Question 14.6: Air at a temperature of T∞ = 370K is flowing normal to a cyl...

Step 1 Assume that the properties of air are calculated at the mean of the cylinder base temperature and the ambient air temperature. This is acceptable since air properties are not very sensitive to temperature and the average temperature of the fin may not be too far from its base temperature. The required mean temperature is T_{m}=\frac{T_{b}+T_{\infty}}{2}=\frac{310+370}{2}=340 K . Air properties are taken from table of properties at this temperature:

Density: \rho_{m}=1.0382 kg / m ^{3}

Kinematic viscosity: \nu_{m}=19.55 \times 10^{-6} m ^{2} / s

Thermal conductivity: k_{m}=0.0293 W / m ^{\circ} C

Prandtl number: P r_{m}=0.7

Thermal conductivity of aluminum, the cylinder material is taken as k_{A l}=207W/m°C

Step 2 Reynolds number based on the cylinder diameter is calculated as

Step 3 Zhukauskas correlation is used now to calculate the convective heat transfer coefficient. For the Reynolds number range that brackets the above value, Table 14.1 gives C = 0.26, m = 0.6, and n = 0.37. Even though the cylinder length is finite, we assume that the flow is largely twodimensional since \frac{L}{D}=\frac{0.075}{0.005}=15 is large. Hence, we have

Table 14.1 Constants in the Zhukauskas correlation

The average heat transfer coefficient then is

Step 4 The non-dimensional fin parameter is calculated now as

Step 5 The fin efficiency is calculated as

Step 6 The heat gain by the cylinder is then calculated as

Step 7 The non-dimensional tip temperature is given by

But the non-dimensional tip temperature is \theta_{L}=\frac{T_{L}-T_{\infty}}{T_{b}-T_{\infty}}. Hence, we have