Question

How do the inefficiencies of the turbine and the compressor affect \((a)\) the back work ratio and \((b)\) the thermal efficiency of a gas-turbine engine?

Short answer

The inefficiencies of the turbine and the compressor directly affect (a) the back work ratio and (b) the thermal efficiency of a gas-turbine engine. The back work ratio increases as the efficiencies of the compressor and turbine decrease, while the thermal efficiency decreases due to the reduction in net work output caused by the inefficiencies of the compressor and turbine.

Step-by-step solution

Define the back work ratio

The back work ratio (BWR) is given by the formula: BWR = \(\frac{W_c}{W_t}\), where \(W_c\) is the work required by the compressor and \(W_t\) is the work produced by the turbine.

Define the thermal efficiency

The thermal efficiency (\(\eta_{th}\)) of the gas-turbine engine is given by the formula: \(\eta_{th} = \frac{W_{net}}{Q_{in}}\) where \(W_{net}\) is the net work output of the engine and \(Q_{in}\) is the heat input to the engine.

Relate the work terms to the efficiency terms

Isentropic efficiency of compressor (\(\eta_{c}\)) is defined as: \(\eta_{c} = \frac{W_{c,ideal}}{W_{c,actual}}\) Isentropic efficiency of the turbine (\(\eta_{t}\)) is defined as: \(\eta_{t} = \frac{W_{t,actual}}{W_{t,ideal}}\)

Analyze the effect of inefficiencies on the back work ratio

From Step 3, we know that \(W_{c,actual} = \frac{W_{c,ideal}}{\eta_{c}}\) and \(W_{t,actual} = \eta_{t} W_{t,ideal}\) The back work ratio in terms of ideal and actual work is: BWR = \(\frac{W_{c,actual}}{W_{t,actual}} = \frac{\frac{W_{c,ideal}}{\eta_{c}}}{\eta_{t} W_{t,ideal}} = \frac{W_{c,ideal}}{W_{t,ideal}} \cdot \frac{\eta_{t}}{\eta_{c}}\) If the compressor and the turbine were both isentropic (ideal), then \(\eta_{c} = \eta_{t} = 1\). In that case, the back work ratio BWR = \(\frac{W_{c,ideal}}{W_{t,ideal}}\) However, when the compressor and turbine are not isentropic (inefficient), the back work ratio will increase due to the decrease in the efficiency of the compressor and turbine. So, the back work ratio is directly affected by the inefficiencies of the compressor and the turbine.

Analyze the effect of inefficiencies on the thermal efficiency

In terms of the actual work done, the net work output can be defined as: \(W_{net} = W_{t,actual} - W_{c,actual} = \eta_{t} W_{t,ideal} - \frac{W_{c,ideal}}{\eta_{c}}\) The thermal efficiency will be now written as: \(\eta_{th} = \frac{\eta_{t} W_{t,ideal} - \frac{W_{c,ideal}}{\eta_{c}}}{Q_{in}}\) If the compressor and the turbine were both isentropic (ideal), then \(\eta_{c} = \eta_{t} = 1\). In that case, the thermal efficiency \(\eta_{th}\) would be given by: \(\eta_{th} = \frac{W_{t,ideal} - W_{c,ideal}}{Q_{in}}\) However, when the compressor and turbine are not isentropic (inefficient), the net work output will decrease, and thus thermal efficiency will also decrease. So, the thermal efficiency is directly affected by the inefficiencies of the compressor and the turbine. In conclusion, the inefficiencies of the turbine and the compressor affect (a) the back work ratio as it increases the back work ratio, and (b) the thermal efficiency as it decreases the thermal efficiency of a gas-turbine engine.