Thermal precipitation and particle conductivity

JOURNAL OF COLLOID AND INTERFACE SCIENCE 22, 107-116

(1966)

Thermal Precipitation and Particle Conductivity 1 EDWARD Y. H. KENG AND CLYDE ORR, JR. Georgia Institute of Technology, Atlanta, Georgia Received May 10, 1965 ABSTRACT

Aerosol particles in a thermal gradient experience a force tending to drive them in the direction of the lower temperature. The force has been described both as strongly dependent and as almost independent of particle thermal conductivity. An analysis of new precipitation data involving both high and low conductivity particles reveals thermal forces to be of almost identical magnitudes regardless of conductivity. INTRODUCTION Aerosol particles within a thermal gradient, as established between two surfaces at different temperature, tend to move in the direction of the lower temperature. The force producing this motion originates from the physical interaction of the molecules of the suspending gas and the particles. Three conditions characterized by the magnitude of the Knudsen number, the ratio of the molecular mean free path X of the gas to the particle radius a, are believed to prevail. Thermal force when the Knudsen number is great is probably produced by the impinging of nonisothermal gas molecules on the particles; the nonisothermal gas molecules existing because of the temperature gradient. This force will be developed even if a particle is uniform in temperature throughout. When the Knudsen number is small, i.e., much less than unity, the thermal force is believed to arise mainly from the net impulse of impinging and departing gas molecules due to the particles' thermal inhomogeneity. Phenomena characteristic of both the high and low Knudsen numbers are believed to share responsibility more or less equally in the intermediate region. Because of the complicated nature of the force mechanism and uncertainties regard-

ing the temperature gradients within the particle and the gas, a large number of thermal or radiometer force equations (1-10) have been proposed. The one which appeared to be most appropriate (11-13) at low Knudsen numbers for a number of years was developed by Epstein (3). However, when high thermal conductivity particles were investigated (14-16), the thermal force as calculated by Epstein's equation was found to give results much lower than experimental ones. The higher the thermal conductivity of the particle, the greater was the discrepancy. According to Epstein's analysis, the temperature gradient on the particle surface is assumed related by thermal transport to that of the gas. This introduces into the resulting force equation the thermal conduetivities of both the particle and the gas. Epstein's equation shows a thermal force Ft directly proportional to the particle radius a, the gas temperature gradient AT, the gas constant R, and the gas viscosity 7; it is inversely proportional to the molecular weight M and the pressure P of the gas; and it is markedly dependent upon the relative value of the thermal conductivity of the gas kg and the particle kp. The expression is Ft

97rR~2a( k~ ) MP ~ + ~ AT

[1] l Investigation supported in part by the NationM Seienee Foundation through Research if the system is considered to be a spherical particle suspended in a motionless homoGrant NSF-G7051. 107 Copyright © 1966 by Academic Press Inc.

108

KENG AND eRR, JR.

geneous gas. The radius of the particle is assumed sufficiently large compared to the molecular mean free path for the fluid to be regarded as a continuum. Believing that Epstein's analysis included incorrect assumptions relating to the boundary conditions for the slip-flow regime and introduced other errors by neglecting the convective terms in the continuum energy ,equation, Brock (9, 10) derived an equation describing thermal force in the slip-flow regime (k/a < 1) for spherical particles. A tangentiM momentum first-order slip co.efficient C~, a temperature jump first-order slip-flow coefficient Ct, and a thermal creep first-order coefficient Ct~ were introduced. The equation obtained is

[2]

where AT is the uniform temperature gradient through the medium. Unfortunately, the coefficients cannot presently be calculated but they are subject to experimental estimation. In this study the coefficients were estimated by a semimacroscopic analysis. By means of these coefficients, calculated and experimental thermal forces are compared. EXPERIMENTAL METHODS AND OBSERVATIONS The precipitator used in this study employed a radiM flow of aerosol between two horizontal circular plates concentrically mounted in such a way that the separation between them could be readily varied by the inclusion of shim pieces (17). The uppermost plate was electrically heated. The lower plate, recessed to accommodate a removable glass collecting disk, was water cooled. The heated plate temperature was controlled by a variable transformer and its temperature was continuously monitored. The cold plate

temperature was regulated by the cooling water flow rate. An aerosol having either air or nitrogen as the suspending medium was introduced through an inlet in the center of the heated, upper plate. It then flowed radially outward between the hot and cold surfaces. The flow rate of the aerosol and the separation of the plates were carefully adjusted so that steady laminar flow was maintained. The particles were precipitated upon the cold plate while the gas was exhausted from the system. The deposit pattern of the aerosol was symmetrical with respect to the center but it was not always a perfectly centered annular ring. The glass collecting plate was coated on both sides with a thin layer of Kel-F 3 (fluorocarbon oil, M. W. Kellog Company) to insure good thermal contact on one side and retention of particles on the others. The temperatures of the hot and cold plates were set for the desired thermal gradient, and the system was allowed to come to equilibrium. A test aerosol was then generated and metered through the precipitator at a constant flow rate. Subsequently, the collecting disk was removed for examination and analysis. Thermal force values were computed from the maximum deposit dimension, the distance (Fig. 1) for particle precipitation being established by those particles traveling from the point (r = r0, y = + L ) , i.e., from the point of entry to the maximum point of deposition. The radius of the deposit was particularly sensitive to aerosol flow rate. A positive displacement mechanism utilizing calibrated reservoirs and a flowing fluid was developed to ensure constant flow at a series of rates. When the deposits were dense and sharply defined, distances were established by direct observation, with attention paid to particles of about 1 g diameter. This size was chosen as standard because it was the smallest that could be conveniently detected microscopically. The glass collecting disk, after being removed from the precipitator, was pIaced on a rectilinear grid having l-ram. divisions. The deposit was oriented with its center at the grid origin, and a series of eight radiM measttrements to the periphery of the deposit was made at 45° increments

THERMAL

PRECIPITATION

AND PARTICLE

109

CONDUCTIVITY

Aerosol Inlet !

2L

J

r

FIG. 1. Geometric system under 30X magnification. The average of these individual measurements was recorded as the deposit radius. The peripheries of less dense deposits were not so easily detected. For these, an alternate technique, which employed a traversing mechanism and a sharply focused pin point of light, was developed. The light, in traversing the surface of the glass collecting disk, brightened noticeably when it encountered even a faint deposit. Radial distances were then measured directly from the vernier scale of the calibrated traversing mechanism. For particularly difficult determinations, examination was made using an optical microscope at magnifications up to 800X; distances were then obtained from a series of four diametric measurements recorded at two or more magnifications. Electron micrographs were employed for detailed study of the more finely divided materials. Thermal forces were computed as described below from the experimental data using an IBM-650 compurer. The most extensive experiments were made with magnesium and aluminum oxide aerosols. They covered a variety of flow rates and temperature gradients from 500 ° to 2500 ° C./cm. at precipitator plate separations of 0.04 and 0.07 cm. Other experiments were conducted using aerosols generated by the exploding wire technique (18). In this way, deposits were obtained with particles of silver, platinum, aluminum, zinc, iron and sodium chloride.

METHOD

OF CALCULATION

Since the flow rate of each aerosol was controlled to produce a steady laminar flow and precipitator plate separations were kept small to minimize entrance effects, a parabolic velocity profile through the precipitator was assumed. It may be described by

dr_ u

dt

3Q 8~r~

1-

[3] ~

'

where Q is the volumetric flow rate of the aerosol, r the radius, and L one-half the plate separation (Fig. 1). The thne, ts, needed for a particle moving from r0 (initial radius) to ~ (deposition radius) was obtained by assuming the falling velocity of the particle to be constant and to be given by Vy = 2 L / t f . On the assumption also that the particle and fluid velocities are identical, integrating Eq. [3] gives ts =

27rL 2 ~ (ra -- ro2).

[4]

Then the falling velocity of particles can be expressed by Vy -

2L _ Q ts ~(rJ -- r02) "

[5]

As the molecular mean free path of the gas, k, becomes comparable to the particle radius, a, the Stokes' law equation must be modified by inclusion of a slip correction factor, S, which is a function of ~/a. The

110

KENG AND ORR, Jl~.

total force acting on a particle can be expressed by F-

Vy_ Z~

6naQ (r~~ - ro~)5 '

[6]

where Z~ is the particle mobility corrected for slip effect and is given b y Z~ = S/6~r~a. T h e gravitational force m a y be resolved f r o m a knowledge of the particle r~dius, a, a n d density, pp, i.e., Fg = 4//33~ra~ppg,

[7]

where g is the gravitational constant. T h e thermal force, F t , acting on a particle is therefore Ft -

6naQ (rd2 _ ro2)S

4 3 ~ra p~ g.

[8]

T h e slip correction factor S, as investigated b y K n u d s e n and W e b e r (19), is expressed b y the empirical relationship S = 1 + ~- [A + Be-C(~/×)], a

[9]

where the best vMues of the empirical constants A, B, and C are, respectively, 1.257, 0.400, and 1.10 according to an analysis b y Davies (20). T h e molecular m e a n free p a t h of the gas, ~, can be calculated b y )~ _

nR T 0.499MP~

_

( ~ r R T ~ ~/2 n 0.499P \ 8 - M - ] ' [10]

where ~ is the m e a n molecular speed given b y ( 8 R T / T r M ) ~/~. A n error analysis was m a d e to validate the ~bove assumptions and simplifications (16). T h e two m o s t significant factors are the entrance effects and the nonlinearity of the temperature profile. F o r the former a reasonable estimate of error was + 3 0 % to - 5 0 %, the lower limit occurring w h e n t h e particle entered the precipitation zone at mid-point. U n d e r this condition the error due to a nonlinear t e m p e r a t u r e profile is greatest. T h e probable combined error of these two effects m a y t h e n be reasonably assumed to be from + 3 0 % to - 7 5 %. W i t h t h e nominal error which m i g h t be ascribed to all other simplifications, it was concluded asp rest unlikely t h a t the actual force could be more t h a n twice the deduced force or less t h a n one-fifth its value.

RESULTS T h e experimental d a t a for m a g n e s i u m oxide and a l u m i n u m oxide are presented in Tables I and II, respectively. Calculated results for these and the other materials are presented in Table I I I . All the d a t a were t a k e n at atmospheric pressure and the calcuTABLE I EXPERIMENTAL D A T A FOR iV[AGNESIUM OXIDE PARTICLES Temperature gradient

Aerosol flow rate

Observed travel of 1# diameter particles (cm.)

(dynes/X 10-s)

841~ 844~ 847~ 844~ 8714 984~ 1127~ 1406~ 875~ 882°

371.3 396.8 425 476 403.2 420.2 402.7 135 324

2.66 3.06 3.06 3.16 3.37 2.96 3.01 2.49 1.72 2.94

2.90 2.40 2.41 2.44 2.42 2.61 2.63 3.58 2.18 2.08

503b 980b 985b 988b 988b 1478b 1963b 2463b

194 180 188 200 201 194 221 198

3.11 1.86 2.11 2.16 2.19 1.76 1.79 1.31

1.04 2.60 2.16 2.21 2.17 3.10 3.46 5.26

(°C./cm.)

(c.e./min.)

400

Calculated thermal force

a Precipitator plate separation, 0.07 cm. b Precipitator plate separation, 0.04 cm. TABLE II EXPERIMENTAL D A T A FOR /~kLUMINUM OXIDE PARTICLES Temperature gradienta

Aerosol flow rates

988 1003 1450 1488 1980

200 330 267 265 200

2.8 3.44 2.9 2.40 1.60

4.08 4.14 5.98 6.14 8.17

1480 1492 1488 1490 1485

182 321 358 400 467

1.94 2.67 2.79 2.84 3.18

6.18 5.53 5.69 6.11 5.65

(°C./em.)

(c.e./min.)

Observed travel Calculated of ltt diameter thermal force particles (cm.) (dynes X 10-s)

a Precipitator plate separation, 0.04 cm.

111

THERMAL P R E C I P I T A T I O N AND PARTICLE CONDUCTIVITY TABLE I I I COMPARISON

Particle material

OF

THE

EXPERIMENTAL

Exploding a concentrated aqueous solution Atomized powder Mg burned in air Atomized pigment Exploding wire Atomized pigment Atomized pigment Exploding wire

Aluminoum oxide Magnesium oxide Iron Platinum Zinc Aluminum Silver

THEORETICAL

VALUES

Absolute thermal force values per temperature gradient, ]Ft/AT I X 10n Experimental Eq. [l] Eq. [2]

Particle thermal conductivity (cal./cm. sec. °K.)

Aerosolization technique

Sodium chloride

AND

(dyne cm./°C.)

(dynecm./°C.) (dyne cm./°C.)

0.0155

3.22

0.046

1.86

0.08 0.09 0.14 0.167 0.265 0.504 0.963

1.89 2.30 2.14 3.35 2.19 2.17 2.86

0. 011 0.0092 0.0058 0.0043 0.0029 0.0015 0.0007

1.85 1.84 1.83 1.84 1.82 1.81 1.84

20

15

• Magnesium Oxide Temperature Gradient:

860 °C/cm.

O Aluminum Oxide Temperature Gradient:

1485 °C/cm.

@ o ~

0

!0 O

!

0 0

I00

,I

I

200

~00

I

400

;00

Aerosol Flow Ratej co/re_in FIG. 2. Conformity of experimental results with Eq. [11]

lations were based on an average particle diameter of 1 u, this being the size upon which attention was concentrated when making measm'ements. The congruity of the experimental data

was checked as follows: Rearranging Eq. [6] yields

2

(,~

(6va~

-

r02) = ~,~)Q,

[111

112

KENG AND ORR, JR. ~o

I

i

I

i

35

3o

o

25

¢G (D

20 I

~o !

!0

/•/

// //

o~

,

i

500

0

Flow Rate:

40 cc/~.n

•

Flow l~ate: 200 cc/min Plate Separation: 0.04 cm

o ratio : oo7

I

f

i

lO00

1500

2000

I

3OO0

2500

Temperature Gradien% °C/am.

FIG. 3. Conformity of experimental results with Eq. [13] which indicates that the quantity (r~2 - r02) should be directly proportional to the aerosol flow rate, Q. T h a t the data verified this linear relationship is shown by Fig. 2. A similar rearrangement of Eq. [8] predicts an essentially linear relationship between the reciprocal of (rd2 - r02) and the thermal force, i.e., that (~d 2 - -

9.02)--1 - -

S

Ft

6nard Q S _ _ 6~a~d Q

[12]

where a~d represents the critical particle radius evaluated at the periphery of the deposit. Since the thermal force is directly proportional to the temperature gradient,

AT (i.e., Ft = ahT, where a is a coefficient depending upon the properties of the gas medium and the particles), Eq. [12] m a y be rewritten m

6~aT~ Q

AT S + --F~. 6~ard Q

[13]

The quantity (r~2 - r02)-1 would be a linear function of AT if a and ar~ are constant. In general, a~ is dependent upon the relative magnitudes of the thermal and gravitational forces and is, therefore, subject to a slight variation with changing temperature gradient. The assumption of constant ~ and a~ is sufficiently valid to permit linear interpo]a-

THERMAL

PRECIPITATION

AND

300ol

PAI~TICLE

CONDUCTIVITY

113

f

2500

2000

15oo

I000

500

00

I

500

I

ZOO0

I

l~oo

i

~oo0

z~ 0

Temperature Gradien% °C/cm

FIG. 4. Data for magnesium oxide aerosols plotted in accordance with Eq. [16]

tion of the experimental data over a limited temperature gradient range. These data are :shown in Fig. 3.

The experimental values of I Ft/ATI (where

Ft is the thermal force acting on the particle

and AT is the average temperature gradient) are of the same order of magnitude regardDISCUSSION OF RESULTS less of their thermal conductivity under the Caution must be exercised in drawing same value of X/a (Table V). critical conclusions regarding thermal force Brock's theory suggests that the thermal magnitudes from precipitation data as ob- force does not depend strongly on the tained here, but the results are sufficiently thermal conductivities of either the particles abundant for confidence to be placed in the or the gas medium. This agrees with the exrelative measurements. The errors due to perimental results. Since there are difficulties method as well as to systematic experi- in evaluating the coefficients of the later mental inaccuracies are basically identical equation (Eq. [2]), rigorous comparison when similar tests are conducted with dif- cannot be made. The most complete data are ferent aerosols. those for magnesium oxide. The experiThe thermal forces for particles of high mental thermal force data were taken at thermal conductivities as computed from sufficiently low ~/a (0.13 < X/a < 0.16) for the experimental results disagree with Cm to be reliably taken as 1,257 (21). The Epstein's theory. They are much greater. following equation (9, 21) can then be

114

KENG AND ORR, JR. I

k

5.0

-a

4.5

O

i

0.15

=

Q = ~00 ec/mln Al~min~m Oxide

@ ~guesium Oxide 4.o

O

3.5 CO O

3.0

©

2.5

O

0 0

! ©

2.0

~a.[2]

1.5

1.0~

•

o . 5 ~ < ~qi[i] )0

500

I000

1500

25oo

2000

Temperature Gradient, °C/cm

FIG. 5. Comparison of experimental results with theoretical equations

employed to evaluate Ct :

Ct -

4Ck~C~ (~ + lbCv'

[14]

where C = 0.499 and ~ is the ratio of the heat capacity of air at constant pressure to that at con~stant volume, Cp/C~,. The value of Ct so calculated is 1.98. According to the analysis, Ct,~ is given by c~

3 { 2~ ~1/2,

= ~ \-~-~]

ured thermal force to the following equation which is obtained by rearranging Eq. [2] and setting E equal to the ratio of the experimental value of Ct,~ to that given by Eq. [15], i.e.

[1 + 3c~ (~)1[1 + 2kz k~

[15]

where R is the gas constant per mole of gas. A value of Ct., thus can be obtained from the experimental results by applying the seas-

+2Ct(~)l,Ft 9~

2R ~112

Ct X

' [16] X

=AIF~I,

THERMAL PRECIPITATION AND PARTICLE CONDUCTIVITY where A depends on AT and X/a for each powder substance. The slope of the line, A I Ft I versus AT in Fig. 4, gives the value of e; it is determined to be 1.17. Brock (9) evaluated ~ by a different arrangement of other investigators' data but the values of e were also close to 1.0. This means that the value of Ct,~ calculated by Eq. [15] is very close to the value obtained by inserting the experimental thermal force into Brock's equation. In other words, if Eq. [15] is accurate, then Brook's equation agrees very well with the experimental results. Equation [2] can also be rearranged to

~1=127ra2~ X

tc~

If the previous value of Cm is used (1.257), and Ct and Ct~ are calculated from Eqs. [14] and [15] the average experimental values of I Ft/ATI may be compared with theory as in Table III. The values of I Ft/ATI calculated by Brook's equation are only approximate because it is impossible to make rigorous calculations of the coefficients C~, Ct, and Ct,~ • Nevertheless, it may be seen that Broek's equation results agree with the experimental data much better than Epstein's equation, which fails completely with the high thermal conductivity particles. In fact, the apparent thermal forces seem to be independent of the thermal conductivity of the particulate material from 0.01 to 0.963 cal./cm, see.°K. This is explained once it is accepted that the thermal force depends more strongly on the coefficients C~, C~, and Ct,, than on the thermal conductivity of the particle. This could also explain why larger forces had been observed for small-particle aerosols generated by the exploding wire technique than those generated by dispersing power of the same substances. The difference in surface conditions influences the thermal force magnitudes even though the thermal conductivity is believed to be the same or similar.

115

Table III also shows the dependence of I Y J A T l o n the thermal conductivity of particles kp. Epstein's equation gives very low values of I Ft/AT I at large kp, whereas Brook's equation values and the available experimental data fall in a narrow region of I Ft/ATIfor all values of kp. The experimental values for low thermal conductivity agree with Epstein's equation. This accounts for the later equations being considered satisfactory previously. Experimental and calculated results for aluminum and magnesium oxides are also plotted in Fig. 5. Brock's equation Shows good agreement whereas Epstein's equation falls far below. It must be noted here that the theoretical lines do not represent the equations exactly owing to the uncertainties of evaluating some of the coefficients and particle properties. However, it is believed that a reasonable estimation has been made and only slight deviations are to be expected. The theoretical results for the aluminum and magnesium oxides are very nearly identical and are represented on Fig. 5 as only one line for each equation. CONCLUSIONS Airborne particles of high thermal conductivity are readily precipitated in a thermal precipitator. The forces on particles of greatly different thermal conductivity are of almost identical magnitudes. Within the limits of reliability with which the several constants can be evaluated, the force agrees with the prediction of Brock (9, 10). ACKNOWLEDGMENT Most of the experimental measurements as reported herein were made by Mr. T. W. Wilson now with E. I. duPont de Nemours and Co., Gibbstown, New Jersey. REFERENCES 1. BAKANOV, S. P., AND DERYAGIN, B. V., Kolloidn. Zh., 21, 365 (1959). 2. 3. 4. 5. 6.

EINSTEIN, A., Z. Physik 27, 1 (1924). EPSTEIN, P. S., Z. Physiic. 54, 537 (1929). HETTNER~G., Z. Physik 27, 12 (1924). HETTNER, G., Z. Physil~ 37, 179 (1926). ]~VBINOWICZ,A., Ann. Physik 62, 695 (1920).

7. WA~DMANS, L., Z. Naturforseh. 14h, 870 (1959). 8. DERJAGUIN, B. V., AND BAKANOV, S. P., Nature 196, 669 (1962).

116

K E N G AND ORR, JR.

9. BROCK, J. R., J. Phys. Chem. 66, 1793 (1962). 10. BROCX, J. R., J. Colloid Sci. 17, 768 (1962). 11. ROSENBLATT, P., AND LAMER, ~. K., Phys. Rev. 70, 385 (1946). 12. SAX~ON, R. L., ~ND RANz, W. E., J. Appl. Phys. 23, 917 (1952). 13. SCHMITT, K. H., Z. Naturforsch. 14A, 870 (1959). 14. SCHADT, C. F., AND CADLE, R. D., J. Colloid Sci. 12, 356 (1957). 15. SCHADT, C. F., AND CAELE, R. D., J. Phys. Chem. 65, 1689 (1961). 16. W~LSON, T. W., McALIsTER, J. A., AND ORR, C., JR., "An Investigation of Thermal

17. 18. 19. 20. 21.

Force with Particular Reference to Materials of High Thermal Conductivity," Final Report, Project B-159, Georgia Institute of Technology, Atlanta, 1961. KETI-ILEY, T. W., GORDON, M. T., AND ORR, C., JR., Science 116, 368 (1952). KAR~ORIS,F. G., ~ND FISH, B. R., J. Colloid Sci. 17, 155 (1962). KNUDSEN, M., AND WEBER, S., Ann. Physik 36, 981 (1911). DAVIES, C. N., Proc. Phys. Soc. 57, 259 (1945). KENNARD, E. H., "Kinetic Theory of Gases," 1st ed. McGraw Hill, New York, New York~ 1938.