Question

1-kg of water that is initially at \(90^{\circ} \mathrm{C}\) with a quality of 10 percent occupies a spring-loaded piston-cylinder device, such as that in Fig. \(\mathrm{P} 4-21 .\) This device is now heated until the pressure rises to \(800 \mathrm{kPa}\) and the temperature is \(250^{\circ} \mathrm{C}\). Determine the total work produced during this process, in kJ.

Short answer

Question: Calculate the total work produced during a process involving a spring-loaded piston-cylinder device containing water with initial conditions of 90°C and 10% quality, and final conditions of 250°C and 800 kPa pressure. Answer: The total work produced during the process is 651.8 kJ.

Step-by-step solution

Determine the initial specific volume of the water

Using the given initial state of the water (temperature \(90^{\circ} \mathrm{C}\) and quality of \(10 \%\)), consult steam tables to find the initial specific volume of saturated liquid \(v_{f}\) and saturated vapor \(v_{g}.\) We can determine the initial specific volume \(v_{1}\) using the relation \(v_{1} = v_{f} +x(v_{g} - v_{f})\).

Determine initial pressure of the water

Consult the steam tables once more to find the initial pressure \(P_{1}\), corresponding to the initial temperature of \(90^{\circ} \mathrm{C}\).

Determine the final specific volume of the water

Using the given final state of the water (pressure \(800 \mathrm{kPa}\) and temperature \(250^{\circ} \mathrm{C}\)), consult steam tables to find the final specific volume \(v_{2}\) of the superheated steam.

Calculate the work produced during the process

For a spring-loaded piston-cylinder device, the work produced \(\Delta W\) can be determined using the formula \(\Delta W = P_{1}\left(v_{1} - v_{2}\right)\). Plug in the determined values of \(P_{1}, v_{1},\) and \(v_{2}\) to calculate the total work produced. Now that we have these steps, let's find the values from steam tables and perform the calculations. From steam tables, for \(90^{\circ} \mathrm{C}\): \(v_{f}=0.001029 \mathrm{m}^{3} / \mathrm{kg}\), \(v_{g}=2.3614 \mathrm{m}^{3} / \mathrm{kg}\), \(P_{1}=70.11 \mathrm{kPa}\). Using the initial quality \(x=10 \% = 0.1\), the initial specific volume \(v_{1}\) can be calculated as: \(v_{1} = v_{f} + x(v_{g} - v_{f}) = 0.001029 + 0.1(2.3614 - 0.001029) = 0.23645 \mathrm{m}^{3} / \mathrm{kg}\). For the final state, when the pressure is \(800 \mathrm{kPa}\) and temperature is \(250^{\circ} \mathrm{C}\), we can find the specific volume \(v_{2}\) from steam tables: \(v_{2} = 0.14355 \mathrm{m}^{3} / \mathrm{kg}\). Now, we can calculate the total work produced during the process: \(\Delta W = P_{1}\left(v_{1} - v_{2}\right) = 70.11 \times 10^{3}(0.23645 - 0.14355) = 651.8 \mathrm{kJ}\). Therefore, the total work produced during this process is \(651.8 \mathrm{kJ}\).

Key concepts

Steam Tables

Steam tables are an essential tool in thermodynamics for engineers and scientists working with steam power systems. These tables provide crucial data on the properties of water and steam at various temperatures and pressures. The tables list values such as specific volume, enthalpy, entropy, and internal energy for both saturated and superheated steam states. Understanding how to use steam tables is critical for determining the conditions of water or steam in a thermodynamic process.

For example, when given the temperature, one can look up the corresponding pressure, specific volume of saturated liquid (\(v_f\)), specific volume of saturated vapor (\(v_g\)), and other properties. During a thermodynamic calculation, as with our piston-cylinder device, we often interpolate between the values provided in the steam tables to find the specific properties at the given state of the water or steam. This step is crucial for accurately determining the work done in a process or the heat transferred.

Piston-Cylinder Device

A piston-cylinder device is a common apparatus used in thermodynamics to study the expansion and compression of gases or vapors. In such a device, a piston moves freely within a cylindrical chamber, allowing the gas or steam inside to do work against the piston as it expands. The device can be equipped with springs or weights to apply pressure, and it is often used to illustrate the fundamental principles of work in thermodynamic cycles.

When the gas or vapor within the piston-cylinder is heated, as in our exercise, the pressure and volume can change, causing the piston to move. The calculation of work in a piston-cylinder system depends on the initial and final states of the substance, as well as the pressure exerted by the piston on the gas or vapor. The total work produced is a key parameter in evaluating the performance of engines and other mechanical systems that rely on thermodynamic cycles.

Specific Volume

Specific volume is a fundamental concept in thermodynamics, defined as the volume occupied by a unit mass of a substance. In the case of our exercise, it is the volume in cubic meters that one kilogram of steam occupies at a certain temperature and pressure. The specific volume is crucial for determining other properties of the system such as density, which is the inverse of specific volume, and is key in calculations involving work and heat transfer.

For a saturated mixture as described at the start of the exercise, the specific volume can be calculated by knowing the quality (the fraction that is vapor) and using the specific volumes of saturated liquid and vapor from the steam tables. This derived value helps us understand how the states of steam change during heating or cooling. The specific volume of a substance will differ depending on whether it is in the liquid, vapor, or superheated state and will inform the calculations for the thermodynamic work involved.

Saturated Steam

Saturated steam is steam that is in equilibrium with liquid water at the same temperature and pressure, meaning it cannot absorb additional heat without changing into a liquid. This state is indicative of a phase change and is a crucial concept when calculating energy transfers in steam systems. Saturated steam has a specific quality, which is the percentage by weight of the vapor phase in the steam. Quality is represented by the letter 'x' and ranges from 0 (completely liquid) to 1 (completely vapor).

In our exercise, the initial quality of the water is 10%, which indicates that the steam consists of 10% vapor and 90% liquid by mass. When steam is heated beyond the saturated phase, it becomes superheated. It then exists solely as a gas, independent of the liquid phase. Calculating the work produced when heating saturated steam until it becomes superheated involves using both the initial and final specific volumes, as exemplified in the provided step-by-step solution.