A metal rod of cross-sectional area 1.0 cm2 is being heated at one end. At one time, the temperature gradient is 5.0°C/cm at cross-section A and is 2.5°C/cm at cross-section B. Calculate the rate at which the temperature is increasing in the part AB = 0.40 J/°C, thermal conductivity of the material of the rod = 200 W/m-°C. Neglect any loss of heat to the atmosphere.

Sol. The rate of heat flow per sec. = dQ/dt = KA dθ/dt The rate of heat flow per sec. = dQ base A/dt = KA dθ base B/dt This part of heat absorbed by the red. Q/t = ms∆θ/dt where dθ/dt = Rate of net temp. variation ⇒ msdθ/dt = KA dθ base A/dt - KA dθ base B/dt ⇒ ms dθ/dt KA(dθ base A/dt - dθ base B/dt) ⇒ 0.4 * dθ/dt = 200 * 1 * 10^-4 (5 – 2.5) °C/cm ⇒ 0.4 * dθ/dt = 200 * 10^-4 * 2.5 ⇒ 0.4 * dθ/dt = 200 * 2.5 * 10^-4/0.4 * 10^-2 °C/m = 1250 * 10^-2 = 12.5 °C/m