Question

On a cool (4.0ºC) Saturday morning, a pilot fills the fuel tanks of her Pitts S-2C (a two-seat aerobatic airplane) to their full capacity of 106.0 L. Before flying on Sunday morning, when the temperature is again 4.0ºC, she checks the fuel level and finds only103.4 L of gasoline in the aluminumtanks. She realizes that it was hot on Saturday afternoon and that thermal expansion of the gasoline caused the missing fuel to empty out of the tank's vent. (a) What was the maximum temperature (in ºC) of the fuelland the tank on Saturday afternoon? The coefficient of volume expansion of gasoline is $\mathbf{9}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathbf{}{\mathbf{K}}^{\mathbf{-}\mathbf{1}}$. (b) To have the maximum amount of fuel available for flight, when should the pilotagefilled the fuel tanks?

Short answer

(a) The maximum temperature of the fuel and the tank are 32°C

(b) The time of filling the fuel tanks is just before the takeoff of the flight.

Step-by-step solution

  Concept of Volume Expansion in volume.

Volume expansion is the expansion of the dimensions of an object with respect to temperature changes and can be expressed as,

$\mathbf{∆}\mathbf{V}\mathbf{=}{\mathbf{V}}_{\mathbf{0}}\mathbf{\beta }\mathbf{∆}\mathbf{T}$ ...(i)

Here, V is the expansion in volume, V₀ is the unexpanded volume, β is the coefficient of volume expansion and T is the absolute temperature.

(a) Determination of maximum temperature of the fuel and the tank.

Volume lost,

$∆\mathrm{V}=\left(106.0‐103.4\right)\mathrm{L}=2.6\mathrm{L}$

This lost volume gives the measure of the difference between volume expansion of the fuel and the tank.

Rearranging equation (i).

$∆\mathrm{T}=\frac{∆\mathrm{V}}{{\mathrm{V}}_0\mathrm{\beta }}=\frac{∆\mathrm{V}}{{\mathrm{V}}_0\left({\mathrm{\beta }}_{\mathrm{fuel}}‐{\mathrm{\beta }}_{\mathrm{Aluminium}}\right)}$

Substitute all the values in the above equation,

$$\begin{gathered} ∆\mathrm{T}=\frac{2.6\mathrm{L}}{\left(106.0\times {10}^{-3}{\mathrm{m}}^3\right)\left(9.5\times {10}^{-4}{\left(°\mathrm{C}\right)}^{-1}-7.2\times {10}^{-5}{\left(°\mathrm{C}\right)}^{-1}\right)} \\ =28°\mathrm{C} \end{gathered}$$

The maximum temperature was,

$28°\mathrm{C}+4°\mathrm{C}=32°\mathrm{C}$

(b) Determination of time of filling the tank.

The tank should be filled just before the takeoff to ensure there is no loss in fuel. This is because there is a measure of volume expansion in both the gasoline and the aluminum tank when the temperature increases. But, the expansion itself depends on the coefficient of volume expansion characteristic to the material, i.e. it is different for gasoline and aluminum. So, when the temperature comes down 4ºC the next morning, there is an air gap left for the apparently lost fuel.