91Ó°ÊÓ

An adiabatic membrane separation unit is used to dry (remove water vapor from) a gas mixture containing 10.0 mole \(\% \mathrm{H}_{2} \mathrm{O}(\mathrm{v}), 10.0\) mole \(\% \mathrm{CO},\) and the balance \(\mathrm{CO}_{2} .\) The gas enters the unit at \(30^{\circ} \mathrm{C}\) and flows past a semipermeable membrane. Water vapor permeates through the membrane into an air stream. The dried gas leaves the separator at \(30^{\circ} \mathrm{C}\) containing \(2.0 \mathrm{mole} \% \mathrm{H}_{2} \mathrm{O}(\mathrm{v})\) and the balance \(\mathrm{CO}\) and \(\mathrm{CO}_{2}\). Air enters the separator at \(50^{\circ} \mathrm{C}\) with an absolute humidity of \(0.002 \mathrm{kg} \mathrm{H}_{2} \mathrm{O} / \mathrm{kg}\) dry air and leaves at \(48^{\circ} \mathrm{C}\). Negligible quantities of \(\mathrm{CO}, \mathrm{CO}_{2}, \mathrm{O}_{2},\) and \(\mathrm{N}_{2}\) permeate through the membrane. All gas streams are at approximately 1 atm. (a) Draw and label a flowchart of the process and carry out a degree of freedom analysis to verify that you can determine all unknown quantities on the chart. (b) Calculate (i) the ratio of entering air to entering gas (kg humid air/mol gas) and (ii) the relative humidity of the exiting air. (c) List several desirable properties of the membrane. (Think about more than just what it allows and does not allow to permeate.)

Step by step solution

01

Draw a Flowchart

A flowchart of the process can be drawn with the given inputs and outputs. On one side of the membrane, the gas mixture comes in at 30 °C with 10% each of H2O(v) and CO, and the balance CO2. It leaves at 30 °C with 2% H2O(v) and the balance CO and CO2. None of the CO, CO2, O2, and N2 transfer across. On the other side, air enters at 50 °C with an absolute humidity of 0.002 kg H2O/kg dry air and leaves at 48 °C. Only the H2O(v) moves across the membrane.

02

Degree of Freedom Analysis

This kind of analysis checks if the system is solvable with the given information. Each flow and composition in the process would be a variable. Then, based on all the equations we can write (like mass balances, energy balances, and the fact that mole fractions should add up to 1), we can count our number of equations. If the number of variables equals the number of equations, we can solve the system.

03

Calculate the Ratio of Entering Air to Entering Gas

This is a mass balance calculation. We know the H2O(v) in the entering gas and exiting gas, so we can find how much transfers across. That must equal the difference in H2O(v) in the entering and exiting air. From those, we can find out the moisture content of the air and hence the air flow.

04

Calculate Relative Humidity of Exiting Air

Relative humidity is the ratio of the actual vapor pressure to the saturation vapor pressure at the same temperature. Using the moisture content and the temperature of the exiting air, we can find its vapor pressure, and then its relative humidity.

05

Desirable Membrane Properties

The task is to list out qualities desired in the membrane. Apart from the one given (that it allows H2O(v) but not CO, CO2, O2, N2), other good properties would be high selectivity, high permeability, and chemical resistance to each component. It should also be mechanically strong and temperature resistant, as well as affordable and durable.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Flowchart

A flowchart is a visual representation of the process that helps to understand how the components move through the system. In this adiabatic membrane separation setup, drawing a flowchart involves outlining the inputs and outputs across the membrane. The flowchart should show:

• On the left side, the incoming gas mixture with 10% water vapor and CO, and the remaining CO extsubscript{2} at 30°C.
• On the right side, the incoming air stream with a specified absolute humidity entering at 50°C.
• Below the membrane, mark what permeates through it, in this case, only H extsubscript{2}O vapor, moving from the gas side to the air side.
• Label the conditions of gases leaving either side, noting the dried gas composition at the same initial temperature and the slightly cooler leaving air.

By mapping these components visually, we gain clarity into how each stream interacts with the membrane, and the expected output from both sides under the operation's conditions.

Degree of Freedom Analysis

Degree of freedom analysis is used to assess if we have sufficient information to solve for all variables in the system. In any process like membrane separation, we count variables such as flows and component compositions. We then determine how many independent equations we can write based on physical laws and process constraints. Here are the steps to perform this analysis:

• Identify the number of unknowns: This includes unknown flow rates and compositions on each side of the membrane.
• Write equations based on mass balances: Here, mass conservation principles, such as the total mass entering and leaving each side being equal, come into play.
• Evaluate other constraints: For instance, mole fractions must sum to one, and energy considerations might impose additional equations.

If the number of unknowns equals the number of independent equations, the system has zero degrees of freedom, indicating it is perfectly solvable with the provided data.

Mass Balance

A mass balance involves keeping track of mass across the process to compute unknowns like flow rates or concentrations. For the adiabatic membrane separation, we follow these principles:

• Analyze the water vapour across the membrane: Determine how much water vapor is transferred by comparing the compositions of the incoming and outgoing streams on both sides.
• Apply the conservation of mass: The amount of each component entering the system equals the amount leaving. This applies individually to each species or as a total flow rate.
• Recalculate unknown flows: Utilize differences in mass balance on either side to derive the flow ratio of incoming humid air to incoming gas, a critical step in determining operational efficiency.

By adhering to these balance equations, we establish a stable method to quantify all relevant stream parameters and ensure no hidden inconsistencies exist.

Relative Humidity

Relative humidity (RH) is an essential factor impacting separation efficiency. It is the ratio of the current vapor pressure of water to the saturation vapor pressure at that temperature. For the air leaving the membrane separator, we calculate RH as follows:

• Identify the vapor pressure: This involves determining what the pressure would be if the air could hold no more water vapor without condensing (saturation).
• Calculate the current moisture content: From the mass balance, determine the final water vapor pressure.
• Express the relative humidity: This is given as \((\frac{\text{Actual vapor pressure}}{\text{Saturation vapor pressure}}) \times 100\%\).

A high relative humidity indicates the air is close to saturation, which is a critical design parameter ensuring efficiency and avoiding condensation issues.

Membrane Properties

Selecting a membrane for separation requires understanding several key material properties. A good membrane enhances process efficiency and lifetime:

• High selectivity: It should preferentially allow water vapour over other gases like CO, CO extsubscript{2}, O extsubscript{2}, and N extsubscript{2} to permeate.
• High permeability: Supports high flow rates of water vapor to maintain operational efficiency.
• Chemical resistance: The membrane must withstand exposure to all gases involved without degrading.
• Mechanical strength: It should withstand operational pressures and temperatures without failure.
• Durability and cost-effectiveness: Long operational life and reasonable manufacturing costs are practical necessities.

By factoring in these properties, one can choose a membrane that enhances the overall process performance, reliability, and cost-effectiveness.