Question

Carbon dioxide gas enters a pipe at \(3 \mathrm{MPa}\) and \(500 \mathrm{K}\) at a rate of \(2 \mathrm{kg} / \mathrm{s} . \mathrm{CO}_{2}\) is cooled at constant pressure as it flows in the pipe and the temperature of \(\mathrm{CO}_{2}\) drops to \(450 \mathrm{K}\) at the exit. Determine the volume flow rate and the density of carbon dioxide at the inlet and the volume flow rate at the exit of the pipe using \((a)\) the ideal-gas equation and \((b)\) the generalized compressibility chart. Also, determine \((c)\) the error involved in the first case.

Short answer

Answer: The steps involved in finding the error between the results obtained using the ideal-gas equation and the generalized compressibility chart for carbon dioxide are: 1. Calculate the specific volumes, densities, and volume flow rates at the inlet and exit conditions using the ideal-gas equation. 2. Refer to the generalized compressibility chart for carbon dioxide and determine the compressibility factor (Z) for both the inlet and exit conditions. 3. Calculate the specific volumes, densities, and volume flow rates at the inlet and exit conditions using the compressibility factor (Z) obtained from the chart. 4. Calculate the errors in the specific volumes, densities, and volume flow rates by comparing the results obtained from the ideal-gas equation and the generalized compressibility chart.

Step-by-step solution

Calculate the density and volume flow rate at the inlet

Using the ideal gas equation, we can find the specific volume, \(v_{1} = \frac{V_{1}}{m}\), at the inlet: $$ v_{1} = \frac{R T_{1}}{P_{1}} $$ Where \(P_{1} = 3 \,\text{MPa}\), \(T_{1}=500 \,\text{K}\), and the gas constant for carbon dioxide \(R = 0.189 \,\text{kPa} \cdot \text{m}^{3} / \text{kg} \cdot \text{K}\). Now, we can find the density, \(\rho_{1} = \frac{1}{v_{1}}\), and the volume flow rate, \(Q_{1} = \frac{m}{\rho_{1}}\), where the mass flow rate, \(m = 2 \, \text{kg/s}\).

Calculate the volume flow rate at the exit

We need to find the specific volume, \(v_{2} = \frac{V_{2}}{m}\), at the exit: $$ v_{2} = \frac{R T_{2}}{P_{2}} $$ Where \(P_{2} = 3 \,\text{MPa}\) (constant pressure) and \(T_{2}=450 \,\text{K}\). Then, we can calculate the volume flow rate at the exit, \(Q_{2} = \frac{m}{\rho_{2}}\), where \(\rho_{2} = \frac{1}{v_{2}}\). #b) Using the generalized compressibility chart #

Determine the compressibility factor, Z

For this step, refer to the generalized compressibility chart for carbon dioxide and find the compressibility factor, \(Z\), for both the inlet and exit conditions (pressure \(P_{1}\) and temperatures \(T_{1}\) and \(T_{2}\)).

Calculate the volume flow rates and density with Z

Now, we can use the compressibility factor to calculate the specific volumes at the inlet and exit conditions: $$ v_{1,Z} = \frac{Z R T_{1}}{P_{1}} $$ and $$ v_{2,Z} = \frac{Z R T_{2}}{P_{2}} $$ Then, we can find the densities, \(\rho_{1,Z} = \frac{1}{v_{1,Z}}\) and \(\rho_{2,Z} = \frac{1}{v_{2,Z}}\), and the volume flow rates, \(Q_{1,Z} = \frac{m}{\rho_{1,Z}}\) and \(Q_{2,Z} = \frac{m}{\rho_{2,Z}}\). #c) Determine the error involved in the first case #

Calculate the error between ideal-gas and generalized compressibility results

Finally, let's calculate the errors in the specific volumes, densities, and volume flow rates: $$ \text{Error in specific volumes} = \frac{v_{1,Z} - v_{1}}{v_{1}} \times 100\% $$ $$ \text{Error in density} = \frac{\rho_{1,Z} - \rho_{1}}{\rho_{1}} \times 100\% $$ $$ \text{Error in volume flow rate at the inlet} = \frac{Q_{1,Z} - Q_{1}}{Q_{1}} \times 100\% $$ $$ \text{Error in volume flow rate at the exit} = \frac{Q_{2,Z} - Q_{2}}{Q_{2}} \times 100\% $$

Key concepts

Volume Flow Rate in Thermodynamics

The volume flow rate is a measurement of the volume of fluid that passes through a given surface per unit time. It is typically measured in cubic meters per second (m³/s) or liters per minute (L/min). In thermodynamics and fluid mechanics, it's a crucial concept as it helps in determining the flow characteristics of a gas or liquid in conduits or over surfaces.

In the context of the given exercise, the volume flow rate of carbon dioxide (CO₂) in the pipe is found by dividing the mass flow rate by the density of the CO₂ gas. To get from mass flow rate to volume flow rate using the ideal gas equation, we can employ the formula:
\[Q = \frac{m}{\rho}\]
where \(Q\) is the volume flow rate, \(m\) is the mass flow rate, and \(\rho\) is the density. It's worth noting that when temperature changes at constant pressure, which occurs as the CO₂ is cooled, the density and hence volume flow rate will also change.

Using the Generalized Compressibility Chart

The generalized compressibility chart is a tool used in thermodynamics to account for the non-ideal behavior of real gases under various pressure and temperature conditions. This chart is based on the compressibility factor (Z), which is a measure of the deviation of a real gas from ideal gas behavior.

When using the chart, you locate the gas in question's reduced pressure and reduced temperature to obtain the Z-factor. This factor is then used to correct the ideal gas equation:
\[v = \frac{ZRT}{P}\]
where \(v\) is the specific volume corrected for real gas behavior. In the exercise, this chart is used to calculate the volume flow rate and density of CO₂ at both the inlet and the exit of the pipe, offering a more accurate result than assuming ideal gas behavior. This is particularly important for gases like CO₂, which can significantly deviate from ideal gas behavior under certain conditions.

Properties of Carbon Dioxide

Physical and Chemical Properties

Carbon dioxide (CO₂) is a colorless and odorless gas at room temperature and pressure. It has a molecular weight of 44.01 g/mol, and it's an important greenhouse gas. In the exercise, we focus on the properties of CO₂ under specific conditions: a high pressure of 3 MPa and elevated temperatures. At these states, CO₂'s behavior deviates from the predictions of the ideal gas law, hence the need for a more sophisticated approach like using the generalized compressibility chart.

Behavior Under Pressure and Temperature Variations

Under varying temperatures at constant pressure, the properties of CO₂, such as density and volume flow rate, change. These changes are pivotal for calculations in thermodynamics to ensure processes such as cooling in pipes are accurately analyzed and understood, affecting the overall performance of systems handling CO₂.

Error Analysis in Thermodynamics

Error analysis in thermodynamics involves determining the inaccuracies or deviations that arise when making thermodynamic calculations. Sources of error can include measurement inaccuracies, assumptions in models (like assuming ideal gas behavior), and empirical correlations.

In the exercise, error was calculated to understand the difference between using the ideal gas law and a more precise method, such as one involving the generalized compressibility factor. The errors calculated for specific volume, density, and volume flow rate allow students to grasp the significance of these deviations and the importance of choosing the correct method for calculations. In professional practice, such error analysis is fundamental to enhancing the reliability and safety of thermodynamic systems.