Question 11.30: To increase the heat transfer rate from the box in Illustrat...
To increase the heat transfer rate from the box in Illustrative Problem 11.17, a vertical fin is added to each of the four sides (see Figure 11.36). The fins are aluminum, 15 cm high and 0.15 cm thick, and protrude 3 cm from the box. Find the new heat loss rate.
The fins are vertical plates with the same characteristic length as the sides of the box, so the fin’s heat transfer coefficient is the same as in Problem 11.17: h_{c} = 5.102 W/m2 °C.
Here P = 0.303 m, k = 204.2 W/m °C per Table 11.1, d = 0.03 m, A_{f}= 0.0369 m^{2} (4 fins) and a = 0.000225 m^{2}. The value of m in Equation 11.48a is thus
m=\sqrt{\frac{P h_{c}}{k a}} \sqrt{\frac{0.303 \times 5.102}{204.2 \times 0.000225}}=5.80 m ^{-1}and md = 0.174. The tanh of md = 0.172 and (tanh md)/md = 0.99. Thus, the fin heat transfer rate is
Q_{f}=\eta h_{c} A_{f}(\Delta t)=0.99 \times 5.102 \times 0.0369 \times 25=4.66 WThis rate added to the rate for the box brings the total heat transfer rate to 18.81 watts, which is an increase of 33% due to the usage of fins.
Note that when fins are added, the surface area of the box exposed to convection is diminished. In this example, the reduction is negligible.