Question

9.43 An ideal air-standard Brayton cycle operates at steady state with compressor inlet conditions of 300 K and 100 kPa and a fixed turbine inlet temperature of 1700 K. For the cycle, (a) determine the net work developed per unit mass flowing, in kJ/kg, and the thermal efficiency for a compressor pressure ratio of 8. (b) plot the net work developed per unit mass flowing, in kJ/kg, and the thermal efficiency, each versus compressor pressure ratio ranging from 2 to 50.

Answer 1

The net work output from the cycle = 520.67
`(KJ)/(kg)`

The efficiency of Brayton cycle = 0.448

Step-by-step explanation:

Compressor inlet temperature
`T_(1)` = 300 K

Turbine inlet temperature
`T_(3)` = 1700 K

Pressure ratio
`r_(p)` = 8

For the compressor the temperature - pressure relation is given by the formula,

⇒
`(T_(2) )/(T_(1) )` =
`r_(p)^{(\gamma - 1)/(\gamma) }`

⇒
`(T_(2) )/(300) = 8^{(1.4 - 1)/(1.4) }`

⇒
`T_(2)` = 543.42 K

This is the temperature at compressor outlet.

Same relation for turbine we can write this as,

⇒
`(T_(3) )/(T_(4) )` =
`r_(p)^{(\gamma - 1)/(\gamma) }`

⇒
`(1700 )/(T_(4) )` =
`8^(0.2857)`

⇒
`T_(4)` = 938.5 K

This is the temperature at turbine outlet.

Now the work output from the turbine
`W_(T)` =
`m C_(p) (T_(3) - T_(4) )`

Put all the values in above formula we get,

⇒
`W_(T)` = 1 × 1.005 × ( 1700 - 938.5 )

`W_(T) = 765.3 (KJ)/(kg)`

This is the work output from the turbine.

Now the work input to the compressor is
`W_(C)` =
`m C_(p) (T_(2) - T_(1) )`

Put all the values in above formula we get,

⇒
`W_(C)` = 1 × 1.005 × ( 543.42 - 300 )

⇒
`W_(C)` = 244.63
`(KJ)/(kg)`

This is the work input to the compressor.

Net work output from the cycle
`W_(net) = W_(T) - W_(C)`

⇒
`W_(net)` = 765.3 - 244.63

`W_(net) = 520.67(KJ)/(kg)`

This is the net work output from the cycle.

The thermal efficiency is given by

`E_(cycle) =1 - \frac{1}{r_(p)^{(\gamma - 1)/(\gamma) } }`

`E_(cycle) =1 - \frac{1}{8^{(1.4 - 1)/(1.4) } }`

`E_(cycle) = 0.448`

This is the efficiency of Brayton cycle.

(b). the graph between plot the net work developed per unit mass flowing and the thermal efficiency, each versus compressor pressure ratio ranging from 2 to 50 is shown in the image below.