Kinetics analysis and model validation
The kinetic analysis of the changes in Mw of the hydrolyzed dextran was modelled using eq. (1). The rate constant, k , is related to the concentrations of substrate and enzyme 34, which could be obtained by fitting with experimental data as presented inTable S1 . The simulated results in reasonable agreement with the experimental data (Figure 3a ). The obtained value ofk decreased with increasing concentration of dextran substrate (Figure 3b ). The curve-fitting results show that the values ofk follow an exponential relationship with dextran concentration (cd ), with a correlation coefficient R2>0.99 as shown in Figure 3b , and its values can be expressed with following equation:
\(k=0.0001e^{-0.017c_{d}}\) (38)
The Mw of dextran can be predicted when the dextranase concentration is fixed as 0.05 g/L by combing eqs. (1) and (38).
To validate the developed model, the simulated permeate flux and the oligodextran concentration in the permeate were compared with the experimental data at different operating pressures as presented inFigure 4, where the simulated permeate flux can be seen to agree well with the experimental data. Increasing operating pressure enhanced the permeate flux as a result of higher driving force. It is noteworthy that the permeate flux decreased significantly in the initial stage but also gradually increased with prolonged operation time. This was a result of the dextran being hydrolyzed into small molecules which could penetrate through the membrane into the permeate and thus alleviate concentration polarization. Similarly, the simulated permeate concentration was in relatively good agreement with the experimental results (Figure 4b ). The relatively large discrepancy in the initial 30 min may have resulted from non-uniform distribution of membrane pore size and the wider molecule weight distribution of dextran substrate 35, which allowed some smaller dextran molecules to penetrate through the large membrane pores into the permeate in the initial filtration stage. The permeate concentration increased as the operation proceeded due to generation of a large amount of smaller oligodextran molecules following sufficient hydrolysis. However, increasing operating pressure reduced the permeate concentration because the denser fouling layer, the shorter the retention time for dextran hydrolysis, and with enhanced water convection transport at higher pressure, oligodextran rejection improved and its concentration in the permeate fell 36. These observations therefore indicate that the developed model is an efficient tool for simulating the filtration characteristics of an EMR.
Figure 3 . Effect of dextran concentration on (a) the variation of average Mw with reaction time and (b) the values of rate constant (k ) in a batch system with free enzyme. The simulated results are plotted as curves, while the experimental results are presented as discrete symbols.
Figure 4. Experimental (solid line) and simulated (discrete symbols) profiles of (a) permeate flux and (b) permeate concentration versus operation time with membrane pore size of 10 nm and membrane porosity of 0.8. Substrate concentration: 50 g/L, enzyme concentration: 0.05 g/L, agitation speed: 160 rpm, and feeding mode: water feeding.