Question

A copper rod initially at room temperature ${20}^{\circ }\text{C})$ of non-uniform cross-section is placed between a steam chamber at ${100}^{\circ }\text{C}$ and ice-water chamber at $0^{\circ }\text{C}$

Identiity the correct statement (s):

• A. Initially the rate of heat flow $\left(\frac{dQ}{dt}\right)$ will be maximum at section $B$
• B. Initially the rate of heat flow $\left(\frac{dQ}{dt}\right)$ will be minimum at section $C$
• C. At steady state the temperature gradient $\left|\left(\frac{dT}{dx}\right)\right|$ will be maximum at the section $B$
• D. At steady state the rate of change of temperature $\left(\frac{dT}{dt}\right)$ for all sections will be equal to zero.

Answer 1

Answer: (B), (C), (D)

At steady state the rate of change of temperature $\left(\frac{dT}{dt}\right)$ for all sections will be equal to zero.

Initially due to temperature difference $\Delta T={100}^{\circ }\text{C}$, maximum amount of heat current $\left(\frac{dQ}{dt}\right)$ will enter through section $A$ but the metal will absorb some amount of heat during flow of heat current from $A$ to $C$.
Thus the value of heat current $\left(\frac{dQ}{dt}\right)$ leaving section $C$ will be minimum.
$\therefore$ option (b) is correct.

At steady statement the rate of heat flow (heat current ) is constant.
$\Rightarrow \frac{dQ}{dt}=kA\left|\frac{dT}{dx}\right|$
or $\left|\frac{dT}{dx}\right|=\frac{\left(\frac{dQ}{dt}\right)}{kA}$
$\left[\right(\because \frac{dQ}{dt}=\text{constant}$ & $k=$ constant for material)]
$\left|\frac{dT}{dx}\right|\propto \frac{1}{A}$
Hence temperature gradient is inversity proportional to area.
$\because$ Area of cross - section at section $B$ is minimum
$\Rightarrow \left|\frac{dT}{dx}\right|$ is maximum at section $B$.

At steady state the rate of heat accumulation at any cross -section is zero.
Hence, the rate of change of temperature w.r.t time at any cross-section will be zero.
$\Rightarrow \frac{dT}{dt}=0$
$\therefore$ option (c) & (d) are correct.

$\begin{array}{c}\text{Why this question}?\\ \text{Tip: At steady state the rate of heat outflow}\\ \text{and heat inflow at any cross-section is equal,}\\ \text{hence no heat accumulation occuss at steady state.}\end{array}$