where S is the intersectional area of the non-cylindrical pore, and L

is the circumference of the pore。 The flow-rate of water through the filter medium can thus be governed by the Hagen–Poiseuille Equa-

The volumes of algae and water filtrate are linked to the volume

fraction by Eq。 (10)

For  better filtration efficiency, a thin  layer of  pre-coat of    fine

particulates (e。g。 diatomaceous earth) may need to be  deposited on  the surface  of  the filter  medium。 In  such  cases, an  additional

resistance is included in Eq。 (15) by using a correction factor, a。

70 P。  Shao  et  al。 / Chemical  Engineering  Journal  268  (2015)     67–75

3。3。 Transport of filtrate water through the algal cake-layer and the filter medium

The resistances of the algal cake-layer and the filter medium are in-series, therefore these two resistances are additive and the flow- rate of the filtrate water can finally be formulated   as,

dV Dp

through direct image analysis of the algal cake-layer, the effect of clogging, if any, on pore size can thus be assessed by a simplified model as developed in this work。 Thus-achieved pore size is an equivalent pore size。

This equivalent pore size of the cake-layer is evaluated by assuming that the porous algal cake-layer is comprised of an ensemble of cylindrical pores (2r) that pose a same hydraulic resis-

tance for water transport as the network pathway does。 Therefore,

In practice, the driving force Dp is generally kept as a   constant

during filtration。 Integrating of Eq。 (16) under a constant pressure difference gives

where a in this case stands for the tortuosity of the network path- way。 Inserting Eq。 (7)into Eq。 (21)yields the equivalent pore radius, r

Based on Eq。 (17), for a given filtration time, t, the volume of the filtrate water can be evaluated。 The capacity of a filter for process- ing the algae feed is thus determined by Eq。   (18),

A requirement for algae-dewatering using vacuum pressure is that the applied pressure difference must be large enough to over- come resistance arising from the surface tension of water。 There- fore, the minimum pressure required for displacing the water   out

ð23Þ

where T is the filter rotation cycle time, which is three times as long

as the filtration time (i。e。 t = T/3) since only one-third of the filter as shown in Fig。 1 is working for the filtration, while the rest of the time is for cake-layer-dewatering and cake-layer-discharging。

3。4。 Compressibility of the algal cake-layer and equivalent pore size of the  cake-layer

Algae deform under the applied pressure in filtration, giving rise to compression of the algal cake-layer。 Compression of the cake-

r

where r is the surface tension of water, and h is the contact angle of water on the surface of algae。

4。 Energy and capital costs, and process   economics

Assuming the flow-rate of the harvested algae to be dewatered is Ffeed, the number of filters required for processing the algae is thus

F

layer reduces the porosity, which can be expressed   as,

where k is the compression index, B is a constant, and p is the com- pressive pressure。 An extensive literature survey showed that there had been no reported data on the compressibility of algal cake- layer。 Therefore, the compression properties of a bio-cake [30,31], consisting mostly of yeast cells, and expected to have similar com- pression properties to that of algae, were adopted to investigate the impacts of compression on water transport through an algal cake-layer。 The obtained pressure-dependent mean porosity of the bio-cake is shown in Eq。 (20),