Question

Water is the working fluid in an ideal Rankine cycle. Saturated vapor enters the turbine at \(18 \mathrm{MPa}\). The condenser pressure is \(6 \mathrm{kPa}\). Determine (a) the net work per unit mass of steam flowing, in \(\mathrm{kJ} / \mathrm{kg}\). (b) the heat transfer to the steam passing through the boiler, in \(\mathrm{kJ}\) per \(\mathrm{kg}\) of steam flowing. (c) the thermal efficiency. (d) the heat transfer to cooling water passing through the condenser, in kJ per kg of steam condensed.

Short answer

Net work per unit mass = h1 - h2 - (h4 - h3). Heat transfer to steam in boiler = h1 - h4. Thermal efficiency = (h1 - h2 - (h4 - h3))/(h1 - h4). Heat transfer to cooling water = h2 - h3.

Step-by-step solution

- State Identification for the Inlet

Identify the state at the inlet of the turbine (State 1). Given that saturated vapor enters the turbine at a pressure of 18 MPa, from the steam tables, find the enthalpy and entropy at this state: State 1: Pressure, P1 = 18 MPa Saturated vapor => h1 and s1.

- State Identification for the Outlet

Identify the state at the outlet of the turbine (State 2). Given the condenser pressure is 6 kPa, find the enthalpy at this state using the fact that the enthalpy at the outlet corresponds to an isentropic expansion process: State 2: Pressure, P2 = 6 kPa Isentropic Expansion => s2 = s1 From the steam tables, interpolate to find h2 with the given s1 and P2.

- State Identification for Boiler Inlet

Identify the state at the boiler inlet (State 3). Assume the fluid exits the condenser as saturated liquid. Thus, State 3: Pressure, P3 = Condenser Pressure = 6 kPa Saturated liquid => h3 and s3.

- State Identification for Boiler Outlet

Identify the state at the boiler outlet (State 4). Use the given information to determine that the boiler heats this water back to saturated vapor. Thus, State 4: Pressure, P4 = Boiler Pressure = 18 MPa Saturated vapor => h4 and s4.

- Net Work Calculation

Determine the net work per unit mass of steam flowing:The turbine work per unit mass, W_t = h1 - h2 The pump work per unit mass can be approximated by the increase in enthalpy of the working fluid from State 3 to State 4: W_p = h4 - h3 Hence, the net work per unit mass, W_net = W_t - W_p.

- Heat Transfer in Boiler

Calculate the heat transfer to the steam passing through the boiler:Q_in = h1 - h4.

- Thermal Efficiency Calculation

Determine the thermal efficiency of the cycle:η = frac{W_net}{Q_in} .

- Heat Transfer in Condenser

Determine the heat transfer to the cooling water passing through the condenser:Q_out = h2 - h3.

Key concepts

Thermodynamic cycles

The Rankine cycle is a type of thermodynamic cycle used in steam power plants. It involves water undergoing various phase changes to convert heat energy into mechanical work. Understanding this cycle is crucial for analyzing power generation systems. The cycle consists of four main processes:
1. Isentropic expansion in the turbine.
2. Isobaric heat rejection in the condenser.
3. Isentropic compression in the pump.
4. Isobaric heat addition in the boiler.
By studying this cycle, you can understand how energy is transferred and transformed in power plants, enabling you to analyze their efficiency and performance.

Isentropic process

An isentropic process is one where the entropy remains constant. In the Rankine cycle, the expansion in the turbine and compression in the pump are considered nearly isentropic. This simplifies the calculations and helps in understanding ideal performance.
During isentropic expansion in the turbine, the steam's pressure drops while its volume increases, doing work on the turbine blades. Conversely, in isentropic compression within the pump, water is pressurized to the boiler level without significant temperature change. To find specific enthalpies during these processes, you need to use steam tables, looking up values for given pressures while keeping entropy constant.

Thermal efficiency

Thermal efficiency is a measure of how effectively a thermodynamic cycle converts heat into work. For the Rankine cycle, it's calculated as the ratio of the net work output to the heat input:
\( \text{Thermal Efficiency} = \frac{W_{\text{net}}}{Q_{\text{in}}} \).
Where:
\( W_{\text{net}} \) is the net work done by the cycle and
\( Q_{\text{in}} \) is the heat added in the boiler.
The higher the thermal efficiency, the better the power plant is at converting heat from fuel into usable work. In practice, improving thermal efficiency can be achieved by increasing the temperature and pressure of the steam entering the turbine or by reducing losses at various stages of the cycle.

Heat transfer calculations

Heat transfer calculations are essential in understanding the energy exchanges within the Rankine cycle. These calculations determine how much heat is added in the boiler and removed in the condenser:
- The heat added in the boiler is determined by \( Q_{\text{in}} = h1 - h4 \). This is the energy required to turn water from a liquid to a vapor.
- The heat rejected in the condenser is given by \( Q_{\text{out}} = h2 - h3 \). This is the energy removed to condense the steam back into a liquid.
Accurate heat transfer calculations ensure efficient plant operation and help identify where energy losses might occur.

Steam tables usage

Steam tables are crucial for solving problems related to the Rankine cycle. They provide the necessary thermodynamic properties of water and steam at various pressures and temperatures. Here's how to use them:
- To find specific enthalpies (h) and entropies (s), locate the pressure or temperature values in the tables.
- For example, to find the enthalpy at the turbine inlet (State 1) where the steam is at 18 MPa and is a saturated vapor, look up the corresponding h and s values.
- If the state point lies between table values, interpolation is required.
Proper use of steam tables allows you to determine essential properties accurately, making your Rankine cycle analysis more reliable.