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Chapter 2: Problem 45

Beginning with a differential control volume in the form of a spherical shell, derive the heat diffusion equation for a one-dimensional, spherical, radial coordinate system with internal heat generation. Compare your result with Equation 2.29.

Short Answer

Expert verified
The heat diffusion equation for a one-dimensional, spherical, radial coordinate system with internal heat generation is derived using a differential control volume in the form of a spherical shell. By applying the conservation of energy, calculating energy inflow, outflow, and generation, and simplifying the equation, we obtain the following heat diffusion equation: \[\rho c\frac{dT}{dt} = \frac{1}{r^2}\bigg(\frac{d}{dr}(r^2 k \frac{dT}{dr})\bigg) + \frac{Q}{k}\] This equation is consistent with Equation 2.29, confirming the correctness of the derivation.

Step by step solution

01

Establish the Initial Conditions

To derive the heat diffusion equation for a one-dimensional, spherical, radial coordinate system with internal heat generation, we will first set up the initial conditions. Consider a spherical shell of radius r and shell thickness dr with a constant volumetric heat generation rate, Q. The sphere has a given thermal conductivity k and a density 蟻.
02

Define the Conservation of Energy

To determine the heat diffusion equation, we will analyze the control volume according to the conservation of energy. This requires calculating the energy balances (inflows, outflows, heat generation, and storage).
03

Calculate the Energy Inflow, Outflow, and Generation

The heat diffusion rate entering the sphere at the radius r can be determined utilizing Fourier's law of heat conduction: \[q_{in} = -kA\frac{dT}{dr}\] where A is the surface area of the inner sphere and is equal to \(4\pi r^2\). Similarly, the heat diffusion rate leaving the sphere at a radius (r + dr) can be determined as: \[q_{out} = -kA\frac{dT}{dr}\Big|_{r+dr}\] The heat generated in the volume element can be assessed as: \[Q_{gen} = QV\] where V is the volume of the spherical shell and is equal to \(4\pi (r^2 dr)\).
04

Calculate the Rate of Energy Storage

The rate of energy storage in the spherical shell is given by the following expression: \[E_{storage} = \rho Vc\frac{dT}{dt}\] where c is the specific heat capacity.
05

Apply Energy Conservation

Applying the conservation of energy to the control volume, we have: \[q_{in} - q_{out} + Q_{gen} = E_{storage}\]
06

Substitute Expressions

Substitute the expressions obtained in Steps 3 and 4 into the conservation of energy equation from Step 5: \[-kA\frac{dT}{dr} - \Big(-kA\frac{dT}{dr}\Big|_{r+dr}\Big) + Q(4\pi r^2 dr) = \rho (4\pi r^2 dr)c\frac{dT}{dt}\]
07

Rearrange and Simplify

Isolate the term \(kA\frac{dT}{dr}\Big|_{r+dr}\) and divide through the initial term: \[\frac{kA\frac{dT}{dr}\Big|_{r+dr} + kA\frac{dT}{dr}}{r^2 dr} = \frac{Q(4\pi r^2 dr) + \rho (4\pi r^2 dr)c\frac{dT}{dt}}{r^2 dr}\] Simplify the equation: \[\frac{k\frac{dT}{dr}\Big|_{r+dr} + k\frac{dT}{dr}}{r^2} = \frac{Q}{k} + \rho c\frac{dT}{dt}\] Finally, we have the derived equation for one-dimensional, spherical, radial coordinates with internal heat generation: \[\rho c\frac{dT}{dt} = \frac{1}{r^2}\bigg(\frac{d}{dr}(r^2 k \frac{dT}{dr})\bigg) + \frac{Q}{k}\]
08

Compare Result with Equation 2.29

The derived equation is consistent with the given Equation 2.29. Therefore, this confirms the derivation correctness for the heat diffusion equation for a one-dimensional, spherical, radial coordinate system with internal heat generation.

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Key Concepts

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

Understanding Spherical Coordinates
Spherical coordinates are a system of defining the position of a point in three-dimensional space using three numbers: the radial distance (r), the polar angle (胃), and the azimuthal angle (蠁). In the context of heat diffusion, we focus on radial symmetry, which simplifies the system to a one-dimensional model since the temperature is only a function of the radial distance (r) from the center.

Picture yourself standing at the center of a sphere, with the radial distance being how far you move away from that center in any direction. As we assess heat diffusion in such coordinates, the equations take on a distinctive form that considers the curvilinear nature of the space. This leads to additional terms in the heat diffusion equation to account for the geometry-specific factors that influence heat transfer.
Fourier's Law of Heat Conduction
Fourier鈥檚 law of heat conduction is a foundational principle that describes how heat energy moves through a material. It states that the rate of heat transfer through a material is proportional to the negative gradient of the temperature and the area through which heat is flowing. Mathematically, we express this law as:
\[q = -kA\frac{dT}{dx}\]
where q is the heat transfer rate, k is the thermal conductivity, A is the surface area, and \(\frac{dT}{dx}\) is the temperature gradient. In a spherical coordinate system, because of the symmetrical nature, the area A can change with the radius, thus introducing a unique consideration to the heat conduction equation specific to spherical systems.
Conservation of Energy in Heat Transfer
The conservation of energy principle is pivotal in thermodynamics and heat transfer. In essence, it dictates that energy cannot be created or destroyed; it can only change forms. Within a control volume, like our spherical shell, we apply this principle to equate the net rate of heat inflow and generation to the rate of energy storage and outflow.

For our spherical coordinate system, every point in the sphere experiences the same energy transformations, assuming radial symmetry. The conservation of energy helps us to set up a balance equation which, when solved, gives us the heat diffusion equation tailor-made for spherical systems, accounting for any internal heat generation.
Internal Heat Generation Effects
Internal heat generation is a term that comes into play when the material or medium we're studying generates heat on its own, apart from any heat transferred to it from its surroundings. This could be due to chemical reactions, radioactive decay, or electrical resistance, among other causes.

In our spherical shell, internal heat generation, represented by Q, adds an additional source term to the heat diffusion equation. It is crucial to account for this source of heat to accurately describe the temperature distribution and understand how the system's temperature changes over time. Not considering this internal heat generation would lead to an incomplete and potentially incorrect analysis of the heat transfer processes within the sphere.

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Most popular questions from this chapter

To determine the effect of the temperature dependence of the thermal conductivity on the temperature distribution in a solid, consider a material for which this dependence may be represented as $$ k=k_{o}+a T $$ where \(k_{o}\) is a positive constant and \(a\) is a coefficient that may be positive or negative. Sketch the steady-state temperature distribution associated with heat transfer in a plane wall for three cases corresponding to \(a>0\), \(a=0\), and \(a<0\).

A hot water pipe with outside radius \(r_{1}\) has a temperature \(T_{1}\). A thick insulation, applied to reduce the heat loss, has an outer radius \(r_{2}\) and temperature \(T_{2}\). On \(T-r\) coordinates, sketch the temperature distribution in the insulation for one-dimensional, steady-state heat transfer with constant properties. Give a brief explanation, justifying the shape of your curve.

A method for determining the thermal conductivity \(k\) and the specific heat \(c_{p}\) of a material is illustrated in the sketch. Initially the two identical samples of diameter \(D=60 \mathrm{~mm}\) and thickness \(L=10 \mathrm{~mm}\) and the thin heater are at a uniform temperature of \(T_{i}=23.00^{\circ} \mathrm{C}\), while surrounded by an insulating powder. Suddenly the heater is energized to provide a uniform heat flux \(q_{o}^{\prime \prime}\) on each of the sample interfaces, and the heat flux is maintained constant for a period of time, \(\Delta t_{o}\). A short time after sudden heating is initiated, the temperature at this interface \(T_{o}\) is related to the heat flux as $$ T_{o}(t)-T_{i}=2 q_{o}^{\prime \prime}\left(\frac{t}{\pi \rho c_{p} k}\right)^{1 / 2} $$ For a particular test run, the electrical heater dissipates \(15.0 \mathrm{~W}\) for a period of \(\Delta t_{o}=120 \mathrm{~s}\), and the temperature at the interface is \(T_{o}(30 \mathrm{~s})=24.57^{\circ} \mathrm{C}\) after \(30 \mathrm{~s}\) of heating. A long time after the heater is deenergized, \(t \geqslant \Delta t_{0}\), the samples reach the uniform temperature of \(T_{o}(\infty)=33.50^{\circ} \mathrm{C}\). The density of the sample materials, determined by measurement of volume and mass, is \(\rho=3965 \mathrm{~kg} / \mathrm{m}^{3}\). Determine the specific heat and thermal conductivity of the test material. By looking at values of the thermophysical properties in Table A.1 or A.2, identify the test sample material.

An electric cable of radius \(r_{1}\) and thermal conductivity \(k_{c}\) is enclosed by an insulating sleeve whose outer surface is of radius \(r_{2}\) and experiences convection heat transfer and radiation exchange with the adjoining air and large surroundings, respectively. When electric current passes through the cable, thermal energy is generated within the cable at a volumetric rate \(\dot{q}\). (a) Write the steady-state forms of the heat diffusion equation for the insulation and the cable. Verify that these equations are satisfied by the following temperature distributions: Insulation: \(T(r)=T_{s, 2}+\left(T_{s, 1}-T_{s, 2}\right) \frac{\ln \left(r / r_{2}\right)}{\ln \left(r_{1} / r_{2}\right)}\) Cable: \(T(r)=T_{s, 1}+\frac{\dot{q} r_{1}^{2}}{4 k_{c}}\left(1-\frac{r^{2}}{r_{1}^{2}}\right)\) Sketch the temperature distribution, \(T(r)\), in the cable and the sleeve, labeling key features. (b) Applying Fourier's law, show that the rate of conduction heat transfer per unit length through the sleeve may be expressed as $$ q_{r}^{\prime}=\frac{2 \pi k_{s}\left(T_{s, 1}-T_{s, 2}\right)}{\ln \left(r_{2} / r_{1}\right)} $$ Applying an energy balance to a control surface placed around the cable, obtain an alternative expression for \(q_{r}^{\prime}\), expressing your result in terms of \(\dot{q}\) and \(r_{1^{*}}\) (c) Applying an energy balance to a control surface placed around the outer surface of the sleeve, obtain an expression from which \(T_{s, 2}\) may be determined as a function of \(\dot{q}, r_{1}, h, T_{\infty}, \varepsilon\), and \(T_{\text {sur- }}\) (d) Consider conditions for which \(250 \mathrm{~A}\) are passing through a cable having an electric resistance per unit length of \(R_{e}^{\prime}=0.005 \Omega / \mathrm{m}\), a radius of \(r_{1}=15 \mathrm{~mm}\), and a thermal conductivity of \(k_{c}=200 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). For \(k_{s}=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \quad r_{2}=15.5 \mathrm{~mm}, \quad h=25\) \(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}, \varepsilon=0.9, T_{\text {o }}=25^{\circ} \mathrm{C}\), and \(T_{\text {sur }}=35^{\circ} \mathrm{C}\), evaluate the surface temperatures, \(T_{s, 1}\) and \(T_{s, 2}\), as well as the temperature \(T_{o}\) at the centerline of the cable. (e) With all other conditions remaining the same, compute and plot \(T_{o}, T_{s, 1}\), and \(T_{s, 2}\) as a function of \(r_{2}\) for \(15.5 \leq r_{2} \leq 20 \mathrm{~mm}\).

A plane wall has constant properties, no internal heat generation, and is initially at a uniform temperature \(T_{i \cdot}\) Suddenly, the surface at \(x=L\) is heated by a fluid at \(T_{\infty}\) having a convection coefficient \(h\). At the same instant, the electrical heater is energized, providing a constant heat flux \(q_{o}^{\prime \prime}\) at \(x=0\). (a) On \(T-x\) coordinates, sketch the temperature distributions for the following conditions: initial condition \((t \leq 0)\), steady-state condition \((t \rightarrow \infty)\), and for two intermediate times. (b) On \(q_{x}^{\prime \prime}-x\) coordinates, sketch the heat flux corresponding to the four temperature distributions of part (a). (c) On \(q_{x}^{n}-t\) coordinates, sketch the heat flux at the locations \(x=0\) and \(x=L\). That is, show qualitatively how \(q_{x}^{\prime \prime}(0, t)\) and \(q_{x}^{\prime \prime}(L, t)\) vary with time. (d) Derive an expression for the steady-state temperature at the heater surface, \(T(0, \infty)\), in terms of \(q_{o}^{\prime \prime}\), \(T_{\infty}, k, h\), and \(L\).
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