4. 4 : Heat Transfer by Convection from a Sphere

Example 4:Heat Transfer by Convection from a Sphere A 200 W heater has a spherical casing of diameter 0.2 m. The heat transfer coefficient for conduction and convection from the casing to the ambient air is obtained from \(\text{Nu} = 2 + 0.6 \text{Re}^{1/2} \text{Pr}^{1/3}\), with \(\text{Re}=10^4\) and \(\text{Pr}=0.69\). The temperature of the ambient air is 30oC and the thermal conductivity of air is \(k\) = 0.02 W/m.K.

  1. Find the heat flux from the surface at steady state.

  2. Find the steady state surface temperature of the casing.

  3. Find the temperature of the casing at steady state for stagnant air. Why is this situation physically infeasible? (GATE-2001)


Solution: Heat flux (\(q\)) from the surface is given by \[q = \frac{Q}{A} = \frac{Q}{4\pi r^2} = \frac{200}{4\times\pi\times 0.1^2} = 1591.5 \text{ W/m$^2$}\] Nusselt number (\(\text{Nu}\)) for convection: \[Nu = 2 + 0.6 \text{Re}^{1/2} \text{Pr}^{1/3} = 2 + 0.6\times(10000)^{1/2}\times(0.69)^{1/3} = 55.02\] Convective heat transfer coefficient (\(h\)): \[h = \frac{\text{Nu}\cdot k}{D} = \frac{55.02\times 0.02}{0.2} = 5.502 \text{W/(m$^2$.K)}\] For heat transfer by convection, \[q = h(T-T_\infty)\]


Substituting the known quantities, we get \[\begin{aligned} 1591.5 &= 5.502\times(T-30) \\ \Longrightarrow \qquad T &= 319.3^{\circ}C \end{aligned}\] i.e., The steady state surface temperature of spherical casing is 319.3.

If the air is stagnant, then \(\text{Re} = 0\). This leads to \(\text{Nu}=2\). Therefore, the heat transfer coefficient for this condition becomes, \[h = \frac{\text{Nu}\cdot k}{D} = \frac{2\times0.02}{0.2} = 0.2 \text{ W/(m$^2$.K)}\] Using this value of \(h\), for the heat flux of 1591.5 W/m\(^2\), we get \[\begin{aligned} q &= h(T-T_\infty) \\ 1591.5 &= 0.2\times(T-30) \\ \Longrightarrow \qquad T &= 7987.5^{\circ}C \end{aligned}\]

This situation (the condition of stagnant air) cannot be maintained for a long time, as explained below:

Surface temperature of 7987.5oC leads to reducing the density of nearby air sharply, as \(\rho\propto T^{-1}\) (as from the ideal gas relation, we have \(\rho \propto P/(RT)\)). This leads to setting up of convection currents, and hence the increase of Nusselt number, thereby reducing the surface temperature.