Thermal Expansion Concept Page - 2

Formula

Relation between linear,superficial and volume expansion

Following relationships exist between various expansion coefficients:
β=2α$\beta =2\alpha$
and
γ=3α$\gamma =3\alpha$
so 
α:β:γ=1:2:3$\alpha :\beta :\gamma =1:2:3$

Definition

Change in volume of isotropic solids on heating

For isotropic materials the volumetric thermal expansion coefficient is three times the linear coefficient:
αV=3αL${\displaystyle {\alpha }_V=3{\alpha }_L}$
This ratio arises because volume is composed of three mutually orthogonal directions. Thus, in an isotropic material, for small differential changes, one-third of the volumetric expansion is in a single axis.

Example

Thermal volume expansion of a cube

Example:
Consider a cube of side l$l$ at 0oC$0^{o}C$ and coefficient of thermal expansion αv${\alpha }_v$. Find the length of its side at temperature of 10oC${10}^{o}C$.
Solution:
V2=V1(1+αvΔT)$V_2=V_1(1+{\alpha }_v\mathrm{\Delta }T)$
V2=l3(1+10αv)$V_2=l^3(1+10{\alpha }_v)$
l2=V1/32=l(1+10αv)1/3$l_2=V_2^{1/3}=l(1+10{\alpha }_v{)}^{1/3}$

Definition

Linear Expansion

Linear Expansion : 
It refers to a fractional change in size of a material due to a change in temperature.

It is represented as L=Lo(1+αΔT)$L=L_o(1+\alpha \mathrm{\Delta }T)$
                     where Lo$L_o$ = Original Length
                                L=$L=$Expanded Length
                                α=$\alpha =$Coefficient of Linear Expansion
                                ΔT=$\mathrm{\Delta }T=$Temperature Difference 
           

Example

Linear expansion of a composite system of rods attached end to end

Two rods P and Q of different metals, having the same area A$A$ and the same length L$L$, are placed between two rigid walls as shown in figure. The coefficients of linear expansion of P and Q are α1${\alpha }_1$ and α2${\alpha }_2$ respectively, and their Young's moduli are Y1$Y_1$ and Y2$Y_2$ . The temperature of both rods is now raised by T$T$ degrees. The new length of the rod P is:
We have,F=AY(ΔL′)L$F={\displaystyle \frac{AY(\mathrm{\Delta }{L}^{\prime })}{L}}$
FP=FQ=AY1(Lα1T−ΔL)L=AY2L(Lα2T+ΔL)$F_P=F_Q={\displaystyle \frac{A{Y}_1(L{\alpha }_{1}T-\mathrm{\Delta }L)}{L}}={\displaystyle \frac{A{Y}_2}{L}}(L{\alpha }_{2}T+\mathrm{\Delta }L)$
⇒ΔL=LTY1+Y2[Y1α1−Y2α2]$\Rightarrow \mathrm{\Delta }L={\displaystyle \frac{LT}{Y_1+Y_2}}[Y_1{\alpha }_1-Y_2{\alpha }_2]$
LP=L1=L+ΔL$L_P=L_1=L+\mathrm{\Delta }L$
=L+LTY1+Y2[Y1α1−Y2α2]$=L+{\displaystyle \frac{LT}{Y_1+Y_2}}[Y_1{\alpha }_1-Y_2{\alpha }_2]$
=L+LTY1+Y2[Y1α1−α2Y2]$=L+{\displaystyle \frac{LT}{Y_1+Y_2}}[Y_1{\alpha }_1-{\alpha }_2{Y}_2]$
=L+LT[α1Y1+α1Y2−Y2α1−α2Y2Y1+Y2]$=L+LT[{\displaystyle \frac{{\alpha }_1{Y}_1+{\alpha }_1{Y}_2-Y_2{\alpha }_1-{\alpha }_2{Y}_2}{Y_1+Y_2}}]$
=L+Lα1T−LY2T(y1+Y2)[α1+α2]$=L+L{\alpha }_{1}T-{\displaystyle \frac{L{Y}_{2}T}{(y_1+Y_2)}}[{\alpha }_1+{\alpha }_2]$
=L+Lα1T−LAY1Y1Y2(Y1+Y2)AT[α1+α2]$=L+L{\alpha }_{1}T-{\displaystyle \frac{L}{A{Y}_1}}{\displaystyle \frac{Y_1{Y}_2}{(Y_1+Y_2)}}AT[{\alpha }_1+{\alpha }_2]$
=L[1+α1T−FAY1]$=L[1+{\alpha }_{1}T-{\displaystyle \frac{F}{A{Y}_1}}]$

Definition

Linear expansion of bimetallic system of rods

Two straight metallic strips each of thickness t$t$ and length L$L$ are rivetted together. Their coefficients of linear expansions are α1${\alpha }_1$ and α2${\alpha }_2$. If they are heated through temperature Δθ$\mathrm{\Delta }\theta$, the bimetallic strip will bend to form an arc of radius r$r$. Find r.θ=lr$\theta ={\displaystyle \frac{l}{r}}$
Also, θ=dldr=l2−l1r2−r1=lαΔTt$\theta ={\displaystyle \frac{dl}{dr}}={\displaystyle \frac{l_2-l_1}{r_2-r_1}}={\displaystyle \frac{l\alpha \mathrm{\Delta }T}{t}}$
∴lr=l2−l1r2−r1=lαΔTt$\therefore {\displaystyle \frac{l}{r}}={\displaystyle \frac{l_2-l_1}{r_2-r_1}}={\displaystyle \frac{l\alpha \mathrm{\Delta }T}{t}}$
⇒r=t(α1−α2)ΔT$\Rightarrow r={\displaystyle \frac{t}{({\alpha }_1-{\alpha }_2)\mathrm{\Delta }T}}$

Example

Advantage of thermal expansion of solid

Following are the advantages of thermal expansion:(1) If we find it difficult to remove the stopper from a glass bottle, we can heat the neck of the bottle. Now the neck of the bottle expands and the stopper comes out easily.(2) The principle of thermal expansion is used in fixing iron rim with the wooden wheel firmly.(3) Rivets are used to hold steel plates together very tightly. A very hot rivet is pushed through the two plates and its end is hammered over. When the rivets cools down it pulls the two plates together very tightly.

Example

Disadvantages of thermal expansion of solids

(1) Changing of shape and dimensions of objects such as doors.(2) Wall collapsing due to bulging.(3) Cracking of glass tumbler due to heating.(4) Bursting of metal pipes carrying hot water or steam are some of the disadvantages of thermal expansion of matter.

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