Question

Aluminium rivets used in airplane construction are made slightly larger than the rivet holes and cooled by “dry ice” (solid CO₂) before being driven. If the diameter of a hole is 4.500 mm, what should be the diameter of a rivet at 23.0ºC if its diameter is to equal that of the hole when the rivet is cooled to -78.0ºC, the temperature of dry ice? Assume that the expansion coefficient remains constant at the value given in Table 17.1.

Short answer

The diameter value of the rivet is$4.511\mathrm{mm}$.

Step-by-step solution

Concept of Linear Expansion in solids.

The linear change in length when the temperature change is given by the linear expansion expression,

$\mathbf{L}\mathbf{=}{\mathbf{L}}_{\mathbf{0}}\left(1+\alpha ∆T\right)$ ...(i)

Here, L is the expanded length, L₀ is the original length, α is the coefficient of expansion of the material and T is the absolute temperature.

 Determination of the diameter value of the rivet following the conditions mentioned.

For aluminium, the coefficient of linear expansion,

$\alpha =2.4\times {10}^{-5}{\left(C°\right)}^{-1}$

Take the diameter of rivet at -78.0ºC as D₀and the diameter at 23.0ºC is D. Then,

$$\begin{gathered} \mathrm{D}={\mathrm{D}}_0+∆D \\ ={\mathrm{D}}_0\left(1+\alpha ∆T\right) \end{gathered}$$

Substitute all the values in the above equation,

$$\begin{gathered} \mathrm{D}=\left(0.4500\mathrm{cm}\right)\left(1+2.4\times {10}^{-5}{\left(\mathrm{C}°\right)}^{-1}\right)\left(23.0\mathrm{C}°-\left(-78.0\mathrm{C}°\right)\right) \\ =0.4511\mathrm{cm} \\ =4.511\mathrm{mm} \end{gathered}$$

Thus, the expanded diameter value of the rivet is $4.511\mathrm{mm}$