A large tank A of water is kept at a constant temperature $\theta_0$.conductor of length l, area of cross-section A and thermal conductivity K.

Question

A large tank $A$ of water is kept at a constant temperature $\theta_0$. It is connected to another body $B$ of specific heat $c$ and mass $m$ by a conductor of length $l$, area of cross-section $A$ and thermal conductivity $K$. Initial temperature of body $B$ is $\theta_1$. Find the variation of temperature $\theta$ of the body $B$ at time $t$.

Answer 1

SOLUTION : $d t$, let a heat $d Q$ flow through the conductor rod.

Then $\frac{d Q}{d t}=\frac{\theta_0-\theta}{t} K A$

The rate of increase in thermal energy of the tank $B$ is

$ \frac{d U_B}{d t}=m c \frac{d \theta}{d t} $

Since $\frac{d Q}{d t}=\frac{d U_B}{d t}$ because no work is done in change in volume, we have

$m c \frac{d \theta}{d t}=K\left(\frac{\theta_0-\theta}{l}\right) A \text { or } \frac{d \theta}{\theta_0-\theta}=\frac{K A}{m l c} d t$

Integrating both sides, we get

$\int_{\theta_1}^\theta \frac{d \theta}{\theta_0-\theta}=\frac{K A}{m l c} t \text { or } \ln \frac{\theta_0-\theta}{\theta_0-\theta_1}=\frac{-K A}{m l c} t$

or $\left(\theta_0-\theta\right)=\left(\theta_0-\theta_1\right) e^{\frac{-K A t}{m l c}}$

or $\theta=\theta_0-\left(\theta_0-\theta_1\right) e^{\frac{-K A}{m l c} t}$