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There is a lot of rubbish here. Proponents of part tank filling base their calculations on the weight difference between a full and half full tank. For a 48 litre tank, that is 24 litres or about 20kg. However, if you take the average level in the tank, the difference is only 12 litres or 10 kg. This will of course increase the fuel needed to accelerate the vehicle but only by about 0.2ml to get the vehicle to 60 mph (assuming overall engine/transmission efficiency of about 50%). By comparison, it takes 15 to 20 ml to get each tonne of vehicle to that speed. That extra 0.2ml of fuel gives the extra 10 kg of vehicle weight extra kinetic energy, so it coasts for a greater distance when you take your foot of the accerator. The net result, under worst conditions, is an unmeasurable increase in consumption - or no increase at all if you minimise braking.
However, the half tank filler must visit the filling station 2 or 3 times for 1 visit by the full tanker. Because of the possible need to wait and the fact that you have to travel slowly in low gear (high consumption) you use a relatively large amount of fuel when visiting the forcourt without making any progress on your journey. Depending on your vehicle, this is likely to be at least 20 to 30 ml which vastly outweighs any saving you might make by part filling the tank. You also waste your own time and be unable to keep an accurate check on consumption.
The most efficient approach is to always fill the tank and to run the vehicle to below quarter full. You will not pay more for fuel overall, you just pay in larger installments, but fewer of them.
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