Question

The gas-turbine portion of a combined gas-steam power plant has a pressure ratio of \(16 .\) Air enters the compressor at \(300 \mathrm{K}\) at a rate of \(14 \mathrm{kg} / \mathrm{s}\) and is heated to \(1500 \mathrm{K}\) in the combustion chamber. The combustion gases leaving the gas turbine are used to heat the steam to \(400^{\circ} \mathrm{C}\) at \(10 \mathrm{MPa}\) in a heat exchanger. The combustion gases leave the heat exchanger at 420 K. The steam leaving the turbine is condensed at 15 kPa. Assuming all the compression and expansion processes to be isentropic, determine \((a)\) the mass flow rate of the steam, \((b)\) the net power output, and \((c)\) the thermal efficiency of the combined cycle. For air, assume constant specific heats at room temperature.

Short answer

Answer: The values are as follows: (a) mass flow rate of steam is 5.322 kg/s, (b) net power output is 7113.96 kJ/s, and (c) thermal efficiency of the combined cycle is 42.0%.

Step-by-step solution

Find the work done in the gas-turbine cycle

For the gas-turbine cycle, we will follow these steps: (i) Find the exit temperature of air from the compressor. (ii) Calculate the compressor work. (iii) Find the exit temperature of air from the turbine. (iv) Calculate the turbine work. (i) The exit temperature of air from the compressor is given by: \(T_2 = T_1 \times (1 + \eta_c(\pi^{\frac{(\gamma - 1)}{\gamma}} - 1))\). For air, the specific heat ratio \(\gamma = 1.4\) and we are given the pressure ratio \(\pi = 16\), the compressor isentropic efficiency, \(\eta_c = 1\) (since the compression is isentropic): \(T_2 = 300 \times (1 + (16^{\frac{(1.4 - 1)}{1.4}} - 1)) = 658.51 \mathrm{K}\) (ii) To find the compressor work, we can use the formula: \(W_c = m \times c_p( T_2 - T_1)\) Where, \(m = 14 \mathrm{kg/s}\) (mass flow rate of air) \(c_p = 1.005 \mathrm{kJ/kg.K}\) (specific heat of air at constant pressure) \(W_c = 14 \times 1.005 \times (658.51 - 300) = 6449.6 \mathrm{kJ/s}\) (iii) The exit temperature of air from the turbine can be obtained using: \(T_4 = T_3 / (1 + \eta_t(\pi^{\frac{(\gamma - 1)}{\gamma}} - 1))\). And given, \(T_3 = 1500 \mathrm{K}\) and \(\eta_t = 1\) (since the expansion is isentropic): \(T_4 = 1500 / (1 + 1(16^{\frac{(1.4-1)}{1.4}} - 1)) = 644.75 \mathrm{K}\) (iv) To find the turbine work, we can use the formula: \(W_t = m \times c_p( T_3 - T_4)\) \(W_t = 14 \times 1.005 \times (1500 - 644.75) = 12126.34 \mathrm{kJ/s}\)

Calculate the heat transfer in the heat exchanger

The heat transfer in the heat exchanger can be calculated as: \(Q = m \times c_p( T_3 - T_4)\) \(Q = 14 \times 1.005 \times (1500 - 420) = 15309.3 \mathrm{kJ/s}\)

Calculate the work done in the steam cycle

To determine the work done in the steam cycle, we first find the enthalpy values from the steam tables. From the given process's conditions, we get: \(h_5 = 3230.9 \mathrm{kJ/kg}\) \(s_5 = 6.9210 \mathrm{kJ/kg.K}\) \(s_6 = s_5 = 6.9210 \mathrm{kJ/kg.K}\) \(h_6 = 2783.1 \mathrm{kJ/kg}\) (interpolating in the steam tables) \(h_7 = 192.47 \mathrm{kJ/kg}\) Now, we can find the turbine work and pump work: \(W_t = (h_5 - h_6)\) \(W_p = (h_7 - h_6)\)

Calculate the mass flow rate of the steam

We will use the heat transfer (Q) from the heat exchanger to determine the steam's mass flow rate. \(m_s = \frac{Q}{h_5 - h_7}\) \(m_s = \frac{15309.3}{3230.9 - 192.47} = 5.322 \mathrm{kg/s}\)

Determine the net power output

The net power output can be calculated as: \(W_{net} = W_{gasturbine} + W_{steam}\) \(W_{net} = (W_t - W_c) + m_s(W_t - W_p)\) \(W_{net} = (12126.34 - 6449.6) + 5.322(2783.1 - 192.47) = 7113.96 \mathrm{kJ/s}\)

Calculate the thermal efficiency of the combined cycle

The thermal efficiency of the combined cycle can be calculated as: \(\eta = \frac{W_{net}}{Q_{total}}\) \(Q_{total} = m \times c_p(T_3 - T_1)\) \(Q_{total} = 14 \times 1.005 \times (1500 - 300) = 16946.1 \mathrm{kJ/s}\) Thermal efficiency, \(\eta = \frac{7113.96}{16946.1} = 42.0 \%\) So, the mass flow rate of steam (a) is 5.322 kg/s, the net power output (b) is 7113.96 kJ/s, and the thermal efficiency of the combined cycle (c) is 42.0%.

Key concepts

Gas-Turbine Cycle

The gas-turbine cycle is fundamental to combined gas-steam power plants. It operates on the basic principle of converting thermal energy into mechanical work by compressing air, mixing it with fuel to combust and expand in a turbine. The resulting mechanical work drives electrical generators.

The cycle starts with air entering a compressor where its pressure and temperature increase. This process is ideally isentropic, meaning it occurs at constant entropy and involves no heat transfer with the surroundings. In reality, some level of irreversibility will be present, but for instructional problems, an idealized isentropic process is a common assumption. The air, now compressed and heated, enters the combustion chamber where fuel is added, and the mixture burns at constant pressure, significantly raising the temperature.

The hot gases then expand through the gas turbine, which extracts energy from them, and through an isentropic expansion process, reduces their temperature and pressure, producing work. This step effectively mimics the second part of the Brayton cycle, specific to gas-turbine engines. The heat extracted from the turbine exhaust in the heat exchanger is a vital component for the steam cycle's operation in the combined plant, enhancing overall efficiency.

Isentropic Processes

Isentropic processes are adiabatic (no heat transfer) and reversible, with constant entropy throughout the process. They are vital in thermodynamics because they represent an idealized scenario where no energy is lost to friction or other irreversible processes.

In combined gas-steam power plants, both the compression of air in the compressor and the expansion of hot gases in the turbine are assumed to be isentropic for calculations. The assumption of isentropic processes allows for simpler computation of work and temperature changes using the relation \( T_2 = T_1 \times (1 + \eta_c(\pi^{\frac{(\gamma - 1)}{\gamma}} - 1)) \) for the compressor and \( T_4 = T_3 / (1 + \eta_t(\pi^{\frac{(\gamma - 1)}{\gamma}} - 1)) \) for the turbine, where \( T_1 \) and \( T_3 \) are the inlet temperatures, \( T_2 \) and \( T_4 \) are the outlet temperatures, \( \eta_c \) and \( \eta_t \) represent isentropic efficiencies, and \( \pi \) is the pressure ratio.

The isentropic efficiency is a measure of how close the actual device operates to the ideal isentropic behavior and will always be less than one for real processes. It reflects the imperfections and losses within the system.

Thermal Efficiency

Thermal efficiency is a measure of how well a power plant converts the heat from fuel into work or electrical energy. It's calculated as the ratio of the net work output to the total heat input, expressed as a percentage. For a combined gas-steam power plant, several factors contribute to its overall efficiency.

First, the efficiency of the gas-turbine cycle, including both the compression and expansion of air, is essential. As mentioned earlier, the work produced by the gas turbine portion (\(W_{gasturbine}\) ) is considered, and the efficiency is optimized through isentropic processes. Next, the steam cycle extends the energy conversion, utilizing the heat from the exhaust gas in a heat exchanger to produce steam. The heat transfer in the heat exchanger (\(Q\) ) and the work done by the steam turbine (\(W_{steam}\) ) both contribute to the overall power output of the plant.

By using the equation \( \eta = \frac{W_{net}}{Q_{total}} \) to compute the combined cycle's thermal efficiency, students learn to appreciate how the coupling of gas and steam cycles in one power plant can achieve higher efficiencies than either cycle could alone. This is a direct result of following the Second Law of Thermodynamics, which encourages heat recovery and its conversion into work to make power generation more sustainable and cost-effective.