On the precision of sedimentation balance measurements

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_{Delft Universityof Technology}

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Department of Civil Engineering

Hydraulic and Geotechnical EngineeringDivision Hydromechanics Section

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On the precision of sedimentation

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balance measurements

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C. Kranenburg

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Report no. 1 - 92

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Hydromechanies Section

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Department of Civil Engineering

Delft University of Technology

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Delft, The Netherlands

1992

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CONTENTS Abstract 1.

2.

Introduetion 1 2 2 2 3 4 6 8 9 9 9

-Precision of the distribution function

f

2.1. Standard deviation

2.2. Locally linear least-squares fit of

.y

2.3. Covariance of

:y

and

g

2.4. Number of data values and resolution 3. Simulation 4. Conclusions Acknowledgements References Notation Figures

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Abstract

The settling veloeities of fine sediments can be determined using a sedimentation balanee essentially consisting of a settling tube at the bottom of which the weight of the sediment deposited is measured as a function of time. The distribution of settling veloeities then is calculated from an equation, which is known as Oden's formula. In this note the influence of random errors in the measured weight on the precision of the calculated distribution of settling veloeities is analyzed. It is shown that the number of digitally sampled data values needed to calculate the distribution function increases with the ratio of time to sampling time interval, and that a compromise harbe sought between precision and resolution.

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1 1. Introduction

The distribution of settling veloeities in a sample of particulate material cao be determined using a sedimentation balance. This instrument essentially consists of a settling tube at the bottom of which the deposited sediment is weighed as a function of time. Several methods to introduce the sample into the settling tube exist. The method employed for samples of fine sediments (mud) is the following: after the sample has been added to the fluid (usua1ly water) in the settling tube, it is mixed thoroughly over the total height of the fluid column. When the mixing is completed, the sediment particles start to settle and the time-dependent weight of the deposited sediment is measured (e.g., Kuijper et al., 1992).

The cumulative distribution function of the settlingtimes' follows from the measured weight. Let -y(t) be the measured weight at time t nondimensionalized by the measured weight for t - 00. At an arbitrary time t all material having a settling time less than t has settled.

The derivative d-y(t)/dt represents the partic1es with settling times greater than t. Of these partic1es a fractional amount td-y(t)/dt has settled after the start at t =0 of the measurement. At time t the fractional amount ftt) of deposited material with settling times less than t therefore is

f(t)

=

-y(t) - t d-y(t)

dt (1.1)

which expression is known as Oden's equation. The function ftt) is the cumulative distribution of the settling times.

In practical applications'ï and d-y/dt have to be estimated from measured data. Such estimates

are denoted by an overbar SQ that

/(1) = ~(t) - t d~(t)

dt (1.2)

1The settling time is defined here as the height of the fluid column divided by the

settling velocity . Settling timestand settling veloeitiesware related by

wc

=h, where

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2

Some aspects of the accuracy of the method outlined are analysed in this note.

2. Precision of the distribution function

_.,

f

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2.1 Standard deviation

Inaccuracies in the function

f

calculated using Bq. 1.2 are caused by systematic and random errors in the measured weight. Since systematic errors, such as those caused by temperature variations or evaporation, depend on the specific properties of the instrument used, only random errors in digitally sampled values of the weight are considered herein. The measured weight including a random error is denoted by '}-. Possible random errors in the measured times t are assumed to he negligible.

Denoting the derivative d;Y/dt by

g,

the standard deviation of the measured function

f

as given by Bq. 1.2 then is given by

(2.1) 2 2 2

2 C

=

s

-

+ t

s

-

-

t- -'Y 8 "ti

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wheres,is the standard deviation of a quantity z, and C:Yiis the covarianee of

;Y

and

g

.

The fractional weight 'Y is defined as the ratio of weight to final weight. The latter weight can he determined with sufficient precision through averaging so that the error in '}- at time t approximately equals the error in the weight at that time divided by the final weight. The random process is assumed to he stationary, which implies that the standard deviation

s

of

'Y

the measured fractional weights '}-is independent of time.

Bq. 2.1 shows that if Sj is to he less than some preselected value, the standard deviation s-

,

has to he 0(r1) for large times (the covariance is shown in Section 2.3 to

vanish). The calculation of the derivative

g

from measured data therefore requires special attention.

2.2 l.ocal1y linear least - SQuaresfit of

.y

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3

by a linear relation, thereby determining the two constants in such arelation by means of a least-squares fit.

Suppose that in a certain time interval the weight is sampled digitally according to

j=±I,±2, ... ±n (2.2)

i

=

1,2, ...

where tiis the settling time at which

ï

=ïi is to he calculated,

n

is an integer, and At a sampling time interval that does not depend onj (but may he made dependent on i).

A locally linear approximation

:y

can he written as

(2.3)

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The sum Qof squared deviations is given by

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(2.4)

The least-squares fit is obtained by settingOQ/01i =0 and OQ/ogi =0, which gives' (e.g., Bendat and Piersol, 1971)

(mean value)

(2.5)

- 3

Eli."

g. = J'Y .

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n(n +1)(2n +1)

'-

-

11

1+)

These equations provide only estimates of

11

and

gl'

Before calculating standard deviations

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of these quantities, their covariance is considered. 2.3. Covariance of ~ and

g

Denoting a random error in a measured weight 'V•. by x.. so that ~ = 'V .

+

x. , 11+) 1+) I i+) Ii+) 1+)

Eqs. 2.5 show that the covariance

c

-

-

is given by

'YI (2.6)

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where the pointed brackets denote ensemble averaging. Assuming that the errors x.. are

1+)

uncorrelated, this average becomes

/I /I /I

(L

xi+)

L

jxi+)

=

L

j(x;+))

) • -/I i»-/I ) • -/I

(2.7)

For a stationary random process (x;+j) is independent of i

+

j, and as a consequence the covariance vanishes (because Ej vanishes). If the process is instationary, the covariance as given by Eq. 2.6 will beO(lIn~ because

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/I

L

j(xt))

<

n2(x2)max )--/1 (2.8)

Consequently, the covariance will then be small if n ,.. 1.

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2.4 Number of data values aod resolutionThe standard deviations of

:y{

and

g{

cao be calculated as indicated by Bq. 2.1. Assuming again that the random errors are uncorrelated, one obtains

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-

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5 2 1 2 s:

= --

S 'Y 2n+ 1 'Y

(2.9)

s~

=

3__ ~ S2 n(n +1)(2n +1)4112 'Y

Substituting these relationships in Bq. 2.1 and using the result of Section 2.3 give

2

Sj

=

1

2n+l (2.10)

Assuming a gaussian distribution of random errors in the calculated function

f,

error bars cao he constructed once Sj is known.

Defining a relative precision of the function

f

by

(2.11)

where k is a coefficient to he chosen, it is found that the first term in brackets in Bq. 2.10 is negligible, if

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[ 1

]4/3

k- ,..1 411 (2.12)

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This condition will he satisfied in most cases (e.g., k ~ 1 and 1/411 ,.. 1). Assuming furthermore that n is large so that ntn

+

1) - (2n

+

1)2/4, Bqs, 2.10 and 2.11 give as an estimate of the number of digitally sampled data values

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[ 12]113 [ 1

]2/3

2n+l - - --k2 411 (2.13)

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This expression shows that, in order to obtain a certain precision (represented by the coefficient k),t!te number of data values has to increase with tI~ t.

The time interval, T, over which the averaging process takes place is given by T = 2n~t, or

(n",l and rlt

< ~)

(2.14)

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The filtering timeTincreases with~tlt. It also increases askdecreases so that the function}' is increasingly smoothed for increasing precision. Consequently, a practical compromise has

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to be sought between precision and resolution.

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Some values of 2n

+

1 and TIt are listed in Table 2.1.

2n+ 1 Tlt t ~t

_k=1

_k=5

_k=25

_k=1

k=5

k=25

1()2 _{49 17 5 0.49 0.17 0.058} 1()3 229 79 27 0.23 0.08 0.027

lQ4

1063 363 125 0.11 0.04 0.012

Hf

4933 1687 577 0.05 0.02 0.006

Table 2.1. Values of 2n

+

1 and TIt.

3.

Simulation

The approach discussed was checked by simulating the measuring procedure numerically. In particular the influence of the linear fit to the data is difficuit to assess otherwise. The simulation was made by prescribing an analytical expression for the weight function 1(t), to which (small) random numbers were added that represented the measuring errors. These random numbers were taken from a uniform distribution with zero mean. The weight 1(t) was given by

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7 t Ito 'Y(t)

=

---""""1

[1 +(t Itot]) (3.1)

in which tois a time constant, and {3 a coefficient representing the width of the distribution of settling times (the width decreases as(3 increases). Eq. 3.1 implies realistic behaviour for

t - 0 and t - 00. The distribution function

f

given by Eqs. 1.1 and 3.1 is

(3.2)

Approximate values

l

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of the distribution function were calculated using Eqs. 1.2, 2.5 and 2.13 for preselected values of the coefficient {3, timeto, relative precision k and the sampling

time interval dt. This interval was prescribed according to an expression of the form

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(i

=

1,2, ... )

(3.3)

were exis a coefficient (0 ~ ex ~ 1). The case where ex= 0 represents a constant sampling time interval, which requires a large number of data values for large t(see Table 2.1). The sampling time intervals are proportional to the sampling times ti for ex

=

1. The number of data values required and the parameter rlt then do not depend on tj'

Some results of the simulations made are shown in Figs. 1-7. The parameters that were varied are the precision k (k

=

5 and k

=

25) and those listed in Table 3.1. Parameters not varied were: to = 200s and ti = 50s. The same set of random numbers was used in all simulation runs.

The figures show the functions 'Y and f, and the deviations of =

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-

f of the

approximate distribution function from the exact function. The deviations ofwere normalised by Sj

=

ks-y. Ideally, the quantity oj7Sj shouldheof order (plus or minus) one, and should not show a systematic dependenee on the precision k or time.

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8 Run no. a {3 _At, (s) 1 0 1 1 2 IIz 1 1 3 1 1 1 4 0 3 1 5 IIz 3 1 6 1 3 1 7 1 3 5

Table 3.1. Values of a, (3 and ~tl'

As a whole, Figs. 1b-7b indeed seem to bear out the above arguments. The erratic nature of the deviations of shown in these figures indicates that these deviations are caused

mainly by the random errors in "I, and that the influence of the linear approximation is minor. This conclusion becomes even more obvious when comparing Figs. 1b and 4b, 2b

and 5b, and 3b and 6b. In each pair of figures the deviations for k

=

25 are the same on the plotted scale, and those fork =5 differ only little in most cases. Apparently the differences in the function 'Y(t) have little influence. Figs. 6b and 7b show some exceptions.The filtering timesT for k =5(0.21tand0.36t,respectively) are large in these cases so that, with respect to Fig. 3b, the influence of the linear approximation does become noticeable.

The small deviations in Fig. 7b for k = 25 are due to the rounding off upwards to one of the integer

n

given by Bq. 2.13.

4

.

Conclusions

The following conclusions can he drawn from this work:

The locally linear least-squares fit is suited to accurately calculate the distribution of settling times from measured data.

The required number of data values increases with precision and ratio of time to sampling time interval.

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9

The resolution decreases with precision. As a consequence, a compromise between resolution and precision has to he sought.

Acknowled~ements

This work was funded inpart by the Commission of the European Communities, Directorate Genera! for Science, Research and Development under MAST Contract no. 0035 (G6 Morphodynamics research programme). The writer is indebted to Mr. A.M. den Toom who carried out the numerical simulations.

References

l.S. Bendat and A.G. Piersol, Random Data: Analysis and Measurement Procedures. Wiley-Interscience, New Vork, 1971.

C. Kuijper, I.M. Comelisse and I.C. Winterwerp, Methodologyand accuracy of measuring physico-chemical properties to characterize cohesive sediments. Cohesive Sediments Report no. 39, Delft Hydraulics, 1992.

Notation

CyZ covariance of quantitiesy and

z

f

cumulative distribution function of settling times

g = d')'/dI, derivative of weight function

h height of fluid column

iJ

integers

k = Sj/s..,' relative precision

n integer

Q sum of squared deviations

s

,

standard deviation of quantity

z

t settling time

to time constant

w fall velocity

x random error in measured weight

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10 'Y óf At

weight divided by final weight

=

f -

J,

error in function

f

sampling time interval

= 2nAt, filtering time interval

T

approximate value measured value linear approximation

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0.75

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Y,f

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1

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0.50 0.25

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O.O~----_.---r---'_--0.25 0.50 1.0 2.0 ---.----_.----4.0 8.0 --._----16 .---32 -.----64 -.128

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t/tO

Fig. 1a. Weight

Y

and distributian functian f, run na. 1.

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1.0 _____ k 5 , I, 1 , 1 , 1 , 1 , 1 , 1 , 1 -, 1 , ... 1/ , - _ 1 , ./~

,

~

,

1 , ______________k 25

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0.5

5f

ksy

_0.0

I

-0.5 .;

\

_

'\'; 1 \ 1 \

"

\

\

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I

, I , I , 1 , 1 , 1 , 1 , I , I , I , I , I

't'

,

,

,---

-

-,

"-

_,

I ,'\ I I I \ I \I

_

...

/

/

,,

--1.0~----_.---r_----,_----_r----_.---._----._----_.----_. 0.25 0.50 1.0 2.0 4.0 8.0 16 32 64 128

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_{Fig. 1b. Errors in distribution function f, run}t/tO_{no. 1.}

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Y,f

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1

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0.75 0.50 0.25 o.o,_----.----.----.---.----.----.---.----.----~ 0.25 0.50 1.0 2.0 4.0 8.0 16 32 64 128 t/tO

F

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g. 2a. Weight

Y

and distribut

i

on

function

f, run no. 2

.

5f

ksy

I

1.0 _____ k 5 ______________ k 25

/"~

/ \

(

\"

"/

\

/\

J,' \ ...,,'

'Ic,

\ '.. /

\

" \ '7 ", \ ,''''', O.O~~----~----~,-,~~'----~~L-- ..

~~=~'----~

..

~

---\ I \ I ' ...." " \,' " \ ""_.JO \

,

"

~

\'

, ,

\./

..

\'

','

-0.5 -0.5 \.00--1.0,_----.----.----.---.----.----,,----.----.----~ 0.25 0.50 1.0 2.0 4.0 8.0 16 32 64 128 t/tO

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0.75

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_Y,t

0.50

r

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0.25

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.

·

1

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O.O~----~---.---.---.----_.---.---.---r----~ 0.25 0.50 1.0 2.0 4.0 8.0 16 32 64 128 t/tO

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g

.

3a

.

Weight

y

and distribution

function

f

,

run no

.

3

.

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0.5

5f

ksy

0.0

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-0.5

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1.0

.

"

I \ I \ I \ '\. I \

"

'V"

"

""

"

,

\

/,

25 ,'\

,

\

,

\

,

\

,

\

,

\

,

\

,

\

,

\

,

\ ~ y--_____ k 5 ______________ k \

'

\

'

\

'

\

'

\_\

,

" \

,

\_\

,

" or \ \ \ \ \ \_---\ I \ I \ I \ I \ I \

,

\ I \

,

\ I \ I \ I \ I

"

, -1.0~----_.---r_----,_----_.----_.---~----,_----~----~ 0.25

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0.50 1. 0 2.0 4.0 8.0 16 32 64 128 t/tO

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g.

3b

.

Errors

in distribution

function

f,

run

no

.

3

.

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0.75

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Y/f

0.50

)

0.25

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O.O~~---.---r---r---.---.---.---.---.---, 0.25 0.50 1.0 2.0 4.0 8.0 16 32 64 128 t/tO

F

i

g.

4a. Weight

y

and distribution

funct

i

on

f,

0.5

5t

ks

y _0.0

r

-0.5

run no. 4.

1.0 25 _____ k 5 ,

", ,

, ,

,

,

I ,

,

,

,

,

-

-'-'

...

,

,

'/

'

,

,

I , '/

,

'/

,

______________ k <,

- -

_\

,'\

\

'

\

,

\ I \ ... \

,

...

,

\

,

...

,

_/

\

,

_\ ...

,

\

,

_...

"

\

,

-

_/

\

,

"

\

,

"

v .../

,

,

,

,

,

I

,

_,

_,

,

,

,

, I , I ,

, ,

I

, ,

't'

..._... ... -1.01---r---r---r---.---.---.---.---.----~ 0.25 0.50 1.0 2.0 4.0 8.0 16 32 64 128 t/tO

F

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g

.

4b

.

Errors

in distribution

function

f,

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0.25 0.75 0.50

Y,t

r

o.o~----~--~----~----~----~--~----~----~--~

0.25 0.50 1.0 2.0 4.0 8.0 16 32 64 128 t/tO

Fig. 5a. Weight

Y

and distribution

function

f, run no

.

5.

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0.5

5f

ksy

_0.0

I

-0.5 1.0

..

_..

_....

... / ,/

....

_____ k 5 ______________k 25 '\

,

,-'I

\

,

\

,

\

I

\

,

\

/

\

\

/\

'...

/

\

\'

...

-

..

..

...

"

,

'\\ / ,'V\

,

\

,

\ \

,

\

,

\

,

,

,

\,

"

"

\, \

,

\

,

\,

,

\

\

--1.0,_----.----.---.----.---.----.~--_.----~--~ 0.25 0.50 1.0 2.0 4.0 8.0 16 32 64 128 t/tO

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y,f

1

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5f

ksy

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0.75 0.50 0.25 O.O~~---'---'---''---.---.---.---r---.---' 0.25 0.50 1.0 2.0 4.0 8.0 16 32 64 128 t/tO

F

i

g.

6a

.

Weight

y

and distribut

i

on

function

f,

run no

.

6

.

1.0

/\

/

\

k = 5

i

\.1 --- k 25

1\

r,

' ,

l\.'

"

\

, I , ,

\

, "

I,

1',

I, ,',

,

,

I

"

,

,

,

"

"

/

"

_"

,,,

_""

/

_{I \}

,

O

.

0~~----~,---~,~----r---~----7'~--

-

--

--

~~--+,---

--\_{, ,}

"

_'-7

,--

--

\_,

,

,

'I

,

"

,I

",'

I'

,

, _

"

_"

,,'

_,

_,

,

_{, ,}

_,

, I,'

,

, ,

"

,

0.5

/

/

- -

--0.5 -1.0~----_r----~----_,r_----,_----,_----ïï----_r----_.----_, 0.25 0.50 1.0 2.0 4.0 8.0 16 32 64 128 t/tO

F

i

g.

6b

.

Errors

in distribution

function

f,

run no

.

6.

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0.75 0.50 Y,f

1

0.25 0.04=--~~--~----~--~----~--~~--~----~--~ 0.25 0.50 1.0 2.0 4.0 8.0 16 32 64 128 t/tO ~

F

ig

.

7a

.

Weight

yl

~nd distribution

funct

i

on

f.

I

\

I

\

I

\

run no.

7

.

1.0 I

I

\ \ k

=

5

1

I \ k = 25 I \

I

,',

\

I

,

" "

,

,

\

~,~---'

,

o

.

o~~~---~,_~_

~

---.--~/-=~~---~~

--

r--,

/

'/

0.5

5f

ksy

-0.5 -1.0~---._--_.----._--_.---._---._--_.---._--_, 0.25 0.50 1.0 2.0 4.0 8.0 16 32 64 128 t/tO

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