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_1``(lt theta_(0))`. A metal rod of length `L`, area of cross section `A` and thermal conductivity `K` connects the two vessels. Find the time taken for the temperature of the water in the smaller vessel to become `theta_(2)``(theta_(1)lt theta_(2)lt theta_(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
`DeltaQ=(KA)/(L)(theta_(0)-theta)dt` . This heat increases the temperature of the water of mass m to `theta+d theta where `DeltaQ=msd theta` . From (i) and (ii),
(KA)/L(theta_(0)-theta)dt=msd theta` . or, `dt=(Lms)/(KA)(d theta)/(theta_(0)-theta` . or, `int_(0)^(T)dt=(Lms)/(KA)int_(theta_(1))^(theta^(2))(d theta)/(theta_(0)-theta)` . where T is the time requierd for the temperature of the water to become `theta_(2)` . Thus, `T=(Lms)/(KA) ln `(theta_(0)-theta_(1))/(theta_(0)-theta_(2))` .