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Chapter 1: Problem 29

Water is the working fluid in an ideal Rankine cycle with reheat. Superheated vapor enters the turbine at \(8 \mathrm{MPa}\), \(440^{\circ} \mathrm{C}\), and the condenser pressure is \(8 \mathrm{kPa}\). Steam expands through the first- stage turbine to \(0.5 \mathrm{MPa}\) and then is reheated to \(440^{\circ} \mathrm{C}\). Determine for the cycle (a) the rate of heat addition, in \(\mathrm{kJ}\) per \(\mathrm{kg}\) of steam entering the first-stage turbine. (b) the thermal efficiency. (c) the rate of heat transfer from the working fluid passing through the condenser to the cooling water, in kJ per \(\mathrm{kg}\) of steam entering the first-stage turbine.

Short Answer

Expert verified
Heat addition per kg: sum of enthalpies from heating and reheating. Efficiency: calculated from difference in added and rejected heat. Heat rejection: enthalpy difference in condenser.

Step by step solution

01

Identify Inlet Conditions and Extract Properties

Look up the properties of steam at the entrance of the first-stage turbine: Inlet Pressure: 8 MPaInlet Temperature: 440掳CFrom steam tables or a Mollier diagram, find the enthalpy (\(h_1\)) and entropy (\(s_1\)).
02

Calculate State After First Expansion

Steam expands isentropically to 0.5 MPa. So,\(s_2 = s_1\). From steam tables, find the enthalpy (\(h_2s\)) and entropy at 0.5 MPa for the same entropy value.
03

Identify Conditions After Reheat

After reheat, pressure is 0.5 MPa and temperature is raised to 440掳C. Find the enthalpy (\(h_3\)) at these conditions.
04

Calculate State After Second Expansion

Steam expands isentropically again to the condenser pressure of 8 kPa. \(s_4 = s_3\). Find the enthalpy (\(h_4s\)) and entropy at 8 kPa for the same entropy value.
05

Identify Condenser Exit Conditions

After expansion, steam enters the condenser. The exit condition of the condenser can be assumed as saturated liquid at 8 kPa. Find the enthalpy (\(h_5\)) at this state.
06

Calculate Heat Added during Initial Heating and Reheat

Calculate heat added during the initial heating process:\[q_{in1} = h_1 - h_3\]Calculate heat added during reheat process:\[q_{in2} = h_3 - h_2\]The total heat added is:\[q_{in} = q_{in1} + q_{in2}\]
07

Calculate Thermal Efficiency

Thermal efficiency can be calculated using:\[畏 = 1 - \frac{q_{out}}{q_{in}}\]Where the heat rejected (\(q_{out}\)) is:\[q_{out} = h_5 - h_4\]
08

Heat Transfer from Working Fluid in Condenser

Heat transfer from the working fluid passing through the condenser is equal to the heat rejected:\[q_{out} = h_5 - h_4\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

isotropic expansion
Isotropic expansion, or more correctly known as isentropic expansion, refers to a process during which entropy remains constant. In the context of the Rankine cycle, steam undergoes this type of expansion in the turbine. This means the process is ideally adiabatic (no heat transfer) and reversible. During the isentropic expansion from a high pressure to a lower pressure (like from 8 MPa to 0.5 MPa), the entropy stays the same. This concept simplifies the calculation of steam properties at different stages since we only need the initial entropy to find the properties at the final stage.
thermal efficiency
Thermal efficiency is a measure of how well a cycle converts heat into work. For the Rankine cycle, it is given by \[ \eta = 1 - \frac{q_{out}}{q_{in}} \], where \[ q_{in} \] is the total heat added to the system and \[ q_{out} \] is the heat rejected in the condenser. Maximizing thermal efficiency is a key objective in designing efficient power plant cycles. In our problem, \[ q_{in} \] includes heat added in both the boiler and the reheat stages, while \[ q_{out} \] is primarily from the condenser. This way, we can see how much of the input energy is converted into useful work to drive turbines and generators.
condenser heat transfer
The condenser in a Rankine cycle removes heat from the steam to turn it back into liquid water. This process occurs at constant low pressure. The amount of heat removed in the condenser can be calculated as \[ q_{out} = h_5 - h_4 \], where \[ h_4 \] is the enthalpy after the second turbine expansion and \[ h_5 \] is the enthalpy of the saturated liquid at condenser pressure. Efficient removal of this heat is critical because it affects the overall thermal efficiency of the cycle. The lower the condenser temperature, the more heat can be removed, improving efficiency.
reheat process
The reheat process occurs in between the two turbine stages. After partial expansion in the first turbine, the steam is reheated to a higher temperature (440掳C in our problem) at a constant pressure of 0.5 MPa. This increases the steam's enthalpy and prevents the steam from becoming too wet (having too many water droplets) during the final expansion to the condenser. The additional heat added during reheat is given by \[ q_{in2} = h_3 - h_2 \], raising the efficiency and work output of the cycle.
enthalpy calculations
Enthalpy, a measure of total energy in the system, plays a vital role in analyzing thermodynamic cycles. You need enthalpy values at various stages of the Rankine cycle to calculate heat added (\[ q_{in} \]) and heat rejected (\[ q_{out} \]). For example, \[ h_1 \] is the enthalpy at the entrance of the first turbine, and \[ h_3 \] is after reheating. Tools like steam tables help to extract these values based on pressure and temperature. Accurately calculating enthalpy at each state is essential for determining the work done and thus, the cycle's efficiency.

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Most popular questions from this chapter

Water is the working fluid in an ideal regenerative Rankine cycle. Superheated vapor enters the turbine at \(8 \mathrm{MPa}\), \(440^{\circ} \mathrm{C}\), and the condenser pressure is \(8 \mathrm{kPa}\). Steam expands through the first- stage turbine to \(0.5 \mathrm{MPa}\), where some of the steam is extracted and diverted to an open feedwater heater operating at \(0.5 \mathrm{MPa}\). The remaining steam expands through the second-stage turbine to the condenser pressure of \(8 \mathrm{kPa}\). Saturated liquid exits the feedwater heater at \(0.5 \mathrm{MPa}\). Determine for the cycle (a) the rate of heat addition, in kJ per \(\mathrm{kg}\) of steam entering the first-stage turbine. (b) the thermal efficiency. (c) the rate of heat transfer from the working fluid passing through the condenser to the cooling water, in \(\mathrm{kJ}\) per \(\mathrm{kg}\) of steam entering the first-stage turbine.

\( The absolute pressure inside a tank is \)0.4\( bar, and the surrounding atmospheric pressure is \)98 \mathrm{kPa}\(. What reading would a Bourdon gage mounted in the tank wall give, in \)\mathrm{kPa} ?$ Is this a gage or vacuum reading?

In a steam power station working on the ideal Rankine cycle with regeneration, steam enters the turbine at 150 bar, \(600^{\circ} \mathrm{C}\). One open feedwater heater is used in the plant. Some steam from the turbine enters the open feedwater heater at a pressure of 12 bar. The pressure in the condenser is \(0.1\) bar. Determine the thermal efficiency of the cycle and the fraction of steam extracted from the turbine.

An open storage tank is placed at the top of a building. The tank contains water up to a depth of \(1.5 \mathrm{~m}\). Calculate the pressure at the bottom of the tank. It is given that atmospheric pressure is \(101.3 \mathrm{kPa}\) and density of water is \(1000 \mathrm{~kg} / \mathrm{m}^{3}\)

Early commercial vapor power plants operated with turbine inlet conditions of about 12 bar and \(200^{\circ} \mathrm{C}\). Plants are under development today that can operate at over 34 MPa, with turbine inlet temperatures of \(650^{\circ} \mathrm{C}\) or higher. How have steam generator and turbine designs changed over the years to allow for such increases in pressure and temperature? Discuss.
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