Question

A steam power plant operates on the simple ideal Rankine cycle between the pressure limits of \(10 \mathrm{kPa}\) and \(5 \mathrm{MPa},\) with a turbine inlet temperature of \(600^{\circ} \mathrm{C} .\) The rate of heat transfer in the boiler is \(300 \mathrm{kJ} / \mathrm{s}\). Disregarding the pump work, the power output of this plant is \((a) 93 \mathrm{kW}\) \((b) 118 \mathrm{kW}\) \((c) 190 \mathrm{kW}\) \((d) 216 \mathrm{kW}\) \((e) 300 \mathrm{kW}\)

Short answer

a) 28 kW b) 123 kW c) 205 kW d) 657 kW

Step-by-step solution

Find the enthalpy values at the turbine entrance and exit

Use the pressure limits and turbine inlet temperature to find the steam saturation conditions. From the steam tables, find the enthalpy values (\(h_2\)) at \(5 \mathrm{MPa}\) and \(600^{\circ}\mathrm{C}\) (turbine inlet) and (\(h_3\)) at \(10\mathrm{kPa}\) (turbine outlet condition). Make sure to consider the given states to refer to steam tables correctly.

Find the work output per unit mass of steam

Calculate the work output per unit mass of steam going through the turbine using the enthalpy values from Step 1. The work output per unit mass (W_turbine) is equal to the difference in enthalpies at the entrance and exit of the turbine: \(W_{turbine}=h_2-h_3\).

Calculate the mass flow rate of steam

Using the given rate of heat transfer in the boiler (\(Q_{boiler}=300\mathrm{kJ/s}\)) and the enthalpy difference between the boiler and condenser (essentially the change in enthalpy during the Rankine cycle), calculate the mass flow rate of steam (\(\dot{m}\)) using the formula \(\dot{m} = \frac{Q_{boiler}}{h_1 - h_4}\). Note that \(h_4\) is the liquid enthalpy at \(10 \mathrm{kPa}\) (the pressure of the exit of the pump) and \(h_1\) is the enthalpy at the exit of the boiler (equals to the entrance of the turbine, \(h_2\)). Hence, the mass flow rate can be calculated as \(\dot{m} = \frac{300}{h_2 - h_4}\).

Calculate the power output

Multiply the mass flow rate of steam obtained in Step 3 by the work output per unit mass calculated in Step 2 to obtain the total power output of the plant: \(Power Output = \dot{m} \times W_{turbine}\). Compare this value to the given options and select the closest one as the correct answer.

Key concepts

Steam Power Plant

A steam power plant is a thermal power station that converts heat energy into electrical energy by using steam as the working fluid. The heart of the steam power plant is the Rankine cycle, which is an idealized thermodynamic cycle. This cycle involves four key processes: water is pumped to high pressure, then boiled into steam in a boiler, the steam is expanded through a turbine generating power, and finally, it is condensed back into water in a condenser.

The efficiency of the steam power plant hinges on the temperature and pressure at which steam is produced in the boiler and then expanded in the turbine. In a typical scenario, such as the textbook exercise, steam is generated at high pressure and temperature and then expanded to low pressure in the turbine, which turns the thermal energy into mechanical energy, and subsequently into electrical energy via a generator.

Enthalpy Calculation

Enthalpy, in thermodynamics, is a quantity that represents the total heat content of a system. It's symbolized by the letter H and is a combination of the system's internal energy and the product of its pressure and volume. Calculation of enthalpy is crucial in various processes within a steam power plant, especially for determining the state of the steam at different stages in the Rankine cycle.

In our exercise, the enthalpy values at the turbine inlet and outlet need to be obtained from steam tables or thermodynamic software by considering the pressure and temperature parameters. These values are essential for calculating the work generated by the turbine since the work is equivalent to the decrease in enthalpy of the steam as it expands through the turbine.

Mass Flow Rate

The mass flow rate is a measure of the amount of mass moving through a given point per unit time. In a steam power plant, it refers to the mass of steam flowing through the boiler and turbine per second. It's typically measured in kilograms per second (kg/s).

Determining the mass flow rate, symbolized as \( \dot{m} \), is essential in calculating the total power output of the plant because it directly relates the amount of steam used to the heat added in the boiler. By knowing the rate of heat transfer in the boiler and the enthalpy change as the steam flows through the boiler and the turbine, we can calculate the mass flow rate using the relationship \( \dot{m} = \frac{Q_{boiler}}{h_1 - h_4} \).

Heat Transfer in Boiler

In a steam power plant, heat transfer in the boiler is a major phase where water is transformed into steam. This phase is where most of the energy input into the cycle occurs. The amount of heat transferred, typically measured in kilojoules per second (kJ/s), also known as the rate of heat transfer, is responsible for the boiling of water into steam at high pressure and temperature.

The exercise provided states the rate of heat transfer in the boiler as 300 kJ/s. This information, along with the enthalpy values, allows us to calculate the total heat added to the water in the Rankine cycle, which is instrumental in later determining other parameters such as the mass flow rate of steam and the overall power output of the plant.

Power Output Calculation

The terminal goal of a steam power plant is to convert the thermal energy produced in the boiler into electrical energy, which is reflected as the power output. Calculating the power output involves finding the product of the mass flow rate of steam and the work done per unit mass of steam by the turbine.

In the exercise, once the enthalpy change and the mass flow rate are known, we multiply them together (\( Power Output = \dot{m} \times W_{turbine} \) to find the total power output. It's a direct measure of the plant's capability to generate electricity and is expressed in kilowatts (kW) or megawatts (MW). The power output shows the efficiency and performance of the power plant, indicating how well the plant converts heat energy into electrical energy.