Solution:

Given data:

Pressure ratio (PR) = 8
Temperature at compressor inlet (T1) = 300 K
Temperature at turbine inlet (T3) = 1300 K

Assumptions:

1. The Brayton cycle is an ideal cycle.
2. The gas turbine power plant operates on the air-standard assumptions.
3. The compressor and turbine are isentropic.

(a) Gas temperature at the exits of the compressor and the turbine:

The Brayton cycle consists of four processes: isentropic compression, constant pressure heat addition, isentropic expansion, and constant pressure heat rejection. The process 1-2 is isentropic compression, and the process 3-4 is isentropic expansion. Therefore, the temperatures at the exits of the compressor (T2) and the turbine (T4) can be calculated using the isentropic relations as follows:

- Isentropic compression (process 1-2):

The pressure ratio (PR) is given as:

PR = P2/P1

where P1 is the inlet pressure and P2 is the outlet pressure. Since the process is isentropic, we can use the isentropic relation for pressure and temperature as follows:

P2/P1 = (T2/T1)^(k/(k-1))

where k is the ratio of specific heats of the working fluid.

Substituting the given data, we get:

8 = P2/P1
T1 = 300 K
k = 1.4

Solving for T2, we get:

T2 = T1*(PR)^((k-1)/k) = 300*(8)^(0.4) = 705.6 K

Therefore, the gas temperature at the exit of the compressor is T2 = 705.6 K.

- Isentropic expansion (process 3-4):

Using the same isentropic relation as above, we can calculate the temperature at the exit of the turbine (T4) as follows:

PR = P3/P4

P3 = P2
P4 = P1

T3 = 1300 K

k = 1.4

PR = P2/P1 = P3/P4

T4 = T3*(PR)^((k-1)/k) = 1300*(1/8)^(0.4) = 457.8 K

Therefore, the gas temperature at the exit of the turbine is T4 = 457.8 K.

(b) Back work ratio:

The back work ratio (BWR) is the ratio of the work required to drive the compressor to the work produced by the turbine. It is given by:

BWR = (work required by compressor - work produced by turbine) / work produced by turbine

Since the compressor and turbine are isentropic, the work done by them is given by:

- Work done by compressor:

Wc = h2 - h1

where h1 and h2 are the specific enthalpies at the inlet and exit of the compressor, respectively.

Using the air-standard assumptions, we can assume that the working fluid is air and that it behaves as an ideal gas. Therefore, the specific enthalpy of the air can be calculated using the specific heat at constant pressure (cp) as follows:

h = cp*T

Using the values of T1 and T2, we can

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