Gas turbine (mean line) analysis question

[Master1022]

Homework Statement

A single stage gas turbine for power generation is designed for the following overall duty:
- Power output is 4.2 MW, and mass flow rate is 70 kg/s
- Inlet stagnation pressure = 3 bar, inlet stagnation temperature = 1000 K
- Rotational speed = 3000 rpm
- No inlet swirl: $\alpha_1 = 0$
- Operating fluid is air, $\gamma$ = 1.4, $c_p$ = 1004 J/kgK
- The total-to-total isentropic efficiency $\eta_{TT} = 0.93$
Perform a mean line analysis and find the following:
(a) Total temperature and total pressure at the turbine outlet

Relevant Equations

Euler work equation

Hi,

I was attempting another turbomachinery question and am struggling with a few concepts. The question is:
"A single stage gas turbine for power generation is designed for the following overall duty:
- Power output is 4.2 MW, and mass flow rate is 70 kg/s
- Inlet stagnation pressure = 3 bar, inlet stagnation temperature = 1000 K
- Rotational speed = 3000 rpm
- No inlet swirl: $\alpha_1 = 0$
- Operating fluid is air, $\gamma$ = 1.4, $c_p$ = 1004 J/kgK
- The total-to-total isentropic efficiency $\eta_{TT} = 0.93$
Perform a mean line analysis and find the following
(a) Total temperature and total pressure at the turbine outlet"

Attempt:
From the power and mass flow, we can get the specific work done (per second):
$$\dot w_s = \frac{P}{\dot m} = \frac{4.2 \times 10^6}{70} = 6 \times 10^4$$
Then we can use the steady flow energy equation to write ($\dot q = 0$):
$$- \dot w_s = \Delta h_0 \rightarrow - \dot w_s = c_p \Delta T_0$$
I am confused of how to utilize the total-to-total isentropic efficiency. If we are told what the machine was designed for, has the efficiency already been accounted for in the power, or do we need to include it separately. I.e. is the 4.2 MW output after accounting for the 93% isentropic efficiency, or is that the 'ideal 100% efficient' condition?

Thank you for any help.

 

[KataKoniK]

Hi there,

I can provide some guidance on how to approach this question. First, let's define some terms to make sure we're on the same page. Total-to-total isentropic efficiency refers to the ratio of the actual work output to the work output in an ideal, isentropic process. In other words, it takes into account any losses or inefficiencies in the system.

In this case, the 4.2 MW output is the actual power output of the turbine, after taking into account the 93% efficiency. So, we do not need to include it separately in our calculations.

To find the total temperature and total pressure at the turbine outlet, we can use the equation you provided:
- \dot w_s = c_p \Delta T_0
However, we need to take into account the efficiency in our calculation. So, the equation becomes:
- \dot w_s = \eta_{TT} c_p \Delta T_0
Solving for \Delta T_0, we get:
- \Delta T_0 = \frac{- \dot w_s}{\eta_{TT} c_p}
Plugging in the values given in the question, we get:
- \Delta T_0 = \frac{- 6 \times 10^4}{0.93 \times 1004} = - 63.9 K
This is the change in total temperature across the turbine. To find the total temperature at the outlet, we need to add this to the inlet stagnation temperature:
- T_{0,out} = T_{0,in} + \Delta T_0 = 1000 + (-63.9) = 936.1 K
Similarly, we can find the total pressure at the outlet using the equation:
- \frac{P_{0,out}}{P_{0,in}} = \left(\frac{T_{0,out}}{T_{0,in}}\right)^\frac{\gamma}{\gamma - 1}
Plugging in the values, we get:
- \frac{P_{0,out}}{3} = \left(\frac{936.1}{1000}\right)^\frac{1.4}{0.4}
Solving for P_{0,out}, we get:
- P_{0,out} = 2.85 bar

I hope this helps and clarifies any confusion you had about the