Analysis and optimization of a multieffect multistage flash distillation system—Part I. Process analysis

ANALYSIS FLASH

ANO UPTt.MfZATKIN OF A MULTiEFFECT MULTiSTAGE DlSTItLATlUN SYSTEM--PART_ i. PROCESS ANAtYSfS

A ~t~~~t~~l mxfrt of 8 multicfikct multistage ftnsh distithtion systmn that nn be used in pr-s optimintion studies is dewziopcd. The mod& CXXWS the heat input to the brine heater, the my& floss SHEin uch eikt. the cmcentrrttion and temperature of the flashing brine icaving c?ch ctfcct, snd that number of stages in each C&X to bc treated as variabfcs in an optimization study. ItH: mod& is bck%wf to be aO2wate t?ttoUgh to dwibc the $%‘oaSS yet simpic CtWUgh lo bc used in tfvc sim~~t~on, design, and uptimitltion of the process. SYMBOLS

heat transfer area, ft’ constant used in the Cfaus~us~i~~yron equation, tb,/ft2 salt concentration of flashing brine at lontian K. wt. :a; (C,h outlet salt concentration at the nth effect. wt. “,; Kpo ct? - salt con~nt~t~~n in the feed sea water, wt. y. 5 - heat capacity per pound, Btuflb, “F feed flow rate, fb/hr F H-f, H-2. and W-3 - heating actions in the first, second and third effects, respectiveiy HR-1, HR-2 and HR-3 - heat recovery sections in the first, second and third effects respectively h unit enthalpy per pound of liquid, BtuJIb unit enthalpy per pound of water vapor, Btuffb N, JI, J,and .Tj - circulation pumps in the first, second and third effects, respectively - ffow rate of flashing brine, Ibjhr L M,, M2 and MJ - mixing points in the first, second‘and third e&ets, respectivefy IV,, N, and N, - ntimber of stages in the first, second and third effects, respective& - vapor pressure of pure water, ib,jft’ - vapor pressure of aqueous solution, ibf/ftr ; heat transfer rate, Btujhr 9 - heat input per unit time in the brine heater, Btu!ht 9, R - ideal gas constant, BtuJlb, “R 336 ~~~Si~~, 4 (tws, 336-360 A B

-

MULTWFFECT

R,,

Rz and

R, -

flow

respcctivcly,

MULTISTAGE

rates of recycle

FLASH DISTILLATION

SYSTEM

brine in the first, second

337 and third effects,

Ib/hr

flow rate of cooling water, Ib/hr r,,r,andr, - recycle ratio in the first. second and third effect respectively R-I, R-2, and R-3 - heat reject sccticns in the first, second and third &Eects, respectively TC - condensing temperature, “R tempcraturc of flashing brine at location A, “R CT,,, TJ - temperature of non-fiashing brine, ‘R Steam temperature at the brine heater, “R T. (AT) - 1; - T,, “F (AI) - T, - T,. ‘F U - overall heat transfer coeficient. Btu/(ht ft2 ‘F) c” - vapor rate, Ib!hr 1%;. Wi and W’, - rate ofcondrnsrite formation in the tirst. second and third ctfccts, rcspectivcly, lb/hr Z&Y - total water production rate, Ib/hr . R4

Greek

Letters - Tr-- T,,“F the average of I in the nth eficct ;l,Vl,,.vapor prcssurc dcprcssion of brine streams, Ib,/ft’ P -- fractional increase in pumping power required due to friction ‘I/ 4. - latent heat vaporization, Btujlb -- density of water, Ib/cu.ft. P Subscript Locations in MEMS System as shown in Fig. I condensate stream flashing brine stream nonflashing brine stream inside condenser tubes the nth eflcct

A.B,C.U,E.F,G’,H,k”,Z

;

1

i II

-

1. INTRODUCTION The

present study is directed to the system analysis of a multieffect multistage (MEMS) flash distillation press. The MEMS flash distillation process is a rather recent development in the flash distillation technology, which appears to have several advantages over the singleeffect multistage @EMS) flash distiltation process fI&3,4), The objective of this paper is to develop a mathcm~tical model which can accurately describe the process and at the same time is simple enough to bc used in an optimization study. Therefore, a so-called ‘*Macro-Stage Model” of the system is developed. -A MEMSdistillation system may be considered as a series combination of several Desahakm,

C (1968) 336-360

338

L. I-. FAN,

C. \‘. CHEXG,

C. L. HWAFiG

L. E. ERICIISON

AND

K. D.

KfANC

SEMS d~st~~la~u~ systems with proper arra~~~rn~n~ fur the recycte loops, G& component SEMS system is called an effect. In developing the equations far the mathematical model, each erfect: i.e., a component SEMSsystem, is considered as a unit. Each cllkt is first ,Issumed to consist of an infinite number of physical QP micro stages (infinite --stags operation); that is, differential equations rather than diRemn= equations are set UP and solved to obtain the idealized performance equations. Sin= an actoat ?lant consists of a fhitC nUi%b@ of stages, tipprupriate r,xxr~&on terms are added to the idczdired ~rfo~an~e~qnatio~sIn the dev~ro~rn~~t of the modei, the heat input to the brine heater, the rccycte flow rate in each effect, the ~o~~ntrat~on and t~rn~ratu~ of the flashins brine stream kaving each c~%xL and the number of stage’ in each e%xt are considered to be variabfts that wilf. be importgot in the su~qu~nt o~tirn~~t~un study, Thenfore, the model is devetoped with this in mind and it r&:ftccts the effects of changes in these variables on the Watcrr production cost. 2. THE DEVELGPNW

LEADihrG TO THE FLAStf

I3fSTILIATtON

PROCm

devefopmrnts Beading to the MEMS system can best be nnd~~toud by making four ~rn~a~s~oRs ~~~~,~,~*~~~~.~~~ which are (1) flash distitfatiun and regular disdttntion, (2) mu&istage operzition ard single stage operation, (3) a recyckd P~O+XZ~S and a ..u~~*t~rough*. process. and 14) the MEMS system and the SEMS systemTZZC m~rtant

The main advantage of flash distillation ;1s compared with a regular distillation proozssis the better eontrot of scale formatiorr in the former. In the flash distillation

process, heated saline water is released into a closed vessel which is maintained at a lower pressure than thtr vapor pressure of the solution. Since the vapor simply flashes os the aarm liquid, the r~sufting prkpitatcs ff~m in the &guid and not cm the heat trZtnsfer surface. Since: the f&h distillation process gnsatfy reduces scale f~r~tj~n~ its development is viety important in d~~~~natio~.

~~pgrar~on cs single stage operatim me operation in a rehung chamber belongs ta the so_CatIed “*completely mixed” or “back-mirr” type due ta the vigorous mixing resufting from the flashing. Themfore, chamber is equal in its composition with the solution leaving the the solution in of the feed to the chamber. In si @ stage operation, the difference in concentration flash chamber and the flashing brine in that chamber is large. Becauseof this, a farge amount of free energy of mixing is wasted and th& thermodynamic eftkiency of the operation is tow. RZmuhistage ~~r~t~~~~ the concentration of B&hing brine increases ~r~~~s~iyc~~ as it passes through the system. Thou@ the c~n~ntr~tion is uniform in each stitge, there is a change in c~~~~trat~o~ of flashing brine from stage to stage, The concc~.%frrlristage

MULTtEFFECX

MULTtSTAGE

FLASH

DlSTtLLATtON

SYSTEM

339

tration difference of solutions entering and leaving a stage is much less than that in single stage operation and the wasted free energy of mixing is correspondingly reduced. The Iow thermodynamic efficiency described above shows up in practice in terms of a higher boiling point elevation of the ffashitg brine and a lower effective temperature difference, for heat transfer. The use of muttistage operation is thus an important development in improving the therm~ynamic efficiency of the Bash distillation process. A ret-)-clerl process cs a “once-through”

process The main reasons for using a recycle stream are to increase the total heat capacity of the flashing brine stream and to increase the percentage conversion of the fresh feed into product water (6). Due CO the high latent heat of vaporization of water and the low heat capacity of the aqueous solution. the solution cools offconsiderably when only a small fraction of the solution has been flash evaporated_ In the flash distiilation process, the highest flashing temperature is limited by the scale formation probIem and the lowest temperature is limited by the temperature of the sea water which is used as the coolant. The percentage conversion of a flash distiIIation process without recycte operated within the temperatirre range mentioned above is less than 20:<_ Since the saline water feed stream has to be pumped and pretreated, a low percentage conversion of feed water into product water will result in a poor overall economy for the process. One way cf improving this situation is to recycle part of the discharge solution and thus increase the total heat capacity or !he flashing brine stream. This greatly improves the process economy and the percentage conversion based on the amount of fresh feed. The incorporation of recycle operation into the single effect multistage flash distiltation system gives rise to an operation which is calfed singte effect multistage {SEMS) flash operation with recycle. The MEMSsysresr

~7srhe SEXfS

system

The recycling of brine described abo-ie causes an irreversible effect because two fluids with a large difference in conoen.ration are mixed. This mixing increases the concentration of flashing brine in the system and this in tuh increases the boiling point elevation. Fig. I shoss a mulrietfect, ztultistage, flash distillation system (MEMS system) in which the system is c?iyk!ed iztto three effects, each etkct consisting of manystages, The recycle loop is arranged so that the brine solution leaving an effect is recycled within the same effect. There is still mixing of two fluid streams with diiferent concentrations in cash effect due to the recycfe operation, but the irreversibility etfect is Iess than in the SEMS system. Williamson, et. al. have summarized the advantages of the MEMS system as compared to the SEMS system as follows (1). The M.EMS system lends itself to better control and optimization of operating variables. This results in a reduced DesaIimrion. 4

(1%8) 336-360

340

L. T. FAN,C.

Y. CflESG,

C. L. HWAXG

L. E. ERICKSON

AND

E;. 0.

KIANC

heat transfer area requirement through the entire temperature range of operation anda pressure drop from stage to stage which is more neariy constant. These can be accomplished by using a higher recycle ratio and a larger number of stages in the high temperature end of the plant than in the low temperature end. 3.

PROCESS

DESCRIPTtOS OF THE MEMSSWXE!!

Fig. i illustrates

a three-efkzr multistage dash system. In order to facilitate the locations in the system are denoted by fetters, Z9 A, B, C, D, etc. and the system is divided into various sections which are denoted by HR-1, R-l, HR-2, etc.The first effect consists of a brine heater H-l, a heat recovery section, HR-1, and a cooling section, R-1. The second effect consists of a brine heater H-2, a heat recovery section, HR-2, and a cooling section, R-2. The third effect consists of a brine heater H-3, a heat recovery section, HR-3, and a cooling section, R-3. The section between tocations 83 and C serves a double purpose;it is the cooling section for the first etkt, R-l, and the brine heater for the second effect, H-2. Similarly, the szcrion between tozations E and F serves as the cooling section for the second effect, R-7, and the brine heater for the third effect, H-3. Sea water is used as a coofant in R- 3. F and L represent the fiow rate of feed brine and Bashing brine, respectively, The feed brine and recycle brine are jointly referred to as the non-flashing brine stream. TI, Tj, and T, represent the temperature of the flashing brine, non-flashing brine, and condensate, respectively. Subscript notation will be used to indicate the Jiscussion, some critisai

Des&nation, 4 (1968) 33+369

MULTIEFFECT

MULTISTAGE

FLASH DISTILLATION

341

SYSTEM

location. For example, (T&, ( T,)B, and (T’),,resprent the temperature of the flashing brine at location F, temperature of the non-flashing brine at location B and the temperature of the condensate at location H respectively. Rt, Rt, and *: represent the recycle flow rate in the first, second, and third effect, respectively, a& iv,, @V,, and EV, represent the condensate produced in the first, second, and third effect respectively. R, represent the cooling water (sea water) used in the third effect. The sea water feed is heated in R-3 and then acidified and degasified to remove CO2 andother dissolved gases. After being heated successively in HR-3, H-3, SiR-2, H-2, KR-1 it is mixed with recycle brine R, to form a brine stream which is heated in brine heater K-l and then introduced into the first effect as the flashinr, brine (L),,. The flashing brine at location C is divided into two streams. One stream, (Llc, is fed into the second effect and the other stream, R,, is recirculate1 by the recycle pump, J,. heated in KR-1 and then mixed with the feed .s:ream 2: the mixing point, MI. As has been described, the combined stream is he?_,‘-5 in brine heater K-I and introduczzd into ti,; fkst effect as f?ashing b6r.k cL.jA. Similarly the flashing brine at locations

F and K are

ciikdd

i?c

;Lj,and

R,,

and (L),and

R,,

respectivciy.

(Ljf

is

fed into the third effect and (L.jI is discharged from the third efkct as the reject brine from the system. Stream R, is recirculated by pump J2, heated in KR-2and K-2, mixed with (L), at mixing point M2 and introduced to the second efi‘rct as (LJD. Similarly R:, is recirculated by pump Js, heated in HR-3 and H-3, mixed with (LjF nt mixing point M,, and introduced to the third effect as (L),. The feed brine and :he recycle brine are heated in each stage by the water vapor evaporated from the flashing brine in that stage. It is possible to arrange the Howsystem so that the temperatures of the feed brine and the recycle brine are equal at any location. In the following discussion, such an arrangement is assumed_ As has been descrikd, the feed brine and the recycle brine are referred to jointly as the nonflashing brine and its temperature is denoted by TP The recycfe brine, I?, , which is a part of rhe flashing

brine at location

C, is introduced

into thecondensing

location B where it becomes a part of the non-flashing following relation should hold

Similarly,

brine stream.

chamber

Therefore,

at

the

(TJ)8 = (TI)c.

0)

CT,), = U&

(2)

U&

(3)

we can write

and = Gf,fx.

When two solutions, which differ in temperature and/or composition, are mixed together, the mixing is thermod)namicalJy irreversible. The two streams (solutions) mixed at hf ,, have a difference in composition; however, by suitably locating point B, the temperature (T,)B of the recycle solution R2, can be adjusted to (T/),, the temperaDesdination. 4 (1968) 336-360

342

I_. 7. FAX, C. f. CIItXG, C.

L. HWASG

L. E. ERICXSDS AXD 2;. D. KlA?t’G

of stream (l..)c. In this way the thermodynamic irreversibility due to mixing czm be minimized. The same statement can be made for tw3 streams mixed at Af 2 and for those at .!I,. in rhis study. isothermal mixing is assumed at M ,. &f2. and hi3 and the ture

heat cf mtxinl;r of the sodium

chloride-water

system is neglected.

Thus,

Simifarlg,

The unit cnthafpq of diiutc salt solutions is assumed to be a function of temperature but independent of composition. This assumption is justified because of the small heat of mixing and the rather limited concentration range (approximately 3,57; to 7”,) encountered irl this process. A strtgc lbirhin each ctfcct consists of it flnshing chamber and a condcnscr char&w ant’. a demister \chich separates the two chambers. When the flashing brine lcaving one stage (the tn - 11th stage) is released into the next stage (the rtth stage), water vapor is Rashcd out of the solution. The water vapor then passes through the demister on its xx-q tu the condenser chamber. uhere it is condensed to heat the brine. i.e. the feed and rcc@ streams.

4.

PRDC-ESS ANAL’iSiS

OF THE BtE%lSSYSTEBt

Thecost of water production in a desalination plant depends on both the operating cost and the capital cost of the plant (8). In the present anal_vsis and the subwqucnt optimization study. only the costs which are significantly affected by changes in the variables of the optimization study are included. Operating costs which must be considered include the cost of pumping and pretreating the fed brine, the cost of providing cooling water, the steam cost, and the pumping cost associated with r&rcuIating the brine solution within eacheffect. Capital costs include the cost of heat transfer area, the cost of the recircuiation pumps, and the outer she!1 casts for the system. Other costs, which do not greatly affect the process optimization study, such as piant site cost. tabur cost. overhead and insurance costs may be added later. Since one of the main objectives of the present study is to develop a model of the MEMS system that can be used to optimize the MEMS process, only those costs that greatly

affect the design

of the process ace included. relations among the operating

Quantitative variables are derived in the next section. The performance of a MEMS system is represented by a temperaturecomposition diagram in Fig. 2. The general approach is to obtain idealized pcrformance equations by assuming infinite stage operation in each elect and apply carrection terms for the finiteness of the number of stagesand g’-W-k’, show how the temperature of the - The fines, a’- b’ -c’* d’--e’-f’ iksuf~rion,

4 (1968) 336-360

MULTIEFFECT MULTISTAGE FL/VW DISTILLATION

SYSTEM

343

flashing brine, T,. decreases as the concentration of the brine, C,, increases in an infinite stage operation in the first, second and third elfccts. respectively. The relations representing these lines are derived in Section 5-4. The concentration gaps. between Zand A, C and D. and F and CI, are caused by the mixing of the recycle brine and the flashing brine streams, The mixing relations arc derived in Section 5-2 The stepped lines along the Iincs. cr’-&‘-c’. &-L-f and g‘- h’-- k’ represent the temperature of the flashing brine in the first, second, znd third efkcts respectively in an actual process where the number of stages in each clkt is finite. ‘The lines, a’-b”-c”, ti”--e--f”, and g”-_h’-k”, show the relation between the condensate temperature. T,, and the flash brine composition, C,. at various locations in the system for an infinite stage operation. The stepped lines along a -- b * c”. iI’-e---j’, and ,~“--/t”-- k” again represent an actual finite stage operation. The vertical distance between the two sets of lines described in the last two paragraphs represents (T, - 7;) at various locations in the system. This dit’fcrence is due to the boiling point elevation of the llashing brine and the pressure dilE?renct! across the dcmirtcr at each location. This temperrtture difference. which will k denoted by II. varies throughout an etfctt. This variation is mainly due to the varying composition. which in turn atfects the boiling point elevation. The average values of I in the first. second, and third tfTects are denoted by (z),.,,.. (z)~.~,, and (rrfJ dl rrspectivcly. SimilarIy, the lines 0” - h’ - cm, d” - e” - f’ and g” - h” - k” show the relation bctwcen non-flashing brine temperature and flashing brine composition. As illustrated in the iigure, the temperature di;Tercnce (T; - rj), in itn infinite stage operation is nearly constant within a heat recovery section. Therefore. (AT),.,, tA’0z.x. and CA77,,, wilt be used to represent the (Tf - 7,) values within HR-1, HR-2, and HR-3 stctions respectively for an infinite stage operation. The figure also shows that the value of (r,Tj) varies gradually from (AT)t+, to (AT)2., in section R-2; and from (AT)3_r to in section R-l; from (AT-),,, to (AT),,, (T,jK -(q., in section R-3. The expressions relating CAT),.,, (AT-),,, and (AT), T to the operating conditions and parameters characterizing the systems are derived in !Section 5-6. The steps shown in the figure, which arc due to the firmertess of the number of stages in individual cffccts, result in losses of effective temperature difference for heat transfer_ The expressions for these losses in temperature difference for heat transfer are derived in Section 5-7. Heat transfer areas are evaluated for individual sections separately or sevcraf sections jointly. The heat loads in the various sections arc derived in Section S-3. The average values of temperature dilYerence for heat transfer in the various sections are catcufated using the relations derived in Section S-5. From these, the relations pumping s-11.

for the heat transfer area requirements are derived in Section 5-9. The head required for each of the recirculation pumps is derived in Section

Desalioarion. 4 (1968) 336460

L. T. FAX, C. Y. CHENG, C. L. HWANG,

344

5. QWAh~TATlVE

RELATfONS

L, E. ERICKSON Alr;D K. D. KfANG

AMONG

THE OPERATING

VARIABLEZ5

q~ant~tat~~~e relations among the operating variables derived in this section wiiI be used to develop a rn~t~ern~t~~J model of the MEMS process that can be used for various purposes such as sjrnulat~ur~, design, and ~ptjrn~~cion af the process.

The

S-i.

Fli3w

rims

Recycte ratios, rr. r2 and f3 are defined by the rebtions .

tt = -,Rt F

The

flow

are related

R, r2

=a*

f3

=g.

rates of the

brine streams leaving individual effects, (t&-, (A& and fLjx, to their con~~tr~t~ons by the fallowing equations.

where CO is the salt ~u~~~tr~t~on in the feed. TJxerefure, the flow rates of the remn bc represented by the following relations.

cycfe brine streams

The fiow rate of cooling water, R,. is given in Section I-3, The flow r;rtes of the brine streams entering individual Akts are given. respectively, by the following reJations. O--IA = F-t-R,

=

F(1-3” q)*

Gh

=

fL)E + R1 =

F &(l

f&

=

g+),+

F-----co w,),

The flow rates of the condensate

R, =

streams

@I + r2ft

WJ

(1-k r3-

are given by the f~JJow~g

00

retstians.

MULIIEFFECT

5-2.

Mixing

MULTISTAGE

FLASH

DISTILLATIOK

345

SYSl-E~f

relnri0rrs

AS shown in Fig. 1, the mixing of the brine streams ?akes pkc at Jbf, &f, - By making salt material ba!ances at the mixing points, we obtain

at the mixing

point,

M, . Similarly,

at ,%I2 and at JV,,

,

&f2,

and

we obtain

and

5-3. Hear taixcis The heat load. (I,, in section H-l, is a very important operating variable. A large value of y, means a large steam cost but it also means a large temperature difference for heat transfer and consequently a low plant cost. The heat load in the remaining sections can be calculated by either one of the following two ways. (I) Method A. This method takes the sum of the enthalpy changes for the non-flashing brine in a section. This is equivalent to taking the sum of the enthalpy changes for the condensate and the hashing brine in the section. (2) Method B. This is an approximate calculation in which the heat toad in a section is taken as the latent heat required for condensate production in that section. .%fethod A. For this mct’nod, it is convcnicnt to evaluate heat toads for sections R-l and HR-2 and sections R-2 and HR-3 combined. in section HR-1, the feed brine stream, F, and the recycle brine stream, RI, are heated from temperature (rJ), to temperature (Tj)A. And since (Tj), = (T/)c, the heat load qr is given by

In sections

R-l

and HR-2,

since (T,), = (TfL and Vi), viz. q2, is given by

F and R2 are heated from (T/j, to (I,),. Therefore, = U’Jc . the heat load in these two sections combined,

Lksahation.

1 (1968)

336-360

346

L. T. FAN, C. Y. CHENG,

Similarly,

C. L. HWANG

the heat Ioad in sections

and the heat load in section

R-3,

44 = (F + R+,C<%

L. E. ERICKSON

R-2 and HR-3

AND

K. D. KIANC

viz. q3, is given by

combined,

viz. q&l is

- (h,L..l

= (F + R+W,)r

-

(h,Lil .

(21)

Recall that ( Tj),, = (.TI)Kand (A,),, = (hl), . By applying an enthalpy balance around the whoic system, w obtain (F + RJ)(h,L + 41 = &(hj)H + J4Q

+ G WU&.

Since F = Lx + XW, the enthaipy

(he), = UQl -@M,,

and

(all, = CTf>, - (7&,

around the whole system becomes

balance

4, = (F + R,)@,),

-

thj)KJ

(22)

-CzrVCz>Kcp*

From Eqs. (211, (22), and (15), we obtain 44 = qr+(ZIV)(&C,

=

qs-+

F

(C,),

-

co

CC,),

C~),C* *

(23)

Sea water is used as cooling medium in section R-3. Since the cooling water must be pumped to the desalting plant, this adds to the water production cost. The cooling water

from

rate am

which

be obtained

from

Eq. (22) as

we obtain 4.. F F j=@- + (4rG

& -=

XIV

Y.

1

Fc, (1

(h,), -

--

(h,)K

A?_] +Gf)K -

z

uyf)K - vyjk

F

Xc)V

1

co .

I-

f25)

tc,>K

Merhod B. In this approximate calculation the heat loads are evaluated for pairs of sections HR-I and R-i, HR-2 and R-2, and HR-3 and R-3. If the heat lkQllfkaGon, 4 (1968) 336-360

MULTIEFFECT

MULTISTAGE

FLASH

DlSTILLATlON

347

SYSTEM

load due to the flashing of the condensate stream is neglected, the heat load in HR-1 and

R-l

combined,

qs,

iszgiven

[F - (L),]i.

qs =

Similarly,

by

the heat Iaad in HR-2

= F[l

-&]i-

and R-2 combined,

qa, is given by (27)

and the heat load in HR-3

and R-3 combined,

q7 = 5-4.

Adiabatic

flashing

is given by

q,,

co *

G F [ --(C,),

(Cf), 1

(33)

d.

reiutions

of brine &shed in each stage depends on the flow rate and temperature drop of the flashing brine stream. The relation will first be derived for infinite stage operation and a correction term will then be introduced for finite stage operation_ In Fig. 3, which represents a flashing chamber of an infinite stage system, L, C,, Tf and it, are the quantity, concentration, temperature, and unit enthalpy of the flashing brine stream respectively. Let nV be the quantity of water vapor evaporated, and let H, be the unit enthalpy of the vapor. The following relations described the transfer of material and energy for this flashing chamber. A total material balance , gives Ihe

quantity

L=

A salt balance

Lt

dLt

or

dL=-dV.

-

=

which may also be written

balance

(Li-dL)(Cf+dCf)

c,
or

as dL -=---_ L

dC.l

+ d/z,) + HgW

or

(30)

c,

is

L. tr, = (L f dL)(h,

L dh, = (If, - h&IL.

Since dh, = C,dl; and H, - h, cqn be approximated porization, 2, the above equation becomes LC,dT-

By combining

(29)

gives LC,

The enthaipy

dV

this equation

with equation

=

by the latent

heat of va-

Ad L.

(30),

we obtain DesdimIion,

4 (1968)

336-360

L. -I’. FAN, C. Y. CHESG,

348

C. L. WVASG

C f”p Assuming.

that

two iocations

C,!i

L. E. ERICKSON AND K. 0. WANG

CiL __T

-_I_dCf

L

is constant

and

Cf

integrating

the above

equatiorl

between

any

1 and 2, wt‘ have (31)

This is the equation Fig 2. The

stepped

lines

for the tines, along

R* - h’ - ct, tf’ - e’ -I’_

u’ - b’ - c’, cf’ - t*‘-f’,

and

and

.p’ - h’ - k’.

g’ - Ir’ - k’ show

in

how

f WD ---

, -

3

--

Y

c

F

2 N

I; 5:

m 1

%

-

Jo

Fig. 2. Teempcraturcs, 7;. T,

G

bc le cf and T, vs concentration of flashing brine C (Sfhcmatic).

Conce3tratior:

of

flashing

Demlitza~ion, 4 (1968) 336-36

1.. r. FAX, c.

350 Sincg

the

f&d

brine

t’. CHEW, and

C. L. tiWANG

L. E. ERICKSnN

AND K;. D. KIANG

the recycle bl;nc receive the heat of condensation of the

water vapor, WC have

-dlV

=

(F + R,)dhj _. A

Substituting equation (35) into (34) equation yields (F + R,)dhj = (F + R,)dh, + ~~?!i!$.tt!!.&!!~

,

(36)

On rearranging, WC obtain

1

= 1- !k-T.v!!! 1‘fh, dh,.

i

(37)

h, - h, Since __._-A < 1, equation (37) can bc approximated

dh, = d/r,

or

hy (38)

c17, = JT, .

Integrating this between any two locations 1 and 2 and rcnrranging give

This derivation

(39)

= (Tf)J - (7,): a

CT,), - CT,),

leads to the conclusion that A7’ is nearly constant in the heat rc-

covery sections. In the heat rejection section. 3 derivation

similar to that described above lcrtds

to the following result for infinite stage operation. (F $, R,)& SinccbothF-t

R,,,

R,andF-t

arz constant. the equation CLII~bc mtcgratcd to gkc

(F -+ R,Ci’7; This

quation

(4’)

= (F f R,, ,)l!T,.

--

(F

-t- R,,

,),%‘I;.

(41)

can be applied to section R-1 to give

(F + R&T,),

-(T&j

= (F + R&V,),

- (r,)c].

(42)

Substituting equation (4) into this equation, it can bc rewritten as (b + R&W,),

- (T,)a]

= (F + R,)[(T,),

- V,),].

(43)

Ske (T,), - (Tjjs = (AT),., and (T,)c -(T,),. L-:(T,a-- (Ti), = (AT)2,nr cquation (43) becomes (the second subscript, CC, is to emphasize the infinite stage optration) (F + R,)(L\T),,,

= (F + R,)(AT),

.,..

(441

Stmilarly, vie can write Dcsulinution, 4 (1968) 336-360

M~!LTIEF~ECT ML’LTISTACiEFLASH DISTILLATION SYSTEM

351

= (F + R,NAT)~,,

(F + R,l(AU,.,

(45)

and (F +

R,MAT)J,,

= (F + R,NAT),.,

=

iF+ R,)(Vfh - (7&c),

(46)

where R4 is the quantity of cooling water used in section R-3 and (AT),_, =

U,), - U,),. In the heat rejection sections, the temperature difference, Tr - Tr, is not con&ant. In section R-l i it varies gradually from (AT), ,sc to (AT12,,; in section R-2, it varies gradually from (AT)2S,, to (AT),,,: and in section R-3 it varies gradually ,from (A’b,r to (AT),,,.

tcmpcrature dilkrcnce.

The

T, - 1;. in HR-I.

‘IS = (F + R,)[(hJ, Since

viz.

(AT),.,

can be derived by

rrn cnt’rgy bulancx to section A- % (see Fig. I). This gives

qtplgrng

R, = r,F and (h,), -(h,J,=

- (h,lA].

C,,[tT,),, - (Ti),Jr the above equation becomes

- U,),l

cl, = 01 + r,)C,[U,),

- F( I + r,)C,(AT),.,

from which UC obtain (AT),,.

=

----. ((1P

ll.

I-rJ

iw

.

*

The tcmpcraturc d~tlcrcnws.

( 7, - T,), in H R 2 and HR-3 for infinite stage opcratwn. bit. (A7’)2,, . and (c\T)~ , . rcspcctlvel~. can bc derived in a similar manner *ctlon\ C‘ Z aI!d F Z. rcspcctively. Since hy nlilhillg ilfl rncrgy h~lI;lllL~ ilr0Ulld

\hj),

= (h,)c,

WC can make an energy balance around C-Z q, -i- (I: -t- R2Wt,),

Since (IV), = F - (Lk, (I,

=

-(F

R,C‘,(A7‘),~,

+ tW,.(h,),.

the ahove energy halance becomes

+ R2)(h,l,: + :L&- + R&h,)c

= R&h,),

=

= ((Lk + R,)l(h,),

to obtain

- (h,)J +

+ q[(h,J,

- (h,kl

+ (F - t&k.)c - [(h/k -

(WI

FC’&AT),., - FC,@), + (&C,.(@)C

= (F + R2)C,(A?-)2 r, - {F - (UC.) c,(JI)c where (a)~ = (T”), - (T,), , Therefore, Dtmhation,

4 (1968)

336-360

FAN, C. Y. CHENG, c. L. HWANG,

L. f.

352

Similarty,

we can make an energ!

@7-)3.m

where

(2JF =

Equations

(T,),

-

L.

E. ERICKSON

balance about

AND K. D. WANG

F-Z and obtain

section

(W

=

(Tcb.

(47). (38). and (39) can bc written

in a generalized

form as

where (C,- f)O represents Co. CC,), , and (C,), respectively for n = 1,2 and 3. In show that equation (SO), (40 = 0. (z)! = (x), . and (r): = (z)~. These equations AT= 7-,- ?, in each effect is related to the heat input in the brine heater per unit of feed. 4,/F, and the recycle ratio, r. in the etkct. High qJF and tow )* favor high AT. When a MEMS system is operated so that it has a higher recycle ratio at the high temperature end than at the low temperature end, then (AT),., ((AT),.,: < (A%, . These relations wilt be used in deriving the effective (Ar) for heat transfer in each etfect (see section

5-7.

Finite

Referring

5-8).

stagt’ opercliolr

to a stage in section

Ar for heat transfer

in the infinite

HR-2 of Fig. 2, it can be seen that the effective stage operation is given by

uw‘. +.._^XY _-- = 2

u,,, =

x1’.

For finite stage operation, the condensation temperature stage, and the effective At becomes (WV +- xy)j,. +J Therefore,

At for heat tranrfer

is constant

within

each

the toss in the effective

is uw i- -.=. XY .____._ 2

L’W -!- x1’ = WV - DW_ ___--2.. .__----2 2

In other words, the loss in At for heat transfer is half of the temperature drop from stage to stage. Since the average temperature drop from stage to stage is given by Eqs. (31). (32). and (33) for the first, second and third effects, respectively, the averabe loss in effective At for heat transfer in each effect is lkalinction,

4 (1%8) 336-W

ML;LTiEFFECY

hlULTISTACE

FLASH DISTtLLATION

353

SYSTE~I

(51)

(53) The equations derived in this section will be used in describing heat transfer m the next section. 5-g. E-*crit~ At _for hetrr rrnn#*r If w let r, be the steam tcmpcrature, then Ar at the inlet the outlet = T, - CT,)., . and the average AI for heat transfer H-I, is (At& = r, - I[(:cTj)A +tTf)Af* Since (A?‘),.,

- (Tj)_,. the above

= (Tf),

equation

the etfcctive At for

= Ts-((Tj)A+ At at in the brine heater,

can be rearranged

to give

The maximum value of (Tr)A must pnerally be limited in order to control scale formation. in this study, the values of T, and
LiheHisc. for sections HR-2 and (Arl,. respatively, are

and HR-3,

the effective

AI’S for heat transfer,

‘.rru -

(A%

=

.t

(cf)F

1 f r -Jo “CCX

The above equations

1

(&*,,

show the effects of recycle

ntio

-

..-,.__~~__U’/)R -. *

and number Desahurion.

(S6)

2N,

-5- I (Trlc __...-_.._ _.____.-_ ._-3 C,(a),

(At),

Kfh - v.f)F _-- - ‘_.‘__.__..,

(Ar)a

(57)

of stages on

4 (1968). 336-360

354

L. T. FAN, C. Y. CHENG, C. L. HWAKG

the ckGaive give rise to

L. E. ERICKSON AND

K. D. KIANG

Ar for heat transfer. A larger recycle ratio and a small number of stages a larger

effective

AZ for heat transfer. However. in addition to their the influence of the recycle ratio on heat economy and heat load and the influence of the number of stages on fluid flow. With all these considered WC are to find an optimum set of values for the recycle ratio and number of stages irt each effect, which alfows adequate pressure drop for Ruid flow, and gives adequate At for heat transfer_ The preaun? drop required for maintaining adequaa kid flow between two successive stages is due to the differen= in the vapor pressure of fiashing brine in the two stages. Sk: the change in vapar pressure with temperature of flashing t,rine is lower at the low temperature end of the MEMS‘system, a larger temperature decrease px stage should be provided at the low temperature end. This will cause a large (AzL,,, at the low temperature end. In order to compensate this effect and maintain an adequate At for heat transfer a higher recycle ratio should be used

influences on Ar, we have to consider

at this end. Conversely, a tower recycle ratio and a larger number of stages should be used at the high temperature end. En the present analysis, the Row of fluid problem is not included due to the lack of avaitable information. Therefore, the conclusion obtained in the above discussion is not borne out in the present study. When such inforniation is available, it should be included in the process analysis and optimization studies. me

effective At for

be obtained

from

heat

transfer

the following

(Ar) in (R -

1) =

in the heat rejection section of each cifcct can

relations

3 [(AT),

(see Fig. 2).

+ (A%]

(Al) in (R - 2) = S C(AT)I + (AT),]

- (a)#, in (R - (I%

1) - (At),.,.,,

in (R - 2) - (AOLI.~,.

These relations will not be used in the later formulations, since with some approximation, the heat transfer area requirement for R-i wifi be evaluated conveniently in combination with either HR-t or HR-2 and the heat transfer area requirement for R-2 wiil be evaluated in combination with either HR-2 or HR-3. Referring to Fig_ 2, the At for heat transfer in the heat rejection section of the third effect, (At),, can be obtained by taking the average value for (T, - T,), and due to finiteness of the number of stages. subtracting from it (x), and the (AfL Thus

we can

write

(Aht - , = =

MULTIEFFECT

MULTISTAGE

FLASH DISTILLATION

355

SYSTEM

Heat transfer area Equations for the heat loads and A? for heat transfer in the various sections have been devefoped in Sections 5-3 and 5-8 respectively. These are used to calculate

5-9.

the heat transfer

areas by the equation

By substituting Eq. (54) into A,, can be obtained as

Eq.

f59), the heat transfer

_. __-_._._.--%i..__.__

T* - (T,),

area

._._ .__-.__

As described

in Section

5-3, two m&hods

can be used

H-l,

viz.

,

q,lF + 2 -~7i--+--;;IS P

in section

1

in evaluating

heat loads.

In the following, heat transfer areas are evaluated by both methods. Mcrhod A. fn Section 5-3, heat ioads have been evaluated for the combined sections of R-t and HR-2 and for the combined sections of R-2 and HR-3. Thcrefore, heat transfer areas wiit also be evaluated for these pairs of sections. For simplicity, the (Arjl and (As), derived in section 5-S will be used as the driving forces in evaluating heat transfer areas for the pair of sections R-Land HR-2 and that of sections R-2 and HR-3 respectiveiy. This simplifying assumption is justified

because the heat transfer area costs for the heat rejection sections, R-l and R-2, are much less significant than the costs of rhe heat recove@ sections. HR-2 and HR-3. Therefore. a somewhat approximate evaluation of the heat transfer area costs for the heat rejection sections does not significantly influence the overall water production cost. By substituting Eqs. (18) and (55) into Eq. (59). the heat transfer arca in section HR-I. A ,, can be obtained as

A,

=

FU + r,!CptU;l, -U&l ._ ^_.____.

---.---.

.-..-.

-.-.___

_.-.._

~(f’itF f,j

P - _.__..___-

_--.

.

(61)

Similarly, the heat transfer area in sections R-l and HR-2, AZ, that in sections R-2 and HR-3, A, and that in section R-3, A,, can be obtained as

Desafinution, 4 (1968) 336-360

356

L. 1. FAN, C. Y. CHESG,

C. L. HWASG

L. E. ERICK.SOS AND K. D. lilAKG

(6-Q :kf ctkod B. In Settion 5-3, heat ioads ha~c ixrn evaluated for pairs of sections WE:-1 and R-l, HR-2 and R-2 and HR-3 and R-3. Therefore, heat transfer areas wilt also be evaluated for these pairs of sections, For simplicity the (dtjl, QSzfr and

(A;), dcrised in Section 5-8 will be wed as the driving forces for the pairs of se&m HR-I. and R-t, HR-2 and R-2 and HR-3 and R-3 rcspectiveIy. 13~substituting Eq. (26) and (55) into equation (5% the heat transfer area in sections HR-1 and R-t,

!jimifarly

the heat transfer

AS, can be obtained as

area in sectiuns HR-2 and R-2, A6, can be obtained as

and the heat trmsfcr area in sections HR-3 and R-3, AT, can be obtained as

MULTIEFFECT

MULTISTAGE

FLASH DlSflLLATION

357

SY!XEM

S-10. Boiling point efetralion Y is defined as the difference between the temperature of the flashing brine and that of the condensate in .a stage. (z),,“, is the average value of z in the nth clfcct. The value

of z in a stage

depends

on the boiling

point

elevation

of the brine and the

drop in condensation temperature due to the demister pressure drop. Due to lack of information. the drop in condensing temperature due to the demister pressure drop is assumed to be 1°F in each effect. More accurate information may be incorporated when it becomes available. Fig. 5 shows the boihrrg paint efevation of an aqueous sodium chloride solution at various brine compositions and at various boiling temperatures(6). Empirical expressions for the relations between the boiling point elevation and the composi-

tion may be obtained by curve titting. The average value of the boiling point elevation in each effect may be evaluated at an average temperature which may be assumsumed

to be constant

Fig. 5. Boiling

equations evatuated

point

in the design

&v&On

and optimization

1% y’, Nacl

in sa!t

calculations.

solution at 125’F.

for the boiling point elevation for the first, second at 225=F, 17S”F, and 129F, respectively.

(ix),,, are expressed iis functions second and third eirect rcspectivefy,

(=I I .40

=

l-01+

175’F

Therefore,

and XS’F.

and third eifects

of the average brine composition by the following

1

are

in the first,

equations.

(C,)., + (C,),

m--------,

the

(68)

2 Desalination,

4 (1968)

336-360

L. T. FAS,

358

C. Y. CIIESG,

C. L. HWASG

cc,,,

(2) a? .ur = 1.0075 + &

In rhc abo~ equation. dcmisttr pressure drop.

1°F

has

L. E. ERICKSON

bcxn nddcd

to

AND

+ (C,), 2

take

R. D.

KlANG

(69)

’

into account

the effect of

punp, J,. takes in recycle brime R, of concentration (C,), at wmpenlure (T,t,:. pressurizes it to a pressure suniciently high so that the recycle brine does not boil within the heating tiubcs. The highest temperature to which this head (AP,)!p required for J, brine stream is heated is (?‘,)A. Thus. the pumping The recirculation

can

be evaluated

as, (71)

where tP)_, is the vapor pnzssurc of rhe brine at concentration (C,)., and temperature (T,),, (P)= is the vapor pressure of the brine at concentration (_C,)c and temperature (Tfjc, and (AP), is the pressure drop due to friction.

The above

equation

may bc written

as (74

where

qI is the fractional

excess

pumping

head

required

to overcome

the friction

IOSSeS. me vapor prcssurc of brine at a given temperature is Icss than the vapor prcssu~e of pure water at the same temperature due to the vapor pressure depression of the solution. Thus,

and cP~)~ are the vapor pressures of pure water at remperatures IT,),, respectively, and @I), and (,j?)c arc vapor pressure dcprcssions of the brine streams at temperature CT,),. and concentration (C,),, and at temperature (7,)c and concentration (C,)c respectively. where

(Pa),,

and (T,),

If we assume that t/Q4 = C&- we obtain (P),, -(P )c = (PO)” - (PO)=, and equation (72) becomes

Within

of water

the operating

temperature

may be represented

range of the MEMS

system,

the vapor pressure

by lnP”

. = -&+D Desu!ination,

4 (1968) 336-360

LIULTIEFFECT ML’LTIST,\C;l: FLAW

DlSTtLLAflOS

where R is the Intent heat of vaporimtion arranging, we obtain l%” = (&‘(,,)-~:Rr

and D is an integration

were

evaluated

B =

eD.

The

constants

IAP), --P Simiiariy,

have

Substituting

1.523 x lO’Ib,/ft’.

= $(i

ihc punping

constant.

On rc-

= B(~)-“.“T

been

Eq. (71) into f IJf)

359

SYSTEM

(74) as

i. = I000

Btu,‘lb,,

5

=

Eq. (73) gives

(ezp[ - &-I -

uxp [ -

&I;

.

(75)

hcsds for pumps J2 and .I, are respectively Ci6)

and

(AI-‘), --._ C’

(77)

The m~them3tic~l model which is developed in this work can be used in simulation. design. and optimization of the muitietkct, multistage. flash distillarion system. When the water production requirement and economic situation xc spxiticd, this model can be used to either simulate or optimize the MEMS process. As soon ;IS values for the heat input to the brine heater per unit of feed. qllF, the recycle ratios, leaving individual etfects. (T,),. (T’l,, rl* r2. and r,, the flashing brine temperatures and (TfbK. and the numbcr of stages in individual elfccts, ,V,, lVzr and Ma are spccifed. the values of other variables (the concentration of Ilashing brine streams leaving each effect, the heat loads. the various tcmperitture ditkrcncrs and corrections, the heat transfer areas, and the pumping rcquircments) crtn be determined using the model. If the unit costs of steam, feed brine, power, cooling water, heat transfer area, outer shrlt material, pumps, the value of money, etc. are specified, the water production costs can be estimated. One specific example has been treated using this model and values of the optimization variables which minimize the water production cost have been determined (12, 1.3). ACKSOWLEDGMENTS

This study

lnteriot,

was supported by the Office of Saline Water. U.S. Department Grant No. 14-01~0001-523, and Grant No. l+Ol-CKKll-1283.

of

REFERENCES 1. W. R. Wtru~wo~, Froceedi~s

F. W.

G~LSER~

AND T. R. SCANLAN. Xultiefftct

First rnIerI~aIio~a1S~~~s~u~

muttistsge t&h distiIIation 3-9, f 965. Washington

on Ft’ufer Desaiin~I~on, October

D. C. 2 (1967j 779. 2. E. A. C~WAUADER,

in 1964 Saline Water Conversion Report, O.S.W., pp. I37141 (196% R, TOWEND. AJsmces in C/r.emi.w~Series, 27. (1960) 147. Principles of Desalinoriorr. Edited by K. S. Spicglcr.Audemic Ress. New York,

3. D. 8. BRICE AND

4. R. S.

SILVER.

N. Y. (1966).

C.

Desafinaiion, 4 (1968) 336-360

MO

L. T. FAN, C. Y. CHENG, C. L. HWANG

L. E. ERICKSON AND

K. D. RUNG

5. Survey of Current

6. 7. 8.

9.

IO.

l!. 12.

13.

Economics of Natural and Dcsahcd Water in the United States, pp. l&32, Edison Electric Institute, New York. N.Y. (1%5). K. S. SPI-EGLEU.Salt Wurer Punfication. pp. 52-60, p. 151. John Wiley, New York, N.Y. (1962). A. ~RAsU.t., Pruc. Insrirurion of .\fech. &grs.. 176 (1960) 312. R. E. Gww~, Expcriena in multistage flash process. ProceedingsFirsl Internationa/Sytnposium on Fk’aterDesahtation. Chxober 3-9. 1965, Washington. D. C., 3 (1967) 213. F. C. A_ A. VAS BERKEL.3. W. VAN HASSELT~?r(n3. H. VAN DERTORREN. Eaperienczs with large sea water Bash esapwators. Proceditgs First f~ter~a~io~I Symposium on Water Desalination, Ocrobcr 3-Y. r%5. Washington. D. C., 3 (1%7) 189. Be&cl Corporation. Cost Studies of Large Multi-stage Flash Saline Water Conwrsion Plants. Rcscarch and Dcvclopment Progress Report No. 116, Of&c of Saline Water. Washington. _ D. C. (196-t). D. W. CLELLAVD ~\ND J. hf. Smv~nr. Dc.whtation. t (1966) 61. L.T.FA~.C.Y.CIIESG.C.L.HWASG.L.E.ERICYX)NAHDK.D.KIASG.A~~~~~~~ and Optimiation of a Multicffcct Multistage Flash Evaporation S>~tem. Special Report No. 74. Kansas Engineering Experiment Station. Manhattan, Kansas, 1967. t. T. FAS, C. Y. CHESG. C. L. HWASG. L. E. ERUX~N ASD K.D. KIA?~‘G.Analy;sis and optimization of a multieRect multistage Rash evaporation sjstem - Part II. Optimization. &vaiinc;rion, 4 (I9681 361. ~esaIinatiaa.4 (1968) 336-360