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Chapter 2: Problem 106

The temperature of a plane wall during steady onedimensional heat conduction varies linearly when the thermal conductivity is constant. Is this still the case when the thermal conductivity varies linearly with temperature?

Short Answer

Expert verified
Answer: No, the temperature of a plane wall during steady one-dimensional heat conduction does not vary linearly when the thermal conductivity varies linearly with temperature. Instead, the relationship between temperature and position is exponential.

Step by step solution

01

Heat Conduction Equation

The heat conduction equation is given by Fourier's law, which states that the heat flux (q) through a solid is proportional to the temperature gradient: q = -k * (dT/dx) where q is the heat flux, k is the thermal conductivity, and dT/dx is the temperature gradient.
02

Steady State Condition

Since the problem states that the wall is at steady state, this means that the heat flux (q) is constant throughout the wall and does not change over time.
03

Linear Variability of Thermal Conductivity

We are given that the thermal conductivity (k) varies linearly with temperature (T): k = k₀ + βT, where k₀ is the initial thermal conductivity and β is the constant of proportionality.
04

Substitute Thermal Conductivity Equation

Substitute the equation for the thermal conductivity (Step 3) into Fourier's law (Step 1): q = -(k₀ + βT) * (dT/dx)
05

Solve the Differential Equation

Now we need to solve the above differential equation to determine the relationship between temperature and position (x): (q / (-k₀ - βT)) = dT/dx Now, perform integration to solve for T: ∫(dT / (k₀ + βT)) = -∫dx Let's substitute k₀ + βT = u and d(u) = β(dT), then the equation becomes: -∫(1/β) * du/u = ∫dx Now integrate: (-1/β) * Ln(u) = -x + C₁ Ln(u) = βx + C₂ (C₂ = βC₁) u = e^(βx) * e^C₂ k₀ + βT = A * e^(βx) (A = e^C₂)
06

Express Temperature as a Function of Position

We already have T in the equation. We just need to express it as a function of x: T(x) = (A * e^(βx) - k₀) / β
07

Conclusion

The relationship between temperature and position (T(x)) is not linear when thermal conductivity varies linearly with temperature, as evident from the exponential nature of the solution. Thus, the temperature of a plane wall during steady one-dimensional heat conduction does not vary linearly when the thermal conductivity varies linearly with temperature.

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Most popular questions from this chapter

A long homogeneous resistance wire of radius \(r_{o}=\) \(5 \mathrm{~mm}\) is being used to heat the air in a room by the passage of electric current. Heat is generated in the wire uniformly at a rate of \(5 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}\) as a result of resistance heating. If the temperature of the outer surface of the wire remains at \(180^{\circ} \mathrm{C}\), determine the temperature at \(r=3.5 \mathrm{~mm}\) after steady operation conditions are reached. Take the thermal conductivity of the wire to be \(k=6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\).

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A 2-kW resistance heater wire whose thermal conductivity is \(k=10.4 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot \mathrm{R}\) has a radius of \(r_{o}=0.06\) in and a length of \(L=15\) in, and is used for space heating. Assuming constant thermal conductivity and one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem during steady operation. Do not solve.

A spherical communication satellite with a diameter of \(2.5 \mathrm{~m}\) is orbiting around the earth. The outer surface of the satellite in space has an emissivity of \(0.75\) and a solar absorptivity of \(0.10\), while solar radiation is incident on the spacecraft at a rate of \(1000 \mathrm{~W} / \mathrm{m}^{2}\). If the satellite is made of material with an average thermal conductivity of \(5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and the midpoint temperature is \(0^{\circ} \mathrm{C}\), determine the heat generation rate and the surface temperature of the satellite.

Consider uniform heat generation in a cylinder and a sphere of equal radius made of the same material in the same environment. Which geometry will have a higher temperature at its center? Why?
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