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Deposition of particles on rough surfaces during turbulent gas-flow in a pipe
DEPOSITION DURING
OF PARTICLES ON ROUGH SURFACES TURBULENT GAS-FLOW IN A PIPE L. W. B. BROW&X
Departmenf of Mechanical Engineering, University of Newcastle, N.S.W. 2308, Australia (first
rewired
30 Nocembrr
1972 and infinalform
15 Vorember 1973)
Abstract--The importance of surface roughness on the deposition of particles from a flowing gas stream is well known but there has been little theoretical work that helps predictions to be made. The theory presented here follows the general approach of Davies (1966) who considered deposition on smooth surfaces. To take account of surface roughness extra distances have been added to the particle-capture distance and, by shifting the position of the origin of the velocity profile. it has been possible to use the same equation for the radial velocity fluctuations. The effect of surface roughness on the friction velocity is dealt with by empirical equations. Details of the analysis are given together with a computer solution procedure. Results of calculations are compared with the experimental results of other workers and a reasonable degree of agreement is shown. The effect of small changes in surface roughness on deposition velocity are shown to be so large that there is difficulty in making precise predictions of deposition rates and results from apparently similar experiments may vary widely. NOMENCLATURE a
a* c C’
2. co
d D e
i 9 k k’ K L ni, P PLI R R’ R, R, S, .% r 11” u t r+
particle radius dimensionless particle radius = a u*/v particle concentration (mass per unit volume) dimensionless particle concentration = C/C, particle concentration at distance A from pipe wall dimensionless particle concentration at A* from pipe wall = Cd-/Co centre-line particle concentration pipe dia. molecular diffusion coefficient displacement in origin of velocity profile-Fig. 1 and equation (15) dimensionless displacement in origin = eu*/v Fanning friction factor for pipe slip correction factor-equation (22) acceleration due to gravity average height of pipe surface roughness (or equivalent surface roughness) dimensionless height of pipe surface roughness = ku*/v Boltzmann’s constant-equation (21) pipe length mass deposition rate of particles per unit area of pipe surface gas pressure pressure drop in pipe pipe radius dimensionless pipe radius = Ru*/v Reynolds number of pipe flow = dU/v Reynolds number of Row around a free-fall particle when its terminal velocity is reached = Zaar.‘r particle stopping distance dimensionless stopping distance = $u*/v gas temperature friction or shear velocity-equation (3) mean velocity of gas radial fluctuation of velocity dimensionless radial fluctuation of velocity = c/u* 801
L. W. B. BROWYE radial fluctuation of velocity at distance A from pipe wall dimensionless radiat fluctuation of velocity at distance 4’ from pipe wall = V, u+ terminal velocity of particle under free-fall conditions velocity of deposition of particles dimensionless velocity of deposition of particles = C:u* distance from pipe wall dimensionless distance from pipe wall = J u*!lv capture distance--equation (6) (measured from origin of velocit> profile) dimensionless capture distance = Au*/v eddy diffusion coefficient kinematic viscosity of gas density of gas density of particles S.D. of average height of pipe surface roughness dimensionless standard deviation of average height of pipe surface roughness = G~W*v particle relaxation time-equation (I?) dimensionless particle relaxation time = TIP’ v dimensionless sum of various distances from pipe wall-equation (18) drag coefficient of particles under free-fall conditions-equation (IO).
INTRODUCTION
A most useful review of the deposition of dust from turbulent gas streams has been given by Kneen and Strauss (1969). Among subsequent papers are those of Beal (1970a). Montgomery and Corn (1970) and Beal (1970b). Wells and Chamberlain (1967) and Chamberlain (1967) found that particle deposition was increased markedly by surface roughness. However, the work of Chamberlain (1967) appears to have been the only attempt to predict deposition rates on rough surfaces. The purpose of this paper is to present an approach suitable for estimating deposition rates on rough surfaces in specific cases, and to show the zffect on deposition rates uhen various parameters are changed. The general approach is the same as that used by Davies (1966), but includes the effect of surface roughness. Not included in this analysis are the effects of electrical charge, particle shape, re-entrainment, surface stickiness gravitational settling and other physical or chemical properties of the particles and surface.
pofile
Fig. I. Assumed model for particle capture and particle concentration
SO3
Deposition of particles from a gas stream MODEL
USED
IN
ANALYSIS
Figure 1 shows the model used in this analysis. At some distance, A, from the wall, the radial component, v~, of the turbulent velocity will be such that a particle with that velocity will be unable to stop before its surface touches the upper parts of the wall surface roughness. It is assumed that when a particle reaches this position it will remain on the surface: ni, = VAX,.
i.e. Now. by definition
(1) So that
or dimensionlessly I’? = 0; ’ cc.
(3
Thus in order to evaluate V it is necessary to determine A, u and C. In determining these quantities the friction velocity is required and consequently its evaluation will be discussed first. ESTIMATION
OF
FRICTION
VELOCITY
u*
The friction velocity is related to the Fanning friction factor by
The Fanning friction factor is related to the pressure drop in a pipe by
Thus by carrying out measurements of the pressure drop, equations (3 and 4) can be used to determine the friction velocity. Where this is inconvenient, use can be made of empirical formulae for the Fanning friction factor for rough pipes. Colebrook and White (1937) and Colebrook (1939) correlated the whole flow region from hydraulically smooth to completely rough with an equation of the type
Using the more recent equations of Bennet and Myers (1962) for smooth flow (k+ 0) and completely rough flow (R, + x ) the following values were obtained A1 = 2.14 A2 = 4.06 A3 = 4.73
i.e. 2.14 - 4.06 log,,
(5)
Ya4
L. h’.
B.
6KOWSE
There is no analytical solution to equation (5) and it is necessary to evaluatef‘by cal methods. e.g. by using the interval halving technique. ESTIMATION
OF
CAPTURE
DISTAKCE
numeri-
il
From Fig. 1 (where it is assumed that contact with any roughness element results in capture). A=Sd+a-+~+~L-e
t.6)
Or Ai = s;
f cl+ + k- +
G;
-
e-.
(7)
1. Estimation of‘s, The stopping distance, S,, is the distance travelled by a particle when given an initial velocity of c,. This is obtained indirectly by estimating the terminal velocity, or. of the particle during free-fall conditions. Davies (1945) fitted experimental data with the following equations: Rr = ‘3 - 2.3363 x 10-‘(ll/R+)2 + 2.0154 x 10-6(1LRt)3 - 6.9105 x 10-g($R$)J 24
(8)
limits of applicability, $R+ < 120,
log,,(R7.)= - 1.29536 f 0986(log,,
$R+) - O.O46677(log,, $R+)?
+ 0~0011235(log,, IC/R;)3.
(9)
where (IO)
and R,=-
Znc, 1’ .
(11)
The limits of applicability are 120 < cj/R$ -e 45 x lo-‘. Thus for a particle of given radius and density, and for a gas of given density and viscosity, equation (10)can be used to determine II/R& equations (8) or (9) can be used to determine R,, and equation (11) can be used to determine the terminal velocity L’~. The relaxation time. r, is the time for a particle projected into a gas. with a given initial velocity, to reach l/exp 1 of its initial velocity (Fuchs. 1964). If a particle was given an initial velocity equal to its above terminal velocity. then (12) It is readily shown that the stopping distance is the initial velocity multiplied by the relaxation time, i.e. s, = CA.% or s; = r;.r-. (13) the evaluation of which is discussed later. This requires a knowledge of L~L\,
Deposition of particles from a _easstream
2. Estimation
so5
of k and a,
These are best based on actual measurement. Alternatively an equivalent surface roughness height. k, can be estimated and or; estimated from Csk= 0,17k or 0: = O.l7k+. 3. Estimation
(l-1)
ofe
As early as 1951 it was suggested that a boundary layer on a rough wall behaves as if its origin is located some distance below the crests of the roughness elements (Moore. PhD Thesis, quoted by Perry et al., 1969). This means that the effect of the surface roughness on the velocity profile outside the sub-layer is to shift the origin to some point below the crests of the roughness. Other workers have supported this view, e.g. Rotta (1962), Perry et aI. (1969) and Grass (1971). A definitive equation for determining the ‘-displacement in origin” is not yet available but the data of Grass (1971) are correlated by the following equation, e/k = 0.53 + 0.0034k”
(15)
Or e+
=
0.53k’
+ OGl34k+‘.
(16)
Grass’s data are for channel flow but it is not unreasonable to assume that near the surface a similar mechanism exists in pipe flow and that equation (15) can therefore be used. ESTIMATION
OF
RADIAL
VELOCITY
FLUCTUATION
V~
Davies (1966) correlated existing experimental data for turbulent flow in a smooth pipe and obtained YC c+ = J’c. Provided Y+ is measured from a shifted origin, as discussed above, the same equation can be applied to flow in a rough pipe. Thus CT- =
AC A+ + 10’
(17)
However, A+ itself requires a knowledge of us [see equations (7 and 13)], so that c_\‘and Ai must be evaluated simultaneously. Substitution of equation (13) into equation (7), and the result into equation (17) gives
unf*5+
VA?= + CA-
+ I#)+
‘Tf + $’ + 10
(18)
where 4” = a+ + k’ + 0: - e+. This leads to a quadratic equation in v:+ with a solution 1 v&t =-- 1 - Q’flO)+ 2(
J[(l-“+yy+q
(1%
L. W. B. BROWSE
so6 ESTIMATION
OF
CONCENTRATIOK. FROM
C,. WALL
.AT CAPTLRE
DIST.AXCE
For one-dimensional mass transfer in turbulent flow, the mass flux perpendicular surface is constant and can be expressed bq dC riz, = (D i E) -_ dy
to a
From equation (1) this is
or
(20) Now KTF
D _=\
(21)
a\
where F = 1+
gC6.32 +2.01 exp(-2190Pa)]
(22)
PincmofHg a in cm and
Equations (21 and 23) are experimental correlations equation (20) from y+ = A’ to ,v+ = R’ gives
of Davies (1945, 1966). Integration of
i.e.
where
r
R-
I=
,A-
dy’ (Dlv + E/V)’
(24)
Thus c,‘:
= I-
v+i.
(25)
C,? could be expressed in terms of v~‘:and I by using equation (2). However. as shown. I” can now be calculated.
507
Deposition of particles from a _easstream CALCULATION
OF
VELOCITY
OF
DEPOSITION.
V
The substitution of equation (25) into equation (2) gives
i.e. I’-
=.
1 l/V,‘: + I’
(35)
The integral I can be evaluated numerically. A computer solution for C’can thus be obtained using the following procedure: (1) Input data-pipe dia., n pipe equivalent surface roughness, k mean flow velocity. a gas temperature, T gas pressure, P gas density. p (or can be included in calculations) gas kinematic viscosity, v particle radius, 0 particle density, pp. (2) Calculate the Fanning friction factor f using equation (5) and the friction velocity tl* using equation (3). Use equations (8-12) to calculate the particle relaxation time r, and consequently r+. Calculate 4’ as defined by equation (18). This requires the use of equations (14 and 16). (3 Calculate 0,‘: using equation (19). (6) Calculate A’ using equations (13 and 7). (7) Use equations (21-23) to determine the integrand of (24) and calculate I using a numerical method. (8) V and consequently Scan now be found using equation (26).
RESULTS To
see
OF
SOME
CALCULATIONS
the effect of varying different parameters, the above approach was used to determine deposition velocities under the following conditions: particle dia.-0.01-10.0 pm; particle densities-05-9.0 gem- 3 ; Reynolds numbers-500&40000; surface roughness ratios (k/Q--O-0.002. The gas was air at 20°C and 760 mm of Hg pressure. The pipe dia. was 1.5 cm. The results of these calculations are shown in Figs. 2-10. Features worth.noting are: (a) the very large effect of surface roughness, except on particles of size larger than about 10 pm dia.; (b) the small effect of particle density until the particle diameter exceeds about 2 pm dia.; (c) the rate of increase of deposition velocity with Reynolds number is greater for the rough surfaces, except when the particles are large, e.g. 10 pm dia.
SOY
L. W. B. BROWSE
Particle
dia..
pm
Fig. 2. Effect of particle size on deposition velocity for various surface roughnesscs. R, = ICJOO~: d = 15cm; p = 1Ggcm‘3. (a) k’J = 0; (b) ic:n = OooO1; (c) k.!d = 04EU5; id) k.J = @C%l: rz! kid = 04N2.
IO.' Port&e
dia.,
I
ptm
Fig. 3. Effect of particle size on deposition velocity for various surface roughaesses. R, fl: W)@: d = [-~cm; pp = tT)gcm-‘. (a) k/d = 0; (b) k/d = 04001; fc) k!d = OTH~OS:fdi k’d = @@II: i@ k/if = 0402.
Deposition of particics from a gss stream
Particle dia.
pm
Fig. 4. Effect of particle size on deposition velocity for various surface roughnesses. R, = 1OooO: d = t.jcm; pp = 2,0gcm-3. (a) kid = 0: (b) k,d = 09001: (c)kid = 03005: (d) 1;d = 0.001: (c) k ‘2 = 04B’.
,,i.,,,,,,,
2
3
4
Reynolds number, Rex lO’4
Fig. 5. Effect of Reynolds number on deposition velocity for various surface roughmsses. Particle dia. = 0.1 pm; d = 1‘jcm; pp = l~0gcm-3. (a) k:d = 0: (b) kjd = O+XJO1: (c) k d = OQOO5: (d) k/d = 0.001: (e, k/d = OGO2. *.E.
%8--c
310
I
Reynolds num
2
her,
3
4
Re ~10~~
Fig. 6. Effect of Reynolds number on deposition velocity for various surface rouglmesses. Parti& dia. = l,Oilm; ri = 1.5cm: pP = 14gcm-3. (a) k’d = 0; (b) k,frf= 04001: (cf k’(i = 04005: [d) k, li = OGOI: (e) k,‘d = 04302.
I
2
3
4
Reynolds number, &xIO’~
Fig. 7. Effect of Reynolds number on deposition velocity for various surface roughnesses. Parficiz dia. = 1Opm: d = l.Scm; p,, = I-0gcmV3. (a) k/d = 0; (b) kid = 04401: (c) k/d = O-OOO5;Cdl k/d = 0.001 : (e)kid = OGO2.
Deposition of particles from a gas stream
811
IO
5
-
E
-
;
-
.g IO.’ =
x
-
e
s ._
d
$
-
C
“0 if
0
C b
lo”_
10-a
’
I
’
I
’
1234s67e9
’
’
’
’
’
’
’
’
’
’
’
Particle density, g CJcm-’ Fig. 8. Effect of particle density on deposition velocity for various surface roughnesses. Particle dia. = 0.1 pm; d = I.5 cm; R, = 10000. (a) k/d = 0; (b) k/d = OGOOl: (c) k/d = 04005; (d) k/d = OGOI : (e)k/d = OGN.
Particle density.
g g cmw3
Fig. 9. Effect of particle density on deposition velocity for various surface roughnesses. Particle dia. = 2pm; d = 1.5cm; R, = lOONI. (a) k/d = 0; (b) k/d = OGOOl; (c)k/d = 04005; (d) k/d = OGOI: (c)k/d= 0402.
L. LV.B.
Particle
BROWSE
density,
p, 9 cm3
Fig. 10. Effect of particle density on deposition velocity for various surface roughnesses. Particle dia. = 5 htm: n = I.5 cm; R, = 10000. (a) k,!d = 0: (b) k/d = 09001; (c) k,‘d = 04X305; (d) k’d = 0031;
COMMENTS
(5) k d = oax?.
OS
CURVE
SHAPES
The shapes of the curves in Figs . 2-4 can be explained as follows: equations (26 and 24) show that the deposition velocity depends on three factors: (1) the radial velocity fluctuation at the capture distance; (2) the molecular diffusion coefficient; (3) the eddy diffusion coefficient. The functional relationship between these quantities is not easy to explain qualitatively owing to the fact that an integration from the capture distance to the centre of the tube is involved. The capture distance for particles in a smooth tube is not greatly different from the particle radius, since the stopping distance for the low radial velocities near the wall is quite small. Equation (17) shows that. since 10 $ A-, the radial velocity fluctuations can be obtained by dividing the capture distance by 10.
Thus
cl-
n-
z 10’
If ri- were the only factor involved then the deposition velocity would decrease linearly as the particle size decreases-see equation (26) The effect of the integral I [equations (24 and 26)J on the deposition velocity can be determined bv noting that for rT = 5. E/V 2 0.1. This is much larger than the largest value
$13
Deposition of particles from a gxj stream
of D,'v.(D/v t 3 x 10e3 for a O.Ol-itrn dia. particle). Thus I can be split into two parts:
where
For small particles, D/vwill be much larger than E v throughout most of the region of 4’+ = a+ to 5. Since D r l/a, equation (21). I will increase as the particle size is increased in this region. For large particles, E/V will predominate in the region of _r- = a- to 5. increasing as the particle size increases. Thus I will decrease with increasing particle size. The total effect of us and 1 on the deposition velocity depends on the relative value of l/ rc and I, being equal to the inverse of the sum of these two quantities-see equation (26). From calculations, 1 is predominant for particles larger than about 0.1 pm dia. The shape of curve (a) in Figs. 2-4 can thus be explained. For particles of dia. 10 pm down to 0.1 pm I predominates and consequently for the larger particles with the eddy diffusion predominant the deposition velocity will decrease as the particle size decreases, while for the smaller particles the molecular diffusion predominates and the deposition velocity will increase as the particle size decreases. The change-over point occurs at a particle size of about 4 pm dia. For particles smaller than about 0.1 ,um dia. the radial velocity fluctuation predominates and the deposition velocity decreases as the particle size decreases. In the case of rough surfaces, the capture distance values are much larger. Even for the least rough case considered (curves (b) of Figs. 2-4): A’ = a+ + 2.2 x IO-‘. Thus for small particles AC is approximately constant and so ci- is at a much higher value than with the smooth surface. The inverse l/c;-. will be small and consequently 1 will predominate for all particles. This means that the inflexion that occurs with particles of about 0.1 pm dia. for smooth surfaces does not occur in the case of rough surfaces.
COMPARISON
WITH
EXPERIhIESTAL
RESULTS
As Kneen and Strauss (1969) pointed out, for a number of reasons the data of various workers are not all directly comparable. Information is often incomplete; different authors have used different definitions of some terms; there is doubt as to the relation between the particle shape factor and the nominal particle diameter; different types of test rigs have been used by different workers. The recent extensive work of Montgomery and Corn (1970) provides the most suitable comparison for the approach presented here. Their experimental set-up involved blowing air plus aerosol along a 30 ft (9.14 m) length of 6.065 in (15.405 cm) i.d. seamless aluminium pipe, the final 10 ft (3.05 m) of which was used to evaluate deposition rates. A summary of the results of their last ten runs (obtained after removing from the aerosol an unbalanced electrical charge that had spoiled the initial runs) is contained in Table 1. Also shown in
814
L. W. B. BROWSE Table 1. Comparison between theory and experiment Montgomery and Corn (1970k-Experimental
Run NO.
17 18 19 20 21 22 23 24 2j 26
Particle dia.
Particle densitv &cm-j)
hm)
16-J
1.89 1.70 1.72 1.18 1.13 I.14
1.22 044 2.16
I.33 I.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33
FIOW
(Reynolds number) 366000 253OOO 98200 48OOO 358OOO 256OOU 1OOOOO 50 600 351OOO 359OOO
This Paper-Theoretical Deposition velocity (ems
Deposition velocity (ems-‘1
!1
Pipe roughness ratio (k 6) (OOOOl) (OOOOjt oOQ342 0514 .3,92 O179 I.90 oGI173 OOQO44 OO103 0.2OJ OOOO2 1 OOO144 0.0329 OOO260 0.396 360 OOO161 O146 i.81 OOOOj5 OOlOO 0.70; oOOO27 OOO161 O&68 oOO372 0.29 1 3.21 OOO640 0.695 1.31 (0)
3.7j 0.14O 0.0172 0.0067 1.52 0.0666 0.0223 0.0041 OGO72 0646
Table 1 are the deposition values for various pipe surface roughness values, obtained using the approach suggested here. From the table it can be seen that for a roughness ratio of around OQOO1the theoretical and experimental values are in reasonable agreement. Calculations showed that, except for run 17, the Montgomery and Corn results are obtained provided a surface roughness ratio in the range 1.7 + 0.9 x 10T4 is assumed. Montgomery and Corn do not indicate the roughness of their test section but they do give experimentally obtained friction factors. These are compared in Table 2 with friction factors obtained using equation (5) and a surface roughness ratio of 1.7 x 10e4. It can be seen that the values are comparable except for run 20, where, because the trend is incorrect (loner Reynolds numbers should give higher friction factors), it is assumed that the experimental result is in error. Table 2. Comparison of friction factors
Montgomery and Corn (1970)-Experimental Run Fanning friction NO. factor x 10’ 17 18 19 20 21 22 23 24 25
26
3.3 3,6 3.7 2.8 3.6 3.j 3.2 4.1 3.5 3.6
Theoretical pipe roughness ratio 1.7 x lo-’ 0 Fanning friction factor x 10’ 3.5
3.:
3.7 4.5 5.3 3.j 3.7 4.j j.2 3.5 3.j
3.9 4.6 jA 3.7 -3.9 1.6 j.3 3.7 3.7
Also included in Table 2 are friction factors computed from equation (5) for roughness ratio 0. These show slightly better agreement with the experimental results, indicating that the flow in Montgomery and Corn’s pipe was aerodynamically smooth. For surfaces that are only slightly rough, the projection of the roughness elements into the laminar boundary layer hardly affects the flow (as witnessed by the very small change
Deposition of partrcles from a gas stream
315
in friction factors between smooth surfaces and slightly rough surfaces) and yet, for particles less than about 5 /m-r,the aspersity heights are not small compared with the particle diameter. This means that a particle that is a certain distance away from a smooth surface is subject to a much lower radial velocity than if it were the same distance away from an aspersity of a rough surface-due to the displacement in origin of velocity profile effect. These considerations explain why deposition rates are much higher for only slightly rough surfaces as compared with smooth surfaces. CONCLUSlONS
The conclusions are summarized as follows: 1. Using smooth surface theory the prediction of the experimental deposition velocities of Montgomery and Corn (1970) are out by as much as 1000-1, whereas by making reasonable assumptions regarding the surface roughness and using the theory presented here, agreement with the experimental results is obtained. 2. The rough surface theory and the trends indicated in Figs. 2-10 are of importance for any worker who is designing a system for sampling particulates or aerosols from the atmosphere or elsewhere. 3. The particle deposition rates are extremely sensitive to roughness even when this is too small to produce aerodynamically rough flow. This is particularly so for particles which give minimum deposition rates on smooth tubes, e.g. particles below 0.05 pm dia. and 1-4 pm dia. for the conditions of Figs. 2-4. This means, from a practical point of view, that precise predictions of deposition rates are hardly possible-especially for certain sizes of particles. The problem of obtaining accurate values of equivalent surface roughness makes prediction even more difficult. This also suggests that, in practice, widely different deposition rates can be obtained, even from apparently similar equipment, due to slight differences in surface roughness. 4. The prediction technique has yet to be verified for aerodynamically rough flow. 5. Other effects are also important in determining deposition rates and further work is needed to enable these to be included in a more complete theory. These effects include physical shape or particle, relative stickiness between particle and surface and electrical charge differences between particles and surface. ~cknowledgeme,tr-This investigation was undertaken at Warren Spring Laboratory, Department of Trade and Industry. Stevenage. England. during a 12 months’ visit of the author as a guest research worker.
REFERENCES Beal S. K. (197Oa) Deposition of particles in turbulent flow on channel or pipe walls. IV&. Sci. Engng 40, 1. Beal S. K. (1970b) Deposition of polydisperse aerosols in turbulent flow. Atmospheric Em-ironment 4,439. Bennett C. 0. and Myers J. E. (1962) Momentum Heat and Mass Transjb, pp. 135-139. McGraw-Hill, New York. Chamberlain A. C. (1967) Transport of Lycopodium spores and other small particles to rough surfaces. Proc. R. Sot. A. 296 (1444). 45. Colebrook C. F. (1939) Turbulent flow in pipes. with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Chem. Engrs 11 (4). 133. Colebrook C. F. and White C. M. (1937) Experiments with fluid friction in roughened pipes. Proc. R. Sot. A 161, 367. Davies C. N. (19-15)Definitive equations for the fluid resistance of spheres. Proc. Phys. Sot. 57 (322). 259. Davies C. N. (1966) Deposition from moving aerosols. In .-terosol Science, pp. 393-446. Academic Press. New York. Davies C. N. (1966b) Deposition of particles from turbulent flow through pipes. Proc. R. Sot. A. 289, 235. Fuchs N. A. (196-t) .Mechanics of Aerosols. p. 7. Pergamon Press. Oxford.
816
L W. B. BKOULI
Grass A. J. (1971) Structural ieatures oi turbulent dew
orer
smooth and rough boundaries. J. fiaui .Lfrch. 50
(2). 33.
Kneen T. and Strauss M’.f 19691Deposition of dust from turbulent yas streams. .At~nospheric Enc-w~nnw~tr3. 55 .Montgomery T. L. and Corn hf. f 1970) Aerosol deposition in a pipe with turbulent airflow. J. .~svsol SCI. 1. IS5
Perry A. E.. Schofield LV, H. and Joubert P. N. (1969) Rough wall turbulent boundary layers. J. Fiud .LIddr. 37(I). 386-387. Rotta J. C. (1962) Turbulent boundary layers in incompressible Row. Progress in .Arronauricni Scirncer, Voi 1. p. 75. Macmillan. New ‘r’ork. \Vells A. C. and Chamberlain .A. C. t I9671 Transport of small particles to vertical surhces. &I:. J -Ipp/_ P~:x 18, 1793.
OF PARTICLES ON ROUGH SURFACES TURBULENT GAS-FLOW IN A PIPE L. W. B. BROW&X
Departmenf of Mechanical Engineering, University of Newcastle, N.S.W. 2308, Australia (first
rewired
30 Nocembrr
1972 and infinalform
15 Vorember 1973)
Abstract--The importance of surface roughness on the deposition of particles from a flowing gas stream is well known but there has been little theoretical work that helps predictions to be made. The theory presented here follows the general approach of Davies (1966) who considered deposition on smooth surfaces. To take account of surface roughness extra distances have been added to the particle-capture distance and, by shifting the position of the origin of the velocity profile. it has been possible to use the same equation for the radial velocity fluctuations. The effect of surface roughness on the friction velocity is dealt with by empirical equations. Details of the analysis are given together with a computer solution procedure. Results of calculations are compared with the experimental results of other workers and a reasonable degree of agreement is shown. The effect of small changes in surface roughness on deposition velocity are shown to be so large that there is difficulty in making precise predictions of deposition rates and results from apparently similar experiments may vary widely. NOMENCLATURE a
a* c C’
2. co
d D e
i 9 k k’ K L ni, P PLI R R’ R, R, S, .% r 11” u t r+
particle radius dimensionless particle radius = a u*/v particle concentration (mass per unit volume) dimensionless particle concentration = C/C, particle concentration at distance A from pipe wall dimensionless particle concentration at A* from pipe wall = Cd-/Co centre-line particle concentration pipe dia. molecular diffusion coefficient displacement in origin of velocity profile-Fig. 1 and equation (15) dimensionless displacement in origin = eu*/v Fanning friction factor for pipe slip correction factor-equation (22) acceleration due to gravity average height of pipe surface roughness (or equivalent surface roughness) dimensionless height of pipe surface roughness = ku*/v Boltzmann’s constant-equation (21) pipe length mass deposition rate of particles per unit area of pipe surface gas pressure pressure drop in pipe pipe radius dimensionless pipe radius = Ru*/v Reynolds number of pipe flow = dU/v Reynolds number of Row around a free-fall particle when its terminal velocity is reached = Zaar.‘r particle stopping distance dimensionless stopping distance = $u*/v gas temperature friction or shear velocity-equation (3) mean velocity of gas radial fluctuation of velocity dimensionless radial fluctuation of velocity = c/u* 801
L. W. B. BROWYE radial fluctuation of velocity at distance A from pipe wall dimensionless radiat fluctuation of velocity at distance 4’ from pipe wall = V, u+ terminal velocity of particle under free-fall conditions velocity of deposition of particles dimensionless velocity of deposition of particles = C:u* distance from pipe wall dimensionless distance from pipe wall = J u*!lv capture distance--equation (6) (measured from origin of velocit> profile) dimensionless capture distance = Au*/v eddy diffusion coefficient kinematic viscosity of gas density of gas density of particles S.D. of average height of pipe surface roughness dimensionless standard deviation of average height of pipe surface roughness = G~W*v particle relaxation time-equation (I?) dimensionless particle relaxation time = TIP’ v dimensionless sum of various distances from pipe wall-equation (18) drag coefficient of particles under free-fall conditions-equation (IO).
INTRODUCTION
A most useful review of the deposition of dust from turbulent gas streams has been given by Kneen and Strauss (1969). Among subsequent papers are those of Beal (1970a). Montgomery and Corn (1970) and Beal (1970b). Wells and Chamberlain (1967) and Chamberlain (1967) found that particle deposition was increased markedly by surface roughness. However, the work of Chamberlain (1967) appears to have been the only attempt to predict deposition rates on rough surfaces. The purpose of this paper is to present an approach suitable for estimating deposition rates on rough surfaces in specific cases, and to show the zffect on deposition rates uhen various parameters are changed. The general approach is the same as that used by Davies (1966), but includes the effect of surface roughness. Not included in this analysis are the effects of electrical charge, particle shape, re-entrainment, surface stickiness gravitational settling and other physical or chemical properties of the particles and surface.
pofile
Fig. I. Assumed model for particle capture and particle concentration
SO3
Deposition of particles from a gas stream MODEL
USED
IN
ANALYSIS
Figure 1 shows the model used in this analysis. At some distance, A, from the wall, the radial component, v~, of the turbulent velocity will be such that a particle with that velocity will be unable to stop before its surface touches the upper parts of the wall surface roughness. It is assumed that when a particle reaches this position it will remain on the surface: ni, = VAX,.
i.e. Now. by definition
(1) So that
or dimensionlessly I’? = 0; ’ cc.
(3
Thus in order to evaluate V it is necessary to determine A, u and C. In determining these quantities the friction velocity is required and consequently its evaluation will be discussed first. ESTIMATION
OF
FRICTION
VELOCITY
u*
The friction velocity is related to the Fanning friction factor by
The Fanning friction factor is related to the pressure drop in a pipe by
Thus by carrying out measurements of the pressure drop, equations (3 and 4) can be used to determine the friction velocity. Where this is inconvenient, use can be made of empirical formulae for the Fanning friction factor for rough pipes. Colebrook and White (1937) and Colebrook (1939) correlated the whole flow region from hydraulically smooth to completely rough with an equation of the type
Using the more recent equations of Bennet and Myers (1962) for smooth flow (k+ 0) and completely rough flow (R, + x ) the following values were obtained A1 = 2.14 A2 = 4.06 A3 = 4.73
i.e. 2.14 - 4.06 log,,
(5)
Ya4
L. h’.
B.
6KOWSE
There is no analytical solution to equation (5) and it is necessary to evaluatef‘by cal methods. e.g. by using the interval halving technique. ESTIMATION
OF
CAPTURE
DISTAKCE
numeri-
il
From Fig. 1 (where it is assumed that contact with any roughness element results in capture). A=Sd+a-+~+~L-e
t.6)
Or Ai = s;
f cl+ + k- +
G;
-
e-.
(7)
1. Estimation of‘s, The stopping distance, S,, is the distance travelled by a particle when given an initial velocity of c,. This is obtained indirectly by estimating the terminal velocity, or. of the particle during free-fall conditions. Davies (1945) fitted experimental data with the following equations: Rr = ‘3 - 2.3363 x 10-‘(ll/R+)2 + 2.0154 x 10-6(1LRt)3 - 6.9105 x 10-g($R$)J 24
(8)
limits of applicability, $R+ < 120,
log,,(R7.)= - 1.29536 f 0986(log,,
$R+) - O.O46677(log,, $R+)?
+ 0~0011235(log,, IC/R;)3.
(9)
where (IO)
and R,=-
Znc, 1’ .
(11)
The limits of applicability are 120 < cj/R$ -e 45 x lo-‘. Thus for a particle of given radius and density, and for a gas of given density and viscosity, equation (10)can be used to determine II/R& equations (8) or (9) can be used to determine R,, and equation (11) can be used to determine the terminal velocity L’~. The relaxation time. r, is the time for a particle projected into a gas. with a given initial velocity, to reach l/exp 1 of its initial velocity (Fuchs. 1964). If a particle was given an initial velocity equal to its above terminal velocity. then (12) It is readily shown that the stopping distance is the initial velocity multiplied by the relaxation time, i.e. s, = CA.% or s; = r;.r-. (13) the evaluation of which is discussed later. This requires a knowledge of L~L\,
Deposition of particles from a _easstream
2. Estimation
so5
of k and a,
These are best based on actual measurement. Alternatively an equivalent surface roughness height. k, can be estimated and or; estimated from Csk= 0,17k or 0: = O.l7k+. 3. Estimation
(l-1)
ofe
As early as 1951 it was suggested that a boundary layer on a rough wall behaves as if its origin is located some distance below the crests of the roughness elements (Moore. PhD Thesis, quoted by Perry et al., 1969). This means that the effect of the surface roughness on the velocity profile outside the sub-layer is to shift the origin to some point below the crests of the roughness. Other workers have supported this view, e.g. Rotta (1962), Perry et aI. (1969) and Grass (1971). A definitive equation for determining the ‘-displacement in origin” is not yet available but the data of Grass (1971) are correlated by the following equation, e/k = 0.53 + 0.0034k”
(15)
Or e+
=
0.53k’
+ OGl34k+‘.
(16)
Grass’s data are for channel flow but it is not unreasonable to assume that near the surface a similar mechanism exists in pipe flow and that equation (15) can therefore be used. ESTIMATION
OF
RADIAL
VELOCITY
FLUCTUATION
V~
Davies (1966) correlated existing experimental data for turbulent flow in a smooth pipe and obtained YC c+ = J’c. Provided Y+ is measured from a shifted origin, as discussed above, the same equation can be applied to flow in a rough pipe. Thus CT- =
AC A+ + 10’
(17)
However, A+ itself requires a knowledge of us [see equations (7 and 13)], so that c_\‘and Ai must be evaluated simultaneously. Substitution of equation (13) into equation (7), and the result into equation (17) gives
unf*5+
VA?= + CA-
+ I#)+
‘Tf + $’ + 10
(18)
where 4” = a+ + k’ + 0: - e+. This leads to a quadratic equation in v:+ with a solution 1 v&t =-- 1 - Q’flO)+ 2(
J[(l-“+yy+q
(1%
L. W. B. BROWSE
so6 ESTIMATION
OF
CONCENTRATIOK. FROM
C,. WALL
.AT CAPTLRE
DIST.AXCE
For one-dimensional mass transfer in turbulent flow, the mass flux perpendicular surface is constant and can be expressed bq dC riz, = (D i E) -_ dy
to a
From equation (1) this is
or
(20) Now KTF
D _=\
(21)
a\
where F = 1+
gC6.32 +2.01 exp(-2190Pa)]
(22)
PincmofHg a in cm and
Equations (21 and 23) are experimental correlations equation (20) from y+ = A’ to ,v+ = R’ gives
of Davies (1945, 1966). Integration of
i.e.
where
r
R-
I=
,A-
dy’ (Dlv + E/V)’
(24)
Thus c,‘:
= I-
v+i.
(25)
C,? could be expressed in terms of v~‘:and I by using equation (2). However. as shown. I” can now be calculated.
507
Deposition of particles from a _easstream CALCULATION
OF
VELOCITY
OF
DEPOSITION.
V
The substitution of equation (25) into equation (2) gives
i.e. I’-
=.
1 l/V,‘: + I’
(35)
The integral I can be evaluated numerically. A computer solution for C’can thus be obtained using the following procedure: (1) Input data-pipe dia., n pipe equivalent surface roughness, k mean flow velocity. a gas temperature, T gas pressure, P gas density. p (or can be included in calculations) gas kinematic viscosity, v particle radius, 0 particle density, pp. (2) Calculate the Fanning friction factor f using equation (5) and the friction velocity tl* using equation (3). Use equations (8-12) to calculate the particle relaxation time r, and consequently r+. Calculate 4’ as defined by equation (18). This requires the use of equations (14 and 16). (3 Calculate 0,‘: using equation (19). (6) Calculate A’ using equations (13 and 7). (7) Use equations (21-23) to determine the integrand of (24) and calculate I using a numerical method. (8) V and consequently Scan now be found using equation (26).
RESULTS To
see
OF
SOME
CALCULATIONS
the effect of varying different parameters, the above approach was used to determine deposition velocities under the following conditions: particle dia.-0.01-10.0 pm; particle densities-05-9.0 gem- 3 ; Reynolds numbers-500&40000; surface roughness ratios (k/Q--O-0.002. The gas was air at 20°C and 760 mm of Hg pressure. The pipe dia. was 1.5 cm. The results of these calculations are shown in Figs. 2-10. Features worth.noting are: (a) the very large effect of surface roughness, except on particles of size larger than about 10 pm dia.; (b) the small effect of particle density until the particle diameter exceeds about 2 pm dia.; (c) the rate of increase of deposition velocity with Reynolds number is greater for the rough surfaces, except when the particles are large, e.g. 10 pm dia.
SOY
L. W. B. BROWSE
Particle
dia..
pm
Fig. 2. Effect of particle size on deposition velocity for various surface roughnesscs. R, = ICJOO~: d = 15cm; p = 1Ggcm‘3. (a) k’J = 0; (b) ic:n = OooO1; (c) k.!d = 04EU5; id) k.J = @C%l: rz! kid = 04N2.
IO.' Port&e
dia.,
I
ptm
Fig. 3. Effect of particle size on deposition velocity for various surface roughaesses. R, fl: W)@: d = [-~cm; pp = tT)gcm-‘. (a) k/d = 0; (b) k/d = 04001; fc) k!d = OTH~OS:fdi k’d = @@II: i@ k/if = 0402.
Deposition of particics from a gss stream
Particle dia.
pm
Fig. 4. Effect of particle size on deposition velocity for various surface roughnesses. R, = 1OooO: d = t.jcm; pp = 2,0gcm-3. (a) kid = 0: (b) k,d = 09001: (c)kid = 03005: (d) 1;d = 0.001: (c) k ‘2 = 04B’.
,,i.,,,,,,,
2
3
4
Reynolds number, Rex lO’4
Fig. 5. Effect of Reynolds number on deposition velocity for various surface roughmsses. Particle dia. = 0.1 pm; d = 1‘jcm; pp = l~0gcm-3. (a) k:d = 0: (b) kjd = O+XJO1: (c) k d = OQOO5: (d) k/d = 0.001: (e, k/d = OGO2. *.E.
%8--c
310
I
Reynolds num
2
her,
3
4
Re ~10~~
Fig. 6. Effect of Reynolds number on deposition velocity for various surface rouglmesses. Parti& dia. = l,Oilm; ri = 1.5cm: pP = 14gcm-3. (a) k’d = 0; (b) k,frf= 04001: (cf k’(i = 04005: [d) k, li = OGOI: (e) k,‘d = 04302.
I
2
3
4
Reynolds number, &xIO’~
Fig. 7. Effect of Reynolds number on deposition velocity for various surface roughnesses. Parficiz dia. = 1Opm: d = l.Scm; p,, = I-0gcmV3. (a) k/d = 0; (b) kid = 04401: (c) k/d = O-OOO5;Cdl k/d = 0.001 : (e)kid = OGO2.
Deposition of particles from a gas stream
811
IO
5
-
E
-
;
-
.g IO.’ =
x
-
e
s ._
d
$
-
C
“0 if
0
C b
lo”_
10-a
’
I
’
I
’
1234s67e9
’
’
’
’
’
’
’
’
’
’
’
Particle density, g CJcm-’ Fig. 8. Effect of particle density on deposition velocity for various surface roughnesses. Particle dia. = 0.1 pm; d = I.5 cm; R, = 10000. (a) k/d = 0; (b) k/d = OGOOl: (c) k/d = 04005; (d) k/d = OGOI : (e)k/d = OGN.
Particle density.
g g cmw3
Fig. 9. Effect of particle density on deposition velocity for various surface roughnesses. Particle dia. = 2pm; d = 1.5cm; R, = lOONI. (a) k/d = 0; (b) k/d = OGOOl; (c)k/d = 04005; (d) k/d = OGOI: (c)k/d= 0402.
L. LV.B.
Particle
BROWSE
density,
p, 9 cm3
Fig. 10. Effect of particle density on deposition velocity for various surface roughnesses. Particle dia. = 5 htm: n = I.5 cm; R, = 10000. (a) k,!d = 0: (b) k/d = 09001; (c) k,‘d = 04X305; (d) k’d = 0031;
COMMENTS
(5) k d = oax?.
OS
CURVE
SHAPES
The shapes of the curves in Figs . 2-4 can be explained as follows: equations (26 and 24) show that the deposition velocity depends on three factors: (1) the radial velocity fluctuation at the capture distance; (2) the molecular diffusion coefficient; (3) the eddy diffusion coefficient. The functional relationship between these quantities is not easy to explain qualitatively owing to the fact that an integration from the capture distance to the centre of the tube is involved. The capture distance for particles in a smooth tube is not greatly different from the particle radius, since the stopping distance for the low radial velocities near the wall is quite small. Equation (17) shows that. since 10 $ A-, the radial velocity fluctuations can be obtained by dividing the capture distance by 10.
Thus
cl-
n-
z 10’
If ri- were the only factor involved then the deposition velocity would decrease linearly as the particle size decreases-see equation (26) The effect of the integral I [equations (24 and 26)J on the deposition velocity can be determined bv noting that for rT = 5. E/V 2 0.1. This is much larger than the largest value
$13
Deposition of particles from a gxj stream
of D,'v.(D/v t 3 x 10e3 for a O.Ol-itrn dia. particle). Thus I can be split into two parts:
where
For small particles, D/vwill be much larger than E v throughout most of the region of 4’+ = a+ to 5. Since D r l/a, equation (21). I will increase as the particle size is increased in this region. For large particles, E/V will predominate in the region of _r- = a- to 5. increasing as the particle size increases. Thus I will decrease with increasing particle size. The total effect of us and 1 on the deposition velocity depends on the relative value of l/ rc and I, being equal to the inverse of the sum of these two quantities-see equation (26). From calculations, 1 is predominant for particles larger than about 0.1 pm dia. The shape of curve (a) in Figs. 2-4 can thus be explained. For particles of dia. 10 pm down to 0.1 pm I predominates and consequently for the larger particles with the eddy diffusion predominant the deposition velocity will decrease as the particle size decreases, while for the smaller particles the molecular diffusion predominates and the deposition velocity will increase as the particle size decreases. The change-over point occurs at a particle size of about 4 pm dia. For particles smaller than about 0.1 ,um dia. the radial velocity fluctuation predominates and the deposition velocity decreases as the particle size decreases. In the case of rough surfaces, the capture distance values are much larger. Even for the least rough case considered (curves (b) of Figs. 2-4): A’ = a+ + 2.2 x IO-‘. Thus for small particles AC is approximately constant and so ci- is at a much higher value than with the smooth surface. The inverse l/c;-. will be small and consequently 1 will predominate for all particles. This means that the inflexion that occurs with particles of about 0.1 pm dia. for smooth surfaces does not occur in the case of rough surfaces.
COMPARISON
WITH
EXPERIhIESTAL
RESULTS
As Kneen and Strauss (1969) pointed out, for a number of reasons the data of various workers are not all directly comparable. Information is often incomplete; different authors have used different definitions of some terms; there is doubt as to the relation between the particle shape factor and the nominal particle diameter; different types of test rigs have been used by different workers. The recent extensive work of Montgomery and Corn (1970) provides the most suitable comparison for the approach presented here. Their experimental set-up involved blowing air plus aerosol along a 30 ft (9.14 m) length of 6.065 in (15.405 cm) i.d. seamless aluminium pipe, the final 10 ft (3.05 m) of which was used to evaluate deposition rates. A summary of the results of their last ten runs (obtained after removing from the aerosol an unbalanced electrical charge that had spoiled the initial runs) is contained in Table 1. Also shown in
814
L. W. B. BROWSE Table 1. Comparison between theory and experiment Montgomery and Corn (1970k-Experimental
Run NO.
17 18 19 20 21 22 23 24 2j 26
Particle dia.
Particle densitv &cm-j)
hm)
16-J
1.89 1.70 1.72 1.18 1.13 I.14
1.22 044 2.16
I.33 I.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33
FIOW
(Reynolds number) 366000 253OOO 98200 48OOO 358OOO 256OOU 1OOOOO 50 600 351OOO 359OOO
This Paper-Theoretical Deposition velocity (ems
Deposition velocity (ems-‘1
!1
Pipe roughness ratio (k 6) (OOOOl) (OOOOjt oOQ342 0514 .3,92 O179 I.90 oGI173 OOQO44 OO103 0.2OJ OOOO2 1 OOO144 0.0329 OOO260 0.396 360 OOO161 O146 i.81 OOOOj5 OOlOO 0.70; oOOO27 OOO161 O&68 oOO372 0.29 1 3.21 OOO640 0.695 1.31 (0)
3.7j 0.14O 0.0172 0.0067 1.52 0.0666 0.0223 0.0041 OGO72 0646
Table 1 are the deposition values for various pipe surface roughness values, obtained using the approach suggested here. From the table it can be seen that for a roughness ratio of around OQOO1the theoretical and experimental values are in reasonable agreement. Calculations showed that, except for run 17, the Montgomery and Corn results are obtained provided a surface roughness ratio in the range 1.7 + 0.9 x 10T4 is assumed. Montgomery and Corn do not indicate the roughness of their test section but they do give experimentally obtained friction factors. These are compared in Table 2 with friction factors obtained using equation (5) and a surface roughness ratio of 1.7 x 10e4. It can be seen that the values are comparable except for run 20, where, because the trend is incorrect (loner Reynolds numbers should give higher friction factors), it is assumed that the experimental result is in error. Table 2. Comparison of friction factors
Montgomery and Corn (1970)-Experimental Run Fanning friction NO. factor x 10’ 17 18 19 20 21 22 23 24 25
26
3.3 3,6 3.7 2.8 3.6 3.j 3.2 4.1 3.5 3.6
Theoretical pipe roughness ratio 1.7 x lo-’ 0 Fanning friction factor x 10’ 3.5
3.:
3.7 4.5 5.3 3.j 3.7 4.j j.2 3.5 3.j
3.9 4.6 jA 3.7 -3.9 1.6 j.3 3.7 3.7
Also included in Table 2 are friction factors computed from equation (5) for roughness ratio 0. These show slightly better agreement with the experimental results, indicating that the flow in Montgomery and Corn’s pipe was aerodynamically smooth. For surfaces that are only slightly rough, the projection of the roughness elements into the laminar boundary layer hardly affects the flow (as witnessed by the very small change
Deposition of partrcles from a gas stream
315
in friction factors between smooth surfaces and slightly rough surfaces) and yet, for particles less than about 5 /m-r,the aspersity heights are not small compared with the particle diameter. This means that a particle that is a certain distance away from a smooth surface is subject to a much lower radial velocity than if it were the same distance away from an aspersity of a rough surface-due to the displacement in origin of velocity profile effect. These considerations explain why deposition rates are much higher for only slightly rough surfaces as compared with smooth surfaces. CONCLUSlONS
The conclusions are summarized as follows: 1. Using smooth surface theory the prediction of the experimental deposition velocities of Montgomery and Corn (1970) are out by as much as 1000-1, whereas by making reasonable assumptions regarding the surface roughness and using the theory presented here, agreement with the experimental results is obtained. 2. The rough surface theory and the trends indicated in Figs. 2-10 are of importance for any worker who is designing a system for sampling particulates or aerosols from the atmosphere or elsewhere. 3. The particle deposition rates are extremely sensitive to roughness even when this is too small to produce aerodynamically rough flow. This is particularly so for particles which give minimum deposition rates on smooth tubes, e.g. particles below 0.05 pm dia. and 1-4 pm dia. for the conditions of Figs. 2-4. This means, from a practical point of view, that precise predictions of deposition rates are hardly possible-especially for certain sizes of particles. The problem of obtaining accurate values of equivalent surface roughness makes prediction even more difficult. This also suggests that, in practice, widely different deposition rates can be obtained, even from apparently similar equipment, due to slight differences in surface roughness. 4. The prediction technique has yet to be verified for aerodynamically rough flow. 5. Other effects are also important in determining deposition rates and further work is needed to enable these to be included in a more complete theory. These effects include physical shape or particle, relative stickiness between particle and surface and electrical charge differences between particles and surface. ~cknowledgeme,tr-This investigation was undertaken at Warren Spring Laboratory, Department of Trade and Industry. Stevenage. England. during a 12 months’ visit of the author as a guest research worker.
REFERENCES Beal S. K. (197Oa) Deposition of particles in turbulent flow on channel or pipe walls. IV&. Sci. Engng 40, 1. Beal S. K. (1970b) Deposition of polydisperse aerosols in turbulent flow. Atmospheric Em-ironment 4,439. Bennett C. 0. and Myers J. E. (1962) Momentum Heat and Mass Transjb, pp. 135-139. McGraw-Hill, New York. Chamberlain A. C. (1967) Transport of Lycopodium spores and other small particles to rough surfaces. Proc. R. Sot. A. 296 (1444). 45. Colebrook C. F. (1939) Turbulent flow in pipes. with particular reference to the transition region between the smooth and rough pipe laws. J. Inst. Chem. Engrs 11 (4). 133. Colebrook C. F. and White C. M. (1937) Experiments with fluid friction in roughened pipes. Proc. R. Sot. A 161, 367. Davies C. N. (19-15)Definitive equations for the fluid resistance of spheres. Proc. Phys. Sot. 57 (322). 259. Davies C. N. (1966) Deposition from moving aerosols. In .-terosol Science, pp. 393-446. Academic Press. New York. Davies C. N. (1966b) Deposition of particles from turbulent flow through pipes. Proc. R. Sot. A. 289, 235. Fuchs N. A. (196-t) .Mechanics of Aerosols. p. 7. Pergamon Press. Oxford.
816
L W. B. BKOULI
Grass A. J. (1971) Structural ieatures oi turbulent dew
orer
smooth and rough boundaries. J. fiaui .Lfrch. 50
(2). 33.
Kneen T. and Strauss M’.f 19691Deposition of dust from turbulent yas streams. .At~nospheric Enc-w~nnw~tr3. 55 .Montgomery T. L. and Corn hf. f 1970) Aerosol deposition in a pipe with turbulent airflow. J. .~svsol SCI. 1. IS5
Perry A. E.. Schofield LV, H. and Joubert P. N. (1969) Rough wall turbulent boundary layers. J. Fiud .LIddr. 37(I). 386-387. Rotta J. C. (1962) Turbulent boundary layers in incompressible Row. Progress in .Arronauricni Scirncer, Voi 1. p. 75. Macmillan. New ‘r’ork. \Vells A. C. and Chamberlain .A. C. t I9671 Transport of small particles to vertical surhces. &I:. J -Ipp/_ P~:x 18, 1793.