## Brayton Cycle – Problem with Solution

Let assume the closed **Brayton cycle**, which is the one of most common **thermodynamic cycles** that can be found in modern gas turbine engines. In this case assume a** helium gas turbine** with single compressor and single turbine arrangement. One of key parameters of such engines is the maximum turbine inlet temperature and the compressor pressure ratio (PR = p_{2}/p_{1}) which determines the thermal efficiency of such engine.

In this turbine the high-pressure stage receives gas (point 3 at the figure) from a heat exchanger:

- p
_{3 }= 6.7 MPa; - T
_{3}= 1190 K (917°C)) - the isentropic turbine efficiency is
*η*= 0.91 (91%)_{T}

and exhaust it to another heat exchanger, where the outlet pressure is (point 4):

- p
_{4}= 2.78 MPa - T
_{4,is}= ?

Thus the compressor pressure ratio is equal to PR = 2.41. Moreover we know, that the compressor receives gas (point 1) at the figure:

- p
_{1 }= 2.78 MPa; - T
_{1}= 299 K (26°C) - the isentropic compressor efficiency η
_{K}= 0.87 (87%).

The heat capacity ratio, , for helium is equal to** =c _{p}/c_{v}=1.66**

- the heat added by the heat exchanger (between 2 → 3)
- the compressor outlet temperature of the gas (T
_{2,is}) - the real work done on this compressor, when the isentropic compressor efficiency is η
_{K}= 0.87 (87%) - the turbine outlet temperature of the gas (T
_{4,is}) - the real work done by this turbine, when the isentropic turbine efficiency is η
_{T}= 0.91 (91%) - the thermal efficiency of this cycle

**Solution:**

1) + 2)

From the first law of thermodynamics, the net heat added is given by **Q _{add,ex} = H_{3 }– H_{2}** [kJ] or Q

_{add}= C

_{p}.(T

_{3}-T

_{2s})

**,**but in this case we do not know the temperature (T

_{2s}) at the outlet of the compressor. We will solve this problem in intensive variables. We have to rewrite the previous equation (to include

**η**

**) using the term (**

_{K}**+h**) to:

_{1}– h_{1}Q_{add} = h_{3} – h_{2} = h_{3} – h_{1} – (h_{2s} – h_{1})/η_{K} ** **[kJ/kg]

Q_{add }**= **c_{p}(T_{3}-T_{1}) – (c_{p}(T_{2s}-T_{1})**/**η_{K})

Then we will calculate the temperature, **T _{2s}**, using p, V, T Relation (from Ideal Gas Law) for adiabatic process between (1 → 2).

In this equation the factor for helium is equal to =c_{p}/c_{v}=1.66. From the previous equation follows that the compressor outlet temperature, T_{2s}, is:

**Q _{add }= **c

_{p}(T

_{3}-T

_{1}) – (c

_{p}(T

_{2s}-T

_{1})

**/**η

_{K}) = 5200.(1190 – 299) – 5200.(424-299)/0.87 = 4.633 MJ/kg – 0.747 MJ/kg =

**3.886 MJ/kg**

3)

The work done on the gas by the compressor in the isentropic compression process is:

*W*_{C,s}* = c*_{p}* (T*_{2s}* – T*_{1}*) = 5200 x (424 – 299) = 0.650 MJ/kg*

The real work done on the gas by the compressor in the adiabatic compression is then:

*W*_{C,real}* = c*_{p}* (T*_{2s}* – T*_{1}*). η*_{C}*= 5200 x (424 – 299) / 0.87 = 0.747 MJ/kg*

4)

The turbine outlet temperature of the gas,** T _{4,is}**, can be calculated using the same

**p, V, T Relation**as in 2) but between states 3 and 4:

From the previous equation follows that the outlet temperature of the gas, T_{4,is}, is:

5)

The work done by gas turbine in the isentropic expansion is then:

*W*_{T,s}* = c*_{p}* (T*_{3}* – T*_{4s}*) = 5200 x (1190 – 839) =** 1.825 MJ/kg*

The real work done by gas turbine in the adiabatic expansion is then:

*W*_{T,real}* = c*_{p}* (T*_{3}* – T*_{4s}*) . η*_{T}*= 5200 x (1190 – 839) x 0.91 =** 1.661 MJ/kg*

6)

As was derived in the previous section, the **thermal efficiency** of an ideal Brayton cycle is a function of **pressure ratio** and **κ**:

therefore

*η*_{th }*=* 0.295 = **29.5%**

The thermal efficiency can be also calculated using the work and the heat (without η_{K}):

*η*_{th,s }*= (**W*_{T,s}* – W*_{C,s}*)*** / **Q

_{add,s}= (1.825 – 0.650) / 3.983

**0.295 =**

*=***29.5%**

Finally, the thermal efficiency including isentropic turbine/compressor efficiency is:

*η*_{th,real }*= (**W*_{T,real}* – W*_{C,real}*)* ** / **Q

_{add}= (1.661 – 0.747) / 3.886

**0.235 =**

*=***23.5%**

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