Question

Consider a 210-MW steam power plant that operates on a simple ideal Rankine cycle. Steam enters the turbine at \(10 \mathrm{MPa}\) and \(500^{\circ} \mathrm{C}\) and is cooled in the condenser at a pressure of \(10 \mathrm{kPa}\). Show the cycle on a \(T\) -s diagram with respect to saturation lines, and determine ( \(a\) ) the quality of the steam at the turbine exit, \((b)\) the thermal efficiency of the cycle, and \((c)\) the mass flow rate of the steam.

Short answer

Answer: To determine the quality of the steam at the turbine exit, we use the isentropic efficiency formula, \(x_2 = \frac{s_2 - s_{f_2}}{s_{g_2} - s_{f_2}}\). To find the thermal efficiency of the cycle, we apply the formula \(\eta_{th} = \frac{W_{net, out}}{Q_{in}} = \frac{(h_1 - h_2) - (h_4 - h_3)}{(h_1 - h_4)}\). After determination of enthalpy and entropy at each key point, we can calculate the specific quality and thermal efficiency values for this Rankine cycle.

Step-by-step solution

1. Draw the T-s diagram with respect to saturation lines for the Rankine cycle

Sketch a T-s diagram to represent the simple ideal Rankine cycle. It has four main points: (1) entering the turbine, (2) exiting the turbine, (3) exiting the condenser, and (4) exiting the pump. To draw the T-s diagram, you should first set the turbines' input requirements: \(10 \mathrm{MPa}\), \(500^{\circ} \mathrm{C}\) and then set the condensers' output requirements: \(10 \mathrm{kPa}\).

2. Determine the enthalpy and entropy at each key point of the cycle

We need to calculate the enthalpy and entropy at the four key points of the cycle, designated \(h_1\), \(h_2\), \(h_3\), and \(h_4\). - At point 1 (turbine inlet): Using the given input data, we can look up the enthalpy and entropy values \(h_1\) and \(s_1\) in the steam tables. \(h_1 = 3379.2 \mathrm{kJ/kg}\), \(s_1 = 6.5859 \mathrm{kJ/kg \cdot K}\). - At point 2 (turbine outlet): Since the process is isentropic, \(s_2 = s_1\). From this, we can find \(h_2\) in the steam table for the given condenser pressure. - At point 3 (condenser outlet): We can find the value of \(h_3\) in the steam table, which is the saturated liquid state for the given condenser pressure \(10 \mathrm{kPa}\). - At point 4 (pump outlet): For the isentropic process, we can find \(h_4\) by using \(h_4 = h_3 + v_3 (P_4 - P_3)\) and the specific volume and pressures from the steam table.

3. Calculate the quality of the steam at the turbine exit

To find the quality of the steam at the turbine exit (point 2), we can use the following formula for isentropic efficiency: \(x_2 = \frac{s_2 - s_{f_2}}{s_{g_2} - s_{f_2}}\) where \(x_2\) is the quality of the steam, \(s_2\) is the entropy at point 2, \(s_{f_2}\) is the entropy of the saturated liquid at the condenser pressure, and \(s_{g_2}\) is the saturated vapor entropy at the condenser pressure.

4. Evaluate the thermal efficiency of the cycle

The thermal efficiency of the cycle can be determined using the following formula: \(\eta_{th} = \frac{W_{net, out}}{Q_{in}} = \frac{(h_1 - h_2) - (h_4 - h_3)}{(h_1 - h_4)}\) where \(W_{net, out}\) is the net work output and \(Q_{in}\) is the heat input of the cycle.

5. Determine the mass flow rate of the steam in the cycle

To compute the mass flow rate of the steam, we should use the following formula: \(\dot{m} = \frac{P}{(h_1 - h_2) - (h_4 - h_3)}\) where \(P\) is the power output of the cycle (given as \(210 \mathrm{MW}\)).