A General Method for Predicting Particulate Collection Efficiency of Venturi Scrubbers Kailash C. Goel’’ Advance Engineering Branch, Atomic Energy of Canada limited, Chalk River, Ontario, Canada
Kenneth G. T. Hollands Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada
The differential equations representing the droplet velocity and particulate concentration in a venturi scrubber are written in a form similar to those developed by Boll. The equations lead to a minimum set of dimensionless groups describing the system. General solutions to the dimensionless equations are obtained by numerical integration, and, in the case of the droplet velocity, presented in graphical form. In the case of particulate concentration, the large set of dependent dimensionless variables prevents a complete presentation in graphical form, and consequently a correlation of the numerical results was sought. It was found that the particulate collection efficiency can be correlated against the irreversible pressure loss associated with the “lost work” performed on the droplets by the gas stream. On this basis, charts are presented from which overall collection efficiency can be estimated. Predictions based upon the use of the charts are compared with the experimental data of Calvert et al.
Introduction The venturi scrubber is widely used for the collection of particles from industrial exhausts. It combines a low first cost with easy maintenance and is capable of high efficiency, even for small particles. The basic physical processes at work in the venturi scrubber have been thoroughly reviewed by Boll (1973). The dominant mechanism of particulate collection is inertial impaction of the particles onto the droplets. This inertial impaction is afforded by the high relative velocity between the gas stream and the droplets, particularly near the point of injection of the liquid. The irreversible drag force work resulting from this high relative velocity is largely responsible for the high pressure loss through the device. Current design practice is based on overall empirical correlations, one relating the tot,alnumber of collection units to the overall pressure drop, another relating the overall pressure drop to the liquid flow rate and gas throat velocity. However, it would appear that an optimized scrubber design procedure must be based upon a more fundamental understanding of its operation. Consequently, there have been a number of studies, (Field, 1952; Gieseke, 1963; Calvert, 1970; Boll, 1973), whose purpose was to predict the performance (i.e., pressure loss and efficiency) of this scrubber from first principles. The most recently reported, that due to Boll (1973), is the first completely comprehensive theory in that it (i) models the whole venturi, including the entrance nozzle, the throat and the diffuser sections; and (ii) uses exact formulas to describe the aerodynamic forces and particle impaction on the drops. Agreement between pressure losses measured on practical venturi scrubbers and predicted by this theory is excellent over a wide range of design parameters. Agreement between measured and predicted particle collection efficiencies has not been found so satisfactory; in some cases the theory underpredicts, in others it overpredicts the measured efficiency. This disagreement is put down to maldistribution of liquid on the one hand and the effects of water vapor condensation on the Address correspondence to this author a t Central Nuclear Services, Ontario Hydro, Toronto, Ontario, Canada. 186
particles on the other. Inaccuracy in estimation of mean droplet size is also most probably involved (Boll et al., 1974). Undoubtedly, future improvement both in the model and in venturi design will make the model more exact. Boll’s analysis left the governing equations in their primitive (dimensional) form and general solutions were not given. In an earlier study (Hollands and Goel, 1975), the authors nondimensionalized these governing equations so as to determine the minimum set of dimensionless groups governing the system, numerically integrated the resulting differential equations, and presented the general solutions in the form of graphs. However, this latter treatment was restricted to the liquid velocity and pressure loss and did not include the particulate collision efficiency. The purpose of the present paper is to extend the above treatment so as to include generalized solutions for the particle collection efficiency as well. Analysis The physical model of the venturi scrubber pertinent to the present study is sketched in Figure la. A gas stream a t some initial velocity and with some initial particulate concentration enters some combination of convergent, straight, and divergent ducts. The ducts can be either circular or rectangular in cross section; however, in the case of rectangular ducts, only the duct width is allowed to converge or diverge, the breadth being assumed constant (see Figure 2). An exception is the case of rectangular venturis in which both the breadth and width converge or diverge by equal amounts. For this case the analysis is applicable with j = 2 (see eq 4). The liquid is introduced into one of the ducts (usually the first) in the form of droplets of uniform diameter, uniformly distributed across the cross section of the duct, with some initial velocity (usually zero). The subsequent flow in each of the ducts is assumed one-dimensional and adiabatic, with both droplets and particulate continuing to be uniformly spread across the ducts at any cross section. Both liquid and gas phases are assumed incompressible. The droplet size is assumed invariant with the downstream distance, and the particle velocity is assumed equal to the gas velocity except in the immediate vicinity of the droplets where inertial impaction governs the deposition of the particles onto the droplets. The
Ind. Eng. Chem.. Fundam., Vol. 16, No. 2. 1977 t
7 = (K/(K
.--i (0)
($9)
Figure 1. (a) Physical model of a venturi scrubber. (b) Model for a single duct.
+ 0.7))2
(6)
A more complex expression for 7 than eq 6 was used by Boll (1973). Use of the same complex expression is not practical in the present analysis. Nevertheless the use of eq 6 does not seem an oversimplification considering the many uncertainties that surround the target efficiency (such as the effects of droplet distortion, droplet-droplet interaction, droplet Reynolds number, etc.) and the modification incorporated below. Equation 6 is expressed in terms of the target efficiency at the inlet, 70,and the relative velocity between the phases, ur, as was done in the case of the drag coefficient (Hollands and Goel, 1975). Thus
Use of eq 7, like that of eq 5, ensures that 7 can be given its most accurate value a t the start of the duct where main drag interaction and particle collection usually occurs. Defining for convenience, the number of collection units, N,, as N,=-In( a ) Rectangular Oluct
C
co
eq 3 may be written as
( b ) Circular Duct
Figure 2. Two types of ducts treated in the analysis.
liquid volume fraction at any cross section is assumed to be small. All of these assumptions are implicit to Boll’s model. As a first step in the analysis, a single duct is examined (Figure lb). Gas and liquid enter to known velocities ug0 and u10, respectively, and particulate enters at known concentration, C O . Although sketched as divergent, the duct may in fact be convergent ( p < O), straight ( p = 0), or divergent ( p > 0). The problem is to develop general expressions for the liquid velocity, gas velocity, and particulate concentration at the end of the duct. Provided this can be done, the larger problem of determining the particulate concentration at the end of the combination of ducts shown in Figure l a can clearly be readily handled. The basic equations governing this model have been developed by Boll (1973);rewritten here after some rearrangement they are
_ d u l-- 3 k g Q ( u g - ~ l )Jug dz
4
dc dz
-::=
pi
d
~II
u1
_3_Q i_ v.I_ ug-uiIC 2Qg d
VI
(2)
(3)
Treating for the moment, the single duct, Figure l b , the cross-sectional area A is given by A = Ao(l
+ z tan P/ro)
(4)
The boundary conditions are: a t z = 0 , u l = ulo; ug = u , ~ ; c = CO.
The drag coefficient, CD, in eq 2 is given by the same expression as that used in the previous study (Hollands and Goel, 1975)
CI)= CDO( ~ ~ r o ~ / ~ ~ r ~ ) l ’ z
(5)
where N,, represents the number of collection units in the entire duct. The governing equations are now nondimensionalized using us0 as the characteristic velocity scale and (2/3) ( d / C ~ o(pl/pg) ) u & ~ as the characteristic length scale (see Hollands and Goel, 1975). If in addition, eq 1 , 4 , 5 ,and 7 are substituted into eq 2 and 8, there results the dimensionless set of equations
and
The boundary condition for eq 9 is Vl(0) = VI0
Solution Liquid Velocity. Clearly, eq 9 must be solved for VI before eq 10 can be integrated. An analytical solution of eq 9 does not appear possible in general although one does exist for the special case of S = 0. Consequently, numerical integration has been resorted to and the general solution so obtained is given in the form of graphs in the work of Hollands and Goel (1975) for both circular and rectangular ducts. Figure 3 shows this solution for the special case where Vlo = 0. (To save space in the paper, charts are presented only for circular ducts. However, the subsidiary relations and the application procedure described below are applicable to both circular and rectangular ducts with appropriate values of j . Charts for rectangular ducts are available elsewhere (Goel, 1975; Hollands and Goel, 1975). The special solution given in Figure 3’ is denoted by ulo(Z,S).The general solution denoted by VI,(L,Z,Vlo) can be obtained from this special solution by using the following subsidiary relation
The form of expression chosen for the target efficiency, 7, in eq 3 is that representing the experimental data of Walton and Woolcock (1960) and correlated by Calvert (1970) Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
187
3.0
the device and it may seem resonable>o attempt to correlate the computational results on the same basis. The overall pressure loss is due to the fluid friction on the wall plus forces expended in the accelerating of the fluid. As pointed out before (Hollands and Goel, 1975), the latter contribution may be subdivided into two components-that component actually resulting in an increase in kinetic energy of the liquid, and a component representing the thermodynamic “lost work” expended in accelerating the droplets. The latter component, called the “irreversible pressure loss”, is defined as
20
10
>
0-
075
050
025
which is written in dimensionless form as 2
z
3
4
I 6
8
IO’
2
3 4 5 6 8
Io‘
Figure 3. Chart for obtaining Vl0for a circular duct.
I
I
N>L \
3
dPi dZ
dV dZ
-= 2MVr A
where dVl/dZ is given by eq 9. Generalized charts for evaluating Pi will be presented shortly. Specific Number of Collection Units, n,. Physically the irreversible pressure loss is caused by the drag force exerted by the gas stream on the droplets due to the relative velocity between the phases. This relative velocity is also the driving force to cause particulate collection by inertial impaction. Thus the irreversible pressure loss is the unavoidable penalty one has to pay as a result of particulate collection, even though the apparent overall pressure loss in a venturi scrubber includes contributions due to other factors as well-namely, wall friction, which in principal can be avoided, and changes in kinetic energies of gas and liquid, which in principal can be recovered. Now the (local) specific number of collection units at any point in a venturi scrubber is defined as the number of collection units obtained in a differential element at that point per unit irreversible pressure loss across the differential element. It can be shown to be given by (see Appendix A)
n, =
I
9
1 CD - p g 2
(13) vr2
In order to nondimensionalize eq 13, we have at the inlet of a duct
nco =
I
“4
0
Figure 4. Chart for obtaining L H for a circular duct.
where L His a dimensionless “hypothetical length” parameter which is plotted as a function of Vlo and S in Figure 4. Details regarding eq 11and Figure 4 can be found in Goel (1975) and Hollands and Goel (1975). Number of Collection Units. With the above mentioned solution for VI,eq 10 can be integrated directly by numerical integration to determine N,, for any L. However, N, (or more precisely, the group (N,,/M) (CDO/~O)) is a function of four independent variables-namely L , S, Vlo, and 90, so that it is not possible to give a general representation of its full dependence without resorting to voluminous tables. Consequently, an attempt has been made, with some success, to correlate the numerical results so as to reduce the number of independent variables. For this purpose, the following definitions of “Irreversible Pressure Loss” and “Specific Number of Collection Units” have been found useful. Irreversible Pressure Loss. As mentioned earlier, empirical studies on venturi scrubbers have tried to correlate the number of collection units with the overall pressure loss across 188
Ind. Eng. Chern., Fundarn., Vol. 16, No. 2, 1977
90
(14) 1 CDO- pg uro2 2 Combining eq 13 and 14 and ,substituting for CD and 9 from eq 5 and 7, respectively, we get
An average specific number of collection units, E,, for any duct is defined as the ratio of total number of collection units in the duct divided by the total irreversible pressure loss across the duct. Thus
ce n, = N Pie
or combining eq 10,12,14,and 16 we may write
where Pieis the dimensionless irreversible pressure loss across the entire duct and can be given by integrating eq 12b from Z = 0 to Z = L. Equation 17 may also be written using eq 15 as
IO‘
k P O
lo-‘
1.0
0.1
0.0
Figure 6. Correlation of fi, as a function of
IV,l/ v,
v,and
40.
Figure 5. Variation of n,/n,o with 1 V,/V,ol and VO.
Correlation. The key to the correlation is the fact that the ratio nc/nd is roughly equal to ( 1 V,/VfiI)-l (Figure 5). (This can also be shown equivalent to saying that 7 over a wide range of K . This simple form of variation of 7 with K , although very approximate, has been used in the past by some investigators, e.g., Field (1952), to derive means of correlating the experimental data). Substituting nc/nd I VfiO/V,I in eq 18 and using eq 12b, we get
-
“average relative velocity” for a duct, eq 20 can be shown to be expressible as
-
Pie v, = 2MV1,
where
is given by
< L,) (if L > L,)
VI, = VI, - Vlo
n co
J
= 2V1, ‘le
V, dV1
and since V, and dVI are always of the same sign (as can be seen from eq 9), this can also be written as
where
(20)
(if L
- VI, - Vlo
(234 (23b)
and L , is the value of Z it which VI has its “peak” value, Vl,, for the value of S considered. VI, and L , can be obtained by using the broken lines shown in Figure 3 and eq 11.This peak value occurs only for divergent ducts ( S > 0). For other ducts, eq 23a is applicable. Charts for Pi,.The “irreversible pressure loss”, PI,, for a duct is given by simultaneous solution of eq 9 and 12b and is, in general, a function of L , S , and Vlo. However, a special solution when Vlo = 0, denoted by P,O(Z,S)is plotted in Figure 7 from which the general solution, denoted by P,,(L,S, Vlo), may be obtained by use of the following subsidiary relation
Ple(L,s,VlO) = (1 - SLH)-21 x [Pio((L LH)(1 - SLH)’l2, s(1- sLH)-’/’-’) - PF(LH(1 - SLH)”2, s(1- sLH)-’/2-1)]
+
is called the “average relative velocity”. Equation 19 gives im approximate expression for iicand is based upon the assumption that nc/nco= I V,o/V,I. In actual practice, as seen from Figure 5 , nc/ncois a more complex function of I V,/Vfil and 70.Therefore,gne may expect iic/nd also to be some general function of Vr/Vr0 and 70 rather than that given by eq19. This general dependence of iic/nco on T,/Vd and 70is plotted in Figure 6 which can be considered an integrated version of Figure 5 over the length of a duct. Details of the preparation of this graph are given in Appendix B. Figure 6 has been .tested against numerical solution of the full set of equations, i.e., eq 9 and 10 for numerous values of L, S , V~O, and 70for bloth circular and rectangular ducts. Provided 70 < 0.9, it has ‘beenfound that Figure 6 gives values of iicwhich are within 51% of the values obtained by precise numerical integration of‘the full equations. Slightly larger errors (of the order of 20%) may be expected for values of 70 > 0.9. Average Relative Velocity, In order to calculate the
vr.
(24)
where L H is plotted as a function of Vlo and S in Figure 4. Details for the derivation of eq 24 are similar to those for eq 11 and can be found elsewhere (Goel, 1975).
Application Rearranging eq 16, the number of collection units in a duct may be given from the following simple design equation
-
-
(Wnco) nco P i e (25) where (iic/nco)is given by Figure 6; n,o is given by c 14; and P,, is given by eq 24 together with Figure 7. The aii’jication Nce =
of eq 25 to calculate the overall particulate collection t riciency of the full venturi scrubber is described in this section. It is assumed that all the dimensions necessary to specify the venturi geometry and the location of the nozzles for injecting the scrubbing liquid are known. In addition, the variInd. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
189
where n is the number of ducts and
20
1
I5 10
0 75 0
are the changes in pressure due to changes in kinetic energies of liquid and gas, respectively. This considerably simplifies the combined procedure of calculating the overall collection efficiency and overall pressure loss in a venturi scrubber.
.05
0 25
0
2
2
Figure 7. Chart for obtaining Pi0 for a circular duct. ables: 61, Qg, pi, pg, PI, pg, u, pp, d,, etc., are also assumed specified. As a first step, the droplet size is calculated by use of an empirical correlation for droplet size produced by pneumatic atomization, such as the Nukiyama-Tanasawa correlation (1940), applied a t the point of liquid injection. Equation 25 is then applied to the first duct in which the two phases flow (e.g., the convergent duct which lies downstream of the point of liquid injection in Figure lb). The gas velocity at the inlet of this duct, ug0 (and, in fact, anywhere along the scrubber), can be determined directly from the application of eq 1,the cross-sectional flow area at any point along the venturi assumed specified. The liquid velocity at the inlet of this duct is assumed zero unless the liquid is injected with some momentum in the axial direction. This fixes the relative velocity at the beginning of the duct, and from this, the initial drag coefficient, CDO, and target efficiency, 70can be determined. The set of dimensionless groups, S and L for this duct can thereby be determined so that the liquid velocity at the end of the duct, Vie, and its irreversible loss, Pi,,can be obtained by use of Figures 3,4, and 7 together with eq 11and 24. Then the average relative velocity, for the duct is calculated by using eq 21-23 and the average specific number of collection units, Ti,/n,o, are read from Figure 6. The total number of collection units, N,,, for the duct is then calculated from eq 25. Having determined the number of collection units for the first duct as above, one then proceeds to find the same for the next consecutive duct. For this purpose, since the inlet liquid velocity at the start of the next duct is the same as that at the end of the preceding duct, the relative velocity at the inlet of this next duct is also fixed. Then the above procedure (excepting the calculation of the drop size which is assumed the same for all the ducts) is applied to this duct to yield ultimately the number of collection units for this duct. After the process has been repeated for all the individual ducts, the total number of collection units for the full venturi scrubber, N,,T, is given by their sums in the individual ducts, so that the overall particulate collection efficiency is readily calculated from the relation
v,,
E = 1 - exp(- N c , ~ )
(26)
Incidentally, it can be shown (see Goel, 1975)that the values of Pi, calculated above for use in calculating the number of collection units for all the ducts can also be used to calculate the pressure loss due to accelerating the fluid by using the following simple relation
190
Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
Comparison with Experimental D a t a Despite its practical importance, accurate data on the particulate collection efficiency of venturi scrubbers are not sufficient to permit an exhaustive comparison with the theory. This is primarily due to the difficulty in making accurate measurements on small scale units and the high cost involved in taking measurements on large equipment. Moreover, the present theory cannot be compared with measurements on small scale units because in such units a considerable fraction of liquid may flow as a film on the walls (which is ineffective for particulate collection) rather than a spray of droplets. The present theory does not account for the liquid films on the wall. However, as the venturi diameter is increased, the fraction of liquid flowing along the walls decreases and the theory becomes more valid for large scale scrubbers which are used in practice. Due to the above difficulty, the present theory could be compared only with the experimental data of Calvert et al. (1972) (Figure 8). These authors took some very careful measurements on a medium size venturi (4 X 12 in. rectangular throat). For the sake of this comparison, the Nukiyama-Tanasawa (1940) equation was used for calculating the drop diameter produced in the venturi; Calvert’s (1970) expression (correlating the experimental data of Walton and Woolcock, 1960) used for 70; and CW was calculated using the “standard curve” due to Lapple and Shepherd (1940). From Figure 8 the agreement between present predictions and the data of Calvert et al. (1972) seems only fair a t best. In addition a trend to ouerpredict t h e efficiencies a t low gas velocities and underpredict a t high gas velocities is apparent. The same trend was also observed by Behie and Beeckman (1973). The reasons causing this trend are not known a t present. One possibility is associated with the use of the Nukiyama-Tanasawa equation for predicting the drop diameter produced by pneumatic atomization. For example, Boll et al. (1974) reported that the Nukiyama-Tanasawa equation underpredicts the drop diameter for venturi scrubbers a t low gas velocities and overpredicts a t high gas velocities. This. implies that the collection efficiency will be overpredicted at low velocities and vice versa since the smaller droplets are more efficient particulate collectors than the larger ones. A reliable method for predicting the average droplet size produced in the venturi is not known a t present. Another reason for the discrepancy between measured and predicted particulate collection efficiencies is the effect of several factors not included in the theory. The important factors are the maldistribution of liquid across the cross section, condensation of water vapor on the particulates causing agglomeration, particulate growth due to hygroscopic effects, liquid film on the walls which is ineffective for particulate collection, etc. Considering the complexities involved in accounting for all or even some of the above mentioned factors as well as the difficulty in taking accurate measurements on particulate collection, the present theory appears suitable for design purposes and for optimization studies of venturi scrubbers. In general, from the comparison with the data of
/
gas containing partlculatea ( c kg/m3)
e % A /
0
-
droplet of diameter d
Figure 9. Diagram to illustrate the specific number of collection units.
spect to the droplet be u,. The differential change in particulate concentration in the gas stream as it flows past the droplet is given by
or " A /
Experimental
E
Figure 8. Comparison of overall collection efficiency predictions with data of Calvert et al. (18172).
Calvert et al. (1972) given above and with the data of other investigators presented by Boll (1973), it may be concluded that the present theory will predict the total number of collection units in a venturi scrubber within about f 5 0 % over a wide range of design and operating variables. The approximate range of variables over which the theory has been compared with the experimental data (including the comparisons given by Boll, 1973) is: throat velocity, 100-250 ft/s (30-80 m/s); liquid flow rate, 0-30 ga1/(1000 ft3 of gas) ( M x 0-3 for air and water); particulate size, 0.1-5 pm; venturi throat dimensions, 1 %in.~ (3 cm) diameter to 6 X 34 in (2.4 X 13.4 cm) rectangular. All tests were performed using air and water. The predictions made by the theory can be improved by using a more accurate expression for the drop diameter produced by pneumatic atomization. The charts are independent of the method of calculating the drop diameter. In addition, as mentioned before, the predictions will be more accurate for large-scale venturi scrubbers and particularly for those scrubbers in which the liquid is injected so as to produce a uniform liquid distribution across the venturi throat. Conclusions Charts have been presented (Figures 3 , 4 , 6 , and 7) by use of which the particulate collection efficiency of a venturi scrubber can be estimated. These can be combined with a similar set of charts presented before (Hollands and Goel, 1975) for the pressure loss to yield a complete design method for a venturi scrubber. The development of these charts is based upon an application of first principles to the operation of a venturi scrubber. Because of the very large number of variables entering into the performance of a venturi scrubber, for the purpose of design, such an approach is to be preferred over methods based upon empirical correlation of measured performance data. Predictions made by using the charts have been compared with the data of Calvert et al. (1972). Appendix A Specific Number of Collection Units. Consider a unit volume of gas containing the dust particulates (with a concentration of c kg/m3) flowing past a liquid droplet of diameter d (see Figure 9). Let the relative velocity of the gas with re-
Further, because of the relative velocity, the droplet will be accelerated (if u, is positive) or decelerated (if u, is negative) by the gas due to drag forces. Some "lost work" is associated with this acceleration or deceleration of droplet due to the irreversible drag force and is the (actual) penalty one has to pay as a result of particulate collection due to inertial impaction. The irreversible pressure loss for the situation shown in Figure 9 may be given by
The specific number of collection units is now defined as the number of collection units per unit irreversible pressure loss, or
Equation A.l constitutes the fundamental definition of the specific number of collection units. Appendix B Derivation of 70) Relationship. Figure 6 shows the approximate dependence of Ti, on and 70.The derivation of this plot is given in this Appendix. In order to evaluate the integral in eq 17, it is usefu! to change the variable of integration from Pi to where VI is defined as
iic(v,,
v,
where x is a dummy variable for integration. Since V, and dV1 are always of the same sign, eq 12b and B.l may be combined to yield dPi = 2M I V,I IdVll = 2M I V,I dQ1
(B.2)
Substitution of eq B.2 into eq 17 yields
(B.3) The integral in eq B.3 may be evaluated directly once the functional dependence of IV,/V,oI on is specified. Inspection of eq B.l and 9 shows that this dependence is fixed once S and V,O are fixed. Typical variations of I V,/V,0l with Q1 (not to scale) are sketched as solid lines in Figure 10, for different values of S . For S = 0, the relationship is a simple linear one: I V,/V,ol = 1 - Q1. Although the relationships for other values of S are more complicated, it may be possible to represent them by a linear relationship of the form: Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
191
1-
APPROXIMATE
EXACT
t
IVrl
-
"
"t
-
Figure 10. Schematic sketch of exact and approximate J V,J profiles in a duct.
Nomenclature A = local duct cross-sectional area, m2 c = particulate concentration in the gas stream, kg/m3 of gas C, = Cunningham correction factor to Stokes' law for particle slip between gas molecules, dimensionless CD = drag coefficient of droplet, dimensionless d = drop diameter, m d = particle diameter, m I f = overall particulate collection efficiency, defined by eq 26, dimensionless G = constant in eq B.4, dimensionless j = 1 for rectangular ducts, = 2 for circular ducts and for rectangular ducts whose width and breadth converge or diverge by equal amounts, see eq 4, dimensionless K = inertial impaction parameter, dimensionless, = C, pDd p 2 1 u g - u11/9igd 1 = length of duct (see Figure 11, m L = dimensionless length of duct = 1. (3/2) * ( C ~ o / d )(pg/pi) ((u,o
Table I
x
00
0.1 0.2
1.0 0.99 0.98 0.97 0.95
0.3 0.4
R0.5
wn
x
0.6
0.92
0.7
0.88
0.8 0.9
0.83
0.95
0.65
function of Vlo and S
L' = liquid flow rate, ga1/(1000 f t 3 of gas) L , = value of L corresponding to V I =VI,, shown as a locus
0.77
provided the constant G is properly chosen. The approximate profiles described by eq B.4 are shown as dashed lines in Figure 10. Suppose we choose G so that the area under this approximate representation of I Vr/Vrol as a function of VI (dashed lines in Figure 10) is equal to the area under the true relationship (solid lines), G can be shown to be given by
of dotted lines in Figure 3, dimensionless m = mass flow rate, kg/s M = liquid to gas mass flow rate ratio, ml/m,, dimensionless n, = specific number of collection units, see eq 13, Pa-l N , = number of collection units, dimensionless A p = pressure loss in accelerating the fluid through the scrubber, Pa pi = irreversible pressure loss associated with "lost work", given by eq 12a, Pa Pi = dimensionless irreversible pressure loss associated with lost work, = pi/(1/2)pg ugo2, given by eq 12b Q = volumetric flow rate, m3/s r = radius, (for circular ducts) and half width (for rectangular ducts), m Re = drop Reynolds number, dimensionless, = l u g - u l l d
S Combining eq B.3-B.5 and simplifying, we get
5 - =
- ulo)/ueo)1/2
L H = dimensionless parameter, plotted in Figure 4 as a
JVrhiVrO
[l
dx
+ (1 - 4) (x-1
- 1)]2
/
= cfimensionless parameter characterizing slope of the given duct = (tan Plro) * (2/3) ( ~ / C D O(u,o/(ugo ) -UIO))~/* u = velocity, m/s u, = relative velocity, = ug - ul, m/s V = dimensionless velocity = u / u g 0 = dimensionless relative velocity, = V, - VI V1 = J &, 1 dx I, dimensionless Virn = peak value of V Igiven by broken line in Figure 3 V, = an average dimensionless relative velocity defined by eq 20 v r h = hypothetical relative velocity, given by eq B.8, dimensionless x = dummy variable for integration z = axial coordinate, (see Figure l),m 2 = dimensionless axial coordinate = t (3/2) ( C ~ o / d )
vr
no, x-Il2
Pg/F
vrh'vro
x dx
which may be evaluated in closed form giving
n co
(Pg/Pl)
where v,h/vrO
=1
+ 2 ( v r / v r O - 1)
(B.7)
(B.8)
where A is slightly less than unity by an amount which depends upon v0. The variation of X with 70which has been found optimal is given in Table I. Equations B.6 and B.8 combined with Table I yield Figure 6. 192
Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
((ugo
- UlO)/V,O)'/*
.
Greek Symbols
and y o is given by I/& - 1.0. Clearly in this case, ric/nCois a function of 70 and vr/Vro and when E, is calculated by using eq B.6 and B.7 and compared to the exact value obtained by numerical solution of eq 17, the results are in fairly close agreement over a wide range of the parameters L , S, 90, Vie, and j . However, it is found that the agreement is improved if eq B.7 is modified slightly to read. V,h/V,O = 1 + 2X(Vr/Vr0 - 1)
-
/3 = half angle of divergence (see Figure l),deg y = 1 / 4 - 1.0, dimensionless
= target efficiency or inertial impaction efficiency of particles on drops, dimensionless = factor given in Table I, dimensionless p = viscosity, P e s p = density, kg/m3 pp = density of the material of the particulate, kg/m3 a = surface tension between gas and liquid, dyne/cm T = time, s 7
Subscripts e = a t the exit of the duct f = a t the exit of the venturi g = ofthegas i = a t the inlet to the venturi
1 = of theliquid 0 = at the inlet of the duct t = a t the throat of tlhe venturi T = total Superscript 0 = for the special case when
Vlo
is assumed to be zero
Literature Cited Behie, S. W., Beeckmans, J M., Can. J. Chem. Eng., 51, 430 (1973). Boll, R H., Ind. Eng. Chem., Fundam., 12, 40 (1973). Boll, R . H., Flais, L. R., Maurer, P. W., Thompson, W. L., JAPCA, 24, 934 ( 1974). Calvert, S.,AlChEJ., 16, 392 (1970). Calvert, S.,Lundgren, D., Mehta, D. S.,JAPCA, 22, 529 (1972). Field, R. B., Ph.D. Thesis, Uriiversity of Illinois, Urbana, Ill., 1952.
Gieseke, J. A., Ph.D. Thesis, University of Washington, Seattle, Wash., 1963. Goel, K. C.,Ph.D. Thesis, University of Waterloo, Waterloo, Ont.. Canada, 1975. Hollands, K. G. T., Goel, K. C., Ind. Eng. Chern., Fundam., 14, 16 (1975). Lapple, C.E.,Shepherd, C. B., lnd. Eng. Chem., 32 (5), 605 (1940). Nukiyama, S.,Tanasawa, Y., Trans. SOC.Mech. Eng. (Tokyo), 4, 5, 6 (19381940) (translated by E. Hope, Defence Research Board, Department of National Defence Canada, Ottawa, 1950). Walton, W. H., Woolcock, A., in "Aerodynamic Capture of Particles," E. G. Richardson, Ed., Pergamon Press, New York, N.Y., 1960. Received for review September 18,1975 Resubmitted June 16,1976 Accepted November 29,1976 T h i s w o r k was supported financially by a grant a n d fellowship f r o m t h e N a t i o n a l Research Council o f Canada.
Extraction Parametric Pumping with Reversible Reaction Shlgeo Goto' and Masakazu Matsubara Department of Chemical Engineering, Nagoya University, Chikusa, Nagoya, Japan
Liquid-liquid extraction parametric pumping can be applied to a first-order reversible reaction system in order to obtain1 a conversion higher than that for chemical equilibrium. On the assumption of phase equilibrium, staged parametric pumps are investigated both for a discrete transfer model and for a continuous flow model. The efficiencies of extraction parametric pumping with reversible reaction are compared with chemical equilibrium and the effects of each system parameter on the efficiencies are evaluated. The results indicate that parametric pumpling should be quite effective if the system parameters can be chosen Suitably.
Since Wilhelm et a]. (1966) introduced the idea of parametric pumping, this new technique has been chiefly applied to adsorption processes to separate components of a fluid (Sweed and Wilhelm, 1969; Gregory, 1974; Chen et al., 1974). The adsorption parametric pumping seems to be suitable by two main reasons, that is: (1)the adsorbate equilibrium relationship between liquid and solid phases is greatly changed due to temperature change, and (2) the solid adsorbent can easily be held stationary while a liquid can flow past the solid adsorbent in a packed bed. Recently, Wankat (1'973)extended the parametric pumping to liquid-liquid extraction by devising three techniques for holding one liquid phase stationary while pumping the other liquid back and forth past the stationary phase. The first technique is to coat thle stationary liquid phase on an inert solid support. This can be operated in the same fashion as the adsorption parametric pumping. The second is to put both liquids into a continuous contact column which is inverted 180' every half-cycle. When this method was tried experimentally, the moving phase could always flow due to gravity but no separation was achieved because there was considerable axial mixing. The third is to use a staged system such as a staged column without downcomers inverted every halfcycle, a series of mixer-settlers and a horizontal helix. We can expect that the parametric pumping must be much useful for the combined process of separation and reaction because the conversions of the desired products can be greatly improved by selective sDeparationsof reactants and products. Tam and Miyauchi (1973) investigated the adsorption parametric pumping accompanied by a reversible reaction in a packed bed and the calculated results showed that the parametric pumping effect considerably improved the conversion. Also, Apostolopoulos I( 1975) studied the adsorption para-
metric pumping with reversible reaction and developed a "near-equilibrium'' approach. In the previous papers, the present authors studied the reaction with extraction which was often referred to as the extractive reaction (Goto and Matsubara, 1972a,b, 1974). In this paper the extraction parametric pumping introduced by Wankat (1973) is applied to the system accompanied by a first-order reversible reaction to investigate numerically to what extent the conversion can be improved. Only the staged system which is the third technique of Wankat (1973) for holding one liquid stationary is considered because this seems to be easiest for performing experimental work in the future.
Staged Parametric Pump Consider a staged parametric pump in Figure 1operating in the direct mode where the temperature of the system can be changed immediately when the flow direction of the moving phase is changed. The stages are numbered from 1 to N starting at the hot reservoir. The hot reservoir is numbered zero and the cold reservoir N 1. For simplicity, we assume that the dead volumes of the hot and cold reservoirs are negligible. Also, the parametric pump is assumed to start from the hot half of the cycle where the moving phase flows from the cold reservoir to the hot reservoir through N stages while the system is hot. Then, the temperature is changed from hot to cold and the direction of the flow is reversed. Both solvents in the moving and the stationary phases are assumed to be immiscible with each other. The volumes of both phases in each stage, VM and V s are assumed to be constant. The first-order reversible reaction is taking place only in the stationary phase of each stage.
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