Question

Consider a simple ideal Rankine cycle and an ideal Rankine cycle with three reheat stages. Both cycles operate between the same pressure limits. The maximum temperature is \(700^{\circ} \mathrm{C}\) in the simple cycle and \(450^{\circ} \mathrm{C}\) in the reheat cycle. Which cycle do you think will have a higher thermal efficiency?

Short answer

Answer: The ideal Rankine cycle with three reheat stages has a higher thermal efficiency than the simple ideal Rankine cycle.

Step-by-step solution

The Rankine cycle is a thermodynamic cycle that describes the operation of heat engines, particularly steam-based power plants. It consists of four processes: (1) isentropic compression of a working fluid, (2) constant pressure heat addition, (3) isentropic expansion of the fluid, and (4) constant pressure heat rejection. Efficiency for the cycle is given by \(\eta = 1 - \frac{Q_L}{Q_H}\), where \(Q_H\) is the amount of heat added and \(Q_L\) is the amount of heat rejected. #Step 2: Simple Rankine cycle efficiency calculation#

For the simple Rankine cycle, the maximum temperature is \(700^{\circ} \mathrm{C}\) or \(973.15\ \mathrm{K}\). Since both cycles operate between the same pressure limits, let \(T_{L}\) be the lower temperature for both cycles. The thermal efficiency of a simple Rankine cycle can be approximated as \(\eta_{simple} = 1 - \frac{T_{L}}{T_{H,simple}}\), where \(T_{H,simple}\) is the maximum temperature in the simple Rankine cycle. #Step 3: Rankine cycle with reheat efficiency calculation#

For the ideal Rankine cycle with three reheat stages, the maximum temperature is \(450^{\circ} \mathrm{C}\) or \(723.15\ \mathrm{K}\). The efficiency of a Rankine cycle with reheat can be improved by reducing the average temperature at which heat is rejected. The thermal efficiency of an ideal Rankine cycle with reheat can be approximated as \(\eta_{reheat} = 1 - \frac{T_{L}}{T_{avg,reheat}}\), where \(T_{avg,reheat}\) is the average temperature between the maximum and minimum temperatures in the reheat cycle. Since there are three reheat stages, \(T_{avg,reheat} = \frac{3 \times T_{H,reheat} + T_{L}}{4}\), where \(T_{H,reheat}\) is the maximum temperature in the reheat cycle. #Step 4: Compare the efficiencies of both cycles#

Now that we have expressions for the thermal efficiency of both cycles, we need to compare them to determine which one will have higher efficiency. \(\eta_{simple} = 1 - \frac{T_{L}}{T_{H,simple}}\) \(\eta_{reheat} = 1 - \frac{T_{L}}{T_{avg,reheat}} = 1 - \frac{T_{L}}{\frac{3 \times T_{H,reheat} + T_{L}}{4}}\) To assess which cycle will have a higher efficiency, we can compare \(T_{H,simple}\) and \(T_{avg,reheat}\). Since \(T_{H,simple} = 973.15\ \mathrm{K}\) and \(T_{H,reheat} = 723.15\ \mathrm{K}\), we have: \(T_{avg,reheat} = \frac{3 \times 723.15 + T_{L}}{4}\) \(T_{H,simple} > T_{avg,reheat}\) because \(\frac{T_{L} + 3 \times T_{H,reheat}}{4} < T_{H,simple}\) Since the denominator of \(\eta_{simple}\) is larger than that of \(\eta_{reheat}\), the efficiency of the Rankine cycle with reheat stages will be higher than the simple Rankine cycle. So, the ideal Rankine cycle with three reheat stages has a higher thermal efficiency than the simple ideal Rankine cycle.

Key concepts

Understanding Thermal Efficiency of the Rankine Cycle

Thermal efficiency plays a vital role in the performance of the Rankine cycle, which is at the heart of most steam-based power generation systems. In essence, thermal efficiency measures how effectively a heat engine converts the heat it receives into work. For the Rankine cycle, the formula for thermal efficiency can be expressed as \(\eta = 1 - \frac{Q_L}{Q_H}\), where \(Q_H\) is the heat absorbed during the constant pressure heating process and \(Q_L\) is the heat ejected during the condensation process.

When evaluating thermal efficiency, the temperatures at which heat exchange occurs are critical factors. The higher the inlet temperature of the steam to the turbine, the greater the potential for high efficiency, because the cycle is approaching the ideal Carnot efficiency. However, materials and practical limitations usually set a maximum allowable temperature for steam entering the turbine.

Several strategies are employed to enhance the efficiency of Rankine cycles, including increasing the maximum temperature of the cycle, decreasing the minimum temperature, or using multiple stages of reheat, a method which allows the steam to be reheated in between expansion stages. This can significantly increase a cycle's thermal efficiency by reducing the average temperature at which heat rejection occurs.

The Simple Rankine Cycle

A simple Rankine cycle is the fundamental model for steam-based thermal power plants. Its simplicity lies in its four key components: pump, boiler, turbine, and condenser. The cycle follows these steps: isentropic compression of the working fluid (usually water) in a pump, constant pressure heating of the fluid in a boiler (turning it into high-pressure steam), isentropic expansion of the steam in a turbine (producing work), and constant pressure cooling in a condenser (condensing the steam into water).

The efficient operation of this cycle is dependent on the thermodynamic properties of the working fluid at different stages, particularly the temperature and pressure of the steam at the turbine inlet. Theoretically, a higher inlet temperature results in a higher efficiency, as it would lessen the ratio of heat rejected to heat gained. However, in a practical simple Rankine cycle, the materials used in the construction of turbines and other components limit the maximum temperature and, by extension, the maximum efficiency achievable.

Advantages of the Rankine Cycle with Reheat Stages

The Rankine cycle with reheat differs from the simple Rankine cycle by the addition of one or more stages in which the steam is reheated after partial expansion. The process involves expanding the steam through a turbine to a certain point, then reheating it to a high temperature before passing it through additional turbine stages. Reheating can improve efficiency by reducing the moisture content of steam at the final stages of expansion, leading to less erosion of turbine blades and improved thermodynamic performance.

The reheat technique also reduces the average temperature at which heat is rejected. Using our exercise as an example, the Rankine cycle with three reheat stages at a maximum temperature of \(450^\circ \mathrm{C}\) or \(723.15 \mathrm{K}\) exhibits a higher efficiency compared to a simple Rankine cycle with a maximum temperature of \(700^\circ \mathrm{C}\) or \(973.15 \mathrm{K}\) due to the lower average temperature for heat rejection.

Why Does the Reheat Cycle Have Better Efficiency?

By reheating the steam, the cycle reduces the exergy destruction in the heat rejection process and brings the Rankine cycle closer to the ideal reversible process. It's essentially a way to mitigate inherent inefficiencies in the steam expansion process, which in the example given, translates to the Rankine cycle with reheat stages being more efficient than the simple Rankine cycle, despite operating at a lower maximum temperature.