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Question
A metal rod of cross sectional area 1.0 cm2 is being heated at one end. At one time, the temperatures gradient is 5.0°C cm−1 at cross section A and is 2.5°C cm−1 at cross section B. Calculate the rate at which the temperature is increasing in the part AB of the rod. The heat capacity of the part AB = 0.40 J°C−1, thermal conductivity of the material of the rod = 200 W m−1°C−1. Neglect any loss of heat to the atmosphere
Solution
Let the temperatur
Let the temperatures at the ends A and B be TA and TB, respectively.
Rate of flow of heat at end A of the rod is given by
`(dQ_A)/(dr)= KA. d/dt (T_A)`
Rate of flow of heat at end B of the rod is given by
`(dQ_B)/dt ( T_B)`
Heat absorbed by the rod = ms∆T
Here, s is the specific heat of the rod and ∆T is the temperature difference between ends A and B.
Rate of heat absorption by the rod is given by
`(dQ)/dt =ms (dT)/(dr)`
∴ ` ms (dT)/(dr)= (KA. dT_A
)/(dl)`
`⇒ (0.4). (dT)/ dt= 200 xx 1xx 10^-4xx (5-2.5) `
`dt/dr=12.5^circ` C/sec
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