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10 February 2012

Brayton cycle: efficiency and net work evaluations

Vai alla versione in italiano pubblicata il 17 Dicembre 2011

As already discussed, the efficiency of the Brayton cycle can be evaluated using the following relationship

yield = 1 - (Pminimum/Pmaximum)[ (gamma - 1) / gamma ]

However this formula requires some instructions for use because otherwise it leads to false conclusions.
From the algebraic point of view the relation Pminimum/Pmaximum can assume any value between 0 and 1.
The ratio is 1 when Pminimum=Pmaximum and gradually decreases to 0 increasing Pmax (or even decreasing Pminimum).
The image shows the evolution of the yield as a function of Pminimum/Pmaximum for the three types of gas.


Curves show that the yield increases with decreasing Pminimum/Pmaximum.
In practice, at the same Pminimum, increasing Pmaximum means getting better returns.
Graphs confirm the account already made that the best performance occurs with monatomic gas, then with the diatomic gas and the polyatomic gas follows.
Erroneously, the formula shows accessible yields close to 100% or at least shows possible returns higher than that of Carnot.
Neither of these statement are true.
Let's see what happens to the Brayton cycle by keeping constant some operating parameters and by varying only the pressure ratio Pminimum/Pmaximum.
We consider as reference parameters those used in the example presented at the end of the previous post, and we assume constant

TA = 300K
TC = 600K
VD = 3m3
Pminimum = 100kPa

Here are collected in the same PV diagram some Brayton cycles at different pressure ratios.


The graph goes from an extreme situation in which

Pminimum/Pmaximum = 1

which is represented by the horizontal line at P = 100kPa (absence of the adiabatic processes and presence of two overlapping isobaric process) to another extreme situation in which

Pmaximum = 550kPa

which is represented by the brown curve (absence of isobaric processes and presence of two overlapping adiabatic processes).
Between these two extreme situations, three intermediate cases in which there is the classic Brayton cycle.
The light blue cycle which is located above the brown curve is always a Brayton cycle but substantially different from earlier: the useful work of this cycle is negative.
Ie it is an example in which the Brayton cycle serves as a heat pump and not as heat engine.

The formula that calculates the yield of the Brayton cycle as a function of Pminimum/Pmaximum is valid as long as the value of Pminimum/Pmaximum is greater than the value at which we observe the disappearance of the isobaric processes.

This limit value depends on the operating conditions and gas type.
To clarify the concept, consider this graph that shows together efficiency and net work for the three gas types.


The image allows us to make some important considerations.
The net work has a trend different from that of the yield.
It is null when the pressure ratio is equal to 1 and increases progressively up to a maximum decreasing the pressure ratio.
The value of the pressure ratio at which corresponds the maximum net work is greater than the point at which the isobaric processes disappear.
Once the maximum has reached, with further reduction of the pressure ratio, the net work decreases rapidly to zero when the pressure ratio causes the disappearance of the isobaric precesses.
It is curious that the yield formula applied to the Brayton cycle in which the isobars are absent gives as result the Carnot efficiency

Yield = 1 - (Pminimum/Pmaximum)[ (gamma - 1) / gamma ] = 1 - TA / TC

Note that the graph has three black arrows indicating the point of the three yield curves in which 50% is reached (which is equal to the Carnot efficiency at the operating conditions considered for the analysis in this post).

DEMONSTRATION
Since the cycle is composed of two overlapping adiabatic processes, it means that

Pminimum*VDgamma = Pminimum*VAgamma = Pmaximum*VCgamma = Pmaximum*VBgamma

Pminimum*VAgamma = Pmaximum*VCgamma

VCgamma / VAgamma = Pminimum/Pmaximum

(VC / VA)gamma = Pminimum/Pmaximum

VC / VA = (Pminimum/Pmaximum)1/gamma

Furthermore

Pminimum*VAgamma = Pmaximum*VCgamma

Pminimum*VA*VA(gamma - 1) = Pmaximum*VC*VC(gamma - 1)

n*R*TA*VA(gamma - 1) = n*R*TC*VC(gamma - 1)

TA*VA(gamma - 1) = TC*VC(gamma - 1)

TA/TC = VC(gamma - 1) / VA(gamma - 1) = (VC / VA)(gamma - 1)

As seen above

VC / VA = (Pminimum/Pmaximum)1/gamma

therefore

TA/TC = [(Pminimum/Pmaximum)1/gamma](gamma - 1) = (Pminimum/Pmaximum)[ (gamma - 1) / gamma ]

Substituting this relationship in the yield equation you obtain

yield = 1 - TA / TC

which is the Carnot efficiency since TA and TC are respectively the minimum and the maximum temperatures of the cycle.

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