Question

A gas-turbine power plant operates on a simple Brayton cycle with air as the working fluid. The air enters the turbine at 120 psia and \(2000 \mathrm{R}\) and leaves at 15 psia and \(1200 \mathrm{R} .\) Heat is rejected to the surroundings at a rate of 6400 \(\mathrm{Btu} / \mathrm{s},\) and air flows through the cycle at a rate of \(40 \mathrm{lbm} / \mathrm{s}\) Assuming the turbine to be isentropic and the compresssor to have an isentropic efficiency of 80 percent, determine the net power output of the plant. Account for the variation of specific heats with temperature.

Short answer

Answer: To find the net power output of the Brayton cycle, follow these steps: 1. Calculate the isentropic work done in the turbine (\(W_{T,s}\)) using ideal gas relations and specific heats of air. 2. Determine the actual work done in the turbine (\(W_{T}\)). 3. Calculate the work done by the compressor (\(W_{C}\)) using isentropic efficiency. 4. Find the net work output of the plant by subtracting the compressor work input from the turbine work done, i.e., \(W_{net} = W_{T} - W_{C}\).

Step-by-step solution

Calculate the Isentropic Work done in the Turbine (\(W_{T,s}\))

We'll use the relation between specific heat at constant pressure (\(C_p\)), temperature and work output:\(W_{T,s} = m*C_p*(T_{1} - T_{2s})\) where \(T_{2s}\) is the temperature at the outlet of the turbine under the isentropic process. As we assume the turbine to be isentropic and account for the variation of specific heat with temperature, we can use the specific heat data tables for air to determine \(C_p\) at both conditions \(T_{1} = 2000\, \text{R}\) and \(T_{2} = 1200\, \text{R}\). After that, we can average them to get the average specific heat, and hence compute the isentropic work done by the turbine.

Calculate the Actual Work done in the Turbine (\(W_{T}\))

Since the turbine is isentropic, the actual work done in the turbine is equal to the isentropic work done i.e., \(W_{T} = W_{T,s}\)

Calculate Work done by the Compressor (\(W_{C}\))

Using the isentropic efficiency, we can express the work done by the compressor as: \(W_{C} = \frac{W_{T}}{\eta_{c}}\) where \(\eta_{c}\) is the isentropic efficiency of the compressor (80%). Hence, we can compute the work done by the compressor.

Calculate the Net Work Output of the Plant

The net work output of the plant is equal to the difference between the turbine work done and the compressor work input i.e., \(W_{net} = W_{T} - W_{C}\).