Analysis and optimization of a reverse osmosis water purification system Part I. Process analysis and simulation

Desulina~ion - Elrvicr

ANALYSIS

AND

OPTIMIZATION

I. PROCESS

L. 7. FAN.

ANALYSIS

C. Y. CHENG.

The Inrritule fir

Amsterdam

OF

- Printed in The Netherlands

A REVERSE

OSMOStS

WATER

SYSTEM

PURIFICATION PART

Publishing Company,

AND SIMULATION

L. Y. S. HO. C. L. HWANG

S~~enrs Dt*.s&n and Opfitnizarion,

AS:~ L. E. ERICKSON

Kansa, Srare Clnilersiry, dftmlruttan, Kamas

(lJX.4.) (Rwciwd

April 7, 196Sf

A mathematical model of a reverse osmosis water purification system that can be used in process optimization studies is devetoped. A boundary layer flow model is used to relate water production rate to the operating pressure, Reynolds number, and membrane area. Cost equations that relate the capital and operating costs to the design variables are a!so developed. These relations are then used in economic analysis of several rebersc osmosis systems. The results of computer simulation of single s1:qe processes we presented. The model is believed to be accurate enough to describe the process and yet simple enough to be used in the simulation. design. and optimization of the proces?.

A J-L B,. B,, B,. B,. B,. and B,

-

cross-section area normal separator unit, ft’

-

constants defined in section 4-1

to the streamline

of any membrane

c, C FB

-

c, G

-

3.05 x IO” Krf/Sc”3 0, constant electrical-power cost. S/‘kW-hr cost of the feed brine per unit watcc production unit pretreatment cost of feed brine unit cost of the material for constructing membrane

cr

-

unit, S/Ib, total cost per pound of fresh water produced

C Pf

-

flow

c Pi1

-

c

PR

-

PT

-

circulation pump installed cost, $/kW turbine installed cost, S/kW

-

diameter of the tubes in the membrane

C”

c

d

separator

%/lb=

work exchanger installed cost, II;/kW high pressure pump installed cost, S/kW

separator

unit

Desafinarion, 5 (1968) 237-265

238

iD Pff5n

L. T. FAN Pl

-

cost of cost of cosf of capital

-

molecular diffusion coetticient of sait, cm’/sec pump work of high pressure pump at fgth stage energy recoxry from the blowdown turbine at the end prcce9.

F,

-

t J=_A

-

-

pump work of circuhtlon pump alt rlth stage energy requirement for flow-work exchanger knning friction tactor vo!umetric flus of water through the membrane. ft3/ft’-hr volumetric ffow rate of fresh water product through the membrane of frth stage. ft3/ft’-hr volumetric tlovv rate of salt component through the membrane of jrth stage, ft3;ft’-hr pumps membrane constant, ft3;ft’-hr-psi length total number of tubes within each me,nbranc separator unit recycle pump at 11th stage total number of stages in the sequence of the process pressure pressure within membrane separator chamber of 11th stage, psi atmosphere presstire, id.7 psi P” - PO = pressure difference across the membrane at nth

-

stage, psi moss flow rate of brine

(DPR!”

tD,r), DPF

DfJ fE,), t Ed,

(Es)n &

1‘ F

-. -

%

the the the cost

al.

high pressure pump at the 11th stage. S circulation pump at the rlth st3gc. S turbine at the nth stage, S of a flow-work e&changer, S!lb,_

soktion

discharged

of the

from the ttth stage.

Ib,!hr 40

R

& Re,

-

G-

-

Li fi

-

s. S” SC

f.

mass flow rate of feed saline water, IbJhr radius of the tubes within the membrane separator unit recycle ratio of nth stage average Reynolds number at rrth stage. membrane area of any membrane separator unit, ft’ Schmidt number turbine maximum veiocity within the membrane separator unit mean velocity within the membrane separator unit = 03 U (for turbulent fiow)

= +r/

-

mean velocity within, the membrane separator chamber of nth stage

-

friction

velocity Desalinarion, 5 (1968)

237-265

OPT!ZfIZATION

OF RO WATER

-

-

-

I.

I&TROD1

mass total Ib,/hr mass stage mass

PURIFIChTIOS

SYSTEM

1. PROCESS

ANALYSIS

239

flow rate of fresh water produced from trth stage, Ib,Jhr mass flow rate of fresh water produced from the system,

of the sheil-and-tube membrane separator unit of rzth lb, fraction of salt concentration of feed saline water nveraee mass fraction of salt concentration within membrane separator chamber of the rrth stage average mass fraction of salt concentration at the nembrane surface mass fraction of sah concentration of reject and brine distance normal to membrane boundary reciprocal of _\;r

momentum transport boundary-layer thickness mass transport boundary-layer thickness liinsmatic viscosity = ii/~. sq. cmkec viscosity of brine solutton density of brine solution. Ib,lft3 density of material of construction. Ib,Tt3 osmotic pressure of the brine solution at the membrane surface of the rrth stage shear stress at the wall friction !oss factor motor efftciency pump ethcienc_v turbine efftciency capitalization charge of initial cost per hour in stream, S!hr allowable stress of the material of construction, psi inefficiency of flow work exchanger

ICTIOS

Because of its simplicity and the potential low energy requirement, reverse osmosis has been drawing widespread attention_ Current research activities have been directed toward the development of membranes which combine good salt rejection, high water transmission at reasonable pressure, and long membrane techniques (2. 4, 5), the life (f-4). The improvemen t of membrane fabrication improvement of cell design and construction (5, 6). and the mathematical and experimental investigation of salt build-up on the membrane surface (7-23) have However. only rather limited attention also received considerable attention. Desalinution,

5 (1965) 237-265

240

L. T. FAX

has been directed to the anal\sis

et al.

and optimization

of the entire system (7. 14). to model and optimize an entire reverse osmosis system. A boundary layer tlow model is used to relate water production rrtte to the operating pressure. Reynolds number. and membrane area. Cost equations that relate the capital and operating costs to the design variabIes are atso developed. These relationsare then used in economic optimization studies of several reverse osmosis systems (17. IS!?). One sjstrm that is considered in this study utIlizcs multistage operation with recycle 6thin each *:tage. A single sta.?.e operation with recjcie has previously been suggested and studied (7. I-#). The recycle stream has been prokidcd to maintain a high flow rate in the cell for reducing the eiiect of salt build-up on the membrane surface. Even though recycling is thermodynamically ineffkient because of the mixing of solutions of different compositions. it is economically desirable The present study utilizes system analysis procedures

e\en in single stage systems because of reduction of the pretreatment cost. misin_c described above increases the aterage concentration and therefore

The

the osmotic pressure. The inefkienc~ can be reduced by utilizing multistage operation. A parallel situation arises in multistage flash distillation_ Recently. a multieffect multistage flash distillation process has been introduced to replace the socalled single-effect multistage flash distillation process (22. 23. 24. 2.5). .4nother reverse osmosis system that is considered in this study utilizes a “flow-work exchanger” to pressurize the feed solution and depressurize the reject brine. The “flow-work exchanger” has been recently introduced in connection with a freezing process based on the high pressure inversion of the order of melting points (15. 16). As pointed out there (16). both the reduction of the magnitude of the shaft work and the improvement of pumping effkiency are important factors in the reduction of energy loss in a pumping and depressurization operation. For a simultaneous pressurization of a condensed fluid and depressurization of another condensed fluid, shaft work for the operations can be greatly reduced by the tlowwork exchanger. The technical and economic feasibiiit> of adopting a flow++& eschaneer in a reverse osmosis process is therefore a part of this study. The mathematical modeling and economic analysis are based on prior work by the General Dynamic Corporation on a single stage reverse osmosis process with tube-type cell of 10 mgpd plant capacity (7). The model is based on boundary la>er theory and one dimensional diffusion theory. The performrtnce equations for the optimization studies have been developed by extending both the results of the modeling study and the cost equations developed by the General Dynamic Corporation. Performance equations developed in the present work have been employed in the subsequent optimi~tion study (17. IS). This study illustrates some of the techniques useful in formuhtting a model of a complicated practical process system. The model is believed to be accurate enough to describe the process, yet simple enough to be used in the simulation. design, and optimization of the process. Dcsulitrarion. 5 (1968)

237-265

Oi”TIX~IZATIOX

OF RO WATER

2. I~ESCRIPTIOS

stage

hS[>

In this section. reverse osmosis

osmosis \arishfes

system

,W..\LYSIS

PLlRlFlCATlOS

OF Till:

SYSTE\I

RtVERSE

1. PROCt:SS ASXLYSIS

OSblOSlS SYSTEhl

the description and analysis of the single system are presented_ Flow relationships

are derived

and

‘41

the quantitative

reIntions

stage and multiin the reverse

among

the operatin_g

are also derived.

Lonsdaie er al. hn\e made a process zlnalysk and economic study for a single stage tube type reverse osmosis system (7). Fig. I is ;I rekiscd diagram of the se.1 water conversion system discussed in tlieir report. Referring to the figure, sea water is brought through it prefilter and pump to the operating pressure by zt high pressure pump. This pressurized sea water is assumed to be mised nt the operating pressure with rtxirculating brine in brine reservoir. The brine is cctrried through the membrane separator unit by means of ;i separate circulation pump. and a portion of the brine is continuously blown down through a recovery turbine to waste. Product water is rccosered at atmospheric pressure and stored in the fresh water rcbtrvoir. Fig. Z is ;I simphfied diagram of Fig. I obtained by eliminating the prefiltcr. brine reservoir and fresh water reservoir. Chis figure is more convenient

Fig. 1.

Schematic diagram of revcrsc osmosis se water comersion

system.

Dmalitmiotr.

5 (I 968) 237-265

L. T. FAX ~1

Fig. 2.

Con%eRtioR:tt

to uw than the original

single

St3@2

a/.

operatian with recycfe.

figure and will be used in the discussion

of multistage

operation_ Recently a Wow-work exchanger” has been introduced to improve the thermodynamic efficiency of the simultaneous pressurization of one condensed fluid and deprcszurizntion of another condensed fluid (16). A single stage reverse osmosis system which adopts a flow-work exchangr in the simultaneous pressurization of the feed brine and the depressurization of the reject brine is illustrated schematically in Fig. 3. it shows that the reject brine and an equivalent volume of feed brine exchange flow work. and a hi_gh pressure pump J, is used to pressurize the remaining part of the feed brine. The basic relations which quantitatively describe the performance of a shell and tube type reverse osmosis system include (I) the flux equation which relates the fresh water production rate to the pressure differential across the membrane and the Reynolds number of the brine flowing in the tubes; (2) an equation which relates the reject brine concectration to the inlet brine concentration and operating variables such as the pressure diffcrentiat across the membrane, Re_unolds number.

brine Fig. 3. A single stage operation

with rccyclc adopting

a ffow-work e~changcr. Resaharion, 5 ( 1968) 237-265

OPTIXllZATIOS

OF RO WATER

PURIFICATIOX

SYSTEM

I. PROCFSS

AXALYSIS

243

and membrane area to feed ratio: and (3) an equation relating the Reynolds number. recycle ratio. and system variables such as the tube diameter and number of tubes in each cell.

The derivation of the water flux equation is complicated by the salt build up, or concentration polarization effect, at the membrane surface. Sherwood cv of. (fO) and Gill and Tien (II. 12, f3) have made a theoretical analysis of the concentration build-up at the membrane surface. Lonsdale and Merten (7. 8. 9) have obtained a water flus equation. and their equation has been fitted to the data of Loeb and ISourirajan (7). StartinS with assumptions si,nii:lr to those made by Lonsdaie et d.. a detailed derivation of the tluh equation is presented here. Main

flow >

1 Bulk brine i solution

veiocity proftfe

~onc~fffrofron profile 4

~

Fresh

‘.~. cont.

of salt

water

outlet

:r/

Velocitv of fluid

Fig. 4. Velocity and salt conccnlraIion

gradients in boundary laycradjacenr to a mcmbranc.

in a reverse osmosis tube, water passing through the membrane is suppited to the membrane boundary by bulk flow of solution normal to the membrane. Salt is carried along with the water. If a steady state is to be maintained without an accumulation of the salt on the membrane, this salt must diffuse back into the main bulk solution. A salt concentration gradient is established near the membrane boundary such that the net salt dux through the membrane is zero. The situation near the membrane boundary is shown in Fig. 4. The volumetric flux of water through a membrane of constant permeability has been reported by Mcrten (8) to be

DesdittaGttr.

5 ( 1968) X7-265

L. T. FAN et at,

244

where AP is the pressure difference across the membrane. and II, is the osmotic pressure of the brine solution at the membrane surface. The relation between the osmotic pressure and the brine concentration was investigated t&fal. (9). These investigators found that the expression 12,loo_i’,.

I-I, =

where _CSis the mass data of Tribus L’Ial. A salt mater’al plane parallel to the -0,

by Merten

g

=

F

? where _r is the in the solution. with the water is the amount If.? e I,

(21

fraction of salt at the membrane surface. fits the experimcntaf (19). balance inside the concentration Fw.mbrane gives

boundary

layer adjacent

to

__:L_..

(3)

1 - .c-

distance from the membrane surface and _Cis the fraction of salt The right hand side of Eq. 3, represents the amount of salt carried to the membrane surface by the bulk flow. and the left hand side of salt that diffuses back to the bulk solution.

3 = _q 1 - 3

(4

Eq. 3 can be simplified to dC - _- = -_F - d_r. _r 0, If we assume D, to be independent of _Cand integrate this equation across the boundary layer from 0 to.b,, the thickness of the concentration boundary layer, we obtain

(6) In the absence

of chemical

reaction

the ratio

between

the concentration

boundary-layer thickness, 6,, and the momentum transport boundary-layer thickness, 6, has been shown to be a constant Schmidt number: that is. (.&I),

and is dependent

6, = &(sc)-+,

(7)

where SC is the Schmidt number

S~=-L=:-E-_ Da

only on the value of the

PD,

which is defined as

O-9 Dtwdinarion, 5 (1968) 237-26.5

OPTlhlIZATlON

OF RO WATER PURIFICATION

The thickness is given as (21)

of the lnminar

where v is the kinematic

viscosity,

SYSTEM 1. PROCESS ANALYSIS

sub-layer

for turbulent

and C. is the friction

245

flow through

pipes

velocity defined as

L’. = ;rO/P

(10)

in which T,, is the shear stress at the wali. For turbulent flow through shear stress, me, can be calculated from the following equation (I). T, = o.zK5

pipes, the

pU:~J’(VIR)“.

(11)

where p is the density of the fluid and (/the maximum velocity which is proportional to the mean veIocity ti. The ratio C/C.!equals 0.8 for turbulent flow through pipes. R is the radius of the pipe and d = 2R denotes the diameter of the pipe. introducing d into Eq_ I1 and then substituting Eqs. 10 and 11 into Eq. 9 yields ‘j

25.2

=

- --.-

n.-...

25.2 tf =2

----.--

(tidpjjI)7f”

Re’f”

where Re is the Reynolds number. Substituting Eq. 12 into Eq. 7. we obtain thickness of the concentration boundary layer as

the

(13) Substituting In

Eq. 13 into Eq. 6. we obtain

F .5 = ._. ____

_e

Da

25.2 d

.__3.__ _

_

(Scl” (Re)f’B

By using an approximation

in Eq. 14 and solving for F, we obtain F =

(SC)“~ ( Re)7/8 ($- - I) -._>_

D

__e__

(16)

The approximation given by Eq. 15 is fairly good if the vahte of ~$52 is below t -4. However, if the value of _&f-Cis above 1.4, say, in the range of i -6 - 2.2 (see (28)), the approximation has to be modified to In+-_0.7(+-

-

1) Dcsahation. 5 (1968) 237-26.5

L. I-. FAN et al.

246

factor. This modification will not change the procedures given here and the general conclusion of this work. Substituting Eqs. 16 and 2 into Eq. 1. and solving for .Q yield where 0.7 is a correction

+

KAP

. .._ __

.k?!-

(Re)“” __. ___._. .._._-_-_.. _-. _...-__.

(SC)"~

25.2 i;?,lW

_c

f\

._._ (SC)’

Substituring

(Re)’

Eq. ‘7 into Eq. 16. we obtain

per unit arca of membrane

f

’

= _._ _

(17)

li .

the volumetric

tlow rate

of fresh wdtcr

as

K(AP - _12.1OO.C-) . ____ _... __. _ 25.2 11

..-

OC

(1% H here

(30) Eqs. 19 and 20 are used to establish the performance equation which relates the exit brine concentration to the design and operating variables. The use of the results by Sherwood and Brian (fO). and Gil1 and Tien (II, I-‘, 13) leads to more eIaborate flux equations than equations (19) and (20). Other performance equations which relate the brine concentration and Reynolds number to the decision variables are derived later in connection with a multistage 2-2.

operation.

Multistage

rmmr

omwsis

s_wtwi

One of the designs proposed in this study would consist of several reverse osmosis shell and tube separator units connected in series. Fig. 5 illustrates a general multistage operation using a conventional way of pressurization and depressurization and Fig. 6 illustrates a general multistage operation adopting a ffow-work exchanger. This multistage sequential system with brine recycle at each stage is proposed in order to reduce thecost associated with salt build-upat the membrane surface. As mentioned in the introductory section. it is undesirable thermodynamically to mix a recycle stream of high salt concentration with another stream of a much Iower salt concentration. The multistage sequential system would minimize this thermodynamic undesirability. Desalination,

5 ( 1968) 237-265

OPTIMIZATIOS

I?

OF RO \\‘ATER

t’URIFICATIOS

(Re), lS,QJ

III

r

I

Fig. 5.

1. PROCESS

P2 (Re),(S&J

II I

I II (as

3

I

2

I

! c

4

W,

w2

%

A gerrera’ mulrlsl~gc

247

ASALYSIS

Px ( Re), (S&J

(X,)0 90

SYSTEM

%

1

opration

with a convcntwn~l

way cqf prcssuri7ation

nnd

dcprcssuri7arwn.

The possibility of including a pump between each of the stages is suggested in the proposed design because the energy requirements will probably be less if the pressure is incrertscd from stngc to stage to compensate for the increase in osmotic pressure due to the increase in salt concentration: that is, the energy costs ~41 he minimized if the plant is operated as near to an ideaI reverse osmosis process 3s possible. Howc\cr. each of these additional pumps wilt also have a capital cost asbociatcd Hith it. This added cost must be economically justified. An economic balance bettbeen the energy and capitol costs must be achieved by a proper selection of those variables which affect the water production costs. An optimization study must be carried out in order to determine the optimal design und the minimum cost of product water. Fig. 7 is a block diapram of the mathematical model of n three-stage reverse osmosis system. It shows that the performance of the system is controlled by (I) the pressure differential across the membrane in each stage. vi=. AP,. AP2, and AP,. in the first. second, and third stases respectively. (2) the Reynolds number in each stage. rk (Re),. (Re),, and (Rcf3 in the first, second. and third stages

A

*,

~3,

Mu

q, R2

M r

1

A

7

ot

.

Jz

WI

w2

Flow -

i 6

I

.

I

T

Pt

WI4

1

%a

s.R

JI

I

,

,,

Ph

s

PI I

qrr-,%a

24

I Work

exchanger

QN _

Feed trine Fig. 6. A general multistage operation

\\ith a tlow-Hark exchnngp. &sulinurion.

5 (1968; 237-265

t.

WI

FAN ct

al.

W3

w2

Fig. 7. Mathematical

T.

model of 4 multistage

rc\crsc

osmosis system.

and (3) the ratio of membrane area to feed in each stage, r*k. S,/q,,, and S,/q, in the first. second, and third stages respectively. Figs. S, 9. and 10 illustrate mcdifications of the general multistage process. Fig 8 illustrates a multistage operation with stagewise pressurization but without recycle. .I,, Jz, and J, are high pressure pumps for the stepwise pressurization and 7 is the btow-down turbine. The number of tubes in each sta_ee is adjusted to

respectively.

S,!q,

obtain

the

mass

rate

WI

in minjmizin~

salt

build-up.

Fig.

9 iIlustrates

a

W3

w2

FIS. 8. Multistage

. W,

required

operation

with stepwisc prtxsurimtion

t

1

w2

w3

without

rtxyclc.

Fig. 9. Mu!tista@z operation with recycle but without stepwise pressurization. multistage operation with recycle in each stage but without intermediate pressurization. Fig. IO illustrates a muttistage operation without step-wise pressurization and recycle in each stage. The membrane area and tube number are varied from one stage to another. The flux equation. the equation for estimating brine concentration. and the relation between the Reynolds number and recycle ratio in each stage are presented in the next three sections.

WI

W3

W?

Fig. IO. Multistage

operation

without

stcpwisc pressurization

and stagewise reqclc.

Desalination,

5 (I 968) 237-265

OPTI~IIZATION

OF RO WATER

PURIFICATIOB

Flux equation at the rlth stcge In order to simplify the analysk then

be obtained

from

Eqs.

19 and

SYSTEM 1. PROCW

we assume

ANALYSIS

i’, = _I-,. The flux equation

249

can

20 as

Referring to Fig. 5. the concentration of the brine leaving eacn stage can be related LO the concentration of the brine entering thst stage aud rhe control variables. By detining (1”. IV,, and S, respectively as thEmass rate of brine leaving the nth stage. rate of fresh water production in the nth stage, and membrane area in the 11th stage, the following material balance equations can be obtained.

and

From

Eqs. 22 and

_y,

=

_.._ xi)

Substituting

23. we obtain so

A-,-

the following

relation

for .I-,.

,

I ..__._. ._... -._. ..._. s,F” p (S,/q,)

(24)

1

Eq. 21 into

_y, = -._.--so

-

Eq. 24 yields

so s, _ * _.- _._..- -- -__. ..- .^_ -_. -.. -.-_ KJAP,, - 12.100s,~ x-“-r ._- __-_ - -.-- --._ P I t C,,(sJ(Re),,““)

The above equation relates s,, to _v,,_, and also to the control variables (Re),, LIP,,. and SJq,. Eq. 25 is one of &he performance equations. By defining a reciprocal composition. _,;1 = I/s,, Eq. 25 may be rewritten in a convenient form as AP, y,, -

12.IOO

(26) ?, + _c!L_ (Re,)“’ The performance

equation

Eqs. 25 and 26 by replacing s,,

and

for single

stage

_u,_ 1T S,,fqo and

operation

can be obtained

AP, respectively

from

by _Td. xe, Sfgo,

AP.

Desulirratiotr, 5 ( 1968)237-265

250

L.

T.

FAX

et al.

Referring to Fig. 5. the amount of fluid flowing into the membrane tubes at the rith stage is equal to q._ 1 (I + R,). where R, is the recycle ratio. By delining the recycle ratio R,, as the ratio of amount of brine recycled in the lath stage to the amount

of brine entering

there are r& membrane equation.

the n:h stage from the preceding

stage. and assuming

that

tubes in parallel in the rzth stage. we can obtain the foilowing

(In-., (I + K”) = I~I, (lkl’j4)

J+

(271

where (I is the dia.,neter of the tube and ri, is the average flow velocity in the tube. The Reynolds number can be related to the recycle ratio by solving Eq. 27 for the average velocity li and substituting this result into the definition of the Reynolds number. Rp_

=

d l&p _._

__

=

Cl Substituting

Re, =

cf~ 4y,_ , ( I_-_.;+ R,) -._.

.__.

P

q,_ L = (qosofs,_

,m,n,t-

=i:

“Iqe-t (1 + 4) - -- -- --* p?zI, I-Id

(28)

,) into Eq. ZS, we abtain

._._!fl?.% . .._ (I + R,). /U?1,s, _ , dll

(2%

The relation between the Reynolds number Re and the recycle ratio in a one-stage operation can be obtained from Eq. 29 as (30) 3.

ECOSOUIC

ANALYSIS

OF REVERSE

OSMOStS SYSTEM

The cost of water production in a desalination plant depends primarily on the capital cost of the plant and the cost of operating it. To select the optimum values of the design variables, it is necessary to have mathematical relations that relate the design variables to the unit cost of water production. In the present study. reverse osmosis systems are classified into the following types: 1. Single stase operation without the use of a flow-work exchanger. 2. Single stage operation with the use of a flow-work exchanger. 3. Multistage operation without the use of a flow-work exchanger. 4. Multistage operation with the use of a flow-work exchanger. In the first and third classes presszlrization of feed brine is accomplished by a high pressure pump, and depressurization of reject brine is accomplished by a turbine. In the second and fourth classes, a flow-work exchanger is used for the simultaneous depressurization of reject brine and the pressurization ofan equivalent volume of feed brine. Desalination.5 (1%8) 237-265

OPTIMIZ4TIOX

OF RO WATER

PURIFICATIOh

SYSTEM

I. PROCIISS

ANALYSIS

251

An economic analysis and an optimization study are made for each of the four reverse osmosis proccsscs described above. Much of the desired information can be obtained by comparing the results of the optimization studies. By comparing class 1 with class 3 and clars Z with class 4. WC can deduce the quantitative advantages obtainable by multistage operations. By comparing class 1 with class 2 and class 3 with class -L we can deduce the quantitative advantages obtainable due to the adoption of a flow-work exchanger. Furthermore, by comparing class I with class 4. we can obtain the quantitative adbantagc due to both multistage operation and the adoption of a flow-\\ork exchanger. In section 3-1. a cost analysis is made for a $!nerai multistage reverse osmosis process which does not use a flow work exchanger and in section 3-2. a similar analysis is made for a general multistage process which uses a flow-work exchanger. The cost equations derived in these sections are used in the subsequent optimization

studies

presented

in (17.

IS). The cost equations

system. ~2. classes 1 and 2. can easily be obtamed stages of the multistae sy*tcm to one.

for the single stage

by reducing

the number

of

operation of the system can be described as follows: Referring to Fig. 5, feed brine. cfo. is pressurized by J, and is introduced with the recycled brine. qoR,, into the first stage where fresh water is produced at the rate of I!-,. A part of the discharge brine. 4,. is pressurized by J1 and fed to the second stasc while the other part. yoR,. is recycled through recycle pump. M,. At the Nth stage. J, is used to pressurize a quantity of brine, f&E. from P,_I to P,; a quantity of fresh witer. ti;,. is produced: a quantity of brine qn-, R,, is recycled; and the remainder of the brine, q,,, is fed to the (n-k I)th stage at concenxx tration s,. At the last stage. a quantity of reject brine, qa. at concentration is discharged through turbine J.,_ The capita1 costs which need to be considered are a. The costs of the high pressure pumps. J,, J,, . .._ J.,, b. The cost of the turbine, T,. c. The costs of the recirculation pumps, M,, hfL, __., M,., d. The cost of the membrane separator units. The operating costs which WC Affected by changes in the design variables are a. The cost of the feed brine. yO. (this includes the cost for pumping the sea water to the plant site and pretreatment costs), b. The energy costs for the high Fressure pumps, J,, J,. ___.Js. c. The cost gain due to energy recovery at reject-brine turbine, TaV, d. The energy costs for the recycle pumps. M,, M2, _._, J%fx The cost to be minimized in the optimization study is the cost per unit quantity of fresh water produced from the entire system. Therefore, the operating costs as well as the capital costs should eventually be expressed on this basis. The

Dmalination.5 (1968) 237-265

252

L. I-. FAN t‘I ai.

For the operating costs, we may find the desired costs by first finding the individual costs on a one hour basis and then dividing them by the hourly water production rate from the entire system. rig’. H;. The capital costs also have to be allocated to a unit quantity of fresh water produced. Following the procedures recommended by OSW f.o.b. costs arc multiplied by 1.3 to obtain installed costs. Interest on the depreciated plant investment is assumed to be 4%. Depreciation is taken over 20 years. These give a total charge of 7.3“< per year for amortization on the initial investment. When 2 O! is added for tases and insurance. and when a load factor of 330-on-stream days per year is used. these assumptions give a capitalization charge. I&.of 1 I .9f x lo-* of the initia1 cost per hour on stream time. In other words I i.9J x 10V6 x ~ln~tial cost of the piant) should be charged to each hour of operation. Therefore the capital cost allocated to a unit amount of fresh water produced can be catcufated by t(I (%!t$.cost~ f where 4 is the capitalization

charge.

Cost of rk feed brine per unit waler pi-ud~irtiun The cost of the feed brine per unit water production C FB

=-

c,

can be given by (31)

~~~~~~~)

where C, is the unit cost of feed brine, Since

4OSO = y,ss qJtt;

required

= (yo -

i$j)s,.

in the above expression

can be given as (32)

Substituting

&ergs

this expression

costs jiw

into Eq. 31 gives

the AigIr prtwttrr

puirtps per mil

i~tet- prodticiion

In a normal operation, it is expected that the operating pressure should increase from the (n- I)th stase to the trth stage, Le., P, shou.d be greater than P r. However. during the search for the optimum condition, we may encounter siktions where P,, becomes less than P,_ *. If such a situation happens, .i, is considered as a turbine. For a normal operation, .I, is used to increase the pressure of the brine from Desalination,

5 (1968)

237-265

OF RO WATER

OPTlhllZATlON

P n-1

to Pm_ Therefore.

PlJRlFlCATION

SYSTEM

the power requirement,

I. l’ROCEX3

ANALYSIS

(E,),, psi-ft3/hr,

253

can be given by

where of=, qe, and qr are the motor efficiency. pump efficiency, and friction ioss factor respectively. and (in_ I is the mass flow rate of brine leaving the (fz- 1)th stage and therefore, entering the r;th stage. From the material balance relations. we obtain the following relation between flow rate and brine concentration. (I IV-1_

=

__

_.--so -..-

---‘----.

-r, - , [ 1 - (so/s,)]

‘Yf

Thus. the energy requirement of the brine concentration as

per unit water production

(Es), _ 1 +~,AP,,-AP,_, _ _.___.______. “5%

‘I*

P

VP

can be given in terms

-x0 _.- _-_.. .. ...-......._xn-,[t

-

(-Q/X,

(35)

,f

The energy

costs pet unit of water production can be obtained by simply plying the above equation by the unit power cost, C, S/kW-hr. and the conversion factor. When Pm is less than Pm_,, J. is considered to be a turbine. Under condition. the energy supplied, (E,),, psi-ft’ihr, becomes a negative value can be represented by

m&iproper such a which

(36) where qt is the efficiency of the turbine. The energy recovered production is obtained by dividing Eq. 36 by It>, that is,

(Et), = _...--..

Yf

-‘Im’lt

t’

.y,- II - C~ol.%U

(37)

1

In the subsequent optimization whether the sign of AP,-AP,_ Energy

study. either Eq. 35 or Eq. 37 is used according or negative.

cost recowrahk front the turbine, TV T>, is used to depressurize the reject

the energy recoverable

to

t is positive

The turbine,

Therefore

per unit water

per unit time, psi-ft3/hr,

brine from P,,. to PO. is given by (38)

iJesalinatiun. 5 (1968) 237-265

and the

energy

recorerabic

per unit water production

is given by

OPCIMIZATION

OF RO WATER

PURIFICATION

(DP,,), = h,,

tY”- I* (Ed,, P”),

(DpR)i,

(Yn-

=

fPR

I.

4,.

(‘53)~

SYSTEM

1. PROCESS

ANALYSIS

255

pm;.

and

of the unavailability of these cost relations, to be proportional to the horsepower ra:ings. Thus, Because

(DPA

=

CPI,

(El),.

(D,,),

=

cm

(EL\L.

the costs are assumed

and

l? PI-

=

CPT EI.

The capital costs alfocnted to a unit quantity the high pressure pump, J,. the recirculation pump. respcctikeiy

where (E,),/

It;.

(E,),’

It>

of fresh water produced for M,,, and the turbine, T>, are

and El/ II-, arc given by Eqs. 37.44, and 39 respectively.

Capitul costs for the rmmhrune separcrtor uttit The cost equation described by Lonsdale. et al. (7) is used in this work. The weight of the membrane separator unit required tit the trth stage is given by

Lonsdaie er al. (7) assumed that the capital cost for the separator is proportional to its weight ( CI’s),. Thus. the capital cost of the separator which is allocated to a unit quantity of water production is given by (46) where C, is the cost of the separator

per unit weight, and ( FF’&fiF’~ can be given by

Desalination.5 I 1968) 237-265

L. T.

256 By substituting

Eq. 32 into Eq. 37 and using the relation

pF,S,

FAN

et 01.

= IV,. we obtain

H’ater productiorz cost ner potaxf of fre.sh ISater pradwed The cost of producing one pound of fresh water, C,. can now be obtained

as

3-Z. C5st ana&sis of a tmdtistuge reverse osmosis system wirh the me of a fiow-wwk exchatzger As stated previously. Fig_ 6 illustrates a multistage reverse osmosis system with a flow-work exchanger. In operating the system a part of the feed. (qO - qa) = W,. is pressurized by a high pressure pump from PO to PI and the rest of the feed, qs, is pressurized by a fiow-work exchanger which utilizes the high pressure reject brine. The reject brine is depressurized from P,, to P, in the turbine r.% and is depressurized from P, to PO tn the flow-work exchanger. The capital costs which need to be considered in an optimization study of a reverse osmosis system with a flow work exchanger include a. The costs of the high pressure pumps. J ,. Jz, .._. I,. b. The costs of the recirculation pumps. M,. M,, _._, Ms. c. The costs of the membrane separators, d. The cost of the flow-work exchanger, e. The cost of the turbine, T,_ The costs of the first high pressure pump, J,, the turbine, Ts, and the flowwork exchanger have to be derived. The other cost items are identical to those obtained in section 3-1 for the class 3 system. The operating costs which are affected by the design variables are: a. The cost of the brine feed, q,,. b. The cost which can be recovered through the depressurization of reject brine from pressure Ps to pressure P, in the turbine T,, c. The energy costs for operating the high pressure pumps, J,, f2, . . . . J&, d. The energy costs for operating the recirculation pumps, M,, M1. _.., M,, e. The energy costs for operating the flow-work exchanger_ Energy costs for the high pressure pumps per unit water productiotl be the energy requirements in operating the high pressure

Let (E,),

pumps.

Desalination, 5 (1968) 237-265

OPTlXllZATlOS

OF

RO

WATER

PURIFICATION

SYSTEM

I.

PROCESS

257

ANALYSIS

(E,), is identical for class 3 and class 4, for I: # 1. Therefore, only (E,), needs to be derived in this section. In the high pressure pump, i,, a part of the feed brine equal to qO -_Y_, = IVJ is pressurized from P, to P,. Therefore.

and

(Eat, -- ^-.- =

AP,_.-...1 ._-+rt/ .-. . .._. - _

rr>

.’

WQ

Tim)Ip

Expressions for (E,),Jllj on the sign of AP.-AP,_ plying these expressions

for !I # I, are given by either Eq. 35 or 37 depending by multiby the energy cost factor C,. ,. The cost of the energy can now be obtained

Energy cost rrmm*rahlc from the turbine, TV The turbine. T,, is used to depressurize the reject brine from Ps to P,. Therefore. the energy recoverable per unit time is given by E2 =

?J,?J*

(1 -

,lr)

ps PI - ..- ----

q\-

P

which leads to

E2 ._-.. =

IV,

calculated

(1 -

,,/)

Py P, _“._.- --.. P

Ener.y The

)Im’lr

cost for operating

_;-:YJ.\_‘ .\

rlrc rqde

-0 pump at the t&t stage

(E&j

CV,

cost of the energy required to operate each of the recycle pumps can be by using Eq. 44 and :he energy cost factor C,..

Ener.qv cost for 0perafin.q

the jlow-lwrk

esclratrger

The flow-work exchanger exchanges flow work between a quantity, qs, of the reject brine and the feed brine from pressure 9, to pressure P,. Therefore, the flow work, psi-ft3/hr, exchanged per hour is

Letting the inefficiency

factor be <, the energy requirement

and the energy requirement

per unit of product

EJ is given by

water is fksufimm-on. 5 (I-)

237-265

25s

L. T. FAN cl

E, _.._ =

‘Yf

Cupid

= API __._.:‘_o. p I,, - _i; -

c,

__

_.

(52)

__

for the high pwsrure

costs

ai.

ptct~rp.~ recircularioon pumps.

a&

mrbitie

The capita1 CCISSallocated to a unit quantity of fresh water production for the high pressure pump. recircukttion pump, and turbine are respectively given by

Capitnl

cost

yf tlw_#hw ork esckarrgcr

The capital cost of a flow-work rate and pressure. D PF

=

/pt.

(Y.v

=

CP,

DpF can be related

to the flow

P.d-

In the present study, Dpt. is assumed work eschanged, that is.

DP,

exchanger

to be proportional

to the amount

of fiow-

L

constant. where C,, is a proportionality quantity of fresh water produced is thus

The capital

cost allocated

to a unit

where E,,‘W, is given by Eq. 52. f f ‘arer productim

~0.~~per pcmtd

of*_ fredz

warm produced

The cost for producing one pound of fresh water, CP can be obtained by

4.

COMPUTER

SIMULATIOX

OF THE SKGLE

STAGE PROCESS

A computer simuiation study is very helpful in showing the performance of a system when the independent variables of the system are systematically varied. Finding the optimum by means of the simulation alone becomes extremely difficult,

however, when the number of independent variables exceeds three. In the single stage reverse

osmosis

system,

the degree

of freedom

of the

Drsolinarion. 5 ( 1968) 237-265

OPTI~lIZATION

OF RO WATER

PURIFICATION

SYSTEM

1. PROCESS

ANALYSIS

259

system is three. If the independent batiables chosen are the reject brine concenthe membrane area to feed ratio. S/qO. and the Reynolds number, Re, tration. sJ. the pressure differential becomes a dependent variable. However. since the overall effect of the

Reynolds number on water production cost is rather minor. only a simulation study changing two independent variables witf be carried out here. It will be shown later that a higher Reynolds number is favorable from the stnndpoint of eliminating

concentration polarization at the membrane surf;,t:e but it also gives rise to a higher pum,pmg cost_ The net effect of the Reynolds number on water production cost is minor. provided that the tlow is maintamed within the turbulent region. With this consideration we assign a filed value. 10,000, for the Reynolds number. In this section the results of simulation studies of reverse osmosis systems with and without a flow-worh exchanger are reported_ In each single stage system considered, the independent variables arc the reject brine concentration. sJ, and the membrane area to feed ratio. S,‘q,. -f- 1. T/w r~i;rfprttt~ttt~~~ rqttuticms fhr rltc sittglu srttgc pro<;ess a) Pressure differential, AP. The pressure differential From Eq. 26, we obtain

has to be calculated

from the independent

variables.

+ 12,100.

(54)

b) Cost equation for a single stage reverse osmosis process without exchanger. By substituting Eqs. 33. 37, 39, -U. and 4Y into Eq. 49, setting rearranging

the resulting

cost

be represented

C, can

Cr = B, Ap

+

B

5

_ ._

sd

+

-+. .’ -k yo S‘J C

__ 9

so

it can

be shown

the water

IV =

1, 2nd

production

- [B2 ( Re)’ * + B3 AP + B,(AP)” s(-J

-!-?_.

_vd -

‘3

(55)

so

where the constants, B,, &. BJ. B,. and B,. C,, C,. 11~~q,, and &IV Jm- @% c,,,, c,,, CPR. section 3-i. These constants will be summarized c) Cost equation for a single stage reverse exchanger. By substituting Eqs. 33, 50. 37. 51_ 44.48. and rearranging the resulting expression it can C, can be represented

that

by

._.!?._ s&$ - s,

“O.. __AfJ -

expression

a flow-work

are related to D,, K. L-. p7 rl, LID, tlI which have been introduced in in part d. osmosis process with a flow-work and 52 into Eq. 53, setting N = 1. be shown that the water production

by Dmulittatiorr, 5 (1968) 237-265

260

c, =

D, =

E, AP $ ,-;

,-(

2,

f&(Re)‘.*

1.5 x io-“cmZ@c

-#- tf3 AP + B,(AP)““]

= 2.7 x 62.4 ib,jft3, density of aluminum = 15,000 psi, allowable stress of aluminum = 11.94 x 10-6hr-’

Pm

ii

= 0.86 x IO-’ ft3/ft’-fir-psi

Zw

T

= 8 x 10-3 tmz:sert

P

= 62.4 Ib,,‘ft3

d

=

1:24 ft

Jr C PF = H) S/‘kW f.0.b. c PI1 = 100 S:kW f-0-b. C PT = 50 S/kW f&x

8

CPR = SO $fkW f.o.b.

i_ o=

Cs = 4.4 Siib,,, of aluminum =, = 0.007 S,‘kW-hr cq = 2.4 x 10-6 S/lb, The Schmidt

qnr ‘Ir ‘Ir 5

= 0.9 =qlp = 0.8 = 0.1 = 0-I

number and curtstztnt C becwrte

3.05 x 10s Kd c -_-- __---c(SC)“% 0,

The constams.

f.o.b.

E; 2320,

B,, &. B,, B,, f3s, and B6. berome = 0. t I36 x IO- ’ S/psi-fib,

B,

& = 0.023 (~C,,

f C,) p

-

= 0.6330 x 10-“S~ftz-hr

= OS2 x IO-’ %[psi-ft*-hr Desatrhation, 5 t 1%8) 237-265

OPTIMIZATION

B

4

OF RO WATER

PURIFICATIOX

!!-!_k?.!t~!tmd = 0.1773

=

SYSTEM

x

low6

I. PROCESS

S/(psi)*-ft’-hr

B5 = ($CPr

- C,} qPq,, (1 - ~7~1.‘~= -0.3494

B, = (JlC,,,

+
I

261

L/D

\!C,

Table

ANALYSIS

and Fig. I I summarize

single staze reverse osmosis

x 10-a S/lb,-psi

the rest&s of the simulation

system without

a flow-work

how water production cost is affected by the membrane the reject brine concentration.

study for the They show

exchanger.

area to feed ratio and by

TABLE I

_ _-_-._ .__-_

__ _.___-. ., --

._..- __-._.. -

. .. ...-.-

‘..,Skfo x4

~

\-.

0.05

. ..--

__

0.07

____

001

Thzf production cosf. 0.8821 08466

O.tU5

00s

0.6947

o.os5

0.09

___.

O.li

___

C,‘lQiM

.gaL

uafef

..____

0.13

- .. -

0.15 ._-_

0.17

---

0.19

pnxiadceci

0.8+x7

0.8580

0.8796

0.9061

0.9359

0.9678

0.6-)2-J

0.6408 0.5814

0.62X 0.5565

0.6191 0.54Sl

0.6250 0.5485

0.6359

0.6502

0.6667

OS!%

0.5636

0.575:!

0.6237

OSS80

0.5297

0.5185

0.5163

0.5197 0.5036

0.5266 0.5092

0.5359 0.5173

0.06 V.065

0.6173 0.6170

0.5$1?6 0.5360

0.5182 0.5141

0.5052 OSVVI

0.4955

0.4967

0.5016

0.5090

0.0-I 0.075

0.6199 0.6248

0.5472 0.5506

0 5143 0.5169

0.4994

0.?943

0.4950

0.4995

0.5065

0.08 0.09 0.10 __._

0.6308 0.5555 0.5212 V.M5V 0.5679 V.5328 0.6607 0.5823 0.5367 _ _____._____ -_.-_.-----_--_--

The overal

optimum

operating

operating

0.5015 0.5055 0.5167 0.5307

condition

pressure

0.5016

0.496I 0.4965 0.4998 05001 0.51IO 0.511.S 0.525I 0.5258 .-._~_----__

is summarized =

1500

0.5077 0.5112 0.5228 OS38I

as

psi

concentration of the reject solution membrane area to feed ratio

= 7% = 0.135 f@/lb,ihr

overall water production

=

cost

0.5008 v.5@%3 0.5157 0.5305

49.43 c/l000 gallons.

Table II and Fig. 12 summarize the results of the simulation study for the single stage reverse osmosis system with a flow-work exchanger. The inefficiency factor of the flow-work exchanger is assumed to be 10%. The overall optimum operating condition is summarized as operating pressure = 1450 psi con~ntration of the reject solution = 5.5% = 0.09 ft2/lm,/hr membrane to feed ratio overall water production cost = 38.72 C/i000 gallons. It is interesting to note that the optimal operating condition is shifted Desulittufiott,

5 (

1968)237-265

L. 7.

262

i * 006

001

00s sf&

FIN.

I I. Simulation

0.11 Memmfle

0 13 orea

to

0 )?

0.13

feed

study of one stage uithout

FAS

et

al.

04s

rotto the use of ilow-uork

exchanger.

si_enificantly by the improvement in the energy recovery system. Of the calculated saving in water production cost. 49.43$ - 38.7% = 10.71~ a part of the saving is directly due to the use of a flow-work exchanger, but the remainder should be attributed to the shifting of the optimum operating condition. If the latter system (the system with flow-work exchanger) were operated under the condition corresponding to the optimal operating condition of the former system (the system without the flow-work exchanger), the water production cost would have been ~-43~~~~ galions. The difference, 40.43~ - 323.72$ = I.71 c/l000 gations. should be credited to shifting the operating condition to the new optimal condition. This study shows the importance of an optimization study in properly crediting an improvement introduced in a process. The ultimate improvement can only be determined by evaluating various alternatives of the process. Dcw~iinarion,

5 (I 968) 237-265

OPTIMfZATION

TABLE

OF RO WATER

PURIFICATIOS

SYSTEM

I. I’ROCESS

AXALYSIS

263

II

SIh,ULA-,TO’;

STUDY

_____.____

.

..-

OF OXE

STAGE

PROCESS

__. ._ __-.--_-_

--.

\VITH --_

FLOW-WORK

E?(CH.\XtER --.

..-----

-

_.-.._

._

0.17

0.15 ._--.---.

0.5030 0.4192 0.4113 0.4195 O.J326

0.04 0.015 0.05 0.055 0.06 0.065

0.4472

0.07

0.4622 0.4770

0.075 0.0s

0.49t4

0.09

0 5192

0.10 _. _ _-_-_----

0.5454 -- -- .

0.5214 O.JI’JS 0.3935 0.3925 0.4003 0.4100 0.4209 0.4323 0.4439 0.4669 0.4894 _______- .~.._

0.5494 0.42f4 0.3927 0.3872 0.3902 0.3969 i).JOSS 0.1150 0.4250 0.4456 0.466-1

0 5YfY 0.3356 0.3996 0.3soO 0.3901 0.3948 0.4017 0.4100 0.4191 0.43Y3 0.4583 - _-_-_-._.

.-

-.._._ _ 0.19 ----

0.65X

O&BY

0.4727 0.4243 0.4080 0 4039 0.3057 0.4107 0.4175

0.4936

0.4255

0.3335

0.4396 0.4203 0.4145 0.4152

0.4195 0.425x

0.4135 0.4512 0.362’) 0.4708 -. ._ -. -.

0.7277 0.5 I53 0.4559 0.4339 0.4266 0.4263 0.429Y 0.4359

0.4434 0.4610 0.480s

‘! as4 053 \

o!G!-

.

031 0.30 3 ,\

0

\

d-.. 009

007

009 0 I, s’/s., Membrone

Fig. 12. Simulation

0 13

orea

0.15

to feed

0 I?

a IS

ratlo

study of one stage with the use of flow-work

exchanger.

Desalirrariotr. 5 ( 1968) 237-265

L. T. FAN et

264

This study was supported by the OfTice of Saline Water, U.S. Department Interior, Grant No. 14-01~ooO1-523, and Grant No. I+01-OOOI-1283.

al.

of

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QPTI&ffZATfOS Of= RO WATER PURiFfCATfOX SYSTEhf I. PROCfZSSANALYSIS 21~

22.

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5 (1968 j 237-265