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Chapter 2: Problem 18

In a nuclear reactor, heat is generated uniformly in the 5 -cm-diameter cylindrical uranium rods at a rate of $7 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}\(. If the length of the rods is \)1 \mathrm{~m}$, determine the rate of heat generation in each rod. Answer: \(137 \mathrm{~kW}\)

Short Answer

Expert verified
Answer: The approximate rate of heat generation in each uranium rod is \(137 \mathrm{kW}\).

Step by step solution

01

Understanding the problem statement

We are given the following information: - Heat generation rate: \(7 \times 10^{7} \mathrm{W/m^3}\) - Diameter of the cylindrical rod: \(5 \mathrm{cm}\) - Length of the rod: \(1 \mathrm{m}\) We need to find the rate of heat generation in each uranium rod.
02

Finding the volume of the cylindrical rod

To find the volume of the cylindrical rod, we can use the formula for the volume of a cylinder: \(V=\pi r^2 h\) Where: - \(V\) is the volume of the cylinder - \(r\) is the radius of the cylinder - \(h\) is the height (or length) of the cylinder Given the diameter of the rod is \(5 \mathrm{cm}\), we can find the radius: \(r = \frac{d}{2} = \frac{5}{2} = 2.5 \mathrm{cm}\) We need to convert the radius to meters: \(r = 0.025 \mathrm{m}\) Since the length of the rod is given as \(1 \mathrm{m}\), we can directly use this value as the height (\(h\)). Now, we can find the volume of the cylindrical rod: \(V=\pi (0.025 \mathrm{m})^2 (1 \mathrm{m}) = \pi (0.000625 \mathrm{m^2})(1 \mathrm{m}) = 0.001963495 \mathrm{m^3}\)
03

Calculating the rate of heat generation in each rod

Now that we have the volume of each cylindrical rod, we can find the rate of heat generation in each rod by multiplying the volume by the given heat generation rate: Rate of heat generation in each rod = (volume of rod) \(\times\) (heat generation rate) Rate of heat generation in each rod = \((0.001963495 \mathrm{m^3}) (7\times10^{7} \mathrm{W/m^3}) = 137456.97 \mathrm{W}\) We can express our final answer in kilowatts (kW) as follows: 137456.97 W \(\approx 137 \mathrm{kW}\) So, the rate of heat generation in each rod is approximately \(137 \mathrm{kW}\).

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Most popular questions from this chapter

A spherical shell with thermal conductivity \(k\) has inner and outer radii of \(r_{1}\) and \(r_{2}\), respectively. The inner surface of the shell is subjected to a uniform heat flux of \(\dot{q}_{1}\), while the outer surface of the shell is exposed to convection heat transfer with a coefficient \(h\) and an ambient temperature \(T_{\infty}\). Determine the variation of temperature in the shell wall, and show that the outer surface temperature of the shell can be expressed as $T\left(r_{2}\right)=\left(\dot{q}_{1} / h\right)\left(r_{1} / r_{2}\right)^{2}+T_{\infty}$

A 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.

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 long solid cylinder of radius \(r_{o}=4 \mathrm{~cm}\) and thermal conductivity \(k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Heat is generated in the cylinder uniformly at a rate of $\dot{e}_{\mathrm{gen}}=35 \mathrm{~W} / \mathrm{cm}^{3}$. The side surface of the cylinder is maintained at a constant temperature of \(T_{s}=80^{\circ} \mathrm{C}\). The variation of temperature in the cylinder is given by $$ T(r)=\frac{e_{\mathrm{gen}} r_{o}^{2}}{k}\left[1-\left(\frac{r}{r_{o}}\right)^{2}\right]+T_{s} $$ Based on this relation, determine \((a)\) if the heat conduction is steady or transient, \((b)\) if it is one-, two-, or threedimensional, and \((c)\) the value of heat flux on the side surface of the cylinder at \(r=r_{a r}\)

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.
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