A cable whose surface temperature is 100 ◦C has a diameter of 10 cm. Its center is at a depth of 1 m under soil whose thermal conductivity is known to be 1 W/m◦C. Determine how much heat it will lose per meter length if the soil surface temperature is 30^◦C. Make a plot of the temperature variation

Question

A cable whose surface temperature is 100 °C has a diameter of 10 cm. Its center is at a depth of 1 m under soil whose thermal conductivity is known to be 1 W/m°C. Determine how much heat it will lose per meter length if the soil surface temperature is 30 °C. Make a plot of the temperature variation as a function of distance vertically above the center of the cable.

Accepted Answer

Step 1 Nomenclature used in this problem follows that in Fig. 5.24. The given data is summarized as under: [latex]a = 1 m, r_{0} = 5 cm[/latex] or 0.05 m, k = 1 W/m °C, [latex]T_{0} = 100 °C[/latex] and [latex]T_{s}= 30 °C[/latex]. Using this data we calculate the conduction shape factor, following Eq. 5.122 as

[latex]S_{Buried cable} =\frac{2\pi }{\ln \left(\frac{2a}{r_{0} } ight) }[/latex] (5.122)

[latex]S=\frac{2 imes \pi }{\ln \left(\frac{2 imes 1}{0.05} ight) } =1.7033[/latex]

Step 2 The heat loss per meter of cable may then be calculated, using Eq. 5.121 as

[latex]q=k(T_{0} -T_{s} ) imes S[/latex] (5.121)

[latex]q = 1 × 1.7033 × (100 − 30) = 119.2 W/m[/latex]

Step 3 In order to make the plot specified in the example, the requisite data is generated using the following additional definitions (compare with Eq. 5.111).

Let

[latex]C=e^{\acute{C} } where \acute{C} =\frac{T_{0} -T_{s}}{\left(\frac{q}{4\pi k} ight) }[/latex] (5.111)

[latex]C_{T} =e ^{C_{\acute{T} }} where C_{\acute{T} } = \frac{T -T_{s}}{\left(\frac{q}{4\pi k} ight) }[/latex]

In terms of [latex]C_{T}[/latex] , the coordinate of the center of the isotherm located along the y axis is given by

[latex]y_{T}=a\left[\frac{C_{T}+1}{C_{T}-1} ight][/latex]

The corresponding radius of the isotherm is given by

[latex]r_{T}=\frac{2a\sqrt{C_{T}} }{C_{T}-1}[/latex]

It is easily seen from Fig. 5.24 that the height of the isotherm above the center of the cable is [latex]d_{T} = 1 − y_{T} + r_{T}[/latex] . The calculations are now shown in Table 5.4.

Table 5.4 Temperature depth data for Example 5.8

Temperature, T	100	90	80	70	60	50	40	30
Distance, [latex]y_{T}[/latex] ,m	0.05	0.081	0.134	0.217	0.341	0.517	0.742	1

Step 4 Figure 5.25 shows the variation of the temperature of the soil vertically above the axis of the cable.

This example is typical of what happens in power stations. The results also are applicable to a buried pipe conveying hot liquids (such as hot water) and vapors (such as steam).

Question 5.8: A cable whose surface temperature is 100 ◦C has a diameter o...

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