Ind. Eng. Chem. Fundam., Vol. 17, No. 3, 1978
Literature Cited Finlay, I. C.,McMillan, T., Proc. 1. Mech. Eng., 182 (3H), 277 (1968). Goldstein, s., Ed., “Modern Developments in Fluid Dynamics”, Vol. 11, Dover, New York, N.Y., 1965. Goldstein, M. E., Yang, W.J., Clark, J. A., J. Heat Transfer, 89, 185-193 (1967). Hodgson, J. W., Saterbak, R. T. Sunderland, J. E.J. Heat Transfer, 90,457-463 (1968).
189
Hodgson, J. W., Sunderland, J. E., lnd. fng. Chem. Fundam., 7 , 567-572 (1968). Kosky, P. G., lnt. J. Heat Mass Transfer, 19, 539-543 (1976). ~p, w., ~ J, FluidMech,, ~ ~ , 37, 565-575 (1969). Michael, D, H,, N
Received f o r review July 20, 1977 Accepted M a y 1,1978
Aerosol Coagulation and Diffusion in a Turbulent Jet Patrick Delattre and Sheldon K. Friedlander’ California lnstitute of Technology,Pasadena, California 9 1 125
A method has been developed for predicting the properties of an aerosol in a turbulent jet in which coagulation is taking place. The analysis leads to simple expressions for the number concentration and volume-average particle size at any point in the jet. In experiments designed to test the theory, jets containing sulfuric acid and stearic acid vapors at low concentration in air were mixed with ambient air. Particle size and concentration were measured on the jet axis at various distances from the orifice with an electrical aerosol analyzer. The slope of the experimental curve relating mean particle volume to distance was 1.2 to 3.1 times the value predicted theoretically. A complete calculation requires data on the particle size and concentration at the jet orifice. Methods of determining these parameters are discussed.
Aerosol formation by condensation and coagulation in a turbulent jet occurs in emissions from industrial sources, automobile tailpipes, and in natural processes. The vaporladen jet entrains colder surrounding air and becomes supersaturated with respect to the vapor on the edge of the mixing zone (Hidy and Friedlander, 1964). If the supersaturation is sufficiently high, the vapor will nucleate producing high particle concentrations which are carried by the turbulent eddies into the mixing zone. Three processes then occur simultaneously: further nucleation, particle growth by condensation, and coagulation. As the jet spreads, the vapor concentration decreases by dilution and condensation. Thereafter the aerosol dynamics is controlled by coagulation and turbulent diffusion. Many workers have studied condensation in a turbulent jet. Some studies (Amelin, 1948; Higuchi and O’Knoski, 1960) focused on the mixing zone to test the predictions of the Becker-Doring theory of nucleation. Green and Lane (1964) quote an equation given by Langmuir in 1942, without derivation, for the size of the particles produced by condensation of supersaturated vapors in a turbulent jet. Sutugin and Fuchs (1968) gave a theoretical treatment of the formation of a condensation aerosol taking into account the three simultaneous processes described above. They used a numerical method to calculate the changes in the concentration of vapor molecules and clusters along the jet. Their calculation predicts the particle size distribution a t any time during aerosol formation. They concluded that coagulation is the main mechanism determining the particle size, and that inaccuracy in the calculation of the nucleation rate should not significantly affect the final result. We have developed a method for predicting the characteristics of an aerosol in a turbulent jet when the aerosol dynamics is controlled by turbulent diffusion and coagulation. The analysis leads to simple mathematical expressions for the number concentration and volume average particle size. The results of the theoretical analysis have been tested by experimental measurements. 0019-7874/78/1017-0189$01.00/0
Theory We consider a steady circular turbulent jet (Figure 1) in which aerosol particles are coagulating as mixing occurs. The equation of conservation of species for the time-average number concentration, takes the following form in cylindrical coordinates
m,,
aN, (1) L dt Jcoag where E and iij are mean gas velocity components in the axial and radial direction, respectively, and is the eddy diffusion coefficient for the aerosol particles. Diffusion in the axial direction is neglected, and the origin of the coordinate system is taken at the nozzle. The eddy diffusion coefficient is defined by
ax
rdr
where the primes refer to the fluctuating components. I t is assumed that t is independent of the particle size distribution. Equation 1 is the usual expression for diffusion in a turbulent jet to which an additional term [dN,ldt]co,,, representing the loss due to coagulation, has been added. Since the Kolmogorov scale is much larger than the particle diameter for particles in the free molecule range, turbulent coagulation can be neglected compared with Brownian coagulation. The first important assumption in the theoretical analysis is that the size distribution is locally self-preserving. Lai et al. (1972) derived an expression for coagulation of a uniform isothermal self-preserving aerosol
where the constant a = 6.67, k is the Boltzmann constant, T is the absolute temperature, pp is the particle density, and V is the aerosol volume fraction.
0 1978 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 17, No. 3, 1978
when x/d
> 20. Making the second important assumption that
fimr2is constant, the quantity A defined by eq 12 is constant A = B(l
-
+ Rmf2)
(12)
and eq 6 may be written
The volumetric concentration of aerosol V is conserved. Its distribution is determined by a differential equation of the form Figure 1. Experimental design: turbulent jet showing cylindrical coordinate system, velocity components, and disposition of sampling probe, diluter, and electrical aerosol analyzer. Equation 3 can be modified to include the effect of interparticle forces, which increase the rate of coagulation (Fuchs and Sutugin, 1965; Graham and Homer, 1973)
(14)
As a trial solution it is assumed that N, can be described by a relationship of the form
7 7
N,=
V
+ a)P
(CX
Substitution in eq 13 gives
aV where G is a constant (Graham and Homer, 1973) characteristic of the chemical composition of the particles and an integral function of the dimensionless size distribution. It is convenient to define a new quantity B such that
In the following analysis it is assumed that the temperature is constant when the distance from the orifice is larger than XI,defined as the distance from the orifice where the aerosol dynamics is controlled by coagulation and turbulent diffusion. Substituting eq 4 and 5 in eq 1leads to
This expression is identical with the equation of conservation for V, eq 14, provided that the last two terms are equal and opposite. As shown below, it is found experimentally that for sufficiently large values of x v = vo--E l (17) u oc
where V, is the aerosol volume fraction and U , the gas velocity at the orifice. The coefficient c is introduced because the spread of matter occurs somewhat more rapidly than that of momentum. The last two terms in eq 16 are equal and opposite if 6
P=;
Writing and and
-
N,' N,'== N,
(8)
C = - 5A - - v,1 6 U,c Substituting eq 18 and 19 in eq 15, the distribution of the number concentration becomes
.. V
Assuming that p and &,' strongly correlated
are small quantities and are
- - Q12
fim'2
= fim{Q1
(10)
Expanding eq 9 in a binomial series and keeping only the second-order terms after time averaging Vl/6N, 11/6 = ~ 1 / 6 ~ , 1 1 / (1 6 + p) (11) The intensity of the concentration fluctuations along the jet axis was measured by Becker et al. (1967) with the isothermal air jet marked by an oil smoke, and was found to approach 0.22 when x/d > 20. (d is the orifice diameter.) The intensity of the temperature fluctuations measured by Wilson and Danckwerts (1964) for a nozzle temperature of 225 "C, approach 0.18
(20) 5 v 1 -AL-x+a 6 U,c where a is a constant determined by the boundary conditions. For x = 0, by eq 20 a = vO5I6,where v, = V,/N,(O). Since eq 20 is not expected to hold for x 0, vo should be interpreted as the apparent particle volume at the orifice. Decay laws for the concentration of a nonreactive component and of the temperature along the jet axis are summarized by Becker et al. (1967). The usual form for these laws is r d -=cl(21)
-
ro
X + X ,
where r is concentration or temperature, r0 is the value at the nozzle, and x , is the abscissa of the apparent origin. C1 varies from 3.5 for a hot jet to 5.7 for an isothermal jet.
Ind. Eng. Chem. Fundam., Vol. 17, No. 3, 1978
191
The usual form of the decay laws for the axial gas velocity is (Wilson and Danckwerts, 1964; Hinze, 1975)
-
U d -=c2u, x + x , '
where Cz varies from 5 for a hot jet to 6.4 for an isothermal jet. The coefficient c defined by eq 17 is determined on the jet axis by c = -C2 (23) C1 when x, and xo' can be neglected, compared with x . The total flow of particles through a cross section of the jet a t a distance x from the jet orifice is
(24) Substituting eq 17 and 20 in eq 24, and assuming that c is constant across the jet
KVO -Q
= CU,(CX
+ u)6'5
where K is the kinematic momentum of the jet = 7rd2U02/4, and C is defined by eq 19. With these assumptions it is possible to calculate the flow of particles across the jet, the local total concentration of particles, the local volume average particle size, and the local size distribution a t any distance from the nozzle.
Experimental Measurements General Considerations. Aerosols generated as described below were sampled along the jet axis as shown in Figure 1. Since the particle concentration was high, coagulation occurred in the sampling probe. Particle diffusion losses in the probe were calculated to be less than 10%. The sample was then diluted to quench coagulation and avoid saturating the aerosol analyzer. The actual concentration sampled in the jet was calculated from the measured concentration by integrating eq 4. The size of the sampling probe was 4 mm inside diameter, much smaller than the diameter of the jet in the sampling region to avoid disturbing the jet. The average concentration N,,of the gas sampled is given by
where q s is the volumetric sampling rate, and r l is defined by qs =
Jr'27rur d r
The experimental conditions were such that
N,, N,,,,
(28)
Experiments were carried out with sulfuric acid and stearic acid aerosols. The calculation of G in eq 4 required the knowledge of the Hamaker constant for these two chemical species. The Hamaker constant for sulfuric acid can be calculated from an expression due to Fowkes (1967) or Lifshitz (Visser, 1972), giving respectively, 4 X and 5.36 X l O - l 3 erg. The calculation was based on optical dispersion data from the Smithsonian Critical Tables (1964) as discussed by Gregory (1969) and Papazian (1971), and the static dielectric constant given by Gillespie and Cole (1956). The Hamaker constant for stearic acid is 4.7 X erg in a vacuum (Papazian, 1971). The constant G calculated as in Graham and Homer (1973) was 1.9 for both sulfuric acid and stearic acid.
U
Figure 2. Aerosol generator: (A) nitrogen cylinder; (B) rotameter; (C) pressure gauge; (D)heating section; (E) diffusion cell; (F) heating section; (G) thermocouple well in liquid pool; (H), (I), (J) thermocouple wells; (K) Nichrome heating wire; (L)glass wool insulation; (M) aluminum foil; (N) powerstats.
The Aerosol Generator. Figure 2 shows a schematic diagram of the important features of the aerosol generator. Pressurized nitrogen was supplied to the generator from the cylinder (A). The flow rate was measured by a calibrated rotameter (B), and the measured value corrected to take into account the pressure measured at the intake of the rotameter (C). The nitrogen, which was heated along (D), flowed over a diffusion cell (E).The cell inside surface area was 0.78 cm2 and the cell height was 4 cm, giving a ratio of cell area over cell height less than 0.3 as recommended by Fortuin (1956). The temperature of the liquid was measured by a chromel-alumel thermocouple inserted in a glass indentation (G) in the core of the pool. Thermocouples measured the gas flow temperature a t different locations (H), (I),and (J). The nitrogen flow and the liquid were heated by different insulated nichrome coils (K) surrounded by glass wool (L) and aluminum foil (M). A bank of powerstats (N) was used to regulate the heaters. The orifice diameter was 0.035 f 0.0025 cm, and its length was 2.5 cm. The nozzle itself was heated by an insulated coil, and the temperature a t the nozzle exit was 230 "C. There was no condensation of vapor in the nozzle. The vaporization rate of the liquid in the diffusion cell was calibrated by measuring with a cathetometer the displacement of the meniscus in the reservoir as a function of time. Good agreement was found with the classical formula for the vaporization rate (Fortuin, 1956) for sulfuric acid when the temperature of sulfuric acid did not exceed 250 "C, using 96% concentrated sulfuric acid. The unit was placed in a hood in which the velocity of the air was about 10 cm s-l. A typical jet velocity at the nozzle was 2.2 lo4 cm s-l, so the jet could be considered a free jet discharged in quiet air. The air was not filtered or dried. The relative humidity in the room was monitored during the experiments. Diluter. The bulk of the aerosol passed through an absolute filter with a large surface area (Gelman Glass Fiber Filter). A small part flowed through a capillary and mixed with the rest of the filtered gas stream. The dilution ratio of the unit was calculated by measuring the flow rates through the filter and through the capillary. (The capillary inserted at the center of the filter was 3.2 mm long and 0.175 mm in diameter.) The flow rate through the capillary was determined from the
192
Ind. Eng. Chem. Fundam., Vol. 17, No. 3, 18"8
Table I. Experimental Results nozzle conditions aerosol gas. volume velocity fraction, (cm s-l),
uo
vo
sampling conditions axial gas aerosol aerosol velocity particle volume (cm s-l), concnfraction, ii ( ~ m - ~N) ,, Sulfuric Acid 3.5 x 102 2.0 x 10'0 2.45 X 2.35 X lo2 8.7 x 109 1.63 X 2.5 X 10'0 1.0x 103 3.4 x 10-8 8.1 x 109 5.5 x 102 2.2 x 10-8 3.5 x 102 3.3 x 109 1.25 x 3.5 x 102 1.05 X 1O'O 1.12 x 10-8 2.35 X lo2 5.1 x 109 7.95 x 10-9 4.6 x 109 7.6 x 10-9 1.5 x 109 2.63 x 10-9 2.25 X 1O1O 1.1 x 10-8 1.6 x 109 3.5 x 102 1.0x 10-9 2.35 X lo2 7.9 x 108 6.6 x 10-lo
ambient sampling relative point ("C), humidity, (cm),
gas temp T O
%
X
1.8 x 10-6 1.8 x 10-6 1.15 x 1.15 X 10-6 1.15 X 8.15 x 10-7 8.15 x 10-7 8.15 x 10-7 2.15 x 10-7 9.75 x 10-8 9.75 x 10-8 9.75 x 10-8
2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2
x 104 x 104 x 104 x 104 x 104 x 104 x 104 x 104 x 104 x 104 x 104 x 104
230 230 230 230 230 230 230 230 230 230 230 230
90 90 50 90 90 90 90 90
12
50 50 50 50
12
1.15 x 10-6 1.15 x 10-6 1.15 x 10-6 6.7 x 10-7 2.85 x 10-7 2.85 x 10-7
2.2 2.2 2.2 2.2 2.2 2.2
x 104 x 104 x 104 x 104 x 104 x 104
230 230 230 230 230 230
50 50 50 50 70 70
2 12
18 4
7.5 12 12
18 19.5 n
L
12 18
v
Stearic Acid 18 7.5 4
7.5
-
3.5 x 102 2.35 X lo2 5.5 x 102 1.0x 103 5.5 x 102
pressure drop across the capillary. The dilution ratio of the unit was thus found to be 6400. The particle losses by diffusion in the capillary were calculated to be about 17%for 100-8, particles and 8%for 150-8, particles. Axial Velocity. The velocity a t the jet axis was measured at various distances from the nozzle orifice with a hot wire anemometer (Data Metrics Model 800 L). For a nozzle temperature equal to 230 "C, the axial velocity decreased in the following way d zaxis -- 5.25 u o
X
100 < x / d
< 500
(29)
Particle Size Measurements. The particle size distribution and number concentration were measured with an electrical aerosol analyzer (Therm0 Systems Inc.). For particle diameters between 100 and 240 A, essentially all particles carry one unit of charge, and for larger particles between 240 and 750 A, the charging is nonuniform and discrete. The instrument was used in the range of particle sizes where the matrix of response of the analyzer to a monodisperse aerosol is close to unity. According to Liu and Pui (1975) this corresponds to sizes in the range 100 to 300 A. The sheath air in the analyzer was dried before it entered the instrument to limit the growth of the sulfuric acid particles in the analyzer by condensation of water vapor. The analyzer was originally factory calibrated and checked with polystyrene latex particles at a few sizes. It was later checked against another analyzer, on a University of Minnesota van, which had more recently been factory calibrated. Agreement within f10% for each channel was found between the two instruments in measuring the size distribution of the atmospheric aerosol. The analyzer provided data on the particle concentration and on the volume fraction of the aerosol a t the diluter. The actual concentration and average particle size a t the point of sampling on the jet axis were calculated using eq 4, taking into account diffusion losses in the capillary and in the sampling probe.
4.35 x 1010 1.6 x 109 8.65 X lo8 3.2 x 109 4.5 x 109 2.1 x 109
1.2 x 10-7 1.46 x 9.0 x 10-9 1.5 X 1.2 x 10-8 6.3 x 10-9
aerosol slopes particle ratio size [C](exptl)/ (A), d, [Cl(th.) 133 153
1.15
141
173 193
2.6
124
143 146 150 97 106 117 174 258 271 208 172 179
1.24
2.96
3.14
2.66
Experimental Results a n d Discussion The aerosol was sampled on the jet axis a t various distances from the orifice, 60 < x/ d < 500. The jet was well defined over this range. Data for sulfuric acid and stearic acid are presented in Table I. The first four columns describe the experimental conditions of each run. The average particle size a t different locations on the jet axis, column 9, was calculated from the volume fraction, column 8, measured with the electrical aerosol analyzer, and from the particle concentration, column 7 , calculated from the concentration measured after the diluter. The density of the sulfuric acid particles was assumed to be 1.3, the density of an aqueous sulfuric acid solution in equilibrium with a 50% relative humidity. The variation of V d V with x is shown in Figure 3. The line (a) was computed by a linear regression analysis. The slope of this line was found to be 6.4 corresponding to C1 = 4.46. This can be compared with the results of Wilson and Danckwerts (1964) for temperature distributions with similar nozzle temperatures for which C1 was found to be 4.2, but closer to the nozzle where the temperature was higher. The variation of Uo/uwith x is also shown in Figure 3, line (b). The coefficient c defined by eq 23 was found to be 1.24. The distribution of aerosol volume concentration in the turbulent jet flow (Figure 4) was similar to the profile of the mean concentration found by Becker et al. (1967). The variation of [V/Nm]5'6 = [CI5l6 with x is shown in Figure 5 for sulfuric acid at 50% and 90% relative humidities and in Figure 6 for stearic acid. The points fall approximately along a straight line as expected from eq 20. The slope C of these lines can be calculated from eq 19. As shown in column 10 of Table I, the experimental values of C exceed the calculated values by a factor of 1.24 to 3.14. Graham and Homer (1973) found about the same discrepancy between experiment and theory in their studies of coagulating lead aerosols. The intercept of these lines was shown above to correspond to the "apparent particle size" a t the orifice. The apparent particle size was independent of the jet vapor loading (within the error associated with the measurements of the particle sizes) and was about 65 8,at 90% relative humidity and 100 A
193
Ind. Eng. Chem. Fundam., Vol. 17, No. 3, 1978
Sulfuric
Sulfuric Acid Vo = 9.75.16' x V o = 2.15*10-7
l,V0/V
b)Uo/u
Uo' 2.2.10%msec-1
-
0
0
4
Tie,
V o = 8,15*10-7 V o = 1.15*10-6 v o = 1.8.10-6
3O
-
b
Vo = 2 . 8 5 W 7
A
v,
o
vo=
50%
(0) V o = 1.15*10-6
x (b)
cm
-
Stearic Acid
14
R.H
230 "C
d,,, = 0.035
Acid Aerosol = 4800
(a)
Vo =9.8.16'
0
R H = 90% ( c l v, =8.iwo-7
A
(d) Vo = l.6.10-6
I
= 6.7.10-7
1.15.10-7
60 40 4 114
7:
8 228
6
14
12
IO
16 456
18
20
x
1)
-
Figure 3. Aerosol volume concentration and gas velocity along the 0
jet axis.
'=.'--
I
f
X
1
I
2
57
4 114
1
1
6
8 228
1
IO
I
12
'
14
I
16
456
1
I8
1 1
2 0 x !cm) x /d
Figure 5. Volume average particle size along the jet axis for the sulfuric acid aerosol plotted according to theory (eq 20). X
Sulfuric Acid Aerosol R.H. i 4 5 % vo = 9.8 *lo-'
u0
= 2.2 * 10' cm
Stearic Acid Aerosol !Re),,, = 4800 R H = 50%
d = ,035cm x/d s 3 4 2
(51= 1 0 5 a -Becker, Hottel 8
0.8
*!a)
v , = I . I ~ . I ~ ~
Williams ( 1 9 6 7 1
\
0.4t 0. 2
0
1
I
I
2
57
4 114
6
8 228
IO
12
14
16 456
18
20x(crn) x/d
Figure 6. Volume average particle size along the jet axis for the stearic acid aerosol plotted according to theory (eq 20).
at 50% relative humidity for sulfuric acid (Figure 5) and about 165 8, for stearic acid (Figure 6). The diameters of clusters produced by self-nucleation of vapor of pure sulfuric acid and stearic acid were calculated to be 18 and 28 A, respectively, from the Becker-Doring theory of nucleation for a nucleation rate of 1 cm-3 s-l. The spontaneous nucleation of sulfuric acid is enhanced by the presence of water vapor (Mirabel and Katz, 1974) and the size of the clusters decreases with increasing water vapor concentration. Thus the apparent particle size, as well as the cluster size, decrease with increasing ability of the vapor to self-nucleate. In general the value of the constant a is not known. I t is interesting to estimate the order of magnitude of the error when the constant is set equal to zero. This approximation should hold best at large values of x for chemical species of extremely low vapor pressure, such as certain salts. If a = 0 the ratio of the calculated particle concentration to experimental particle concentration ( N r n ) & d / ( K ) e x p t l varies from 10 to 5 when x varies from 4 to 18 cm for V, = 1.15 X The ratio varies from 100 to 12 when x varies from 4 to 18 cm for V, = 9.75 X 10-8. In both cases, the aerosol was sulfuric acid at 50% relative humidity. The ratio - of the experimental to calculated particle size ( G ) e x p t l / ( d p ) c & d varies from 1.95 to 1.5 for V, = 1.15 X and from 4.75 to 1.8for V, = 9.75 X 10-8 when x varies from 4 to 18 cm. The improvement of the
agreement between theory and experiment with increasing distance from the nozzle results from the asymptotic character of coagulation. The discrepancies between calculation and experiment when a = 0 are of the same order of magnitude as the discrepancies found by Sutugin and Fuchs (1968) between their numerical computations and measured concentrations and particle sizes. Application to Auto Catalyst Emissions. This theory can be used to estimate the particle size and concentration of sulfuric acid particles produced by a catalytic converterequipped vehicle (Pierson, 1976). It is recognized that the actual mixing conditions in a car exhaust are complex and the model of a turbulent jet in stagnant air is an oversimplification. The volume fraction of sulfuric acid aerosol in the automotive exhaust is taken to be 5 X 10-9 (Miller et al., 197613). The exhaust gas velocity is set equal to 20 m s-l, the exhaust gas temperature to 230 OC,and the exhaust pipe diameter to 7 cm. Taking a particle density of 1.4 we calculate the average particle size and concentration in the jet a t a distance of 4 m from the nozzle corresponding to a dilution ratio equal to 10. The average particle concentration across the jet is calculated to be 2.1 X lo8 cm-3 and the average particle size 170 8, taking a = 2 X 10-16, as in Figure 5 (c), and assuming this constant to be independent of the nozzle gas velocity. Miller et al. (1976b) measured the size and concentration of particles produced in a dynamometer study of a catalyticconverter-equipped vehicle. After a rapid mixing of the ex-
194
Ind. Eng. Chem. Fundam., Vol. 17, No. 3, 1978
haust gases with 10 parts of filtered air, they measured a particle concentration of.about 108 cm-3 and particle sizes between 200 and 400 A. Whitby et al. (1976) measured the size and concentration of sulfuric acid particles produced by vehicles equipped with a catalytic converter along a test track in a sulfate dispersion experiment. They consider (Miller et al., 1976) that the exhaust gases were diluted 500 times and measured particle concentrations of about 7 f 1.6 X lo6 and particle sizes between 178 and 316 A. With the assumption that 4 m behind the car, after dilution by a factor of 10, the exhaust is quickly dispersed, and then diluted 50 times, quenching the coagulation, we calculate the particle concentration to be about 4.2 X lo6 ~ m and - the ~ particle size 170 A. Our estimate is consistent with these two measurements in the laboratory and in the field.
Summary and Conclusions The local number concentration and volume average particle size of a coagulating aerosol in a turbulent jet can be estimated from the expression
N- =
v v,
The local aerosol volume fraction, is determined from the distribution of the mean concentration of a noncoagulating (conserved) species. The correction factor for intermolecular forces, G , calculated from theory, was found tobe 1.9 for both sulfuric acid and stearic acid. The apparent particle volume at the orifice u, = V,JN,(O), must be known to carry out the calculation. When the aerosol is pre-formed in the gas before it issues from the orifice, u, can be determined if N,(O) (and V,) are known. Otherwise, the apparent particle size must be determined experimentally or estimated from nucleation theory. In measurements with sulfuric acid, the apparent particle size at the orifice was 65 at 9096 relative humidity and 100 8, at 50% relative humidity, independent of V , over the limited range of the data. More experiments are needed to determine the effect of the jet exit velocity on u,. Experimentally measured values of when plotted vs. x fall approximately on a straight line consistent with the above expression. The slope of the curve is 1.2 to 3.1 times the theoretical value, indicating that coagulation takes place somewhat more rapidly than theory predicts.
(vm)5/6
Acknowledgment This work was supported in part by the French Foreign Office and by Environmental Protection Agency Grant No. R802160. The contents do not necessarily reflect the views and policies of the Environmental Protection Agency. We also wish to express our appreciation to the Pasadena Lung Association, which provided funds for some of the instrumentation. Nomenclature a = constant defined by eq 20 A = constant defined by eq 12 B = constant defined by eq 5
c = constant defined in eq 23, dimensionless C = constant defined by eq 19 C1 = constant in eq 21, dimensionless C2 = constant in eq 22, dimensionless
d = orifice diameter, cm G = correction factor for intermolecular forces k = Boltzmann's constant K = kinematic momentum of jet, cm4/s2 N , = total number concentration, cm-3 p = exponent in eq 15, dimensionless q s = volumetric sampling rate, cm3/s Q = total flow of particles over a cross section of the jet, s-1 r = radial distance measured from jet axis, cm t = time, s T = absolute temperature, K u = velocity in axial direction, cm/s velocity at orifice, cm/s -uU=, =average particle volume = V / N , , cm3 V = aeroso! volume fraction, dimensionless w = velocity in radial direction, cm/s x = distance along jet axis, cm Greek Symbols a = dimensionless constant in coagulation theory = 6.67 r = concentration or temperature t = eddy diffusion coefficient, cm2/s pp = particle density, g/cm3 - = denotes time average value
' = denotes fluctuating quantity
-
= denotes dimensionless fluctuating quantity o = subscript denotes conditions at orifice
Literature Cited Amelin, A. G., Kolloid. Zh., IO, 168 (1948). Becker, H. A., Hottel, H. C., Williams, G. C., J. FluidMech., 30,285 (1967). Fortuin, J. M. H., Anal. Chim. Acta, 15,521 (1956). Fowkes, T. M., in "Surfaces and Interfaces", J. J. Burke, N. L. Reed, V. Weiss, Ed., Vol. 1, p 197, Syracuse University Press, Syracuse, N.Y., 1967. Fuchs, N. A., Sutugin, A. G., J. Colloid Sci., 20, 492 (1965). Gillepsie, R. J., Cole, R. H., Trans. Faraday SOC., 52, 1325 (1956). Graham, S.C., Homer, J. E., faraday Symp. Chem. SOC., I , 85 (1973). Green, H. L., Lane, W. R., "Particulare Clouds: Dusts, Smokes and Mists", 2nd ed, E. and F. N. Spon Ltd, London, 1964. Gregory, J., Adv. Colloid lnterface Sci., 2, 396 (1969). Hidy, G. M., Friedlander, S.K., AlChEJ.. 10, 115 (1964). Higuchi, W. I., O'Konski, C. T., J. Colloid. Sci., 15, 14 (1960). Hinze, J. O., "Turbulence", 2nd ed, McGraw-Hill, New York, N.Y., 1975. Lai, F. S., Friedlander, S.K., Pich, J., Hidy. G. M., J. Colloid lnterface Sci., 39, 395 (1972). Liu, 6. Y. H., Pui, D. Y. H., J. Aerosol Sci., 6, 249 (1975). Miller, D. F., Levy, A., Pui, D. Y. H., Whitby, K. T., Wilson, W. E., Jr.. J. Air Polluf. Control Assoc., 26, 576 (1976a). Miller, D. F., Trayser, D. A., Joseph, D. W., Battelle Columbus Laboratories, "Size Characterizationof Sulfuric Acid Aerosol Emissions", Report 760041 Given at SAE Annual Meeting in Detroit, Mich., Feb 23, 1976b. Mirabel, P., Katz, J. L.. J. Chem. Phys., 80, 1138 (1974). Papazian, H. A., J. Am. Chem. Soc., 93,5634 (1971). Pierson, W. R., CHEMTECH, 6, 332 (1976). "Smithsonsian CRITICAL Tables", p 531, 1964. Sutugin, A. G., Fuchs, N. A., J. Colloidinterface Sci., 27, 216 (1968). Visser, J., Adv. Colloid lnterface Sci., 3,331 (1972). Whitby, K. T. et al., in GM/EPA Sulfate Dispersion Experiment-EPA-6001376-035, April 1976. Wilson, R. A. M.. Danckwerts, P. V., Chem. Eng. Sci., 19,885 (1964).
Received for review July 22,1977 Accepted May 8,1978
