What Factors Make the Efficiency of a Rankine Cycle Lower than that of the Carnot Cycle

When it comes to understanding cyclical thermodynamics, the Carnot cycle reigns as the ultimate benchmark. It's a purely theoretical concept that defines the maximum efficiency a heat engine can achieve, given the constraints of its working temperatures. In real-world applications, however, the Rankine cycle stands as a practical alternative, albeit with lower efficiency compared to the Carnot cycle. This article delves into the reasons behind this difference, focusing on the efficiency calculations and key principles governing these cycles.

Theoretical Benchmark: The Carnot Cycle

The Carnot cycle is more efficient because heat is added at the highest temperature possible during the cycle. This feature makes it a purely theoretical cycle, often referred to as a reversible cycle, since it operates in a state of perfect thermodynamic equilibrium. The efficiency (eta;) of a Carnot cycle is given by the formula:

eta;Carnot 1 - (T2 / T1)

Where T1 is the temperature of the hot reservoir (maximum temperature) and T2 is the temperature of the cold reservoir. This equation clearly indicates that the Carnot cycle achieves its maximum efficiency when T1 is at its highest value.

The Practical Limitations: The Rankine Cycle

In contrast, the Rankine cycle introduces more practical considerations into the efficiency equation. In the Rankine cycle, heat is added through a series of stages at progressively higher temperatures, eventually reaching the maximum temperature that the system can handle. This means that the mean temperature of the heat input (Tm) is lower than the maximum temperature (Tmax) of the Carnot cycle. The efficiency of the Rankine cycle is given by:

eta;Rankine 1 - (T2 / Tm)

While T2 is still the temperature of the exhaust, Tm represents the average temperature of the heat input. As Tmax is always greater than Tm, the Carnot cycle theoretically achieves the highest possible efficiency.

Understanding Reversibility and Irreversibility

A key difference between the Carnot and Rankine cycles lies in their degrees of reversibility. The Carnot cycle is an ideal, reversible cycle that maintains thermodynamic equilibrium throughout its operations. This idealized nature makes it incapable of practical application, but it serves as a benchmark for efficiency. In contrast, the Rankine cycle, while still an ideal cycle, introduces practical limitations such as irreversible processes like entropy increases during the combustion of fuel or phase changes in the working fluid.

Revisiting the equation for lost work potential due to irreversibility, it is defined by:

Lost work cycle entropy times; cold reservoir temperature

These irreversibilities mean that the Rankine cycle cannot achieve the maximum efficiency possible as defined by the Carnot cycle. To understand the impact of these irreversibilities, it's important to recognize the entropy changes that occur during each phase of the Rankine cycle, particularly during the expansion and condensation processes.

Theoretical to Practical: From Carnot to Rankine

The journey from the theoretical Carnot cycle to the practical Rankine cycle involves several modifications. The Carnot cycle initially serves as a hypothetical model for the maximum efficiency, but it is impractical due to its reversible processes. Modifications are made to adapt the Carnot cycle to practical constraints, leading to the development of the Carnot vapor cycle. Further modifications in the Carnot vapor cycle result in the Rankine cycle, which provides the basis for modern steam power plants.

The Rankine cycle, while less efficient than the Carnot cycle, is more practical due to its ability to handle real-world processes. The cycle includes reversible and irreversible processes, with the latter leading to losses in efficiency. However, the Rankine cycle is an invaluable tool for engineers and thermodynamicists in understanding and improving the efficiency of power generation systems.

Conclusion

The choice between using the Carnot cycle as a theoretical benchmark and the Rankine cycle in practical applications is a reflection of the trade-off between ideal conditions and real-world feasibility. While the Carnot cycle represents the theoretical maximum efficiency, the Rankine cycle is the more practical choice, offering a balance between efficiency and operational simplicity.

Further Reading

For a deeper understanding of these cycles and their applications, following the link provided can be extremely beneficial. It offers a more comprehensive explanation and practical examples that can help clarify the principles discussed in this article.