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Chapter 2: Problem 20

Consider a small but known volume of metal that has a large thermal conductivity. (a) Since the thermal conductivity is large, spatial temperature gradients that develop within the metal in response to mild heating are small. Neglecting spatial temperature gradients, derive a differential equation that could be solved for the temperature of the metal versus time \(T(t)\) if the metal is subjected to a fixed surface heat rate \(q\) supplied by an electric heater. (b) A student proposes to identify the unknown metal by comparing measured and predicted thermal responses. Once a match is made, relevant thermophysical properties might be determined, and, in turn, the metal may be identified by comparison to published property data. Will this approach work? Consider aluminum, gold, and silver as the candidate metals.

Short Answer

Expert verified
(a) Given a large thermal conductivity and fixed surface heat rate \(q\), we derive a differential equation for the temperature of the metal as a function of time, \(T(t)\), using the energy conservation equation. The derived equation is: \[ \frac{\partial T(t)}{\partial t} = \frac{q}{\rho V c_p} .\] (b) The proposed approach of comparing measured and predicted thermal responses may provide some insight into identifying the unknown metal among aluminum, gold, and silver. However, it should be considered as a preliminary step, and more specific and controlled methods should be employed to confirm the metal's identity, such as checking for other distinctive properties or using analytical techniques.

Step by step solution

01

(a) Deriving the differential equation for temperature as a function of time

We are given that the metal has a large thermal conductivity and a fixed surface heat rate \(q\). To derive a differential equation for the temperature of the metal as a function of time \(T(t)\), we can use the energy conservation equation. The energy conservation equation states that the rate of change of internal energy in a volume is equal to the rate at which heat is supplied minus the rate at which energy leaves the volume. Mathematically, this can be represented as: \[ \frac{\partial U}{\partial t} = q - \dot{Q}, \] where \(U\) is the internal energy, \(t\) is the time, and \(\dot{Q}\) is the rate at which energy leaves the volume. For the given problem, the internal energy in a volume of metal can be defined as the product of density \((\rho)\), volume \((V)\), specific heat \((c_p)\), and temperature change \((T)\): \[U=\rho V c_p T(t) .\] To find the rate of change of internal energy with respect to time, we can differentiate the equation above with respect to time: \[ \frac{\partial U}{\partial t}=\rho V c_p\frac{\partial T(t)}{\partial t} .\] Now, we can substitute this back into the energy conservation equation: \[ \rho V c_p \frac{\partial T(t)}{\partial t} = q - \dot{Q} .\] Since we are neglecting spatial temperature gradients, there will be no energy loss from the volume. Hence, \(\dot{Q}\) can be set to 0: \[ \rho V c_p \frac{\partial T(t)}{\partial t} = q .\] Thus, the derived differential equation for the temperature of the metal as a function of time is: \[ \frac{\partial T(t)}{\partial t} = \frac{q}{\rho V c_p} .\]
02

(b) Evaluating the proposed approach for identifying the metal

To determine whether or not this approach will be effective in identifying the unknown metal among aluminum, gold, and silver, we can evaluate how different the thermophysical properties are among these candidate metals. If the candidates have significantly different thermophysical properties (e.g., density, specific heat, and thermal conductivity), then comparing the measured and predicted thermal responses may be an effective method to identify the unknown metal. However, if the candidates have similar thermophysical properties, then this method may not be as reliable in identifying the unknown metal. In practice, the thermophysical properties of aluminum, gold, and silver have noticeable differences, which could potentially lead to distinct thermal responses. However, it is important to consider other factors, such as experimental measurement uncertainties or the potential presence of alloys or impurities, which can significantly impact the accuracy of this approach. Overall, the approach may provide some insight into the identity of the unknown metal but should be considered as a preliminary step before using more specific and controlled methods to confirm the metal's identity, such as checking for other distinctive properties (e.g., conductivity, color, and oxidation behavior) or employing analytical techniques (e.g., X-ray diffraction, mass spectrometry).

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Key Concepts

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

Thermal Conductivity
Thermal conductivity is a material property that describes how well a material can conduct heat. It is often denoted by the symbol \( k \) and is expressed in units of \( \, W/(m \, K) \). Materials with high thermal conductivity, like metals such as aluminum, gold, and silver, are excellent at transferring heat.

Understanding thermal conductivity is crucial in many applications, from designing energy-efficient buildings to managing heat in electronic components.
  • A high thermal conductivity means the material can transfer heat efficiently through itself.
  • Conversely, low thermal conductivity indicates poor heat conduction, which is typical for insulators like rubber or plastic.
When dealing with problems involving thermal conductivity, one assumes that the material can equilibrate temperature efficiently, allowing us to simplify many thermal calculations.
Temperature Gradient
The temperature gradient is the rate of change of temperature with respect to distance within a material. It is a vector quantity and is typically denoted by \( abla T \).

The significance of the temperature gradient in thermal conduction is that it drives the flow of heat from regions of high temperature to regions of low temperature.
  • In the absence of a temperature gradient, heat transfer would not occur.
  • A steep temperature gradient signifies a significant temperature change over a short distance.
In the exercise, neglecting the spatial temperature gradient simplifies the problem, implying the material's internal temperature is relatively uniform.
Internal Energy
Internal energy is the total energy contained within a system, primarily in terms of kinetic and potential energies at the atomic or molecular level. In thermodynamics, it is a state function and often represented as \( U \).

In the context of thermal problems, the change in internal energy corresponds to changes in temperature and phase of a material.
  • The increase in internal energy typically results from external heat addition or work done on the system.
  • Decrease would be when the system loses heat or does work on the surroundings.
The exercise links internal energy to the temperature change in metal via the equation \( U = \rho V c_p T(t) \), facilitating understanding of how heat input affects the thermal state of the metal.
Thermophysical Properties
Thermophysical properties are the physical properties of materials that define their behavior under thermal changes. These include thermal conductivity, specific heat capacity, density, and sometimes other properties like thermal expansion coefficient.

Each of these properties plays a critical role in determining how a material will respond to heat.
  • Thermal conductivity: Indicates how quickly heat will spread through the material.
  • Specific heat capacity \((c_p)\): Reflects how much heat is needed to raise a material's temperature.
  • Density \((\rho)\): Impacts the amount of material available to store or conduct heat.
In identifying unknown metals like aluminum, gold, and silver, understanding these properties can help in predicting thermal response, potentially aiding in material identification.

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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 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 \(0 \leq r \leq r_{2}\).

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?

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 \(L\).

Consider a one-dimensional plane wall with constant properties and uniform internal generation \(\dot{q}\). The left face is insulated, and the right face is held at a uniform temperature. (a) Using the appropriate form of the heat equation, derive an expression for the \(x\)-dependence of the steady-state heat flux \(q^{\prime \prime}(x)\). (b) Using a finite volume spanning the range \(0 \leq\) \(x \leq \xi\), derive an expression for \(q^{\prime \prime}(\xi)\) and compare the expression to your result for part (a).
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