Question

Water is the working fluid in an ideal Rankine cycle. Superheated vapor enters the turbine at \(8 \mathrm{MPa}, 480^{\circ} \mathrm{C}\). The condenser pressure is \(8 \mathrm{kPa}\). The net power output of the cycle is 100 MW. Determine for the cycle (a) the rate of heat transfer to the working fluid passing through the steam generator, in \(\mathrm{kW}\). (b) the thermal efficiency. (c) the mass flow rate of condenser cooling water, in \(\mathrm{kg} / \mathrm{h}\), if the cooling water enters the condenser at \(15^{\circ} \mathrm{C}\) and exits at \(35^{\circ} \mathrm{C}\) with negligible pressure change.

Short answer

Find specific enthalpies from steam tables, compute work and heat values, and use them to determine thermal efficiency and mass flow rates.

Step-by-step solution

Determine enthalpies at key points

Using steam tables, find the specific enthalpies at the critical points around the Rankine cycle. At state 1 (turbine inlet), for superheated vapor at 8 MPa and 480°C, use steam tables to find the specific enthalpy, h1. Similarly, find the specific enthalpy at state 2 (turbine exit) for 8 kPa, which is usually taken to be the saturated vapor enthalpy, h2. At state 3 (condenser exit), find the saturated liquid enthalpy at 8 kPa, h3. Finally, find the specific enthalpy at state 4 (pump outlet) after isentropic compression.

Calculate work done by turbine and pump

Use the specific enthalpies to find the work done by the turbine (W_turbine) and pump (W_pump). The work done by the turbine is given by \[ W_{turbine} = h_1 - h_2 \] The work done by the pump is given by \[ W_{pump} = h_4 - h_3 \]

Compute the heat added in the boiler

The heat added (\this Q_in\this) in the boiler is given by the specific enthalpy difference between the inlet and outlet of the boiler:\[ Q_{in} = h_1 - h_4 \]

Determine thermal efficiency

The thermal efficiency (\this \frac{\text{Net Work Output}}{\text{Heat Added}}\this) of the Rankine cycle can be calculated using:\[ \eta = \frac{W_{net}}{Q_{in}} = \frac{(W_{turbine } - W_{pump})}{Q_{in}} \]

Calculate mass flow rate of working fluid

Find the mass flow rate of the working fluid using the net power output and the net work done per unit mass of the fluid. Given the net power output is 100 MW, find the mass flow rate \this \frac{\text{Power}}{\text{Net Work}}\this:\[ \dot{m} = \frac{100,000}{W_{net}} \]

Determine the mass flow rate of cooling water

Use the energy balance in the condenser to find the mass flow rate of cooling water. The heat rejected in the condenser is given by \( Q_{out} = h_2 - h_3 \). Using the cooling water temperature change, the mass flow rate of cooling water (\this m_c\this) can be found by:\[ Q_{out} = m_c \times C_p \times (T_{exit} - T_{inlet}) \]

Key concepts

ideal Rankine cycle

The Rankine cycle is a model that predicts the behavior of steam-based power plants. It's composed of four key processes: isentropic expansion in the turbine, isobaric heat rejection in the condenser, isentropic compression in the pump, and isobaric heat addition in the boiler. An 'ideal Rankine cycle' assumes no losses like friction or heat leakage, making it a perfect theoretical model. Understanding this cycle helps you grasp how power plants convert heat into mechanical work efficiently.

thermal efficiency

Thermal efficiency is a measure of a cycle's ability to convert the heat added into useful work. For the Rankine cycle, it's calculated by the ratio of the net work output to the heat added in the boiler. The formula is: \[ \eta = \frac{W_{net}}{Q_{in}} \] Here, \( W_{net} \) is the net work output (turbine work minus pump work) and \( Q_{in} \) is the heat added in the boiler. The higher the thermal efficiency, the more efficient the cycle is at converting heat into work.

enthalpy calculation

Enthalpy calculations are critical in analyzing the Rankine cycle. Enthalpy, denoted by \( h \), is a measure of total energy content in a fluid. In the problem, the enthalpies at different states of the cycle (turbine inlet, turbine exit, condenser exit, and pump outlet) are needed:

• \( h_1 \): Enthalpy at turbine inlet
• \( h_2 \): Enthalpy at turbine exit
• \( h_3 \): Enthalpy at condenser exit
• \( h_4 \): Enthalpy at pump outlet

These values are obtained from steam tables and are crucial for calculating work and heat transfer in the cycle.

turbine work

The work done by the turbine is the energy extracted from the steam as it expands. This can be calculated using the enthalpies at the turbine inlet and exit: \[ W_{turbine} = h_1 - h_2 \] Turbine work is a major part of the net work output, and understanding it helps in determining the overall efficiency and power output of the cycle.

pump work

Pump work is the energy required to compress the working fluid, bringing it back to the boiler pressure. This work is calculated with the enthalpies of the fluid at the pump inlet and outlet: \[ W_{pump} = h_4 - h_3 \] The work done by the pump is usually much smaller compared to the turbine work, but it's crucial for maintaining the continuous flow of the working fluid in the cycle.

heat transfer

Heat transfer in the Rankine cycle occurs in two main components: the boiler and the condenser.

• In the boiler, heat is added to convert water into superheated steam (\[ Q_{in} = h_1 - h_4 \]).
• In the condenser, heat is removed to convert the steam back into liquid (\[ Q_{out} = h_2 - h_3 \]).

Efficient heat transfer in these components is essential for maximizing the cycle's thermal efficiency.

mass flow rate

The mass flow rate of the working fluid is determined by using the net power output of the cycle and the net work done per unit mass: \[ \dot{m} = \frac{100,000}{W_{net}} \] This helps in understanding how much fluid is cycled per unit time to achieve the desired power output. Additionally, the mass flow rate of the cooling water in the condenser is calculated by the heat balance, utilizing the specific heat capacity of water and the temperature change: \[ Q_{out} = m_c \times C_p \times (T_{exit} - T_{inlet}) \] Knowing these flow rates helps in designing and choosing the appropriate size for components like the pump and turbine.