Conversion of energy forms

The various conversion processes

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Different ways to create excess energy

Reminder of the energy conservation law

With :

Mechanical engine amplifying inlet energy

Let us consider the case of a machine operating at a stable and constant rotation speed, and for which no potential, kinetic or internal energy variation can be recorded between times t and t ', t ' being later than t. The produced work is consumed by a load.

We suppose that no heat is exchanged, except of worsened heat, between the machine and the external medium. Then, it comes :

With these assumptions, equation (1) becomes :

ΔQ _{out losses} + ΔW _int + ΔW _out + ΔW _in = 0

ΔW _out  = - ΔQ _{out losses} - ΔW _int - ΔW _in

By convention, the work brought to the input of the machine [ ΔW _in ], and disbursed for the operator, is counted negatively. The work provided at the output of the machine is counted positively.

| ΔW _out | = - ΔQ _{out losses} - ΔW_int +  | ΔW _in |

In some identified cases when machines are cleverly engineered, the work of internal forces can be negative and is added to the work provided at the inlet of the machine. But of course, part of the internal work created by friction losses still remains positive, that reduces the outlet work of the machine.

ΔW_int < 0

ΔW _{int friction losses} = - ΔQ _{out losses} > 0 => the friction losses are entirely converted into worsened heat.

And the coefficient of energy [COE] is equal to :

COE = Σ E _out / Σ E _in = [ | ΔQ _{out losses} | + | ΔW _int | + | ΔW _in | ] / | ΔW _in |

The useful outlet energy is equal to :  | ΔW _{out useful} | = | ΔW _int | + | ΔW _in |

The efficiency is equal to ratio : η = Σ E _{out useful}  / Σ E _in = [ | ΔW _int | + | ΔW _in | ] / [ | ΔQ _{out losses} | + | ΔW _int | + | ΔW _in | ]

Then : η = | ΔW _{out useful} | / [ | ΔQ _{out losses} | + | ΔW _{out useful} | ]

And :

COE = Σ E _out / Σ E _in = [ | ΔQ _{out losses} | + | ΔW _{out useful} | ] / | ΔW _in |

COE = [ | ΔW _{out useful} | * (1-η) / η  +  | ΔW _{out useful} | ] / | ΔW _in |

COE = [ (1-η) / η  + 1 ] * | ΔW _{out useful} | / | ΔW _in

COE = (1 / η)  | ΔW _{out useful} | / | ΔW _in

Let's calculate COP ratio

COP = Σ E _{out useful} / Σ E _{in operator} = | ΔW _{out useful} | / | ΔW _{in =} [ | ΔQ _{out losses} | + | ΔW _int | + | ΔW _in | ]  /  | ΔW _in |

We can then write COE as follows :

We see in this example of energy conversion process, not using a thermodynamic Carnot type cycle, that COE coefficient defines the energy amplification of the machine, including worsened heat rejected at outlet, while the COP coefficient defines the ratio of net energy recovered to energy expended by operator at input.

Case of heat pump - relationship between efficiency, COP and COE

We suppose that heat pump operates at stable run. One can write :

There is no energy amplification of the machine : | ΔW _in | = 0.

With these assumptions, equation (1) becomes :

| ΔQ _out | + | ΔQ _{out losses} | - | ΔQ _in | - | ΔW _in | = 0   => 

| ΔQ _out | + | ΔQ _{out losses} | = | ΔQ _in | + | ΔW _in |

COE = Σ E _out / Σ E _in = [ | ΔQ _out | + | ΔQ _{out losses} | ] / [ | ΔQ _in | + | ΔW _in | ] = 1

Let's calculate COP coefficient :

COP = Σ E _{out useful} / Σ E _in = | ΔQ _out | / | ΔW _in | = [ | ΔQ _in | + | ΔW _in | -  | ΔQ _{out losses} | ] / | ΔW _in |

The efficient of the machine is equal to : η = | ΔQ _out | / [ | ΔQ _{out losses} | + | ΔQ _out | ] = | ΔQ _out | / [ | ΔQ _in | + | ΔW _in | ]

And : COP = COE - η . | ΔQ _{out losses} | / | ΔQ _out |

Amplify the energy within a machine => COE (coefficient of energy) > 1

In some identified cases when machines are cleverly engineered, the work of internal forces can be negative and it adds to the work provided at the inlet of the machine. But of course, part of the internal work created by friction losses still remains positive, that reduces the outlet work of the machine.

Then : | ΔW_int  | < 0

and | ΔW_{int friction losses} | > 0

Let's write the complete equation in that case :

| ΔW_outlet|   = | ΔW_inlet | - | ΔW_{int friction losses} | +  | ΔW_inlet |

COE = | ΔW_outlet |  / | ΔW_inlet | = [ | ΔW_inlet | + | ΔW_int | - | ΔW_{int friction losses} | ] / | ΔW_inlet |

COE = 1 + [ | ΔW_int | - | ΔW_{int friction losses} | ] / | ΔW_inlet |

We will assume that | ΔW_int | is much greater than | ΔW_inlet |. | ΔW_int | / | ΔW_inlet | = k >>1. Furthermore, as usual, | ΔW_{int friction losses} | is negligible compared to | ΔW_inlet |. Then :

COE = 1 + [ | ΔW_int | - | ΔW_{int friction losses} | ] / | ΔW_inlet | = 1 + k -  | ΔW_{int friction losses} | / | ΔW_inlet | 

COE = 1 -  | ΔW_{int friction losses} | / | ΔW_inlet | + k =  η + k > 1

The internal friction losses drop down the efficiency of the machine and then the global COE whereas the other internal forces increases the COE over unity.