In a simple single stage gas turbine plant, if T1 is the minimum temperature and T3 is the maximum temperature then what is the work ratio in terms of rP? 

Concept:

Work ratio: 

• It is defined as the ratio of net work to turbine work.
• The gas turbine is work on the Brayton cycle.
• This cycle consist of  two adiabatic process and two isobaric process.

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\(Work Ratio=\frac{W_{net}}{W_T}=\frac{W_T-W_C}{W_T}\)

Turbine work is given by,

WT = Cp(T3 - T4)

Compressor work is given by,

WC = Cp(T2 - T1)

Therefore, \(Work ~Ratio=\frac{C_p(T_3-T_4)-C_p(T_2-T_1)}{C_p(T_3-T_4)}\)

\(\Rightarrow1-\frac{T_2-T_1}{T_3-T_4}\)

\(Work~ratio=1-\frac{T_1(\frac{T_2}{T_1}-1)}{T_4(\frac{T_3}{T_4}-1)}\)

Here, \(\frac{T_2}{T_1}=\frac{T_3}{T_4}=r_p^{\frac{\gamma -1}{\gamma}}\)

Therefore, \(Work~ratio=1-\frac{T_1}{T_4}\)

\(Work~ratio=1-\frac{T_1}{T_4}\times\frac{T_3}{T_3}\)

\(Work~ratio=1-\frac{T_1}{T_3}\times\frac{T_3}{T_4}\)

\(Work~ratio=1 - \frac {T_1}{T_3} (r_p)^\frac {\gamma - 1}{\gamma}\)