literature Cited
( 1 ) Barker, J. A,, Fock, LV., Smith, F., Phys. Fluzds 7 , 897 (1964). ( 2 ) Danon, F., Pitzer, K. s.,J . Chem. Phys. 36, 425 (1962). (3) Guttman, I., Meeter, D. A , , Dept. of Statistics, University of Wisconsin, Tech. Rept. 37 (1964). (4) Hirschfelder, J. 0.. Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liquids,” corrected printing with notes added, IYiley, New York, 1964. ( 5 ) Marquardt,D. L., J Soc.Znd. Ap,bl.Mafh. 2,431 (1963). (6) Meeter, D. A., Ph.D. thesis, Dept. of Statistics, University of Wisconsin, 1964. ( 7 ) O’Connell, J. P., Prausnitz, J. M., AD1 Auxiliary Publica-
tions Project, Library of Congress, Washington, D. C., Document 8432, 1965. ( 8 ) O’Connell, 3. P., Prausnitz, J. M., in “Advances in Thermophysical Properties at Extreme Temperatures and Pressures,” pp. 19-31, Purdue University, Lafayette, Ind., 1965. ( 9 ) Sherwood, A. E., private communication, June 1965. (10) Sherwood, A. E., Prausnitz, J. M., J . Chem. Phys. 41, 429 (1964). (11) Tee, L. S.,Gotoh, S.,Stewart, W. E., IND.ENG. CHEM. FUNDAMENTALS 5 , 356 (1966). RECEIVED for review September 17, 1965 ACCEPTEDApril 7, 1966
BOUNDARY LAYER EFFECTS IN REVERSIE OSMOSIS DESALINATION W I L L I A M N. G I L L Clarkson College of Technology, Potsdam, N . Y .
DALE ZEH AND C H I T l E N Syracuse University, Syracuse, N . Y .
Predicted average production rates based on a detailed analysis of brine-side liquid-phase diffusional behavior of continuous reverse osmosis in a parallel plate channel are compared with the experimental data of Merten, Lonsdale, and Riley. The agreement obtained is good and tends to bolster confidence in the method of analysis, which accounts for nonlinear effects,
osmosis, or ultrafiltration, is an attractive process for desalination because of its simplicity, and comprehensive research programs are under way to study various aspects of this process. T h e liquid-phase brine-side diflusion problem is interesting, because it involves nonlinear effects which polarize salt concentration a t the liquid-membrane interface; this increases the effective osmotic pressure and thereby reduces the system productivity. Merten, Lonsdale, and Riley (5) recently reported experimental results for a continuous reverse osmosis system and correlated their data on the basis of both fully developed turbulent flow and ma’js transfer in the system studied. However, they indicated a need for more exact theory. Thus the purpose of the present note is to show that their results can be reached reasonably well by theory based on different and, in view of the low h’Rs range investigated, seemingly more reasonable assumptions than they set forth. T h e experimental qjystem studied by these investigators consisted of a membrane chamber 0.254 cm. high, in which rectangular membranes 2.54 by 7.62 cm. were employed. Thus the aspect ratio was 10 to 1 and the channel lengthheight ratio was 30 to 1. \Yith a n aspect ratio of 10, for simplicity, this system can be considered as flow between infinite parallel plates and on this basis all except the limiting experiments ( U = 200 cm. per second) on Figure 1 of (5) were run in the range of about 12 to 2700. T h e data for membrane 2, which were taken in the region where the throughput is more sensitive to the brine velocity, correspond to XRe5 120 and thus on the basis of the calculation of Collins and Schowalter ( 7 ) the hydrodynamic entrance accounted for only EVERSE
about 0.8 to 8.0y0of the total channel length. Under ordinary circumstances one would expect these data to be analyzed on a fully developed laminar flow basis. Furthermore, since If,, N 530, the diffusion boundary layer will be much thinner than the momentum boundary layer and one can simplify matters by using a linearized expression for the velocity field near the membrane surface, as was done for the case of constant flux by Dresner (2). The diffusion equation for the boundary layer can be written
where u = (rw/lL)y, u =
uw
- -1 -d r w y-2 p dx 2
Flow through the membrane is described by -vW
=
AAP
- ?r0C+(x,O)= AAP[1 - BzC+(x,O)]
(3)
Therefore for fully developed flow, the problem is described by the following dimensionless nonlinear system of equations:
bC+
3 P. - = +[I
3.
- B2C+(.,O)]
2c + 3 p} bC+ + aab2 +
C+(O,co) = 1
bC+
- (.,a) 3P VOL. 5
(4) (54
= 0
NO. 3
AUGUST 1 9 6 6
367
where
and Equation 5c is obtained by equating the expression for the water flux equal to zero a t the membrane surface, y = 0, and combining the result with Equation 3 as was done to obtain Equation 5b of ( 3 ) . A method for solving this nonlinear system of equations was developed recently (4). Let m
c+ = 1 + C ei(p)d
(7)
i=l
so that
&’’+ 3 p2ei’ - sip& =
(B2 - 1)O’i-1
+ B2
i -2 j=1
Oj(0)O’t-,-l (8)
and the boundary conditions are &’(a) =
and -O,’(O) = 1
- BP,i
0
(W
= 1 i--2
-e,(o)
= (1
- 2 ~ ~ ) e ~ - ~-( B02) C
~,(O)LL,-~(O),
jol
iI 2 Ob)
The solution for el, can be obtained exactly and is
Higher order functions must be obtained by numerical integration and, for B:! = 0.33, the parameters necessary to calculate the water produced were found to be
e,(o)
=
x
0.4918, e2(o) = -0.3755 10-1, e4(o) = 0.3683
x
10-3, ea(0) = -0.3374 x e5(o) = 0.3227 10-2
10-2,
x
T h e average flux for the entire membrane, designated F1 by Merten et al., is given by
For the case of fully developed flow between parallel plates 3 U/.l
r* = -
R
and
Therefore F1 is given by
Figure 1 compares the laminar flow theory with the membrane 1 experiments of Merten et al. and with their Equation 4 which was derived to correlate the data. A linear plot is used 368
I&EC FUNDAMENTALS
here, since it shows clearly that most of the experimental data on this membrane were obtained in the region where the productivity of the system is very insensitive to the brine velocity in the channel. Therefore, these data are useful for obtaining membrane constants but not very useful for theory testing. If A = 0.76 X 10-5 gram/sq. cm. sec. atm., which is the value reported by Merten, Lonsdale, and Riley (5), the laminar result runs parallel to but somewhat below the turbulent correlation until one reaches U ‘V 5.5 cm. per second and then the throughput predicted (5)falls off very much more rapidly than the laminar results. If A = 0.823 X 10-5 gram/sq. cm. sec. atm., which is the membrane constant calculated by present theory on the basis of fitting the experimental value a t 32 cm. per second, the two 16 approaches yield essentially coincident results down to U cm. per second, where the turbulent correlation again falls off most rapidly. Both theories agree very well with experiment, but since the data on membrane 1 are all in the region of flat throughput response to changes in U (minimum U for membrane 1 is 5.7 cm. per second), Figure 1 does not provide a definitive test of the correspondence between theory and experiment. I t does, however, suggest that experiments in the low velocity range would provide such a test, since the laminar and turbulent theories differ substantially in this region. Even though the present theory and that of (5) predict essentially the same average throughput for the experiments on membrane 1, the detailed behavior of the system predicted by these two approaches is markedly different. Thus if accurate experiments on local behavior were available, definitive comparisons could be made in these terms. Such data are difficult to obtain accurately. Fortunately, Merten et al. performed low velocity experiments on membrane 2, and here one can see very marked differences in the correspondence of our solution and that of (5) with the experimental data. Such differences are best illustrated by a semilog plot, as shown on Figure 2. T h e throughput corresponding to the lowest experimental velocity used, U = 0.2 cm. per second, is predicted to be 0.7 X gram/sq. cm.-sec. by Equation 4, whereas the present theory predicts F1 = 3.04 X 10-4 gram/sq. cm.-sec. and the experimental value is 3.16 X gram/sq. cm.-sec. Thus, for U = 0.2 cm. per second the turbulent results are too low by a factor of about 4.5, whereas the relative difference between laminar theory and experiment is less than 4%. T h e other three experimental points for membrane 2 also are obviously in much better agreement with the present work, and the average difference between theory and experiment is about 5%, which is probably within experimental error. For the various runs made on membrane 1 the ratio of the inlet to outlet water flux varies from about 1.15 to 1.35not very large flux variations along the conduit. However, for the membrane 2 experiment, the ratio varies from approximately 1.5 to 2.4. To determine the value of A , Merten et al. assumed that polarization was zero in the high velocity test runs a t 200 cm. per second. The present results suggest that the polarization is different from zero even in these cases, since the constants which seem to correlate the data best are slightly larger than those reported. However, these effects are reasonably small, less than 10%. T h e experiments of Merten et al. were not sufficiently detailed to test definitively the local aspects of the laminar flow theory. However, the rather good agreement between observed and predicted over-all results enhances confidence in both the experiments and the theory.
- e*-* &----’.-
c
-
‘ A=0.823~10‘~ (Gr/cm? sec.atmJThis
Ii
%:I1 I
A.0.76
x
,L----=-
work.
(Gr/cm? sec,atmJMerten et 01, Eq.14
V cdsec. Figure 1.
Relation of average flux to bulk mean channel velocity for membrane Linear plot 7.62-cm. channel length, l / R = 6 0
3.9 !
/
/ 1.5
I
/
/
/
Log Figure 2.
v
Relation of average flux to bulk mean channel velocity for membrane Semilog plot 7.62-cm. channel length, l / R = 6 0
Nomenclature
A = B2 = C+ = D = F = L = NB, = NPe = AP =
R U u
membrane constant +/AP ratio of point to inlet concentration, p 2 / p 2 ( 0 , y ) diffusion coefficient average mass flux over-ill length of conduit Schmidt number, Y / D Peclet number, ~ R U / V pressure drop across membrane = half distance between plates of channel = bulk mean channel velocity = local axial velocity
u uw x
y
local transverse velocitv transverse velocity a t wall = axial coordinate = transverse coordinate
= =
GREEKLETTERS = dimensionless coordinate defined by Equation 6 /3 r = gamma function u = dimensionless coordinate defined by Equation 6 r W = wall shear stress ?yo = osmotic pressure of inlet brine solution p = absolute viscosity Y = kinematic viscosity VOL. 5
NO. 3
AUGUST I 9 6 6
369
=
pt
solution density
= salt concentration = defined by Equation 7
i2
literature Cited
(1) Collins, M., Schowalter, LV. R., Phys. Fluids 5 , 1122 (1962). ( 2 ) Dresner, L., “Boundary Layer Build-Up in Demineralization of Salt Water by Reverse Osmosis,” Oak Ridge Natl. Lab., ORNL-3621 (May 1964). ( 3 ) Gill, W. N., Tien, Chi, Zeh, D. W., IND.ENG.CHEM.FUNDAMENTALS 4,433 (1965).
( 4 ) Gill, W. N., Tien, Chi, Zeh, D. W., Office of Saline Water, U. S. Dept. Interior, “Analysis of Continuous Reverse Osmosis Systems for Desalination,” Quart. Rept., Jan. 1, 1965-March 31, 1965, Contract 14-01-0001-401; Znt. J . Heat M a s s Transfer, in press. (5) Merten, U., Lonsdale, H. K., Riley, R. L., IND.ENG.CHEM. FUNDAMENTALS 3,210 (1964).
RECEIVED for review August 16, 1965 ACCEPTEDMarch 31, 1966 Work supported by the Office of Saline LVater, U. S.Department of the Interior, under Grant 14-01-0001-664.
CORRELATION OF HEAT AND MASS TRANSFER DATA FOR HIGH SCHMIDT AND REYNOLDS NUMBERS DAVIS W.
HUBBARD’AND E. N. LIGHTFOOT
Department of Chemical Engineering, University of Wisconsin, Madison, Wis. Turbulent mass transfer in rectangular ducts was studied via diffusion-controlled reduction of potassium ferricyanide in excess caustic. Observations were made a t Schmidt numbers from 1700 to 30,000 and Reynolds numbers from 7000 to 60,000 b y variation of temperature, caustic concentration, and flow rate. Results agreed more closely with the Chilton-Colburn analogy than with available semi-empirical expressions, and tend to cast doubt on the latter. Values of b In Nu/b In Sc varied from an average of 0.367 a t Re = 60,000 to 0.300 a t Re = 7000,and values of b In Nu/d In f from about 0.8 a t Sc = 1700 to about 0.3 at Sc = 30,000.
THE goal of this work was to provide a critical test of “semiempirical” models currently used to describe heat, mass, and momentum transfer. Such models are useful, if valid, because they permit estimation of temperature and concentration profiles, and therefore of heat and mass transfer rates, from friction factor measurements (24). They have been used in addition for extrapolating available heat and mass transfer data to new operating conditions, to different boundary conditions (27, 29), and even to different geometries (7, 25). They have also proved useful for predicting the effects of homogeneous chemical reactions on mass-transfer coefficients (25,
37). Unfortunately, there is reason to doubt the validity of a t least some of these expressions, because they conflict with each other and they are not fully supported by available data. For example, most are in disagreement with the well-known Chilton-Colburn relation :
for large A. The asymptotic forms of some of the more commonly used expressions are listed in Table I. I t is immediately clear from this table that both the predicted f and A dependence of the Nusselt number vary to a n appreciable extent. Another reason for concentrating attention on these asymptotic expressions is that they are simply related to the basic assumptions made in developing the corresponding turbulence model. Thus, a value of ‘/2 for exponent b arises from use of the von KArmAn hypothesis for velocity profiles and of the Prandtl analogy between eddy diffusivities for momentum, heat, and mass, in the region near the confining wall. The exponent c follows from assuming eddy diffusivities
Table 1.
Asymptotic Forms of Some Commonly Used Heat and Mass Transfer Correlations
Author
Like this empirical correlation, however, the heat and mass transfer correlations derived from the semi-empirical models have the form
These functions are normally complicated, but they always yield a simple asymptotic solution of the form {Limit, A +
a
]
(3)
Present address, Department of Chemistry and Chemical Engineering, Michigan Technological University, Houghton, Mich.
(4) Nu = R e f 2
Deissler
(5) NU = 0.248 -Re
Levich Lin, Moulton, and Putnam
I&EC FUNDAMENTALS
( 73) Nu
N
Nu
rv
AI13
flA1I4
ReflA’j‘ Rev‘TA1/3 9
(75)
Nu = (14.5)(2 d 3 ) a’ Re f l A 1 j 3
Vieth, Porter, Nu and Sherwoodb (37)
=
3d5- (1.77)3Re f2 A1j3 * X 2*
(%)
1
370
A’usselt Number
Chilton and Colburna
(2)
St = St(j,A)
St = @Ac
Ref.
a
Normally used for all A f
1/2.
Claimed useful f o r all A
5
’/2.
