Question

Consider a steam power plant operating on the ideal Rankine cycle with reheat between the pressure limits of \(30 \mathrm{MPa}\) and \(10 \mathrm{kPa}\) with a maximum cycle temperature of \(700^{\circ} \mathrm{C}\) and a moisture content of 5 percent at the turbine exit. For a reheat temperature of \(700^{\circ} \mathrm{C}\), determine the reheat pressures of the cycle for the cases of \((a)\) single and \((b)\) double reheat.

Short answer

Question: Calculate the reheat pressures for a steam power plant operating on the ideal Rankine cycle with single and double reheat, given the following parameters: pressure limits of 30 MPa and 10 kPa, maximum cycle temperature of 700°C, moisture content at turbine exit of 5%, and reheat temperature 700°C.

Step-by-step solution

Recall the ideal Rankine cycle with reheat

The ideal Rankine cycle consists of four processes: 1-2: Isentropic compression in the pump 2-3: Constant pressure heat addition in the boiler 3-4: Isentropic expansion in the turbine 4-1: Constant pressure heat rejection in the condenser In the reheat cycle, the steam is first expanded in the high-pressure turbine and then sent back to the boiler to be reheated before being expanded again in the low-pressure turbine. This process is mainly done to reduce the moisture content at the turbine exit and improve the cycle efficiency. We will analyze the single and double reheat cases.

Calculate enthalpy and entropy values at different points of the cycle

For this step, we will use the given information for pressure limits and temperature. We will also use the steam tables to find the corresponding values of enthalpy and entropy at different points of the cycle. State 1: Condenser exit (10 kPa, saturated liquid) \(h_1 = h_f @ P_1\) \(s_1 = s_f @ P_1\) State 2: Pump exit (30 MPa, isentropic compression) \(s_2 = s_1\) Using steam tables, find \(h_2\) @ \(P_2\) and \(s_2\) State 3: Boiler exit (30 MPa, 700°C) Find \(h_3\) and \(s_3\) @ \(P_3\) and \(T_3\) from steam tables State 4: Turbine exit (10 kPa, \(x = 0.95\)) Here, \(x\) is the quality of the steam, which represents the moisture content. The quality value of 0.95 means 5% moisture content at the turbine exit. \(s_4 = s_3\) Find \(h_4 = h_f + x*(h_g - h_f)\) and \(s_4\) @ \(P_4\) and \(x\) from steam tables

Determine the reheat pressures for single and double reheat

For single reheat, we perform the following calculation: State 5: single reheat exit (700°C, reheat pressure to be determined) Entropy must remain constant when the steam is reheated (\(s_5 = s_4\)). Using steam tables, find the reheat pressure \(P_5\) @ \(T_5 = 700 °C\) and \(s_5\). For double reheat, we first find the conditions at the intermediate pressure and temperature: State 5: double reheat first exit (intermediate pressure, 700°C) \(s_5 = s_4\) Using steam tables, find \(P_5\) @ \(T_5 = 700 °C\) and \(s_5\) State 6: double reheat second exit (700°C, second reheat pressure to be determined) \(s_6 = s_5\) Using steam tables, find the reheat pressure \(P_6\) @ \(T_6 = 700 °C\) and \(s_6\) To summarize, we have calculated the entropy and enthalpy values at different points of the ideal Rankine cycle with reheat. We then used these values to find the reheat pressures for single and double reheat cases.

Key concepts

Thermodynamic Cycles

Thermodynamic cycles are the cornerstones of energy systems, particularly in power generation. A thermodynamic cycle can be understood as a series of processes that a fluid undergoes where it eventually returns to its original state. The performance of these cycles directly influences the efficiency and output of power plants.

One such cycle is the Rankine cycle, which is widely used in steam power plants. The cycle involves heating a fluid to a high-pressure vapor, then allowing it to expand and do work (like turning a turbine), and finally condensing it back to a liquid to start the process over. The key benefit of the Rankine cycle is its ability to convert heat into work with high efficiency, which is why it's so prevalent in electricity generation.

The Rankine cycle with reheat, as described in the original exercise, adds a reheat step to further dry the steam and improve the efficiency of the cycle. Reheating in a thermodynamic cycle reduces the moisture content in the steam at the turbine exit, preventing turbine blade erosion and preserving the energy content of the steam.

Steam Power Plant Efficiency

When discussing steam power plant efficiency, we are mainly concerned with how effectively a power plant converts the heat energy contained in the steam into mechanical work and eventually electricity.

The efficiency of a steam power plant is greatly influenced by the cycle it operates on. In the Rankine reheat cycle, the efficiency is enhanced by reheating the steam between turbine stages. The reason for this is that by reheating, we are able to re-expand the steam at a higher average temperature, which is consistent with the Carnot principle that efficiency is increased by higher heat source temperatures and lower heat sink temperatures.

Aside from the thermodynamic improvements, efficiency can also be influenced by mechanical factors such as the design of the turbine and insulation of the pipes to reduce heat loss. Addressing these key areas leads to a more efficient power plant, which generates more electricity from the same amount of fuel, saving costs and resources.

Isentropic Processes

Isentropic processes are idealized processes in which entropy remains constant. In thermodynamics, these processes are significant because they represent the most efficient transformations possible – no energy is lost due to friction, unrestrained expansion, or other kinds of inefficiencies.

In the context of the Rankine cycle, both the compression in the pump and the expansion in the turbine are ideally isentropic, meaning they take place without any change in entropy. Realistically, these processes cannot be perfectly isentropic because some energy will be lost due to factors like friction and heat transfer to the surroundings. However, engineers strive to make these processes as close to isentropic as possible to increase the overall efficiency of the power plant.

In the solution provided, when assessing the efficiency or analyzing the reheat conditions of the steam, the assumption of isentropic processes simplifies calculations and serves as a benchmark for the real performance of the steam power plant. This simplification is crucial for understanding the fundamental thermodynamic concepts before applying practical considerations that deviate from the ideal case.