Chapter 2: Problem 141
Consider a third-order linear and homogeneous differential equation. How many arbitrary constants will its general solution involve?
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How do you distinguish a linear differential equation from a nonlinear one?
A spherical tank is filled with ice slurry, where its inner surface is at \(0^{\circ} \mathrm{C}\). The tank has an inner diameter of \(9 \mathrm{~m}\), and its wall thickness is \(20 \mathrm{~mm}\). The tank wall is made of a material with a thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where $k_{0}=0.33 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=0.0025 \mathrm{~K}^{-1}\(, and \)T\( is in \)\mathrm{K}$. The temperature outside the tank is \(35^{\circ} \mathrm{C}\), and the convection heat transfer coefficient is \(70 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Solar radiation is incident on the tank's outer surface at a rate of $150 \mathrm{~W} / \mathrm{m}^{2}$, where the emissivity and solar absorptivity of the outer surface are \(0.75\). Determine the outer surface temperature of the tank.
A cylindrical fuel rod $(k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}) 2 \mathrm{~cm}$ in diameter is encased in a concentric tube and cooled by water. The fuel rod generates heat uniformly at a rate of $100 \mathrm{MW} / \mathrm{m}^{3}$, and the average temperature of the cooling water is \(75^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of $2500 \mathrm{~W} / \mathrm{m}^{2}\(. \)\mathrm{K}$. The operating pressure of the cooling water is such that the surface temperature of the fuel rod must be kept below \(200^{\circ} \mathrm{C}\) to prevent the cooling water from reaching the critical heat flux (CHF). The critical heat flux is a thermal limit at which a boiling crisis can occur that causes overheating on the fuel rod surface and leads to damage. Determine the variation of temperature in the fuel rod and the temperature of the fuel rod surface. Is the surface of the fuel rod adequately cooled?
Write an essay on heat generation in nuclear fuel rods. Obtain information on the ranges of heat generation, the variation of heat generation with position in the rods, and the absorption of emitted radiation by the cooling medium.
A spherical container with an inner radius \(r_{1}=1 \mathrm{~m}\) and an outer radius \(r_{2}=1.05 \mathrm{~m}\) has its inner surface subjected to a uniform heat flux of \(\dot{q}_{1}=7 \mathrm{~kW} / \mathrm{m}^{2}\). The outer surface of the container has a temperature \(T_{2}=25^{\circ} \mathrm{C}\), and the container wall thermal conductivity is $k=1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. Show that the variation of temperature in the container wall can be expressed as $T(r)=\left(\dot{q}_{1} r_{1}^{2} / k\right)\left(1 / r-1 / r_{2}\right)+T_{2}$ and determine the temperature of the inner surface of the container at \(r=r_{1}\).
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