Question

A gas-turbine power plant operates on the simple Brayton cycle between the pressure limits of 100 and 1600 kPa. The working fluid is air, which enters the compressor at \(40^{\circ} \mathrm{C}\) at a rate of \(850 \mathrm{m}^{3} / \mathrm{min}\) and leaves the turbine at \(650^{\circ} \mathrm{C}\). Using variable specific heats for air and assuming a compressor isentropic efficiency of 85 percent and a turbine isentropic efficiency of 88 percent, determine \((a)\) the net power output (b) the back work ratio, and \((c)\) the thermal efficiency.

Short answer

Question: Determine the net power output, back work ratio, and thermal efficiency of a gas-turbine power plant operating on the simple Brayton cycle, given the specified conditions, flow rate, and isentropic efficiencies for the compressor and turbine. Answer: To find the net power output, back work ratio, and thermal efficiency of the gas-turbine power plant, you need to follow these steps: 1. Calculate the entropy change during the compression and expansion process using isentropic efficiencies. 2. Use the specific enthalpies at the inlet and outlet of the compressor and turbine to calculate the net power output. 3. Calculate the back work ratio using the work output of the turbine and the work input to the compressor. 4. Calculate the thermal efficiency using the net work done and the heat added during the heat addition process. Once the calculations are performed based on the given parameters and variable specific heats for air, you will have the net power output, back work ratio, and thermal efficiency of the gas-turbine power plant.

Step-by-step solution

Find the entropy change during the compression process

Using the given isentropic efficiency of the compressor (85%), we can calculate the entropy change during the compression process as follows: \[\eta_{c} = \frac{h_{2s} - h_{1}}{h_{2} - h_{1}}\] \[\Delta s_{c} = s_{2s} - s_{1}\] To find the actual temperature (T2) and specific enthalpy (h2) at the exit of the compressor, we can use the isentropic relation and interpolate the value of T2 using specific heat tables for air. \[T_{2s} = T_{1} \left(\frac{P_{2}}{P_{1}}\right)^{(\gamma - 1) / \gamma}\]

Find the entropy change during the expansion process

Using the given isentropic efficiency of the turbine (88%) and the actual temperature at the turbine exit (650°C), we can calculate the entropy change during the expansion process as follows: \[\eta_{t} = \frac{h_{3} - h_{4}}{h_{3} - h_{4s}}\] \[\Delta s_{t} = s_{3} - s_{4s}\] To find the actual specific enthalpy (h4) at the exit of the turbine, we can use the isentropic relation for the expansion process. \[T_{4s} = T_{3} \left(\frac{P_{4}}{P_{3}}\right)^{(\gamma - 1) / \gamma}\]

Calculate the net power output

We can now use the specific enthalpies at the inlet and outlet of the compressor and turbine to calculate the net power output as follows: \[W_{net} = \dot{m}(h_{3} - h_{4} - h_{2} + h_{1})\]

Calculate the back work ratio

Using the work output of the turbine and the work input to the compressor, we can calculate the back work ratio: \[BWR = \frac{W_{c}}{W_{t}} = \frac{h_{2} - h_{1}}{h_{3} - h_{4}}\]

Calculate the thermal efficiency

Finally, we can calculate the thermal efficiency of the power plant using the net work done and the heat added during the heat addition process (enthalpy change from state 2 to state 3). \[\eta_{th} = \frac{W_{net}}{Q_{in}} = \frac{h_{3} - h_{4} - h_{2} + h_{1}}{h_{3} - h_{2}}\] By following these steps and using the given parameters and variable specific heats for air, we can determine the net power output, back work ratio, and thermal efficiency of the gas-turbine power plant operating on the simple Brayton cycle.

Key concepts

Isentropic Efficiency

Isentropic efficiency is a measure of how closely a real-world process, such as compression or expansion in a turbine or compressor, approaches the ideal isentropic process. Technically, it is the ratio of the actual work to the ideal work. For example, in a compressor with an isentropic efficiency of 85%, it means that the compressor does its job with only a 15% loss compared to the ideal scenario.

Considering a Brayton cycle, which is an idealized process for gas-turbine engines, the isentropic efficiencies of the compressor and turbine play pivotal roles in determining the performance of the cycle. The higher the isentropic efficiency, the closer the process is to being reversible and ideal, meaning less energy is wasted.

Specific Enthalpy

Specific enthalpy is a property of a substance that measures the total heat content per unit mass. It's an extensive property commonly used in thermodynamics to express the energy transfer in processes involving heat and work. In the context of a Brayton cycle, specific enthalpy values for air at various points in the cycle are critical for calculating the net power output and thermal efficiency. Precisely, the changes in specific enthalpy between the states - from entering the compressor, exiting the compressor, entering the turbine, and finally exiting the turbine - must be carefully evaluated.

These changes are essential for analyzing the cycle's performance, as the work done by the turbine and the work required by the compressor directly relate to the specific enthalpies at their respective entry and exit points.

Thermal Efficiency

Thermal efficiency is a dimensionless performance metric for engines and power plants that indicates the ratio of useful output energy to input heat energy. It's a key indicator of how effectively a system converts heat into work. In the Brayton cycle, the thermal efficiency provides insight into the effectiveness of converting the heat added to the gas into mechanical energy. This efficiency is influenced by factors such as the properties of the working fluid and the isentropic efficiencies of the compressor and turbine. A higher thermal efficiency means that less fuel is needed to produce the same amount of work, which is valuable for economic and environmental reasons.

Back Work Ratio

The back work ratio (BWR) is a term used in thermodynamics to describe the proportion of the turbine's work output that is used to drive the compressor. In simple terms, it's a measure of how much energy produced by the turbine is consumed by the compressor and it remains an important factor in the net power output of the Brayton cycle. A high BWR signifies that a significant portion of the energy generated by the turbine is being used to run the compressor, which may indicate lower efficiency. Conversely, a lower BWR implies that most of the turbine's work output is available for external use, leading to higher overall efficiency.

Entropy Change

Entropy change quantifies the degree of disorder or randomness in a system. During a thermodynamic process, entropy change is tied to the heat transfer and can be used to define the irreversibility of a process. In an isentropic process, ideally, there would be no change in entropy. However, during the real processes in a Brayton cycle, entropy will increase due to inefficiencies such as friction and heat loss. Calculating the entropy change during compression and expansion allows understanding the deviation from the ideal process and aids in the evaluation of isentropic efficiency. By assessing the entropy change, engineers can design more efficient turbines and compressors by minimizing irreversibilities within the system.