Chapter 2: Problem 2
Is heat transfer a scalar or a vector quantity? Explain. Answer the same question for temperature.
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A spherical container of inner radius \(r_{1}=2 \mathrm{~m}\), outer radius \(r_{2}=2.1 \mathrm{~m}\), and thermal conductivity $k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\( is filled with iced water at \)0^{\circ} \mathrm{C}$. The container is gaining heat by convection from the surrounding air at \(T_{\infty}=25^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(h=18 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}\). Assuming the inner surface temperature of the container to be \(0^{\circ} \mathrm{C},(a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the container, (b) obtain a relation for the variation of temperature in the container by solving the differential equation, and \((c)\) evaluate the rate of heat gain to the iced water.
Consider a chilled-water pipe of length \(L\), inner radius \(r_{1}\), outer radius \(r_{2}\), and thermal conductivity \(k\). Water flows in the pipe at a temperature \(T_{f}\), and the heat transfer coefficient at the inner surface is \(h\). If the pipe is well insulated on the outer surface, \((a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the pipe, and (b) obtain a relation for the variation of temperature in the pipe by solving the differential equation.
Consider a large plane wall of thickness \(L=0.3 \mathrm{~m}\), thermal conductivity \(k=2.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and surface area \(A=12 \mathrm{~m}^{2}\). The left side of the wall at \(x=0\) is subjected to a net heat flux of \(\dot{q}_{0}=700 \mathrm{~W} / \mathrm{m}^{2}\), while the temperature at that surface is measured to be $T_{1}=80^{\circ} \mathrm{C}$. Assuming constant thermal conductivity and no heat generation in the wall, \((a)\) express the differential equation 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 temperature of the right surface of the wall at \(x=L\). Answer: (c) \(-4^{\circ} \mathrm{C}\)
A flat-plate solar collector is used to heat water by having water flow through tubes attached at the back of the thin solar absorber plate. The absorber plate has an emissivity and an absorptivity of \(0.9\). The top surface \((x=0)\) temperature of the absorber is \(T_{0}=35^{\circ} \mathrm{C}\), and solar radiation is incident on the absorber at $500 \mathrm{~W} / \mathrm{m}^{2}\( with a surrounding temperature of \)0^{\circ} \mathrm{C}$. The convection heat transfer coefficient at the absorber surface is $5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, while the ambient temperature is \(25^{\circ} \mathrm{C}\). Show that the variation of temperature in the absorber plate can be expressed as $T(x)=-\left(\dot{q}_{0} / k\right) x+T_{0}\(, and determine net heat flux \)\dot{q}_{0}$ absorbed by the solar collector.
A heating cable is embedded in a concrete slab for snow melting. The heating cable is heated electrically with joule heating to provide the concrete slab with a uniform heat of \(1200 \mathrm{~W} / \mathrm{m}^{2}\). The concrete has a thermal conductivity of \(1.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). To minimize thermal stress in the concrete, the temperature difference between the heater surface \(\left(T_{1}\right)\) and the slab surface \(\left(T_{2}\right)\) should not exceed \(21^{\circ} \mathrm{C}\) (2015 ASHRAE Handbook-HVAC Applications, Chap. 51). Formulate the temperature profile in the concrete slab, and determine the thickness of the concrete slab \((L)\) so that \(T_{1}-\) \(T_{2} \leq 21^{\circ} \mathrm{C}\).
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