Question

Air is used as the working fluid in a simple ideal Brayton cycle that has a pressure ratio of \(12,\) a compressor inlet temperature of \(300 \mathrm{K},\) and a turbine inlet temperatureof \(1000 \mathrm{K} .\) Determine the required mass flow rate of air for a net power output of \(70 \mathrm{MW}\), assuming both the compressor and the turbine have an isentropic efficiency of \((a) 100\) percent and \((b) 85\) percent. Assume constant specific heats at room temperature.

Short answer

Question: Calculate the required mass flow rate of air for a simple ideal Brayton cycle with a pressure ratio of 12, compressor inlet temperature of 300 K, and turbine inlet temperature of 1000 K. The net power output is 70 MW. Calculate for two cases: isentropic efficiency of 100% and 85%. Answer: To find the required mass flow rate of air for the given Brayton cycle, first calculate the work done by the compressor and turbine based on isentropic efficiency. The net work done by the cycle can be found by subtracting the compressor work from the turbine work, while the heat supplied can be calculated from the temperature difference in the combustion chamber. Next, determine the thermal efficiency of the cycle, and finally find the mass flow rate using the net power output. For the case with isentropic efficiency of 85%, adjust the compressor and turbine work calculations accordingly and repeat the steps.

Step-by-step solution

Calculate compressor work

We are given that the isentropic efficiency of the compressor and turbine are both 100%. This indicates ideal conditions (no losses) so the work done by the compressor can be found using the formula: $$ W_{c} = c_{p} \cdot (T_{2s} - T_{1}) $$ where \(c_{p}\) is the specific heat at constant pressure (\(1.005 \frac{kJ}{kg\cdot K}\) for air), \(T_{1}\) is the compressor inlet temperature (\(300K\)), and \(T_{2s}\) is the isentropic exit temperature. We can find \(T_{2s}\) from the pressure ratio and the isentropic relation: $$ \frac{T_{2s}}{T_{1}} = \left(\frac{P_{2}}{P_{1}}\right)^{(\gamma-1)/\gamma} $$ where \(\gamma\) is the specific heat ratio (\(1.4\) for air) and the pressure ratio \((P_{2}/P_{1}) = 12\). Solving for \(T_{2s}\), we get: $$ T_{2s} = T_{1} \cdot \left(\frac{P_{2}}{P_{1}}\right)^{(\gamma-1)/\gamma} $$

Calculate turbine work

Similar to the compressor work, we can find the turbine work using isentropic efficiency (which again is 100%): $$ W_{t} = c_{p} \cdot (T_{3} - T_{4s}) $$ where \(T_{3}\) is the turbine inlet temperature (\(1000K\)), and \(T_{4s}\) is the isentropic exit temperature. We can find \(T_{4s}\) from the pressure ratio and the isentropic relation: $$ \frac{T_{3}}{T_{4s}} = \left(\frac{P_{3}}{P_{4}}\right)^{(\gamma-1)/\gamma} $$ Since it is a Brayton cycle, \(P_{3}/P_{4} = P_{2}/P_{1} = 12\). Solving for \(T_{4s}\), we get: $$ T_{4s} = T_{3} \cdot \left(\frac{P_{3}}{P_{4}}\right)^{(\gamma-1)/\gamma} $$

Calculate net work and heat supplied

The net work done by the cycle can be found by subtracting the work done by the compressor from the work done by the turbine: $$ W_{net} = W_{t} - W_{c} $$ The heat supplied to the cycle can be found by calculating the heat added in the combustion chamber: $$ q_{in} = c_{p} \cdot (T_{3} - T_{2}) $$ since T_2 = T_2s when isentropic efficiency is 100%.

Calculate thermal efficiency

The thermal efficiency of the Brayton cycle can be found using the formula: $$ \eta_{th} = \frac{W_{net}}{q_{in}} $$

Calculate mass flow rate

Now we can find the required mass flow rate of air by using the net power output: $$ \dot{m} = \frac{P_{out}}{W_{net}} $$ where \(P_{out}\) is the net power output \((70MW)\). For the case with isentropic efficiency = 85%, follow the same steps as above, but adjust compressor and turbine work calculations based on the following modified equations: $$ W_{c, actual} = \frac{W_{c}}{isentropic\ efficiency_{compressor}} $$ and $$ W_{t, actual} = W_{t} \cdot isentropic\ efficiency_{turbine} $$ Then repeat steps 3 to 5 with these new values of compressor and turbine work.

Key concepts

Understanding Thermodynamics in the Brayton Cycle

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In the context of the Brayton cycle, thermodynamics lays the groundwork for understanding how energy is transformed in a gas turbine engine, commonly used in jet aircraft and power plants. The Brayton cycle describes the workings of a continuous combustion cycle where air is compressed, then heated, and finally, the expanded gas is used to perform work such as turning a turbine.

The efficiency of the Brayton cycle is heavily dependent on how well these processes—compression, heating, and expansion—are managed. The key principle here is energy conservation, meaning the work produced by the turbine minus the work used to compress the air equals the net work output, and therefore the power generated by the system. Emphasizing realistic scenarios, considering factors like isentropic efficiency—noted in the exercise—is crucial for practical applications of the Brayton cycle in thermodynamics.

Isentropic Processes in Compressors and Turbines

An isentropic process is an idealized thermodynamic process that is both reversible and adiabatic, meaning no heat is exchanged with the environment, and the entropy remains constant. In the Brayton cycle, both the compression and expansion of gases occur ideally as isentropic processes.

Real-world deviations from isentropic efficiency—such as in the exercise scenarios where the efficiency is given as 100% and 85%—lead to losses in the system's performance. The formula to calculate work done during these processes involves the specific heat at constant pressure (\(c_p\)), the temperatures before and after the process, and the isentropic efficiency. The exercise makes use of the relationship between temperature and pressure during these ideal processes to compute the net power output and thermal efficiency, highlighting the significance of isentropic processes in thermodynamic cycles.

Mass Flow Rate Calculation

The mass flow rate is a critical concept in the Brayton cycle, representing the amount of mass flowing through the system per unit time. It's an essential factor for calculating the net power output, and hence for determining the amount of air required for the system to produce a specific output of power.

In the given exercise, the mass flow rate calculation hinges on the net work output, which is deduced from the difference between the turbine's work and the compressor's work. Given a net power output, calculating the mass flow rate involves dividing this power by the net work per mass unit of the working fluid. It's important to correct for actual efficiencies, as systems rarely operate at 100% isentropic efficiency. These adjustments can alter the required mass flow to achieve the same power output, as illustrated in the provided solution steps.

Thermal Efficiency of the Brayton Cycle

Thermal efficiency in the Brayton cycle is a measure of how well the cycle converts the heat input into useful work. It is defined as the net work done by the cycle divided by the heat added to the air flowing through the system. The Brayton cycle's thermal efficiency is critically important in determining how effectively a power plant or jet engine converts fuel into power.

The exercise illustrates how to calculate thermal efficiency using the specific heat of air, the temperatures at various points in the cycle, and the net work output. The efficiency is impacted by various factors, including the compression ratio, the specific heat of the working fluid at constant pressure, and the isentropic efficiencies of the compressor and turbine. The formula provided brings to light how the thermal efficiency can be optimized by manipulating these variables within the cycle.

Calculating Net Power Output in Power Generation

Net power output is the usable power available after subtracting the power consumed by the system itself, such as the power required to operate a compressor in the Brayton cycle. It's the figure of paramount interest for practical applications like electricity generation or providing thrust in jet engines.

In the exercise, finding the net power output involves understanding that it's a direct result of the net work per mass unit of the working fluid and the mass flow rate through the system. The net work is obtained by the difference in work produced in the turbine and work consumed by the compressor. This net power output calculation is vital for sizing equipment and forecasting the performance of power plants, ensuring that power generation aligns with operational demands and efficiency targets. The example given serves as a foundation for students to grasp how energy is converted into work, and what factors influence the output of a thermodynamic cycle.