147
Ind. Eng. Chem. Fundam. 1984. 23, 147-153
conclusions can be reached: (a) The rates of growth and solution of sodium sulfate decahydrate are controlled by the rates of heat and mass transfer near the crystal surface. (b) The rate of solution of anhydrous sodium sulfate is also controlled by the rate of mass transport from the crystal surface. (c) The rate of growth of the (111) face of the anhydrous crystals is limited by the rate of incorporation of the ions into the crystal lattice a t the surface.
Acknowledgment We gratefully acknowledge the support of the Department of Energy in the form of Research Grant No. DEFG02-79ET-00088. The work reported here has been taken in part from the Master of Chemical Engineering Thesis (June, 1981) of David Rosenblatt. Nomenclature A = volume shape factor B = surface area shape factor C = concentration of sodium sulfate in solution, g-mol/cm3 C, = total concentration in solution, g-mol/cm3 C, = heat capacity of the solution, cal/(g "C) D = diffusivity of sodium sulfate in solution, cm2/s F = ratio of the salt flux to the total material flux g = gravitational constant, cm/sz h = heat transport coefficient, cal/(cm2 s "C) AHf = heat of fusion of sodium sulfate decahydrate, cal/g-mol AH8 = heat of solution of anhydrous sodium sulfate, cal/g-mol k = mass transfer coefficient, cm/s 5 = crystallization rate coefficient, (cm/s)/ (g-mol L)1.5 L = characteristic crystal dimension, cm m = crystal mass, g Ns = flux of salt to the interface, g-mol/(cm2 s) N , = flux of water to the interface, g-mol/(cm2 s) r = equivalent crystal radius, cm R = gas constant, cal/(g-mol K) T = temperature of the solution, K t = time, s V = bulk solution velocity past the crystal, cm/s X = mole fraction of sodium sulfate in solution
Gr = p:ge3/3c(AC)/p2, Grashoff number Nu = hL/K, Nusselt number Pe = (Re)(sc),Peclet number Per = (2r/D)(dr/dt), Peclet number due to radial velocity Re = eVp,/p, Reynolds number Sc = p/p,D, Schmidt number Sh = kL/D, Sherwood number Greek Letters cy = fractional solution specific volume change pT = temperature coefficient of fractional volume change, O C - ' /3c = concentration coefficient of fractional volume change, (g-mol/ L)-l K = solution thermal conductivity, cal/(cm s "C) pc = crystal phase density, g/cm3 pm = crystal phase molar density, g-mol/cm3 ps = solution density, g/cm3 p = solution viscosity, g/(cm s), P Subscripts i = refers to interface properties b = refers to bulk solution properties e = refers to equilibrium properties Registry No. Sodium sulfte decahydrate, 7727-73-3;sodium sulfate, 7757-82-6.
Literature Cited Brian, P. L. T.; Hales, H. B. AIChE J . 1969, 75, 419-425. Bruton, W.; Cabrera, H.; Frank, F. C. Phil. Trans R . SOC. London 1951, AZ43, 299-358. Frank, F. C. "On the Kinematic Theory of Crystal Gyowth and Dissolution Process" in "Growth and Perfection of Crystals", International Crystal Conference", Doremus, R.; Turnbuii, J., Ed.; New York, Wiley: 1958; pp 4 11-41 8. Hayakawa, T.; Matsuoka, M. Heat Transfer Jap. Res. 1973, 2 , 104-115. Hixson, A. W.; Knox, K. L. Ind. Eng. Chem. 1951, 4 3 , 2146. Liu, V. Y.; Tsuei, H. S.; Youngqulst, 0. R. Chem. Eng. frog. Symp. S e r . 1971, 170. McCabe, W. L.; Stevens, R. P. Chem. Eng. f r o g . 1951, 4 7 , 168-174. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. "Mass Transfer"; McGraw-Hill: New York, 1975. Wetmore, F. E. W.; LeRoy, D. J. "Principles of Phase Equilibria"; McGraw-Hill: New York, 1951.
Received for review November 9, 1981 Accepted October 10, 1983
Interstitial Flow Intensification within Packed Granular Bed Filters: Experiments and Theory Robert W. L. Snaddon' and Peter W. Dletz General Electric Company, Corporate Research and Development, Schenectady, New York 1230 1
The mechanical collection efficiency of a granular bed filter is determined by the flow within the bed. I n spite of the demonstrable sensitivity of existing models to their proposed flow structures, most do not attempt accurate characterization of the flow. I n particular, interstitial flow intensification and flow separation are usually neglected. In the present paper an analytic model and collection measurements are combined to examine these effects. A potential flow model is developed in which the magnitude of the flow intensification is characterized by a single "intensification parameter". The experimental data are then employed to develop a unique correlation of this parameter with Reynolds number.
Introduction In recent years considerable interest has been expressed in the use of packed granular bed filters for separating fine particles and droplets from gas streams. This interest stems from the inherent simplicity, reliability, and low cost 0196-431318411023-0147$01.50/0
of these devices as well as their ability to remove fine submicron particulates, especially when augmenting agencies such as electrostatics are employed as shown by Kallio e t al. (1979), Kallio and Dietz (1981), Dietz (1981), Zahedi and Melcher (1976), and Zahedi and Melcher 0 1984 American Chemical Society
148
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984 CRITICAL
- --- TRAJECTORY ./
/
I
Figure 1. Flow intensification within packed beds.
(1977). Several models have been developed in efforts to understand the physics of collection and to predict performance (Tardos et al., 1978). Most of these models do, however, have the tendency to underpredict performance, particularly at higher Reynolds numbers. It is this which has spurred the investigation described here. The most challenging aspect of the modeling process is undoubtedly the stipulation of an appropriate flow field; see, for example, Tardos et al. (1978), Happel (1958), Kuwabara (1959), Lamb (1932), LeClair and Hameliec (1968), and Ranz and Wong (1952). In reality, granule to granule contacts and random packing configurations give rise to complex flow fields. Accurate and detailed solutions have been attempted by Sorensen and Stewart (1974) and Karabelas et al. (1973) for a few cases of regularly packed spheres, but their special packing and the associated high cost of computing collection efficiencies render these flow solutions impractical for most filter studies. Instead, the trend has been to employ the unit-cell techniques first adopted by Happel (1958) and Kuwabara (1959) for predicting pressure drops in packed beds. Rather than attempting to describe the exact flow around a set of bed granules, this approach seeks some average representation of the activities within the bed. While this has led to simple, flexible flow models, there is often little physical basis for the assumed flow patterns, especially at higher Reynolds numbers. From the experimental work of Van der Merwe and Gauvin (1979) and Jollis and Hanratty (1966), it is expected that, in real filters, the acceleration of the flow through the voids and the accompanying flow separation a t higher Reynolds numbers will lead to intensified flows which impinge on the upstream face of bed granules (see Figure 1). Although these effects must have a profound effect on collection, it appears that with the exception of Alexander (1978), most workers have chosen to ignore them. It is felt that these omissions may explain the underpredictions mentioned above. In this paper we describe a unit-cell model and experiments which seek to examine and understand these effects.
Theory The Unit-Cell Model. In this approach the packed bed is viewed as an assemblage of unit cells each comprising a spherical bed granule surrounded by a spherical shell of gas (see Figure 2). The fine particles are assumed to be evenly distributed in the gas as it enters the cell. The rate at which these particles are remvoed from the gas which flows within the cell is determined by analyzing their trajectories with an equation of motion. The effects of neighboring bed granules are modeled by imposing the initial particle trajectory conditions and flow boundary conditions at the cell boundary, the radius ( 1 ) of which is related to granule radius (a) and void fraction ( a ) by 1 = a(1 - a)-1/3
(1)
It is assumed that all fine particles whose trajectories pass within one fine particle radius of the bed granule w ill
be collected. The critical trajectory for a particle of given
I/>
‘‘ \ 4 -/’ Figure 2. The unit-cell model.
size is that which just passes within one particle radius of the bed granule. Once this is known, the “single-sphere’’ collection rate, r, can be computed from
r
=
L“;
(E)27d2nosin 0 d0
u,
(2)
where 0, is the angular position at which the critical trajectory enters the cell, u, is the fine particle velocity, and no is the inlet concentration of fine particles. The single-sphere collection efficiency, q’, is defined as the ratio of this collection rate to the total rate at which particles pass into the cell, i.e. q‘ =
r/noUox12
(3)
where Uo is the superficial gas velocity in the bed. The total bed efficiency, q , can then be obtained by equating the change in particle flux across a thin layer of the bed to the collection rate of the enclosed bed granules. Integration over the length of the bed, L , yields 7 = 1 - exp {-3/4(12/a3) (1 - a)q’L)
(4)
The Equation of Motion for Fine Particles. The following simple form of the equation of motion for a spherical fine particle of radius, rp, and mass density, ( p , ) is employed in this model. 4/3ar,3pp
dv
= 6aprp(u - u ) / c
Here, u is the local gas velocity, v is the fine particle velocity, p is the gas viscosity, and c is a correction term to the Stokes drag law which accounts for noncontinuous effects when the size of the fine particles is comparable with the mean free path of the gas molecules; see Davies (1945). The assumptions implicit in this simple formulation are as follows. (1)Stokes drag law with the correction factor c is valid. (2) The mass of the displaced gas is so am11 that bouyancy and momentum transfer effects can be ignored. (3) The presence of the fine particles does not alter the streamlines. (4) Long-range forces (e.g., those resulting from gravitational or electrical fields) are negligible. ( 5 ) The f i e particles are large enough so that diffusion effects can be ignored. The Flow Model. With reference to Figure 1 and the previous description of intensification effects, it is clear that the constrictions in packed beds must give rise to flows which impinge on the upstream face of bed granules with velocities greater than the superficial velocity. The flow structure and the relative magnitudes of local velocities with respect to the superficial velocity will not change under laminar conditions. However, boundary layer separation and the concomitant transition to turbulent flow
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984 149
a t higher Reynolds numbers will give rise to recirculating flows and changes in effective void fraction. Measurements made by Van der Merwe and Gauvin (1971) and Jollis and Hanratty (1966) in packed beds show that the transition to turbulence occurs a t Reynolds numbers around 150. Here Reynolds number is defined by
Re = 2p,Uoa/p
(7)
where @J is the velocity potential and the radial and tangential components of velocity are given by u, =
I
I
l o t
(6)
and the degree of flow intensification is expected to increase with Reynolds number when Re > 150. With the exception of high concentrations of fine particles (C0.5 pm, say) where diffusion transport is significant, mechanical collection is characterized by geometric and inertia effects and is thought to occur predominantly around the upstream faces of the bed granules. Based on Ranz and Wong’s (1952) model of a free jet impinging on a sphere, Alexander (1978) developed an empirical approach to correlate a jetting factor with Reynolds number. Here, intensification was represented simply by a change in effective gas velocity whereas the preceding arguments call for changes in flow structure at Re > 150. In this paper a potential flow model is proposed which allows the simulation of varying degrees of flow intensification by altering the flow around the upstream faces of bed granules. The elements of this model are as follows. For potential flow, the flow field is described by Lapace’s equation
A2@ = 0
I
4
-
n
0
H
2
e Figure 3. Radial velocity profiles at the cell boundary.
while higher centerline velocities result as p is increased. The relationship between @ and 4 is obtained by applying the conservation of mass to the cell. This yields
6 = 72 cos-1 (1- 2/p)
(12)
The orthogonal properties of the Legendre polynomials are used to obtain an expression for this boundary condition which is convenient for evaluating the constants in eq 9; i.e., they allow the velocity profile to be expressed as a series over the interval -a12 to a12 in the following way.
a*
dr
1) P,(cos 8) d cos 8) P, cos 8 (13)
and
For an axisymmetric flow, the general solution to eq 7 in spherical coordinates is given by
The constants A, and P, are then easily evaluated on a digital computer using standard integral and recurrence relationships. Solution. Solution is made more manageable by normalizing lengths, velocities and time in the following way.
m
cP(r,8)= C (Anrn+ B,r-(n+l))P,(cos0) n=l
(9)
where P, is the nth-order Legendre polynomial, A , and B, are constants, and n = 1, 2, 3, ..., m. Intensification effects are incorporated into the model by appropriate choice of boundary conditions. In prior use of the unit-cell potential flow models, the boundary conditions used to evaluate the flow field have been u, (r = a) = 0
(loa)
u = u*uo
(144
v = v*u,
(14b)
r = r*a
(144
1 = l*a
(14d)
t = t*a/Uo
(144
Normalized, the flow field equation yields the following two expressions for the fluid velocities.
and
m
u, (r =
I ) = - Uo cos 8
(lob)
Here it can be seen that the maximum radial velocity occurs along the centerline of the cell and never exceeds Uo,the superficial velocity. However, since we expect higher radial velocities as the flow accelerates through the voids, the following alteration to eq 10b is proposed in order to simulate the intensification effects.
u,(r = 1 ) = -DUO cos 0 (0 I0 1 6, I1) =O
(6C 8 1 a/2)
(11)
This gives “step function” radial velocity profiles a t the cell boundary as shown in Figure 3. The original solution is retained when p, the intensification factor, is set to 1
u,* =
C (nA,*r*(”-l)- ( n + 1)B,*r*-(n+2)]P, (cos 8)
“=l
m
ug* =
C (A,*r*(,-l) +Bn*r*-(n+2)J(n/sin8) n=1
(COS
e P,(COSO)
X
- p,-l(cos
e)! (i5b)
n = l , 3 , 5,...,
To visualize the effects of the new boundary conditions, the streamlines corresponding to constant values of the dimensionless stream function, \k/ cfoa2, equispaced at intervals of A(*/ Uoa2)= 0.1 are shown in Figure 4. As p is increased the flow is constrained to enter the cell in a narrowing jet centered around the axis.
150
Ind. Eng. Chern. Fundarn., Vol. 23, No. 2, 1984 10
';z 0 8 > 1
0
2
w
0 u u 06 w w
a W
I la1 r? = 1
ibl
P
a =
2
:
04
_I
8m
02
10'
102
10
10
--= STOKES NUMBER (St =IC+)
Figure 5. Single-spherecollection efficiencies vs. Stokes number ( a = 0.5).
,
up_____IC/
0
Id,
= 4
IJ
= E
VACUUM
Figure 4. Streamlines around the upstream faces of bed granules.
When normalized, the equation of motion reduces to the following two coupled second order differential equations. d2r* ug*2/r* = (u,* - v,*)/St (164 dt*2
?
SECTION
TEST
I
The dimensionless Stokes number is given by
St =
2 CPprp2U0
-___ 9 I.ta
and is a measure of the ratio of the inertial to viscous drag forces acting on a fine particle. To compute the trajectories of fine particles using eq 15 and 16, it is necessary to specify initial conditions. Here it is assumed that as they enter the cell, the velocities of the fine particles are the same as the local fluid velocities, i.e. v*(r* = l*) = u* (r* = I*) (18) The manner in which filter performance predictions are altered by the intensification factor can be examined by calculating single-sphere efficiencies. To do this it is necessary to specify values of the interception parameter, rp/a. Recalling that the single-sphere efficiency appears as an exponent in the expression for the total bed efficiency (eq 4), it is clear from Figure 5 that varying the degree of intensification has a profound effect on performance predictions. This is particularly true a t higher Stokes numbers where inertia effects dominate. The role of interception can also be assessed from results in Figure 5. In most practical granular bed filters rp/a will be of the order of 0.001 or smaller and interception will play a small, if not negligible part in the collection. The collection which occurs a t low Stokes numbers when rp/a is increased does not, however, result from interception alone. Indeed, Kallio et al. (1979) have shown that the minima which occur just before the inertia regime (i.e., as Stokes numbers are increased) result from the fine particles moving away from the collecting surface as the streamlines curve around the bed granule. This observation illustrates the fact that the contributions to collection
I
I
A-
TSICHARGE NEUTRALIZER CLIMET AEROSOL GENERATOR I
Figure 6. Schematic layout of test facility.
arising from different mechanisms are coupled in a nonlinear fashion and they cannot always be viewed as a simple superposition of individual contributions. The question now arises as to the appropriate intensification factor to be used under different flow conditions. At the present time, the detailed understanding required for a theoretical formulation does not exist and it has been necessary to adopt an empirical approach. The experiments are described in the following section. Experimental Section Equipment. The layout of the 50.8-mm diameter model granular bed test facility used in this study is shown schematically in Figure 6. The entrance section, test section, and freeboard region were manufactured from Plexiglas tubing. The packed test beds were supported between two open mesh metal screens. This arrangement was mounted vertically and was preceded by an annular absolute filter. Monodisperse polystyrene latex microspheres were used as test aerosols. These were dispersed from a methanol solution by use of a Climet 295 aerosol generator. After dispersion, the aerosol
Ind. Eng. Chem. Fundam., Vol. 23, No. 2, 1984
151
DILUTION AIR
I-
fO46m
4
Figure 7. The sampling probe. was passed through a Thermo-Systems, Inc. (TSI) 3012, 85Kr radioactive neutralizer to remove any charges on the aerosolswhich might have arisen from natural electrification during dispersion. The Climet 295 dispersed the aerosols into a prefiltered air stream of constant flowrate which was fed into the lower end of the absolute filter. Vacuum was used to draw the air and test aerosol through the test section, the annular filter facilitating simple control of test flowrates via a single valve downstream. The test section flowrates and the dilution flowrates to the sempling probes were measured with calibrated flowmeters and pressure gauges. Average face velocities up to 1.8 ms-l could be achieved with this arrangement. Aerosol sampling was accomplished with two probes arranged as shown in Figure 7. The probes were manufactured from 6.35-mm diameter stainlew steel tubing. To mimimize wall losses, the probe length was kept to a minimum, and the aerosol stream was only passed through a single 90’ bend. Also, sampling rates were adjusted to maintain near-isokineticconditions at the probe tip by adding dilution air via the T-piece mixer shown in Figure 7. The inlet and outlet aerosol concentrations were measured simultaneouslywith two Climet 208 optical particle counters. The resolution of the counters was improved by connecting the photomultiplier output of each to separate Tracor Northern, TN, 1705 pulse height analyzers. These pulse height analyzers display number of counts against pulse height (particlesize) at a resolution of 0.3 pm/channel at a particle size of 1pm and 0.004 pm/channel at a particle size of 8 pm. With 1010 channels and a range of 0.3 to 20 pm, an almost continuous display of counts vs. size was available while the automated facilities allowed quick and easy integration of counts over any given size range. Experimental Procedure. In small-scale model beds, consideration must be given to edge effects which arise from the changes in void fraction which occur in regions adjacent to containing surfaces. To account for these and to ensure good agreement between experimental conditions and model input parameters,the samplingprobe tips were located on the centerline of the test section and the void fraction and face velocity in the central core of the bed were obtained in the following fashion. Two different spherical bed media were used. These were aluminum spheres with diameters of 2.90 rt 0.00 mm and glass beads with diameters of 2.62 i 0.35 mm. Care was taken with the bed packing procedure to ensure reproducible packing densities. Bed void fractions were measured for different depths of bed using a liquid displacement technique. The contributions to these measured void fractions arising from edge effects were then estimated with data from Chandrasekhara and Vortmeyer (1979). This yielded core void fractions for the aluminum and glass of 0.36 and 0.32, respectively. The role of edge effects on the velocity distribution in cyclindrical packed beds has been investigated by Fanhein and Stankovic (1979). They were able to correlate velocity profiles measured by several workers with the ratio of granule radius to bed radius (a/Rb)for a / R b > 0.02. Their predictions are shown in Figure 8. Edge effects are more pronounced at higher values of a / R b because the nonuniform void fractions near the walls extend farther into the bed. In this work the outlet probe was located 25.4 nm above the bed and the core face velocity was taken to be the centerline velocity predicted by the above correlation. Four different polystyrene latices were used. These has mean diameters of 1.049,2.02,3.07, and 4.31 wm and standard deviations
0.0 L 0.6
0.2
3.4
C.6
RADIAL POSITION
0.8 ir
1
RbI
Figure 8. Velocity distributions at the outlet of the bed after Fanheim and Stankovic (1979). of 0.0587, 0.0135,0.13, and 0.26 pm, respectively. Noise due to ambient particulates being a problem at smaller sizes and excessive dropout a problem at larger sizes, the above range of sizes represents those which could be easily generated, sampled, and counted in this system. The nonideal characteristics of the optical particle counting systems resulted in broadening of the displayed distributions on the pulse height analyzers. Nevertheless, the displayed distributions for the four test aerosols were still sufficiently separated to allow them to be generated and counted simultaneously, and the number of particles of a given size sampled in a given time was obtained by integrating the number of counts over the width of the distribution. The inlet and outlet probes and particle counting systems were run simultaneously at identical settings. Collection efficiency data were compiled from averages of five measurements made at the same test conditions. Testa were performed to check the agreement between the inlet and outlet sampling systems when the bed and supporting screens were removed. It was found that regular checks on the dilution and sampling flowrates were required to avoid mismatch or bias. With these precautions, the data were reproducible to within 1% for the 1.049-pm particles to within 8% for the 4.31-pm particles.
Results and Discussion If the proposed physical description of the flow field and anticipated intensification effects is plausible, one would expect different values of p to give good agreements between model predictions and measurements under different flow regimes. More specifically, one would expect a single value of 0to give a good match over the laminar regime, and starting with the onset of turbulence at a Reynolds number around 150,we would expect increasing values of p to give a good match at higher, turbulent Reynolds numbers. The validity of this interpretation of the hydrodynamics is examined in this section by comparing theoretical predictions with the experimental data. Figures 9a and b show data from laminar flow (Re = 79) and turbulent flow (Re = 309) measurements. Here it is immediately obvious that higher values of 0are required to achieve a reasonable match for the turbulent case. A simple “least-squares” algorithm was used to determine the values of p which provided the best match to the data at different test conditions and the data are shown in Figure 10. These data suggest a trend which is consistent with the views put forward above. The appropriate best fit value of 0 for Re < 100 appears to be constant at 5, while increasing values of 0 are indicated for Re > 200. Referring to the discussion of the boundary conditions for the flow model, the laminar flow result seems to imply
Ind. Eng. Chem. Fundam., Vol. 23, No. 2 , 1984
152
FINE PARTICLE RADIUS l r m l
1
I
1
T7
1
I
/
'*-'
0.8
P
0.0
I
,
,
,
,
I
,
1.0 FINE PARTICLE RADIUS (pmi
0.'
10
Figure 9. (a) Total bed efficiency, Re = 79; (b) total bed efficiency, Re = 309. I
10,
A 2.9 mm ALUMINUM I3 2.9 mm ALUMINUM 0 2.62 mm GLASS
B
2.62 mm GLASS
I
I
I
I l l
L = 50.8 L = 25.4 L = 50.8 L = 25.4
I
I
I
l
l
l
mn rnm
0
0
A A
I3
I
I
I
I
I
rn
1 ' 1 ) -
v
1
1
'
1
I
"
I 1 1 1 1 1 1
I
I
I
I
l
l
l
,
0.2
0.0
n i
,
1
i n
,
in
FINE PARTICLE RADIUS [ p m )
Figure 11. (a) Total bed efficiencies after Alexander (1978), Re = 66; (b) total bed efficiencies after Alexander (1978), Re = 612.
l
A
mm mm
0
0.
I
Re = 612 a =2mm Uo = 2.4 rns-' = 1.85 X 1 0 a, = lo3 k g K 2 L = 2 5 rnm 0.36
_.I ,I,1, 1
I
b
m
1.0,
l
Figure 10. "Best fit" intensification factors vs. Reynolds number.
that local velocities within the bed could be as high as 5 times the superficial velocity. This is reasonable if one considers the reduction in the average flow cross section within the bed and the no slip conditions at the granule surfaces. Under turbulent conditions the ratio of maximum local velocity to superficial velocity is not expected to increase indefinitely with increased Reynolds numbers. Instead it is anticipated that at very high Reynolds numbers the ratio will be limited by the length scale of the smallest voids, thus resulting in the values of fi saturating at approximately 13 for Reynolds numbers somewhere in the region of 1000 or greater. Behavior in this region is not, however, of great practical significance since pressure drops for typical applications would be prohibitively high. In this study, constraints on the system pressure drop and on the maximum size of the bed granules prevented investigations of this limiting condition. The results presented so far tie in well with the physical arguments and they demonstrate the importance of flow
intensification effects. The data obtained from the experimental study are, however, limited and to extend the examination of the predictive capability of the collection model, model predictions are compared with measurements made by Alexander (1978). Here, filter performance was investigated by passing air and a polydisperse dioctylphthialate (DOP) aerosol through packed beds of glass beads. The liquid aerosol was generated by an atomizer nebulizer and the collection efficiencies were measured using an Anderson 2000 cascade impactor. In Figure 11,results for two different Reynolds numbers are compared with model predictions using values of p taken from Figure 10. It can be seen that the predictions for the inertia dominated collection (large aerosols, large Stokes numbers) are good, particularly if they are compared with those available from a straightforward potential flow model (P = 1). However, the lack of agreement at low Stokes numbers is attributed to the fact that the model does not include diffusion or electrostatic collection mechanisms. These mechanisms have their most profound effects at low Stokes numbers and the discrepancies between experiment and theory at small particle sizes are thought to arise from a combination of diffusion and residual electrostatic forces. Conclusions Experiments and theory indicate that interstitial flow intensification play an important role in the collection of fine particles with granular bed filters. A unit cell potential flow model has been developed which allows the flow around bed granules to be altered to simulate varying degrees of intensification through an intensification factor, p. Physically, this factor is representative of the ratio of the maximum interstitial velocity to the superficial velocity. Experiments show that a value of 0 = 5 is required to give good predictions of collection in the laminar regime
153
Ind. Eng. Chem. Fundam. 1984, 23, 153-158
while, beginning with the onset of flow separation a t Reynolds numbers of about 150, higher values are required for higher Reynolds number flows. The trends are consistent with physical arguments and the model is expected to prove useful in the study of Titer performance, especially since many additional effects such as electrostatic enhancement can be examined by the simple addition of force terms to the equation of motion.
7' = single-sphere collection efficiency 8, = angular position defining critical trajectory p = gas viscosity, kg s-l p g = gas density, kg rn-, pp = fine particle mass density, kg rn-, \k = stream function, rn-, s-l
Nomenclature
Alexander, J. C. Ph.D. Thesis, Department of Electrical Engineering, MIT, Cambridge, MA, 1978. Chandrasekhara, B. C.; Vortmeyer D. Waerme Stoffubemrag. 1979, 72, 105-1. 1. 1. Davies, C. N. Proc. Phys. SOC. 1945, 5 7 , 259-270. Dietz, P. W. J . Aerosol Sci. 1981, 72, 27. Fanhein, R. W.; Stankovic, L. M. Chem. Eng. Sci. 1979, 3 c , 1350. Happel, J. AlChE J . 1958, 4(2), 197. Jollis, K. R.; Hanratty, T. J. Chem. Eng. Sci. 1988, 27, 1185-1190. Kallio, G. A.: Dietz, P. W. Presented at the 7th TDFM Convention on "Gas Borne Particles", Oxford, England, June 1981. Kallio, G., Dietz, P. W.;Gutfinger, C. "Proceedings of the Symposium on the Transfer and Utilization of Particulate Control Technology", Denver, CO. 1979. Karabelas, A. J.; Wagner, T. H.; Hanratty, T. J. Chem. Eng. Sci. 1973, 28, 673-682. Kuwabara, S. J . Phys. SOC.Jpn. 1959, 74, 527. Lamb, H. "Hydrodyamics", 6th ed.; Cambridge University Press; Cambridge, England, i932.Leclair, B. P.; Hameliec, A. E. Ind. Eng. Chem. Fundam. 1988, 7 , 542. Ranz, W. E.; Wong, J. 8. Ind. Eng. Chem., 1952, 44(6), 1371. Sorensen, J. P.; Stewart, W. E. Chem. Eng. Sci. 1974, 29, 819. Tardos, G. I.; Abuaf, N.; Gutflnger, C. J . Air Pollut. Control Assoc. 1978, 28(4\. 355. .,~ Vander Merwe, D. F.; Gauvin, W. H. AiChE J . 1971, 17(3), 5-19. Zahedi, K.; Melcher, J. R. Ind. Eng. Chem. Fundam. 1977, 76, 240. Zahedi. K.; Melcher, J. R . J . Air Pollut. Control Assoc. 1978, 26(4). 345.
A,, B, = constants used in eq 9 and 15 c = correction factor for noncontinuous effects L = granular bed length, m P, = nth-order Legendre polynomial Re = Reynolds number S t = Stokes number Uo = superficial/face gas velocity, m s-l a = bed granule radius, m 1 = unit-cell radius, m no = inlet concentration of fine particles, mW3 r, 8 = polar coordinates rp = fine particle radius, m t = time, s u = local gas velocity, m s-l ur, ug= radial and tangential components of local gas velocity, m s-l u = fine particle velocity, m s-l ur, ug = radial and tangential components fine particle velocity, m 1*, r*, u*, etc. = normalzied parameters
Greek Letters = bed void fraction @ = intesification factor r = single-sphere collection rate, s-l 7 = total bed collection efficiency CY
= velocity potential, m2s-l s-l 4 = angle defining edge of step velocity profile L i t e r a t u r e Cited
~
Received for review May 6, 1982 Accepted August 25, 1983
This work was performed under the U S . Department of Energy Contract No. DE AC01-79-ET15490.
Crystallization of Calcium Carbonate Accompanying Chemical Absorption Hideharu Yagi, Akira Iwazawa, Rikio Sonobe, Toshinao Matsubara, and Haruo Hikila +
Department of Chemical Engineering, University of Osaka Prefecture, 4-804 Mozu-Umemachi, Sakai, Osaka, 59 I , Japan
Crystallization of CaCO, accompanying chemical absorption of COPas a single gas and as a mixture with SO, into aqueous solutions of Ca(OH), was studied in a stirred vessel with a flat gas-liquid interface. The nucleation and growth rates determined by CSD analysis of crystals formed in a MSMPR crystallizer were related to several operating conditions. The type and shape of the crystals varied with the concentration of Ca(OH),, but their mean size was only slightly influenced by the concentration of Ca(OH), and by the mean residence time of suspension. The presence of SO, considerably reduced the growth rate of CaCO, crystals under the conditions of a slight decrease in the rate of COPabsorption.
Introduction
The gas-liquid reaction which produces a sparingly soluble material is an important process in the chemical, electronic, and metallugical industries. Well-known examples of its large-scale use include wet gas desulfurization, the production of calcium carbonate as a rubber filler, and ferrite production by the goethite method. The control of crystal size and shape is important for the subsequent separation process (Sohnel and Matejckova, 1981) and the quality of the product. It is probable, though it has not been confirmed experimentally, that the rate of gas absorption influences, and is influenced by, the habit of 0196-4313/84/1023-0 153$01.50/0
crystals produced by the gas-liquid reaction. In spite of its importance, little information is available on the characteristics of this process. The present work deals with the crystallization of calcium carbonate, CaC03, produced by the chemical absorption of a single or mixed gas in an aqueous solution of calcium hydroxide, Ca(OH),. Although many theoretical and experimental works on gas absorption have been reported and CaC0, is known to form different crystal varieties in different circumstances, the crystallization kinetics and the behavior of CaCO, crystals produced by this process have not been studied. A stirred vessel with a flat 0 1984 American Chemical Society
