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A finite difference solution for reverse osmosis in turbulent flow
Deafiintatian - Else&r
Publishing Company,
A FINITE DIFFERENCE BULENT FLOW S. SRINIVASA~‘* Department
Amsterdam - Printc.8 in The Netherlands
SOLUTION
FOR REVERSE OSMOSIS IN TUR-
ASD CHI TlEN
o/Chemical
(Received in rcvird
E~girreering ad
form
.\fetatiurg.rp Syucuse
Universif_v, Syracuse. N. Y. f U.S.&)
November 2ft. 1968)
A finite difference solution for reverse osmosis in turbulent flow is presented. The cast of a flat duct made of two parallel semi-permeable membranes is considered. The pertinent diffusion equation with the eddy difksivity based on Deissler’s expression is solved using a two-step linearized finite difference scheme. taking into account the nonlinear effect due to the varying water flux across
membrane along the axial direction. Numerical conditions are given for a variety of cases.
results relating concentration
SY MSOLS
R
-
BZ
-
membrane constant (Po.)rAp
half the distance between membranes
I?
-
b+
=
b;
-
value of b+ at inlet
-
bulk concentration
=B
=
4 CI c
-
c+
=
=w
-
c;
_Y G H
l
=
-
cdcr satt concentration salt concentration
at intet
c/c* wall concentration CM/C1
salt diffusion coefficient friction factor defined by Eq. W-8) defined by Eq. (III-Q)
Presentaddress:Departmentof Chemicai Engineering,University of Dayton, Dayton, Ohio. Deshtation.
7 (1%9/70)
5 ! -74
S. SFUNlVASAN AND CHI TEN
position
index (_\T+direction) position index (z’ direction) defined by Eq. (IV-6) t?uJ\ 4 IV,,,. Reynolds number based on equivalent N,, at inlet = bL’,/v N RCCat inlet
diameter
defined as -!?Y f Schmidt number, r;lD osmotic pressure osmotic pressure at inlet fraction of water withdrawn axial component of velocity u/(MWf r.l+ at Y+ = 26 butk vefocity bulk velocity at inlet transverse veto&y &*., wail velocity wall velocity at X+ = 0 axial coordinate -~/C(W%,,)I transverse coordinate 1.\’ -t,* ” ( P ) (f/b+) -1-
= (J-/b)
1 B,
ah
defined by Eq. (I-l) pressure drop across the membrane increments in .A-+and z* directions eddy diffusivity weighting factor (= OS)
lksaiina&n.
7 (t969/70) 5 1-74
FlN1l-E
DIFFERENCE
V
P Tw
-
SOLUTION
FOR
RO IN TURBULENT
FLOW
kinematic viscosity density wall shear stress
INTRODUClIOh
A number of investigations have been reported in the literature in recent years relating the performance of reverse osmosis systems with system configurations and operating variables. The earlier work of Merten (7) described the concentration polarization phenomenon in reverse osmosis operation based on onedimensional dilfusion and film model. A more exact analysis was performed by Sherwood, Brian, Fisher and Dresner (9). Based on Berman’s expression for the velocity profile, the two-dimensional diffitsion equation was solved in a manner similar to that of the classical Craetz-Nusselt problem assuming constant longitudinal water flux across the membrane. It was pointed out in a subsequent study (5) that this rather arbitrary assumption of constant water flux can be removed and a perturbation solution valid for short axial length was obtained_ A finite difference solution for the same problem has also been carried out (I). For a more general t~vo-dimension~1 case. Gill, lien and Zeh (6) obtained an asymptotic solution to the diffusion equation, in which the velocity profile within the concentration boundary layer was approximated by a linear expression because of the relatively high value of the Schmidt number of the water-salt system. The approximate method of integral solution has also been found useful for the solution of reverse osmosis problems and studies on the simultaneous development of velocity and concentration profiles for a number of cases have already been made (IO. I I). Although different assumptions have been made in these studies, the flow conditions of the works cited above (5, 6, 9-11) were assumed to be laminar. The assumption of laminar flow for reverse osmosis process is well understood since almost all of these earlier studies were concerned primarily with d~aiination of sea water. For sea water with an average salt concentration of 3.5 %, the osmotic pressure is around 360 psi. In terms of the membranes available today as well as those anticipated in the near future, the operating pressure for a reverse osmosis system should be about 1000 psi in order to have reasonable production capacity. In view of this relatively high operating pressure and also owing to the fact that thp: distance between adjacent membranes is extremely narrow. the flow within a membrane duct would necessarily be laminar. The same consideration. however, does not remain valid if fairiy dilute systems are to be treated such as the treatment and renovation of waste water. In turbulent flow, mass transfer rates are usually enhanced by the added eddy diffusion. Consequently, concentration polarization in the turbulent Row case would be somewhat Iess severe and so does its detrimental effect_ Turbulent flow Desalination, 7 ( 1%9/70) 5 l-74
54
S. SRINIVASAN AND CHt TlEN
in reverse osmosis may very well be more advantageous
even mith added pressure
drops. For reverse
osmosis
flow, Dresner (J) obtained the following
in turbulent
expression;:
U-1) Similar
above
expressions ha\.e also been given by Sherwood and co-workers (8). The expression, undoubtedly valuable in a qualitative sense. does not yiefd
precise information required for a better understanding of the process because of the substantial
difference
operating conditions.* interpretation
between
the basic assumptions
Consequently,
used and
the actual
great caution must beexercised for the proper
of the results of these
expressions. For example, an average value of u, needs to be defined since in practice c, does not remain constant but decreases along the axial direction. The object of this investigation is to present a detailed analysis on the mass transfer characteristics of reverse osmosis in turbulent flow. Specificaliy, the system considered is that of paraftel membrane sheets with salt solution flowing in between. A finite difference solution for the pertinent diffusion equation using Deissteis expression for eddy diffusivity is obtained using a two-step linearized finite difference scheme. The use of finite difference is primarily
due to the fact
that comparativeljr fewer assumptions are involved in this method. Furthermore, the finite difference method is applicable to both large and small longitudinal distances and, therefore, less restrictive as compared with the series solution employed for the Iaminar case. s,,~,,,t,,t,,t,,t.,~t,t.,,~l,,t
-___--.______._ Cx-
. ... *,,,*,.
1
2b
,
‘I
G
MEMBRANE
Fig 1. Schematic diagram of reverse osmosis duct. FORMULATlON OF PROBLEM A
solution -----
schematic description of the physical problem is given in Fig. 1. Salt with an initial concentration cr and osmotic pressure (PO,), under pressure
l An implicitassumptionin the one-dimensionalmodel is that theconvectiveterm in the diffusion equation cart be negfcctcd. An order of magnitude analysis given in the appendix indicates that this is incorrect.
Lkafktatioff* 7 f 1969170) 5l-74
FINITE DIFFERENCE SftLUTlON FOR RO IN TURBULENI-
FLOW
- 55
is forced through a flat duct whose walls are made of parallel semipermeable membranes. The distances between the membranes is 26. The flow is assumed to be turbuknt. Because of symmetry, it is only necessary to consider haif of the conduit. The pertinent equation of change which describes the transport by diffusion and convection, in dimensionless form. can be written as follows:
(H-2)
(W3a)
The boundary conditions are:
c+=fatx’=U, -
dZ+
osz+
51
= Oat=+ = 1, X4 > 0
(II-S) J
(H-6)
(U-7) where IV%is the Schmidt number and is taken to be 560. In Eq. (H-7), it is implicitly assumed that the membrane rejects the sait completely. The water fiux. u,, across the membrane is given by the phenomenologicai law u,,. =
ACAP - W,,lJ
(H-8)
where A is the membrane constant, AP is the pressure drop across the membrane and P, is the osmotic pressure. Assuming that the conceutrat~on variation of osmotic pressure is linear, one has>
S. SRINWASAN
56
c
=
A(Ap)
1 _
_%??
AP
f
The dimensionless Of Y=
quantity,
f?
cI
AND CHI TtEN
1
(H-9)
ut, is then
c 1 Bz c,,+ ._.A... - _.___-_. - -_--1 Bz 1 Bl L’W,
(II-IO)
and
4 = (F,,hfAf’
(II-1 I)
The fraction of water recovered, Qfr may be obtained by integrating the wail velocity with respect to the axial distance. Q’ is given by the following equation: (M-12)
For the solution of Eq. (II-I), it is necessary to describe the velocity as the eddy diffusivity which is part of the term, A. ft is assumed that permeation velocity. L:,, is of much smaller magnitude as compared velocity, U. its effect may be ignored and the velocity expressions given (3) may be used. These are: 9’
u+ =
dj++ -----.---.-f o I + (0.124)’ ~+p+ [I - exp (-
-0.1242 u+y’)J
field as wet1 because the to the axial by Deissler
(11-13a)
)‘+ < 26 and y’
> 26
(II-13b)
where u: is taken to be the value of tl+ at J+ = 26. The transverse velocity component. v+. is assumed to be independent of c+ and equal to the vatue of u:_ The eddy diffusivity term, Jr, is: 5 = (0.124)* u+r* a’
-
ditions
E = 0.36y+ v
(f
-
[l - exp(-
5).
0.t242 u+)t+)],
r+ -C 26
(IL14a)
y+ > 26
Eqs. (If-l) together with Eqs. (ILt3a. b)-(fi-14a, b) and the boundary conof Eqs, (II-9-o-(1-7), give a complete description of the physical problem Desafina!ion, 7 (I 969170) 5 1-74
RSITE
MFFERENCE
SOLUTION FOR RO 183TURBULENT
57
FLOW
and the solution of this system of equations represents the main object of the present investigation. It should be mentioned that the parameter, b+, present in Eq. (U-l) varies along the axial direction because of the continuous permeation of pure water across the membrane. The quantity. h”, can be related to the equivalent Reynolds number, iv,,,, as follows: (U-15) .
Furthermore. the friction factor, f. can be expressed in terms of &,, the well known Blasius equation. f = 0.079 [A&-J - ‘I4
by
(11-16)
Since water is being withdrawn from the system. &,, decreases continuously. As a consequence, the parameter, b’, the velocity profile, u”, as well as the eddy diffusivity expressions all vary along the axial direction. The manner in which these variations arc accounted for will be described in the following section. SUMERICAL
SOLUflON
A two-step linearized finite difference scheme which has been proved successful in the past (I) is used for the solution of Eq. (11-l) with the boundary conditions of Eqs. (ll-5)-(II-7). A brief description of the procedure is given as follows. The first step of the computation schemeconsists of obtaining an approximate concentration profile for the advance longitudinal position in a marching process. i.e. to obtain concentration value at the i + 1st X+ position using the concentration values at the preceding mesh points. For this purpose. a corresponding difference equation of Eq. (11-I) is obtained by approximating the various derivatives as follows: dc+ -= 8.X” dC’
B=+=
+
Ci+ ---
+ cS.i
1-j -
(fit-l)
Ax +
l ci+,.j+i _---
-
2Az’
+ ci+l.i-1 --_
(111-2)
_ ~l+l/tCi++l.i+l- CAj+l/ __
2
+
Aj-ljZlCl*+* e.m--d
j +
(AZ”)’
Aj-112
c,+,l./-* (111-3)
The mesh network is shown in Fig. 2. Substituting Eqs. (III-lt(III-3) into Eq.
the foflowing expression is obtained: Dedhafion,
7 (t%9/70) 51-74
S. SRINIVASAN
Fig. 2. M&
2ah; +
-
I’,. j -
b;,. +
ui : J.
[NRC,] --_--VI.\
i?k,‘-l tAj+,,,) (AZ’)2 L___^-._..--
(hj+
c.
TEEN
ai2
f
-e
c+l.f+1
-
hj-$[2)
(AZ +)2
Ax+
_
CHI
net work.
iNR,J CNR=J + -.
+
AND
I
INRLI IINRQ~ + --_-261=+ (111-4)
Eq. (W-4) for j = 2, 3. . s n + 1, yields n equations with n + 2 unknowns, Le. 1 corresponds to the membrane wall and c:*,,j.j= 1,2...n + l,nf2(j= = n -t I corresponds to the axis of the channel). However, the boundary coni dition of Eq. (H-6) yields CT+, .n+2 = c++ x.sa Also, from Eq. (II-7). one has, 1
AZ+
-
CNR~.,I CNscl$+,.a &.,
-
-&e:c,.x=0
(W-5)
Here, ~t.+~., is approximated by u:, I. Thus, one now has n + I equations for the n f I unknown concentrations in the advance xi position. The system of linear tridiagonal equations is solved by Thomas method (2). This method is equivalent to plain Gaussian elimination, but it is known toavoid the error growth associated with the back solution of the elimination method. The first approximations for CT+, , I through CT+ ,,*+ t are thus obtained. Knowing c: + ,, I, -GT may be obtained from Eq. (II-IO). The bar over uf + I _, indicates that the value is only the first approximation. The second step involves tlie formulation of the Crank Nicholson equation for Eq. (II-l). This, after some rearrangement, is given as:
FiNfTE DIFFERENCE SOLUTION
+
b+j 4j
___YK~_
FOR RO IN TURBULENT
U%c,l --.---
+
c*&+I,, ---.--
+
Jbi2)\ -
(AZ +)2
59
FLOW
-.._
1
cl++ 1, j
where, (III-7)
G=(l
f-f =
-
‘iI’
&+I
cc,:,*,- cifj-II lJ*
[Aj+
ctj+
1
-
tAj+
(W-8)
I/Z
+
Aj- I,*)
Ctj
f
Aj-
112
cLj-
II
(M-9) and E* is is the weighting coefficient. For the present case. a vafue of z* = 0.5 corresponding to Crank Nichotson weighting is used. Similar to the previous step, together with the boundary conditions of Eq. (11-6) and Eq. (W-5), Eq. (fit-6) defines n + I equations with n i- I unknowns. i.e. ci+ , _j. j = I, 2 . . . n + I. The value of vi+, , which appears _. :_ in Eq. (111-S) isapproximated by the arithmetic average of Y: +$,, and VT. ,_ This system of II + I equations in the unknowns. cf +, . , through CT+, _+ ,, is then solved -_._ and_ the new concentration values are used to obtain a second approximation which is then used for another round of calculations. This iteration of V:+l.It procedure may be carried out as many times as required, although in almost all cases, three iterations were found adequate. The concentration values from the Iast iteration performed are taken to be the correct values. This, in turn, are the vaiues used in the calculations for the next advance _yf position. The variation in /VRcr: Reynofds number based on equivalent diameter, is given by, CNtcrl..
= CNtcrJ, (1 - Q’)
(III-
LO)
The subscript
I refers to the value at the inlet. Q‘ is obtained from Eq. (11-12). Using Eqs. (H-15) and (II-l@, it can be shown that,
@+I,+= b,+ (1
-
(III-1 1)
Q+)“’
bf is calculated for each step and the current value of bc is used in the computation scheme. As pointed out earlier, the velocity and eddy diffusivity expressions involve b* and hence need to be recalculated as b+ varies along the longi~d~na~ Desahation.
7 (1969/70) 51-74
S. SRINIVASAN AND CHl TEN
60 direction.
in the computation performed in this work, these values were recatfive percent increase in the water recovery instead of after every step. This is believed to be a reasonable balance between computer time involved culated for every
in recalculating
the quantities
and adherence
to the physical
model.
RESULTS AND DISCUSSION
The numerical comp&tations were performed on an IBM System1360 computer. The following combination of parameters was considered.
digitaf
b: = 100,200,300 (These correspond
to inlet Reynolds
numbers
[A&],
of 5967, 13175 and 20941,
respectively.) NRc,
=
0.006,0.015
82 = 0. l/4, 113, l/2 to very dilute solutions with high operating pressures.) = 0 corresponds For each set of specified value of b:, NRC, and B2, the quantities c”,, c:. Q‘, of sc. The results for cz. t*.~.X+ are hC and NRcr: were calculated as functions presented graphically in Figs. 3 to 8. Figs. 9 and 10 show typical results for 0: and Q’ as functions of _\r+for different values of B, forb: = 200and Iliaecy= 0.015.
(B,
Fig. 3. Dimensionless
concentration
It was stated in the expected to be reduced with of salt buildup due to the number) is clearly indicated
previous
LISa3function of dimensionless
section
longitudinal
that concentration
distance.
polarization
is
the increased eddy diffusion. The extent of reduction increase of turbulence (in terms of inlet Reynolds in Figs. I1 and 12 in which values of c”, were ptotted Deralinotion.
7 (1969/70)
51-74
FINITE DIFFERENCE
SOLU-l+ION FOR RO IN ~RBULE~
FLOW
D OIYEWflONLESS
LOUCfTUOlNAL
OISTIWCE.
I’
Fig. 4_ Dimensionless concentration as a function of dimensionkss longitudinal distance.
Fig. 5. Dimensionless concentration as a function of dimcnsionbs
longitudinal distana.
x+ with specified values of NRC, and B2 for different values of [iV&j,. Because of the presence of the Reynolds number, in the definition of x+, all the abscissae in these figures were referred to a common value ot’ [Nx& The significant reduction of c”, with the increase of inlet Reynolds number becomes quite obvious. Furthermore, the decrease becomes more pronounced with the decrease cf the value of B, as we11 as the decrease of Nat,. As an exampie, at X+ = 20 (based on b: = 300), the reduction of the excess salt concentration buildup
against
Desalinalion. 7 (19&9]70) 5 I-74
S. SRWIVASAN
62
Fig. 6. Dimensionfess
concentration
as a function of dimensionless
AND CWI ‘I-EN
longitudinal distant.
. DIMENSIONLESS
LONGfTUDilllL
Fik. I. Dimensionless
DISTANCE.
concentration
s+
as o function of dimensionless &SdiMtiOlt.
bngitudinat
distance
7 ( 1969170) 5 I-74
FINITE DIFFERENCE SOLUTlON FOR RO IN TURBULEbX
D~YENSIOHLESS
LONGINOINAL
OISTANCE.
FLOW
63
x-
Fig 8. Dimcnsionkss concentration as a function of dimensionless longitudinal distance.
OIYENStONLESS
WWGITUOIHAL
Fig 9. Dimcnsionks
OISTAKCE. P
wall v&city
VS.dimensionless axial distamx.
(defined as c”, - I) is approximately 53 % with a 3.5 fold increase of inlet Reynolds number (from 5967 to 20941) for the case of A& = 0.015 and B, = 0.5. The corresponding reduction for the case of JVRs, = 0.006 and B, = 0.25, however, was found to be 82%. The effect of the parameters B, and NRC, on the production capacity of the reverse osmosis systems is indicated in Tables I and Ii. Based on the definitions given in the nomenclature, it can be seen that the actual amount of pure water Lksu/4kztioll, 7 ( 1969170) 5 1-74
S. SRCNlVASAN AND CfiI TtEN
Fig. IO. Fraction of water: recovery VS.dimensiontess longitudinal distance.
Fig. 1i. ERkct of inlet Reynolds number on eanantration
polarization.
produced is proportional to the product of Q* and [h&&, which was found to be increasing with the increase of [iVRcr],+ Also, this increase is most pronounced for lower values of B2 and higher values of NRcWrwhich, physically speaking. corresponds to the case of very dilute feed stream operating at relatively high pressures with high capacity membranes. It is interesting to note that these conditions are most likely to be met if reverse osmosis is to be developed as a practical means for treatment of waste water. DesalinaGo~. 7 (1969170) 51-74
t24,12
_-.
.w.-"..-..^__.
0.2899 3819 --
---
0,222t 4650
20.0 p iii?_I-_
271.63 0.1278 0.2198 2084 f212 0.2830 2684 0.3209 3043
fj
*
?
..-
0.01294 122.72 O*OiA t 607.50 0.1246 1187 0.1841 1752 0,2406 2282
_-
a1804 0.2423 2311 0.09986 3192 1316
-e-1_--_-
%+--*+_
0.06676 0.1783 2637 OJ259 3733 1398
--
0.00934 123.0 0.0467 612.24 0.09174 IZOI, 0.1353 1782 0.1746 2353
-----vm.-.-._--“-
IS,0 10.0 5.0
--
0.0286
-
0.02139 28f.76
-1
--.”
0.01397 292,54
..--... -
0.0294 6t5,24 0.05826 1220 0.087 1822 0.1147 2402
0.~592~
_..._- ._ _.....+“__
*1-.-e.__^
-.*..G
---.-p__
591.38
-
.“..--
--.....__
0,1717 0*26$2 1024 1582 0,3tas 1852 0.332 1981
0.04174 249.0
r
0,191 ll3tJ 0.2734 1631 0.34iY 2081
om9t
0,1)2022 t20.65
I-
CNnoA=9484 e-P-". [N&j = 5967 Q' Chdr(Q+~ Q’ INd,(Q+-I ---
b,+ i= IS0
1.0
4
TABLE11
1.0 5.0 10,O f"5.0 20.0
b: =200
[Nf&J, = 20941 [NRr,J,= 13175 ..a-Q" k%kl,~Qf~ Q" ChLii?)
b: = 300
z:0,006BI= 0.25 NRe,v
TABLEI
-
S. SRINWASAN AND CHI TIEN
1
.--_
. ..--_
_-__I
_.-._.
Fig. 12 Effectof inlet Reynoldsnumber on conocntration pofarizttion.
The results of this work are afso compared with those based on the onedimensional model. The one-dimensional diffusion equation based on complete salt rejection at the surface of the membrane can be written as: (IV-I)
Using Deissler’s expression for eddy diffusivity, is obtained (8):
Eq. (II-14a). the following result
(IV-2) A mass balance on the salt indicates that, cgf=_= CB c1
1 1-
Q’
In terms of the dimensionless may be written as,
(IV-3) quantities defined in the present work, E$. (IV-2)
Desdim~ion.7 (1%9/70) 51-74
FlNlTE
DiFFERENCE
SOLUTION
FOR RO IN TURBULENT
FLOW
67
where, @Z__
1
(IV-5a)
1 -B2
.{lV-Sb) ;
B = rrBl
Fig. 13. Comparison between present inestigation diffusion model.
and that based on one dimensional
. FRACTtON
0’ OFWATER RECWERY.
Fik 14. Comparison between present investigation and that based on one dimensional diffusion model. FIq. (W-4) defines the relationship between Q’ and c”, for a given combination 6, and B2. In order to obtain the numerical results of c”, US.Q’, a value of &,_* + is specified and the resulting implicit equation in c*, can be solved using of Q Newton Raphson iteration scheme, Comparison between the results of the finite difference solution and those of Eq. (W-4) are given in Figs. 13 and 14 for two cases, bt = 200, A&=, = 0.006. B2 = 0.5, and 6: = 150, NRC, = 0.015 and B, = Desalimlion, 7 (
1969/70) 5 I-74
68
S. SRINWASAN
AND
CHI
TtEN
0.5. in terms of excess salt buildup (Le. c*, - I), the one-dimensional model yields much lower values and tends to underestimate the extent of concentration polarization_ Furthermore, this underestimation increases as one proceeds downstream. A conventional way of describing the mass transfer characteristic is in terms of the mass transfer coefficient which can be defined as follows:
k
_=-
(W-6)
Accordingty, the Sherwood number, Nst,, becomes Jv,
!!i&s,.-
=
b
pb
=o
(W-7)
Kc3 - 4)
Substituting Eqs. (H-7) and (W-3) into Eq. (W-7), one has, NSh
!f!! = CbJ
=
CJLI (u:> Cc:, (1 (I
cugdr
JOC
. ii
100 too too 200 200 200
7 1) 9
0
q
I :
300 300
-
Q’)cT
(W-8)
Et 0 025 03 0 025 as 0 025 OS
I
20 DIYEMSIOMLESS
Q')
I
40 60 UwrGllUOlWL
m
OISTANCE.
a+
Rb
Fig. ISa. Shcxwood number vs. dimensionless Iongitudinal distance.
Numerical values of the local Sherwood number 2s 2 function of x+ are shown in Figs. 1% and 15b for a variety of cases. For smail lon~tudina~ distances Desahzfion.
7 (1%9/70) 51-74
F~M3-E DIFFERENCE
Fig
SOLUTION
1%~ Stiood
FOR RO 1N TURBULENT
number vs. dinunsionkss
FLOW
fongitudinal
69
distance.
(x+ I IS), A&, was found to be essentially independent of &. For X’ > 20, A& decreases with the increase of B,. However, the effect to Bz on NsL is much fess as compared with that on other pertinent parameters (for example, compare Fig. 3 with Fig. 15). For a numerical treatment such as given in this work. the question of predicting the results for conditions not specifically covered in the numerical cakulation always presents some difficulty because of the absence of the closedform expression. Owing to the large number parameters involved in this work (i.e., A&,_ bt, B, and N,,), only limited cases can be considered- The prediction of ali pertinent quantities by direct interpofation, therefore, becomes difficult. On the other hand, the most important quantity from a practical point of view is that of the wall concentration, c”,. It is believed that a reasonable procedure for the prediction of CL is through the use of expression for A$,, given by Eq. (W-8). combining this expression with Eqs. (II-IO) and (U-12) yields,
flmahation.
7 (1969f70) 51-74
70
S. SRINlVASAN AND CHI TIEN
For a given set of parameters, the Iocal Sherwood number as a function of _v+ can be estimated by interpolation based on the data given in Figs. 1%. b. Once this information becomes available, the relationship of c: us. x+ can be obtained from Eq. (IV-g). The main advantage of this procedure lies in the fact of the less dependency of Nsb on the parameters, 6:, NRcc and B, as discussed previously. Consequently, one can predict values of Nsb by interpolation with more ease. It should also be mated that in the preser?r work, the Schmidt number is taken as 560 (i.e. NaCI-H,O System). However, if the we&established relationship of Nsb being proportional to the cubic root of N,, is assumed to be also valid for the present problem, the performance of a reverse osmosis system in turbutent flow with systems other than NaCI-H,O can be easily predicted by this procedure. ACKNOWLE3xiEMEN-r
This work was performed under Grant No. WP-00968432, Federal Water Pollution Control Administration, Department of Interior. The authors also wish to thank the Syracuse University Computing Center for the use of its facilities. APPENDIX ORDER OF MAGNITUDE
ANALYSfS OF DlFFUSION
EQUATXON IN NRBULEh?
FLOW
OF
REVERSE OSSiOSS OPERATION
Equation (II-I) is
The magnitude
of u within
the concentration
lished as
= (b,+)(i
-
(t)
Q+)-'
boundary
layer can be estab-
FiNlTE DIFFERENCE SOLUTtOON FOR RO IN TURBULENT
FLOW
7L
since b+ is defined as (r,,,/f7)*/~(6) by Eq. (II-Q)of the manuscript and the relationship of b+ and 6: is given by Eq. (III-1 1) of the manuscript. For the estimation of the magnitude of (6/b), the result of one-dimensional model can be used, or
Combining
these expressions, one has,
= (f),r,(%_)(b,N~3
=
(:)(-&-)
(b)
N&It3
OI-
Note that NRcr:is defined as four times that of NRt as was done in the manuscript. From Eq. (II-IS) one has
0’
fz(32)
(NRA’
so that _
(NRC,) 4
Also, from Equations (III-IO) and (III-l 1) WR&“=
(NRA
(b‘),(I
(b+)x+=
(1
-
-
Q’)
Q+)7’R
Hence, 6
(T-)
-
(N;c)f
(&)[
&--‘(l
_
LQ+)3/4
For the present case, N RG..= O_OI5 De&ha&m,
7 (1969170) W-74
72
S.SRMIVASAN AND CHI TIEN
(b*)f = tm (N,)
= 560
Q-+=
0.35
the value of (IV,& the manuscript,
can be found by combining
Equations (f.&I5) and {I#-46) of
With these numerical data, the order of magnitude of(W) 0.0177, LE.,
can be estimated as
P
(T ) - (10-q Consequently, u+ -
one has, 1.81 or (IO’)
The concentration derivative, <&“@x*) can be obtained from the numericaf data of this work. Lt was found that at C’z = 1.9, (of _Y” = 60)
‘l%e magnitude of the term, (&[Czj, can be estimated by using the botindary condition using Eq. (H-7) of the manuscript, or
ac+ = - W,,,,,XNs,XKiXC,+) a2 By EC&(II40) of the manuscript, -
73
For the present &se, 8, = t/t, c+_ = 1.9
The magnitude of Y+ can be taken as that of YL. i.e.. v+ - (lo-‘) The magnitude of f&‘/&z*)
is given as
So the relative magnitude between the first and second terms on the left side of the diffusion equation (Eq. (II- 1)) can be evaluated: TIte First Term
Consequently,
both terms ,are of comparable
that one can neglect
the axial term without
magnitude
introducing
and does not appear
substantial
error.
REFERENCES
1. P. L. T. BFUAN,Id. Eng. C/tent. Fun~u~~e~r~rs,4 (l%S) 439. 2. G. H. BRUCE, D. W. PEACEMN AND J. RACHFORD, JR., Pm&m 198 (1953) 79.
Transactkwts cf.l.M.E.,
&xahaffon,
7 f 1969/70) 5 l-74
S. SRlNlYWAN
74 8. T.
9
K.SHERWOOD.
T_ K Smwmo.
AND CHI TlEN
P. L. T. BRIAN AND R. E. Ftstwt. M.I.T. Rep? No 295-l DSR9336(1963). P. L T. &UAN. R. E. FIWER AND L. DREW~, hf. &kg. Chem, Ftmda-
menrds, $ (1965) 113. 10. S. sRllWM.fi&X. Cm TIEB AND W. N. GILL, Chem &. Sri., 22 (l%7) 417. I I. S. SRW%*ASAN ARXX CM TIEN, Rcwrsc Osmosis Desalination in Tubular Membrane Duct, Drsa&Wtion, 3 ( 1%7f 5.
Publishing Company,
A FINITE DIFFERENCE BULENT FLOW S. SRINIVASA~‘* Department
Amsterdam - Printc.8 in The Netherlands
SOLUTION
FOR REVERSE OSMOSIS IN TUR-
ASD CHI TlEN
o/Chemical
(Received in rcvird
E~girreering ad
form
.\fetatiurg.rp Syucuse
Universif_v, Syracuse. N. Y. f U.S.&)
November 2ft. 1968)
A finite difference solution for reverse osmosis in turbulent flow is presented. The cast of a flat duct made of two parallel semi-permeable membranes is considered. The pertinent diffusion equation with the eddy difksivity based on Deissler’s expression is solved using a two-step linearized finite difference scheme. taking into account the nonlinear effect due to the varying water flux across
membrane along the axial direction. Numerical conditions are given for a variety of cases.
results relating concentration
SY MSOLS
R
-
BZ
-
membrane constant (Po.)rAp
half the distance between membranes
I?
-
b+
=
b;
-
value of b+ at inlet
-
bulk concentration
=B
=
4 CI c
-
c+
=
=w
-
c;
_Y G H
l
=
-
cdcr satt concentration salt concentration
at intet
c/c* wall concentration CM/C1
salt diffusion coefficient friction factor defined by Eq. W-8) defined by Eq. (III-Q)
Presentaddress:Departmentof Chemicai Engineering,University of Dayton, Dayton, Ohio. Deshtation.
7 (1%9/70)
5 ! -74
S. SFUNlVASAN AND CHI TEN
position
index (_\T+direction) position index (z’ direction) defined by Eq. (IV-6) t?uJ\ 4 IV,,,. Reynolds number based on equivalent N,, at inlet = bL’,/v N RCCat inlet
diameter
defined as -!?Y f Schmidt number, r;lD osmotic pressure osmotic pressure at inlet fraction of water withdrawn axial component of velocity u/(MWf r.l+ at Y+ = 26 butk vefocity bulk velocity at inlet transverse veto&y &*., wail velocity wall velocity at X+ = 0 axial coordinate -~/C(W%,,)I transverse coordinate 1.\’ -t,* ” ( P ) (f/b+) -1-
= (J-/b)
1 B,
ah
defined by Eq. (I-l) pressure drop across the membrane increments in .A-+and z* directions eddy diffusivity weighting factor (= OS)
lksaiina&n.
7 (t969/70) 5 1-74
FlN1l-E
DIFFERENCE
V
P Tw
-
SOLUTION
FOR
RO IN TURBULENT
FLOW
kinematic viscosity density wall shear stress
INTRODUClIOh
A number of investigations have been reported in the literature in recent years relating the performance of reverse osmosis systems with system configurations and operating variables. The earlier work of Merten (7) described the concentration polarization phenomenon in reverse osmosis operation based on onedimensional dilfusion and film model. A more exact analysis was performed by Sherwood, Brian, Fisher and Dresner (9). Based on Berman’s expression for the velocity profile, the two-dimensional diffitsion equation was solved in a manner similar to that of the classical Craetz-Nusselt problem assuming constant longitudinal water flux across the membrane. It was pointed out in a subsequent study (5) that this rather arbitrary assumption of constant water flux can be removed and a perturbation solution valid for short axial length was obtained_ A finite difference solution for the same problem has also been carried out (I). For a more general t~vo-dimension~1 case. Gill, lien and Zeh (6) obtained an asymptotic solution to the diffusion equation, in which the velocity profile within the concentration boundary layer was approximated by a linear expression because of the relatively high value of the Schmidt number of the water-salt system. The approximate method of integral solution has also been found useful for the solution of reverse osmosis problems and studies on the simultaneous development of velocity and concentration profiles for a number of cases have already been made (IO. I I). Although different assumptions have been made in these studies, the flow conditions of the works cited above (5, 6, 9-11) were assumed to be laminar. The assumption of laminar flow for reverse osmosis process is well understood since almost all of these earlier studies were concerned primarily with d~aiination of sea water. For sea water with an average salt concentration of 3.5 %, the osmotic pressure is around 360 psi. In terms of the membranes available today as well as those anticipated in the near future, the operating pressure for a reverse osmosis system should be about 1000 psi in order to have reasonable production capacity. In view of this relatively high operating pressure and also owing to the fact that thp: distance between adjacent membranes is extremely narrow. the flow within a membrane duct would necessarily be laminar. The same consideration. however, does not remain valid if fairiy dilute systems are to be treated such as the treatment and renovation of waste water. In turbulent flow, mass transfer rates are usually enhanced by the added eddy diffusion. Consequently, concentration polarization in the turbulent Row case would be somewhat Iess severe and so does its detrimental effect_ Turbulent flow Desalination, 7 ( 1%9/70) 5 l-74
54
S. SRINIVASAN AND CHt TlEN
in reverse osmosis may very well be more advantageous
even mith added pressure
drops. For reverse
osmosis
flow, Dresner (J) obtained the following
in turbulent
expression;:
U-1) Similar
above
expressions ha\.e also been given by Sherwood and co-workers (8). The expression, undoubtedly valuable in a qualitative sense. does not yiefd
precise information required for a better understanding of the process because of the substantial
difference
operating conditions.* interpretation
between
the basic assumptions
Consequently,
used and
the actual
great caution must beexercised for the proper
of the results of these
expressions. For example, an average value of u, needs to be defined since in practice c, does not remain constant but decreases along the axial direction. The object of this investigation is to present a detailed analysis on the mass transfer characteristics of reverse osmosis in turbulent flow. Specificaliy, the system considered is that of paraftel membrane sheets with salt solution flowing in between. A finite difference solution for the pertinent diffusion equation using Deissteis expression for eddy diffusivity is obtained using a two-step linearized finite difference scheme. The use of finite difference is primarily
due to the fact
that comparativeljr fewer assumptions are involved in this method. Furthermore, the finite difference method is applicable to both large and small longitudinal distances and, therefore, less restrictive as compared with the series solution employed for the Iaminar case. s,,~,,,t,,t,,t,,t.,~t,t.,,~l,,t
-___--.______._ Cx-
. ... *,,,*,.
1
2b
,
‘I
G
MEMBRANE
Fig 1. Schematic diagram of reverse osmosis duct. FORMULATlON OF PROBLEM A
solution -----
schematic description of the physical problem is given in Fig. 1. Salt with an initial concentration cr and osmotic pressure (PO,), under pressure
l An implicitassumptionin the one-dimensionalmodel is that theconvectiveterm in the diffusion equation cart be negfcctcd. An order of magnitude analysis given in the appendix indicates that this is incorrect.
Lkafktatioff* 7 f 1969170) 5l-74
FINITE DIFFERENCE SftLUTlON FOR RO IN TURBULENI-
FLOW
- 55
is forced through a flat duct whose walls are made of parallel semipermeable membranes. The distances between the membranes is 26. The flow is assumed to be turbuknt. Because of symmetry, it is only necessary to consider haif of the conduit. The pertinent equation of change which describes the transport by diffusion and convection, in dimensionless form. can be written as follows:
(H-2)
(W3a)
The boundary conditions are:
c+=fatx’=U, -
dZ+
osz+
51
= Oat=+ = 1, X4 > 0
(II-S) J
(H-6)
(U-7) where IV%is the Schmidt number and is taken to be 560. In Eq. (H-7), it is implicitly assumed that the membrane rejects the sait completely. The water fiux. u,, across the membrane is given by the phenomenologicai law u,,. =
ACAP - W,,lJ
(H-8)
where A is the membrane constant, AP is the pressure drop across the membrane and P, is the osmotic pressure. Assuming that the conceutrat~on variation of osmotic pressure is linear, one has>
S. SRINWASAN
56
c
=
A(Ap)
1 _
_%??
AP
f
The dimensionless Of Y=
quantity,
f?
cI
AND CHI TtEN
1
(H-9)
ut, is then
c 1 Bz c,,+ ._.A... - _.___-_. - -_--1 Bz 1 Bl L’W,
(II-IO)
and
4 = (F,,hfAf’
(II-1 I)
The fraction of water recovered, Qfr may be obtained by integrating the wail velocity with respect to the axial distance. Q’ is given by the following equation: (M-12)
For the solution of Eq. (II-I), it is necessary to describe the velocity as the eddy diffusivity which is part of the term, A. ft is assumed that permeation velocity. L:,, is of much smaller magnitude as compared velocity, U. its effect may be ignored and the velocity expressions given (3) may be used. These are: 9’
u+ =
dj++ -----.---.-f o I + (0.124)’ ~+p+ [I - exp (-
-0.1242 u+y’)J
field as wet1 because the to the axial by Deissler
(11-13a)
)‘+ < 26 and y’
> 26
(II-13b)
where u: is taken to be the value of tl+ at J+ = 26. The transverse velocity component. v+. is assumed to be independent of c+ and equal to the vatue of u:_ The eddy diffusivity term, Jr, is: 5 = (0.124)* u+r* a’
-
ditions
E = 0.36y+ v
(f
-
[l - exp(-
5).
0.t242 u+)t+)],
r+ -C 26
(IL14a)
y+ > 26
Eqs. (If-l) together with Eqs. (ILt3a. b)-(fi-14a, b) and the boundary conof Eqs, (II-9-o-(1-7), give a complete description of the physical problem Desafina!ion, 7 (I 969170) 5 1-74
RSITE
MFFERENCE
SOLUTION FOR RO 183TURBULENT
57
FLOW
and the solution of this system of equations represents the main object of the present investigation. It should be mentioned that the parameter, b+, present in Eq. (U-l) varies along the axial direction because of the continuous permeation of pure water across the membrane. The quantity. h”, can be related to the equivalent Reynolds number, iv,,,, as follows: (U-15) .
Furthermore. the friction factor, f. can be expressed in terms of &,, the well known Blasius equation. f = 0.079 [A&-J - ‘I4
by
(11-16)
Since water is being withdrawn from the system. &,, decreases continuously. As a consequence, the parameter, b’, the velocity profile, u”, as well as the eddy diffusivity expressions all vary along the axial direction. The manner in which these variations arc accounted for will be described in the following section. SUMERICAL
SOLUflON
A two-step linearized finite difference scheme which has been proved successful in the past (I) is used for the solution of Eq. (11-l) with the boundary conditions of Eqs. (ll-5)-(II-7). A brief description of the procedure is given as follows. The first step of the computation schemeconsists of obtaining an approximate concentration profile for the advance longitudinal position in a marching process. i.e. to obtain concentration value at the i + 1st X+ position using the concentration values at the preceding mesh points. For this purpose. a corresponding difference equation of Eq. (11-I) is obtained by approximating the various derivatives as follows: dc+ -= 8.X” dC’
B=+=
+
Ci+ ---
+ cS.i
1-j -
(fit-l)
Ax +
l ci+,.j+i _---
-
2Az’
+ ci+l.i-1 --_
(111-2)
_ ~l+l/tCi++l.i+l- CAj+l/ __
2
+
Aj-ljZlCl*+* e.m--d
j +
(AZ”)’
Aj-112
c,+,l./-* (111-3)
The mesh network is shown in Fig. 2. Substituting Eqs. (III-lt(III-3) into Eq.
the foflowing expression is obtained: Dedhafion,
7 (t%9/70) 51-74
S. SRINIVASAN
Fig. 2. M&
2ah; +
-
I’,. j -
b;,. +
ui : J.
[NRC,] --_--VI.\
i?k,‘-l tAj+,,,) (AZ’)2 L___^-._..--
(hj+
c.
TEEN
ai2
f
-e
c+l.f+1
-
hj-$[2)
(AZ +)2
Ax+
_
CHI
net work.
iNR,J CNR=J + -.
+
AND
I
INRLI IINRQ~ + --_-261=+ (111-4)
Eq. (W-4) for j = 2, 3. . s n + 1, yields n equations with n + 2 unknowns, Le. 1 corresponds to the membrane wall and c:*,,j.j= 1,2...n + l,nf2(j= = n -t I corresponds to the axis of the channel). However, the boundary coni dition of Eq. (H-6) yields CT+, .n+2 = c++ x.sa Also, from Eq. (II-7). one has, 1
AZ+
-
CNR~.,I CNscl$+,.a &.,
-
-&e:c,.x=0
(W-5)
Here, ~t.+~., is approximated by u:, I. Thus, one now has n + I equations for the n f I unknown concentrations in the advance xi position. The system of linear tridiagonal equations is solved by Thomas method (2). This method is equivalent to plain Gaussian elimination, but it is known toavoid the error growth associated with the back solution of the elimination method. The first approximations for CT+, , I through CT+ ,,*+ t are thus obtained. Knowing c: + ,, I, -GT may be obtained from Eq. (II-IO). The bar over uf + I _, indicates that the value is only the first approximation. The second step involves tlie formulation of the Crank Nicholson equation for Eq. (II-l). This, after some rearrangement, is given as:
FiNfTE DIFFERENCE SOLUTION
+
b+j 4j
___YK~_
FOR RO IN TURBULENT
U%c,l --.---
+
c*&+I,, ---.--
+
Jbi2)\ -
(AZ +)2
59
FLOW
-.._
1
cl++ 1, j
where, (III-7)
G=(l
f-f =
-
‘iI’
&+I
cc,:,*,- cifj-II lJ*
[Aj+
ctj+
1
-
tAj+
(W-8)
I/Z
+
Aj- I,*)
Ctj
f
Aj-
112
cLj-
II
(M-9) and E* is is the weighting coefficient. For the present case. a vafue of z* = 0.5 corresponding to Crank Nichotson weighting is used. Similar to the previous step, together with the boundary conditions of Eq. (11-6) and Eq. (W-5), Eq. (fit-6) defines n + I equations with n i- I unknowns. i.e. ci+ , _j. j = I, 2 . . . n + I. The value of vi+, , which appears _. :_ in Eq. (111-S) isapproximated by the arithmetic average of Y: +$,, and VT. ,_ This system of II + I equations in the unknowns. cf +, . , through CT+, _+ ,, is then solved -_._ and_ the new concentration values are used to obtain a second approximation which is then used for another round of calculations. This iteration of V:+l.It procedure may be carried out as many times as required, although in almost all cases, three iterations were found adequate. The concentration values from the Iast iteration performed are taken to be the correct values. This, in turn, are the vaiues used in the calculations for the next advance _yf position. The variation in /VRcr: Reynofds number based on equivalent diameter, is given by, CNtcrl..
= CNtcrJ, (1 - Q’)
(III-
LO)
The subscript
I refers to the value at the inlet. Q‘ is obtained from Eq. (11-12). Using Eqs. (H-15) and (II-l@, it can be shown that,
@+I,+= b,+ (1
-
(III-1 1)
Q+)“’
bf is calculated for each step and the current value of bc is used in the computation scheme. As pointed out earlier, the velocity and eddy diffusivity expressions involve b* and hence need to be recalculated as b+ varies along the longi~d~na~ Desahation.
7 (1969/70) 51-74
S. SRINIVASAN AND CHl TEN
60 direction.
in the computation performed in this work, these values were recatfive percent increase in the water recovery instead of after every step. This is believed to be a reasonable balance between computer time involved culated for every
in recalculating
the quantities
and adherence
to the physical
model.
RESULTS AND DISCUSSION
The numerical comp&tations were performed on an IBM System1360 computer. The following combination of parameters was considered.
digitaf
b: = 100,200,300 (These correspond
to inlet Reynolds
numbers
[A&],
of 5967, 13175 and 20941,
respectively.) NRc,
=
0.006,0.015
82 = 0. l/4, 113, l/2 to very dilute solutions with high operating pressures.) = 0 corresponds For each set of specified value of b:, NRC, and B2, the quantities c”,, c:. Q‘, of sc. The results for cz. t*.~.X+ are hC and NRcr: were calculated as functions presented graphically in Figs. 3 to 8. Figs. 9 and 10 show typical results for 0: and Q’ as functions of _\r+for different values of B, forb: = 200and Iliaecy= 0.015.
(B,
Fig. 3. Dimensionless
concentration
It was stated in the expected to be reduced with of salt buildup due to the number) is clearly indicated
previous
LISa3function of dimensionless
section
longitudinal
that concentration
distance.
polarization
is
the increased eddy diffusion. The extent of reduction increase of turbulence (in terms of inlet Reynolds in Figs. I1 and 12 in which values of c”, were ptotted Deralinotion.
7 (1969/70)
51-74
FINITE DIFFERENCE
SOLU-l+ION FOR RO IN ~RBULE~
FLOW
D OIYEWflONLESS
LOUCfTUOlNAL
OISTIWCE.
I’
Fig. 4_ Dimensionless concentration as a function of dimensionkss longitudinal distance.
Fig. 5. Dimensionless concentration as a function of dimcnsionbs
longitudinal distana.
x+ with specified values of NRC, and B2 for different values of [iV&j,. Because of the presence of the Reynolds number, in the definition of x+, all the abscissae in these figures were referred to a common value ot’ [Nx& The significant reduction of c”, with the increase of inlet Reynolds number becomes quite obvious. Furthermore, the decrease becomes more pronounced with the decrease cf the value of B, as we11 as the decrease of Nat,. As an exampie, at X+ = 20 (based on b: = 300), the reduction of the excess salt concentration buildup
against
Desalinalion. 7 (19&9]70) 5 I-74
S. SRWIVASAN
62
Fig. 6. Dimensionfess
concentration
as a function of dimensionless
AND CWI ‘I-EN
longitudinal distant.
. DIMENSIONLESS
LONGfTUDilllL
Fik. I. Dimensionless
DISTANCE.
concentration
s+
as o function of dimensionless &SdiMtiOlt.
bngitudinat
distance
7 ( 1969170) 5 I-74
FINITE DIFFERENCE SOLUTlON FOR RO IN TURBULEbX
D~YENSIOHLESS
LONGINOINAL
OISTANCE.
FLOW
63
x-
Fig 8. Dimcnsionkss concentration as a function of dimensionless longitudinal distance.
OIYENStONLESS
WWGITUOIHAL
Fig 9. Dimcnsionks
OISTAKCE. P
wall v&city
VS.dimensionless axial distamx.
(defined as c”, - I) is approximately 53 % with a 3.5 fold increase of inlet Reynolds number (from 5967 to 20941) for the case of A& = 0.015 and B, = 0.5. The corresponding reduction for the case of JVRs, = 0.006 and B, = 0.25, however, was found to be 82%. The effect of the parameters B, and NRC, on the production capacity of the reverse osmosis systems is indicated in Tables I and Ii. Based on the definitions given in the nomenclature, it can be seen that the actual amount of pure water Lksu/4kztioll, 7 ( 1969170) 5 1-74
S. SRCNlVASAN AND CfiI TtEN
Fig. IO. Fraction of water: recovery VS.dimensiontess longitudinal distance.
Fig. 1i. ERkct of inlet Reynolds number on eanantration
polarization.
produced is proportional to the product of Q* and [h&&, which was found to be increasing with the increase of [iVRcr],+ Also, this increase is most pronounced for lower values of B2 and higher values of NRcWrwhich, physically speaking. corresponds to the case of very dilute feed stream operating at relatively high pressures with high capacity membranes. It is interesting to note that these conditions are most likely to be met if reverse osmosis is to be developed as a practical means for treatment of waste water. DesalinaGo~. 7 (1969170) 51-74
t24,12
_-.
.w.-"..-..^__.
0.2899 3819 --
---
0,222t 4650
20.0 p iii?_I-_
271.63 0.1278 0.2198 2084 f212 0.2830 2684 0.3209 3043
fj
*
?
..-
0.01294 122.72 O*OiA t 607.50 0.1246 1187 0.1841 1752 0,2406 2282
_-
a1804 0.2423 2311 0.09986 3192 1316
-e-1_--_-
%+--*+_
0.06676 0.1783 2637 OJ259 3733 1398
--
0.00934 123.0 0.0467 612.24 0.09174 IZOI, 0.1353 1782 0.1746 2353
-----vm.-.-._--“-
IS,0 10.0 5.0
--
0.0286
-
0.02139 28f.76
-1
--.”
0.01397 292,54
..--... -
0.0294 6t5,24 0.05826 1220 0.087 1822 0.1147 2402
0.~592~
_..._- ._ _.....+“__
*1-.-e.__^
-.*..G
---.-p__
591.38
-
.“..--
--.....__
0,1717 0*26$2 1024 1582 0,3tas 1852 0.332 1981
0.04174 249.0
r
0,191 ll3tJ 0.2734 1631 0.34iY 2081
om9t
0,1)2022 t20.65
I-
CNnoA=9484 e-P-". [N&j = 5967 Q' Chdr(Q+~ Q’ INd,(Q+-I ---
b,+ i= IS0
1.0
4
TABLE11
1.0 5.0 10,O f"5.0 20.0
b: =200
[Nf&J, = 20941 [NRr,J,= 13175 ..a-Q" k%kl,~Qf~ Q" ChLii?)
b: = 300
z:0,006BI= 0.25 NRe,v
TABLEI
-
S. SRINWASAN AND CHI TIEN
1
.--_
. ..--_
_-__I
_.-._.
Fig. 12 Effectof inlet Reynoldsnumber on conocntration pofarizttion.
The results of this work are afso compared with those based on the onedimensional model. The one-dimensional diffusion equation based on complete salt rejection at the surface of the membrane can be written as: (IV-I)
Using Deissler’s expression for eddy diffusivity, is obtained (8):
Eq. (II-14a). the following result
(IV-2) A mass balance on the salt indicates that, cgf=_= CB c1
1 1-
Q’
In terms of the dimensionless may be written as,
(IV-3) quantities defined in the present work, E$. (IV-2)
Desdim~ion.7 (1%9/70) 51-74
FlNlTE
DiFFERENCE
SOLUTION
FOR RO IN TURBULENT
FLOW
67
where, @Z__
1
(IV-5a)
1 -B2
.{lV-Sb) ;
B = rrBl
Fig. 13. Comparison between present inestigation diffusion model.
and that based on one dimensional
. FRACTtON
0’ OFWATER RECWERY.
Fik 14. Comparison between present investigation and that based on one dimensional diffusion model. FIq. (W-4) defines the relationship between Q’ and c”, for a given combination 6, and B2. In order to obtain the numerical results of c”, US.Q’, a value of &,_* + is specified and the resulting implicit equation in c*, can be solved using of Q Newton Raphson iteration scheme, Comparison between the results of the finite difference solution and those of Eq. (W-4) are given in Figs. 13 and 14 for two cases, bt = 200, A&=, = 0.006. B2 = 0.5, and 6: = 150, NRC, = 0.015 and B, = Desalimlion, 7 (
1969/70) 5 I-74
68
S. SRINWASAN
AND
CHI
TtEN
0.5. in terms of excess salt buildup (Le. c*, - I), the one-dimensional model yields much lower values and tends to underestimate the extent of concentration polarization_ Furthermore, this underestimation increases as one proceeds downstream. A conventional way of describing the mass transfer characteristic is in terms of the mass transfer coefficient which can be defined as follows:
k
_=-
(W-6)
Accordingty, the Sherwood number, Nst,, becomes Jv,
!!i&s,.-
=
b
pb
=o
(W-7)
Kc3 - 4)
Substituting Eqs. (H-7) and (W-3) into Eq. (W-7), one has, NSh
!f!! = CbJ
=
CJLI (u:> Cc:, (1 (I
cugdr
JOC
. ii
100 too too 200 200 200
7 1) 9
0
q
I :
300 300
-
Q’)cT
(W-8)
Et 0 025 03 0 025 as 0 025 OS
I
20 DIYEMSIOMLESS
Q')
I
40 60 UwrGllUOlWL
m
OISTANCE.
a+
Rb
Fig. ISa. Shcxwood number vs. dimensionless Iongitudinal distance.
Numerical values of the local Sherwood number 2s 2 function of x+ are shown in Figs. 1% and 15b for a variety of cases. For smail lon~tudina~ distances Desahzfion.
7 (1%9/70) 51-74
F~M3-E DIFFERENCE
Fig
SOLUTION
1%~ Stiood
FOR RO 1N TURBULENT
number vs. dinunsionkss
FLOW
fongitudinal
69
distance.
(x+ I IS), A&, was found to be essentially independent of &. For X’ > 20, A& decreases with the increase of B,. However, the effect to Bz on NsL is much fess as compared with that on other pertinent parameters (for example, compare Fig. 3 with Fig. 15). For a numerical treatment such as given in this work. the question of predicting the results for conditions not specifically covered in the numerical cakulation always presents some difficulty because of the absence of the closedform expression. Owing to the large number parameters involved in this work (i.e., A&,_ bt, B, and N,,), only limited cases can be considered- The prediction of ali pertinent quantities by direct interpofation, therefore, becomes difficult. On the other hand, the most important quantity from a practical point of view is that of the wall concentration, c”,. It is believed that a reasonable procedure for the prediction of CL is through the use of expression for A$,, given by Eq. (W-8). combining this expression with Eqs. (II-IO) and (U-12) yields,
flmahation.
7 (1969f70) 51-74
70
S. SRINlVASAN AND CHI TIEN
For a given set of parameters, the Iocal Sherwood number as a function of _v+ can be estimated by interpolation based on the data given in Figs. 1%. b. Once this information becomes available, the relationship of c: us. x+ can be obtained from Eq. (IV-g). The main advantage of this procedure lies in the fact of the less dependency of Nsb on the parameters, 6:, NRcc and B, as discussed previously. Consequently, one can predict values of Nsb by interpolation with more ease. It should also be mated that in the preser?r work, the Schmidt number is taken as 560 (i.e. NaCI-H,O System). However, if the we&established relationship of Nsb being proportional to the cubic root of N,, is assumed to be also valid for the present problem, the performance of a reverse osmosis system in turbutent flow with systems other than NaCI-H,O can be easily predicted by this procedure. ACKNOWLE3xiEMEN-r
This work was performed under Grant No. WP-00968432, Federal Water Pollution Control Administration, Department of Interior. The authors also wish to thank the Syracuse University Computing Center for the use of its facilities. APPENDIX ORDER OF MAGNITUDE
ANALYSfS OF DlFFUSION
EQUATXON IN NRBULEh?
FLOW
OF
REVERSE OSSiOSS OPERATION
Equation (II-I) is
The magnitude
of u within
the concentration
lished as
= (b,+)(i
-
(t)
Q+)-'
boundary
layer can be estab-
FiNlTE DIFFERENCE SOLUTtOON FOR RO IN TURBULENT
FLOW
7L
since b+ is defined as (r,,,/f7)*/~(6) by Eq. (II-Q)of the manuscript and the relationship of b+ and 6: is given by Eq. (III-1 1) of the manuscript. For the estimation of the magnitude of (6/b), the result of one-dimensional model can be used, or
Combining
these expressions, one has,
= (f),r,(%_)(b,N~3
=
(:)(-&-)
(b)
N&It3
OI-
Note that NRcr:is defined as four times that of NRt as was done in the manuscript. From Eq. (II-IS) one has
0’
fz(32)
(NRA’
so that _
(NRC,) 4
Also, from Equations (III-IO) and (III-l 1) WR&“=
(NRA
(b‘),(I
(b+)x+=
(1
-
-
Q’)
Q+)7’R
Hence, 6
(T-)
-
(N;c)f
(&)[
&--‘(l
_
LQ+)3/4
For the present case, N RG..= O_OI5 De&ha&m,
7 (1969170) W-74
72
S.SRMIVASAN AND CHI TIEN
(b*)f = tm (N,)
= 560
Q-+=
0.35
the value of (IV,& the manuscript,
can be found by combining
Equations (f.&I5) and {I#-46) of
With these numerical data, the order of magnitude of(W) 0.0177, LE.,
can be estimated as
P
(T ) - (10-q Consequently, u+ -
one has, 1.81 or (IO’)
The concentration derivative, <&“@x*) can be obtained from the numericaf data of this work. Lt was found that at C’z = 1.9, (of _Y” = 60)
‘l%e magnitude of the term, (&[Czj, can be estimated by using the botindary condition using Eq. (H-7) of the manuscript, or
ac+ = - W,,,,,XNs,XKiXC,+) a2 By EC&(II40) of the manuscript, -
73
For the present &se, 8, = t/t, c+_ = 1.9
The magnitude of Y+ can be taken as that of YL. i.e.. v+ - (lo-‘) The magnitude of f&‘/&z*)
is given as
So the relative magnitude between the first and second terms on the left side of the diffusion equation (Eq. (II- 1)) can be evaluated: TIte First Term
Consequently,
both terms ,are of comparable
that one can neglect
the axial term without
magnitude
introducing
and does not appear
substantial
error.
REFERENCES
1. P. L. T. BFUAN,Id. Eng. C/tent. Fun~u~~e~r~rs,4 (l%S) 439. 2. G. H. BRUCE, D. W. PEACEMN AND J. RACHFORD, JR., Pm&m 198 (1953) 79.
Transactkwts cf.l.M.E.,
&xahaffon,
7 f 1969/70) 5 l-74
S. SRlNlYWAN
74 8. T.
9
K.SHERWOOD.
T_ K Smwmo.
AND CHI TlEN
P. L. T. BRIAN AND R. E. Ftstwt. M.I.T. Rep? No 295-l DSR9336(1963). P. L T. &UAN. R. E. FIWER AND L. DREW~, hf. &kg. Chem, Ftmda-
menrds, $ (1965) 113. 10. S. sRllWM.fi&X. Cm TIEB AND W. N. GILL, Chem &. Sri., 22 (l%7) 417. I I. S. SRW%*ASAN ARXX CM TIEN, Rcwrsc Osmosis Desalination in Tubular Membrane Duct, Drsa&Wtion, 3 ( 1%7f 5.