/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Consider a cylindrical nuclear f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 14

Consider a cylindrical nuclear fuel rod of length \(L\) and diameter \(D\) that is encased in a concentric tube. Pressurized water flows through the annular region between the rod and the tube at a rate \(\dot{m}\), and the outer surface of the tube is well insulated. Heat generation occurs within the fuel rod, and the volumetric generation rate is known to vary sinusoidally with distance along the rod. That is, \(\dot{q}(x)=\dot{q}_{o} \sin (\pi x / L)\), where \(\dot{q}_{o}\left(\mathrm{~W} / \mathrm{m}^{3}\right)\) is a constant. A uniform convection coefficient \(h\) may be assumed to exist between the surface of the rod and the water. (a) Obtain expressions for the local heat flux \(q^{\prime \prime}(x)\) and the total heat transfer \(q\) from the fuel rod to the water. (b) Obtain an expression for the variation of the mean temperature \(T_{m}(x)\) of the water with distance \(x\) along the tube. (c) Obtain an expression for the variation of the rod surface temperature \(T_{s}(x)\) with distance \(x\) along the tube. Develop an expression for the \(x\)-location at which this temperature is maximized.

Short Answer

Expert verified
In this problem, the local heat flux $q^{\prime \prime}(x) = \frac{\dot{q}(x)}{A}$, and after integrating over the length and perimeter of the fuel rod, the total heat transfer $q$ is obtained. The mean temperature of the water, $T_{m}(x)$, can be found using energy balance, and the rod surface temperature variation, $T_s(x)$, can be obtained using Newton's law of cooling. To find the x-location where the rod surface temperature is maximized, differentiate $T_s(x)$ with respect to x and set the derivative equal to zero, then solve for the x-location.

Step by step solution

01

Obtain expression for the local heat flux

Local heat flux is the rate of heat transfer per unit area, so we'll have to divide the volumetric generation rate by the rod cross-sectional area to have an expression for local heat flux. Use Fourier's law of conduction: \[q^{\prime \prime}(x) = -k\nabla T\] for a cylindrical geometry, we get: \[q^{\prime \prime}(x) = -\frac{d}{dr}\left(k\frac{d T}{dr}\right)\] since we are given a one-dimensional conduction source term, \(\dot{q}(x) = \dot{q}_{o} \sin\left(\frac{\pi x}{L}\right)\), we can rewrite Fourier's law of conduction for one-dimensional case as: \[ q^{\prime \prime}(x) = \frac{\dot{q}(x)}{A}\], with A being the cross-sectional area.
02

Obtain expression for the total heat transfer

Integrate the local heat flux expression over the length and perimeter of the fuel rod to obtain the total heat transfer from the fuel rod to the water: \[q = \int_{0}^{L} \int_{0}^{2 \pi R} q^{\prime \prime}(x) r dr d\theta\]
03

Obtain expression for the mean temperature of the water with distance x

To find the mean temperature of the water with distance x, use energy balance: \[ \dot{m} c_{p} \left(\frac{dT_{m}}{dx}\right) = -\int_{0}^{2\pi R} q^{\prime \prime}(x)rdr d\theta\]
04

Obtain expression for the rod surface temperature variation with distance x

Use Newton's law of cooling for the convection: \[q^{\prime \prime}(x) = h \left( T_s(x) - T_{m}(x) \right)\]
05

Find the x-location at which the rod surface temperature is maximized.

We have the expression for the rod surface temperature in terms of local heat flux and mean temperature of the water. To find the x-location where the surface temperature is maximized, we need help of calculus. Differentiate the expression for the rod surface temperature with respect to x and set the derivative equal to zero. Solve for the x-location.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nuclear Fuel Rod
A nuclear fuel rod is a slender, often cylindrical, component used in nuclear reactors. It contains nuclear fuel, such as uranium or plutonium, which undergoes nuclear fission to generate heat. This heat is crucial for producing steam that drives turbines for electricity generation. In nuclear reactors, multiple fuel rods are grouped together to form a fuel assembly, and they are immersed in a coolant, usually water, which carries away the heat.

The focus of the exercise is a cylindrical nuclear fuel rod with heat generation varying along its length. Understanding the distribution of heat along the rod helps design monitoring schemes and ensure safe reactor operation. Efficient heat transfer from the fuel rod to the coolant is vital for maintaining structural integrity and preventing overheating.
Sinusoidal Heat Generation
Sinusoidal heat generation refers to the variations in the rate of heat production along the nuclear fuel rod. In this scenario, the volumetric heat generation rate inside the rod changes as a sine wave along its length. This sinusoidal variation is characterized mathematically by \(\dot{q}(x) = \dot{q}_{o} \sin (\pi x / L)\), where \(\dot{q}_{o}\) is a constant and \(x\) is the position along the rod.

Such sinusoidal heat distribution might represent real-world fluctuations in heat production due to reactor dynamics. Evaluating how heat generation varies along the rod helps in analyzing parameters like temperature distribution, which is essential for maintaining efficient heat transfer and preventing thermal stress. By understanding this concept, you can apply it to predict the behavior of heat in more complex geometries.
Convection Coefficient
The convection coefficient \(h\) represents how effectively heat is transferred from the rod's surface to the moving fluid, usually water in this case. This coefficient influences the rate of heat transfer through convection, which is a primary method of removing heat from the nuclear fuel rod. The value of \(h\) depends on factors like fluid velocity, viscosity, and surface conditions of the rod.

Using Newton's Law of Cooling, the heat flux at the surface \(q^{\prime \prime}(x)\) can be described by the formula: \[q^{\prime \prime}(x) = h \left( T_s(x) - T_{m}(x) \right)\],where \(T_s(x)\) is the surface temperature of the rod, and \(T_m(x)\) is the mean temperature of the coolant.

This relationship helps us understand how the heat transfer mechanism works and ensures that the rod's temperature is regulated effectively. In reactor design, careful evaluation of \(h\) is crucial to maintain safe operating temperatures.
Cylindrical Geometry
Cylindrical geometry is essential to understanding the heat transfer characteristics of a nuclear fuel rod. The rod's shape influences how heat travels through and away from it. In cylindrical coordinates, heat conduction is primarily a radially outward process, given the geometry of the rod.
  • The local heat flux \(q^{\prime \prime}(x)\) is obtained by dividing the volumetric heat generation rate by the cross-sectional area of the rod.
  • This value represents how heat is conducted across the rod's surface.
For an exact understanding, applying Fourier's law of conduction in cylindrical coordinates is necessary.

Integrating the expression over the rod's surface and length provides the total heat transfer. Analyzing and understanding cylindrical coordinates allows precise calculations of temperature distributions and gradients, critical for ensuring the rod's structural integrity under operational conditions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An ideal gas flows within a small diameter tube. Derive an expression for the transition density of the gas \(\rho_{c}\) below which microscale effects must be accounted for. Express your result in terms of the gas molecule diameter, universal gas constant, Boltzmann's constant, and the tube diameter. Evaluate the transition density for a \(D=10-\mu \mathrm{m}\)-diameter tube for hydrogen, air, and carbon dioxide. Compare the calculated transition densities with the gas density at atmospheric pressure and \(T=23^{\circ} \mathrm{C}\).

Heat is to be removed from a reaction vessel operating at \(75^{\circ} \mathrm{C}\) by supplying water at \(27^{\circ} \mathrm{C}\) and \(0.12 \mathrm{~kg} / \mathrm{s}\) through a thin-walled tube of \(15-\mathrm{mm}\) diameter. The convection coefficient between the tube outer surface and the fluid in the vessel is \(3000 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\). (a) If the outlet water temperature cannot exceed \(47^{\circ} \mathrm{C}\), what is the maximum rate of heat transfer from the vessel? (b) What tube length is required to accomplish the heat transfer rate of part (a)?

One way to cool chips mounted on the circuit boards of a computer is to encapsulate the boards in metal frames that provide efficient pathways for conduction to supporting cold plates. Heat generated by the chips is then dissipated by transfer to water flowing through passages drilled in the plates. Because the plates are made from a metal of large thermal conductivity (typically aluminium or copper), they may be assumed to be at a temperature, \(T_{s, c p^{-}}\) (a) Consider circuit boards attached to cold plates of height \(H=750 \mathrm{~mm}\) and width \(L=600 \mathrm{~mm}\), each with \(N=10\) holes of diameter \(D=10 \mathrm{~mm}\). If operating conditions maintain plate temperatures of \(T_{\text {s.tp }}=32^{\circ} \mathrm{C}\) with water flow at \(\dot{m}_{1}=0.2 \mathrm{~kg} / \mathrm{s}\) per passage and \(T_{m, i}=7^{\circ} \mathrm{C}\), how much heat may be dissipated by the circuit boards? (b) To enhance cooling, thereby allowing increased power generation without an attendant increase in system temperatures, a hybrid cooling scheme may be used. The scheme involves forced airflow over the encapsulated circuit boards, as well as water flow through the cold plates. Consider conditions for which \(N_{\mathrm{cb}}=10\) circuit boards of width \(W=350 \mathrm{~mm}\) are attached to the cold plates and their average surface temperature is \(T_{s, \text { do }}=47^{\circ} \mathrm{C}\) when \(T_{s, \text { ep }}=32^{\circ} \mathrm{C}\). If air is in parallel flow over the plates with \(u_{\infty}=10 \mathrm{~m} / \mathrm{s}\) and \(T_{\infty}=7^{\circ} \mathrm{C}\), how much of the heat generated by the circuit boards is transferred to the air?

A mass transfer operation is preceded by laminar flow of a gaseous species B through a circular tube that is sufficiently long to achieve a fully developed velocity profile. Once the fully developed condition is reached, the gas enters a section of the tube that is wetted with a liquid film (A). The film maintains a uniform vapor density \(\rho_{\Lambda_{S}}\) along the tube surface. (a) Write the differential equation and boundary conditions that govern the species A mass density distribution, \(\rho_{A}(x, r)\), for \(x>0\). (b) What is the heat transfer analog to this problem? From this analog, write an expression for the average Sherwood number associated with mass exchange over the region \(0 \leq x \leq L\). (c) Beginning with application of conservation of species to a differential control volume of extent \(\pi r_{o}^{2} d x\), derive an expression (Equation 8.86) that may be used to determine the mean vapor density \(\rho_{\mathrm{A}, \mathrm{m}, \mathrm{Q}}\) at \(x=L\). (d) Consider conditions for which species \(\mathrm{B}\) is air at \(25^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) and the liquid film consists of water, also at \(25^{\circ} \mathrm{C}\). The flow rate is \(\dot{m}=2.5 \times 10^{-4} \mathrm{~kg} / \mathrm{s}\), and the tube diameter is \(D=10 \mathrm{~mm}\). What is the mean vapor density at the tube outlet if \(L=1 \mathrm{~m}\) ?

Water at \(20^{\circ} \mathrm{C}\) and a flow rate of \(0.1 \mathrm{~kg} / \mathrm{s}\) enters a heated, thin-walled tube with a diameter of \(15 \mathrm{~mm}\) and length of \(2 \mathrm{~m}\). The wall heat flux provided by the heating elements depends on the wall temperature according to the relation $$ q_{s}^{\prime \prime}(x)=q_{s, o}^{\prime \prime}\left[1+\alpha\left(T_{s}-T_{\mathrm{ref}}\right)\right] $$ where \(q_{s, \rho}^{\prime \prime}=10^{4} \mathrm{~W} / \mathrm{m}^{2}, \alpha=0.2 \mathrm{~K}^{-1}, T_{\text {ref }}=20^{\circ} \mathrm{C}\), and \(T_{s}\) is the wall temperature in \({ }^{\circ} \mathrm{C}\). Assume fully developed flow and thermal conditions with a convection coefficient of \(3000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Beginning with a properly defined differential control volume in the tube, derive expressions for the variation of the water, \(T_{m}(x)\), and the wall, \(T_{s}(x)\), temperatures as a function of distance from the tube inlet. (b) Using a numerical integration scheme, calculate and plot the temperature distributions, \(T_{m}(x)\) and \(T_{s}(x)\), on the same graph. Identify and comment on the main features of the distributions. Hint: The \(I H T\) integral function \(D E R\left(T_{m}, x\right)\) can be used to perform the integration along the length of the tube. (c) Calculate the total rate of heat transfer to the water.
See all solutions

Recommended explanations on Physics Textbooks

Energy Physics

Read Explanation

Atoms and Radioactivity

Read Explanation

Thermodynamics

Read Explanation

Magnetism

Read Explanation

Physical Quantities And Units

Read Explanation

Force

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.