Determination of the thermal conductivity of a metal by Searle’s method

As shown in the above Figure 30.1, let:

• $d$: mean diameter of the bar
• $l$: distance between thermometer $T_{1}$ and $T_{2}$
• $\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}$: readings of the thermometers $T_{1}, T_{2}, T_{3}, T_{4}$ in the steady state
• $m_{w}$: mass of water collected in the beaker
• $t$: time during which water is collected
• $c_{w}$: specific heat capacity of water
• $k$: thermal conductivity of the material of the bar

Then $\frac{dQ}{dt} = kA\cdot \frac{d\theta}{dt}$

$$\frac{m_{w}c_{w}(\theta_{3}-\theta_{4})}{t}=k\pi(\frac{d}{2})^{2}\frac{(\theta_{1}-\theta_{2})}{l}$$

1. First open the wooden box and using the vernier calliper measure two diameters $(d_{1}, d_{2})$ of the bar in two directions normal to each other to get the mean diameter.
2. Also using the calliper measure distances $(l_{1})$ from inside and $(l_{2})$ from outside between the thermometer $T_{1}$ and $T_{2}$ with the help of the outer jaws and the inner jaws of the calliper.
3. Close the box now to provide heat insulation.
4. Insert the thermometers $T_{1}$ and $T_{2}$ into the respective holes containing mercury to ensure good thermal contact.
5. Use the upper inlet of the steam chamber to admit steam so that the chamber will be completely filled with steam throughout the experiment.
6. Connect the water outlet of the constant pressure apparatus to the $T_{4}$ thermometer chamber so that the incoming stream of water would oppose and meet directly the heat conducted along the bar.
7. Note down the readings of the four thermometers at five minute time intervals.
8. When all these four readings become constant indicating steady state record the four readings $\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}$. If the difference between the readings of $T_{2}$ and $T_{4}$ is not sufficient, adjust the height of the water pressure head to obtain a sufficient difference.
9. Finally collect water from the $T_{3}$ outlet into an initially weighed $(m_{0})$ beaker for a time $(t)$ which is measured by a stop clock. Collect about 500 ml of water and measure the mass of the beaker with the water $(m_{1})$.
10. Taking the specific heat capacity of water as 4200 J kg⁻¹ K⁻¹ calculate $k$ as explained in the theory.