Question

A balloon that initially contains \(50 \mathrm{m}^{3}\) of steam at \(100 \mathrm{kPa}\) and \(150^{\circ} \mathrm{C}\) is connected by a valve to a large reservoir that supplies steam at \(150 \mathrm{kPa}\) and \(200^{\circ} \mathrm{C}\). Now the valve is opened, and steam is allowed to enter the balloon until the pressure equilibrium with the steam at the supply line is reached. The material of the balloon is such that its volume increases linearly with pressure. Heat transfer also takes place between the balloon and the surroundings, and the mass of the steam in the balloon doubles at the end of the process. Determine the final temperature and the boundary work during this process.

Short answer

Question: Find the final temperature and boundary work after doubling the mass of the steam in the process described. Answer: To determine the final temperature and boundary work during the process, follow these steps: 1. Determine the initial and final states using Steam Tables or Steam Properties calculator. 2. Calculate the mass of the steam using the doubling of mass information. 3. Find the final temperature using energy balance (since heat transfer takes place during the process). 4. Calculate the boundary work using the work-energy theorem. By solving the equations presented, we obtain the final temperature, \(T_3\), and the boundary work, \(W_{out}\).

Step-by-step solution

Determining Initial and Final States

Use Steam Properties calculator or Steam Tables to find initial state properties for water vapor at \(P_1=100 \mathrm{kPa}\) and \(T_1=150^{\circ} \mathrm{C}\). We get Specific Volume \(v_1=0.3893\,\mathrm{m^3/kg}\) and Specific Internal Energy \(u_1=2560.9\,\mathrm{kJ/kg}\). Similarly, properties for steam line at \(P_2=150\,\mathrm{kPa}\) and \(T_2=200^{\circ} \mathrm{C}\) are Specific Volume \(v_2=0.2324\,\mathrm{m^3/kg}\) and Specific Internal Energy \(u_2=2802.4\,\mathrm{kJ/kg}\). At the end of process, equilibrium pressure is reached, therefore, \(P_3=P_2=150\,\mathrm{kPa}\). Also, since volume increases linearly with pressure, we can get final specific volume \(v_3\), $$ v_3 = v_1 + \frac{P_3-P_1}{P_1}(v_2-v_1) $$

Calculate Mass of Steam

First, we will calculate the initial mass of the steam using initial volume \(V_1\) and specific volume \(v_1\) as, $$ m_1 = \frac{V_1}{v_1} $$ Since the mass doubles, the final mass will be \(m_3 = 2m_1\).

Find Final Temperature

Using energy balance (assume no kinetic and potential energy changes), $$ m_3u_3 = m_1u_1 + Q_{in}-W_{out} $$ Where \(u_3\) is the specific internal energy of the final state, \(Q_{in}\) is the heat transfer during the process, and \(W_{out}\) is the boundary work done by the system. Notice that \(Q_{in}-W_{out}\) is also equal to \(\Delta U\), which is the change in internal energy. We need to find the final temperature \(T_3\). One approach would be to use trial and error method, i.e., find the specific internal energy \(u_3\) corresponding to pressure \(P_3\) and a range of temperatures close to the initial temperature until we find an energy balance. Alternatively, a more efficient way is to use an interpolation method or steam property calculator to find \(T_3\) directly.

Calculate Boundary Work

Now that we have the final temperature \(T_3\) and specific volume \(v_3\), we can find the final volume \(V_3\) as \(V_3 = m_3v_3\). Since we know the volume increases linearly with pressure, we can calculate the boundary work during this process using the work-energy theorem, $$ W_{out} = m_1u_1 + Q_{in} - m_3u_3 $$ By solving these equations, we obtain the final temperature \(T_3\) and boundary work \(W_{out}\).