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Chapter 2: Problem 136
Consider a third order linear and homogeneous differential equation. How many arbitrary constants will its general solution involve?
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The heat conduction equation in a medium is given in its simplest form as $$ \frac{1}{r} \frac{d}{d r}\left(r k \frac{d T}{d r}\right)+\dot{e}_{\text {gen }}=0 $$ Select the wrong statement below. (a) The medium is of cylindrical shape. (b) The thermal conductivity of the medium is constant. (c) Heat transfer through the medium is steady. (d) There is heat generation within the medium. (e) Heat conduction through the medium is one-dimensional.
In subsea oil and natural gas production, hydrocarbon fluids may leave the reservoir with a temperature of \(70^{\circ} \mathrm{C}\) and flow in subsea surrounding of \(5^{\circ} \mathrm{C}\). As a result of the temperature difference between the reservoir and the subsea surrounding, the knowledge of heat transfer is critical to prevent gas hydrate and wax deposition blockages. Consider a subsea pipeline with inner diameter of \(0.5 \mathrm{~m}\) and wall thickness of \(8 \mathrm{~mm}\) is used for transporting liquid hydrocarbon at an average temperature of \(70^{\circ} \mathrm{C}\), and the average convection heat transfer coefficient on the inner pipeline surface is estimated to be \(250 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The subsea surrounding has a temperature of \(5^{\circ} \mathrm{C}\) and the average convection heat transfer coefficient on the outer pipeline surface is estimated to be \(150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the pipeline is made of material with thermal conductivity of \(60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), by using the heat conduction equation (a) obtain the temperature variation in the pipeline wall, \((b)\) determine the inner surface temperature of the pipeline, \((c)\) obtain the mathematical expression for the rate of heat loss from the liquid hydrocarbon in the pipeline, and \((d)\) determine the heat flux through the outer pipeline surface.
A spherical vessel has an inner radius \(r_{1}\) and an outer radius \(r_{2}\). The inner surface \(\left(r=r_{1}\right)\) of the vessel is subjected to a uniform heat flux \(\dot{q}_{1}\). The outer surface \(\left(r=r_{2}\right)\) is exposed to convection and radiation heat transfer in a surrounding temperature of \(T_{\infty}\). The emissivity and the convection heat transfer coefficient on the outer surface are \(\varepsilon\) and \(h\), respectively. Express the boundary conditions and the differential equation of this heat conduction problem during steady operation.
How do you recognize a linear homogeneous differential equation? Give an example and explain why it is linear and homogeneous.
Consider a short cylinder of radius \(r_{o}\) and height \(H\) in which heat is generated at a constant rate of \(\dot{e}_{\text {gen. }}\). 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.
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