Feinman, J., private communication, U. S. Steel Research Laboratorv, Monroeville, Pa., 1973. Houghton, G., Proc. Roy. SOC.,Ser. A 272, 33 (1963). Houghton, G., Can. J . Chem. Eng. 44, 90 (1966). Jameson, G. J., Chem. Eng. Sci. 21, 35 (1966). Jameson, G. J., Davidson, J. F., Chem. Eng. Sci. 21, 29 (1966). Newton, I., “Principia,” Book 11, Sect. VII, Prop. XL, Excpts. 1-14, 1719. Poisson, 5.D., Mem. Acad. Sci. Paris 11, 521, 566 (1832).
Stokes, Sir G. G., Trans. Cambridge Phil. SOC.9, 8 (1850). Taylor, G. B., Sproule, D. O., Trans. Roy. SOC.Can. 23, 91
(1929). Tchen, C . M., Ph.D. Thesis, Technical University of Delft, 1947. Torobin, L. B., Gauvin, W. H., A.Z.Ch.E. J. 7, 615 (1961). Tunstall, E. B., Houghton, G., C h a . Ens. Sci.23, 1067 (1968). RECEIVED for review March 31, 1972 ACCEPTEDJune 8, 197’3
Forced Convection Mass Transfer in Hyperfiltration at High Fluxes David G. Thomas Oak Ridge National Laboratory, Oak Ridge, Tenn. 37830
Markedly reduced concentration polarization appears to occur in hyperfiltration a t Schmidt numbers of - 6 0 0 in fully developed turbulent flow when the transpiration velocity (product water nux) exceeds -0.3 cm/min. Analysis in terms of a simple eddy diffusion model indicates that for Schmidt numbers greater than 100, the thickness of the concentration boundary layer i s appreciably reduced when the ratio of transpiration velocity to friction velocity, v/u*, is greater than 36,95/N~,N8,f.
A
concentration gradient from the membrane into the bulk of the feed solution (commonly referred to as concentration polarization) develops in hyperfiltration (reverse osmosis) because the membrane is more permeable to solvent than to solute; in practice a completely solute impermeable membrane is not encountered. When the membrane is impermeable to both solute and solvent no concentration gradient occurs. Previous studies of convective transfer processes in the presence of transpiration have been largely devoted to heat transfer in the gas phase. For instance, for a turbulent boundary layer with uniform free stream velocity, uniform blowing (or suction) along a plate of uniform temperature, hfoffatt and Kays (1968) recommended the empirical heat-transfer correlation originally suggested by Mickley, et al. (1954) , and Spalding (1960) St/Sto
=
In (1
+ B)
B
E)
dc
dY
+ vc = (1 - R)vc,
396 Ind. Eng. Chem. Fundam., Vol. 1 2 , No. 4, 1973
and the Blasius friction factor for tubes
0.079
f=NRel/(
(4)
Brian (1966) evaluated eq 2 and obtained
(1)
where Sto is the Stanton number for zero transpiration, B = m”/G St, m” is the mass flux through the plate (positive into the main stream), and G is the free stream mass velocity. Clearly eq 1 cannot be applied to hyperfiltration since, for hyperfiltration, when the transpiration velocity is zero the Stanton number is zero. Contributions to the theory of concentration polarization in hyperfiltration have been made by Merten (1963), Sherwood, et al. (1965), Brian (1965), and Johnson, et al. (1966). In their analyses one or more of the following assumptions were made: steady state turbulent flow, high Schmidt number (and therefore negligible longitudinal mass transport) , dilute solutions, and a n imperfectly rejecting membrane. With all of these assumptons Brian (1966) gave as the expression for the salt flux, N
N = -(D+
where is the molecular diffusivity, e is the eddy viscosity, c is the concentration, v is the transpiration velocity, R is the intrinsic membrane rejection, and c, is the concentration at the membrane-solution interface. Using the relation for eddy viscosity proposed by Vieth, et al. (1963)
(2)
where Robs = 1 - ( c W / c t ) ,R = 1 - (cW/c,), c, is the product concentration, c t is the bulk solution concentration, u is the axial velocity, N R ~ is the Reynolds number, and N s c is the Schmidt number. If, instead of eq 3, Deissler’s (1955) relation for the eddy diffusivity
1=
nZu+y+[l
- exp(-nzu+y+)l
(6)
V
is truncated by expanding the exponential and retaining the lowest order term in y+, then from eq 2 and 4 and using the recommended value of n = 0.124
Gill, et al. (197’1), have shown that for Reynolds numbers greater than 10,000 and a Schmidt number of 560, truncation of the eddy diffusivity model equation (as done in developing eq 7) resulted in less than 5% error in mass-transfer coefficient
compared to the value calculated using the full eddy diffusivity model equation. .lfter a careful analysis of high Schmidt number mass transfer data, Hubbard and Lightfoot (1966) recommended as the most suitable expression for correlating turbulent flow mass transfer in pipes
15
10
:1‘
05
L
0 2
-4fter rearrangement to this form, eq 5 becomes 01
0
I
I
I
I
I
2
4
6
8
1
I 0
l
Z
l
4
The mass-transfer Stanton number is defined as S t = k,/u
(9)
and from eq 8, 8a, and 9, the mass-transfer coefficient, k,, for hyperfiltration a t low fluxes with a n imperfectly rejecting membrane is
The value of the concentration polarization, $, in terms of the observed and intrinsic rejection is
The concentration polarization equatioiis have been tested experimentally b y Sherwood, et al. (1967), using a rotating cylinder apparatus and salt solutions with Schmidt numbers of 608 and 750 a t fluxes from 0.025 to 0.038 cm/min, and by Shor, et al. (1968), lvith a tubular geometry, Schmidt numbers of 600 and 815, and fluxes of 0.2 and 0.3 cm/min. The results were consistent with solutions of eq 2 for the appropriate geometry. The present paper describes results of experiments a t fluxes substantially larger than those attained in previous studies, fluxes suificiently large that concentration polarization equations previously derived apparently no longer apply. In addition, low flux results are presented for a wider range of velocities and Schmidt numbers than previously available.
.
Figure 1 Extrapolation of experimental results to obtain intrinsic rejection using limiting expressions for high and low fluxes (eq 39 and 5)
membranes, particularly the hydrous zirconium oxide-poly(acrylic acid), [Zr(IV)-PXA] composite membrane developed by Johnson and coworkers (1972). The cellulose acetate membranes were mounted either in a 0.63 x 6.3 x 63 cm parallel plate channel or on the inside of a 1.43-cm diameter X 425-cm long tube. Dynamic membranes were formed on the inside of a 1.37-cm diameter X 51-cm long tube. Sampling along the length of a tube showed that product composition was within 10% of the asymptotic value for tube lengths greater than 15 diameters from the entrance. Additional details of the equipment may be found elsewhere (Sheppard and Thomas, 1970,1971). Data Acquisition and Analysis
Experimental Section
The hyperfiltration measurements were made in a highpressure, stainless steel loop whose distinguishing features were separate pumps for pressuring feed and for circulating the pressurized fluid through the test sections; low flow rates were measured with a high-pressure rotameter, and higher flow rat’eswere measured with one of three calibrated Venturis mounted in parallel. The salt-rejecting membranes were either neutral fixed cellulose acetate type (Lonsdale, 1966) or ion-exchange type dynamic membranes (Kraus, et al., 1967). In the latter case, intrinsic rejection is a function of salt conceiitratioii a t the membraiie-solution interface and a suitable allowance has to be made in reducing date to account for the decrease in intrinsic rejection as concentration polarization increases when axial velocity is decreased (Shor, et al., 1968). Although there was some overlap, most of the “low” flus results were obtaiiied with cellulose acetate membranes while the “high” flus results were obtained with dynamic
Evaluation of the rate of mass transfer in hyperfiltration requires determination of the ratio [(l - Robs)/Robs]/ [ ( l - R)/R] or its exact equivalent (e, - c,)/(et - e,). The product composition, ,c, is readily determined. Since the test sections mere so short and production rate was so small relative to the total amount of solution circulating, the concentration of the turbulent core, e t , can be taken as the concentration of the circulating stream with no loss of accuracy. Determination of the concentration a t the membrane solution interface, e,, is difficult because the detector must have the same water and salt permeability as the surrounding membrane. Other possibilities include determination of the concentration profile in the vicinity of the membrane, then extrapolation of the data to the wall to obtain c,, or calculation of the value of e, from the intrinsic rejection and the product concentration. Since the concentration boundary layer is -0.001 cm thick for the Schmidt numbers of interest, development of probes small enough to measure the concentration profile was not considered to be feasible and s~ i t Ind. Eng. Chern. Fundam., Vol. 1 2 , No. 4, 1973
397
Table I. Evaluation of Intrinsic Rejection by least Squares Using Both the High Flux (Eq 39) and l o w Flux (Eq 5) Relations
10-2
( 1 - R)/R (Eq 391
Eq 5
Flux, crn/sec
Nso ( 1 - R)/R
0.089 580 0.1434 0.303 690 0.1183 0.590 690 0.3886 1 . 2 0 760 0.4842 a Standard deviation.
I
10
I
I
~
Eq
(1
uo
0.0067 0.0041 0.0131 0.0079
- R)/R
39
(1
ua
0.1464 0.1236 0.4115 0.4989
0.0045 0.0035 0.0100 0.0070
- R)/R (Eq 5)
1.021 1.045 1059 1.031 =
I lli;
I
I
I I I I Ill
I
1
10-2
I I I IIlJ ~
05
3
5
10)
2
5
to4
2
5
to5
2
5
106
R E Y N O L D S NUMBER 02
1-E
Figure 3. Hyperfiltration j factors at low flux (solid line i s “best fit” line; dashed lines define the limits of the present r e dts)
01
0 05
1
2 31
13.~
2
]Ill/ 5
1s-2
! 2
1
1 1 Ill,/ 5
10-1
I
2
I
I I I I I 5
1
NoCI C3trCEIITRATlON l e q ~ 1 ~ 0 l e n l s / l 1 l e r 1
Figure 2. Illustration of concentration dependence of rejection of poly(acry1ic) acid membranes: o,sheet membrane (Sachs and Lonsdale, 1970); 0, dynamic membrane (Sachs, et a/., 1969); dynamic membrane, present study, A = 0.85 cm/min, V = 0.42 cm/min
was necessary to determine the intrinsic rejection of the membranes. Two variations of the same procedure have been used to determine the intrinsic rejection: Sherwood, et al. (1967), took data a t two different velocities and solved the equivalent of eq 5 for both the numerical constant and the intrinsic rejection, while Shor, et al. (1968), used a ;lope-intercept procedure (Le., extrapolation to infinite axial velocity) to obtain the intrinsic rejection. The latter procedure was chosen for this study. Some representative results are shown in Figure 1 for fluxes from 0.089 to 1.2 cm/min (31.5 to 425 gal/ft2 day). The lower portion of the figure is a plot of log [(l - Roba/ Robs)]us. v/us 4, coordinates suggested by eq 5 while the upper portion is a plot of log [(I - Robs)/Robs]us. v1/2/u13/10 coordinates suggested by the limiting form of the equation developed in a subsequent section of this paper for high fluxes. Since the flus, Schmidt number, tube diameter, and temperature in any one test are constant or nearly constant, the principal independent variable is the velocity; the exponents on velocity in the two theories have so nearly the same value (0.75 us. 0.8125) that extrapolation using either eq 5 or 39 should give substantially the same results. T o illustrate this, a least-square analysis was made for the results shown in Figure 1 (see Table I). The ratio (1 - R ) / R determined using the two different equations was always within 6% and the two results were always within the 95% confidence limits of each other. Although probably not significant, it is interesting to note that the u values calculated using eq 39 are somewhat smaller than those using eq 5 . 398
Ind. Eng. Chern. Fundarn., Vol. 12, No. 4, 1973
Although Zr(1V)-PAA membranes are ion-exchange type, their decrease of intrinsic rejection with concentration is less than predicted. If the water and salt are completely coupled (as espected in ion-exchange type membranes) and if fluxes are sufficiently high to obtain asymptotic membrane rejection, then (1 - R ) should equal the distribution coefficient for salt between the membrane and the solution at the membrane-solution interface (Johnson, et al., 1966). Then ioneschange theory predicts that data for rejection at different univalent ion concentration should approach unit slope when plotted as log (1 - R ) us. log concentration (Johnson, et al., 1972). Concentration dependence of NaCl rejection for two Zr(IV)-PAX membranes is shown in Figure 2 for membranes with fluxes of 0.42 and 0.85 cm/min. Also included in Figure 2 are data for a dynamically formed PA4 membrane (Sachs, et al., 1969) and for a PAA precast sheet membrane [Sachs and Lonsdale, 1970) ; the flux from each membrane was -0.05 cm/min. The slopes of all curves shown on Figure 2 are -0.5, considerably less than the value of unity expected from theory; the reason is not well understood (Johnson, et al., 1972). Of more importance for the present purposes is the small value of the concentration dependence of rejection. This means that there is a relatively small variation in intrinsic rejection which must be accounted for (Shor, et al., 1968; Thomas, 1972b) as concentration polarization changes as velocity is changed. low Flux Results
Flux and rejection of conventional cellulose acetate membranes were measured using either the flat plate or the tubular geometries. Feeds were usually -0.05 -11‘ solutions of NaCl or MgC12 a t temperatures of 10 to 3OoC to obtain Schmidt numbers from 600 to 1850; in one test HC1 was used as feed to obtain a Schmidt number of 280. Axial velocities in these tests were varied from 3 to 1000 cm/sec to permit accurate evaluation of the intrinsic rejection. The membranes used had fluxes from 0.027 to 0.089 cm/min, considered “low” for the purposes of this paper even though current commercial units are being operated with fluxes in the range 0.03 to 0.06 cm/ min. Results are presented as j factor us. Reynolds number
10-2
T
b L I N el 01 195'; a W O R R ' S AhD V+WTMAN 113281 D B E R N I R D O A N D E l A N 119451 7 E A G L E AND FERGLSON 119301 WEBB, ECKERT A N 9 GCLDSTEIN 119711
>
IC
2
5
3o2
2
5
103
'.
, 2
S t d l DT OF PRA\D-.
5
d
2
5
1c5
REYNOLDS NUMBER, 2
3
N-VBER
Figure 4. Comparison of low flux hyperfiltration results with previously available data
in the upper portion of Figure 3; the solid curve is the estimated "best fit" and the dashed lines are drawn to include substantially all of the experimental results. Data from other investigations are shown in the lower portion of Figure 3 together with the dashed lines for the limits of the present results. The j factor curves bear a striking resemblance to ones for heat transfer (McAdams, 1953) ; for Reynolds numbers less than 2000-3000 the laminar flow curve has a slope of - 2 / 3 and for Reynolds numbers greater than 10,000 the results are not inconsistent with a ~ y R e - ' ' ~ dependence. Departure from laminar flow seems to be relatively sharply defined and there is a smooth transition from the end of laminar flow to turbulent flow. Although the maximum deviation of results in a single run was -*lo%, in turbulent flow the total deviation of the present results was f 3 i and -33% with somewhat greater deviation in the results of other investigations. The extent of the deviation is not totally unexpected since 1lcAdams (1953) notes that the best of the expressions for turbulent forced convection heat transfer correlate available results to +40%. The mean line through the present results suggests a coefficient of 0.03 in eq 8, rather than the commonly accepted value of 0.04. I n order to obtain results a t as high and as low Schmidt numbers as possible with the available equipment, 11gC12 solution was run a t 10°C and HC1 was run a t 27"C, resulting in Schmidt numbers of 1850 and 280, respectively. Results are shown in Figure 4 as Stanton number 2's. Schmidt number for Reynolds number of l o 4 ; also shown are data for both heat and mass transfer from the literature as well as curves for Deissler's (1955) and Colburn's (1933) correlations. The present results are not inconsistent n-ith results of prior studies; in the range of Schmidt numbers of interest (3002000) there is little difference in the Colburn and Deissler correlations. The hyperfiltration J factors and Stanton numbers shown in Figures 3 and 4 are on the lower edge of the band of previous results for heat and mass transfer. -4lthough the reason for this is not known it may be due to surface roughness embossed in the membrane by its support structure. Since the concentration boundary layer a t these Schmidt numbcm is thinner than the hydrodynamic boundary layer, the membrane could be hydrodynamically "smooth" and still have sufficient irregularities in the surface for the salt concentration to build up in relatively stagnant regions as i t is rejected by the membrane. With partially rejecting membranes this means that the product from the stagnant regionb would be more concentrated than from the well stirred regions; with
Figure 5. Illustration of magnitude of correction due to concentration dependence of rejection of PAA membranes. (Ns,= 760, v = 1.20 cm/min, R , = 0.68): J~ intrinsic rejection independent of concentration; 0,intrinsic rejection a function of concentration; - - -,low flux results
the present method of data reduction the net result ~ o u l d be a n apparent increase in concentration polarization or a n apparent reduced mass-transfer coefficient. Results supporting this possibility have been obtained by placing qmall wires in contact with the membrane surface to generate stagnant regions (Thomas, 1972a). High Flux Results
.
M ass Transfer K i t ,h the d e ve 1o p ni e n t o f d y I 1a m i c a 11y formed hydrous zirconium oxide-poly(acry1ic acid) [%rlIV)PAA] composite membrane by Johnson, et al. (1972), it became possible to obtain hyperfiltration data a t fluxes a n order of magnitude greater than attainable with cellulose acetate membranes. It soon became evident that concentration polarization a t high flux (-1 cm/miii) was much less than predicted by eq 5, Le., the slopes of the curves show1 in the lower portion of Figure 1 should all be nearly the same (within +loyo with the variation being largely due to temperature differences). For example, the results for a flus of 1.2 cm/min (Figure 1) were recalculated in j-factor form making appropriate corrections for the ion-exchange characteristics of the membrane. The results are shown as open circles in Figure 5 together with the same results (open squares) rvhich were not corrected for the ion-exchange characteristics of the membrane. Also shown is a dashed line indicatiiig the locus of the low flux results, i.e., fluxes from 0.027 to 0.089 ern !mill. The true j factors, shown as round symbols, are -307, greater than the uncorrected results, a relatively small correction compared to the difference between the high flus and low flux results. A411of the present high flux results for fully developed turbulent flow are shown in Figure 6 as log { [(l - Robs)/Robil/ [(I - R ) / R ]] us. log [ 2 j u ; ~ ~ ~ ~ ~ for ' . ~comparison ~ ~ ~ / ~ j ~ ] ; some low flux results are also included. For fluxes up to -0.3 cm/min the results are in sat'isfactory agreement lvith the semitheoretical curve, eq 5 . For fluxes greater than -0.3 cm/min the results are progressively farther beloiv the semitheoretical curve as the flux increases. K h e n the flus is constant, the difference between the seniitheoretical curve and the experimental results increases as the value of A = 25. u.\iRe1"4z\i~c2'S/~ increases. For instance, a t the highest flus, when the value of A is 0.8, the experimental value of [(I Robs)/Robs]/[(l - R ) / R ]is 0.57 of the value expected from eq 5 while when A = 2.2 the experimental result for [(l Ind. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
399
THEORY iSHERWOOD e! 01 19651
\
061
01
Figure
02
03
04
0 5 06
A = 25
5 NRJt4
08
IO
/ 1
20
30
NscZt3
6. Comparison of high flux results with the theory
of Sherwood, et a/.(1 965)
to the wall begins to exert a delaying effect on transition at (NRe)w=: 0.2, and a t ( N R ~=) ~2 the transition Reynolds number on mean axial velocity is an order of magnitude greater than the asymptotic value a t low fluxes. Although the theoretical calculations of Hains (1971) are not strictly relevant to the experiments, they are shown in Figure 7 as solid squares. T o make them comparable to the experimental results, they are normalized to the zero flux critical Reynolds number of 2800 obtained in this study. Somewhat surprisingly the normalized theoretical calculations are in rather good agreement with the experimental results. Interestingly, the ratio of v/u a t which there is a n effect on transition Reynolds number is close to the value of v/u, above which laminar boundary layers are stabilized. Ailsothe masimum wall Reynolds number obtained in this study (-2) is about that atwhich Berman (1958) predicted a separation velocity profile would be observed. High Flux Correlation
v (cm/min) ( 0 60
0 01
4
cm d i m tube)
01
I
I
l l ' 1 l I I
1
I
I
4
I I l!lll
Robs)/Robs]/[(l - R ) / R ]is only 0.2 of the semitheoretical value. Laminar-Turbulent Transition. The stabilizing effect of suction on laminar boundary layers has been extensively studied (Schlichting, 1968). Theoretical calculations indicate that when the value of the volume coefficient, u/ua, is greater than 1.18 X the boundary layer on a flat plate with zero pressure gradient is maintained in laminar flow. Although not of primary interest in this study, some indirect data were obtained which might be interpreted in terms of the effect of velocity normal to the wall on the transition Reynolds number. I n mass-transfer studies transition is readily identifiable when results are plotted in j-factor form as in Figure 3. Laminar flow results fall along a line with a slope of - 2 / ~ ; at the end of the laminar flow regime there is a rather abrupt increase in the j factor accompanying the transition from laminar to turbulent flow. Accurate definition of the transition requires rather closely spaced data; since study of transition was incidental in the present case, spacing of the data was often not optimum for a n accurate evaluation of the transition Reynolds number. Reynolds numbers a t transition as estimated from j factorReynolds number plots are shown in Figure 7 as a function of wall Reynolds number, ( N R J ~= vD/u, where v is the velocity normal to the wall, Le., the product water flux. The results were obtained with tubes of three different diameters, 0.6, 1.0, and 1.4 cm; flux in cm/min is identified along the top of the figure for an 0.6-cm diameter tube. Velocity normal 400 Ind. Eng. Chem. Fundom., Vel. 12, No. 4, 1973
Background. There have been extensive analytical and experimental studies of momentum and heat transfer through turbulent boundary layers on a flat plate with suction (Cebeci and Mosinskis, 1971; Julien, et al., 1971; Moffat and Kays, 1968; Simpson, et al., 1969, 1970). Much less work has been done on flow in tubes n i t h suction. Weissberg (1955) measured turbulence intensities near the tube wall in the presence of suction and observed a reduction in turbulence levels when the ratio of suction velocity to friction velocity, u/u*, exceeded -0.05. Kinney and Sparrow (1970) developed a n analytical model for tube flow with suction. Incorporated in their analysis was a mixing length model with a modified van Driest (1956) damping factor in the wall region. Merkine, et al. (1971), point out that the functional form of the damping factor developed by Kinney and Sparrow (1970) yields a turbulence level which increases with suction, in contrast to the results of Weissberg's measurements. The van Driest modification of the mixing length, I , is 1
=
~ p [ l- exp(-y+/A+)]
(12)
where K = 0.4, y + = y u * / u , and A+ = 26 give good correlation of experimental data in the absence of suction. The term in square brackets on the right is referred to as the damping factor and was introduced to account for the decreased turbulence in the immediate vicinity of a solid surface. To account for the effect of suction a t the wall, Nerkine, et al. (1971), suggest the following formulation, based on earlier work by Kays, etal. (1970)
A + = 4.42/(0.17 - v + )
(13)
This definition of the damping factor predicts a decrease of turbulence level with suction as well as giving good agreement with the limited amount of experimental data available for flow of air in tubes with wall suction. High Schmidt Number Hyperfiltration Results. T h e results shown in Figures 3 and 4 confirm the general validity of the approach and assumptions involved in the derivation of eq 5 and 7 for hyperfiltration a t low fluxes. However, the results shown in Figure 6 indicate that a t fluxes of the order of 1 cm/min there is a marked departure of experimental measurements from the semitheoretical curve with the extent of the departure increasing as the flux increases. Even though the flus appears to be large enough to affect the rate of mass transfer, the transverse velocity is still quite small; Le., v/u = is the largest value of transverse to mean velocity achieved in this study with fully developed turbulent
flow. Since the transverse velocity is so small compared with the mean velocity, there seems little doubt that, for high Schmidt numbers, the assumptions leading to the salt transport equation (eq 2) are equally valid for the entire raiige of fluxes covered in this study. Nor can a n explanation for the flus effect be found in the proposed expressions for damping factor. I n the present study the maximum ratio of transverse to friction velocity, v/u* = v+, is far too small to affect the value of A + in the damping factor term, eq 12 and 13. h plausible explanation of the flus effect on the rate of mass transfer is that a steady transverse velocity becomes important when its magnitude is of the order of the magnitude of transverse velocity fluctuations in the region where the concentration profile is changing rapidly. This may be illustrated with Figure 8 which shows zero-flus concentration profiles calculated from Deissler’s (1955) model equations for Schmidt numbers of 1 t o 1000. The upper portion of the figure shows experimental measurements of the root-mean-square value of the transverse velocity fluctuations (Fidman, 1953, 1960; Laufer, 1954; Orlov, 1966) as a function of distance from the wall. The ij’/y* results for y + > 2 were extrapolated to smaller values of y + assuming the limiting relation: ij‘ a y 2 (Townsend, 1956). The maximum value of flus (expressed as v/u*) attained in this study was -loT3. From Figure 8, a value of D’/u* = occurs a t a y + of -0.3. For a Schmidt number of -700 and y + = 0.3, the calculated value of (c, - c)/(c, - c t ) in the absence of a transverse velocity is -0.2. I n other words, the mean and fluctuating transverse velocities are in fact approsimately of the same order of magnitude in the region where the concentration is changing rapidly. .Ipplying these ideas in reverse, when the Schmidt number is reduced to unity, the value (c, - c)/(c, - c e ) = 0.2 occurs a t a y + of 5 ; a t y + = 5 the value of ij‘/u* is -0.1. According to the plausibility argument advanced above, this means that a value of the transverse velocity (expressed as D/u*) of -0.1 is required for there to be a significant effect in low Prandtl or Schmidt number situations. Weissberg’s (1955) measurements of turbulence intensity near the wall of a porous tube indicate a 20% reduction in the value of zi’/u* a t v/u* 0.09 while the boundary layer heat-transfer measurements of Moffat and Kays (1968) show a 4-%fold increase in Stanton number when v/u* 0.09. Finally, a value of v/u* of -0.1 would markedly affect the value of the damping coefficient, eq 13. Thus, the amount of wall suction required to produce a n observable effect on the rate of heat or mass transfer appears to be related to the thickness of the concentration or temperature boundary layer and t h a t thickness of course is related t o the Prandtl or Schmidt number. Recognizing that the plausibility arguments advanced above are based on estimates of concentration profile and 0’ fluctuations when transverse velocity is zero, the order of magnitude agreements of the predictions a t high and low Schmidt numbers suggest t h a t i t might be of interest to develop a n eddy diff usivity model which incorporates the features outlined above. Eddy Diffusivity Model for High Schmidt Numbers with Transverse Velocity. A rather consistent picture of motion in the vicinity of the viscous sublayer has been developed as a result of studies using diverse techniques (e.g., Corino and Brodkey, 1969; Kim, et al., 1971; Morrison, et al., 1971; Sirkar and Hanratty, 1969, and references therein). Although motion in the wall region is, in one sense, random in nature, in another sense, events occur with a preferred range of fre-
-
-
!
5 2 10-1
5
I+- *
10-2
5 2 to-3 1
5 2 0
1
5 2 10-2 10-2 2
5
10-1 2
5
1
2
5
to
2
5
(02
Y+
Figure 8. Zero-flux concentration profiles and v ’ fluctuations in vicinity of wall: 0, Laufer ( 1 954);A, Orlov ( 1 966); 0, Fidman ( 1 953, 1960)
quency and scale in a recognizable sequence. -illthough these events occur randomly in the wall layer, the first phase of the sequence appears to be the formation of a transverse spatial periodicity in the velocity field; this phase is followed b y breakup stage characterized by a n eruption on “burst” of fluid away from the wall which possesses a n identifiable frequency. Based on these studies, Kim, et (11. (1971), suggest that the flow structure in the wall region might better be described as a two-part model than as one kind of eddy structure. Despite the complicated flow patterns t h a t have been identified, simple models of the eddy diffusivity have been notably successful in correlating forced convection heat and mass transfer results a t high Prandtl or Schmidt numbers; a combination of the two approaches will be followed in the subsequent sections. The limiting form of the eddy diffusivity as y + 0 may be obtained from a Taylor series espansion of the axial and radial turbulence velocity components (Notter and Sleicher, 1969, and references therein) leading to lim
y-+o
f =
+
~ ~ ( y + ) 3~ ~ ( y + ) 4.
y
(14)
where the magnitudes A I and A 2 were not specified but in general could be a function of Schmidt or Prandtl number. For large Schmidt numbers the concentration boundary layer is sufficiently thin that data may be correlated with expressions derived from the assumpton that
E
=
A(y+)m
Y
(15)
An immediate consequence of eq 15is that
Because of the difficulty in making measurements close to the wall from which t / v can be evaluated directly, the usual procedure is to infer the value of m from measurements of Stanton number over a range of Schmidt numbers. The uncertainties associated with this approach are illustrated in Figure 4 which shows results from the present study together with representative data from the literature. The dashed line (ColInd. Eng. Chem. Fundam., Vol. 1 2 , No. 4, 1973
401
20
I
1
I I IIIII
I
I
Einstein and Li (1958) and Sternberg (1961,1965) have developed rudimentary models of the viscous sublayer assuming, respectively, that the sublayer is “active” or “passive” (Kistler, 1962).Both models predict that the extent of the viscous wall influence on a fluctuation of frequency, W , is of the order
I I I I I I
0
0
‘ O F
y
- d&
(21)
Because the present analysis is concerned with concentration fluctuations, i t seems appropriate to substitute the molecular diffusivity, 53,for Y. Then assuming that the concentration boundary layer scales with N s ~ - ‘ ” (consistent with the assumption of rn = 4 in eq 15), eq 21 can be rearranged to give
&)
u + = wv
burn, 1933) has a slope - 2 / / 3 while the solid line (Deissler, 1955) has a slope - a / 4 . I n the Schmidt number range of interest in the present study, there is little difference between the two lines; however, for N s c > lo4 the Colburn relation clearly gives a superior fit to the data. Some investigators (Hubbard, 1968;Sirkar and Hanratty, 1969) have proposed that the limiting value of the Stanton number for m = 3 can only be observed when N s c > l o 4 and that use of m = 4 is indicated for smaller values of N s c . For convenience in later developments, m = 4 will be assumed for the present study. The value of the numerical coefficient in eq 15 can be estimated by expanding the exponential in eq 6, assuming that the concentration boundary layer is thinner than the momentum boundary layer and retaining the lowest order terms in y +. TheresultisA = 2.37 X Hinze (1959)reviews simple physical arguments which suggest that the eddy diffusivity might be modeled as
provided the scale of motion responsible for turbulent diffusion is small compared to the scale of the sublayer, and that the scale of motion is small compared to the distance over which 0’ varies significantly. An estimate of the length scale ratios may be obtained from the data in the upper portion of Figure 8 V’
-
N
10-2(?/+)2
U*
and combination of eq 14, 17, and 18 with m 2.37 X to give
=
2
NSol/2 1
Equation 22 suggests that the characteristic frequency may be proportional to Nso-’/’. Landau and Lifshitz (1959)have proposed that because of the linearity of the equations in the viscous sublayer, the characteristic frequency should be scaled with the kinematic viscosity, Y , and the friction velocity, u*.Monin and Yaglom (1971)have questioned the rigor of these arguments but did not propose an alternative. Laufer and Narayanan (1971) and Kim, et al. (lgil),suggested that the characteristic frequency associated with the viscous sublayer of a turbulent boundary layer scaled with the outer flow parameters, u, and 6, L e . , w ~ / u , = constant. Although few measurements have been made of frequency spectra in the wall region of pipes, those that are available were made using a variety of sensors, Le., ones sensitive to fluctuations in temperature, velocity, and mass transfer. As a result, spectral data are available for a wide range of Prandtl and Schmidt numbers. Average values of the frequency, W
W
=
la
u E , ! v i d w / L m E,(w)dw
were determined for pipe flow data from the literature which covered a range of Schmidt numbers from 0.72 to 2400. From a plot of WY/U*~ us. N s c a t roughly constant Reynolds number i t was apparent that the dimensionless frequency was not inconsistent with the Nso-”’ relation suggested by eq 22. However, W + was clearly a function of the Reynolds number as it was in turbulent boundary layer studies. Therefore, the characteristic frequency was scaled with pipe diameter and mean velocity following Laufer and Narayanan (1971).The results are shown in Figure 9 as (wD/u) NsC112 US. NRe; although the data are limited it appears that
WD Nso‘/l
4 and A =
N
6
(23)
U
or
If Y / U * is taken as the characteristic length scale of the viscous sublayer, then d: appears to be sufficiently small when y + < 1 for eq 17 to be a useful model equation for eddy diffusivity. Hinze (1959) points out that the length scale, 2,in eq 17 and 19 may be interpreted as the scale over which a fluid element moves substantially in only one direction. If it is assumed that the convective action of the flux acts only during the relatively quiescent time between bursts, then a “suction scale” (= I,) can be formed from the flux, v, and W , the characteristic frequency for the wall region
I , = v/w 402
Ind. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
(20)
Combining eq 17,20,and 23a gives
I= Y
=
);(
(.e+ + L+)
(5)
(6:+
+ $)
as a model equation for eddy diffusivity when the Schmidt number is large and when there is a component of velocity normal to the wall.
Concentration Polarization a t High Fluxes. Partial integration of eq 2 with boundary conditions c = c, at y = 0 and c = c t a t y = gives (Sherwood, et al., 1967)
where c , = (1 - R ) ca = product salt concentration. Combining eq 18, 19, 23a, and 25 with the integral on the right side of eq 26 gives
04
03
32
where a = l/Nsc, b = 8.33 X 1 0 - 4 N ~ e N s c 1 ” and f~~ q = 2.37 X lop4. The functional form of the integral depends on the ratio b2/4ay. When (b2- 4ay) < 0, the value of the integral becomes 1
sin
CY -
(y+)2
In
01
,?
1
+ 2ry+ cos 4 2 + rz +
2
10
5
2
2
5 b2 43q
Figure 10. Flux function
CY
2 cos - tan-’ 2 where cos CY = -b/2 z/G and r and substitution in eq 26
=
Denoting the term in curly brackets as
( ~ / q ) . ”After ~ evaluation
~
+
fib 1 then since h
=
0 when b2 = 4ay
lim fz(v+)
denoting the term in curly brackets as
11
~
- d b + h - d b - h h
+o
=
l/di = 5.699 Nsc1’4
and ____
and defining h
=
d b 2 - 4ay
lim f1(v+)
=
and finally the value of v/u* when b2
h+0
and
V U*
From the definition of Robs and R , the term on the left of eq 28 is
and in the limit of b + 0, eq 28 is the same as eq 7 . Values of as a function of b2/2aq are shown in the upper portion of Figure 10. When (b2- 4ay) > 0 , integration of eq 27 gives
fl(v+)
where
=
4aq is
36.95 NReh’scf
Values of [b”?f2(v+)] - 1 as a function of b2/4apare shown in the lower portion of Figure 10. Results previously shown in Figure 6 are replotted in Figure 11 as [(I - Robs/Robs]/[(l - R)/R] us. A , where A , is either the argument of eq 28 or 33 depending on the value of v, N s c and N R ~The . solid line is the theoretical curve. Whereas in Figure 6 there was a pronounced flux effect not accounted for by eq 5 or 7 , the present results cluster around the curve derived using the model eddy diffusivity equation (eq 24). The maximum deviation of the results shown in Figure 11 is substantially the same as for the l o x flus results shown in Figure 3. Discussion
b d =2
1 - - d b 2 - 4ay; g 2
=
b 2
1 - db2 - 4aq 2
+
With this value for the integral, eq 26 becomes
where h
=
Extensive measurements a t fluxes less than 0.1 cm/min confirm the general theory of salt buildup a t membrane surfaces in hyperfiltration developed for large Schmidt numbers assuming that the transverse velocity had no effect o n the flow field (Johnson, et al., 1966; Sherwood, et al., 1965). For fully developed turbulent flow Schmidt numbers of -600 and ratios of transpiration velocity to mean axial velocity less than -3 X lo+, the present results are consistent with
d b 2 - 4aq and use has been made of the relations -
u*
=
udj/2; f
0.079 = -~
NR~”‘
The magnitude of the deviation among different sets of measurements (as indicated by the range of values for the numeriInd. Eng. Chem. Fundam., Vol. 12, No.
4, 1973 403
-
be u/u* 0.1, not too different from the value of 0.08 at which Weissberg (1955) observed a significant effect. A concentration polarization equation was developed using the eddy diffusivity model. When v+ > 36.95/N~,Ns,f the equation can be put in a form similar to eq 38
6
5
600 1720 690 690 690 630
\
033 0019 059 017 030 0 11
THEORY
1
01
02
03
04
0 5 06
0.8
I O
a” Figure 1 1 . Comparison of high flux results with present theory
cal coefficient) is about the same as observed with the best correlations for turbulent forced convection heat transfer. The mean value for the coefficient is somewhat greater than the value 25 expected from the heat, mass, and momentum transfer analogies. It is believed that this difference is primarily due to increased concentration polarization occurring in minute depressions in the membrane surface. Perhaps the mosl important outcome of the experimental studies was the observation that for fully developed turbulent flow, Schmidt numbers of -600 and ratios of transpiration velocity to mean axial velocity greater than -3 X lo+, measurements began to deviate from eq 38 with the deviation increasing as the transpiration velocity (product water flux) increased. Aside from its scientific interest, this observation may have important practical implications since a t high fluxes the concentration polarization is apparently much less than predicted from prior theories. Since the maximum value of the ratio, v/u, was i t seems unlikely that the transverse velocity would have a significant effect on the mean velocity profile and with the large Schmidt numbers used in this study, the assumption of negligible longitudinal mass transport appears reasonable. The small value of v/u also appears to rule out use of modified van Driest damping factors which are currently popular among investigators studying the effect of transpiration on turbulent boundary layers. Consequently a tentative model eddy diffusivity equation is proposed based on the hypothesis that high fluxes cause a thinning of the concentration boundary layer. Further, that this thinning occurs when the transpiration velocity is the same order of magnitude or greater than the fluctuating velocity normal to the wall in the region where the concentration is changing rapidly. A key part of the tentative model is the experimental fact that the characteristic frequency in the wall region decreases as the square root of the Schmidt number. As a result the ratio of transpiration velocity to friction velocity, v/u*, a t which flux effects influence turbulent mass transfer is given by v/u* = 36.95 N R e Ns,f. T o simplify the development of the eddy diffusivity model, a form suitable for high Schmidt number conditions ( N s , > 200) was used. Consequently the flux effect criterion cannot be applied to conditions typical of those encountered using gases as fluids (Prandtl or Schmidt number of the order of unity). Severtheless, the criterion suggests that for gases the transpiration velocity required for a n observable effect would 404
Ind. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
where 1 < kl < 4 2 and kl is a function of b2/4aq, Figure 10. Since, in a given experiment, v, ( N R e ) r , and N s c are substantially constant, the velocity dependence of concentration polarization a t high fluxes (eq 39) is substantially the same as that given by eq 38. The principal effect of operation at high fluxes is to diminish the effect of the Schmidt number; i.e., a t high fluxes the velocity normal to the wall exerts a controlling influence on the concentration boundary layer thickness. Based on the limited results shown in Figure 11, the experimental value of the numerical coefficient in eq 39 is 311 f 25%, a somewhat smaller variation than that for the low flux results, eq 38. Although the agreement of the experimental results with eq 39 is encouraging, the concepts underlying the eddy diffusivity model must still be regarded as tentative. Additional studies with high flus membranes are required to test the general applicability of the concentration polarization equation. Perhaps of more importance, the correlation of the present esperimental results with a n eddy diffusion model incorporating a frequency dependent flux term suggests the importance of additional fundamental studies of wall region phenomena a t y + less than one to serve as a guide for development of theoretical models to more accurately describe mass transfer at high Schmidt numbers. Nomenclature
*4 A+ A” Ai An a
B b
= = = = = = = = =
coefficient, eq 15 damping factor argument of concentration polarization equations coefficient in eq 14 coefficient in ea 14 .Ysc-’ suction or blowing parameter, ri~”/G S t
8.33 x 1
0
-
4 U+ ~
~
~
~
~
concentration = concentration a t membrane-solution interface Ca = concentration of product water CW = concentration of turbulent core Ct De = equivalent diameter D = diameter a, = molecular diff usivity E, = spectrum function j l ( v + ) = [cos&2)],sin (Y .fz(~+) = ( d b h - d b - h ) / h = Fanning friction factor, ( D A p , 4 L ) / ( p u 2 / 2 ) = free stream mass velocity G H = distance between parallel planes h = coefficient, d b 2 - 4aq I = j factor, St Ssoz kc = mass-transfer coefficient k = cross-flow parameter, (v/u,) (Hu,/v) 1 = mixing length 1, = suction length scale, v / w 6: = length scale, eq 17 2+ = liondimensional length scale, ~ u , / u hff = mass flu.; through porous plate m = exponent, eq 15 s = salt flu.; S R ~= Reynolds number, D u / v ( & V R ~=) ~ wall . Reynolds number, D v / v SsC = Schmidt number, u / D A p l L = presbure gradient P = constant, 2.37 X C
s
+
~
~
~
iiitriiiric rejection. 1 - (c,/c,) observed rejection, 1 - ( c , / c t ) ii-n :rl) -1,
Stniitoii iiumber u-ith suction or blowing: Stanton number for impermeable surface iiieiui axial velocity r.ni.8. longitudinal velocity fluctuation frictioii velocity, udj> center line velocity noiidimeiisioiial velocity, u ( y ) / u * free stream velocity truiispiration velocity iiornial to wall r.1ii.s. radial-velocity fluctuatioii Iioiidiniensional traiispiratioii velocity, DIU* distance normal to wall iioiidinieiisioiial distance normal to wall, yu,/u
(IYbZ).
GRI.:I:I(LETTLRS CY
=
6
=
e
=
K
=
V
=
%.
=
w w
= +
=
Kistler, A. L., “MBcanique de la Turbulence, Editions du Centre National de la Recherche Scientifique,” Paris, pp 287-297, 1962. Kraus, K. A,, Shor, A. J., Johnson, J. S., Desalination 2. 243 (1967). Landau, L. D., Lifshitz, E. M., “Fluid Mechanics,” p 162, Pergamon Press, London, 1959. Laufer, J., NACA Report 1174, 1954. Laufer, J., Narayanan, M. A. B., Phys. Fluids 14, 182 (1971). Lin, C. S., Denton, E. B Gaskill, H. S., Putnam, G. L., Ind. Ena. Chem. 43. 2136 11951). Liu. H.. A.1.Ch.E. J . 13. 644’(1967) Lonsdale, H. K., “Desalination -bir Reverse Osmosis,” Ulrich Merten, E d , MIT Press, Cambridge, Mass., 1966. McAdams, W. H., “Heat Transmission,” 3rd ed, McGraw-Hill, New York, S . Y., 1953. Merkine, L., Solan, A., Winograd, Y., J . Heat Transfer 93, 242 (1971). Merten,’U., IND. ENG.CHEM.,FUNDAM. 2, 229 (1963). Meyerink, , ~ - - -E. ~S. C., Friedlander, S. K., Chem. Eng. Sci. 17, 121
cos-’(--bj2d/aq) boundary layer thickness eddy diffusivity coefficient, 0 . 4 kinematic viscosity colicelitration polarization, (c,/ct) - 1 characteristic frequency of viscous sublayer iiondimeiisioiial frequency, W V / U , 2
literature Cited
Armistead, 1:. -4., Keyes, J . J., J . Heat Transfer 90, 13 (1968). Berman, A. Y., Proc. U . S. Int. Conf. Peacejul Uses At. Energy, 2nd 4, 3-51 (1958). Bernardo, E., Eian, C. S., NA4CAReport WR E-136, 1945. Brian, P. L. T., First International Symposium on Water Ilesalination, Washington, l).C., Paper SWD/79, 1965. Brian, P. L. T., in “Desalination by lteverse Osmosis,” U. JIerten, Ed., Chapter 5, MIT Press, Cambridge, Mass., 1966. Cebeci, T., llosinskis, G . J., J . Heat Transfer 93, 271 (1971). Colburn, A. P., Trans. AIChE 29, 174 (1933). Corino, E. R., Brodkey, It. S., J . Fluid Jlcch. 37, 1 (1969). Ileissler, 11. G,, NACA lieport 1210, 1935. Eagle, A. E., Ferguson, 11. AI., Proc. Roy. Soc., Ser. A 127, 540 (1930). Einstein, II. A., Li, H., Truns. Amer. SOC.Civil Eng. 123, 293 (19,X). Fidman, B. A., ( 1 9 3 , 1960), cited by Lyatkher, V . M., Sou. Phys. Dokl. 13 ( 5 ) , 3S9 (1968).
Fortuna, G., Hanratty, T. J., Int. J . Heat Jlass Transfer 14, 1499 (1971’1. Friend, k. L.,lIetzner, A. B., A.I.Ch.E. J . 4, 393 (1958). Gill, W. N., Ilerzansky, L. J., Doshi, M. R., “Surface and Colloid Science,” 1V, E. Matejevic, Ed., Wiley, Kew York, N. Y., 1971. Hains. F. 11.. I’hus. Fluids 14. 1620 (1971 1. Hamer, E. Ad.G.: Office of Sahne-Water Report 424, 1969. Harriott, P., Hamilton, li. M.,Chem. Eng. Sei. 20, 1073 (1965). Hinhe, J. O., “Turbulence,” pp 20-22, McGraw-Hill, Kew York, s.Y . , 1959. Hubbard, I). R., A.I.Ch.E. J . 14, 354 (1968). Hubbard, D. W., Lightfoot, E. N., ISD.ENG.CHI:M.,FLXDAM.5 , 370 (1966). Johnson, J. S., Ilresner, L., Kraus, K. A., “Principles of Deqalination,” K. S. Spiegler, Ed., Chapter 8, Academic Press, Kew York, S. Y., 1966. Johnson, J. S., Minturn, R. E., Wadia, P., J . Electroanal. Chem. 37, 267 (1972). Julien, H. L., Kays, W. hl., Moffat, R. J., J . Heat Transfer 93, 373 (1971). Kays, W. Sl., Nofiat, .:1 J., Thielbahr, W. H., J . Heat Transfer 92, 49!) (1970). Kim, I
