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Chapter 2: Problem 30

Consider a medium in which the heat conduction equation is given in its simplest form as $$ \frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial T}{\partial r}\right)=\frac{1}{\alpha} \frac{\partial T}{\partial t} $$ (a) Is heat transfer steady or transient? (b) Is heat transfer one-, two-, or three-dimensional? (c) Is there heat generation in the medium? (d) Is the thermal conductivity of the medium constant or variable?

Short Answer

Expert verified
Answer: The heat transfer process is transient. Explanation: The presence of the term \(\frac{\partial T}{\partial t}\) denotes a time-dependent temperature change, indicating a transient heat transfer process. (b) Is the heat transfer process in the medium one-, two-, or three-dimensional? Answer: The heat transfer process is one-dimensional. Explanation: The given equation contains only the radial derivative term \(\frac{\partial}{\partial r}\), representing one-dimensional heat transfer in the radial direction. (c) Is there heat generation in the medium? Answer: No, there is no heat generation in the medium. Explanation: The given equation does not contain any term representing heat generation (such as Q). (d) Is the thermal conductivity of the medium constant or variable? Answer: The thermal conductivity of the medium is constant. Explanation: The given equation does not explicitly include the thermal conductivity (k). However, the thermal diffusivity term 伪 is related to thermal conductivity. Since the equation is given in its simplest form, we can assume that the thermal conductivity (and thus, thermal diffusivity) is constant.

Step by step solution

01

Steady or Transient Heat Transfer

To determine if the heat transfer is steady or transient, we need to look at the time-dependency term in the equation. The term \(\frac{\partial T}{\partial t}\) represents the rate of change of temperature with respect to time. Since it is present in the equation and not equal to zero, the heat transfer is transient, which means the temperature changes with time.
02

One-, Two-, or Three-Dimensional Heat Transfer

As the given equation only contains the radial derivative term \(\frac{\partial}{\partial r}\), it is describing a one-dimensional (1D) heat transfer process in the radial direction.
03

Heat Generation

There is no term representing heat generation in the given equation (often represented by a term like Q). Therefore, we can conclude that there is no heat generation in the medium.
04

Constant or Variable Thermal Conductivity

The given equation does not include the thermal conductivity (usually denoted by k) explicitly. However, the term \(\alpha\) represents the thermal diffusivity, which is related to thermal conductivity through the relationship: $$\alpha = \frac{k}{\rho c_p}$$, where \(\rho\) is the density and \(c_p\) is the specific heat capacity. Since the equation is given in its simplest form, we can assume that the thermal conductivity (and therefore, the thermal diffusivity) is constant.

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Key Concepts

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

Transient Heat Transfer
Transient heat transfer refers to situations where the temperature in a system changes with time. This is different from steady-state heat transfer, where the temperature remains constant over time. In the context of the heat conduction equation, the term \( \frac{\partial T}{\partial t} \) indicates that the temperature \( T \) changes as time \( t \) progresses. If this term is present and not zero, as in the given exercise, the heat transfer is non-steady, or transient. This means conditions like initial temperature distribution or environmental changes impact the temperature over time.
  • The concept of transient heat transfer is crucial in scenarios where systems require time to reach equilibrium.
  • Examples include cooling of electronic devices or heating of building materials.
  • Understanding the transient nature helps in predicting how long it takes for a system to achieve thermal stability.
In most practical applications, transient heat transfer problems consider initial conditions, such as the initial temperature distribution. This factor plays a significant role in defining how temperatures evolve with time.
Radial Heat Transfer
In many engineering applications, heat transfer occurs in a radial direction, meaning it moves outward from a center point or inward toward it. The equation from the exercise features radial derivatives \( \frac{\partial}{\partial r} \), showing that the heat transfer is radial. This one-dimensional radial flow is prevalent in cylindrical or spherical systems, where heat moves either radially inwards or outwards.
  • An example of radial heat transfer is the process in pipes or reactors, where heat flows from the center towards the outer surface.
  • This is crucial in the design of insulation materials or thermal liners to manage energy losses.
  • Analyzing radial heat transfer helps in precise temperature distribution predictions in systems where geometry plays a significant role.
The radial heat transfer approach simplifies complex multi-dimensional scenarios into more manageable one-dimensional problems. This is particularly useful in designing efficient thermal management systems.
Thermal Diffusivity
Thermal diffusivity is a material-specific property indicating how quickly heat spreads through it. It combines three key parameters: thermal conductivity, density, and specific heat capacity. Mathematically, it is expressed as \( \alpha = \frac{k}{\rho c_p} \), where \( k \) is thermal conductivity, \( \rho \) is the density, and \( c_p \) is the specific heat capacity. Thermal diffusivity represents the material鈥檚 ability to conduct thermal energy relative to its ability to store it.
  • A high thermal diffusivity means heat transfers rapidly through the material, indicating efficient thermal conduction.
  • Conversely, low thermal diffusivity suggests the material retains heat, slowing down temperature changes.
  • Applications include designing materials for electronics that require rapid heat dissipation.
Understanding thermal diffusivity is essential for selecting appropriate materials in systems requiring efficient heating or cooling. In the exercise, the constant \( \alpha \) implies uniformity in material properties, aiding in simplified calculations for heat distribution analysis.

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Most popular questions from this chapter

Consider a long rectangular bar of length \(a\) in the \(x-\) direction and width \(b\) in the \(y\)-direction that is initially at a uniform temperature of \(T_{i}\). The surfaces of the bar at \(x=0\) and \(y=0\) are insulated, while heat is lost from the other two surfaces by convection to the surrounding medium at temperature \(T_{\infty}\) with a heat transfer coefficient of \(h\). Assuming constant thermal conductivity and transient two-dimensional heat transfer with no heat generation, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve.

Consider a steam pipe of length \(L=30 \mathrm{ft}\), inner radius \(r_{1}=2\) in, outer radius \(r_{2}=2.4\) in, and thermal conductivity \(k=7.2 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\). Steam is flowing through the pipe at an average temperature of \(300^{\circ} \mathrm{F}\), and the average convection heat transfer coefficient on the inner surface is given to be \(h=12.5 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\). If the average temperature on the outer surfaces of the pipe is \(T_{2}=175^{\circ} \mathrm{F},(a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the pipe, \((b)\) obtain a relation for the variation of temperature in the pipe by solving the differential equation, and \((c)\) evaluate the rate of heat loss from the steam through the pipe.

A large steel plate having a thickness of \(L=4\) in, thermal conductivity of \(k=7.2 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\), and an emissivity of \(\varepsilon=0.7\) is lying on the ground. The exposed surface of the plate at \(x=L\) is known to exchange heat by convection with the ambient air at \(T_{\infty}=90^{\circ} \mathrm{F}\) with an average heat transfer coefficient of \(h=12 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\) as well as by radiation with the open sky with an equivalent sky temperature of \(T_{\text {sky }}=480 \mathrm{R}\). Also, the temperature of the upper surface of the plate is measured to be \(80^{\circ} \mathrm{F}\). Assuming steady onedimensional heat transfer, \((a)\) express the differential equation and the boundary conditions for heat conduction through the plate, \((b)\) obtain a relation for the variation of temperature in the plate by solving the differential equation, and \((c)\) determine the value of the lower surface temperature of the plate at \(x=0\).

Consider a cylindrical shell of length \(L\), inner radius \(r_{1}\), and outer radius \(r_{2}\) whose thermal conductivity varies in a specified temperature range as \(k(T)=k_{0}\left(1+\beta T^{2}\right)\) 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 the heat transfer rate through the shell.

A 6-m-long 3-kW electrical resistance wire is made of \(0.2\)-cm-diameter stainless steel \((k=15.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). The resistance wire operates in an environment at \(20^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(175 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) at the outer surface. Determine the surface temperature of the wire \((a)\) by using the applicable relation and \((b)\) by setting up the proper differential equation and solving it. Answers: (a) \(475^{\circ} \mathrm{C}\), (b) \(475^{\circ} \mathrm{C}\)
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