GREEKLETTERS
Marshall. W. R.. Jr.. Chem. Enp. Propr. Monopr. Ser.. 50 (2) (1954). Mayer, E., J . Amer. Rocket SOC., 31, 1783 (1961). Narasimhan, M. V., Narayanaswamy, K., J . Indian Inst. Sci., 45 (41, 83 (1963). Nukiyama, S., Tanasawa, Y., Trans. SOC.Mech. Eng. Tokyo, 4, 86, 138 (1938). Nukiyama, S., Tanasawa, Y., ibid., 5, 63, 68 (1939). Nukivama., S.., Tanasawa. Y.. ibid.. 6. 11-7. 11-18 (1940). Pars,-D. C., “Introduction to Dynamics,” Cambridge University Press, Cambridge, England, 1953. Ramakrishnan, S., Kumar, R., Kuloor, N. R., Chem. Eng. Sci., 24, 731 (1969). Weiss, M. A., Worsham, C. H., J. Amer. Rocket SOC., 29, 252 (1959). Wetzel, R. H., Marshall, W. R., Jr., AIChE, National Meeting, Washington, D. C., March 1954. Wigg, L. D., J . Inst. Fuel, 37, 500 (1964). I
= contact angle, degrees 6 = substitution given in the text y = surface tension of the liquid, dynes/cm wf = viscosity of the liquid, g/cm-sec pa = air density, g/cm3 p1 = liquid density, g/cm3 0 = nozzle angle, degrees (Y
literature Cited
Castlemen, R. Jr., Bur. h’bnd. (u.8)J . Res., 6, 369 (1931). Clare, H., Radcliff, A., J . Inst. Fuel, 27, 510 (1954). Garner, F. H., Henny, V. E., Fuel, 32, 151 (1953). Golitzine, N., Sharp, C. R., Badham, L. G., National Aeronautical Establishment of Canada, Report 14-ME, 186 (1951). Gretzinger, J., Marshall, W. R., Jr., AIChE J . , 7, 312 (1961). Hrubecky, H. F., J . Appl. Phys., 29, 592 (1958).
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RECEIVED for review July 27, 1970 ACCEPTED March 3, 1971
Scale-Up of Agitated Vessels Mass Transfer from Suspended Solute Particles Donald N. Miller Engineering Department, E . I . du Pont de Nemours & Co., Inc., Wilmington, Del. 19898
Mass transfer from solute particles suspended in agitated systems can occur by forced convection, free convection, and radial diffusion. Prior theoretical developments for each of these mechanisms are reviewed and evaluated. Experimental measurements on the rates of solution of benzoic acid pellets freely suspended in water were made in 1-, lo-, and 100-gal. tanks. The vessels were geometrically similar, fully baffled, and equipped with four-blade flat-paddle agitators. Based on these measurements, supplemented with data from the literature, appropriate procedures are recommended for identifying the controlling mechanism and for scale-up of mass transfer performance.
Interphase mass transfer is often a rate limiting factor that must be reliably predicted in the design of agitated vessels. Presuming that geometric similarity is maintained, there remains the problem of predicting the agitator speed that will duplicate rate performance. This problem was the subject of considerable study in the past. There are many theoretical approaches recommended in the literature for correlating data and predicting mass transfer performance in agitated vessels. I n this review, the relative merits of these various approaches will be discussed for the important rate process-mass tranfer from suspended solute particles. Further, based on experimental information obtained both directly in support of this study and from the literature, improved correlation and scale-up procedures are recommended. General theoretical Background
There are three possible mechanisms by which mass transfer can occur from suspended solids-forced convection, free convection, and radial diffusion. The first and
last occur as the predominant mechanisms in different two-phase flow regimes. Free convection can exist as a parallel mechanism in either. At one extreme, the solid particles are relatively large and heavy and a difference in motion or “slip” velocity exists in the vertical direction between solid and fluid phases. This difference or “slip” velocity is the terminal settling velocity of the solid particles. Where the fluid moves in downflow, the solute particles will precede the fluid by the amount of their terminal settling velocity; where the fluid moves in upflow the particles will lag the fluid by this amount. The “slip” between phases produces a laminar boundary layer in the fluid at the phase interface; the effective thickness of which is dependent on the magnitude of “slip” and the particle size. The major resistance to mass transfer occurs through this laminar film. Rate of mass transfer depends on the molecular diffusivity of the solute and the viscosity of the fluid as well as laminar film thickness. This forced convection mechanism will be referred to in Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
365
subsequent discussion as mass transfer in the “slip” velocity or large particle flow regime. At the other extreme, the solid particles are relatively small and light, and the suspended phase tends to move with no slip along with the circulating fluid. I n this case, there is no boundary layer development and mass transfer occurs by radial diffusion. The fluid in such a system can be in either laminar or turbulent flow. An effective or turbulent diffusivity can be used in the latter situation to characterize mass transfer. The radial diffusion mechanism will be referred to in subsequent discussion as mass transfer in the small particle flow regime. The Frossling equation (Frossling, 1938) covers the suspended solids mass transfer range from large to small particle flow regimes and can be modified to account for free convection as a contributing mechanism: NSh,
=2
+ 1.10 NRe,l/2NS:$3
(1)
where
Ns~= , k = D, = D = NRe, =
us = u = Nsc =
kD,/ D, particle Sherwood number mass transfer rate constant, cm/sec particle diameter, cm molecular diffusivity, cm2/sec DPus/v,particle Reynolds number particle “slip” velocity, cmisec kinematic viscosity, cm2/sec u/D, Schmidt number
The two terms to the right of the equal sign represent the mass transfer contributions of radial diffusion and forced convection. The first is the expression for diffusion from a fixed sphere based on film theory (Langmuir, 1918). The second term NSh,
= 1.10 NRe6”Nd3
= 0.1983 ( P i V’)-1’4v3r4
Experimental
To evaluate scale-up procedures for the large particle flow regime, measurements were made of rates of solution from benzoic acid pellets freely suspended in water in 1-, lo-, and 100-gal tanks. Data to support evaluation of scale-up procedures for the small particle flow regime were obtained from the study by Harriott (1962). In both studies, mass transfer by free convection was a negligible effect. The three tanks used in the large particle study were scaled geometrically in the ratios 1:2:4.5. Design details are shown in Figure 1. Dimensions are listed in Table I. Each tank is fully baffled with four longitudinal baffles at 90” spacing. Agitators are four-blade, flat paddles. Bottom-dished heads are consistent with standard ASME design specifications. The experimental runs were all made with liquid levels a t the upper overflow nozzles, liquid depth-to-tank
(3)
is the relationship for liquid film controlled mass transfer derived from boundary layer theory (Miller, 1964). The coefficient 1.10 in Equation 3 was established in a study of mass transfer from fixed solid spheres in agitated vessels (Miller, 1967). I t is consistent with the constant found by Garner and Suckling for mass transfer from the upstream flow area on a fixed solid sphere of solute (Garner and Suckling, 1958). With turbulent flow in mechanically agitated tanks, there are frequent erratic changes in the directions of local fluid movement. There are also instabilities in the gross circulatory flow patterns, particularly in the transition zone between downflow core and upflow annular regions. As a result, there may be very little wake development associated with either suspended or fixed solute particles in these systems. The primary variable governing the extent to which radial diffusion controls mass transfer in a suspended solids system is particle size. It is interesting to compare the particle-size range within which the first term becomes dominant in Equation 1 with the turbulence microscale. The turbulence microscale, 7, is a measure of the size of eddies a t the small eddy equilibrium end of the turbulence energy spectrum. This region is thought to be most important for mass transfer. As defined by Shinnar and Church (1960), 7
This expression assumes that the energy input per unit volume by mechanical agitation, P / V, is an order of magnitude higher in the vicinity of the impeller than in the overall bulk liquid. I n nonviscous agitated systems, 11 values fall typically in the range between 10 to 150 microns. In reviewing mass transfer data obtained in the small particle flow regime by Harriott (1962), the author determined that radial diffusion becomes dominant as the mass transfer mechanism at particle sizes of 205 microns and under. Boundary layer development seems unimportant as a mass transfer consideration when the solute particles approach the turbulence microscale dimension.
(4)
where
P = agitator power input, hp V = volume, io3 gal 366 Ind. Eng. Chern. Process Des. Develop., Vol. 10, No. 3, 1971
TOP VIEW
G-lA-
-+f
0.06A rad
0.06A rad
SIDE VIEW
Figure 1. Vessel design
Table II. Vessel Dimensions (Harriott, 1962)
Table I. Vessel Dimensions (this Work) Nominal vessel size, gal
1
Vessel holdup, gal
10
100
0.2
0.4
4
4 5% 2 0.4 0.4
Dimensions, in. 6 834 12 4
J K
%
12 17% 24 8 1%
27 39% 54 18 3%
%2
%6
%4
% % % %6 ?4
%
D
E I
1'%6
1
2% 9/32
1
2%
diameter ratios being 1.5:l. Agitator power inputs ranged from 0.05 to 19 h p per lo3 gal. Agitator speeds were in the range 170 to 490 rpm in the 1-gal tank; 103 to 290 rpm in the 10-gal tank; and 25 to 168 rpm in the 100-gal tank. USP benzoic acid in % r-in. cylindrical pellets was used. Several weighed additions of pellets were made during each run. Water was introduced a t the bottom center and allowed to overflow through a screen a t the upper side nozzles. Water velocities in the feed nozzles were kept low relative to the impeller tip speeds so that there was no detectable effect of water input rate on mass transfer performance. Time intervals sufficient to allow several displacements of the tank contents were allowed between pellet additions and sampling. Each effluent sample was analyzed for benzoic acid by electrometric titration. Rate constants were calculated from the following relationship:
h = Q c / [ K ( c *- c ) ]
(5)
where
Q c*
2% I % ,2 0.3, 0.4 0.4
30
4 6%
8 8% 2, 3, 4 0.4, 0.6, 0.8 0.8
21 20 7 1.4 2
2 0.4
0.4
1'%6
34 96
A B
2
0.4 Dimensions, in.
= liquid flow rate, cm3/sec = solute saturation concentration, g-moles/cm3
c = solute bulk concentration, g-moles/cm3 = mean pellet surface area, cm2
K
Solubility, c*, as a function of temperature is available in the literature (Bourgoin, 1878). The initial total solute surface area, calculated from a pellet count and measured dimensions, was adjusted to compensate for changes owing to shrinkage during each run. The several values of rate constants measured in each tank a t fixed impeller speeds were averaged for correlation. Harriott's test work was done in three vessels-a 4-in. diam round-bottom resin flask in which liquid holdups from 0.2 to 0.4 gal were tested; an 8-in. diam, 2-gal flat-bottom tank; and a 21-in. diam, 30-gal flat-bottom tank. Known dimensions are given in Table 11. Each tank was fully baffled. The agitators were sixblade turbines. Power inputs ranged from 0.07 to 122 hp per lo3 gal. Agitator speeds were in the range 127 to 1500 rpm in the 4-in. flask; 85 to 1500 rpm in the 8-in. tank; and 200 rpm in the 21-in. tank. I n obtaining data applicable to the small particle flow regime, Harriott measured neutralization rates of ionexchange beads suspended in dilute caustic. Narrow-size
fractions of standard grades of Dowex-50-WX8, a moderately cross-linked, strongly acid resin, were used. Mean particle diameters for the small particle data ranged from 12.7 to 205 microns. Liquid film controlled mass transfer rates were followed by measuring the decrease in p H or conductivity with time. Rate constants were calculated from the following relationship :
12 = [V(ln ( l / c l ) - In (1/c2) l ] / A A t = [ V ( p H i - p H n ] ] / A A t (6) where
V = liquid holdup, cm3 el, c2 = initial and final solute concentrations, g-moles/cm3 A = external resin surface area, cm2 at = exposure time, sec pH1, pHn = initial and final pH's Mass transfer resistance within the resin particles had a negligible effect on overall rate. Forced Convection
The Gilliland-Sherwood Sherwood equation NSh,
Correlation.
The
Gilliland-
= C7Nbe,N&
(7)
where Ci, a, and b = constants NSh, = k d / D , impeller Sherwood number N h = d 2 n / v , impeller Reynolds number d = impeller diameter, cm n = impeller speed, rps was used in several previous studies to characterize suspended solids mass transfer. The constant b is theoretically % based on penetration theory and % based on boundary layer theory (Miller, 1964). I n most of the earlier studies, penetration theory was presumed to apply (Barker and Treybal, 1960; Bieber, 1962; Hixon and Baurn, 1941; Humphrey and Van Ness, 1960; Sykes and Gomezplata, 1967); in at least one, boundary layer theory was assumed (Marangozis and Johnson, 1962). Using Equation 7 with boundary layer theory, the data taken in this study correlate with
N s = ~ 8.48 N R ~ P ~ ~ ~ N s E ~ (8) ~~ These results are shown plotted in Figure 2 . The mean percent deviation of the experimental values of NShNSc-' ' from those calculated with Equation 8 is 13.80%. T h e square of the multiple correlation coefficient R i for the fit is 0.790. This number can be interpreted as the fractional improvement afforded by the correlation over the use of a mean value for all of the points. The coefficient Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971 367
would be 1.0 for perfect correlation, zero for no improvement over the mean. The Calderbank and Moo-Young Correlation I. Calderbank and Moo-Young have proposed the use of a Reynolds number in Equation 7 based on turbulence theory (Calderbank and Moo-Young, 1960). Assuming a homogeneous, isotropic field of turbulence, a Reynolds number can be derived by relating dynamic pressure owing to turbulence [Di 3(2.00gP/ V)'3p1 ] to viscous shear stress [I'y2.00
gPl V')' 1. N R ~=, 1.123 g' 'D; ' ( P /V)' 6 / v ' *pl
(9)
In work with sieve plate columns, these authors found no dependence of mass transfer on bubble size. Several correlations of the type previously discussed do not include particle size as a variable. Based on these observations, and NSh, = kD,/D for N S h in substituting NRe for N R ~ in Equation 7, Calderbank and Moo-Young chose the constant a such that D, was eliminated as a parameter. The constant b was taken to be %, consistent with boundary layer theory. The resulting equation is
h = Cm[g(P/V ) v / p ] ' 4NS;L
(10)
where
Clo = a constant Using Equation 10, the data of this study correlate with
h = 0.222 [g(P/V ' ) v / p ] ' 4NscL
(11)
Results are plotted in Figure 3. The mean percent deviation of experimental h's from those calculated from Equation 11 is 28.4%. The square of the adjusted multiple correlation coefficient is 0.277. Middleman's Correlation. Middleman has suggested a correlation based on Kolmogoroffs theory for D, >> TI (Middleman, 1965). Kolmogoroff postulates that if solute particles are free to move in a turbulent field and if d >> D, >> 7 , the dynamics of the particles depend only on D, and P f V . A density difference between phases would also affect particle dynamics, but this possible variable has been neglected. Presuming the dependency of mean fluid velocity close to a particle on D, and P l V ' , by dimensional analysis,
us = C,2Dj/3(P/ V')',' 3
(12)
where
C12 = a constant For impeller Reynolds numbers above 50,000
P/V
=
CI3d2n3
(13)
where C1, = a constant.
Substituting Equation 13 in Equation 12 and Equation 12 in the Reynolds number of the Frossling Equation 1, the following relationship can be obtained by rearrangement:
( N s ~-, 2)Nscl = C l d v ~ e 1 ' 2 ( D p / d ) 2 ' 3(14) where
Cla = a constant The data of this study correlate with CI4= 0.719. Results are plotted on Figure 4. The mean percent deviation of experimental values from those calculated via Equation 14 is 28.3%. The square of the multiple correlation coefficient is 0.350. The Levich Mass Transfer Coefficient. Levich derives an expression for mass transfer based on a three-layer film structure at the phase interface (Levich, 1962). Moving from the main turbulent stream toward the phase interface, he differentiates between a turbulent boundary layer, a viscous sublayer, and a diffusion sublayer. Mass transfer occurs by turbulent diffusion in the outer two layers and by molecular diffusion in the inner film. The effective or turbulent diffusivity, De, is defined as
De = Cisuil
(15)
where
Cli = a constant ul = velocity of turbulence eddies 1 = scale of turbulence eddies, cm Effective diffusivity De in the turbulent boundary layer is taken to be linearly dependent on distance from the phase interface
D,, = CiGulmy
(16)
where
0.01
103
. 103
0003
.:
-5
0
Y
0.001
Pt *
+
102 104
x 105
Vessel (Go1 1
1 10 100
0.0003 0.001 106
fp_v'i
0.01 vg
0. 1 2/3
(cm /sec )
NRe
Figure 3. k vs. [(P/V')vg/p]''4NSc-2
368 Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
CI6
= a constant
u: = a characteristic velocity of turbulence eddies in
the main stream flow, cm/sec
Correlotion suggested b y
y = distance from the phase interface, cm
Middl ernan 1965
Two possible dependencies are derived for the viscous sublayer,
De = C i 7 ~ h y ~
(17)
De = C ~ R U ~ Y ~
(18)
N P
r
zn
I
and
D,
NSh, = CH(CLI) ' 'NRe,NS;
(19)
where = a constant Co = drag coefficient
C19
h? = D/Dp{ 2
Figure
4. (Nshn -2)NSc-'
300
+ C ~ ~ ( C 'NRe,Ns; D)' '}
(20)
where
h? = the Levich mass transfer rate constant, cm/sec Using drag coefficients calculated for cylinders with a 1:1 length-to-diameter ratio (Wieselberger, 1922) and a value of 0.055 for C1g, reference values of the Levich mass transfer rate constant were calculated for all the experimental runs using Equation 20. The experimental h's ratioed to the corresponding h l values range from 0.229 to 1.204. These ratios increase within this range with increasing impeller Reynolds numbers, power inputs, and agitator speeds. The Calderbank and Moo-Young Correlation 11. The concept of relative motion between phases induced by a difference in density was advanced by Calderbank and Moo-Young as an alternate to their correlation discussed earlier (Calderbank and Moo-Young, 1961). The Grashof number INGr = Dig(Pp - P ) / V 2 P (21) where = particle density, g/cm3
is substituted in the Frossling Equation 1 for the particle Reynolds number. These authors make the assumption consistent with their earlier development that mass transfer in the large particles regime is independent of particle size. An exponent for the Grashof number is chosen such that D, is eliminated for Sherwood numbers much larger than two. With this treatment, their correlation reduces to
NSh, = 2
+ 0.31 N
R ~
(22)
vs. NRel '(Dp/dL)2
N R =~D&pp - p ) / Dvp, Raleigh number As Calderbank and Moo-Young point out, this expression can be derived in a t least two other ways. Allen (1900) has correlated terminal settling velocities of small solid spheres with ut =
The exponent % on the Schmidt number occurs with based on the use of Equation 17; this exponent is Equation 18. Inserting provision for the small particle end of the mass transfer spectrum and rearranging, the following expression can be obtained from Equation 19:
where
100
30
= effective diffusivity in the viscous sublayer,
cm2/sec Cli, CL8= constants Measurements of the damping of turbulence in the viscous sublayer have not been accurate enough as yet to permit selection of the more appropriate equation. Combining these relationships, Levich shows that
pp
10
3
where
?4 { D Y ( p p - P ) ' / V P
}'
(23)
where ut = terminal settling velocity, cm/sec
When this expression is substituted for "slip" velocity in the particle Reynolds number of the Frossling Equation 1, it reduces to Equation 22 with a constant of 0.28 rather than 0.31. There are a number of publications (Acrivos and Taylor, 1962; Bowman et al., 1961; Friedlander, 1961) in which the following equation is derived for mass transfer from spheres at Reynolds numbers less than 1.0 and Peclet numbers above 1000:
N s =~ 0.99 NpA '
(24)
where
Np, = D,ut/D, Peclet number Stokes' Law for terminal settling velocity applies in the low particle Reynolds number range:
(25) Ut = DXPP - P ) / l 8 V P Substituting Equation 25 into Equation 24, the Equation 22 relationship is obtained with a constant of 0.38 rather than 0.31. Rate constants h; were calculated using Equation 22 for all the experimental conditions explored in this study. The experimental k's were then ratioed to these values. The h/h? values range from 0.322 to 2.28. As with the h / h t ratios, there is a significant trend toward increased h / k: values with increasing impeller Reynolds numbers, power inputs, and agitator speeds. I n a recent paper, Brian and Hales (1969) suggested that the Middleman and Calderbank and Moo-Young I1 correlations be combined, NSh being equated to weighted functions of each. Their data (Brian et al., 1969), obtained for particle diameters in the range of l / l h to % in., show best coincidence with the Middleman correlation weighted predominantly. For larger, heavier particles, as with the % r-in. particles in this study, a shift toward greater dependence on the Calderbank and Moo-Young I1 correlation can be expected. Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
369
constants B = 2, C26 = 1.10, c = %, and d = % were used in the correlation of data in this study. Using Equations 26 and 1, rate constants kF were calculated for each experimental condition. The ratios h/kF range from 0.1958 to 1.375 and show the increase with increasing impeller Reynolds numbers, power inputs, and impeller speeds that was noted previously. A correlation of suspended solids mass transfer with power input per unit volume has been proposed by Harriott (1962), and is commonly used in industrial design. The k / k $ values are shown plotted vs. P / V in Figure 5. The least mean squares fit to the points is represented by
Harriott’s Correlation. The more general terminal settling velocity expression, for which Equation 25 is a special case, is Ut =
{ 4 DPg(P, - P ) / 3 PC$
(26)
Particle Reynolds numbers for the large particle mass transfer regime are typically much larger than one. The use of Equation 26 for “slip” velocity in the Frossling Equation 1 provides a correlation for suspended solids mass transfer that is not theoretically restricted to the Stokes’ flow region. Keey and Glen (1964) discuss the adequacy of the Frossling equation in correlating mass transfer from fixed solid spheres to a fluid in forced convection. They conclude that the constants in the general relationship NSh, =
B
+CZBN~,N$~
k / k $ = 0.502 P / V’023’
(27) The mean percent deviation of the experimental k / k $ ’ s from those predicted by Equation 27 is 27.2%. The square of the adjusted multiple correlation coefficient is 0.496. Data Correlation in This Study. Note in Figure 5 that the small tank results are grouped above and the large tank results below the correlating curve. Equal power input per unit volume is not a satisfactory scale-up criterion for suspended solids. P / V a t NRe > 50,000 is a function of the primary variables impeller speed, n, and diameter, d , in the fixed relationship dzn3(see Equation 13). To segregate the true effects of these primary variables on the mass transfer coefficient, a least mean squares fit was made to the following equation and the adjusted multiple correlation coefficient calculated. In k l k j = a0 + al In n + az In d (28)
(264
where
B, -226, c, and d = constants are functions of the extent to which there is downstream wake development and the degree of penetration of turbulence into the boundary layer. The use of certain correlations is suggested for specific particle Reynolds number ranges. These reduce to the Equation 26 constants for the overlapping N%p ranges indicated below: NRe,
0.1 to 10 1 to 80 100 to 9000 1000 t o 90,000
c?6
0
0.98 0.56 0.58 0.245
5 2 0
d
C
% % %
% ?4 ?4
0.62
0.31
Ref.
Ward et al., 1962 Kramers, 1946 Aksel’rud, 1953 Steinberger and Treybal, 1960
where
Levich’s Equation 19 is suggested for correlating data a t NRe, values above 10,000. As mentioned earlier, the coefficient CZ6= 1.10 was established in a study of mass transfer from fixed solid spheres in agitated vessels (Miller, 1967). The fact that this constant checks the value obtained by Garner and Suckling (1958) for the upstream flow area on a fixed solid sphere of solute suggests that there may be little or no wake development in agitated tanks. The effects of turbulence in the boundary layer seem to show up as required adjustments to the Equation 26 constants only a t particle Reynolds numbers above 10,000. The particle Reynolds numbers in this study ranged from 300 to 650. Taking all these factors into consideration, the
UO,ai,a2 =
constants
The last term involving d was then dropped and the procedure repeated. Results are summarized below: Trial a0
al a2
Mean % dev
Riq
1
2
-3.23 0.536 -0.1434 12.07 0.890
-3.62 0.626 12.22 0.881
Comparison of R?i values for the two trials shows that the impeller diameter has no significant effect on k / k $ . Impeller speed is the significant variable. Nienow presents
3.0
10
0.5 r u \ Y Y
0.1
0.05I 0.01
I
I
I I I l I l l
I
I
I
I
IIIIII
0. 1 PIVYH
I l I l l l l
P
10 ~ 0 3) ~ ~ 1
Figure 5 . k/ki-* vs. P/V’ 370
I
1.0
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
I
I
I I I l l l l
100
further experimental support for this conclusion in a recent paper (Nienow, 1969). A plot of k / k $ vs. n' is shown in Figure 6. The correlating equation is
k/k$ = 0.0267 (TZ')'~''
(29)
The ratios u s / u tcorresponding to k / k $ can be similarly correlated. The us values can be back-calculated from Equation 1, substituting the experimental values of k ; u,'s are obtained from Equation 26. The correlation with impeller speed is u s / u t= 0.000644 ( T Z ' ) ~ ' ' ~ (30) A summary of the various correlating equations covered in the preceding discussion is presented in Table 111. Reduced rate constants based on the Levich ( k / k L " ) and second Calderbank and Moo-Young ( k / k : ) correlations were fitted to impeller speed consistent with Equation 29. As indicated in Table 111, correlation of k / k $ with impeller speed provides the closest fit to the experimental data. The mean deviation of experimental points is 12.22% and the square of the multiple correlation coefficient is 0.881. The influence that impeller speed has on large particle mass transfer is associated with its effect on the longitudinal solids concentration gradient. As impeller speed is reduced from a high level of agitation, the vertical gradient in solids concentration becomes more and more pronounced until a layer of clear liquid appears under the top liquid surface. The clear layer depth increases with further reduction in impeller speed, and a point is eventually reached a t which the solids are no longer fully suspended. A gradient in solids concentration can affect mass transfer in several ways:
settling could not have been a factor if solids distributions were axially uniform. Pronounced gradients did develop at lower impeller speeds, however, and settling velocities were undoubtedly affected in the lower, more concentrated regions of the tanks. Impingement and slowed movements of the pellets along the bottom tank walls also contributed to the impeller speed dependency. Several attempts have been made to characterize suspended solids gradients in agitated systems. Weisman and Efferding report a relationship to predict the height of the slurry-clear liquid interface above the impeller midplane (Weisman and Efferding, 1960). Kneule (1956) and Zwietering (1958) developed correlations to predict the minimum impeller speed required for full solids suspension. Each of these developments utilizes power input per unit volume-or the equivalent d'n3 fixed relationship-as a correlating variable. Since the experimental rate measurements correlate best with impeller speed alone, these relationships are not applicable as indicators of expected mass transfer performance. Free Convection
Free convection can occur as a parallel mass transfer mechanism in the transition between forced convection and radial diffusion, if there is a substantial density difference between bulk solution and solution a t the phase interface. Circulation induced by gravity produces the mass transfer effect. Garner and Keey (1958) have established in fixed pellet
By creating a collateral solution concentration gradient, thus reducing the driving force for mass transfer in the lower part of an agitated system. Through hindered settling effects. As the result of reduced settling velocity in the bottom of a tank where particles are slowed or temporarily deposited out a t the wall.
3.0
U
1 .o
Y
\ Y
0.5
0.15
Differences in the vertical bulk solution concentration gradients in this study were small compared with saturation concentrations. The first effect listed above, therefore, was not significant. Solids loadings were kept low enough (0.006 to 1.2%) on all the experimental runs that hindered
IO
1000
100 n'(rpm )
Figure 6. k / k $ vs. n'
Table 111. Comparison of Mass Transfer Correlations
Large Particle Flow Regime
Correlation
Marangozis and Johnson, 1962 Calderbank and Moo-Young I, 1960 Middleman, 1965 Harriott, 1962 Levich, 1962 Calderbank and Moo-Young 11, 1961 This study
Mean dev of experimental points, Yo
Square of multiple correlation coeff R i
NSh, = 8.48 N R ~ P ~ ~ ' N ~ ~ ~ ~
13.80
0.790
k = 0.222 [g(P/V')v/p]'N%-?
28.4
0.277
(NSh,')N%-' = 0.719 N%' '(D,/d)' k = 0.0329 D/Dp(2 + 0.055 (CD)' 2Nb,N& 3)(n')06'4 k = 0.0450 D/D,(2 + 0.31 Nd3)(n')O6l2
28.3 27.2 13.88 12.63
0.350 0.496 0.848 0.872
k = 0.0267 D/Dp(2 + 1.10 N& 'NsA 3)(n')0626
12.22
0.881
Relationship tit to the data of this study
k = 0.502 D/Dp(2 + 1.10 N& 'N& 3 ) ( P / V ' ) 0 2 3 1
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
371
studies, using benzoic acid and water, that free convection does not occur a t particle Reynolds numbers above 750. Below this limit, free convection effects can be increasingly important. I n parallel, the free and forced convection mechanisms interact with the result that overall mass transfer is inhibited. The interference is least when the fluid is in upflow and greatest in downflow past a solute particle. I t is clear that free and forced convection effects are not additive (Acrivos, 1958) although Barker and Treybal report some success with this approach in correlating fixed pellet mass transfer data (Barker and Treybal, 1960). If free convection predominates over forced convection, the second term to the right of the equal sign in Equation 1 should be replaced with
N s ~=, 0.60 NG:'~N&'
the small particle flow regime reported in the literature that derive from boundary layer theory (Acrivos and Taylor, 1962; Bowman et al., 1961; Frankel and Acrivos, 1968; Friedlander, 1957, 1961; Yuge, 1956). The starting point in all these developments is the equation for steadystate diffusion from a solid sphere into a fluid, where there is a constant relative translational velocity between phases. Expressed in spherical coordinates,
uve = ( 2 / N p , ) v 2 e uve
DhP/v2p
Ap =
N R ~for , forced convection > 0.30 NG:"N%-'
(32)
By this test, forced convection was the dominant mass transfer mechanism in all the experimental measurements made for this study in the large particle flow regime. With NReDvalues vanishingly small such that the radial diffusion contribution predominates in Equation 1, free convection can also exist as a parallel mass transfer mechanism. Its relative importance can be weighed in this situation by comparing NSh, calculated from Equations 2 and 31. Radial Diffusion
Correlations Based on Boundary Layer Theory. There are a number of theoretical expressions for mass transfer in
Boundary conditions for Equation 33 are: e = 1 a t R = m,ande=OatR=l. Stoke's equations are used for the fluid velocity components in Equation 33. Results, therefore, are applicable only for particle Reynolds numbers less than one-the so-called "creeping flow'' region
uR= -[I - % ( I / R ) + '/2 (1m3)1 cos B
(34)
U, = [I - 3/4 (1/R) - '/4 ( 1 / R 3 ) ]sin B
(35)
I t is further assumed that the solution for c can be expressed in one or another arbitrarily chosen power series in R and 8. Analytical solutions can then be obtained for extreme values of Peclet number. For Np, values above 1000, as has been mentioned earlier,
I ly
103 Experimental-Harrioti,
1962
102
Nshp
10
1
10-1
102
I01
100
Npe
Figure 7. N s ~vs. , N,,
372 Ind. Eng. Chem. PI'ocess Des. Develop., Vol. 10, No. 3, 1971
sin e)(a/
e = (c* - c ) / ( c * - c , , ) , reduced solute concentration c, = solute concentration at a position far removed from the phase interface, g-moles/cm3 R = 2 r/D,,, reduced radial distance r = radial distance from the center of the sphere, cm 8 = angular displacement from the axis through the stagnation point UR = u,/u,, reduced radial velocity component U, = ut ( u - , reduced angular velocity component u , = fluid velocity a t a position far removed from the phase interface, cm/sec
(31)
bulk and interface solution density difference, g/cm3 The relative importance of free and forced convection can be gaged by comparing NSh, calculated via Equations 3 and 31. If these two expressions for NSh, are set as equal, they can be solved for a limiting value of NRe, above which forced convection is dominant.
(u,/R)(ae/ae) / + (l/R2 ae)(sin s(ae/ae) /
= u R ( a e i a R )+
0 2 8 = ( l / R ' ) ( a / a R ) (R'(ae/aR)
where
N G ~=
(33)
where
104
ierfc [(n+ l ) ( r ’ - r ) / ( D / ~ ) ~ ~ ]-CO) I ( c * (36) is obtained. At Npe’s approaching zero, solutions differ depending on the power series assumption and the techniques and simplifications used in solution. Several of the solutions are shown plotted in Figure 7. The Frankel solution (Frankel and Acrivos, 1968) is unique in that the additional complexity of constant shear developed by rotation of a freely suspended sphere is taken into consideration. Four numerical solutions of Equation 33 obtained by Yuge (1956) in the intermediate range 10 > Np, > 0.5 are also plotted. All expressions approach as a lower limiting condition. In addition to the calculated curves and points, Harriott’s data (Harriott, 1962) on the neutralization of 12.7- to 205-micron beads of Dowex-50-WX8 resin are plotted in Figure 7. Corresponding Nsh,’scalculated from Equation 31 for these data showed free convection to be unimportant as a contributing mass transfer mechanism. The experimental points decrease with decreasing Peclet number in the range 1.0 > Np, > 0.1 and are two to four times higher than the theoretical values. I n calculating the NSh,points, Harriott’s molecular diffusivity 1.93 x lo-’ cm2 per sec was used. This number was calculated from information on ionic mobility. Too low a molecular diffusivity would explain the observed discrepancy between measured and calculated data. On the other hand, it seems reasonable to expect a higher mass transfer characteristic in a highly turbulent fluid medium even though there is no net movement between solute and solution phases. Penetration Theory. I n a turbulent field surrounding a solute pellet, there is a continuous circulation of eddies. These can attach and detach from the phase interface, picking up dissolved solute in the interval attachment. Fick’s second law applies for mass transfer by molecular diffusion in this situation. Making the assumption that the field surrounding a single solute pellet is bounded by the radial distance, r, from the center of the sphere, the following mass transfer expression can be derived based on Fick’s second law:
where
N
= k(c* - eo), average mass flux across the phase boundary, g-moles/cm2sec
The 7 in Equation 36 represents the time of exposure of an eddy a t the phase interface. For all the experimental conditions covered in this study, the first term closely approximates the full summation in Equation 36, and
k = 1.128 ( D / T ) ” ~
(37)
Substituting this relationship in Nsh,in the Frossling Equation l and rearranging T
= Dl(O.887 k - 0.973 N R , ~ ’ ~ ’DID,)’ NS~
(38)
using Harriott’s experimental k’s along with the appropriate “slip” velocities calculated from Equations 26 and 30, values of exposure time, T , were calculated from Equation 38. These showed a broad, erratic scatter in the range 0.0035 to 0.550 second. There was no correlation with power input per unit volume. I t is apparent that mass transfer in the small particle flow regime is not consistent with penetration theory. Film Theory with Turbulent Diffusivity. Alternative to the penetration concept, mass transfer can be considered t o occur simply by radial diffusion into a surrounding turbulent fluid. Mass transfer with turbulence occurs more rapidly than in a molecular diffusion process and the rate can be characterized by a turbulent or effective diffusivity, De. Making the appropriate substitution in the Frossling Equation 1 and rearranging,
De = (kDp - 1.10 DNR,,”’NS;’~)/~
(39)
Using Harriott’s experimental k’s and the appropriate “slip” velocities from Equations 26 and 30, values of D, were calculated from Equation 39. Referred to Harriott’s calculated molecular diffusivity for ionized sodium hydroxide in dilute aqueous solution (1.93 x lo-’ cm2 per sec), these values correlate with P / V’ as follows:
D e / D = 3.08 (PIV)01294
(40)
Figure 8 shows a plot of this relationship. The mean percent deviation of experimental from calculated D e /D ratios by Equation 40 is 16.37%. The square of the adjusted multiple correlation coefficient is 0.604. Film theory with turbulent diffusivity is the most suitable basis for correlating mass transfer in the small particle flow regime.
e
n
0.1
1
10
100
300
P/V’ (H P /lo3 Gal ) Figure 8.
De/D vs:P/V’ Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
373
Conclusions
The following procedures are recommended for scaleup of suspended solids mass transfer performance: Forced Convection. Use Equations 1, 26, and 30. If the solute particle size is not changed, duplication of performance from one scale to the next requires that impeller speed be held constant. Free Convection. Use Equation 1 with the substitution of Equation 31 for the second term to the right of the equal sign. In the large particle flow regime, there are two conditions that can produce deviations from this relationship: The development of axial particle and solution concentrations will moderate mass transfer performance. If there is parallel forced convection, the two mechanisms will interact to suppress mass transfer. Quantitative formulations for the effects of these conditions are not available. If performance a t one scale is known, however, it can be duplicated at another scale if certain restrictions are allowed. For example, deviations associated with the first condition can be obviated by duplicating the axial concentration profile. This is possible if Equations 26 and 30 are applied to match uB/ut.To eliminate deviations associated with the second condition, it is necessary to keep both particle size and the axial concentration gradients the same. I n both cases, with no change in particle size, duplication of mass transfer performance requires that impeller speed be held constant. Radial Diffusion. Use Equation 1 with the substitution of 2 D,/D from Equation 40 for the first term to the right of the equal sign. In this case, if solute particle size is not changed in scale-up, duplication of performance requires that agitator power input per unit volume be kept constant. Practical limitations on agitator design become more restrictive as the scale of operation is increased. I t is important, therefore, to plan any experimental work done in smaller equipment for scale-up purposes so that the useful ranges of agitator speed and power input are not exceeded. Nomenclature
A = pellet surface area, cm2
K
= mean pellet surface area, cm2
a, ao, al, a2 = a constant
b, bo, b l , b2 = a constant CO = drag coefficient cl, c2 = initial and final solute C7, GO,C ~ ZC13, , CIS,C U , CIS,CIS,C X = constants c = bulk solute concentration, g-mole/cm3 c* = saturation concentration, g-mole/cm3 c, = solute concentration a t a position far removed from the phase interface, gmoles/cm3 D = molecular diffusivity, cmz/sec D, = effective diffusivity, cm2/sec De, = effective diffusivity in the turbulent boundary layer, cm2/sec D, = particle diameter, cm d = impeller diameter, cm g = acceleration of gravity, cm/sec* k = mass transfer rate constant, cm/sec k? = mass transfer constant based on the Levich correlation, cm/sec 374
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
mass transfer constant based on the Calderbank and Moo-Young correlation 11, cm/sec mass transfer constant based on the Frbssling equation, cm/sec scale of turbulence eddies, cm D;gap/u‘p, Grashof number Dput/D,Peclet number D&(p, - p ) / DVP,Raleigh number d h / u , impeller Reynolds number D,u,/v, particle Reynolds number 1.123 g’ ‘Di ‘ ( P /VI)’‘ / V 1 ’p’ ‘, a Reynolds number u / D ,Schmidt number kd/D , impeller Sherwood number kD,/ D, particle Sherwood number mean mass flux, g-mole/cm2 sec impeller speed, rps impeller speed, rpm agitator vower input. hv pH1, pH2 =-initial i n d finaipH’sQ = volumetric flow rate, cm3/sec R = 2 r / D p ,reduced radial distance Ri = adjusted multiple correlation coefficient r = radial distance from the center of the sphere, cm r‘ = outer bounds of fluid surrounding a sphere of solute, measured from the center, cm At = exposure time, sec U R = u r / u , , reduced radial velocity component U , = u,!u,, reduced angular velocity component u = fluid velocity, cm/sec us = “slip” velocity, cm/sec ut = terminal settling velocity, cm/sec ui = velocity of turbulence eddies normal to a phase interface, cm/sec u , = fluid velocity a t a position far removed from the phase interface, cm/sec uL. = a characteristic velocity of turbulence eddies in bulk stream flow, cm/sec V = volume, cm3 V = volume, io3 gal y = distance from a phase interface, cm Greek Symbols ‘1=
turbulence microscale, cm
0 = angular displacement e = (c* - c)/(c* c - ) , reduced solute concentration
+
viscosity, poise kinematic viscosity, cm2/sec = fluid density, g/cm3 = particle density, g/cm3 = time of exposure of an eddy a t a phase interface, sec
c1= u =
P PP 7
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Symposium on Distillation,” Brighton, England, May 1960. Calderbank, P. H., Moo-Young, M. B., Chem. Eng. Sci., 16, 39 (1961). Frankel, N. A., Acrivos, A., Phys. Fluids, 11 (9), 1913 (1968). Friedlander, S. K., AIChE J., 3 (l),43 (1957). Friedlander, S. K., ibid., 7 (2), 347 (1961). Frossling, N., Gerlands Beitr. Geophys., 52, 170 (1938). Garner, F. H., Suckling, R. D., AIChE J., 4, 114 (1958). Garner, F. H., Keey, R. B., Chem. Eng. Sci., 9, 119 (1958). Harriott, P., AIChE J., 8, 93 (1962). Hixon, A. W., Baum, S. J., Ind. Eng. Chem., 33, 478 (1941). Humphrey, D. W., Van Ness, H. C., AIChE J . , 6, 289 (1960). Keey, R. B., Glen, J. B., Can. J . Chem. Eng., 42, 227 (1964). Kneule, F., Chem. Eng. Tech., 28, 221 (1956). Kramers, H., Physica (Utrecht), 12, 61 (1946). Langmuir, I., Phys. Reu., 12, 368 (1918). Levich, V. G., “Physicochemical Hydrodynamics,” Trans-
lation by Scripta Technica, Ind., Prentice-Hall, Englewood Cliffs, N. J., 1962. Marangozis, J., Johnson, A. I., Can. J . Chem. Eng., 40, 231 (1962). Middleman, S., AIChE J . , 11, 751 (1965). Miller, D. N., Ind. Eng. Chem., 56 ( l o ) , 18 (1964). Miller, D. N., Chem. Eng. Sci., 22, 1617 (1967). Nienow, A. W., Can. J . Chem. Eng., 47, 248 (1969). Shinnar, R., Church, J. M., Ind. Eng. Chem., 12 (3), 253 (1960). Steinberger, R. L., Treybal, R. E., AIChE J . , 6 (2), 227 (1960). Sykes, P., Gomezplata, Can. J . Chem. Eng., 45, 181 (1967). Ward, D. M., Trass, O., Johnson, A. I., ibid., 40, 164 (1962). Weisman, J., Efferding, L. E., AIChE J., 6 (3), 419 (1960). Wieselberger, C., Phyi. Z., 23, 219 (1922). Yuge, T., Sci. Per. Res. Inst., Tohoku Univ., Ser. B., 6 , 143 (1956). Zwietering, T. N., Chem. Eng. Sci., 8, 224 (1958). RECEIVED for review March 5, 1970 ACCEPTED October 6, 1970
Membrane Separation of Gases Using Steady Cyclic Operation Donald R. Paul Department of Chemical Engineering, University of Texas, Austin, Tex. 78712
Membrane permeation for separation of gas mixtures is usually considered only in terms of steady-state operation. The steady-state separation factor for two species is the product of the diffusion coefficient and solubility coefficient ratios-i.e., cyss =
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