Question

An ideal Brayton cycle has a net work output of \(150 \mathrm{kJ} / \mathrm{kg}\) and a back work ratio of \(0.4 .\) If both the turbine and the compressor had an isentropic efficiency of 85 percent, the net work output of the cycle would be \((a) 74 \mathrm{kJ} / \mathrm{kg}\) \((b) 95 \mathrm{kJ} / \mathrm{kg}\) \((c) 109 \mathrm{kJ} / \mathrm{kg}\) \((d) 128 \mathrm{kJ} / \mathrm{kg}\) \((e) 177 \mathrm{kJ} / \mathrm{kg}\)

Short answer

Answer: (c) 109 kJ/kg

Step-by-step solution

1. Analyzing the problem

To find the net work output, we will need to determine the compressor work and the turbine work in the cycle. We can find the work without considering the isentropic efficiency first, and then compare the results with the isentropic efficiency.

2. Finding the compressor work

Let's denote the compressor work as \(W_{c_{actual}}\). We know that the back work ratio is 0.4, which means that \(W_{c_{actual}} = 0.4 W_{net}\) where \(W_{net}\) is given as \(150\textrm{ kJ/kg}\). So, we can find the compressor work by: \(W_{c_{actual}} = 0.4(150)\) \(W_{c_{actual}} = 60 \textrm{ kJ/kg}\)

3. Finding the turbine work

Let's denote the turbine work as \(W_{t_{actual}}\). According to the Brayton cycle, the net work output can be expressed as the difference between the turbine work and the compressor work. So, we can find the turbine work by: \(W_{t_{actual}} = W_{net} + W_{c_{actual}}\) \(W_{t_{actual}} = 150 + 60\) \(W_{t_{actual}} = 210 \textrm{ kJ/kg}\)

4. Finding the work considering the isentropic efficiency

Now let's consider the isentropic efficiency. Since both the compressor and the turbine have an isentropic efficiency of 85%, we can express the work as: \(W_{c_{isentropic}} = \frac{W_{c_{actual}}}{\textrm{Isentropic Efficiency}}\) \(W_{c_{isentropic}} = \frac{60}{0.85}\) \(W_{c_{isentropic}} = 70.588 \textrm{ kJ/kg}\) \(W_{t_{isentropic}} = W_{t_{actual}} \times \textrm{Isentropic Efficiency}\) \(W_{t_{isentropic}} = 210 \times 0.85\) \(W_{t_{isentropic}} = 178.5 \textrm{ kJ/kg}\)

5. Finding the net work output with the isentropic efficiency

Using the isentropic work values for compressor and turbine, we can calculate the net work output of the cycle considering the isentropic efficiencies: \(W_{net_{isentropic}} = W_{t_{isentropic}} - W_{c_{isentropic}}\) \(W_{net_{isentropic}} = 178.5 - 70.588\) \(W_{net_{isentropic}} \approx 107.912 \textrm{ kJ/kg}\) The value is close to \(109 \textrm{ kJ/kg}\), so the correct option is (c).