Text of the exercise is incomplete. Complete version:
"A 9-cm-diameter spherical ball whose surface is maintained at a temperature of 110?C is suspended in the middle of the room at 20?C. If the convection heat transfer coefficient is 15 W/m2.C and the emissivity of the surface is 0.8, determine the total rate of heat transfer from the ball."
Solution
The ball transfers heat to the room by radiation and convection. First of all, let's calculate the area of the ball's surface. Given the radius r=9 cm=0.09 m, the area is
[tex]A=4 \pi r^2 = 0.102 m^2[/tex]
1) Let's start with radiation. The radiative heat transfer rate is given by Stefan-Boltzmann law:
[tex]P_r=\epsilon \sigma A (T^4-T_0^4)[/tex]
where [tex]\epsilon=0.8[/tex] is the emissivity of the ball, A=0.102 m^2 is the area of the ball's surface, [tex]\sigma = 5.67 \cdot 10^{-8} W/(m^2 K^4)[/tex] is the Stefan's constant, [tex]T=110^{\circ}=383 K[/tex] is the temperature of the ball and [tex]T_0=20^{\circ} = 293 K[/tex] is the temperature of the room.
So, using these values, we get:
[tex]P_r = 65.5 W[/tex]
2) Let's calculate the heat transfer rate by convection. This is given by
[tex]P_c = h A (T-T_0)[/tex]
where [tex]h=15 W/m^2[/tex] is the heat transfer coefficient. Using again [tex]A=0.102 m^2[/tex], T=383 K and T0=293 K, we find
[tex]P_c = 137.7 W[/tex]
3) Therefore, the total heat transfer rate is
[tex]P=P_r + P_c = 65.5 W + 137.3 W=203.2 W[/tex]
