Powering Transport - some back-of-envelope calculations

David MacKay

I love back-of-envelope calculations, and I often throw them into my public lectures.

Here is one:
Question: If we powered all the cars on a road using biofuels grown on the verge of the road, how wide would the verge have to be?
Assumptions:Comments
one lane of cars
60 miles per hour (typical speed on the open road)
30 miles per imperial gallon (average for new cars in Europe today)
1200 litres of biofuel per hectare per year (typical for biofuels grown in Europe)
80 metres car-spacing A reasonably-safe stopping distance?

Answer:
(60 miles per hour) / ( (30 miles per imperial gallon) * (1200 litres per hectare per year) * (80 metres ) )
= 8 kilometres!
google's calculator can do these calculations for you)

Let's do another calculation. Imagine that we switch all the cars over from today's liquid-powered cars to today's electric cars, and power them with wind turbines. How far apart would the wind turbines be?


Assumptions:Comments
one lane of cars
60 miles per hour
20 kWh per 100 km (the Tesla can do 15 kWh per 100 km, so this may be a pessimistic figure)
0.5 MW per turbine (assuming a 2-MW turbine with a load factor of 25%, which is typical in Europe)
80 metres car-spacing

The power required per unit length of road is
(60 miles per hour) * (20 kWh per 100 km) / (80 metres )
= 0.24 MW per km.
(you can choose the units in which google calculator returns its answer)
So the spacing between turbines, to power one perpetual lane of cars, would be
( 0.5 MW per turbine ) / (0.24 MW per km) = 2 km per turbine.

If you would prefer to express this in terms of the 'effective width of the verge', assuming the verge were 'all windfarm', we can use my estimate of the power per area of wind farms of 2.5 W/m2. (See figure 3 of my paper 'Could energy intensive industries be powered by carbon-free electricity?' for data supporting this number.) We get:
Windfarm verge width = (0.24 MW per km) / (2.5 W/m**2) = 100 metres,
which is 80 times less land than the biofuel solution; and most of that 100 metre strip would not be occupied by the wind turbines, so it would still be available for other non-wind-harvesting uses such as agriculture. The wind-farm solution uses less land because, under the above assumptions, electric vehicles are more energy-efficient than liquid vehicles, and because the power per unit area of biofuels in Europe is very small, even compared to the power-per-area of windfarms, which, at 2.5 W/m2, is not huge.

What about solar panels?
solar-motorwayS
Solar panels on a roof alongside a motorway in Belgium. Picture: AFP/GETTY
If the verge were occupied by solar panels, how wide would the verge need to be to power all the cars?
We have got most of the numbers already, from the wind/electric-car assumptions. We need to introduce just one more number, namely the average power per unit area of solar parks in Europe. This can be found in my paper 'Solar energy in the context of energy use, energy transportation, and energy storage': 5 W/m2, for real solar parks in Germany and England. (Higher powers per unit area, such as 10 W/m2, are achieved by solar parks in more sensible sunny locations such as Spain and California.)

Solar park verge width = (0.24 MW per km) / (5 W/m**2) = 50 metres.
For comparison, a typical single lane on a motorway is 3.65 metres wide.
[Note however that solar parks in Europe produce seasonally intermittent power that would not be well matched to transport demand, which is quite steady. In UK and Germany, the average midwinter power output of a solar park is roughly ten times less than the midsummer output.]


Summary: how big would the verge be?
Biofuel verge liquid-fuelled cars8000 m
Wind park verge electric cars100 m(or one 2MW turbine every 2km)
Solar park verge electric cars50 m

All the standard disclaimers apply — answers depend on assumptions; assumptions could be changed; lots of innovations are possible in crops / crop processing / vehicle design, such that the assumptions might be changed. Yes, I know! Indeed all those possibilities are one motivation for doing back-of-envelope calculations, to make clear which assumptions are the one we really want to make progress on, to make a material difference.


David MacKay FRS is a member of the 2014 Longitude Committee. He was the Chief Scientific Advisor at the Department of Energy and Climate Change from 2009 to 2014, and is Regius Professor of Engineering at the University of Cambridge. He is well known as author of the popular science book, Sustainable Energy — without the hot air.


David MacKay FRS
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Last modified: Mon Jun 23 17:31:02 BST 2014