Can the Efficiency Be Greater Than One?

This is one of the perennial top search terms for this blog.

Anticlimactic answer: Yes, because input and output are determined also by economics, not only by physics.

Often readers search for the efficiency of a refrigerator. Its efficiency, the ratio of output and input energies, is greater than 1 because the ambient energy is free. System’s operators are interested in the money they pay the utility, in relation to the resulting energy for cooling.

If you use the same thermodynamic machine either as a refrigerator or as a heat pump, efficiencies differ: The same input energy drives the compressor, but the relevant output energy is either the energy released to the ‘hot side’ at the condenser or the energy used for evaporating the refrigerant at the ‘cool side’:

The same machine / cycle is used as a heat pump for heating (left) or a refrigerator or AC for cooling (right). (This should just highlight the principles and does not include any hydraulic details, losses etc. related to detailed differences between refrigerators / ACs and heat pumps.)

For photovoltaic panels the definition has sort of the opposite bias: The sun does not send a bill – as PV installers say in their company’s slogan – but the free solar ambient energy is considered, and thus their efficiency is ‘only’ ~20%.

Half of our generator, now operational for three years: 10 panels, oriented south-east, 265W each, efficiency 16%. (The other 8 panels are oriented south-west).

When systems are combined, you can invent all kinds of efficiencies, depending on system boundaries. If PV panels are ‘included’ in a heat pump system (calculation-wise) the nominal electrical input energy becomes lower. If solar thermal collectors are added to any heating system, the electrical or fossil fuel input decreases.

Output energy may refer to energy measured directly at the outlet of the heat pump or boiler. But it might also mean the energy delivered to the heating circuits – after the thermal losses of a buffer tank have been accounted for. But not 100% of these losses are really lost, if the buffer tank is located in the house.

I’ve seen many different definitions in regulations and related software tools, and you find articles about how to game interpret these guidelines to your advantage. Tools and standards also make arbitrary assumptions about storage tank losses, hysteresis parameter and the like – factors that might be critical for efficiency.

Then there are scaling effects: When the design heat loads of two houses differ by a factor of 2, and the smaller house would use a scaled down heat pump (hypothetically providing 50% output power at the same efficiency), the smaller system’s efficiency is likely to be a bit lower. Auxiliary consumers of electricity – like heating circuit pumps or control systems – will not be perfectly scalable. But the smaller the required output energy is, the better it can be aligned with solar energy usage and storage by a ‘smart’ system – and this might outweigh the additional energy needed for ‘smartness’. Perhaps intermittent negative market prices of electricity could be leveraged.

Definitions of efficiency are also culture-specific, tailored to an academic discipline or industry sector. There are different but remotely related concepts of rating how useful a source of energy is: Gibbs Free Energy is the maximum work a system can deliver, given that pressure and temperature do not change during the process considered – for example in a chemical reaction. On the other hand, Exergy is the useful ‘available’ energy ‘contained’ in a (part of a) system: Sources of energy and heat are rated; e.g. heat energy is only mechanically useful up to the maximum efficiency of an ideal Carnot process. Thus exergy depends on the temperature of the environment where waste heat ends up. The exergy efficiency of a Carnot process is 1, as waste heat is already factored in. On the other hand, the fuel used to drive the process may or may not be included and it may or may not be considered pure exergy – if it is, energy and exergy efficiency would be the same again. If heat energy flows from the hot to the cold part of a system in a heat exchanger, no energy is lost – but exergy is.

You could also extend the system’s boundary spatially and on the time axis: Include investment costs or the cost of harm done to the environment. Consider the primary fuel / energy / exergy to ‘generate’ electricity: If a thermal power plant has 40% efficiency then the heat pump’s efficiency needs to be at least 2,5 to ‘compensate’ for that.

In summary, ‘efficiency’ is the ratio of an output and an input energy, and the definitions may be rather arbitrary as and these energies are determined by a ‘sampling’  time, system boundaries, and additional ‘ratings’.

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