Question

An \(\quad\) insulated, vertical piston-cylinder device initially contains \(10 \mathrm{kg}\) of water, \(6 \mathrm{kg}\) of which is in the vapor phase. The mass of the piston is such that it maintains a constant pressure of \(200 \mathrm{kPa}\) inside the cylinder. Now steam at \(0.5 \mathrm{MPa}\) and \(350^{\circ} \mathrm{C}\) is allowed to enter the cylinder from a supply line until all the liquid in the cylinder has vaporized. Determine ( \(a\) ) the final temperature in the cylinder and \((b)\) the mass of the steam that has entered.

Short answer

Answer: To find the final temperature in the cylinder and the mass of the steam that has entered, follow these steps: 1. Identify the initial properties of water in the liquid and vapor phases in the cylinder using a saturated water table at a constant pressure of 200 kPa. 2. Calculate the heat required to vaporize all liquid water in the cylinder. 3. Determine the properties of the steam entering the cylinder using a superheated steam table with a given pressure of 0.5 MPa and temperature of 350°C. 4. Calculate the mass of steam needed to provide the required heat. 5. Find the final mass of water in the cylinder by adding the initial mass and the mass of steam that entered. 6. Determine the final specific volume in the cylinder and use a saturated water table to find the corresponding temperature at a constant pressure of 200 kPa. The final temperature and mass of steam that entered the cylinder can be obtained by performing these calculations.

Step-by-step solution

Identify the initial properties in the cylinder.

To determine the initial situation, we know that the total mass of water in the cylinder is 10 kg, with 6 kg of this mass in the vapor phase. The pressure in the cylinder is kept constant at 200 kPa. Using these values, we can find the initial properties of water in each phase using the saturated water table at 200 kPa pressure. We'll find the initial specific volume (v), internal energy (u), and enthalpy (h) for both liquid and vapor phases in the cylinder.

Determine the heat required to vaporize all liquid water.

To calculate the heat required, we can use the following formula for the heat added during the vaporization process: \(Q = m_l \times (h_{v_f} - h_{l_f})\) Where \(Q\) is the heat added, \(m_l\) is the mass of liquid water, and \(h_{v_f}\) and \(h_{l_f}\) are the initial enthalpy values of the vapor and liquid phases, respectively. Now, we can plug in the values to get the heat required.

Identify the properties of the steam entering the cylinder.

We are given that the steam entering the cylinder has a pressure of 0.5 MPa and a temperature of \(350^{\circ} \mathrm{C}\). Using these values, we can find the specific volume (\(v_g\)), internal energy (\(u_g\)), and enthalpy (\(h_g\)) for the entering steam using the superheated steam table.

Calculate the mass of steam needed to provide the required heat.

To find the mass of steam that entered the cylinder (\(m_g\)) during the process, we can use the equation: \(Q = m_g \times (h_{in} - h_{i_v})\) Where \(h_{in}\) is the enthalpy of the incoming steam and \(h_{i_v}\) is the initial enthalpy of the vapor phase in the cylinder. Now, we can solve for the mass: \(m_g = \frac{Q}{(h_{in} - h_{i_v})}\)

Calculate the final properties in the cylinder.

The final mass of water in the cylinder is the sum of the initial mass plus the mass of steam that entered: \(m_{final} = m_l + m_{i_v} + m_g\) To find the final temperature in the cylinder, we'll use the relation of specific volume, mass, and total volume: \(v_{final} = \frac{m_{final}}{m_l \times v_{l_f} + m_{i_v} \times v_{i_v} + m_g \times v_g}\) Now, we can use the saturated water table to find the temperature corresponding to the final specific volume and the constant pressure of 200 kPa. This temperature is the final temperature in the cylinder.

Key concepts

Piston-Cylinder Device

The piston-cylinder device is a fundamental component used in thermodynamic studies to illustrate a variety of processes, including vaporization. Essentially, it is a closed system consisting of a cylinder fitted with a sliding piston. This mechanism allows for the volume inside the cylinder to change while accommodating the movement of the piston.

In our exercise, the piston-cylinder device starts with water in a partially vaporized state, under a constant pressure maintained by the piston's mass. This setup is crucial in understanding how heat addition (or removal) can induce a change of state in a substance within a controlled environment. Importantly, the constant pressure plays a pivotal role in the vaporization process, as it dictates the temperatures at which boiling occurs and guides the use of thermodynamic tables.

Enthalpy of Vaporization

The enthalpy of vaporization, symbolized as \( h_{fg} \), is a thermodynamic property representing the energy per unit mass required to transform a substance from the liquid phase to the vapor phase at a constant pressure and temperature. In the given exercise, calculating the enthalpy of vaporization is key to understanding how much energy is needed to vaporize the remaining liquid water.

The enthalpy change involved during vaporization (\( h_{v_f} - h_{l_f} \) in the equation) gives us the heat (\( Q \) ) added to the system. With this knowledge, students can then calculate the total heat required to fully vaporize the liquid water in the piston-cylinder device under the given conditions.

Saturated Water Table

The saturated water table contains valuable data for various properties of water at different pressures and temperatures where water coexists in equilibrium as both liquid and vapor - the saturated state. These tables are an indispensable resource in our exercise for determining the properties of water in each phase at the specified pressures.

The initial state of water in the exercise is deduced by looking up the corresponding values for specific volume, internal energy, and enthalpy at 200 kPa in the saturated water table. This initial state information is a stepping-stone to further calculations, such as the heat required for vaporization and the end state after the vaporization process is complete. Familiarity with how to use these tables is crucial for students exploring thermodynamics.

Superheated Steam Properties

Superheated steam is steam at a temperature higher than its saturation temperature for the corresponding pressure. In the exercise, understanding superheated steam properties is vital since the cylinder receives steam from a supply line at an elevated temperature and pressure. These properties include specific volume (\( v_g \)), internal energy (\( u_g \)), and most importantly for this calculation, enthalpy (\( h_g \) ).

By referring to superheated steam property tables, we can find the enthalpy of incoming steam, which determines how much mass of this steam is needed to achieve complete vaporization of the liquid inside the cylinder. Knowing to consult the appropriate data for superheated steam further reflects one's comprehensive understanding of steam tables and the thermodynamic properties associated with changing phases under different conditions.