Question

Steam at \(1 \mathrm{MPa}\) and \(300^{\circ} \mathrm{C}\) is throttled adiabatically to a pressure of 0.4 MPa. If the change in kinetic energy is negligible, the specific volume of the steam after throttling is \((a) 0.358 \mathrm{m}^{3} / \mathrm{kg}\) (b) \(0.233 \mathrm{m}^{3} / \mathrm{kg}\) \((c) 0.375 \mathrm{m}^{3} / \mathrm{kg}\) \((d) 0.646 \mathrm{m}^{3} / \mathrm{kg}\) \((e) 0.655 \mathrm{m}^{3} / \mathrm{kg}\)

Short answer

Answer: The specific volume of the steam after throttling is approximately 0.375 m³/kg.

Step-by-step solution

Determine the initial conditions of the steam

Using the given information, the initial pressure and temperature of the steam are \(P_1 = 1\ \mathrm{MPa}\) and \(T_1 = 300^\circ\mathrm{C}\).

Look up initial enthalpy and specific volume using steam tables

Refer to steam tables (such as the ones found in thermodynamics textbooks or online) to find the initial enthalpy \(h_1\) and the specific volume \(v_1\) at the given initial conditions. For steam at \(P_1 = 1\ \mathrm{MPa}\) and \(T_1 = 300^\circ\mathrm{C}\), the enthalpy \(h_1 = 3051.2\ \mathrm{kJ/kg}\) and the specific volume \(v_1 = 0.233\ \mathrm{m^3/kg}\).

Determine the final pressure of the steam

Steam is throttled to a final pressure, \(P_2 = 0.4\ \mathrm{Mpa}\).

Use the isenthalpic process to determine the final specific volume

Since the throttling process is isenthalpic, the final enthalpy \(h_2\) is equal to the initial enthalpy, i.e., \(h_2 = h_1\). Hence, \(h_2 = 3051.2\ \mathrm{kJ/kg}\). Now, find the specific volume \(v_2\) at \(P_2 = 0.4\ \mathrm{MPa}\) and \(h_2 = 3051.2\ \mathrm{kJ/kg}\) using the steam tables. The specific volume at these conditions is approximately \(v_2 = 0.375\ \mathrm{m^3/kg}\).

Select the correct answer

The specific volume of the steam after throttling is found to be \(v_2 = 0.375\ \mathrm{m^{3}/kg}\). So, the correct alternative is (c) \(0.375\ \mathrm{m^{3}/kg}\).

Key concepts

Isenthalpic Process

An isenthalpic process is a thermodynamic process in which the enthalpy of the working fluid remains constant. This is a typical characteristic of a throttling process, often encountered in refrigeration cycles and high-velocity gas flows. In the context of our example, when steam is throttled, there's no heat exchange with the surrounding environment, and all the energy changes are reflected in the velocity and therefore momentum of the steam rather than in temperature.

When steam passes through a throttle, it experiences a drop in pressure without any work being done on or by the system and without any heat transfer. Since the enthalpy (\(h\) stays the same before and after throttling, the process is isenthalpic. Despite the pressure drop, the specific energy content per mass unit of the fluid does not change. This concept is crucial for designing components like flow valves and is instrumental in various engineering applications, such as the operation of steam turbines.

Steam Tables

Steam tables are invaluable tools used in thermodynamics to find the properties of water and steam at various temperatures and pressures. These tables provide data for specific volume, enthalpy, entropy, and other thermodynamic properties. In solving problems, such as our exercise on throttling, steam tables allow us to determine the condition of steam (whether it's superheated, saturated, or a mixture) based on initial information like temperature and pressure.

For those dealing with thermodynamics, whether in academia or industry, referring to steam tables or using software based on them is part of everyday work. They enable precise calculations in designing and analyzing the performance of boilers, turbines, and condensers, among others. When using steam tables, always ensure that you are referencing the correct category of steam and reading the values correctly to avoid errors in your calculations.

Specific Volume

Specific volume is a property used to describe how much space a certain mass of substance occupies and is given by the volume divided by the mass. It's the inverse of density, indicating the expanse a unit mass will spread out into under certain conditions. In our exercise dealing with steam, the specific volume tells us the volume one kilogram of steam would occupy after the throttling process.

The specific volume is particularly critical in any calculation involving phase changes and can provide insights into the behavior of substances under different pressures and temperatures. In the context of a throttling process, understanding the changes in specific volume despite constant enthalpy can explain phenomena like the cooling effect in refrigerators and the performance of steam in turbines.