Two tanks are connected by a tube 12 mm in diameter and 0.8 m long with a removal barrier separating the two. One tank contains nitrogen and the other contains carbon dioxide. The... Two tanks are connected by a tube 12 mm in diameter and 0.8 m long with a removal barrier separating the two. One tank contains nitrogen and the other contains carbon dioxide. The pressure and temperature are maintained at 450 kPa and 35°C throughout the system. The diffusivity is 0.08 cm²/s (for nitrogen DAB DBA). At t=0, the barrier between the two tanks is opened. At some point, the system reaches steady state and the mole fraction of N2 reaches 0.8 on one side and 0.55 on the other. Determine at this time: i) The rates and directions of mass transfer of CO2 and N2 ii) The velocities of each species iii) The molecular average velocity of the mixture. Assume mass transfer resistance is only in the tube which is 0.8 m long. The system is a perfectly well-mixed IDEAL gas. Equimolar counter diffusion takes place. yCO2 = 0.25, yN2 = 0.75, T = 25°C, P = 200 kPa.
Understand the Problem
The question is asking for the calculation of mass transfer rates and velocities of nitrogen and carbon dioxide under specified conditions. Given the parameters such as diffusivity, temperatures, and pressures, we will use the concepts of mass transfer and gas diffusion to derive the required answers.
Answer
To determine the velocities, use $v_N = \frac{D_N}{R T_N}$ and $v_{CO_2} = \frac{D_{CO_2}}{R T_{CO_2}}$. Specific calculated values depend on the parameters given.
Answer for screen readers
The velocity of nitrogen is given by $v_N = \frac{D_N}{R T_N}$, and for carbon dioxide, it’s $v_{CO_2} = \frac{D_{CO_2}}{R T_{CO_2}}$. The specific calculated values will depend on the provided parameters.
Steps to Solve
- Identify Key Parameters
First, list the parameters provided in the problem. This will typically include diffusivity ($D$), temperatures ($T$), pressures ($P$), and molar masses ($M$) of nitrogen and carbon dioxide.
- Calculate Molar Masses
Determine the molar mass of nitrogen ($M_N$) and carbon dioxide ($M_{CO_2}$). The molar mass of nitrogen is approximately $28 , \text{g/mol}$, and for carbon dioxide, it is about $44 , \text{g/mol}$.
- Calculate the Diffusion Coefficients
Use the provided diffusivity or, if needed, calculate it using correlations such as the Chapman-Enskog equation when applicable for different gases at the given conditions.
- Apply Fick’s Law of Diffusion
Use Fick's law to calculate the mass transfer rates. Fick's first law states that the rate of flux ($J$) of a species is proportional to the concentration gradient:
$$ J = -D \frac{dC}{dz} $$
- Calculate Velocity of Diffusion
To find the velocity ($v$) of the gases, use the formula:
$$ v = \frac{D}{R T} $$
where $R$ is the universal gas constant and $T$ is the absolute temperature in Kelvin.
- Substitute the Known Values
Plug in the identified constants and parameters into the equations you've set up in the previous steps to find the desired mass transfer rates and velocities.
The velocity of nitrogen is given by $v_N = \frac{D_N}{R T_N}$, and for carbon dioxide, it’s $v_{CO_2} = \frac{D_{CO_2}}{R T_{CO_2}}$. The specific calculated values will depend on the provided parameters.
More Information
The calculations of mass transfer rates are vital in processes like chemical manufacturing and environmental engineering, where the control of gas-phase reactions is important.
Tips
- Forgetting to convert temperatures to Kelvin before using them in calculations.
- Confusing the sequence of operations; make sure to apply the laws of diffusion correctly and substitute parameters properly.
- Misidentifying molar masses or diffusivities, which can lead to incorrect calculations.
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