Brayton Cycle

Reading Problems 12.2 → 12.4 12.16, 12.17, 12.18, 12.29, 12.36, 12.116, 12.117

Introduction

The gas turbine cycle is referred to as the Brayton Cycle or sometimes the Joule Cycle. The actual gas turbine cycle is an open cycle, with the intake and exhaust open to the environment.

• can use different fuels

• simple in construction, easy to maintain

• can handle large volumes of gases

• small weight-to-power ratio

Definitions

Back Work Ratio: the ratio of the compressor work to the turbine work

Open Cycle Gas Turbine Engines

1 • compressor power requirements vary from 40-80% of the power output of the turbine (re- mainder is net power output), i.e. back work ratio = 0.4 → 0.8

• high power requirement is typical when gas is compressed because of the large specific volume of gases in comparison to that of liquids

Idealized Air Standard Brayton Cycle

• closed loop • constant pressure heat addition and rejection • ideal gas with constant specific heats

2 Brayton Cycle Efficiency

The efficiency of the cycle is given by the benefit over the cost or

T ! 4 − 1 Wnet QL mc˙ p(T4 − T1) T1 T1 η = = 1 − = 1 − = 1 − ! Q Q mc˙ (T − T ) T T3 H H p 3 2 2 − 1 T2

If we use the isentropic equations with the ideal gas law, we see that

T P !(k−1)/k P !(k−1)/k T T T 2 = 2 = 3 = 3 ⇒ 4 = 3 T1 P1 P4 T4 T1 T2

and

T T η = 1 − 1 = 1 − 4 T2 T3

If we define the pressure ratio as:

P2 P3 rp = = P1 P4

(1−k)/k η = 1 − (rp)

3 Maximum Pressure Ratio

Given that the maximum and minimum temperature can be prescribed for the Brayton cycle, a change in the pressure ratio can result in a change in the work output from the cycle.

Brayton Cycle with Reheat

4 • total work is increased • but additional heat input is required • net efficiency may or may not increase

Compression with Intercooling

• the work required to compress in a steady flow device can be reduced by compressing in stages • cooling the gas reduces the specific volume and in turn the work required for compression • by itself compression with intercooling does not provide a significant increase in the effi- ciency of a gas turbine because the temperature at the combustor inlet would require addi- tional heat transfer to achieve the desired turbine inlet temperature

5 How Can We Improve Efficiency?

We know the efficiency of a Brayton cycle engine is given as

W˙ W˙ − W˙ η = net = turbine compressor Q˙ H Q˙ H

There are several possibilities, for instance we could try to increase W˙ turbine or decrease W˙ compressor.

Recall that for a SSSF, reversible compression or expansion

W˙ Z out = v dP ⇒ keep v ↑ in turbine, keep v ↓ in compressor m˙ in

6 This can be achieved through the use of intercooling and reheating.

W˙ T − W˙ C (↓) Compressor −→ η = , overall (↓) Q˙ H,T otal(↑)

W˙ T (↑) − W˙ C ) Turbine −→ η = , overall (↓) Q˙ H,T otal(↑)

The conclusion is the intercooling and/or reheating by themselves will lower η. We have to find a way to reduce Q˙ H

Brayton Cycle with Regeneration

7 • a regenerator is used to reduce the fuel consumption to provide the required Q˙ H

• the efficiency with a regenerator can be determined as:

! Tmin (k−1)/k η = 1 − (rp) Tmax

• for a given Tmin/Tmax, the use of a regenerator above a certain rp will result in a reduction of η

withanideal regenerator

withouta regenerator

betterwitha betterwithout T1 / T3 =0.2 regenerator aregenerator

T1 / T3 =0.25

T1 / T3 =0.3

rp,cr rp

8 Brayton Cycle With Intercooling, Reheating and Regeneration

T Tmax QH,R QH

7s 9s

Qreg

Qreg 2s 4s Q Tmin L QL,I s

9 Compressor and Turbine Efficiencies

Isentropic Efficiencies

h2,s − h1 cp(T2,s − T1) (1) ηcomp = = h2 − h1 cp(T2 − T1)

h3 − h4 cp(T3 − T4) (2) ηturb = = h3 − h4,s cp(T3 − T4,s)

Wnet QH − QL QL cp(T4 − T1) (3) ηcycle = = = 1 − = 1 − QH QH QH cp(T3 − T2)

(4) Calculate T2s from the isentropic relationship,

!(k−1)/k T2,s P2 = . T1 P1

Get T2 from (1).

(5) Do the same for T4 using (2) and the isentropic relationship.

(6) substitute T2 and T4 in (3) to find the cycle efficiency.

10 PROBLEM STATEMENT:

Air enters the compressor of a gas-turbine power plant, at 290 K, 0.1 MP a. The ratio of the maximum to minimum pressure in the cycle is 4.0 and the maximum cycle temperature is 1200 K. Compressor and turbine isentropic efficiencies are 0.85. The compression process occurs in two stages, each having a pressure ratio of 2.0 with intercooling to 300 K in between. A 75% effective regenerator reduces fuel costs.

a) Determine the net work transfer [kJ/kg] b) Determine the thermal (first law) efficiency.

QL Regenerator

Combustion

QH

Wnet Compressor2 Compressor1 Turbine Intercooler QL,I

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