Reverse Osmosis and Ultrafiltration - American Chemical Society

30 Determination of Gel Volume Deposited on Ultrafiltration Membranes J. L. GADDIS1, D. A. JERNIGAN1,3, and H. G. SPENCER2 1

Department of Mechanical Engineering, Clemson University, Clemson, SC 29631 Department of Chemistry, Clemson University, Clemson, SC 29631

Downloaded by STANFORD UNIV GREEN LIBR on July 1, 2012 | http://pubs.acs.org Publication Date: January 1, 1985 | doi: 10.1021/bk-1985-0281.ch030

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A method is developed to interpret the transient flux response of an ultrafiltration membrane to yield an estimate of gel volume deposited. Experiments of this type are presented for polyvinyl alcohol (PVA) ultrafiltration in step pressure changes up to pressures of 4.2 MPa. Thicknesses of 20 μm are typically indicated for this combination.

Any pressure-driven membrane process tends to produce permeated solvent more or less in proportion to the pressure applied to the membrane. As the pressure increases the permeate production at some point fails to maintain proportionality to the pressure increase — a form of diminishing return. Michaels (I) proposed for the common case of separation of large molecules that the retained molecules, being unable to diffuse sufficiently against the incoming solvent flow, reach a concentration at which the mixture provides hydraulic resistance to the solvent, termed as "gel". Blatt and coworkers (2) and later Shen and Probstein C^) studied the development of concentration near the membrane and the necessary distribution of flux along the channel subject to the gel concept of constant concentration. By simple extension of such concepts a finite membrane element may be shown to produce a flux-pressure result as shown in the solid line of Figure 1. Up to point "A" the permeate flux is limited by the membrane permeability to solvent flow. Beginning above point "A" there is at first a small, then growing from downstream to upstream, coating of gel on the membrane as the pressure is increased. Since the gel thickness increases along the channel, the upstream portions are more productive. The fraction of non-gel-affected membrane surface shrinks rather quickly as the pressure is raised. Therefore the average flux reflects rather strongly the tendency of the gel coating. Some investigators suspected that the gel concept might be incorrect and that the effect might be only osmotic resistance. Trettin and Doshi (4^) have illuminated the subject and show that ^Current address: CARRE, Inc., Seneca, SC 29678 0097-6156/85/0281-0415$06.00/0 © 1985 American Chemical Society

In Reverse Osmosis and Ultrafiltration; Sourirajan, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

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both effects are possible. Observations of time-dependent fluxes by Reihanian et al 05) indicate that the filter cake layer or compressed gel layer tends to resist diffusive tendencies. The transients reported herein are all associated with increasing pressure steps and not decreasing steps which do fail to mirror the increasing pressure effect. The pressures are felt to be high enough to neglect osmotic pressure effects.


The Hypothesis If a step change in pressure is applied to an ultrafiltration membrane operating at steady state it should eventually respond from point "0" to point "2" on Figure 1. Point "0" is typically in the non-permeable region so part of the membrane is covered with flux retarding, concentrated solute. Point "2" has a greater extent and thickness of concentrated solute, or gel. If the pressure step is applied suddenly, the gel does not form instantly and hence the flux increases along a line of temporarily constant permeability shown by the heavy dashed line. Because there is no absolutely perfect step, there is a rounding of the curve as the membrane-gel combined resistance builds. Then, while pressure is kept constant at P2, the flux accomodates to the steady point 2. This process is sketched on a time history of flux in Figure 2. Prior to time, t0, the flux is constant at J0. (The overbar on the symbol denotes spatial average over the module). At time t 0 the pressure is raised to P Q and maintained. The average flux increases along with the pressure until the growing resistance of gel coat forces it to diminish to the steady point 2. The "wall" denotes herein the interface between the concentration boundary layer and membrane or gel. At the wall there is, due to bulk motion, a solute flux Jc w passing toward the gel, where J is the local solvent flux and c w is the concentration of solute at the wall, which in this case is the gel concentration. The diffusion of solute away from the gel may be described as D 3c/3y|w where D is a suitably defined diffusion coefficient and 3c/3y|w is the solute concentration derivative evaluated at the wall. The net solute flux between these two must be interpreted as a deposition of solute onto the gel surface.

whenever steady state is reached the gel mass is constant and the right hand side is a familiar boundary condition, mg is the mass of gel per unit area. Values of mg, J, and c w are both time and streamwise coordinate dependent, and the following simplifying assumptions are presumed. Initially there is a region from 0 to x* having no gel (mg = 0 ) , c w < c g , and J = Jp where Jp is the flux limited by permeability and pressure. Following this is a region x > x having a layer of gel growing (_3) according to x + l'3, J decaying according to x~l'3s an x the gel grows monotonically and the diffusion, by quasiequilibrium arguments, is again time invariant. The pressure cannot negotiate a true step change and so the event that signals the start of the assumed quasi-equilibrium period must be reckoned. We have used herein the time at which the average flux reaches the asymptotic value. This time is the t^ of Figure 2. Probably the end of pressure rise would be better in keeping with the nature of the interpretive assumptions, since only at that point will the gel begin to cover the region to x = x p . However, the actual pressure trace was not recorded in coordination with the flux trace. Only a few seconds, representing about 1 percent of the total active duration of experiment, separate the two events. Thus the timing errors are estimated to be of small consequence. In equation (1) during the transient operating period we integrate from x = 0 to I and from time t^ to t2» There results

The diffusion at quasi steady state must equal Jc w in the region 0 to Xp, so the first and third integrals cancel each other. The value of D 3c/3y|w equals J2^ w at each position according to the assumption. This substitution is made in the fourth integral on the right. The left side is simply the mass of gel per unit of width, mg, accumulated during the time interval. Equation (2) may be rewritten

Further it is convenient to add and subtract a term c g and divide the equation by c g to produce Vg - mg/Cg, the gel volume per unit of width. •*-see the Appendix for a partial justification.


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Gel Volume Deposited on UF Membranes

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Using J as the spatial mean value of flux


gives

This expression is useful because one only measures mean fluxes on channels. The Vg is the gel volume deposited per unit area in the transient from tQ to t£. Graphically, Equation (5) represents the shaded area of Figure 2. Resistances of the membrane, Rm, and gel, R g , are additive in the form

Here R is the resistance to flow P/J and P is the pressure difference across the membrane; osmotic pressure is assumed negligible. With only increases in pressure considered the trend of Figure 1 shows clearly that R2 > Ri and ^ is not pressure dependent so Rg2 > R gl* Data for resistance increase are derived from steady state values only as

Upon the determination of resistance increase from Equation (6) and the volume from equation (5), it is possible to compute a resistivity, R, as

This resistivity is u/K where u is the viscosity and K the specific permeability of Darcy's law (see Reference 6 ) . To compute the restivity, the pressure on the newly formed gel layer should be J2AR on the thickness Vg, on average. The quotient of these is the average pressure gradient through the gel layer, so

This resistivity is approximate since it is the ratio of spatially averaged quantities. A more careful estimate can be computed by knowledge of the extent and thickness distribution of the gel layer. Some calculations of this type have been made for a range of hypothetical step changes. These calculations though not presented tend to show systematic shifts of 15 to 20 percent from the value given by equation (7). For the purpose of this discussion the cursory evaluation of Equation (7) is considered adequate.


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Experimental Setup A schematic of the ultrafiltration system employed for this study is illustrated in Figure 3. The reservoir maintained the feed solution and accepted the return of the membrane retent (concentrate) and permeate. A positive displacement pump with an attached pneumatic accumulator and by-pass loop pressurized the system. Bourdon tube gauges measured both the pump exit and module supply pressure. Temperature was monitored by a thermocouple placed on the concentrate line close to the module. Tap water flowing countercurrent through a concentric heat exchanger provided temperature control. Permeate from the module flowed through a 0.34-mm diameter orifice across which pressure difference was measured with a 1 psi pressure transducer. The pressure transducer signal was recorded continuously by a strip chart recorder and by an x-y plotter during transient operation. The orifice-transducer-recorder system was calibrated from steady state data measured with a graduated cylinder and a stopwatch. The concentrate flow was measured with a float type flowmeter calibrated for each fluid concentration and temperature tested. The membrane module consisted of a hydrous zirconium oxide (7_) membrane formed on the 12.5-mm internal diameter of a 0.91 m long, sintered stainless steel support tube obtained from Mott Metallurgical Corporation in Farmington, Connecticut. The support tube had a 0.5-micron pore size (rating of the manufacturer). Buttwelded to each end of the support tube was a 0.3»m length, 15.8-mm internal diameter, stainless steel tube. The permeate chamber surrounding the membrane tube was a 2-inch, schedule 40, stainless steel pipe sealed with tube fittings to the solid tubing on either end of the membrane tube. The permeate chamber was unusually large to accommodate some features not germane to this investigation. Other details of the equipment may be found in Reference 8. Procedure Solutions of 2 percent and 4 percent mass concentration of polyvinyl alcohol (PVA) at 60°C and 80°C were tested in the ultrafiltration loop with flow rates from 3.8 to 15.1 liters per minute. The PVA was DuPont Elvanol T-25 which is a highly hydrolyzed, high molecular weight blend commonly used for textile warp sizing. Solvent water was RO permeate from laboratory tap water. A typical procedure follows. The pump was energized with the reservoir filled with 57 liters of demineralized water. The pump discharge pressure was adjusted to 4.8 MPa using the by-pass loop valve and the upstream module valve. Membrane pressure was maintained low until data was to be taken. The solution to be tested was mixed in the feed reservoir. The 2 percent and 4 percent concentration of PVA solutions contained 1140 grams and 2280 grams, respectively, of dry powder per 57 liters of water. PVA was slowly added to the reservoir to prevent coalescence. Mixing the PVA was accomplished at 60°C or 80°C depending on



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the operating temperature of the upcoming run. The desired test temperature was maintained by adjusting the cooling water valve in order for the heat exchanger to remove excess heat generated by the pump. The by-pass loop valve and valves upstream and downstream of the module were adjusted simultaneously to maintain the 4.8 MPa pump discharge, pressure the desired flow rate, and initial module pressure. After steady state was reached these three valves were quickly coordinated to raise the pressure to the new level while maintaining the pump discharge pressure and throughflow rate. The temperature varied only slightly since the pump power was maintained constant. The permeate flux was recorded on an X-Y plotter versus time and also on a strip chart recorder. The plotter scale was used for flux record and the recorder maintained a record of sequence of runs. When the data for the final pressure setting was recorded, the feed solution was pumped into a storage tank. The feed reservoir was filled with 20 liters of water to rinse the ultrafiltration membrane at 80°C for 2 hours. During this rinse a low module pressure and high recirculation rate were maintained. With rinsing complete and system drained, the feed solution was transferred back into the reservoir to start another run. Results and Discussion Figure 4 shows the steady state average flux obtained at various operating pressures and shear levels (indicated by flow rate). These data for 2 percent and 60°C are typical of those obtained. In each pressure excursion data were only taken in asending pressure to avoid hysteresis effects. The membrane was washed between series. Data obtained at such steady operating conditions were used to provide the orifice calibration of Figure 5. The strip chart R value was an arbitrary scale on the X-Y Plotter used to record transient flux data. Figure 6 is a tracing of an actual X-Y (R-time) trace obtained when the pressure was raised from 2.1 MPa to 2.8 MPa at 7.6 dm3/min flow at 2 percent concentration and 60°C. The transient period for the pressure rise was not measured but normally was consumated within 5 to 10 seconds. The transient period for the flux was much longer, usually 100 to 200 seconds. By using the steady state calibration, Figure 5, the arbitrary scale of the recorder was replaced by permeate flow rate to provide an average flux history. A minor complication emerged due to temperature transients. The procedure described produced ideally constant pump power so that no temperature change would be expected. Indeed, it was found that the temperature was usually constant to within 1°C. However, the large permeate chamber, when subjected to temperature changes, would experience expansion or contraction of the fluid sufficient to cause excursions such as shown in Figure 7. In extreme cases, the runs were disqualified because their interpretation was deemed unreliable. For the most part, the temperature variations were 0.3°C or less, producing traces like Figure 7. Care was exerted to control the final temperature to within 0.1°C of its initial value and to include the temperature-induced wiggle completely within the




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Figure 3.

Figure 4. Membrane.

Schematic of Test Apparatus.

Steady State Flux Data for 2% PVA at 60°C in a Tubular



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Figure 5.

Permeate Flow Meter Calibration.

Figure 6. Recorded Flow Trace During a Step From 2.1 MPa to 2.7 MPa at 2%, 60°C.


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analysis. Thus any rise was compensated by a corresponding fall in permeate flow. In addition to runs having large temperature variations some runs were not analyzed because no gel was formed and others because of flux levels too low to measure with confidence. There remained 29 individual runs which were analyzed according to Equation (5) and Equation (6). The volume of gel deposited rather than its volume per unit area is presented in Figure 8. A considerable scatter is definitely noted to the data such that any results should be considered provisional in nature. The data isolated in the triangular region all belong to the subset of 4 percent concentration, 60°C. The flow rates under these conditions were the lowest of all the concentration-temperature pairs run and the flow rates are therefore subject to the highest influence of experimental uncertainty. The line shown in Figure 8 is suggested as a possible trend for the data. Such a line passing through tha origin has constant resistivity; the particular line has R = 1.9 x 10 1 " Pa-s/m2. For an average viscosity of 4 x 10"^ Pa-s, this resistivity corresponds to a Darcy law permeability of 2.1 x 10~20 m 2 . According to existing correlations for permeability of particles, for example CarmanKozeny (_5), and for a void fraction of 0.4, the mean particle diameter would be 4.5 nm. This does not mean that PVA forms a packed bed of 4.5 nm particles, but rather that the gel layer has the resistivity of such a packed bed. The mean thicknesses of deposit shown on Figure 8 range generally up to 2.2 x 10~5 m or in one case to 3.8 x 10"^ m for the 0.0357 m 2 test section. The mass of gel is anticipated to be 0.2 to 0.6 g at a volume of about 1 cm3. Detection of such small mass amounts by analysis of mass depletion is clearly impractical. In principle one can analyze the residue washed off following deposition as was reported in Reference 8. However no attempt was made to determine gel mass and hence concentration. The experimental uncertainty of the procedure is fairly significant compared with the scale of results. Coarsely, the flow rates may be determined to within ± 0.005 cm 3 /s. If this uncertainty is applied over the 100-second integration period the volume uncertainty would be = 0.5 cm 3 . This certainly would account for the scatter tendency shown in Figure 8. The increase in resistance is much more accurately determined. Most values should lie within ± 10% of the value indicated. Conclusion Subject to the approximations made, a method has been advanced which allows determination of gel volume in a membrane system. Experiments with polyvinyl alcohol solutions illustrate the use of the method. Transient flux responses to step pressure changes required hundreds of seconds to be consummated; volumes of gel of the order of cubic centimeters on a test section of 357 cm were registered. The thickness of the gel layer was frequently 20 um and produced resistances to flow of 5 x lO-^ Pa-s/m.



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Figure 7. Flux History in Step From 2.7 MPa to 4.2 MPa Showing the Effect of Temperature Variation.

Figure 8.

Plot of Gel Volume Versus Increase in Resistance.


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Appendix


This appendix considers justification of the assumption that the concentration rises rapidly at the membrane to the gel concentration level in response to sudden flux increases. The slowest point to respond should be the most upstream point, x p , which is included in treatment herein. The accepted differential equation is, assuming constant diffusion coefficient D and assuming the velocity normal to the membrane surface is equal to permeate flux:

Auxiliary conditions are

plus appropriate initial conditions. The dimensional variables are marked with a tilde and may be expressed in terms of non-dimensional variables by

Thus

Introduce n = 1 - e" v such that

Break the solution into a sum c*(x,n) + c'(n,t), treating x as a parameter in c 1 . The transient response as t -*• °° does not vanish but tends to (1 - n)B. Treating B as B(x) plausibly allows solution to c*(x, n) which together with (l-n)B forms the steady state solution. The equation for c1 and its boundary conditions are

The solution should be related strongly to the solution of the simplified form


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This simplified form produces solutions which are lower in amplitude and almost twice as responsive than the precursive form, as judged from numerical studies. The solutions to diffusive slab problems with constant concentration boundary conditions by experience respond (to 95 percent) within t

Reverse Osmosis and Ultrafiltration - American Chemical Society

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