Predicting the performance of RO membranes

DESALINATION Desalination 132 (2000) 181-187

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Predicting the performance of RO membranes Nader M. A1-Bastaki*, Abderrahim Abbas Department of Chemical Engineering, College of Engineering, University of Bahrain, PO Box 32038, Isa Town, Bahrain Tel. + 973 782-122; Fax +973 694-844; e-mail: naderbsk@eng, ubo. bh

Received 5 July 2000; accepted 19 July 2000

Abstract

Theoretical and experimental recoveries are compared for spiral-wound and hollow-fiber membranes. The effect of ignoring concentration polarization and pressure drops is studied. Ignoring concentration polarization and pressure drops results in significant overestimation of the recovery. With spiral-wound membranes pressure drops were found to be less significant. In addition, a solution method based on integration was used for hollow-fiber membranes. Each of the water and salt fluxes is def'med as an implicit function of two dimensions, namely length and radius, and this function is integrated over the whole of the membrane. The integration method resulted in recovery predictions closest to the average of the experimental data. The agreement between predicted and experimental salt rejection was less pronounced, the theoretical values being higher than the experimental. Keywords: Reverse osmosis; Desalination; Hollow-fiber membrane; Spiral-wound membrane; Modeling

1. Introduction

Theoretical models for RO membranes have been developed by many workers during the past three decades [1-10]. In developing a theoretical model to predict the performance of RO membranes, one of the first things to consider is the choice of the mass transfer model describing the flux o f water and salt through the membrane. Another issue of importance is the concentration polarization, which is particularly significant in *Corresponding author.

spiral-wound membranes. Several equations have been employed to calculate the mass transfer coefficient for the concentration polarization film adjacent to the membrane. In addition, the pressure drop on both the permeate (low pressure) and the shell (high pressure) side can be significant, particularly on the permeate side o f hollowfiber membranes. With spiral-wound membranes these pressure drops are considered less significant and are ignored in many cases. In the present work a comparative study was performed on the modeling of both spiral-wound and hollow-fiber RO membranes. The objective

Presented at the Conferenceon Membranesin Drinkingand Industrial Water Production,Paris, France, 3-6 October 2000 InternationalWaterAssociation,EuropeanDesalinationSociety,AmericanWaterWorksAssociation,JapanWaterWorksAssociation 00119164/00/$- See front matter© 2000 ElsevierScienceB.V. All rightsreserved PII: S 0 0 1 1 - 9 1 6 4 ( 0 0 ) 0 0 1 4 7 - 8

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N.M. AI-Bastaki, A. Abbas / DesaBnation 132 (2000) 181-187

was to compare the consequences of enforcing simplifying assumptions, particularly ignoring the concentration polarization and the pressure drops on the accuracy o f the model. Moreover, a comparison was made between the results of a simplified method of solution and a method based on integration for hollow-fiber membranes.

The solution diffusion mass transfer model can be employed to calculate the water flux, Jw, and the salt flux, J~, as follows [ 10]:

Pe ) -

J~ = B(CMC e )

¢ _ CM - Cp _ exp(v~ / k) C B - Cp

(ZM -- Z p )]

(1) (2)

where A is the pure water permeability constant, B is the solute permeability constant, P is the pressure, ~ is the osmotic pressure and C is the concentration. The subscript B indicates the bulk solution on the high pressure side, subscript P refers to the permeate and subscript M refers to the membrane surface (on the high pressure side). The concentration o f the permeate, Cp, in Eq. (2) can be calculated from Js and the total volumetric flux, Vw:

(3)

Ce = J , / vw

(4)

Due to the presence of concentration polarization, the concentration of the salt at the membrane wall, CM, is higher than that in the bulk o f the feed side stream, CB. The equation used for calculating CM was derived from a consideration o f the convective mass transfer of the salt towards the membrane by the feed water and a diffusive mass transfer of the salt away

(6)

Here, Re and Sc are the Reynold's and the Schmidt numbers and e is the void fraction in a hollow-fiber bundle. For a spiral-wound membrane the following equation can be used to determine k [12]: Sh = 0.04 Re °75 SC 0"333

(7)

The total permeated water, QP, the remaining brine, Qs, and the brine concentration, CB, at any point can be calculated from the following equations: Qp = vw • s .

(8)

Q8 = QF - Qp

(9)

C B = (QFCF - QeCe )/ QB

Jw + J , vw - - Pp

(5)

where the mass transfer coefficient k can be calculated from empirical relations such as [11]: Sh = 1.09 ReO.333Sc0.33 E

2. Background

Jw = A[(PB -

from the membrane. The resulting equation is:

(10)

The membrane area, SM, for the spiral-wound membrane is W.z and for the hollow fiber is (1/h)rd~z where (I/h) is the area per unit volume. Another factor that must be considered in solving Eqs. (1) and (2) is the pressure drop on the product side, Pc, and the on the feed side, PB. For a spiral-wound membrane, each of the permeate and feed side flows can be considered as a flow between two parallel plates with a length L, a width W and a spacing t. Hence the pressure drop on the feed side can be calculated as follows:

N.M. Al-Bastaki, A. Abbas / Desalination 132 (2000) 181-187

AP8 =12QBzt2/( Wt3 )

(11)

where/2 is the water viscosity and QB is the bulk flow rate on the feed side. For the hollow-fiber membranes the flow on the feed side is across a densely packed bed of fibers. Several equations have been suggested for the pressure drop in feed side. The equation selected here is an equation which is widely used for packed beds, namely, Ergun's equation:

150

de

+1.75

e

p

d e j (12)

.(R-R,) where R is the radius of the fiber bundle, e is the porosity of the packed bed, p is the density of water, us is the superficial velocity for an empty bed and de is the particle diameter, taken as 1.5 of the fiber diameter [8]. The permeate side pressure drop in a hollowfiber membrane can be calculated by considering a capillary flow with a continuous input of water as a result of the flux, v~ [8]:

dep _ 81JQp dz rtr,'

(13)

For an average constant flux, this equation can be integrated to give the following: 16/1

2

APe =--Z-ro VwZ r,

(14)

where ri and ro are the inside and outside radii of the fiber. The solution of the above system of equations can be achieved by an iterative technique as shown in the flow diagram in Fig. 1. The objective is to calculate the total flux, Vw(Z), at any point along the z direction in a spiral-wound membrane and the flux Vw(Z,R) at a point (z,R) in

183

a hollow-fiber membrane. The procedure shown in Fig. 1 starts with assuming a value for flux. Then the mass transfer coefficient, k, is calculated from Eq. (6) or (7). The concentration polarization factor is calculated from Eq. (5). For the first trial the bulk brine concentration, CB, Can be taken as the feed concentration, C~; and the permeate concentration, Cp, can be ignored. Next the brine flow rate and concentration, QB and CB, are calculated from Eqs. (9) and (10). The pressure drops on the shell side are calculated from Eqs. (1 1) or (12) and on the permeate side from Eq. (14). Then Eqs. (1) and (2) are used to calculate the water and salt fluxes, Jw and Js and a new estimate of the volumetric flux Vw and the permeate concentration, Ce, are obtained from Eqs. (3) and (4). The calculations are repeated until the new and old values of the flux are equal. The Mathcad 8 package was used to write a computer program to do the above iterative calculations. The program contains a subroutine called vw(z,R) and a second one called Js(z,R) that returns the values of these variables at any length or radius for the membrane. The total flux, Vw, and the salt flux, J,, calculated by the above procedure are treated as variables that depend on the distance traveled inside the membrane and, hence, is a function of z in spiral-wound membranes and of z and R in hollow-fiber membranes. The calculations can be done using various degrees of simplifications. Three cases are considered below. Case 1: The concentration polarization and the pressure drops are ignored. The pressures on the feed side and permeate side, P~ and Pc, will be assumed to be constants. Case 2: The concentration polarization as well as the pressure drops will be included using the method described above (Fig. 1). However, the whole module is taken as a single element and the calculations are made using constant averaged pressure drops and velocities [10]. Case 3: The flux of the water, v~(z,R), and the salt, J,(z,R), are considered as functions that

N.M. Al-Bastaki, A. Abbas / Desalination 132 (2000) 181-187

184

Specifiy z 1

Calculate k (eq.6 or 7)

Vw,

I

| and qb(eq.5), Pv(eq.11 or 12) and PB (eq. 14) •

and R | Assume a | value for [

I

J

Calculate Jw, Js, Cp and CB from equations 1, 2, 4, and 10

Fig. 1. Flow diagram showing the iterative procedure to calculate the total solution permeated, v~, as a function of the distance traveled inside the membrane (z for spiralwound and (z, R) for hollow-fiber membrane).

alculate a new v a l u e ~ r Vwfrom equation 32

/ Yes

f | No | ~i

Replace old value of v~ with the new value

1

implicitly depend on the dimensions z and R. The dependence o f these two functions on z and R are expressed in terms of the subroutine described by Fig. 1. The functions vw and J, were integrated over the whole membrane module. A differential area, dSM, was defined as follows for a hollowfiber membrane (see Fig. 2):

Ro

..... s

I

. ..

--~

t

N

2,, %

+

'

I

I

||

(15)

I

!i

I

|

| I

Ri

S

The total product flow rate, Qp, and permeated salt flow rate, Q,, were calculated by integrating each o f the functions dQp=vw(z,R) dSM and dQ, = J~(z,R) dSM, each over the whole of the membrane area. In Mathcad8 it is possible to apply double and triple integrals over a function that is not given explicitly but is expressed in terms o f a subroutine. Using Mathcad8 the following integrations were made to calculate the product flow rate:

R

%•

%% %% %%

I

tI tI

S s St 7 ~ ' SS

Fig. 2. Schematic diagram of the cross section of a hollow-fiber membrane showing an element dR used for membrane area calculation.

A(M. Al-Bastaki, A. Abbas / Desalination 132 (2000) 181-187 Ro

R~

100 ~

L

Qp = ~ ~(1/h)(2zc)vw (z,R)RdzdR

185

(16)

- -

] / - - - Case3. integrated | • Experimental, Ohya [13] |

95

Case2, averaged

90

0

85 <~

and

Case1. simple

~

•

80" 75-

Ro

Q, = f ~(1/h)(Zzc)J,(z,R)RdzdR Rj

~

L

(17)

~-

65-

°

.

°

•

° / / I

1 1

60-

0

55

The recovery, A, and salt rejection, SR, were calculated using the following relations:

5045 22

23

24

25

26

27

28

29

Feed Pressure. bars

A = Qp

(18)

QF

SR = 1 - Ce

(19)

CF

3. Experimental procedure The experiments were performed at different pressures (from 20 to 50 bars) using a singleelement FilmTec SW30-2521-A seawater spiralwound membrane. The feed water was prepared by mixing sodium chloride in distilled water. All the experiments were performed at room temperature (25°C) and a feed concentration of 10,000mg/1 (ppm). The experimental set-up consisted of a high-pressure pump, the membrane, pressure and flow meters and feed and product tanks. A more detailed description of the experimental set-up and procedure can be found elsewhere [13].

4. Results and discussion Fig. 3 shows the results of the three theoretical cases discussed earlier for a B9 hollow-fiber membrane along with the experimental results by Ohya et al. [14]. The simplest theoretical case ignores the concentration polari-

Fig. 3. Comparison of theoretical recovery predictions using the different cases with the experimental values of Ohya [13]. zation and the pressure drops inside the fiber and on the feed (shell) side. Although this method is suitable for a quick estimation of the performance, it can be seen that it over-estimates the recovery due to ignoring the factors that negatively influence the driving force for the water flux, (AP-A~r). The second case takes into account these factors but uses an averaged value of the velocities and pressure drops for the whole membrane. Fig. 3 shows that this second case somewhat under-estimates the recovery. The recovery calculated by integration [Eqs. (16) and (18)] is clearly closest to the average curve of the experimental results. Fig. 4 shows similar results for the salt rejection. However, all of the theoretical cases seem to over-estimate the salt rejection. The results for spiral-wound membranes is shown in Fig. 5, compared with the experimental results of the present work. As with the previous results, the simple case (no polarization or pressure drops) leads to over-estimations of the permeate flow rates accounting for the concentration polarization and the pressure drops lead to a very good agreement with the experimental results. The calculations were also repeated ignoring the pressure drops but including the

186

N.M. Al-Bastaki, A. Abbas / Desalination 132 (2000) 181-187

system consists of multiples of larger spiral wound membranes (longer channels), which are arranged in series.

100 95 I

90 o~ rY ~o

~

• ,

°

°

.

°

°

°

° °

•

85

5. Conclusions

8O 75 70

, --

Case 1, Simple

]

Case2. averaged

/

J

- Case3, integrated

•

EXperimental, Ohya [13] |

Average of e x ~ n m e ~ j _ 65

22

•

•

23

24

25

26

27

28

29

Pressure, bars

Fig. 4. Comparison of the salt rejection results for

different theoretical cases with the experimental results of Ohya [ 13]. 50 40

•

Experimental With polarization and pressuredrop~

~ 3o

" " / w " "" ""i

6. Symbols

2o

E 12.

Prediction of the recovery can be significantly affected by the simplifying assumptions. Significant over-estimation of the recovery for the hollow-fiber and spiral-wound membranes were found. For the spiral-wound membrane, ignoring the pressure drops was less significant. Defining the flux as an implicit function of length and radius (z and R) and integrating this function over the whole o f the membrane resulted in the closest agreement with the average of the experimental results of the recovery. Such results were less evident when the salt rejection was considered

A 10

CB 0 20

30

40

50

CM

Pressure, bars

Fig. 5. Comparison of experimental permeate flow rate for the spiral-wound membrane with theoretical values.

Ce

D 1/h concentration polarization. This case also resulted in a good agreement with the experiment, which indicates that for a small single element, the pressure drops are not significant due to the short path and the relatively large channels (compared to the capillary channel in hollow fibers). As a result, the concentration polarization is the main cause of reducing the driving force for the element used. However, this conclusion cannot be necessarily extended to the case when the

Js Jw k L Pe Pp Q8 QF

--

Pure water permeability constant, kg H20/m2.pa.s --Salt concentration in the highpressure side (shell side), kg/m 3 - - C o n c e n t r a t i o n at the membrane wall, kg/m 3 - - Concentration of the permeate, kg/m 3 - - Diffusivity of the salt in water, m2/s - - Membrane area per unit volume of fiber bundle - - Salt flux, kg salt/m2.s - - Water flux, kg H20/m2.s - - Mass transfer coefficient, m/s - - Fiber length, m - - Pressure on the shell side of the fiber bundle, Pa - - Pressure in the fiber bore, Pa - - Brine (reject) flow rate, m3/s - - Feed flow rate, m3/s

N.M. Al-Bastaki, A. Abbas / Desalination 132 (2000) 181-187 -

Re R, Ro

-

--

-

--

ri

-

ro Sc Sh

----

SM Vw

W z

-

-

-

--

-

-

-

Permeate (product) flow rate, m3/s R e y n o l d ' s number (2roUs p/Ix) Inside radius o f the fiber bundle, m Outside radius of the fiber bundle, m Inner fiber radius, m Outer fiber radius, m Schmidt number (~/pD) Sherwood number (2kro/D) 2 Membrane area, m Permeation velocity, m/s Membrane width, m Distance along the membrane module axis, m

Greek A E

P P

-----

Recovery (QP/QF) Porosity o f the fiber bundle Viscosity, kg/m.s Density, kg/m 3

R

e

f

e

r

[1] [2] [3] [4] [5] [6] [7] [8] [9] [I0] [11] [12] [13] [14]

e

n

c

e

187

s

M.S. Dandavati, M.R. Doshi and W.N. Gill, Chem. Eng. Sci., 30 (1975) 877. [2] J.M. Dickson, J. Spencer and M.L. Costa, Desalination, 89 (1992) 63. W.N.Gill and B. Bansal, AIChE J., 19(4) (1973) 823. J.J. Herrnans, Desalination, 26 (1978) 45. S. Kimura and S. Sourirajan, AIChE J. 13(3) (1967) 497. H. Ohya and S. Sourirajan, AIChE J. 15(6) (1969) 829. R. Rautenbach and W. Dahm, Desalination, 65 (1987) 259. M. Sekino, J. Membr. Sci., 85 (1993) 241. Y. Taniguchi, Desalination, 25 (1978) 71. N.M. AI-Bastaki and A. Abbas, Desalination, 126 (1999) 33. R.E. Treybai, Transfer Operations, 3rd ed., McGraw Hill, New York, 1980. G. Belfort, Membrane Processes, Academic Press, New York, 1984. N.M. AI-Bastaki and A. Abbas, Sep. Sci. Yechnol., 33 (1998) 2531. H. Ohya, H. Nakajima, K. Takagi, S. Kagawa and Y. Nigishi, Desalination, 21 (1977) 257.