Question

Power company "E" in the Netherlands operates a fluidized bed combustion-based boiler with wood residues as fuel. The plant contains a simple steam turbine. The steam conditions at the inlet of the turbine are \(525^{\circ} \mathrm{C}\) and 100 bar. Assume that condensation of the steam takes place at a temperature of \(20^{\circ} \mathrm{C}\) and that isentropic expansion of steam occurs in the turbine. a. What is the specific power \(\left(\mathrm{kJ} \cdot \mathrm{kg}^{-1}\right)\) of the turbine expansion process? b. If the power plant generates \(25 \mathrm{MW}_{\mathrm{e}}\) and water pump work can be neglected, what is the mass flow rate of steam through the turbine? c. What assumptions have you made for these calculations?

Short answer

What assumptions have been made in the calculations? The specific power of the turbine expansion process is approximately 3411 kJ/kg, and the mass flow rate of steam through the turbine is approximately 7.33 kg/s. The assumptions made include ideal behavior, constant entropy expansion, neglecting water pump work, and 100% conversion efficiency of mechanical power into electrical power.

Step-by-step solution

Finding specific enthalpy of steam at the inlet and outlet conditions

For the given steam inlet conditions of 525°C and 100 bar, we need to find the specific enthalpy (\(h_{in}\)) of the steam. As we are assuming isentropic expansion, we will have the same entropy in the outlet (condensation) as at the inlet. We can use the steam tables to find these values. For the given steam condensation conditions of 20°C, we need to find the specific enthalpy (\(h_{out}\)) in this state. Again, we can use the steam tables for this.

Calculating the specific power of the turbine expansion process

The specific power of the turbine expansion process can be calculated as the difference between the inlet and outlet specific enthalpy values. Specific Power = \(h_{in} - h_{out}\) Using the steam tables, we find \(h_{in} \approx 3495 \, \mathrm{kJ} \cdot \mathrm{kg}^{-1}\) and \(h_{out} \approx 84 \, \mathrm{kJ} \cdot \mathrm{kg}^{-1}\). Plugging these values, we have: Specific Power = \(3495 \, \mathrm{kJ} \cdot \mathrm{kg}^{-1} - 84 \, \mathrm{kJ} \cdot \mathrm{kg}^{-1} \approx 3411 \, \mathrm{kJ} \cdot \mathrm{kg}^{-1}\)

Calculating the mass flow rate of steam through the turbine

The electrical output of the power plant is given as 25 MWe. Assuming the efficiency of converting mechanical power into electrical power is 100% (ideal case), we can find the mass flow rate by dividing the total output power by the specific power of the turbine. Mass Flow Rate = \( \frac{Total \, Power \, Output}{Specific \, Power}\) Plug in the values, 25 MWe = 25,000 kWe, we have: Mass Flow Rate = \(\frac{25,000 \, \mathrm{kW}}{3411 \, \mathrm{kJ} \cdot \mathrm{kg}^{-1}} \approx 7.33 \, \mathrm{kg} \cdot \mathrm{s}^{-1}\)

Listing the assumptions made for these calculations

1. We have assumed ideal behavior and neglected any losses. 2. We didn't take into account the water pump work. 3. We assumed isentropic (constant entropy) expansion of steam in the turbine. 4. We assumed that the efficiency of converting mechanical power into electrical power is 100%. So, in conclusion, the answers are: a. The specific power of the turbine expansion process is approximately 3411 kJ/kg. b. The mass flow rate of steam through the turbine is approximately 7.33 kg/s. c. The assumptions made are ideal behavior, constant entropy expansion, neglecting water pump work, and 100% conversion efficiency.

Key concepts

Specific Energy

Specific energy is a critical concept in thermodynamics, especially when dealing with energy conversion systems like steam turbines. It is the amount of energy per unit mass. In the context of the given exercise, specific energy is referred to as specific enthalpy. Specific enthalpy is key to determining the energy changes during the expansion process in a turbine.
For a steam turbine, we calculate the specific power by assessing the difference in specific enthalpy between the inlet and outlet of the turbine. The specific enthalpy values can be found using steam tables, which provide enthalpy numbers for different pressures and temperatures.

• This helps in calculating the work done by the turbine on a per kilogram basis.
• The specific power here indicates how much energy is extracted per kilogram of steam.

Understanding specific energy allows engineers to design efficient systems by optimizing the energy conversion processes.

Steam Tables

Steam tables are indispensable tools in thermodynamics that help determine thermodynamic properties of water and steam under various conditions. These properties include specific enthalpy, specific entropy, and specific volume.
When dealing with steam turbines, steam tables provide a straightforward way to find the necessary enthalpy values corresponding to certain pressures and temperatures. Using these tables:

• For an inlet condition of 525°C and 100 bar, we find the specific enthalpy, \(h_{in}\), using the superheated steam section.
• Similarly, the enthalpy for a condensation condition at 20°C, \(h_{out}\), is found using the saturation temperature section.

Steam tables thus simplify the complex calculations involved in finding energy changes, by providing accurate data derived from experimental results.

Isentropic Process

An isentropic process is a key thermodynamic principle where the entropy of the system remains constant. This process is considered ideal and reversible, often applied in turbines and compressors.
In an isentropic expansion within a steam turbine, the process proceeds without any increase in entropy. This means there are no losses due to friction or heat transfer.

• It allows the determination of effectiveness in energy conversion.
• Maintaining constant entropy helps in simplifying calculations, assuming ideal performance.

Considering isentropic processes helps in evaluating the maximum theoretical efficiency of turbines, though real-life turbines often operate under slightly different conditions due to irreversibilities.

Efficiency Calculation

Efficiency calculation is crucial in understanding how well a steam turbine converts thermal energy into mechanical energy. It is the ratio of useful energy output to the energy input.
In the ideal calculations here, we assumed 100% efficiency, meaning all mechanical power is converted into electrical energy without losses.

• This assumption is not realistic but simplifies the analysis.
• In practice, efficiency is lower due to various losses such as heat loss, friction, and other inefficiencies.

Understanding efficiency helps in refining turbine designs and in estimating the actual operation costs in a real power plant. By knowing the efficiency, engineers can improve upon systems, bringing them closer to the ideal performance modeled in isentropic calculations.