Conclusions
The boundary-layer assumptions, used to reduce the coupled equations of motion, energy, and continuity which describe laminar natural convection heat transfer from a vertical plate, have been experimentally shown to be valid for both water and mercury in the moderate (IOs to lo9) Grashof number range. The experimental dimensionless temperature profiles agreed with the similarity solution and with the more recent first-order perturbation analysis. The plate temperature varied as the 1/5 power of the distance up the plate, as predicted by Sparrow and Gregg. The Kusselt number for both water and mercury agreed well with theory. The boundary-layer for mercury was about seven times thicker than for water and could be fairly accurately predicted by the Von Karman-Pohlhausen integral solution to the boundarylayer equations. Nomenclature Q
h, Gr,* k
Nu, Pr 4
T
T,
T, x
Y
= thermal diffusivity, sq. ft./sec. = coefficient of thermal expansion,
a
= gravity force. ftJsec.2 = local heat-transfer coefficient, B.t.u./hr. sq. ft. F. = modified Grashof number based on 2, gpp4/kv2,
dimensionless = thermal conductivity, B.t.u./hr. ft. F. = local Xusselt number, h,X/k, dimensionless = Prandtl number, v/a,C,/k, dimensionless = heat-flux rate, B.t.u./hr. sq. ft. = temperature, O F. = wall temperature, O F. = bulk temperature, O F. = coordinate measuring distance along plate from leading edge, ft. = coordinate measuring normal distance from plate, ft.
@
O
6( 5 ) 7) V
p
= = = =
-
(l/p)(ap/aT),
F.-I
boundary layer thickness, ft. similarity variable, y/z[Gr,*/5]1’5, dimensionless kinematic viscosity, lb., sq. ft./sec. density, lb. m/ft. see3.
literature Cited
Chang, K. S.,Akins, R. G., Bankoff, S. G., IND. EXG.C H m . F U N D A M E ~6,T26-37 A L S (1966). Chang, K. C., Akins, R. G., Burris, L., Bankoff, S. G., “Free Convection of a Low Prandtl Number Fluid in Contact with Uniformly Heated Yertical Plate,” Argonne National Laboratory, ANL-6836 (1964). Dotson, J. P., Heat Transfer from a Vertical Plate by Free Convection,” h1.S. thesis, Purdue University, 1954. Julian, D. V.,“Experimental Study of Natural Convection Heat Transfer from a Uniformly Heated Vertical Plate Immersed in hlercury,” Ph.D. thesis, Kansas State University, 1967. Lorenz, L., Wiedemanns Ann. 13, 582 (1881). Prandtl L., “Uber Flussigkeitsbewegun bei sehr kleiner Reibung,” Proceedings of Third International hlathematical Congress, Heidelberg, 1904; reprinted in Tier Abhdl. zur HydroAerodynamik, Gottingen, 1927; NACA TN-462 (1955). Sauriders, 0. A., Proc. Roy. SOC.(London) A172, 55-71 (1939). Schecter, R. S., Isbin, H. S., A.Z.Ch.E. J . 4, 81-9 (1958). Sparrow, E. hI., “Free Convection with Variable Properties and 1-ariable \Tall Temperatures,” Ph.D. thesis, Harvard University, 19FiB. Sparrow, E. M., Laminar Free Convection on a Vertical Plate with Prescribed Konuniform Wall Heat Flux or Prescribed Nonuniform Wall Temperature,” NACA TN 3608 (1955). Sparrow, E. hl., Gregg, J. L., Trans. A S M E 78, 435-40 (1956) Yang, K. T., Jerger, E. W.,J . Heat Transjer, Trans. A S M E C86, 107-15 (1964).
RECEIVED for review J d y 31, 1968 ACCEPTEDJune 23, 1969 Work partially supported by N.S.F. Grant GK-2114.
P A R T I C L E DEPOSITION F R O M T U R B U L E N T STREAMS BY M E A N S OF T H E R M A L F O R C E R. L E E B Y E R S A N D S E Y M O U R CALVERT’ Center for Air Environment Studies, The Pennsylvania State University, University Park, Pa. 16802 Particle deposition from turbulent streams of hot gases was studied to determine what factors control particle collection. Collection efficiency was found as a function of particle size over the range of 0.3 to 1.3 microns. Diffusion was not a contributing factor in particle deposition; the only significant mechanism proved to be thermophoresis. When the mean free path to particle radius ratio, the Knudsen number, increased from small to large values, collection efficiency rose gradually at first and then increased more rapidly at about a Knudsen number of 0.2. This phenomenon i s explained by the increasing contribution of the free molecular thermal force as the Knudsen number approaches unity. The effects of flow parameters and size of collector were determined. A mathematical model, based on existing theories of thermophoresis, was developed to predict collection efficiency. Agreement between the model and experiment was good, considering the accuracy with which the slip flow coefficients and constants are known. The transition thermal force is related theoretically to the free molecular thermal force by an exponential term containing the inverse of the Knudsen number and constant, T . Results suggest that T i s not a constant but is temperature-dependent, because of its basic relationship to the momentum accommodation coefficient.
HE motion of small particles suspended in a gas containing Ta thermal gradient has been the subject of many investigators. The force responsible for this motion, called the thermal force, has been studied primarily by observing the movement of a particle suspended in a stagnant gas. This technique has permitted the experimental testing of various Present address, University of California, Riverside, Calif. 646
l&EC
FUNDAMENTALS
proposed theories. As a result, the thermal force is fairly well understood. It has been applied in the development of a dust sampler which permits sampling a small quantity of dust-laden gas at essentially 100% collection efficiency. The use of these samplers has been discussed by Fraula (1956), Gordon and Orr (1954), and Kitto (1952). The design of this device is such that the gas passes through
Table I.
Investigator
Expressions for Thermal Force
Thermal Force Expression
Cawood
Ft
Waldmann, Derjaguin and Bakanov
F t = -dT/dx
=
-32rP2u 15VB
X/rp >> 1
=
Derjaguin and Yalamov
-4rP2(pX/T)dT/dx
Brock
(vrp2)( k , / k , + ctVr,) ( c r n b ’ ~ p ) + 3cnA/?p)(1 + 2 k g / k p + 2CtX/rp) (dT/dz) F t = (-) - g k , t k , + 2ct(X/rp)kp dT,dx (-127) -
(1
2k,
t k, t 2ct(X/r,)kp
Underestimates thermal force by approximately a factor of s/iT
free molecular regime
X/r,
Ft
Comments
>> 1,
Xlr,
Epstein Brock
Range of Applicability
-1/2(ar,*)(pX/T)dT/dz
1
F t = Ft* exp ( - r r p / X )
the collection channel a t a very low velocity, so that the flow is completely laminar. This investigation has been undertaken to determine the possibility of using the thermal force as a means for cleaning large quantities of hot gases under conditions of turbulent flow. Xo attempts to study quantitatively the factors contributing to thermal deposition from turbulent streams have been reported in the literature. Ratchell and maggener (1964) experimentally determined the shearing force which would strip particles from a surface layer formed by thermal forces. A hypothetical model for aerosol deposition on a conduit wall was set forth by Postma (1961), but no experimental verification of the model was reported, The object of this research is directed toward establishing how the thermal force effects particle deposition from hot gases in turbulent flow. Both turbulent diffusion and thermophoresis may be controlling steps in particle transport. It is also necessary to verify and extend the basic definition of particle motion by thermal force alone. The collection efficiency due to thermophoresis as a function of particle size has not been previously reported. It has been shown that the thermal force is not appreciable for particles greater than 10 microns in diameter. In fact, the efficiency of the thermal precipitator begins to fall with increase in radius above 2 microns. Of particular interest is particulate matter less than 2 to 3 microns, because of the difficulty of collection in this range encountered by other techniques. Thus, because of the limitations of the magnitude of the thermal force, as well as practical considerations, attention is focused on particles in this size range. Theoretical Expressions
Early observations of a dust-free space surrounding a hot body were made by Tyndall. Quantitative description of the force responsible for this effect has been offered by many (Brock, 1962a, b, 1967, 1969; Cawood, 1936; Davies, 1966b; Derjaguin and Bakanov, 1959, 1962; Derjaguin et al., 1965; Epstein, 1929; Waldmann, 1959). The description of the thermal force is determined by the dimensionless parameter, A / T ~ ,called the Knudsen number. Table I summarizes the theoretical expressions for the thermal force and the conditions for which they are applicable. Figure 1 gives a quantitative
Shows good agreement with experimental results
0.2 is indicative of the fact that the slip flow mechanism is becoming increasingly less important than the free molecule mechanism, the latter giving rise to a thermal velocity which is independent of the size of the particle. Also shown in Figure 5 is the predicted efficiency as calculated from Brock’s equation. The theory and the experimental results indicate the same trend as a function of particle size. Temperature. The temperature of the entering gas influences collection efficiency in two respects. First, the temperature gradient in the laminar sublayer is directly proportional to the difference between the inlet and wall temperatures (see Equation 9 ) . Thus, the higher the inlet temperature the higher the temperature gradient; as a result, the efficiency can be expected to increase. Second, the mean free path, and hence the Knudsen number, is directly proportional to the absolute temperature. As the temperature rises, the value of the Knudsen number increases, which shifts the thermal force mechanism toward the free molecule regime where the force is largest. Again, the result expected is higher collection efficiency. Figure 7 shows that experimental efficiencies increase with gas temperature. Theoretical temperature dependence is indicated by the dashed lines and though the magnitude is not the same as the experimental value, the trend is the same. Gas Velocity. Figure 8 represents the collection efficiency obtained a t three different gas velocities. The curves show a slight decrease in efficiency with increasing velocity. The theoretical curves in Figure 8 also predict a small dependence of efficiency on gas velocity. From Equation 12 the velocity effect can be explained primarily by the relative contribution of the Reynolds number in the numerator, the heat transfer coefficient in the denominator, and the average velocity in the exponential term. Length-Diameter Ratio. It would seem reasonable that the longer the test section the higher the collection efficiency. The data in Figure 9 show, however, that collection efficiency does not increase in proportion to the increase in length of the
A . L / D = 115.0 6. L / D = 75.0 C. L / D = 38.5
D =0.026 F% T; = 900"F Q = 8.4CFM
0
1
Figure 7.
I
0.2
1
0.4
I 0.6
I 0.8
Porticle Diometer, p
I
I 1.0
/.2
I
I
1
0.a
0
04
0.6
Figure 9. efficiency
I 1.0
I
12
Diometer, p
Porticie
Effect of temperature on collection efficiency
1
08
Effect of length-diameter ratio on collection Diameter constant
0
1
I
Particle
Figure 8.
I
1
I
Diome ter, p
Effect of gas velocity on collection efficiency
collector. Since the temperature gradient decreases rapidly as a function of the distance downstream from the entrance, the driving force is greatest near the entrance and collection is correspondingly greater near the entrance. As a result, a point is reached beyond which an increase in the length of the collector has very little effect on the collection efficiency. Effect of Diffusion on Particle Deposition. Diffusion could play a role in particle deposition from turbulent streams. The model developed here assumes that diffusion would not play an important role a t any point in the flow cross section. T o establish the validity of this assumption, the following calciilations and experiments were carried out.
Friedlander and Johnstone (1957) present some theoretical equations from which deposition of aerosols from turbulent flow through a pipe can be calculated. Using laboratory conditions of room temperature air a t 8.3 cfm in a 0.312-inch diameter tube yielding a Reynolds number of 13,760, collection efficiency for a 1.0-micron particle was estimated by their equations to be approximately 0.2%. At the same flow conditions but with a temperature gradient imposed on the gas, the experimental collection efficiency was 1301,. The assumption that particle deposition from turbulent flow is negligible compared to deposition by the thermal force is obviously justified. The negligible contribution of turbulent deposition was also shown experimentally. An aerosol was passed through the test section a t room temperature a t 8.3 cfm. After one hour of sampling, the collecting tube was removed from the test section and cut in half axially. The exposed collecting surface is shown in Figure 10 (tube A ) . The absence of any deposit is clearly obvious. This means that the mechanisms that might have caused particle deposition under these experimental conditions were not strong enough to remove particles from the stream. Since turbulent diffusion is one one such possible mechanism, by the above reasoning, it can be concluded that diffusion was not contributing to particle deposition. The above experimental conditions do not duplicate exactly the flow conditions when a hot aerosol was used, the difference between the two cases being a higher average temperature in the laminar sublayer in the hot aerosol case. However, the small difference in these two temperatures (about 50" F.) is not great enough to cause a significant change in particle diffusivity. Thus, the conclusion based on the room temperature run can be applied a t high temperature. For comparison, tube B in Figure 10 shows the type of deposit formed when thermal forces were present. This deposit is typical of those found when using hot aerosols. As an additional check on the role of diffusion, samples were taken a t room temperature by total filter. Using the VOL.
8
NO.
4
NOVEMBER
1969
651
velocities. Discrepancies of this order of magnitude have also been found b y Jacobsen and Brock (1965), Rosenblatt and LaMer (1946), and Schadt and Cadle (1957). In any case, since the theoretical considerations for the transition t o 1:ontinuurn regimes have been based, a t best, on rough app roximations, quantitative agreement with experimental shn..lA nn+ hn -e dat; n.IyuIu vu 0 ,..,,cted. To show the reliability of the experimental method and the precision of the data, experimental and theoretical efficiencies (based on Brock's slip flow equation) were substituted into Equation 11, which was rearranged so that the righbhand side contained only the Knudsen number, X/rP; the left-hand side waa designated as the "correlation factor." A plot of the correlation factor us. the Knudsen number for all the experimental conditions- that is, 0.3 to 1.0 micron for pantide diameter, 300" to 600" F. for the average g'astemperatui8, and 8700 to 14,400 for the Reynolds number biwed on avera ge flow conditions-is shown in Figure 12. The ,vA.Lyu, .^I*u.lvugh scattered from one experimental rnn to the next, do not show a discernible trend with temperature or velocity. The experimental points show the same dependence on the Knudsen number as the theoretical points; a t Knudsen number less than 0.2 the experimental values of the correlation factor cluster very closely around the theoretical values predicted by Brock's slip flow equation. 1
Figure 10. Particle deposit on tube wall Q = 8.3 cfm, Ti = T, = room temperature Q = 8.3 cfm, r, = 90Oo F., r, = 800 F.
A. B.
2
-2 20
c
i-
Temperature Dependence of Transition Range Thermal Force. In the transition range Brock has found the thermal
force to he approximated by:
.;D
-8
9
PG*exp (-n,/X)
Ft =
In fitting the data of this work to Equation 15, it was found that T was not constant hut varied with Knudsen number. Table I1 shows this result for a typical run. However, the experimental results of other workers have shown that r does not change with particle size. It was concluded that some other factor in this work, which itself is a function of the particle size, must be influencing r.
c
Ti = 675 -F 0 =6.2 CFM
h = 0,072~ Experiment ----Brock
----LwjaSuin 0
I 0.2
Figure 1 1.
Equotion Equation
I
I
0.6 Porlicle Diometer, 0.4
(15)
I
I
0.8
I. 0
1.2
[
p
Comparison of results with existing theory
same linear velocity as ia the above low temperature run, the weight of the outlet sample averaged 99.52% of the inlet sample. Again., the results indicate thkt no significant deposition occurred, confirming the observation that diffusion is not a contributing factor in psrticle deposition. The lack of any measurable deposit in the low temperature runs just described rules out not only contribution of diffusion to particle deposit~ionhut also the concern that electrostatic effects might he contributing to particulate deposition.
O',t
900 450 9w 900
N-
0-
E 2-
*-
14 98 625
."
27.37 = 0.r-
sw
8.39
675, 900 900 900 9w
615 8.74
8.50 830 8.20
CFM
..
:. I
.? n
Experimental Results Compared to Existing Theory. Figures 5 , 7 , and 8 compare predicted efficiencies aa calculated by Brock's equation with experiment. Figure 11 compares
experimental collection efficiency with predicted efficiencies based on the equations of Brock and Derjaguin. The Derjaguin and Yalamov equation (see Table I) was derived for large aerosol particles and therefore is not applicable in the range of this investigation. However, if the Derjnguin equation is plotted and extrapolated along with the experimental curve into the continuum regime, it is obvious that the Derjaguin predictions for collection efficiency correspond to thermal velocities three to four times the experimental 651
I & EC
FUND AMENT ALS
MERlMENTAL RESUTS
( W
0.I
, 02
THEOREmCAL PRWKTIONS QASED ON QROCK'S EQUATION
I
03
Knudsen Number.
!
I
04
05
AhD
J
06
Figure 12. Correlation of theoretical and experimental results
Table 11.
Experimental Variation of Tau with Knudren Number
T, = 675" F. Q
=
6.2 cfm L I D = 38.5 X = 0.072
Knudsen
Efficiency,
D,. p
No.,X ! r p
70
7
0.35 0.40 0.45 6.50 0.55 6.50 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.oo 1.05 1.10
0.411 0.360 6.320 0.288 0.262 0.240 0,222 0.206 0.192 0.180 0.169 0.160 0.152 0.144 0.137 0.131
19.41 17.88 16.53 15.33 14.26 13.29 12.42 11.63 10.90 10.24 9.63 9.06 8.54 8.05 7.60 7.18
0.1400 0.1864 0.2189 0,2423 0.2591 0.2714 0.2804 0.2868 0.2914 0.2946 0.2967 0.2979 0.2985 0.2987 0.2984 0.2978
Tau average
=
0.
p
r 0.
0
0.250
The direction to look for this factor was suggested by the theoretical derivation of Equation 15 given by Brock (1969). This derivation led to an expression for T for polyatomic gases : T = 0.06 0 . 0 9 ~ ~ 0.28a(1 - /3k,/k,) (16)
+
+
The ratio k,/kp is one of the order of 0.005 for this work. Since 0 < p < I , the term P(k,k,) is negligible compared to unity and Equation 16 becomes: T
= 0.06
+ 0.3701
(17)
Thus T becomes a linear function of a. The value of T must change according to changes Rhich might occur in a. Since the smallest particle in these measurements is of the order of 1000 times greater in diameter than the gas molecules, all particles, regardless of size, would appear to the gas molecules as a flat surface. Thus, it is not reasonable to expect that the accommodation coefficient, and hence T, should change with particle size. However, if, for each experimental run, the average value of T was plotted against the average temperature in the laminar sublayer, a definite trend between the two was evident. This relationship, shown in Figure 13, indicates that T decreases with temperature. Using a least squares polynomial fit for these data points, the following relation was obtained for T as a function of temperature: 7
= -0.118 X IO+t
+ 0.38
(18)
where t is in degree Fahrenheit. Combining Equations 17 and 18 yields a as a linear function of temperature over the range of 130' to 190' F.: a = -0.32 X
+ 0.86
(19)
Ideally this relation should be subject to experimental and/or theoretical confirmation. It was not possible with the experimental method used for measuring the thermal force, independently to establish a relation between a and r. Little seems to be known theoretically about the temperature dependence of a. Jackson and Mott (1932) and Brown (Brown, 1957; Kaminsky, 1965b) have shown a! to have a complicated temperature dependence based on crude models. These models are limited in application to the interaction of noble gas atoms on metal surfaces. Eucken and Bertram (Eucken and Bertram, 1936; Kaminsky, 1965a) have ex-
I I I 1 I 80 /20 160 Zoo 40 Average brninar Sublayer Ternperdure, "F
< Figure 13.
2 0
T vs. average laminar sublayer temperature
pressed the temperature dependence of a by the relation: In (1/a
- Ea - 1) = Q+ constant RT
(20 )
where Q and E, are determined from relaxation and adsorption times, respectively. This relation, which predicts a decrease in the accommodation coefficient with temperature, has shown good agreement with the data of Hunsmann (Hunsmann, 1923; Kaminsky, 1965a) and Rowley and Evans (1935). Amdur and Guildner (1957) reported accommodation coefficients for noble gases, hydrogen, and oxygen on gas-covered metallic surfaces and found a linear dependence of the accommodation coefficient on temperature, in degrees Centigrade: L Y ~= awo[1 - . (( t - 25O)] (21 1 where { is characteristic of the gas-metal system used. On the basis of Equations 20 and 21, though they pertain to gas-metal interaction rather than gas-crystal interaction, i t appears that Equation 19 shows the right dependence of the accommodation coefficient on temperature. I n addition to the semitheoretical equations just mentioned, the experimental results of Blodgett and Langmuir (1932) and Rolf (1944) have shown a relation between the accommodation coefficient and temperature which is similar to that given by Equation 19. Figure 14 compares this work and the data of others.
It is believed that this temperature dependence of the accommodation coefficient explains the variation of the constant T in Equation 15. Fuchs and Yankovskii (1958) and later Keng and Orr (1966) found that in moving aerosol streams, which are experiencing a thermal force, small particles precipitate out earlier than larger particles. This being the case, a small particle will deposit out nearer the entrance of the test section than larger particles. Thus, the average temperature which a smaller particle experiences in the laminar sublayer before deposition will be higher than the average temperature a larger particle experiences. AS a VOL.
8
NO. 4
NOVEMBER
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653
1.
H2
on W fsurmce slight& go5 covered) Bbdgetl
Experimental efficiencies in the slip flow region proved to be higher than those predicted by the mathematical model. Agreement was good, considering the assumptions and approximations contained in the thermal force theory and in the model itself. In the transition region the thermal force is theoretically related to the free molecular thermal force by an exponential term containing the inverse of the Knudsen number and a constant, 7. Results of this work suggest that 7 is not a constant but is bemperature-dependent, because of its basic relationship to the momentum accommodation coefficient Particle deposition under the conditions of this work was found to be the result of thermal forces alone, turbulent diffusion not being a contributing factor.
a Longmuir
2 . H e m Pt isunbce goo5 covered)
Rolf
3.
H2 m W
I oxygen covered I
B1odgstt 61 Longmulr
4. A i r m NoC1 This Work
I
Nomenclature
c cz
cm
= average concentration a t any point, particles/cu = average inlet concentration, particles/cu. ft. = coefficient of viscous slip, 1.2
.ft.
= average outlet concentration, particles/cu. ft. ct = temperature jump coefficient clrn = thermal velocity coefficient, ft./hr. R. C, = heat capacity of air a t constant pressure, B.t.u./lb.
co
I
I
100
200
1
1 300
400
' F.
I
I
500
6OO
i
*R Temperature dependence of accommodation Temperofure,
Figure 1 4. coefficient
result, the accommodation coefficient, and hence 7, will be smaller for small particles than for larger particles. The variation of 7,which at first sight appeared to be a function of particle size alone, can be explained on the basis of the temperature dependence of the accommodation coefficient, which in turn relates to particle size in terms of the relative temperature which particles of varying size experience before deposition. Summary and Conclusions
The object of this research was to establish how thermophoresis effects particle deposition from hot gases in turbulent flow. Both turbulent diffusion and thermal force were considered in particle transport. A mathematical model based on esisting theory was developed to predict particle collection. An aer os01 of sodium chloride, generated by atomization, was heated as high as 900' F. and passed through a copper tube in a water-cooled jacket. Under conditions of turbulent flow a large thermal gradient was generated in the laminar sublayer and provided the necessary driving force for thermal deposition. Particle collection was determined by taking inlet and outlet samples a t the test section using a Goetz aerosol spectrometer (1962). Based on the size fractionation principle of this instriiment, it was possible to determine collection efficiency as a function of particle diameter. Collection efficiency was also determined as a function of aerosol temperature, flow rate and length-diameter ratio of the collector. Particle deposition by thermal force is influenced mostly by particle size and aerosol temperature, which in turn determines the thermal gradient. Collection efficiency ranges from about 30% for particle diameter around 0.3 micron down to about 5% for particle diameter around 1.2 microns. The higher the absolute temperature of the aerosol stream, the higher will be the thermal gradient and the collection efficiency. 654
l&EC
FUNDAMENTALS
C, D D, E,
= = = =
R
= ideal gas law constant, 10.731
t
T Ta
= temperature, degrees = absolute temperature, ' R. = absolute temperature evaluated a t edge of laminar
T;
= absolute
Cunningham correction to Stokes law diameter of tube, ft. diameter of particle, ft. activation energy based on mean adsorption time, 1b.r ft. ED^ = point collection efficiencj for particles of size D,, % ' Eo = over-all collection efficiency,yo f = Fanning friction factor F t = thermal force, 1b.f = heat transfer coefficient, B.t.u./hr. sq. ft. F. h k , = thermal conductivity of gas, B.t.u./hr. ft. ' F. k, = thermal conductivity of particle, B.t.u./hr. ft. ' F. L = total length of test section, it. P = pressure, lb.f/sq. ft. P D p = penetration, C O / C , , for particle size D, = activation energy based on relaxation time, 1b.t ft. Q = distance measured radially outward from a point r of symmetry as origin rp = radius of particle, ft. lb. cu. ft. lb.r/sq. in. lb. mole ' R.
sublayer, ' R. temperature evaluated a t test section inlet, ' R. T, = absolute temperature a t wall, ' R. T, = absolute temperature measured a t any axial distance from inlet, ' R. ug = velocity of a gas molecule, ft./hr. V = volume, cu. ft. 7 = average velocity, ft./hr. Va = velocity a t edge of laminar sublayer, ft./hr. V t = thermal velocity, ft./hr., = weight of aerosol a t test section inlet, 1b.f TI,.' Wo = weight of aerosol a t test section outlet, 1b.r z = distance, ft. = distance measured from tube wall, ft. y
GREEKLETTERS a
p
=
momentum accomnlodation coefficient
= thermal accommodation coefficient
r]
= laminm sublayer thickness, ft. = temperature coefficient of momentum accommodation coefficient, O CY1 = viscosity, lb./hr. ft.
K
= translational
X
= B.t.u./hr. ft.’ F. = Mean free path, ft.
6
t
part of gas thermal conductivity,
p
= symbol representnig particle size in microns = 3.1415 = gas density, lb./cu. ft.
7
= coefficient relating transition thermal force to free
p
...
‘lr
molecular thermal force
DIMENSIONLESS GROUPS Kn = Knudsen number, X/r, Re
y+
= Reynolds number, Dvp/q = dimensionless quantity, (y Re f l j ) / D
Derjaguin, B. V., Yalamov, Yu, J. Colloid Sei. 20, 555 (1965). Epstein, P. S., 2. Physik 64,537 (1929). Eucken, A., Bertram, A., 2. Physik. Chem. B.31,361 (1936). Fraula. H. C.. Brit. J . Ind. Med. 13, 196-201 (1956). Friedlander, S . K., Johnstone, H. F., Ind. Eng. Chem. 49, 1151 (1957). Fuchs, h. A,, Yankavskii, S., Dokl. Akad. Nauk, SSSR 119, 1117 (1958). Goetz, A.,APCA J . 12, 479 (1962). Gordon, &I. T., Orr, C., Jr., Air Repair 4, 1-4 (1954). Hunsmann, W.. Z . Elektrochem. Anoew. Phusik. Chem. 44. 211 (1923). ‘ Jackson, J. M., Mott, N. F., Proc. Combridge Phil. SOC.28, 136 (1932). Jacobsen, S.; Brock, J. R., J . Colloid Sei. 20, 544 (1965). Kaminsky, M., “Atomic and Ionic Impact Phenomena on Metal Surfaces,” p. 57, Springer, New York, 1965a. Kaminsky, M., “Atomic and Ionic Impact Phenomena on h4etal Surfaces,” p. 58, Springer, New York, 1965b. Keng, E. Y. H., Orr, C., Jr., J . Colloid Interface Sci. 22, 107-16 (1966). Kitto, P. I-I.,J . Chem. Metall. Mining SOC.52, 284-306 (1952). McAdams, W. R., “Heat Transmission,’’ McGraw-Hill, New Ynrk. - -...I lWi4
Literature Cited
Amdur, I., Guildner L., J . Am. Chem. SOC.79, 311 (1957). Blodgett, K. B., Langmuir, I., Phys. Rev. 40, 78 (1932). Brock, J. R., “Experiment and Theory for the Thermal Force in the Transition Region,” in press, 1969. Brock, J. R., J . Colloid Interface Sci. 23, 448-52 (1967). Brock, J. R., J . Colloid Sci. 17,768 (1962a). Brock, J. R J . Phys. Chem. 66, 1793 (196213). Brown, R. %., “Thermal Accommodation of Helium on Beryllium,” thesis, University of Missouri, 1957. Cawood, W., Trans. Faraday SOC.32, 1069 (1936). Davies, C. N., “Aerosol Science,” p. 181, Academic Press, New York, 1966. Derjaguin, B. V., Bakanov, S. P., Dokl. Akad. Nauk S S S R 147, NO. 1 139-42 (1962). Derjagdin, B. V.,Bakanov, S. P., Kolloid Zh. 21, 377 (1959). Derjaguin, B. V., Bakanov, S. P., Rabinovich, J. I., Proceedings 1st National Conference on Aerosols, Czech. Academy of Science, p. 197, 1965.
O’Brien, J. E., Wastes Eng. 33, (1962). Postma, A. K., “Stydies in illicrometrics,” Part 11, Atomic Energy Commission Research and Development Rept. HW-70791 (1961). Rolf. P.. Phws. Rev. 65. 185 (19441 Rosenbfatt, ‘P.,LaMer; V. K.,Phis. Rev. 70,385-95 (1946). Roivley, H. IT.,Evans, W. V., J . Am. Chem. Soc. 57,2059 (1935). Schadt, C. F., Cadle, R. I)., J . Colloid Sci. 12, 356 (1957). Taheri, W., Ph.1). thesis, Pennsylvania State Universitv. “. University Park, Pa., 1967. Taheri, M., Barton, R., Annual Conference, A.I.H.A., 1967. Wachtell, G. P., Waggener, J. P., “Flow Stahility of Gas-Solids Suspensions,” p. 34, Franklin Institute, Philadelphia, 1964. Waldmann, L., Z. Naturforsch. 14a,589 (1959). RECEIVED for review April 17, 1968 ACCEPTED June 25. 1969 Investigation supported by Public Health Service research grant AP00320 from the National Air Pollution Control Administration.
T U R B U L E N T M I X I N G OF T W O SPECIES W I T H AND W I T H O U T CHEMICAL REACTIONS H. L . T O O R Carnegie-Mellon University, Pittsburgh, Pa. 16213 The mean of the product of the fluctuations of the concentrations of two species in a turbulent mixture is plus or minus 1 times the product of the RMS concentration fluctuations of the individual species when the molecular diffusivities of the two species are equal. In an ideal mixer the mean of the product of the fluctuations is simply related to the decay law of the mixer and the relationship is identical to that found for very rapid irreversibe second-order chemical reactions when the reactants are fed in stoichiometric proportions. Limited experimental data for slow irreversible reactions are consistent with the hypothesis that the mean of the products of the fluctuations with a slow reaction is independent of the speed of the reaction.
HEN chemical reactions take place in a turbulent fluid in which the concentrations are nonhomogeneous, the timeaverage rate of reaction depends upon the fine scale concentration field if the reactions are not firstcorder. The second-order irreversible react,ion rA=-k(cA
+
cA)(cB$.cB)
(1 )
on a time-average basis becomes ?A
= -k ( C A c B
+
Z B )
(2 )
when the reaction is isothermal. Thus the prediction of the
=.
time-average rate of reaction, given the time-average conThis is centrations, necessitates a knowledge of apparently a difficult problem in general, since FB presumably depends upon both the turbulent mixing and the chemical reaction. Two highly simplified models (Kattrtn and Adler, 1967; Mao and Toor, 1969) allow some predictions of this quantity, but neither gives much insight into the physical situation. Although & can be positive or negative, depending upon whether the reactants are premixed or not, the influence of the term is much more profound in the latter case, for with VOL.
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NO.
4
NOVEMBER
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