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Concentration polarization at the phase boundary of an external tubular reverse osmosis membrane
Desnlina~inn - Elscvier Publishing Company,
CONCENTRATiON AN
EXTERNAL
Amsterdam
POLARIZATION TUBULAR
AT
REVERSE
- Printed in The Netherlands
THE
OSMOSIS
PHASE
BOUNDARY
OF
MEMBRANE
C. K. TSAO F?zc Cafholc
University
of
Ammha.
Washi8gton,
D.C.
(U.S.A.)
(Rcaivcd April 16, 1970; in revisedform June 7, 1970)
SUMMARY
The salt concentration polarization at the phase boundary of external reverse osmosis membranes supported by horizontal and vertical tubes in stationary saline water at sea depth has been studied in an approximate way. Distributions of sah concentration and water flux along the perimeter of horizontal tubular membrane channe1 and along the tube of vertical tubular membrane channel have been found and presented. For horizontal tubes it is found that the salt concentration at the upper stagnation point, the radius and the membrane constant are the factors which affect the salt concentration and water flux distributions. For vertical tubes the membrane constant is the main factor for salt concentration and water flux distributions. SYMBOLS
-
membrane constant. g/cm2-see-atm diffusivity of salt in water, cm’/sec water flux. g/cm*-see gravitational acceteration, pressure difference, atm salt flow rate. g@c radial distance, cm tube radius, cm length velocity component
-
cm&z*
in O-direction,
cm/set
velocity component in r- or y-direction, cm/set velocity component in z-direction, cm/set salt concentration ratio = j&p*_ distance normal to phase boundary, cm axial distance, cm velocity function, cm/sfx diffusion boundary layer thickness, cm Desalination.
8 (1970)
243-258
244
ef
C. K. TSAO
-
8
-
P I’
-
JL P
-
viscous boundary layer thickness, cm angle measured from the upper viscosity, g/cm-sax kinematic viscosity. cm’/sec
stapation
point
osmotic pressure, atm sea water density, glee = p. + p,
Subscripts s Y
-
e
-
0
-
salt
water ambient sea water tube surface
ISfRODUCTlON
The serious problem encountered in a reverse osmosis system is the buildup of salt concentration at the membrane surface_ The salt concentration polarization has been analyzed by several investigators for the cases of Sows between parallel membranes, through a channel and inside round tubes (l-6.7). The concentration polarization on the outer surface of a round tubular membrane channel has not been studied. Of particular interest in connection with this geometry is the salt concentration which will occur when there is no flow over the tube. Such a condition could exist if the membrane were used to desalt sea water at ocean depth in the absence of ocean currents. As water permeates through the membrane from outside to the inside of a tubular channel, the salt rejected will build a concentration gradient on the phase boundary- Due to the salt density difference in the boundary layer and in the ambient sea water. natural convection will_occur near the phase boundary. Hence the satt concentration at the phase boundary is not constant but varies along the ,sarfaw. it is the purpose of this study to analyze. in an approximate way. the salt concentration distribution in the boundary layer of the horizontal and the vertical tubes (Fig. 1) in stationary sea water. The membrane is assumed to be impervious to salt. HORIZONTAL
TUBE
We consider the natural convection due to concentration polarization over a horizontal tube. For a horizontal tube, the concentration polarization becomes serious on the upper half of the tube. The analysis is mainly for that part. If the length of the tube is much larger than the diameter, the flow can be considered as twodimcnsional. The tangential component momentum equation is as follows,
Desafimnion, 8 (1970) 243258
CONCENTRATION
POLARIZATION
AT PHASE
245
BOUNDARY
Fig. 1. Gcortwtricd configuration.
+P
(
a2t4 d 5 ( r ) ar2 +z
Let us first examine
term,
+
i -r*
a%
+
a02
this equation
2 do --
(1)
r2 d0 )
by comparing
the magnitude
the inertia force (pujr) (c~u/c%I) and the friction force p(d2u;8r2).
convection
takes place in the diffusion
boundary
of two typical The
natural
layer where the salt concentration
varies. Therefore the variation of tl with respect to r is proportional to u,,,/S where U, is the maximum tangential velocity and 6 is the diffusion boundary layer thickness. For this reason, the ratio of the inertia force and the friction force is
written as; (IO)
Desalination, 8 (1970) 243~2!S8
C. K. lSA0
246 inertia force . ..---W----_~ PU:IS friction force /u&P2
_
PU, 6z a- D \’ W
where s is a characteristic length along the perimeter and the relation d x (Ds/u,,,)* is used. For sea water the typical values of D and v are: D = 1.61 x IO’s cm2/sec equal to and s = 0.009 cm2/sec. Then the ratio is approximately inertia force friction force So for the present viscous
Cz 1.79 x IQ-J
case the inertia effect is negligibly
effect. From boundary
terms on ihe left hand order of magnitude of is lower than the term -one concludes that Eq. -2 I( _.!;_
layer theory (RI), it is easy to show
with the
that the inertia
side of Eq. 1 are of the same order of magnitude. But the the last three viscous terms on the right hand side of Eq. 1 p(u’2r(l&2) by the order of 6 or S2. From this examination 1 can be simplified to the form
m
= -(p-pP,)gsinO
It can also be written d2U = Ic ay2 Since the water
as
(p -
p,)
g sin 0
concentration
P. - pm where p. is a function for p.,
PI -
small in comparison
Pee =(Pso-Psd(l
is essentially
constant
we can write
of both 0 and y. We assume
p -
the following
pz
=
relation
13)
-fy
where S, the diffusion boundary layer thickness and p,,,. the salt concentration at the membrane surface, are functions of 0 only. Introducing Eq. 3 into Eq_ 2a gives
.
with the boundary
conditions:
when f = 0 and y = 6, u = 0. Integrating
Eq. 4
twice, we obtain
Desdination,8 (1970)243258
CONCENTRATION
PSO
U=:
POLARIZATION
Am
-
u
AT PHASE BOUNDARY
gsinOg{
(1 -
-$)
-
247
(1 -
5)‘)
(5)
The flow rate of the salt constituent
in the boundary
layer region is
d 0-,
=
f
Substituting
Q,=-
P,udv 0
Eqs. 3 and 5 into 6, nnd integrating,
~f,gsinO 6(XIr 3
I)
($+&
)
is the salt concentration flux FW is determined by the equation
The condition
A(P-
52
osmosis
X,X)
the water
(8)
of zero salt l%.tx at the membrane
_$%F,+D
(7)
ratio. In revem
where x = pJp,,
F,=
there results
is expressed by
(-g_Lo=o
(9)
With Eqs. 3 and 8, Eq. 9 gives the boundary
layer thickness
Introducing 6 into Eq. 7, $3. becomes x pA3
‘* sin 0 v
(X 1)” - I? ,x)Jx”
.
112
~
9 560 )
(11)
The salt balance in a volume element Sr,AOof unit thickness can be expressed as
where dQJrad6 is the rate of salt concentration of water’through the membrane and -D(3p$,y)
increase as a result of permeation is the mass diffusion through the Desahafio?r. 8 (1970) 243-258
248
c. K. TSAO
diffusion boundary layer. The mass diffusion may be approximated s D&,--p,,)/& and thus Eq. 12 is written as
by - Ddp,/$~
(12a) substituting
Eqs. 10 and 11 into EZq.12. finally we have
~P;‘,D’P~,~~~~~-_(~ --7
\*A3
-
(P -
x+1
II3
+ % -I-._-+ 3)
56
n ,xpx3
560x
n&x
1
8pED3 -- USA 9
_
VA3
cos
o
_-?;_+_’
( _
560
)
I)’ x&Y
(x (P -
The boundary condition is at 0 = 0, x = x0 = &,f~.,)~. Eq. 13 is thedifferential equatiotl for x which is solved numerically by finite difference method. At 0 = 0, there is a singular point which can be avoided by integration. (For detail, see (5%) VERTICAL.
TUBE
If the tube is not too long and the radius is not too small the diffusion boundary layer thickness 6 of salt on a vertical tube is. in general, smaller than the radius
of the tube. in this case the curvature
of the tube affects the fiow field
this simplification. the equations for the natural convective flow over the outer surfXe of a vertical tube are essentially the same as those for the flow over a flat plate which are as follows: insignificantly.
au
-----t-~=O
Under
dw
(14)
at
(15)
Desnlimtlon. 8
(1970) 243-258
CONCENTRATION
POLARIZATiON
249
AT PHASE BOUNDARY
The boundary conditions are given as follows, whenz=O*u=w=O,p,=ps,
when .v = 0, 0 = UC&-f), w = 0, PI = A&)
(171
when r = S, w = 0. p. = p,* From Eq. 14 we have Y u‘-
-
I
CYW
7
Z
dg +
(W
u.
0
Substituting Eq. 18 into Eq. 15 and integrating over ~7from 0 to 6, we find
= -V(~)o+~~(P.-PAdi.
(19) 0
We assume p,
-
ps4
=
(Jt -
l)P,,
( \
1 -
2
$-
)
(20)
and
lnttoducing obtain
Eqs. 20 and 21 into Eq. 19 and carrying out the integration
-&-$r2a, = - -$ f y&c
-
1)
we
(22)
Similarly from Eq. 16 with Eqs. 18, 20 and 21 we have
Desalinafion.
8 (1970)
243-258
CONCEbiiTiON
0
POLARIZATION
30
00
00
AT PHASE BOUNDARY
I20
ofGRE~
Fig. 2. r&y =s 2 cm, A =
concentration ntio-Horhontal 1.0 x IO-* p;lcm~-w-arm
cm
251
IO
-
tube.
fiuxwhich
has the largest value when x = 1 (Eq. 9, x can not be less than unity). Also when x =r 1, there is no salt concentration gradient which usually plays an important role in bahmzing the salt concentration polarization. AH these sonditions are favorable For the fast increase ofthz salt concentration on themcmbrane surface near the upper stagnation point. In Fact, at the stagnation poitt zero concentration polarization exists at the beginning and finaily the salt concentration will reach a certain definite value. When _r, is large, For example x0 = 2.5. x decreases very fast near 0 = 0. With high X, value, the rate of salt ~once~t~tion increase is low and the salt concentration gradient on the me&mbrane surface is high. Even when the vetocity is zero, the ~on~ntration gradient is high enough to diffuse back a large amount of salt into the ambient sea water. This indicates that it may not be easy for the salt to reach a high concentration on the upper stagnation point. From Fig. 2, one may conclude that for this particular tube the most probable value of x0 will be between 1.5 and 2.0. The exact vague can be determined From experiment. Fig. 3 gives the diffusion boundary fayer thickness which is related to X- by Eq. 10. In the case of x0 = 1.5 and 2.0, the variation of the thickness is Desalindon,
8 ( 1970) 2443-258
252
C. K. -ISA0
Desahkation.
8 (1970) 243-258
CONCENTRATION
POLARIZATION
AT PHASE BDUNDARY
253
Desalination,
8 ( 1970) 24.7-258
6. K. TSAO
CONCEPITRATION
f’OLARfZATfON I
AT PHASE BOUNDARY
255
I
I
I
25 x
I
0
PO
Fig. 9. Salt concentration
0
20
Fig. 10. Water flux-Vertical
40
SsL-
ratio-Vertical
tube.
40
60
z I
‘n
Zlm)-
,
O0
80
100
100
tube.
moderate in the range 0 = IO” to 120”. Fig. 4 is the water flux along the perimeter.
From Eq. 8, F,,, is a linear function of x. The salt concentration ratio and the water fiux for x0 = 1.5 and r, = I,2 and 3 cm are shown in Figs. 5 and 6 respectively. These plots show that a horizontal tube with small radius has a lower salt concentration ratio and a kigker water flux than those with larger radius. Fig. 7 is the Dedinarion,
8 (1970) 243-258
256
C. K. TSAO
0
20 F5g. f 1.
40
e0
zec.,
100
--so
Diffusion boundary layer thickness-Verticat
tube.
a0 Fig. 12. Velocity function-Vertical
salt
and Fig. 8 and
F,
tube.
the water
for tubes
increase with the increase of membrane constant A.
However, the percentage of increase in water flux is larger than that in salt concentration. For example, at 0 = 60” when A increases from i-0 x IO- 5 to 1.25 x Daahotion, 8 (1970) 243-258
CONCENTRATION
POLAIUZATlON
AT
PHASE
BOUNDARY
257
10m5, F%,increases approximately 18 y. while x increases only 3.5 % (not shown in the figure). Salt concentration ratio, water flux, boundary layer thickness and velocity function of vertical tubes 100 centimeters long for various membrane constants are shown in Figs. 9, IO, I I and 12 respectively. The salt concentration increases very rapidly with the increase of z near the top. The reason is the same as in the case of x0 = 1 for horizontal tube. But for a vertical tube, the salt concentration ratio is always equal to unity at the top and therefGre it always has a steep increase near the top. Because of this fast increase of salt concentration near y the top, the water flux (Fig. IO) decreases rapidly in the first quarter of the length _ of the tube. For instance. for the curve A = 1.0 x IO-’ the water flux is reduced to half of its initial value when z = 32.5 cm. The velocity function r (z) gradually increases along the tube. The continuous increase of the flow velocity will detain. to a certain degree, the buildup of the salt concentration at the membrane surface. Consequently. concerrtration increases very slowly in the last 60 centimeters of the vertical tube. The diffusion boundary layer thickness at .z = 100 cm is about 0.08 cm which is less than the tube radius. So it is reasonable to treat the present problem of natural convection on a vertical tube as that on a vertical plate. CONCLUSION
In this preliminary analysis we find the distributions of salt concentration at the phase boundary and water flux through the membrane of the horizontal and vetiical external tubular reverse osmosis systems in stationary sea water. From the practical point of view we catculate the total water flux (gjsec) for both horizontal and vertical tubes (Table 1) by integrating the water flux F,,.over the area. From the table we can see that for the present case the horizontai tube is more effective in desalination than the vertical tube. We may express the total fresh wa:er ffux in TABLE TOTAL A
=
I FRESH
WATER
FLUX
(G/SEC)
1.0 x tO-6 g/cm*-ss-aim
P=8Oatm
rr,=24atm
-
: 3
Horizontal rube (x0 = 1.5)
Vertical trtbc
0.26948 0.50964 0.73612
0.17191 0.34381 0.51572
0.10779 0.20386 0.29445
0.07807 0.15614 0.2342 1
-Desahation,
8 (1970) 243-258
2%
c. ti. I-SAO
gallons per day. For exampfe, for a horizontal tube of radius of 1 cm and length of 100 cm, the total fresh water fiux is about 6.1 gal/day. Comparing this figure with the figures in Table 1 of reference (I), the 6.1 gallons per day per lOOcm length for I cm radius is quite reasonable. ACKNOWLEDGEMENT
This study is supported Center, Annapolis Division.
1,
2.
3. 4. 5. 6.
7. 8.
9. 10.
by the Naval Ship Research
and Development
P. L. T. BRIAN,Proc. First itr?ernafitmd Sjmposium on Wor~r Drs&wriion, Washington, D,C_ Ortmkr 3-9, l%S, 1 (1967) 349. T. K. SHERWOOD. P. L. T_ BRIAN. IL E. I%S?ER AND L_ DRESNER. ht. Ekg. Chem. Firndnmcnrals, 4 tt%s 113. iv, Grtt, C. T&ES A?QZ D_ W. &3& fbkk, 4 (t%s)423. P. L. T. ~RIMU. #Jid.. 4 (1965) 439. R. Es FASUF.FK, T. K. SHLRWOCID A&V F’+L. T. BRtAh‘, O+rr of &-ah? Wurer, Res. De&p. P#gr. Rep. No. M1, Scpr t965. P. I,., T. BRIAN- Ibid., No. 145, Sept. 1955. S. SRIMVASAN.CHI ZEN AND W. N. C&L. 1bM, No. 243, March $967, U. M~mtcu. Prof. firsr Inremariotd Symposium on Water Lksdinntion, Washinpon. D.C., OCldw 3-9.1965, 1 (1967) 175. C. K. T-0, Mod&d~ Reps. 58, Naval Ship Ra. and Develop. Center,Annapolis Division, f%s. H. Scmtcnn~c. &runa%ryt?~~=r 7%e0~, McGraw-Hill, New York, N.Y.
CONCENTRATiON AN
EXTERNAL
Amsterdam
POLARIZATION TUBULAR
AT
REVERSE
- Printed in The Netherlands
THE
OSMOSIS
PHASE
BOUNDARY
OF
MEMBRANE
C. K. TSAO F?zc Cafholc
University
of
Ammha.
Washi8gton,
D.C.
(U.S.A.)
(Rcaivcd April 16, 1970; in revisedform June 7, 1970)
SUMMARY
The salt concentration polarization at the phase boundary of external reverse osmosis membranes supported by horizontal and vertical tubes in stationary saline water at sea depth has been studied in an approximate way. Distributions of sah concentration and water flux along the perimeter of horizontal tubular membrane channe1 and along the tube of vertical tubular membrane channel have been found and presented. For horizontal tubes it is found that the salt concentration at the upper stagnation point, the radius and the membrane constant are the factors which affect the salt concentration and water flux distributions. For vertical tubes the membrane constant is the main factor for salt concentration and water flux distributions. SYMBOLS
-
membrane constant. g/cm2-see-atm diffusivity of salt in water, cm’/sec water flux. g/cm*-see gravitational acceteration, pressure difference, atm salt flow rate. g@c radial distance, cm tube radius, cm length velocity component
-
cm&z*
in O-direction,
cm/set
velocity component in r- or y-direction, cm/set velocity component in z-direction, cm/set salt concentration ratio = j&p*_ distance normal to phase boundary, cm axial distance, cm velocity function, cm/sfx diffusion boundary layer thickness, cm Desalination.
8 (1970)
243-258
244
ef
C. K. TSAO
-
8
-
P I’
-
JL P
-
viscous boundary layer thickness, cm angle measured from the upper viscosity, g/cm-sax kinematic viscosity. cm’/sec
stapation
point
osmotic pressure, atm sea water density, glee = p. + p,
Subscripts s Y
-
e
-
0
-
salt
water ambient sea water tube surface
ISfRODUCTlON
The serious problem encountered in a reverse osmosis system is the buildup of salt concentration at the membrane surface_ The salt concentration polarization has been analyzed by several investigators for the cases of Sows between parallel membranes, through a channel and inside round tubes (l-6.7). The concentration polarization on the outer surface of a round tubular membrane channel has not been studied. Of particular interest in connection with this geometry is the salt concentration which will occur when there is no flow over the tube. Such a condition could exist if the membrane were used to desalt sea water at ocean depth in the absence of ocean currents. As water permeates through the membrane from outside to the inside of a tubular channel, the salt rejected will build a concentration gradient on the phase boundary- Due to the salt density difference in the boundary layer and in the ambient sea water. natural convection will_occur near the phase boundary. Hence the satt concentration at the phase boundary is not constant but varies along the ,sarfaw. it is the purpose of this study to analyze. in an approximate way. the salt concentration distribution in the boundary layer of the horizontal and the vertical tubes (Fig. 1) in stationary sea water. The membrane is assumed to be impervious to salt. HORIZONTAL
TUBE
We consider the natural convection due to concentration polarization over a horizontal tube. For a horizontal tube, the concentration polarization becomes serious on the upper half of the tube. The analysis is mainly for that part. If the length of the tube is much larger than the diameter, the flow can be considered as twodimcnsional. The tangential component momentum equation is as follows,
Desafimnion, 8 (1970) 243258
CONCENTRATION
POLARIZATION
AT PHASE
245
BOUNDARY
Fig. 1. Gcortwtricd configuration.
+P
(
a2t4 d 5 ( r ) ar2 +z
Let us first examine
term,
+
i -r*
a%
+
a02
this equation
2 do --
(1)
r2 d0 )
by comparing
the magnitude
the inertia force (pujr) (c~u/c%I) and the friction force p(d2u;8r2).
convection
takes place in the diffusion
boundary
of two typical The
natural
layer where the salt concentration
varies. Therefore the variation of tl with respect to r is proportional to u,,,/S where U, is the maximum tangential velocity and 6 is the diffusion boundary layer thickness. For this reason, the ratio of the inertia force and the friction force is
written as; (IO)
Desalination, 8 (1970) 243~2!S8
C. K. lSA0
246 inertia force . ..---W----_~ PU:IS friction force /u&P2
_
PU, 6z a- D \’ W
where s is a characteristic length along the perimeter and the relation d x (Ds/u,,,)* is used. For sea water the typical values of D and v are: D = 1.61 x IO’s cm2/sec equal to and s = 0.009 cm2/sec. Then the ratio is approximately inertia force friction force So for the present viscous
Cz 1.79 x IQ-J
case the inertia effect is negligibly
effect. From boundary
terms on ihe left hand order of magnitude of is lower than the term -one concludes that Eq. -2 I( _.!;_
layer theory (RI), it is easy to show
with the
that the inertia
side of Eq. 1 are of the same order of magnitude. But the the last three viscous terms on the right hand side of Eq. 1 p(u’2r(l&2) by the order of 6 or S2. From this examination 1 can be simplified to the form
m
= -(p-pP,)gsinO
It can also be written d2U = Ic ay2 Since the water
as
(p -
p,)
g sin 0
concentration
P. - pm where p. is a function for p.,
PI -
small in comparison
Pee =(Pso-Psd(l
is essentially
constant
we can write
of both 0 and y. We assume
p -
the following
pz
=
relation
13)
-fy
where S, the diffusion boundary layer thickness and p,,,. the salt concentration at the membrane surface, are functions of 0 only. Introducing Eq. 3 into Eq_ 2a gives
.
with the boundary
conditions:
when f = 0 and y = 6, u = 0. Integrating
Eq. 4
twice, we obtain
Desdination,8 (1970)243258
CONCENTRATION
PSO
U=:
POLARIZATION
Am
-
u
AT PHASE BOUNDARY
gsinOg{
(1 -
-$)
-
247
(1 -
5)‘)
(5)
The flow rate of the salt constituent
in the boundary
layer region is
d 0-,
=
f
Substituting
Q,=-
P,udv 0
Eqs. 3 and 5 into 6, nnd integrating,
~f,gsinO 6(XIr 3
I)
($+&
)
is the salt concentration flux FW is determined by the equation
The condition
A(P-
52
osmosis
X,X)
the water
(8)
of zero salt l%.tx at the membrane
_$%F,+D
(7)
ratio. In revem
where x = pJp,,
F,=
there results
is expressed by
(-g_Lo=o
(9)
With Eqs. 3 and 8, Eq. 9 gives the boundary
layer thickness
Introducing 6 into Eq. 7, $3. becomes x pA3
‘* sin 0 v
(X 1)” - I? ,x)Jx”
.
112
~
9 560 )
(11)
The salt balance in a volume element Sr,AOof unit thickness can be expressed as
where dQJrad6 is the rate of salt concentration of water’through the membrane and -D(3p$,y)
increase as a result of permeation is the mass diffusion through the Desahafio?r. 8 (1970) 243-258
248
c. K. TSAO
diffusion boundary layer. The mass diffusion may be approximated s D&,--p,,)/& and thus Eq. 12 is written as
by - Ddp,/$~
(12a) substituting
Eqs. 10 and 11 into EZq.12. finally we have
~P;‘,D’P~,~~~~~-_(~ --7
\*A3
-
(P -
x+1
II3
+ % -I-._-+ 3)
56
n ,xpx3
560x
n&x
1
8pED3 -- USA 9
_
VA3
cos
o
_-?;_+_’
( _
560
)
I)’ x&Y
(x (P -
The boundary condition is at 0 = 0, x = x0 = &,f~.,)~. Eq. 13 is thedifferential equatiotl for x which is solved numerically by finite difference method. At 0 = 0, there is a singular point which can be avoided by integration. (For detail, see (5%) VERTICAL.
TUBE
If the tube is not too long and the radius is not too small the diffusion boundary layer thickness 6 of salt on a vertical tube is. in general, smaller than the radius
of the tube. in this case the curvature
of the tube affects the fiow field
this simplification. the equations for the natural convective flow over the outer surfXe of a vertical tube are essentially the same as those for the flow over a flat plate which are as follows: insignificantly.
au
-----t-~=O
Under
dw
(14)
at
(15)
Desnlimtlon. 8
(1970) 243-258
CONCENTRATION
POLARIZATiON
249
AT PHASE BOUNDARY
The boundary conditions are given as follows, whenz=O*u=w=O,p,=ps,
when .v = 0, 0 = UC&-f), w = 0, PI = A&)
(171
when r = S, w = 0. p. = p,* From Eq. 14 we have Y u‘-
-
I
CYW
7
Z
dg +
(W
u.
0
Substituting Eq. 18 into Eq. 15 and integrating over ~7from 0 to 6, we find
= -V(~)o+~~(P.-PAdi.
(19) 0
We assume p,
-
ps4
=
(Jt -
l)P,,
( \
1 -
2
$-
)
(20)
and
lnttoducing obtain
Eqs. 20 and 21 into Eq. 19 and carrying out the integration
-&-$r2a, = - -$ f y&c
-
1)
we
(22)
Similarly from Eq. 16 with Eqs. 18, 20 and 21 we have
Desalinafion.
8 (1970)
243-258
CONCEbiiTiON
0
POLARIZATION
30
00
00
AT PHASE BOUNDARY
I20
ofGRE~
Fig. 2. r&y =s 2 cm, A =
concentration ntio-Horhontal 1.0 x IO-* p;lcm~-w-arm
cm
251
IO
-
tube.
fiuxwhich
has the largest value when x = 1 (Eq. 9, x can not be less than unity). Also when x =r 1, there is no salt concentration gradient which usually plays an important role in bahmzing the salt concentration polarization. AH these sonditions are favorable For the fast increase ofthz salt concentration on themcmbrane surface near the upper stagnation point. In Fact, at the stagnation poitt zero concentration polarization exists at the beginning and finaily the salt concentration will reach a certain definite value. When _r, is large, For example x0 = 2.5. x decreases very fast near 0 = 0. With high X, value, the rate of salt ~once~t~tion increase is low and the salt concentration gradient on the me&mbrane surface is high. Even when the vetocity is zero, the ~on~ntration gradient is high enough to diffuse back a large amount of salt into the ambient sea water. This indicates that it may not be easy for the salt to reach a high concentration on the upper stagnation point. From Fig. 2, one may conclude that for this particular tube the most probable value of x0 will be between 1.5 and 2.0. The exact vague can be determined From experiment. Fig. 3 gives the diffusion boundary fayer thickness which is related to X- by Eq. 10. In the case of x0 = 1.5 and 2.0, the variation of the thickness is Desalindon,
8 ( 1970) 2443-258
252
C. K. -ISA0
Desahkation.
8 (1970) 243-258
CONCENTRATION
POLARIZATION
AT PHASE BDUNDARY
253
Desalination,
8 ( 1970) 24.7-258
6. K. TSAO
CONCEPITRATION
f’OLARfZATfON I
AT PHASE BOUNDARY
255
I
I
I
25 x
I
0
PO
Fig. 9. Salt concentration
0
20
Fig. 10. Water flux-Vertical
40
SsL-
ratio-Vertical
tube.
40
60
z I
‘n
Zlm)-
,
O0
80
100
100
tube.
moderate in the range 0 = IO” to 120”. Fig. 4 is the water flux along the perimeter.
From Eq. 8, F,,, is a linear function of x. The salt concentration ratio and the water fiux for x0 = 1.5 and r, = I,2 and 3 cm are shown in Figs. 5 and 6 respectively. These plots show that a horizontal tube with small radius has a lower salt concentration ratio and a kigker water flux than those with larger radius. Fig. 7 is the Dedinarion,
8 (1970) 243-258
256
C. K. TSAO
0
20 F5g. f 1.
40
e0
zec.,
100
--so
Diffusion boundary layer thickness-Verticat
tube.
a0 Fig. 12. Velocity function-Vertical
salt
and Fig. 8 and
F,
tube.
the water
for tubes
increase with the increase of membrane constant A.
However, the percentage of increase in water flux is larger than that in salt concentration. For example, at 0 = 60” when A increases from i-0 x IO- 5 to 1.25 x Daahotion, 8 (1970) 243-258
CONCENTRATION
POLAIUZATlON
AT
PHASE
BOUNDARY
257
10m5, F%,increases approximately 18 y. while x increases only 3.5 % (not shown in the figure). Salt concentration ratio, water flux, boundary layer thickness and velocity function of vertical tubes 100 centimeters long for various membrane constants are shown in Figs. 9, IO, I I and 12 respectively. The salt concentration increases very rapidly with the increase of z near the top. The reason is the same as in the case of x0 = 1 for horizontal tube. But for a vertical tube, the salt concentration ratio is always equal to unity at the top and therefGre it always has a steep increase near the top. Because of this fast increase of salt concentration near y the top, the water flux (Fig. IO) decreases rapidly in the first quarter of the length _ of the tube. For instance. for the curve A = 1.0 x IO-’ the water flux is reduced to half of its initial value when z = 32.5 cm. The velocity function r (z) gradually increases along the tube. The continuous increase of the flow velocity will detain. to a certain degree, the buildup of the salt concentration at the membrane surface. Consequently. concerrtration increases very slowly in the last 60 centimeters of the vertical tube. The diffusion boundary layer thickness at .z = 100 cm is about 0.08 cm which is less than the tube radius. So it is reasonable to treat the present problem of natural convection on a vertical tube as that on a vertical plate. CONCLUSION
In this preliminary analysis we find the distributions of salt concentration at the phase boundary and water flux through the membrane of the horizontal and vetiical external tubular reverse osmosis systems in stationary sea water. From the practical point of view we catculate the total water flux (gjsec) for both horizontal and vertical tubes (Table 1) by integrating the water flux F,,.over the area. From the table we can see that for the present case the horizontai tube is more effective in desalination than the vertical tube. We may express the total fresh wa:er ffux in TABLE TOTAL A
=
I FRESH
WATER
FLUX
(G/SEC)
1.0 x tO-6 g/cm*-ss-aim
P=8Oatm
rr,=24atm
-
: 3
Horizontal rube (x0 = 1.5)
Vertical trtbc
0.26948 0.50964 0.73612
0.17191 0.34381 0.51572
0.10779 0.20386 0.29445
0.07807 0.15614 0.2342 1
-Desahation,
8 (1970) 243-258
2%
c. ti. I-SAO
gallons per day. For exampfe, for a horizontal tube of radius of 1 cm and length of 100 cm, the total fresh water fiux is about 6.1 gal/day. Comparing this figure with the figures in Table 1 of reference (I), the 6.1 gallons per day per lOOcm length for I cm radius is quite reasonable. ACKNOWLEDGEMENT
This study is supported Center, Annapolis Division.
1,
2.
3. 4. 5. 6.
7. 8.
9. 10.
by the Naval Ship Research
and Development
P. L. T. BRIAN,Proc. First itr?ernafitmd Sjmposium on Wor~r Drs&wriion, Washington, D,C_ Ortmkr 3-9, l%S, 1 (1967) 349. T. K. SHERWOOD. P. L. T_ BRIAN. IL E. I%S?ER AND L_ DRESNER. ht. Ekg. Chem. Firndnmcnrals, 4 tt%s 113. iv, Grtt, C. T&ES A?QZ D_ W. &3& fbkk, 4 (t%s)423. P. L. T. ~RIMU. #Jid.. 4 (1965) 439. R. Es FASUF.FK, T. K. SHLRWOCID A&V F’+L. T. BRtAh‘, O+rr of &-ah? Wurer, Res. De&p. P#gr. Rep. No. M1, Scpr t965. P. I,., T. BRIAN- Ibid., No. 145, Sept. 1955. S. SRIMVASAN.CHI ZEN AND W. N. C&L. 1bM, No. 243, March $967, U. M~mtcu. Prof. firsr Inremariotd Symposium on Water Lksdinntion, Washinpon. D.C., OCldw 3-9.1965, 1 (1967) 175. C. K. T-0, Mod&d~ Reps. 58, Naval Ship Ra. and Develop. Center,Annapolis Division, f%s. H. Scmtcnn~c. &runa%ryt?~~=r 7%e0~, McGraw-Hill, New York, N.Y.