Question

Water initially at 200 kPa and \(300^{\circ} \mathrm{C}\) is contained in a piston-cylinder device fitted with stops. The water is allowed to cool at constant pressure until it exists as a saturated vapor and the piston rests on the stops. Then the water continues to cool until the pressure is \(100 \mathrm{kPa}\). On the \(T\) -v diagrams sketch, with respect to the saturation lines, the process curves passing through both the initial, intermediate, and final states of the water. Label the \(T, P\) and \(v\) values for end states on the process curves. Find the overall change in internal energy between the initial and final states per unit mass of water.

Short answer

Answer: To find the overall change in internal energy, we need to first find the internal energy values of the initial and final states from the steam tables. Then, we can calculate the overall change by subtracting the initial internal energy from the final internal energy per unit mass of water, which is given by the formula: \(\Delta u = u_3 - u_1\).

Step-by-step solution

Identify the initial state and locate it on the \(T\)-v diagram.

We are given that the water is initially at \(200 \, \mathrm{kPa}\) and \(300^{\circ} \mathrm{C}\). Using the saturation temperature table, we find that the saturation temperature at \(200 \, \mathrm{kPa}\) is \(T_{sat} = 120.23^{\circ} \mathrm{C}\). Since the initial temperature is higher than the saturation temperature, we can conclude that the water is initially in the superheated vapor region.

Find the intermediate state and locate it on the \(T\)-v diagram.

When the water cools down at constant pressure, it will eventually reach the saturated vapor state. Thus, the intermediate state will be at the same pressure as the initial state (\(P = 200 \, \mathrm{kPa}\)) but at the saturation temperature, which we found in step 1 to be \(T_{sat} = 120.23^{\circ} \mathrm{C}\).

Find the final state and locate it on the \(T\)-v diagram.

The final state is reached when the pressure drops to \(100 \, \mathrm{kPa}\). Since the water is still cooling down, we know that the temperature will also decrease. So, the final state will be below the saturation line at \(P = 100 \, \mathrm{kPa}\).

Sketch the process curves on the \(T\)-v diagram.

We now need to sketch the process curves on the \(T\)-v diagram. The initial state is in the superheated vapor region, the intermediate state is on the saturated vapor line, and the final state is below the saturation line at \(P = 100 \, \mathrm{kPa}\). First, sketch the process curve that connects the initial and the intermediate states, which is a constant pressure cooling process. Then, sketch the process curve that connects the intermediate and the final states, which is a cooling process with the piston stopped.

Calculate the overall change in internal energy.

To find the overall change in internal energy, we need the internal energy values for the initial, intermediate, and final states. We can find these values from the steam tables: For the initial state (State 1): \(u_1\) is the internal energy of superheated vapor at \(200 \, \mathrm{kPa}\) and \(300^\circ \mathrm{C}\). For the intermediate state (State 2): \(u_2\) is the internal energy of saturated vapor at \(200 \, \mathrm{kPa}\). For the final state (State 3): \(u_3\) is the internal energy of water at \(100 \, \mathrm{kPa}\) and some temperature below the saturation temperature. Now, calculate the overall change in internal energy per unit mass of water: \(\Delta u = u_3 - u_1\). Note that \(u_2\) is not used in this calculation since we are only interested in the initial and final states.