Figure shows a large tank of water at a constant temperature `theta_(0)` and a small vessel containing a mass `m` of water at an initial temperature `theta(lttheta_(0))`. A metal rod of length `L`, area of cross section `A` and thermal conductivity `K` connect the two vessels. Find the time taken for the temperature of the water in the smaller vessel to become `theta_(2)(theta_(1)lttheta_(2)lttheta_(0))` . Specific heat capacity of water is ` s` and all other heat capacities are negligible.

Explanation

Suppose, the temperature of the water in the smaller vessel is `theta` at time t, In the next time interval dt, a heat `DeltaQ` is transferred to it where
`Delta Q = (KA)/(L ) (theta_0 - theta) dt" "…(i)`
This heat increases the temperature of the water of mass m to `theta+ d theta` where
`Delta Q= ms d theta" "...(ii)`
From (i) and (ii)
`(KA)/(L ) (theta_0- theta) dt = ms d theta`
or, `dt = (L ms)/(KA ) (d theta)/( theta_0 - theta) or, underset(0) overset( T) ( int) dt = (Lms)/(KA)=underset(theta_1) overset(theta_2) (int) (d theta)/(theta_0- theta)`
where T is the time required for the temperature of the water to become `theta_2`.
Thus, `T=(Lms)/(KA ) 1n (theta_0- theta_1)/(theta_0- theta_2)`