Question

Consider a combined gas-steam power plant that has a net power output of \(450 \mathrm{MW}\). The pressure ratio of the gas-turbine cycle is \(14 .\) Air enters the compressor at \(300 \mathrm{K}\) and the turbine at \(1400 \mathrm{K}\). The combustion gases leaving the gas turbine are used to heat the steam at \(8 \mathrm{MPa}\) to \(400^{\circ} \mathrm{C}\) in a heat exchanger. The combustion gases leave the heat exchanger at \(460 \mathrm{K}\). An open feedwater heater incorporated with the steam cycle operates at a pressure of 0.6 MPa. The condenser pressure is 20 kPa. Assuming all the compression and expansion processes to be isentropic, determine ( \(a\) ) the mass flow rate ration of air to steam, ( \(b\) ) the required rate of heat input in the combustion chamber, and ( \(c\) ) thermal efficiency of the combined cycle.

Short answer

To solve this problem, follow these steps: Step 1: Determine the gas-turbine cycle parameters and output using the given pressure ratio and temperatures at the inlet and discharge of the compressor and the turbine. Calculate the temperatures T2 and T4 using the isentropic relations, then find the net work developed by the gas-turbine cycle. Step 2: Calculate the mass flow rate ratio of air to steam (\(\dot{m}_a/\dot{m}_s\)) using the net power output of the combined cycle and the work output from step 1. Step 3: Analyze the steam cycle parameters, using the provided operating pressure of the open feedwater heater, condenser pressure, and the temperature of the steam and combustion gases. Step 4: Find the required rate of heat input in the combustion chamber (Q_in) using the specific heat capacities and temperatures of the combustion gases and steam. Step 5: Calculate the thermal efficiency of the combined cycle using the net power output and rate of heat input in the combustion chamber found in the previous steps. Following these steps, you will be able to find the mass flow rate ratio of air to steam, the required rate of heat input in the combustion chamber, and the thermal efficiency of the combined cycle for the given gas-steam power plant.

Step-by-step solution

Determine the gas-turbine cycle parameters and output

To find the mass flow rate ratio of air to steam, we first need to determine the output of the gas-turbine cycle. We can do this using the pressure ratio and the temperatures at the inlet and discharge of the compressor and the turbine. - The pressure ratio of the gas-turbine cycle is given as \(r_p = 14\) - The air enters the compressor at \(T_1 = 300 \mathrm{K}\) - The air enters the turbine at \(T_3 = 1400 \mathrm{K}\) Using the isentropic relations for the ideal gas turbine cycle, we can find \(T_2\) and \(T_4\): \(T_2 = T_1 \times (r_p)^{(\gamma - 1)/\gamma} = 300 \mathrm{K} \times (14)^{(1.4 - 1)/1.4} \approx 767.23 \mathrm{K}\) \(T_4 = T_3 / (r_p)^{(\gamma - 1)/\gamma} = 1400 \mathrm{K} / (14)^{(1.4 - 1)/1.4} \approx 535.91 \mathrm{K}\) Now, we can find the net work developed by the gas-turbine cycle: \(W_{cycle} = W_t - W_c = C_p(T_3 - T_4) - C_p(T_2 - T_1) = C_p[(T_3 - T_4) - (T_2 - T_1)]\)

Determine the mass flow rate ratio of air to steam

We are given the net power output of the combined cycle as \(450 \mathrm{MW}\). We can use this value and the work output from step 1 to find the mass flow rate ratio of air to steam, denoted as \(\dot{m}_a/\dot{m}_s\): \(\dfrac{\dot{m}_a}{\dot{m}_s} = \dfrac{450 \times 10^6 \mathrm{W}}{W_{cycle}}\)

Analyze the steam cycle parameters

We are given the operating pressure of the open feedwater heater as \(P_{FWH} = 0.6 \mathrm{MPa}\) and the condenser pressure as \(P_{cond} = 20 \mathrm{kPa}\). We also know that the combustion gases leaving the gas turbine are used to heat the steam at \(8 \mathrm{MPa}\) and the steam is heated to a temperature of \(T_{steam} = 400^\circ \mathrm{C} = 673.15 \mathrm{K}\). The combustion gases leave the heat exchanger at \(T_{comb,gas} = 460 \mathrm{K}\).

Determine the required rate of heat input in the combustion chamber

We can use the specific heat capacities and the temperatures of the combustion gases and the steam to find the required rate of heat input in the combustion chamber, denoted as \(Q_{in}\): \(Q_{in} = \dot{m}_a (C_{p,gas}(T_3 - T_1) - C_{p,steam}(T_{steam} - T_{comb,gas}))\)

Calculate the thermal efficiency of the combined cycle

Finally, we can determine the overall thermal efficiency of the combined cycle (\(\eta_{combined}\)) using the net power output and the rate of heat input in the combustion chamber found in previous steps: \(\eta_{combined} = \dfrac{450 \times 10^6 \mathrm{W}}{Q_{in}}\) Using these steps, you can calculate the mass flow rate ratio of air to steam, required rate of heat input in the combustion chamber, and the thermal efficiency of the combined cycle for the given gas-steam power plant.

Key concepts

Isentropic Processes

Isentropic processes are essential to understanding the workings of a combined gas-steam power plant. These are idealized thermodynamic processes that occur at constant entropy, meaning there are no changes in the entropy of the system throughout the process. In isentropic processes, no energy is wasted as heat transfer, making these processes reversible and highly efficient.

In the context of a gas-turbine cycle in power plants, isentropic compression in the compressor and expansion in the turbine are desired for maximum efficiency. However, in real-life applications, some degree of entropy increase is inevitable, leading to less efficiency than the ideal scenario. We can use isentropic relations, such as the isentropic process formula for temperature change, \(T_2 = T_1 \times (r_p)^{(\gamma - 1)/\gamma}\), where \(r_p\) is the pressure ratio and \(\gamma\) is the specific heat ratio, to calculate temperatures after these processes.

Gas-Turbine Cycle

The gas-turbine cycle, also known as the Brayton cycle when referring to gas turbines, is a sequence of processes used to produce power in an engine or power plant. This cycle comprises four main steps: air is taken in and compressed in the compressor; the compressed air is then heated and ignited in the combustion chamber; the high-pressure combustion gases expand through the turbine, producing work; and finally, the gases are released to the atmosphere or used in additional processes.

The step-by-step solution provided illustrates the gas-turbine cycle's operation by determining the temperature changes of the air as it goes through the compression and expansion processes, which are assumed to be isentropic.

Thermal Efficiency

Thermal efficiency is a measure of how well a power plant converts heat into work. In terms of a combined gas-steam power plant, it's the ratio of the net power output to the rate of heat input in the combustion chamber. Higher thermal efficiency indicates more efficient energy conversion and less waste. Mathematically, it is expressed as \(\eta_{combined} = \frac{450 \times 10^6 \mathrm{W}}{Q_{in}}\), where the numerator refers to the power output and the denominator is the rate of heat added. Improving thermal efficiency is always a goal in power plant design, as it reduces both fuel consumption and emissions.

Heat Exchanger

A heat exchanger in a power plant is a system used to transfer heat between two or more fluids without mixing them. In a combined gas-steam power plant, the heat exchanger uses the heat from the exhaust gases of the gas turbine to increase the steam's temperature before it enters the steam turbine. The effectiveness of a heat exchanger is critical for the overall efficiency of the plant, as it captures and reuses energy that would otherwise be wasted. Proper design and operation of heat exchangers contribute to higher thermal efficiency by utilizing as much of the available thermal energy as possible.

Mass Flow Rate

The mass flow rate is a fundamental concept in both the gas and steam cycles of a power plant, representing the amount of mass flowing through a system per unit time. It is a crucial factor in the calculation of power output as well as in determining the energy transfer in heat exchangers. In the exercise, the mass flow rate ratio of air to steam (\(\dot{m}_a/\dot{m}_s\)) is computed to understand how much air and steam are needed to deliver the desired power output. This ratio impacts the design and operation of the plant since it determines how much air must be compressed and how much steam must be heated for efficient function.

Feedwater Heater

A feedwater heater is a component in a steam cycle that preheats the water (feedwater) before it enters the boiler or steam generator, using steam extracted from the turbine. This process improves the cycle's efficiency by reducing the amount of fuel needed in the boiler to bring the water up to the required temperature. Feedwater heaters can be classified as either open or closed systems; in this exercise, an open feedwater heater operates at a specific pressure, where the feedwater mixes directly with extracted steam, transferring the heat more effectively.

Ideal Gas Relations

Ideal gas relations are equations that describe the behavior of an ideal gas - a hypothetical gas that perfectly fits all the assumptions of the kinetic-molecular theory. While real gases do not always behave as ideal gases, these relations provide a close approximation for many engineering calculations involving gases. Equations such as the ideal gas law (\(PV = nRT\)), and formulas for isentropic processes involving ideal gases, play a central role in calculating parameters such as temperatures, pressures, and volumes in different parts of the engine or power plant cycle.