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Chapter 2: Problem 20

Write down the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and heat generation in its simplest form, and indicate what each variable represents.

Short Answer

Expert verified
Answer: The equation is given by: \[ \rho C_p \frac{\partial T}{\partial t} = k \frac{\partial^2 T}{\partial x^2} + q_g \] Where the variables represent: - \( \rho \): density of the material (kg/m鲁) - \( C_p \): specific heat of the material (J/kg路K) - \( T \): temperature (K) - \( t \): time (s) - \( k \): thermal conductivity (W/m路K) - \( x \): coordinate along the x-axis (m) - \( q_g \): heat generation per unit volume (W/m鲁)

Step by step solution

01

Write down the general heat conduction equation

The general three-dimensional transient heat conduction equation is given by: \[ \rho C_p \frac{\partial T}{\partial t} = k\left(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\right) + q_g \] Where: - \( \rho \) : density of the material (kg/ m^3) - \( C_p \) : specific heat of the material (J/ kg 路 K) - \( T \) : temperature (K) - \( t \) : time (s) - \( k \) : thermal conductivity (W/m路K) - \(x, y, z\) : the coordinates along the Cartesian axis (m) - \(q_g\) : heat generation per unit volume (W/m^3)
02

Simplify the equation for one dimension and constant thermal conductivity

As the problem considers the one-dimensional case for a plane wall with constant thermal conductivity, we only need to consider the temperature change along one axis (let's say the x-axis). Therefore, we can simplify the heat conduction equation as follows: \[ \rho C_p \frac{\partial T}{\partial t} = k \frac{\partial^2 T}{\partial x^2} + q_g \] This is the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and heat generation in its simplest form.
03

List each variable and its representation

The variables in the equation are: - \( \rho \) (density of the material): represents the mass of the material per unit volume, given in kg/m^3 - \( C_p \) (specific heat of the material): represents the amount of heat energy required to raise the temperature of a unit mass of the material by 1 degree Celsius, given in J/kg路K - \( T \) (temperature): the temperature of the material at a given point, given in Kelvin (K) - \( t \) (time): the time at which the temperature is evaluated, given in seconds (s) - \( k \) (thermal conductivity): represents the ability of the material to conduct heat, given in W/m路K - \( x \) (coordinate along the x-axis): the position along the plane wall where the temperature is evaluated, given in meters (m) - \( q_g \) (heat generation per unit volume): represents the rate of heat generation within the material per unit volume, given in W/m^3

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Key Concepts

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

Thermal Conductivity
Understanding thermal conductivity is crucial to solving problems in heat transfer like the one given in the exercise. Thermal conductivity, represented by the symbol \( k \), is a material property that indicates how easily heat can pass through a material. The units of \( k \) are W/m路K, which stands for watts per meter per degree Kelvin. Materials with high \( k \) values are good heat conductors (like metals), whereas materials with low \( k \) values are insulators (like wood or foam).

When we say that a material has constant thermal conductivity, we mean that \( k \) remains the same regardless of changes in temperature, which is an approximation often used for a limited temperature range. This assumption simplifies the heat equation because the thermal conductivity can be moved outside the derivative in the equation, making it easier to solve.
Heat Generation
Heat generation within a material is described by the term \( q_g \) in the heat conduction equation. It is the rate at which heat is produced per unit volume, typically measured in watts per cubic meter (W/m^3). Sources of heat generation can include chemical reactions, electrical currents, or radiation absorption. This term is crucial in applications where heat isn't just transferred but also created within the boundaries of the system, such as in electrical devices or reactors.

In the context of the exercise's equation, heat generation is considered to be uniformly distributed throughout the material. When heat is generated within a body, it adds to the heat that flows through the material by conduction, thereby influencing the temperature distribution and making it an essential aspect of transient heat conduction analysis.
One-Dimensional Heat Conduction
One-dimensional heat conduction refers to heat transfer in only one spatial dimension. In real-world scenarios, heat conduction can occur in three dimensions, but many problems, like the one presented in the exercise, assume that heat transfer predominantly happens in a single direction. This approximation simplifies the mathematical description greatly and is often valid when the temperature gradient in the other two directions is negligible compared to the one being studied.

In one-dimensional problems, we typically focus on how temperature changes with time at different locations along a single axis, typically denoted as \( x \) in Cartesian coordinates. By considering heat transfer in a plane wall, the heat equation further simplifies. In the given solution, the equation produced is a one-dimensional transient heat conduction equation, essential for predicting how temperature within the wall changes over time given the material's properties, initial conditions, and any heat generation.

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Most popular questions from this chapter

Consider a large plane wall of thickness \(L=0.05 \mathrm{~m}\). The wall surface at \(x=0\) is insulated, while the surface at \(x=L\) is maintained at a temperature of \(30^{\circ} \mathrm{C}\). The thermal conductivity of the wall is \(k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and heat is generated in the wall at a rate of \(\dot{e}_{\text {gen }}=\dot{e}_{0} e^{-0.5 x / L} \mathrm{~W} / \mathrm{m}^{3}\) where \(\dot{e}_{0}=8 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}\). Assuming steady one-dimensional heat transfer, \((a)\) express the differential equation and the boundary conditions for heat conduction through the wall, \((b)\) obtain a relation for the variation of temperature in the wall by solving the differential equation, and (c) determine the temperature of the insulated surface of the wall.

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 \(T_{\infty 1}=27^{\circ} \mathrm{C}\) and \(h_{1}=5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) 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 equation 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.

Consider a \(1.5\)-m-high and \(0.6-\mathrm{m}\)-wide plate whose thickness is \(0.15 \mathrm{~m}\). One side of the plate is maintained at a constant temperature of \(500 \mathrm{~K}\) while the other side is maintained at \(350 \mathrm{~K}\). The thermal conductivity of the plate can be assumed to vary linearly in that temperature range as \(k(T)=\) \(k_{0}(1+\beta T)\) where \(k_{0}=18 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\beta=8.7 \times 10^{-4} \mathrm{~K}^{-1}\). Disregarding the edge effects and assuming steady onedimensional heat transfer, determine the rate of heat conduction through the plate. Answer: \(22.2 \mathrm{~kW}\)

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 \(r=r_{1}\).

A pipe is used for transporting boiling water in which the inner surface is at \(100^{\circ} \mathrm{C}\). The pipe is situated in a surrounding where the ambient temperature is \(20^{\circ} \mathrm{C}\) and the convection heat transfer coefficient is \(50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The pipe has a wall thickness of \(3 \mathrm{~mm}\) and an inner diameter of \(25 \mathrm{~mm}\), and it has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where \(k_{0}=1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=0.003 \mathrm{~K}^{-1}\) and \(T\) is in \(\mathrm{K}\). Determine the outer surface temperature of the pipe.
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