Question

Two \(\mathrm{kg}\) of a two-phase, liquid-vapor mixture of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) exists at \(-40^{\circ} \mathrm{C}\) in a \(0.05 \mathrm{~m}^{3}\) tank. Determine the quality of the mixture, if the values of specific volume for saturated liquid and saturated vapor \(\mathrm{CO}_{2}\) at \(-40^{\circ} \mathrm{C}\) are \(v_{\mathrm{f}}=0.896 \times 10^{-3} \mathrm{~m}^{3} / \mathrm{kg}\) and \(v_{\mathrm{g}}=3.824 \times 10^{-2} \mathrm{~m}^{3} / \mathrm{kg}\) respectively.

Short answer

The quality of the mixture is approximately 0.645.

Step-by-step solution

Write Down Given Data

Identify and note all the given data from the problem:\( m = 2 \, \text{kg} \) (mass of CO2)Volume of tank, \( V = 0.05 \, \text{m}^3 \)Specific volume of saturated liquid CO2 at -40°C, \( v_f = 0.896 \times 10^{-3} \, \text{m}^3 / \text{kg} \)Specific volume of saturated vapor CO2 at -40°C, \( v_g = 3.824 \times 10^{-2} \, \text{m}^3 / \text{kg} \)

Calculate Specific Volume of Mixture

The specific volume of the mixture, \( v \), can be found using the formula: \( v = \frac{V}{m} \).\[ v = \frac{0.05 \, \text{m}^3}{2 \, \text{kg}} = 0.025 \, \text{m}^3 / \text{kg} \]

Use Quality of Mixture Formula

The quality of the mixture, \( x \), is given by the formula: \[ v = v_f + x (v_g - v_f) \] Substituting the values: \[ 0.025 = 0.896 \times 10^{-3} + x (3.824 \times 10^{-2} - 0.896 \times 10^{-3}) \]

Solve for Quality, x

First, simplify the equation: \[ 0.025 = 0.000896 + x (0.03824 - 0.000896) \] \[ 0.025 = 0.000896 + x (0.037344) \] Now, isolate \( x \) on one side of the equation: \[ 0.025 - 0.000896 = x (0.037344) \] \[ 0.024104 = x (0.037344) \] \[ x = \frac{0.024104}{0.037344} \] \[ x \approx 0.645 \]

Key concepts

specific volume

In thermodynamics, specific volume is a fundamental property. It represents the volume occupied by a unit mass of a substance. The formula for specific volume is: \( v = \frac{V}{m} \). Here, \( V \) is the total volume and \( m \) is the mass.

Specific volume is particularly important in analyzing phases of a substance such as liquids and vapors. It helps determine the behavior of substances when they change phases.
In the given exercise, specific volume of the CO2 mixture is found to be 0.025 \( \text{m}^3/\text{kg} \). This is calculated by dividing the tank volume (0.05 \( \text{m}^3 \)) by the mass (2 kg).

saturated liquid

A saturated liquid is a liquid that is about to evaporate. At this state, the liquid has absorbed as much heat as possible without changing phase. This is known as the saturation point.

For the exercise, at -40°C, the specific volume for saturated liquid CO2 is given as \( v_f = 0.896 \times 10^{-3} \text{m}^3 / \text{kg} \). It is crucial in determining other properties of the liquid-vapor mixture.
The heat required to change the phase from liquid to vapor at this temperature causes the specific volume to increase.

saturated vapor

A saturated vapor is a vapor that is about to condense. At this stage, the vapor is in equilibrium with its liquid phase.

In the exercise, the specific volume for saturated vapor CO2 at -40°C is \( v_g = 3.824 \times 10^{-2} \text{m}^3 / \text{kg} \). This value is larger than that of the saturated liquid because vapor occupies more volume than liquid.
Knowing this property is essential for calculating the quality of the mixture and other thermodynamic properties.

two-phase mixture

A two-phase mixture contains both liquid and vapor phases in equilibrium. Understanding the quality of the mixture helps to quantify the proportion of liquid and vapor present.

The quality \( x \) is the ratio of the mass of vapor to the total mass: \( x = \frac{\text{mass of vapor}}{\text{total mass}} \).
This concept is used in the formula \( v = v_f + x (v_g - v_f) \), where the specific volume of the mixture is a function of the specific volumes of the saturated liquid and vapor, and the quality.
In the exercise, solving for quality \( x \) involves inserting values into the equation and simplifying. It shows how specific volumes relate to the amount of liquid and vapor phases in a mixture.