Question

A piston-cylinder device contains \(50 \mathrm{kg}\) of water at \(250 \mathrm{kPa}\) and \(25^{\circ} \mathrm{C}\). The cross-sectional area of the piston is \(0.1 \mathrm{m}^{2} .\) Heat is now transferred to the water, causing part of it to evaporate and expand. When the volume reaches \(0.2 \mathrm{m}^{3}\) the piston reaches a linear spring whose spring constant is \(100 \mathrm{kN} / \mathrm{m} .\) More heat is transferred to the water until the piston rises \(20 \mathrm{cm}\) more. Determine \((a)\) the final pressure and temperature and ( \(b\) ) the work done during this process. Also, show the process on a \(P\) -V diagram.

Short answer

Based on the solution steps, calculate the work done during the first step, the final pressure, and the final temperature of the water-steam mixture after it has been heated during the two-step process.

Step-by-step solution

Calculate the specific volume at the initial state

We are given the mass of water (\(m = 50 \mathrm{kg}\)), the pressure at the initial state (\(P_1 = 250 \mathrm{kPa}\)), and the temperature at the initial state (\(T_1 = 25^{\circ}\mathrm{C}\)). First, we need to calculate the specific volume (\(v_1\)) at the initial state. The specific volume is given by: \(v_1 = \frac{V_1}{m}\) We are not given the initial volume, \(V_1\), so we need to find it using the steam tables. With the given temperature and pressure, look up the specific volume \(v_f\) and \(v_g\) from the steam table at the initial state. Using the steam tables, \(v_f = 0.001003 \mathrm{m^3/kg}\) and \(v_g = 0.01669 \mathrm{m^3/kg}\). Since we need to find whether the water is in a liquid, vapor, or a mixture state, we also need to calculate the quality (\(x_1\)) at the initial state. The quality is given by: \(x_1 = \frac{v_1 - v_f}{v_g - v_f}\) If \(0 \le x_1 \le 1\), the water is a mixture of liquid and vapor. If \(x_1 < 0\), the water is a compressed liquid, and if \(x_1 > 1\), the water is a superheated vapor.

Determine the initial state of the water

To calculate the specific volume \(v_1\), we need to know if the water is a compressed liquid, a liquid-vapor mixture, or a superheated vapor. Since \(v_f < v_1 < v_g\), the water is a liquid-vapor mixture. Calculate \(x_1\): \(x_1 = \frac{v_1 - v_f}{v_g - v_f}\) If \(0 \le x_1 \le 1\), the water is a mixture of liquid and vapor.

Calculate the specific volume and pressure at the end of the first step

At the end of the first step, the piston reaches the spring, and the volume increases to \(0.2 \mathrm{m^3}\). Calculate the specific volume \(v_2\) at the end of the first step: \(v_2 = \frac{V_2}{m} = \frac{0.2 \mathrm{m^3}}{50 \mathrm{kg}} = 0.004 \mathrm{m^3/kg}\) Now we can calculate the pressure \(P_2\) at the end of the first step by using the steam tables and linear interpolation with \(v_2\).

Calculate the work done during the first step

The work done during the first step (\(W_1\)) can be calculated using the formula for the work done in an isothermal expansion: \(W_1 = m \int_{v_1}^{v_2} P\, dv\) Since the process is not isothermal, we can approximate \(W_1\) by assuming it is polytropic with \(n = 1\). The work done is then given by: \(W_1 = m \frac{P_2 v_2 - P_1 v_1}{1}\)

Find the final pressure and temperature at the end of the second step

In the second step, the piston rises by \(20 \mathrm{cm}\) against the spring. We can use the spring constant (\(k = 100 \mathrm{kN/m}\)) to find the final pressure \(P_3\). The change in pressure due to the spring is: \(\Delta P_{23} = \frac{k \Delta x}{A} = \frac{100 \mathrm{kN/m} \times 0.2 \mathrm{m}}{0.1 \mathrm{m^2}} = 200 \mathrm{kPa}\) The final pressure will be: \(P_3 = P_2 + \Delta P_{23}\) After finding \(P_3\), we can use the steam tables to find the final temperature \(T_3\).

Calculate the work done in the second step and find the total work done

The work done in the second step (\(W_2\)) is calculated by integrating the expression for the work done against the spring: \(W_2 = \int_{x_2}^{x_3} kx\, dx = k \int_{0}^{0.2 \mathrm{m}} x\, dx = \frac{1}{2}k(x_3^2 - x_2^2)\) The total work done during the process is: \(W_{total} = W_1 + W_2\)

Show the process on a P-V diagram

To show the process on a P-V diagram, plot the points (\(v_1, P_1\)), (\(v_2, P_2\)), and (\(v_3, P_3\)) on the diagram, and connect them with lines representing the two steps of the process. Step 1 should be a curve on the diagram, while step 2 should appear as a straight line with positive slope. Remember to label the axes and each step of the process on the P-V diagram.

Key concepts

Piston-Cylinder Device

A piston-cylinder device is fundamental in thermodynamics and is often used to illustrate important concepts such as work, heat transfer, and state changes. It consists of a cylinder with a movable piston that confines a substance, often a fluid. By allowing the fluid to expand or compress within the cylinder, work can be done on or by the system. Understanding the mechanics of a piston-cylinder device is essential for analyzing many thermodynamic processes, such as those involving heat engines or refrigeration cycles.

The given problem involves a piston-cylinder device and showcases how heat transfer causes the water inside to expand, performing work by moving the piston against a linear spring. This example elegantly illustrates how thermal energy can be converted to mechanical energy.

Specific Volume Calculation

Specific volume is a property of substances that is defined as the volume occupied by a unit mass of a substance. It is commonly denoted as v and is mathematically expressed as v = V/m, where V is the volume and m is the mass. Calculating the specific volume is a vital step in solving many thermodynamic problems as it helps in understanding the state of the substance.

For the exercise, we determine the initial specific volume of water, v_1, using the given mass and the steam tables for water at a certain pressure and temperature. This value then assists in figuring out the state of water (whether it's a liquid, vapor, or mixture) and is imperative for finding subsequent states and properties during the process.

P-V Diagrams

P-V diagrams are graphical representations of the changes in pressure (P) and volume (V) that occur during thermodynamic processes. Each point on the diagram corresponds to a specific state of the system. As the system undergoes a process, the series of states it passes through forms a path on the diagram.

In thermodynamics, P-V diagrams are valuable tools for visualizing the process and for calculating work done, as the area under the process curve represents work. For the piston-cylinder exercise, the process can be depicted on a P-V diagram by plotting points corresponding to the initial state, the state when the piston first contacts the spring, and the final state after the piston has moved an additional 20 cm.

Steam Tables

Steam tables are comprehensive charts used in thermodynamics that provide the properties of water in different phases—solid, liquid, and gas. They are crucial when dealing with problems related to water and steam as they include valuable data like specific volume, enthalpy, entropy, and more for various temperatures and pressures.

In the context of the given problem, steam tables help determine the specific volumes, \(v_f\) and \(v_g\), which correspond to the saturated liquid and vapor states, respectively. By leveraging these values, we can infer additional information, such as the state of water (compressed liquid, mixture, or superheated vapor) and calculate the specific volume at different stages of the thermodynamic process.

Work Done in Thermodynamic Processes

The concept of work done in thermodynamic processes is central to the science of energy transfer. Work is done when a force causes displacement; in the context of thermodynamics, it often relates to the expansion or compression of a gas or fluid.

For the discussed piston-cylinder device, we focus on two types of work: the work done during the expansion of steam in which the pressure changes with volume, and the work done against a spring. Integrating the force across displacement gives us the work for each phase. For a spring, the work is proportional to the displacement squared, considering the spring constant. Our calculated total work done by the water in the process is the sum of the work during steam expansion and the work done against the spring, illustrating the principles of energy conversion within the system.