The Bimetallic Strip

A bimetallic strip is made from two thin strips of metal that have different coefficients of linear expansion, as figure shows.

(a)

(b) Heated (c) Cooled

(a) A bimetallic strip and how it behaves when (b) heated and (c) cooled

Often brass \(\left[\alpha=19 \times 10^{-6}\left(\mathrm{C}^{\circ}\right)^{-1}\right]\) and steel \(\left[\alpha=12 \times 10^{-6}\left(\mathrm{C}^{\circ}\right)^{-1}\right]\) are selected. The two pieces are welded or riveted together. When the bimetallic strip is heated, the brass, having the larger value of \(\alpha\), expands more than the steel. Since the two metals are bonded together, the bimetallic strip bends into an arc as in part \(b\), with the longer brass piece having a larger radius than the steel piece. When the strip is cooled, the bimetallic strip bends in the opposite direction, as in part c.

The expansion in length is called linear expansion. The expansion in area is called area expansion. The expansion in volume is called volume expansion

\( \frac{\Delta l}{l}=a_{1} T \)

(a) Linear expansion

\(\frac{\Delta A}{A}=2 a_{1} T\)

(b) Area expansion

(c) Volume expansion Illustration :

Show that the coefficient of area expansion, \((\triangle A / A) / \Delta T\). of a rectangular sheet of the solid is twice its linear expansively, \(\alpha_{1}\)

Consider a rectangular sheet of the solid material of length a and breadth \(b\) (Fig.). When the temperature increases by \(\Delta T\), a increases by \(\Delta a=a \alpha_{1} \Delta T\) and \(b\) increases by \(\Delta b=b \alpha_{1} \Delta T\). From figure the increase in area

\(\Delta A=\Delta A_{1}+\Delta A_{2}+\Delta A_{3} \)

\( \Delta A=a \Delta b+b \Delta a+(\Delta a)(\Delta b) \)

 \(=a \alpha_{l} b \Delta T+b \alpha_{1} a \Delta T+\left(\alpha_{1}\right)^{2} a b(\Delta T)^{2} \)

\( =\alpha_{1} a b \Delta T\left(2+\alpha_{1} \Delta T\right)=\alpha_{1} A \Delta T\left(2+\alpha_{1} \Delta T\right)\)

Since \(\alpha_{1} ; 10^{-5} \mathrm{~K}^{-1}\), the product \(\alpha_{1} \Delta T\) for fractional temperature is small in comparison with 2 and may be neglected. Hence,

\( \left(\frac{\Delta \mathrm{A}}{\mathrm{A}}\right) \frac{1}{\Delta \mathrm{T}}=2 \alpha_{1} \)

Volume Thermal Expansion

The volume of a normal material increases as the temperature increases. Most solids and liquids behave in this fashion. By analogy with linear thermal expansion, the change in volume \(\Delta \mathrm{V}\) is proportional to the change in temperature \(\Delta \mathrm{T}\) and to the initial volume \(\mathrm{V}_{0}\), provided the change in temperature is not too large. These two proportionalities can be converted into equation \(\Delta \mathrm{V}=\gamma \mathrm{V}_{0} \Delta \mathrm{T}\) with the aid of a proportionality constant \(\gamma\), known as the coefficient of volume expansion. The algebraic form of this equation is similar to that for linear expansion, \(\Delta \mathrm{L}=\alpha \mathrm{L}_{0} \Delta \mathrm{T}\).

Common Unit for the coefficient of volume Expansion : \(\left(\mathrm{C}^{\circ}\right)^{-1}\)

The unit for \(\gamma\), like that for \(\alpha\), is \(\left(C^{\circ}\right)^{-1}\). Values for \(\gamma\) depend on the nature of the material. The values of \(\gamma\) for liquids are substantially larger than those for solids, because liquids typically expand more than solids. Given the same initial volumes and temperature expansion is three times greater than the coefficient of linear expansion : \(\gamma=3 \alpha\).

If a cavity exists within a solid object, the volume of the cavity increases when the object expands, just as if the cavity were filled with the surrounding material. The expansion of the cavity is analogous to the expansion of a hole in a sheet of material. Accordingly, the change in volume of a cavity can be found using the relation \(\Delta \mathrm{V}=\gamma \mathrm{V}_{0} \Delta \mathrm{T}\), where \(\gamma\) is the coefficient of volume expansion of the material that surrounds the cavity.