mathematical modeling of reverse osmosis systems

DESALINATION

ELSEVIER Desalination 160 (2004) 29-42 www.elsevier.com/locate/desal

Mathematical modeling of reverse osmosis systems

Abstract

-J-he present investigation pertains to modeling of seawater desalination system. A simulation model was developed and veriJied for a small-scale reverse osmosis system. The proposed model combines material balances on the feed tank, membrane moclule and product tank with membrane mass transfermodels. Finally a comprehensive simulation model has been developed incorporating the effect ofmass transfer inhibition The model is non-linear differential equation representing the feed concentration as a function of operating time and space. The solution of the simultaneous differential equations was obtained using the fourth order Runge-Kutta method, due to self starting and stability. The model was veritied using the experimental data from the literature [17,24]. Parameter sensitivity was carried out to select the proper step size. The simulation was run for over 1000 h enabling a prediction of operational perfomrance at high overall system recoveries.

K~~vow!s: Mathematical modeling; Reverse osmosis: Mass transfer; Inhibition: Concentration polarization

1. Introduction

Membrane separation systems are gaining popularity in the food and bioprocessing industries due to its less energy requirement-, negligible denaturation of food product, retention of aroma and flavours. This technique has also got numerous applications in processing industries such as chemical, nuclear. biotechnology, petroleum and petrochemical industries. Reverse osmosis is the

*Corresponding author.

most popular technology for seawater desalination. During the last two decades hundreds of reverse osmosis seawater desalination plants have been built worldwide. Each year the plant sizes and cost-effectiveness have increased. Recently the reverse osmosis has achieved growing acceptance as an economical and viable alternative to multi- stage flash distillation (MSF) process for desalting seawater [.1.2,5]. A number of investigators have carried out the work on different aspects ofreverse osmosis seawater desalination. Few models for

001 l-91 WOG/S- See front matter 0 2004 Hsevier Science J3.V. All rights reserved PII: SO0 I l-9 164(03)006 16-7

solvent and solute fluxes through membranes have been developed and analyzed neglecting the effect ofmass transfer inhibition. Concentration polarization and fouling of the membrane are the two serious problems that would prevent the use of RO into many of the processes. Both of these phenomena arc flux inhibiting boundary layer effects and that -fouling is almost always a result of*concentration polarization. Concentration polarization may be defined as the presence ofa higher concentration of rejected species. at the surface of a membrane than in the bulk solution, due to the convective transport ofboth solute and solvent [ 10,331. It is generally considered a totally revers- ible effect that c.an be described in a first approximation. by a variety of analytical models. The fouling in the membrane is the condition, in which membrane undergoes plugging or coating by some element in the stream being treated, in such a way that its output or its flus is reduced. A model for colloidal membrane fouling has been reported by Green and Belfort [9], which allows the theoretical prediction of membrane flux declination. A lot of work has been done on the membrane foulants by the various investigators and a detailed analysis is available in the literature [ 15.161. The measure- ment and control in reverse osmosis desalination have been studied by Mindler and Epstein [22].

The reduction of concentration polarization is important for the improvement of the performance of osmotic type membrane, which will lead to reduction in the fouling ofthe membrane. Several measures to reduce the concentration polarization to control the fouling have been adopted and proposed. The techniques used to reduce the concentration polarization are increasing flow rate, assembling an intensifier for turbulent flow, impulse methods, agitating methods, periodic depres- surization of membrane tube, flow reversal, pre- coating of membrane surfaces. enzyme immobili- zation, modification of membrane polymeric structure, mechanical and ultrasonic vibration o! membranes have been tried also but with little success 13.71. The turbulence promoter acts to

reduce concentration polarization and therefore fouling, by increasing the friction factor and bulk velocity. A model has been developed by Chiolle et al. [6] for the reverse osmosis with the turbulence promoting nets for the parallel wall channels module. The model developed by Drioli and Bellucci [8] shows the effect of the interaction of the concentration polarization and solute-membrane interaction on the pressure driven membrane when working with multicomponent solution. The modification ofthe membrane polymeric structure plays an important role in the reduction of concentration polarization through the tluidized bed was developed by van der Waal [:20]. Bhattacharyya et al. [25] developed a Galerkin finite elements program to compute the concentration profile throughout a reverse osmosis membrane module to predict the performance of the module. The finite element method allowed rapid evaluation of various membrane module configurations, such as tapered cell geometry and channels containing spaces. Based on the available work., a comprehensive simulation model has been developed for seawater desalination system.

2. Models for solvent and solute transport in reverse osmosis

Various models and mechanism for the solvent and solute transport through reverse osmosis membrane have been developed and proposed by a number of investigators [12,19]. The flow of solvent through the membrane is defined in terms of flux:

J,,, = <Qj~s,> Clv,:, (1)

The solvent flus of the permeate depends on the hydraulic pressure applied across the membrane, minus the difference in the osmotic pressure of the solutions of the feed and permeate side of the membrane [2].

J, =A@-An) (2)

?I

While the solute tlux depends on the concen- t-ration gradient

J, = B,AC (3)

AC=C, -C, (4)

The membrane rejection is defined as the difference between the feed concentration and permeate concentration

R=(q -c& =[l-(c,/ci)j (5)

From the solvenr and soIute flux Eqs. (2) and (3) it can be shown that the rejection is the function of pressure and concentrations. Since the solvent flux is dependent on pressure, an increase in pressure will increase solvent flux at constant solute flux. Consequently the percentage of the re.jection will increase. Thus combining tlux models and relating it with rejection, it can be shown that the permeate concentration is equal to material balance around the membrane

C, = C,, (Js / Jw ) (6) so that the re.jection R is given by

R=l.Q-(J,C,,lJ,,C,j (7)

Substituting the expressions for the fluxes in the expression of rejection we have:

From this expression5 it appears that if pressure drop is increased to a large value then rejection approaches towards unity. However this camiot be achieved due to the limitation of membrane. Nevertheless one can reach almost up to the desired level. The model presented above is [he model for ideal mass transfer which does not give the exact picture of the reverse osmosis system.

The simple process case of continuous mode of operation if once through as shown in Fig. 1 is run most easily. Llnderthis type of operation, feed

Cv Qr RETENIA~ Semi-Batch

ROMJZMBUNE

FEED

TANK

c,. v,

Continuau

Fig. 1. hIodes of reverse osnlosis system operation

characteristics remain the same and the retentate or concentrate is collected separately, as is the permeate. if an initial feed volume is used, feed is run to exhaustion. fn the absence of mass transfcl inhibition, rqjection. flux and stream concentrations ideally remain the same with time. ‘The single pass recovery for this type of operation relates permeate production to feed rate

Recovery (2) = QP / Qf (9

In a semi-batch, unsteady state mode ofopera- tion, as was the basis for simulation [ 181. retentate is recycled to the feed tank and permeate is collected separately. This process is essentially a closed loop concentrating system. As the operation time increases, the volume of permeate collected increases. The permeate produced at any instant of time is called the instantaneous permeate.

The permeate or product collected in the product tank over a span of time is called the average product. Since the permeate is remov-ed continuously from the feed, the volume ofthe feed decreases, the feed becomes more and more concentrated with time. The feed in this type of process can also be referred to as the concentrate.

As feed volume diminishes and concentration increases, the system will operate as if it were running in sequential increments of increasing concentration. in a semi-batch, steady state mode. This type of process allows the system to run at varying levels of recovery and large-scale simulation can be operated. Recovery is defined in terms of an overall system recovery as the total quality of product generated up to a given time divided by the initial feed volume:

Recovery (X) = VP / VfO (10)

At some point in the operation the system must be stopped as the feed becomes so concentrated that the flux drops significantly, due to a large increase in the osmotic pressure of the feed. If the permeate flows in a semi-batch, unsteady-state system is returned to the feed tank, the mode of operation is termed “semi-batch, steady state”. Although the flow pattern is not an operational mode, it is used to study the characteristics ofthe system. Since both permeate and retentate stream are recycled to the feed tank, the feed volume and concentration do not change with time.

The system material balances, together with these mass transfer models, were used to simulate system operation. Correlation of flux, solute concentrations and re.jection with operating time and overall system recovery are functions of the model. This model also predicts operational performance c.haracteristics of the system at various times and recoveries. The effects of pressure, feed concentration, volume and membrane characteristics on separation efficiency can also be described. The material balances can be formulated for the system: A material balance made on the product tank yields

QpCp = d(V&m )I dt (11)

QPCP = (dVp /dt)C,, +(dC,, ldt)V, (12)

BoundaT conditions: at t = 0, V, = 0, C = C’ pl” P

The change in the volume ofthe permeate with time is the production rate of the membrane.

dV,/dt=Q, (131

By substitution of this in Eq. (12) we get

QpCp = Q&m + (dc, ldt) V,

or dC, 1 dt = Qp (CP - C, )/ VP (14)

The material balance around the membrane module is

In this balance an assumption is made that in this system the concentration within the membrane does not change greatly with spatial distribution. A mean permeate concentration from the membrane module was used.

Similarly the balance around the feed tank becomes

Q/G. -Q&J, = d(V’C,)/dt (16)

In the model it was assumed that the feed tank was well mixed. Thus the concentration o-f the feed to the membrane equals the concentration in the feed tank. Therefore, at any instant in time, t, c 1’ = c

Thdfcombination of Eqs. (15) and (16) with substitution of CTtf as C, gives

- QpCp = (dV/, / dt)C, + (dC, / dt ) V$ (17)

The change in the feed volutne with time can be taken as the production rate, so

-dV,/dt=Q, (18)

lntegrating with boundary condition at t = 0, I/r, = y/v

J’ji = Vfo - Qpt (19)

Substituting this value into Eq. ( 17) we get

- QpCp = -QpCr + (v,, - Qg) (dc, ldt) c-w

Rearrangement of Eq. (30) gives

dc., ldt = Qp (c, - C, )b(v,o - Q,t ) (21)

To get the solution of Eq. (21) we need the relationship between Q!, and C,> with the expression for c’, in terms of CL,

To achieve this we have to get the relationship between osmotic pressure and feed concentration, which is done through the v$an’t Hoff‘expression

x:=(+dV)RT (32)

Eq. (22) shows that the osmotic pressure of solution increses with the increasing concentration and temperature directly. The osmotic pressure coefficient must be determined for different solutions. It has been determined by various researchers to be less than unity and slightly increases with increasing solution concentration 1171. if the solute is not known or it is complex, wc have to use mass concentration instead of molar concentration. For convenience: this model assumed a constant temperature and incorporated the other constant Y which simplifies osmotic pressure to solute concentration coefticient.

n=YC (23)

AZ = YAC

The value of Y was assumed t-o be constant over the operating range of the solute concentration. Incorporation ofEq. (23) into the expression for the solute flux Eq. (2) yields:

J,,. = A,, lap - \fl(c/ - C, )] (24)

J, =B,b, -cpl (3)

J.s = Jw kp /Cwp ) (6) Combining the above equations. we get:

J& = J,.C,

Approximating the equation through ,4%AI’lB, >z=- As,Y/B, and CT! >> CT!, for high rejection,

w2

(37)

By substitution of the expression for C’,, in Eq. (24) we get the expression for flux in terms of ‘;

J, = 4, [@ - yCf + {(YC, )/(a3 -a&‘,, )}] (281

substitution of Eq. (38) into Eq. ( 1)

Putting the expression for C,;] and Opq Eqs. (27) and (28). into the expression for the concentration change with time into Eq. (2 1 ), we get:

where the model constants are

a, = bW’~C,,

Eq. (30) is the non-linear differential equation, which can be solved numerically. The solution of this equation gives the relationship between the operating time and concentration of feed. Concen- tration of feed is a function ofoperating time. Once the C, is calculated at any time, permeate concentration, rejection and flux can be determined. The overall recovery can be obtained by using feed and permeate concentrations. In the mode of operation used. the system is essentially closed: that is the mass of the solute in the initial feed must equal the total ofthe various process streams and tanks at any instant of time. The overall mass balance is

The overall recovery is expressed in terms of C,&, CTi and t;jO,.:

VP lVfO = (c,. -c,J/(c, -qm) (33)

The equation for total dissolved solid (TDS) concentration in the product tank can be obtained by substituting Eq. (27) into Eq. (14)

dC,Jdf= b‘D(c, -qm.)llvp or

dC,, ldt = [a, - a,C, + {a,Cf /(a3 - a,C,)}]

E(cdh -~&C,1~ (34) k/5 k, - c,o )@f - C,” II

Eqs. (30) and (34) can be solved simultaneously with the help of fourth order Runge-Kutta technique.

For the determination of model constant the six model constants and two initial conditions were used in the simulation program. The initial conditions are feed concentration cfi and feed volume I$. Membrane surface area 5’(, and operating pressure gradient AP are two model constants that represent design variables, the solvent (water) concentration is C’

For distilled witer n, = 0, so .{,> = AIV AI’. The slope of the plot between J,,, vs. AP determines A,,,. The solute permeability constant was determined by operating the system at several different concentration of the feed at constant pressure. Since .J,= B\ACY and */,,,Cl,/CWj, = Us (C, - C:,), a plot of JwCplC,, vs. (CIc- C’J yields B,y.

The osmotic pressure coefficient can also be determined experimentally on the reverse osmosis system. Since A,,, was previously found for pure water, the relationship

bNJ,uv)l =M,-c,)] can be plotted to determine yf.

3. Model for reverse osmosis system with concentration polarization

The model developed in this study depicts the ideal mass transfer. does not include the concentration polarization and fouling of the membrane, which causes the significant decline in the solvent flux. To avoid the concentration polarization, creating turbulence in the feed velocity is one of the remedies. There is certain feed velocity, say critical velocity, above which flux declination is slower than at lower velocities. The model developed by Slater et al. [ 171 considered the feed concentration changes with time globally.

To accomplish the concentration polarization and incorporate the feed velocity in the model we considered the feed concentration changes with time locally. So that the time and space dependence of feed conc.entration will be considered in this model. Cr = C/. (time, space)

Differentiating C’::, partially with respect to time and space coordinalc

dC, = (aC, / at)dt + (aC, / ay )dy . or

dcpidt=acpiat+(ac,iay)ub (35)

Using Eq. (2 i) for material balance around the feed tank with the concentration as a function of time and space, we get

dc, ldt = [e, k’, - C, )dv,o - Q,t))l (36)

Substitution of Eq. (35) into Eq. (36) gives

(ac, iat)+ (ac, /a&7, =

Qp {(c, - C, )b(v,o - Q,t )I ’ Or

ac, / at = bp icy - C, )bb’p - Qpd - 4 bc, / 4

(’ 7) 3

Eq. (37) shows the time as well as spatial dependence of the feed concentration. To solve this equation we have to get the expression for the spatial dependence of feed concentration. For a membrane under steady state condition. when the solution flows through the system parallel to the membrane surface at a given rate, both the solute and the solvent are forced to pass through the membrane owing to the action ofthe pressure difference, as depicted in Fig. 2. The solvent can pass through the membrane completely but most of the solute accumulates at the surface due to the rejection caused by the membrane. Thus a concentration gradient is built between the membrane surface and bulk solution, which makes the solute diffuse back towards the bulk solution. The higher the solute concentration at the rncmbrane surface the lower will be solute permeation rate

Fig. 9. Solute transfer on the boundary layer of the Inem- brane.

of the solvent. This urlfavourable phenomenon is called concentration polarization [ 10,241.

The differential equation for mass conservation in the process of reverse osmosis is as follows:

J, = F,,,C,,, - D (dC,, ldy) (38)

The solute flus is equal to the difference between the total amount of solute Rowing towards the membrane surface and the portion of solute diffusing back from the membrane s&ace to bulk solution.

dC,ldy-[(Fw/D)c,] =-(J,y lD)

This is a firsi order ordinq di Rerential equation which can be solved through the particular integral and complementary function.

C, = P wbl exd@,, / D)vl + C, (39)

where CL and p are integration constants.

Boundary conditions at J’ = 0 c,, = c‘, and at y = 6 Cl, = C’:,

C, - C, = Pexp[a] (40)

C,,, - C, = P expblexp[(ic,,, / D>Sl (41) Dividing Eq. (4 I) by (40) in order to get con-

centration polarization ratio

(Cw -C, )/(C, -C, )= expk ~~>~I (42) using the approximation limit (JI, << C, < c’,*, so that we can neglect C,> from Eq. (42).

C, lC, = exp[(F, /II)&]. or

C, lC, = exp[F, lk] (43)

k=D/6,

C, = C, exp[F, /k] (44)

Substituting these values into Eq. (39) we get:

C, = (C, - C, )expk / kxy / @I + C, (45)

‘I’0 calculate wall concentration we have to calculate the mass transfer coefficient and F,“. The value ofF>,, can be determined experimentally for a given membrane and operating condition. To estimate the value of k several investigators have proposed a scheme to determine the mass transfer coefficient. Majority of them does not include the major parameter particularly the action of F,v in radial tlow rate, i.e. solvent flux. The calculation ofk through the expression developed earlier [ 14- 161 gives the following:

j, =(k/Ub)Sc2’3

for the cases of turbulent flow in round tubes

j,=f/2 k = (~T&)/(~SC~‘~) (46)

The substitution 0fEq. (46) into Eqs. (43) and (45) gives us

C, /C, = wM (47)

with y= (2FwSc2” /fog).

Brian [ 16,231 approximated the average value of concentration polarization in the tubular membrane with the following equation:

C, lC, = l.333exp[(y/0.75)(y/6)lCP

CQ = (C, -C, )exddd>l+ C, (48) Partial differentiation o,f Eq. (48) with respect

to space coordinate gives:

JC, QY = ((c, - C, I/ ~bxdh @I (4%

Substitution of Eq. (49) into Eq. (37) will give the expression for feed concentration changes with time in the presence of concentration polarization and bulk velocity.

Xv /at = e-{U,(C, -C,)lG)yexp[y(ylG)] (50)

with 8 = QP (C, -C,)@/,, - Q,t). Concentration polarization can be minimized

by the turbulent promotion, with the increased friction factor and bulk velocity.

In order to simulate this mass transfer process and to determine the mass transfer coefficient when both the longitudinal and radial flows exist simultaneously 14,201, Xuesong Wang [24] developed a new type of diffusion current method for the determination of k. lie deduced the empirical formulae of the mass transfer coefficient for one and two dimensions as follows: For lam inar flow one dimension

Sh = 1 .66Re0,36 SC’.~~ (d, / L)o.42

two dimensions

(51)

Sh =4.72Re”.36 SC’.~~(~, !L)o.42X”.25 (52)

Concentration polarization ratio for laminar flow with two dimensions can be obtained by substitution of Eq. (52) into Eq. (43)

C,, / C, = exp[h]

A = [((,,l?V”.75)/(Reo.36 SCOJ~)~LIA,)~.~Z]

Substitution of the value of k in Eq. (45) gives

c, = k - c, )expMY / a (53)

Differentiating Eq. (53) with respect to space coordinate, we will get:

JC, / dy = {(C, - C, )/ G}exp[h(y / S)] (54)

Substituting Eq. (54) into Eq. (37). we will get:

acJat=e-{u,(c~ -c,)/6}~exp[h(y/6)](55)

This represents the system equation when the concentration ofthe feed is taken as the function of time and space with the value ofmass transfer coefficient k. Similarly the system equation can be obtained for turbulent flow:

Turbulent flow:

one dimension

Sh = 0.073 Re0.74 SC’.~~ (d, / L)o.32

two dimensions

(56)

Sh = 0.147Re0.74 SC’.~~(LI, /L)o.32Xo l9 (57)

The concentration polarization ratio in the case ofturbulent flow is

C, /C, = ewhl (58)

writh q = [((6.8~“.s1)/(Reo.74 ,sc’.‘~ )XL/~~ ,,.Q] .

The equation for feed concentration is

Cw = (Ch - C, )expMy / @I+ C, (59)

Dit‘ferentiating Eq. (59) with space coordinate:

dc, / dy = k, - C, )/ Sh ddy / @I (60)

Substitution ofEq. (60) into the master Eq. (37) gives:

ac, 1 at = 0 - (uh (c, - c, )/ S)ll exp[q(y / S)] (6 I )

From the experiment carried out by Xuesong Wang 1241 a dimensionless empirical equation for the mass transfer coefficient for a tube type fluidized bed under laminar flow conditions was obtained as follows:

Sh, = 2.58Rey SC’.*~(~, /L)“.08Xo/025 ((2,

C, lC, = exp[z] (63)

T = [(0.388V$‘98)/(Re~36 SC~‘~)~L/~, )oos]

The expression for the space dependence of feed concentration can be obtained by subslitution of Eq. (63) in Eq. (45).

Cp = (C, - Cp )exp[& / S)] + C, W

Differentiating Eq. (64) with respect to space coordinate gives:

aC, /+={(C, -C,)/6bexp[r(~~/S)] (65)

By substitution of Eq. (65) into the master Eq. (37) for the system we get:

ilC, /at = e-(u,(C, -C,)/Gbexp[z(yl6)] (66)

Eq. (66) represents the master system equation which includes the spatial as well as time dependence of feed concentration with inclusion of feed velocity and concentration polarization by a remedial factor to reduce concentration polarization, with the help of l-luidized bed.

Once the above equation is solved then the value of solvent flux with compaction correction factor. recovery, rejection and concentration of permeate can be calculated by substituting the value of Cc,, in the corresponding equations, dis- cussed earlier.

4. Models verification

Experimental data for aqueous salt (NaCl) solution form the literature [17&l were used to

38 K Jand Ed al. ,’ Iksniincrfion I60 (2004) 29.---/2

verify the model. The constants and initial conditions for the simulation are given as:

c;? = 2.00 kghd VfO = 1.50 13, Cl? - 1.0~ lti k&l?; s, = 0. I8 1 111:. AZ' = 4.02x 10" kg/m h’, A,,, = 4.2~ 10 I7 h/m, Bc= l.l2rlO-.lm/h, Y = 1.02x IO” m2/h2, Re = 30 -1 OjT ~71f/Z., = 0.0 15, 0.62 mm < d I: 0.9 mm, 450 < SC < 2650, 0.55 <E -C 0.9&0.0623 <X,< 1.455

5. Results and discussion

Fig. 3 shows the variation of permeate flux with time. Initially as operating time increases, the permeate tlux decreases slowly. After the attain- ment of around 400 h, the flux decreases sharply. The simulation results suggest that the permeate flux decreases slowly at first, but as time advances, the concentration gradient increases and the flux falls more sharply. Fig. 4 shows the simulation results of solute concentration in feed with time. As predicted by the model, the feed concentration increases linearly with time at a slow rate during the early hours (i.e. up to 400 h) of operation. Beyond this, the feed concentration increases exponentially with time. During the initial period of operation of the reverse osmosis system, an initial feed volume was used. Retentate concentration is always incrementally greater than the feed concentration; and the retentate is recycled to the feed tank as shown in Fig. 1. The feed volume decreases continuously with time because of permeate production, hence feed concentration increases with time. If I/;, was larger? the curve would remain linear for a longer time as described earlier.

Both plots of permeate and average product concentrations, as predicted by the model, is sho\vn in Fig. 5. The permeate concentration increases gradually with time (around up to 400 h of operation) and beyond this time the permeate concentration increases rapidly with time. At the end of simulation, the permeate concentration reaches to a value of I.027 kg/m: after 1000 h. The average

17

IS

13

7

^,

0 100 200 300 400 500 600 700 800 900 loo0

Time, t (hr)

Fig. 3. Simulation results of permeate flux vs. time.

0 loo FJO 300 400 54Jo 600 7cn I)00 900 loo0

Tim* t (b0

Fig. 4. Simulation results of feed concentration vs. time.

1.2 ..-

a

b.6 8 c E ‘: .j”’ ; P $0.2

I e”

0 0 loo 200 300 400 500 ml 700 800 %M ml0

Time, t (br)

Fig. 5. Simdation resdts of permeate concentration and average permeate concentration vs. time.

permeate concentration also increases more gradually, reaching 0.178 kg/m’ due to the effects of dilution in the product tank.

6. Simulation results of reverse osmosis system with concentration polarization

Fig. 6 shows the variation ofpermeate flux vs. time with concentration polarization. The permeate flux gradually increases up to 450 h reaching 16.00 kg/m2h. After this period, the flux decreases rapidly up to 650 h of operation and then it decreases linearly. In Fig. 3. the flux decreases up to around 450 h without incorporating the concentration polarization, whereas in Fig. ii the flux increases gradually due to change in feed concentration. After 650 h of operation. the change in flux is slow with concentration polarization.

The variation of feed concentration with time has been shown in Fig. 7. The feed concentration decreases almost linearly at a slow rate with time up to around 400 h of operation of the system.

2 ^ .

0 100 ii 300 400 SO0 600 700 SW 900 1000

Timt, t (ar)

Fig. 6. Simulation results of permeate flnx vs. time with concentration polarization.

25

0 100 200 300 400 500 6Mf 700 800 900 10

Time, t (br)

Fig. 7. Simulation results oE feed concentration vs. rime with concentration polarization.

After this period, the feed concentration increases exponentially with time. In Fig. 4, the feed concentration got increased up to 400 h of operation without concentration polarization, which is the contrast with concentration polarization.

Fig. 8 shows the variation of permeate concentration and average production concentration with time. The permeate concentration C?P remains almost constant with time up to around 450 h of operation as is the case without concentration polarization. Beyond time t - 450 11, the CP increases linearly at a high rate. This trend was observed almost same as explained in Fig. 5. The variation of average permeate concentration with time remains same as observed for the case of without concentration polarization.

7. Conclusions

The following conclusions can be drawn from the present study:

0 loo .200 300 4lm 500 600 700 flcn 900 loo0

Time, t (iv)

Fig. 8. Simulation results of permeate concentration and average permeate concentration vs. time with concen- tlation polarization.

1. The model developed without concentration polarization was effectively used for the prediction of feed concentration, permeate concentration, rejection and flux as a function of operating time.

2. The proposed model developed with con- sideration of spatial dependence of solute feed concentration (hence the permeate concentration, average permeate concentration and rejection as a function oftime and space) on simulation showed good agreement with the actual reverse osmosis process and improved the permeate flux.

3. In the proposed model the bulk feed velocity shows the important role which is the remedial factor for improvement of permeate flux.

Symbols

u,-n, - Constants in model equations 4” - Solvent permeability constant. h/m B.% - Solute permeability constant. m!h c - Concentration, kg/m’ c, - Solute bulk concentration flow, kg/m) C; - Solute feed concentration, kg/m’ C,, - Initial solute feed concentration, kg/m? c;, - Solute concentration in the feed tank,

kg/m” c c:”

- Solute permeate concentration, kg/m’ ,,Y” - Average permeate concentration, kg/m3

CT - Solute retentate concentration, kg/m” C/J - Concentration as a flinction of time and

space, kg/m’ q,, - Solute wall concentration, kg/m:’ c w/r - Solvent permeate concentration (water):

kg/m 3 D - Diffusion coefficient, m/h de - Equivalent diameter, (Wtl;)i’(H+ WY), m 4,. - Equivalent diameter of fluidized bed,

$

d,/[ 1 + (3/2) (1 --E)(djc$J]t m - Diameter of the fluidized particles, m - Tube diameter of the fluidized bed. m

.p’ - Volumetric flux rate, m-ilm2li - Friction factor

H - Height of the channel, m j,, - Chilton-Colbumj factor, (/&?‘)lU,,

A’. .Jand et al. ,” Iksaiinrrlion 160 (2004) 29. -/2

--......” Solute flux: kg/111~11

- Solvent (water flux). kg/m% - Mass transfer coefficient: m/h

References

A.M. Al-Mudaiheem and H. Miyamura, Con- struction and commissioning of Al Jobail Phase 11 desalination plant. DesaIinat-ion, 55 (1985) IL.1 I. S.E. Aly, Combined RO/VC desalination system. Desalination, 58 (1986) 85-97. D. Mahlab, N. Ben Yosef and G. Belfort, Concen- tration polarization profile for dissolved species in unstirred batch hyperfiltration (reverse osmosis). II. Transient case. Desalination, 34 ( 1978) 297-303. K.B. Bird,WE. Stewart and E.N.Li~htFoot.~I’ransport Phenomena. Wiley International Edition. Yew York. 1960,663 p. D.C. Brandt, Seawater reverse osmosis: an economic alternative to distillation. Desalination. 52 (1985 ) 177-186

Greek

.-‘..-- Length of the channel or height of the bed, m

- Number of moles of species i --...-- Membrane pressure gradient, kg/,/m hZ - Volumetric feed tlow rate, m’/h - Volumetric permeate flow rate, m’/h .--...- Volumetric retentate flow rate, m’/h - Solute rejection - Reynolds number, (pql,fjh!y,) _--- Reyolds number (tluldlzed bed) - Universal gas constant, kg mZ/h2 k - Membrane surface area. m* ----.. Schmidt num bcr (p/p n) - Sherwood number (u’, k/D) - Sherwood number (fluidized bed)

- Absolute temperature, K - Time, h ....-.-- HuIk feed velocitv: m/h - Feed volume. m’ - Initial feed volume, my’ -..--. Feed tank volume. m’ - Permeate volume. m’ - W number (F,q,~~jl>) ..--.-- Channel width, m - Overall system recovery - Space coordinate, m ---. Single-pass system recovery

- Thickness of the boundary layer. m - Voidage ..---- Absolute viscosity, kg h/m’ - Density. kg/m’ - Osmotic pressure, kg/m Ii’ ......-- Osmotic pressure coefticient, m*/h’ - Osmotic pressure to solute concen-

tration ratio, m2/h’ .....-.... X number (dC F:, p/p) - X, number (fluidized bed). (4,. F,, p/p)

31

A. Chiolle, G Gianotti, M. Gramondo and G Parrini, Mathematical model of reverse osmosis in paraliel- wall channels with turbulence promoting nets. Desalination. 26 ( 1978) 3--- 16. J.E. Cruver, 1JS Department of Interior, Office of Saline Water Res. Dcvel. Progr. Rept., NTIS: PB 223 181, No. 882, Sept. 1973. E. Drioli and F. Bellucci, Concentration polarization and solute-membrane interactions affecting pressure driven membrane processes. Desalination, 26 (1978) 17.--36. G. Green and G. Belfort, Fouling of ultrafiltration membranes: lateral migration and the particle trajectory model, Desalination, 35 (1980) 129--147. H. Ohya. Reverse Osmosis Method, Cltrafiltration Method. 1. Theory. Sachu Publisher, Japan. 1976. NM. Jackson, ;Llodel to simulate the structure and performance of cellular polymeric membranes: structure, flux and filtration characteristics. Filtr. Separ., Jan (199.5) 69. G. Jonsson, Overview of theories for water and solute transport in UF!RO membranes. Desalination, 35 (198Oi2L38. Y. Kurokawa, M. Lurashige and K. \iui. A visco- elastic model for initial flux decline through reverse osmosis membrane. Desalination, 52 (1981) 9-- 14. E3.R. Min, AL. Gill and W.K. Gill,.4 note on fluoride removal by reverse osmosis. Desalination, 49 ( 1984) 89-93. DE Potts, R.C. i\hlert and S.S. Wang. .A critical review of fouling of reverse osmosis membranes. Desalination. 36 (198 1) 135-264.

[161 B. Ramachandaran and B.M. Mishra, Mechanism and transport through reverse osmosis membranes. Lecture Notes: Membrane and Its Application, Orga- nized by Dept. Chem. Eng. and Dept. Biochem. Eng. and Biotechnology, IlT Delhi, Feb. 13-l 5 (1996) 27.

1171 C.S. Slater. J.M. Zielinski, R.G. Wendel and C.Ci. L&in, Modeling of a small scale reverse osmosis systems. Desalination, 52 (1985) 267-284.

[18] M. Soltanieh and W. Gill, Review ofR0 membranes and transport tnodels. Chem. Eng. Comm., I2 ( 198 1) 279,

[19] S. Sourirajan and T. Matsura, RO:‘Ultrafiltration Principles. Ottawa, Canada, NRC. 1985.

[.?(I] M.J. van der Waal. P.M. van der Veiden, J. Koning, CA. Smolders and W.P.M. van Swaay, Use of flui- dised beds as turbulence promotors in tubular mem-

brane systems. I.>esalination, 22 (19’77) 465 -483. PI] W.J. Weber. Physicochemical Processes for Water

Quality Control, Wiley, New York. 1972. [22] A.B. Mindler and A.C. Epstein, Measurements and

control in reverse osmosis desalination. Desali- nation. 59 (1986) 343-379.

[23] K. Jamal, Modeling of reverse osmosis membrane system, M. Tech. dissertation. AMU Aligarh, 1996.

[24] X. Wang, Mass transfer and the fluidized bed inien- sification ofreverse osmosis. Desalination, 62 ( 1987) 2 1 l-220.

[25] D. Bhattacharyya. S.L. Back and R.I. Kermode, Prediction of concentration polarization and flux behavior in reverse osmosis by numerical ‘analysis. J. Membr. Sci., 48 (1990) 23 1.

mathematical modeling of reverse osmosis systems

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