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Chapter 2: Problem 156

The heat conduction equation in a medium is given in its simplest form as $$ \frac{1}{r} \frac{d}{d r}\left(r k \frac{d T}{d r}\right)+\dot{e}_{\text {gen }}=0 $$ Select the wrong statement below. (a) The medium is of cylindrical shape. (b) The thermal conductivity of the medium is constant. (c) Heat transfer through the medium is steady. (d) There is heat generation within the medium. (e) Heat conduction through the medium is one-dimensional.

Short Answer

Expert verified
(a) The medium is of cylindrical shape. (b) The thermal conductivity of the medium is constant. (c) Heat transfer through the medium is steady. (d) There is heat generation within the medium. (e) Heat conduction through the medium is one-dimensional. Answer: There is no incorrect statement among the provided options. All the statements (a), (b), (c), (d), and (e) are correct based on the given heat conduction equation.

Step by step solution

01

Analyze the given heat conduction equation

The given heat conduction equation is: $$ \frac{1}{r} \frac{d}{d r}\left(r k \frac{d T}{d r}\right)+\dot{e}_{\text {gen }}=0 $$ In this equation: - r is the radial distance from the center of the medium - k is the thermal conductivity of the medium - T is the temperature - \(\dot{e}_{\text{gen}}\) is the heat generation rate per unit volume
02

Checking statement (a)

(a) The medium is of cylindrical shape. Since the equation uses radial distance 'r' and has a term \(\frac{1}{r}\), it implies that the medium is of cylindrical shape. So this statement is correct.
03

Checking statement (b)

(b) The thermal conductivity of the medium is constant. We see that the thermal conductivity 'k' appears as a constant in the equation. It does not depend on the radial distance or temperature. Therefore, this statement is also correct.
04

Checking statement (c)

(c) Heat transfer through the medium is steady. There is no time-dependent term in the given heat conduction equation, which implies that the heat transfer through the medium is steady. Thus, this statement is correct.
05

Checking statement (d)

(d) There is heat generation within the medium. The term \(\dot{e}_{\text {gen }}\) represents the heat generation rate per unit volume within the medium. Since it appears in the equation, this means that there is heat generation within the medium. This statement is also correct.
06

Checking statement (e)

(e) Heat conduction through the medium is one-dimensional. The given heat conduction equation only contains a radial derivative, \(\frac{dT}{dr}\), which suggests that the heat conduction through the medium occurs only in the radial direction and is one-dimensional. So, this statement is correct as well. Since all the statements (a), (b), (c), (d), and (e) are correct, the exercise appears to have a mistake or typo in the given options. There is no wrong statement among the provided options.

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Most popular questions from this chapter

A spherical metal ball of radius \(r_{o}\) is heated in an oven to a temperature of \(T_{i}\) throughout and is then taken out of the oven and allowed to cool in ambient air at \(T_{\infty}\) by convection and radiation. The emissivity of the outer surface of the cylinder is \(\varepsilon\), and the temperature of the surrounding surfaces is \(T_{\text {surr }}\). The average convection heat transfer coefficient is estimated to be \(h\). Assuming variable thermal conductivity and transient one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve.

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Heat is generated in a long wire of radius \(r_{o}\) at a constant rate of \(\dot{e}_{\text {gen }}\) per unit volume. The wire is covered with a plastic insulation layer. Express the heat flux boundary condition at the interface in terms of the heat generated.

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