Air at 1 atmospheric pressure and 27 °C blows across a 12 mm diameter sphere at a a small heater inside the sphere maintains the surface temperature at 77 °C. With k = 0.026 W/m (kelvin) and with (Nu) = 31.4, the heat loss by the sphere would be

Concept:

Heat loss through the convection is given by

Q_loss = h × A × (∆T)

Nusselt number \(Nu = \frac{{h{l_c}}}{k}\)

Where l­_c is the characteristic length, in case of heater it is equal to the diameter of the sphere

Calculation:

Given, Nu = 31.4, k = 0.026 W/mK, D = 12 mm ⇒ r = 6 × 10-3 m , Surface temperature T_s = 27°C and atmospheric temperature T_∞ = 77°C

\(Nu = \frac{{h{l_c}}}{k}\)

⇒ \(h = \frac{{Nu\times{k}}}{l_c}\)

\(h = \frac{{31.4 \times 0.026}}{{0.012}} = 68.03\;W/{m^2}K\;\)

Surface area of sphere A = 4 × π × r² = 4 × 3.14 × (6 × 10^-3)²

Q_loss = 68.03 × 4 × 3.14 × (6 × 10^-3)² × (77 – 27) = 1.54 J/s