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Chapter 2: Problem 21

Write down the one-dimensional transient heat conduction equation for a long cylinder with constant thermal conductivity and heat generation, and indicate what each variable represents.

Short Answer

Expert verified
#Short Answer# The one-dimensional transient heat conduction equation for a long cylinder with constant thermal conductivity and internal heat generation is: \(\frac{1}{r}\frac{\partial}{\partial r}(r \frac{\partial T}{\partial r}) + \frac{q_g}{k} = \rho C_p \frac{\partial T}{\partial t}\) where T represents the temperature, r is the radial distance from the center of the cylinder, t is the time, k is the constant thermal conductivity, \(q_g\) denotes heat generation per unit volume, 蟻 symbolizes material density, and \(C_p\) stands for the specific heat capacity of the material.

Step by step solution

01

1. Heat Conduction Equation for a Long Cylinder

For a long cylinder, we will use the cylindrical coordinate system. The one-dimensional heat conduction equation, considering internal heat generation and constant thermal conductivity, is given by: \(\frac{1}{r}\frac{\partial}{\partial r}(r \frac{\partial T}{\partial r}) + \frac{q_g}{k} = \rho C_p \frac{\partial T}{\partial t}\) where: T: Temperature (of the cylinder) r: Radial distance from the center of the cylinder t: Time k: Thermal conductivity of the material (constant) \(q_g\): Heat generation per unit volume \(\rho\): Density of the material \(C_p\): Specific heat capacity of the material
02

2. Explanation of Variables

Each variable in the heat conduction equation represents the following: - T (Temperature): It describes the temperature distribution within the cylinder as a function of radial distance and time. - r (Radial distance): It represents the distance from the center of the cylinder. We are only considering the one-dimensional heat transfer in the radial direction. - t (Time): It represents the time variable and helps us to understand how the temperature distribution is changing with time. - k (Thermal conductivity): It is a material property that determines how efficiently heat is conducted through the material. For a constant thermal conductivity, this value does not change with temperature. - \(q_g\) (Heat generation per unit volume): It represents the rate at which heat is generated within the material per unit volume. For a constant heat generation, this value does not change with temperature, distance, or time. - 蟻 (Density): It is a material property that represents the mass per unit volume of the material. - \(C_p\) (Specific heat capacity): It is a material property that measures the amount of heat required to raise the temperature of a unit mass of the material by one degree Celsius/Kelvin.

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Key Concepts

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

Cylindrical Coordinate System
When dealing with objects like long cylinders, it's essential to use a coordinate system that matches their geometry. Enter the cylindrical coordinate system. This system is perfect for problems with radial symmetry, meaning they look the same along their length. Rather than the traditional x, y, z coordinates, we switch to r, 胃, z.
The primary focus here is on the radial distance (r) from the center of the cylinder, especially when examining heat flow.
  • **Radial Distance (r):** Measures how far you are from the cylinder's center.
  • **Angular Coordinate (胃):** Represents the rotational angle around the cylinder's axis, but often not needed in radial-only problems.
  • **Axial Distance (z):** Goes along the cylinder's length but isn鈥檛 used in purely radial problems.
The advantage of this system is its alignment with the natural symmetry of cylindrical objects, simplifying the mathematical description of phenomena like heat conduction.
Transient Heat Conduction
Transient heat conduction deals with how temperature changes over time. This form of heat conduction is not in a steady state; instead, it varies.
For a cylinder, this means the temperature inside changes as time passes, influenced by factors like heat generation or varying external conditions.
The governing equation for transient heat conduction in a cylinder involves partial derivatives, which account for changes in temperature both over time and across different radial positions.
  • The left side of the equation includes terms describing spatial changes in temperature.
  • The right side focuses on how temperature evolves with time.
This equation helps predict how heat will spread within the cylinder, taking into account internal heat sources and other boundary conditions.
Thermal Conductivity
Thermal conductivity (k) is a material property that measures a material's ability to conduct heat. Different materials transfer heat more or less efficiently, depending on this property.
In the context of our equation, a constant thermal conductivity means the material conducts heat uniformly, regardless of temperature changes.
  • **High Thermal Conductivity:** Materials like metals that easily transfer heat.
  • **Low Thermal Conductivity:** Insulators, such as rubber or wood, that resist heat flow.
Understanding thermal conductivity is crucial for predicting how quickly heat can move through a material, directly affecting the thermal response of objects like our cylinder.
Specific Heat Capacity
Specific heat capacity (Cp) is another important material property. It indicates how much energy is needed to raise the temperature of a specific amount of material by one degree.
This property helps determine how the cylinder heats up or cools down as energy is added or removed.
  • A material with high specific heat can store more heat without a significant change in temperature.
  • A material with low specific heat heats up or cools down quickly with added energy.
Knowing the specific heat capacity allows us to determine how much energy is needed for temperature changes, vital for energy efficiency and thermal regulation in various applications.

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Most popular questions from this chapter

Consider a 20-cm-thick large concrete plane wall \((k=0.77 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) subjected to convection on both sides with \(T_{\infty 1}=22^{\circ} \mathrm{C}\) and \(h_{1}=8 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the inside, and \(T_{\infty 2}=8^{\circ} \mathrm{C}\) and \(h_{2}=12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the outside. Assuming constant thermal conductivity with no heat generation and negligible radiation, \((a)\) express the differential equations and the boundary conditions for steady one-dimensional heat conduction through the wall, \((b)\) obtain a relation for the variation of temperature in the wall by solving the differential equation, and \((c)\) evaluate the temperatures at the inner and outer surfaces of the wall.

A solar heat flux \(\dot{q}_{s}\) is incident on a sidewalk whose thermal conductivity is \(k\), solar absorptivity is \(\alpha_{s}\), and convective heat transfer coefficient is \(h\). Taking the positive \(x\) direction to be towards the sky and disregarding radiation exchange with the surroundings surfaces, the correct boundary condition for this sidewalk surface is (a) \(-k \frac{d T}{d x}=\alpha_{s} \dot{q}_{s}\) (b) \(-k \frac{d T}{d x}=h\left(T-T_{\infty}\right)\) (c) \(-k \frac{d T}{d x}=h\left(T-T_{\infty}\right)-\alpha_{s} \dot{q}_{s}\) (d) \(h\left(T-T_{\infty}\right)=\alpha_{s} \dot{q}_{s}\) (e) None of them

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 \(\dot{e}_{\text {gen }}=35 \mathrm{~W} / \mathrm{cm}^{3}\). 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 $$ T(r)=\frac{\dot{e}_{\text {gen }} r_{o}^{2}}{k}\left[1-\left[1-\left(\frac{r}{r_{o}}\right)^{2}\right]+T_{s}\right. $$ Based on this relation, determine \((a)\) if the heat conduction is steady or transient, \((b)\) if it is one-, two-, or three-dimensional, and \((c)\) the value of heat flux on the side surface of the cylinder at \(r=r_{o^{*}}\)

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 \(15^{\circ} \mathrm{C}\), the vessel's outer surface experiences convection heat transfer with a coefficient of \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). To prevent thermal burn on skin tissues, 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.

Consider a spherical shell of inner radius \(r_{1}\), outer radius \(r_{2}\), thermal conductivity \(k\), and emissivity \(\varepsilon\). The outer surface of the shell is subjected to radiation to surrounding surfaces at \(T_{\text {surr }}\), but the direction of heat transfer is not known. Express the radiation boundary condition on the outer surface of the shell.
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