Question

Consider a simple ideal Rankine cycle with fixed turbine inlet conditions. What is the effect of lowering the condenser pressure on

Short answer

#Short Answer# When the condenser pressure in a simple ideal Rankine cycle is lowered, it results in less work done on the pump and more work done by the turbine. Consequently, the net work output increases, and the heat rejected in the condenser decreases. Therefore, the cycle efficiency improves when the condenser pressure is lowered, as seen when comparing two different condenser pressures, P1 (higher pressure) and P2 (lower pressure).

Step-by-step solution

Draw a schematic diagram and label the state points

Create a diagram of the simple ideal Rankine cycle with a pump, a boiler, a turbine, and a condenser with labeled state points (1, 2, 3, and 4) representing the different phases in the cycle.

Write the expressions for the work output, heat exchange, and cycle efficiency

Write down the following expressions for an ideal Rankine cycle: - Work done by the turbine, \(W_T\): \(W_T = h_3 - h_4\) - Work done on the pump, \(W_P\): \(W_P = h_2 - h_1\) - Heat added in the boiler, \(Q_{in}\): \(Q_{in} = h_3 - h_2\) - Heat rejected in the condenser, \(Q_{out}\): \(Q_{out} = h_4 - h_1\) - Net work output, \(W_{net}\): \(W_{net} = W_T - W_P\) - Cycle efficiency, \(\eta\): \(\eta = \frac{W_{net}}{Q_{in}}\)

Analyze the effect of lowering the condenser pressure

For a lower condenser pressure, the following changes are observed in an ideal Rankine cycle: - Enthalpy at state point 1 (\(h_1\)) decreases due to lower condenser pressure, resulting in a decrease in \(W_P\). - Enthalpy at state point 4 (\(h_4\)) also decreases due to lower condenser pressure, while \(h_3\) remains nearly constant. This results in an increase in \(W_T\). - As a result, the net work output, \(W_{net}\) increases. - Heat rejected in the condenser, \(Q_{out}\) decreases due to lower condenser pressure and reduced enthalpy difference between state points 4 and 1. - The heat added in the boiler, \(Q_{in}\), remains constant as it is independent of the condenser pressure. - As \(W_{net}\) increases and \(Q_{in}\) remains constant, the cycle efficiency, \(\eta\), increases when the condenser pressure is lowered.

Compare the results for two different condenser pressures

Considering two different condenser pressures, P1 (higher pressure) and P2 (lower pressure), and calculating the work, heat interactions, and cycle efficiency for both cases, it can be concluded that: 1. The work done on the pump is lower for P2 compared to P1. 2. The work done by the turbine is higher for P2 compared to P1. 3. The net work output increases when the condenser pressure is lowered from P1 to P2. 4. The heat rejected in the condenser is lower for P2 compared to P1. 5. The cycle efficiency increases when the condenser pressure is lowered from P1 to P2. In summary, lowering the condenser pressure in a simple ideal Rankine cycle positively affects the net work output and the cycle efficiency, while reducing the heat rejected in the condenser.

Key concepts

Condenser Pressure

In a Rankine cycle, condenser pressure plays a vital role in the overall efficiency and output of the system. The condenser is where the working fluid is cooled and condensed, usually after it has passed through the turbine. By lowering the condenser pressure, the saturation temperature of the vapor is reduced, which allows the condenser to reject heat at a lower temperature.

When the condenser pressure is decreased, the enthalpy at the exit of the condenser (state point 4) decreases mainly due to the lower saturation temperature. This reduction in enthalpy at point 4 results in greater net turbine work output since the difference between the turbine's inlet and outlet enthalpy (states 3 and 4) becomes larger. Moreover, a lower pump work input is needed to pressurize the working fluid at state 1 because of the reduced pressure, leading to an increase in the cycle's net work. Such an improvement in the net work output also enhances the cycle's efficiency, making the operation of the system more economical and energy efficient.

An additional benefit of lowering the condenser pressure is the reduction in heat rejected to the environment. Since the driving force for heat transfer in the condenser is the temperature difference between the vapor and the cooling medium, a lower condenser temperature means fewer heat losses. However, it's essential to be aware that too low of a condenser pressure can result in operational issues such as increased risks of cavitation in the pump and challenges in maintaining reliable condensation.

Turbine Work Output

The work produced by the turbine, commonly referred to as turbine work output, is a crucial measure of the effectiveness of the Rankine cycle. It is determined by the difference in enthalpy between the turbine inlet and outlet (states 3 and 4). This work is the primary source of electrical power generation in the cycle.

When exploring the impacts of condenser pressure, if the pressure is lowered, this difference in enthalpy increases. It means that each kilogram of working fluid can produce more work as it expands through the turbine since it is starting with higher energy content at the inlet and releasing it to a lower pressure and temperature at the outlet. Fundamentally, the greater the enthalpy drop across the turbine, the more work is output for the same flow rate of steam.

However, increasing the turbine work output must be balanced against material limits and efficiency losses at very low pressures. Turbine blades are optimized for certain pressure and temperature ranges, and extremely low pressures can lead to inefficiencies due to moisture in the steam or decreased blade performance. Ensuring the turbine operates within its designed parameters is critical for maximizing work output.

Cycle Net Work

The concept of cycle net work is defined as the total work output of the cycle after accounting for the work input required to operate the pump. In the Rankine cycle, the main goal is to maximize the net work output as this determines the amount of usable energy that can be converted into electricity.

The formula for calculating the net work output is given by subtracting the work done on the pump from the turbine work output. Intuitively, a high net work output signifies a more efficient and effective power cycle. When the condenser pressure is lowered, the result is an increased turbine work output coupled with a decrease in pump work input, both of which contribute positively to the net work of the cycle. This is an integral benefit as it directly correlates to the amount of electricity that can be generated for a given amount of heat energy supplied to the boiler. Thus, managing condenser pressure is a critical factor in optimizing the Rankine cycle for maximum electricity production and operational profitability.

Heat Exchange in Thermodynamics

Understanding heat exchange in thermodynamics is crucial when analyzing the efficiency of thermal systems like the Rankine cycle. Heat exchange refers to the transfer of thermal energy from a higher temperature system or substance to a lower temperature one.

In the Rankine cycle, heat is added to the working fluid in the boiler, transforming it into high-pressure steam. This heat input is a major component of the cycle's energy flow and its efficiency. The amount of heat added must be balanced with the amount of heat rejected in the condenser to maintain continuity of the thermodynamic process. A critical aspect to consider is the quality of the thermal energy transferred. For instance, high-quality energy input in the form of superheated steam can better convert heat energy into work, leading to improved efficiency.

By lowering the condenser pressure, less heat is rejected and more is utilized for useful work, as previously discussed. These considerations regarding heat exchange underscore the importance of temperature and pressure regulation in maximizing the cycle's performance. A good understanding of how heat exchange affects the operation and efficiency of the Rankine cycle is essential for optimizing the thermodynamic processes within power generation systems.