sequence of this is that it, in effect, permits one to use infinite dilution activity coefficients obtained from the chromatographic data to predict activity coefficients at finite concentrations through the medium of the group contributions.
GREEKLETTERS
fl
=
an empirical constant in eq 15
y z = liquid phase activity coefficient of component i $ %= x z V z / % c i V z= volume fraction of component i
Nomenclature
A I3 C&HiR HkG
a structural group in the molecules A,B, and hl a structural group in the molecule A,B, a t e r m i n e q 15 = PA - VA/V, = partial residual energy of mixing of cpmponent i = group contribution of group k to AHiR 1 = number of groups of type -1in the molecule A l L = refers to the liquid solvent A i 711. = number of groups of type A in the molecule A,B, n = number of groups of type 13 in the molecule X,B, nk,i = number of groups of type k in molecule i PA = contribution of group A to ln (PIOVY) PB = contribution of group 13 to ln (Pi0VN) Pto = vapor pressure of component i R = gas constant ASiFH = Flory-Huggins partial excess entropy of mixing for component i T = temperature V A = contribution of group A to V i = coiitribution of group 13 t o V i VB V i = liquid molar volume of component i = liquid molar volume of component L VL = chromatographic net retention volume VN = nioles of L in the chromatographic column Tt’ = mole fraction of component. i in t’heliquid phase = gas compressibility factor = group fraction of group in a solution of d and 13 ZA Z k = group fraction of group k in a solution of groups = = =
2
SUPERSCRIPTS
0
*
= =
at infinite dilution in the standard state
literature Cited
Deal, C. H., Ilerr, E. L., Ind. Eng. Chem. 60, 28 (1968). Deal, C. H., Ilerr, E. L., Inst. Chem. Eng. Syrnp. Ser. N o . 3.2 3, 40 (1969). Harris, H. G., Prausnitz, J. PI., J . Chrornatogr. Sci. 7, 685 (1969). Kwantes, A., Rijnders, G., in “Gas Chromatography 1958,” p 125, 11. H. Desty, Ed., Butterworths, London, 1958. Petricek, J. L., M.S.Thesis, The University of Nebraska, Lincoln, Kebr., 1966. Pierotti, C. J., Deal, C. H., Derr, E. L., Ind. Eng. Chem. 51, 95 i l 9 . 3 \ I-
Porter, P. E., Deal, C. H., Stross, F. H., J . Amer. Chem. SOC. 78, 2999 (1956) Scheller, W. A., IND. EXG.CHI~M., FUNDAM. 4 , 459 (1965). Wilson, G. W., Deal, C. H., IND.ENG.CHEM.,FUNDAM. 1, 20 (1962). Y&ngJ’G. C., M.S.Thesis, The University of Nebraska, Lincolii, Yebr., 1967. for review October 26, 1970 ACCEPTED September 14, 1971
ItEcicIveD
Dispersion of Chbrged Particles in a Turbulent Air Stream under Transverse Flow Conditions Shan K. Sunejal and Darsh T. Wasan” Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Ill. 60616
Equations have been developed to predict the diffusion of electrically charged particles in a turbulent air stream within a circular straight porous pipe with uniform transverse flow at the walls. The calculated concentration profiles agree well with those determined experimentally in the absence of electric charge on the particles. In the experiments, an aerosol of 5-11 particles of potassium chloride was introduced from a point source located at the axis of a fully developed turbulent flow of air in a 6-in. diameter pipe, and timeaveraged concentration distributions were measured at four distances downstream. Average velocities covered a range from 9.55 to 23.0 ft/sec (corresponding to entrance Reynolds numbers of 26,800-66,000) with transverse injection velocities ranging from 0 to 0.1 70 ft/sec. The highest injection volume corresponded to 29.0% of that of the main stream. The plume width of the dispersing aerosol i s affected b y the transverse flow. The effective dispersion coefficient and the radial mass flux for charged and uncharged particles increase with air injection and decrease with suction through the pipe walls. Also, for increasing values of aspect ratio (length/diameter), they increase with injection and decrease with suction.
A n understanding of the way gases and particulate matter diffuse from a point source in turbulent streams is important for the aerodynamic design of air pollution control equipment - Present address, Research and Development Department Amoco Chemicals Corporation, Stal~dardoil Research Center: Naperville, Ill. 60540.
and for numerous other applications in industry, in agriculture, and in public health. The movenient of particles in a turbulent field has been the subject of many studies, such as those by Tchen (1947), Hughes and Gilliland (1952), Corsin and Lumley (1956) , Lie (1956), So0 (1956), Lumley (1957), Friedlander (1957), Hinze Ind. Eng. Chem. Fundom., Vol. 11, No. 1, 1972
57
a\
MAlN FLOW
hJECTlW
SECTllW
TUBE
Figure 1 .
Schematic diagram of the system analyzed
(1959), So0 and Tien (1960), Kada and Haiiratty (1960), Csanady (1963), and Chao (1964), and has been discussed at length by So0 (1967). However, most of these researchers were concerned with vertical or horizontal flow in pipes and ducts; none of them considered the effects of transverse flow. Consequently, we have analyzed the dispersion of particles from a point source in a turbulent field under transverse flow (Suneja, 1970). Experiments were carried out with a n aerosol of potassium chloride particles injected into a turbulent air stream within the core of a straight, circular, porous-walled horizontal pipe. A uniform transverse flow was created by either injection or aspiration of air through t h e walls. This paper presents both our experimental data and theoretical analysis for the concentration distributions of the particles. It also presents theoretical expressions for the eddy dispersion coefficient in the pipe core and the radial mass flux of the particles as a function of the air injection or aspiration velocity and the aspect ratio. I n addition, because fine particles tend to carry an electrical charge, we included charge measurements in our experiments, and we developed theoretical expressions to predict how the magnitude of such charges is likely to affect the concentration profiles. We have treated the particulate phase as a quasi-continuum, a n approach that has been used by several others to study the transport of particles in a moving field (see Soo, 1967). For example, Rounds (1955) used such an approach as the basis for a theoretical solution to the problem of particles diffusing downward from an elevated source. Because his solution is valid only for neutral stability, Godson (1958) later extended it by approximate methods to cover any stability. Similar relevant work has been reported by Hay and Pasquill (1957), who measured the vertical distribution of airborne particles that were released a t a height of about 500 ft, and by Xorseth and Mitchell (1963), who studied the dilution of aerosols flowing through a duct. Our experiments were run a t Reynold numbers of 26,800, 49,400, and 66,000, corresponding to average velocities of 9.55, 17.5, and 23.0 ft/sec. Air injection velocities ranged from 0 to 0.17 ft/sec, the highest of which corresponded to about 29% by volume of the main stream entering the porous pipe in the longitudinal direction. Theoretical Analysis
The system on which our analysis is based is shown in Figure 1, where the radial and longitudinal coordinates are also indicated. To describe the effects of transverse flow, three expressions are needed : the concentration of t h e particles, the eddy diffusivity for material transport, and the radial mass flux. For their development, the follow-ing assumptions are made. (1) The fluids (in this case, air) are incompressible and isothermal. (2) After passing through the calming section (Figure l ) , the main turbulent flow is 58 Ind.
Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
fully developed at the entrance t o the porous pipe. (3) Similar fluids are flowing in the axial direction and through the porous walls. (4) The transverse flow is uniform throughout the length of the porous section; that is, the radial velocity a t the wall does not vary with the longitudinal distance. Fundamental Equations. F o r the system shown in Figure 1, with flow in t h e subsonic range, t h e carrier fluid may be treated as incompressible. However, t h e dependence of the density of the particulate phase (pp) on the transport processes must be taken into account. We have the following continuity relation for the fluid phase bu
-
bx
+ -1 a
-
r br
(vr)
=
0
The equationq for the x and r components of the momentum of the fluid phase in the presence of transverse flow, as given by So0 (1967) and by Stukel and So0 (1969), are
pu
bu
-
dX
+ pv bv
-
br
bP br
= - - -
KF(v -
0,)
- b-s bX
(3)
where p and pp are the densities of the fluid and particulate phases, respectively, K is the effectiveness parameter, F is the reciprocal of the relaxation time for momentum transfer between a particle and the fluid, and 7 is the shear stress of the fluid. The contiiiuity equation for the particulate phase is given by
(4) and t,he equations for the x and r components of the momentum of the particulate phase in the presence of transverse flow are
(5)
Here, up and up refer to the longitudinal and radial velocities of the particulate phase, E, and E , represent the longitudinal and radial coniponents of the electric field, and rP refers t o the particle shear stress. Finally, the electric field E is given by the Poisson equation
bE, b @E,) = Y 2 P - + -1 bx r br mp eo
(7)
where y/mp represents the electric charge per unit mass of particles and eo refers to the permittivity of the carrier fluid. Now, for a dilute suspension such as we are considering in this study, the effect of the particles on the velocity profile of the carrier fluid can be neglected, and the “effectiveness” term on the right-hand side of eq 2 and 3 can be dropped. T o solve eq 4-7, we add and subtract ubpp/bx and vbpp/br in eq 4 and rearrange to get PP u b-
dX
PP + v 3br
=
b - -[,fJp(Up
dx
- u)l-
l b - - [rpp(vp - v ) ] =
r br
-
bJ,, 1a - - - (rJpr) (8) bx r br ~
If the effect of the self-field of the particles is accounted for, Fick’s law takes the form (see Soo, 1967; Stukel and Soo, 1969) (9)
where 01, is the eddy dispersion coefficient of the particles aiid J,, aiid J,, refer to the particulate flux 111 the x and r directions, respectively. Substituting eq 9 and 10 into eq 8, we obtain
Table 1. Maximum Variation of $ B / u with r / R for 4 = 0 and r / R = 0 to 0.7 at RerN = 26,800 M a x d e v o f # p / u from its mean value
X/D
A = 0“
A = 5.0 cm2/sec
A = 0.3 cmZ/sec
1 2 4 8
19,88% *9.88% *9.88% *9,88%
*3.07% &1.98% =t2.68% +4.51%
*7 82%
Here, the values of
for 6
CY,
CY
- -
u*D -
=
3=6.67% *4.58% +3.48%
0 were calculated from
+
0.0315(1 - r 2 / R 2 ) ( l 2 r 2 / R 2 )
which is Ikichardt’s equation as reported by Groenhof (1970).
T o obtain the concentration distribution of the particulate phase, we must solve eq 11 with appropriate boundary conditions. I n this system, the concentration gradient in t h e x direction is much less than t h a t in t h e r direction, with the result t h a t
b)
0, for all x)
where eq 16 implies t h a t the diffusing particles do not reach t h e tube walls, and therefore treats the fluid as infinite in t h e radial direction, eq 17 implies that the concentration profiles are symmetrical about the pipe axis, eq 18 implies t h a t t h e concentration of the particles is zero a t the injection location outside the injection tube, and eq 19 implies t h a t the electric field is symmetrical about the pipe axis. Concentration of Particles. The solution of eq 1.5, with u = 0 aiid g = 0 and the above boundary conditions (16-18), is PP(Z,T) = -~Q
r exp(-r2/X)
The integral on the right-hand side of eq 23 is a n incomplete
y(l,r2/M)
Boundary Conditions. F o r t h e solut’ionof eq 14 ai?a 15, t h e relevant boundary conditions are
0
0 to
y function, y(l,r2/M). Therefore, eq 23 can be rewritten as
and
E,
(20)
4XffPCO
B y analogy, we assume that the solution of eq 15 with the above boundary conditions (eq 16-19) is of the form
where Q is the amount of particles injected per unit time and ill is a function of x alone, which will be determined later. To obtain an expression for the electric field E,, we substitute eq 21 into the right-hand side of eq 14
Equation 26 is the required solution of eq 14. T o solve eq 15, we substitute eq 21 into it, thus obtaining
+
We now set aP (gM/2mpF)(E,/r) equal to &, which is the effective eddy dispersion coefficient of the electrically charged particles. Then eq 27 simplifies to
Substituting eq 21 into eq 28, multiplying both sides by exp ( r 2 / J f ) / uand , simplifying gives
where t’he terms ~vithiiit’hefirst’ pair of brackets are functions of x alone, while those wit,hin the secoiid pair are functions of both z aiid r , provided the term GP/u does not vary with the radial posit’ion.The validity of this assumption is enhanced by the presence of a n electric charge on the particles (see Table I), since t’he quantity (qJ!/2mpF) ( E T / r )decreases with an Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1, 1972
59
increase in r / R and so makes the $p profile flatter than that of ap.Thus, eq 29 gives rise to two equations
have the same value in both the presence and absence of transverse flow. On that basis, eq 35 becomes
2A1: (1 + 2 uco
and
4 1
u-.
(37)
URY
where Equation 30 must be solved to obtain the required expression for Jf (x).The assumption that $p/u does not vary with radial position in the core region permits the relationship
where $pc is the effective eddy dispersion coefficient of the charged particles a t the pipe axis. Substituting eq 32 into eq 30 gives
42 b2 24 = -___
mp2FeoippCo)
for isokinetic iiijectioii of the tracer. I n eq 37, b2 represents the correction due to the finite size of the injection tube. Because the internal diameter of t h a t tube is quite small, b2 is generally much less than other terms in the right-hand side of eq 37 and so can be neglected. Substituting eq 37 into eq 21 gives PP _ PPC
(33) exp whose solution requires a prior knowledge of the functional form of $pc (that is, cypo (gJI/Zm,F)(E,/r). In analogy to the functional form for cypea, we assiiine that c y p o is of the form Pucawhere /3 and a are not functions of x a d uc varies with z. Substituting puCafor apo,E , expression from eq 14, and ua,& for ucas discussed elsewhere (Suneja, 1970) in eq 33 gives
[
-rZua,uoo 4 ~ ~ , ~ -~ 1,75$x/D) c ~ ~ ~ ( l~ A x u , , , ( ~ 2+2/D)
+
+
+
dM dx
-
20, ~
Ru,,
Jf
=
4/3fca--1U,,a--1
where PPC
=
~
Q
_
[
uc
2eoFIIS,2u,,z
(34)
Equation 37 is a first-order differential equation with variable coefficients, and its solution is
(1 - 1,754;)
+
(1
+ 24:)]
q2
I-
2Q
1
In (1
and Q is the mass injection rate of the particles. For isolrinetic injection, Q = d 2 u c o ( p p c o ) t . T o compute the concentration profiles pp/ppo from eq 38, one needs the value of cyco, which is a function of the entrance Reynolds number alone and is obtained from the data for no transverse flow (Groenhof, 1970). Our data (Suneja. 1970) indicate the following relationship between cyoo and ReIN for entrance Reynolds number up to 80,000
- 4$x/D) aoo=
0.005 !!R e 1 P 5 P
(35) I n this solution, bhe parameters /3 and a must be determined. However, P can be determined on the basis that this solution must reduce to eq 20 for t'he case of u, = 0. This gives
P
= cypoo/Ucoa
Ind. Eng. Chem. Fundam., Vol. 11,
No. 1, 1972
(40)
Effective Eddy Dispersion Coefficient for Material Transport of Charged Particles. For calculating t h e effective eddy dispersion coefficient for charged particles, eq 28 can be rearranged in t h e form
(36)
where cypoo is the eddy dispersion coefficient of uncharged particles a t the pipe axis in the absence of transverse flow. I n our experiments, the average size of the particles is about 5 microns. So0 (1967) and others have previously shown that particles in this size range in the absence of a n y electzical charge have the same eddy diffiisivity as a gas, so that cypco =: a,,, where cyco is the eddy diffusivity of a gas a t the pipe axis with no transverse flow. The data obtained for gaseous dispersion withoiit transverse flow (Kada and Hanratt'y, 1960; Suneja, 1970) indicate that a = 0.875. Since no change is introduced in the basic charact,eristics of homogeneity and isotropy of the turbulence i n the core region when fluid is injected or withdrawn through the pipe walls, the paramet'er a should 60
e;
1
uav uco _ ____-
(39) 4(;)
b2
(38)
4?Tzffpco 'avo
+
1
br The system considered here is such that the radial concentration gradients (ie., dp,/dr) go through large variations in radial direction, so that the values calculated from eq 41 involve large errors (due to inaccuracies in the values of bp,/br) as we move away from the pipe axis. However, consistent with the assumption made earlier in this analysiq, the eddy dispersion coefficient in the core region can be considered essentially constant and the same as that a t the pipe axis. SO taking the limit of eq 41 for T = 0, we obtain
CWF%ESSm A I R
Inserting the expression for pp from eq 38, 39, and 40 into eq 42 gives
[(1
- 3 . 5 4 + 4#-X
(1
- 1.754:)
(1
- 4q$)
4
FILTER
+ T l O N
0-4
N S: FI
+
5 ( 1 - 1.75$:)
uc dz
(1+
FILTER
PI
VI
)40;
(43)
CI
Figure 2.
- FLOW IMIUTC? - PRESSURE IWlCATO?
-
VELCCIN IMICATO?
- CQLTE'IWTIM
IMICATC?
A schematic view of the experimental equipment
1
Equation 43 implies t,liat t,he effective eddy dispersion coefficient increases with fluid injection aiid decreases with suctioii through the porous n-alls of the pipe. It also indicates that the prmeiice of a unipolar electiic charge on the particles further increases the dispersion coefficient. Tising eq 43, t h e loiigitudiiial profiles foi the effective eddy dispersion coefficient can be calculated 111 the presence of mass injection or suction for aiiy desiietl value of the entrance Reynolds number, the transverse flox parameter (0)' and the electric charge parameter ( A ) . Radial M a s s Flux of Particles (AVD7).T h e mass flux in t,he radial direction is given b y t h e following expression of continuity equation
When the expression for S,,is inserted into eq 44, the e y r e s sioii for N,, reduces to the form
a,>
+ p,, bz
-
Then, when the expression for the form
pp
dr
(45)
is inserted, eq 45 reduces to
where JI is given by eq 37, Ailso,as shown elsewhere (Suneja, 1970), in the presence of transverse flow, the turbulent velocity profile in the core region is given by the expression
The point mass flux in the preseiice and absence of transverse flow is calculated from eq 46 and 47. Equipment
Figure 2 is a schematic diagram of the equipment. The major components were a wind tunnel that iiicluded a porouswalled t,est section through which air was injected or withdrawn t o induce transverse flowv,an aerosol geiieiator aiid an injection tube, aiid a n aerosol sampling and measuring system. Wind Tunnel. T h e wind tuiiiiel consisted of t'hree parts: a n aluminum entrance section, a porous aluiidum test section, a n d a mild steel discharge section. T h e entrance sectioii was 6.065 in. ill i.d. a n d 30 ft, or about 60 diamet,ers, in length. I t included a n 11-ft converging-divergiiig
AEROSU.
TO ATN.
G R O W G!ASS STOPPER
TWNSPMENl M l K XI AEROSOL I N J E C T I m
AEWSCC GWEPATICN C W E R
APACITME W E R 5 LITERS C A P A C l M l
SATUR4TEC KCL S a u l I c m I N WATER
fu@I N ,
Figure 3.
OPENIW)
Aerosol generation equipment
section that' coiitaiiied a honeycomb plat,e and a fiber glass pad t'o ensure ail essentially fully developed turbulent velocity profile at t h e entrance of t h e test section. Air was supplied to t h e tuiiiiel from a blower. The test, sect,ioii was 2 ft in length aiid consisted of a n inner aluiidum pipe, 6.025 in. in i d . , aiid an outer aluminum pipe, 8 in. in i d . , fitted coaxially with special flanges. The air for transverse flow was injected or withdrawn through 14 ports placed symmetrically around the circumfereiice of the aluminum pipe. The discharge section consisted of three unit's: a n aluminum pipe, 6 ft long aiid 6.065 in. in i.d., which supported a t'raversing mechaiiism for t'he aerosol sampling syst'em; an orifice meter that was used to set' the Reynolds number aiid was therefore equipped with ail orifice plat'e followed by an 18411. length of pipe; and a diverging sect'ioii, 3 ft in length. Aerosol Generator a n d Injection Tube. T h e aerosol generator is s h o r n in Figure 3 . Filtered compressed air was passed through a sat'urated aqueous solution of KC1 and then into a 5-1. flask where, during a mean residence time of about 30 sec, most of t h e entrained water was evaporated t o produce a "dry'! aerosol of I i C l particles. The average particle size was 5 p (range of 1-15 p ) , as determined by measurements with an optical microscope. T h e average elect'rical charge was 2.4 elementary charges per 5-p particle, as determined by a Wesix ion spectrometei and a Keithley electrometer. The dryness of the Rerosol !vas monitored at the exit of the flask with a water-sensitive paper, Ozalid 105 SZ niaiiufact'ured by Ozalid (a division of General Aniline aiid Film Corp.) . The injection t,ube was made of thin-walled copper tubiiigan L-shaped piece, 5/16-ii1. i.d. and 3/8-i11. o.d., soldered to a straight 1-in. piece. 3/10-i1i. i d . and ~/lG-in.o.d. This tube was positioned so that, its center \?-as on the center line of the mind twine1 and its tip was a t the front edge of the test secInd. Eng. Chem. Fundam., Val. 1 1 , No. 1, 1972
61
_1
u
140
1,O
0,8
0.6 (hmc-)
0.4
0
0.2
f
-1-
0,2
f-
0.4 (&E
0.6
0.8
1.0
CENTER)
Figure 6. Particle concentration profiles vs. radial distance
EXPERIEZMAL DATA
'1.0
0.8
0,6 &Or
Figure 5.
0.4 hTER)L R
0.2
0
-1-
02
0.4
0,6
0.8
0
1.0
t(PgO(ECEMER)
Particle concentration profiles vs. radial distance
Figure 7. Particle concentration profiles v5. radial distance
tion. It was small enough so that it did not disturb the flow in the tunnel to an appreciable degree, but was large enough t o introduce enough aerosol into the stream to permit reliable concentration measurements downstream. Aerosol Sampling and Measuring System. T h e aerosol sampling system comprised a hollow probe attached t o a n assembly of six 150-ml glass tubes connected in series, each containing 50 ml of demineralized water. T h e last t u b e in t h e series was connected t o a vacuum pump. T h e probe consisted of a 3-in. length of copper tubing, 1/4-in. o.d., soldered to a 27-in. length of copper tubing, 3/8-in. 0.d. So that saniples could be taken at any radial or axial position in the test section, the smaller end of the probe was attached to a vertical pillar, 51/4 in. high, that moved axially along a 30-in. track mounted on the discharge section of the wind tunnel. Samples withdrawn through the probe were bubbled through the water in the tubes, where the KC1 particles were dissolved. T o determine KCl concentration, the contents of the six tubes were combined and their conductivity was measured with a conductivity cell consisting of a probe, an impedance bridge, and a cathode-ray oscillograph that had been calibrated with known solutions of KC1. Measurements were sensitive down to 1 ppm, and reproducibility was within about 8%. Two problems mere inherent in the sampling: (1) the submerged oiifices in the aerosol generator usually became clogged with solid KCI after about 4 hr of operation; (2) as others have observed (Soo, 1967), some of the charged KC1 particles initially tended to deposit on the walls of the injection and sampling tubes, but the deposition decreased as the walls became coated with KCI. Consequently, to ensure reasonably uniform injection and sampling of the aerosol, all runs were terminated after 3 hr. Figure 4 shows the results
of two calibrat'ion runs in which 10-min samples were collected directly at the outlet of the injection tube to determine typical variations in KC1 concentration with time. These variations were not considered great enough to necessitate the use of correction factors for the data. Temperature Monitoring System. To ensure t h a t there were no temperature gradients in the air stream a n d t o provide a basis for determining fluid properties, temperatures were monitored with copper-constantan thermocouples that were attached t o supports a t t h e front end of t h e sampling probe and connected t o a potentiometer pyrometer.
62 Ind.
Eng. Chem. Fundam., Vol. 1 1 , No. 1, 1972
Procedures
Before each run, the sampling probe and injection tube were washed thoroughly with demineralized water and dried, and the blower in the wind tunnel was started and allowed to run until the temperature of the equipment and air stream reached equilibrium. The valve after the blower was then adjusted to give the desired Reynolds number, the sampling probe was set a t the desired axial posit'ion, and t.he air injection, if any, was started. Kext, the vacuum pump on the sampling system was started and the aspiration rate adjusted so that sampling was near isokiiietic. Finally, the aerosol injection was started and the velocity a t the injection tube outlet was adjusted to equal that of the main air stream. Because it was necessary t,o inject an aerosol with a constant KCI concentration for all Reynolds numbers, the aerosol was always generated at a rate corresponding to the highest Reynolds number (66,000) and, depending on the run, any excess was bled off from the 5-1. drying flask (see Figure 3). During each run, samples were withdrawn for 1 hr each at thiee different points in the test section.
04
'L
I
018
0:6
A
/ ?O @€imCtxTm)~
Figure 8.
-1-
'
bh4 f
I-
4'
016
I
0.:
'
t
O!
O1.0
(aewEcom*)
Particle concentration profiles vs. radial distance
lAd{ .6 ~BELCWemf- -} ,
I .8
I
Figure 10. tration
I
.4
1
I
.2
I
I 0
' I
CL\ i \ I
-
.2
\
,4
I
,6
I ,8
, 1.0
+(h comn)
Effect of electrical charge on particle concen-
1.0 0,8
sp 0,6 0
\ c i a 0.4
0,2
'1,O
0,8
0.6
0.4
(BELarCWrEn)
Figure
0.2
-f -1-
0
0,2
f
0,4
0.6
0,8
1.0
(PBOMtomR)
9. Particle concentration profiles vs. radial distance
Results and Discussion
Figures 5-9 compare the dimensionless aerosol concentration profiles a t various x / D and injection parameter values and three different Reynolds numbers. The solid lines are the curves predicted theoretically by eq 38, while the points represent our experimental data. T h e experimental profiles are centered with respect t o the maximum observed concentration. I n the theoretical predictions, the eddy diffusivity of the particles was taken to be the same as t h a t of a gas, which is quite true for small particles (average 5 p ) . Also, since the charge on the particles was negligible, the electric charge term, A in eq 38, was taken to be zero. Equation 38 predicts t h e dimensionless concentration profiles within the experimental error a t all aspect ratios; however, the discrepancy is greatest at a ratio of 1.0, probably because of t'he disturbance introduced b y the injection tube. These curves also show that the plume width of the dispersing aerosol is affected by t'he transverse flow. For example, for a Reynolds number of 26,800, a n aspect ratio of 4, and no air injection (6 = 0), the width of t h e plume is about 50% of the radius, whereas for the same Reynolds number and aspect ratio but with 6 = -0.0183, the plume width is only 45% of the radius. Thus, the material injected radially through the walls of the porous pipe tends t o "squeeze" the dispersing particles toward the pipe axis-at a fixed entrance Reynolds number, the higher t h e injection rate, the narrower the plume width. Figure 10 shows the effect of electric charge on the dimensionless concentration distribution for various aspect ratios a t an entrance Reynolds number of 49,400 and one injection parameter value. The solid curves represent a hypothetical value of 9.3 cm2/sec (that is, 0.01 ft2/sec) for the electric charge parameter, while the dotted curves represent uncharged particles at aspect ratios of 1.0 and 3.0. The curves
.
X/D
Figure 1 1 Theoretically calculated particulate concentration vs. aspect ratio for fluid injection at the pipe wall
for the chaiged particles are flatter, and the plume widths are broader, presumably because repulsive forces cause dispersion of charged particles of similar polarity. Particle concentrations (ie., pp) calculated from eq 38 and 39 and nondimensionalized by dividing b y ( p p e o ) z l ~ c O(that is, the particle concentration at the injection point) are plotted versus x / D in Figures 11-13 for ReIN = 26,800; t h e solid lines represent uncharged particles, while the dotted lines represent particles with a charge corresponding to a value of A of 9.3 cm*/sec (that is, 0.01 ft*/sec). This corresponds t o a chaige of 3.9 (lom4)C on a 5-p particle. For r/R less than b/R, the dimensionless particle concentration is unity at the injection tube (Le., x / D = 0 ) and decreases monotonically as x / D increases. By contrast, for a radial distance greater than the injection tube radius from the pipe axis, the concentration is zero at x / D = 0 and goes through a maximum at some value of x / D . This maximum value increases as the radial distance from the pipe axis increases. Also, the charged particles attain a concentration maximum at a smaller value of x / D . The effective eddy dispersion coefficient values in the pipe Ind. Eng. Cham. Fundam., Vol. 1 1 , No. 1, 1972
63
10 10
h
10
10 1
0
n
\
X
U Y U
0
2
6
4
8
10
X/D
Figure 14. Eddy dispersion coefficient in the pipe core vs. aspect ratio
X/D
Figure 12. Theoretically calculated particulate concentration vs. aspect ratio for no fluid injection or suction
X
3U
-
0
2
4
6
8
10
r/R
Figure 15. Predicted enhancement factor for radial mass flux of tracer vs. radial distance from pipe axis
Figure 1 3. Theoretically calculated particulate concentration vs. aspect ratio for fluid suction at the pipe wall
core in the presence and absence of transverse flow and of electric charge on the particles, as calculated from eq 43, are plotted us. x / D in Figure 14, where the solid lines represent uncharged and the dotted lines represent charged particles. The effect of the charge is t o increase the effective dispersion coefficient and decrease its rate of change with x / D . Injection also increases the dispersion coefficient whereas 64
Ind. Eng. Chem. Fundam., Vol. 11, No. 1, 1 9 7 2
suction decreases it. Moreover, with an increase in x / D , these values increase for injection and decrease for suction. Figure 15 displays the ratio of radial mass fluxes in the presence and absence of fluid injection us. the dimensionless radial distance from the pipe axis for both charged and uncharged particles for ReIN = 26,8000 and various values of x / D . The results show that the radial flux enhancement factor (NpJ(.Vpr)4= 0) is larger than unity in the presence of injection and less than unity in the presence of suction.
the enhancement factor first increases with a n increase in x / D , goes through a maximum, and then starts decreasing. Conclusions
-c
X/D
Fiaure - 16. Predicted enhancement factor for radial mass flux of tracer vs. aspect ratio
The following conclusions can be drawn from the present study. (1) Theory and experimental data are in good agreement for the dispersion of solid particulate matter bearing negligible charge. (2) The concentration profiles of charged and uncharged dispersing particulate matter in the central core of a pipe in the presence and absence of fluid injection or suction can be calculated by using the proposed expressions (eq 38 and 39). (3) The effect of the electric charge on the particulate concentration has been correlated with the aspect ratio, entrance Reynolds number, amount of aerosol injected, and the amount of an inert gas transpired through the porous walls. (4) The eddy dispersion coefficient of both charged and uncharged particles increases with fluid injection and decreases Kith suction through the pipe walls (see Figure 14). Also for increasing values of aspect ratio, i t decreases in the case of suction and increases in the case of injection. (5) The turbulent dispersion coefficient of unipolarly charged particles is greater than t h a t of uncharged particles (see Figure 14). Hence the charge on t h e particles should be minimized t o reduce the deposition rates on containing walls. (6) The radial mass flux of both charged and uncharged particulate matter increases with fluid injection and decreases with fluid suction. ( 7 ) The present experimental technique may be used to achieve on-line dynamic dilution of a n aerosol stream before t s concentration is measured with a n instrument. Nomenclature
This is due to the fact that the radial gradients of particulate concentration increase with injection (note steeper profiles of Figures 6, 8, and 9 compared to those of Figures 5 and 7) and decrease with suction, and the radial flux depends directly on the radial gradient. Figure 15 also shows that in the presence of injection a t a particular value of x / D , the enhancement factor first remains constant with a n increase in the dimensionless radial distance from the pipe axis, then starts increasing, and finally reaches a constant value. However, this constant asymptotic value is less for charged than for uncharged particles. This is due to t h e fact t h a t the charge on the particles increases the radial flux by the same magnitude both in the presence and absence of transverse flow, thereby reducing the enhancement factor as defined above and plotted in Figures 15 and 16. (This IS best comprehended b y noting that ( N 1 N 3 ) / ( A r q -V3) is less than XI/K*, where f i 3 represents the additive flux due to particulate charge, and hriand K Zrepresent the radial fluxes for uncharged particles in the presence and absence of transverse flow, respectively.) I n the presence of suction a t a paiticular value of x / D , the enhancement factor decreases monotonically for uncharged particles, whereas for charged particles i t increases slightly, goes through a maximum, and then decreases to zero with a n increase in the dimensionless radial distance from the pipe axis. Finally, Figure 16 displays the radial mass flux enhancement factor for charged and uncharged particles vs. x / D for ReIu = 26,800 and various values of the dimensionless radial distance from the pipe axis. I n the presence of injection, a t a particular value of r / R , the enhancement factor first increases with a n increase in x / D , goes through a maximum, and then starts decreasing. For suction, hoffever, this maximum is reduced if the particles carry a charge; except near the pipe axis,
+
+
= inside radius of injection tube, cm = diameter of porous pipe, cm
longitudinal and radial components of electric field, respectively, V/cm = dimensionless longitudinal velocity ( = u/u,,) = reciprocal of relaxation time for momentum transfer between a particle and the fluid, sec-1 = effectiveness parameter, dimensionless = mass of a particle, g = radial mass flux of particles, g/sec om2 = charge on a particle of mass mp, C = mass injection rate of particles (7rbZucoppafor isokinetic injection), g/sec = radius of porous pipe, cm = longitudinal and radial velocities of fluid phase, cm/sec = longitudinal and radial velocities of particulate phase, cm/sec = average velocity a t the entrance t o the porous section, cm/sec = radial velocity, cm/sec = radial velocitv at t h e wall. cm/sec 2, r = longitudinal and r'adial coordinates, cm Note: F o r intercoilversion of units (em2 of fluid) v=~ _ g of particle _ sec2 C =
GREEKLETTERS = eddy diffusivity of particles in the absence of
eiectric charge, cmi/sec = permittivity of carrier fluid, C/(V)(cm of
eo p, p p 7,
4 +p
rP
fluid) or F/icm of fluid) = density of fluid and particulate phases, respectively, g/cma = shear stress of fluid and particles, respectively, dyn/cm2 = transverse flow parameter (= U , / U , ~ ~ ) , dimensionless (negative for fluid injection and positive for suction) = eddy dispersion coefficient of particles in the presence of electric charge, cm2/sec Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1, 1972
65
SUBBCRIPTS refers to pipe axis = refers to the absence of transverse flow = refers t o particulate phase = refers t o a quantity at x / D = 0 =
C 0
P i
Literature Cited
Abramowitch, M., Stegun, I. A., Ed., “Handbook of Mathematical Functions,” Table 267, pp 978-983, Dover Publications. New York. N. Y.. 1967. Chai, 8. T., Oesteir. Ing.’Arch. 18, 7 (1964). Corsin, S., Lumley, J., Appl. Sci. Res., Sect. A 6, 114 (1956). Csanady, G. T., J . Atmos. Sci. 20,201 (1963). Friedlander. S. K.. A.I.Ch.E. J . 3. 381 (19573. Godson, W.’ L., Archiv. M e t e d . Geophys. Bioklimatol., Ser. A 10,
Lie, V. C., J . Meleorol. 13, 399 (1956). Lumley, J. L., Ph.D. Thesis, John Hopkins University, Baltimore. Md.. 1957. Norsetb, H. G., Mitchell, R. I., Ann. N . Y . Acad. Sci. 195, 88 11963). Rounds,‘W., Jr., Trans. Amer. Gwphys. Union 36, 395 (1955). Soo, S. L., Chem. Eng. Sci. 5,57 (1956): Soo, S. L., “Fluid Dvnamics of MultiDhase Systems,” Blaisdell Publishing Co., Waltham, Mass., 1967. Soo, S. L., Tien, C. L., J . Appl. Mech., Trans. A S M E 27, 5 (1960). Stukel, J. J., Soo, S . L., Powder Technol. 2, 278 (1969). Suueja, S. K., Ph.D. Thesis, Illinois Institute of Technology, Chicago, Ill., 1970. Tchen, C. RI., Dissertation, Delft, Martinus Nijhoft, The Hague, 1947.
305 (19.58).
Groenhof, H. C., Chem. Eng. Sci. 25, 1005 (1970). Hay, J. S., Pasquill, F., J . Fluid Mech. 2,299 (1957). Hinze, J. L., “Turbulence,” McGraw-Hill, New York, N. Y., 1959. Hughes, R. R., Gilliland, E. R., Chem. Eng. Progr. 48,497 (1952). Kada and Hanratty, A.1.Ch.E. J. 6 , 624 (1960).
RECEIVED for review February 22, 1971 ACCEPTEDSeptember 22, 1971 Financial support was provided by the Fine Particles Section of IIT Research Institute.
Theory of Coalescence by Flow through Porous Media Lloyd A. Spielman*l and Simon 1. Goren Department of Chemical Engineering, University of California, Berkeley, Calif, 9.4720
Our equations describing coalescence by flow through porous media are solved to yield specific predictions for oil-in-water dispersions. It i s shown that for small incoming volume fractions of suspended oil and oil-towater viscosity ratios which are not extremely large, the steady-state removal of droplets of a given size i s independent of the sizes and amounts of other droplets introduced; the aqueous pressure drop also is predicted to be independent of the incoming drop size distribution. Solutions are also obtained which are intended to describe the operation of semipermeable porous barriers. A correlation for the dimensionless filter coefficient as a function of a single dimensionless adhesion number i s suggested when capture i s by London forces in porous media of similar geometry and wettability. The theory i s largely supported by the fibrous mat coalescence experiments of the following paper.
Coalescence of liquid-liquid dispersions induced by flow through porous media has proved effective in a variety of industrial applications, including the treatment of aqueous wastes containing finely suspended oils. We recently summarized the most important applications and relevant literature (Spielman and Goren, 1970b). One of the chief barriers to greater use of porous coalescence was found to be the need for extensive pilot study because lack of understanding of this complex process has prevented the formulation of trustworthy general design equations. To overcome this, we combined concepts of two-phase flow through porous media with concepts of water and aerosol filtration to give a theoretical framework for predicting pressure drop, degree of phase separation, and some complicated capillary phenomena occurring in the separation of liquid-liquid dispersions b y flow through porous media. Application of the theory requires the Division of Engineering and Applied Physics, Harvard University, Cambridge, Mass. 02138. 66
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
capillary pressure, relative permeabilities, and filter coefficient be known as functions of the local oil saturation and other relevant parameters. I n this paper we obtain approximate solutions t o the equations presented earlier (Spielman and Goren, 1970b). Our aim is twofold: first t o provide interpretation of the fibrous mat coalescence experiments reported in the following paper (Spielman and Goren, 1972), and secondly to illustrate the use of the theoretical framework in establishing scaling criteria so that pressure drop and filtration data obtained from relatively small-scale apparatus may be used with confidence to design coalescers for specific industrial applications. Throughout the present treatment, oil is again arbitrarily taken as the suspended phase both for concreteness and because that was the actual situation in the experiments of the following paper, In most of the present work we assume the volume fraction of oil dispersed in the inlet aqueous phase is very small and the viscosity of the oil is not too large compared with that of the continuous phase. When these approximations
