loading
top sectiondowncomercolumnriserHeriserdraft tubegas spragerbottom sectionair
Figure 1. Schematic of the AALR
2.2 Liquid recirculating velocity
The liquid recirculation in airlift reactors is caused by the hydrostatic pressure difference between the riser and the downcomer. If we neglect the pressure drop caused by gas acceleration, which is less than 1% of the total pressure drop,11 the pressure drop Pls due to the difference in gas holdup between the riser and the downcomer at steady state equals to ΔPloss, the frictional loss, i.e.:
Pls???Ploss
The hydrostatic pressure difference is expressed by:
(3)
??lsd?1??gd???g?gd??lsr?1??gr???g?gr? Pls?gHd??????(4)
Since ?lsd and ?lsr are much larger than ?g in the air-water-silica sands system, thus Equation (4) can be simplified as:
5
Pls?gHd?lsd?1??gd???lsr?1??gr?
??(5)
The total frictional loss in airlift reactors is:12
??Ploss????Pf?????P?????P?????P?rfdftfb (6)
where (-ΔPf)r and (-ΔPf)d are the frictional losses in the riser and downcomer, respectively. As described by Verlaan et al.,13 (-ΔPf)r and (-ΔPf)d were given by:
1?lsrkfrVlr2 212??lsdkfdVld 2????P?fr(7) (8)
???P?fdwhere kfr and kfd are frictional loss coefficients in the riser and downcomer, respectively.
(-ΔPf)t and (-ΔPf)b in Equation (6) are frictional losses due to the reverse flows at the top and bottom sections of airlift reactors, respectively. According to Livingston and Zhang,1 (-ΔPf)t and (-ΔPf)b can be expressed by:
1kft?lsrVlr2 ft22kfb?lsdVld ???Pf?b?12???P??(9) (10)
The relationship between the liquid recirculating velocity and the superficial liquid velocity in the reactor is as follows:
Vlr?Ulr
1??gr??srUld
1??gd??sd(11)
Vld?(12)
Because the liquid volumetric flow rate through the riser is equal to that through the downcomer, the following relationship holds:
UlrAr?UldAd
Combining Equations (3)-(13) gives:
(13)
2??k?kk?k??????Ar?12?frftfdfb?gHd??1????1???U???????? (14) gdlsrgr?lrlsrlsd22??lsd?2A?1??gr??sr??1??gd??sd??d????
6
2.3 Gas holdup
To obtain the liquid recirculating velocity from Equation (14), the gas holdup in the riser and downcomer and the superficial gas velocity in the riser must be known at first.
The three-phase model of IALRs developed by modifying the Zuber and Findlay’s14 two-phase drift-flux model due to Bando et al.15 is adopted in the present investigation to estimate the gas holdup in the riser and downcomer. This model is represented by:
?gr?Cr?Ugr?Ulr?Usr??UbtCd?Ugd?Uld?Usd??UbtUgdUgr (15)
?gd? (16)
where Ubt is the bubble terminal velocity, Cr and Cd are the distribution parameters. The value of Cr or Cd is an index of the flow pattern, equal to 1 when the flow distribution is flat. In the present study, Equation (15) is used to estimate the values of Ubt by fitting to the experimental data of the gas holdup in the riser.
Generally, it is assumed that in multiphase systems the particles move at their terminal settling velocities relative to the actual velocity of the liquid phase, thus the vertical velocities of the solid particles in the riser and downcomer relative to the reactor wall are given by:
Vsr?Ulr?Ust
1??gr??srUld?Ust
1??gd??sd(17)
Vsd?(18)
In the same way to obtain the relationship between the liquid recirculating velocity and the superficial liquid velocity in the reactor, we have:
Usr?Vsr?sr Usd?Vsd?sd
(19) (20)
The gas volumetric flow rate through the riser is equal to that through the downcomer plus
7
the aeration rate from the gas sparger:
?AUgr?Ugd?d?Ar?Qgi ???Ar(21)
In order to solve Equations (15)-(21) for gas holdup and the superficial gas velocity, the relationship between the gas holdup in the riser and that in the downcomer is needed. In the AALR of an air-water system, Bello16 experimentally obtained the following relationship:
?gd?0.89?gr
(22)
In view of the similarity between the reactor used in the present investigation and that of Bello, Equation (22) is adopted. Combining Equations (15) and (22) gives:
UgrArQgi?AdAd
???ArQgi?Ar??sdUlrAr?Cd??Ugr??????U?Ubt??Ulr?1????A??sdst????AAAdd?dgdsdd? ???Ugr?0.89???srUlrCr?Ugr?Ulr???srUst??Ubt??1??gr??sr??(23)
Because ?g??l??s?1, hence
UgrArQgi?AdAd
???AQ?A?UlrA?Cd??Ugrr?gi??Ulrr???r???sdUst??Ubt???AdAd?Ad???ld?sd?Ad? ??Ugr?0.89??UlrCr?Ugr?Ulr???srUst??Ubt??lr?sr???(24)
For the present study, the solids are very small in size, so there is no significant difference between the solids concentrations in the riser and that in the downcomer. Consequently, we consider that the solid particles mix uniformly with the liquid in the reactor when calculating the
8