Question

Steam enters the turbine of a vapor power plant at 100 bar, \(520^{\circ} \mathrm{C}\) and expands adiabatically, exiting at \(0.08\) bar with a quality of \(90 \%\). Condensate leaves the condenser as saturated liquid at \(0.08\) bar. Liquid exits the pump at 100 bar, \(43^{\circ} \mathrm{C}\). The specific exergy of the fuel entering the combustor unit of the steam generator is estimated to be \(14,700 \mathrm{~kJ} / \mathrm{kg}\). No exergy is carried in by the combustion air. The exergy of the stack gases leaving the steam generator is estimated to be \(150 \mathrm{~kJ}\) per \(\mathrm{kg}\) of fuel. The mass flow rate of the steam is \(3.92 \mathrm{~kg}\) per \(\mathrm{kg}\) of fuel. Cooling water enters the condenser at \(T_{0}=20^{\circ} \mathrm{C}, p_{0}=\) \(1 \mathrm{~atm}\) and exits at \(35^{\circ} \mathrm{C}, 1 \mathrm{~atm}\). Develop a full accounting of the exergy entering the plant with the fuel.

Short answer

The net exergy entering the plant with the fuel is 14,550 kJ.

Step-by-step solution

- Understand exergy definition

Exergy is a measure of the maximum useful work possible during a process that brings the system into equilibrium with a heat reservoir. For the power plant, exergy is carried by both the fuel and the stack gases.

- Fuel exergy calculation

Begin by noting that the specific exergy of the fuel entering the combustor unit is given as 14,700 kJ/kg. This value is the exergy content that the fuel adds to the plant.

- Exergy loss in stack gases

The exergy of the stack gases is given as 150 kJ per kg of fuel. This is the portion of the fuel exergy that is lost to the environment.

- Total mass flow rate of the fuel

The plant's steam mass flow rate per kg of fuel is given as 3.92 kg. We consider this along with the fuel exergy and stack gas exergy to find total exergy input and output.

- Calculate the total exergy input

Multiply the specific exergy of the fuel by the mass flow rate: Exergy input = 14,700 kJ/kg * 1 kg = 14,700 kJ

- Calculate the exergy loss in stack gases

Multiply the exergy of stack gases by the mass flow rate: Exergy loss in stack gases = 150 kJ/kg * 1 kg = 150 kJ

- Compute the net exergy entering the plant

Subtract the exergy loss from the total exergy input: Net exergy = 14,700 kJ - 150 kJ = 14,550 kJ

Key concepts

exergy definition

Exergy is a fundamental concept in thermodynamics that measures the maximum useful work obtainable as a system reaches thermal equilibrium with its surroundings. Unlike energy, which is always conserved, exergy quantifies the quality and potential of energy to perform work. In vapor power plants, exergy is a crucial metric because it accounts for losses and inefficiencies in the system.
For instance, in the given exercise, exergy accounts for the fuel's ability to produce work and the losses due to irreversibilities, such as heat dissipation.

• It encapsulates both energy and entropy principles.
• It helps in identifying performance improvements.
• It is essential in environmental impact assessments.

Understanding exergy helps engineers optimize processes by minimizing exergy destruction or losses.

exergy calculation

Calculating exergy involves several steps that make use of the specific exergy values provided for various parts of the plant. In the context of a vapor power plant, let's break down the steps of the calculation:

Step 1: Fuel Exergy
The specific exergy of the fuel entering the combustor is given as 14,700 kJ/kg. This quantifies the maximum useful work the fuel can provide.

Step 2: Stack Gas Exergy
Next, we account for the exergy present in the stack gases. In this exercise, it is 150 kJ per kg of fuel, representing a loss to the environment.

Step 3: Mass Flow Considerations
The total mass flow rate of steam per kg of fuel is given as 3.92 kg, which helps in calculating the total exergy input and losses.

Formula Recap:
For net exergy entering the plant:
\[ \text{Net Exergy} = \text{Fuel Exergy} - \text{Stack Gas Exergy} \]

Plugging in the numbers:

Net exergy = 14,700 kJ - 150 kJ = 14,550 kJ.

adiabatic expansion

Adiabetic expansion is a process in which steam expands within the turbine without exchanging heat with its surroundings. This assumption simplifies the analysis and calculations since we focus on work interactions only.

• The absence of heat transfer means any increase in the steam's specific volume comes at the cost of its internal energy.
• As steam expands adiabatically, it loses pressure and temperature but gains velocity.

In the exercise, steam enters the turbine at a high pressure of 100 bar and a temperature of 520°C.
By the time it exits, the steam pressure has dropped to 0.08 bar with a quality of 90%, reflecting a significant reduction in exergy usable for work.

mass flow rate

Understanding mass flow rate is crucial to the overall exergy analysis, as it directly affects energy balance and efficiency metrics.
For the vapor power plant, the mass flow rate of the steam per kilogram of fuel is given as 3.92 kg.

• This tells us how much steam is generated for each unit of fuel burned.
• It aids in calculating the distributed exergy both into and out of the plant.

In mathematical terms:
\[ \text{Total Exergy Input} = \text{Fuel Exergy} \times \text{Mass Flow Rate} \]
Given our numbers:
\[ \text{Exergy Input} = 14,700 \frac{kJ}{kg} \times 1 \frac{kg}{fuel} = 14,700 kJ\]
This mass flow rate helps convert specific quantities (like per kg of fuel) into total system metrics, giving a better picture of overall system performance.

energy loss

Energy loss in a vapor power plant primarily occurs due to exergy destruction and some unavoidable efflux.
In the provided exercise, the energy loss is quantified by the exergy in the stack gases:

• The given exergy loss is 150 kJ per kg of fuel.
• These gases carry away exergy that could otherwise have been used for useful work.

Understanding these losses is critical for improving system efficiency.
By identifying and minimizing energy losses, engineers can devise strategies to retain more exergy within the power cycle, ultimately leading to a more efficient plant operation.

For instance, strategies for reducing these losses include:

• Enhancing insulation.
• Reducing frictional losses.
• Optimizing component design.

Every unit of exergy conserved directly translates to increased total efficiency.