Question

Consider a simple ideal Rankine cycle. If the condenser pressure is lowered while keeping turbine inlet state the same, \((a)\) the turbine work output will decrease. \((b)\) the amount of heat rejected will decrease. \((c)\) the cycle efficiency will decrease. \((d)\) the moisture content at turbine exit will decrease. \((e)\) the pump work input will decrease.

Short answer

Question: Evaluate the effects of lowering the condenser pressure in a simple ideal Rankine cycle while keeping the turbine inlet state the same. Indicate if the following statements are true or false: (a) The work output of the turbine will decrease. (b) The amount of heat rejected will decrease. (c) The cycle efficiency will decrease. (d) The moisture content at the turbine exit will decrease. (e) The pump work input will decrease. Answer: Based on our analysis, the correct answers are: (a) False (b) True (c) False (d) True (e) False

Step-by-step solution

Analyzing the turbine work output

We should first analyze how lowering the condenser pressure affects the turbine work output. If the condenser pressure is lowered but the turbine inlet state remains the same, the difference in pressure across the turbine will increase. As a result, the turbine will be able to perform more work on the working fluid, so the work output of the turbine should increase in this case. Therefore, statement \((a)\) is false.

Evaluating heat rejection

Next, let's consider the effect on the amount of heat rejected. Heat is rejected during the condensation process, where the working fluid loses energy by transferring it to the cooling medium. Lowering the condenser pressure implies that the temperature of the working fluid will decrease, so the difference in temperature between the fluid and the cooling medium will decrease. A smaller temperature difference reduces the rate of heat transfer, and hence, the amount of heat rejected decreases. Thus, statement \((b)\) is true.

Analyzing cycle efficiency

The efficiency of the Rankine cycle is given by the ratio of the net work output to the total heat input. Increasing the turbine work output, decreasing the heat rejected, and keeping the heat input the same would mean that the net work output increases, giving rise to a higher cycle efficiency. Therefore, statement \((c)\) is false.

Evaluating moisture content

The moisture content at the turbine exit is related to the saturation temperature and pressure of the working fluid in the condenser. If the condenser pressure is lowered, the saturation temperature of the working fluid will also decrease. As a result, the working fluid will become more superheated at the turbine exit, which means that the moisture content will decrease. Therefore, statement \((d)\) is true.

Analyzing pump work input

Lastly, let's consider how lowering the condenser pressure affects the pump work input. The pump work input is determined by the liquid-vapor saturation line in the P-h diagram. If the condenser pressure is lowered, the pressure difference across the pump will increase, implying a larger work input for the pump. Therefore, statement \((e)\) is false. In conclusion, the correct answers to the exercise are: - \((b)\) the amount of heat rejected will decrease. - \((d)\) the moisture content at the turbine exit will decrease.

Key concepts

Turbine Work Output in the Rankine Cycle

Understanding turbine work output is essential for analyzing the performance of the Rankine cycle. In a Rankine cycle, the turbine is responsible for converting the thermal energy from steam into mechanical work, which can then be converted into electricity. The work output of the turbine is influenced by the pressure difference between the inlet and outlet of the turbine. When the condenser pressure is decreased while keeping the turbine inlet state unchanged, the pressure differential across the turbine increases. This results in more available energy to be converted into work, thus boosting the turbine's work output.

In practice, turbines are designed to operate efficiently within specific pressure ranges, and such changes in condenser pressure may have further implications on the overall system efficiency. Additionally, the work done by the turbine is a critical component in determining the efficiency of the entire power plant, as it directly affects the net work output, which is a part of the efficiency calculation for the Rankine cycle.

Heat Rejection in Condensers

The process of heat rejection in condensers is integral to the functionality of the Rankine cycle. The condenser's role is to convert exhaust steam from the turbine back into liquid water, which in turn is pumped back to the boiler. The effectiveness of heat rejection is dependent on the temperature differential between the steam and the cooling medium within the condenser.

When the condenser pressure is lowered, it results in a reduced saturation temperature for the working fluid, minimizing the temperature difference with the cooling medium and thus decreasing the heat transfer rate. Efficient heat rejection is crucial because residual heat remaining in the working fluid could lower the cycle efficiency. By effectively rejecting heat, the working fluid is more thoroughly condensed, allowing the cycle to operate more efficiently by reducing the energy required for the subsequent phase of reheating the fluid.

Moisture Content at Turbine Exit

Moisture content at the turbine exit is an important factor in the operation of steam turbines and the Rankine cycle's efficiency. High moisture content in the steam can cause erosion and damage to the turbine blades, while also reducing the energy conversion efficiency. Moisture content is closely related to the thermodynamic state of steam, which includes pressure and temperature.

When condenser pressure is decreased, the saturation temperature of the steam drops, causing the steam to be more superheated at the turbine exit. As a result, the steam has a lower moisture content, which is beneficial for the turbine's performance and longevity. Maintaining an appropriate moisture level is critical; not only does it protect the turbine from physical damage, but it also ensures that the maximum amount of energy is extracted from the steam. Hence, adjustments in the condenser pressure are one way to control moisture content at the turbine exit, optimizing both the cycle's safety and efficiency.