Question

An ideal reheat Rankine cycle with water as the working fluid operates the boiler at \(15,000 \mathrm{kPa}\), the reheater at \(2000 \mathrm{kPa}\), and the condenser at \(100 \mathrm{kPa}\). The temperature is \(450^{\circ} \mathrm{C}\) at the entrance of the high-pressure and low pressure turbines. The mass flow rate through the cycle is \(1.74 \mathrm{kg} / \mathrm{s} .\) Determine the power used by pumps, the power produced by the cycle, the rate of heat transfer in the reheater, and the thermal efficiency of this system.

Short answer

Answer: The thermal efficiency of the system is approximately 40.6%.

Step-by-step solution

Determine enthalpies at each state point

To find the enthalpies at each point, we first have to determine specific properties of water at the given pressure and temperature conditions. We can use steam tables to find these properties. State 1 (Condenser exit): Since the fluid exits the condenser as a saturated liquid, we can find the enthalpy at state 1 using the given pressure: \(h_1=h_{f@100kPa}=339.49 \mathrm{kJ/kg}\) State 2 (After first pump): The fluid is compressed isentropically, thus we find enthalpy at state 2: \(s_2=s_1=s_{f@100kPa}=1.0096 \mathrm{kJ/(kgK)}\) Since we have an isentropic process between states 1 and 2, we use the isentropic relation and refer to the steam table for the specific enthalpy at 15,000 kPa: \(h_2=h_{2s} \approx 353.75 \mathrm{kJ/kg}\) State 3 (Boiler exit): The fluid exits the boiler at the stated temperature of 450°C. We can look up the enthalpy at this temperature and pressure of 15,000 kPa: \(h_3=h_g@450^{\circ}C,15000kPa=3276 \mathrm{kJ/kg}\) State 4 (High-pressure turbine exit): The fluid expands isentropically through the high-pressure turbine, so we have: \(s_4=s_3=s_g@450^{\circ}C,15000kPa=6.5857 \mathrm{kJ/(kgK)}\) Using this entropy value and the reheater pressure information, we can find the enthalpy value for the fluid at the exit of the high-pressure turbine and the entrance of the reheater: \(h_4=h@2000kPa, s_4 \approx 2854.5 \mathrm{kJ/kg}\) State 5 (Reheater exit): The fluid exits the reheater at 450°C (stated above). We can look up the enthalpy at this temperature and pressure of 2,000 kPa: \(h_5=h_g@450^{\circ}C,2000kPa=3177.2 \mathrm{kJ/kg}\) State 6 (Low-pressure turbine exit): The fluid expands isentropically through the low-pressure turbine, so we have: \(s_6=s_5=s_g@450^{\circ}C,2000kPa=7.3961 \mathrm{kJ/(kgK)}\) Using this entropy value and the condenser pressure information, we can find the enthalpy value for the fluid at the exit of the low-pressure turbine: \(h_6=h@100kPa, s_6 \approx 2399 \mathrm{kJ/kg}\) State 7 (After second pump): The fluid exits the cycle at state 7 after passing through the second pump. We have an isentropic process between states 6 and 7, so: \(s_7=s_6 \Rightarrow h_7=h_{f@100kPa} \approx 2409.4 \mathrm{kJ/kg}\)

Calculate the power used by the pumps

To calculate the power used by the pumps, we can use the mass flow rate and the changes in enthalpy for the two pumps: \(W_{pump1} = m\cdot(h_2-h_1)\) \(W_{pump1} = 1.74\mathrm{kg/s}\cdot(353.75\mathrm{kJ/kg}-339.49\mathrm{kJ/kg}) \approx 24.76\mathrm{kW}\) \(W_{pump2} = m\cdot(h_7-h_6)\) \(W_{pump2} = 1.74\mathrm{kg/s}\cdot(2409.4\mathrm{kJ/kg}-2399\mathrm{kJ/kg}) \approx 18.17\mathrm{kW}\) Total power used by pumps: \(W_{pumps} = W_{pump1}+W_{pump2} \approx 42.93\mathrm{kW}\)

Calculate the power produced by the turbines

To calculate the power produced by the turbines, we can use the mass flow rate and the changes in enthalpy for the two turbines: \(W_{HPturbine} = m\cdot(h_3-h_4)\) \(W_{HPturbine} = 1.74\mathrm{kg/s}\cdot(3276\mathrm{kJ/kg}-2854.5\mathrm{kJ/kg}) \approx 732.53\mathrm{kW}\) \(W_{LPturbine} = m\cdot(h_5-h_6)\) \(W_{LPturbine} = 1.74\mathrm{kg/s}\cdot(3177.2\mathrm{kJ/kg}-2399\mathrm{kJ/kg}) \approx 1355.86\mathrm{kW}\) Total power produced by turbines: \(W_{turbines} = W_{HPturbine}+W_{LPturbine} \approx 2088.39\mathrm{kW}\)

Calculate the heat transfer rates in the boiler and reheater

To calculate the heat transfer rates, we can use the mass flow rate and the changes in enthalpy for the boiler and reheater: \(Q_{boiler} = m\cdot(h_3-h_2)\) \(Q_{boiler} = 1.74\mathrm{kg/s}\cdot(3276\mathrm{kJ/kg}-353.75\mathrm{kJ/kg}) \approx 5085.67\mathrm{kW}\) \(Q_{reheater} = m\cdot(h_5-h_4)\) \(Q_{reheater} = 1.74\mathrm{kg/s}\cdot(3177.2\mathrm{kJ/kg}-2854.5\mathrm{kJ/kg}) \approx 562.37\mathrm{kW}\)

Calculate the thermal efficiency of the system

To calculate the thermal efficiency, we can use the formula: \(\eta_{th} = \frac{W_{turbines}-W_{pumps}}{Q_{boiler}+Q_{reheater}}\) \(\eta_{th} = \frac{2088.39\mathrm{kW}-42.93\mathrm{kW}}{5085.67\mathrm{kW}+562.37\mathrm{kW}} \approx 0.406\) The thermal efficiency of the system is approximately 40.6%.

Key concepts

Understanding Thermal Efficiency in the Rankine Cycle

Thermal efficiency serves as a key indicator of a power plant's performance, implying how well the system converts heat into work. When discussing the Rankine cycle, which is typical in steam turbines, thermal efficiency provides insight into how much of the heat input from the boiler is effectively transformed into usable mechanical energy. Essentially, it represents a ratio of the net work output (turbines' work output minus the work input required by the pumps) to the total heat input (heat absorbed by the boiler and the reheater).

The formulation for thermal efficiency is encapsulated by the expression: \[\theta_{th} = \frac{W_{turbines} - W_{pumps}}{Q_{boiler} + Q_{reheater}}\]
Where the numerator represents the useful work output, and the denominator quantifies the input heat energy. In the context of an ideal reheat Rankine cycle, where steam is expanded in two stages with reheating in between, the efficiency is improved in comparison to a single-stage expansion. This occurs because the expansion process approaches isentropic conditions more closely and reduces the moisture content of the steam at the final stages, leading to enhanced turbine performance and longevity.

In the given problem, the thermal efficiency is calculated to be approximately 40.6%, which is to say, about 40.6% of the heat supplied to the system is converted into work, while the rest is lost. By understanding and calculating this efficiency, we can gauge the effectiveness of the power plant and seek ways to improve its performance through engineering modifications such as increasing the boiler temperature or incorporating multiple reheating stages.

Exploring the Reheat Cycle in Rankine System

The innovative concept of the reheat cycle is an enhancement of the traditional Rankine cycle, tailored to boost the cycle's efficiency and control the moisture content at the end of the expansion process. Reheating involves returning partially expanded steam to a boiler (the reheater), where it is reheated at a constant pressure to the initial temperature before being expanded again in a low-pressure turbine.

In our ideal reheat Rankine cycle scenario, the steam exits the high-pressure turbine at Point 4, is reheated back to its original high temperature at Point 5, and is sent through the low-pressure turbine. This two-stage expansion with an intermediate reheating phase permits a greater extraction of energy from the steam and hinders the erosion of turbine blades which can be caused by excessive moisture. Moreover, this method also allows for a larger volume flow rate in the low-pressure stages of the turbine, which can be leveraged to reduce the cost and size of the low-pressure turbine equipment.

The reheater's heat transfer rate is crucial as it re-energizes the steam, restoring its potentials for the forthcoming work-producing expansion in the low-pressure turbine. In the solved exercise, the rate of heat transfer in the reheater is found to be approximately 562.37 kW, a testament to the importance of the reheater in elevating the overall efficiency of the power plant. Beyond increasing efficiency, reheating aligns the expansion process with an isentropic ideal more closely, thus lead to a more favorable efficiency outcome.

Enthalpy Changes: The Workhorse of Heat Engines

The metric of enthalpy, symbolized as 'h' and measured in kilojoules per kilogram (kJ/kg), is integral to thermodynamics and characterizes the total heat content of a system. In the Rankine cycle, enthalpy signifies the energy states that the working fluid (usually water or steam) undergoes as it circulates through the system components (boiler, turbine, condenser, and pump).

The alterations in enthalpy between two states (initial and final) of the fluid signify the energy absorbed or released during processes. For instance, when steam flows through a turbine, the decrease in its enthalpy corresponds to the work derived from its expansion. Similarly, enthalpy increases in the boiler and reheater reflect the heat added to the system.

Consideration of enthalpy is central to solving our exercise. By consulting steam tables at each state point—where defined pressures and temperatures prevail—we determine specific enthalpies, allowing us to assess the energy exchanges within the cycle. These values are used to calculate the rate of work performed by the pumps and the turbines, as well as the heat transfer rates in the boiler and reheater.
In summary, enthalpy serves as a pivotal measure in the thermal engineering domain, where being capable of determining and manipulating it is key to designing systems with desirable energy efficiency and output.