Question

A piston-cylinder device initially contains \(50 \mathrm{L}\) of liquid water at \(40^{\circ} \mathrm{C}\) and 200 kPa. Heat is transferred to the water at constant pressure until the entire liquid is vaporized. (a) What is the mass of the water? (b) What is the final temperature? (c) Determine the total enthalpy change. (d) Show the process on a \(T\) -v diagram with respect to saturation lines.

Short answer

Question: During a constant-pressure process, liquid water in a piston-cylinder device undergoes a phase change from liquid to vapor. The initial pressure is 200 kPa, the initial volume is 1 m³, and the initial temperature is 40°C. Determine (a) the mass of the water, (b) the final temperature, (c) the total enthalpy change, and (d) represent the process on a T-v diagram with respect to saturation lines. Answer: Follow the step-by-step solution provided to determine (a) the mass of the water, (b) the final temperature, (c) the total enthalpy change, and (d) to represent the process on a T-v diagram with respect to saturation lines. Consult steam tables for water to find specific volumes and enthalpies at the initial and final states, and use the given initial conditions and saturation temperatures for the calculations.

Step-by-step solution

Calculate the initial specific volume and the mass of the water

We are given the initial volume, temperature, and pressure of the water. We can use the initial temperature and pressure to find the specific volume of water in the liquid phase. Consult the steam tables for water and use the given temperature (\(40^{\circ} \mathrm{C}\)) and pressure (200 kPa) to find the specific volume, \(v_1\), of liquid water. Once we have the specific volume, we can calculate the mass of the water using the formula: $$ mass (m) = \frac{total\ volume}{specific\ volume} $$

Find the final temperature

Since the entire liquid is vaporized at constant pressure, the final temperature corresponds to the saturation temperature at the given pressure (200 kPa). Again, consult the steam tables for water and find this final temperature, which we can denote \(T_2\).

Calculate the specific enthalpies and the total enthalpy change

Using the initial temperature and specific volume, find the initial specific enthalpy (\(h_1\)) of liquid water from the steam tables. Then, using the final temperature (\(T_2\)) and the given constant pressure, find the final specific enthalpy (\(h_2\)) of vaporized water in the steam tables. Now, we can calculate the total enthalpy change, \(\Delta H\), using the mass of the water and the specific enthalpies at the initial and final states: $$ \Delta H = m (h_2 - h_1) $$

Draw the T-v diagram for the process

In the T-v diagram, plot the initial state (\(T_1\), \(v_1\)) as a point in the liquid-phase region of the diagram. Then, plot the final state (\(T_2\), \(v_2\)), which should be in the vapor-phase region. Draw an arrow connecting the initial and final states, which represents the constant pressure process. Also, plot the saturation lines (liquid-vapor phase boundaries) on the diagram. The resulting T-v diagram should help visualize the process of heating the water at constant pressure until it turns completely into vapor.