Filling up at dawn

I recently read in a free daily paper that it is advisable to fill up the car tank early in the morning rather than at noon because the gasoline will be usually cooler at that time of the day and thus denser, so we are pumping more grams of gas per liter paid. Does this really make a difference?

The volumetric thermal expansion coefficient of gasoline is 950·10^-6/°C, which means that when the gasoline is colder we obtain an excess mass of

(m_cold − m_hot)/m_hot = 950·10^-6ΔT.

For a rather large ΔT of 20 °C the gain would be around 2%. Take into account, however, that station tanks are buried a couple of meters underground, which damps the temperature oscillations experienced at the surface. This damping effect can be estimated as:

ΔT(z) = ΔT_surface·e^−z/D,
D = √(2K/ω),

where K is the soil thermal diffusivity, with typical values for natural soils ranging from 0.2 to 0.8 10^-6m²/s. Setting z = 2 m and ω = 2π rad/day, we have

1.9·10^-12ΔT_surface ≤ ΔT_{station tank} ≤ 1.4·10^-6ΔT_surface,

that is, the temperature of the station tank is basically constant across the day. So, filling up the car tank in the cool hours of the day does not seem to make any difference.

There is a measurable gain, though, when we compare car tank filling during different seasons of the year. Here, the temperature oscillations are higher and, most importantly, the damping effect of the soil is much lesser. Taking

ΔT_surface = 30 °C,
z = 2 m,
ω = 2π rad/year,

we have

7.3 °C ≤ ΔT_{station tank} ≤ 14.8 °C,

which translates to

0.7% ≤ (m_cold − m_hot)/m_{hot t} ≤ 1.4%.

The potential impact as a cost-saving measure is, in any case, very unspectacular.