See Our New JEE Book on Amazon

**Problem:**
A cylinder of radius R made of a material of thermal conductivity K_{1} is surrounded by a cylindrical shell of inner radius R and outer radius 2R made of material of thermal conductivity K_{2}. The two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is?
(IIT JEE 1988)

- $K_1+K_2$
- ${K_1 K_2}/{(K_1 +K_2)}$
- ${(K_1 +3 K_2)}/{4}$
- ${(3K_1 + K_2)}/{4}$

**Answer:** The answer is (C) i.e., effective thermal conductivity of the system is ${(K_1 +3 K_2)}/{4}$.

**Solution:**
Let $\Delta T=(T_h-T_l)$ be the temperature difference between the two ends and $x$ be the length of the cylinders.

The rate of heat flow through the cylinder of radius $R$ and thermal conductivity $K_1$ is \begin{align} \frac{\mathrm{d}Q_1}{\mathrm{d}t}&=K_1A_1 \frac{\Delta T}{x}\\ &=K_1(\pi R^2)\Delta T/x.\nonumber \end{align} The rate of heat flow through the cylindrical shell of inner radius $R$, outer radius $2R$, and thermal conductivity $K_2$ is \begin{align} \frac{\mathrm{d}Q_2}{\mathrm{d}t}&=K_2A_2 \frac{\Delta T}{x}\\ &=K_2(4\pi R^2- \pi R^2)\Delta T/x \nonumber\\ &=3K_2\pi R^2\Delta T/x.\nonumber \end{align} The rate of heat flow through the system from one end to the other end is \begin{align} \label{moa:eqn:1} \frac{\mathrm{d}Q}{\mathrm{d}t}&=\frac{\mathrm{d}Q_1}{\mathrm{d}t}+\frac{\mathrm{d}Q_2}{\mathrm{d}t}\nonumber \nonumber\\ & = (K_1+3K_2)(\pi R^2) \Delta T/x\nonumber\\ & =\left(\tfrac{K_1+3K_2}{4}\right)(4\pi R^2)\Delta T/x \nonumber\\ &=K_\text{eff}A_\text{eff}\Delta T/x. \end{align} From above equation, the effective thermal conductivity is $K_\text{eff}=(K_1+3K_2)/4$.

- Heat Conduction
- The end Q and R of two thin wires, PQ and RS, are soldered (IIT JEE 2016)
- Two blocks connected in series and parallel ( IIT JEE 2013)
- Two rods connected in series and parallel (IIT JEE 2014)
- Two conducting cylinders of equal length but different radii connected in series (IIT JEE 2018)
- Temperature at the junction of the three identical rods (IIT JEE 2001)
- Three rods forms an isosceles triangle. Temperature at the vertex? (IIT JEE 1995)
- Evaporation of water and melting of ice kepts at two ends of a metal rod AB of length 10x (IIT JEE 2009)
- MCQ on a composite block is made of slabs A, B, C, D, and E (IIT JEE 2011)
- A point source of heat of power P kept at the centre of a spherical shell (IIT JEE 1991)
- A double pane window used for insulating a room thermally from outside (IIT JEE 1997)
- An electric heater is used in a room of total wall area 137 m
^{2}( IIT JEE 1986)