Diffusion-layer structure in reverse osmosis channel flow

Desolinarion - Ekwier

Publishing

DIFFUSION-LAYER

Company,

STRCJCTURE

Amsterdam

- Printed in The Netherlands

iN REVERSE

OSMOSIS

CHANNEL

FLOW+ T. 3. HENDRICKS

AND

F. A. WILLIAMS

Departmentof Armspace and ~~echonisof Engkering frr Jo&a, Co!*

(U.S.A.)

Sciences, Urriversity of Cuiifomia, San Diego.

(Rec.&al October 2!3,1970)

SUMMARY

Salt concentrntion

profiles

in brine adjacent

to the membrane

during reverse

osmosis were measureit with electrical conductivity microprobe5 for fully-developed two-dimensional. chennel fiow in a closed-return water tunnel. Cellulose acetate membranes were emptoyed with solutions of N&NO,, NaN03, NaCI, Na,SO, and MgS04, at hydrostatic pressures from 10 to 40 atm and with Reynolds Rumhers from 137 to 1365. The excess concentration was found to vary exponentially with distance normal to the membrane. An integral technique, developed on the basis of an exponential-profile assumption, was found to provide reasonable agreement between theoretical and experimental concentration profiles. This integral theory gives good values for the wall concentration and can be applied relatively quickly in design calculations.

A

-

0

-

B

-

b

-

c

-

D

-

d

-

h

-

K

-

L

-

Membrane constant Scale of nonhomogeneity Nondimensionai slope of exponential profile A Constant Salt concentration Diffusion coefficient Characteristic e-folding thickness of diffusion iayer Channel half-height Cell constant, Eq. (2) Leakage resistance

____-_._-_ This fescatch was sponsored by the Office of Saline Water, U.S. Dept. of Interior. We wish to

l

thank U. Mcxten, Gulf General Atomic, for supplying mcmbmnes help with the theory.

and M. K_ Liu, U.C.S.D., for

Desolinu~ion, 9 (1971) 155-180

-I-. J. HENDRICKS AND F. A. WILLIAMS

156 III P

-

R Re r SC s t u u 0 IL’,

-

s

-

Y z

-

L4,ih

-

6

-

rl v <

-

I:

-

P

-

a

-

SW -

Measured resistance Polarization resistance Salt rejection coefficient Reynolds number

for membrane

Probe radius Schmidt number Solution ciectrical conductivity Time Streamwise bulk velocity Streamwise vcjocity component Velocity component normal to membrane Volume flow per unit surface area through membrane Distance parallel to membrane in flow direction Distance normal to membrane Nondimensional excess salt concentration. Eq. (5)

Hydrostatic pressure difference across membrane Osmotic pressure difference across membrane Ratio of initial osmotic pressure difference to difference, AnJAp = RR~IAJI Nondimensional distance normal to membrane, Kinematic viscosity a’/9, nondimensional streamwise distance Osmotic pressure Liquid

hydrostatic

pressure

Eq. (21)

density

Nondimensional Wall shear

streamwise

distance,

Eq. (4)

SUbSCfipts -

End of hydrodynamic boundary-layer entrance development Characteristic for development of diffusion layer Merging of diRusion layers from opposite sides of channel

-

An experimental vabte Bufk brine conditions Conditions in the permeate’fiuid A theoretical value Conditions in the brine at the surface

region

of the membrane

fk5&mtton,

9 (1971) 155-180

DIFFUSION-LAYER

1.

STRUCTURE

IN X0

CHANNEL

FLOW

157

INTRODUCTION

“Concentration polarization” can constitute a limiting factor in the design of desalination equipment employing reverse osmosis. Experimental studies of this phenomenon in continuous-flow systems have been reported by Merten (I), by Sherwood f-3) and by St~thmann (3). Efforts have been made to compare these experimental results with calculations based on theories of salt-layer buildup (2-4). Attempts to make thorough comparisons between theory and experiment have been hampered by the fact that no qunntitarive measurements of concentrrtion profiles within the diffusion layer of the brine have been available; only the fluxes of water and salt through the membrane have been measured. Profile measurements can be useful not only in providing tests of fluid-mechanical thearies but also in checking reintions for membrane fluxes. Profile measurements performed with conductivity microprobes in laminar channel-flows are reported herein. In the folIowing section a description is given of the experimental apparatus, inst~mentat~on and measurement technique. Experimental results are presented in Section 3. A new integral method for czdculating concentration profiles theoretically is developed in Section 4, which also contains other new theoretical results. Section 5 presents a comparison of various theore?ical and experimental results. The satisfactory agreement and the simp!icity of the integral theory underscore the utility of this theory. 2.

EXPERIMENT

The apparatus consisted of a water tunnel encased within a water-filled pressure vessel. A block diagram of the system is shown in Fig. 1. The closedreturn tunnel included a circulation system, a flow-smoothing section, filters and a test section equipped with a precision stage for positioning an elcctricai conductivity microprobe. The test section was a channel 193 cm in length and 21 cm in width, with an adjustable height 2h which could not exceed 8 mm. Except for some M&O, tests with height of 3.2 mm, all measurements were made for the maximum channel height. The primary considerations in the choice of dimensions were to provide a diffusion layer of thickness compatible with the spatial resolution of the microprobe and to produce essentially two-dimensional flow by eliminating edge effects. Membnnes, hacked by filter paper and stainless steel screen supports, could be seated to both the upper and lower faces of the channel. The low ratio of wall velocity lo bulk velocity caused the concentration profile on the lower face to be insensitive to the presence of the membrane on the upper face; therefore, for ease of operation, most of the data was taken with a single membrane on the lower face of the channel. Diffusion layers adjacent to membranes on the upper face of the

Desalinafion, 9 (1971) 155-l tKi

chaxmei were infiueq!d discusKd herein.

apprceiably by buoyant convecIian,

which will not be

Asymmnric cellulose acetate membranes. supplied by Gulf General Atomic, wfn: used in all runs. Rejection was roughly 98% for NaCl at 60 atm, and the membrane constant A for undamaged membranes ranged from 1.3 x iOS5 to 2.1 x IO-‘cm/sec.acm. Upstream from the test section in the channd was an impermeable-walled dcvclopmcnt. section 29 cm in length, for cstab!ishing a velocity profile corrcspending to fully-developed channel flow. This development szction was prazded by a smooth contra&on from 3.2 cm and a set of cahning screens, to dampen nonuniformitks in the flow. Dye ffow Ses?s showed that at the lower c&me! flow rates, the Bow was iaminar. pfan~‘paraIlet and approximately paraWir in velocity : profik -. Circulation was provided by a submerged pump. A se2 of pressure-compeasate+, constant flow valves control&d the Bow rate. Reynotds numbers, Rt, based on bti velocity and on twice the’&annel height, of i37,410,819,1365,2547,4O!M and 6270 were available. This range spans both laminar and turbulent conditions,

BrrrrLinmlon, 9 (197Q 15Slsd

DIFFUSIOX-LAY&R

STRUCTURE IN RO CHAN%L

FLOW

159

since the transition Reynolds number for a rectangular duct is approximately 1600 (Ref. (5)). A bypass line passed the channel solution through a 5~ filter. removed heat gcncratcd by the circulation pump. received make-up solution and mixed the circulating solution with the brine from the permeable section_ An air-driven piston pump provided pressurization and make-up solution. The accuracy of pressure regulation was +0_2 atm, and applied pressures & up to 100 atm were available. A glycerindlled, mechanically driven stage, mounted upon the upper plate of the channel at one of five axial locations, positioned the microprobe at various points in the diffusion layer. All results reported herein were obtained with the probe located at a distance x = 180 cm downstream from the leading edge of the membrane. Micrometers driven by anti-backlash worm gears enabled microprohe travel normal to the membrane to be adjusted with accuracy ranging from 1 to 5/l_ Microprobes were constructed from platinum wire 45~ in diameter by first etching the wire SO a fine q-xx. The etched wire w-as covered by an electrically insulating coating, except in the neighborhood of the tip, which was coated with platinum black. Tip diameters range from 5 to 20~ for useabte probes, since the structure of the diffusion layer cannot be resolved well for larger diameters, and since the strength of the probe became marginally acceptable at smaller diameters. These Iimitations are appreciab!y more stringent than those encountered by Liu (6) in batch-cell studies. A microprobe mounted in the probe-positioning stage is electrically isolated from the rest of the apparatus, which serves as the second electrode. The AC resistance between these two electro&s was measured with an accuracy better than one part in 500, by using an impedance bridge (il’tustrated in Fig_ 2) which employed an oscilloscope to detect phase-sensitive error voltages and to check for nonlinear&s that sometimes appeared in concentrated solutions. The resistance depends on the. probe geometry and on the electrical conductivity-hence salt concentration--of the solution surrounding the tip. Calibrations of measured resistance J%ZWYSKS solution conductivity s were performed by positioning the probe tip in the center of the channel and circulating a series of uniform solutions of known conantration through the channel while recording M. Calibration data were fitted by the method of least squares to the formula M = [(P +

K/s)-’ + L-‘I-‘,

in which L represents the parallel leakage resistance of the stage, P the series “polarization” resistance of the surface layer at the probe tip, and K the “cell constant”. Typic& best-fit values were L = 550 kf& P = 30 l&l and K = 100 cm- I_ For a hotiogcneous solution, if the probe were a sphere of radius r and the second electrode a concentric sphere of radius u, then an elementary calculation would give

Desalination, 9 (1971) 155-180

T. J. HENDRICKS

AND F. A. WILLIAMS

Fig. 2 Block diagram of impedanm bridge.

which, with the best-fit K, yields r ;2: 8~1for r/u 4 1, in good agreement with microscopic observations of tip radii. Study has ken given to the effects of concentration gradients and membrane proximity on measured resistance (7). Eq_ (2) suggests that such effects will be of order r/a, where a is eitber the distance over which s changes by a few percent or the distance from the probe tip to the membrane. In fact. the effects appear to be smaller than r/a. A linear expansion of the concentration profile about the tip location produces cancelling corrections to K, thereby suggesting rhat the effat

of nonuniformity

is of order (+/u)‘, which is less thon 1 o/Oin the

present experiments. The value of K for a spherical probe in a uniform solution adjacent to a perfstly insulating plane wall can be calculated by the method of images; if r, represents the distance from the center of the spbeze to the wall. then it is found that K=
r~~~~~4~r)

(3?

at ‘large values of a]r, A’ = 1_443/(4nr) for a = I (contact with the wall), K = 2/(&r) for a = 0 (center of sphere at the wall), and K = co for o ,( -r (sphere completely imbedded in the wall). Fig. 3 shows a representative profile of resistance versus stage travel J* for a microprobe near the membrane in the unpressuked DaoliMrion,

9 (1971) 1X5-180

DIFFUSION-LAYER

0

SIMJCTURE

5

IN

RO CHANNEL

IO

I5 Y

Fig.

3.

161

FLOW

20

25

(P)

Theoretical and cxpcrimc~tal resistance profiles adjacent to

a

membrane in a

uniform solution.

channel containing a uniform solution. The broken line is a theoretic4 curve cakutated from E& (3), and the experimental points illustrate that the probe must be roughiy two radii from the membrane for the resistance to increase by 10%. The probe shape may contribute to the result that the observed wall effect is less than the theoreticai, since a semi-infinite cylindrical probe with radial current lines, parallel to the membrane, would begin to exhibit a significant influence of the wall on the measured resistance only as the probe began to protrude into the membrane. Fig. 3 also illustrates the change in probe characteristics produced by driving the probe into the membrane. Since membrane contact generally renders tne microprobe useless, the increase in measured resistance in the vicinity of the wall provided a valuable indication of membrane location, which depended somewhat on applied pressure and past history of the system. For a few of the larger microprobes, a wall correction was applied by assuming that the ratio of the local conductivity to the free-stream ~nductivity was the ratio of the local resistance measured under high-slpeed turbuient flow conditions (under which the diffusion layer was of negligible thickness) to the IocaI resistance measured under the laminar fiow conditions of interest (7) Justification for this correction is given in Ref. (7). Desrilinotion,9 (1971) 155-150

162

T. J. HENDRICKS

AND

F. A. WILLIAMS

In the experimental procedure, first the system was pressurized and the desired salt concentration established. The channel was then allowed to come to thermai equihbrium and the microprobe to stabilize for a period of two to four hours. The probe was then moved toward the membrane under turbulent flow conditionq and the resistance monitored to determine the approximate location of the membrane and to obtain the wail correction factor for larger microprobes. A desired taminar ftow con~tion was then established, and resistancewasmeasured at selected normal distances from the membrane. One or two inward and outward passes of the probe were made, the local resistances being averaged for computing the concentntiin profile. Another laminar-flow Reynolds number was then established. and the profile measurement was repeated. Following the profile measuremenis, the channel Row rate was increased to promote rapid mixing, and the probe was tjren calibrated by increasing the solution salinity stepwise. recording resistance at each step. until the resistance reached the lowest value obtained in sampling the profile. Concentration profiles were then obtained from Eq. (f) by using tabulated dependcnces of s on concentration (7). 3. RESULTS

Concentration profiles were obtained for the conditions listed in Table 1, where ii denotes the ratio of the osmotic pressure difference across the membrane in the absence of a diffusion layer to the hydrostatic pressure difference, and o represents the nondimensional streamwise distance, defined as cf = AAp(3xhfD’U)‘“.

(4)

Fig. 4 shows a typical concentration profile for NHIN03. Here y is the ncrmal distance from the membrane, and 2 is the nondimensional excess conantration, defined as 2 = c/cc) - 1,

(3

with c the local salt concentration and co the bulk concentration. The error bars on the experimental points are an estimation of the statistical error in balancing the bridge and in positioning and calibrating the probe.. Errors in 2 are greatest at small values of y due to wall effects and at large values of y due to sensitivity of the instrument. Experimental points indicated by dots correspond to averages of two passes, whife the crosses represent four. The labeled curves are theoretical results which will be discussed in Section 5. The heavy solid line is an attempt to fit the excess concentration data to a function of the form z = Z,eWYjd).

(6)

whem dis a characteristic relaxation length for the profile. The data will necessarily DesaIim:ion,

9 (1971)

IS-180

DIFFUSION-LAYER

TABLE RANGE

STRUCTURE

IN RO CHANNEL

FLOW

163

I OF CO?.iiITIOW

UWXR

WHIM

CONCENTRAY’ION

PROFIJIS

WERE

MWURLD

lN CHANhZL

FLOW

sofr

-. NHaNOt

20

_-___.--

337

3.78 2.62 2.08 1.75 1.79

0.0166 0.0153 0.0167 0.015s 0.0293

410 819

;z

0.03 17

0.79

34

137

2.55

0.0270 0.56 IO 0.65

0.79 0.70

40

137 819 137

8.22 5.70 4.39 3.01 2.39

0.0085 0.0088

0.88 0.88

O-G141

0.85

0.0209 0.0182

6.8s 0.85

410

10

NaNO3

20

819 1365 137

410

819 NaCl

NaGOd

M

20

10 MgSO2

to

20

10 l

Two diffacnt

-

0.84 0.84 0.84 0.84 0.79

137

4.28

0.0249

410

2.95

0.0347

137

4.28

0.83

0.968 0.968 0.S

137 410 819 I37 410

4.95 3.66 2.72 2.41 1.73

0.0279 0.0382 0.0290 0.0774 0.125

0.99 0.99 C.99 0.98 O-98

2.3s. 5.M 1.63, 3.60 UC, 2.85 t.O9,2.40 4.70, to.40 3.25, 7.20 2.58, 5.70 2.17,4.80 SC

0.10,0.033 ox@, 0.027 0.09,0_02 0.06,0.02 0.04.0.027 0.C4.o.os7 0.03, 0.033 0.03, 0.02 0.033

0.98

137 410 819 3365 137 410 at9 1365 137

values of u and c5 wmpond

to tests with

two

different

0.98 0.98 0.98 0.99 0.99 0.99 0.99 0.98

membranes.

depart from this Iinear log-normal fit at the centerline of the channel where 2 goes to zero. The tendency of the data to deviate from the fit for small values of y is a result of the infiuence of the membrane on measured resistance. The exact position of the membrane is also uncertain, and it is estimated that theactual position of the membrane could be displaced to the left by as much as 50~ in extreme cases. This error in the zero point of the ordinate is equivalent to displacing the data up ward on the plot by the amount indicated by the solid bar on the 2 axis. Desufiruhm,

9 (1971) 155-t 80

T. J. HENDRICKS

164

AND

F. A. WILLIAMS

While Fig. 4 corresponds to a small value of S, Fig. 5 shows some results for relatively la& values of 6. The curves labeled I, 2 and 3 have osmotic pressures of 10.3. 11.9 and 12.7 atm, respectively. It is seen that the experimental profiles are ilpproximsced well by Eq. (6) for both smaI1 snd large values of & Eq_ (6) was found to hold over more than an order of magnitude in Z for all salts, over the entire range of b and D investigated herein. I

200

Fig. -?. C~ncent~~~ion

By comparing effluent

solutions,

I

1

400

600

profile

the bulk the rejection

shows representative curves rejection ratio at low values The break in the curve occurs 1600 and demonstrates the

1

I

BOO y fm~crce.)

in channel flow,

NHaN03,

I

1000

3p

-

1400

=0P 21.6

atm,

He

== 137,

c,, and c,, of the circulating and ratio co/c,, can be obtained experimentally. Fig. 6

concentrations

of rejection ratio versus Reynolds number. The low of Re is attributable to concentration polarization. approximately at the transition Reynolds number of enhanced mixing rate caused by turbulence. For

Re > 1600 diffusion layers were too thin to be measured with conductivity microprobes. Curves of rejection ratio tend to fiatten at the highest Reynolds numbers and are expected to approach I I( 1 - R), where R is the intrinsic rejection coefficient

of the membrane. Values of R obtained from this asymptote ranged from 0.7 to values exceeding 0.97 for various salts at various values of b (see Table I, in which the last 15 entries for R are approximate estimates). Intrinsic rejection coefficients Desalinafion.

9 (1971) 155-180

DIFFUSION-LAYER

STRUCTURE

IN RO CHANNEL

165

FLOW

‘Ov

I

1

I

I

5

/

Ok_I__L-----

’

1

I I

I

800

loots

I2001

I400

y ~mlCrons)

Fig. 5. Concentration profiles Large fb.

in channelilow, NHINOJ. Jp = 13.6 ntm, Re -2 :37.

obtained in this manner were employed in theoretical calculations OF profiles. It may be remarked that values of 2, inferred from Fig. 6 by using the simple theoretic4 rekttionship rP;‘ec = (1 - R) (1 -i- Z,) are in rough agreement with values obtained directly by using conductivity microprobes, as indicated in Table If; the average value of the ratio of the inferred to measured 2, is 0.9.

TABLE

II

COMPARLSOS

OF INFLRREU

AND

MUSlJREI>

EXCESS

CON’CENTRATlON

AT ML

MEblBRAM

SURFACE

FOR

SHahO3 _____

-._--.-._-----..-

--.. == 2f.7

- ..--

Re

Jp

-

htferred Z,, Meawrrd __--. _- - -_----__--

819

0.9 0.95

410 137

I.1 2.2

1365

-_~---____-__-_---

uirn

i?tp =

Zpc

-10.0

arm

Inferred ZK

Measured ZM

_--_--

-_-

0.8

0.3

-

0.95 1.8

0.35 0.4

0.3 0.7

2.8

0.5

0.85

Desalination, 9 (197 1) 155-l 80

T. J. HENDRICKS

*

21.5

otm

x

100

otm

o

X

Average

AND F. A. WILLIAMS

values

500

loo0

so00

io,ooo

Re Fis. 6. Depmdmce of apparentrejation ratio on Reynoldsnumber, NHaNOa. Since the rejection coefficient for NaNO, was found to be approximately to that for NH,N03 at comparable values of Ap and 6. profile differences for these two salts must be traceable to the difference in the diffusion coefficient D. On the other hand, since the value of I) is approximately the same for NaNO,

equal

and NaC1, profile differences for these two s&s result from different rejection coefkients. At given vaiues of CTand 6, NaNOs. whose diffusion coefficient is approximately 0.81 times that for NH,N03. exhibited an appreciably lower value of d than did NH,NO,; thus, the slope of the excess concentration profile in&eases as the diffusion coefficient decreases. The value of the rejection co&icier& on the other hand, affects wall concentration, without greatly influencing the slope; that an incrwse in R increases 2, was inferred by observing that at essentially the same values of c and 6, NaCl with R = 0.968 possessed a value of Z,. which was approxin.ately 25% higher than that of NaNO, with R = 0.85. The data shown in Fig. 7 correspond to the largest value of o for which accurate profile measurements were made. That in Fig. 8 is for N&I at sea water concentration: in this case the osmotic pressure, 35.9 atm, exceeded the applied pressure, and therefore water could be transported out of the brine only because of imperfect rejection. It was estimated that R = 0.5 for Fig. 8. The divalent anion in Na,SO.+ leads to very high rejection. It also tends to Drs&u~~im, 9 (1971) M-180

167

DIFFUSION-LAYER STRUCTURE IN RO CHANNEL FLOW

.

I

I

I

I

I

1

I

I

a H

f f

C

I

0

200

I 400

I

I

600

800

1 loo0

1

1200

-.

y krxrons) Fig. 7.

Concentration profile in channel flow, NaNOh Ap = 40.8 atm, Re -= 137. Large o.

produce time-dependent microprobe characteristics which introduce systematic errors into measured concentration profiies. Although these errors were not too great for Na,SO, solutions, they became extremely troublesome for the doubly divalent saIt MgSO,, as may be inferred from representative results shown in Fig. 9. It is seen that experimentat wall concentrations sometimes differed by a factor of three under identical conditions of channel flow. The major cause of the large errors was the fact that with the microprobe in a uniform MgSO., solution, the measured resistance increased irregularly with time. In comparison with the nitrates, where drifts of this kind were typically a few hundreths of a kilo-ohm ia ten minutes, changes of a few kilo-ohms in ten minutes were not uncommon in MgSO,. These small probes were much more susceptible to drift than the larger probes used in Ref. (6). An additional source of error in 2, for MgSO, was enhancement of the influence of uncertainty in the wall position, caused by the high profile slopes which stemmed from the Iow diffusion coefficient. 4.

THEORY

A number of theories have been developed

for concentration Dcsalinafion,

polarization

in

9 (1971) 155-l 80

T. J. HENDRICRS

168

AND F. A. WILLIAMS

IO 5-

I -Z

05-

--

1 -2.2

‘0

ab ;; 01-

,001’

0

I xx)

F’ra. II Concentration

I 400

profik

I 1 600 600 y Imlcrorls)

in channel

flow. NaCl.

I

I 1200

Ap = 21.6 attn. Re =

137.3fR

z- 1.

channel or tube flow (4,8-17). None should be compared directly with the present experimental results because none consider streamwise variations in wall flux and incomplete rejection for arbitrary values of 6. Hence we must either amplify existing theories or develop new ones. The character of the development of the diffusion layer in channel flow can be understood by first noting that there is a characteristic salt diffusion time t,, which is related to the diffusion coefficient D and to the characteristic wall velocity vc aecording IO I = Djv,‘. The characteristic wall velocity is the product of the membrane constant and the hydrostatic pressure difference across the membrane, = AAp. Associated with the velocity vc and the time rc is a characteristic k thickness of the concentration layer yc = vtfC = Djv,. Fig. IO illustrates characteristic lengths associated with tube or channel geometries. For a tube or channel with characteristic lateral dimension H and characteristic streamwise velocity V. the velocity gradient at the wall is of the order of U/U. The streamwise velocity within the polarization iayer is therefore of order U y,/H_ This velocity multiplied by 1, shoutd give the characteristic streamwise distance _x-, required for development of the polarization layer. Substitution of previous relations for y, and fc yields .rc = D2U/vc3H. Tube or channel fiows are known to have a characteristic entrance length .I-*for the velocity De.wdh.don,

9 (1971)

155-180

DIFFUSION-LAYER

169

STRUCTURE IN RO CHANNEL FLOW

016

A

I

l

d

.014

6

Q

A

012

0

0 010

A OA

0

9’ $_

002l-

0” 0

4

e

12

16

20

24

28

3.2

36

40

44

48

2w

Fig. 9. Profile slope Or-‘) and excessconcentration al the wall for Mg!50~ in channel flow, 20.4atm, h = 1.6mm.

3p -

_ Edge of velocity J t

\botindary layer \

Fig. 10. Ilfustration of lengths in channei flow problem_

profile to become fully developed, given by sg = H Re, where the Reynolds number is Re = OH/V, v being the kinematic viscosity. It is seen that xc/se = S~L)~/u,~lf ‘, where Sc E Y/D is the Schmidt number. Desalination, 9 ( 197 1) 1% 180

170

T. J. HENDRICKS AND F. A. WILWAMS

From the results of Ref. (6) for the unstirred batch cell, it is clear that for channel-flow, if R > 1 - B. then the polarization layer shmld remain of constant thickness yc after it develops. However, if R > 1 - 5, then the polarization layer should spread toward the center of the channel as a diffusion wave, with front location y proportional to the square root of time t. viz., y = (Dt)** Setting y = H andt = s,lCJ in this formula produces an expression for the axial distance required for the diffusion waves on either side of the channel to merge, .v, = UH’/D. It is seen that sD!.rc = (D/ocH)‘. With the representative numerical values Ap = IO atm. cl = IO--’ cm/set. atm and D = 10H5 cm’jsec., we have uc = 1OeJ cm/set., fc = 10’ sec. and yc = 10-r cm. Further setting H = I cm and U = 10 cm/set., with v = IO” cm’/sec., we find sc = lo3 cm, x,, = IO6cm and sg = 10’ cm. These numbers are very rough: in fact, usually H -z 1 cm so that xR < sc. As a basis for developing quantitative theories for the development of the diffzsion layer, we note that in channel-flow desalination systems, if buoyancy forces are suppressed, then it is quite reasonable to assume that the brine density, viscosity and diffusion cuefficienr are constant and that streamwise diffusion and streamwise viscous momentum transfer are negligible. With pressure in the interior of the channel assumed to be independent of the coordinate p normal to the wall, the equations for conservation of mass, momentum and salt become (/a), respcctively, r7ujd.v + Sujdp = 0

=

d

(7)

jdxj2h - r,,.jptt,

@)

and u iicfijx + v t?cjZ_v-

D d’cldy’

= 0.

(9)

Here u and u are streamwise and transverse components of velocity, s and y are the corresponding coordinates, the origin of the coordinate system has been placed at the leading edge of the membrane on one of the walls of the channel whose height is 2h, and f, = p’@u jay),= 0

(101

is the focal wall shear. The system is assumed to be symmetrical about the plane Y = h. In the form of the momentum equation appearing in Eq. (I%),the axial pressure gradient has been eliminated through integration over the channel cross section. Desalino~ion,

9 (1971) 13S-180

DIFFUSION-LAYER

Boundary

STRUCTURE IN RO CHANNEL

conditions

u(x, 0) = 0,

FLOW

171

at the wall for these equations

G(X, 0) = -u,,

are (11)

and D(&,Q),=

e = - R cfx,O)L*,,

where % = A Ahp[l - 6c(x, O)/col

(13)

Eq. (12) is a salt-flux conservation condition applied across the membrane, and Eq. (13) is the usual membrane equation. Upstream boundary conditions are c(0, _r) = ce = ccnstant,

u(0, s.) = ue(y),

(14)

where the sgecified function u&) is zs constant if the membrane originates at the channel entrance and if the vefocity profile is uniform at the channel entrance. Symmetry conditions are Sc/dp = dufdy = 1: = 0

at i’ = 11.

(1%

On physical grounds, this set of boundary conditions should determine a unique solution to Eqs. (7), (8) and (9). In most analyses it is assumed that the polarization layer is confined to a sufficiently narrow vicinity of the wail for the equations

to constitute a sufficiently accurate representation of the velocity field for use in Eq. (9). This approximation is usually excelfent for streamwise distances of order sc but necessarily breaks down for streamwise distances of order .~a. If Eq. (16) is adopted, then to calculate the structure of the polarization layer, the only additional information that is needed from the velocity field is the function T,.(S). In reverse osmosis wall velocities are generally small enough that, away from the hydrodynamic entrance region, the simple Poiseuille formula ?- = 3pvL;/h = constant

(17)

can be used. where the bulk velocity is Zh

Within the context of the approximation appearing condition for Eq. (9), in place of Eq. (1 S), is

aclap= 0

atr=

co.

in Eq. (16). a suitable boundary

(19) D~solinotiun. 9 (1971) 155-I 80

172

T. J. ffENDRICKS

AND

F. A. WILLlAMS

Except in a linearized analysis (7) which will not be discussed herein, Eqs. (16) through (19) have been used in all theories for fully-developed channel flow.

Thus, we assume that x 4 snThe published theoretical work which corresponds most cIoscIy to our experimental conditions is the series expansion approach of Gill, Tien and Zeh (13). Solutions were obtained as an expansion in powers of the nondimensional streamwise coordinate CT.The analysis was performed only for R = I, and the . corresponding results are labeled Ml in Fig. 4. Liu (19) has extended the analysis to R # I. To first order, he found that i where

Z = crR( I - S)[F(-

l/3, 2/3; -#/a3)/F(3/2)

F is the contluent hypergeomctric

- q/c].

(20)

function, F is the Gamma function, and

Q = _s AAp/D

(21)

is a nondimensional distance normal to ihe membrane. lie integrated a two-point boundary-vaiue problem numericaIly to obtain terms in higher powers of G. up to cr’_ It is these more complete numerical results that we have labeled M. IfR _= 1 - 6, ther in analogy with results given in Ref. (6). the asymptotic solution valid for s, + s < sD can be written easily. Roughly half of the present experimental results appear to obey the condition R 5 I - 6. For d = 0 and R = 1, we find that theoretically as 4 *x3. 2 = [03/9 + b - ,t(l + ~j’)-je-q_

(22)

where b is an arbitrary constant (or order unity) whose value cannot be determined by considering only the asymptotic regime. On physical grounds. one might expect that Eq (22) would be reasonably good for S = 0 and R < I (but close to unity) in the range of axial distance specified by I + cr’j9 G I/( I - R). For R -G1 - 6. the asymptotic solution is Z = [R/(1

-

RN evf

-sCl

-

S/(1 -

RM),

(23)

which possesses no x dependence whatever_ Eq. (23) describes the rejection-loss phenomenon (6) for channel flow. Values of 0 for a few of our experimental results are large enough to make comparison with Eqs. (22) and (23) interesting. Since the experiments show that except near the center of the channel, the excess concentration profiles can be represented accurately as an exponential function of the distance from the wall, it is quite reasonable to develop an integral method based on an exponential profile approximations Such a method would appear to offer an ideal balance between simplicity and speed of caleulat.ion on the one hand. and accuracy on the other, for use in practical design calculations. This method is developed here under the approximations appearing in Eqs. (16) and (17). In assuming a fully developed velocity field, the present formulation is less complete than the integral method reported earlier (4)_ In assuming that the Desulinuflon. 9 (1971) 155-l 80

DIFFUSION-LAYER

STRUCTURE

IN RO CHANNEL

FLOW

173

thickness of the polarization layer is small compared with the channel half-height, the present results (as well as those of Ref. (4)) will tend to become inaccurate at large longitudinal distances. The exponential profile approximation is entirely compatible with more detailed representations of the velocity field than those given in Eqs. (16) and (17). However, the velocity fietd is usually fuliy developed in practice and is always fully developed in our experiments, and selciom in our experiments does the thickness of the polarization layer become comparable with the channel half-height. Under these conditions, the present results should be more accurate than those of Ref. (4). Moreover. the simplicity of the present approach makes it easy, with minor modifications, to introduce more complicated wall conditions, such as a rejection coefficient dependent on wall concentration. With the nondimensional variables 2, ~7and < z 03/9, Eqs. (9) and (11 j through (16) yield the problem +zlag - (i - s - sz,)az;aq= d2zfi$, @zlw##-o = -R(I + Z,)(l-ii - 6Z,),

Z=O

at

<=Q

z’z/c’rl = 0

at

q = co,

(24)

)

where Z, zz Z(T.0). The final boundary condition replaces the condition &Z/all = 0 at ?J = hAAp/D because for lypical experimental conditions IrilAjr/D > 10. We assume an excess concentration profile of the form a59

q) = GW

which automatically boundary condition

(25)

expC - B(t>Sl.

satisfies aZ/aq = 0 at rr = co, provided that B(r) > 0.The given in the middle line of Eq. (24) yields

B(e) = R(I + Z,)(I

- 6 - &Z&Z,.

(26)

In the spirit of the integral method, we now require that the differential equation be satisfied in an integral sense. Substitution of Eq. (25) into the equation and integration from rt = 0 to rt = co readily yields d(ZJB”)/d<

= [R + (R - 1)ZJl

- i’i - 6Z,),

(27)

in which Eq. (26) can be used to obtain the single first-order, ditferential equation

ordinary

R2[R Z&l _ _-._. - d.-..-2. - 6Z,Y _ ____ +(Ri_--.--.. l)Z,-J( ----- 1 i.-----

dzw -z-

nonlinear,

=

Z,U(l

- 6) + (1 - 2&z,

+ SZJ

According to the initial condition in Eq. (24), Z,,.(O) = 0 is the initial condition for the integration of Eq. (28). Eq. (28) can be inteSrated analytically. However, the algebraic complexity of the result makes it more efficient to perform a numerical integration. Theequation can very easily and economically be programmed for a forward integration. From the result of the integration, B(r)can be calculated by Eq. (26). and the complete Desalination.

9

(1971) 15%tR0

T. J. HENDRICKS

174

AND F. A. WILLIAMS

concentration ficld can then be calculated from Eq. (25). Original physical variables are retrieved through Eqs. (4). (5) and (21). If one is interested only in the velocity through the membrane, the result of the integration of Eq. (28) can be used directly in the formula AAp(i

-

6

SZ,).

(29)

We have employed the Runge-Kutta-Gill fourth-order method to perform the integration. The average time required per run was 3.6 csonds on a CDC 3600. In the figures. results of the integral method based on the exponential profile are labeled 11 for R = 1 and I for imperfect rejection. 5.

COMPARISONS

Differences between theoretical results obtained by the series method and by the intepl technique are likely to reflect errors in the integral technique at Fmall values of b and in the series method at large values of r~.Results such as those shown in Fig. 4 demonstrate good agreement between the two approaches for intermediate and small values of ci. For large values of a, Eq. (23) shows that the underlying exponential-profile approximation of the integral technique is precisely correct if R < I - d; therefore results of the integral technique necessarily become valid as u + co with R < I - 6. These observations suggest that the integral technique will be at its worst as CJ-+ co with R > 1 - S, since the exponential profile obviously does not represent well the diffusion-wave phenomenon which is expected (6) to develop in this limit. Since we have not obtained accurate experimental results for u large with R > 1 - 6, the integral technique can be used to represent proper theoretical results for all of our experiments. On most figures, the only theoretical results shown are those of the exponential-pro~le integral method. Comparison of the curves in Fig. 4 shows that accounting for imperfect rejection in the theory substantially improves agreement between theory and experiment. For small values of S, use of R = 1 in the theory overestimates 2,. since the decrease in 2, produced by transport of salt through the membrane is not taken into account. For sufficiently large values of 6, Fig. 5 shows that use of .R = 1 underestimates 2,; the proper value R = 0.7 produces reasonable agreement between theory and experiment in Fig. 5. while use of R = 0.5 would again underestimate 2,. This illustrates the balance between a wall concentration limited by the difference between applied and osmotic pressures (JZ-+ I)? and one determined primarily by rejection (e.g., 2, --, 0 as R --, 0). In all of the cases tested, it was found that both theoretical and experimental slopes of the concentration profile were relativefy independent of the value of R and that good agreement between theoretical and experimental values of 2, could be obtained only by using the proper value of R in the theoretical calculations.

DIFFUSION-LAYER

STRUCTURE

IN RO CHANNEL

FLOW

175

The value of cr (8.22) in Fig. 7 is too large for profile calculations to be made by the series method. The integral method is seen to produce reasonable agreement when the proper value of R is employed. Since R -=I 1 - 6 for NaNO,, the results in Fig. 7 can be compared with the asymptotic sohttion given in Eq. (23). The value R/(I - I?). appearing in Eq. (23). should be an upper limit for Z, in all cases. This limiting restriction was always obeyed ex~rimentally. The closest approach to the limit occurs in Fig. 7, where R/(1 - R) = 7.3 and 2,. = 6.6. The trend in the departure from the limit for other cases was found to be consistent with the trend in CT.Eq. (23) predicts that d = 198~ for Fig. 7, while the data gives d = 16814.Thus the asymptotic theory agrees with the integral method in predicting a somewhat shallower slope than that observed experimentally. Theory and experiment both show that the slopes are equal for the firsi two NaNO, entries in Table I and shallower for the third. Since 6/A > I for Fig. 8. theoretical calcutations cannot be made with R = I. The good agreement, in this przcutiar case, between experiment and the integral theor+/ $v&h the ;rasonable selection R T-- 0.5, emphasizes the wide range of conditions over which the integral technique can be applied. In spite of the large degree of experimental scatter in Fig. 9, one can discern a qualitative agreement between the integral theory and experiment. Trends caused by changing the membrane constant and by changing the flow rate are mainly in agreement. A comparison of the experimental profiles with the theoretical predictions of the exponential-profile integral model is shown in Fig. 1 I for all the runs with ammonium nitrate, sodium nitrate, sodium chloride. or sodium sulfate solutions for which the rejection coefftcient is well known. The horizontal bar represents the error associated with the uncertainty in the membrane position_ The average value of Z,JZ,, for the nitrate and chloride solutions is 0.96, indicating good agreement with the integral theory. The sodium sulfate values ate shown by dotted lines and fall significantly above a value of unity. Since some difficulty was experienced with drift in this solution. and since drift would cause a shift in this direction. these experimental values are somewhat questionable. Including the Na2S0, data, the average ratio for the excess wall concentration is 1.I 1. The ratio of the slcpes shows that except for one case, the experimental slope is always greater than that predicted by theory. In three cases, the experimental slope is more than twice as steep. Fig. l2a is a plot of the ratio of the experimental relaxation length to the theoretical relaxation length as a function of the channel Reynolds number. Althou~k there is an appreciable amount of scatter, there is a significant trend towards a smaher value for this ratio as the Reynolds number increases. Fig. 12b, a plot of the ratio of the experimental excess wall concentration to the theoretical excess wall concentration, as a function of the Reynolds number, suggests that this ratio may also decrease as the Reynolds number increases, although the effect DfW&If.Jlio~r, 9 (1971) lSS-180

176

I-. J. HENDRICKS

*I

--

---

----

OOw

6I

.81

I.0 I

12 1

I.4 I

AND F. A. WILLIAMS

NH NO,

f-h

NOCI

No+&

16 I

I.8 1

20I

24I

26I

=wc =w,

Fig. 11. Comparison of thcotxkal and cxpcrimenlal conceuttation at the wall for channel flow.

values of profile slope and CXCQS

is certainly less pronounced than that for the relaxation length. The results are consistent with a variety of functional dependences on Reynolds number, one being a Iinear relationship passing through unity at Re = 0. An explanation of Fig. I2 would afirtiori explain the steeper experimental slope. Although disturbance of the fiow field by the presence of the microprobe would produce the observed trends with increasing Reynolds number, estimates of the magnitude of this effect indicate that it is unlikely to be significant; exploratory investigations with very thin probes and with probes slightly canted forward. although not entirely conclusive because of experimental difficulties, nevertheless did not provide convincing evidence for the presence of substantial probe-generated fluid mechanical disturbances. We fee4 reasonably certain that all points in Fig. I2 correspond to smooth, laminar flow. It is possible that a wavy wall, produced by imprinting of the support screen on the surf&e of the membrane, could change the diffusion profile; microscopic examination of the membrane revealed a regular threedimensional wavy surface with a wave length of 254~ and an ampEtude of IO to 25jl. Fig. 13 contains a summary of the nondimensional excess membrane concentration,

Z,, for all the runs of ammonium

and sodium

nitrate, sodium

chloride,

Desdinarion, 9 (1971) 155-180

DIFFUSION-LAYER 2.0 16

STRUCTL’RE

IN RO CHANNEL

177

FLOW

0 F

0

0 NH,NO, 0 NaNOs 0 Nact

b IO

20

I

Re

Fig.

52. Dcpendcnce

of profile slope and of exams concentration

at the wall on Reyuolds

number.

sodium sulfate, that have b 4 1. With the exception of the two sodium nitrate series at R = 0.88 and 0.85, the rate at which the experimental value of Z, increases with increasing c decreases monotonically with decreasing rejection. The deviation for sodium nitrate is probably experimental error, since one series (R = 0.85) lies above the general trend and the other (R = 0.88) below. For sodium sulfate, where some drift problems were encountered, there is also a tendency for the experimental values to lie above the theoretical values, while for the other solutions, the deviations appear more random. In theory, the sodium sulfate curves, which have R + B > 1, should approach (1 - S)fri as Q --+ 00, while the nitrate curves, which have R + S < 1, should approach (I - R)/R. There is and

De5dinution.

9 ( 1971) 1% 180

t. J. HENDRICKS

Fig. 13. Summary of channel-flow

observations

AND

of excess concentmtion

F. A. WlLLlAMS

at the wail.

suggestion of an infiection in the sodium nitrate curve for R = 0.88, which is consistent with an approach to the rejection-loss asymptote. None of the curves contradict the linear behavior that is expected theoretically near tr = 0.

a

6. CONCLUSIDNS

Concentration

4.Y)=

co + (c,

profiles are approximated -

well by an equation of the form

co)tYd;

where c, is the centerline concentration, c, is the concentration at the membrane, y denotes the hormal distance from membrane, and d is the characteristic difiusional relaxation iength. This result has been demonstrated to be good for I I cz 5 8 and for 0.001 5 S/R s 0.8, with R > 0.7. It is expected theoretically to breakdown onlyifbothR s 1 - dandaislarge. An integral method using an exponential profile as input, gives good agreeDcsd4iwtion. 9 (1971) 155-180

DIFFUSION-LAYER

ment

STRUCTURE

with a series expansion

IN RO CHANNEL

method

FLOW

for small

values

179 of cr and with asymptotic

theory for large values of cr with R c I -- ii. It also gives good agreement with experiment for the concentration at the membrane, although the predicted slope is generally less than observed. The difference increases with increasing Reynolds number. Relatively good agreement for fi, is obtained for cases which cannot be cafculated by the series expansion method (including modi~cation for imperfect membranes) or by the asymptotic theory. The integral method may be simply modified to include variable rejection, or more accurate descriptions of the f3o.v field, and requires less compu;er time than the expension method. Within the accuracy of the measurements. the direct measurements of the excess concentration at the membrane agree with the values inferred by measuring the water and salt fluxes. Conductivity microprobes can be used 10 obtain accurate salt concentration profiles in nitrate solutions under laminar flow conditions. ThT-& probes arer difikult ‘10 use in chrtnnef-fiow experiments wi’h magncrium sulfate solutions. Quanti’iat.ve channel-flow expenments are at least an order of magnitude more dificult to perform than ark quantitative batch-cell experiments. However, with the equipment that has now been constructed. channei-flow measurements of SALconcentration profiles for nitrate and chloride solutions have become relatively straightforward.

REFERENCES

t _ U. MERm, 2.

3. 4. 5.

6. 7.

8.

9. IO. ii. 12.

H. K. I!_.o%sDM_EAND R. L. RILEY, Boundary-Layer Effects in Reverse Osmosis, I&. Etzg. Chem. Fwxzkmentals, 3(1964) 210-213. T. Ic. SHERWOOD, P. L. T. BRIAN AHD R. E. FISHER, Desalination by Reverse Osmosis. Ind. Eng. Chem. Fu~da~nru~s. 6(1%7) Z-12. H. STRATHMA~X. Control of Concentration Polarization in Reverse Osmosis Desaiination of Water, Summary Rcpt., Contract 14-01-0001-965. Amicon Corp., ixxington, Mass.. 1968. S. SRINIVASAN. C TIE!! AND W. N. GILL, Simultaneous Development of Velocity and Concentration Profiles in Reverse Osmosis Systems, Chem. En& Sk. 22 (1967) 417-433. H. !k~~cmt~~, Bounchy Lapr 7’heor); 4th Edition. McGraw-Hill. New York, N.Y., 1960. M. K. LIU AND F. A. WILLIAM, Concentration Polarization in an Unstirred Batch Cell: Measurements and Comparison with Theory, fnr. J. Heat Marr Trons/Pr, 13 (1970) 1441-1457. T. J. HENDRICKS AND F. A. WILLIAMS, Boundary Layer Flow Problems in Desalination by Reverse Osmosis, Final Rept.. OtTice of Saline Water Grant No. 14-01-0001-951, University of Calif., San Diego, June 1970. U. MERTEN, Flow Relationships in Reverx Osmosis, 2nd. i5fg. Chem. FundamenfuLr. Z(1963) 229-232. T. K. SHERWOOD, P. L. T. BRXAN. R. E. FLQZER AND L. DRESNER, Salt Concentration at Phase Boundaries in Desalination Processes, fnd. Eng C&em. ~anda~re~~~u~,4( 1965) 113-l I 8. W, N. GILL, C. Tnm 4~zm D. W. ZE& Concentration Polarization Effects in n Reverse Osmosis System. Ind. fig. CIzem. Fitndamenrafs, 411965) 433439. P. L. T. BRAN, Concentration Polarization in Reverse Osmosis Desalination with Variable Rux and IIncomplete Salt Rejection. Ind. h. Chem. Fundamentafs. 4(1%5) 439-445. W. N. GILL, C. Tm ATGIJD. W. Zul, Concentration Polarization in a Reverse Osmosis Sj’stem, ind_ Eirg. Chem. Fundanrmtuls, 5(1%6) 149-150. Desalination, 9 (1971) 155-180

180

T. J. HENDRICKS

AND F. A. WILLIAMS

13. W. N. GILL. C. TIES ASI) D. W. ZEH. Analysis of Continuous Rcrcrsc Osmosis Systems for Desalination, Intern. J. I&WI hfass Transfw. 9(1966) 907-923. 14. C. TIES AYD W. N. GILI_ The Relaxation of Concentration Polarization in a Reverw: Osmosis Desalination System. AIChE J.. IZ(1966) 722-727. IS. W. N. G~Lc.. D. W. 2!~tt ASD C. TIE<. Reverse Osmosis in Annuli, AICirE 1.. 12ff966) 16. 17.

I Is;--I 146. W. N. Gtu.

D. W. ZEII ANDC. TIES, Boundary Lager Effects in RcverscOsmosisDesaiination. inil. &. Ckvn. F~~da~~nr~~s x1%6) 367-370. S. SRI~IV~SAX AXD C. Trt~, Reverse Osmosis De&ination in a Tub&at Membrane Duct. Des&kizlio?~, 3(1%7)

s 16.

18. F. A. WILLIAMS. Linc3rizrd Wl%S) 19.

Analysis

of Constant-Property

Duct

Hews.

J.

Fluid

hfrch.

241-261.

M. K. LIU. unpublished

works.

De.sdina/iun.

9 (1971) 155-180