15–1

15: THERMAL CONDUCTIVITY

Aim (1) To determine the thermal conductivity k of copper using a modified version of Searle's method. (2) Compare values for k obtained using two different techniques; compare theory and experiment. Introduction If two opposite faces of a block of material are maintained at a constant difference in temperature, then the amount of heat energy flowing through the block in unit time, i.e. the power dissipated, is given by:

P =

ΔQΔt

=kA (θ1 − θ2 )

d … (15.1)

where A is the area of the faces, d is the length of the block, θ1 and θ2 are the temperatures of the two faces, and k is a constant, called the thermal conductivity, which depends on the material of the block. Searle's apparatus is used to measure k for a material. A modified version of Searle's apparatus is shown in Figure 15.1. The material whose thermal conductivity is to be determined (a copper bar in our case) is heated electrically at one end and cooled by water circulating through a cavity at the other. The cavity has water inlet and outlet tubes which each have a support tube for a thermometer (Th3 and Th4) so that the rise in temperature of the water passing through the cavity can be determined. In the steady state, the temperature gradient along the bar is measured by means of two thermometers (Th1 and Th2) placed in support tubes fixed to the surface of the bar. The apparatus is lagged to minimise heat losses from the bar. In Searle's method, it is assumed that all the heat enters one end of the bar, is conducted along the bar, and is extracted at the other end. This gives us two methods for calculating the thermal conductivity k from equation (15.1): (A) by determining the rate at which heat is extracted from the bar; and (B) by determining the rate at which the heat is supplied to the bar.

15–2

Figure 15.1. Searle's apparatus.

Method (A): Heat extracted from bar. Assume that T1, T2, T3 and T4 are the readings of the thermometers Th1, Th2, Th3 and Th4 in a steady state, and that a mass M of water (of specific heat C) emerges from the cavity in a time Δt. Then the rate at which the water extracts energy from the bar is:

ΔQΔt

=M (C3T3 − C4T4)

Δt≈

MC (T3 − T4)Δt

, where C =C3 + C4

2 … (15.2).

Substituting equation (15.2) into equation (15.1), and using A = πD2/4 where D is the diameter of the rod, we obtain:

kA =

4MCd (T3 −T 4)πD2 (T1 − T2) Δt

… (15.3).

Method (B): Heat supplied to the bar. Since the bar is electrically heated, we can calculate the rate of supply of energy to the bar. Thus if a current I passes through the heater under a potential difference V, the rate of supply of energy is:

ΔQΔt

= VI … (15.4).

Substituting equation (15.4) into equation (15.1) and rearranging as before, we get

kB =

4VIdπD2 (T1 −T2 )

… (15.5).

Ideally the values for the thermal conductivity of copper obtained by the two methods, kA and kB, should be the same.

15–3

Procedure

Examine the demonstration model of the modified Searle's apparatus which is on display in the laboratory. Draw your own sketch, and identify the parts shown in Figure 15.1.

Temperatures are measured using platinum thermometers. Good thermal contact with the support tubes is achieved by means of heat-sink compound. The thermometers are linked to a computer via a data acquisition box. After start-up the computer displays the MATLAB window. At the command prompt ‘>>’ type ‘tc’ and then press ‘Enter’. After typing your names, press ‘Enter’ again. This will start the data acquisition, which has a sampling rate of 1000 readings per second. The computer displays simultaneously temperature (in °C) for each thermometer at the top right hand corner of the screen at one-second intervals. Temperature readings at the bottom right hand corner of the screen are the average of 60000 readings, updated every minute. The averaged temperature readings are accumulated in the lower part of the graph. The computer sounds a beep at every update. Typically, it takes 30 minutes from a cold start of the equipment for it to achieve thermal equilibrium. Should temperature readings not be constant after such time, check that the water flow rate is sufficient, and that flow rate and power supply are stable.

At no time should the total current to the electrical heater be allowed to exceed 6 A. On arrival in the Laboratory, you will find that the equipment will have been set to a water flow rate of about 120 g min–1 and a power dissipation of about 30 W. Check that the system has reached a steady state, by observing the temperature of the four thermometers. Do not proceed until the system is in thermal equilibrium! This is a good time to read through the rest of the script, and plan your approach to taking measurements. Note that uncertainties are not required until the final stage of the experiment, but you should record your estimates as you proceed.

In particular, think ahead and plan how you will record your measurements—set up a table!

Determine the mass flow rate by weighing the amount of water collected from the cavity in a measured time.

Record the current I through the heater and the voltage V across it, and hence calculate the actual

power dissipation.

Checkpoint 1: ask a demonstrator to review your work

Record four consecutive sets of temperature readings for each of the thermometers Th1 to Th4.

Work out an average value for each of the quantities T1 to T4.

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Now, adjust the needle valve on the flow indicator until the flow rate increases to about 140 g min–1 and increase the current through the heater until the power is about 35 W. When a steady state has again been reached, take new sets of readings for the flow rate, temperatures, and power supplied to the equipment. Repeat the procedure once more. Increase the flow rate by about 20 g min–1 (to about 160 g min–1) and the power by about 5 W (to about 40 W). When all the data has been recorded, switch off the electrical power but leave the water flowing through the equipment.

Checkpoint 2: ask a demonstrator to review your work

Calculate kA using Equation (15.3), and kB using Equation (15.5), for each of the water flow rates. [Values for the specific heat capacity of water C as a function of temperature are shown in Table 15.1. Use the value for C for the temperature nearest the average of T3 and T4.]

From the three values of kA obtained, calculate a mean value and the uncertainty in the mean value.

Repeat the same procedure for kB.

Discussion and Conclusion Your discussion is a significant part of this experiment. You should compare the values of kA and kB with each other, and with the value published in Kaye & Laby. Using Equations (15.3) and (15.5), account for any discrepancies you observe. You will need to think about how any “real world” effects could influence your results. A qualitative discussion is fine, but you should be as detailed as you can in your analysis. As a guide, about two or three paragraphs would be appropriate.

Checkpoint 3: ask a demonstrator to finish marking your experiment!

Table 15.1. Temperature dependence of the specific heat capacity C of water.

°C 0 1 2 3 4 5 6 7 8 9 0

10

20

30

40

4217.4

4191.9

4181.6

4178.2

4178.3

4213.8

4190.4

4181.0

4178.1

4178.4

4210.4

4189.0

4180.5

4178.0

4178.6

4207.4

4187.7

4180.1

4178.0

4178.8

4204.5

4186.6

4179.7

4177.9

4178.9

4201.9

4185.5

4179.3

4177.9

4179.2

4199.6

4184.6

4179.0

4178.0

4179.4

4197.4

4183.7

4178.7

4178.0

4179.6

4195.4

4182.9

4178.5

4178.1

4179.9

4193.6

4182.2

4178.3

4178.2

4180.1

Notes:

15–5

(i) The units of specific heat capacity are J kg–1 °C–1 ; temperatures are in °C. (ii) An example of using this table: if we want to obtain the specific heat C for a temperature of

24°C, we read along the 20 °C row and down the 4 °C column, obtaining a reading of 4179.7 J kg–1 °C–1.