_ i. r nmflUP

NATIONAL BUREAU OF STANDARDS REPORT

17014.

NOTES ON SEDIMENTATION MODELS

By

I 6 Richard Savage

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Acting Director THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT

1103-21-5118 4 June 1952

NBS REPORT

1704

NOTES ON SEDIMENTATION MODELS

by

I. Richard Savage

Approved for public release by the

The publication, re|

, .

, .. . Director of the National Institute of

unless permission is 25.D.C. Such p< Standards and Technology (NIST)

prepared if that 1 on October 9, 2015

n part, Is prohibited

andards, Washington

las been specifically

ort for its own use,

FOREWORD

This note is concerned with the sedimentation of small

spherical particles. In the note it is shown by the use of

moment generating functions (Laplace transforms) that there

is a fundamental relationship between the distribution of

particle sizes and the density of particles (as a function of

ti me ) oThis mathematical relationship has practical applications

for it gives a technique to the experimenter of making inferences

about the distribution of particles from measurements on the

density of particles. This is important since measurement of

density is a much simpler operation than measurement of particle

sizes o

This is a technical report on Research in Applications

of Mathematical Statistics to Problems of the Chemical Corps

carried out in the Statistical Engineering Laboratory of the

National Bureau of Standards (National Bureau of Standards

Project Number 1103-21 -5118) in accordance with War Department

Delivery Order Number CD2-2876®

J, Ho Curtiss

Chief, National Applied

Mathematics Laboratories

Ao V 0 Astin

Acting Director

National Bureau of Standards

NOTES ON SEDIMENTATION MODELS

By

Xo Richard Savage

Summary ; This not© is concerned with the following sedimentation

experiment® A mist is allowed to settle within a closed

tank* while being stirred sufficiently to maintain a homogeneous mixture®

Let f(x) be the density of particles of radius x, and

let N( t

)

be the total density of particles at time t® It is

shown by the use of the theory of the Laplace transform, that

f(x) uniquely determines N(t), and conversely® Further the

transform theory gives an explicit method for finding the

relationship® One of the results of this work is that one

can determine f (x) by simply making a series of density measurements to determine N(t)„ If f(x) were found directly it would

be necessary to measure sizes of small particles which is a

much more difficult job than measuring densities® Another

result is that in fitting curves to observed functions f(x)

and N(t), any knowledge of properties of one of them can be

used also in the fitting of the other®

Finally the note contains several pairs of functions f(x)

and N(t), that can be used in the analysis of this type of

experiment®

Introductions In this note we shall be interested in the

sedimentation of small spheres, and as an added complication

the spheres may also die. The objective is to show the

mathematical relation between the distribution of particle

sizes at the beginning of an experiment and the number of

particles at various times.

S toke { s L aw s Fundamental to any discussion of sedimentation

of small spherical particles in air is Stoke J s Law. The law

states that small particles fall in air at a constant velocity.

Quoting from the 24— Edition of the Handbook of Chemistry

and Physics pp„ 1872-1873 w Stoke »s Law gives the rate of fall

of a small sphere in a viscous fluid. When a small sphere

falls under the action of gravity through a viscous medium

it ultimately acquires a constant velocity*

2gx2 (d

1

-d

2

) 7T~~

- , <3l > dz

where x is the radius of the sphere, d^ and the densities

of the sphere and the medium respectively, and rj the coefficient

of viscosity. V will be in cm. per sec. if g is in cm. per

sec.

2

, x in cm. d^ and dg in g per cm.3 and ^

in dyne-sec. per

cm . 2 or poises. H

The coefficient of viscosity no measures the ‘'internal

* o ~ friction1 * of the air. *q has dimensions of mass - (length

x time). Consider two infinite parallel planes, one of which

- 3 -

is moving in its own plane with constant velocity# Then the

air between the planes will eventually arrive at a steady state#

Here let F be the force acting on a unit area of one of the

planes, v the velocity of air between the planes and x a

direction perpendicular to the planes, then Yj is defined as follows s

Model ; We shall assume that at the beginning of the experiment

a chamber contains spherical particles differing only in size#

In particular f(x)N

Q

is the density of particles of radius x«

Nq is chosed so that 2f(x) = 1* N(t;x) is the density of the

particles in the chamber at time t of size x# It is assumed

that there is enough stirring of the air to keep a homogeneous

mixture in the chamber, but not enough stirring to distrub

the particles that have settled according to Stoke 1 s Law 0 Let p(x) be the probability of the death of a particle

of radius x in a unit of time#

Differential Equation

s

dN(t;x) -- _ p(x)N(t;x) _ vYAN(t;x)/V “TF”

= - N(t;x) [pM+v^/v]

o

Here v is the velocity of particles of radius x as given

by Stoke »s Law, A is the horizontal surface of the chamber

and V the volume® This differential equaltion is essentially

- k -

the same as that for radioactive dqcay 0 Solution s

N { t|x)

/ = f(x)NQ

' t[p(x)+vxA/V]

©

Let N(t) be the density of particles of all kinds at

time t, this is the observale quantitjo Clearly N(t) ~

2N(t?x) where the summation is over all possible x s and if

x can vary continuously the sum of course becomes an integral <> which form we shall work withs

po " t[p(x)+v A/V]

N ( t ) «— C e f (x)dx

N0 *

O

Using the v„ given by Stoke 8 s Law we get:

N(t)„ ( ~TT

:— J

0 o

- t[p(x)+2gx2A(d1 =»d2)/97¥l

e 4

Interpretations The solution N(t) can be given a simple _ iry

probability interpretation for it is the moment generating

function of the random variables

pfx) + 2gx2A(d 1 -d 2 )/9T V

(It should be remembered d^ » d2 > If we set 2gA(d]_“d2) /9y|V

= and if p(x) is constant (p) then the result may be simply

expressed ass

N(t;

“Pt -tx^

e Es

N0

If p is known then N( t

)

determines f(x) by inversion

TTJJformulas for moment generating functions (Laplace transform) 0 And of course if two experiments can be performed one with p

known and the other with the p unknown, the unknown p can be

found without using the inversion formula,.

Methodology ; In many experiments it is possible to make p ~ 0,

In these cases complete knowledge of N(t) determines f(x) and

N

o

conversely® This duality is useful for if we have apriori

information of f (x) the observed N ( t) will confirm this infor®

N

Q

mation and also determine any unknown parameters in f(x) e Likewise an observed N(t

)

will imply precisely what the f(x)

N

o

was® Consequently in the next section we shall give examples

of f^x) and the corresponding N(t) e The examples of f(x)

used there are ones proposed by other writers in this field

(except example 4) 0 Experimentally this duality is useful for if one has

an idea of the nature of f(x) then one can select ^with good

chance of success) the function to fit the observed N(t) 0 0

Of course this duality has the great advantage of relating in

a definite manner the functions f(x) and N(t), and in partic®

N

0

ular it is possible using this method to test hypotheses about

the function f^x) without measuring x, but simply by inaki

a series of density determinations*

Examples s

I* p (distribution)

Boudillon

Studies in Air Hygiene

London (l94y )

x e

Here

N(t) s 1 V (l+2^a2

t ) nf^

This distribution has the interesting property that

N(t;x) has the same fuctional form as f(x)«

2* Weibull (distribution)

Roslin, P. and Rammler, E s J 0 Journal of the Institute of Fuel

(a*b*x) > 0 * Here

oo as. N(t) = Z

r*

0

3* Logarithmic (distribution)

Hect, T., and Choate, S* B.

Journal of the Franklin Institute

W369 '^297

- 7

(o,x) > 0

2 2 2im + 2i o

and N( t) _ 2 e

The examples two and three have the desireable property

that if say radius has the distribution of the example then

also mass and surface area have this same distribution

»

4© A distribution with finite range, and a mode at the

extreme

Let f(x) = (n+l)an+1xn

0 < x < 1/a

“0 x<0orx>l/a

a > 0

Here

9

n+1

a

whereP (m) is the incomplete gamma function©

z

- 8 -

Bibliographic Note : General references to experimental and

theoretical material for this type of problem can be found

in W„ E„ Ranae (AEC S0-1000; April 30, 1950) and Boudillon

(Studies In Air Hygiene , London, 1948) • The Laplace transform

is treated extensively in Widder ( The Laplace Transform, Princeton, 1946)*

ADDENDUM

In this report the following densities have been useds

i\

v rr> the number of particles per unit volume g N(t 9 x)* the

number ©f particles with radius x p@r unit volume at time

t% and N(t)i> the total number of particles per unit volume

at time to In some applications the observable quantities an®

not associated with counts of particles but rather with masses <> In this case we define the following set of densi ties

$

Here ML is proportional to the mass of the particles per unit

volume at the beginning of the experiment® M(tfx) la proportional

to the mass of particles of radius x per unit volume at time

t9 and lift) is proportional t© the total mass of particles

per unit volume at time to

When the M functions ar© observed® instead of the N

functions® th© theory is ©lightly modified;© In this ease®

the convenient ratio to use is M(t)/k o ‘This ratio is the

moment generating function of the random variable /x%

f assuming p ® 0) where x has th© probability density function

M(t$x) a N(t|x)x^

.’lift) » f N(tgx)x^dx ® N 5 J

0

So!

* Consequently, by the usa of ln„SP3ion theory

can rind the function f(x) from the observed ratio »

112 the 8ase whsr® the original probability density

function, fU), 1. of the j

! type, the resulting distribution

18 8189 °f tbls *W curing from the original by having

^f® we degrees of freedom*

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