Particle Collection and Pressure Drop in Venturi Scrubbers Richard H. Boll The Babcock & Wilcox Company, Alliance Research Center, Alliance, Ohio 44601
An a priori mathematical model of venturi scrubber performance is developed and compared with experimental data on pressure drop and particle collection. It comprises simultaneous differential equations of drop motion, momentum exchange, and particle impaction on drops. These are readily integrated by computer for the whole venturi, given its configuration and operating conditions. Provided liquor distribution is reasonably uniform, pressure drop prediction i s quite good for a wide variety of venturi sizes and shapes. Accuracy of prediction of particle collection i s only fair. Discrepancies are thought to be due to either maldistribution of spray liquor or condensation of water vapor. Thus, provided these conditions are avoided, the model can be used to optimize design and operating conditions for specific applications.
T h e venturi scrubber has been used for roughly 20 years to collect particulate material of one kind or another. I t s advantages include (1) high efficiency for relatively small particles, (2) low first cost, and (3) low maintenance costs. The main cost is for operating fans to overcome the venturi’s relatively high pressure drop. If this could be lowered without affecting particle collection, the cost of pollution abatement might be correspondingly reduced. However, due to the experimental difficulties and the multiplicity of design and operating variables, i t is unlikely that this desirable result ail1 be achieved without fundamental understanding of the operation of this piece of equipment. Various investigators (Uoucher, 1953; Feild, 1952; Gieseke, 1963; Johnstone, et al., 1954) have inquired into the mechanism of particulate collection in venturis, and it seems generally agreed that inertial impaction is the only important one. Inertial impaction, in turn, has been the subject of several theoretical and experimental studies. Too, the aerodynamic forces acting on spheres (drops) in a n air stream have been studied rather extensively. Thus, i t is possible to predict the relationship between drop velocity and position within a venturi if the drop size is known. Finally, the drop sizes produced when a liquid is atomized in a high-velocity gas stream have been measured and correlated in several investigations. The literature contains a few attempts to combine this basic information into a theory of venturi-scrubber performance (Calvert, 1968; Feild, 1952; Gieseke, 1963; Kristal, et al., 1957). Pressure drop is included in two instances (Calvert, 1968; Gieseke, 1963), but in no case is consideration extended beyond the venturi throat. Moreover, simplifying approximations are often used to describe aerodynamic forces on drops and/or particle impaction on drops. These approximations are, a t best, of questionable necessity. The main purpose of the present study was to construct and test a mathematical model of venturi scrubber performanceparticle collection and pressure drop-that applies to the whole venturi including entrance nozzle, throat, and diffuser. In other words, this paper develops a theory that can be used to study the effects of changing venturi configuration as well as operating conditions and particle size. Avoidance of needless approuimations, which can becloud interpretation of results, was a secondary purpose. 40 Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
Basic Information
Obviously, a mathematical model can be no more accurate than the underlying correlations upon which i t is based. Since there is some latitude in selecting these, i t is appropriate to consider the reasons for the present selections. Drag Coefficient. Drag coefficient relates the force on a body (drop) to t h e velocity difference between the body and the gas
C,
FD
(~Dd2/4)P~(vg
Vd2/%
(1)
Figure 1 shows drag coefficient data for spheres according to various investigators (Crowe, 1961; Ingebo, 1956; Lapple, 1950; Lapple and Shepherd, 1940; Torobin and Gauvin, 1960). The drop Reynolds number range above about five hundred is not too important because i t is only rarely encountered in venturi scrubbers. (Large drop diameters and high velocities between gas and drop are mutually exclusive on account of drop instability and breakup.) The range below about NR, = 10, too, is not very important because little particle collection or momentum exchange occurs here. The Lapple and Shepherd (1940) curve in Figure 1 actually represents the evaluated mean of several sets of experimental data. It applies to single spheres in steady flow in the absence of free stream turbulence. It is sometimes referred to as the “standard curve.” Ingelso’s (1956) data are of particular interest because some of them were obtained while atomizing liquids in a high-velocity air stream in a manner similar to atomizing in venturi scrubbers. However, he obtained substantially the same results with solid spheres, which implies that the aerodynamic behavior of drops in this size range is indistinguishable from that of solid spheres. Torobin and Gauvin’s (1960) data (Figure I) show the effect of free stream turbulence. The numbers on the curves are relative turbulent intensities, root mean square axial velocity perturbation divided by gas velocity relative to the sphere. The effect of free stream turbulence is mainly to lower the ~ However, only with estraordrag over a limited N R range. dinarily high free stream turbulence does a significant effect occur in the NR, range below about 700. Thus, free stream turbulence seems unlikely to be the cause of Ingebo’s low C D
IO0
10
"0
1.0
0 01
LD
01
10
60
SEPARATION NUMBER
Figure 2. Target efficiency results of various investigators
0.I 0:4
L0.25
0.0 I I
10
100
1000
10,000
NRe
Figure 1 . Drag coefficient data for spheres
values, and i t will only rarely be of significance in venturi scrubbers. Crowe's (1961) data (Figure 1) were obtained using solid spherical particles more or less uniformly distributed across the entire cross section of a shock tube. They scatter about +20% about a mean value that is approsimately 12% above the standard curve. Thus, they confirm t h a t the standard curve applies for a multiplicity of spheres with possibly a small (12%) correction for sphere-to-sphere interaction. This leaves the choice of CD correlation between the standard curve and Ingebo's. I n light of Crowe's data, we believe the standard curve is the better choice for the following reason. Ingebo injected his particles or drops as a small concentrated stream (1/4-in.diameter at most) into the center of a much larger duct (&in. diameter). Thus, the high central concentration of slowly moving drops or particles would have much the same effect as a stationary porous body in the duct. Consequently, some of the gas must have flowed around the mass of drops rather than between them, reducing the actual gas velocity in the vicinity of the individual drops. This would have lowered the actual drop acceleration and reduced the apparent drag coefficient, accounting for the low CD values. This effect was not evident in Crow's data because he distributed his particles uniformly across the flow conduit before flow was started. Target Efficiency. A drop moving through a gas will sweep out a cylindrical volume equal to t h e product of drop cross sectional area, relative velocity between gas a n d drop, and time. However, not all of t h e particles in this gas volume will necessarily impinge upon and (presumably) be collected by t h e drop. As t h e gas flows around t h e drop, t h e smaller particles in t h e outer region of t h e cylinder will be carried around t h e drop. This effect is described by the target efficiency, which is the ratio of the number of particles actually impinging on the drop to the total number in the gas volume swept out. I n principle, target efficiency depends upon three parameters. However, for N R ~ below about 2000, the flow around the particle is alw-ays viscous and the number of important parameters reduces to 2. These may be taken to be N Rand ~ the separation number, $.
x
dx
Figure 3. Apparatus concept
Cc in eq 2 is the Cunningham correction factor on Stokes' law (Ranz and Wong, 1952), which should be included for D, < 1 P.
Figure 2 shows the target efficiency results of several investigators (Dorsch, et al., 1955; Herne, 1960; Langmuir and Blodgett, 1946; Ranz, 1956; Ranz and Wong, 1952; Walton and Woolcock, 1960) for inertial impaction on spheres. Theoretical calculations are feasible only for very high and very low drop Reynolds numbers. It will be noted that the three calculations (Dorsch, 1955; Herne, 1960; Langmuir and Blodgett, 1946) for potential flow and the Walton and Woolcock (1960) experimental data agree quite well in spite of the fact that Walton and Woolcock's A Y ~values e often fell substantially below the criterion for potent'ial flow. I n fact, we take this agreement as justification for selection of one of the potential flow calculations, the Dorsch curve, as representative of venturi scrubbers, wherein the intermediate drop Reynolds number range, 10 < N R < ~ 500, is the important one. The position of the Ranz and Wong (1952) data in Figure 2 attests to the difficulty of measuring target efficiency experimentally and suggests the magnitude of the error. t h a t may be involved in the selection of the Dorsch curve. Drop Size. Probably the most widely quoted correlation of drop sizes produced by venturi-type atomizers is t h a t of Nukiyama and Tanasawa (1940) (see also Xlarshall and Friedman, 1950). Their empirical equation gives the surface-to-volume mean drop size as a function of relat'ive velocity bet'ween gas and liquid, flow ratio of liquid to gas, and liquid properties such as density, viscosity, and surface tension. S o t so widely quoted are the paramet'errange limitations of the original Nukiyama-Tanasawa (S-T) work and the degree to which their results are confirmed by subsequent workers. Table I compares the results of Nukiyama and Tanasawa with those of several subsequent investigators. It will be noted Ind. Eng. Chem. Fundam., Vol. 12, No. 1 , 1973
41
Table I. Comparison
of Results of Various Investigators with Those of N u k i y a m a and T a n a s a w a
N-T 1940
Atomizing velocity ft/sec Liquor-to-gas ratio, ga1/1000 ft3
lewis, et al.,
lewis, ef af.,
Marshall,
1948
1948
1954
240-750
700
1000
0.6-7.5
1-13
1.8-3.5
1-30
9
6.3
30-73
34
0.8-1.2 -0.15 Air
K-2
Base
25-100
Weirs and Worsha rn,
Hrubecky,
1958
1959
Typical commercial venturi scrubber
187-522
300-900
200-1000
150-350
...
1.3-3.8
0.03-1.0
5-30
9
1.0
3.2-11
If
34
30
70
18-22
70
1.0
1.0
0.8
1.0
0.8
1.0
0.1
3.34 Hot combust. gas
2.0 Air
0.13-0.45 Air
6.0 Air
12 Cold combust. gas
io-100
100-150
85-120
135-200
Liquid viscosity,
CP
Liquid surface tension, dyn/cm Liquid density, g/cm3 Gas flow dimension, in. Gas
% ' of N-T drop size
that the later works shoiv mean drop sizes ranging from 25 to 2007& of those predicted by the X-T equation, which provides a rough confirmation of the N-T equation and extends the original parameter range somewhat. All of the results in Table I apply to liquid injection either parallel or opposite to the gas flow. Hrubecky (1958) observed drop sizes approximately txice as large for liquid injection across the gas flow. However, as Nukiyama and Tanasawa pointed out, it is difficult to obtain valid data for this case with small apparatus because the liquid tends to overshoot the gas stream or else impinge on t'he air-conduit wall. The present development uses the N- T equation to predict mean drop size. HoIvever, it is also evident from Table I that this equation has not been confirmed in portions of the parameter ranges of int'erest and, even where i t has been tested, its predicted drop size is subject to a n uncert'ainty of about a factor of 2. Mathematical Model
Consider the apparatus of Figure 3. The compressibility of the gas is neglected. The mass flow rate of gas is Xg. Liquid is injected a t a n arbitrary point, station 1, a t a flow rate, -Tld. The liquid is assumed to be atomized immediately and distributed uniformly across the venturi. I t may have a n initial velocity, 2)dl, that is different from zero if the liquid leaving the liquid nozzles has an L component of velocity. The spatial concentration of liquid drops is nd. This varies with position according to the drop velocity. Particles, however, are assumed to be imbedded in the gas, except during actual capture by a drop. Their concentration is np. For simplicity, we assume that all liquid drops are of the same diameter, D d .Newton's 1alv of motion for these drops is (3) Combining this with eq I , we have
However, to preserve the proper sign of dvd/dt and avoid a potential problem a t S R 0, ~ C-D ~ + m , it is convenient to de42
Ind. Eng. Chem. Fundam., Vol.
12,No. 1, 1973
...
fine a new drag function
I n terms of CDhT,eq 4 becomes
(The equations marked with an asterisk are the final equations of the model.) Note that both CDN and N R are ~ considered to be positive even though (0, - ud) may be negative. The value of cd is, of course, simply the time integral of drop acceleration
( 7 )* , Neglecting mass exchange between gas and liquid and (for the moment) wall friction, a momentum balance around a control volume of differential length, Figure 3, takes the form
The gas and liquid mass flows can be written
M g= "Apg
(9)
m-lfg
(10)
alfd =
Equation 10 is the defining equation for m, the liquor-to-gas flow ratio on a mass basis. Combining eq 8,9, and 10 gives
- _dp___vodvo _ _+ Pv
sc
2/8d21d
(11)
sc
Observation of venturis with a transparent \Val1 invariably reveals a film of liquid on the mall that is moving slowly compared with the gas velocity. Apparently, liquid drops are continually deposited on and reentrained from this film. Thus, there is a momentum loss to the wall that is over and above that due to ordinary friction. On the other hand, wall friction with entrained solid particles is usually less than that predicted by single-phase-flow correlations even using the mixture density as the fluid density (Eoothroyd, 1966). Consequently,
it is not unreasonable t o estimate wall friction in venturi scrubbers according to single-phase flow with fluid density equal to (m 1 ) ~With ~ . this addition, eq 11becomes
+
-dp - %dVO I mu0dud + (m Bo
P&!
+ l)fu,2dx 29D,
so
(12)
For convenience in integrating eq 12, we express position in the venturi in terms of drop velocity and drop time of flight
clz
=
uddt; x = XI
+
1'
uddt
(13)*
The integrated form of eq 12, then, becomes
Equation 14 is valid, of course, only downstream of the point of liquid injection. Equating the rate of loss of particles from the gas stream to the rate of collection of particles within a control volume of length dz, we have
If we assume that the net rate of transfer of drops to the venturi wall is zero, and also that there is negligible drop coalescence within the venturi, then the numbers rate of drop flow must he given by the liquor flow rate and also by the drop concentration and velocity. Thus
Combining eq 15 and 16 to eliminate nd and simplifying
-dn, - l . h m & o n,
Da~dn,
vdl
dx
(17)
If we define a "number of collection units," No, by N , d -In n2 ~
(18)
%?JU
and combine eq 13,17, and 18 and integrate, we obtain
Finally, for ease of numerical calculation, we note that the separation number, $, upon which qt depends, can be viewed as the product of the drop Reynolds number, NE., and a new number, N ,
Equations 6,14, and 19 are the main equations of the present mathematical model. Equations 7, 13, 20, and the definition of drop Reynolds number are obvious but mathematically necessary adjuncts. Although this set of equations has been derived under the tacit assumption that the drops always move downstream with the gas, the equations are essentially correct even when the liquor is injected in the upstream direction. A slight exception occurs in the region between the point of liquor injection (station 1 in Figure 3) and the point a t which the
Figure 4. Prototype venturi nozzle and throot
drops stop and start to move downstream (between stations 0 and 1, Figure 3). Over the length of this region, the wall friction term in eq 14 has negative and positive components according to a sign of vd. These cancel, making the gas density rather than the gas-liquid mixed density the governing factor. However, since the length of duct involved here is very short, the error is very small. Moreover. eq 19 gives two values of N , in this region, neither of which is correct. However, once the drops have moved downstream beyond the injection point, eq 19 gives a single correct value of N,. The mathematical model was programmed for both digital and analog computers. I n both instances CDN was fitted as a function of NEe by a series of linear approximations that reproduced the curve within 1% over the range 0 5 NR* 51000. Similarly, nr was fitted within 1% as a function of .\/$ over the entire range of 0 5 4 5 m . Equation 6 is the key to programming for either machine. Starting from known initial values of v,, vt, and x a t t = 0, one computes a value of N,, finds a value of, ,C from the function fit,and computes drop acceleration according to eq 6. Integration with respect to time, step-by-step or continuous depending upon the machine, then gives v, and x, eq 7 and 13. From x, in turn, one obtains vo according to the venturi configuration. Simultaneously with the integration of eq 6, one integrates eq 14 and 19. T o obtain N , for various particle sizes, one can either repeat the integrations for different N , values (analog computer) or simultaneously integrate several eq 19, each corresponding to a different value of N,. The computation is stopped, of course, when x equals or slightly exceeds the position of the end of the diffuser. Use of the analog machine was particularly convenient when i t was desired to produce plots of pressure drop or particle collection as functions of position in the venturi. The digital machine was more convenient when only overall pressure drop and particle collection was desired. Prediction of Pressure Drop
Pressure-drop profiles were measured in a "prototype" venturi, Figure 4. The cross-sectional dimensions of this venturi a t the throat were 14 X 12 in. A full-sized venturi for Ind. Eng. Chem. Fundom., Vel. 12, No. 1, 1973 43
2.0
I
nXmL POS,T,ON.",I,
Figure 5. Run 53
0.2
THROAT I
I
I
I
I
I
I
I
I
Figure 6, Run 5 3
the pulp and paper or steel industries might have a corresponding throat cross-section dimension of 14 x 120 in. Thus, the prototype venturi represented a 1-ft slice taken out of an otherwise full-scale venturi. Thirteen pressure taps were located along its 15-ft length in the flow direction. One-hundred forty-five air-water tests were made with the prototype venturi. These covered the following ranges of operating variables: liquor-to-gas ratio, 0-12 gaI/(lOOO fts); throat velocity, 150-300 %/see; liquor nozzle angle, 20' downstream to 40' upstream; liquor injection point, 5'/r in. to 156/4 in. above throat; liquor nozzles, 2 sizes. I n addition to the pressure-drop data, photographs were made to record the uniformity of liquor coverage of the venturi throat. The runs with no liquor injection yielded a mean experimental friction factor (Moody) of 0.020. At a typical venturi Reynolds number of 900,000, this corresponds t o a relative roughness of 0.001, or an absolute roughness of about 0.02 in., which seems reasonable for the rusted mild steel construction. However, a value o f f = 0.027 was used in analyzing subsequent data because it seemed t o give a slightly better fit in the shape of the pressuredrop curves. 44 Ind. Eng. Chem. Fundom., Vol. 12. No. 1, 1973
Figure
8. Run 123
The data with liquid injection were analyzed by feeding the exact parameter values for each run into the analog computer and plotting out a predicted pressure-drop profile. This was expressed in terms of the number of throat velocity heads, N Y * d 2 ( - A ~ ) g d ( p ~ u 2 *The ) . experimental data were added to these plots. I n computing throat velocity and gas density, the volume of water vapor produced in saturating the incoming air was added to the air flow on the assumption that adiabatic saturation would occur immediately on liquid injection. This adjustment amounted t o less than 3% of the mass flow of either stream, although i t might reach 10% in commercial practice. The plots and the corresponding photographs were then studied t o discern any trends of quality of prediction with operating variables.
28
.
o,zt I dTH,,,,, , , , , , 0
I
-3-2
-1
0
I
2
3
4
5
6
7
, , , 8
9
10
I1
AXlAL POSlilON,r,ft
Figure
9. Run 45
- 3 - 2
-I
0
I
2
3
4
5
6
7
8
9
1011
AXIAL POSITION.x.Il
Figure 1 1. Run 42
k
Figure 12. Run 42
I n general, the predicted pressure-drop profiles corresponded quite well in shape and magnitude with the measured ones, provided the liquor distribution was uniform across the venturi. This is illustrated in Figures 5-12. I n fact, the only significant trend over the entire range of experimental variables seemed to be a slight tendency t o underestimate A p a t low values of L and overestimate a t high L values. This is further illustrated in Figure 13, which shows some of the overall pressure-drop data and an average curve for the predicted Ap’s. Possibly this trend is due to a slight maldistribution of liquor, where Figures &E represent extremes. Additional overall pressure-drop data were obtained from pilot-scale venturis and from the literature. Figure 14 shows the range of venturi sizes and shapes. Figures 15, 16, and 17 compare the observed pressure drops with the corresponding predictions. It will be noted that Brink and Contant’s (1958) data for 90 spray jets, Figure 15, are predicted within about
=t5y0,whereas the 45-jet data fall about 10% below the theoretical curve. This could be due to slight overpenetration in the case of the smaller number of jets. Our own pilot data appear to scatter somewhat more (Figure 16). However, the actual scatter is not so great as it appears (about *ZO%) because the theoretical prediction is not actually a narrow curve, but a series of points which tend to fall off of the average curve in the same direction as the corresponding experimental points. Ekman and Johnstone’s (1951) data, Figure 17, are mostly predicted within & 12%, but the three points a t the highest L’s are about 25% below the prediction. Again, this might be due to maldistribution of liquor a t the higher liquorto-gas ratios. From these comparisons, Figures 13, 15, 16, and 17, it is concluded that the present model predicts overall pressure drop within *20’7,, for a wide range of venturi sizes and configurations. Figure 18, which compares the theoretical Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
45
2.6
2.2
THEORY (AVERAGE1
2.c
2.4
DlSTRl8UTlON OF LIQUOR
1.8 2.2
0
0
1.6
z
2.0
aW
.
=
0
z
THEORY (AVERAGE)
1.4
$
3 0 v)
1.8
U
1.2
y1
A
Y
a
4
1.6
W
&
0
1.0
1.4
0 : 3/4" NOZZLES
0.8
0: 1 / 2 " N O Z Z L E S
;90 SPRAY JETS
NOTE. DATA POINTS FOR SPRAYING 2 0 4 1.2
UPWARD AT MID E L E V A T I O N
0.6
0.4
x
I
I
I
I
1
2
4
6
8
IO
12
0
0
= 4 5 SPRAY JETS
1.0
14
6
8
10
L,pol / ( I O 0 0 cu.f I.)
Figure 1 3. Comparison of "prototype" pressure-drop data with theory
L,gal/mcf
4.6 4.2
t1 I
BRINK 8 CONTANT
0 0
0
0 6 x 2 VENTURI
A
/
3 x 1-112 VENTURI
PILOTS
00
11875'' D I A
a
JOHNSTONE
Figure 14. Size comparison of venturis
predictions, serves to show t h a t there is actually a difference in pressure drop according to size and/or configuration. T h a t is, the Brink and Contant pressure drops are about 50% higher than Ekman and Johnstone's, and the difference is correctly predicted by the model. The present pressure-drop predictions may be compared with Gieseke's (1963) prediction of pressure drop in a venturi throat of 11/4-in.diameter and 30411. length. His pressure drop equation was substantially the same as the present one. However, he neglected wall friction but included mass transfer from liquid to gas. His theoretical curve is consistently about 15y0below his experimental data, which is probably due to omitting wall friction. On the other hand, neglect of liquid vaporization in our model may seem a n oversimplification. This is not the case, however, provided throat velocity and density are adjusted for the added water vapor. In the first place, the mass of vapor46
0
0 6 x 1-1/2 VENTURI
3"or 6"BREADTH
EKMAN
Ind. Eng. Chem. Fundom., Vol. 12, No. 1 , 1973
14
Figure 15. Comparison of Brink and Contant pressuredrop data with theory
"PROTOTYPE"
1
I.?
/
/
0
THEORY (AVERAGE)
A
i 0.6
I
I
1
1
I
I
4
8
I2
16
20
24
L , gal /mcf
Figure 16. Comparison of pilot pressure-drop data with theory
1.2 r
2.0 (DATA FOR RALNAL OU1 THEORY
1.0
0.8
'
.
.c
z a-
g w
0.6
'
Dp
0.8~
U
YI 3
U Y
0.4
.
5- 8.8 gal/mcf
0
v 4 2 = 2 I9 -486 f t /sec
0.2
'
0.05
(DATA FOR R A D I A L OUTWARD INJECTION)
h I .O
0.1
6.0
( N c ) THEOR
2
0
6
4
8
Figure 19. Comparison of Ekman and Johnstone particlecollection data with theory
10
,
L gal / Il o o o c u f I )
Figure 17. Comparison of Ekman and Johnstone's pressuredrop data with theory L = 10.8gollmcf
I
z
:i
vo2
:2 1 8 f t / s e c
5
3
BRINK 8 CONTANT
1.0
I
I
I
I
I
I
1
1
0.1
1
1
I
I
1.0
4
and
Contant grade-
Dp, MICRONS
Figure 20. Comparison of Brink efficiency data with theory
E K M A N 8 JOHNSTONE
4
R
12
16
20
24
L,go1/(1000cu i t 1
Figure 1 8. Comparison of theoretical pressure-drop for several venturis
misrepresenting the velocity profile of the vapor. However, as reflection on eq 11 will reveal, the vapor must' still contribute a n overall pressure drop of one velocity head per pound of liquid vaporized per pound of gas. Since the liquid that is actually liquid causes pressure drop equivalent to 1.0-1.5 velocity heads per pound of liquid per pouiid of gas (Figure 18, a t L/G 'v 9, m = 1.0 lb of liquid/lb of gas), the difference between vapor and liquid can amount to a t most 3370 less pressure drop due to the vapor. In other words, with 10% liquid vaporization, the present model might overestimate pressure drop by a t most 3%. Prediction of Particle Collection
ized water would hardly exceed 10% of that injected unless the inlet gas temperature exceeded 500'F. Such cases are rare in practice. Secondly, t'lie law of coiiservation of momentum applies whether or not the injected water is vaporized. Thus, we can consider that the vaporized water is included in t,he m factor of eq 11. Then, the error due to vaporization resides in
The model was again tested by feeding into the computer the exact venturi configuration and operating conditions for each experimental run. Where polydisperse particles were concerned, eq 19 was evaluated several times for several assumed values of particle diameter. This permitted the construct'ion of a plot' of S, us. part,icle size. Since S, is Ind. Eng. Chem. Fundam., Vol. 12, No. 1 , 1973
47
10
1
CASE I L
-
5.0
0
-
=
0
4lgal/mcf
v 4 2 = 2 4 2 ft/sec
Ap
0
7 1 . 9 " ~ 2 0( E X P )
Ap = 7 2 d H 2 0 (THEORY)
0 0
0
'z
r"
1.0
10
I
- L -
Vg2
1
CASE 2 -
51.7qal/mcf
=
242ft/SCC
Ap = 90.0"H20 ( E X P )
-
Ap
i
8 1 . 3 " H 2 0 (THEORY)
-
-
0.5
0.I
10 .
Dp. M ICRONS
Figure 22. Comparison of Ghtheroth's grade-efficiency data with theory for low Ap
-
1.0-
-
A.
directly related to efficiency, N , = -In (1 - E ) , a plot of AT, vs. D, amounts to a "grade efficiency curve." Figures 19-22 compare theory and experiment for three venturis under a variety of operating conditions. Figure 19 shows Ekman and Johnstone's (1951) data for radial outward injection. They prepared monodisperse dibutyl phthalate aerosols and collected these in venturis of 13/16-in.throat diameter, Figure 14. The variation in N , shown in Figure 19 is due to variation in throat velocity and liquor-to-gas ratio. It will be noted that the experimental number of collection units is proportional to the theoretical, but ranges from about 15% of theoretical for 0.8-p particles to about 35% of theoretical for 1.0-p particles. Figure 20 compares Brink and Contant's (1958) grade efficiency curve with the theoretical. They collected 80% orthophosphoric acid mist with a 3oyOacid spray using 45 spray jets. Their venturi was of commercial size Lvith a 6 X 34 in. throat, Figure 14. Particle size distributions and total loadings were measured before and after the venturi with a cascade impactor. It will be noted from Figure 20 t h a t the N , data parallel the theoretical curve, but fall about 50% below it. Figures 21 and 22 compare Guntheroth's (1966) experimental grade efficiency data with theoretical predictions. H e collected solid paraffin particles in a venturi of 2-in. throat diameter with water injected radially inward in the throat. Particle size distributions were obtained before and after the venturi by electron microscope sizing of thermally precipitated aerosol samples. The data points of Figures 21 and 22 were 48
Ind. Eng. Chem. Fundam., Vol. 12,
No. 1, 1973
calculated from these size distributions together with his data on overall efficiency. It will be noted that experimental N , values are considerably above the theoretical in Figure 21, which applies for conditions giving pressure drops in excess of 72 in. of nater. In Figure 22, where the pressure drop was only about 20 in. of water, the agreement is quite good. I n Figure 21, note that pressure drop predictions are reasonably accurate, but the prediction in Figure 22 is high by an exceptional40%',.This may be due to the fact that Guntheroth did not give experimental pressure-drop data, but only a correlation from which the "experimental" value had to be back calculated. Discussion
The fact that the present model predicts pressure drop fairly accurately in spite of assuming uniform drop size and uniform liquor distribution may be taken as confirmation of the validity of the choice of mean drop size and drop drag coefficient. Too, overall pressure drop is not very sensitive to mean drop size, a fact that may help to explain the present success in this respect. On the other hand, the accuracy of prediction of particle collection leaves something to be desired. There are several plausible reasons for overestimating S,.Perhaps, as Calvert (1968) hypothesizes, the liquor attains an appreciable velocity before it breaks up into drops of final size. Thus, a n appreciable portion of its sweeping act,ion occurs vihile it is still in t'he form of large, ineffective globs. -ilternatively, maldistribut'ion of liquor could be the main effect. For example, consider a venturi with operating conditions and particle size that would produce an N , value of 10 for uniform liquor distribution. Now suppose there is maldistribution such that 25Yc of the liquor is removed from one half of the venturi and placed in the other half. Because N, is proportional to the liquor-to-gas ratio, the lean half of the venturi will now have an ATc of 5 ( E = 99.33%) and the rich half an A', of 15 ( E = 99.99997%) ,
for a n average whole-venturi efficiency of 99.6?y0 or an N , of 5.7. Further, some of the overestimation of A7, could b e due to underestimating mean drop size, since N, is much more sensitive to drop size than is pressure drop. However, none of t’hesefactors can explain underestimating the experimental efficiency, Figure 21. T o be sure, using a drop size smaller than the Xukiyama-Tanasawa size raises the N , values; but the best that can be attained by this tact’ic is experimental and theoretical X, curves crossing each other at a pronounced angle. Rather, lve believe the explanation in these two cases is probably water vapor condensation in the venturi. For example, a n overall pressure drop of 72 in. of water (Figure 21, case 1) implies a n adiabatic isentropic pressure drop of about’ this magnitude a t the exit of the venturi throat. Such a n expansion corresponds to a temperature drop of 29°F. Giint’herothheated his inlet air to 86°F (30”C), making 57°F the minimum air temperature in the venturi in case 1. Thus, water vapor condensation could have occurred in cases 1 and 2 (Figure 21) in spite of his precaution if the experiments were run in the summertime when dewpoints exceeding 57’F are common. The condensation would, of course, enlarge the particles, markedly improving impaction efficiency. Finally, the fact t h a t Guntheroth’s case 3 (Figure 22) agrees very !vel1 with the present theory suggests t h a t it is sometimes possible to achieve theoretical particle collect’ion in the absence of water vapor condensation. Conclusions
The present mathematical model, eq 6 , 7 , 13, 14, 19, and 20, is readily programmed for automatic computation. When evaluated using the standard curve for drag coefficients and the Kukiyama-Tanasawa equation for mean drop size, i t predicts pressure drop fairly accurately for a wide range of venturi sizes, configurations, and operating conditions. This is especially true if reasonably uniform liquor distribution is obtained. When evaluated using target efficiencies for potential flow (Dorsch, 1955), both under- and overprediction of actual grade efficiency curves occur. Howerer, this may be due to maldistributioii of liquor on the one hand and water vapor coridensatiori on the other. Inaccuracy of mean drop size, too, may be involved to some extent. Thus, the available data do riot contradict the model provided maldistribution and condensatioii are avoided. Xccordingly, the model can be used for study of optimum design and operating conditions in specific applications. Experimeiital coiifirmation of particle-collection predictions is, however, still in order. .\dditional data are needed on actual drop sizes and gradeefficiency curves in commercial-scale venturis. By deliberately producing water rapor condensation, efficient collection of even the smallest particles may be possible with a reasonable, although fairly high. pressure drop. Acknowledgment
The pressure drop profiles and photographs of throat coverage in the “prototype” venturi Ivere obtained by C. A . Leemaii. IT-. Downs obtained the pressure drop data on the “pilot” venturis. Nomenclature
-1
CC
= =
cros*-vxtioiial area of \-enturi, ft* C‘uiiningham correction factor to Stokes’ law for particle +lip betn een gas molecules, dimeiisioiiless
CD CDN
’
=
De
=
D E
= =
f
=
F D
=
gc
=
L m 64 n
s, s, X v h
P
= = =
= =
9
drag coefficient for drops, dimensionless C D N ~= e drag function for drops, dimensionless equivalent diameter of venturi, 4A divided by the flow perimeter, ft diameter, ft efficiency of collecting particles of a given size, dimensionless ;1Ioody friction factor for venturi, dimensionless aerodynamic drag force on drop, lb of force units conversion factor, 32.2 (lb ft)/(sec2 lb of force) liquor-to-gas flow ratio, ga1/(1000 ft3) liquor-to-gas flow ratio, lb of liquorllb of gas mass flow rate, lb/sec numbers concentration in the venturi, (it) -’ -In (1 - E ) = number of collection units, dimensionless ( D p 2 p p C c ) / D d 2 p p= ) particle-size number, dimensionless DdIvg - v d ’ p p / p g = drop Reynolds number, dimensionless - A p / ( p g ~ , 2 2 / 2 g c )= pressure drop expressed as a number of throat velocity heads, dimensionless pressure in the venturi, lb of force/ft2 time of flight of a drop, see velocity, ft/sec position in the venturi, ft
’ =
t
=
v x
= =
GREEKLETTERS qt = target efficiency for impacting particles on drops, dimensionless p = viscosity, lb/(ft see) = density, lb/ft3 P Dp2ppCc(y, - o d ) / l S p p D d ) = separation number, dimensionless A = defined equal to, as distinct from consequentially equal
$
SUBSCRIPTS 0 = a t entrance to venturi (gas) nozzle 1 = a t point of liquor injection 2 = a t endof throat 3 = ateridofdiffuser x = a t generalized position p = pertaining to particles g = pertainingtogas d = pertaining to drop (or liquor) literature Cited
Boothroyd, R. G., Trans. Inst. Chem. Eng. 44, T306 (1966). Boucher, R. 51.G., Znd. Chem. 29,5l (1953). Brink, J. A. Jr., Contant, C. E., Ind. Eng. Chem. 5 0 , 1157 (1958). Calvert, S.,in “Air Pollution,” Vol. 3, Chapter 46, A. C. Stern, Ed., Academic Press, New York, N. Y., 1968. Crowe, C. T., Ph.D. Thesis, University of Michigan, Ann Arbor, PvIich., 1961. Dorsch. R. G.. Saoer. P. G.. Kadow. C. F., .VACA Tech. Y o t e 3 5 8 7 ’ ( ~ 1955 0 ~ j. ’ Ekman, F. O., Johnstone, 11.F., Ind. Eng. Chem. 43, 1358 (1951). Feild, R. B., Ph.D. Thesis, University of Illinois, Urbana, Ill., 14.52
Gieseke, J. A,, Ph.D. Thesis, University of Washington, Seattle, Wash., 1963. Guntheroth, H., Fortschr. Ber. VDI 2. Ser. 3 No. 13 (1966). Herne, H., in “Aerodynamic Capture of Particles,’’ E. G. Richardson, Ed., pp 26-34, Pergamon Press, New York, N. Y., 19An
Hrubecky, H. F., J. Appl. Phys. 29, 572 (1958). Ingebo, R. D., S A C A Tech. Note 3762 (Sept 1956). Johnstone. H. F.. Feild. K.B.. Tassler. 11. C., Ind. Ena. Chem. 46, 1601 (i954j. Kristal, E., Dennis, R., Silverman, L., J . Air Pollut. Contr. Ass. 6,204,(1957). Langmuir, I., Blodgett, K. B., Air Force Technical Report KO. 5418 (Feb 1946). Lapple, C. E., in “Chemical Engineers Handbook,” 3rd ed, pp 1013-1050, J. H. Perry, Ed., hIcGraw-Hill, New York, PI’. Y., 1950. Lapple, C. E., Shepherd, C. B., Ind. Eng. Chem. 32,605 (1940). Lewis, H. C., Edwards, D. G., Goglia, AI. J., Rice, R. I., Smith, L. W., Znd. Chem. 40, 67 (1948). Marshall, W. R., Jr., Chem. Eng. Progr. Symp. Ser. S o . 2 50 (1954). Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
49
AIarshall, W. R., Jr., Friedman, S. J., in “Chemical Engineer’s Handbook,” 3rd ed, J. H. Perry, Ed., Section 13, McGrawHill, Kew York, S . Y., 1950. Kukiyama, S., Tanasawa, Y., Trans. SOC.X e c h . Eng. (Tokyo), 4 , 5 , 6 (1938-1940), translated by E . Hope, Defense Research Board, Department of National Defense (Canada), Ottawa, hlar 18, 1950. Itanz, UT. E., Pa. State Cnzv. Eng. Res. Bull. B-66 (Dec 1956). Hanz, W.E., Wong, J. B., Ind. Eng. Chem. 44, 1371 (1952). Torobin, L. B., Gauvin, W. H., Can. J . Chem. Eng. 38, 189 (1960).
Walton,>W.H., Woolcock, A., in “Aerodynamic Capture of Particles, E. G., Richardson, Ed., pp 129-153, Pergamon Press, Kew York, X. Y., 1960. Weiss, 11.A,, Worsham, C. H., A R S J . 29, 252 (1959). RECEIVED for review September 22, 1971 ACCEPTED October 10, 1972 This paper was presented t o the 69th National AIChE Meeting, Cincinnati, Ohio, May 16, 1971
Ion-Exchange Kinetics for Calcium Radiotracer in a Batch System ling-Chia Huang” and Ku-Yen L i Department of Chemical Engineering, Cheng Kung Lniversity, Tainan, Taiwan, Republic of China
A mathematical model has been proposed for isotopic exchange reactions between ion-exchange resin and ions in liquids. A theoretical equation has been developed to predict the exchange fraction as a function of time. This equation i s derived on the basis of an unsteady-state process taking into account intraparticle diffusion and film diffusion. Experiments were carried out for exchange reactions between calcium chloride solutions and a calcium type ion-exchange resin. It i s found that ihe exchange rate i s controlled mainly b y the intraparticle diffusion if the solution i s vigorously stirred. On the other hand, the film diffusion i s more important than the intraparticle diffusion in the determination of the rate of exchange reaction for cases of low stirring speeds. These experimental results agree well with that predicted b y the theoretical equation proposed in this study.
N u m e r o u s reports (Boyd, et a ~ 1950; , Huang, et al., 1969; Ionescu, et al., 1958; Quershi and Shabbir, 1966; Turse and Rieman, 1961) are available in the lherature concerning heterogeneous isotopic exchange reactions. =2lthough the phenomenon of a n exchange reaction is known to be complex, equations similar to that of Macky (Heitner-Wirguin and Alarkovits, 1963; Lieser, et al., 1965) have been used to study the individual contribution of fast intraparticle diffusion, liquid film diffusion, or surface chemical react’ion. Such a n approach, however, is not always adequate sirice many exchange reactions are controlled by two or more of these mechanisms (Boyd and Soldano, 1953; Huang, et al., 1972; Ionescu, etal., 1958; Quershi and Shabbir, 1966). The present work was undertaken to formulate a mathematical model for unsteady-state heterogeneous isotopic exchange reactions. A theoretical equation has been developed to predict the exchange fraction as a function of dimensionless time taking into account intraparticle diffusion and film diffusion. A number of related problems have been discussed by other jnvestigators, For example, the system of intraparticle diffusion and surface chemical reaction 11-as studied by Ishida and Wen (1968). Smith arid Xmundson (1951) also investigated the probleni of intraparticle diffusion with chemical reaction. Experiments were carried out for isotopic exchange reactions between ion-exchange resins and ions in aqueous solutions. The experimental results will be compared with the theoretical equation proposed in this study. 50
Ind. Eng. Chern. Fundarn., Vol. 12, No. 1 , 1973
Theoretical Development
We consider a heterogeneous exchange reaction in which a liquid solution is immersed in a solid particle through many complicated channels. Although the exchange processes in the particle are complex, it can be assumed that there exists a quasi-homogeneous phase inside the solid particle (Helfferich and Plessel, 1958). Thus, the exchange phenomenon in the particle can be regarded as a diffusion process in a homogeneous phase. Experimental results of Turse and Rieman (1961) indicated that the rate is controlled by chemical reaction if chelates are formed by cations and resin. Kithout formation of chelates, Boyd, et al. (1950), reported that the intraparticle diffusion is a rate-controlling step if the concentration of the solution is high. On the other hand, the rate is determined by the film diffusion if the concentration of the solution is low. Therefore, the mechanisms of the exchange reaction are, in fact, controlled by both the diffusion through the liquid film and the diffusion inside the particle for systems where the chelate complex is not present. This is the case considered in the present study. The intraparticle diffusion is governed by Fick’s second law
where
