Question

The ideal Brayton and Rankine cycles are composed of the same four processes, yet look different when represented on a \(T-s\) diagram. Explain.

Short answer

The T-s diagram for the Brayton cycle has inclined lines for heat addition and rejection due to gas behavior, while the Rankine cycle has horizontal lines due to phase changes.

Step-by-step solution

- Understand the Brayton Cycle

The Brayton cycle is commonly used for gas turbines and consists of four processes: two isentropic (adiabatic and reversible) processes and two isobaric (constant pressure) processes. The steps of the Brayton cycle are: 1) Isentropic compression, 2) Isobaric heat addition, 3) Isentropic expansion, and 4) Isobaric heat rejection.

- Understand the Rankine Cycle

The Rankine cycle is used for steam power plants and also consists of four processes: two isentropic processes and two isobaric processes, but with a key difference. For the Rankine cycle, the steps are: 1) Isentropic compression by a pump, 2) Isobaric heat addition in a boiler, 3) Isentropic expansion in a turbine, and 4) Isobaric heat rejection in a condenser.

- Analyze the Brayton Cycle on a T-s Diagram

On a Temperature-Entropy (T-s) diagram, the Brayton cycle is represented as follows: the isentropic compression (1 to 2) and isentropic expansion (3 to 4) are vertical lines because the entropy remains constant. The isobaric heat addition (2 to 3) and heat rejection (4 to 1) are inclined lines showing an increase or decrease in temperature and entropy.

- Analyze the Rankine Cycle on a T-s Diagram

On a Temperature-Entropy (T-s) diagram, the Rankine cycle appears differently: the isentropic expansion (1 to 2) and compression (3 to 4) are also vertical lines. However, the heat addition (2 to 3) and heat rejection (4 to 1) occur at different entropies because of the phase change of water into steam, resulting in horizontal or almost horizontal lines.

- Compare and Contrast the Cycles

Both cycles have the same types of processes but appear different on a T-s diagram because of the phase changes in the Rankine cycle which result in nearly horizontal lines during heat addition and rejection. In the Brayton cycle, these processes are inclined due to constant pressure heat addition and rejection for a gas.

- Conclude the Differences

In summary, the T-s diagram for the Brayton cycle shows inclined lines for the isobaric processes, while the Rankine cycle shows almost horizontal lines due to phase changes. These differences are due to the working fluid being a gas in Brayton and a combination of liquid and steam in Rankine.

Key concepts

Isentropic Process

The term 'isentropic' refers to a process during which the entropy of the system remains constant. In simpler terms, there is no change in the disorder of the system. This type of process is both adiabatic (no heat transfer) and reversible.

In both the Brayton and Rankine cycles, we see isentropic compression and expansion phases.

For instance:

• Brayton cycle: Isentropic compression occurs in the compressor, while isentropic expansion occurs in the turbine.
• Rankine cycle: Isentropic compression is achieved via a pump (compressing the liquid), and isentropic expansion happens within the turbine.

Understanding isentropic processes is crucial because they help us track energy transformations within thermodynamic cycles without entropy change, making the analysis simpler and results more predictable.

Isobaric Process

An isobaric process is where the pressure remains constant throughout. These processes are key components of both the Brayton and Rankine cycles, but they manifest differently in each.

In the Brayton cycle, the constant pressure processes occur during the following stages:

• Heat addition (combustion): Energy is added to the system without increasing pressure.
• Heat rejection: Excess energy is removed to prepare the system for another cycle.

For the Rankine cycle, the isobaric processes work as follows:

• Heat addition: Occurs in the boiler where water is converted to steam.
• Heat rejection: Takes place in the condenser where steam is condensed back to water.

It's essential to understand isobaric processes because these are where heat exchange occurs, altering the system's energy without affecting pressure.

T-s Diagram

The Temperature-Entropy (T-s) diagram is an incredibly useful tool to visualize and analyze thermodynamic cycles. Each cycle, Brayton or Rankine, shows unique characteristics on a T-s diagram.

For the Brayton cycle, the diagram reveals:

• Vertical lines representing isentropic processes (compression and expansion) because entropy doesn't change.
• Inclined lines for isobaric processes (heat addition and heat rejection), as temperature and entropy both vary with constant pressure.

Conversely, the Rankine cycle shows:

• Vertical lines for isentropic processes (compression by pump and expansion by turbine).
• Almost horizontal lines during heat addition and rejection because of the phase change from liquid to steam (constant temperature but changing entropy).

A well-crafted T-s diagram is crucial for comparing and understanding different thermodynamic processes and cycles.

Phase Change

Phase change describes the transition between different states of matter, such as from liquid to gas or vice versa.

This concept is particularly important in the Rankine cycle where water undergoes phase changes:

• From liquid to steam in the boiler (isobaric heat addition).
• From steam back to liquid in the condenser (isobaric heat rejection).

During these phases:

• Energy is absorbed or released, but the temperature remains constant, showing up as horizontal lines on the T-s diagram.

Unlike the Brayton cycle which deals with gases, the Rankine cycle's working fluid is often water, which undergoes significant phase changes. Understanding phase changes is essential because it highlights the energy required to convert materials into different states without temperature change, marking a key difference in how energy is managed within these cycles.