Question

Steam enters the first turbine stage of a vapor power cycle with reheat and regeneration at \(32 \mathrm{MPa}, 600^{\circ} \mathrm{C}\), and expands to \(8 \mathrm{MPa}\). A portion of the flow is diverted to a closed feedwater heater at \(8 \mathrm{MPa}\), and the remainder is reheated to \(560^{\circ} \mathrm{C}\) before entering the second turbine stage. Expansion through the second turbine stage occurs to \(1 \mathrm{MPa}\), where another portion of the flow is diverted to a second closed feedwater heater at \(1 \mathrm{MPa}\). The remainder of the flow expands through the third turbine stage to \(0.15 \mathrm{MPa}\), where a portion of the flow is diverted to an open feedwater heater operating at \(0.15 \mathrm{MPa}\), and the rest expands through the fourth turbine stage to the condenser pressure of \(6 \mathrm{kPa}\). Condensate leaves each closed feedwater heater as saturated liquid at the respective extraction pressure. The feedwater streams leave each closed feedwater heater at a temperature equal to the saturation temperature at the respective extraction pressure. The condensate streams from the closed heaters each pass through traps into the next lower-pressure feedwater heater. Saturated liquid exiting the open heater is pumped to the steam generator pressure. If each turbine stage has an isentropic efficiency of \(85 \%\) and the pumps operate isentropically (a) sketch the layout of the cycle and number the principal state points. (b) determine the thermal efficiency of the cycle. (c) calculate the mass flow rate into the first turbine stage, in \(\mathrm{kg} / \mathrm{h}\), for a net power output of \(500 \mathrm{MW}\).

Short answer

Sketch the cycle layout, calculate state enthalpies, account for isentropic efficiencies, evaluate work and heat, and determine thermal efficiency and mass flow rate for the specified conditions.

Step-by-step solution

- Sketch the Layout of the Cycle

Create a schematic diagram of the vapor power cycle. Label all key components such as the steam generator, turbines, feedwater heaters (both closed and open), and the condenser. Number the principal state points at significant junctions in the cycle based on the description provided.

- Applying the First Law of Thermodynamics

Apply the first law of thermodynamics to each component in the cycle to establish the energy balance equations. For turbines and heaters, use the relation: \[ \text{Energy loss/gain} = \text{mass flow rate} \times \text{specific enthalpy change} \]

- Calculate State Properties from Steam Tables

Determine specific enthalpies and entropies at key state points using steam tables or Mollier charts, given the pressures and temperatures at each state. Include state points after expansion and reheat stages.

- Accounting for Isentropic Efficiencies

Since each turbine stage has an isentropic efficiency of 85%, use the formula: \[ h_{actual} = h_{isentropic} + \frac{h_{actual} - h_{isentropic}}{\text{isentropic efficiency}} \] to adjust the enthalpies after each turbine expansion.

- Heat Addition and Removal Calculations

Calculate the heat added in the steam generator and reheaters, and the heat removed in the condenser using enthalpy values at appropriate state points: \[ Q_{in} = m \times (h_{in} - h_{out}) \]

- Determine Work Done by Each Component

Calculate the work done by each turbine stage and pumps using: \[ \text{Work}_{turbine} = m \times (h_{in} - h_{out}) \] \[ \text{Work}_{pump} = m \times (h_{out} - h_{in}) \]

- Calculate Net Work Output

Sum up the work outputs and subtract the work input from pumps to get the net work output: \[ W_{net} = \text{Work}_{turbine} - \text{Work}_{pump} \] Ensure to account for all turbines and pumps in the cycle.

- Calculate Thermal Efficiency

Use the formula for thermal efficiency: \[ \text{Thermal Efficiency} = \frac{W_{net}}{Q_{in}} \] to find the efficiency of the cycle.

- Determine Mass Flow Rate

Given the net power output, use the relationship: \[ \text{Power} = \text{mass flow rate} \times W_{net} \] to solve for the mass flow rate (m) into the first turbine stage.

Key concepts

Isentropic Efficiency

Isentropic efficiency is a measure of how close a real-life process comes to an ideal isentropic process, which is both adiabatic and reversible. It is particularly relevant in turbine and pump efficiencies. In the context of our vapor power cycle, each turbine stage has an isentropic efficiency of 85%. To account for this, we use the formula: \[ h_{actual} = h_{isentropic} + \frac{h_{actual} - h_{isentropic}}{\text{isentropic efficiency}} \] The isentropic efficiency helps bridge the gap between the ideal theoretical performance and the actual performance, which includes inevitable losses. It means the actual enthalpy change during expansion or compression is higher than it would be in an ideal scenario. This has to be factored into the energy balance equations to make accurate calculations for work and heat.

First Law of Thermodynamics

The first law of thermodynamics, often called the law of energy conservation, states that energy cannot be created or destroyed, only transformed. For the vapor power cycle, this means we need to account for all the energy changes in the system. The relevant formula is: \[ \text{Energy loss/gain} = \text{mass flow rate} \times \text{specific enthalpy change} \] We apply this law to each component in the cycle:

• Turbine stages convert thermal energy to mechanical work.
• Feedwater heaters adjust the temperature of the water.
• Condensers remove energy and return the steam to liquid.

Using this principle, we can set up energy balance equations for every component to ensure all energy transformations are accounted for, allowing us to determine net work output and thermal efficiency accurately.

Heat Addition and Removal

In a vapor power cycle, heat addition and removal are crucial parts. Heat is added during the steam generation and reheating stages, and removed in the condenser. The formula for heat addition or removal is: \[ Q_{in} = m \times (h_{in} - h_{out}) \] Where m is the mass flow rate and the specific enthalpy change \( (h_{in} - h_{out}) \) depends on the states of the steam. By carefully calculating these values at each state point, we can determine how much energy is entering and leaving the system. This will also help in finding out the total work done and the thermal efficiency. Proper heat management is vital to maximize the cycle's performance and efficiency.

Thermal Efficiency

Thermal efficiency is a measure of how well the vapor power cycle converts heat into work. It is calculated using the formula: \[ \text{Thermal Efficiency} = \frac{W_{net}}{Q_{in}} \] Here, \( W_{net} \) is the net work output (total work done by turbines minus work consumed by pumps), and \( Q_{in} \) is the heat added to the cycle. A higher thermal efficiency means a larger proportion of the input heat is converted into useful work. Knowing the efficiency helps in evaluating the performance of the cycle and determining potential areas for improvement. Efficient cycles reduce energy losses and improve overall performance.

Mass Flow Rate

The mass flow rate is the amount of steam entering the first turbine stage per unit time, often measured in \( \text{kg} / \text{h} \). Given a net power output requirement (e.g., 500 MW), we can find the necessary mass flow rate. The formula relating power, mass flow rate, and work is: \[ \text{Power} = \text{mass flow rate} \times W_{net} \] Rearranging this formula allows us to solve for the mass flow rate: \[ \text{mass flow rate} = \frac{\text{Power}}{W_{net}} \] This calculation ensures that the cycle can meet desired power outputs by adjusting the flow of steam. Correctly calculating the mass flow rate is crucial for designing and operating efficient power systems.