# Is there a fundamental efficiency limit for generators?

Im currently interested in the theoretical efficiency limit of a generator, ie any device transforming kinetic energy to electrical energy. In particular, id be interested in turbines, that convert angular motion into electricity.

In thermodynamics there is the well known Carnot-limit $$1-\frac{T_C}{T_H}$$ that describes the fundamental limit for an ideal heat machine in terms of the difference between a hot reservoirs temperature $$T_H$$ and its cold reservoirs temperature $$T_C$$.

Im looking for something similar, but for motion to electricity conversion. Clearly friction plays a role in practice, but an ideal turbine or generator should not be a heat machine in the Carnot sense. Thus the limit seems inapplicable.

Is there any known fundamental limitation for converting kinetic energy to electric energy, besides friction?

• The Carnot efficiency applies to operation in a cycle, and the (adiabatic) turbine does not operate in a cycle. The maximum shaft work for a turbine is simply the mass flow rate times the change in enthalpy per unit mass of the fluid passing through the turbine (at constant entropy per unit mass of the fluid). Dec 10 '18 at 23:46

No, there is no such fundamental limit (except the obvious one that you can't get energy for free). Large alternators at central power stations, for example, can reach as much as 98% or 99% efficiency, and superconducting generators can get even closer to 100%.

• Thank you, Thorondor and @Tygo Huurnink for your answers! Both answers are interesting but seem somewhat contrary for me. Betz's limit would seem to be a reasonable limit, in the sense i would have imagined. However, the example Thorondor gives seems to clearly violate Betz's limit. From my understanding Betz's law should apply to all machines that use newtonian fluids as a working medium. Thorondor, would you care to remark why Betz limit seems to be inapplicable in the examples you give? Are they relying on some kind of non-newtonian working medium?
– ckrk
Dec 11 '18 at 14:30
• @ckrk Betz's law applies only to the very specific situation of turbines extracting energy from fluid flow using a single rotor made of thin aerodynamic surfaces. To see this, note that in the proof it is assumed that the fluid flows through the turbine at a single constant velocity v. Designs that gradually extract energy using multiple layers of blades, such as this one, can do better than the Betz limit. Dec 11 '18 at 19:21
• I think they mean 80% with respect to Betz's limit. A heterogeneous flow of air will only make it less efficient, as far as I understand it. Dec 11 '18 at 19:35
• @TygoHuurnink To consider a simpler example instead, it's very well known that a screw turbine (i.e. an Archimedean screw used as a turbine) can have an efficiency above the Betz limit. For example, here's a 3D printed one with efficiency of about 80%. Dec 11 '18 at 19:51
• The chat discussion on the Betz limit appears to be gone. I'd just like to add that the Betz limit also assumes a turbine in a free flow that can also flow around the turbine, such as a wind turbine. For a water turbine in a dam that is typically not the case, which is another reason Betz's limit does not apply there. Aug 31 '19 at 10:04

This is a hard question on the face of it, but here are some observations which might help.

First, you are right that an electrical generator and a water-driven turbine are not heat engines that convert random thermal motion into shaft work. This means that the carnot limit does not apply to them.

(As an aside, note that a set of meshed gears conveys power in one form (a particular combination of torque and RPM) to another form (a different combination of torque and RPM) without the carnot limit- but there are other limits which do apply.)

The most fundamental of those limits are the inevitable presence of friction losses and power leakage out of the "engine" and into the environment, where it performs no useful work. In the case of an electrical generator, the friction loss shows up as heat being generated and wasted in the wires carrying the electrical current, and the leakage occurs when some of the magnetic fields generated inside the device protrude out of it (a condition called "flux leakage") and hence play no part in the development of current flow.

In a water turbine, the objective is to turn the kinetic energy content of moving water into extractable shaft work. Here the friction term comes from bearing friction in the turbine wheel and viscosity losses in the water itself, and the leakage term comes from water that squeezes its way past the turbine blades without doing work on the blades.

In the case of the electrical generator, efficiencies of 85% are typical, and in the case of the water turbine, a type called the Francis turbine can extract over 90% of the available kinetic energy in the water. Both conversion efficiencies are far in excess of the carnot limit for a heat engine of comparable power output.