Question

Fifteen kg of carbon dioxide \(\left(\mathrm{CO}_{2}\right.\) ) gas is fed to a cylinder having a volume of \(20 \mathrm{~m}^{3}\) and initially containing \(15 \mathrm{~kg}\) of \(\mathrm{CO}_{2}\) at a pressure of 10 bar. Later a pinhole develops and the gas slowly leaks from the cylinder. (a) Determine the specific volume, in \(\mathrm{m}^{3} / \mathrm{kg}\), of the \(\mathrm{CO}_{2}\) in the cylinder initially. Repeat for the \(\mathrm{CO}_{2}\) in the cylinder after the \(15 \mathrm{~kg}\) has been added. (b) Plot the amount of \(\mathrm{CO}_{2}\) that has leaked from the cylinder, in \(\mathrm{kg}\), versus the specific volume of the \(\mathrm{CO}_{2}\) remaining in the cylinder. Consider \(v\) ranging up to \(1.0 \mathrm{~m}^{3} / \mathrm{kg}\).

Short answer

Initial specific volume: 1.33 m³/kg. Specific volume after addition: 0.67 m³/kg. Leak mass calculation: 30 kg - 20 m³/v.

Step-by-step solution

- Specific Volume Calculation (Initial State)

Determine the specific volume of the \(\text{CO}_2\) in the cylinder initially. The specific volume is the volume divided by the mass of the gas. Initially, there is \(\text{15 kg}\) of \(\text{CO}_2\) in a \(\text{20 m}^3\) container.\[\text{Specific Volume} = \frac{\text{Volume}}{\text{Mass}} = \frac{20 \; \text{m}^3}{15 \; \text{kg}} = 1.33 \; \text{m}^3/\text{kg}\]

- Specific Volume Calculation (After Addition)

After adding \(\text{15 kg}\) of \(\text{CO}_2\), the total mass in the cylinder becomes \(\text{15 kg} + 15 \; \text{kg} = 30 \; \text{kg}\). The volume of the cylinder remains \(\text{20 m}^3\).\[\text{Specific Volume} = \frac{\text{Volume}}{\text{Mass}} = \frac{20 \; \text{m}^3}{30 \; \text{kg}} = 0.67 \; \text{m}^3/\text{kg}\]

- Developing Expression for Mass Leakage

Define \(\text{v}\) as the specific volume of \(\text{CO}_2\) remaining in the cylinder after some has leaked. Use the expression \(\text{v} = \frac{\text{20 m}^3}{\text{Mass Remaining}}\) to find the remaining mass.Rearrange the expression to solve for mass remaining:\[\text{Mass Remaining} = \frac{20 \; \text{m}^3}{v}\]Then, use the initial total mass to find the leaked mass:\[\text{Mass Leaked} = 30 \; \text{kg} - \frac{20 \; \text{m}^3}{v}\]

- Plot Leakage vs Specific Volume

Generate a plot of Mass Leaked versus Specific Volume for values of \(\text{v} \) ranging up to \(\text{1.0 m}^3/\text{kg}\).For every value of \(\text{v}\) from \(\text{0.67 to 1.0 m}^3/\text{kg}\), compute the amount of mass that has leaked using the derived formula:\[\text{Mass Leaked} = 30 \; \text{kg} - \frac{20 \; \text{m}^3}{v}\]Plot these values to visualize the relationship.

Key concepts

Specific Volume Calculation

The concept of specific volume is essential in understanding how a gas behaves in different conditions. Specific volume is defined as the volume occupied by a unit mass of a substance. In simpler terms, it tells us how much space each kilogram of gas takes up.
To calculate specific volume, you use the formula:
\[ \text{Specific Volume} = \frac{\text{Volume}}{\text{Mass}} \]
For instance, if you have a cylinder containing 15 kg of \text{CO}_2 in a 20 m³ space, the specific volume would be:
\[ \text{Specific Volume} = \frac{20 \text{ m}^3}{15 \text{ kg}} = 1.33 \text{ m}^3/\text{kg} \]
After adding another 15 kg of \text{CO}_2 to the cylinder, the total mass becomes 30 kg, and the volume remains the same (20 m³). The new specific volume is:
\[ \text{Specific Volume} = \frac{20 \text{ m}^3}{30 \text{ kg}} = 0.67 \text{ m}^3/\text{kg} \]
This decrease in specific volume indicates that more mass of gas is compressed into the same volume.

Mass Leakage

Mass leakage refers to the loss of gas mass due to a leak, often through a small opening like a pinhole. When a gas leaks from a container, it influences the remaining specific volume and pressure inside the system.
In this exercise, after adding 15 kg of CO2 to the cylinder (making a total of 30 kg), the gas starts to leak. The relationship between mass remaining and specific volume can be expressed through:
\[ \text{v} = \frac{20 \text{ m}^3}{\text{Mass Remaining}} \]
Rearranging this to solve for mass remaining, you get:
\[ \text{Mass Remaining} = \frac{20 \text{ m}^3}{\text{v}} \]
Using the total initial mass, the mass that has leaked can be found by:
\[ \text{Mass Leaked} = 30 \text{ kg} - \frac{20 \text{ m}^3}{\text{v}} \]
As specific volume increases, it indicates that more gas has leaked out because the remaining mass decreases.

Ideal Gas Behavior

The ideal gas law models how gases behave under various conditions of temperature and pressure. This law assumes no interactions between gas molecules and that these molecules occupy no volume.
According to the ideal gas law, the relationship between pressure (P), volume (V), temperature (T), and the number of moles (n) is given by:
\[ PV = nRT \]
where R is the universal gas constant. This exercise uses aspects of this law, particularly how pressure and volume relate in a thermodynamic system.
Although real gases like CO2 do not perfectly follow the ideal gas law, at many conditions (especially low pressure and high temperature), their behavior can be approximated by it. This approximation allows for simpler calculations that are very useful in solving thermodynamic problems.

Thermal Systems Analysis

Analyzing thermal systems involves understanding how energy, in the form of heat, transfers between different regions of a system and how it affects the system's state.
This exercise essentially requires us to look at how a gas’s state changes as it leaks out from a closed cylinder. We analyze the specific volume and mass relationships to determine how much gas has been lost.
Important steps in this analysis include:

• Calculating initial and final specific volumes
• Developing expressions to describe mass leakage
• Understanding how changes in volume and pressure influence gas behavior

By breaking down the problem into these key parts, students can better understand the complex interactions within a thermal system, applying fundamental thermodynamic principles to real-world applications.