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Chapter 2: Problem 6

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 \(a<0\).

Short Answer

Expert verified
Sketch linear for \( a=0 \), convex for \( a>0 \), concave for \( a<0 \).

Step by step solution

01

Understanding Temperature Dependence

The given problem involves a material with thermal conductivity that depends on temperature, represented as \( k = k_{e} + aT \) where \( k_{e} \) is a constant, \( T \) is the temperature, and \( a \) is a coefficient that can be positive, negative, or zero.
02

Case a = 0: Constant Thermal Conductivity

When \( a = 0 \), the thermal conductivity \( k \) becomes constant, \( k = k_{e} \). This means the material behaves as a typical constant-conductivity material, and the temperature distribution across the wall will be linear. We can sketch this as a straight line from the hot side to the cold side of the wall.
03

Case a > 0: Increasing Thermal Conductivity with Temperature

For \( a > 0 \), the thermal conductivity increases with temperature. This implies that higher temperatures facilitate greater heat conduction. The temperature distribution will be concave, starting steep from the hot side and becoming less steep toward the cold side, creating a curve convex to the temperature axis.
04

Case a < 0: Decreasing Thermal Conductivity with Temperature

When \( a < 0 \), the thermal conductivity decreases with an increase in temperature, meaning that heat conduction is less efficient at higher temperatures. The temperature distribution will be convex, with a gentler slope near the hot side and becoming steeper as it approaches the cold side, creating a curve concave to the temperature axis.

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Key Concepts

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

Thermal Conductivity
Thermal conductivity (\(k\)) is an important property of materials. It describes a material's ability to conduct heat. The higher the thermal conductivity, the better the material is at conducting heat. Thermal conductivity can vary with temperature.
  • A constant thermal conductivity implies the material conducts heat evenly across different temperatures.
  • When thermal conductivity depends on temperature, it may either increase or decrease as the temperature changes.
This variability is crucial in determining how heat will move through a material, making it an essential factor in heat transfer problems.
Steady-State Heat Transfer
Steady-state heat transfer refers to the situation where temperatures within a system remain constant over time. This implies there's a balance between incoming and outgoing heat in the system.
- In this state, while heat moves from a region of high temperature to low temperature, the temperatures at specific points in the material do not change with time. - Steady-state is crucial in simplifying complex heat transfer problems, as it avoids the need to consider time-dependent temperature changes.
Considering the temperature-dependent thermal conductivity allows us to predict how effectively heat can be transferred through materials under steady-state conditions.
Plane Wall
A plane wall is a simple geometric configuration used in thermal analysis to study heat transfer. It's essentially a flat, two-dimensional barrier through which energy, in the form of heat, transfers.
The temperature distribution across a plane wall depends on both the material's thermal properties and boundary conditions. In the steady-state heat transfer of a plane wall, the analysis assumes a consistent temperature difference across the wall. This assumption helps simplify calculations by treating each side of the wall as uniform in temperature. This set-up helps introduce more complex heat transfer topics in a manageable way.
Temperature Dependence
Temperature dependence describes how a material's properties change with temperature changes. In the case of thermal conductivity, it can directly affect how heat is transferred. When thermal conductivity depends on temperature, it's usually expressed in a formula such as\( k = k_{e} + aT \), with \(a\) representing the rate of change.
  • If \(a > 0\), the heat conduction enhances at higher temperatures, leading to a concave temperature distribution in the material.
  • If \(a = 0\), the conduction remains constant, resulting in a linear temperature distribution.
  • For \(a < 0\), the conduction decreases with rising temperatures, causing a convex temperature profile.
Understanding these conditions helps in predicting how materials will behave under different thermal circumstances.

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Most popular questions from this chapter

A spherical shell with inner radius \(r_{1}\) and outer radius \(r_{2}\) has surface temperatures \(T_{1}\) and \(T_{2}\), respectively, where \(T_{1}>T_{2}\). Sketch the temperature distribution on \(T-r\) coontinates assuming steady- state, one-timensional concuction with constant properties. Briefly justify the shape of your curve.

A large plate of thickness \(2 L\) is at a uniform temperature of \(T_{t}=200^{\circ} \mathrm{C}\), when it is suddenly quenched by dipping it in a liquid buth of temperature \(T_{\mathrm{w}}=20^{\circ} \mathrm{C}\) Heat transfer to the liquid is characterized by the convection ccefficient \(h\). (a) If \(x=0\) corresponds to the midplane of the wall, on \(T-x\) coordinates, sketch the temperature distributions for the following conditions: initial condition \((t \leq 0)\), steady-state condition \((t \rightarrow \infty)\), and two intermediate times. (b) On \(q_{r}^{*}-r\) coordinates, sketch the variation with time of the heat flux at \(x=L\) (c) If \(h=100 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\), what is the heat flux at \(x=L\) and \(t=0\) ? If the wall has a thermal conductivity of \(k=50 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}_{\text {, what is }}\) is the corresponding temperature gradient at \(x=L\) ? (d) Consider a plate of thickness \(2 L=20 \mathrm{~mm}\) with a density of \(\rho=2770 \mathrm{~kg} / \mathrm{m}^{3}\) and a specific heat \(c_{p}=875 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). By performing an energy balance on the plate, determine the amount of energy per unit surface area of the plate \(\left(\mathrm{J} / \mathrm{m}^{2}\right)\) that is transferred to the bath over the time required to reach steady-state conditions. (c) From other considerations, it is known that, during the quenching process, the heal flux at \(x=+L\) and \(x=-L\) decays exponentially with time according to the relation, \(q^{\prime \prime}=A \exp (-B r)\), where \(t\) is in seconds, \(A=1.80 \times 10^{4} \mathrm{~W} / \mathrm{m}^{2}\), and \(B=4.126 \times\) \(10^{-3} \mathrm{~s}^{-1}\). Use this information to determine the \(\mathrm{en}^{-}\) ergy per unit surface area of the plate that is transferred to the fluid during the quenching process.

An apparatus for measuring thermal conductivity cmploys an electrical heater sandwiched between two identical samples of diameter \(30 \mathrm{~mm}\) and length \(60 \mathrm{~mm}\), which are presced between plates maintained at a uniform tempenature \(T_{n}=77^{\circ} \mathrm{C}\) by a circulating fluid. \(\mathrm{A}\) conducting grease is placed between all the surfaces to ensure good thermal contact. Differential thermocouples are imbedded in the samples with a spocing of \(15 \mathrm{~mm}\). The lateral sides of the samples are insulated to ensure cee-dimensional heat transfer thrueugh the samples. (a) With two saruples of SS316 in the apparatus, the heater draws \(0.353 \mathrm{~A}\) at \(100 \mathrm{~V}\) and the differential thermocouples indicate \(\Delta T_{1}=\Delta T_{2}=25.0^{\circ} \mathrm{C}\). What is the thermal conductivity of the stainless steel sample material? What is the average tempenture of the samples? Compare your result with the therrmal conductivity value reported for this material in Table A.1. (b) By mistake, an Armeo iron sample is placed in the lower position of the apparatus with one of the SS 316 samples from part (a) in the upper portion. For this situation, the heater draws \(0.601 \mathrm{~A}\) at \(100 \mathrm{~V}\) and the differential thermocouples indicate \(\Delta T_{1}=\) \(\Delta T_{2}=15.0^{\circ} \mathrm{C}\). What are the thermal conductivity and average temperature of the Armco iron sample? (c) What is the advantage in constructing the apparatus with two identical samples sandwiching the heuter rather than with a single heater-sample combination? When would heat leakage out of the lateral surfaces of the samples become significant? Under what conditions would you cxpect \(\Delta T_{1} \neq \Delta T_{2}\) ?

Sections of the trans-Alaska pipeline run above the ground and ate supported by vertical steel shafts \((k=25\) W/m \(-\mathrm{K})\) that are \(1 \mathrm{~m}\) long and have a cross-sectional area of \(0.005 \mathrm{~m}^{2}\). Under normal cperating conditions, the temperature variation along the length of a shaft is known to be governed by an expression of the form $$ T=100-150 x+10 x^{2} $$ where \(T\) and \(x\) have units of \({ }^{\circ} \mathrm{C}\) and meters, respectively. Temperature variations are small over the shaft cross section. Evaluate the temperature and conduction heat rate at the shaft-pipeline joint \((x=0)\) and at the shaft-ground interface \((x=1 \mathrm{~m})\). Explain the difference in the heat rates.

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.
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