A copper sphere initially at a uniform temperature T0 is immersed in a fluid. Electric heaters are placed in the fluid and controlled so that the temperature of the fluid follows a periodic variation given by
T∞ − Tm = A sin ωτ
where
Tm = time-average mean fluid temperature
A = amplitude of temperature wave
ω = frequency
Derive an expression for the temperature of the sphere as a function of time and the heat-transfer coefficient from the fluid to the sphere. Assume that the temperatures of the sphere and fluid are uniform at any instant so that the lumped-capacity method of analysis may be used.