Question

A steam power plant operates on an ideal reheat Rankine cycle between the pressure limits of \(15 \mathrm{MPa}\) and 10 kPa. The mass flow rate of steam through the cycle is \(12 \mathrm{kg} / \mathrm{s} .\) Steam enters both stages of the turbine at \(500^{\circ} \mathrm{C}\) If the moisture content of the steam at the exit of the low pressure turbine is not to exceed 10 percent, determine \((a)\) the pressure at which reheating takes place, ( \(b\) ) the total rate of heat input in the boiler, and \((c)\) the thermal efficiency of the cycle. Also, show the cycle on a \(T\) -s diagram with respect to saturation lines.

Short answer

Question: In a reheat Rankine cycle, steam enters the high-pressure turbine at 15 MPa and 500°C. Steam expands through the turbine and then is reheated in the boiler to 500°C. The steam then enters the low-pressure turbine and expands through the turbine to a condenser pressure of 10 kPa. The moisture content in the steam at the turbine exit must not exceed 10 percent. If the mass flow rate of steam through the cycle is 12 kg/s, calculate (a) the pressure at which reheating takes place, (b) the total rate of heat input in the boiler, and (c) the thermal efficiency of the cycle. Sketch the cycle on a T-s diagram with respect to saturation lines. Answer: (a) The pressure at which reheating takes place is 1000 kPa (1 MPa). (b) The total rate of heat input in the boiler is 34,009.28 kW. (c) The thermal efficiency of the cycle is 43.09%.

Step-by-step solution

Review the ideal reheat Rankine cycle

The ideal reheat Rankine cycle consists of 5 main components: a boiler, a high pressure turbine, a reheater, a low pressure turbine, and a condenser. The cycle operates between two pressure limits, involving a high pressure (15 MPa) and a low pressure (10 kPa), and refers to two temperatures namely \(T_{1}\) = 500°C as the temperature at the entrance to both turbines. The cycle can be described by the following steps: 1. The water is pumped from the condenser (state 1) to the boiler (state 2). The boiler pressure increases. 2. The water is heated in the boiler from state 2 to state 3 (saturation vapor). 3. The steam (state 3) expands in the high pressure turbine to a lower pressure and enthalpy (state 4). 4. The steam is reheated in the boiler from state 4 to state 5. 5. The steam (state 5) expands in the low pressure turbine to state 6. The moisture content in the steam should not exceed 10 percent. 6. The steam (state 6) enters the condenser and releases its heat to the cooling water. The steam condenses and goes back to the initial state 1 to begin the cycle again.

Calculate the pressure at which reheating takes place (a)

Since we need to limit the moisture content of the steam at the exit of the low pressure turbine (state 6) to 10%, we can find the corresponding enthalpy of saturated liquid and saturated vapor at the low pressure (P6 = 10 kPa): \(h_{f6}\) = enthalpy of saturated liquid at P6 \(h_{g6}\) = enthalpy of saturated vapor at P6 By using steam Tables at 10 kPa, we get \(h_{f6} = 191.83 \mathrm{kJ/kg}\) \(h_{g6} = 2584.0 \mathrm{kJ/kg}\) Now, by making an energy balance on the low pressure turbine, with the moisture content being 10%: \(h_6 = 0.9 \times h_{g6} + 0.1 \times h_{f6}\) Calculate \(h_6\): \(h_6 = (0.9 \times 2584.0) + (0.1 \times 191.83) = 2349.65 \mathrm{kJ/kg}\) Using the enthalpy at state 5 and state 6, we can find the pressure at which reheating takes place by checking the steam tables and finding the pressure P5 at which the enthalpy of 2349.65 kJ/kg corresponds. P5 = 1000 kPa So, the pressure at which reheating takes place is 1000 kPa (1 MPa).

Calculate the total rate of heat input in the boiler (b)

To find the total rate of heat input in the boiler, we need to calculate the enthalpies at states 2, 3, 4 (exit of high pressure turbine), and 5 (reheating pressure P5 = 1 MPa and temperature \(T_{5}\) = 500°C) using steam tables. \(h_2 = 632.9 \mathrm{kJ/kg}\) (using boiler pressure, 15 MPa) \(h_3 = 3374.1 \mathrm{kJ/kg}\) (at boiler pressure and turbine inlet temperature, 500°C) \(h_4 = 2741.06 \mathrm{kJ/kg}\) (exit of high pressure turbine and reheating pressure, 1 MPa) \(h_5 = 3373.9 \mathrm{kJ/kg}\) (at reheating pressure and turbine inlet temperature, 500°C) The heat input in the first boiler stage (Qin1) equals the enthalpy difference between states 3 and 2: \(Q_{in1} = h_3 - h_2\) Calculate \(Q_{in1}\): \(Q_{in1} = 3374.1 - 632.9 = 2741.2 \mathrm{kJ/kg}\) The heat input in the reheater (Qin2) equals the enthalpy difference between states 5 and 4: \(Q_{in2} = h_5 - h_4\) Calculate \(Q_{in2}\): \(Q_{in2} = 3373.9 - 2741.06 = 632.84 \mathrm{kJ/kg}\) Now, the total rate of heat input in the boiler can be found by multiplying the sum of Qin1 and Qin2 by the mass flow rate (12 kg/s): Total heat input = (Qin1 + Qin2) × mass flow rate Calculate the total heat input: Total heat input = (2741.2 + 632.84) × 12 = 34009.28 kW So, the total rate of heat input in the boiler is 34,009.28 kW.

Calculate the thermal efficiency of the cycle (c)

The thermal efficiency of the cycle is the ratio between the net work done by the cycle and the heat input. The net work done can be calculated as the difference between the work done by the turbines and the work done by the pump. The work done by the high pressure turbine (W_HP) can be calculated as: \(W_{HP} = h_3 - h_4\) Calculate \(W_{HP}\): \(W_{HP} = 3374.1 - 2741.06 = 633.04 \mathrm{kJ/kg}\) The work done by the low pressure turbine (W_LP) can be calculated as: \(W_{LP} = h_5 - h_6\) Calculate \(W_{LP}\): \(W_{LP} = 3373.9 - 2349.65 = 1024.25 \mathrm{kJ/kg}\) The work done by the pump (W_pump) can be estimated using the isentropic work relation: \(W_{pump} = v_1 \times (P_2 - P_1)\) Where \(v_1 = 0.00101 \mathrm{m^3/kg}\) is the specific volume of saturated liquid at state 1, and \(P_1 = 10 \times 10^3 \mathrm{Pa}\) and \(P_2 = 15 \times 10^6 \mathrm{Pa}\) are the low and high pressures. Calculate \(W_{pump}\): \(W_{pump} = 0.00101 \times (15 \times 10^6 - 10 \times 10^3) = 15.12 \mathrm{kJ/kg}\) Now, the net work done (W_net) can be calculated as: \(W_{net} = (W_{HP} + W_{LP}) - W_{pump}\) Calculate \(W_{net}\): \(W_{net} = (633.04 + 1024.25) - 15.12 = 1642.17 \mathrm{kJ/kg}\) Finally, the thermal efficiency (η) can be calculated as: \(\eta = \frac{W_{net}}{Q_{in1}+Q_{in2}}\) Calculate the thermal efficiency: \(\eta = \frac{1642.17}{2741.2 + 632.84} = 0.4309\) So, the thermal efficiency of the cycle is 43.09%.

Sketch the cycle on the T-s diagram

To illustrate the cycle on a T-s diagram, we must first find the entropies (s) at each state. Using the steam tables, we can obtain the following values: - State 1 (P1 = 10 kPa): \(s_1 = 0.6493 \mathrm{kJ/kg}\) - State 2 (P2 = 15 MPa): \(s_2 = 2.1386 \mathrm{kJ/kg}\) - State 3 (P2 & \(T_{1}\)): \(s_3 = 6.2283 \mathrm{kJ/kg}\) - State 4 (P5 = 1 MPa): \(s_4 = s_3 = 6.2283 \mathrm{kJ/kg}\) - State 5 (P5 & \(T_{1}\)): \(s_5 = 6.6859 \mathrm{kJ/kg}\) - State 6 (P1 & moisture content limit): \(s_6 = s_5 - \Delta s_{6}\) We can now plot the cycle on a T-s diagram with respect to the saturation lines. The diagram will consist of: - A vertical line from state 1 to state 2 (low pressure to boiler pressure) representing the pump operation. - A line from state 2 to state 3 (boiler pressure) representing the heat addition in the boiler. - An isentropic (vertical dashed) line from state 3 to state 4 (reheating pressure) representing the high pressure turbine expansion. - A horizontal line from state 4 to state 5 (reheating pressure & \(T_{1}\)) representing the steam reheating. - An isentropic (vertical dashed) line from state 5 to state 6 (low pressure, moisture content limit) representing the low pressure turbine expansion. - A line from state 6 to state 1 representing the heat removal in the condenser. By drawing the aforementioned lines and points in the T-s diagram, the cycle can be represented visually with respect to the saturation lines.

Key concepts

Steam Power Plant

A steam power plant is an energy conversion system that uses the thermal energy of steam to generate electricity. Typically, it includes major components such as boilers, turbines, condensers, and pumps. In the example of the reheat Rankine cycle, the plant combines these components in such a manner to increase the efficiency of the power generation process.

The process begins with water being pumped to high pressure in the boiler, where it turns into steam. The high-pressure steam then expands through a turbine that generates electricity. The steam may undergo reheating to avoid excessive moisture content in the turbine, which can cause turbine blade erosion and reduce efficiency. The steam then continues to expand in a low pressure turbine and finally condenses back into water in the condenser, completing the cycle.

Understanding each component and its role in the cycle is crucial for analyzing the plant's performance and thermal efficiency.

Thermal Efficiency

Thermal efficiency is a measure of how well an energy conversion system converts heat into work or output. In steam power plants, thermal efficiency is the ratio of the mechanical energy output to the thermal energy input. High thermal efficiency is a goal in steam power plants as it indicates a greater proportion of heat is converted to useful work.

In the case of the reheat Rankine cycle, thermal efficiency is optimized by reheating the steam between turbine stages, as it reduces moisture content, and thus, improves the quality of the steam entering the low pressure turbine. By calculating the net work output and total heat input, one can determine the cycle's efficiency, which is a critical indicator of the plant's performance.

Saturation Lines

Saturation lines on a T-s diagram represent the physical boundaries where water transitions between phases at a particular pressure: solid, liquid, and vapor. Every unique pressure has its saturation points, which are classified as the saturated liquid line and the saturated vapor line. These lines limit the conditions where water exists as a mixture of liquid and vapour.

In a reheat Rankine cycle, it’s essential to ensure that the steam exiting the turbine doesn't cross the saturated vapor line significantly, as excessive moisture content can damage turbine blades. By reheating the steam before it enters the low pressure turbine, we avoid condensation that would otherwise occur if the steam were to expand to the condenser pressure in one stage. The content on saturation lines helps students understand phase changes and the importance of maintaining steam quality within a steam power plant cycle.

T-s Diagram

A T-s (Temperature-Entropy) diagram is used to model thermodynamic processes. This diagram plots temperature (T) against entropy (s), showing how a substance undergoes phase changes and temperature variations with respect to heat addition or removal.

For a steam power plant, the T-s diagram is instrumental for visualizing the different stages of the reheat Rankine cycle, including where heat is added in the boiler and reheater, and where it is removed in the condenser. It provides a clear graphical representation of the cycle, allowing for immediate recognition of the sequence of processes, thus enhancing the understanding of thermodynamic principles, especially when one considers the cycle with respect to saturation lines.

Enthalpy Calculations

Enthalpy is a key concept in thermodynamics, linking the internal energy of a system with the system's pressure and volume. It is used to calculate the heat transfer during processes where pressure may vary, which is common in steam power plants.

In a reheat Rankine cycle, enthalpy calculations are vital for determining the heat added in the boiler and reheater, the work done by the turbines, and ultimately, the thermal efficiency of the cycle. The enthalpy values for different points in the cycle are typically obtained from steam tables or specific equations. Step-by-step solutions often involve using these values to make energy balance calculations around individual components of the power plant, assisting in the understanding of the cycle's energy flows and efficiency.