CONCENTRATION POLARIZATION IN REVERSE OSMOSIS UNDER NATURAL CONVECTION
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Concentration polarization in the desalination of salt water by reverse osmosis i s investigated for the case in which the desalination membrane i s a vertical flat plate. An expression for the concentration of salt at the surface of this membrane i s obtained by expanding the solution of the governing laminar boundary layer equations for natural convection about the leading edge of the plate. This expression, together with an equation for the membrane concentration at large distances from the leading edge, can be used then to predict the polarization effect at any point along the membrane.
CONC‘ESTRATION polarization offers one of the main obstacles to the efficiency of reverse osmosis systems, in that the buildup of salt-rich fluid near the surface of the osmotic membrane decreases the effective pressure driving force by increasing the local osmotic pressure of the solution. This problem has been studied in the past primarily for batch (Dresner, 1963; Nakano et al., 1967) and forced convection (Dresner, 1963; Srinivasan et al , 1967) systems. The latter are of particular practical importance because the convective motion past the membrane tends to clean the surface of the salt-rich solution, thereby enhancing the efficiency of the process. However, this can also be accomplished in a natural convection system, in which the fluid motion is imparted by the difference in density betrveen the bulk solution and the heavier fluid near the membrane surface. The equations for natural convection past a vertical flat plate, shown schematically in Figure 1, have been developed previouslj--for example, by Schlichting (1960) and Ostrach (1953). I n the present case the conditions a t the surface of the plate (membrane) require special attention. To begin with, for the perfect semipermeable membrane assumed here, the net flux of salt through the surface must be zero-Le.,
Furthermore, the normal velocity, is no longer zero on the surface, but is related t o the over-all pressure-driving force by means of the standard relation (Merten, 1963) ti’,
The momentum, continuity, and energy equations are next made dimensionless using NAAP/Sc2 for the characteristic velocity parallel to the plate, AAP for the characteristic velocity normal to the plate, D / A A P for the characteristic length in the normal direction, and DN/Sc2AAP for the characteristic length along the plate. The resulting system then becomes
=
e(a)= o
together with the membrane relation
U ( x , o ) = -[I
- a e ( x , 011
(10)
However, since for a typical reverse osmosis system-NaC1 and water, for example-AT = 0(109) and Sc = 0(103), Equations 3 and 4 reduce simply to
The above system cf equations applies, of course, only within the diffusion layer. Outside this region, buoyancy effects are effectively negligible but the inertia terms must be retained (Stewartson and Jones, 1952). The solutions appropriate to each region must then be developed separately and matched within the overlap domain according to the standard requirements of the method of inner and outer
I I
I
d \
8
a u av -+-=o ax a y with boundary conditions u(x, 0)
=
0
Figure 1.
Schematic of vertical plate
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MEAN SPUARE OUTPUT,
e-qansions. For the case of large Schmidt numbers, however, this extra Complication is unnecessary, since it can ewily be deniorirtrated that the solutjon within the diffusion h l e r becomes uncoupled fiom that in the region beyond, and that the appropriate boiindary conditions at the edge of this tiiffusion layer (y+ 03 ) are as shown in Equation 9. The solution to the problem formulated above u as obtained lj! means of the series expailsion for the concentration and velocity profiles about, the leading edge of the plate
e
-
m
z”’5p, ( 7 )
= P1’5 n-0
m
11
-
(14)
P77’5jn’ (‘7)
2.3’5
Downloaded by STOCKHOLM UNIV on September 9, 2015 | http://pubs.acs.org Publication Date: May 1, 1969 | doi: 10.1021/i160030a030
n-0
~>!iich\Then substituted into Equations 6 to 12 yield a system c: coupled ordinary differential equations for the functions 1,-and pn. Although, following the method of Blasius (dchlichting, 1960), this system could have been transformed into one involving only universal functions independent of cy, ir was found simplest to solve the original set numerically for any given cy using the standard Kutta-Merson technique (Fox, 1962). Of particular interest are the values of 0 (2, 0)-i.e., the dimensionless salt concentration at the surface relative to that of the bulk-which, in view of Equation 2, determine the flow rate through the membrane when all the other parameters of the system are known. From Equation 13,
e (z,o) = 21’5
aX”5
Figure 2. Dimensionless surface concentrution along the plate for different values of 01
fact, since e (2, 0 ) --+ l / a and u (z~ 0 ) --+ 0 fnr from the leading edge, the problem reduces there to the standard case of free convection from a solid vertical plate of uriifornr cwicentration. The appropriate similarity solution which applies a t high Schmidt numbers (Acrivos, 1960) yields then that
-
m
c
245pn ( 0 )
0.50
as x-+
m
(15) and, therefore, in view of Equations 8 and 10, that
n-0
in which the coefficients pn(0) must be obtained from t h e solution of the system of ordinary differential equations mentioned above. Equations 6 to 12 can be simplified further if cy>> 1. Thus in terms of the new variables 8 = a0,2 = a5z, = ay,C = v, and d = a3u,Equations 6 to 12 remain unchanged except for the two boundary conditions, Equations 8 and 10, which nom become
Some typical profiles for e (z, 0 ) obtained from these solutions are shown in Figure 2. These curves, together with Equations 15, 18, and 20, provide a clear indication of the polarization effecf, in natural convection past a vertical flat plate and can be used to estimate the efficiency of a reverse osmosis process under such conditions. Nomenclature
and
(17)
A C
C, Again using an expansion similar to Equations 13 and 14 followed by a numerical solution of the resulting system of oidinary differential eq1iatioiw, one arriw- once inore at Ikpation 15. It call be shown that the coefficients in F k p t i o n 15 are polynoinial functions of cy of the form pn ( 0 ) =
C p m , n (0)am
= 0, 1, 2,
*
(18)
m=n
These new coefficieiits prn,,,(0), which are now independent of a , were calculated by using the values of pn (0), obtained from the solution of Equations 6 to 14, and the corresponding set of a’s. The resulting values of pm,,,(0) are shown in Table I. The solutioii developed so far is limited to a small or a t least moderate range of z [outside a small region near the leading edge of radius 0 (Sc2/iV)] and clearly fails when z -+ 00. I n 360
(3+4m4
l&EC
FUNDAMENTALS
C,
= membrane constant, cm./sec.-atm. (typically = salt concentration, moles/liter = salt concentration of bulk solution, moles/liter = salt concentration at membrane surface, m o k y litei
D
= diffusivity of salt in mater, sq. crn./see.
g
= acceleration of qravity, cm./swz = dimensionless group = pCmp/ ( B L V ) ~ = pressure on salt water side of membrane, atm.
N P, Pa
on fresh water (product) sidc of memhrsne, atm. Po,(,) = osmotic pressure of bulk solution = aC,, atm. A P = P, - Pa atm. Sc = Schmidt number = v / D u’ = dimensional velocity parallel to plate wrface, cm./sec. u = dimensionless velocity parallel to plate wrface -u’Sc2/NAAP v’ = dimensional velocity normal toplate swfaw, an /sec. v = dimensionless velocity normal to plate siirface = u‘/AAP z’ = dimensional distance along plate (from leading edge), cm. = pressure
Table I. Coefficients pm,n(0) in Equation 18
_-
11
0 1 2 3 4 5 G
o
1
2
3
4
5
6
1,58317 1.0517G 0.30097 -0.02291 -0.05339 -0.02001 -- 0.00234
- 1.91072 -3.72888 -3.03087 - 1.03920 0.21338 0,41670
1.91259 7.77975 11.78384 9,16252 2,54684
- 1.62654 -11,27606 -29.81984 -41.29691
0.82698 13.02682 59.78568
- 0,20327 - 14,32774
0.11023
y’ y
= dinienqionleqs distance along plate = z’Sc3AAP/vN = dimensional distance normal t o plate, cni. = dimensionlesidistancenormaltoplate = y’ScAAP/v
cy
= diniensionless number = TC,/AP
5
3 17
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m
_--
-
V T
p pH
constant relating concentration and density = (1 - p , / p ) / ( C - C,), lit’ers/niole = similarity parameter = y / ~ ’ ’ ~ = dimerisioiiless concentration = (C - C,)/C, = kinematic viscosity, sq. cm./sec. = constarit relat,ing concent,ration and osmotic pressure, at 111.-Ii t erjniole = solution density, g,’cc. = I ~ l keolut,ion deiisit,j-,g/cc. =
literature Cited
Acrivos, A , , .L.Z.C‘h.h’. J . 6, .is4 (1960). Dresnrr, L.. “Boiintlary Layer Riiildiip i n the Demineralization
of Salt Water by Reverse Osmosis,” Oak Ridge National Laboratory, ORSL Rept. 3621 (1963). Fox, L., “Numerical Solrition of Ordinary and Part,ial Differential Equations,” Pergamon Press, Oxford, 1962. hlerten, u., ISD. EN. CHEM. FUND.AJiENT.4LS 2, 229 (1963). Kakano, Y., Tien, C., Gill, W. N., -4.Z.Ch.E. J . 13, 1092 (1967). Ostrach, S.,“Analysis of Laminar Free-Convection Flow and Heat Transfer about a Flat Plate Parallel to the Ilirection of the Generating Body Force,” National Advisory Committee for ileronaiitics, KAC-4 Rept. 1111 (1953). Schlichting, H., “Boundary Layer Theory,” McGraw-Hill, New York, 1960. Srinivasan, S.,Tien, C., Gill, W. X., Chem. Eng. Sei. 22, 417
(1967).
Stewartson, K., Jones, L. T., J . ileronaut. Sri. 24, 379 (1957).
-4.R. JOHNSON ASDREAS ACRIVOS Stanford t,7niversity Stanford, Calij. 9@06 I~ECEIVED for review December 23, 1968 ACCEPTEDFebriiary 27, 1969 Work supported in part by the Office of Saline Water, U. S. Department of the Interior.
AXIAL MIXING AND SELECTIVITY Two Consecutice First-Order Reactions For reaction systems with linear kinetics, the performance of nonideal reactors can be accurately calculated by a simple numerical integration of the residence time dist ’bution. Experimental residence time distribution data were used to calculate reactor performance for a system of two consecutive first-order reactions. Mixing parameters for the dispersion and the CSTR cascade models were obtained from the variances of the residence time distributions. Model predictions gave excellent agreement with the integration results over a wide range of mixing intensity. Reactor selectivity was strongly dependent on mixing and reaction kinetics.
T m axial dispersion model for fluid mixing in continuous vessels has been the subject of intensive study, particularly as a representation of nonideal chemical reactors. The basis for the model and its mathematical aspects have been thoroughly covered by Levenspiel and Bischoff (1963). Little of the effort’ siiice has been applied to t)he effect of dispersion on the selectivity of multireaction systems. A simple example is the system of two consecutive first-order reactions, A--+13-+ C. It is, however, ai1 important case, since it appears frequently in industrial practice. Since the rate expressions are linear in concentration, analytical solution of the species conservation equations is possible. Kmmers and Westerterp (1963) solved the case of the steady-state isothermal reactor with axial dispersion, using the boundary conditions proposed by Danckwerts (1963).
dC, d2C, u-=D-+rt az
C,(O)
az2
dC,
- D - ( 0 ) = C,, dz
dCi
- ( L )= 0 dz
I n solving the same problem, Tichacek (1963) specified a semi-infinite reactor using B (m ) = 0 as the second boundary condition. Therefore, these t u o solutions give different results at relatively high values of the dispersion coefficient. On the basis of his results for this and other consecutive reaction systems, Tichacek concluded that the yield loss of VOL.
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