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Momentum balance analysis of flux and pressure declines in membrane ultrafiltration along tubular modules
Journal of Membrane Science 241 (2004) 335–345
Momentum balance analysis of flux and pressure declines in membrane ultrafiltration along tubular modules H.M. Yeh∗ , J.H. Dong, M.Y. Shi Department of Chemical and Materials Engineering, Tamkang University, Tamsui 251, Taiwan Received 25 January 2003; accepted 7 June 2004
Abstract Predicting equations for the permeate flux of membrane ultrafiltration in a tubular module were derived from momentum balance with the considerations of declining flow rate and transmembrane pressure along the membrane tube. Correlation predictions are confirmed with the experimental results. The effects of solute concentration, transmembrane pressure, tube radius and tube length on the flux and pressure declines along the membrane tube were thoroughly discussed. © 2004 Elsevier B.V. All rights reserved. Keywords: Permeate flux; Declination; Ultrafiltration; Tubular membrane; Momentum balance
1. Introduction Ultrafiltration of macromolecular solutions has now become an increasingly important industrial process for the concentration, purification, or dewatering of macromolecular and colloidal species in solution. Ultrafiltration is primarily a size-exclusion-based pressure-driven membrane separation process, the pressure applied to the working fluid provides the driving potential to force the solvent to flow through the membrane. The operational pressure is usually in the range of 10–100 psi. In membrane ultrafiltration processes, solutes that are rejected by the membrane accumulate on the membrane surface and form a concentration polarization layer there. At steady state, the quantity of solutes conveyed by the solvent to the membrane is equal to those that diffuse back. A number of mathematical models are available in the literature that attempt to describe the mechanism of transport through membranes. Membrane ultrafiltration of macromolecular solutions is usually analyzed by the following models: (i) the gel polarization model [1–7], (ii) the osmotic pressure model [8–16], and (iii) the resistance-in series model [17–21]. In the gel polarization model, permeate flux
∗
Corresponding author. Tel.: +886-2-9180149; fax: +886-2-26209887. E-mail address: [email protected] (H.M. Yeh).
0376-7388/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2004.06.005
is reduced by the hydraulic resistance of the gel layer. In the osmotic pressure model, permeate flux reduction results from the decrease in effective transmembrane pressure that occurs as osmotic pressure of the retentate increases. In the resistance-in-series model, permeate flux decreases due to the resistances caused by fouling or solute adsorption and concentration polarization. This last method easily describes the relationships of permeate flux with operating parameters. Fane [22] has discussed initial ultrafiltration flux decline in terms of the boundary layer theory, adsorption, and pore plugging. The flow rate of solution declines along the membrane tube due to the permeation of solvent and the transmembrane pressure declines because of the friction loss of fluid flow. Accordingly, permeate flux also declines along the cross-flow modules. The effects of solution concentration, fluid velocity and tube length and diameter on the declines of permeate flux and transmembrane pressure along the hollow-fiber modules [23] and along the tubular ceramic membrane [24] were discussed. In these studies, the momentum balance was taken imprecisely by simply applying Hagen–Poiseuille theory without the consideration of flux loss. In this study, we ultrafiltered macromolecular solutions in a tubular membrane module and we analyzed the decline of permeate flux by taking the mass and momentum balances with the consideration of flux loss, coupled with the use of resistance-in-series model.
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Fig. 1. Flows and fluxes in a tubular membrane for ultrafiltration.
d τs (2πrm ) d (ρu2b ) + ( p) + =0 2 dz dz πrm
2. Theory 2.1. Mass balance Let q(z) be the volumetric flow rate of solution in a membrane tube and νm be the permeate flux by ultrafiltration. Then, a mass balance over a slice dz of the tube, as shown in Fig. 1, gives: dq = −2πrm νm dz
(1)
2.2. Momentum balance For the steady-state operation, the momentum balance within the differential length dz of a membrane tube is [25]:
pressure g auge
pressure gauge
tubular-membrane module
where p (=p − ps ) denotes the transmembrane pressure, and p(z) and ps are the pressures in tube and shell sides, respectively, while the shear stress τ s relates to the friction factor f as τs = (ρu2b /2)f . For laminar flow, f = 2 u , the 16/(2rm ub ρ/µ), and for flow in a tube, q = πrm b above equation can be rewritten as: ρ dq2 d p 8µq + + 4 =0 4 dz dz π2 rm πrm
(2)
It is mentioned in Section 1 that the momentum balance in previous works [23,24] was taken imprecisely without the first term in Eq. (2). 2.3. Resistance-in-series model As mentioned before, membrane ultrafiltration is a pressure-driven process. For a small applied pressure, the permeate flux through a membrane is observed to be proportional to the applied pressure. However, as the pressure is increased, the flux begins to drop below that which would result from a linear-pressure behavior. Eventually, a limiting flux is reached where any further pressure increase no longer results in any increase in flux. Accordingly, the following relations between permeate flux νm and transmembrane pressure p are reached: νm = 0,
for p = 0
(3)
Table 1 Experimental data of permeate flux for pure water with ui = 0.088 m/s
Fig. 2. Flow diagram of experimental apparatus.
( p)exp (×10−5 Pa)
(¯νm )exp (×106 m3 /(m2 s))
0.3 0.5 0.8 1.1 1.4
2.327 3.768 4.637 5.869 6.908
Rm = 1.055 × 1010 (Pa s/m) [9].
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337
Table 2 Experimental data of Dextran T500 aqueous solution Ci (wt.%)
pi (10−5 Pa)
v¯ m (×106 m3 /(m2 s)) ui = 0.059 m/s
ui = 0.088 m/s
ui = 0.118 m/s
ui = 0.147 m/s
0.1
0.3 0.5 0.8 1.1 1.4
1.379 2.377 3.509 4.171 4.840
1.396 2.484 3.947 4.637 5.436
1.405 2.512 4.145 4.949 5.817
1.402 2.521 4.145 5.391 6.256
0.2
0.3 0.5 0.8 1.1 1.4
1.303 2.189 3.187 3.624 4.197
1.329 2.240 3.768 4.171 4.637
1.199 2.287 3.811 4.573 5.141
1.261 2.385 4.145 5.222 5.717
0.5
0.3 0.5 0.8 1.1 1.4
1.030 1.858 2.590 2.883 3.219
1.033 1.873 2.798 3.173 3.401
1.028 1.868 2.810 3.349 3.789
1.023 1.847 2.960 3.664 4.145
1.0
0.3 0.5 0.8 1.1 1.4
0.953 1.464 1.991 2.456 2.642
0.954 1.487 2.047 2.590 2.947
0.949 1.494 2.112 2.674 3.113
0.951 1.507 2.146 2.947 3.436
νm = constant,
p νm = νm,lim ,
for small p
(4)
as p → ∞ (or large enough)
(5)
The resistance-in-series model may be expressed as [26–32]: νm (z) =
p(z) Rm + Rf + φ p(z)
(6)
where Rm denotes the intrinsic resistance of membrane, Rf the resistance due to fouling phenomena, such as solute adsorption, while φ p(z) is the resistance due to the concen-
tration polarization/gel layer, which will be proportional to the amount and specific hydraulic resistance of the compressible layer deposited and may be assumed to be a linear function of transmembrane pressure with φ as a proportional constant. It is easy to check that Eq. (6) satisfies Eqs. (3)–(5), provided that φ = 1/νm,lim . 2.4. Flux decline The permeate flux νm will be solved from Eqs. (1), (2) and (6) with the following inlet conditions: at z = 0, q = qi
(7)
Table 3 The fitting parameter of experimental data Ci (wt.%)
ui (m/s)
(Rm + Rf ) (×10−10 Pa m2 s/m3 )
Rf (×10−10 Pa m2 s/m3 )
φ (×10−5 s/m)
0.1
0.059 0.088 0.118 0.147
1.958 2.013 2.036 2.091
0.903 0.958 0.981 1.036
0.867 0.581 0.399 0.196
0.2
0.059 0.088 0.118 0.147
1.995 2.068 2.424 2.361
0.940 1.013 1.369 1.306
1.173 0.719 0.462 0.202
0.5
0.059 0.088 0.118 0.147
2.488 2.572 2.661 2.783
1.433 1.517 1.606 1.728
1.820 1.446 1.064 0.616
1.0
0.059 0.088 0.118 0.147
2.558 2.657 2.728 2.834
1.503 1.602 1.673 1.779
1.986 1.556 1.317 0.854
Rm = 1.055 × 1010 (Pa m2 s/m3 ).
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and
p = pi
(8)
For mathematical simplicity, we define the following dimensionless groups: 8µLq (9) Q= 4 πrm pi
P =
p
pi
(10)
z (11) L Substitution of Eq. (6) into Eq. (1) with the use of above definition yields:
P dQ = −α (12) dZ 1 + β P Z=
while Eqs. (2), (7) and (8) can be rewritten, respectively, as: γ
d P dQ2 + +Q=0 dZ dZ
at Z = 0, Q = Qi
(13) (14)
and
P = 1
(15)
where α=
16µL2 3R rm
(16)
β=
φ pi R
(17)
γ=
4 p ρrm i 64µ2 L
(18)
8µLqi 4 p πrm i
(19)
R = Rm + Rf
(20)
Qi =
The flow rate of solution declines along the tubular membrane due to membrane ultrafiltration and thus, solvent is permeated through the porous tube wall by transmembrane pressure. Since the permeation rate νm is small compared with the volume flow rate q, we may assume q declines slightly (q ≈ qi > 2πrm L¯νm ) and linearly along the tube by approximately setting p = pi ( P = 1) in Eq. (12) for mathematical simplicity. Accordingly, integration of Eq. (12) from Z = 0 (Q = Qi ) to Z = Z yields: z α 1 Q = Qi − α dZ = Qi − Z (21) 1+β 0 1+β Substituting Eq. (21) into Eq. (13) and integrating from Z = 0 ( P = 0) to Z, one has: 2 α 2
P = 1 + γ Qi − Qi − Z 1+β α − Qi Z + (22) Z2 2(1 + β) or
2αγ
P = 1 + − 1 Qi Z 1+β 2 α α + Z2 − 2(1 + β) 1+β
(23)
Once P is calculated from Eq. (23), the declining fluxes, νm and V (=νm /νm,i ), are readily obtained from Eqs. (6) and (17), i.e.
Fig. 3. Comparison of experimental results with correlation predictions: Ci = 0.1 wt.%.
H.M. Yeh et al. / Journal of Membrane Science 241 (2004) 335–345
339
Fig. 4. Comparison of experimental results with correlation predictions: Ci = 1.0 wt.%.
V =
νm (1 + β) P = νm,i 1 + β P
(24)
V¯ =
where νm,i =
pi R + φ pi
Substitution of Eqs. (23) and (24) into Eq. (26) results in:
(25)
2.5. Average permeate flux Taking the average value of dimensionless permeate flux, V, in Eq. (24), one obtains: 1 1 L ν¯ m = V dz = V dZ (26) V¯ = νm,i L 0 0
1+β β
1 dz × 1− 0 1 + β[1 + (2αγ/(1 + β) − 1)Qi Z
+ {α/(2(1 + β)) − (α/(1 + β))2 }Z2 ] 2A + B 1+β 2 tan−1 √ = 1− √ β 4AC − B2 4AC − B2 B − tan−1 √ (27) , B2 < 4AC 4AC − B2
Fig. 5. Decline of transmembrane permeate with pi as parameter.
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Fig. 6. Decline of transmembrane permeate with Ci as parameter.
=
C =1+β
1 1+β 1− √ 2 βo B − 4AC (2A + B − √B2 − 4AC)(B + √B2 − 4AC) ln √ √ , (2A + B + B2 − 4AC)(B − B2 − 4AC) B2 > 4AC
where
(31)
Therefore, once α, β, γ and Qi are specified, V as well as the average permeate flux ν¯ m is calculated from Eqs. (27), (28) and (25). (28) 3. Experimental
α −γ A=β 2(1 + β)
2αγ − 1 Qi B=β 1+β
α 1+β
2 (29)
(30)
3.1. Apparatus and materials The flow sheet of an ultrafiltration apparatus is shown in Fig. 2. The membrane medium used was a 15 kDa MWCO tubular ceramic membrane (M2 type, Techsep, France,
Fig. 7. Decline of transmembrane permeate with rm as parameter.
H.M. Yeh et al. / Journal of Membrane Science 241 (2004) 335–345
341
Fig. 8. Decline of transmembrane permeate with L as parameter.
inside radius rm = 0.003 m, length L = 0.4 m) made of Zr O2 /carbon. The tested solute was dextran T500 (Pharmacia, Mn = 1 703 000 and Mw = 503 000). The solvent was ion exchange water. The feed solution was circulated by a high-pressure pump with a variable speed motor (L-07553-20, Cole-Parmer Co., Chicago, IL), and the feed flow was measured with a flowmeter (L-03217-34, Cole-Parmer Co.). The pressure was measured with a pressure transmitter (Model 891.14.425, Wika). 3.2. Experimental conditions and procedure The experimental conditions were as follows. The feed concentrations Ci were 0.1, 0.2, 0.5 and 1.0 wt.% dextran
2 ) were 0.059, 0.088, T500; the feed velocities ui (= qi /πrm 0.118 and 0.147 m/s; and the feed inlet transmembrane pressures pi were 30, 50, 80, 110, and 140 kPa. The feed solution temperature in all experiments were kept at 25 ◦ C in a constant-temperature water bath controlled by a thermostat. During a run, both permeate and retentate were recycled back to a large feed tank the feed concentrations were observed to be unchanged, therefore, the effects of concentration polarization and solute adsorption on the membrane are insignificant. The experimental procedure was as follows. First, a fresh tubular ceramic module was used to determine the intrinsic resistance of membrane Rm . Average permeate fluxes for water ν¯ m,w were measured under various transmembrane pressures and flow velocities ui . Then the feedwater was
Fig. 9. Decline of permeate flux with pi as parameter.
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replaced with the tested solution. Average permeate fluxes for dextran T500 solution ν¯ m were measured under all operating conditions (Ci , ui , pi ) at steady state. Values of permeate flux reached steady state within 30–120 m. After each solution run, the membrane module was cleaned by a combination of high circulation and backflushing with pure water. The cleaning procedure was repeated until the original water flux had been restored. 4. Discussion 4.1. Correlation equations for R and φ The experimental data of the permeate fluxes for pure water (¯νm,w )exp and solution (¯νm )exp are presented in Tables 1 and 2, respectively. With the use of Table 1, a straight line of (1/¯νm,w )exp versus 1/( p)exp could be constructed by the least-squares method [29–32]. Thus, the intrinsic resistance of the tubular-membrane module employed in this study can be determined from Table 1 by using the following equation, which can be modified from Eq. (6) by setting Rf = 0 and φ = 0 for pure water: 1 Rm = (¯νm,w )exp ( p)exp
(32)
In the above equation: ( p)exp = 21 (( pi )exp + ( po )exp )
(33)
in which ( pi )exp and ( po )exp are the experimental values of inlet and outlet transmembrane pressures. Under various ui and ( p)exp , the measured value of Rm for the membrane system employed in present study was determined graphically as [9]: Rm = 1.055 × 1010 (Pa s/m)
(34)
Furthermore, if experimental data obtained in ultrafiltration of an aqueous solution is also applied to Eq. (6), then 1 Rm + Rf =φ+ (35) (¯νm )exp ( p)exp Therefore, from a straight line plot of (1/¯νm )exp versus 1/( p)exp at a certain flow velocity ui and feed concentration Ci , the experimental values of φ (the intersection at the ordinate) and (Rm + Rf ) (the slope), as well as Rf , were determined graphically from Table 2 as functions of ui and Ci . The results are presented in Table 3. Finally, the correlation equations for φ and Rf were constructed by following the same procedure performed in the previous works [29–32]. The results are: φ = 1.37 × 105 u−0.02 Ci0.535 (s/m) i
(36)
Rf = 1.74 × 1010 u−0.005 Ci0.244 (Pa s/m) i
(37)
4.2. Comparison of correlation predictions with experimental results The average values of permeate flux ν¯ m may be predicted from Eqs. (27), (28) and (25) with the use of the correlation equations, Eqs. (34), (36) and (37) and the system constants: L = 0.4 m, rm = 0.003 m, and the fluid viscosity [31]: µ = 0.894 × 10−3 exp(0.408Ci ) (Pa s or kg/m s)
(38)
Some correlation predictions for average permeate flux were calculated and the results are compared with the experimental data, as shown in Figs. 3–5. It is seen from these figures that though the predicting values are in qulitative agreement with the experimental results for practically operating conditions, only that at Ci = 1.0 wt.% was in good agreement.
Fig. 10. Decline of permeate flux with Ci as parameter.
H.M. Yeh et al. / Journal of Membrane Science 241 (2004) 335–345
343
Fig. 11. Decline of permeate flux with rm as parameter.
We may conclude that the present simulation model is unsuitable for the solution of low solute concentration. 4.3. Declines of transmembrane pressure and permeate flux The declines of transmembrane pressure and permeate flux along the tube with inlet transmembrane pressure pi , and solution concentration Ci , as well as with tube radius rm and tube length L as parameters were calculated from Eqs. (23) and (24), and the results are shown in Figs. 5–12. It is seen in Figs. 5 and 9 that both P and V decline more rapidly for lower pi due to the small effectively driving force, p, at the tube end. As shown in Figs. 6 and 10, P and V decline rather rapidly for larger Ci because that in this
case, the concentration polarization layer increases rapidly at the tube end. We may find in Figs. 7 and 11 that P and V decline more rapidly when rm decreases. This is because that for smaller rm the thickness of concentration polarization layer is rather larger compared with tube diameters, resulting in high resistance. Finally, as a common sense shown in Figs. 8 and 12, P and V decline more seriously as the length of membrane tube increases. It is concluded that the decline of permeate flux occurs mainly due to the decline of transmembrane pressure and the increase of the thickness of concentration polarization layer along the membrane tubes, and thus it is rather sensitive for lower inlet transmembrane pressure and larger inlet solution concentration, as well as for smaller tube radius and larger tube length.
Fig. 12. Decline of permeate flux with L as parameter.
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5. Conclusion The correlation equations, Eqs. (24) and (27) or (28), for predicting the permeate fluxes of membrane ultrafiltration in tubular-membrane modules, were derived from momentum and mass balances by the resistance-in-series model with the considerations of the declines of transmembrane pressure and flow rate along the fibers. The declines of flow rate, transmembrane pressure and permeate flux along the tube may be predicted from Eqs. (21), (23) and (24), respectively. For predicting the average permeate flux, one may employ Eq. (27) or Eq. (28). Ultrafiltration of Dextran 500 aqueous solution in a tubular ceramic module has been carried out for various feed concentrations, transmembrane pressures and feed flow rates. Correlation predictions qualitatively confirmed with the experimental results, as shown in Figs. 3–5. It is found that the correlation predictions of average permeate flux obtained in present study are more accurate than those obtained in previous works [31,32], in which the feed flow rates in the momentum balance equation, Eq. (2), were merely taken as the inlet value, or as its arithmetic-mean value, while in present study, the decline of feed flow rate through the tube was taken into consideration. The assumption of laminar flow is easy to check by the maximum value of Reynolds number with ui = 0.147 m/s and Ci = 1.0 wt.% as: 2rm ui ρ 2(0.003)(0.147)(1000) = µ 0.894 × 10−3 e−0.408 = 1484 < 2100
(Re)max =
As mentioned earlier, the resistance-in-series model satisfies the three essential conditions of membrane ultrafiltration, Eqs. (3)–(5). Further, as expected, it is shown in Table 3 that the fouling layer resistance Rf as well as the coefficient of the resistance due to concentration polarization φ decreases with the increase of crossflow velocity ui , while increases as the solution concentration Ci increases. Therefore, the present model easily describes the relationships of permeate flux with operating and design parameters, and we believe that this model will also be suitable for most membrane ultrafiltration systems including systems with different kinds of feed solutions, different materials of membrane tubes, and various design and operating conditions.
Acknowledgements The authors wish to express their thanks to the National Science Council of ROC for financial aid under Grant No. NSC 89-2214-E-032. Nomenclature Ci
concentration of feed solution (wt.% dextran T500)
L p ps
p
P q Q rm Rf Rm u V z Z
effective length of membrane tube (m) pressure distribution on the tube side (Pa) uniform permeate pressure on the shell side (Pa) transmembrane pressure, p − ps (Pa) dimensionless transmembrane pressure,
p/ pi volume flow rate in a tubular-membrane module (m3 /s) dimensionless flow rate, defined by Eq. (9) inside radius of membrane tube (m) resistance due to solute adsorption and fouling (Pa s/m) intrinsic resistance of membrane (Pa s/m) fluid velocity in the membrane tube, 2 ) (m/s) q/(πrm dimensionless permeate flux, defined by Eq. (24) axial coordinate (m) dimensionless axial coordinate, z/L
Greek letters α dimensionless group, defined by Eq. (16) β dimensionless group, defined by Eq. (17) φ 1/νm,lim (m2 s/m3 ) γ dimensionless group, defined by Eq. (18) µ viscosity of solution (Pa s) νm permeate flux of solution (m3 /(m2 s)) νm,lim limiting flux (m3 /(m2 s)) Subscripts i at the inlet o at the outlet Superscript − average value
References [1] W.F. Blatt, A. Dravid, A.S. Michales, L. Nelsen, Solute polarization and cake formation in membrane ultrafiltration: causes, consequences, and control techniques, in: J.E. Filnn (Ed.), Membrane Science and Technology, Plenum Press, New York, 1970, p. 47. [2] M.C. Porter, Concentration polarization with membrane ultrafiltration, Ind. Eng. Chem. Proc. Res. Dev. 11 (1972) 234. [3] R.B. Grieves, D. Bhattacharyya, W.G. Schomp, J.L. Bewley, Membrane ultrafiltration of a nonionic surfactant, Am. Inst. Chem. Eng. J. 19 (1973) 766. [4] J.J.S. Shen, R.F. Probstein, On the prediction of limiting flux in laminar ultrafiltration of macromolecular solutions, Ind. Eng. Chem. Fundam. 16 (1977) 459. [5] S. Nokao, T. Nomura, S. Kimura, Characteristics of macromolecular gel layer formed on ultrafiltration tubular membrane, Am. Inst. Chem. Eng. J. 25 (1979) 615.
H.M. Yeh et al. / Journal of Membrane Science 241 (2004) 335–345 [6] A.G. Fane, C.J.D. Fell, A.G. Waters, The relationship between membrane surface pore characteristics and flux for ultrafiltration membranes, J. Membr. Sci. 9 (1981) 245. [7] A.G. Fane, Ultrafiltration of suspensions, J. Membr. Sci. 20 (1984) 249. [8] J.G. Wijmans, S. Nakao, C.A. Smolders, Flux limitation in ultrafiltration: osmotic pressure model and gel layer model, J. Membr. Sci. 20 (1984) 115. [9] A.A. Kozinski, E.N. Lightfoot, Protein ultrafiltration: a general example of boundary layer filtration, Am. Inst. Chem. Eng. J. 18 (1972) 1030. [10] W. Leung, R.F. Probstein, Low polarization in laminar ultrafiltration of macromolecular solutions, Ind. Eng. Chem. Fundam. 18 (1979) 274. [11] R.P. Wendt, E. Klein, F.F. Holland, K.E. Eberle, Hollow fiber ultrafiltration of calf serum and albumin in the pregel uniform-wall-flux region, Chem. Eng. Commun. 8 (1981) 251. [12] S. Nakao, S. Kimura, Models of membrane transport phenomena and their applications for ultrafiltration data, J. Chem. Eng. Jpn. 15 (1982) 200. [13] C. Kleinstreuer, M.S. Paller, Laminar dilute suspension flows in plateand-frame ultrafiltration units, Am. Inst. Chem. Eng. J. 29 (1983) 529. [14] M.J. Clifton, N. Abidine, P. Aptel, V. Sanchez, Growth of the polarization layer in ultrafiltration with hollow-fiber membranes, J. Membr. Sci. 21 (1984) 233. [15] R.P. Gooding Ma, C.H. Gooding, W.K. Alexander, A dynamic model for low-pressure, hollow-fiber ultrafiltration, Am. Inst. Chem. Eng. J. 31 (1985) 1728. [16] H. Nabetani, M. Nakajima, A. Watanabe, S. Nakao, S. Kumura, Effects of osmotic pressure and adsorption on ultrafiltration of ovalbumin, Am. Inst. Chem. Eng. J. 36 (1990) 907. [17] B.H. Chiang, M. Cheryan, Ultrafiltration on skim milk in hollow fibers, J. Food Sci. 51 (1986) 340. [18] M. Assadi, D.A. White, A model for determining the steady state flux of inorganic microfiltration membrane, Chem. Eng. J. 48 (1992) 11.
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[19] H.M. Yeh, T.W. Cheng, Resistance-in-series of membrane ultrafiltration in hollow fiber of tube-and-shell arrangement, Sep. Sci. Technol. 28 (1993) 1341. [20] H.M. Yeh, H.H. Wu, Membrane ultrafiltration in combined hollowfiber module systems, J. Membr. Sci. 124 (1997) 93. [21] H.M. Yeh, J.W. Tsai, Membrane ultrafiltration in multipass hollow-fiber modules, J. Membr. Sci. 142 (1998) 61. [22] A.G. Fane, Ultrafiltration: factors influencing flux and rejection, in: R.J. Wakeman (Ed.), Progress in Filtration and Separation, vol. 4, Elsevier, Amsterdam, 1986. [23] H.M. Yeh, H.P. Wu, J.F. Dong, Effect of design and operating parameters on the declination of permeate flux for membrane ultrafiltration along hollow-fiber modules, J. Membr. Sci. 213 (2003) 33. [24] H.M. Yeh, Decline of permeate flux for ultrafiltration along membrane tubes, Desalination 145 (2002) 153. [25] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, Wiley, New York, 1971, p. 35. [26] B.H. Chiang, M. Cheryan, Ultrafiltration on skim milk in hollow fibers, J. Food Sci. 51 (1986) 340. [27] H. Nabetani, M. Nakajima, A. Watanabe, S. Nakao, S. Kumura, Effects of osmotic pressure and adsorption on ultrafiltration of ovalbumin, Am. Inst. Chem. Eng. J. 360 (1990) 907. [28] M. Assadi, D.A. White, A model for determining the steady state flux of inorganic microfiltration membrane, Chem. Eng. J. 48 (1992) 11. [29] H.M. Yeh, H.H. Wu, Membrane ultrafiltration in combined hollowfiber module systems, J. Membr. Sci. 124 (1997) 93. [30] H.M. Yeh, J.W. Tsai, Membrane ultrafiltration in multipass hollowfiber modules, J. Membr. Sci. 142 (1998) 61. [31] H.M. Yeh, H.Y. Chen, K.T. Chen, Membrane ultrafiltration in a tubular module with a steel rod inserted concentrically for improved performance, J. Membr. Sci. 168 (2000) 121. [32] H.M. Yeh, K.T. Chen, Improvement of ultrafiltration performance in tubular membranes using a twisted wire-rod assembly, J. Membr. Sci. 178 (2000) 43.
Momentum balance analysis of flux and pressure declines in membrane ultrafiltration along tubular modules H.M. Yeh∗ , J.H. Dong, M.Y. Shi Department of Chemical and Materials Engineering, Tamkang University, Tamsui 251, Taiwan Received 25 January 2003; accepted 7 June 2004
Abstract Predicting equations for the permeate flux of membrane ultrafiltration in a tubular module were derived from momentum balance with the considerations of declining flow rate and transmembrane pressure along the membrane tube. Correlation predictions are confirmed with the experimental results. The effects of solute concentration, transmembrane pressure, tube radius and tube length on the flux and pressure declines along the membrane tube were thoroughly discussed. © 2004 Elsevier B.V. All rights reserved. Keywords: Permeate flux; Declination; Ultrafiltration; Tubular membrane; Momentum balance
1. Introduction Ultrafiltration of macromolecular solutions has now become an increasingly important industrial process for the concentration, purification, or dewatering of macromolecular and colloidal species in solution. Ultrafiltration is primarily a size-exclusion-based pressure-driven membrane separation process, the pressure applied to the working fluid provides the driving potential to force the solvent to flow through the membrane. The operational pressure is usually in the range of 10–100 psi. In membrane ultrafiltration processes, solutes that are rejected by the membrane accumulate on the membrane surface and form a concentration polarization layer there. At steady state, the quantity of solutes conveyed by the solvent to the membrane is equal to those that diffuse back. A number of mathematical models are available in the literature that attempt to describe the mechanism of transport through membranes. Membrane ultrafiltration of macromolecular solutions is usually analyzed by the following models: (i) the gel polarization model [1–7], (ii) the osmotic pressure model [8–16], and (iii) the resistance-in series model [17–21]. In the gel polarization model, permeate flux
∗
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is reduced by the hydraulic resistance of the gel layer. In the osmotic pressure model, permeate flux reduction results from the decrease in effective transmembrane pressure that occurs as osmotic pressure of the retentate increases. In the resistance-in-series model, permeate flux decreases due to the resistances caused by fouling or solute adsorption and concentration polarization. This last method easily describes the relationships of permeate flux with operating parameters. Fane [22] has discussed initial ultrafiltration flux decline in terms of the boundary layer theory, adsorption, and pore plugging. The flow rate of solution declines along the membrane tube due to the permeation of solvent and the transmembrane pressure declines because of the friction loss of fluid flow. Accordingly, permeate flux also declines along the cross-flow modules. The effects of solution concentration, fluid velocity and tube length and diameter on the declines of permeate flux and transmembrane pressure along the hollow-fiber modules [23] and along the tubular ceramic membrane [24] were discussed. In these studies, the momentum balance was taken imprecisely by simply applying Hagen–Poiseuille theory without the consideration of flux loss. In this study, we ultrafiltered macromolecular solutions in a tubular membrane module and we analyzed the decline of permeate flux by taking the mass and momentum balances with the consideration of flux loss, coupled with the use of resistance-in-series model.
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Fig. 1. Flows and fluxes in a tubular membrane for ultrafiltration.
d τs (2πrm ) d (ρu2b ) + ( p) + =0 2 dz dz πrm
2. Theory 2.1. Mass balance Let q(z) be the volumetric flow rate of solution in a membrane tube and νm be the permeate flux by ultrafiltration. Then, a mass balance over a slice dz of the tube, as shown in Fig. 1, gives: dq = −2πrm νm dz
(1)
2.2. Momentum balance For the steady-state operation, the momentum balance within the differential length dz of a membrane tube is [25]:
pressure g auge
pressure gauge
tubular-membrane module
where p (=p − ps ) denotes the transmembrane pressure, and p(z) and ps are the pressures in tube and shell sides, respectively, while the shear stress τ s relates to the friction factor f as τs = (ρu2b /2)f . For laminar flow, f = 2 u , the 16/(2rm ub ρ/µ), and for flow in a tube, q = πrm b above equation can be rewritten as: ρ dq2 d p 8µq + + 4 =0 4 dz dz π2 rm πrm
(2)
It is mentioned in Section 1 that the momentum balance in previous works [23,24] was taken imprecisely without the first term in Eq. (2). 2.3. Resistance-in-series model As mentioned before, membrane ultrafiltration is a pressure-driven process. For a small applied pressure, the permeate flux through a membrane is observed to be proportional to the applied pressure. However, as the pressure is increased, the flux begins to drop below that which would result from a linear-pressure behavior. Eventually, a limiting flux is reached where any further pressure increase no longer results in any increase in flux. Accordingly, the following relations between permeate flux νm and transmembrane pressure p are reached: νm = 0,
for p = 0
(3)
Table 1 Experimental data of permeate flux for pure water with ui = 0.088 m/s
Fig. 2. Flow diagram of experimental apparatus.
( p)exp (×10−5 Pa)
(¯νm )exp (×106 m3 /(m2 s))
0.3 0.5 0.8 1.1 1.4
2.327 3.768 4.637 5.869 6.908
Rm = 1.055 × 1010 (Pa s/m) [9].
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337
Table 2 Experimental data of Dextran T500 aqueous solution Ci (wt.%)
pi (10−5 Pa)
v¯ m (×106 m3 /(m2 s)) ui = 0.059 m/s
ui = 0.088 m/s
ui = 0.118 m/s
ui = 0.147 m/s
0.1
0.3 0.5 0.8 1.1 1.4
1.379 2.377 3.509 4.171 4.840
1.396 2.484 3.947 4.637 5.436
1.405 2.512 4.145 4.949 5.817
1.402 2.521 4.145 5.391 6.256
0.2
0.3 0.5 0.8 1.1 1.4
1.303 2.189 3.187 3.624 4.197
1.329 2.240 3.768 4.171 4.637
1.199 2.287 3.811 4.573 5.141
1.261 2.385 4.145 5.222 5.717
0.5
0.3 0.5 0.8 1.1 1.4
1.030 1.858 2.590 2.883 3.219
1.033 1.873 2.798 3.173 3.401
1.028 1.868 2.810 3.349 3.789
1.023 1.847 2.960 3.664 4.145
1.0
0.3 0.5 0.8 1.1 1.4
0.953 1.464 1.991 2.456 2.642
0.954 1.487 2.047 2.590 2.947
0.949 1.494 2.112 2.674 3.113
0.951 1.507 2.146 2.947 3.436
νm = constant,
p νm = νm,lim ,
for small p
(4)
as p → ∞ (or large enough)
(5)
The resistance-in-series model may be expressed as [26–32]: νm (z) =
p(z) Rm + Rf + φ p(z)
(6)
where Rm denotes the intrinsic resistance of membrane, Rf the resistance due to fouling phenomena, such as solute adsorption, while φ p(z) is the resistance due to the concen-
tration polarization/gel layer, which will be proportional to the amount and specific hydraulic resistance of the compressible layer deposited and may be assumed to be a linear function of transmembrane pressure with φ as a proportional constant. It is easy to check that Eq. (6) satisfies Eqs. (3)–(5), provided that φ = 1/νm,lim . 2.4. Flux decline The permeate flux νm will be solved from Eqs. (1), (2) and (6) with the following inlet conditions: at z = 0, q = qi
(7)
Table 3 The fitting parameter of experimental data Ci (wt.%)
ui (m/s)
(Rm + Rf ) (×10−10 Pa m2 s/m3 )
Rf (×10−10 Pa m2 s/m3 )
φ (×10−5 s/m)
0.1
0.059 0.088 0.118 0.147
1.958 2.013 2.036 2.091
0.903 0.958 0.981 1.036
0.867 0.581 0.399 0.196
0.2
0.059 0.088 0.118 0.147
1.995 2.068 2.424 2.361
0.940 1.013 1.369 1.306
1.173 0.719 0.462 0.202
0.5
0.059 0.088 0.118 0.147
2.488 2.572 2.661 2.783
1.433 1.517 1.606 1.728
1.820 1.446 1.064 0.616
1.0
0.059 0.088 0.118 0.147
2.558 2.657 2.728 2.834
1.503 1.602 1.673 1.779
1.986 1.556 1.317 0.854
Rm = 1.055 × 1010 (Pa m2 s/m3 ).
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and
p = pi
(8)
For mathematical simplicity, we define the following dimensionless groups: 8µLq (9) Q= 4 πrm pi
P =
p
pi
(10)
z (11) L Substitution of Eq. (6) into Eq. (1) with the use of above definition yields:
P dQ = −α (12) dZ 1 + β P Z=
while Eqs. (2), (7) and (8) can be rewritten, respectively, as: γ
d P dQ2 + +Q=0 dZ dZ
at Z = 0, Q = Qi
(13) (14)
and
P = 1
(15)
where α=
16µL2 3R rm
(16)
β=
φ pi R
(17)
γ=
4 p ρrm i 64µ2 L
(18)
8µLqi 4 p πrm i
(19)
R = Rm + Rf
(20)
Qi =
The flow rate of solution declines along the tubular membrane due to membrane ultrafiltration and thus, solvent is permeated through the porous tube wall by transmembrane pressure. Since the permeation rate νm is small compared with the volume flow rate q, we may assume q declines slightly (q ≈ qi > 2πrm L¯νm ) and linearly along the tube by approximately setting p = pi ( P = 1) in Eq. (12) for mathematical simplicity. Accordingly, integration of Eq. (12) from Z = 0 (Q = Qi ) to Z = Z yields: z α 1 Q = Qi − α dZ = Qi − Z (21) 1+β 0 1+β Substituting Eq. (21) into Eq. (13) and integrating from Z = 0 ( P = 0) to Z, one has: 2 α 2
P = 1 + γ Qi − Qi − Z 1+β α − Qi Z + (22) Z2 2(1 + β) or
2αγ
P = 1 + − 1 Qi Z 1+β 2 α α + Z2 − 2(1 + β) 1+β
(23)
Once P is calculated from Eq. (23), the declining fluxes, νm and V (=νm /νm,i ), are readily obtained from Eqs. (6) and (17), i.e.
Fig. 3. Comparison of experimental results with correlation predictions: Ci = 0.1 wt.%.
H.M. Yeh et al. / Journal of Membrane Science 241 (2004) 335–345
339
Fig. 4. Comparison of experimental results with correlation predictions: Ci = 1.0 wt.%.
V =
νm (1 + β) P = νm,i 1 + β P
(24)
V¯ =
where νm,i =
pi R + φ pi
Substitution of Eqs. (23) and (24) into Eq. (26) results in:
(25)
2.5. Average permeate flux Taking the average value of dimensionless permeate flux, V, in Eq. (24), one obtains: 1 1 L ν¯ m = V dz = V dZ (26) V¯ = νm,i L 0 0
1+β β
1 dz × 1− 0 1 + β[1 + (2αγ/(1 + β) − 1)Qi Z
+ {α/(2(1 + β)) − (α/(1 + β))2 }Z2 ] 2A + B 1+β 2 tan−1 √ = 1− √ β 4AC − B2 4AC − B2 B − tan−1 √ (27) , B2 < 4AC 4AC − B2
Fig. 5. Decline of transmembrane permeate with pi as parameter.
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Fig. 6. Decline of transmembrane permeate with Ci as parameter.
=
C =1+β
1 1+β 1− √ 2 βo B − 4AC (2A + B − √B2 − 4AC)(B + √B2 − 4AC) ln √ √ , (2A + B + B2 − 4AC)(B − B2 − 4AC) B2 > 4AC
where
(31)
Therefore, once α, β, γ and Qi are specified, V as well as the average permeate flux ν¯ m is calculated from Eqs. (27), (28) and (25). (28) 3. Experimental
α −γ A=β 2(1 + β)
2αγ − 1 Qi B=β 1+β
α 1+β
2 (29)
(30)
3.1. Apparatus and materials The flow sheet of an ultrafiltration apparatus is shown in Fig. 2. The membrane medium used was a 15 kDa MWCO tubular ceramic membrane (M2 type, Techsep, France,
Fig. 7. Decline of transmembrane permeate with rm as parameter.
H.M. Yeh et al. / Journal of Membrane Science 241 (2004) 335–345
341
Fig. 8. Decline of transmembrane permeate with L as parameter.
inside radius rm = 0.003 m, length L = 0.4 m) made of Zr O2 /carbon. The tested solute was dextran T500 (Pharmacia, Mn = 1 703 000 and Mw = 503 000). The solvent was ion exchange water. The feed solution was circulated by a high-pressure pump with a variable speed motor (L-07553-20, Cole-Parmer Co., Chicago, IL), and the feed flow was measured with a flowmeter (L-03217-34, Cole-Parmer Co.). The pressure was measured with a pressure transmitter (Model 891.14.425, Wika). 3.2. Experimental conditions and procedure The experimental conditions were as follows. The feed concentrations Ci were 0.1, 0.2, 0.5 and 1.0 wt.% dextran
2 ) were 0.059, 0.088, T500; the feed velocities ui (= qi /πrm 0.118 and 0.147 m/s; and the feed inlet transmembrane pressures pi were 30, 50, 80, 110, and 140 kPa. The feed solution temperature in all experiments were kept at 25 ◦ C in a constant-temperature water bath controlled by a thermostat. During a run, both permeate and retentate were recycled back to a large feed tank the feed concentrations were observed to be unchanged, therefore, the effects of concentration polarization and solute adsorption on the membrane are insignificant. The experimental procedure was as follows. First, a fresh tubular ceramic module was used to determine the intrinsic resistance of membrane Rm . Average permeate fluxes for water ν¯ m,w were measured under various transmembrane pressures and flow velocities ui . Then the feedwater was
Fig. 9. Decline of permeate flux with pi as parameter.
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replaced with the tested solution. Average permeate fluxes for dextran T500 solution ν¯ m were measured under all operating conditions (Ci , ui , pi ) at steady state. Values of permeate flux reached steady state within 30–120 m. After each solution run, the membrane module was cleaned by a combination of high circulation and backflushing with pure water. The cleaning procedure was repeated until the original water flux had been restored. 4. Discussion 4.1. Correlation equations for R and φ The experimental data of the permeate fluxes for pure water (¯νm,w )exp and solution (¯νm )exp are presented in Tables 1 and 2, respectively. With the use of Table 1, a straight line of (1/¯νm,w )exp versus 1/( p)exp could be constructed by the least-squares method [29–32]. Thus, the intrinsic resistance of the tubular-membrane module employed in this study can be determined from Table 1 by using the following equation, which can be modified from Eq. (6) by setting Rf = 0 and φ = 0 for pure water: 1 Rm = (¯νm,w )exp ( p)exp
(32)
In the above equation: ( p)exp = 21 (( pi )exp + ( po )exp )
(33)
in which ( pi )exp and ( po )exp are the experimental values of inlet and outlet transmembrane pressures. Under various ui and ( p)exp , the measured value of Rm for the membrane system employed in present study was determined graphically as [9]: Rm = 1.055 × 1010 (Pa s/m)
(34)
Furthermore, if experimental data obtained in ultrafiltration of an aqueous solution is also applied to Eq. (6), then 1 Rm + Rf =φ+ (35) (¯νm )exp ( p)exp Therefore, from a straight line plot of (1/¯νm )exp versus 1/( p)exp at a certain flow velocity ui and feed concentration Ci , the experimental values of φ (the intersection at the ordinate) and (Rm + Rf ) (the slope), as well as Rf , were determined graphically from Table 2 as functions of ui and Ci . The results are presented in Table 3. Finally, the correlation equations for φ and Rf were constructed by following the same procedure performed in the previous works [29–32]. The results are: φ = 1.37 × 105 u−0.02 Ci0.535 (s/m) i
(36)
Rf = 1.74 × 1010 u−0.005 Ci0.244 (Pa s/m) i
(37)
4.2. Comparison of correlation predictions with experimental results The average values of permeate flux ν¯ m may be predicted from Eqs. (27), (28) and (25) with the use of the correlation equations, Eqs. (34), (36) and (37) and the system constants: L = 0.4 m, rm = 0.003 m, and the fluid viscosity [31]: µ = 0.894 × 10−3 exp(0.408Ci ) (Pa s or kg/m s)
(38)
Some correlation predictions for average permeate flux were calculated and the results are compared with the experimental data, as shown in Figs. 3–5. It is seen from these figures that though the predicting values are in qulitative agreement with the experimental results for practically operating conditions, only that at Ci = 1.0 wt.% was in good agreement.
Fig. 10. Decline of permeate flux with Ci as parameter.
H.M. Yeh et al. / Journal of Membrane Science 241 (2004) 335–345
343
Fig. 11. Decline of permeate flux with rm as parameter.
We may conclude that the present simulation model is unsuitable for the solution of low solute concentration. 4.3. Declines of transmembrane pressure and permeate flux The declines of transmembrane pressure and permeate flux along the tube with inlet transmembrane pressure pi , and solution concentration Ci , as well as with tube radius rm and tube length L as parameters were calculated from Eqs. (23) and (24), and the results are shown in Figs. 5–12. It is seen in Figs. 5 and 9 that both P and V decline more rapidly for lower pi due to the small effectively driving force, p, at the tube end. As shown in Figs. 6 and 10, P and V decline rather rapidly for larger Ci because that in this
case, the concentration polarization layer increases rapidly at the tube end. We may find in Figs. 7 and 11 that P and V decline more rapidly when rm decreases. This is because that for smaller rm the thickness of concentration polarization layer is rather larger compared with tube diameters, resulting in high resistance. Finally, as a common sense shown in Figs. 8 and 12, P and V decline more seriously as the length of membrane tube increases. It is concluded that the decline of permeate flux occurs mainly due to the decline of transmembrane pressure and the increase of the thickness of concentration polarization layer along the membrane tubes, and thus it is rather sensitive for lower inlet transmembrane pressure and larger inlet solution concentration, as well as for smaller tube radius and larger tube length.
Fig. 12. Decline of permeate flux with L as parameter.
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H.M. Yeh et al. / Journal of Membrane Science 241 (2004) 335–345
5. Conclusion The correlation equations, Eqs. (24) and (27) or (28), for predicting the permeate fluxes of membrane ultrafiltration in tubular-membrane modules, were derived from momentum and mass balances by the resistance-in-series model with the considerations of the declines of transmembrane pressure and flow rate along the fibers. The declines of flow rate, transmembrane pressure and permeate flux along the tube may be predicted from Eqs. (21), (23) and (24), respectively. For predicting the average permeate flux, one may employ Eq. (27) or Eq. (28). Ultrafiltration of Dextran 500 aqueous solution in a tubular ceramic module has been carried out for various feed concentrations, transmembrane pressures and feed flow rates. Correlation predictions qualitatively confirmed with the experimental results, as shown in Figs. 3–5. It is found that the correlation predictions of average permeate flux obtained in present study are more accurate than those obtained in previous works [31,32], in which the feed flow rates in the momentum balance equation, Eq. (2), were merely taken as the inlet value, or as its arithmetic-mean value, while in present study, the decline of feed flow rate through the tube was taken into consideration. The assumption of laminar flow is easy to check by the maximum value of Reynolds number with ui = 0.147 m/s and Ci = 1.0 wt.% as: 2rm ui ρ 2(0.003)(0.147)(1000) = µ 0.894 × 10−3 e−0.408 = 1484 < 2100
(Re)max =
As mentioned earlier, the resistance-in-series model satisfies the three essential conditions of membrane ultrafiltration, Eqs. (3)–(5). Further, as expected, it is shown in Table 3 that the fouling layer resistance Rf as well as the coefficient of the resistance due to concentration polarization φ decreases with the increase of crossflow velocity ui , while increases as the solution concentration Ci increases. Therefore, the present model easily describes the relationships of permeate flux with operating and design parameters, and we believe that this model will also be suitable for most membrane ultrafiltration systems including systems with different kinds of feed solutions, different materials of membrane tubes, and various design and operating conditions.
Acknowledgements The authors wish to express their thanks to the National Science Council of ROC for financial aid under Grant No. NSC 89-2214-E-032. Nomenclature Ci
concentration of feed solution (wt.% dextran T500)
L p ps
p
P q Q rm Rf Rm u V z Z
effective length of membrane tube (m) pressure distribution on the tube side (Pa) uniform permeate pressure on the shell side (Pa) transmembrane pressure, p − ps (Pa) dimensionless transmembrane pressure,
p/ pi volume flow rate in a tubular-membrane module (m3 /s) dimensionless flow rate, defined by Eq. (9) inside radius of membrane tube (m) resistance due to solute adsorption and fouling (Pa s/m) intrinsic resistance of membrane (Pa s/m) fluid velocity in the membrane tube, 2 ) (m/s) q/(πrm dimensionless permeate flux, defined by Eq. (24) axial coordinate (m) dimensionless axial coordinate, z/L
Greek letters α dimensionless group, defined by Eq. (16) β dimensionless group, defined by Eq. (17) φ 1/νm,lim (m2 s/m3 ) γ dimensionless group, defined by Eq. (18) µ viscosity of solution (Pa s) νm permeate flux of solution (m3 /(m2 s)) νm,lim limiting flux (m3 /(m2 s)) Subscripts i at the inlet o at the outlet Superscript − average value
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