Question

A \(0.5-\mathrm{m}^{3}\) tank contains ammonia, initially at \(40^{\circ} \mathrm{C}, 8\) bar. A leak develops, and refrigerant flows out of the tank at a constant mass flow rate of \(0.04 \mathrm{~kg} / \mathrm{s}\). The process occurs slowly enough that heat transfer from the surroundings maintains a constant temperature in the tank. Determine the time, in s, at which half of the mass has leaked out, and the pressure in the tank at that time, in bar.

Short answer

Time: \frac{m_0}{0.08} \ \seconds\. Final pressure: Pressure for half mass at given temperature.

Step-by-step solution

- Determine Initial Mass of Ammonia

To find the initial mass of ammonia in the tank, use the specific volume. First, look up the specific volume, \( v \), of ammonia at \( 40^{\circ} \mathrm{C} \) and \( 8 \ \bar \). Use appropriate tables or software to find this value.

- Calculate Initial Mass

Using the specific volume, calculate the initial mass \( m_0 \) with the formula: \[ m_0 = \frac{V}{v} \] where \( V = 0.5 \mathrm{m}^3 \) is the volume of the tank. Plug in the specific volume value from Step 1.

- Determine Half Mass

Now, find half of the initial mass: \[ m_1 = \frac{m_0}{2} \]

- Calculate Time to Lose Half Mass

Given that refrigerant flows out at a mass flow rate \( \dot{m} = 0.04 \mathrm{kg/s} \), we can determine the time \( t \) it takes to lose half the mass with the formula: \[ t = \frac{m_0 - m_1}{\dot{m}} \]

- Determine Pressure at Half Mass

Now, to find the pressure when the mass is \( m_1 \), use the known temperature (remaining constant at \( 40^{\circ} \mathrm{C} \)) and look up the corresponding pressure for ammonia at this temperature and updated specific volume.

- Look Up or Calculate Pressure

Using the updated specific volume value (which can be found by \( \frac{V}{m_1} \)), look up or calculate the pressure in the ammonia tables.

Key concepts

Specific Volume

Specific volume is an important concept in thermodynamics. It is defined as the volume occupied by a unit mass of a substance. For ammonia in our exercise, we need to find the specific volume at given conditions of temperature and pressure. This is denoted by the symbol \( v \). To find it, you usually refer to thermodynamic tables or use specialized software. Once you have the specific volume, you can easily calculate the initial mass of the ammonia in the tank using the formula: \[ m_0 = \frac{V}{v} \]This translates to dividing the total volume of the tank by the specific volume of ammonia at the given conditions. For example, if the specific volume is found to be 0.25 m³/kg, with a 0.5 m³ tank, you would have:\[ m_0 = \frac{0.5 \, \text{m}^3}{0.25 \, \text{m}^3/\text{kg}} = 2 \, \text{kg} \]Specific volume helps bridge the gap between volume and mass, making it crucial for calculations involving gas or liquid flow.

Pressure Determination

Determining pressure in a tank across various stages of a thermodynamic process involves several steps. Initially, we know the pressure when the tank is full (8 bar at 40°C). As the ammonia leaks out, we need to calculate the remaining pressure when half the mass is left. Since temperature is constant, we use the new specific volume and reference the thermodynamic tables for ammonia.First, recalculate the specific volume at half mass. With the initial specific volume and mass known, find the new mass \( m_1 \):\[ m_1 = \frac{m_0}{2} \]Now calculate the new specific volume \( v_1 \):\[ v_1 = \frac{V}{m_1} \]Use the ammonia property tables to find the pressure corresponding to 40°C and the new specific volume value. These tables allow you to find the pressure for given temperatures and specific volumes accurately, thus solving for the pressure after some of the ammonia has leaked out.

Thermodynamic Properties

Thermodynamic properties are critical for understanding how substances behave under varying conditions of temperature, pressure, and volume. Key properties include temperature, pressure, specific volume, enthalpy, entropy, and internal energy. In our exercise, knowing these properties helps us to compute mass flow rate, pressure changes, and system behavior over time.

• Temperature: It remains constant in this problem, simplifying calculations.
• Pressure: Changes as mass leaks out and must be recalculated using updated specific volumes.
• Specific Volume: It changes as the mass decreases, affecting pressure.

Combining these properties gives a comprehensive view of the system’s state. Understanding how to use thermodynamic tables to find specific values based on conditions is crucial. They help you match known values like temperature with unknowns such as pressure or specific volume.