Question

Using EES (or other) software, investigate the effect of the condenser pressure on the performance of a simple ideal Rankine cycle. Turbine inlet conditions of steam are maintained constant at \(10 \mathrm{MPa}\) and \(550^{\circ} \mathrm{C}\) while the condenser pressure is varied from 5 to 100 kPa. Determine the thermal efficiency of the cycle and plot it against the condenser pressure, and discuss the results.

Short answer

Answer: To determine the effect of varying condenser pressure on the thermal efficiency of a simple ideal Rankine cycle, follow the step-by-step procedure for calculating the thermal efficiency for different condenser pressures. Once the efficiencies are calculated, plot them against the condenser pressure values. Analyze and discuss the trend observed in the plotted graph to understand the relationship between condenser pressure and cycle efficiency, and the reasons behind the observed trends and their impact on the performance of the simple ideal Rankine cycle.

Step-by-step solution

Obtain the inlet conditions for the turbine

The inlet conditions for the steam turbine are given as \(10 \mathrm{MPa}\) and \(550^{\circ} \mathrm{C}\). We will use these as the initial conditions for our Rankine cycle analysis.

Find enthalpy and entropy at the turbine inlet

We need to determine the enthalpy \(h_1\) and entropy \(s_1\) at the given turbine inlet. To do this, use the steam table data for the given pressure and temperature values.

Determine isentropic conditions for the exit of the turbine

The steam expands isentropically in an ideal Rankine cycle. Therefore, the entropy at the exit of the turbine (\(s_2\)) will be equal to the entropy at the inlet of the turbine, \(s_1\). With this entropy value and the given condenser pressures, you can obtain the steam table values for the isentropic conditions at the exit of the turbine.

Calculate work output from the turbine

Now we can determine the work output from the steam turbine using the enthalpies obtained in Steps 2 and 3. The work output (\(W_t\)) can be calculated as: \(W_t = h_1 - h_2\)

Obtain enthalpy at the condenser exit (pump inlet)

The enthalpy at the exit of the condenser (or the pump inlet, \(h_3\)) can be obtained using the steam table values at the saturated liquid state corresponding to the given condenser pressure.

Calculate work input to the pump

Using the values of enthalpy at the pump inlet (\(h_3\)) and the enthalpy at the turbine inlet (\(h_1\)), you can calculate the work input to the pump (\(W_p\)): \(W_p = v_3(P_1 - P_3)\)

Determine heat input to the cycle

The heat input to the cycle (\(Q_in\)) occurs between the pump exit and the turbine inlet. You can calculate the heat input using the enthalpies at the turbine inlet (\(h_1\)) and the pump exit (\(h_4\)): \(Q_{in} = h_1 - h_4\)

Calculate the thermal efficiency

The thermal efficiency of the cycle (\(\eta\)) can be calculated using the heat input (\(Q_{in}\)) and work output (\(W_{net}\)) of the cycle: \(\eta = \frac{W_{net}}{Q_{in}} = \frac{W_t - W_p}{Q_{in}}\)

Plot efficiency against condenser pressure

Now, using the values of condenser pressure ranging from \(5\) to \(100 \mathrm{kPa}\), repeat the steps to calculate the thermal efficiency for the different condenser pressures. Plot the efficiency values against the condenser pressure values.

Discuss the results

Interpret the results from the plotted graph to understand the trend and relationship between condenser pressure and cycle efficiency. Explain the reasons for the observed trends and how they impact the performance of a simple ideal Rankine cycle.