Question

A steam power plant operates on a simple ideal Rankine cycle between the pressure limits of 3 MPa and \(50 \mathrm{kPa} .\) The temperature of the steam at the turbine inlet is \(300^{\circ} \mathrm{C},\) and the mass flow rate of steam through the cycle is \(35 \mathrm{kg} / \mathrm{s}\). Show the cycle on a \(T\) -s diagram with respect to saturation lines, and determine ( \(a\) ) the thermal efficiency of the cycle and \((b)\) the net power output of the power plant

Short answer

Question: Calculate the thermal efficiency and net power output of a steam power plant operating with a simple ideal Rankine cycle. The plant uses steam with a mass flow rate of 30 kg/s, a temperature of 300°C and pressure limits of 3 MPa (turbine inlet) and 50 kPa (exiting condenser).

Step-by-step solution

Draw the T-s diagram with respect to saturation lines

First, we need to sketch the T-s (temperature-entropy) diagram of the Rankine cycle with respect to the given conditions. On the vertical axis, draw a temperature line, and on the horizontal axis, an entropy line. Plot the saturated liquid and vapor lines such that they intersect at the critical point. Then, plot the ideal Rankine cycle as follows: starting at point 1 (saturated liquid before entering the pump), a vertical line will connect point 1 with point 2 (before entering the boiler); from point 2, a horizontal line will connect to point 3 (saturated vapor before entering the turbine); and finally, a vertical line will connect point 3 to point 4 (the condenser outlet).

Determine the enthalpies at different points of the cycle

We need to find the enthalpies at points 1 through 4 to continue our analysis: a) At point 1 (P1 = 50 kPa), use the saturation table to find the enthalpy of the saturated liquid, h1. b) At point 2, use the isentropic pump equation: \(h_2 = h_1 + v_1(P_2 - P_1)\), where \(v_1\) is the specific volume at point 1. Find \(v_1\) from the saturation table. c) At point 3 (P3 = 3 MPa, T3 = 300°C), look up the enthalpy (h3) from the steam table. d) At point 4, assume an isentropic turbine process \(s_4 = s_3\). Use the condition \(P_4 = 50 \, \mathrm{kPa}\) to find the quality of the steam at point 4, \(x_4\). Then, use this quality to calculate h4: \(h_4 = h_{f4} + x_4(h_{g4} - h_{f4})\), where \(h_{f4}\) and \(h_{g4}\) are the enthalpies of saturated liquid and vapor, respectively, at pressure \(P_4\)

Compute the thermal efficiency

To compute the thermal efficiency, we need to determine the heat input and the net work done. Use the following equations: Heat input: \(Q_\text{in} = m(h_3 - h_2)\), Work done by the turbine: \(W_\text{turbine} = m(h_3 - h_4)\), Work done by the pump: \(W_\text{pump} = m(h_2 - h_1)\), Net work done: \(W_\text{net} = W_\text{turbine} - W_\text{pump}\). Calculate the thermal efficiency by the following formula: Thermal efficiency: \(\eta_\text{th} = \frac{W_\text{net}}{Q_\text{in}}\).

Determine net power output

Now that we have found the net work done, we can easily calculate the net power output of the power plant. The net power output is given by the relationship: Net power output = \(W_\text{net} = m \cdot (h_3 - h_4 - h_2 + h_1)\). Plug in the known values and calculate the net power output. Finally, we have the (\(a\)) thermal efficiency of the Rankine cycle and (\(b\)) the net power output of the power plant.

Key concepts

T-s Diagram and Rankine Cycle

A temperature-entropy (T-s) diagram is a visual representation of thermal processes, often used in thermodynamics to analyze cycles like the Rankine cycle, which is fundamental to steam power plants. The T-s diagram plots temperature (T) against entropy (S) and contains key information such as the phase change of water into steam and the different states the working fluid (usually water/steam) undergoes during the cycle.

When illustrating the Rankine cycle on the T-s diagram, we depict four main stages: 1) isentropic compression in a pump, 2) constant pressure heat addition in a boiler, 3) isentropic expansion in a turbine, and 4) constant pressure heat rejection in a condenser. These stages form a loop on the diagram, representing the continuous process of energy conversion from heat into mechanical work, which can then be converted into electricity.

By visualizing these processes, the T-s diagram provides a clear picture of the cycle's efficiency and helps in determining areas for potential improvement. It also aids in understanding how changes in operating conditions, like pressure and temperature, will affect the cycle's performance.

Enthalpy Calculations within the Rankine Cycle

Enthalpy, a measurement of total energy within a system, is a crucial aspect of thermal cycle analysis. Accurately calculating enthalpy at various points in the Rankine cycle allows us to evaluate the performance of thermodynamic processes such as those within a steam power plant.

To find the enthalpies necessary for a Rankine cycle analysis, data is derived from established thermal properties of water and steam. These properties are typically found in steam tables or derived from equations of state. At points throughout the cycle -- entering the pump, entering the boiler, entering the turbine, and leaving the condenser -- the enthalpy is measured or calculated. These values are crucial inputs for further calculations, especially for determining the net work output and the cycle's thermal efficiency.

It is important to follow specific steps, such as calculating the increase in enthalpy due to work done by the pump or the decrease in enthalpy as work is extracted by the turbine, using known constants, specific volume, pressure changes, and qualities of steam.

Thermal Efficiency of the Rankine Cycle

The thermal efficiency of the Rankine cycle is a measure of how effectively a power plant converts heat into work. Higher thermal efficiency indicates that less energy is being wasted in the conversion process, which is advantageous both economically and environmentally.

To calculate the thermal efficiency within the Rankine cycle, we compare the net work output of the cycle to the heat energy input. This involves calculating the heat added in the boiler and the work done by both the turbine and the pump. Typically, the work input by the pump is much smaller than the work output by the turbine, and so it is often seen as a minor subtraction from the turbine work in the efficiency calculation.

Thermal efficiency provides insights into the performance of the cycle and indicates where improvements could be made. By analyzing the efficiency, engineers can make decisions on changes in the cycle that could involve higher initial pressures, higher temperatures, or incorporating features like reheat and regeneration to achieve better efficiency.

Net Power Output of the Power Plant

The net power output is the final, telltale measure of a power plant's performance, reflecting the amount of electrical power that can be delivered to the grid after accounting for all losses within the system. For a Rankine cycle, this is determined by calculating the work produced by the turbine and subtracting the work required to operate the pump.

In determining net power output, the enthalpies at each of the four key points of the Rankine cycle are needed. These, multiplied by the mass flow rate of the steam through the cycle, enable the calculation of the turbine's power output and the pump's power input. The difference between these two values yields the net power output of the power plant, essentially quantifying the power available for external use.

Improving net power output can involve increasing the mass flow rate of the steam, enhancing the efficiency of the turbines and pumps, or optimizing the operating conditions such as pressures and temperatures at which the cycle operates. It's a critical figure not only for operational purposes but also for economic considerations, as it directly correlates to the revenue a power plant can generate.