Relation between TMP and flux

Pressure controlled region

Due to the back transport of particles and macromolecules, no significant membrane fouling occurs if operating flux remains at sub-critical level at low TMP. Under this pressure controlled region, flux is linearly proportional to TMP as shown in the figure below.

Pressure vs Flux

Fig. 1. Conceptual diagram of the relation between TMP and flux in short-term filtration.

Transitional region

Once the operating flux exceeds the critical flux, particles/solutes start to deposit on membrane surface noticeably and create additional permeation barriers. Therefore, the TMP-flux curve departs from the linear line. If the TMP increases further, the cake layer already formed on membrane surface starts to become more compact due to a higher pressure drop across the cake layer, which squeezes the cake layer stronger, especially in the bottom of the cake layer near membrane surface (see cake layer compaction). As a result, specific cake layer resistance per depth increases as TMP increases and the gains in flux becomes smaller.

The relation between cake resistance, Rc, and the specific cake thickness of cake layer,cva , can be described using  specific cake resistance, α, as shown in equation (1).

1    —————————————– (1)

R= Cake resistance (/m )
α = Specific cake resistance (m/kg)
c = Particle concentration (kg/m3)
V= Filtrate volume (m3)
A = Membrane surface area (m2)

The above equation is typically valid for rigid particles since the specific cake resistance, α, does not change significantly when cake layer depth and pressure loss increase. However, when soft particles are filtered, additional equation is required to reflect the changing α with flux/TMP.

For instance, when yeast cells are filtered with a polysulfone membrane with 0.45μm pores, the cake resistance, Rc, increases with increasing TMP (McCarthy, 1998).  The correlation between α and TMP was expressed as equation (1), where α proportionally increases at the TMP range of 30-500kPa as described as equation (2) .


1     ————————————(2)
α0 = specific cake resistance when ΔPT is zero
kc = proportionality constant

Mass transfer controlled region

If TMP increases further beyond the transitional area, then flux gets into the mass transfer controlled region. In this region, the specific cake resistance calculated with equation (2) reaches high enough to offset the gains in driving force (TMP), which means flux remains at almost constant level regardless of TMP increase. Depending on the properties of particles and solutes, cake layer may even collapse and create an excess cake resistance that can reduce flux. Temperature, crossflow velocity, liquid viscosity also can affect the maximum flux obtainable as depicted in the above figure.

Where do iMBR and sMBR stand in the curve?

The MBR with immersed membrane (iMBR) are designed to run at the low transitional region near the pressure controlled region. Suction pressure can be effectively used since cake resistances are not significant. Membrane fouling occurs very slowly, but particle/macromolecule deposition is not avoidable completely. The thin cake layer can eventually trigger sudden TMP rise as discussed here, but the filtration duration can be prolonged at such low fouling condition

On the contrary, the MBR with side stream membranes (sMBR) likely run between the mass transfer controlled region and the high transitional region. Since the high TMP near the module entrance (300-600 kPa) decreases rapidly toward the exit (50-100 kPa) under a high flow velocity (1-4 m/s), entrance lies in the mass transfer region while exit likely lies in the high transitional region.

Experimental data

Fig. 2 shows flux and TMP profiles in chicory juice filtration using a disc membrane with a rotating turbulent promoter. Flux increases linearly at low TMP for tighter membranes with 100 kDa MWCO as indicated by red circles, but it reaches a maximum at 50-60 kPa. On the other hand, flux is already near the maximum for looser membrane with 0.2 μm pores. It is noteworthy that the maximum flux at mass transfer controlled region is not a function of membrane pore size or permeability. In fact, 100 kDa membrane has much lower membrane permeability than 0.2 μm membrane as suggested by the fluxes at low TMP.

TMP vs3

Fig. 2. Flux profiles in TMP stepping experiment for 100kDa and 0.2 μm membranes using a fixed disc membrane with rotating turbulence promoter at 1,000 rpm in chicory juice filtration (Luo et al, 2013).


© Seong Hoon Yoon