Example $3$: Consider the simple power generation system based on an open Brayton cycle. The system consists of a compressor, combustion chamber, and turbine whose performance characteristics are known. The turbinecompressor combination operates at $120$ rps $.$ The objective of the simulation is to determine the power output, $W_{s}kW,$ if $8000$ kW of energy is added at the combustor by burning fuel. The compressor draws air and the turbine exhaust rejects it at atmospheric pressure of $101$ kPa $.$ The entering air temperature is $25C.$ The following simplifications are introduced:

Assume perfect$-$gas properties throughout the cycle and a $c_p$ constant al $)$

Neglect the mass added in the form of fuel in the combustor so that the mass rate of flow $w$ is constant throughout the cycle

Neglect the pressure drop in the combustor so that $p_2=p_3$ and the hight pressure in the system can be designated simply as $p$

Neglect heat transfer to the environment

The performance characteristics of the axial compressor and the gas turbine are shown in the Fig.

Example $3$: Equations can be developed for the curves in the figures:

Solution: To achieve the simulation, the values of the following unknown variables must be determined: $m,p,W_c,t_2,W_t,t_3,$ and $W_s.$

Seven independent equations must be prepared to solve for this set of unknowns.

Four equations are already available from the perfermance characteristics cutves of the compressor and the turbine $($preceding slide$).$ The three other equations comes from the enrgy balances of:

Compressor: $W_c^{}=m^{}{C}_p(t_2-t_1)=m^{}{C}_p(t_2-25)$

Combustor: $8000=m^{}{C}_p(t_3-t_2)$

Turbine: $,W_t^{}=W_c^{}+W_s^{}$

Home Work

Solve the seven equations using the Newton$-$Raphson method to determine the shaft power output of the system Hand written not code at all Or Matlab code only