Question

An asymmetric ultrafiltration membrane is used with the aim of separating dyes from a liquid stream and to achieve a more concentrated dye-water mixture. The feed waste stream arrives at a flow rate of 3.0m³.hour^-1 with concentration of 0.5 kg.m^-3. The aim is to achieve a dye-water concentration leaving the membrane of 20 kg.m^-3, so that it can be reused to dye fabric. The membrane’s operating characteristic was calculated from various experiments:

                                 

where the bulk concentration C has units of  kg.m^-3 and flux is measured in m³.hour^-1.m^-2. We have tubular membranes available from a supplier, but only as 30 m² modules. The modules are placed in parallel, so the effectively can be treated as one large membrane.

1. What is the flow rate of the cleaned water from the system (we can use it elsewhere in our process)?

2. What number of modules would we have to purchase if operated in a single-stage feed-and-bleed configuration?

3. What would be the optimal module arrangement if operated with two stages of feed-and-bleed in series? (hint: optimal should be taken to be the smallest number of modules per feed-and-bleed stage; each stage may have a different area).

4. For the optimal arrangement you choose: specify what will be the total flow of cleaned water (hint: think carefully!)

5. What are your thoughts on putting four feed-plus-bleed stages in series?

Please note: it is in your interest to solve this problem as if it were in a test or exam; i.e. don’t use computer software.

Answer 1

Solution 

1. A single membrane unit is being used: the aim is calculate the total area provided by the parallel modules. As shown in the solutions developed in class:

• From a mass balance around the unit, assuming no dye leaves in the permeate (a reasonable

assumption for ultrafiltration): Q₀C₀ = Q_R1C_R1   

• From a volume balance we have Q₀= Q_P1 + Q_R1 

• The given flux equation 

                                   

• When solving the above equations we find that:

                     

So the flow of clean water from the single unit is Q_P1 = 2.925m³.hr^-1.

2. Use the flux equation and solve for A₁ = 327.7m², provided by 11 modules.

3. We require to find the areas of the two units to determine this. As with the single module system, set up the mass and volume balances for both units. Substitute these into the flux equation for each unit:

                               

These 2 equations involve 3 unknowns: A₁, A₂ and C_R1 . As is the case with optimization, there are always spare degree(s) of freedom to adjust. In this case we adjust C_R1 to different values and find the result of A₁ and A₂ so their total sum is at a minimum.

Some values that were tried:

                 

This tables makes sense: as we increase the intermediate retentate concentration, C_R1, the area required in the first unit to achieve that concentration increases, meaning that less area is required in the second unit. There will be a broad range of valid solutions around C_R1 = 5 kg.m^-3, corresponding to 2 modules in the first unit and 1 module in the second unit.

4. Let the optimal arrangement be the last row in table 5 (note there are many possible, similar solutions), then the total flow of cleaned water is found by calculating the permeate flows from both units.

Note however, that the area of the first unit, 41.9m², is not actually achieved, it is A₁ = 60m². So solve the membrane mass and volume balance equations for this new area to calculate C_R1  (requires an iterative solution, as in the midterm). Then calculate the permeate flow rate Q_P1=

                                     

Repeat this for the second unit using A₂ = 30m² to calculate Q_P2 . Sum the two for the final solution of clean water flow.

Note that C_R2  will exceed the specifications of 20 kg.m^-3 from the overall system.

5. Using more than 2 feed-plus-bleed units will be counterproductive for 3 or more stages. We are already using 3 modules in 2 units, so at best a 3-unit system will use a module each, but at significantly increased capital cost.