Water and air are separated by a mild-steel plane wall. It is proposed to increase the heat-transfer rate between these fluids by adding straight rectangular fins of 1.27-mm thickness and 2.5-cm length, spaced 1.27 cm apart. The air-side and water-side heat-transfer coefficients may be assumed

Question

Water and air are separated by a mild-steel plane wall. It is proposed to increase the heat-transfer rate between these fluids by adding straight rectangular fins of 1.27-mm thickness and 2.5-cm length, spaced 1.27 cm apart. The air-side and water-side heat-transfer coefficients may be assumed constant with values of 11.4 and 256 W/m²·K respectively. Determine the percent change in total heat transfer when fins are placed on (a) the water side, (b) the air side, and (c) both sides.

Accepted Answer

For a l m² section of the wall, the areas of the primary surface and of the fins are

[latex]A_{o}=1 \mathrm{~m}^{2}-79 ext { fins }(1 \mathrm{~m})\left[\frac{0.00127 \mathrm{~m}}{\mathrm{fin}} ight][/latex]

[latex]=0.90 \mathrm{~m}^{2}[/latex]

[latex]A_{f}=79 ext { fins }(1 \mathrm{~m})[(2)(0.025 \mathrm{~m})]+0.10 \mathrm{~m}^{2}[/latex]

[latex]=4.05 \mathrm{~m}^{2}[/latex]

Values of fin efficiency can now be determined from Figure 17.11. For the air side

[latex]L \sqrt{h / k t}=0.025 \mathrm{~m}\left[\frac{11.4 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}}{(42.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})(0.00127 \mathrm{~m})} ight]^{1 / 2}[/latex]

[latex]=0.362[/latex]

and for the water side

[latex]L \sqrt{h / k T}=0.025 \mathrm{~m}\left[\frac{256 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}}{(42.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})(0.00127 \mathrm{~m})} ight]^{1 / 2}[/latex]

[latex]=1.71[/latex]

The fin efficiencies are then read from the figure as

[latex]\eta_{\mathrm{air}} \cong 0.95[/latex]

[latex]\eta_{ ext {water }} \cong 0.55[/latex]

The total heat transfer rates can now be evaluated. For fins on the air side

[latex]q=h_{a} \Delta T_{a}\left[A_{o}+\eta_{f a} A_{f} ight][/latex]

[latex]=11.4 \Delta T_{a}[0.90+0.95(4.05)][/latex]

[latex]=54.1 \Delta T_{a}[/latex]

and on the water side

[latex]q=h_{w} \Delta T_{w}\left[A_{o}+\eta_{f w} A_{f} ight][/latex]

[latex]=256 \Delta T_{w}[0.90+0.55(4.05)][/latex]

[latex]=801 \Delta T_{w}[/latex]

The quantities [latex]\Delta T_{a}[/latex] and [latex]\Delta T_{w}[/latex] represent the temperature differences between the steel surface at temperature [latex]T_{o}[/latex] and the fluids.

The reciprocals of the coefficients are the thermal resistances of the finned surfaces. Without fins the heat-transfer rate in terms of the overall temperature difference, [latex]\Delta T=T_{w}-T_{a}[/latex], neglecting the conductive resistance of the steel wall, is

[latex]q=\frac{\Delta T}{\frac{1}{11.4}+\frac{1}{256}}=10.91 \Delta T[/latex]

With fins on the air side alone

[latex]q=\frac{\Delta T}{\frac{1}{54.1}+\frac{1}{256}}=44.67 \Delta T[/latex]

an increase of 310% compared with the bare-wall case.
With fins on the water side alone

[latex]q=\frac{\Delta T}{\frac{1}{11.4}+\frac{1}{801}}=11.24 \Delta T[/latex]

an increase of 3.0%.

With fins on both sides the heat-flow rate is

[latex]q=\frac{\Delta T}{\frac{1}{54.1}+\frac{1}{801}}=50.68 \Delta T[/latex]

an increase of 365%.
This result indicates that adding fins is particularly beneficial where the convection coefficient has a relatively small value.

Question 17.3: Water and air are separated by a mild-steel plane wall. It i...

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