Question 17.3: Water and air are separated by a mild-steel plane wall. It i...
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.
For a l m² section of the wall, the areas of the primary surface and of the fins are
A_{o}=1 \mathrm{~m}^{2}-79 \text { fins }(1 \mathrm{~m})\left[\frac{0.00127 \mathrm{~m}}{\mathrm{fin}}\right]=0.90 \mathrm{~m}^{2}
A_{f}=79 \text { fins }(1 \mathrm{~m})[(2)(0.025 \mathrm{~m})]+0.10 \mathrm{~m}^{2}
=4.05 \mathrm{~m}^{2}
Values of fin efficiency can now be determined from Figure 17.11. For the air side
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})}\right]^{1 / 2}=0.362
and for the water side
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})}\right]^{1 / 2}=1.71
The fin efficiencies are then read from the figure as
\eta_{\mathrm{air}} \cong 0.95\eta_{\text {water }} \cong 0.55
The total heat transfer rates can now be evaluated. For fins on the air side
q=h_{a} \Delta T_{a}\left[A_{o}+\eta_{f a} A_{f}\right]=11.4 \Delta T_{a}[0.90+0.95(4.05)]
=54.1 \Delta T_{a}
and on the water side
q=h_{w} \Delta T_{w}\left[A_{o}+\eta_{f w} A_{f}\right]=256 \Delta T_{w}[0.90+0.55(4.05)]
=801 \Delta T_{w}
The quantities \Delta T_{a} and \Delta T_{w} represent the temperature differences between the steel surface at temperature T_{o} 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, \Delta T=T_{w}-T_{a}, neglecting the conductive resistance of the steel wall, is
q=\frac{\Delta T}{\frac{1}{11.4}+\frac{1}{256}}=10.91 \Delta TWith fins on the air side alone
q=\frac{\Delta T}{\frac{1}{54.1}+\frac{1}{256}}=44.67 \Delta Tan increase of 310% compared with the bare-wall case.
With fins on the water side alone
an increase of 3.0%.
With fins on both sides the heat-flow rate is
q=\frac{\Delta T}{\frac{1}{54.1}+\frac{1}{801}}=50.68 \Delta Tan increase of 365%.
This result indicates that adding fins is particularly beneficial where the convection coefficient has a relatively small value.
