Question

An ideal Rankine steam cycle modified with two closed feedwater heaters is shown below. The power cycle receives \(75 \mathrm{kg} / \mathrm{s}\) of steam at the high pressure inlet to the turbine. The feedwater heater exit states for the boiler feedwater and the condensed steam are the normally assumed ideal states. The fraction of mass entering the high pressure turbine at state 5 that is extracted for the feedwater heater operating at \(1400 \mathrm{kPa}\) is \(y=0.1446 .\) Use the data provided in the tables given below to (a) Sketch the \(T\) -s diagram for the ideal cycle. (b) Determine the fraction of mass, \(z\), that is extracted for the closed feedwater heater operating at the \(245 \mathrm{kPa}\) extraction pressure. (c) Determine the required cooling water flow rate, in \(\mathrm{kg} / \mathrm{s}\), to keep the cooling water temperature rise in the condenser to \(10^{\circ} \mathrm{C}\). Assume \(c_{p}=4.18 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\) for cooling water (d) Determine the net power output and the thermal efficiency of the plant.

Short answer

Answer: The primary objective when analyzing an ideal Rankine steam cycle with two closed feedwater heaters is to determine the operating parameters of the cycle, such as the mass flow rate of the working fluid, mass fraction extraction for the feedwater heaters, required cooling water flow rate, net power output, and thermal efficiency of the plant. This is achieved by sketching the T-s diagram, identifying the fluid states during each process, and applying mass and energy balance equations to solve for the desired parameters.

Step-by-step solution

Sketch the T-s Diagram for the Ideal Cycle

For the given ideal Rankine Cycle, the working fluid, steam, goes through a series of processes: 1. Isentropic compression in the pump (1-2) 2. Heating and evaporation in the boiler (2-3) 3. Isentropic expansion in the turbine (3-4) 4. Condensation in the condenser (4-1) Now that we know the processes, we can sketch the T-s diagram. Place points 1-4 for each corresponding state and connect them accordingly according to the process order.

Determine the fraction of mass, z, for the closed feedwater heater operating at the 245 kPa extraction pressure.

As given, the fraction of mass, y, that is extracted for the feedwater heater operating at 1400 kPa is 0.1446. Using steam tables, determine the properties at each state (1-4). Then apply mass and energy balance equations for the process to calculate the fraction of mass, z, extracted for the closed feedwater heater operating at the 245 kPa extraction pressure.

Determine the required cooling water flow rate, in kg / s, to keep the cooling water temperature rise in the condenser to 10 °C.

To find the cooling water flow rate, we must first find the heat transfer rate in the condenser. It can be calculated by finding the difference in enthalpy between state 4 and state 1. Once we have the heat transfer rate, we can determine the cooling water flow rate using the given specific heat capacity of the cooling water cp = 4.18 kJ/kg·K and the allowed temperature rise in the condenser.

Determine the net power output and the thermal efficiency of the plant.

The net power output can be calculated by finding the work done by the turbine and the work required by the pump. The difference between these two values, multiplied by the mass flow rate of the steam gives the net power output. To find the thermal efficiency, we then calculate the total heat transfer rate in the boiler and divide the net power output by the total heat transfer rate, then convert the result into a percentage. In summary, start by sketching the T-s diagram for the ideal Rankine Cycle and identifying the working fluid's states during each process. Then, apply mass and energy balance equations to each process to determine the required parameters for the given tasks, such as the fraction of mass z, the cooling water flow rate, net power output, and the thermal efficiency of the plant.

Key concepts

Understanding the T-s Diagram

The Temperature-entropy (T-s) diagram is a graphical representation of a thermodynamic cycle and is a fundamental tool in understanding the Rankine steam cycle. On the T-s diagram, the horizontal axis represents entropy (s) and the vertical axis is temperature (T), allowing us to visualize changes in state points throughout the cycle. Each segment of the cycle corresponds to a process in the Rankine cycle: the vertical line 2-3 represents the heating and evaporation in the boiler, while the nearly vertical line 3-4 shows isentropic (constant entropy) expansion in the turbine. The condensation in the condenser is depicted by the line 4-1, and the isentropic compression by the pump is the line 1-2.

For students, to properly sketch a T-s diagram, it's essential to accurately plot each state point and connect them reflecting the actual processes. Labeling the points of extraction for feedwater heaters as done in the modified Rankine cycle is also crucial as it gives a complete view on where the cycle deviates from the ideal cycle due to real-world applications like feedwater heating.

Applying Mass and Energy Balance

The mass and energy balance principle is critical to solving problems related to the Rankine cycle. It involves ensuring that mass and energy are conserved throughout the system. The mass flow entering the boiler must equal the mass flow exiting the turbine, but in systems with feedwater heaters, the mass balance becomes slightly more complex.

To find the fraction of mass extracted, the mass entering and exiting each feedwater heater must be considered, adhering to the conservation of mass principle. Energy balance is also crucial when evaluating enthalpies at different state points, as the energy added in the boiler must equal the sum of the turbine work, the energy removed in the condenser, and the work input of the pump, again considering the heat exchange in feedwater heaters.

Determining Thermal Efficiency

Thermal efficiency is the measure of how effectively a power plant converts heat into work. It's defined as the ratio of the net work output of the cycle to the total heat input, expressed as a percentage. The higher the thermal efficiency, the better the plant is at converting thermal energy into electrical energy, which reflects on the cost-effectiveness and environmental impact of the power generation process.

For students to determine the thermal efficiency, they need to calculate the net power output of the plant and the amount of heat added in the boiler. The net power output is the work done by the turbine minus the work consumed by the pump. Using the Rankine cycle's state points, calculate enthalpies to find these work quantities, and make sure to consider the mass flow rate of the steam.

Calculating Cooling Water Flow Rate

The cooling water flow rate is an essential parameter in the operation of a Rankine cycle's condenser. It's the amount of water needed to absorb the waste heat from the steam after it exits the turbine, effectively condensing it back to water.

The flow rate is determined based on the heat transfer rate in the condenser and the temperature increase that the cooling water can undergo, which is typically limited to prevent thermal pollution and to maintain efficiency. To calculate this flow rate, the energy extracted by the cooling water (which equals the heat lost by the working fluid) is divided by the product of the specific heat capacity of water and the allowed temperature rise.

Exploring Isentropic Processes

Isentropic processes are ideal processes that occur at constant entropy. These are important in the thermodynamic analysis as they represent no entropy change, thus no irreversibility or loss of work potential. In practice, a real process cannot be perfectly isentropic, but understanding it allows engineers to gauge the efficiency of real processes.

In the Rankine cycle, the expansion in the turbine and the compression in the pump are often assumed isentropic for simplification. This assumption permits the easy calculation of work done by or on the fluid using the enthalpy values at the beginning and end of these processes. However, students must remember that in real-world conditions, some efficiency loss occurs making these processes isentropic to a good approximation rather than perfectly isentropic.