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Chapter 2: Problem 15

Heat flux meters use a very sensitive device known as a thermopile to measure the temperature difference across a thin, heat conducting film made of kapton \((k=0.345 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). If the thermopile can detect temperature differences of \(0.1^{\circ} \mathrm{C}\) or more and the film thickness is \(2 \mathrm{~mm}\), what is the minimum heat flux this meter can detect?

Short Answer

Expert verified
Answer: The minimum detectable heat flux for this meter is 17.25 W/m².

Step by step solution

01

Formula for heat flux

The formula for heat flux (q) is given by Fourier's law: \(q = -k \cdot \frac{\Delta T}{d}\), where q is the heat flux, k is the thermal conductivity, \(\Delta T\) is the temperature difference, and d is the thickness of the film.
02

Plug in given values

We have the given values: \(k = 0.345\ \mathrm{W}/(\mathrm{m}\cdot\mathrm{K})\), \(\Delta T = 0.1^{\circ}\mathrm{C}\) (we can use Celsius as the temperature difference unit, because the conversion factor is a ratio, the offset of the Kelvin scale will cancel out), and \(d = 2\ \mathrm{mm} = 0.002\ \mathrm{m}\). We plug these values into the formula for heat flux: \(q = -0.345\frac{0.1}{0.002}\).
03

Calculate the minimum heat flux

Now, we can calculate the minimum heat flux this meter can detect: \(q = -0.345\frac{0.1}{0.002} = -17.25\ \mathrm{W}/\mathrm{m}^2\).
04

Interpret the result

The minimum detectable heat flux is \(-17.25\ \mathrm{W}/\mathrm{m}^2\). The negative sign indicates that the heat is detected to flow in the opposite direction, which is used mostly for convention. The absolute value of the heat flux, \(17.25\ \mathrm{W}/\mathrm{m}^2\), can be taken as the minimum detectable heat flux this meter can detect.

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Most popular questions from this chapter

A large plane wall, with a thickness \(L\) and a thermal conductivity \(k\), has its left surface \((x=0)\) exposed to a uniform heat flux \(\dot{q}_{0}\). On the right surface \((x=L)\), convection and radiation heat transfer occur in a surrounding temperature of \(T_{\infty}\). The emissivity and the convection heat transfer coefficient on the right surface are \(\bar{\varepsilon}\) and \(h\), respectively. Express the houndary conditions and the differential equation of this heat conduction problem during steady operation.

Consider a function \(f(x)\) and its derivative \(d f l d x\). Does this derivative have to be a function of \(x\) ?

How do you recognize a linear homogeneous differential equation? Give an example and explain why it is linear and homogeneous.

Consider the base plate of an \(800-W\) household iron with a thickness of \(L=0.6 \mathrm{~cm}\), base area of \(A=160 \mathrm{~cm}^{2}\), and thermal conductivity of \(k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The inner surface of the base plate is subjected to uniform heat flux generated by the resistance heaters inside. When steady operating conditions are reached, the outer surface temperature of the plate is measured to be \(112^{\circ} \mathrm{C}\). Disregarding any heat loss through the upper part of the iron, \((a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the plate, \((b)\) obtain a relation for the variation of temperature in the base plate by solving the differential equation, and (c) evaluate the inner surface temperature. Answer: (c) \(117^{\circ} \mathrm{C}\)

Consider a 20-cm-thick large concrete plane wall \((k=0.77 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) subjected to convection on both sides with \(T_{\infty 1}=27^{\circ} \mathrm{C}\) and \(h_{1}=5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the inside, and \(T_{\infty 2}=8^{\circ} \mathrm{C}\) and \(h_{2}=12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the outside. Assuming constant thermal conductivity with no heat generation and negligible radiation, (a) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the wall, (b) obtain a relation for the variation of temperature in the wall by solving the differential equation, and \((c)\) evaluate the temperatures at the inner and outer surfaces of the wall.
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