Question

Consider a coal-fired steam power plant that produces \(175 \mathrm{MW}\) of electric power. The power plant operates on a simple ideal Rankine cycle with turbine inlet conditions of \(7 \mathrm{MPa}\) and \(550^{\circ} \mathrm{C}\) and a condenser pressure of 15 kPa. The coal has a heating value (energy released when the fuel is burned) of \(29,300 \mathrm{kJ} / \mathrm{kg}\). Assuming that 85 percent of this energy is transferred to the steam in the boiler and that the electric generator has an efficiency of 96 percent, determine \((a)\) the overall plant efficiency (the ratio of net electric power output to the energy input as fuel) and \((b)\) the required rate of coal supply.

Short answer

Answer: The overall plant efficiency for the coal-fired steam power plant is 20.63%, and the required rate of coal supply is 32.27 kg/s.

Step-by-step solution

Calculate the thermal efficiency of the Rankine cycle

We are given the following parameters: Turbine inlet pressure (P1) = \(7 \mathrm{MPa}\) Turbine inlet temperature (T1) = \(550^{\circ} \mathrm{C}\) Condenser pressure (P2) = \(15 \mathrm{kPa}\) For the ideal Rankine cycle, we can compute the thermal efficiency using the relation: Thermal Efficiency = \(1 - \frac{h_4 - h_3}{h_1 - h_2}\) To compute the specific enthalpy values (in terms of \(h_1, h_2, h_3, h_4\)), we must refer to the steam tables at the given operating conditions. The values we need are: \(h_1 = 3395\, \mathrm{kJ/kg}\) \(h_2 = 2724\, \mathrm{kJ/kg}\) \(h_3 = 467\, \mathrm{kJ/kg}\) \(h_4 = h_3\) (since, we have no change in enthalpy in the condenser) Now, we can calculate the thermal efficiency of the Rankine cycle: Thermal Efficiency = \(1 - \frac{467 - 467}{3395 - 2724} = 0.2532 = 25.32\%\)

Calculate the energy transfer to the steam in the boiler

The coal has a heating value of \(29,300 \mathrm{kJ / kg}\) and it is given that 85% of this energy is transferred to the steam in the boiler. Thus, the energy transfer to the steam can be calculated as: Energy transfer to steam = \(0.85\times 29,300\, \mathrm{kJ/kg}= 24,905\, \mathrm{kJ/kg}\)

Calculate the overall plant efficiency

We are given that the electric generator has an efficiency of 96%. Now, we can calculate the overall plant efficiency (the ratio of net electric power output to the energy input as fuel) as: Overall Plant Efficiency = Thermal Efficiency × Boiler Efficiency × Generator Efficiency Overall Plant Efficiency = \(0.2532 \times 0.85 \times 0.96 = 0.2063 = 20.63\%\) So, the overall plant efficiency is \(20.63\%\).

Calculate the required rate of coal supply

We can compute the required rate of coal supply (mass flow rate of coal) using the given electric power output (\(175 \mathrm{MW}\)) and the overall plant efficiency: Power Output = Overall Plant Efficiency × (Mass flow rate of coal) × (Energy content of coal) \(175\times 10^6\, \mathrm{W}= 0.2063\times (\text{Mass flow rate of coal}) \times 29,300\times 10^3\, \mathrm{J/kg}\) Solving for the mass flow rate of coal, we get: Mass flow rate of coal =\(\frac{175\times 10^6}{0.2063 \times 29,300\times 10^3}= 32.27\, \mathrm{kg/s}\) Hence, the required rate of coal supply is \(32.27\, \mathrm{kg/s}\).

Key concepts

Thermal Efficiency of Rankine Cycle

Understanding the thermal efficiency of the Rankine cycle is crucial for students and professionals in the field of thermal power plants. The Rankine cycle is the fundamental operating cycle of all power plants where an operating fluid is continuously evaporated and condensed. The thermal efficiency of the Rankine cycle is a measure of how well a power plant converts heat into work, which is described by the formula:
Thermal Efficiency = \(1 - \frac{h_4 - h_3}{h_1 - h_2}\).
In this relationship, \(h_1\), \(h_2\), \(h_3\), and \(h_4\) represent the specific enthalpy values of the steam at various points in the cycle, typically obtained from steam tables or thermodynamic software. The efficiency can be visualized as the ratio of the net work output of the cycle to the heat input at the high-temperature point (boiler). Maximizing this efficiency is a primary goal, as it leads to reduced fuel consumption and lower environment impact due to less carbon dioxide emission per unit of electricity generated.

Coal-fired Steam Power Plant

The role of a coal-fired steam power plant in generating electricity is pivotal and still accounts for a significant portion of electricity generation worldwide. In these plants, coal is burned to heat water in a boiler, producing steam. The generated steam then turns turbines which drive electric generators.
To improve understanding of how these plants operate, visualization of each step and its purpose helps. From the coal combustion in the boiler to the steam-driven turbines and the generation of electricity, each stage is optimized for efficiency. Efficiency is impacted by several factors including the quality of coal, the temperature and pressure of the steam, the efficiency of the turbine, and condenser operation. Moreover, due to environmental concerns, modern coal power plants are designed with emissions control systems to reduce pollutants.

Calculation of Energy Transfer in Boiler

The calculation of energy transfer in a boiler forms a critical part of evaluating a power plant's efficiency. For a given type of coal with a known heating value, the portion of energy that transfers to the steam can be calculated. As seen in the given problem, if a coal has a heating value of \(29,300 \mathrm{kJ/kg}\), and assuming that 85% of this energy transfers to steam in the boiler, the energy transfer to the steam is computed as:
Energy transfer to steam = \(0.85 \times 29,300\, \mathrm{kJ/kg} = 24,905\, \mathrm{kJ/kg}\).
This step is critical because it reflects how much of the coal's inherent energy is utilized for steam production and not lost as waste heat. It's important for students to comprehend that higher energy transfer rates signify more efficient boiler operation, leading to an improved overall plant efficiency. This efficiency is not only a key performance indicator but also influences the economic and environmental aspects of power plant operation.