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 2.2m³.hour^-1 with concentration of 1.2 kg.m^-3. 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. If two membrane modules, each of area 25 m², are simply placed in series, give reasonable estimates of:

1. the dye concentration from the first membrane module?

2. the permeate flow rate from the first membrane module?

3. the dye concentration from the final membrane module?

4. the permeate flow rate from the final membrane module?

5. Then explain whether the above answers seem reasonable.

Please show all calculations, assumptions and relevant details.

Answer 1

Solution : An illustration of the system is

                   

recognizing that it is the retentate from the first module that is fed into the second membrane. We make the assumption, common in ultrafiltration, that the permeate concentration leaving each module is approximately zero. The flux J_v refers to the flux through the membrane, in other words the flux of the permeate. The flux equation is in terms of the bulk concentration in the membrane, which is ultimately the concentration leaving, i.e. C_R1 and C_R2 for module 1 and 2 respectively.

We solve the membrane modules in series, because there are no recycle loops or coupling between the modules. We can cascade down from module to module. Mass balance on module 1:

Grading for this question:

• 5 marks to set up the diagram with the correct flow connections, or it must be very clear from your calculations what the connections would have been

• 2 marks for the dye mass balance across module 1

• 2 marks for the volume balance across module 1

• 2 marks for correctly using the flux equation (correct flow term and concentration term)

• 2 marks for substituting equations into each other to attempt to solve for C_R1

• 2 marks for attempting to solve the non-linear equations by guess-check or calculator 

• 2 marks for the solving for the correct values of C_R1;Q_R1; Q_P1

• 3 marks for setting up the equations for solving module 2

• 2 marks for solving for the actual values in module 2

• 2 marks for interpretation in the final part