Chapter 2: Problem 82
What is heat generation? Give some examples.
Chapter 2: Problem 82
What is heat generation? Give some examples.
All the tools & learning materials you need for study success - in one app.
Get started for freeA circular metal pipe has a wall thickness of \(10 \mathrm{~mm}\) and an inner diameter of \(10 \mathrm{~cm}\). The pipe's outer surface is subjected to a uniform heat flux of \(5 \mathrm{~kW} / \mathrm{m}^{2}\) and has a temperature of \(500^{\circ} \mathrm{C}\). The metal pipe has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where $k_{0}=7.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, \)\beta=0.0012 \mathrm{~K}^{-1}\(, and \)T$ is in \(\mathrm{K}\). Determine the inner surface temperature of the pipe.
A cylindrical nuclear fuel rod \(1 \mathrm{~cm}\) in diameter is encased in a concentric tube \(2 \mathrm{~cm}\) in diameter, where cooling water flows through the annular region between the fuel rod $(k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ and the concentric tube. Heat is generated uniformly in the rod at a rate of \(50 \mathrm{MW} / \mathrm{m}^{3}\). The convection heat transfer coefficient for the concentric tube surface is $2000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. If the surface temperature of the concentric tube is \(40^{\circ} \mathrm{C}\), determine the average temperature of the cooling water. Can one use the given information to determine the surface temperature of the fuel rod? Explain.
Consider a spherical shell of inner radius \(r_{1}\) and outer radius \(r_{2}\) whose thermal conductivity varies linearly in a specified temperature range as \(k(T)=k_{0}(1+\beta T)\) where \(k_{0}\) and \(\beta\) are two specified constants. The inner surface of the shell is maintained at a constant temperature of \(T_{1}\), while the outer surface is maintained at \(T_{2}\). Assuming steady one-dimensional heat transfer, obtain a relation for \((a)\) the heat transfer rate through the shell and (b) the temperature distribution \(T(r)\) in the shell.
Consider a large plate of thickness \(L\) and thermal conductivity \(k\) in which heat is generated uniformly at a rate of \(\dot{e}_{\text {gen }}\). One side of the plate is insulated, while the other side is exposed to an environment at \(T_{\infty}\) with a heat transfer coefficient of \(h\). (a) Express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the plate, (b) determine the variation of temperature in the plate, and (c) obtain relations for the temperatures on both surfaces and the maximum temperature rise in the plate in terms of given parameters.
A spherical vessel is filled with chemicals undergoing an exothermic reaction. The reaction provides a uniform heat flux on the inner surface of the vessel. The inner diameter of the vessel is \(5 \mathrm{~m}\) and its inner surface temperature is at \(120^{\circ} \mathrm{C}\). The wall of the vessel has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where $k_{0}=1.01 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=0.0018 \mathrm{~K}^{-1}\(, and \)T\( is in \)\mathrm{K}$. The vessel is situated in a surrounding with an ambient temperature of \(15^{\circ} \mathrm{C}\), and the vessel's outer surface experiences convection heat transfer with a coefficient of \(80 \mathrm{~W} / \mathrm{m}^{2}\). K. To prevent thermal burns to workers who touch the vessel, the outer surface temperature of the vessel should be kept below \(50^{\circ} \mathrm{C}\). Determine the minimum wall thickness of the vessel so that the outer surface temperature is \(50^{\circ} \mathrm{C}\) or lower.
What do you think about this solution?
We value your feedback to improve our textbook solutions.