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Chapter 2: Problem 20

At a given instant of time the temperature distribution within an infinite homogeneous body is given by the function $$ T(x, y, z)=x^{2}-2 y^{2}+z^{2}-x y+2 y t $$ Assuming constant propertics and no internal heat generation, determine the regions where the temperature changes with time.

Short Answer

Expert verified
Temperature changes with time when \(y \neq 0\).

Step by step solution

01

Understanding the Problem

The temperature distribution within a body is described by a function of spatial coordinates \(x\), \(y\), \(z\), and time \(t\). The goal is to find regions where the temperature changes with time.
02

Identifying Terms Involving Time

We start by examining the function \(T(x, y, z) = x^2 - 2y^2 + z^2 - xy + 2yt\). Among these terms, \(2yt\) is the only one that includes time \(t\).
03

Partial Differentiation with Respect to Time

To determine how temperature changes with time, we take the partial derivative of \(T(x, y, z)\) with respect to \(t\), which gives \(\frac{\partial T}{\partial t} = 2y\).
04

Determining Regions of Time Change

The temperature changes with time where \(\frac{\partial T}{\partial t} eq 0\). Specifically, temperature changes with time in regions where \(y eq 0\).
05

Considering Special Cases

If \(y = 0\), then the derivative \(\frac{\partial T}{\partial t} = 0\), meaning the temperature doesn't change with time along the \(x\)-\(z\) plane (\(y=0\)).

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Key Concepts

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

Temperature Distribution
Temperature distribution is a concept that lets us understand how temperature varies across different points in space within a given material or body. Imagine slicing through a loaf of bread - each slice could represent a temperature at a different location in the loaf. Temperature distribution can be affected by:
  • Material properties: Some materials conduct heat better than others.
  • External conditions: The environment outside can influence how heat spreads inside.
  • Initial temperature conditions: How heat is initially spread out in the material.
In our given problem, the temperature distribution is given by the function \[T(x, y, z) = x^2 - 2y^2 + z^2 - xy + 2yt\]. This function describes how the temperature within a three-dimensional body changes with its spatial coordinates \(x\), \(y\), and \(z\), and time \(t\). Identifying how temperature varies in different regions is crucial for applications like engineering projects, environmental simulations, and thermal analysis.
Partial Differentiation
Partial differentiation is a mathematical technique used to examine how one variable changes while keeping other variables constant. This technique helps us understand the specific part each variable plays in influencing an outcome, like temperature in our case.In our exercise, we apply partial differentiation to the temperature function \[T(x, y, z) = x^2 - 2y^2 + z^2 - xy + 2yt\].

Focusing on the Time-dependent Term

Time is represented by \(t\) in our equation, and the partial derivative with respect to \(t\) tells us how the temperature changes over time while ignoring changes in \(x\), \(y\), and \(z\). For this function, the derivative \[\frac{\partial T}{\partial t} = 2y\] indicates that temperature changes directly depend on the \(y\) coordinate. This outcome shows partial differentiation's power in isolating specific influences on a variable, making it a valuable tool in disciplines like physics and engineering.
Time-dependent Temperature Changes
Understanding time-dependent temperature changes is key to predicting how temperature evolves in a material over time. This involves examining which parts of the body or regions show a variation in temperature as time progresses.In our problem, the term \(2yt\) is critical because it includes the time variable \(t\). When we take the partial derivative with respect to time, \[\frac{\partial T}{\partial t} = 2y\], we find that only \(y\) affects changes over time. Thus, temperature changes occur in regions where \(y eq 0\).

Regions of Constant Temperature

When \(y = 0\), \[\frac{\partial T}{\partial t} = 0\], meaning temperature remains unchanged over time in these areas. This suggests stabilization in temperature along the \(x\)-\(z\) plane. By focusing on these relationships, we gain insights into how temperature dynamics work, which can guide practical decisions in thermal management and design.

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Most popular questions from this chapter

To determine the effect of the temperature dependence of the thermal conductivity on the termperature distribution in a solid, censider a material for which this dependence may be represented as $$ k=k_{e}+a T $$ where \(k\), 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 plane layer of coal of thickness \(L=1 \mathrm{~m}\) experiences uniform volumetric generation at a rate of \(\hat{q}=20 \mathrm{~W} / \mathrm{m}^{2}\) due to slow exidation of the coal particles. Averuged over a daily period, the top surface of the layer transfers heat by convection to ambient air for which \(h=5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(T_{\mathrm{m}}=25^{\circ} \mathrm{C}\), while receiving solar irrudiation in the amoent \(G_{5}=400 \mathrm{~W} / \mathrm{m}^{2}\). Irnatiation from the amosphere may be neglected. The solar absorptivity and emissivity of the surface are each \(\alpha_{S}=\varepsilon=0.95\). (a) Write the steady-sate form of the heat diffusion equation for the layer of coul. Verify that this equation is satisfied by a temperiture distribution of the form $$ T(x)=T_{x}+\frac{\dot{q} L^{2}}{2 k}\left(1-\frac{x^{2}}{L^{2}}\right) $$ From this distribution, what can you say about conditions at the bottom surface \((x=0)\) ? Sketch the temperature distribution and label key features. (b) Obtain an expression for the rate of heat transfer by conduction per unit area at \(x=L\). Applying an energy balance to a control surface about the top surface of the layer, obtain an expression for \(T_{r}\) Evaluate \(T_{4}\) and \(T(0)\) for the prescribed conditions. (c) Daily average values of \(G_{5}\) and \(h\) depend on a number of factors such as time of year, cloud cover, and wind conditions. For \(h=5 \mathrm{~W} / \mathrm{m}^{2}-\mathrm{K}\), compute and plot \(T_{\text {t and }} T(0)\) as a function of \(G_{5}\) for \(50 \leq G_{s} \leq 500 \mathrm{~W} / \mathrm{m}^{2}\). For \(G_{5}=400 \mathrm{~W} / \mathrm{m}^{2}\), compute and plot \(T_{\text {, and }} T(0)\) as a function of \(h\) for \(5 \leq h \leq 50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\).

A plane wall of thickness \(2 \mathrm{~L}=40 \mathrm{~mm}\) and themal conductivity \(k=5 \mathrm{~W} / \mathrm{m}\) - \(\mathrm{K}\) experiences uniform volumetric heat generation at a rate \(q\), while convection heat transer occurs at both of its surfaces \(\left(x=-L_{4}+L\right)\). each of which is exposed to a fluid of temperature \(T_{m}=20^{\circ} \mathrm{C}\). Under steady-state conditions, the termperature distribution in the wall is of the form \(T(x)=a+\) \(b x+c x^{2}\), where \(a=82.0^{\circ} \mathrm{C}, b=-210^{\circ} \mathrm{C} / \mathrm{m}, c=\) \(-2 \times 10^{4} \mathrm{C} / \mathrm{m}^{2}\), and \(x\) is in meters. The origin of the \(x\)-coordinate is at the midplane of the wall. (a) Sketch the temperature distribution and identify significant physical features. (b) What is the volumetric rate of heat generation \(\dot{4}\) in the wall? (c) Determine the surface heat fluxes, \(q_{1}^{\prime \prime}(-L)\) and \(q_{:}^{\prime \prime}(+L)\). How are these fluxes related to the heat generation rate? (d) What are the convection coefficients for the sarfaces at \(x=-L\) and \(x=+L\) ? (e) Obtain an expression for the heas flux distribution, \(q\) " \((x)\). Is the heat flux zero at any location? Explain any significant features of the distribution. (i) If the source of the heat generation is suddenly deactivated \((\dot{q}=0)\), what is the rate of change of energy stored in the wall at this instant? (2) What temperuture will the wall eventually reach with \(\dot{q}=0\) ? How much energy mus be removed by the fluid per unit area of the wall \(\left(1 / \mathrm{m}^{2}\right)\) to reach this state? The density and specific heat of the wall material are \(2600 \mathrm{~kg} / \mathrm{m}^{3}\) and \(800 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively,

A young engineer is asked to design a themal protection barricr for a sensitive clectronic device that might be exposed to irradiation from a high- powered infrarcd laser. Having learmed as a student thut a low thermal conductivity material provides good insulating characteristics, the enginecr specifies use of a nanostructured aerogel, characterized by a thermal conductivity of \(k_{e}=\) 0.0ns W/m - \(\mathrm{K}\), for the protective harrier. The engineer's boss questions the wisdom of selecting the acrogel because it has a low themal conductivity. Consider the sudden laser irradiation of (a) peire aluminum, (b) glass, and (c) aerogel. The laser provides irradiation of \(G=10 \times 10^{6}\) Whm \({ }^{2}\). The absorptivities of the materials are \(a=0.2,0.9\), and \(0.8\) for the aluminum, glass, and acrogel, rerpectively, and the initial ternperature of the barrier is \(T_{i}=300 \mathrm{~K}\). Explain why the boss is concerncd. Hint: All materials experience thetmal expunsion (or contraction), and local stresses that develop within a material are, to a first approximation, proportional io the local temperahure gradicnt.

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 nate per unit volume in the wall and the heat fluxes at the two wall faces \((x=0, L)\).
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