Question

(c) A cruise ship is propelled by a gas turbine power plant that operates on Brayton's cycle, with extreme temperatures of 1000K and 288K. The compression ratio of the propulsion plant is 6, with isentropic efficiencies of the turbine nt = 0.9 and that of the compressor nc = 0.85. For y = 2 = 1.4, determine the, (i) Maximum efficiency of the plant (ii) Performance parameters (13 Marks)

Answer 1

solution :-

(a) maximum efficiency ?

  
T1 = 288 K

  
13 = 1000 K

Compression ratio (r.) = 6

  
nt = 0.90

  
ne = 0.85

$\gamma =\frac{C_{p}}{C_{v}}=1.4$

First find the pressure ratio (rp) :-

  $P_{1}V_{1}^{\gamma }=P_{2}V_{2}^{\gamma }$

  $r_{p}=\frac{P_{2}}{P_{1}}=\left ( \frac{V_{1}}{V_{2}} \right )^{\gamma }$

  
To = (ra)' = 61.4 = 12.286
​​​​​​​

Pressure ratio (rp) = 12.286

Now calculate the temperatures

$T_{2s}=T_{1}\left ( r_{p} \right )^{\frac{\gamma -1}{\gamma }}$

$T_{2s}=288\times \left (12.286 \right )^{\frac{1.4 -1}{1.4 }} \ K = 589.73 \ K$

Isentropic compressor

$\eta _{c}=\frac{T_{2s}-T_{1}}{T_{2}-T_{1}}$

$T_{2}=T_{1}+\frac{T_{2s}-T_{1}}{\eta _{c}}$​​​​​​​

$T_{2}=288+\frac{589.73-288}{0.85} = 642.98 \ K$

T3 = 1000 K

$T_{4s}=\frac{T_{3}}{\left ( r_{p} \right )^{\frac{\gamma -1}{\gamma }}}$

$T_{4s}=\frac{1000}{\left ( 12.286 \right )^{\frac{1.4 -1}{1.4 }}} \ K=488.36 \ K$

Isentropic Turbine

$\eta _{t}=\frac{T_{3}-T_{4}}{T_{3}-T_{4s}}$

$T_{4}=T_{3}-{\eta _{t}}\left ( {T_{3}-T_{4s}} \right )$

$T_{4}=1000-{0.90}\left ( {1000-488.36} \right ) \ K=539.52 \ K$

Now calculate the efficiency

$\eta _{cycle}=\frac{W_{turbine}-W_{compressor}}{Q_{supplied}}$

​​​​​​​​​​​​​​ $\eta _{cycle}=\frac{C_{p}\left (T_{3}-T_{4} \right )-C_{p}\left (T_{2}-T_{1} \right )}{C_{p}\left (T_{3}-T_{2} \right )}$ ​​​​​​​​​​​​​​​​​​​​​ ​​​​​​​​​​​​​​

​​​​​​​​​​​​​​ $\eta _{cycle}=\frac{\left (T_{3}-T_{4} \right )-\left (T_{2}-T_{1} \right )}{\left (T_{3}-T_{2} \right )}$ ​​​​​​​​​​​​​​​​​​​​​

​​​​​​​​​​​​​​ $\eta _{cycle}=\frac{\left (1000-539.2 \right )-\left (642.98-288 \right )}{\left (1000-539.2 \right )}$ ​​​​​​​​​​​​​​​​​​​​​​​​​​​​

​​​​​​​​​​​​​​ $\eta _{cycle}=0.2296$

Maximum efficiency is 22.96 %.

(b) Performance parameters ?

Other parameters

Assuming Cp = 1.005 Kj/kg K

$W_{turbine}=C_{p}\left ( T_{3}-T_{4} \right )=1.005\times \left ( 1000-539.2 \right )$​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

$W_{compressor}=C_{p}\left ( T_{2}-T_{1} \right )=1.005\times \left ( 642.98-288 \right )$​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

$Q_{supplied}=C_{p}\left ( T_{3}-T_{2} \right )=1.005\times \left ( 1000-642.98 \right )$

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