Chapter 2

Consider a large plane wall of thickness \(L=0.8 \mathrm{ft}\) and thermal conductivity $k=1.2 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}$. The wall is covered with a material that has an emissivity of \(\varepsilon=0.80\) and a solar absorptivity of \(\alpha=0.60\). The inner surface of the wall is maintained at \(T_{1}=520 \mathrm{R}\) at all times, while the outer surface is exposed to solar radiation that is incident at a rate of $\dot{q}_{\text {solar }}=300 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}$. The outer surface is also losing heat by radiation to deep space at \(0 \mathrm{~K}\). Determine the temperature of the outer surface of the wall and the rate of heat transfer through the wall when steady operating conditions are reached. Answers: $554 \mathrm{R}, 50.9 \mathrm{Btu} / \mathrm{h}^{\mathrm{ft}}{ }^{2}$

Consider a short cylinder of radius \(r_{o}\) and height \(H\) in which heat is generated at a constant rate of \(\dot{e}_{\mathrm{gen}}\). Heat is lost from the cylindrical surface at \(r=r_{o}\) by convection to the surrounding medium at temperature \(T_{\infty}\) with a heat transfer coefficient of \(h\). The bottom surface of the cylinder at \(z=0\) is insulated, while the top surface at \(z=H\) is subjected to uniform heat flux \(\dot{q}_{H}\). Assuming constant thermal conductivity and steady two-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem. Do not solve.

Consider a small hot metal object of mass \(m\) and specific heat \(c\) that is initially at a temperature of \(T_{i}\). Now the object is allowed to cool in an environment at \(T_{\infty}\) by convection with a heat transfer coefficient of \(h\). The temperature of the metal object is observed to vary uniformly with time during cooling. Writing an energy balance on the entire metal object, derive the differential equation that describes the variation of temperature of the ball with time, \(T(t)\). Assume constant thermal conductivity and no heat generation in the object. Do not solve.

Consider a 20-cm-thick large concrete plane wall $(k=0.77 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ subjected to convection on both sides with \(T_{\infty 1}=22^{\circ} \mathrm{C}\) and $h_{1}=8 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\( on the inside and \)T_{\infty 22}=8^{\circ} \mathrm{C}$ and \(h_{2}=12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the outside. Assuming constant thermal conductivity with no heat generation and negligible radiation, \((a)\) express the differential equations and the boundary conditions for steady one-dimensional heat conduction through the wall, (b) obtain a relation for the variation of temperature in the wall by solving the differential equation, and \((c)\) evaluate the temperatures at the inner and outer surfaces of the wall.

A \(1000-W\) iron is left on the ironing board with its base exposed to ambient air at \(26^{\circ} \mathrm{C}\). The base plate of the iron has a thickness of \(L=0.5 \mathrm{~cm}\), base area of \(A=150 \mathrm{~cm}^{2}\), and thermal conductivity of \(k=18 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The inner surface of the base plate is subjected to uniform heat flux generated by the resistance heaters inside. The outer surface of the base plate, whose emissivity is \(\varepsilon=0.7\), loses heat by convection to ambient air with an average heat transfer coefficient of $h=30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ as well as by radiation to the surrounding surfaces at an average temperature of \(T_{\text {sarr }}=295 \mathrm{~K}\). Disregarding any heat loss through the upper part of the iron, \((a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the plate, \((b)\) obtain a relation for the temperature of the outer surface of the plate by solving the differential equation, and (c) evaluate the outer surface temperature.

A heating cable is embedded in a concrete slab for snow melting on a $30 \mathrm{~m}^{2}$ surface area. The heating cable is heated electrically with joule heating. When the surface is covered with snow, the heat generated from the heating cable can melt snow at a rate of \(0.1 \mathrm{~kg} / \mathrm{s}\). According to the National Electrical Code (NFPA 70), the power density for embedded snow-melting equipment should not exceed $1300 \mathrm{~W} / \mathrm{m}^{2}$. Formulate the temperature profile in the concrete slab in terms of the snow melt rate. Determine whether melting snow at $0.1 \mathrm{~kg} / \mathrm{s}$ would be in compliance with the NFPA 70 code.

A series of ASME SA-193 carbon steel bolts are bolted onto the upper surface of a metal plate. The metal plate has a thickness of \(3 \mathrm{~cm}\), and its thermal conductivity is \(15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The bottom surface of the plate is subjected to a uniform heat flux of $5 \mathrm{~kW} / \mathrm{m}^{2}$. The upper surface of the plate is exposed to ambient air with a temperature of \(30^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The ASME Boiler and Pressure Vessel Code (ASME BPVC.IV-2015, HF-300) limits the maximum allowable use temperature to \(260^{\circ} \mathrm{C}\) for the SA-193 bolts. Formulate the variation of temperature in the metal plate, and determine the temperatures at \(x=0,1.5\), and \(3.0 \mathrm{~cm}\). Would the SA-193 bolts comply with the ASME code?

Consider a steam pipe of length \(L\), inner radius \(r_{1}\), outer radius \(r_{2}\), and constant thermal conductivity \(k\). Steam flows inside the pipe at an average temperature of \(T_{i}\) with a convection heat transfer coefficient of \(h_{i}\). The outer surface of the pipe is exposed to convection to the surrounding air at a temperature of \(T_{0}\) with a heat transfer coefficient of \(h_{\sigma}\). Assuming steady one-dimensional heat conduction through the pipe, \((a)\) express the differential equation and the boundary conditions for heat conduction through the pipe material, (b) obtain a relation for the variation of temperature in the pipe material by solving the differential equation, and (c) obtain a relation for the temperature of the outer surface of the pipe.

Consider a steam pipe of length \(L=35 \mathrm{ft}\), inner radius $r_{1}=2 \mathrm{in}\(, outer radius \)r_{2}=24\( in, and thermal conductivity \)k=8 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}{ }^{\circ} \mathrm{F}$. Steam is flowing through the pipe at an average temperature of $250^{\circ} \mathrm{F}$, and the average convection heat transfer coefficient on the inner surface is given to be $h=15 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2},{ }^{\circ} \mathrm{F}$. If the average temperature on the outer surfaces of the pipe is \(T_{2}=160^{\circ} \mathrm{F},(a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the pipe, \((b)\) obtain a relation for the variation of temperature in the pipe by solving the differential equation, and (c) evaluate the rate of heat loss from the steam through the pipe.

Consider a water pipe of length \(L=17 \mathrm{~m}\), inner radius $r_{1}=15 \mathrm{~cm}\(, outer radius \)r_{2}=20 \mathrm{~cm}$, and thermal conductivity \(k=14 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Heat is generated in the pipe material uniformly by a \(25-\mathrm{kW}\) electric resistance heater. The inner and outer surfaces of the pipe are at \(T_{1}=60^{\circ} \mathrm{C}\) and \(T_{2}=80^{\circ} \mathrm{C}\), respectively. Obtain a general relation for temperature distribution inside the pipe under steady conditions and determine the temperature at the center plane of the pipe.