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Chapter 2: Problem 40

The steady-state temperature distribution in a onedimensional wall of thermal conductivity \(k\) and thickness \(L\) is of the form \(T=a x^{3}+b x^{2}+c x+d .\) Derive expressions for the heat generation rate per unit volume in the wall and the heat fluxes at the two wall faces \((x=0, L)\).

Short Answer

Expert verified
The heat generation rate per unit volume in the wall is given by: \(q(x) = -k(6ax + 2b)\) And the heat fluxes at the two wall faces are: \(q_{0} = -kc\) \(q_{L} = -k(3aL^2 + 2bL + c)\)

Step by step solution

01

Calculate the first and second derivatives of the temperature function

First, we need to calculate the first and second derivatives of the given temperature function with respect to x: \(T(x) = ax^3 + bx^2 + cx + d\) First derivative: \(\frac{dT}{dx} = 3ax^2 + 2bx + c\) Second derivative: \(\frac{d^2T}{dx^2} = 6ax + 2b\)
02

Calculate the heat generation rate per unit volume

Now, we can use the heat conduction equation to find the heat generation rate per unit volume (\(q\)): \(\frac{d^2 T}{dx^2} = - \frac{q}{k}\) Substitute the second derivative of T into the equation: \(6ax + 2b = - \frac{q}{k}\) We need to find 'q', so multiply both sides by -k: \(q = -k(6ax + 2b)\) Thus, the heat generation rate per unit volume in the wall is: \(q(x) = -k(6ax + 2b)\)
03

Calculate the heat fluxes at the two wall faces

We can now use the first derivative and the heat flux equation to find the heat fluxes at the two wall faces (x = 0, L): \(q_{x} = -k\frac{dT}{dx}\) Substitute the first derivative of T into the equation: \(q_{x} = -k(3ax^2 + 2bx + c)\) At \(x = 0\), the heat flux is: \(q_{0} = -k(3a(0)^2 + 2b(0) + c)\) Therefore, \(q_{0} = -kc\) At \(x = L\), the heat flux is: \(q_{L} = -k(3aL^2 + 2bL + c)\) In summary, the heat generation rate per unit volume in the wall is: \(q(x) = -k(6ax + 2b)\) And the heat fluxes at the two wall faces are given by: \(q_{0} = -kc\) \(q_{L} = -k(3aL^2 + 2bL + c)\)

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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 when studying heat transfer in materials. Thermal conductivity, represented by the symbol \(k\), is a measure of a material's ability to conduct heat. It specifies the amount of heat that passes through a unit area of the material with a given temperature gradient per unit time. The higher the value of \(k\), the more efficient the material is at transferring heat. For instance, metals like copper have high thermal conductivity, which is why they feel colder on touch compared to wood, which has a low \(k\) value. In the textbook exercise, the coefficient \(k\) appears in the equation \(q(x) = -k(6ax + 2b)\), depicting the relationship between the material's thermal conductivity and the rate of heat being generated per unit volume within the wall.
Temperature Distribution
Temperature distribution refers to how temperature varies within an object. It鈥檚 often described by a mathematical function, such as \(T=ax^3+bx^2+cx+d\) in the provided exercise, which shows how temperature changes along the one-dimensional wall. Uniform temperature distribution implies that every part of the object is at the same temperature, which is rare in real-world scenarios due to various factors such as heat sources and sinks. Non-uniform temperature distribution, as illustrated in the exercise, is characterized by the temperature changing as a function of position. This is essential for predicting how heat will flow through the material, which is based on the principle that heat moves from regions of higher temperature to regions of lower temperature. Understanding temperature distribution helps in controlling heat within systems, such as in building insulation or electronics cooling.
Heat Flux
Heat flux is the rate of heat energy transfer per unit surface area. It's a vector quantity, which means it has both a magnitude and a direction. In the context of the textbook problem, the heat flux \(q_x\) can be calculated using the formula \(q_x = -k\frac{dT}{dx}\). The negative sign indicates that heat flows from higher to lower temperatures, adhering to the second law of thermodynamics. Heat flux can vary within a material if the temperature distribution is non-uniform. For example, at the wall faces \(x = 0\) and \(x = L\), the heat fluxes, \(q_0\) and \(q_L\), are determined not only by the thermal conductivity but also by how steeply the temperature gradient changes at these specific points. Analyzing heat flux is imperative for designing thermal management systems to ensure materials and structures can handle the anticipated heat loads without failure.

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Most popular questions from this chapter

A steam pipe is wrapped with insulation of inner and outer radii \(r_{i}\) and \(r_{o}\), respectively. At a particular instant the temperature distribution in the insulation is known to be of the form $$ T(r)=C_{1} \ln \left(\frac{r}{r_{o}}\right)+C_{2} $$ Are conditions steady-state or transient? How do the heat flux and heat rate vary with radius?

A plane wall that is insulated on one side \((x=0)\) is initially at a uniform temperature \(T_{i}\), when its exposed surface at \(x=L\) is suddenly raised to a temperature \(T_{s}\). (a) Verify that the following equation satisfies the heat equation and boundary conditions: $$ \frac{T(x, t)-T_{s}}{T_{i}-T_{s}}=C_{1} \exp \left(-\frac{\pi^{2}}{4} \frac{\alpha t}{L^{2}}\right) \cos \left(\frac{\pi}{2} \frac{x}{L}\right) $$ where \(C_{1}\) is a constant and \(\alpha\) is the thermal diffusivity. (b) Obtain expressions for the heat flux at \(x=0\) and \(x=L\). (c) Sketch the temperature distribution \(T(x)\) at \(t=0\), at \(t \rightarrow \infty\), and at an intermediate time. Sketch the variation with time of the heat flux at \(x=L, q_{L}^{\prime \prime}(t)\). (d) What effect does \(\alpha\) have on the thermal response of the material to a change in surface temperature?

Consider a plane wall \(100 \mathrm{~mm}\) thick and of thermal conductivity \(100 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Steady-state conditions are known to exist with \(T_{1}=400 \mathrm{~K}\) and \(T_{2}=600 \mathrm{~K}\). Determine the heat flux \(q_{x}^{\prime \prime}\) and the temperature gradient \(d T / d x\) for the coordinate systems shown.

A cylinder of radius \(r_{o}\), length \(L\), and thermal conductivity \(k\) is immersed in a fluid of convection coefficient \(h\) and unknown temperature \(T_{\infty}\). At a certain instant the temperature distribution in the cylinder is \(T(r)=a+b r^{2}\), where \(a\) and \(b\) are constants. Obtain expressions for the heat transfer rate at \(r_{o}\) and the fluid temperature.

A spherical particle of radius \(r_{1}\) experiences uniform thermal generation at a rate of \(\dot{q}\). The particle is encapsulated by a spherical shell of outside radius \(r_{2}\) that is cooled by ambient air. The thermal conductivities of the particle and shell are \(k_{1}\) and \(k_{2}\), respectively, where \(k_{1}=2 k_{2}\). (a) By applying the conservation of energy principle to spherical control volume \(A\), which is placed at an arbitrary location within the sphere, determine a relationship between the temperature gradient \(d T / d r\) and the local radius \(r\), for \(0 \leq r \leq r_{1}\). (b) By applying the conservation of energy principle to spherical control volume \(\mathrm{B}\), which is placed at an arbitrary location within the spherical shell, determine a relationship between the temperature gradient \(d T / d r\) and the local radius \(r\), for \(r_{1} \leq r \leq r_{2}\). (c) On \(T-r\) coordinates, sketch the temperature distribution over the range \(0 \leq r \leq r_{2}\).
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