Question

Consider a combined gas-steam power plant that has a net power output of \(280 \mathrm{MW}\). The pressure ratio of the gas turbine cycle is \(11 .\) Air enters the compressor at \(300 \mathrm{K}\) and the turbine at \(1100 \mathrm{K}\). The combustion gases leaving the gas turbine are used to heat the steam at \(5 \mathrm{MPa}\) to \(350^{\circ} \mathrm{C}\) in a heat exchanger. The combustion gases leave the heat exchanger at \(420 \mathrm{K} .\) An open feedwater heater incorporated with the steam cycle operates at a pressure of 0.8 MPa. The condenser pressure is 10 kPa. Assuming isentropic efficiences of 100 percent for the pump, 82 percent for the compressor, and 86 percent for the gas and steam turbines, determine ( \(a\) ) the mass flow rate ratio of air to steam, \((b)\) the required rate of heat input in the combustion chamber, and (c) the thermal efficiency of the combined cycle.

Short answer

Answer: In a combined gas-steam power plant, the mass flow rate ratio of air to steam is related to the energy balance through the heat exchanger. The energy transferred from the combustion gases (air side) is used to heat the steam, and this can be represented by the equation: \( \dot{m}_a c_{p,a}(T_3 - T_{4a}) = \dot{m}_s (h_5 - h_4) \) Here, the mass flow rate ratio of air to steam can be calculated as: \(\frac{\dot{m}_a}{\dot{m}_s} = \frac{h_5 - h_4}{c_{p,a}(T_3 - T_{4a})}\)

Step-by-step solution

Calculate temperatures at each stage of the gas turbine cycle

To do this, we will apply the isentropic temperature ratio formula for the compressor and the turbine. The formula is: \(T_2=T_1 \times \left(\frac{P_2}{P_1}\right)^{(\gamma-1)/\gamma}\) The isentropic efficiencies for the compressor and the gas turbine are given as 82% and 86% respectively. We can calculate the actual temperatures as: \(T_{2a} = T_1 + \frac{T_2 - T_1}{\text{Isentropic efficiency}}\) \(T_{4a} = T_3 - \text{Isentropic efficiency} \times (T_3 - T_4)\) Step 2: Determine heat transfer in the heat exchanger

Apply energy balance to the heat exchanger

In the heat exchanger, the energy transferred from the combustion gases is used to heat the steam. So, the energy balance equation can be written as: \( \dot{m}_a c_{p,a}(T_3 - T_{4a}) = \dot{m}_s (h_5 - h_4) \) Step 3: Analyze the steam cycle

Determine properties of steam at various stages

We are given that the open feedwater heater operates at 0.8 MPa, and the condenser pressure is 10 kPa. We can use the steam tables to find the properties of steam at these various stages, including enthalpy and entropy values. Step 4: Determine heat transfer and work in the steam cycle

Apply energy balance and mass flow rate balance to the steam cycle components

Using the steam properties found in the previous step, we can apply energy balance to the components of the steam cycle. We can write energy balance equations for the steam turbine, the feedwater heater, and the condenser. Also, apply the mass flow rate balance to the feedwater heater: \( \dot{m}_{s,2} = \dot{m}_s + \dot{m}_{s,3} \) Step 5: Calculate the required heat input, mass flow rate ratio, and thermal efficiency

Calculate heat input in the combustion chamber

Using the energy balance equations for the heat exchanger and steam cycle components, we can calculate the total heat input, which is the required heat input in the combustion chamber: \(\dot{Q}_\text{input} = \dot{m}_a c_{p,a} (T_3 - T_2a)\)

Calculate the mass flow rate ratio of air to steam

From the energy balance equation in the heat exchanger, solve for the mass flow rate ratio: \(\frac{\dot{m}_a}{\dot{m}_s} = \frac{h_5 - h_4}{c_{p,a}(T_3 - T_{4a})}\)

Calculate the thermal efficiency of the combined cycle

The thermal efficiency of the combined cycle is the ratio of the total power output to the total heat input: \(\eta_\text{combined} = \frac{\dot{W}_\text{net}}{\dot{Q}_\text{input}}\) Where \(\dot{W}_\text{net}= 280 \, \text{MW}\). Calculate the thermal efficiency using the previously calculated values. Those are the steps to analyze and determine the required parameters for a combined gas-steam power plant.

Key concepts

Understanding Isentropic Efficiency

The concept of isentropic efficiency is crucial when analyzing both gas and steam turbines in a combined cycle power plant. It represents the efficiency with which the compressor and turbines convert input energy into work while undergoing adiabatic (no heat exchange) processes. Ideal turbines and compressors are isentropic, meaning they do no generate entropy and their processes are reversible. However, in reality, actual devices are not perfect due to friction and other losses.

In a gas turbine, isentropic efficiency is defined as the ratio of the work output (or input for a compressor) of the actual process to the work output (or input) of the ideal isentropic process. Mathematically, the actual temperature after compression, using the isentropic efficiency, can be calculated as:
\[T_{2a} = T_1 + \frac{T_2 - T_1}{\text{Isentropic efficiency}}\]
In our exercise, the compressor and turbines have given isentropic efficiencies of 82% and 86% respectively. These efficiencies have a direct impact on the performance of the power plant cycle, as they affect the temperatures at the various stages, as well as the overall energy required and efficiency of the power plant.

The Mass Flow Rate Ratio

The mass flow rate ratio is an important factor when analyzing a combined gas-steam power plant. It defines the ratio of the mass flow rate of air (\(\dot{m}_a\)) for the gas turbine to the mass flow rate of steam (\(\dot{m}_s\)) for the steam turbine.

This ratio is integral to achieving energy balance within the heat exchanger, where the hot air from the gas turbine heats the water to become steam for the steam turbine. The respective mass flow rates must be matched to transfer heat efficiently without waste. By balancing the energy equation of the heat exchanger, we can derive:
\[\frac{\dot{m}_a}{\dot{m}_s} = \frac{h_5 - h_4}{c_{p,a}(T_3 - T_{4a})}\]
This calculation, outlined in step 6 of the solution, involves multiple parameters like specific heat capacity and enthalpy changes, and it’s fundamental for the proper operation and design of the plant.

Heat Input Calculation

In any power plant, the heat input calculation is pivotal for understanding the operation and efficiency. It is the total amount of energy added to the system, typically through the combustion of fuel. The rate of heat input is crucial for determining how efficient the power plant is in converting thermal energy into electrical energy.

To calculate the heat input to our combined cycle power plant, the specific heat capacity of air and the temperature difference before and after the combustion process are essential. For the gas turbine, the calculation uses the formula:
\[\dot{Q}_\text{input} = \dot{m}_a c_{p,a} (T_3 - T_2a)\]
This value directly correlates to the fuel consumption and the cost of running the power plant. It also impacts the thermal efficiency, making it a critical factor in the overall economic analysis of the plant’s operation.

Evaluating Thermal Efficiency

The thermal efficiency evaluation of a power plant measures how effectively it converts the heat input into electrical energy. Higher thermal efficiency indicates a more cost-effective plant as it means less fuel is used for the same amount of electricity generated.

The thermal efficiency for a combined gas-steam power plant can be calculated by taking the ratio of the useful work output to the heat input, expressed as:
\[\eta_\text{combined} = \frac{\dot{W}_\text{net}}{\dot{Q}_\text{input}}\]
As per step 7 in our exercise, to find the efficiency, we need to calculate the net power output, which is given, and the total heat input, derived from the previous steps. This efficiency is a key indicator of the power plant's performance and is critically analyzed to improve the plant's cost-effectiveness and minimize environmental impact.

Energy Balance in Power Cycles

The principle of energy balance in power cycles underlines the understanding that energy within a closed system must remain constant, following the first law of thermodynamics. For power plants, this means the energy going into the system (heat input) must be equal to the energy coming out (work output plus losses).

An energy balance must be performed for each main component of the cycle: the compressor, combustion chamber, gas turbine, heat exchanger, steam generator, steam turbine, and condenser. For example, in the heat exchanger of our combined cycle plant, the balance ensures that the energy transferred from the combustion gases matches the energy absorbed by the steam, obeying the equation:
\[\dot{m}_a c_{p,a}(T_3 - T_{4a}) = \dot{m}_s (h_5 - h_4)\]
This balance is not only essential for the design and analysis of power plants but also for their optimal operation and control. It allows engineers to pinpoint inefficiencies and losses, which can then be minimized or recuperated to improve the overall efficiency of the power plant.