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ΔE = ΔU + ΔE_{k} + ΔE_{p} = ΔQ_{ in} + ΔQ_{ out} + ΔQ_{ out losses} + ΔW _{int} + ΔW_{ out} + ΔW_{ in} (1) |
With :
ΔE | : overall energy variation of the system, |
ΔU | : variation of internal energy, |
ΔE_{k} | : variation of kinetic energy, |
ΔE_{p} | : variation of potential energy, |
ΔQ_{ in} | : heat received by the system from external medium, |
ΔQ_{ out } | : heat transfered by the system to external medium |
ΔQ_{ out losses } | : worsened heat (friction losses) transfered by the system to external medium |
ΔW_{ int} | : work of internal forces other than losses generated by energy conversion process, |
ΔW_{ in } | : work of external forces input into the system, |
ΔW_{ out } |
: work output by the system. |
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.
ΔU = ΔE_{k} = ΔE_{p }= 0 |
Then : |
ΔE = ΔU + ΔE_{k} + ΔE_{p} = 0 |
We suppose that no heat is exchanged, except of worsened heat, between the machine and the external medium. Then, it comes :
ΔQ_{ in} = ΔQ_{ out }= 0 |
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.
| ΔW_{ out }| = | ΔQ _{out losses} | + | ΔW_{ int} | + | ΔW_{ in }| |
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}
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 }|
COP = | ΔW_{ out useful} | / | ΔW in |
We can then write COE as follows :
COE = (1 / η) COP > COP > 1 |
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.
We suppose that heat pump operates at stable run. One can write :
ΔU = ΔE_{k} = ΔE_{p }= 0 |
Then : |
ΔE = ΔU + ΔE_{k} + ΔE_{p} = 0 |
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
COE = 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} |
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
1+k > COE = | ΔW_{outlet} | / | ΔW_{inlet} | = η + k > 1 |
with η = efficiency and k much greater than 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.