A diffusion model for a settling non-consolidated dust mass

CATENA

Vol. 12, 373-402

Braunschwcig 1985

A DIFFUSION MODEL FOR A SE'ITLING NON-CONSOLIDATED DUST MASS D. Goossens, Leuven ABSTRACT A physical model, describing and predicting the lateral diffusion of a non-consolidated dust mass that settles down freely in air, is presented. It appears that in the total diffusion process a distinction has to be made between two (in origin totally different) types of diffusion. In the case of the first diffusion type, the settling dust mass acts as a closed system and all the particles fall with the same velocity. In the second diffusion type, the dust mass no longer acts as a closed system and each particle settles with its own fall velocity. Both of these diffusion types are investigated. The result is a set of four equations: the first and the second describe the starting point of the lateral diffusion, the third describes the lateral diffusion of the first type and the fourth describes the lateral diffusion of the second type. The equations are tested in six diffusion experiments and their validity seems to be very high. The physical and mathematical construction of the model allows extension of the formulae so that the frame of the model can be used in several other sedimentological, fluid dynamical or geomorphological research domains. The model shows clearly that the classical sedimentation techniques, which are very frequently used in grain size distribution measurements, can lead to completely wrong results if no extreme care is observed in connection with the concentration of the settling sediment. 1.

INTRODUCTION

One of the most frequently used techniques in determining the grain size distribution of a fine sediment consists in letting the material settle down in a sedimentation cylinder and measuring the fall velocity of the particles. By converting the measured fall velocities into corresponding equivalent diameters it is then possible to get an idea of the grain size distribution of the sediment. Although the free, vertical settling of particles in a stagnant fluid is often considered as a relatively simple process, there are many parameters that have an important impact. A detailed investigation of the grain size distribution of a sediment by means of the sedimentation technique has therefore to take into account the effects of these parameters. A very important parameter is the general shape of the particles. Natural particles are, in general, not perfectly spherical: they can be more or less spherical but also strongly flattened; they can be very well rounded but also be extremely angular. This is very important because the most frequently used sedimentation formulae (for example that of Stokes) are only valid for perfectly spherical particles. The influence of the flattening of the particles on the fall velocity has been investigated by McNOWN & MALAIKA (1950), by SCHULTZ et al. (1954), and, more recently, by DIETRICH (1982). The influence of the flattening on the internal motions (oscillation, rotation and tumbling) of the particles during the fall was examined by KUMAR (1967), STRINGHAM et al. (1969) and MIDDLETON & SOUTHARD (1978). WILLIAMS (1966) and DIETRICH (1982) investigated the effect of the roundness of the ISSN0341

8162

~) Copyright1985 by CATENA VERLAG, D-3302 Cremlingen-Destedt, W Germany

374

GOOSSENS

particles on the fall velocity. The role of the flocculation of particles during the sedimentation process was analysed by KRONE (1963). McNOWN & LIN (1952) and McLAUGHLIN (1959) examined the influence of the sediment concentration in the fluid on the fall velocity of the individual particles. McNOWN & MALAIKA (1950), finally, derived a formula to correct the decrease of the fall velocity of the particles due to wall effects of the sedimentation cylinder. Besides the above described factors there is still another very important parameter that influences the sedimentation process: the diffusion of the settling mass during the fall. Up to now, no detailed investigation of the spontaneous diffusion of a mass of natural particles (the ones used in earth science applications) during its free vertical fall exists. This seems to be a bil astonishing, because the diffusion of a settling mass influences directly the concentration of the sediment in the fluid and consequently also the lZallvelocity of the particles. In the literature concerning atmospheric pollution and the dispersal of polluents in the air, an overwhelming quantity of diffusion models can be found. An excellent review of these diffusion models is given by PASQUILL (1974). It has to be emphasized, however, that all these diffusion models are dealing with ideal particles, i.e. particles that are so tiny that their inertia can be completely neglected compared with the movements of the fluid in which they are immersed. The diffusion process in a cloud consisting of such particles is nearly totally determined by the turbulence characteristics (and the turbulence itself) of the fluid. It is clear that in the case of natural particles such as sand or silt, the inertia of the particles can no longer be neglected. The classical atmospheric diffusion models that are derived for atmospheric pollution problems are therefore not applicable for most of the natural sediments. In this paper, a physical model is presented that describes mathematically the lateral diffusion of a mass of natural dust (loess) when it settles down freely in air. Because the fall velocity of the dust mass plays an important role in the diffusion process it is necessary to investigate briefly the evolution of the fall velocity in the course of the fall first. This will be done in section 2. The diffusion model properly consists of three parts and is treated in section 3. In section 4, finally, the theoretically predicted diffusion of six dust masses is compared with the experimentally obtained diffusion values.

2.

EXPERIMENTAL S E T - U P AND DETERMINATION OF THE FALL VELOCITY

For each of the six experiments, typical calcareous loess from the well-known Kesselt quarry in Belgium was used. The median sedimentation diameter of the loess was 23/~m. In experiment A the total mass M of the dust cloud (the term "mass" will in future only be used in its strictly physical meaning in order to avoid confusion) was 12 g. In experiment B the mass of the dust cloud was 9 g, in experiment C it was 6 g, in experiment D 4 g, in experiment E 2 g and in experiment F 1 g. In each experiment a plastic cylinder with an internal diameter of 2,20 cm was pinched vertically in a hole in a horizontal wooden plank that was attached to a laboratory tower (see fig. 1). Two rails along which a thin wooden board could be moved were constructed on the underside of the plank. The plastic cylinder was pushed far enough through the hole to allow its aperture to be closed exactly by the upper surface of the wooden board. The sediment was then put into the cylinder (without any compaction) and dispersed over the whole section as equally as possible. A thin wire was attached to the wooden board; by pulling strongly and abruptly at the wire in a horizontal direction the board could be displaced nearly instantaneously, so that all the dust particles in the cylinder started to settle down at the same moment. The maximal height at which the plank could be attached to the

DIFFUSION MODEL FOR SETTLING NON CONSOLIDATED DUST

a

375

plank ~

r!U t board i

/.t--- wire

i

i

I

t

i

I

if' i

I

4

,

b

,

,

ii ,

Fig. 1: Scheme of experimental set-up

tower was 4,44 m. Each experiment consisted therefore in a free vertical fall (of Kesselt loess) of 444 cm. The complete fall process was filmed with a highspeed-camera and the time interval between two successive pictures was adjusted to 0,01 s. The diffusion of the settling dust cloud could be measured very simply on the film pictures after a correction for the parallax. The fall velocity was found by measuring the total vertical displacement of the dust cloud between two successive pictures; the point for which this fall velocity is valid was taken as the mean of the displacement interval. Also for the determination of the fall velocity, a parallax correction was used.

376

GOOSSENS

0

,. , H0 cm :. "' ..

-:,

:..

H cm

.... :'..-

A

10

0 '~. :

100

C

B

i\.

200.

200-

.;~..

2OO-

-\

• I\

300"

300-

300-

' ~! u 400-

400.

400-

2~o400

u 2~0 4~0 8~o 8~o CII~/S

0

'

0 n

600

l

" .

Clrn

".

cln

U

8 0 0 cm/s

0

U cnl

•

~

I

2 0

400

1001

. ~.~

300" .f

~-

c.p.

0

I

|

200 400 600

I

e.p.

! I I "U 300 0I I 200 400 600 c r n / s

~) U 2 0 400 600 cm/s I

I

4000

F

--~i

200~ ~ ~--e.p2001~ 300.

800

" "

I 100

600

'U

800 cm/s

Fig. 2: The fall velocity u in function of the fall distance H. A : M = 12 g; B : M = 9 g; C : M = 6g; D : M = 4 g; E : M = 2 g; F : M = 1 g. e.p, = explosion point

cn~

DIFFUSIONMODEL FOR SETTLINGNON CONSOLIDATEDDUST

377

Fig. 2A-F show, for the six different masses, the evolution of the fall velocity u in function of the fall distance H. Note that the origin of the ordinate axis (fall distance) is taken at the top and not at the bottom. From the figures it can be seen immediately that between the starting point of the fall (Pall distance zero) and a certain point (called the explosion point) the fall velocity varies parabolically with the fall distance. At the explosion point a very sharp reduction in the fall velocity occurs in a very short interval. After this interval the fall velocity decreases slowly to a constant value, which corresponds with the normal terminal fall velocity of the individual particles. For the dust clouds of 6 g, 9 g and 12 g the explosion point is situated at a Pall distance larger than 444 cm. Because the maximal fall distance that could be reached in the experiments was restricted to 444 cm it was not possible to determine its correct value for these three dust clouds. It is important to point out that the parabolical increase of the fall velocity with fall dis* tance between the starting point of the fall and the explosion point is caused by the fact that in this interval the dust cloud acts as a closed system: the fall velocity is not determined by the mass of the individual particles but by the total mass of the dust cloud itself. If the dust cloud (considered as a closed system) would settle down in vacuum, the analytic function describing the relation between the fall velocity u and the fall distance H would be

1

u2

tl = 2g

(1)

where g is the fall acceleration (g = 9,81 m/s2). In a fluid with a low density and viscosity (such as air) the relation between the fall velocity and the fall distance will not exactly be the same as described in eq. (1), but the difference will not be great. Moreover, the higher the mass of the dust cloud, the more the relation in vacuum is approximated. Regression analysis shows that, for the six experiments shown in fig. 2, the relationship between the fall velocity u and the fall distance H is best given by the expression H = a •u2

(2)

if H is considered as the dependent and u as the independent variable, and u = b

,/n

ifu is considered as the dependent and H as the independent variable. In eq. (2) and eq. (3), a and b are the proportionality factors. If one wants to use the relationship between u and H as an analytical equation (valid in both directions) and not as an empirical equation (only valid in the direction from independent to dependent variable), one has to take the geometrical mean of the numerical values of the two proportionality factors instead of the numerical value of a or b alone (MEULEPAS 1985). The expressions for u in function of H and H in function o f u that will be used in this paper are therefore: u=

and

I

b

.x/H

378

GOOSSENS

H=

%/-a-a u 2 b

The values of the proportionality factors a and b for the six experiments (obtained by regression analysis) are given in table 1. After the explosion point the fall velocity variation is not completely linear with the fall distance, but the difference is very small so that the use of the linear relationship u = ~ +fltt

(4)

can be seen as a very good approximation. If one wants to use this relationship as an analytical (and not only empirical) equation as mentioned before, the values of the coefficients a and/3 have to be taken so that eq. (4) corresponds with the bisector of the two corresponding straight lines found by regression analysis (MEULEPAS 1985). Table 2 gives these values for a and/3 for the experiments D, E and F (for the experiments A, B and C they could not be determined for reasons mentioned earlier). Tab. 1: NUMERICAL VALUES OF THE PROPORTIONALITY FACTORS a AND b FOR THE SIX EXPERIMENTS Exp. No. A B C D E F

M (g) 12 9 6 4 2 1

a (s2/cm) X 10-4 5,449 5,391 5,751 5,693 5,966 6,697

b (cml/Z/s) 42,343 42,612 41,613 41,408 40,834 38,964

Tab. 2: NUMERICAL VALUES OF THE COEFFICIENTS a AND/3 FOR THE EXPERIMENTS D, E and F Exp. No. D E F

3.

M(g) 4 2 1

a (cm/s) 222,3ll 350,492 235,320

fl(J/s) × 10-~ --4,623 -9,618 --6,983

THE DIFFUSION MODEL

3.1. DIFFUSION M O D E L PART I: D E T E R M I N A T I O N OF THE START OF THE DIFFUSION The physical basic principle of the model is that the diffusion of the settling dust cloud is determined by an overpressure inside the cloud. This overpressure pushes the individual particles apart and therefore causes lateral diffusion. Nearly the whole model is based on one central basic assumption: before the explosion point is reached, the settling dust cloud is considered as an isolated, closed system. This means that changes in the physical environment of

DIFFUSION MODELFOR SETTLINGNON CONSOLIDATEDDUST

379

the cloud (for example changes in pressure or temperature) are not transmitted towards the internal part of the cloud. In reality a perfectly, hermetically isolated system is not possible because of the existence of pores between the dust particles, but the smaller the total pore volume and the smaller the pores themselves, the larger the inertia of the transmission of physical changes will be. When the dust cloud is settling freely along its streamline the well-known equation of Daniel Bernoulli holds: at any time t, the sum of the static pressure and the dynamic pressure equals the total pressure provided that the friction losses are negligible (which for simplicity will be assumed here). So, in that case, PSTAT +pDYN

(5)

= PTOT

The static pressure is the pressure measured parallel with the stream direction, the total pressure the pressure measured perpendicular to it. The dynamic pressure is equal to ' ' f where PF is the bulk density of the fluid (in our case air) and u is the velocity of the dust cloud. Eq. (5) becomes therefore:

PSTAT +

PF " u2 2

PTOT

(6)

The total pressure P T O T is the sum of two pressures: on the one hand the normal atmospheric pressm~ PA of the air and on the other hand the extra-pressure PE caused by the resistance force F W the moving_,body encounters when passing through the fluid. The magnitude of this resistance force F W is equal to

FW

=

CD.P F • u 2 • A 2

(7)

where C D = drag coefficient of the object A -- maximal section of the object, perpendicular to the stream direction if the movement of the body through the fluid is turbulent, which is the case for the dust clouds of the experiments. From eq. (7), the extra-pressure PE can be calculated:

PE

FW A

--

CD P F 2

u2 (8)

Strictly speaking, this expression only holds for the stagnation point, i.e. that point of the body where the velocity sinks to zero. For a spherical body in laminar flow, the position of the stagnation point is constant and situated at the most upstream point of the sphere. In the case of turbulent flow the position of the stagnation point is no longer stable; due to the turbulent eddies it moves constantly over the upstream hemisphere. For the sake of simplicity it is assumed here, that eq. (8) may be used as a mean expression describing the extra-pressure PE for the whole upstream hemisphere of the body. The total pressure PTOT is then equal to

P T O T = PA +

C D ,o F • u 2 2

(9)

380

GOOSSENS

Substitution of (9) in (6) gives: PF PSTAT +

u2 2

PA +

CDPF 2

u2

PF P S T A T = PA + C D PF P S T A T = PA +

u2

PF

u2

2 u2 2

(C D-l)

If the pressure inside the dust cloud is not influenced by changes in pressure outside (isolated system), the pressure in the pores inside the cloud is equal to the normal atmospheric pressure PA. The difference in pressure in,idc :rod oul~idc lhe cloud (written as £Xp) is then: Ap = P A -

PSTAT

AP = P A - - [ P A + PF Ap = , ,

PF " u2 2

(CD-

1)1

u2 2

.(1 -- C D)

(10)

The model will now be treated in a new reference system, where the pressure outside the cloud is equalized to zero. The pressure p inside the cloud is then, in this new reference system, of course equal to PF P=

u2 2

.(1-

C D)

(11)

This expression only holds on the condition that no diffusion has yet occurred, i.e. if the volume of the cloud is equal to the basic volume (the volume at the start of the fall). If diffusion occurs, a correction term has to be added. This correction term can be found as follows. If the dust cloud is seen as an isolated system (basic assumption of the model) and the temperature of the air inside the cloud is constant (as can be assumed), the Boyle-Mariotte expression holds: P-V

1

where p = pressure of the air in the pores of the dust cloud at time t V = total volume of the air inside the cloud at time t

DIFFUSION MODEL FOR SETTLING NON CONSOLIDATED DUST

381

or

p.V=e where e = a constant lfp0 and V0 are the pressure and the volume at the time to when the diffusion starts, one can also write P0V0

= a

Division of both equations gives PO V o

1

pV OF

P --

PO V o V

(12)

~l'he pore volume Vis inversely proportional to the concentration X of the dust particles: \/'

~

_ _

1

X or

). X where 2 = a constant Eq. (12) becomes then:

XO 2 X

P = P0

X P0"

X0

(13)

The concentration Z o f the particles inside the cloud is equal to the sum of the masses m i of the individual particles divided b \ Hae total volume V of the dust cloud: n

X--

}~ m i i=l V

(with n = total number of particles)

382

GOOSSENS

or, more briefly, M

X = -V-

{14)

Substitution of(14)in (13)gives:

P=Po

M T M0

MVo PO' M o . V

(15)

Vo If it is assumed now that the total amount of dust that is taken away by longitudinal diffusion (not lateral diffusion) during the settling process of the cloud is negligibly small (this is true for sufficiently high masses M0 but not for low masses), then M = M0 and eq. (15) can be simplified to

v0 P = PO

V

(16)

This expression holds only for "sufficiently high masses". For "too small masses" it gives an overestimation for p. Where the border lies between "sufficiently high masses" and "too small masses" will be investigated in section 4. From the film pictures it was seen that although the dust clouds started the settling process as cylinders, they rapidly obtained a more or less spherical shape. The volume V can therefore, with a good approximation, be written as V = ~ 4 . # . R3 where R = the radius of the dust cloud at time t If R0 = the radius at time to (the time at which the lateral diffusion starts), eq. (16) becomes: 4

~-.~z. (R0)3 P = P0

_4 . 7 r . R 3 3

(__~0_)3 = P0

(17)

If R' stands for the growth of the radius in the time interval [t 0 ' t], then R = Ro + R' and eq. (17) becomes

P=P0

R0 )3 R 0 + R,

(18)

DIFFUSION MODELFOR SETTLINGNON-CONSOLIDATEDDUST

383

Hence, the right hand member of eq. (11) has to be multiplied by the correction term [ R 0 / ( R 0 + R,)]3 and the final equation becomes:

P =

PF u 2 2

( .(1-C

D)"

R0 R 0+R'

)3 (19)

p in eq. (19) is the difference in pressure inside and outside the dust cloud. Note that this expression only holds for "sufficiently high masses"; for "too small masses" it gives an overestimation for p. Because all the physical parameters in the right hand member of(19) are positive, the sign of p only depends on the term (1 - C D)- This has the following important consequences: 1. If the drag coefficient C D of the settling dust cloud is larger than 1, then p is negative. This means that the pressure inside the cloud is smaller than the pressure outside the cloud so that the cloud will be compressed. Lateral expansion is therefore formally excluded. If the compaction was already optimal at the start of the fall, no change in volume of the cloud will occur. 2. I f C D = 1 then p = 0: no pressure gradient exists and consequently no expansion or compression will occur. 3. If C D < 1, then p is positive. The pressure inside the cloud is larger than the pressure outside the cloud and hence the cloud will expand: lateral diffusion will be the result. It can be concluded that the point at which the lateral diffusion starts is completely described by the condition C D = 1. The drag coefficient thus appears to play an important role in diffusion processes. A mathematical expression for the drag coefficient in the case of our diffusion experiments is therefore necessary if the exact starting point of the lateral diffusion in these experiments has to be predicted. Such an expression can be found as follows. At any time t during the settlement of the dust cloud holds, that the total force F acting on the cloud is equal to the sum of the driving force (the weight G ofthe cloud) and the resistance forces (the buoyancy force FB and the flow resistance force F w ): F=(~+FB+Fw Algebraically written, this becomes 1:

(;

FB

--

C2o)

FW

If it is born in mind that I)F

=M

.6

2) G

=M

.g

3) F B = P F . g . V CDP F • u2 • A 4) F w =

(turbulent resistance)

384

GOOSSENS

t h e n eq. (20) can be transformed to

Md=M.g-,oF.

g

V-

CD,o F - u 2 . A 2

(21)

M is the total mass of the cloud, d its acceleration a n d V its volume. The other terms were already defined earlier in this paper. Because by definition d = d u a n d u = d H (with H the fall distance), this can be rewritdt dt ten as du CD`OF ' u2 ' A M • u ' d--H = M • g -- ,o F • g • V -2

(22)

The left h a n d side of (22) can be specified. In section 2 it was m e n t i o n e d that

U

~/H

~

(23)

before the explosion point is reached, and u=a+fl

' H

(24)

after the explosion point has b e e n reached. If these two equations are differentiated one obtains: du dH

]/ b = V~

du -dH

= fl

1

(first case)

2V FH-

and (second case)

Hence, in the first case, du

M • u • -dH

V

= m.u

=M

b

,/7 ~/b ,/7-

b.M

2,/7

1

2,/7 v~-- ~/ b ,/7

1 2,/-7

DIFFUSION MODEL FOR SETTLING NON CONSOLIDATED DUST

385

and in the second case: du

M u

d--H = M u - f l

Equation (22) then becomes: CD PF

bM

in the first case:

M.g--PF.g.

u2 " A

V--

2ffa

in the second case:

C D 'P F • u2 • A

M • u -fl = M • g - p F • g • V -

Solving for C D : -

in the first case:

in the second case:

b M/ CD =

2 "A u2

PF

2

CD =

" M'g--PF'g"

V

.(M.g-O

V-- M. u

F.g-

2if-a-)

PF-U 2 • A

pF-g • V of the cloud is written as G', these equations

If the immersed weight M.g become finally:

in the first case:

CD =

in the second case:

C D

2 • (G' OF. u2 . A

=

2

• (G'

b M) 2,?-7

-

M

• u -/5')

(23)

(24)

PF •u2 •A With the help ofeq. (23), it is now possible to calculate the time to and the fall distance Ho at which the diffusion starts. It was proved earlier that, at the start of the diffusion, C D = 1 so that

2

=

oF uo2Ao

(G, b M) 7,?-7

where Ao = A at time to Uo = u at time to Solving for Uo gives:

uo=/

P F 2A0.

"(G'-

2 O•-M ~)b

(25)

b')

386

GOOSSENS

In order to write u0 in function of to, one has to consider that on the one hand

~/

u=

.7-

b

(26)

and on the other hand dH dt

u

(27)

so that dH = I

b

at

VG-

or

Integration gives

iT1

dH =

Vb ~7-

i dt + constant

Because at time t = 0 the fall distance H = 0, the constant must be zero and the solution of the equation is

,/-H- =

I

~ ba 7

t

(28)

Substitution of (28) in (26) gives then

u=

b

V

vq-

b u = - -

2,/7

.t

b

vq-

.

t

7 (29)

which is an expression valid for all times t before the explosion point is reached. Eq. (25) can therefore be rewritten as

DIFFUSION MODEL FOR SETTLING NON CONSOLIDATED DUST

2~Fa - t o

=

PF A 0

387

2,f;a--

from which finally:

to =

b2PF

A0

2~-a

(30)

This formula expresses to in function of the total mass M alone, because ,o F and A0 are constants and a, b, and G' are directly (and only) dependent on the value of M. IfH0 has to be expressed in function of M, eq. (28) (with adjusted indices) must be substituted in eq. (30) and after some calculations one finds

H0 =

bPF

A0

b M)

2,~l-I

(31)

3.2. DIFFUSION MODEL PART II: THE DIFFUSION BEFORE THE EXPLOSION POINT IS REACHED From the moment that C D < 1, the pressure inside the dust cloud is larger than the pressure outside the cloud. Because of the pressure gradient the dust grains at the border of the cloud will undergo a lateral outward directed force F, so that lateral diffusion will occur. The magnitude of this pressure force F, acting on a grain is equal to F, = p • S

(32)

in which p symbolizes the overpressure and S a specific section. S has to be considered as an equivalent section, perpendicular to the diffusion direction, which undergoes a same force as the groin in question. S is dependent on the size, the shape and the orientation of the grain, but lor o n e pariicuku, stable o r i e n t e d grain S is c o n s t a n t . If the grains til~2 c o n s i d e r e d to be spherical with a radius equal to their sedimentation equivalent radius r, one can prove (see for instance MARIENS 1976) that in this case the size of the specific section S can be equated to the size of the maximal cross-section through the sphere: F "3

-

,'r

(33)

Hence, eq. (32) becomes F, = p . r2 . ~r

(34)

The total force F"acting on a grain sets the grain in motion. The larger the mass m of the grain, the larger the inertia of the process. Mathematically this is expressed as

388

GOOSSENS • dt = d(m • ~')

in which t symbolizes the time and ~'the (lateral) velocity of the grain (~"is the diffusion velocity). Algebraically written: F . dt = d(m -v)

(35)

The total force F is the sum of two forces: on the one hand the force ~_~caused by the overpressure inside the dust cloud and on the other hand the resistance forceF W the moving grain undergoes from the surrounding fluid:

F=P*+FW or, algebraically, F= F,-

FW

so that eq. (35) can be rewritten as ( F , - FW) • dt = d(m • v)

(36)

This equation holds in principle at any time t during the fall of the dust cloud. Physically spoken, however, it is only meaningfull from the moment the diffusion starts, i.e. from the moment C D has been reduced to less than one. If the time elapsed since the diffusion has been started is written as th, then eq. (36) becomes: (F,-

FW) • dt h = d(m . v)

(37)

The relationship between the total time t elapsed since the beginning of the fall and the diffusion time t h is given by the expression t = tO + th

(38)

Eq. (37) can be written as F,.dt h-F

W.dt h=d(m.v)

The mass m of the grain is constant, so that F , . d t h - F w .dt h = m . d ( v ) The magnitude of the force F , acting on the grain because of the existence of the overpressure is equal to p. S (see cq. (32)): p . S . dt h - F W - dt h = m . d ( v )

(39)

DIFFUSION MODELFOR SETTLINGNON CONSOLIDATEDDUST

389

If the grain is spherical and r is the sedimentation equivalent radius, then S = r 2. n (see eq. (33)). Moreover, if the mineralogical composition of the grain is uniform and homogeneous, the sedimentation equivalent radius is equal to the volume equivalent radius and therefore 4 m = P K , - -f ~ z

r3

where PK = bulk density of the grain Eq. (39) then becomes p.r 2.rc,dt h-F

Wdt

h = P K 7 - 4r t

r 3 • d(v)

(40)

The overpressure was calculated earlier as (see eq. (19)): PF u2 2

P=

{ R0 "(1-CD)\R0+R'

/3 /

so that eq. (40) may be transformed to

PF u 2 2

( .(1-CD

)

R0 ) 3 • r 2 . n . d t h - - F w • dt h RoT~7 4 = pK.7-.~.

(41)

n gives

Division of both members through r 2 PF " u2 2 ( 1 -- C D )

r 3 d(v) •

(

/'

\RoTR'/

• dt h -

FW r2.n

4 = --5 P K

.dt h

.r.d(v)

(42)

From eq. (29) it can be derived that u2 =

? b'- . t2 4a

so that eq. (42) can be rewritten as PF b 2 t 2 8a

t "(1--CD)

R0 )3 • dt h R0+R'

FW 4 r 2 • rc ' d t h = 7 " '°K " r- d(v)

390

GOOSSENS

Elaborating the first term of the left hand member: PF

b2 " t2 8 a

R// " /~}R'

~3 dth

--

,°F b 2 "t 2 8.a CD FW r 2 Jr

R0 R0+R,

)3 dth

4 " dth = -3- " P K " r d(v)

(43)

The drag coefficient CD before the explosion point is reached was found to be equal to (see eq. (23)):

cD=

2 PF

(G,_ b M)

u2 " A

2x/-7-

so that substitution in (43) leads to PF

b2 t 2 R 0+R'"

8 a

dth--

R0 R0+R, Because u 2

FW

dth _

u2"A

2Q--~

4

.dt h=

r2.r:

~

PKrd(v)

t2 o4 •• a• (see eq. (29)), this can be simplified to

Fb2 t2 • 8a

-~3

}

PF

8'a

)3 " R0~R'

1

-dr h -

(

bM

~- • G' FW - - - r2.rc

2x/-a_

R0 ' R0+R,

4 • dth = --y P K r d ( v )

)3 dt h

(44)

The maximal cross-section A of the (spherical assumed, see remark in section 3.1) dust cloud at time t is equal to R 2 • re, where R stands for the radius of the cloud. R is equal to Ro + R' (see eq. (18)), so that at any time t: A = (R 0 + R') 2 . rc Substitution in eq. (44) gives

DIFFUSION MODEL FOR SETTLING NON-CONSOLIDATED DUST

~Fb2t2 t~0 )3 8.a

• R0+R,

1

.dt h -

FW r 2 .rr

b M) R(~0)3 0+R'

(

( R 0 + R') 2.

391

~

• G'

4 " dth = ~

dth

PK.r.d(v)

Because t = to + t h (see eq. (38)): PF b2 8a

( (t0+th

)2

R0 R0+R'

)3 dth

Ro )3 , (G, ~M t (R0+~'~ 2 ~ 7~ R 0 +R'

FW r 2.r~

dth

4 ~ '°Krd(v)

" dth=

If both members are divided by dth, one obtains:

8 a ,

"(t0+th)2

" R0+R'

-

1 (R 0 + R ' ) 2 .~ FW r2 . ~

, .

4 = -3- P K r

b.M 2~r~ d(v) dt h

R0 " R0+R'

(45)

From the experiments it came out (see section 4) that the diffusion velocity V of the grains remained relatively low, so that the diffusion has to be seen as a laminar motion and not as a turbulent motion. This means that, for grains assumed spherical with an equivalent radius r, the resistaflce forceFwis equal to 6 • ~z • r/F • r - ~"(with 7?F the dynamic viscosity of the fluid) as can be derived from the well-known Stokes equation. Substituting this expression in eq. (45) leads to PF b 2 8 a (t0

{ +th)2

'

R0

\ R 0 +R'

)3

1

(G b M)(R0

(R0 + R ' ) 2 • ~r 6 r/F - .v r

4 = -- PKr 3

2~a d(v) dt h

R 0 + R,

392

GOOSSENS

The lateral diffusion velocity v is by definition equal to ;,"a:, so that P F b2 8-a

(t0+th

( R0 )2 " R 0 + R '

1

(R 0 + R') 2 . ~

4

-j- • p K . r-

3

(o

b M)(R0 ;,77

)3

R0 +

dR' -

r

dr.

\dth,] dt h

) '~ gives finally: Division of both members through (' ~"o

PFb 2 8-a . ( t 0 + t h )2 --

--

r

\

R0

1

( b ' M ) • G' 2@--~

(R0+a')2n

dt--h-= -3-

" '

R0

"

dt h

(46)

This differential equation describes completely the lateral diffusion of a grain with an equivalent radius r which is situated in a diffusing dust cloud with a total mass M. For one and the same dust cloud, M (and consequently also a and b) are constant, so that the lateral diffusion distance R' reached after a diffusion time t h is only dependent (besides the nature of the fluid) on the equivalent grain radius r. If the dust cloud is composed of grains of different sizes (as is always the case with natural sediments), a characteristic radius for the whole sample has to be choosen in order to applicate the differential equation to the dust cloud in whole. The most obvious solution consists ila taking the median radius of the dust sample. It will be seen in section 4 that this choise indeed yields very good results on the condition that in eq. (46) R' is everywhere multiplied by a factor two. The final diffusion equation becomes then: PF ' b 2 8-a

.(t0+th)2

12r/F . ( R 0 + 2 R ' / 3 r \ ] 0R

_

( b . M ) 1

(R 0 + 2R') 2 -rt

dR' dt h

• G'

2x/-a

8 .p t = -3K •r

R O+2R' R0

)

3

d (dR'/ \dth/ dt h

(47)

DIFFUSION MODEL FOR SETFLING NON CONSOLIDAI'ED I)UST

303

where r is the median equivalent radius of the dust sample. To obtain a particular solution for the differential equation (which is of order two), two initial conditions are needed. These initial conditions are found very easily. At the moment to the diffusion will start, both the lateral diffusion distance R' and the lateral diffusion velocity dR' are still zero. We therefore have the following two initial conditions: .7(t

1)

ift h = 0, then R' = 0

2)

if t h = 0, then

dR' _ 0 dt h

Solving diffusion equation (47) explicitly is exceedingly dilficult because of its type (non-linear and non-homogeneous differential equation of second order, where the independent variable occurs in a quadratic form in the denominator of one of the terms in the left hand member). Fortunately, computer programs which give very approximative numerical solutions for difficult second order differential equations exist. When testing dilti~sion equation (47) in the case of our experiments, use of such a computer program was made. Attention has to be paid to the fact that diffusion equation (47) is only valid if the total mass M of the dust cloud is "sufficiently high", for the substitution of p through PF u2 2

(. (1--CD)'

R0 R0+R

)3 ,

in step (41) is only allowed under this condition (see also the remark made under eq. (19)). For "too small" values of M, eq. (19) overestimates p and consequently diffusion equation (47) will also overestimate the diffusion. I will come back to this later.

3.3. DIFFUSION MODEL PART III: THE DIFFUSION AFTER THE EXPLOSION POINT HAS BEEN REACHED Because of the diffusion process described in section 3.2., the concentration of the particles in the dust cloud will continuously decrease once the diffusion has started. At a certain moment the concentration has become too low in order that the cloud can still act as a closed system. At this moment (corresponding with the explosion point) the fall velocity of the dust particles is not determined by the total mass M of the cloud anymore. Every particle will now sink with a velocity only determined by its own weight, shape and roundness. At the moment the particles start to sink individually, their fall velocity will be characterized by a suddenly and very strong decrease because before the explosion point, they were all settling with a velocity lying Par above their normal terminal fall velocity. The enormous fluid resistance, which before the explosion point was acting on the cloud as a whole, will now act on each particle individually and will cause the dramatical decrease of the velocity of each particle. The velocity will decrease till the normal terminal fall velocity v m of the particles is reached; from then on it will stay constant. When the dust cloud reaches the explosion point and the velocity of the particles suddenly decreases enormously, it is just as if the cloud is striking to an invisible horizontal

394

GOOSSENS

"resistance plain" which is situated in the air. The dust cloud is also no longer acting as one closed system, so that there can no longer be spoken of a solid object as before. The dust mass becomes now a cloud in the real sense of the word and can best be compared with a mass of heavy gas with a density higher than that of the fluid in which it is immersed. In the literature there can be found several models describing the diffusion occuring when a cylinder-shaped mass of heavy gas sinking in the air suddenly encounters an impenetrable horizontal substratum. A very good review of these models is given by WHEATLY (1985). In all the models, the diffusion is seen as caused by the difference in hydrostatic pressure inside and outside the heavy gas mass. The lateral diffusion distance R1 traversed per time unit t] by the gas particles (or, in other words, the lateral diffusion velocity of the gas) has therefore to be proportional to the difference in hydrostatic pressure between the gas and the surrounding medium: dR 1 - -px.g.h--pM.g.h dt 1 or

dR 1 dt 1

(Px--

PM ) gh

PX is the bulk density of the heavy gas, P M the bulk density of the surrounding medium, g the fall acceleration and h the height of the gas volume at the moment the diffusion starts (see fig. 3). A relatively simple expression based on the above mentioned relationship and frequently used in the study of heavy gases is

,I/ Px-- PM )

dR] dt 1

-

K.

~/[

~o-~-

(48)

.g-h

R

i i i

t J

i i

i i

9x , Tx ,t r

i

i

Pm Tm

)

Fig. 3: Representation of a heavy gas cloud as a cylinder of height h and radius R, within which the temperature is TX and the bulk density is PX' The ambient temperature and density are TM and PM'

DIFFUSION MODEL FOR SETTLING NON-CONSOLIDATED DUST

395

This equation only holds on the condition that the temperature T X of the heavy gas is equal to the temperature T M of the surrounding medium, which is indeed the case for our settling dust clouds. Experiments have pointed out (WHEATLY 1985) that, in the case of ideal particles, the dimensionless proportionality factor K approximates very closely to unity so that dR 1 7oM

dt 1

.g.h

For non-ideal particles (for example the natural dust particles of our experiments) the value of K will be different from unity and eq. (48) must be retained. In order to calculate the lateral diffusion distance Rb one can write:

dR1 = K

~oM

'gh

PM

"gh

• dt 1

which gives after integration:

R1 = K .

'

-t l + c o n s t a n t

(49)

I f h = 0, also R1 = 0. Hence the value of the integration constant is zero and eq. (49) becomes

R 1 = K.

PM

.g.h

(50)

.t 1

If the settling dust clouds of the experiments are considered as spheres (see section 3.2.) which are ringed in by cylinders with radius R and height y = 2R (see fig. 4), then equation (50) remains applicable and h and P X can be specified further: 1) h = 2R b , where R b corresponds with the radius of the spherical cloud at the moment the diffusion starts. 2)

PX

M V -

M 4 -~- .re - ( n b ) 3

where M symbolizes the total mass and V the total volume of the cloud. Substitution of these values in eq. (50) gives

R1

=

K"

I(

g"

M

4 " -~- .zr - ( R b ) 3 ~

M 2Rb

t 1

396

GOOSSENS

and after some simplification:

R 1=

V

K-

2.g.R

b.

(

4-re 'PM'(Rb)3

--1

)

t1

(51)

As was the case for the equation describing the diffusion before the explosion point, also here it came out from the experiments that the total diffusion distance R1 in diffusion equation (51) has to be multiplied by a factor two. Hence, for silty particles, the value of K must be 0,5. The final equation describing the diffusion after the explosion point is reached is then, tbr silty particles:

R1 =

V

2.g.Rb.

(-

4.~z

3 .M PM(Rb)3

--1 2

(52)

y--2R

Fig. 4: Representation of a spherical dust cloud ringed in by a cylinder with radius R and height y = 2R.

The time t 1 in eq. (52) is the total time elapsed since the diffusion started. One has, however, to be very. careful in filling in time values in eq. (52). When the particles of the dust cloud reach the invisible horizontal resistance pkmc ab mentioned earlier, their vclocit5 will suddenly decrease very strongly. The resistance plane, however, is not cornpletely impenetrable: the particles will, after being decelerated very strongly at the top of the plane. pass through it. Their velocity will decrease'till it is equal to the normal terminal fall velocity. So it is better to speak of a resistance zone instead of a resistance plane, but the deceleration of the particles is by far the most expressed at the upper part of the zone. This deceleration of the particles has a very important impact on the numerical value oft1 which has to be used in eq. (52). As a consequence of the deceleration, the total time t x a particle (after the resistance zone is reached) needs to complete a vertical distance x will be much larger than the time it would have needed to complete the same vertical distance x if it would not have encountered the resistance zone yet. The vertical movements of the grains are in the resistance zone so to speak "delayed", or, what means the same, for an equal motion of the particles the time scale

DIFFUSION MODEL FOR SETTLING NON-CONSOLIDATED DUST

397

in which the nlovcnlents occur progresses much fastcr. Eq. (52) does not take into account this elfect, because it is derived from eq. (48) which was developed in circumstances where deceleration processes and resistance zones were not present. Eq. (52) is in our case only applicable if the time tl (i.e. the time elapsed since the start of the diffusion) is expressed in the time scale that was operative before the particles penetrated in the resistance zone. If, for example, one wants to know the diffusion distance R1 after an el/~cctiw, diffusion time of five seconds, it is not correct to equalize tl in eq. (52) to five seconds and solve the equation. One has first to calculate which vertical distance the particles have completed through the resistance zone in five seconds. Then one has to calculate how much time the particles would have needed to complete the same vertical distance if they would not have been decelerated in the resistance zone. This last value has then to be used for tl in eq. (52). If, on the other hand, one is only interested in the course of the lateral diffusion distance R~ in function of the eall distance H, then one calculates first the time the particles would need to reach this fall distance H if they would not be decelerated in the resistance zone. From this value the total time the particles needed to reach the resistance zone is subtracted. The difference gives the total time t~ after the start of the diffusion, but expressed in the first time scale. It is this value for t~ that has to be used in eq. (52). Example: If the initial mass M of the cloud is 4 g, the resistance zone is reached after a fall of 335 cm (see fig. 2D). At that moment, the total time passed since the start of the fall is 0,879 s. If now the lateral diffusion distance of the cloud has to be known after a fall of 380 cm, then the time the particles would have needed to sink 380 cm if they would not have been decelerated in the resistance zone is calculated first. This time is equal to 0,936 s. The time the particles need to sink from 335 cm to 380 cm is thus, expressed in the first time scale: 0,936 s 0,879 s = 0,057 s. It is this value for tl that has to be used in eq. (52).

4. CONFRONTATION OF THE THEORETICALLY PREDICTED DIFFUSION WITH THE EXPERIMENTALLY OBSERVED DIFFUSION AND DISCUSSION Fig. 5 shows, for the six experiments undertaken, the theoretically predicted curves together with the experimentally obtained points. For the prediction of the diffusion before the explosion point eq. (47) was used, there where for the diffusion after the explosion point eq. (52) was taken. It has already been mentioned in section 2 that in the experiments A, B and C (corresponding with a dust mass of 12 g, 9 g and 6 g, respectively) the explosion point is reached at a fall distance of more than 444 cm, i.e. more than the maximal possible height the plank could be attached to the laboratory tower. It follows that for these experiments only eq. (47) could be verified. For the experiments D, E and F (dust mass of 4 g, 2 g and 1 g) the explosion point is reached within the fall distance of 444 cm, so that verification of not only eq. (47) but also ofeq. (52) would be possible. Unfortunately, for experiment E and experiment F (2 g and 1 g, respectively) the diffusion of the individual particles after the explosion point had been reached was no longer clearly visible from the film pictures. It has to be emphasized that the original film pictures, which were of a size of only a few ram, had to be enlarged for more than 25 times which has of course an important impact on the resolution of the pictures. Moreover, the dust mass in these two experiments was very small (2 g and 1 g) so that a strong diffusion (as is predicted by eq. (52)) results in a very low concentration of the dust particles, which is hardly distinguishable on a projected film picture. For experiment D (dust mass of 4 g) on the other hand, the diffusing cloud was reasonably visible on the film so that eq. (52) could here be tested.

398

GOOSSENS

H O-

H o-

<.o-I

cm

100-

(2m

100-

100-

A

--"

C

B

200-

200-

300"

300 -

400-

400"

4

4

Ho

, 8

6

. 10

0

D cm

0

0

-I

~

D

4 6 ~lb

4 6 8 10 cm

H o-

Ho 1

E lOO ~

lOO-

"'"

ooll '

D

F

lOO

;":

"

oo , ,

i;ill

1 4001

-

~

D cm

|

oolli:il ,

D

~

0 2 4 6 8 cm

0 2 4 6 8

D

cm

.:

I

0

I

I

I

I

1~)

I '

'

20

. . . .

310 I

I

I

,

410

. . . .

5101

I

cDm

Fig. 5: The theoretical diffusion as predicted by the model (solid lines) together with the experimentally observed diffusion (dots). The arrow indicates the theoretical starting point of the diffusion. A : M = 12 g; B : M = 9 g; C : M = 6 g; D : M = 4 g; E : M = 2 g; F : M = 1 g. e.p. -- explosion point

DIFFUSIONMODELFOR SETTLINGNON-CONSOLIDATEDDUST

399

It thus follows that for the verification ofeq. (47) all six experiments could be used, there where for the verification of eq. (52) only experiment D could be taken. The total width D of each dust cloud was measured on the projected film pictures with an accuracy of about 0,5 mm, which corresponds with an accuracy of about 1,5 mm in reality. A higher accuracy was impossible because of the restricted resolution. Each measured width was plotted in fig. 5 at the corresponding fall distance H. In order to draw the theoretical curves, first of all the starting point H0 of the diffusion was calculated by solving eq. (31 ). The numerical values of a, b and M (and therefore also G') can be found in table 1. For A0 a value of 3,80 cm 2 (the internal cross sectional area of the plastic cylinder) was used and ['or P F a wflue of 1,22 x 10-3 g/cm 3 was taken. To calculate the diffusion before the explosion point, eq. (47) had to be solved. This was done numerically by asking the computer to calculate (and plot in a graph) the value of R' for 100 points situated in a time interval of 0,5 s alter the start ot the diffusion. The points were equally distributed over the time interval. For a great number of H-values after the starting point H0, the fall time t was then calculated by means ofeq. (28). Subtracting these t-values from that of to (obtained by eq. (30)) gave the corresponding thvalues, which were used in the computer curves to find the corresponding R'-values. These R'-values were added to the Ro-value (equal to 1,10 cm) and the sum was multiplied by two in order to get the total predicted diameter of the diffusing cloud. By connecting all the predicted points by a continuous line, the predicted curves of fig. 5 (before the explosion point) were obtained. To draw the diffusion curve after the explosion point, it suffices to calculate the diffusion distance R1 at only two fall distances because expression (52) is the equation of a straight line. If the starting point of the diffusion is taken as one of these two points, only one single R~value has to be calculated. It is born in mind that the tl-value that has to be used in eq. (52) should be expressed in the time scale operative before the settling particles reach the explosion point (see section 3.3.). From the figures 5A-F it can be deduced that the theoretical diffusion, as predicted by the model, agrees very well with the experimentally observed diffusion. For experiment A (M = 12 g) there are, unfortunately, no experimental D-values available between 240 and 340 cm because the film was strongly underexposed in this interval. The D-values that are available, however, agree nearly perfectly with the predicted values. In experiment B (M = 9 g) the experimental points show a larger scatter than in experiment A, but the predicted curve fits the points very' good. In experiment C the agreement between the predicted and the experimental diffusion values is also remarkably good; only in the short interval between 180 and 220 cm the experimental values are a bit too low. In experiment D not only eq. (47) but also eq. (52) can be verified. Both equations predict the diffusion very well, although in the diffusion interval before the explosion point the experimental values seem to lie a bit under the predicted ones. This phenomenon has to be explained as follows. At the end of section 3.2. it was mentioned that diffusion equation (47) is only valid if the total mass M of the diffusing dust cloud is "sufficiently high", for the substitution ofp (i.e. the overpressure inside the cloud) through ~-Pt u2 Ii CDI (-R(}+ i " ]<'' "' instep (41 ) in the derivation of the diffusion equation is only allowed under this condition. For"too small" values of M, eq. (19) overestimates p and consequently eq. (47) will also overestimate the diffusion. Where the border lies between "sufficiently high" and "too low" values for M is impossible to predict, but from fig. 5 it can be deduced that it has to be situated at a value of about 4 g. For a dust cloud with a mass higher than 4 g eq. (47) can be used without any problems; for a dust mass of 4 g the results are still acceptable, but for a dust mass less than 4 g

400

GOOSSENS

the predicted diffusion deviates far from the observed one. This is very good illustrated in the experiments E and F (2 g and 1 g respectively), where the experimental points lie far under the theoretical curves. Moreover, the question arises if in these two cases there can still be spoken of an effective diffusion; the experimental points seem to scatter around a value of about 2,20 cm, i.e. the diameter the dust cloud had when it started to settle down out of the plastic cylinder.

5.

CONCLUSION

In this paper a theoretical model describing and predicting the lateral diffusion of a vertical settling dust cloud was presented. It was necessary to distinguish between two (in origin totally different) types of diffusion, which both occur during the vertical settlement of the cloud. The border between these two different types of diffusion is situated at the explosion point, i.e. the point where the settling dust mass no longer acts as a closed system but every particle settles individually. The exact position of the explosion point can be read directly from a lhll height - fall velocity diagram but up till now a theoretical prediction of it has not yet been possible. The starting point of the diffusion of the first type is described by eq. (30) and (31). Eq. (47) describes completely the diffusion between the starting point and the explosion point, but is only applicable for dust clouds with a mass higher than 4 g (and with a good approximation also for a mass of 4 g). For dust clouds with a mass smaller than 4 g, eq. (47) overestimates the lateral diffusion. After the explosion point has been exceeded, eq. (52) can be used. It has to be emphasized that the physical and mathematical construction of the first part of the model (describing the behaviour of the dust mass before the explosion point is reached) allows extension of the presented formulae by addition of specific terms without the physical base of the model has to be disturbed. It is therefore possible to apply the model, unchanged or in an only slightly modified and/or adapted way, in several other sedimentological, fluid dynamical or geomorphological research domains. It is, for example, not only usable in the case of dust but also in the case of sand; it is usable in air but also in water; the model presented here was used for vertical motions but it can also be applied in the case of a horizontal or even any arbitrary direction with only small additions of terms in the equations.

REFERENCES

DIETRICH, W.E. (1982): Settling velocities of natural particles. Water Resources Research 18, 16151626.

KRONE, R.R. (1963): A study ofrheologic properties ofestuarial sediment. Hyde Eng. Lab. and Soni Eng. Lab., University of California, Berkely. KUMAR, A. (1967): Effect of surface texture on fall behavior of cylinders and disks in a quiescent fluid. Ph.D. Dissertation, Colorado State University, Fort Collins, Colorado. MARIENS, P. (1976): Natuurkunde deel 2: Mechanische eigenschappen van de materie. Wouters, Leuven, 114 p. McLAUGHLIN, R.T. (1959): Settling properties of suspensions. Proceedings of the A. S.C.E., Paper No. 2311. McNOWN, J.S. & LIN, P.N. (1952): Sediment concentration and fall velocity. Proceedings 2nd Midwestern Conference on Fluid Mechanics, Ohio State University.

DIFFUSION MODEL FOR SETTLING NON CONSOLIDATED DUST

401

McNOWN, J.S. & MALAIKA, J. (1950): Effects of particle shape on settling velocity at low Reynolds numbers. Eos Trans. A.G.U., 31, 74-81. MEULEPAS, E. (1985), personal communication. MIDDLETON, G.V. & SOUTHARD, J.B. (1978): Mechanics of sediment movement. Lecture note for Short Course 3, Eastern Section of the Society of Econ. Paleontology and Mineralogy, Binghampton, N.Y., 248 p. PASQUILL, F. (1974): Atmospheric Diffusion. Ellis Horwood Ltd., Chichester, 429 p. SCHULTZ, E.F., WILDE, R.H. & ALBERTSON, M.L. (1954): Influence of shape on the fall velocity of sedimentary particles. Sediment Ser. Rep. 5, Missouri River Div., Corps of Eng., U.S. Army, Omaha, Nebraska, 161 p. STRINGHAM, G.E., SIMONS, D.B. & GUY, H.P. (1969): The behavior of large particles falling in quiescent liquids. U.S. Geological Survey, Professional Paper 562-C, 33 p. WHEATLY, C.J. (1985): Dispersion of heavy gases, von Karman Institute for Fluid Dynamics, Lecture Series 1985-02, 53 p. WILLIAMS, G.P. (1966): Particle roundness and surface texture effects on fall velocity. Journal of Sedimentary Petrology, 36, 255-259.

LIST OF SYMBOLS USED IN THIS PAPER A

maximal cross-section of the cloud (measured perpendicular to the stream direction) at time t

A o = maximal cross-section of the cloud (measured perpendicular to the stream direction) at time to a numerical factor, only dependent on the dust mass a numerical factor, only dependent on the dust mass drag coefficient diameter of the dust cloud at time t total force acting on the object (cloud or particle) i , , = lateral outward directed force, due to overpressure in the dust cloud F B = buoyancy force F W = fluid resistance force G = weight of the dust cloud G' = immersed weight of the dust cloud g = fall acceleration (9,81 m/s 2) H = Pall distance at time t H 0 = Pall distance at time to h = height of the heavy gas volume at the m o m e n t the diffusion starts K = a constant M = total mass of the dust cloud at time t bd0 = total mass of the dust cloud at time to mass of a dust particle nl total n u m b e r of particles n p = difference in pressure inside and outside the cloud (in second reference system) at time t p0 = difference in pressure inside and outside the cloud (in second reference system) at time to P A = atmospheric pressure P E = resistance pressure PDYN = dynamic pressure P STAT = static pressure P TOT = total pressure A p = difference in pressure inside and outside the cloud (in first reference system) at time t radius of the dust cloud at time t R = R o = radius of the dust cloud at time to R' = lateral diffusion distance of the cloud after diffusion time t h for the first diffusion type lateral diffusion distance of the cloud after diffusion time tt for second diffusion type Rl = R b = radius of the (spherical) cloud at the m o m e n t at which the second type diffusion starts sedimentation equivalent radius of a dust particle r

a

b = CD= D = F =

402

GOOSSENS

S specific section of a dust particle T m = temperature of the medium surrounding the heavy gas T x = temperature of the heavy gas t = total settling time of the dust after start of the l~all to = settling time at which the lateral diffusion starts t h = total diffusion time (first diffusion type) tl = total diffusion time (second diffusion type) U fall velocity at time t fall velocity at time to Uo = Uoo= terminal fall velocity V = total volume of air inside the cloud at time t Vo = total volume of air inside the cloud at time to F = total volume of the dust cloud at time t f0 = total volume of the dust cloud at time to v lateral diffusion velocity a numerical factor, only dependent on the dust mass /~ a numerical factor, only dependent on the dust mass = acceleration g a constant 2 = a constant dynamic viscosity of the fluid qF= bulk density of the fluid PF= PK = bulk density of the dust particle t) M = bulk density of the medium surrounding the heavy gas PX = bulk density of an arbitrary heavy gas X = concentration at time t concentration at time to Xo = =

Address of author: Dirk Goossens, Research assistant N.F.W.O., Laboratory for Experimental Geomorphology, Redingenstraat 16bis, 3000 Leuven, Belgium