measure its diameter. You can measure its volume by placing it in a large measuring cup halfway filled with water, and measuring the change in volume when it is completely immersed in the water. Refrigerate the apple overnight so that it is at a uniform temperature in the morning, and measure the air temperature in the kitchen. Then take the apple out and stick one of the thermometers to its middle and the other just under the skin. Record both temperatures every \(5 \mathrm{~min}\) for an hour. Using these two temperatures, calculate the heat transfer coefficient for each interval and take their average. The result is the combined convection and radiation heat transfer coefficient for this heat transfer process. Using your experimental data, also calculate the thermal conductivity and thermal diffusivity of the apple and compare them to the values given above.