Question

A piston-cylinder device contains \(0.005 \mathrm{m}^{3}\) of (1) liquid water and \(0.9 \mathrm{m}^{3}\) of water vapor in equilibrium at \(600 \mathrm{kPa}\). Heat is transferred at constant pressure until the temperature reaches \(200^{\circ} \mathrm{C}.\) (a) What is the initial temperature of the water? (b) Determine the total mass of the water (c) Calculate the final volume. \((d)\) Show the process on a \(P\) -v diagram with respect to saturation lines

Short answer

Answer: The initial temperature is 158.85°C, the total mass is 7.53059 kg, the final volume is 1.90841 m³, and the process can be represented on a P-v diagram as a straight line connecting initial and final points, starting in the two-phase region and moving to the right, away from the saturation lines.

Step-by-step solution

1. Finding the initial temperature

To find the initial temperature, locate the given pressure (\(600 \mathrm{kPa}\)) in the steam table and look for the saturation temperature corresponding to this pressure. According to the steam table, the saturation temperature at \(600 \mathrm{kPa}\) is \(158.85^{\circ}\mathrm{C}\). Thus, the initial temperature of the water is \(158.85^{\circ}\mathrm{C}\).

2. Finding the total mass of the water

We will first find the mass of each phase (liquid and vapor) and then combine them to find the total mass. For this, we'll utilize the specific volume (\(v\)) of the liquid and vapor phase at the given pressure, which can be found in the steam table. At \(600 \mathrm{kPa}\), saturation properties are as follows: \(v_{f} = 0.00109 \mathrm{m^3/kg}\) (specific volume of liquid) \(v_{g} = 0.3056 \mathrm{m^3/kg}\) (specific volume of vapor) Now, we can find the mass of each phase by dividing the given volume by the specific volume: \(m_{f} = \frac{V_{f}}{v_{f}} = \frac{0.005 \mathrm{m^3}}{0.00109 \mathrm{m^3/kg}} = 4.58716 \mathrm{kg}\) (mass of liquid) \(m_{g} = \frac{V_{g}}{v_{g}} = \frac{0.9 \mathrm{m^3}}{0.3056 \mathrm{m^3/kg}} = 2.94343 \mathrm{kg}\) (mass of vapor) Total mass of water is the sum of the masses of liquid and vapor phases: \(m_{total} = m_{f} + m_{g} = 4.58716 \mathrm{kg} + 2.94343 \mathrm{kg} = 7.53059 \mathrm{kg}\)

3. Calculate the final volume

Now that we know the total mass of water, we can calculate the final volume after heating. First, find the specific volume of the water at the final temperature of \(200^{\circ}\mathrm{C}\) and \(600 \mathrm{kPa}\) using the steam table: At \(200^{\circ}\mathrm{C}\) and \(600 \mathrm{kPa}\), the properties are as follows: \(v_{2} = 0.2535 \mathrm{m^3/kg}\) (specific volume) Next, we can calculate the final volume by multiplying the total mass by the specific volume at the final temperature: \(V_{final} = m_{total} \times v_{2} = 7.53059 \mathrm{kg} \times 0.2535 \mathrm{m^3/kg} = 1.90841 \mathrm{m^3}\)

4. P-v Diagram

To plot the process on a Pressure-Volume (P-v) diagram with respect to saturation lines, start by plotting the initial and final points: - Initial point: \(P_1 = 600 \mathrm{kPa}, V_1 = V_{f} + V_{g} = 0.005 \mathrm{m^3} + 0.9 \mathrm{m^3} = 0.905 \mathrm{m^3}\) - Final point: \(P_2 = 600 \mathrm{kPa}, V_2 = 1.90841 \mathrm{m^3}\) Next, plot the saturation lines on the P-v diagram. The process will be a straight line connecting the initial and final points, as the pressure remains constant throughout the heating process. The process begins in the two-phase region and moves to the right, away from the saturation lines, as the temperature increases and the volume expands.