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The use of independently sealed microporous hollow fiber membranes for oxygenation of water: model development
11
Journal of Membrane hence, 69 (1992) 11-20 Elsevler Science Publishers B V , Amsterdam
The use of independently sealed microporous hollow fiber membranes for oxygenation of water: model development Tariq Ahmed and Michael J. Semmens Unwersrty of Mmnesota, Department of Cwd and Mmeral Engmeermg, 500 P&bury Drwe, S E , Mmneapoh, MN 55455 (USA) (Recemed October 11,1990, accepted m revised form September 12,199l)
Abstract Mass transfer m a hollow fiber membrane aerator was studied Indlvldually-sealed hollow fibers are filled w&h oxygen and immersed m a flowmg stream of water Gas diffuses across the membrane wall and dissolves directly into the water without formmg bubbles Earher studied found that mass transfer performance depended on the oxygen partial pressure mslde the membrane In this paper a model describes changes m gas composltlon along the fiber length that result from the back dlffuslon of gases, such as nitrogen, from the water mto the fiber The results of expenmental studies verify model predlctlons The model estunates an average saturation oxygen concentration along the fiber which predicts the mass transfer performance of the aerator under different condltlons Keywords fiber membranes, mlcroporous and porous membranes, modules, water treatment; transfer
Introduction Bubbleless membrane-aeration was successfully demonstrated using sealed-end hollow fibers pressurized with pure oxygen [ 11. The aerator is a bundle of individually sealed microporous polypropylene hollow fibers Immersed m flowing water, oxygen diffuses across the mlcroporous membrane wall and dissolves directly mto the water. Smce the pore diameter of the membrane wall is very small (0.02-0.05 pm), and since the membrane is hydrophobic, Correspondence to. Michael J Semmens, Umverslty of Minnesota, Department of Clvll and Mmeral Engmeenng, 500 Pillsbury Drwe, S E , Mmneapobs, MN 55455 (USA)
0376-7388/92/$05
oxygen
the membrane pores are not wetted. This encourages a high rate of mass transfer across the membrane, and shifts the mass transfer limltation to the liquid film outside of the membrane. Membrane modules gave effective oxygen transfer with large mass-transfer coefficients and an oxygen transfer efficiency of 100%. In earlier studies [ 1 ] module performance was found to be strongly dependent on the operating pressure. This was expected since the driving force for transfer increases as the oxygen pressure is increased However, assuming the oxygen partial pressure inside the fiber to be uniform along the fiber length and equal to the feed-oxygen partial pressure, calculated
00 0 1992 Elsevler Science Pubbshers B V All nghts reserved
12
T Ahmed and M J Semmens/J
mass transfer coefficients increased with gas pressure Since mass transfer is limited by hquid-film diffusion outside the fibers, the overall mass-transfer coefficient should be mdependent of pressure. Back diffusion of nitrogen from the water into the fiber was thought to have a significant impact on the oxygen partial pressure along the fiber, and to cause the apparent pressure dependence of the measured mass transfer coefficient. Recent papers by C&e et al [ 2,3] compared membrane performance in both flowthrough and dead-end modes Dead-end fiber modules performed worse than the modules having flowing oxygen inside the fibers The authors concluded that the poor performance in the dead-end mode was caused by oxygen having to diffuse through the water vapor that filled the fibers. In the flow-through mode water vapor was rapidly swept away by the flowing gas. Even earlier studies by Schaffer et al [4] suggested the sigmticance of nitrogen back diffusion, which can reduce the oxygen concentration m sealed membranes. We analyse the influence of mtrogen back diffusion on the performance of sealed-end hollow fiber membrane aerators. Theory In the bubbleless membrane aerator (see Ref. [ 11, Fig. 1) , pure oxygen is inside of individually sealed fibers at a pressure low enough to avoid the formation of bubbles. The water to be aerated is pumped through the shell side of the module outside the membrane A large concentration gradient is thus mamtained across the membrane and oxygen diffuses into the flowing water. However, a mtrogen concentration gradient exists across the membrane, and mtrogen dissolved in the water will enter the membrane fibers At steady state, the fiber will contam oxygen, nitrogen and other dissolved gases with composition varying along the fiber length. To
Membrane Scr 69 (1992) 11-20
analyze the impact of nitrogen and other gases on membrane aerators, a model was developed for the gas-phase composition along the length of the fibers. Gas-phase model The following assumptions slmphfy the model: (a) nitrogen diffuses mto the membrane from the water, (b) the total pressure in the fiber is constant, and equal to the sum of individual partial pressures of oxygen and nitrogen at any point, 1.e P (total) =p(O,)
+p(&)
(1)
(c ) water vapor and other dissolved gases may be neglected, (d) the mass transfer coefficients of nitrogen and oxygen are equal The transport of oxygen and mtrogen along a differential length of the fiber is shown in Fig. 1. Oxygen and nitrogen enter the element of fiber by convection and diffusion (1,2) from the upstream fiber element and diffusion of nitrogen across the fiber wall (4). Transport out of the element includes oxygen transfer to the water (3) and convection and diffusion of both oxygen and nitrogen to the downstream fiber (5,6). Since this analysis concerns oxygen transfer to water, which is limited by liquid film resistance, the equations are in terms of equivalent liquid-phase concentrations. These concentrations are the product of the partial pres-
1
Anal Diffusion In 2 Convective Flow In 3 Oxygen Flow Out
5 Axial Diffusion Out 6 Convectwe Flow Out 4 Nitrogen Flux In
Fig 1 Mass balance on a hollow fiber element
T Ahmed and M J Semmens/J
sure of the gas and the appropriate Henry’s law constant Referring to Fig. 1,at steady state the differentral balance of oxygen Inside the fiber leads to: dn, o= --&-R
2kL C, Hs-cIL [
1
(2)
where the axial oxygen flux n, is given by Fick’s law-
Slmllarly for nitrogen, we can write the mass balance and the flux equation as: ()=
dn,
2kL C,
dz
R [ Hz
----
--c,,
1
(4)
and, n2 =
-lE+c d.2
2 u is
Combmmg eqns. (2) and (4) and simplifymg, -cpL
c, =c-Cl Therefore, _~[(&L)C’
+
(g-c,,-c,, >I
Equatrons (2), (3) and (7), subJect to the above boundary conditions, were solved using a Runge-Kutta routme (or a multiple-shooting method) to give the gas concentrations, axial fluxes and gas velocity along the fiber The program required an initial guess of the oxygen concentration at the sealed-end of the fiber, and that kL be specified. The model then back-calculated the correspondmg feed pressure of oxygen The calculated feed pressure was compared with the known pressure and the calculations were repeated until the boundary conditions were satisfied. Selections of kL were based upon our experimentally observations Experimental Hollow-fiber membranes with varrous dlameters were supplied by Celanese Separation Products, Hoechst Celanese, Charlotte, NC. Hollow fiber modules were made by potting fibers into an external shell. A single-fiber contactor was made from a long glass tube with a Y-connector at one end. One end of the fiber was potted into the Y-connector with epoxy (FE-9000, H B. Fuller, St. Paul, MN) and the other end of the fiber was heat sealed. Details of the constructron of similar modules are described by Ahmed and Semmens [ 1 ] and Yang and Cussler [ 51 Several experiments checked some of the assumptions of the model and the model results Three types of modules were constructed for measuring the pressure drop, gas flow velocity and gas composition along the length of the fiber. (i) Measurement of the pressure drop along the ftber: A flow-through system was constructed as m Fig. 3. A differential manometer was connected between the inlet and outlet of a hollow fiber such that the exit end was essentially sealed The pressure drop across 362 cm of the fiber was measured while the system ran The module was used in the apparatus shown
1 (6)
Referring to assumption (b), we can write:
!$
13
Membrane Scl 69 (1992) 11-20
(7)
2
The velocity, ug, of the gas insrde the membrane 1s caused by oxygen diffusing out of the fiber The boundary conditions for the above equation are: at z= 0,Cl = C (oxygen partial pressure measured at the Inlet) and C2= 0, at z=L, nl=O, n2=0 and u,=O (sealed-end).
14
T Ahmed and M J Semmens/J
1,
I’
Fig 2 The module used for measurmg pressure drop along the fiber length
Oxygenated
waterout l
Syringe for Methylene BILE l”,ectlon
Fig 3 The module used to measure gas velocity wlthm the fiber lumen
Oxvaenated
G/ /
oxygen
I
Sampling Pal
supply
Fig 4 The module used to measure the gas composltlon at different points along the fiber
Membrane Scl 69 (1992) 11-20
used for visuahzmg the gas flow velocity inside a hollow fiber under fixed operating comhtions. The module was used in the apparatus shown m Fig. 4 m Ref. [ 11.The system was operated until It achieved steady state in the gas phase; then a small volume of methylene blue solution was injected at the upstream end of the fiber by piercmg the gas supply line with a micro-syringe. The methylene blue solution formed a visible 2 to 4 mm plug inside the fiber. The gas flow swept the plug down the inside of the fiber and its position was recorded with time. The test was repeated several times to verify the reproducibility of the velocity profile. (in) Gas concentratwn measurement wlthln the ftber: In this experiment, a membrane module was made with several, small sections of fiber interconnected through miniature sampling ports as in Fig. 4. During oxygen transfer, gases mside the fiber flowed through each sampling port Each port had a volume of approx. 0 5 cm3 and was made of tygon tubing that could be pierced by a gas tight syringe for sampling The test apparatus shown in Fig. 4 (Ref. [ 1 ] ) was used. After the attamment of steady state, 5 ~1 gas samples were drawn from each sampling port. The gas samples were analyzed imme&ately on a Hewlett Packard 5890A gas chromatograph using a packed column (molecular sieve 5A) and a thermal conductivity detector. Results
m Fig. 4 m Ref. [ 11. Water, havmg a constant oxygen concentration, was pumped from a reservoir to the module by a centrifugal pump (March Mfg., Inc., Model AC-3C-MD) and a &gnal flowmeter (Signet Scientific, Model MK 508) monitored flowrate. A gas cylinder supplied oxygen and a pressure transducer (Contactor Instruments, Mmneapohs, MN, Model P853) measured the differential pressure. (ii) Gas veloc@ measurements mszde the fzber: The second type of module (Fig. 3)) was
The calculated and measured gas pressure drops across the Inside of the fiber at different operating pressures are m Table 1. It shows that the pressure drop due to friction across the 362 cm length of the fiber is neghgible compared to the operating pressure. The calculated pressure drop along the fiber was obtained from the gas velocity predicted by the model when the measured kL was used. An average gas velocity was assumed within each 5 cm fiber length and the
T Ahmed and M J Semmens/J
15
Membrane Scr 69 (1992) 11-20
TABLE 1 Calculated and measured pressure drop m the fiber Operatmg pressure (PSI)
Calculated pressure
Measured pressure
drop (PSI)
drop (PO)
1 2 5 10
0 0 0 0
0 0 0 0
G Q) $
04
2
03
E
022 026 036 050
Measured press asa%ofop pressure
184 12 184 1 16
018 024 092 116
Fiber Dla 0 04 cm OperatNlg Pressure 2 7 PSI Assumed kL 0 009 cm/sac . Observed _ Model Result
05------1
20
40
60
80
100
20
40
60
80
100
120
Length (cm) Fig 6 The predIcted changes m gas velocity mthm a hollow fiber with different assumed values of kL The calculatlon assumes the fibers are operated at an oxygen pressure of 3 0 psig
z g 12
$;I.< 0
0
120
Length (cm) Fig 5 An example of the observed and predicted velocity profile m a fiber operated with an oxygen feed pressure of 2 7 ps1g
frictional loss was estimated by the HagenPoiseuille equation. The agreement between the calculated and measured pressure drops was good at operating pressures below l-3 psr, but the measured values were consrstently and increasingly larger than the calculated values at higher operating pressures. The gas flow velocity inside the fiber, measured by visualization with methylene blue solution, was greatest when the gas entered the fiber and decreased rapidly towards the sealed end of the fiber (see Fig. 5). The gas velocrty data are plotted against the length of the fiber. The solid line m the figure was predicted by the model when an appropriate kL value was assumed, and the agreement IS excellent. As expected, the velocity strongly depends on the value of the mass transfer coefficient selected Figure 6 illustrates the variation in the gas flow
f
10
z?i
08
P z
06
$ n
Operating
5 PSI
04
i:y-Jzzz 0
Pressure
O
100
200
1 0
Length (cm) Fig 7 An example of the observed and predicted changes m oxygen partial pressure along a hollow fiber operated with an oxygen feed pressure of 10 pslg
velocrty profile predicted by the model when the value of kL increases from 0.008 cm/set to 0.03 cm/set. The curves, calculated for a 200-pm fiber, show that as kL increases the gas velocrty increases proportionately The measurements of the gas concentratron profile msrde the fiber were made at an operating pressure of only about 1 psrg, since at higher gas pressures the oxygen leaked out around the synnge needle durmg samplmg. The concentration of oxygen measured along the length of the fiber, IS shown in Fig 7. Once again, the model predrction represented by the solid line is shown for comparison; the agreement is very good. The partial pressure of oxygen inside the fiber was observed to decrease
16
T Ahmed and M J Semmens/J
from the absolute feed pressure of 1.07 atm. to a value that approaches equilibrium with the concentration of dissolved oxygen outside the fiber. In Fig. 7, aerated tap water was used and the water was close to saturation, so the pressure leveled off at about 0.2 atm. Figure 7 shows that the high partial pressure of oxygen desirable for rapid oxygen transfer across the fiber wall was only maintained m the first 35% of the fiber length. This loss of drivmg force has led previous investigators to dismiss sealed end fibers. C&6 et al. [2] argued that better oxygen transfer could be achieved by pumping oxygen continuously through the inside of the hollow fibers to flush water vapour from the fibers. In reality, removing accumulated nitrogen appears more Important. Having verified the model’s ability to predict the observed gas composition and the velocity profiles axially along a hollow fiber, the model was used to find the influence of different design parameters and operating conditions. A sensitivity analysis evaluated the influence of kL, fiber length and diameter etc. on the gas composition along the fiber. The model was insensitive to each of these parameters. The masstransfer coefficient was assumed to be 0.008 cm/set unless otherwise stated. Oxygen and nitrogen concentration profiles and axial fluxes were calculated in a 1 m long 400 pm fiber operated at an oxygen feed pressure of 5 psig; the results are shown m Figs. 8, 9. The oxygen concentration declines along the length of the fiber as the nitrogen concentration increases. Interestingly, the curves in Fig. 8 are calculated for an influent water that contams no dissolved oxygen, and the oxygen pressure at the sealed end of the fiber therefore approaches zero. In this scenario the nitrogen pressure at the sealed end of the fiber will approach the feed oxygen pressure. Figure 9 shows that the axial flux of oxygen decreases gradually along the length of the fiber, whereas the axial flux of nitrogen increases to a maximum
-2 E tii
Fiber Dia Operaimg
s
E ’
s s
0
0 04 cm Pressure
5 psi
llc__l Nllrogen
H
00
Membrane Scl 69 (1992) 11-20
0
Oxygen
0
20
40
60
80
100
120
Length (cm)
Fig 8 Calculated concentration profiles for oxygen and nltrogen along a hollow fiber supplied with pure oxygen at 5 0 ps1g
08
Faber Dta Operatmg
0 04 cm Pressure
5 psi
Length (cm) Fig 9 Calculated fluxes of nitrogen and oxygen along the axis of a hollow fiber supphed with pure oxygen at 5 pslg
and then decreases toward the sealed end. Knowing the concentrations of oxygen and nitrogen along the inside of fiber, the gas fluxes across the membrane can be calculated. These calculations assumed that the dissolved oxygen concentration outside the fiber was always zero, and the dissolved nitrogen concentration was always saturated with respect to air (0.79 atm. partial pressure of nitrogen). The results are shown in Fig. 10. The calculated nitrogen fluxes along the fiber length are interesting since they change sign approximately half way down the fiber’s length At the feed end of the fiber, the supply of pure oxygen encourages nitrogen to enter the fiber. Entry by back diffusion will occur along the fiber as long as the partial pressure of nitrogen within the fiber 1sless than external partial pressure of approximately 0.79
T Ahmed and M J Semmens/J
6
17
Membrane Set 69 (1992) 11-20
Fiber Dla 0 04 cm Operatmg Pressure5 psi
4oE4
$ 08 0 u,
06
u 04
2OE4'
0
I
I
20
40
I 60
80
I 100
120
Length (cm) Fig 10 Calculated fluxes of nitrogen and oxygen across the membrane wall of a hollow fiber supplied with pure oxygen at 5 pslg
atm. This transfer occurs continuously and the
nitrogen that enters the fiber is swept down the fiber length by the flow of oxygen. As the oxygen diffuses out of the fiber, the gas phase becomes enriched in nitrogen and at some point along the fiber a nitrogen partial pressure of 0.79 atm is attamed. At this point, there is no net flux of mtrogen across the membrane wall. However, as the gas continues to move down the fiber’s length the nitrogen pressure mcreases and the nitrogen begins to diffuse out of the fiber and back into the water. The net accumulation of nitrogen along the length of the fiber shows it to be zero which is consistent with the assumption of steady state. The Influence of the operating pressure on the gas composition within the fiber was examined using the model. The oxygen supply pressure was varied between 2 and 24 psi; the calculated pressure profiles of oxygen along the fiber are m Fig. 11 as a function of the &menslonless fiber length. The operating pressure profoundly affects the oxygen profile along the fiber As the pressure increases a more favorable profile arises with a greater length of the fiber occupied by a high partial pressure of oxygen. This is explained by noting that nitrogen back diffusion is controlled by the nitrogen concentration gra&ent across the membrane and the liquid film diffusion, which is dependent only upon the water flow conditions out-
Fig 11 Calculated oxygen concentration profiles along a hollow fiber membrane supplied with pure oxygen at dtiferent pressures
side the fiber. As the oxygen pressure increases, the driving force for oxygen transfer increases and the diffusion-mduced gas velocity mslde the fiber increases significantly. By comparison, the nitrogen back diffusion rate does not increase ngmficantly, and so a longer fraction of the fiber length is required to accumulate the higher pressures of nitrogen Operation at higher oxygen pressures 1sclearly very desirable since it dramatically increases the effective driving force (oxygen concentration gradient across the membrane) Our analysis of mass transfer calculates kL by assuming that the oxygen partial pressure 1s constant and equal to the supply pressure. With the data in Fig. 11, it IS possible to correct the calculated kL to reflect the true pressure profile To this end, a mean partial pressure for each operating pressure was obtamed by integrating the pressure of oxygen over the fiber length. These “effective” pressures were then correlated to the actual operating pressure as follows: pnl =fPo
(8)
where Pm is the mean oxygen pressure withm the fiber, f is a correction factor and P,, is the oxygen supply pressure. The calculated correction factors are in Table 2, and the followmg correlation arose from these data.
18
T Ahmed and M J Semmens/J
Membrane SCL 69 (1992) 11-20
1000~
TABLE 2
I
Calculated values of the correction factor Operatmg pressure
Correction factor
(PSl)
2 4 6 10 24
0 0 0 0 0
327 400 506 546 712
f=O 26 (P,)’ 31
10'
(9)
This equation can be used to calculate a corrected average C* value for a specific operating condition. For example if the oxygen supply pressure is kept at 2 psig., the correction factor is f= 0.322 and the mean oxygen pressure is 0.64 PWZ.
Ahmed et al. [l] showed that observed values of kL increased as the partial pressure of oxygen increased (see Ref. [ 11, Fig. 11) These calculations of kL assumed that a constant oxygen supply pressure was maintained along the length of the fiber, although it is clear from the discussion above that this assumption was mcorrect. Accordingly, these data were recalculated usmg eqn. (9) to correct for the effect of pressure. The resulting data are shown in Fig. 12 The corrected correlations are (Fig. 13) Sh = 0 143 Re” 83,
103
104
Re
(10)
Sh=0.0184 ReOW3c” 33 The correlation coefficient ( r2 = 0.843) shows appreciable improvement. The new correlation was compared with literature correlations (see Ref [ 11, Table 2 ) . The correction was partially successful m reducing the scatter of data (Fig 13) but kL remains pressure dependent.
Discussion
This analysis of the gas transfer m sealedend fibers has shown the importance of back
Fig 12 The data shown m Fig 11 m Ref [ 11 followmg correction for the chanqng oxygen concentration profiles at different operating pressures
drffuslon of dissolved gases from the water to the fibers. In aerating natural waters, the high equilibrium partial pressure of dissolved nitrogen can dramatically affect the effectiveness of gas transfer Other gases may accumulate m a similar way, but they are less rmportant because of their lower partial pressures. Water vapor is an important exception, as discussed below. If other gases are transferred, such as SOz, the presence of both oxygen and nitrogen will be important. Water has a low vapor pressure but will accumulate within sealed-end fibers. Condensation has been observed m earlier studies and m our stu&ed. C8te et al. [2,3] speculated that condensation was caused by temperature dlfferences; however, the gas model developed here may offer a more likely explanation. We believe that water vapor enters and accumulates within ,__“”
1
y.o14317’x*oE295s
R”2=0843 I
1
IOCO-: f
loo,
l”,l,ooo Re
Fig 13
T Ahmed and M J SemmensjJ
Membrane SCL 69 (1992) 11-20
the fibers m the same way as other dissolved gases. For the water to escape from the fiber at the sealed end, the partial pressure inside must exceed the external vapor pressure, which 1s not possible without condensation Water will therefore continuously accumulate within sealed fibers and the performance of the fiber will deteriorate accordmgly. This problem must be overcome to make sealed fibers practical. The effect of pressure on hollow fiber membranes 1s not yet fully understood. The model developed here successfully predicted the gas velocrty and composition changes within the sealed fibers but was only verified at low operating pressures. Additional studies are needed to validate the model at higher operating pressures. When the model was employed to correct the data for operating pressure differences, It was only partially successful m reducmg the data.
Conclusions
A model 1s presented to predict changes in gas composltlon along the length of a sealedend fiber supplied with pure oxygen, while the fiber 1s immersed m flowing water stream The model accurately accounts for the back drffuslon of nitrogen from the water mto the fiber, and predicts experimentally measured conditions wlthm the fiber. The model also explams the observed accumulation of condensate within sealed fibers observed by earlier investigators The gas composition model was used to investigate the apparent dependence of masstransfer coefficrent on the pressure of oxygen supplied to the fibers. The model was unable to fully correct the observed pressure dependence of the mass transfer m the fibers Further work IS needed to account for this effect.
19
Acknowledgements The Mineral s&y of funded mlsslon
work was conducted at the Clvll and Engineering Department of UmverMinnesota, Minneapolis and partially by Metropolitan Waste Control Com(MWCC), St. Paul, MN.
List of symbols
c
total gas concentration (mg/l) concentration gas phase oxygen (mg/l) concentration gas phase nitrogen c2 (mg/l) water phase oxygen concentration c1L (mg/l) C2L water phase nitrogen concentration (mg/l) water phase oxygen concentration in C* equlhbrmm with the gas phase (mg/ 1) d fiber diameter (cm ) equivalent diameter of the module de (cm) D dlffusivlty of oxygen or nitrogen m water (cm’/sec) Henry’s law coefficient for oxygen HI (-) Henry’s law coefficient for mtrogen & (-) J gas flux in or out of the fiber (mg/ cm2-set) K overall mass transfer coefficient (cm/set ) kG, k,, kL mass transfer coeff. of gas, membrane and liquid respectively (cm/ set) L length along the fiber fiber length (cm) LO n number of fiber flux of oxygen (mg/cm'-set ) nl flux of nitrogen ( mg/cm2-sec ) n2 R radius of fiber (cm)
Cl
20
t
“P UL
V z
T Ahmed and M J Semmens/J
time (set ) velocity of gas in the fiber (cm/set) velocity of water (cm/set) volume of reservoir (1) distance along the fiber (cm)
Greek V
kmematlc vlscoslty
( cm2/sec)
Drmensum2ess numbers Re
v&e Reynolds number = -
Sh
kvd Sherwood number = y
SC
Schmidt number = i
Membrane Scr 69 (1992) 11-20
References
1 T Ahmed and M J Semmens, Use of sealed end hollow fibers for bubbleless membrane aeration Expemmental studies, J Membrane Scl ,69 (1992) l-10 2 P L C&, J -L. Bemllon, A Huyard and J -M Faup, Bubble-free aeration usmg membranes Process analysis, Water Pollut Control Fed, 60(11) (1988) 19861992 3 P L CXti, J L Berslllon and A Huyard, Bubble-free aeration using membranes Mass transfer analysis, J Membrane Scl ,47 (1989) 91 R B Schaffer, F V Ludzack and M B Ettmger, Sewage treatment by oxygenation through permeable plastic films, J Water Pollut Control Fed ,32 (9) (1960) 939941 Ming-Chlen Yang and E L Cussler, Deslgnmg hollowfiber co&actors, AI&E J ,32( 11) (1986) 1910-1916
Journal of Membrane hence, 69 (1992) 11-20 Elsevler Science Publishers B V , Amsterdam
The use of independently sealed microporous hollow fiber membranes for oxygenation of water: model development Tariq Ahmed and Michael J. Semmens Unwersrty of Mmnesota, Department of Cwd and Mmeral Engmeermg, 500 P&bury Drwe, S E , Mmneapoh, MN 55455 (USA) (Recemed October 11,1990, accepted m revised form September 12,199l)
Abstract Mass transfer m a hollow fiber membrane aerator was studied Indlvldually-sealed hollow fibers are filled w&h oxygen and immersed m a flowmg stream of water Gas diffuses across the membrane wall and dissolves directly into the water without formmg bubbles Earher studied found that mass transfer performance depended on the oxygen partial pressure mslde the membrane In this paper a model describes changes m gas composltlon along the fiber length that result from the back dlffuslon of gases, such as nitrogen, from the water mto the fiber The results of expenmental studies verify model predlctlons The model estunates an average saturation oxygen concentration along the fiber which predicts the mass transfer performance of the aerator under different condltlons Keywords fiber membranes, mlcroporous and porous membranes, modules, water treatment; transfer
Introduction Bubbleless membrane-aeration was successfully demonstrated using sealed-end hollow fibers pressurized with pure oxygen [ 11. The aerator is a bundle of individually sealed microporous polypropylene hollow fibers Immersed m flowing water, oxygen diffuses across the mlcroporous membrane wall and dissolves directly mto the water. Smce the pore diameter of the membrane wall is very small (0.02-0.05 pm), and since the membrane is hydrophobic, Correspondence to. Michael J Semmens, Umverslty of Minnesota, Department of Clvll and Mmeral Engmeenng, 500 Pillsbury Drwe, S E , Mmneapobs, MN 55455 (USA)
0376-7388/92/$05
oxygen
the membrane pores are not wetted. This encourages a high rate of mass transfer across the membrane, and shifts the mass transfer limltation to the liquid film outside of the membrane. Membrane modules gave effective oxygen transfer with large mass-transfer coefficients and an oxygen transfer efficiency of 100%. In earlier studies [ 1 ] module performance was found to be strongly dependent on the operating pressure. This was expected since the driving force for transfer increases as the oxygen pressure is increased However, assuming the oxygen partial pressure inside the fiber to be uniform along the fiber length and equal to the feed-oxygen partial pressure, calculated
00 0 1992 Elsevler Science Pubbshers B V All nghts reserved
12
T Ahmed and M J Semmens/J
mass transfer coefficients increased with gas pressure Since mass transfer is limited by hquid-film diffusion outside the fibers, the overall mass-transfer coefficient should be mdependent of pressure. Back diffusion of nitrogen from the water into the fiber was thought to have a significant impact on the oxygen partial pressure along the fiber, and to cause the apparent pressure dependence of the measured mass transfer coefficient. Recent papers by C&e et al [ 2,3] compared membrane performance in both flowthrough and dead-end modes Dead-end fiber modules performed worse than the modules having flowing oxygen inside the fibers The authors concluded that the poor performance in the dead-end mode was caused by oxygen having to diffuse through the water vapor that filled the fibers. In the flow-through mode water vapor was rapidly swept away by the flowing gas. Even earlier studies by Schaffer et al [4] suggested the sigmticance of nitrogen back diffusion, which can reduce the oxygen concentration m sealed membranes. We analyse the influence of mtrogen back diffusion on the performance of sealed-end hollow fiber membrane aerators. Theory In the bubbleless membrane aerator (see Ref. [ 11, Fig. 1) , pure oxygen is inside of individually sealed fibers at a pressure low enough to avoid the formation of bubbles. The water to be aerated is pumped through the shell side of the module outside the membrane A large concentration gradient is thus mamtained across the membrane and oxygen diffuses into the flowing water. However, a mtrogen concentration gradient exists across the membrane, and mtrogen dissolved in the water will enter the membrane fibers At steady state, the fiber will contam oxygen, nitrogen and other dissolved gases with composition varying along the fiber length. To
Membrane Scr 69 (1992) 11-20
analyze the impact of nitrogen and other gases on membrane aerators, a model was developed for the gas-phase composition along the length of the fibers. Gas-phase model The following assumptions slmphfy the model: (a) nitrogen diffuses mto the membrane from the water, (b) the total pressure in the fiber is constant, and equal to the sum of individual partial pressures of oxygen and nitrogen at any point, 1.e P (total) =p(O,)
+p(&)
(1)
(c ) water vapor and other dissolved gases may be neglected, (d) the mass transfer coefficients of nitrogen and oxygen are equal The transport of oxygen and mtrogen along a differential length of the fiber is shown in Fig. 1. Oxygen and nitrogen enter the element of fiber by convection and diffusion (1,2) from the upstream fiber element and diffusion of nitrogen across the fiber wall (4). Transport out of the element includes oxygen transfer to the water (3) and convection and diffusion of both oxygen and nitrogen to the downstream fiber (5,6). Since this analysis concerns oxygen transfer to water, which is limited by liquid film resistance, the equations are in terms of equivalent liquid-phase concentrations. These concentrations are the product of the partial pres-
1
Anal Diffusion In 2 Convective Flow In 3 Oxygen Flow Out
5 Axial Diffusion Out 6 Convectwe Flow Out 4 Nitrogen Flux In
Fig 1 Mass balance on a hollow fiber element
T Ahmed and M J Semmens/J
sure of the gas and the appropriate Henry’s law constant Referring to Fig. 1,at steady state the differentral balance of oxygen Inside the fiber leads to: dn, o= --&-R
2kL C, Hs-cIL [
1
(2)
where the axial oxygen flux n, is given by Fick’s law-
Slmllarly for nitrogen, we can write the mass balance and the flux equation as: ()=
dn,
2kL C,
dz
R [ Hz
----
--c,,
1
(4)
and, n2 =
-lE+c d.2
2 u is
Combmmg eqns. (2) and (4) and simplifymg, -cpL
c, =c-Cl Therefore, _~[(&L)C’
+
(g-c,,-c,, >I
Equatrons (2), (3) and (7), subJect to the above boundary conditions, were solved using a Runge-Kutta routme (or a multiple-shooting method) to give the gas concentrations, axial fluxes and gas velocity along the fiber The program required an initial guess of the oxygen concentration at the sealed-end of the fiber, and that kL be specified. The model then back-calculated the correspondmg feed pressure of oxygen The calculated feed pressure was compared with the known pressure and the calculations were repeated until the boundary conditions were satisfied. Selections of kL were based upon our experimentally observations Experimental Hollow-fiber membranes with varrous dlameters were supplied by Celanese Separation Products, Hoechst Celanese, Charlotte, NC. Hollow fiber modules were made by potting fibers into an external shell. A single-fiber contactor was made from a long glass tube with a Y-connector at one end. One end of the fiber was potted into the Y-connector with epoxy (FE-9000, H B. Fuller, St. Paul, MN) and the other end of the fiber was heat sealed. Details of the constructron of similar modules are described by Ahmed and Semmens [ 1 ] and Yang and Cussler [ 51 Several experiments checked some of the assumptions of the model and the model results Three types of modules were constructed for measuring the pressure drop, gas flow velocity and gas composition along the length of the fiber. (i) Measurement of the pressure drop along the ftber: A flow-through system was constructed as m Fig. 3. A differential manometer was connected between the inlet and outlet of a hollow fiber such that the exit end was essentially sealed The pressure drop across 362 cm of the fiber was measured while the system ran The module was used in the apparatus shown
1 (6)
Referring to assumption (b), we can write:
!$
13
Membrane Scl 69 (1992) 11-20
(7)
2
The velocity, ug, of the gas insrde the membrane 1s caused by oxygen diffusing out of the fiber The boundary conditions for the above equation are: at z= 0,Cl = C (oxygen partial pressure measured at the Inlet) and C2= 0, at z=L, nl=O, n2=0 and u,=O (sealed-end).
14
T Ahmed and M J Semmens/J
1,
I’
Fig 2 The module used for measurmg pressure drop along the fiber length
Oxygenated
waterout l
Syringe for Methylene BILE l”,ectlon
Fig 3 The module used to measure gas velocity wlthm the fiber lumen
Oxvaenated
G/ /
oxygen
I
Sampling Pal
supply
Fig 4 The module used to measure the gas composltlon at different points along the fiber
Membrane Scl 69 (1992) 11-20
used for visuahzmg the gas flow velocity inside a hollow fiber under fixed operating comhtions. The module was used in the apparatus shown m Fig. 4 m Ref. [ 11.The system was operated until It achieved steady state in the gas phase; then a small volume of methylene blue solution was injected at the upstream end of the fiber by piercmg the gas supply line with a micro-syringe. The methylene blue solution formed a visible 2 to 4 mm plug inside the fiber. The gas flow swept the plug down the inside of the fiber and its position was recorded with time. The test was repeated several times to verify the reproducibility of the velocity profile. (in) Gas concentratwn measurement wlthln the ftber: In this experiment, a membrane module was made with several, small sections of fiber interconnected through miniature sampling ports as in Fig. 4. During oxygen transfer, gases mside the fiber flowed through each sampling port Each port had a volume of approx. 0 5 cm3 and was made of tygon tubing that could be pierced by a gas tight syringe for sampling The test apparatus shown in Fig. 4 (Ref. [ 1 ] ) was used. After the attamment of steady state, 5 ~1 gas samples were drawn from each sampling port. The gas samples were analyzed imme&ately on a Hewlett Packard 5890A gas chromatograph using a packed column (molecular sieve 5A) and a thermal conductivity detector. Results
m Fig. 4 m Ref. [ 11. Water, havmg a constant oxygen concentration, was pumped from a reservoir to the module by a centrifugal pump (March Mfg., Inc., Model AC-3C-MD) and a &gnal flowmeter (Signet Scientific, Model MK 508) monitored flowrate. A gas cylinder supplied oxygen and a pressure transducer (Contactor Instruments, Mmneapohs, MN, Model P853) measured the differential pressure. (ii) Gas veloc@ measurements mszde the fzber: The second type of module (Fig. 3)) was
The calculated and measured gas pressure drops across the Inside of the fiber at different operating pressures are m Table 1. It shows that the pressure drop due to friction across the 362 cm length of the fiber is neghgible compared to the operating pressure. The calculated pressure drop along the fiber was obtained from the gas velocity predicted by the model when the measured kL was used. An average gas velocity was assumed within each 5 cm fiber length and the
T Ahmed and M J Semmens/J
15
Membrane Scr 69 (1992) 11-20
TABLE 1 Calculated and measured pressure drop m the fiber Operatmg pressure (PSI)
Calculated pressure
Measured pressure
drop (PSI)
drop (PO)
1 2 5 10
0 0 0 0
0 0 0 0
G Q) $
04
2
03
E
022 026 036 050
Measured press asa%ofop pressure
184 12 184 1 16
018 024 092 116
Fiber Dla 0 04 cm OperatNlg Pressure 2 7 PSI Assumed kL 0 009 cm/sac . Observed _ Model Result
05------1
20
40
60
80
100
20
40
60
80
100
120
Length (cm) Fig 6 The predIcted changes m gas velocity mthm a hollow fiber with different assumed values of kL The calculatlon assumes the fibers are operated at an oxygen pressure of 3 0 psig
z g 12
$;I.< 0
0
120
Length (cm) Fig 5 An example of the observed and predicted velocity profile m a fiber operated with an oxygen feed pressure of 2 7 ps1g
frictional loss was estimated by the HagenPoiseuille equation. The agreement between the calculated and measured pressure drops was good at operating pressures below l-3 psr, but the measured values were consrstently and increasingly larger than the calculated values at higher operating pressures. The gas flow velocity inside the fiber, measured by visualization with methylene blue solution, was greatest when the gas entered the fiber and decreased rapidly towards the sealed end of the fiber (see Fig. 5). The gas velocrty data are plotted against the length of the fiber. The solid line m the figure was predicted by the model when an appropriate kL value was assumed, and the agreement IS excellent. As expected, the velocity strongly depends on the value of the mass transfer coefficient selected Figure 6 illustrates the variation in the gas flow
f
10
z?i
08
P z
06
$ n
Operating
5 PSI
04
i:y-Jzzz 0
Pressure
O
100
200
1 0
Length (cm) Fig 7 An example of the observed and predicted changes m oxygen partial pressure along a hollow fiber operated with an oxygen feed pressure of 10 pslg
velocrty profile predicted by the model when the value of kL increases from 0.008 cm/set to 0.03 cm/set. The curves, calculated for a 200-pm fiber, show that as kL increases the gas velocrty increases proportionately The measurements of the gas concentratron profile msrde the fiber were made at an operating pressure of only about 1 psrg, since at higher gas pressures the oxygen leaked out around the synnge needle durmg samplmg. The concentration of oxygen measured along the length of the fiber, IS shown in Fig 7. Once again, the model predrction represented by the solid line is shown for comparison; the agreement is very good. The partial pressure of oxygen inside the fiber was observed to decrease
16
T Ahmed and M J Semmens/J
from the absolute feed pressure of 1.07 atm. to a value that approaches equilibrium with the concentration of dissolved oxygen outside the fiber. In Fig. 7, aerated tap water was used and the water was close to saturation, so the pressure leveled off at about 0.2 atm. Figure 7 shows that the high partial pressure of oxygen desirable for rapid oxygen transfer across the fiber wall was only maintained m the first 35% of the fiber length. This loss of drivmg force has led previous investigators to dismiss sealed end fibers. C&6 et al. [2] argued that better oxygen transfer could be achieved by pumping oxygen continuously through the inside of the hollow fibers to flush water vapour from the fibers. In reality, removing accumulated nitrogen appears more Important. Having verified the model’s ability to predict the observed gas composition and the velocity profiles axially along a hollow fiber, the model was used to find the influence of different design parameters and operating conditions. A sensitivity analysis evaluated the influence of kL, fiber length and diameter etc. on the gas composition along the fiber. The model was insensitive to each of these parameters. The masstransfer coefficient was assumed to be 0.008 cm/set unless otherwise stated. Oxygen and nitrogen concentration profiles and axial fluxes were calculated in a 1 m long 400 pm fiber operated at an oxygen feed pressure of 5 psig; the results are shown m Figs. 8, 9. The oxygen concentration declines along the length of the fiber as the nitrogen concentration increases. Interestingly, the curves in Fig. 8 are calculated for an influent water that contams no dissolved oxygen, and the oxygen pressure at the sealed end of the fiber therefore approaches zero. In this scenario the nitrogen pressure at the sealed end of the fiber will approach the feed oxygen pressure. Figure 9 shows that the axial flux of oxygen decreases gradually along the length of the fiber, whereas the axial flux of nitrogen increases to a maximum
-2 E tii
Fiber Dia Operaimg
s
E ’
s s
0
0 04 cm Pressure
5 psi
llc__l Nllrogen
H
00
Membrane Scl 69 (1992) 11-20
0
Oxygen
0
20
40
60
80
100
120
Length (cm)
Fig 8 Calculated concentration profiles for oxygen and nltrogen along a hollow fiber supplied with pure oxygen at 5 0 ps1g
08
Faber Dta Operatmg
0 04 cm Pressure
5 psi
Length (cm) Fig 9 Calculated fluxes of nitrogen and oxygen along the axis of a hollow fiber supphed with pure oxygen at 5 pslg
and then decreases toward the sealed end. Knowing the concentrations of oxygen and nitrogen along the inside of fiber, the gas fluxes across the membrane can be calculated. These calculations assumed that the dissolved oxygen concentration outside the fiber was always zero, and the dissolved nitrogen concentration was always saturated with respect to air (0.79 atm. partial pressure of nitrogen). The results are shown in Fig. 10. The calculated nitrogen fluxes along the fiber length are interesting since they change sign approximately half way down the fiber’s length At the feed end of the fiber, the supply of pure oxygen encourages nitrogen to enter the fiber. Entry by back diffusion will occur along the fiber as long as the partial pressure of nitrogen within the fiber 1sless than external partial pressure of approximately 0.79
T Ahmed and M J Semmens/J
6
17
Membrane Set 69 (1992) 11-20
Fiber Dla 0 04 cm Operatmg Pressure5 psi
4oE4
$ 08 0 u,
06
u 04
2OE4'
0
I
I
20
40
I 60
80
I 100
120
Length (cm) Fig 10 Calculated fluxes of nitrogen and oxygen across the membrane wall of a hollow fiber supplied with pure oxygen at 5 pslg
atm. This transfer occurs continuously and the
nitrogen that enters the fiber is swept down the fiber length by the flow of oxygen. As the oxygen diffuses out of the fiber, the gas phase becomes enriched in nitrogen and at some point along the fiber a nitrogen partial pressure of 0.79 atm is attamed. At this point, there is no net flux of mtrogen across the membrane wall. However, as the gas continues to move down the fiber’s length the nitrogen pressure mcreases and the nitrogen begins to diffuse out of the fiber and back into the water. The net accumulation of nitrogen along the length of the fiber shows it to be zero which is consistent with the assumption of steady state. The Influence of the operating pressure on the gas composition within the fiber was examined using the model. The oxygen supply pressure was varied between 2 and 24 psi; the calculated pressure profiles of oxygen along the fiber are m Fig. 11 as a function of the &menslonless fiber length. The operating pressure profoundly affects the oxygen profile along the fiber As the pressure increases a more favorable profile arises with a greater length of the fiber occupied by a high partial pressure of oxygen. This is explained by noting that nitrogen back diffusion is controlled by the nitrogen concentration gra&ent across the membrane and the liquid film diffusion, which is dependent only upon the water flow conditions out-
Fig 11 Calculated oxygen concentration profiles along a hollow fiber membrane supplied with pure oxygen at dtiferent pressures
side the fiber. As the oxygen pressure increases, the driving force for oxygen transfer increases and the diffusion-mduced gas velocity mslde the fiber increases significantly. By comparison, the nitrogen back diffusion rate does not increase ngmficantly, and so a longer fraction of the fiber length is required to accumulate the higher pressures of nitrogen Operation at higher oxygen pressures 1sclearly very desirable since it dramatically increases the effective driving force (oxygen concentration gradient across the membrane) Our analysis of mass transfer calculates kL by assuming that the oxygen partial pressure 1s constant and equal to the supply pressure. With the data in Fig. 11, it IS possible to correct the calculated kL to reflect the true pressure profile To this end, a mean partial pressure for each operating pressure was obtamed by integrating the pressure of oxygen over the fiber length. These “effective” pressures were then correlated to the actual operating pressure as follows: pnl =fPo
(8)
where Pm is the mean oxygen pressure withm the fiber, f is a correction factor and P,, is the oxygen supply pressure. The calculated correction factors are in Table 2, and the followmg correlation arose from these data.
18
T Ahmed and M J Semmens/J
Membrane SCL 69 (1992) 11-20
1000~
TABLE 2
I
Calculated values of the correction factor Operatmg pressure
Correction factor
(PSl)
2 4 6 10 24
0 0 0 0 0
327 400 506 546 712
f=O 26 (P,)’ 31
10'
(9)
This equation can be used to calculate a corrected average C* value for a specific operating condition. For example if the oxygen supply pressure is kept at 2 psig., the correction factor is f= 0.322 and the mean oxygen pressure is 0.64 PWZ.
Ahmed et al. [l] showed that observed values of kL increased as the partial pressure of oxygen increased (see Ref. [ 11, Fig. 11) These calculations of kL assumed that a constant oxygen supply pressure was maintained along the length of the fiber, although it is clear from the discussion above that this assumption was mcorrect. Accordingly, these data were recalculated usmg eqn. (9) to correct for the effect of pressure. The resulting data are shown in Fig. 12 The corrected correlations are (Fig. 13) Sh = 0 143 Re” 83,
103
104
Re
(10)
Sh=0.0184 ReOW3c” 33 The correlation coefficient ( r2 = 0.843) shows appreciable improvement. The new correlation was compared with literature correlations (see Ref [ 11, Table 2 ) . The correction was partially successful m reducing the scatter of data (Fig 13) but kL remains pressure dependent.
Discussion
This analysis of the gas transfer m sealedend fibers has shown the importance of back
Fig 12 The data shown m Fig 11 m Ref [ 11 followmg correction for the chanqng oxygen concentration profiles at different operating pressures
drffuslon of dissolved gases from the water to the fibers. In aerating natural waters, the high equilibrium partial pressure of dissolved nitrogen can dramatically affect the effectiveness of gas transfer Other gases may accumulate m a similar way, but they are less rmportant because of their lower partial pressures. Water vapor is an important exception, as discussed below. If other gases are transferred, such as SOz, the presence of both oxygen and nitrogen will be important. Water has a low vapor pressure but will accumulate within sealed-end fibers. Condensation has been observed m earlier studies and m our stu&ed. C8te et al. [2,3] speculated that condensation was caused by temperature dlfferences; however, the gas model developed here may offer a more likely explanation. We believe that water vapor enters and accumulates within ,__“”
1
y.o14317’x*oE295s
R”2=0843 I
1
IOCO-: f
loo,
l”,l,ooo Re
Fig 13
T Ahmed and M J SemmensjJ
Membrane SCL 69 (1992) 11-20
the fibers m the same way as other dissolved gases. For the water to escape from the fiber at the sealed end, the partial pressure inside must exceed the external vapor pressure, which 1s not possible without condensation Water will therefore continuously accumulate within sealed fibers and the performance of the fiber will deteriorate accordmgly. This problem must be overcome to make sealed fibers practical. The effect of pressure on hollow fiber membranes 1s not yet fully understood. The model developed here successfully predicted the gas velocrty and composition changes within the sealed fibers but was only verified at low operating pressures. Additional studies are needed to validate the model at higher operating pressures. When the model was employed to correct the data for operating pressure differences, It was only partially successful m reducmg the data.
Conclusions
A model 1s presented to predict changes in gas composltlon along the length of a sealedend fiber supplied with pure oxygen, while the fiber 1s immersed m flowing water stream The model accurately accounts for the back drffuslon of nitrogen from the water mto the fiber, and predicts experimentally measured conditions wlthm the fiber. The model also explams the observed accumulation of condensate within sealed fibers observed by earlier investigators The gas composition model was used to investigate the apparent dependence of masstransfer coefficrent on the pressure of oxygen supplied to the fibers. The model was unable to fully correct the observed pressure dependence of the mass transfer m the fibers Further work IS needed to account for this effect.
19
Acknowledgements The Mineral s&y of funded mlsslon
work was conducted at the Clvll and Engineering Department of UmverMinnesota, Minneapolis and partially by Metropolitan Waste Control Com(MWCC), St. Paul, MN.
List of symbols
c
total gas concentration (mg/l) concentration gas phase oxygen (mg/l) concentration gas phase nitrogen c2 (mg/l) water phase oxygen concentration c1L (mg/l) C2L water phase nitrogen concentration (mg/l) water phase oxygen concentration in C* equlhbrmm with the gas phase (mg/ 1) d fiber diameter (cm ) equivalent diameter of the module de (cm) D dlffusivlty of oxygen or nitrogen m water (cm’/sec) Henry’s law coefficient for oxygen HI (-) Henry’s law coefficient for mtrogen & (-) J gas flux in or out of the fiber (mg/ cm2-set) K overall mass transfer coefficient (cm/set ) kG, k,, kL mass transfer coeff. of gas, membrane and liquid respectively (cm/ set) L length along the fiber fiber length (cm) LO n number of fiber flux of oxygen (mg/cm'-set ) nl flux of nitrogen ( mg/cm2-sec ) n2 R radius of fiber (cm)
Cl
20
t
“P UL
V z
T Ahmed and M J Semmens/J
time (set ) velocity of gas in the fiber (cm/set) velocity of water (cm/set) volume of reservoir (1) distance along the fiber (cm)
Greek V
kmematlc vlscoslty
( cm2/sec)
Drmensum2ess numbers Re
v&e Reynolds number = -
Sh
kvd Sherwood number = y
SC
Schmidt number = i
Membrane Scr 69 (1992) 11-20
References
1 T Ahmed and M J Semmens, Use of sealed end hollow fibers for bubbleless membrane aeration Expemmental studies, J Membrane Scl ,69 (1992) l-10 2 P L C&, J -L. Bemllon, A Huyard and J -M Faup, Bubble-free aeration usmg membranes Process analysis, Water Pollut Control Fed, 60(11) (1988) 19861992 3 P L CXti, J L Berslllon and A Huyard, Bubble-free aeration using membranes Mass transfer analysis, J Membrane Scl ,47 (1989) 91 R B Schaffer, F V Ludzack and M B Ettmger, Sewage treatment by oxygenation through permeable plastic films, J Water Pollut Control Fed ,32 (9) (1960) 939941 Ming-Chlen Yang and E L Cussler, Deslgnmg hollowfiber co&actors, AI&E J ,32( 11) (1986) 1910-1916