Question

The heat transfer from a 2 m diameter sphere to a 25 °C air stream over a time interval of one
hour is 3000 kJ. Estimate the surface temperature of the sphere if the heat transfer coefficient is
10 W/m2 K

Answer 1

Question #79207

The heat transfer from a 2 m diameter sphere to a 25 °C air stream over a time interval of one hour is 3000 kJ. Estimate the surface temperature of the sphere if the heat transfer coefficient is 10 W/m²K

Answer:

The convective heat transfer ($Q = 3000\ \mathrm{kJ} = 3000000\ \mathrm{J}$) is given by:

$$Q = h A \left(T _ {w} - T _ {f}\right) \tau ,$$

where $h = 10\ \mathrm{W/m^2K}$ is the heat transfer coefficient,

$$A = \pi d ^ {2}$$

is the surface area of the sphere,

$d = 2\ \mathrm{m}$ is the diameter of the sphere,

$T_{w}$ is the surface temperature,

$T_{f} = 25^{\circ}\mathrm{C}$ is the temperature of the air,

$\tau = 1\ \mathrm{hr} = 3600\ \mathrm{s}$ is the period of time.

So, we can derive $T_{w}$ from (1) as follows:

$$T _ {w} = T _ {f} + \frac {Q}{h A \tau},$$

Substitute into (2) and (3):

$$A = \pi \cdot 2 ^ {2} = 12.566\ \mathrm{m ^ {2}},$$ $$T _ {w} = 25 + \frac {3000000}{10 \cdot 12.566 \cdot 3600} = 31.6^{\circ}\mathrm{C}.$$