Ind. Eng. Chem. Res. 1994,33, 1771-1779
1771
Dynamic Properties of Ultrafiltration Systems in Light of the Surface Renewal Theory Andrzej Koltuniewicz' and Andrzej Noworyta Technical University of Wroclaw, Institute of Chemical Engineering, ul. Norwida 416, 50 373 Wroclaw, Poland
Flux decline in pressure-driven membrane processes is caused by concentration polarization and fouling. Application of the surface renewal concept provides an opportunity to describe these phenomena. To this end, the Danckwerts surface-age distribution function has to be modified by assuming that the age of the oldest surface element cannot exceed the duration of the process. In this manner the variation of surface-age distribution during the process has been included to correspond to the effect of the building up of the concentration polarization layer during the initial period of ultrafiltration. When the modified version of surface-age distribution is applied to determine the average flux, the relationship obtained takes into account flux decline as the effect of concentration polarization layer development. The practical significance of the model resides in its ability to describe the dynamic behavior of the plant: it can be applied to the optimization of operation modes, thus enabling the membrane cleaning strategy to be determined (i.e. process duration, backflushing, or pulsation frequency, etc.). In particular, it is of great importance in membrane largescale continuous plants.
Introduction The main task in designing ultrafiltration processes is to ensure maximum yield and rejection with minimum costs, with as long as possible membrane lifetimes. The majority of economical factors used during the designing and controlling membrane processes are based on the permeate flux. Capital costs for larger plants (>lo0 m2) are proportional to membrane area. Operating costs, i.e., capital charge, power, membrane replacement, membrane cleaning,labor, and maintenance,are also dependent on membrane surface area and consequently on permeate flux (Cheryan, 1986). Therefore the optimal values of permeate flux should be carefully determined during design with respect to membrane and feed properties as well as operating conditions. One of the main problems being solved in large-scale systems is to ensure the stationary conditions of the membrane process. Even in cases when all operating parameters are kept on a constant level, permeate flux declines systematically with time. This is a consequence of reversible and irreversible changes in the membrane and its closest vicinity. Concentration polarization is the primary reason for the flux decline during the initial period of operation (Fane et al., 1983,1985,1987;Abulnor, 1988;Aimar, 1988,1989; Finnigan, 1989; Fell, 1990). Accumulation of the solute retained on a membrane surface leads to increasing permeate flow resistance R, at the membrane wall region. Concentration polarization can be controlled in a crossflow membrane module by means of construction (Finnigan, 1989; van den Berg and Smolders, 1988) or velocity adjustment (Fane, 1987;van den Berg and Smolders,1988; Aimar, 1989; Milisic, 1986; Tarleton, 19881, pulsation (Finnigan, 1989;Bauser, 1982;van den Berg and Smolders, 1988; Milisic, 19861, ultrasound (Milisic, 1986; Athaide and Govind, 19871, or electric field (van den Berg and Smolders, 1988; Athaide and Govind, 1987). Another reason for flux decline is membrane fouling, which is considered as a group of physical, chemical, and biological effects leading to irreversible loss of membrane permeability. Attempts to analyze the fouling phenomenon have shown that the main factors are adsorption of 0888-5885/94/2633-1771$04.50/0
some feed components (Fane, 1983, 1985; Aimar, 1988; Fell, 1990; Suki, 1984; Bauser, 1982; Hanemaaijer, 1989), clogging of the pores (Hanemaaijer, 1989; Gilron, 1987; Schippers, 1980, 1981), and deposition of solids on the membrane surface (Fane, 1983,1985; Suki, 1984; Gilron, 1987; Audinos, 1989; Le and Gollan, 1989) accompanied by crystallization and precipitation (Hanemaaijer, 1989; Gilron, 1987) or compaction of the membrane structure (Turker, 1987; Schippers, 19811, chemical interaction between membrane material and components of the solution (Fell, 1990; Suki, 1984; Turker, 1987; Reihanian, 1983) gel coacervation, and bacterial growth. Fouling can be reduced by proper selection of the membrane material and/or by membrane pretreatment using surfactants (Fane, 1985, 1987; Fell, 1990; Hanemaaijer, 1989; Reihanian, 1983;Belfort, 1979;Kim, 1989), polymers (Fane, 1987; Fell, 1990; Bauser, 1982; Hanemaaijer, 1989; Le and Gollan, 1989; Belfort and Marx, 1979), and enzymes (Howell, 1981). Besides these methods, fouling can be limited by adjusting the environment parameters (Fane, 1983;Matthiasson, 1983;van den Berg, 1988; Aimar, 1989; Belfort and Marx, 1979) and by using all the means that reduce concentration polarization. All the effects mentioned above take place simultaneously with different intensities with respect to the system and time of operation (Fane, 1983,1985;Abulnour, 1988; Aimar, 1988; Finnigan and Howell, 1989; Fell, 1990). Therefore it is practically impossible to separate the concentration polarization effects from the fouling. Besides, the division of membrane permeability changes into the reversible and irreversible seems to be rather conventional and connectedwith the applied cleaning method (Abulnour,1988;Fell, 1990;van den Berg, 1988). However, some exceptions occur due to the irreversible chemical changes in membrane material (Fell, 1990; Suki, 1984; Turker, 1987; Reihanian, 1983). In this paper the flux decline is described as a result of concentration polarization layer development. The model is based on Danckwerts's surface renewal theory (Danckwerts, 1951). 0 1994 American Chemical Society
1772 Ind. Eng. Chem. Res., Vol. 33, No. 7,1994
Theory The application of Danckwerts's surface renewal theory to membrane processes has been presented elsewhere (Koltuniewicz, 1992), taking into account steady-state conditions. It has been stated that the surface renewal model is more realistic than the commonly used film model, since the structure of the wall region is not stable. The main arguments that justify the random nature of the mass transport at the membrane boundary layer are listed below. 1. Inertial forces, that are caused by membrane roughness, are capable of picking up the fluid's elements at the turbulent flow regime. 2. Drag forces, that affect the movement of fluid elements at the wall region, are resultants of random components perpendicular to the membrane surface. These forces are caused by pore sizedistribution and their random arrangement on the surface. They can stimulate the chaotic movement of the fluid elements in the turbulent as well as in the laminar flow. 3. Lifting forces, that are caused by the 'lateral migration" of the colloidal particles and fluid elements at the wall region, were reported in many papers. They have been used as an explanation for "flux paradox", i.e. excessive permeate flux that could not be described by the film model. In this paper the new potential for the surface renewal theory to predict permeate flux decline during buildup of the concentration polarization layer has been shown. According to the surface renewal model, the membrane is not covered by a uniform concentration polarization layer, as it was assumed in the film model, but rather by a mosaic of small surface elements with a different age, and therefore with a different permeate flow resistance. Hydrodynamical impulses can sweep away any element randomly, and at that moment a new element starta building up a layer of retained solute at the same place on the membranesurface. The ageof the element is the period of time that has passed since the last hydrodynamic impulse. The local permeate flux can then be considered as age-dependentbecause of the increasing flow resistance due to retained solute accumulation. It is assumed that the local flux can be approximated by the flux observed during batch ultrafiltration under steady-state conditions. The appropriate flux-age relationship has been verified experimentally (Koltuniewicz, 1990)
J ( t ) = (J,- J*)e"'
+ J*
(2)
where f ( t )is the surface-age distribution function. The 'age distribution function" has been derived here on the basis of the following assumptions: 1. The ages of all elements are within the range 0 < t < t, in any time of process duration t,, which leads to the conclusion that
Jotpf(t) dt = 1
3. Hydrodynamical impulses are random events with a uniform probability distribution on the entire membrane surface and throughout the time of process t,. The intensity of the hydrodynamical impulses, which results in local restoration of membrane permeability at any surface element, can be expressed by the rate of surface renewal (5). This parameter denotes the contribution of the surface renewed per unit time for some age fraction (t, t dt). It has been assumed (followingDanckwerta) that the rate of surface renewal is equal to each age fraction and that it is time-independent.
+
S =
f ( t )dt - f ( t
+ dt) dt
dtl dt
(5)
The rate of surfacerenewal depends on hydrodynamical conditions in the membrane vicinity, such as shear stresses or turbulence. A differential equation (eq 5) gives the surface-age distribution functionf(t) (Danckwerts,1951). On the basis of the fust assumption (contrary to Danckwerts's model that has been applied in a former paper (Koltuniewicz, 1992)), one could obtain the modified age distribution function that is dependent on the time of process duration t, as follows:
This distribution varies during process time t,, since the upper age limit for the oldest element increases. Therefore this model enables the concentration polarization layer development to be described during the time of permeation. After a sufficiently long period of time, the proposed distribution converges with Danckwerts's distribution: (7)
(1)
where Jo is the pure solvent (water) permeability, J* is the value which the flux approaches after stabilization, and parameter A expresses the rate of flux decline. The effective permeate flux that passes through the total membrane surface can be considered as an average value from local fluxes that flow through the infinitesimal elements of the membrane surface with different ages:
S = JotpJ(t)f ( t )dt
2. Surface age distribution is only the result of the hydrodynamical impulses. In cases where there are no impulses, the age of any element is equal to the process duration t,, and the surface age distribution can be expressed by Dirac's function:
(3)
Danckwerta (1951) evaluated the error that results from admitting the infinitesimal value of the oldest surface element in his distribution as less than lo%, unless the condition is fulfiied that st > 1.5. Nevertheless, his evaluation had been performed for mass transfer during absorption,based on a particular solution of diffusionmaw transport according to the second Fickian law, and could not be arbitrarily applied to membrane processes. A more generalized surface-age distribution function has been developed by Kolek (1978),where the interfacial area was considered time-dependent. When the effective permeate flux is determined after integration (eq 2) with instantaneous flux calculated from eq 1and the surface-age distribution function from eq 6, the general relationship is obtained
Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1773
lo
p ermea ti on
cleaning
Figure 1. Diagram of typical flus-time dependency during cyclic operations in large-scale ultrafiltration systems.
This equation makes it possible to determine the flux decline caused by the development of the concentration polarization layer during the process duration t,. It is easy to prove that eq 8 is the generalization of the model presented earlier for stationary conditions (Koltuniewicz, 1992). The asymptotic value of the flux after a long period of cross-flow ultrafiltration (3dcan be derived directly from eq 8 as a limit:
During process design and control, one should be able to predict the average value of the flux during permeation time (t,) by taking into account the flux decline.
Jav= LjtpJ(tp) dt, t,
O
Large-scale membrane systems operate in a cyclic mode, where the CIP (clean in place) operation alternates with the normal run (Figure 1). The fouling effect could then be included as a reduction of pure water permeability during the subsequent cycles of the process Jo(t). The rate of such reduction may depend on the cleaning method (i.e., cleaning time tc and cleaning agents). Membrane lifetime ( 7 ) determines a number of such cycles according to the formula
The average value of the permeate stream after n cycles of operation can then be expressed as follows:
When permeation times (t,J and cleaning times ( t d ) in each ith cycle remain constant, a simpler relationship holde:
Figure 2. Flux decline during dead-end ultrafiltration (u = 0).
The flux decline in a single cycle J(t,) is considered to be a result of the combined effects of concentration polarization and fouling. However, it was noticed in the actual experiments that the fouling effect was negligible. The possibility of describing the dynamic behavior and properties of the ultrafiltration systems (the large-scale in particular) can obviously be important in (1)automatic process control, (2) optimization of process parameters and mode, including cleaning, sanitizing, pretreatment,
1774 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 Xtp11O6
Im/sl 16
14 2
0
161
134 0252
3
A
161
85
6641 0168
209
1399
I
432
12 0622 6641 0018
5956 2542
274
10
8
6
4
2
tplrl
Ittdmf Irnlsl 16
14
12
6
o
40
134 0252
7
A
40
85
165
0168 5192
1399 13635
0622 165
0018 1479
2542 2430
10
8
6
4
2
0
200
400
600
800
1000
1200
-
1400
1600
1800
2000
2200
2400 2600 tp [SI
Figure 3. Flux decline during cross-flow ultrafiltration. Lines were determined by eq 8 and points from the experiment. The broken lines 0 5 ) (a, top) for high-pressure and (b, bottom) for low-pressure conditions. denote J., i.e. the asymptotic values of the flux (for t ,
etc., and (3) intensification of the process yield by the use of pulsation, ultrasound, back-flushing,and other methods. Experimental Section
The surface renewal model was verified during the ultrafiltration of skim milk in a hollow fiber module (Romicon HF-16-43-PM50). The main parameters of the module were as follows: membrane surface area = 1.4 m2, number of fibers = 682, fiber length = 0.60 m, fiber diameter = 1.1 mm. Composition of the powdered skim milk used during the experiment was 51.2 % lactose, 35.7 % proteins, 8.3 76 mineral salts, 0.8% fat, and 4.0% water. The concentrations of dry mass in the skim milk prepared were CO= 10,13.4,20,34.6,50, and 85.5 g/L. The constant value of concentration in the recirculating retentate was maintained by returning the total permeate
back to the loop. Temperature was kept level a t T = 303 K by use of an ultrathermostat and heat exchanger in the recalculation loop. Transmembrane pressure in the module was controlled automatically in the range hpm = 0-160 kF'a by use of a manostat system. The experiments were carried out in two modes: (1) unsteady-state mode without recirculation of retentate u = 0; (2)cross-flow mode with velocity in the range u = 0-0.622m/s. During the unsteady-state experiments, flux decline was observed continuously (Figure 2) under conditions of constant concentration, pressure, and temperature. The results of the experiments enabled parameters A and J* in eq 1 to be determined for various concentrations and pressures. Flux decline during cross-flow ultrafiltration was registered continuously for approximately 6 h for fixed
Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994 1775 Table 1. Comparison of the Parameter A Measured and Calculated (eq 15) for Selected Operating Conditions CO(kg/mg) hpm ( P a )
10 13.4 20 34.6 50 85
20 100 180 20 100 180 20 100 180 20 100 180 20 100 180 20 100 180
10BA,,
(8-1)
0.00225 0.01108 0.02070 0.00254 0.01343 0.02123 0.00300 0.01634 0.02840 0.00400 0.02500 0.04000 0.00500 0.02560 0.05100 0.00750 0.03900 0.07000
10BAd (8-9 0.00219 0.01099 0.01979 0.00239 0.01298 0.02338 0.00325 0.01628 0.02932 0.00444 0.02222 0.04010 0.00547 0.02738 0.04929 0.00742 0.03712 0.06682
re1 error 2.70 -0.85 4.80 -1.9 3.4 -9.1 -1.7 0.8 -3.1 -9.9 13.1 -0.2 -8.5 -6.5 3.4 1.1 5.4 4.8
operation conditions, i.e. for velocity (u), concentration (Co), pressure (APTR), and temperature. Some results presented in Figure 3a and Figure 3b were limited to the range of time t, when the effect of flux decline was spectacular. The experiments were carried out in order to verify the model of flux decline (eq 8). By the end of these observations the permeate flux approached the asymptotic values JB, a state that corresponds to stable operating conditions. After each run, the cleaning procedure was applied to restore the original membrane permeability. The following cleaning agents were applied: 0.1 N NaOH solution, 0.1 N HC1 solution, and an enzyme solution. Ultrapure water was used after each stage of cleaning to check the permeability of the membrane. There was no systematic reduction of membrane permeability observed during the 6-month duration of the experiments.
Determination of the Model Parameters Permeability of the membrane (Jo) was checked before each run for various transmembrane pressures APm. The constant value of flow resistance R m = 2.42 X los kPa s/m was maintained by use of the cleaning procedure. This means that the relationship between pure water permeability and pressure is linear:
Dead-end ultrafiltration of the skim milk was carried = 20,40,60,80,100, out for nine various pressures (MTR 120,140,160, and 180) and six concentrations (Co = 10, 13.4, 20, 34.6, 50, and 85). It was observed that the rate of flux decline (A) was dependent on both of those parameters (Figure 2). The relevant correlation between accumulation rate A in eq 1 and process parameters was worked out as
A = 0.298 X 104C,0.687APTR
(15)
The accuracy of this correlation can be evaluated on the basis of the results in Table 1,where the calculated and measured A values were compared for selected values of pressure and concentration. The other correlations were prepared for calculations of the equilibrium flux on the basis of the same experiments. A comparison similar to that for A is presented in Table 2. Parameter J* in eq 1expresses the minimum value that the permeate flux reaches asymptotically after
Table 2. Comparison of the Parameter JI Measured and Calculated (eq 18) for Selected Operating Conditions
lVJ*, CO(kdm3)
10 13.4 20 34.6 50
85
A l h (Pa)
20 100 180 20 100 180 20 100 180 20 100 180 20 100 180 20 100 180
(mVm2s) 0.267 0.273 0.282 0.196 0.197 0.187 0.106 0.104 0.110 0.048 0.055 0.060 0.028 0.032 0.037 0.020 0.020 0.021
lVJ*& (m3/mls) 0.2560 0.2560 0.2560 0.179 0.179 0.179 0.109 0.109 0.109 0.056 0.056 0.056 0.035 0.035 0.035 0.018 0.018 0.018
re1 error 4.3 6.6 10.1 9.5 8.1 -6.1 -2.7 -4.6 0.8 -14.3 -1.8 7.1 -20.1 -8.6 5.7 11.1 11.1 16.6
a long period of dead-end ultrafiltration (Figure 2). It was found that parameter J * is not dependent on pressure (see Table 2) but strongly decreases with concentration according to the relationship J* = 3.875 X lo4
c,'.21 One can expect that for very dilute solutions J * should approach the pure solvent permeability: J*(Co) = J, for C, = 0
(16a)
On the other hand, for the case of high concentrations, one can predict J * on the basis of the osmotic pressure model J*(Co) = 0 for APTR - ?r(Co)= 0
(16b)
The similar condition for the gel polarization model is J*(Co) = 0 for C, = C, As it may be expected from eq 16a, the equilibrium flux should be pressure-dependent for dilute solutions. This was not the case observed during the experiments in the range of the operating parameters. Therefore eq 16 may be used only for interpolation within the range of experiments. Nevertheless, the extrapolation to higher values of concentrations gives good results, as it is shown in Table 5. In the experiments, the majority of information about hindering the flux during solute accumulation has been obtained in simple dead-end experiments. Similar results for flux predictions during ultrafiltration of BSA solutions and kaolin suspensions have been presented elsewhere: for dead-end ultrafiltration (Koltuniewicz and Noworyta, 1990) and flux limitation during cross-flow UF (Koltuniewicz, 1992). The rate of surface renewal (s) was determined by use of eq 9 on the basis of asymptotic values of permeate flux
(Jd.
The asymptotic values of the flux were measured after a long period (-6 h) of cross-flow ultrafiltration. Parameters Jo,A, and J * in eq 9 were consecutively determined, as it was shown above. The experimental and computa-
1776 Ind. Eng. Chem.
Res., Vol. 33, No. 7, 1994
Table 3. Selected Values of Parametera Uwd during Determination of the Rate of Surface Renewal experimentaldata calculated data apTR co U J J~ (W14) J* (eq 18) A (eq 15) s (eq 9) (Pa) (kg/ma) (m/s) (106mVm% (106mVm%) (106ma/m%) (1081/81 (108 1/81 1.110 2.610 0.179 1.390 3.010 0.252 0.179 2.596 2.031 20 13.4 0.443 3.728 8.264 2.542 4.179 0.622 1.000 1.703 0.179 1.40 2.021 0.282 0.056 4.445 2.000 2.631 8.264 20 34.6 0.443 2.480 3.043 0.622 1.000 1.100 0.179 1.360 1.330 0.252 0.018 7.400 2.110 20 65.0 0.443 1.790 8.264 2.127 2.340 0.622 1.090 3.562 0.179 1.370 4.371 0.252 0.179 23.364 1.910 6.115 74.38 180 13.4 0.443 2.540 7.461 0.622 1.070 2.073 0.179 1.470 2.570 0.252 0.056 40.000 2.000 180 34.6 0.443 3.648 74.38 2.480 4.497 0.622 1.244 1.260 0.179 1.350 1.548 0.252 0.018 66.600 2.010 180 85.0 0.443 2.219 74.36 2.560 2.753 0.622
tional data used for determining the rates of surface renewal are specified in Table 3. The obtained resulta show that the rate of surface renewal is not dependent on concentration and pressure but is strongly dependent on velocity. This relationship was determined for the range of velocity u = 0-0.622 m/s, i.e., for laminar flow:
s (eq 17) (1081/81 1.090 1.380 2.040 2.580 1.090 1.390 2.040 2.580 1.090 1.380 2.040 2.580 1.090 1.380 2.040 2.580 1.090 1.380 2.040 2.580 1.090 1.380 2.040 2.580
j .io6 Imlrl 4 .B
44
40
s = 3.48~O.~' X lo3
(17)
It is supposed that the constant is dependent on con-
3.6
figuration of the module. 3.2
Verification of the Model Inserting parameters Jo,A, J*, and s calculated from the above correlations (eqs 14-17) into eq 9 gives the relationship between permeate flux and process paramMIX,and u)in stationary conditions. As it eters Ja(CO, is shown in Figure 4 and Table 5, the limiting flux phenomenon can be accurately described by use of the model. The model holds throughout the entire range of pressure, velocity, and concentration variations, making it useful for process optimization. Some additional calculations were done in order to compare the surfacerenewal model with some models that are commonly used (Table 4). Todetermine the mass-transfer coefficient k on the basis of the film model, the calculations were performed using the Leveque equation Sh = l.86Re0.89S~0.89(dh/1)0.85 (18) and the Chiang equation (1982) Sh = 0.087Reo~~Sc0*33
28
24
ZC
1.6
1.;
0.1
01
Figure 4. Flux-pressure relationship during cross-flow ultrafiiltration. Lines were determined by eq 9 with parameters calculated from eqe 14-17. Points correspond to measured values.
(19)
It should be emphasized that the Chiang correlation has been carried for skim milk ultrafiltration in the same type of hollow fiber HF-15-43-PM50 as described in this paper. The values of the mass-transfer coefficients obtmned from these correlations contributed in the
determination of the flux based on the gel model and the osmotic model. According to the gel polarization model, the flux was calculated from the formula J = k ln(CG/Co)
(20)
Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 1777 Table 4. Comparison the Permeate Flux: Measured and Calculated from Surface Renewal Model and Due to Four Variants of Film Model, i.e. Gel Model and Osmotic Model with Relevant Mass-Transfer Coefficients Calculated According to Leveque = 180 kPa, T = 293 K Formula (eq 18) and Chiang Formula (eq 19). A& permeate flux J (1P ma/%) operating parame gel model osmotic model CO(kdm) u (mls) ex, surf renewal model Leveaue Chiana Leveaue Chiana 3.540 3.562 5.140 10.938 5.424 13.4 0.179 11.308 4.371 5.754 13.603 0.252 13.956 4.350 6.062 19.518 0.443 19.622 6.090 6.115 6.932 7.277 24.253 7.461 7.753 0.622 23.953 7.430 8.123 7.312 2.067 7.026 3.600 2.073 3.304 34.6 0.179 9.277 2.561 8.745 4.023 2.570 3.699 0.252 13.073 12.548 4.831 3.648 4.455 3.639 0.443 15.592 4.497 4.984 15.999 4.486 5.391 0.622 3.907 1.241 3.713 1.871 1.244 1.746 85.0 0.179 4.622 2.091 1.548 1.955 4.823 1.545 0.252 6.632 2.511 2.215 2.219 2.355 6.816 0.443 8.242 8.357 2.803 2.747 2.753 2.635 0.822 Table 5. Comparison of Experimental Data (from Cheryan, 1986) Concerning Flux during Ultrafiltration of Skimmed Milk in a Hollow Fiber UF Module (Romicon HF-15-43-PM5O)with the Values Calculated According to Surface Renewal Model (in Parentheses) permeate flux J (l/m2h) transmembrane pressure APm velocity u (m/s) 30 kPa 70 kPa 110 kPa 150 kPa 1.97 15 24 27.5 29 (22.03) (21.58) (17.86) (20.65) 23 21 23 1.53 14 (18.82) (18.48) (15.69) (17.9) 17.5 16 17.5 1.11 13 (15.37) (15.15) (13.22) (14.69) 13 0.71 11 12.5 13 (11.57) (11.45) (10.31) (11.18) 8.0 8.5 8.5 0.34 7.5 (7.22) (7.06) (7.16) (6.7)
where the constant gel concentration for skim milk was assumed to be CG = 22% (Cheryan, 1986). Due to the osmotic model, the transmembrane pressure is reduced by osmotic pressure differences at both sides of the membrane:
J=
- AdC,)
(21)
Rm
The osmotic pressure depends on the wall concentration according to the virial expansion. The virial coefficients for milk proteins were taken from Jonsson (1984). By the assumption of perfect retention, the osmotic pressure difference can be expressed as follows: AT = 0.44CW- 0.00017C:
+ 0.000 079C:
(atm) (22)
On the other hand, the wall concentration is dependent on hydrodynamic conditions a t the wall region that is represented by the mass-transfer coefficient:
J = k ln(Cw/Co) The fluxes were calculated on the basis of five various models, and then they were compared with the experimt,ital data in Table 4. The results from the surface renewal model were very close to those obtained from the osmotic and gel models combined with the Leveque equation. Flux limiting described by the surface renewal model has been compared with data reported in the literature (Cheryan and Chiang, 1984). Although the temperature 60 OC and concentration (19.1%) were beyond the
experimental range that are actually presented in the paper, the same module type and the same solution type (skim milk) encouraged the author to make such a comparison (see Table 5). The flux dimensions in Table 5 and the fixed values of velocities correspond to the literature data that were presented in the form of the diagram of Figure 4.9 by Cheryan (1986). The temperature effect has been involved in accordance with the linear flux temperature relation reported by Madsen (1977). The data converge very well with surface renewal model calculations (see Table 5) that were carried for the same conditions. The main potential of the presented model is its possibility of describing the dynamic behavior of ultrafiltration systems by use of eq 8. The comparison of flux decline observed during each run of cross-flow ultrafiltration with values obtained from eq 8 gives very good results (see Figure 3). In order to compare the effects of UTR, and u ) on the particular operating parameters (CO, flux decline, only those runs were selected that were under the conditions with extreme values (minimum and maximum) of process parameters. Stabilization times of permeate flux (see broken lines in the figures, that denote J,) observed during the experiments were within the range of 1000-2000 s. The shortest stabilization time was observed for the maximum values of processing parameters (see line 3 in Figure 3a), and the longest one was for the minimum values of CO, MTR, andu (see line 6 in Figure 3b). Concentration exerts a more distinct effect on the dynamics of building up the concentration polarization layer than pressure and velocity do (compare lines 3,4,7,8 and 1,2,5,6in Figure 3aand Figure 3b, respectively). The practical significance of the description of the dynamics of membrane systems during ultrafiltration is discussed in the next paragraph.
Conclusions The model has been examined experimentally during skim milk ultrafiltration. The experimental results converged very well with the model calculations. Besides, good agreement was observed between results calculated on the basis of the surface renewal model and the film model in flux limiting conditions. This statement is of great importance for the full range of predictions of the flux. The flux can be determined on the basis of literature data about limiting flux ( J b )combined with small-scale batch experiments, where parameters Jo,A, and J* must be determined. Eventually the rate of the surface renewal can be calculated according to eq 8, that can be expressed
1778 Ind. Eng. Chem. Res., Vol. 33, No. 7, 1994
U
- -4
I' = 0168 10"m'Im'~
12
m I 6 I.
2
I
b
4 2
P
84
bo0
(00
U4
1WO
1Iw
1'00
Mw
100
ZWO
2Xd
2t, 111
Z(00
Figure 5. Relationship between average permeate flux during cyclic operation (Jav)and the time of permeation in single cycle (tp)(see eq 13). Each curve corresponds to another value of cleaning time (tA. Parts a-d have been drawn for various velocities (u) and concentration (CO).
as follows:
J b - J* s=AJ, - J h The the flux limiting and flux decline phenomena can be predicted on the basis of the surface renewal model. It gives an opportunity to select the optimal set of operating parametes during dead-end and cross-flowultrafiltration. The main object of this paper was the prediction of the dynamic properties of ultrafiltration systems on the basis of the surface renewal theory. Knowledge about flux decline during cross-flow ultrafiltration could be helpful during experiments to evaluate the time needed to reach stable conditions. Another use for the model is the possibility of separating the concentration polarization effects from the effects of irreversible fouling. The most important profit that can be practically achieved from predicting flux decline in large-scale ultrafiltration plants lies in the possibility of increasing the yield. As it was mentioned in the Theory section, the average yield depends on the arrangement of cyclic operations (see Figure l),including permeation time (t,) as well as cleaning time (tc). The minimum cleaning time can be considered as a constant for any given membranesolution system and operation conditions, whereas the permeation time can be selected arbitrarily. As it is shown in Figure 5b-d, there are some optimum values of t, that give the maximum average permeate flux J,, for a fixed set of operation parametera (including tc) in the cyclic mode of operation. Figure 5a-d shows that average flux (Jav)can be several times greater than the flux Jawith
steady-state conditions. This occurs when shorter periods of permeation "stand in the way" of the full development of the concentration polarization layer. Further reduction of permeation time (tp) below the optimum value results in a decrease of the average flux during cyclic operation (Jav) because of an excessively high contribution of "unproducible" cleaning time (t,) during the process. As it is shown in the Figure 5 a 4 , the shorter the time occupied by cleaning &), the greater the value of the flux (J& which could be obtained during the cyclic mode of operation. Nevertheless, this conclusion has to be restricted only to cases where cleaning operations are fully effective. Therefore for any given system the time of cleaning tc can be adjusted only by the application of a different cleaning method. In light of these observations, it can be seen that the tremendous potential of flux increase (during the cyclic operation) lies in the cleaning strategy. Well-known methods can be more advantageous when less timeconsuming operations, that reduce concentration polarization (such as pulsations, back-flushing, reverse flow), are applied frequently, whereas the full cleaning procedures may be applied with lower frequency than usual, to prevent fouling effects. Another advantage may be the reduction of energy consumption by decreasing velocity and pressure. Comparisons of Figure 5a with Figure 5b, and Figure 5c with Figure 5d, have shown that the maximum values of average flux for cyclic operations (JaJdepend only slightly on velocity. Therefore it should be concluded that the proper selection of operating time t, could be more effective in some conditions than an increase in velocity. Besides, it can be observed that the effect of process time t, on the
Ind. Eng. Chem. Res., Vol. 33, No. 7,1994 1779 average flux Jev is greater for higher concentrations with conditions of the same velocity. The final effect should be considered by taking into account all operating parameters including the fouling effect during the total lifetime of the membrane. The application of the presented model seems helpful in the management of process strategy.
Nomenclature A = rate of flux decline (Us) C = solute concentration in feed (kg/m3) F = membrane surface area (m) f ( t ) = surface-age distribution function (Us) J = permeate flux (m/s) J(t) = instantaneous flux passing through element of membrane surface during cross-flow ultrafiltration as well as the flux during batch ultrafiltration (m/s) JO= initial value of the flux (m/s) J* = flux observed after infinite time of batch ultrafiltration (m/s) Ja = average flux in steady-state conditions of cross-flow ultrafiltration (m/s) &t,) = instantaneous flux after time t of cross-flow UF (m/s) Sa"= average flux obtained in cyclic mode of operation (m/s) AZm ' = transmembrane pressure (kPa) R, = hydraulic resistance of the membrane (Pa s/m) R, = hydraulic resistance of the wall region (Pa s/m) s = rate of surface renewal ( U s ) t = current time ( 8 ) t, = time of permeation (s) t, = time of membrane cleaning (s) u = velocity (m/s) Greek Letters T = membrane lifetime (s) 6 = Dirac's function .rr = osmotic pressure (Pa)
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Received for review March 8, 1994 Accepted March 31, 1994. e
Abstract published in Advance ACS Abstracts, May 1,1994.
