It has been suggested in the high-rise, densely populated cities of the near future that there is the potential to generate energy from rain falling on the rooves of these tall building. It is shown from first principles, by means of a simple calculation using some basic equations in physics, that this is not an economically feasible concept. Some suggestions will be made to correct the British government’s erroneous assumptions when it comes to governing energy strategy through its Electricity Market Reform policies.
It does not seem feasible to me to produce electricity from rainfall on a city. The first thought that came to my mind when this idea was suggested is that water is generally pumped to the top of high buildings and held in storage tanks to be available for when it is needed to be used - in order to maintain satisfactory water pressure. The second thought that I had was that there would not be a significant enough amount of water falling on a building’s roof to generate more hydro energy than this standard engineering process consumes. Further thought led me to believe that there is more potential for a “pumped storage” system for energy storage using a water tank at the top of a tall building to store electricity than there is for a sustainable hydropower scheme.
Now the economics of the situation are really governed by the physics. So let us consider the equation for Gravitational Potential Energy (GPE), which is GPE = mgh (which is the mass multiplied by the effect of gravity of the water (or its weight) multiplied by the height of the building). This would then be converted into Kinetic Energy (KE) when the water drops down. So GPE = mgh = KE (= 1/2mv^2)
This kinetic energy could be used to turn a turbine with efficiency of say at least 50% given the inefficiencies such a system would have to overcome.
Now, let’s do a calculation: 50cm rain falling on 1 hectare (10,000 sq metre), and then falling a height of 1000 metres (taller than the Burj Khalifa in Dubai but in the future it could be possible), a typical high-rise city of the future, how many kWh can be generated?
So 0.5m of rain (a huge amount - more than a year's worth!)
We have the volume of rain given by area multiplied by depth of total rainfall = 0.5*10000=5000m^3, 1 m^3 water = 1 ton
GPE = mgh = 5000 ton of water * 1000 (to convert to kg) * 10 ( m/s^2, the effects of to 1 significant figure) * 1000 (m, height of the buildings) = 50 000 000 000 kg m^2/s^2 = 50 billion joules
The unit we are after is kWh (kilo watt hours).
1 watt is 1 joule per second.
So we need to divide the figure for the KE by the efficiency factor which we have already agreed and then by 1000 to convert to kW and then divide by the number of seconds in 1 hour:
50 000 000 000 / 2 (efficiency of system) / 1000 / 60 / 60 = 6944 kWh.
6944 kWh is enough electricity to power one energy-efficient US household.
To pump water to top of the building storage tank would require more energy than this would generate, as the pumps are not as efficient as the turbines and lose energy to heat for example.
In pumped storage where water is pumped to a higher reservoir during off peak electricity prices and then during peak demand, where electricity has a higher market price, it is released through turbines into a lower reservoir turning the GPE into electricity again. The energy efficiency of a system such as this one (for example in Dinorwig Power Station in Wales) is about 70%.
It is clear that in these cities of tomorrow with 1 hectare plots of 1 km tall buildings there would need to be sufficient care taken in regard to energy policy. There would not be enough room for hydro power to be effective, and there would be a great problem of pollution from conventional power sources. Therefore it is clear that a high density power generating option that did not pollute the environment would be needed to meet the demand safely, reliably and securely. This means nuclear power is needed in the particular situation that we have looked at in discussing this idea.
Furthermore it is clear from the things that we have considered that there needs to be some way of storing electricity for when it is needed as there clearly is fluctuation in demand. Therefore the option of pumped storage capacity would be a valuable addition to any complete energy policy strategy.
The question of pollution is a big one to contend with as most emissions in cities such as the one we have considered come from the exhaust emissions of combustion engines in cars. This can be curtailed by simply switching to a hydrogen-powered fuel economy for transport, with hydrogen produced from renewable and nuclear power and stored in tanks at filling stations until it is needed. This option is about 30% efficient but it would allow for greater utility in the transport sector.
William Gaskell is an Old Etonian Physicist. Educated at Eton College, Windsor, he now holds a Bachelor of Science (honours) degree in Physics from Bristol University, and a Master of Science degree in Physics and Technology of Nuclear Reactors with Merit from the University of Birmingham. He has developed a special interest in all aspects of nuclear power, energy generation and infrastructure and aspires to following his father and grandfather to Trinity College, Cambridge to undertake a PhD qualification in the physics, economics and policy of his interests.