Mechanical Engineering Principles

222 MECHANICAL ENGINEERING PRINCIPLES

different strips of metal riveted together).

When heated, brass expands more than steel,

and since the two metals are riveted together

the bimetallic strip is forced into an arc

as shown in Figure 20.1(b). Such a move-

ment can be arranged to make or break

an electric circuit and bimetallic strips are

used, in particular, in thermostats (which are

temperature-operated switches) used to con-

trol central heating systems, cookers, refrig-

erators, toasters, irons, hot-water and alarm

systems.

Brass Steel

(a) (b)

Figure 20.1

(vi) Motor engines use the rapid expansion of

heated gases to force a piston to move.

(vii) Designers must predict, and allow for, the
expansion of steel pipes in a steam-raising
plant so as to avoid damage and consequent
danger to health.

20.3 Expansion and contraction of

water

Water is a liquid that at low temperature displays an
unusual effect. If cooled, contraction occurs until,
at about 4°C, the volume is at a minimum. As the
temperature is further decreased from 4°Cto0°C
expansion occurs, i.e. the volume increases. When
ice is formed, considerable expansion occurs and it
is this expansion that often causes frozen water pipes
to burst.
A practical application of the expansion of a
liquid is with thermometers, where the expansion
of a liquid, such as mercury or alcohol, is used to
measure temperature.

20.4 Coefficient of linear expansion

The amount by which unit length of a material
expands when the temperature is raised one degree

is called thecoefficient of linear expansionof the

material and is represented byα(Greek alpha).

The units of the coefficient of linear expansion

are m/(mK), although it is usually quoted as just

/K or K−^1. For example, copper has a coefficient

of linear expansion value of 17× 10 −^6 K−^1 ,which

means that a 1 m long bar of copper expands by

0.000017 m if its temperature is increased by 1 K

(or 1°C). If a 6 m long bar of copper is subjected

to a temperature rise of 25 K then the bar will

expand by (6× 0. 000017 ×25) m, i.e. 0.00255 m

or 2.55 mm. (Since the kelvin scale uses the same

temperature interval as the Celsius scale, achange

of temperature of, say, 50°C, is the same as a change

of temperature of 50 K).

If a material, initially of length L 1 and at a

temperature oft 1 and having a coefficient of linear

expansionα, has its temperature increased tot 2 ,then

the new lengthL 2 of the material is given by:

New length=original length+expansion

i.e. L 2 =L 1 +L 1 α(t 2 −t 1 )

i.e. L 2 =L 1 [1+α(t 2 −t 1 )] ( 20. 1 )

Some typical values for the coefficient of linear

expansion include:

Aluminium 23× 10 −^6 K−^1 Brass 18 × 10 −^6 K−^1

Concrete 12 × 10 −^6 K−^1 Copper 17 × 10 −^6 K−^1

Gold 14 × 10 −^6 K−^1 Invar (nickel-

Iron 11–12× 10 −^6 K−^1 steel alloy) 0. 9 × 10 −^6 K−^1

Steel 15–16× 10 −^6 K−^1 Nylon 100 × 10 −^6 K−^1

Zinc 31 × 10 −^6 K−^1 Tungsten 4. 5 × 10 −^6 K−^1

Problem 1. The length of an iron steam

pipe is 20.0 m at a temperature of 18°C.

Determine the length of the pipe under

working conditions when the temperature is

300 °C. Assume the coefficient of linear

expansion of iron is 12× 10 −^6 K−^1.

Length L 1 = 20 .0 m, temperature t 1 = 18 °C,

t 2 = 300 °Candα= 12 × 10 −^6 K−^1

Length of pipe at 300°C is given by:

L 2 =L 1 [1+α(t 2 −t 1 )]

= 20 .0[1+( 12 × 10 −^6 )( 300 − 18 )]

= 20 .0[1+ 0 .003384]= 20 .0[1.003384]

= 20 .06768 m

i.e. an increase in length of 0.06768 m or 67.68 mm