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On turbulent deposition of particles to rough surfaces
J. Aerosol Sci., Vol. 17, No. 6, pp. 903-920, 1986.
0021 ~8502/86 $3.00+0.00 Pergamon Journals Ltd.
Printed in Great Britain.
ON TURBULENT DEPOSITION OF PARTICLES TO ROUGH SURFACES ALEXANDER ORON a n d CHAIM GUTFINGER Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel
(First received 14 November 1985; and in final form 14 February 1986) Abstract--An analytical method is presented for calculation of particle deposition from turbulent streams to rough surfaces with fibrous roughness elements. Particle deposition is considered as a twostage process consisting of filtration inside the roughness layer and turbulent transport outside it. Velocity profiles are developed for these two regions and matched continuouslyand smoothly. These are used to calculate the transport processes in each region. The overall particle deposition velocity is obtained by addition of resistances of the roughness layer and of the domain above it. Comparison with experiments of Wells and Chamberlain (1967) shows that the theory correlates quite well with the experimental data.
NOMENCLATURE
p
Particle concentration Drag coefficient Diameter Particle diffusivity Fiber height Dimensionless fiber height, hu,/v Particle flux Deposition velocity Mixing length Pressure Particle radius Pipe radius Reynolds number, 2RUo/v Stopping distance, given by equation (1) Dimensionless stopping distance, Su./v Schmidt number, v/D Parameter of the solution, equation (35) Velocity ~_ Friction velocity, (~w/p)2 Average velocity in pipe flow r.m.s, velocity Coordinate Coordinate Dimensionless coordinate, yu,/v Solidity, solid fraction Filtration coefficient Eddy viscosity Filtration efficiency Kinematic viscosity Dimensionless coordinate, y/h Density
a
V~/V'
T
Relaxation time Dimensionless velocity, given by equation (28)
C
Co d D h h+ J k 1
P rp R
Re S S+ Sc t U U,
Uo 1)' X
y y+ ri
v
Subscripts Average At the outer edge of the transition region f Fluid h Aty=h P Particle + Dimensionless
av
c
As 17:6-B
903
904
ALEXANDER ORON and CHAIM GUTFINGER
INTRODUCTION Computation of particle deposition from turbulent gas streams is very important because of many applications in various fields of engineering, technology and health physics that deal with gas-solid two-phase flows. These applications include pneumatic transport of particles in pipes (Owen, 1969), particle deposition in vegetation canopy (Thorn, 1975), design of sampling devices and air cleaning installations and particle deposition in human lungs (Wilson, 1947). When turbulent flow of a gas with solid particles takes place near a solid wall, the turbulent mechanism is the main mechanism of particle deposition. There is a possibility that some additional mechanisms are involved in the deposition process, e.g. Brownian diffusion (in the case of submicron particles), gravitational settling (in the case of heavy particles deposited on horizontal surfaces) and electrical effects (electrically charged particles in the presence of an external electric field). Beside these effects, when a wall is rough, the roughness elements provide an additional mechanism for removal of the particles from the turbulent gas stream. In the field of theoretical investigations Friedlander and Johnstone were the first to explain how solid particles deposit on a smooth wall. They define a concept of'the stopping distance' of a particle. It is a distance that a particle travels in free flight due to its initial velocity through the fluid until stopping. For a particle of a diameter dp, density pp and initial velocity Vpo the stopping distance is given by: 2
S = dn Vpo "--,PP 18v p
(1)
where v is the dynamic viscosity of the fluid and p its density. When molecular diffusion is negligible, the dominant mechanism of particle deposition from a turbulent stream is the one due to flow field fluctuations in the direction perpendicular to the wall. According to the model of Friedlander and Johnstone (1957), these turbulent fluctuations bring a particle within one stopping distance from the wall and the deposition process is completed by the free flight of the particle toward it. McCoy and Hanratty (1977) considered the relative effects of turbulent fluctuations and molecular diffusivity on deposition of particles, and found that the effect of molecular diffusivity is negligible for particles with dimensionless relaxation times, z +, higher than 0.15, where r + is defined as
1.pp.(dpu, z+=18
)a.
(2)
p \--~-
pnd2/18u,
The relaxation time, z = is the time in which the force acting on a rigid particle reaches the e- 1 part of its initial value, thereby diminishing the particle velocity by a factor o f e -1. The deposition velocity of the particles in a given region is defined as the ratio of particle flux to the change of particle concentration through this region: J k = --. Ac
(3)
This quantity, which has dimensions of velocity, is put in dimensionless form as k+ -
k It,
,
(4)
where u, = x / / ~ , is the friction velocity, zw being the shear stress on the wall. Friedlander and Johnstone assumed that the turbulent diffusivity of a particle en which arises from the relation d6
-
v'c'
= ~PUy
(5)
Turbulent deposition of particles
905
is equal to the eddy viscosity, e, of the fluid which is given by Lin et al. (1953): e = v(y+/14.5) 3
0 <~ y+ <<. 5
e = v(0.2y+ -0.959)
5 ~< y+ ~< 30
e = 0.4vy+
30 ~< y+ ~< 100.
(6)
The problem of very small particle deposition, i.e. the mass transfer problem, has been solved by Levich (1962) under the assumption that there are two important mechanisms, namely that of molecular diffusion and that of turbulence. His method, similar to that of Friedlander and Johnstone (1957), consists essentially of integration of the flux equation: dc d = (D + ep)~yy,
(7)
leading to an expression for the deposition velocity k + = const. So- 2/3,
(8)
which is similar to the experimental results of Shaw and Hanratty (1977), who found k+ = 0.089-Sc- 0.704,
(9)
where Sc = v/D is the Schmidt number and D the particle molecular diffusivity. Liu and Ilori (1974) assumed that the turbulent mass diffusivity of the particle, el, and the turbulent mass diffusivity of the fluid, e, are not the same and suggested an expression: e~ = e + v'2~,
(10)
where v' is the turbulent intensity of the flow field in the direction of the wall and z is the particle relaxation time, given above. Inertial effects are characterized by the additional term v'2 z, which vanishes as the particle relaxation time decreases, i.e. as its diameter decreases. In that case, the particle follows all the turbulent fluctuations of the fluid. The relaxation time term is very important when the particle is not small and does not immediately react to flow field fluctuations. Another example of non-identical expressions for e and ep is that of Kitamoto and Takashima (1974), who suggest: e,=e
l+2az+
~
+a2z2+
~y+-
,
(11)
where a = v~/v'. In contrast with the large quantity of experimental and theoretical papers dealing with particle deposition on smooth surfaces, the number of papers on rough surface deposition is fairly limited. Levich (1962) has solved the problem of mass transfer to a fully rough surface (h+ >> 1) under the assumption of a linear velocity profile inside the roughness layer: Y u(y) = u, ~,
(12)
where h is the height of roughness elements. His result, in terms of dimensionless deposition velocity, is: k+ = const Reh-1/2 Sc -3/4, (13) where Reh = hu./v is the Reynolds number based on the roughness height. Some recent papers have shown that this result is not very accurate, and that the deposition rate is proportional to Sc-2/3 (Yaglom and Kader, 1974; Fernandez de la Mora and Friedlander, 1982; Hahn et al., 1985). There are two main approaches to the problem of particle deposition onto a rough surface. One of them, which can be termed as a geometric one, is based on the derivation of the 'effective stopping distance', defined by Friedlander and Johnstone (1957) and constitutes an
906
ALEXANDER ORON and CHAIM GUTFINGER
extension of their theory. Browne 11974) has defined a 'capture distance' that takes into account the roughness of the wall and has generalized the theory of Davies (1966). E1Shobokshy and Ismail (1980) and EI-Shobokshy (1983) have incorporated into the theory of Browne an expression of Liu and Ilori (1974) for a particle eddy diffusivity. These theories have been verified for hydraulically smooth flows (h, < 5). Sehmel (1980) has reviewed the general field of particle deposition from the atmosphere to rough surfaces for the case of h+>>5. A second method for solving the problem consists of treating particle deposition as a twostage process, that of turbulent diffusion from the turbulent core to the roughness layer and that of deposition inside it. Davidson and Friedlander (1978) have solved the problem of deposition of heavy particles from the atmosphere onto a rough surface (tall grass) under the influence of gravity. Gutfinger and Friedlander (1985) have presented the method of computing particle deposition rates in the case of a fully developed flow in a pipe with walls roughened by a fibrous layer. The central point of their work is that the fibrous roughness layer is an additional resistance for particle deposition. Comparing their theory with the data of Wells and Chamberlain (1967) has shown that for particles of 0.65#m diameter the two-stage theory does indeed correlate with the data quite well. For larger particles, however, the two-stage theory gave results that were worse than the simple single-stage eddy viscosity extension of the Friedlander and Johnstone model. A close scrutiny of their work reveals that some of their assumptions are excessively restrictive. These are listed below. 1. The expression for the eddy viscosity of the flow does not account for the existence of the fibers on the wall and therefore can at best hold for the case of very low fiber densities. 2. The turbulent diffusivity of the particles is assumed to be identical to that of the fluid, limiting their theory to particles of very low inertia. 3. In spite of the requirement of a very dilute fiber canopy as stated in assumption 1, it is assumed that the filtration process is essentially limited to the tip of the fibers. The present paper solves the problem of particle deposition to rough surfaces without making use of the above assumptions. We feel, therefore, that the solution herewith presented can serve as an extension of the solution of Gutfinger and Friedlander to dense rough surfaces and does not suffer from the limitations imposed on the solution by the other assumptions. F O R M U L A T I O N OF THE P R O B L E M We solve the problem of deposition of suspended particles from a turbulent fully developed pipe flow to rough walls. We limit our treatment to rough surfaces which consist of fibers of identical height h and diameter df, and which do not penetrate into the turbulent core of the flow, i.e.
h÷ < 30.
(14)
The last assumption is not restrictive. If the fibers do protrude into the core of the flow, the two-stage process is reduced to a single-stage process consisting of the roughness layer only. The resistance to particle deposition is negligible in the turbulent core because of perfect mixing in it. The following assumptions are incorporated in the mathematical model: (a) The solid particles are of spherical shape. (b) The concentration of the particles is low enough as not to affect the flow field. Likewise, we neglect any interaction between the particles. (c) The particles and the pipe walls are not charged. Likewise, there is no external electric field, precluding any electrostatic effects. (d) We neglect the effect of the gravitational force on the deposition process. In this context we can assume the deposition as taking place on vertical walls.
Turbulent deposition of particles
907
(e) The flow is fully developed turbulent pipe flow, hence the radial component of the mean velocity vanishes. (f) We ignore effects ofreentrainment andbounce-offoftheparticles on thewall. In other words, the condition of removing a particle from the flow is its reaching the wall. (g) The particles are assumed to be large enough the effect of molecular diffusion on the deposition process could be neglected. This corresponds roughly to particle sizes of > 0.5 #m. We now present the two-stage mathematical model of particle deposition on a rough wall. First, we consider particle deposition inside the roughness layer. Then we treat the turbulent diffusion process in the fluid outside the roughness layer. The overall deposition velocity is obtained by addition of resistances of these two regions. VELOCITY PROFILES INSIDE AND OUTSIDE THE ROUGHNESS LAYER
The velocity profile inside the roughness layer For a one-dimensional model in which the only non-vanishing velocity component is u, and flow field homogeneity in the directions x and z is assumed, the time-averaged Navier-Stokes equations reduce to:
~y(-U'V')+v ~
/15)
- p ,gx'
where tit are the time-averaged velocity components ui and u~ the deviation from it, caused by turbulent fluctuations. The left-hand side of this equation can be identified as ~3z/cgyand the right-hand side as the total drag force per unit volume of the fluid fo, z being the local shear stress. Now equation (15) is written as: 0r cgy = fv,
(16)
and expression fv in terms of a drag coefficient, Co. Equation (16) may be rewritten as (Jackson, 1981):
~O~ = p ~2C o u2,
(17)
where 2 = nAI/S, with n being the number of fibers with a frontal area AI on the area S free of them. The shear stress z may be expressed in terms of molecular and eddy viscosity as:
• =p
(0u du)(du =p
+12
Ty
'
(18)
where I is the Prandtl mixing length and e the eddy viscosity. Let us assume that the mixing length is independent of y. This assumption has already appeared in the literature (Inoue, 1963; Jackson, 1981) and has led to the exponential velocity profle inside the roughness layer. Substituting equation (18) in equation (17) leads to: d2u 2 du d2u 2C o V~y2 + El dy d 7 - - 2 - h -u2'
(19)
with the boundary conditions: y=h;
u=uh
(20)
y=0;
u=0.
(21)
908
ALEXANDERORONand CHA|MGUTF|NGER
The local gas velocity Un isn't yet known and will be found later. Equation (19) does not contain explicitly the independent variable, y, hence, its order can be lowered by integrating it once, following the procedure outlined in Kamke (1959), resulting in:
v ( du ) 2 2 2 ( d u ) 3 2 Tyy v(du)2 E = ~ ~yy
where
2~_~°u3
dy s
=--
(22)
+E,
2 2 ( d u ) 3 y=o =0 + 3 l d-yy = const.
Equation (22) will be solved with the initial condition of equation (20). We first find an expression for 2Co/6h in equation (22). If n is the number of fibers on a surface of area A, then the total area occupied by the fibers on the surface is nrcd}/4. On the other hand, ifa is the solidity of the filter and S is the area free of fibers, then the total area of the fibrous surface is (S + rindS~4) and the following equality may be written down for the fiber occupied area: 4
- c~S+ T
]"
Hence, S=
1 - ~ rmd} •
(23)
4
It is evident that Af = hdf and therefore
2 - nAy _ S
4oe
h
(24)
re(1 - ~) dj."
Lee and Liu (1982) have shown that for a fiber filter, a dimensionless drag coefficient = ½CDReI is invariant in a fairly wide range of velocities and is a function of filter solidity, a. They plot the dimensionless drag coefficient, ~, experimentally obtained, as a function of the solidity. This function is monotonically increasing with the solidity of the fiber layer. From the definition of f~ it can be written that 2Or
Co dfu, -
.
(25)
Now, the coefficient ~ h ° in equation (22) can be computed from equations (24) and (25): 2C D 4a~ v 6~h- = 3n(1 - a)'u,d}"
(26)
)~Co is substituted into equation (22):
The term ~
+St
u,d}
+E.
(27)
Equation (27) is rewritten in dimensionless form by defining a dimensionless velocity q5 as a function of a dimensionless distance from the wall ~ = y/h.
U U,
--
resulting in:
= 4,(0,
(28)
(d )2 (d )3 +b
a7
-e,
(29)
Turbulent deposition of particles
909
where 3n(1 - ~ ) (
a=
ds ) 2
G~
~
,)5
b = rc(1-~t)h+'(h 2~ta
(30)
(~
)5
e = a[~b'(0)] 2 + bl-~b'(0)] a.
(31) (32)
This equation expresses a dependence of the velocity on geometric characteristics of the roughness layer. Equation (29) can be rewritten as: t~ -----#a~b '2 + bqb'3 - e.
(33)
This equation is of the form =
0(~'),
(34)
for which a parametric solution is known (Kamke, 1959): ~= 1+
f ' O'(t) dt tl
(35)
t
= o (t).
(36)
Comparing equations (34) and (36) we note that the parameter t has the physical meaning of
de
t = d-~-"
(37)
The function g'(t) in equation (35) is found by differentiating equation (33) with respect to the variable t = $', resulting in: = 1+
f t tl
2a + 3bt 3(at 2 + bt 3 _ e)2/3 dt
(38) (39)
dp = ~/at 2 + bt 3 - e.
Here tl is the value of the parameter t corresponding to ~ = 1. It is determined by computing the stress at y = h. From a force balance it is obtained that T = zw (1 - y / R ) , where R is the tube radius. Since for the roughness layer h <~ R, we may approximate 17w ~ , "t'h.
NOW, zw=zh=p(v+e)~y
du y=h
u, ,~12U2,2 =PV-~q+e h2,1,
therefore:
2 vu, 12u2"71, u, =--h-t1 +--~-which leads to
1
1)2 ,40,
,,
The only other term not determined as yet in the expression for the velocity inside the roughness layer, equations (38) and (39), is the constant e. We rewrite it from equation (32), setting ~'(0) = to: e = at~ + bt~,
910
ALEXANDER ORON and CHAIM GUTFINGER
IO.O00i
1.000
/ o
0.100
to(a= 0,175) to(a =0.271)
.
0.010
tota=0.5} I 2
0.001 0
I 4
I (5
t 8
L-_ I0
DIMENSIONLESS HEIGHT h+
Fig. 1. Values of the parameter t as a function of dimensionless roughness height for different roughness layer solidities.
which upon substitution into equations (38) and (39) yields the final expression for the velocity profile inside the roughness layer given by
3bt _ t3)] 2/xdt = 1 + f ' 3[a(t2 _ t22a) ++ b(t3 I1
q~ = , ~ ( t 2 - t2o) + b(t 3 - t3).
(41) (42)
The parameter to corresponds to ~ = 0 where ~b = 0. Hence, from equation (41) we obtain:
f t, (2a + 3bt)dt to 3[a( t2 - ~ o 2 ) ~ - t3)] 2/3 = 1.
(43)
Figure 1 presents plots of to and t~ as functions of h+. The velocity profile inside the roughness layer is now defined by equations (41) and (42) with the constants t~ and to given by equations (40) and (43), respectively. This velocity profile satisfies the non slip condition, u Ir =o = 0.
Velocity and eddy viscosity in the region outside the roughness layer According to the results of Monin and Yaglom (197 l) the mean velocity distribution in the region y > h is given by
u(y) _ 1 In y -- d, u,
x
(44)
Yo
where x is the von Karman constant approximately equal to 0.4, d is the displacement height and y0 is a constant of an order of magnitude h. This expression is valid for the fully rough flows (h+ > 70). Since we are dealing with transitionally rough flows (h+ < 30), we need a similar expression for the mean velocity in this region. We develop it using the approach of Monin and Yaglom. Let us first list the parameters which the flow velocity and its derivatives depend on. These are the fluid density, p, the stress on the wall Tw, the distance from it, y, the kinematic viscosity
Turbulent deposition of panicles
911
of the fluid, v, and the height of the roughness layer, h. From these parameters it is possible to form two dimensionless groups, h + and h/y. So, dy
_ u,f
Ky
h+
(45)
,
where the funetionfis a correction of the logarithmic velocity profile due to the existence of the roughness layer. By expanding f ( : , h + ) in a Taylor series, it is obtained that d__u_u= u, [F(h+)+ h G(h+)], dy r(Y - Yx) y - yl
(46)
where F and G are functions that will be defined later and y~ is a parameter corresponding to the displacement height of Monin and Yaglom. The mean velocity u(y) is obtained by integration of equation (46):
u(y) _ F(h+ ) in y - y: u, K yo
hG(h+ ) ~(y - yl)'
(47)
where y0 is a constant of integration that satisfies continuity of the mean velocity at y = h. Like the y~ of Monin and Yaglom, our y~ has a physical meaning of the effective displacement of the wall due to the existence of the roughness layer. Under the assumption that l = K(y- Yl) above the roughness layer, we express the eddy viscosity, e, in this region by du e = 12 ~y = x(y -- Yl )u,F(h+) + xhu.G(h+ ),
(48)
and, in dimensionless form:
- = ~cF(h+)(y+ - y~ +) + ~ch+G(h+).
(49)
v
Equations (47) and (49) provide expressions for the velocity and eddy viscosity profiles above the roughness layer. In order to uniquely determine these profiles the functions F(h+) and G(h+) have to be found. This is accomplished by matching the shear stress, found from equation (46) with that in the roughness layer at y = h, and the velocity profile at h + = 70 with that of Monin and Yaglom. Hence, at y = h,
l -u- ,~q= u ,
[ F ( h + ) + h _ yh 1 G(h+ )l '
(50)
where the left-hand side of equation (50) was found by differentiating equation (39) and setting ~b'(1) = tl at the edge of the roughness layer, y = h. Thus, the function G(h+) is expressed by:
G(h+)=hhYl
-F(h+)+
t~ -
~+
-h t: .
(51)
We now substitute the expression for G(h +) from equation (51) into the expression for the velocity, equation (47),
Now comparing the expression for the velocity profile, equation (52) with that of Monin and Yaglom, equation (44), we note that the matching requirement at h+ /> 70 results in
F(h+) = 1, h+ >1 70
(53)
912
ALEXANDER ORON and CHAIM GUTFINGER
and
l
F(h+)-~t, =0, h+ >~70.
(54)
Substitution of equation (53) into equation (54) results in h tl = ~ ,
h. >~70.
(55)
Equation (55) can also be derived from equation (40) by letting h+ --, co. This fact attests to the consistency of the present model. Grass (1971) has performed experimental measurements of the mean velocity in flows over smooth and rough surfaces. From Fig. 4 in his work, which plots the dimensionless velocity, u +, versus the natural logarithm of the dimensionless distance from the wall, lny +, it is seen that the slopes of the lines for rough surfaces are identical to the slope of the line corresponding to flow over a smooth surface. His experiments cover data from smooth surfaces and up to h+ = 84.7. Thus, Monin and Yaglom's statement that F(h+) = 1 is essentially correct over the whole range ofroughnesses from smooth and up to h+ = 84.7. We therefore extend the validity of equation (53) over the whole range of interest of h+ and write down the final expression of the velocity profile outside the roughness layer by amending equation (52) to read u
u+ = - - = u.
In y - y l
Yo
Yl
1-
tt
•
(56)
Y - Yl
Similarly the expression for the eddy viscosity, equation (49) can be rewritten as:
We now consider the question of where the outer edge of the transition zone is. The velocity distribution in it is the disturbed logarithmic profile. The second term in equation (56) decreases with increasing y. Hence it is logical to assume that the outer edge of the transition zone is that of the smooth wall 'logarithmic region'. Based on experimental data, Reynolds (1974) suggests that this region extends to y = 0.2R, R being the pipe radius. The magnitude of the constant l/h is found from an assumption of its continuity at y = h:
From the experimental results of Grass (1971) it can be obtained that the shifting of the logarithmic profile of the mean velocity due to an existence of the roughness layer is 0.6h approximately for transitionally rough flows, so Y~ = 0.6. h
(59)
Substitution of equation (59) into equation (58) leads to: 1 h
0.16.
(60)
This value was used in equations (31), (40), (56) and (57) for the respective calculations inside and outside the roughness layer. Plots of dimensionless velocity profiles inside and outside the roughness layer are presented as functions of the dimensionless coordinate, ~ = y/h, in Figs 2 and 3, respectively. The curves were calculated for a roughness layer solidity of 0.0963, a value for which experimental filtration data were reported by Lee and Liu (1982). The parameter in Figs 2 and 3 is the dimensionless height of the roughness layer, h+. Figure 3 was drawn for the whole domain ~ ~>0, and therefore Fig. 2 may be viewed as an enlargement of the lower left corner of Fig. 3.
Turbulent deposition of particles
913
2.5
"O-
2.0
>I-
U q
1.5 =,
/
h+=5 ~ 1 /
°~ I.O
/
0.5
L 0
I
I I I 0.5 I.O DIMENSIONLESS COORDINATE ~=y/h
I
1.5
Fig. 2. Dimensionless velocity inside the roughness layer (¢ = 0.0963).
10.0 -o>-
o kO > ¢O ~O Z
_o
o.o_
//,//
\ 0 i:0o
lad B
2.5
0
,5 I0 DIMENSIONLESS COORDINATE~=y/h
15
Fig. 3. Dimensionless velocity outside the roughness layer (~ = 0.0963).
As seen in Fig. 2, the velocity inside the roughness layer approaches zero smoothly as ~ --* 0. At the edge of the roughness layer, ~ = 1, the velocity profile inside the roughness region calculated from equations (41) and (42) connects smoothly and continuously to that in the outside region, equation (56). Figure 4 shows the velocity profiles inside the roughness layer for several values of solidity, ~. As seen from this figure, the velocity drops sharply in the region close to the tips of the roughness elements for higher values of solidity, ~. In contrast with this behavior, at very low solidities (~ = 0.005) the velocity distribution becomes approximately linear. This is
ALEXANDER ORON and CHAIM GUTFINGER
914
3.5~
3.0 a=O.O05
\
-,e>.. 2.5 io o IJJ > 2.0 co co w
J z
o co z w Q
1.5
1.0
a =0.0963 ~=0.175~ e:0.271
0.5
0
0.5 1.0 DIMENSIONLESS COORDINATE~=y/h
Fig. 4. Dimensionless velocity inside the roughness layer for several values of fiber layer solidity (h+ = 10).
I.O
~
-e>i,-
PRESENT THEORY THEORY OF GUTFINGER & FRIEDLANOER
//
/ ~
/ /
/
w >
/ ~, o.~
z
z w
0
0.5 DIMENSIONLESS
COORDINATE
1.0 ~'=ylh
Fig. 5. Velocity profile inside the roughness layer. Comparison with the solution of Gutfinger and Friedlander (1985) (h+ = 10, ct = 0.0963).
qualitatively similar to the velocity distribution developed by Levich (1962), equation (12), for turbulent flow near a rough wall. In Fig. 5 we compare the velocity profile inside the roughness layer calculated from equations (41) and (42) with that derived by Gutfinger and Friedlander (1985). The comparison is performed for the relatively dilute roughness layer of solidity (ct = 0.0963). As
Turbulent deposition of particles
915
seen from the figure, the solution given in the present paper results in a more moderate velocity decay inside the roughness layer.
PARTICLE C O N C E N T R A T I O N AND D E P O S I T I O N VELOCITY After solving the hydrodynamic problem we now derive an expression for the particle deposition rate. We divide the whole domain of the flow into three regions: 1. The roughness layer (0 ~< y ~< h). 2. The transition zone which extends from the tips of the fibers to the edge of the logarithmic region, which, as mentioned above, lies at a distance of approximately y = 0.2R. 3. The turbulent core of the flow (3: > 0.2R). If kc, k,, and k, denote the deposition rates in the turbulent core, transition region and the roughness layer, respectively, the overall deposition rate, ktot, is obtained from
1 1
kt°t = I
1"
(61)
Similarly, the dimensionless overall deposition rate is expressed as: kt°t+ =
1 1
1
1
(62)
k;+' +k~-+ + k,--f First, we derive the deposition rate in the roughness layer, k,. In this region the following diffusion-convection equation is satisfied:
Oc
Oc
Oc
~c )
u~x + V-~y = ~---~((D + e,)-~x ) + ~---~((D + ~p) -~y
/~(y)c,
(63)
where x is the longitudinal variable, y is the lateral variable, i.e. the distance from the wall, u, v are components of the time averaged velocity, D is the molecular diffusion coefficient and ,p is the turbulent diffusivity of the particles. The term fl(y) is the absorption coefficient, under the assumption that the rate of particle removal by the roughness elements is proportional to the local particle concentration. This is, indeed, the physical meaning of the last term in equation (63). Lee and Liu (1982)presented the absorption coefficient in the form /
/~(y) =
•
~u(y),
(64)
where
4=r/ fl - ~dr( l - or)'
(65)
and ~/isthe overall filtrationefficiencyof a single fiber of diameter d/in the roughness layer. W e now assume the pipe flow and the concentration profile to be fully developexl and neglect the diffusiveflux in the x-direction compared with that in the y-direction.W e also neglect the convective terms, as is usually done in turbulent mass transfer. This leads to:
dy
(D + ep) ~y
= flu(y)c,
(66)
with the boundary conditions
c=0
aty=0
(67)
C=Ch
aty=h.
(68)
916
ALEXANDERORON and CHAIM GUTFINGER
Since the velocity u(y) inside the roughness layer is written in the parametric form, equations (38) and (39), we must replace the y-dependent equation (66) by the t-dependent equation. This leads to the following equation: d [{1 ~p'~ t dc7 9'(t)g(t) g-- + - - 1 ~-~-,, --::l = flhh+ - - ~ , t ~c v J g (tl Ot l L\
(69)
with the boundary conditions ( = 1, t = t ~
(70)
(=0,
(71)
t=t0,
( being the particle concentration normalized by ch. The difference between the turbulent diffusivity of the flow and that of the particles was discussed in the Introduction. Many authors assumed that they are equal, i.e. ep = e. Our opinion is that it is important to incorporate into the model a difference between them that would account for the effect of particle inertia. This effect is very pronounced in the very close proximity of the wall for relatively large particles, i.e. those of diameter of above 1/~m. In the following derivation we use the model of Kitamoto and Takashima, (1974a, b) because of its simplicity and the reasonable results which can be obtained. According to this model,
,,
v
\dy+/ /
\ay+/
where z + is the dimensionless particle relaxation time and a = V'p/V' is the relation between the r.m.s, particle velocity and the r.m.s, flow velocity. Expressing du +/dy + in equation (72) in terms of t with the help of equation (37), we obtain:
--
h+t
l + 2 a r + ~+ +a2~z+ t~_ .
(73)
We follow Gani~ and Mastanaiah (1981) and assume a semi-empirical expression for a in the form 1 a = , (74) 1 + 529Re -°5°9 z+ R+ where R is the pipe radius. This expression is based on experimental data of Martin and Johanson (1965). It gives a good estimation for (r in the range 2 x 104< Re < 1.5 x 106. In order to obtain the concentration profile d we have solved equation (69) with the boundary conditions, equations (70) and (71). The solution was obtained numerically using a Runge-Kutta procedure. Figure 6 shows the normalized concentration profiles inside the roughness layer for particles of diameters 0.65, 1.1 and 2.1 #m. These values correspond to the experimental data of Wells and Chamberlain (1967). These data lie in the inertia-interception dominating domain of particle diameters; a larger diameter leads to an increase in the single fiber overall filtration efficiency. This results in lower concentrations for larger particles. The deposition flux in the roughness layer is computed from: de f h Jh = (O + el,) ~y y = o + o fl(y)c(y)dy.
(75)
Here, the first term expresses a particle flux at the wall, while the second one accounts for particle deposition on the roughness elements. In terms of the variable t the expression for the flux is rewritten as:
[to
r/l
ep'~ d¢?]'
4~/
(h)~,,,g(t)g'(t)d(t)dt]"
(76)
Turbulent deposition of particles
917
I.O
0.75 Z 0 n.. I.-Z w L) Z 0 ¢.~
p.m
0.5
u3 LU J Z 0
I/a.m
Z LU
F, 0.2!
N' 0
I 0.25
I 0.5
I 0.75
I 1.0
DIMENSIONLESS COORDINATE ~'=y/h
Fig. 6. Dimensionless concentration profile inside the roughness layer (~ -- 0.0963, h+ = I0).
The particle deposition rate in the roughness layer, k. can be calculated from
k, =--=u,Jh ch
F
~
Lh+o (to)
[ ( 1 +-v-~P)]dE ~ + 4~q (h)ft'9(t)9'(t)c(t)dt],t ~ ,=,o rt(1 - =) ~ to (77)
or, in dimensionless form,
I( 1 ep ) dE1 "~ n(14o~q k,+ - Jh to - 0t) (ff__f) ftl 9(t)g'(t)E(t)dt. t u.ch h+g'(to) S-c + v -d-i t=,o to
(78)
For the transitional zone the resistance to particle transfer is computed by integrating the flux equation: d = (D+~p) dc d-~, (79) leading to
k,+ =
k~ 1 -- = , u, [" 0.2R÷ dy+
(80)
|.Jh~+S+__1+ ~_e_~ Sc v where S is the stopping distance of the particle. We assume here that a particle which approaches within a distance S from the tips of the fibers reaches the roughness layer. The turbulent particle diffusivity is given by equation (72) which, together with equation (57), results in:
918
ALEXANDER ORON and CHAIM GUTFINGER
where du÷/dy+is found from equation (56). Equation (81) is valid when the fibers go beyond the viscous sublayer (h+ > 5). When the flow is hydraulically smooth, i.e. h+ < 5, we have to divide the region outside the roughness layer into two regions: the viscous region (h+ < y+ < 5) and that of 5 < y~ < 0.2R+. Thus, the resistance of the transition zone in this case is obtained from: 1 _ 1 f h5 dy+ kt u, . +s+ 1 % Sc v
+ 1 fs0,2R, -----.dY+ u,
1 Sc
(82)
% v
The dimensionless deposition velocity is obtained from: 1
k"=fh5
dy+
I0.2/~ dy+
++St Sc 1 ~+
-{- j5
--
1
Sc +
(83)
% v
The second integral in equation (83) is calculated with the help of equation (81), while in the first integral we estimate the expression %/vby means of equation (72) with the expression for s/v taken from the distribution of Lin et al. (1953): ~
(y+)3.
v
(84)
lq5.5
The expression du +/dy÷ is computed by differentiating equation (56). In order to compute the particle deposition velocity in the turbulent core we follow Friedlander and Johnstone (1957) and use the Reynolds analogy by assuming that the turbulent Schmidt number of a particle is equal to unity:
Sc,-
Ep
- 1.
(85)
Accordingly, J c,v - cc = .z2 (Uo - uc),
U,
(86)
c~ and u~ being the concentration and the velocity at the outer edge of the transition zone, respectively. The velocity uc can be obtained from equation (56) at y = 0.2R. The particle deposition velocity in the turbulent core is given by: kc -
j Car - - Ce
-
u2,
(87)
U o - - Uc'
or, in dimensionless form: u, kc+ = Uo-U~"
(88)
The overall particle deposition rate is computed from equation (62) with equations (78), (80) [or (83)] and (88). C O M P A R I S O N WITH EXPERIMENTS In the literature there are few works which can supply experimental rough surface deposition rates in the range of 0 < h + < 30. There are data on grass-covered surfaces, however these correspond to h + >> 30. One of the experimental works that covers the range of the present paper is that of Wells and Chamberlain (1967). It provides data on deposition rates of tri-cresyl-phosphate particles to vertical rough surfaces. The experiments were performed in an annular space between two concentric tubes with filter-paper lined walls.
Turbulent deposition of particles
919
The height of filter's fibers was 100 microns. The measurements were performed with particles of diameters 0.65, 1.1 and 2.1 #m and flow Reynolds numbers 4000-45,000. The range is equivalent to the range 1.1 < h+ < 10. It should be noted that Wells and Chamberlain did not report the diameter of the fibers dI and the solidity of the fiber layer 0t. This missing information was taken from the work of Lee (1977) corresponding to his Dacron filter B for which 0t = 0.0963 and d I = 12.9 #m. That work also provided the single fiber overall filtration efficiencies used in computing the deposition rates. Figure 7 compares the results of the present theory with the experimental data of Wells and Chamberlain. It should be remembered that the missing values (ds and ct) were taken arbitrarily, hence the comparison has a symbolic character. This comparison shows that the present theory gives reasonable results for particle deposition onto fibrous surfaces. Specifically, a good fit is achieved with the experimental data for particles of diameters 1.1 and 2.1 #m. For the 'small' particles of 0.65 #m the fit is satisfactory for Re > 35,000 (h+ > 7.5). It should also be noted that for small dimensionless roughness, h+ (corresponding to low Reynolds numbers), the deposition rate is approximately equal for all tested particles. This can be explained by the fact that the overall deposition rate in this case is accounted for by the very high resistance of the viscous sublayer, which is almost identical for these particles. Compared to this high resistance, the resistances of the roughness layer and of the transition zone are negligible. As the dimensionless roughness height, h +, increases, the resistance of the viscous sublayer decreases and finally vanishes, when the roughness elements protrude beyond it. In this case, the resistance of the roughness layer becomes the most important and affects the overall deposition velocity. Now, for larger particles, the resistance of the roughness layer is lower due to the higher single fiber filtration efficiency. For smaller particles there is a higher chance of escape through the fibers, hence the resistance of the fiber layer is higher and the deposition rate is lower than for larger particles. The resistance of the roughness layer for the transport of small particles decreases very slowly with increasing of h ÷, so it is dominant in the computation of the overall deposition rate. For large particles, the resistance of the roughness layer decreases sharply with increasing h +. In this case, the transition region resistance dictates the overall deposition velocity. h+ 5 I
I0 I o
I0.00
~
°
.1/
.~"
+
.....~
>. 1.00, I-.
q
,y
hi >
,.EORY
/~
Z
--
,f
o OJO, I,-
0
/
w
0.01
0.0(
x
0
---dp=2.1 p.m
- - ~
t
dp=l. I p m
EX_PER~_ ATA o
dp=21Fm
p
+
dp= I.I p.m
x
x
dp=O.65 p.rn
I
I0000
I
20000
I
30000
I
40000
I
50000
REYNOLDS NUMBER Re
Fig. 7. Comparison of the theoretical results with the experimental data of Wells and Chamberlain (1967) (a = 0.0963). AS 1 7 : 6 - C
920
ALEXANDER ORON and CHAIMGUTFINGER SUMMARY
This paper has presented a theoretical model of particle deposition to rough surfaces from turbulent streams. The flow domain was divided into two regions: the roughness layer and the region above it. First the hydrodynamic problem was solved for the region inside the roughness layer. By assuming that the mixing length of the flow is independent of the transverse coordinate, a one-dimensional Navier-Stokes equation was written down and solved in terms of a parametric solution. The velocity profile obtained is a function of fiber layer solidity, its geometry and the flow Reynolds number. For small solidities the velocity distribution is approximately linear (Fig. 4) and qualitatively similar to the velocity profile developed by Levich [equation (12)]. Using an approach similar to that of Monin and Yaglom, profiles of the velocity and of the eddy viscosity were derived for the region outside the roughness layer. The velocity profiles inside and outside the roughness layer are connected smoothly and continuously, as seen in Fig. 3. The velocity profiles found in the first part of the paper were used in the solution of the mass transfer problem. Inside the roughness layer a diffusion-convection-absorption equation was solved, particle concentrations obtained (Fig. 6) and a resistance to particle deposition computed. The region outside the roughness layer was treated by means of a standard procedure of integration of the mass flux equation. The particle turbulent diffusivity was obtained using a corrected Kitamoto-Takashima expression. The overall particle deposition rate was obtained by addition of resistances of the roughness layer and the region outside it. The theory was used to correlate the experimental deposition data of Wells and Chamberlain (1967). The agreement between theory and experiments was found to be reasonable. REFERENCES
Browne, L. W. B. (1974) Atmos. Envir. 8, 801. Davidson, C. I. and Friedlander, S. K. (1978) J. Geo. R-O A. 83, 2343. Davies, C. N. (1966) In Aerosol Science, (Edited by Davies, C. N.). Academic Press, London. EI-Shobokshy, M. S. (1983) Atmos. Envir. 17, 639. EI-Shobokshy, M. S. and Ismail, I. A. (1980) Atmos. Envir. 14, 297. Fernandez de la Mora, J. and Friedlander, S. K. (1982) Int. J. Heat Mass Transfer, 25, 1725. Friedlander, S. K. and Johnstone, H. F. (1957) Ind. Engn a Chem. 49, 1151. Ganie, E. M. and Mastanaiah, K. (1981) Int. J. Multiphase Flow, 7, 401. Grass, A. J. (1971) J. Fluid Mech. 50, 233. Gutfinger, C. and Friedlander, S. K. (1985) Aerosol Sci. Technol. 4, 1. Hahn, L. A., Stukel, J. J., Leong, K. H. and Hopke, P. K. (1985) J. Aerosol Sci. 16, 81. Hinze, J. O. (1975) Turbulence: An Introduction to its Mechanism and Theory. McGraw-Hill, New York. Inoue, E. (1963) J. met. Soc. Japan 41, 317. Jackson, P. S. (1981) J. Fluid Mech. 111, 15. Kamke, E. (1959) Differential#leichungen. Leipzig. Kitamoto, A. and Takashima, Y. (1974a) Bull. Tokyo lnst. Technol. 121, 41. Kitamoto, A. and Takashima, Y. (1974b) Bull. Tokyo Inst. Technol. 121, 67. Lee, K. W. (1977) Filtration of submicron aerosols by fibrous filters. Ph.D. thesis, University of Minnesota. Lee, K. W. and Liu, B. Y. H. (1982) Aerosol Sci. Technol. 1, 35. Levich, V. G. (1962) Physiochemical Hydrodynamics. Prentice-Hall, Englewood Cliffs, NJ. Liu, B. Y. H. and Agarwal, J. K. (1974) J. Aerosol Sci. 4, 227. Liu, B. Y. H. and Ilori, T. A. (1974) Envir. Sci. Technol. 8, 351. Lin, C. S., Moulton, R. W. and Putman, G. U (1953) Ind. Engng Chem. 45, 653. Martin, G. Q. and Johanson, L. N. (1965) AIChE J. 11, 29. McCoy, D. D. and Hanratty, T. J. (1977) Int. J. Multiphase Flow 3, 319. Monin, A. S. and Yaglom, A. M. (1971) Statistical Fluid Mechanics: Mechanism ofTurbulenee, Vol. 1. MIT Press, Cambridge, MA. Owen, P. R. (1969) J. Fluid Mech. 39, 407. Reynolds, A. J. (1974) Turbulent Flows in Engineering. John Wiley, London. Sehmel, G. A. (1980) Atmos. Envir. 14, 983. Shaw, D. A. and Hanratty, T. J. (1977) AIChE J. 23, 28. Thom, A. S. (1975) In Vegetation and the Atmosphere, (Edited by Monteith, J. L.) Vol. 1. Academic Press, London. Wells, A. C. and Chamberlain, A. C. (1967) Br. J. appl. Phys. 18, 1793. Wilson, I. B. (1947) J. Colloid Sci. 2, 271. Yaglom, A. M. and Kader, B. A. (1974) d. Fluid Mech. 62, 601.
0021 ~8502/86 $3.00+0.00 Pergamon Journals Ltd.
Printed in Great Britain.
ON TURBULENT DEPOSITION OF PARTICLES TO ROUGH SURFACES ALEXANDER ORON a n d CHAIM GUTFINGER Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel
(First received 14 November 1985; and in final form 14 February 1986) Abstract--An analytical method is presented for calculation of particle deposition from turbulent streams to rough surfaces with fibrous roughness elements. Particle deposition is considered as a twostage process consisting of filtration inside the roughness layer and turbulent transport outside it. Velocity profiles are developed for these two regions and matched continuouslyand smoothly. These are used to calculate the transport processes in each region. The overall particle deposition velocity is obtained by addition of resistances of the roughness layer and of the domain above it. Comparison with experiments of Wells and Chamberlain (1967) shows that the theory correlates quite well with the experimental data.
NOMENCLATURE
p
Particle concentration Drag coefficient Diameter Particle diffusivity Fiber height Dimensionless fiber height, hu,/v Particle flux Deposition velocity Mixing length Pressure Particle radius Pipe radius Reynolds number, 2RUo/v Stopping distance, given by equation (1) Dimensionless stopping distance, Su./v Schmidt number, v/D Parameter of the solution, equation (35) Velocity ~_ Friction velocity, (~w/p)2 Average velocity in pipe flow r.m.s, velocity Coordinate Coordinate Dimensionless coordinate, yu,/v Solidity, solid fraction Filtration coefficient Eddy viscosity Filtration efficiency Kinematic viscosity Dimensionless coordinate, y/h Density
a
V~/V'
T
Relaxation time Dimensionless velocity, given by equation (28)
C
Co d D h h+ J k 1
P rp R
Re S S+ Sc t U U,
Uo 1)' X
y y+ ri
v
Subscripts Average At the outer edge of the transition region f Fluid h Aty=h P Particle + Dimensionless
av
c
As 17:6-B
903
904
ALEXANDER ORON and CHAIM GUTFINGER
INTRODUCTION Computation of particle deposition from turbulent gas streams is very important because of many applications in various fields of engineering, technology and health physics that deal with gas-solid two-phase flows. These applications include pneumatic transport of particles in pipes (Owen, 1969), particle deposition in vegetation canopy (Thorn, 1975), design of sampling devices and air cleaning installations and particle deposition in human lungs (Wilson, 1947). When turbulent flow of a gas with solid particles takes place near a solid wall, the turbulent mechanism is the main mechanism of particle deposition. There is a possibility that some additional mechanisms are involved in the deposition process, e.g. Brownian diffusion (in the case of submicron particles), gravitational settling (in the case of heavy particles deposited on horizontal surfaces) and electrical effects (electrically charged particles in the presence of an external electric field). Beside these effects, when a wall is rough, the roughness elements provide an additional mechanism for removal of the particles from the turbulent gas stream. In the field of theoretical investigations Friedlander and Johnstone were the first to explain how solid particles deposit on a smooth wall. They define a concept of'the stopping distance' of a particle. It is a distance that a particle travels in free flight due to its initial velocity through the fluid until stopping. For a particle of a diameter dp, density pp and initial velocity Vpo the stopping distance is given by: 2
S = dn Vpo "--,PP 18v p
(1)
where v is the dynamic viscosity of the fluid and p its density. When molecular diffusion is negligible, the dominant mechanism of particle deposition from a turbulent stream is the one due to flow field fluctuations in the direction perpendicular to the wall. According to the model of Friedlander and Johnstone (1957), these turbulent fluctuations bring a particle within one stopping distance from the wall and the deposition process is completed by the free flight of the particle toward it. McCoy and Hanratty (1977) considered the relative effects of turbulent fluctuations and molecular diffusivity on deposition of particles, and found that the effect of molecular diffusivity is negligible for particles with dimensionless relaxation times, z +, higher than 0.15, where r + is defined as
1.pp.(dpu, z+=18
)a.
(2)
p \--~-
pnd2/18u,
The relaxation time, z = is the time in which the force acting on a rigid particle reaches the e- 1 part of its initial value, thereby diminishing the particle velocity by a factor o f e -1. The deposition velocity of the particles in a given region is defined as the ratio of particle flux to the change of particle concentration through this region: J k = --. Ac
(3)
This quantity, which has dimensions of velocity, is put in dimensionless form as k+ -
k It,
,
(4)
where u, = x / / ~ , is the friction velocity, zw being the shear stress on the wall. Friedlander and Johnstone assumed that the turbulent diffusivity of a particle en which arises from the relation d6
-
v'c'
= ~PUy
(5)
Turbulent deposition of particles
905
is equal to the eddy viscosity, e, of the fluid which is given by Lin et al. (1953): e = v(y+/14.5) 3
0 <~ y+ <<. 5
e = v(0.2y+ -0.959)
5 ~< y+ ~< 30
e = 0.4vy+
30 ~< y+ ~< 100.
(6)
The problem of very small particle deposition, i.e. the mass transfer problem, has been solved by Levich (1962) under the assumption that there are two important mechanisms, namely that of molecular diffusion and that of turbulence. His method, similar to that of Friedlander and Johnstone (1957), consists essentially of integration of the flux equation: dc d = (D + ep)~yy,
(7)
leading to an expression for the deposition velocity k + = const. So- 2/3,
(8)
which is similar to the experimental results of Shaw and Hanratty (1977), who found k+ = 0.089-Sc- 0.704,
(9)
where Sc = v/D is the Schmidt number and D the particle molecular diffusivity. Liu and Ilori (1974) assumed that the turbulent mass diffusivity of the particle, el, and the turbulent mass diffusivity of the fluid, e, are not the same and suggested an expression: e~ = e + v'2~,
(10)
where v' is the turbulent intensity of the flow field in the direction of the wall and z is the particle relaxation time, given above. Inertial effects are characterized by the additional term v'2 z, which vanishes as the particle relaxation time decreases, i.e. as its diameter decreases. In that case, the particle follows all the turbulent fluctuations of the fluid. The relaxation time term is very important when the particle is not small and does not immediately react to flow field fluctuations. Another example of non-identical expressions for e and ep is that of Kitamoto and Takashima (1974), who suggest: e,=e
l+2az+
~
+a2z2+
~y+-
,
(11)
where a = v~/v'. In contrast with the large quantity of experimental and theoretical papers dealing with particle deposition on smooth surfaces, the number of papers on rough surface deposition is fairly limited. Levich (1962) has solved the problem of mass transfer to a fully rough surface (h+ >> 1) under the assumption of a linear velocity profile inside the roughness layer: Y u(y) = u, ~,
(12)
where h is the height of roughness elements. His result, in terms of dimensionless deposition velocity, is: k+ = const Reh-1/2 Sc -3/4, (13) where Reh = hu./v is the Reynolds number based on the roughness height. Some recent papers have shown that this result is not very accurate, and that the deposition rate is proportional to Sc-2/3 (Yaglom and Kader, 1974; Fernandez de la Mora and Friedlander, 1982; Hahn et al., 1985). There are two main approaches to the problem of particle deposition onto a rough surface. One of them, which can be termed as a geometric one, is based on the derivation of the 'effective stopping distance', defined by Friedlander and Johnstone (1957) and constitutes an
906
ALEXANDER ORON and CHAIM GUTFINGER
extension of their theory. Browne 11974) has defined a 'capture distance' that takes into account the roughness of the wall and has generalized the theory of Davies (1966). E1Shobokshy and Ismail (1980) and EI-Shobokshy (1983) have incorporated into the theory of Browne an expression of Liu and Ilori (1974) for a particle eddy diffusivity. These theories have been verified for hydraulically smooth flows (h, < 5). Sehmel (1980) has reviewed the general field of particle deposition from the atmosphere to rough surfaces for the case of h+>>5. A second method for solving the problem consists of treating particle deposition as a twostage process, that of turbulent diffusion from the turbulent core to the roughness layer and that of deposition inside it. Davidson and Friedlander (1978) have solved the problem of deposition of heavy particles from the atmosphere onto a rough surface (tall grass) under the influence of gravity. Gutfinger and Friedlander (1985) have presented the method of computing particle deposition rates in the case of a fully developed flow in a pipe with walls roughened by a fibrous layer. The central point of their work is that the fibrous roughness layer is an additional resistance for particle deposition. Comparing their theory with the data of Wells and Chamberlain (1967) has shown that for particles of 0.65#m diameter the two-stage theory does indeed correlate with the data quite well. For larger particles, however, the two-stage theory gave results that were worse than the simple single-stage eddy viscosity extension of the Friedlander and Johnstone model. A close scrutiny of their work reveals that some of their assumptions are excessively restrictive. These are listed below. 1. The expression for the eddy viscosity of the flow does not account for the existence of the fibers on the wall and therefore can at best hold for the case of very low fiber densities. 2. The turbulent diffusivity of the particles is assumed to be identical to that of the fluid, limiting their theory to particles of very low inertia. 3. In spite of the requirement of a very dilute fiber canopy as stated in assumption 1, it is assumed that the filtration process is essentially limited to the tip of the fibers. The present paper solves the problem of particle deposition to rough surfaces without making use of the above assumptions. We feel, therefore, that the solution herewith presented can serve as an extension of the solution of Gutfinger and Friedlander to dense rough surfaces and does not suffer from the limitations imposed on the solution by the other assumptions. F O R M U L A T I O N OF THE P R O B L E M We solve the problem of deposition of suspended particles from a turbulent fully developed pipe flow to rough walls. We limit our treatment to rough surfaces which consist of fibers of identical height h and diameter df, and which do not penetrate into the turbulent core of the flow, i.e.
h÷ < 30.
(14)
The last assumption is not restrictive. If the fibers do protrude into the core of the flow, the two-stage process is reduced to a single-stage process consisting of the roughness layer only. The resistance to particle deposition is negligible in the turbulent core because of perfect mixing in it. The following assumptions are incorporated in the mathematical model: (a) The solid particles are of spherical shape. (b) The concentration of the particles is low enough as not to affect the flow field. Likewise, we neglect any interaction between the particles. (c) The particles and the pipe walls are not charged. Likewise, there is no external electric field, precluding any electrostatic effects. (d) We neglect the effect of the gravitational force on the deposition process. In this context we can assume the deposition as taking place on vertical walls.
Turbulent deposition of particles
907
(e) The flow is fully developed turbulent pipe flow, hence the radial component of the mean velocity vanishes. (f) We ignore effects ofreentrainment andbounce-offoftheparticles on thewall. In other words, the condition of removing a particle from the flow is its reaching the wall. (g) The particles are assumed to be large enough the effect of molecular diffusion on the deposition process could be neglected. This corresponds roughly to particle sizes of > 0.5 #m. We now present the two-stage mathematical model of particle deposition on a rough wall. First, we consider particle deposition inside the roughness layer. Then we treat the turbulent diffusion process in the fluid outside the roughness layer. The overall deposition velocity is obtained by addition of resistances of these two regions. VELOCITY PROFILES INSIDE AND OUTSIDE THE ROUGHNESS LAYER
The velocity profile inside the roughness layer For a one-dimensional model in which the only non-vanishing velocity component is u, and flow field homogeneity in the directions x and z is assumed, the time-averaged Navier-Stokes equations reduce to:
~y(-U'V')+v ~
/15)
- p ,gx'
where tit are the time-averaged velocity components ui and u~ the deviation from it, caused by turbulent fluctuations. The left-hand side of this equation can be identified as ~3z/cgyand the right-hand side as the total drag force per unit volume of the fluid fo, z being the local shear stress. Now equation (15) is written as: 0r cgy = fv,
(16)
and expression fv in terms of a drag coefficient, Co. Equation (16) may be rewritten as (Jackson, 1981):
~O~ = p ~2C o u2,
(17)
where 2 = nAI/S, with n being the number of fibers with a frontal area AI on the area S free of them. The shear stress z may be expressed in terms of molecular and eddy viscosity as:
• =p
(0u du)(du =p
+12
Ty
'
(18)
where I is the Prandtl mixing length and e the eddy viscosity. Let us assume that the mixing length is independent of y. This assumption has already appeared in the literature (Inoue, 1963; Jackson, 1981) and has led to the exponential velocity profle inside the roughness layer. Substituting equation (18) in equation (17) leads to: d2u 2 du d2u 2C o V~y2 + El dy d 7 - - 2 - h -u2'
(19)
with the boundary conditions: y=h;
u=uh
(20)
y=0;
u=0.
(21)
908
ALEXANDERORONand CHA|MGUTF|NGER
The local gas velocity Un isn't yet known and will be found later. Equation (19) does not contain explicitly the independent variable, y, hence, its order can be lowered by integrating it once, following the procedure outlined in Kamke (1959), resulting in:
v ( du ) 2 2 2 ( d u ) 3 2 Tyy v(du)2 E = ~ ~yy
where
2~_~°u3
dy s
=--
(22)
+E,
2 2 ( d u ) 3 y=o =0 + 3 l d-yy = const.
Equation (22) will be solved with the initial condition of equation (20). We first find an expression for 2Co/6h in equation (22). If n is the number of fibers on a surface of area A, then the total area occupied by the fibers on the surface is nrcd}/4. On the other hand, ifa is the solidity of the filter and S is the area free of fibers, then the total area of the fibrous surface is (S + rindS~4) and the following equality may be written down for the fiber occupied area: 4
- c~S+ T
]"
Hence, S=
1 - ~ rmd} •
(23)
4
It is evident that Af = hdf and therefore
2 - nAy _ S
4oe
h
(24)
re(1 - ~) dj."
Lee and Liu (1982) have shown that for a fiber filter, a dimensionless drag coefficient = ½CDReI is invariant in a fairly wide range of velocities and is a function of filter solidity, a. They plot the dimensionless drag coefficient, ~, experimentally obtained, as a function of the solidity. This function is monotonically increasing with the solidity of the fiber layer. From the definition of f~ it can be written that 2Or
Co dfu, -
.
(25)
Now, the coefficient ~ h ° in equation (22) can be computed from equations (24) and (25): 2C D 4a~ v 6~h- = 3n(1 - a)'u,d}"
(26)
)~Co is substituted into equation (22):
The term ~
+St
u,d}
+E.
(27)
Equation (27) is rewritten in dimensionless form by defining a dimensionless velocity q5 as a function of a dimensionless distance from the wall ~ = y/h.
U U,
--
resulting in:
= 4,(0,
(28)
(d )2 (d )3 +b
a7
-e,
(29)
Turbulent deposition of particles
909
where 3n(1 - ~ ) (
a=
ds ) 2
G~
~
,)5
b = rc(1-~t)h+'(h 2~ta
(30)
(~
)5
e = a[~b'(0)] 2 + bl-~b'(0)] a.
(31) (32)
This equation expresses a dependence of the velocity on geometric characteristics of the roughness layer. Equation (29) can be rewritten as: t~ -----#a~b '2 + bqb'3 - e.
(33)
This equation is of the form =
0(~'),
(34)
for which a parametric solution is known (Kamke, 1959): ~= 1+
f ' O'(t) dt tl
(35)
t
= o (t).
(36)
Comparing equations (34) and (36) we note that the parameter t has the physical meaning of
de
t = d-~-"
(37)
The function g'(t) in equation (35) is found by differentiating equation (33) with respect to the variable t = $', resulting in: = 1+
f t tl
2a + 3bt 3(at 2 + bt 3 _ e)2/3 dt
(38) (39)
dp = ~/at 2 + bt 3 - e.
Here tl is the value of the parameter t corresponding to ~ = 1. It is determined by computing the stress at y = h. From a force balance it is obtained that T = zw (1 - y / R ) , where R is the tube radius. Since for the roughness layer h <~ R, we may approximate 17w ~ , "t'h.
NOW, zw=zh=p(v+e)~y
du y=h
u, ,~12U2,2 =PV-~q+e h2,1,
therefore:
2 vu, 12u2"71, u, =--h-t1 +--~-which leads to
1
1)2 ,40,
,,
The only other term not determined as yet in the expression for the velocity inside the roughness layer, equations (38) and (39), is the constant e. We rewrite it from equation (32), setting ~'(0) = to: e = at~ + bt~,
910
ALEXANDER ORON and CHAIM GUTFINGER
IO.O00i
1.000
/ o
0.100
to(a= 0,175) to(a =0.271)
.
0.010
tota=0.5} I 2
0.001 0
I 4
I (5
t 8
L-_ I0
DIMENSIONLESS HEIGHT h+
Fig. 1. Values of the parameter t as a function of dimensionless roughness height for different roughness layer solidities.
which upon substitution into equations (38) and (39) yields the final expression for the velocity profile inside the roughness layer given by
3bt _ t3)] 2/xdt = 1 + f ' 3[a(t2 _ t22a) ++ b(t3 I1
q~ = , ~ ( t 2 - t2o) + b(t 3 - t3).
(41) (42)
The parameter to corresponds to ~ = 0 where ~b = 0. Hence, from equation (41) we obtain:
f t, (2a + 3bt)dt to 3[a( t2 - ~ o 2 ) ~ - t3)] 2/3 = 1.
(43)
Figure 1 presents plots of to and t~ as functions of h+. The velocity profile inside the roughness layer is now defined by equations (41) and (42) with the constants t~ and to given by equations (40) and (43), respectively. This velocity profile satisfies the non slip condition, u Ir =o = 0.
Velocity and eddy viscosity in the region outside the roughness layer According to the results of Monin and Yaglom (197 l) the mean velocity distribution in the region y > h is given by
u(y) _ 1 In y -- d, u,
x
(44)
Yo
where x is the von Karman constant approximately equal to 0.4, d is the displacement height and y0 is a constant of an order of magnitude h. This expression is valid for the fully rough flows (h+ > 70). Since we are dealing with transitionally rough flows (h+ < 30), we need a similar expression for the mean velocity in this region. We develop it using the approach of Monin and Yaglom. Let us first list the parameters which the flow velocity and its derivatives depend on. These are the fluid density, p, the stress on the wall Tw, the distance from it, y, the kinematic viscosity
Turbulent deposition of panicles
911
of the fluid, v, and the height of the roughness layer, h. From these parameters it is possible to form two dimensionless groups, h + and h/y. So, dy
_ u,f
Ky
h+
(45)
,
where the funetionfis a correction of the logarithmic velocity profile due to the existence of the roughness layer. By expanding f ( : , h + ) in a Taylor series, it is obtained that d__u_u= u, [F(h+)+ h G(h+)], dy r(Y - Yx) y - yl
(46)
where F and G are functions that will be defined later and y~ is a parameter corresponding to the displacement height of Monin and Yaglom. The mean velocity u(y) is obtained by integration of equation (46):
u(y) _ F(h+ ) in y - y: u, K yo
hG(h+ ) ~(y - yl)'
(47)
where y0 is a constant of integration that satisfies continuity of the mean velocity at y = h. Like the y~ of Monin and Yaglom, our y~ has a physical meaning of the effective displacement of the wall due to the existence of the roughness layer. Under the assumption that l = K(y- Yl) above the roughness layer, we express the eddy viscosity, e, in this region by du e = 12 ~y = x(y -- Yl )u,F(h+) + xhu.G(h+ ),
(48)
and, in dimensionless form:
- = ~cF(h+)(y+ - y~ +) + ~ch+G(h+).
(49)
v
Equations (47) and (49) provide expressions for the velocity and eddy viscosity profiles above the roughness layer. In order to uniquely determine these profiles the functions F(h+) and G(h+) have to be found. This is accomplished by matching the shear stress, found from equation (46) with that in the roughness layer at y = h, and the velocity profile at h + = 70 with that of Monin and Yaglom. Hence, at y = h,
l -u- ,~q= u ,
[ F ( h + ) + h _ yh 1 G(h+ )l '
(50)
where the left-hand side of equation (50) was found by differentiating equation (39) and setting ~b'(1) = tl at the edge of the roughness layer, y = h. Thus, the function G(h+) is expressed by:
G(h+)=hhYl
-F(h+)+
t~ -
~+
-h t: .
(51)
We now substitute the expression for G(h +) from equation (51) into the expression for the velocity, equation (47),
Now comparing the expression for the velocity profile, equation (52) with that of Monin and Yaglom, equation (44), we note that the matching requirement at h+ /> 70 results in
F(h+) = 1, h+ >1 70
(53)
912
ALEXANDER ORON and CHAIM GUTFINGER
and
l
F(h+)-~t, =0, h+ >~70.
(54)
Substitution of equation (53) into equation (54) results in h tl = ~ ,
h. >~70.
(55)
Equation (55) can also be derived from equation (40) by letting h+ --, co. This fact attests to the consistency of the present model. Grass (1971) has performed experimental measurements of the mean velocity in flows over smooth and rough surfaces. From Fig. 4 in his work, which plots the dimensionless velocity, u +, versus the natural logarithm of the dimensionless distance from the wall, lny +, it is seen that the slopes of the lines for rough surfaces are identical to the slope of the line corresponding to flow over a smooth surface. His experiments cover data from smooth surfaces and up to h+ = 84.7. Thus, Monin and Yaglom's statement that F(h+) = 1 is essentially correct over the whole range ofroughnesses from smooth and up to h+ = 84.7. We therefore extend the validity of equation (53) over the whole range of interest of h+ and write down the final expression of the velocity profile outside the roughness layer by amending equation (52) to read u
u+ = - - = u.
In y - y l
Yo
Yl
1-
tt
•
(56)
Y - Yl
Similarly the expression for the eddy viscosity, equation (49) can be rewritten as:
We now consider the question of where the outer edge of the transition zone is. The velocity distribution in it is the disturbed logarithmic profile. The second term in equation (56) decreases with increasing y. Hence it is logical to assume that the outer edge of the transition zone is that of the smooth wall 'logarithmic region'. Based on experimental data, Reynolds (1974) suggests that this region extends to y = 0.2R, R being the pipe radius. The magnitude of the constant l/h is found from an assumption of its continuity at y = h:
From the experimental results of Grass (1971) it can be obtained that the shifting of the logarithmic profile of the mean velocity due to an existence of the roughness layer is 0.6h approximately for transitionally rough flows, so Y~ = 0.6. h
(59)
Substitution of equation (59) into equation (58) leads to: 1 h
0.16.
(60)
This value was used in equations (31), (40), (56) and (57) for the respective calculations inside and outside the roughness layer. Plots of dimensionless velocity profiles inside and outside the roughness layer are presented as functions of the dimensionless coordinate, ~ = y/h, in Figs 2 and 3, respectively. The curves were calculated for a roughness layer solidity of 0.0963, a value for which experimental filtration data were reported by Lee and Liu (1982). The parameter in Figs 2 and 3 is the dimensionless height of the roughness layer, h+. Figure 3 was drawn for the whole domain ~ ~>0, and therefore Fig. 2 may be viewed as an enlargement of the lower left corner of Fig. 3.
Turbulent deposition of particles
913
2.5
"O-
2.0
>I-
U q
1.5 =,
/
h+=5 ~ 1 /
°~ I.O
/
0.5
L 0
I
I I I 0.5 I.O DIMENSIONLESS COORDINATE ~=y/h
I
1.5
Fig. 2. Dimensionless velocity inside the roughness layer (¢ = 0.0963).
10.0 -o>-
o kO > ¢O ~O Z
_o
o.o_
//,//
\ 0 i:0o
lad B
2.5
0
,5 I0 DIMENSIONLESS COORDINATE~=y/h
15
Fig. 3. Dimensionless velocity outside the roughness layer (~ = 0.0963).
As seen in Fig. 2, the velocity inside the roughness layer approaches zero smoothly as ~ --* 0. At the edge of the roughness layer, ~ = 1, the velocity profile inside the roughness region calculated from equations (41) and (42) connects smoothly and continuously to that in the outside region, equation (56). Figure 4 shows the velocity profiles inside the roughness layer for several values of solidity, ~. As seen from this figure, the velocity drops sharply in the region close to the tips of the roughness elements for higher values of solidity, ~. In contrast with this behavior, at very low solidities (~ = 0.005) the velocity distribution becomes approximately linear. This is
ALEXANDER ORON and CHAIM GUTFINGER
914
3.5~
3.0 a=O.O05
\
-,e>.. 2.5 io o IJJ > 2.0 co co w
J z
o co z w Q
1.5
1.0
a =0.0963 ~=0.175~ e:0.271
0.5
0
0.5 1.0 DIMENSIONLESS COORDINATE~=y/h
Fig. 4. Dimensionless velocity inside the roughness layer for several values of fiber layer solidity (h+ = 10).
I.O
~
-e>i,-
PRESENT THEORY THEORY OF GUTFINGER & FRIEDLANOER
//
/ ~
/ /
/
w >
/ ~, o.~
z
z w
0
0.5 DIMENSIONLESS
COORDINATE
1.0 ~'=ylh
Fig. 5. Velocity profile inside the roughness layer. Comparison with the solution of Gutfinger and Friedlander (1985) (h+ = 10, ct = 0.0963).
qualitatively similar to the velocity distribution developed by Levich (1962), equation (12), for turbulent flow near a rough wall. In Fig. 5 we compare the velocity profile inside the roughness layer calculated from equations (41) and (42) with that derived by Gutfinger and Friedlander (1985). The comparison is performed for the relatively dilute roughness layer of solidity (ct = 0.0963). As
Turbulent deposition of particles
915
seen from the figure, the solution given in the present paper results in a more moderate velocity decay inside the roughness layer.
PARTICLE C O N C E N T R A T I O N AND D E P O S I T I O N VELOCITY After solving the hydrodynamic problem we now derive an expression for the particle deposition rate. We divide the whole domain of the flow into three regions: 1. The roughness layer (0 ~< y ~< h). 2. The transition zone which extends from the tips of the fibers to the edge of the logarithmic region, which, as mentioned above, lies at a distance of approximately y = 0.2R. 3. The turbulent core of the flow (3: > 0.2R). If kc, k,, and k, denote the deposition rates in the turbulent core, transition region and the roughness layer, respectively, the overall deposition rate, ktot, is obtained from
1 1
kt°t = I
1"
(61)
Similarly, the dimensionless overall deposition rate is expressed as: kt°t+ =
1 1
1
1
(62)
k;+' +k~-+ + k,--f First, we derive the deposition rate in the roughness layer, k,. In this region the following diffusion-convection equation is satisfied:
Oc
Oc
Oc
~c )
u~x + V-~y = ~---~((D + e,)-~x ) + ~---~((D + ~p) -~y
/~(y)c,
(63)
where x is the longitudinal variable, y is the lateral variable, i.e. the distance from the wall, u, v are components of the time averaged velocity, D is the molecular diffusion coefficient and ,p is the turbulent diffusivity of the particles. The term fl(y) is the absorption coefficient, under the assumption that the rate of particle removal by the roughness elements is proportional to the local particle concentration. This is, indeed, the physical meaning of the last term in equation (63). Lee and Liu (1982)presented the absorption coefficient in the form /
/~(y) =
•
~u(y),
(64)
where
4=r/ fl - ~dr( l - or)'
(65)
and ~/isthe overall filtrationefficiencyof a single fiber of diameter d/in the roughness layer. W e now assume the pipe flow and the concentration profile to be fully developexl and neglect the diffusiveflux in the x-direction compared with that in the y-direction.W e also neglect the convective terms, as is usually done in turbulent mass transfer. This leads to:
dy
(D + ep) ~y
= flu(y)c,
(66)
with the boundary conditions
c=0
aty=0
(67)
C=Ch
aty=h.
(68)
916
ALEXANDERORON and CHAIM GUTFINGER
Since the velocity u(y) inside the roughness layer is written in the parametric form, equations (38) and (39), we must replace the y-dependent equation (66) by the t-dependent equation. This leads to the following equation: d [{1 ~p'~ t dc7 9'(t)g(t) g-- + - - 1 ~-~-,, --::l = flhh+ - - ~ , t ~c v J g (tl Ot l L\
(69)
with the boundary conditions ( = 1, t = t ~
(70)
(=0,
(71)
t=t0,
( being the particle concentration normalized by ch. The difference between the turbulent diffusivity of the flow and that of the particles was discussed in the Introduction. Many authors assumed that they are equal, i.e. ep = e. Our opinion is that it is important to incorporate into the model a difference between them that would account for the effect of particle inertia. This effect is very pronounced in the very close proximity of the wall for relatively large particles, i.e. those of diameter of above 1/~m. In the following derivation we use the model of Kitamoto and Takashima, (1974a, b) because of its simplicity and the reasonable results which can be obtained. According to this model,
,,
v
\dy+/ /
\ay+/
where z + is the dimensionless particle relaxation time and a = V'p/V' is the relation between the r.m.s, particle velocity and the r.m.s, flow velocity. Expressing du +/dy + in equation (72) in terms of t with the help of equation (37), we obtain:
--
h+t
l + 2 a r + ~+ +a2~z+ t~_ .
(73)
We follow Gani~ and Mastanaiah (1981) and assume a semi-empirical expression for a in the form 1 a = , (74) 1 + 529Re -°5°9 z+ R+ where R is the pipe radius. This expression is based on experimental data of Martin and Johanson (1965). It gives a good estimation for (r in the range 2 x 104< Re < 1.5 x 106. In order to obtain the concentration profile d we have solved equation (69) with the boundary conditions, equations (70) and (71). The solution was obtained numerically using a Runge-Kutta procedure. Figure 6 shows the normalized concentration profiles inside the roughness layer for particles of diameters 0.65, 1.1 and 2.1 #m. These values correspond to the experimental data of Wells and Chamberlain (1967). These data lie in the inertia-interception dominating domain of particle diameters; a larger diameter leads to an increase in the single fiber overall filtration efficiency. This results in lower concentrations for larger particles. The deposition flux in the roughness layer is computed from: de f h Jh = (O + el,) ~y y = o + o fl(y)c(y)dy.
(75)
Here, the first term expresses a particle flux at the wall, while the second one accounts for particle deposition on the roughness elements. In terms of the variable t the expression for the flux is rewritten as:
[to
r/l
ep'~ d¢?]'
4~/
(h)~,,,g(t)g'(t)d(t)dt]"
(76)
Turbulent deposition of particles
917
I.O
0.75 Z 0 n.. I.-Z w L) Z 0 ¢.~
p.m
0.5
u3 LU J Z 0
I/a.m
Z LU
F, 0.2!
N' 0
I 0.25
I 0.5
I 0.75
I 1.0
DIMENSIONLESS COORDINATE ~'=y/h
Fig. 6. Dimensionless concentration profile inside the roughness layer (~ -- 0.0963, h+ = I0).
The particle deposition rate in the roughness layer, k. can be calculated from
k, =--=u,Jh ch
F
~
Lh+o (to)
[ ( 1 +-v-~P)]dE ~ + 4~q (h)ft'9(t)9'(t)c(t)dt],t ~ ,=,o rt(1 - =) ~ to (77)
or, in dimensionless form,
I( 1 ep ) dE1 "~ n(14o~q k,+ - Jh to - 0t) (ff__f) ftl 9(t)g'(t)E(t)dt. t u.ch h+g'(to) S-c + v -d-i t=,o to
(78)
For the transitional zone the resistance to particle transfer is computed by integrating the flux equation: d = (D+~p) dc d-~, (79) leading to
k,+ =
k~ 1 -- = , u, [" 0.2R÷ dy+
(80)
|.Jh~+S+__1+ ~_e_~ Sc v where S is the stopping distance of the particle. We assume here that a particle which approaches within a distance S from the tips of the fibers reaches the roughness layer. The turbulent particle diffusivity is given by equation (72) which, together with equation (57), results in:
918
ALEXANDER ORON and CHAIM GUTFINGER
where du÷/dy+is found from equation (56). Equation (81) is valid when the fibers go beyond the viscous sublayer (h+ > 5). When the flow is hydraulically smooth, i.e. h+ < 5, we have to divide the region outside the roughness layer into two regions: the viscous region (h+ < y+ < 5) and that of 5 < y~ < 0.2R+. Thus, the resistance of the transition zone in this case is obtained from: 1 _ 1 f h5 dy+ kt u, . +s+ 1 % Sc v
+ 1 fs0,2R, -----.dY+ u,
1 Sc
(82)
% v
The dimensionless deposition velocity is obtained from: 1
k"=fh5
dy+
I0.2/~ dy+
++St Sc 1 ~+
-{- j5
--
1
Sc +
(83)
% v
The second integral in equation (83) is calculated with the help of equation (81), while in the first integral we estimate the expression %/vby means of equation (72) with the expression for s/v taken from the distribution of Lin et al. (1953): ~
(y+)3.
v
(84)
lq5.5
The expression du +/dy÷ is computed by differentiating equation (56). In order to compute the particle deposition velocity in the turbulent core we follow Friedlander and Johnstone (1957) and use the Reynolds analogy by assuming that the turbulent Schmidt number of a particle is equal to unity:
Sc,-
Ep
- 1.
(85)
Accordingly, J c,v - cc = .z2 (Uo - uc),
U,
(86)
c~ and u~ being the concentration and the velocity at the outer edge of the transition zone, respectively. The velocity uc can be obtained from equation (56) at y = 0.2R. The particle deposition velocity in the turbulent core is given by: kc -
j Car - - Ce
-
u2,
(87)
U o - - Uc'
or, in dimensionless form: u, kc+ = Uo-U~"
(88)
The overall particle deposition rate is computed from equation (62) with equations (78), (80) [or (83)] and (88). C O M P A R I S O N WITH EXPERIMENTS In the literature there are few works which can supply experimental rough surface deposition rates in the range of 0 < h + < 30. There are data on grass-covered surfaces, however these correspond to h + >> 30. One of the experimental works that covers the range of the present paper is that of Wells and Chamberlain (1967). It provides data on deposition rates of tri-cresyl-phosphate particles to vertical rough surfaces. The experiments were performed in an annular space between two concentric tubes with filter-paper lined walls.
Turbulent deposition of particles
919
The height of filter's fibers was 100 microns. The measurements were performed with particles of diameters 0.65, 1.1 and 2.1 #m and flow Reynolds numbers 4000-45,000. The range is equivalent to the range 1.1 < h+ < 10. It should be noted that Wells and Chamberlain did not report the diameter of the fibers dI and the solidity of the fiber layer 0t. This missing information was taken from the work of Lee (1977) corresponding to his Dacron filter B for which 0t = 0.0963 and d I = 12.9 #m. That work also provided the single fiber overall filtration efficiencies used in computing the deposition rates. Figure 7 compares the results of the present theory with the experimental data of Wells and Chamberlain. It should be remembered that the missing values (ds and ct) were taken arbitrarily, hence the comparison has a symbolic character. This comparison shows that the present theory gives reasonable results for particle deposition onto fibrous surfaces. Specifically, a good fit is achieved with the experimental data for particles of diameters 1.1 and 2.1 #m. For the 'small' particles of 0.65 #m the fit is satisfactory for Re > 35,000 (h+ > 7.5). It should also be noted that for small dimensionless roughness, h+ (corresponding to low Reynolds numbers), the deposition rate is approximately equal for all tested particles. This can be explained by the fact that the overall deposition rate in this case is accounted for by the very high resistance of the viscous sublayer, which is almost identical for these particles. Compared to this high resistance, the resistances of the roughness layer and of the transition zone are negligible. As the dimensionless roughness height, h +, increases, the resistance of the viscous sublayer decreases and finally vanishes, when the roughness elements protrude beyond it. In this case, the resistance of the roughness layer becomes the most important and affects the overall deposition velocity. Now, for larger particles, the resistance of the roughness layer is lower due to the higher single fiber filtration efficiency. For smaller particles there is a higher chance of escape through the fibers, hence the resistance of the fiber layer is higher and the deposition rate is lower than for larger particles. The resistance of the roughness layer for the transport of small particles decreases very slowly with increasing of h ÷, so it is dominant in the computation of the overall deposition rate. For large particles, the resistance of the roughness layer decreases sharply with increasing h +. In this case, the transition region resistance dictates the overall deposition velocity. h+ 5 I
I0 I o
I0.00
~
°
.1/
.~"
+
.....~
>. 1.00, I-.
q
,y
hi >
,.EORY
/~
Z
--
,f
o OJO, I,-
0
/
w
0.01
0.0(
x
0
---dp=2.1 p.m
- - ~
t
dp=l. I p m
EX_PER~_ ATA o
dp=21Fm
p
+
dp= I.I p.m
x
x
dp=O.65 p.rn
I
I0000
I
20000
I
30000
I
40000
I
50000
REYNOLDS NUMBER Re
Fig. 7. Comparison of the theoretical results with the experimental data of Wells and Chamberlain (1967) (a = 0.0963). AS 1 7 : 6 - C
920
ALEXANDER ORON and CHAIMGUTFINGER SUMMARY
This paper has presented a theoretical model of particle deposition to rough surfaces from turbulent streams. The flow domain was divided into two regions: the roughness layer and the region above it. First the hydrodynamic problem was solved for the region inside the roughness layer. By assuming that the mixing length of the flow is independent of the transverse coordinate, a one-dimensional Navier-Stokes equation was written down and solved in terms of a parametric solution. The velocity profile obtained is a function of fiber layer solidity, its geometry and the flow Reynolds number. For small solidities the velocity distribution is approximately linear (Fig. 4) and qualitatively similar to the velocity profile developed by Levich [equation (12)]. Using an approach similar to that of Monin and Yaglom, profiles of the velocity and of the eddy viscosity were derived for the region outside the roughness layer. The velocity profiles inside and outside the roughness layer are connected smoothly and continuously, as seen in Fig. 3. The velocity profiles found in the first part of the paper were used in the solution of the mass transfer problem. Inside the roughness layer a diffusion-convection-absorption equation was solved, particle concentrations obtained (Fig. 6) and a resistance to particle deposition computed. The region outside the roughness layer was treated by means of a standard procedure of integration of the mass flux equation. The particle turbulent diffusivity was obtained using a corrected Kitamoto-Takashima expression. The overall particle deposition rate was obtained by addition of resistances of the roughness layer and the region outside it. The theory was used to correlate the experimental deposition data of Wells and Chamberlain (1967). The agreement between theory and experiments was found to be reasonable. REFERENCES
Browne, L. W. B. (1974) Atmos. Envir. 8, 801. Davidson, C. I. and Friedlander, S. K. (1978) J. Geo. R-O A. 83, 2343. Davies, C. N. (1966) In Aerosol Science, (Edited by Davies, C. N.). Academic Press, London. EI-Shobokshy, M. S. (1983) Atmos. Envir. 17, 639. EI-Shobokshy, M. S. and Ismail, I. A. (1980) Atmos. Envir. 14, 297. Fernandez de la Mora, J. and Friedlander, S. K. (1982) Int. J. Heat Mass Transfer, 25, 1725. Friedlander, S. K. and Johnstone, H. F. (1957) Ind. Engn a Chem. 49, 1151. Ganie, E. M. and Mastanaiah, K. (1981) Int. J. Multiphase Flow, 7, 401. Grass, A. J. (1971) J. Fluid Mech. 50, 233. Gutfinger, C. and Friedlander, S. K. (1985) Aerosol Sci. Technol. 4, 1. Hahn, L. A., Stukel, J. J., Leong, K. H. and Hopke, P. K. (1985) J. Aerosol Sci. 16, 81. Hinze, J. O. (1975) Turbulence: An Introduction to its Mechanism and Theory. McGraw-Hill, New York. Inoue, E. (1963) J. met. Soc. Japan 41, 317. Jackson, P. S. (1981) J. Fluid Mech. 111, 15. Kamke, E. (1959) Differential#leichungen. Leipzig. Kitamoto, A. and Takashima, Y. (1974a) Bull. Tokyo lnst. Technol. 121, 41. Kitamoto, A. and Takashima, Y. (1974b) Bull. Tokyo Inst. Technol. 121, 67. Lee, K. W. (1977) Filtration of submicron aerosols by fibrous filters. Ph.D. thesis, University of Minnesota. Lee, K. W. and Liu, B. Y. H. (1982) Aerosol Sci. Technol. 1, 35. Levich, V. G. (1962) Physiochemical Hydrodynamics. Prentice-Hall, Englewood Cliffs, NJ. Liu, B. Y. H. and Agarwal, J. K. (1974) J. Aerosol Sci. 4, 227. Liu, B. Y. H. and Ilori, T. A. (1974) Envir. Sci. Technol. 8, 351. Lin, C. S., Moulton, R. W. and Putman, G. U (1953) Ind. Engng Chem. 45, 653. Martin, G. Q. and Johanson, L. N. (1965) AIChE J. 11, 29. McCoy, D. D. and Hanratty, T. J. (1977) Int. J. Multiphase Flow 3, 319. Monin, A. S. and Yaglom, A. M. (1971) Statistical Fluid Mechanics: Mechanism ofTurbulenee, Vol. 1. MIT Press, Cambridge, MA. Owen, P. R. (1969) J. Fluid Mech. 39, 407. Reynolds, A. J. (1974) Turbulent Flows in Engineering. John Wiley, London. Sehmel, G. A. (1980) Atmos. Envir. 14, 983. Shaw, D. A. and Hanratty, T. J. (1977) AIChE J. 23, 28. Thom, A. S. (1975) In Vegetation and the Atmosphere, (Edited by Monteith, J. L.) Vol. 1. Academic Press, London. Wells, A. C. and Chamberlain, A. C. (1967) Br. J. appl. Phys. 18, 1793. Wilson, I. B. (1947) J. Colloid Sci. 2, 271. Yaglom, A. M. and Kader, B. A. (1974) d. Fluid Mech. 62, 601.