Research ArticleModel of Close Packing for Determination of the MajorCharacteristics of the Liquid Dispersions Components

Kiril Hristov Kolikov1 Dimo Donchev Hristozov2

Radka Paskova Koleva1 and Georgi Aleksandrov Krustev3

1 Department of Mathematics and Informatics Plovdiv University ldquoPaisii Hilendarskirdquo 24 Tsar Assen Street 4000 Plovdiv Bulgaria2Department of Physics University of Food Technologies 26 Maritsa Boulevard 4000 Plovdiv Bulgaria3 Department of Physics Plovdiv University ldquoPaisii Hilendarskirdquo 24 Tsar Assen Street 4000 Plovdiv Bulgaria

Correspondence should be addressed to Radka Paskova Koleva rpkolevagmailcom

Received 6 February 2014 Accepted 27 June 2014 Published 17 July 2014

Academic Editor Khashayar Ghandi

Copyright copy 2014 Kiril Hristov Kolikov et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We introduce a close packing model of the particles from the disperse phase of a liquid dispersion With this model we find thesediment volumes the emergent and the bound dispersion mediumWe formulate a new approach for determining the equivalentradii of the particles from the sediment and the emergent (different from the Stokes method) We also describe an easy manner toapply algebraic method for determining the average volumetric mass densities of the ultimate sediment and emergent as well as thefree dispersionmedium (without using any pycnometers or densit]meters)Themasses of the different components and the densityof the dispersion phase in the investigated liquid dispersion are also determined bymeans of the established densitiesWe introducefor the first time a dimensionless scale for numeric characterization and therefore an index for predicting the sedimentation stabilityof liquid dispersions in case of straight andor reverse sedimentation We also find the quantity of the pure substance (withoutpouring out or drying) in the dispersion phase of the liquid dispersions

1 Introduction

The volumetric mass density is an important variable whichparticipates in determining the quality of the liquid disper-sions as discussed elsewhere [1 2]

The liquid dispersions (LDs) consist of dispersion med-ium (most often water) and dispersion phase In the dis-persion phase the LDs there are particles with larger andparticles with smaller volumetric mass density than theaverage density of the dispersion medium which we shallcall heavy particles and light particles The heavy particles inthe conditions of homogeneous gravitational and centrifugalfield are displaced downwards (in direction of the fieldintensity vector) in containerswith LD and this phenomenonis called straight sedimentation or settling The light particlesare displaced upwards in containers with LD and this phen-omenon is called reverse sedimentation (creaming) or emer-gence In the suspensions there are more heavy particles andin the emulsions there are light particles After reaching the

condition of complete phase separation of the LD we haveultimate (maximal) sediment and emergent

The scientific studies of the LD are developed mainly intwo directions

(1) determining the size and the density of the particles inthe dispersion phase of LD and their distributionaccording to size and analyzing the process of theirsettling that is sedimentation analysis as discussed byHodakov and Udkin [3]

(2) studying the ability of LD in unchanged externalimpacts to maintain for definite time the homoge-neous distribution of the dispersion phase in the dis-persion medium that is studying the sedimentationstability as discussed by Hodakov and Udkin [3]

For sedimentation analysis and determining the sedi-mentation stability of LD there are widely used visual andoptical methods and devices including taking samples from

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 615236 10 pageshttpdxdoiorg1011552014615236

2 The Scientific World Journal

K

a

hf

hs

hrarrg

(a)

K

a

hf

hs

h

(b)

Figure 1 Vertical axial section of straight square prismatic container with separated components of the liquid dispersion (a) model of perfectclose packing of the particles from the dispersion phase (b) model of nonperfect close packing of the particles from the dispersion phase

the different levels of the volume of the tested LD as discussedby Westwood and Kabadi [4] measuring the mass of thesedimented particles and applying electricity or some radi-ation as discussed by Puttmer et al [5] The optical methodscan achieve a very high degree of sensitivity and automationbut they have a serious disadvantagemdashthe tested LD mustbe transparent for the applied length waves Therefore thetested sample is often dilutedThe dilution however changesthe properties of the LD This problem is avoided by usingthe liquid separated by centrifugation Moreover the opticaldevices (spectrophotometers) have expensive equipment

In this paper we introduce a model of close packing ofthe particles from the dispersion phase of the LD With thismodel we determine the volume of the sediment and theemergent of the LD

The empirical formula of Stokes for finding the radius ofthe equivalent spherical particles from the dispersion phaseis well known as discussed by Hodakov and Udkin [3] Weformulate a new much simpler approach for finding theaverage Stokes radii of the particles from the dispersion phaseWe also offer a simple algebraic method for determinationunder given temperature and external pressure of the averagevolumetric mass density of the LD componentsmdashof the max-imum sediment and the maximum emergent as well as thedispersion medium without using pycnometers as discussedby Westwood and Kabadi [4] or densitometers as discussedby Puttmer et al [5] With the determined densities we findthe componentsmasses and also the volumetric density of thedispersion phase of the tested LD

Based on the densities of the components we defined forthe first time also a dimensionless scalemdashnumeric character-istic for characterizing the quality therefore for predictingthe sedimentation stability of liquid dispersions

There is not a physical theory yet for the quantitativedetermination of the dispersion phase and the dispersionmedium of LD The most used method is drying which isa long process resulting also in the destruction of the testedsubstance

Through the mass of the LD the densities and the linearsizes of its components we determine the percentage ofthe pure substance in the dispersion phase as well as thepercentages of the bound and the free dispersion medium ofLD

All these results are obtained without complex methodsand devices We use straight prismatic or cylindrical trans-parent containers where we pour homogenized LD The fullcontainers are placed under the influence of gravitationalor centrifugal filed under constant temperature and externalpressure

We assume that the size of the container is large enoughso the wall-adjacent effect can be avoided

2 Model of Close Packing of the Particles fromthe Dispersion Phase of Liquid Dispersion

We shall call the distribution of the particles from the disper-sion phase assumed to be spherical and touching adjacentparticles after the complete separation phase a model of theclose packing of the dispersion phase

Figure 1 shows axial section of straight containers 119870which we assume have square foundations

We assume that at the initial moment in the containerhomogenized LD is poured and that it is placed in conditionsof static homogeneous gravitational or centrifugal filed underconstant temperature and external pressure In the case ofcentrifugal filed because of the relatively small sizes of 119870compared with those of a centrifuge we can assume that thecentrifugal filed in 119870 is practically homogeneous

After the phase separation of LDs sharp macroscopic flatborder is formed between the components of the dispersionphase with separated dispersion medium in 119870

In themodel of the close packing of the particles from thedispersion phase offered by us it is assumed that their shapeis slightly different from spherical particles which for mostof the LD is satisfactorily met as discussed by Hodakov and

The Scientific World Journal 3

Udkin [3] Moreover as a result of the homogenizing of theLD and the conditions in which the sedimentation takesplace we can also assume sequential particle distributionalong the horizontals in the maximum sediment and emer-gent

The concentration gradient of the dispersion particlesduring the formation of the sediment and the emergentalso significantly contributes to this distribution along thehorizontals (As end result this gradient becomes 0)

After the ultimate separation a part of the dispersionmedium is located among the particles from the dispersionphase of the sediment and the emergent This dispersionmedium (in most cases water) is called bound (nonfree) Thedispersion medium separated outside the sediment and theemergent is called unbound (free) We shall emphasize thatunlike the free dispersion medium the bound dispersionmedium cannot be separated even during centrifugationbecause between the particles of themediumand the particlesof the dispersion phase there are huge quantum-mechanical(close) interactionsmdashmuch larger than any centrifugal

It should be taken into account that by approaching thesurfaces of two particles the forces of attraction between themgrow relatively slowmdashfor dispersion interactions by the law11205913 where 120591 is the distance between the particles Contrari-

wise the forces of repulsion are practically insignificant toenough short distance but by further reduction of 120591 theseforces are increasing rapidly by the low 1120591

119899 where 119899 asymp 10As a result by approaching the surfaces of the particles toequilibrium state of a close packing the majority of the workis for the forces of attraction and the depth of the minimumof the potential curve is close in absolute value to the workof the forces of attraction that is molecular component ofthe energy Furthermore the energy and the interaction forcebetween the particles are determined not only by the distancebetween the particles and the value of the complex constant ofHamaker but also by the size and the shape of the interactingparticles discussed by Shchukin et al [6]

Let the square of the basis 119870 have an internal side length119886 and let the height of the LD in119870 be ℎWith ℎ we designatethe height of the ultimate sediment and with ℎ

119891we designate

the height of the ultimate emergent (Figures 1(a) and 1(b))We shall find the number of the particles from the

dispersion phase We assume that an ultimate (maximal)sediment (emergent) in close packing along the length 119886 canbe applied 119894 number of dispersion particles and each of theseparticles has an average Stokes radius 119903 = 119903

119904(119903 = 119903

119891) We

also assume that the height ℎ = ℎ119904(ℎ = ℎ

119891) can be applied 119895

number of such particles

(a) Considering the model of perfect close packing of theparticles from the dispersion phase due to the largevalues of 119894 and 119895 we can ignore the difference in oneparticle through row or column

Then the number of the particles in the sediment (emer-gent) is 119899 = 119894

2119895 Moreover because of simple geometrical

considerations it follows that ℎ = (119895 minus 1)radic3 + 2119903 = 119895radic3119903 +

(2 minus radic3)119903

Due to the large values of numbers 119895 and 119903 ≪ 119895 we canassume that ℎ = 119895radic3119903 that is

119886 = 2119894119903 ℎ = 119895radic3119903 (1)

(b) Considering themodel of nonperfect close packing ofthe particles from the dispersion phase

119886 = 2119894119903 ℎ = 2119895119903 (2)

Therefore we assume the average values of 119886 and ℎ that is

119886 = 2119894119903 ℎ = (1 +radic3

2) 119895119903 (3)

3 Volume of the Sediment andthe Emergent in Liquid Dispersion

According to formula (3) for the volume 119881 = 119881119904(119881 = 119881

119891)

of the full ultimate sediment (emergent) in 119870 we have 119881 =

1198862ℎ = 2119894

2119895 (2 + radic3)119903

3 where 119903 is the radius of the Stokesparticles of the dispersion phase

On the other hand119881 = 1198811015840+11988110158401015840 where1198811015840 is the volume of

the dispersion phase in this sediment (emergent) and11988110158401015840 is thevolume of the bound dispersion medium of the sediment(emergent) The following equation is valid

1198811015840=4

3120587 11990331198942119895 (4)

Therefore 11988110158401015840 = 119881 minus 1198811015840 = 21198942119895(2 + radic3 minus (23)120587)1199033 Fromthe obtained expressions for1198811198811015840 and11988110158401015840 we determine that

1198811015840

119881=

2120587

3 (2 + radic3)

asymp 0 56

11988110158401015840

119881= 1 minus

2120587

3 (2 + radic3)

asymp 0 44

(5)

Then

1198811015840asymp 0 56119886

2ℎ 119881

10158401015840asymp 0 44119886

2ℎ (6)

From the equations in (6) it follows that 1198811015840 asymp 1 311988110158401015840Let 1205881015840 = 120588

1015840

119904(1205881015840= 1205881015840

119891) be the average volumetric mass

density of the particles from the dispersion phase in the ulti-mate sediment (emergent) In addition let 12058810158401015840 be the averagedensity of the dispersion medium (bound or free) in theLD Then according to formulae (6) for the masses 1198981015840 and11989810158401015840 respectively in the dispersion phase and the bound dis-

persion medium in the ultimate sedimentemergent in119870 wehave

1198981015840= 11988110158401205881015840asymp 0 56119886

2ℎ1205881015840 119898

10158401015840= 1198811015840101584012058810158401015840asymp 0 44119886

2ℎ12058810158401015840

(7)

where ℎ = ℎ119904(ℎ = ℎ

119891)

4 The Scientific World Journal

In (7) if 1205881015840 = 1205881015840

119904 then 1205881015840 gt 120588

10158401015840 and if 1205881015840 = 1205881015840

119891 then

1205881015840lt 12058810158401015840

In our studies we obtain approximated formulas in whichthe practically negligible distance between the particles isignored that is we assume that the particles of the dispersionphase by ultimate sedimentation are closely fitted to eachotherThis does not affect the calculations because we do notconsider only the case when the particles are perfectlyarranged closely to each other

4 Determination of the Average StokesRadii of the Particles in the Sedimentand in the Emergent

Using (3) we can find the numbers 119894 = 119894119904(119894 = 119894

119891) or 119895 =

119895119904(119895 = 119895

119891) and hence the average Stokes radius of the parti-

cles from the sediment (emergent) To this end we count thedispersion particles symmetrically around the center of themasses of the ultimate sediment (emergent) along the verti-cal in a chosen linear unit of distance (This is done becauseeven in monodisperse system the dispersion particles are notof absolutely the same size) Then the obtained number ismultiplied by the number of single distances contained in ℎThis is how we find 119895 and then according to (3)

119894 =

119886 (1 + (radic32))

2ℎ119895 (8)

Hence from (3) for the average Stokes radii of theparticles from the sediment or the emergent we have

119903 =ℎ

(1 + (radic32)) 119895

=119886

2119894 (9)

If the LD in 119870 is polydisperse the obtained radii 119903 of theparticles from the sediment or the emergent are determinedin average values

With this we offer a new method different from Stokesrsquofor determination of the average Stokes radii in dispersionparticles in LD The determination of the 119899 = 119894

2119895 of the dis-

persion particles from the ultimate sediment or emergent inLD in 119870 provides important information also for the foodmedical chemical and so forth industries

5 Algebraic Method for Determiningthe Average Densities and the Masses ofthe Components of a Liquid Dispersion

The densities of the ultimate sediment and emergent as wellas the free dispersionmedium can be definedwith the widelyused pycnometers as discussed by Westwood and Kabadi [4]But to this end these three components of the LD must bemechanically separated This requires the ultimate emergentto be removed from the container first and then pouringout the free dispersion medium and finally there is access tothe ultimate sediment After that the three substances sam-ples are taken which are placed in the pycnometer Thepycnometer method for measuring the densities is easy to

K1 K2 K3

m1

R R R

h1

m2h2

hf3

hf2

hf1

m3 h3

hs1ms1 ms2 ms3hs2 hs3

mf2

mf3

mf1

Figure 2 Vertical axial section of three equal straight cylindricalcontainers with separated components of liquid dispersions

apply and inexpensive but the measurement takes a lot oftime

The measurement can be also done in a noncontact waybymeans of ultrasound densitometers as discussed by Puttmeret al [5]These devices however use complex equipment andare expensive

We offer an algebraic method with significantly simplifiedmeasurements It employs containersmdashprismatic or cylindri-cal At the beginning themasses of the LDmust bemeasuredwhich need to be equal but taken in different proportionsfrom the dispersion phase The LD is placed in the respectivecontainers under constant temperature and external pressureAfter the formation of the ultimate sediment and emergentand the free dispersion medium between them in theconditions of gravitational or centrifugal field we performelementary linear measurements of the geometrical sizes ofthe separated particles in the containers with LD (Figure 2)The densities and the masses of the components are deter-mined with simple algebraic operations

As it is not necessary to count the particles in the ultimatesediment and emergent in this case we can use straightcylindrical containers Let119870

119894(119894 = 1 2 3) be such containers

which for convenience we assume are same with radii of themain planes of 119903 as they are presented in the vertical axialsections in Figure 2The three containers contain LDwith thesame components of the dispersion medium and the disper-sion phase taken in different proportions (These proportionscan be achieved for example by minimal dilution withdispersion medium of the given LD that is by increasingthe relative share of the dispersion medium in the differentcontainers)

Let us assume that at the initial moment 119905 = 0 the testedLD is in homogenized condition that is the dispersion phaseof this dispersion is evenly distributed in the whole volume ofthe containers Moreover the containers are placed underspecific constant temperature and external pressure

In the conditions of gravitational or centrifugal fieldcomes a moment of time 119905 gt 0 when in LD of119870

119894(119894 = 1 2 3)

appears a condition of ultimate sediment and emergent withconstant heights of their volumes Thus within time 119905 com-plete phase separation is achieved that is the volumes and themasses of the formed ultimate sediment and ultimate emer-gent in 119870

119894(Figure 2) no longer changemdashthe heavy particles

have settled and the light particles have emerged

The Scientific World Journal 5

Some cases of LD are possible when free dispersionmedium and only ultimate sediment or only ultimate emer-gent are formed Some more complicated cases are alsopossible when more than one sediment andor emergent areformed

The values of the average density of the substance in theultimate sediment 120588

119904of the average density of the substance in

the ultimate emergent 120588119891and the density of the dispersion

medium 120588 = 12058810158401015840 in the containers119870

119894(119894 = 1 2 3) are the same

Characteristics of the ultimate sediment and emergentare the lights of their heights ℎ

119904119894and ℎ

119891119894(119894 = 1 2 3) in 119870

119894

which are constant after certain moment of time 119905 Withℎ119894(119894 = 1 2 3) we designate the heights of the free dispersion

medium and with 119898119904119894 119898119891119894 119898119894(119894 = 1 2 3) we designate the

masses of the ultimate sediment and emergent and the freedispersion medium of LD in 119870

119894from Figure 2 Then we can

write

119898119904119894= 1205871198772ℎ119904119894120588119904 119898119891119894= 1205871198772ℎ119891119894120588119891 119898119894= 1205871198772ℎ119894120588

(119894 = 1 2 3)

(10)

where 119877 is the radius of the internal bases of the containersWith 119872

119894(119894 = 1 2 3) we designate the total masses of LD

which fill up 119870119894 We consider these masses to be known

(measured) According to the law on conservation of masseswe have the following system of three linear equations

119898119904119894+ 119898119891119894+ 1198980119894= 119872119894 119894 = 1 2 3 (11)

If 119867119894= ℎ119904119894+ ℎ119894+ ℎ119891119894 and 120588

119894is the average volumetric

mass density of LD in119881119894 then it is known that 119896

119894= 1198721198941205871198772=

119867119894120588119894(119894 = 1 2 3) Then from (10) and (11) the system follows

ℎ119904119894120588119904+ ℎ119891119894120588119891+ ℎ119894120588 = 119896119894 119894 = 1 2 3 (12)

By solving (12) the three densities are determinedmdashof theultimate sediment of the ultimate emergent and of thedispersion medium

120588119904=1198631

119863 120588

119891=1198632

119863 120588 =

1198633

119863 (13)

where11986311986311198632 and 119863

3 according to the Cramers rule are

determinants of third orderSince the proportions of the components in LD are differ-

ent then the equations in the system (11) are not equivalentand the system (12) is consistent But if we pour the sameLD even in three different in size and shape straight con-tainers then the equations in (11) will be proportional Thenthe system (12) will be inconsistent

From (10) and (13) we can determine themasses119898119904119894119898119891119894

119898119894 as well as the mass 119898

119901119894= 119898119904119894+ 119898119891119894

of the dispersionphase for any 119894 Then we also find the density 120588

119901of the

dispersion phase in the tested LD

120588119901=

ℎ119904119894120588119904+ ℎ119891119894120588119891

ℎ119904119894+ ℎ119891119894

(14)

If we need to determine only the density 120588119901of the

dispersion phase and the density 120588 of the dispersionmedium

two containers 1198701and 119870

2are enough and a system with two

linear equations

119898119901119894+ 1198981= 1198721lArrrArr (ℎ

119904119894+ ℎ119891119894) 120588119901+ ℎ119894120588 = 119896119894 119894 = 1 2

(15)

Thus

120588119901=1198631

119863 120588 =

1198632

119863 (16)

where 119863 1198631 and 119863

2 according to the Cramers rule are

determinants of second orderIf we know the density of one of the ingredients for

example 120588 for distilledwater then also two containers1198701and

1198702are enough Then 120588

119904and 120588

119891are also determined with a

system with two linear equations

119898119904119894+ 119898119891119894+ 119898119894= 119872119894lArrrArr ℎ

119904119894120588119904+ ℎ119891119894120588119891= 119896119894minus ℎ119894120588

119894 = 1 2

(17)

which is solved analogically to system (12)In case when the density 120588 of the dispersion medium is

known andwe have only ultimate sediment or ultimate emer-gent it is sufficient to use only one container 119870 = 119870

1 Then

we have only one linear equation

119898119904+ 119898 = 119872 or 119898

119891+ 119898 = 119872 (18)

that is

120588119901=119896 minus ℎ120588

ℎ119904

or 120588119901=119896 minus ℎ120588

ℎ119891

(19)

where 120588119901= 120588119904or 120588119901= 120588119891and 119896 = 119872120587119877

2Because119898

119904= 1198981015840+ 11989810158401015840(119898119891= 1198981015840+ 11989810158401015840) where1198981015840 is the

mass of the dispersion phase and11989810158401015840 is the bound dispersionmedium in the ultimate sediment (emergent) and accordingto formula (19) we can find the density 1205881015840 of the dispersionphase

1205881015840=

120588119901minus 0 44120588

0 56 (20)

In this case when the tested LD has sediments and emer-gents consisting of several components then the density ofeach component can be determined through a measurementwith more containers In the general case we use a systemconsisting of 119899 linear equations and 119899 unknowns which ispresent in the following way

119901

sum

119896=0

119898119904119896 119894+

119897

sum

119895=0

119898119891119895 119894+ 119898119894= 119872119894 119894 = 1 2 119899 (21)

where 119899 = 119901+119897+1 From (21) the unknown densities 120588119904119896 119894 120588119891119895 119894

and 120588 are determined

From (13) (14) (16) and (19) using the method shownin Kolikov et al [7] we can determined the maximumabsolute inaccuracies (errors) Δ120588

119904 Δ120588119891 Δ120588119901 and Δ120588 and the

maximum relative inaccuracies (errors) Δ120588119904120588119904 Δ120588119891120588119891

Δ120588119901120588119901 and Δ120588120588 discussed by Kolikov et al [8]

6 The Scientific World Journal

6 Characterization Scale forSedimentation Stability

There are different methods for evaluation of the sedimenta-tion stability 119878 of LD One of the most popular methods is thevisual methodThe visual methodmeasures the observed dis-placement over a period of time of the horizontal separationline between the clear and the muddy part of the tested LD asdiscussed elsewhere [9 10]Themethod employs transparentcontainer and a clock

From the instrumental methods most popular ones indetermining the sedimentation stability are the opticalabsorptionmethodsmdashthroughmeasuring the intensity of thelight passing through the LD as discussed elsewhere [11 12]

According to our method the evaluation of 119878 is donethrough the densities of the components of the LD and it isnot necessary to measure the time and optical transmissionof LD is no required Moreover we introduce scale forcharacterization of the sedimentation stability 119878

Let one particle of the dispersion phase with volume 119881and density 120588

119901be located in dispersion medium with density

120588 in the conditions of gravitational field Then the value ofthe resultant force of its weight and the Archimedean force is

119865 = 119881119892 (120588119901minus 120588) (22)

where 119892 is gravitational constant (In (22) we neglect the fric-tion force which is very little for small velocities of theparticle)

Therefore if the particle is heavy that is 120588119901gt 120588 hArr

120588120588119901lt 1 then119865 gt 0 andwe have straight sedimentation (set-

tling) And if the particle is light that is 120588119901lt 120588 hArr 120588

119901120588 lt

1 then 119865 lt 0 and we have reverse sedimentation (emer-gence)

Thus we suggest the evaluation of the sedimentationstability 119878 to be done through the following dimensionlessphysical variables 119868

119904= 120588120588

119904 119868119891= 120588119891120588 (120588119904 120588 = 0) and hence

through the variable

119868 = max (119868119904 119868119891) = max(

120588

120588119904

120588119891

120588) (23)

The number 119868 is the sedimentation stability index in adimensionless scale for characterization of 119878 Since 0 lt 119868

119904

119868119891le 1 then 0 lt 119868 le 1 that is the scale is the interval (0 1]

The more the index 119868 of an LD is closer to 1 the more stablethe LD is and the closer the index 119868 is to 0 the more unstablethe LD is Thus if 119868 = 1 the LD is absolutely sedimentarystable that is there is no phase separation

We shall notice that if the sediment is absent in the LDthen 119868 = 119868

119891isin (0 1] and if the emergent is absent then 119868 =

119868119904isin (0 1]

7 Determining the Percentage ofthe Substance in the Sediment andthe Emergent

The amount of the substance in the dispersion phase is amajor factor which determines the quality of many products

The drying is widely used method for testing the dry matterin different suspensions To this end a sample is taken fromthe homogenized LD which is placed in a suitable containerweighted with a scale and placed in a drier for drying untilconstant weight is obtained This weight is divided into theinitial (not dried) full weights of the sample and thenmultiplied by 100 The obtained value is called dry substancepercentage of LDThe disadvantages of thismethod are its rel-atively long duration and the spending of significant thermalenergy for drying a unit of weight from the tested substanceMoreover the heating usually to 105∘C required by thestandard results in destruction of the treated substance and itcan only be used for suspensions

We offer a different method which does not have theseflaws

If 119870 is a straight cylindrical container filled-up with LDwith weight 119872 and internal radius of the bases 119877 thenaccording to formula (7) the masses of the dispersion phaseand the bound dispersion medium in the ultimate sedimentor emergent are respectively 1198981015840 = 0 561205871198772ℎ1205881015840 and 11989810158401015840 = 0441205871198772ℎ12058810158401015840 Then knowing the geometrical parameters of the

used container as well as the heights masses and densitieswe have in percentage the amount of the substance 1198751015840 in theultimate sediment or emergent and of the bound dispersionmedium 119875

10158401015840 respectively

1198751015840= (

1198981015840

119872sdot 100) = (

0 561205871198772ℎ1205881015840

119872sdot 100) (24)

11987510158401015840= (

11989810158401015840

119872sdot 100) = (

0 441205871198772ℎ12058810158401015840

119872sdot 100) (25)

The percentage 119875 of the free dispersion medium can alsobe calculated with our method namely

119875 = (

1205871198772(119867 minus ℎ

119904minus ℎ119891) 12058810158401015840

119872sdot 100) (26)

Let us denote that unlike the method of drying until con-stant weight through formula (24) the percentage 1198751015840 of thetested substance is determined with the participation of thebound dispersion medium We take into account only theamount of the pure dispersion phase in the specific sedimentor emergentThrough drying complete dryness of the samplecannot be achieved in practicemdashin the sediment therealways remains some percentage bound dispersion mediumTherefore we cannot expect to have coinciding percentage ofdry substance determined according to the two methods

Let us consider a case of sediment If 1198751015840119904is the percentage

of dry substance in the sediment determined with the dryingmethod then for practical reasons we can write downmodification of formula (24) namely

1198751015840

119904= 1205761198751015840

119904asymp (120576

0 561205871198772ℎ1199041205881015840

119872sdot 100) (27)

Here the variable 120576 is a conversion factor which canchange within the limits 1 lt 120576 lt (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904 where

The Scientific World Journal 7

Table 1 Linear dimensions masses and densities of the components of liquid dispersions

Exp 119877 [cm] ℎ1199041

[cm] ℎ1[cm] ℎ

1199042[cm] ℎ

2[cm] 119872

1[g] 119872

2[g] 120588

119904[gsdotcmminus3] 120588 [gsdotcmminus3]

1 13 24 63 36 76 5607 7397 169 1032 13 24 61 36 76 5517 7433 173 1023 13 24 61 35 78 5509 7439 172 1024 13 24 62 36 77 5574 7495 174 1025 13 24 62 36 77 5551 7458 171 1026 13 24 62 36 76 5557 7383 163 106Avg 13 24 62 36 77 5553 7434 170 103

K1 K2

m1

R R

h1

m2h2

hs1ms1

ms2 hs2

Figure 3 Vertical axial sections of two identical straight cylindricalcontainers with ultimate sedimentation

1198981015840

119904+ 11989810158401015840

119904= 119881

1015840

1199041205881015840

119904+ 11988110158401015840

11990412058810158401015840 is the total weight of the

sediment (dispersion phase and bound dispersion medium)The unobtainable value 120576 = 1 corresponds to the drysubstance in the ultimate sediment in case of complete dryingof the sample in119870 that is without bounddispersionmediumas with formula (24) we offer If the ultimate sediment is notsubjected to drying then the value 120576 = (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904asymp

1 + (120588101584010158401205881015840

119904) lt 2 Thus for the conversion factor in formula

(27) we have 1 lt 120576 lt 2

8 Example

We present models of two dispersion system suspensionsfrom calcium carbonate powder respectively 700 g and1100 g measured with electronic scale with accuracy of 001 gand distilled water respectively 50mL and 65mL measuredwith graduated cylinders with accuracy of 05mL In order toevaluate the accuracy of our method we assume the densitiesof CaCO

3and the distilled H

2O to be unknown

In order to determine the densities respectively 120588119904of the

ultimate sediment and 120588 of the dispersion medium we usetwo identical cylindrical containers 119870

1and 119870

2 with radius

of the internal basis 119877 = 0 013m (Figure 3)In the two containers we pour the two homogenized sus-

pensions (their components are in different proportions)Weperform 6 observations with room temperature 20∘C

Table 1 presents the values of the measured variables119877 ℎ1199041 ℎ1 1198721(for container 119870

1) and 119877 ℎ

1199042 ℎ2 1198722(for

container1198702) during the 6 observations

For densities 120588119904and 120588 according to formula (15) we find

with the system of linear equations

10038161003816100381610038161003816100381610038161003816

1198981199041+ 1198981= 1198721

1198981199042+ 1198982= 1198722

lArrrArr

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1199041120588119904+ ℎ1120588 =

1198721

1205871198772

ℎ1199042120588119904+ ℎ2120588 =

1198722

1205871198772

(28)

Its solutions are

120588119904=1198631

119863=

1

1205871198772

1198721ℎ2minus ℎ11198722

ℎ1199041ℎ2minus ℎ1ℎ1199042

120588 =1198632

119863=

1

1205871198772

ℎ11990411198722minus1198721ℎ1199042

ℎ1199041ℎ2minus ℎ1ℎ1199042

(29)

Then for the densities of the ultimate sediment and thedispersion medium we have respectively the average values120588119904asymp 1 70 gsdotcmminus3 and 120588 asymp 1 03 gsdotcmminus3 which are also

presented in Table 1For the index of the sedimentation stability we find

119868 =120588

120588119904

asymp 0 61 (30)

The value of 119868 according to our scale (01] shows that thetested LD has a little over the average sedimentation stability

According to the model discussed by Kolikov et al [7]we shall determine the maximum absolute inaccuracies Δ120588

119904

and Δ120588 for the established densities 120588119904and 120588 as well as the

maximum relative inaccuraciesΔ120588119904120588119904andΔ120588120588 IfΔ119872

119894Δℎ119894

Δℎ119904119894(119894 = 1 2) andΔ119877 are absolute inaccuracies of119872

119894 ℎ119894 ℎ119904119894

and 119877 then from (29) it follows that

Δ120588119904

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(31)

8 The Scientific World Journal

Table 2 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198631|

10038161003816100381610038161003816100381610038161003816

1198631

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

1 444 75 169 1056 1393 119 143 785 65 1065 1284 11542 372 644 173 1039 1400 115 143 777 624 1055 1315 9903 263 457 174 1037 1401 115 147 792 619 1061 1357 7044 384 663 173 1050 1411 117 145 788 635 1073 1332 10205 384 664 173 1045 1405 117 145 782 630 1073 1332 10226 408 661 162 1047 1390 117 143 807 658 1004 1231 628Avg 376 640 171 1046 1400 117 144 789 636 1055 1309 920

Table 3 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198632|

10038161003816100381610038161003816100381610038161003816

1198632

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

1 444 459 103 1056 1393 045 068 610 407 247 371 1582 372 380 102 1039 1400 045 068 625 417 245 367 1573 263 268 102 1037 1401 045 068 605 415 245 357 1574 384 394 103 1050 1411 045 068 618 411 247 371 1585 384 390 102 1045 1405 045 068 620 413 245 367 1576 408 433 106 1047 1390 045 068 584 390 254 382 163Avg 376 387 103 1046 1400 045 068 610 409 247 369 158

Δ120588

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(32)

where Δℎ1= 0 056 cm Δℎ

2= 0 067 cm Δℎ

1199041= Δℎ1199042

=

Δ119877 = 0 cm Δ1198721= 0 27 g and Δ119872

2= 0 30 g

Considering the calculations fromTables 2 and 3 we havethe values of the maximum absolute inaccuracies Δ120588

119904asymp

0 43 gsdotcmminus3 and Δ120588 asymp 0 18 gsdotcmminus3From here we find the values of the maximum relative

inaccuracies

Δ120588119904

120588119904

=043

170asymp 025

Δ120588

120588=018

103asymp 017 (33)

Thus obtained value of 120588 at 20∘C temperature is withinthe error for the density of the distilledwater which shows theaccuracy of our model

Then we find the percentage 1198751015840 of the dry substance inthe sediment and the percentages 11987510158401015840 and 119875 of the bounddispersionmedium in the sediment and of the free dispersionmedium accordingly We use formulae (20) (24) (25) and(26) and the data from Table 1 The results are shown inTable 4

Because of the approximations in the calculations thesum of the values of 1198751015840 11987510158401015840 and 119875 is around 100

9 Discussion

Our model of close packing of the particles from the dis-persion phase of the studied liquid dispersion offers a newmethod for finding the volumes and the masses of the sedi-mented substances in LD as well as the average Stokes radiiof the particles of these substances Moreover this study isdistinguishedwith simplicity and its application is very cheap

Our simple algebraicmethod offers an increased accuracyin determining the average densities and weights of the ulti-mate sediment and emergent of the dispersion medium andof the dispersion phase in the liquid dispersionThefinding ofthese densities is achieved without the use of pycnometers orother more complex methods This prevents the destructiveeffects in the LD and the vibrations practically have no effecton the measurements results unlike the methods existinguntil now

Our method of determining the coefficient of the quanti-ties of substances in the components of LD does not requirepouring out the liquid from the container in order tomeasurethemasses of the ultimate sediment and of the free dispersionmedium (water) The known method of pouring out anddrying is destructive Using the drying method on LD it isgenerally even impossible to determine the percentage of thesubstance amount in the emulsion The method we proposeovercomes this disadvantage

A great advantage and particularly valuable for onemethod is that it has numerical characteristicmdashscale for eval-uation of a given variable The index 119868 which we introduce in(23) is a dimensionless scale for characterization of the sedi-mentation stability 119878 naturally following from the law on themovement of the dispersion particle from the dispersionphase of LD It provides a very easy way to evaluate 119878 of

The Scientific World Journal 9

Table 4

Exp1 2 3 4 5 6

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

119872 (g) 5607 7397 5517 7433 5509 7439 5574 7495 5551 7458 5557 7383120588119904(gcm3) 169 173 172 174 172 163

120588 = 12058810158401015840 (gcm3) 103 102 102 102 102 106

1205881015840 (gcm3) 221 229 227 23 227 2071198751015840 () dry matter 2811 3197 2973 3296 2939 3174 2942 3284 2917 3256 2656 300011987510158401015840 () bound dispersion

medium in the sediment 1029 1169 1037 1153 1038 1121 1026 1143 1030 1149 1069 1207

119875 () free dispersion medium 6146 5620 5985 5535 5994 5675 6021 5561 6046 5589 6275 5790

one LD in straight andor reverse sedimentation through thedensities 120588

119904 120588119891 and 120588 which are the solution of the system of

linear equations of type (11)The index 119868 isin (0 1] provides the possibility not only to

evaluate and to predict the sedimentation stability but alsoto compare different LDs according to their sedimentationstability Moreover the tests of the different LDs must beperformed in constant temperature and external pressure

10 Conclusion

The method which we offer does not require special equip-ment The phase separation of LDs can be achieved in theconditions not only of gravitational (earthrsquos) but also of cen-trifugal field In the case of centrifugal field the time interval 119905for determining the ultimate sediment and emergent can bemuch smaller compared with time for treating the LD withthe acceleration of gravity 119892

The advantage of our physicomathematical model forfinding the Stokes radii in the dispersion phase is that it doesnot use complicatedmathematical analysis for structuring thesedimentation curve

The three variables 1198751015840 11987510158401015840 and 119875 characterizing in per-centage the respective quantity of substance in the ultimatesediment or emergent bound dispersion medium in the sed-iment or the emergent and the free dispersion medium areintroduced in the literature for the first time with this paper

And with the sedimentation stability index 119868 whichwe introduce it is possible to demonstratively and withoutparticular difficultiesmake qualitative analysis of suspensionsand emulsionsmdashproducts of the food medical cosmeticchemical and so forth industries Such products are forexample different suspensions and emulsions such as foodsand drinks medications and cosmetic products paints andvarnishes and cement and lime solutionsThis is particularlyand technically useful in the creation of products withpredetermined sedimentation stability

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The obtained results are made with the support of Fund ldquoSci-entific Studiesrdquo of the Bulgarian Ministry of Education andScience under the Contract T 12 This paper is publishedwith the financial support of the Project BG051PO00133-05-001 ldquoScience and Businessrdquo financed by the HumanResources Development Operational Programme of theEuropean Social Fund

References

[1] D Dakova D Christozov and M Beleva ldquoBarycentric methodof determining the physical parameters of a single-phase parti-cle in liquid disperse systemsrdquo Journal of Colloid and InterfaceScience vol 256 no 2 pp 477ndash479 2002

[2] R Chanamai and D J McClements ldquoDepletion flocculation ofbeverage emulsions by gum arabic andmodified starchrdquo Journalof Food Science vol 66 no 3 pp 457ndash463 2001

[3] G S Hodakov andU P Udkin SedimentationAnalysis of HighlyDispersed Systems (in Russian) Chemistry Moscow Russia1981

[4] B M Westwood and V N Kabadi ldquoA novel pycnometer fordensity measurements of liquids at elevated temperaturesrdquoTheJournal of Chemical Thermodynamics vol 35 no 12 pp 1965ndash1974 2003

[5] A Puttmer R Lucklum B Henning and P HauptmannldquoImproved ultrasonic density sensor with reduced diffractioninfluencerdquo Sensors and Actuators A Physical vol 67 no 1ndash3 pp8ndash12 1998

[6] E D Shchukin A V Percov and E A Amelina ColloidalChemistry pp 306ndash309 Yurait Moscow Russia 6th edition2012 (Russian)

[7] K Kolikov G Krastev Y Epitropov and D Hristozov ldquoAna-lytically determining of the absolute inaccuracy (error) ofindirectly measurable variable and dimensionless scale char-acterising the quality of the experimentrdquo Chemometrics andIntelligent Laboratory Systems vol 102 no 1 pp 15ndash19 2010

[8] K Kolikov G Krastev Y Epitropov and A Corlat ldquoAnalyticallydetermining of the relative inaccuracy (error) of indirectlymeasurable variable and dimensionless scale characterising thequality of the experimentrdquoCSJM vol 20 no 1 pp 314ndash331 2012

[9] M H Ly M Aguedo S Goudot et al ldquoInteractions betweenbacterial surfaces and milk proteins impact on food emulsionsstabilityrdquo Food Hydrocolloids vol 22 no 5 pp 742ndash751 2008

10 The Scientific World Journal

[10] J P F Macedo L L Fernandes F R Formiga et al ldquoMicro-emultocrit technique a valuable tool for determination ofcritical HLB value of emulsionsrdquo AAPS PharmSciTech vol 7no 1 pp E1ndashE7 2006

[11] M Corredig and M Alexander ldquoFood emulsions studied byDWS recent advancesrdquo Trends in Food Science and Technologyvol 19 no 2 pp 67ndash75 2008

[12] J M Quintana A Califano and N Zaritzky ldquoMicrostructureand stability of non-protein stabilized oil-in-water food emul-sionsmeasured by opticalmethodsrdquo Journal of Food Science vol67 no 3 pp 1130ndash1135 2002

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CatalystsJournal of

2 The Scientific World Journal

K

a

hf

hs

hrarrg

(a)

K

a

hf

hs

h

(b)

Figure 1 Vertical axial section of straight square prismatic container with separated components of the liquid dispersion (a) model of perfectclose packing of the particles from the dispersion phase (b) model of nonperfect close packing of the particles from the dispersion phase

the different levels of the volume of the tested LD as discussedby Westwood and Kabadi [4] measuring the mass of thesedimented particles and applying electricity or some radi-ation as discussed by Puttmer et al [5] The optical methodscan achieve a very high degree of sensitivity and automationbut they have a serious disadvantagemdashthe tested LD mustbe transparent for the applied length waves Therefore thetested sample is often dilutedThe dilution however changesthe properties of the LD This problem is avoided by usingthe liquid separated by centrifugation Moreover the opticaldevices (spectrophotometers) have expensive equipment

In this paper we introduce a model of close packing ofthe particles from the dispersion phase of the LD With thismodel we determine the volume of the sediment and theemergent of the LD

The empirical formula of Stokes for finding the radius ofthe equivalent spherical particles from the dispersion phaseis well known as discussed by Hodakov and Udkin [3] Weformulate a new much simpler approach for finding theaverage Stokes radii of the particles from the dispersion phaseWe also offer a simple algebraic method for determinationunder given temperature and external pressure of the averagevolumetric mass density of the LD componentsmdashof the max-imum sediment and the maximum emergent as well as thedispersion medium without using pycnometers as discussedby Westwood and Kabadi [4] or densitometers as discussedby Puttmer et al [5] With the determined densities we findthe componentsmasses and also the volumetric density of thedispersion phase of the tested LD

Based on the densities of the components we defined forthe first time also a dimensionless scalemdashnumeric character-istic for characterizing the quality therefore for predictingthe sedimentation stability of liquid dispersions

There is not a physical theory yet for the quantitativedetermination of the dispersion phase and the dispersionmedium of LD The most used method is drying which isa long process resulting also in the destruction of the testedsubstance

Through the mass of the LD the densities and the linearsizes of its components we determine the percentage ofthe pure substance in the dispersion phase as well as thepercentages of the bound and the free dispersion medium ofLD

All these results are obtained without complex methodsand devices We use straight prismatic or cylindrical trans-parent containers where we pour homogenized LD The fullcontainers are placed under the influence of gravitationalor centrifugal filed under constant temperature and externalpressure

We assume that the size of the container is large enoughso the wall-adjacent effect can be avoided

2 Model of Close Packing of the Particles fromthe Dispersion Phase of Liquid Dispersion

We shall call the distribution of the particles from the disper-sion phase assumed to be spherical and touching adjacentparticles after the complete separation phase a model of theclose packing of the dispersion phase

Figure 1 shows axial section of straight containers 119870which we assume have square foundations

We assume that at the initial moment in the containerhomogenized LD is poured and that it is placed in conditionsof static homogeneous gravitational or centrifugal filed underconstant temperature and external pressure In the case ofcentrifugal filed because of the relatively small sizes of 119870compared with those of a centrifuge we can assume that thecentrifugal filed in 119870 is practically homogeneous

After the phase separation of LDs sharp macroscopic flatborder is formed between the components of the dispersionphase with separated dispersion medium in 119870

In themodel of the close packing of the particles from thedispersion phase offered by us it is assumed that their shapeis slightly different from spherical particles which for mostof the LD is satisfactorily met as discussed by Hodakov and

The Scientific World Journal 3

Udkin [3] Moreover as a result of the homogenizing of theLD and the conditions in which the sedimentation takesplace we can also assume sequential particle distributionalong the horizontals in the maximum sediment and emer-gent

The concentration gradient of the dispersion particlesduring the formation of the sediment and the emergentalso significantly contributes to this distribution along thehorizontals (As end result this gradient becomes 0)

After the ultimate separation a part of the dispersionmedium is located among the particles from the dispersionphase of the sediment and the emergent This dispersionmedium (in most cases water) is called bound (nonfree) Thedispersion medium separated outside the sediment and theemergent is called unbound (free) We shall emphasize thatunlike the free dispersion medium the bound dispersionmedium cannot be separated even during centrifugationbecause between the particles of themediumand the particlesof the dispersion phase there are huge quantum-mechanical(close) interactionsmdashmuch larger than any centrifugal

It should be taken into account that by approaching thesurfaces of two particles the forces of attraction between themgrow relatively slowmdashfor dispersion interactions by the law11205913 where 120591 is the distance between the particles Contrari-

wise the forces of repulsion are practically insignificant toenough short distance but by further reduction of 120591 theseforces are increasing rapidly by the low 1120591

119899 where 119899 asymp 10As a result by approaching the surfaces of the particles toequilibrium state of a close packing the majority of the workis for the forces of attraction and the depth of the minimumof the potential curve is close in absolute value to the workof the forces of attraction that is molecular component ofthe energy Furthermore the energy and the interaction forcebetween the particles are determined not only by the distancebetween the particles and the value of the complex constant ofHamaker but also by the size and the shape of the interactingparticles discussed by Shchukin et al [6]

Let the square of the basis 119870 have an internal side length119886 and let the height of the LD in119870 be ℎWith ℎ we designatethe height of the ultimate sediment and with ℎ

119891we designate

the height of the ultimate emergent (Figures 1(a) and 1(b))We shall find the number of the particles from the

dispersion phase We assume that an ultimate (maximal)sediment (emergent) in close packing along the length 119886 canbe applied 119894 number of dispersion particles and each of theseparticles has an average Stokes radius 119903 = 119903

119904(119903 = 119903

119891) We

also assume that the height ℎ = ℎ119904(ℎ = ℎ

119891) can be applied 119895

number of such particles

(a) Considering the model of perfect close packing of theparticles from the dispersion phase due to the largevalues of 119894 and 119895 we can ignore the difference in oneparticle through row or column

Then the number of the particles in the sediment (emer-gent) is 119899 = 119894

2119895 Moreover because of simple geometrical

considerations it follows that ℎ = (119895 minus 1)radic3 + 2119903 = 119895radic3119903 +

(2 minus radic3)119903

Due to the large values of numbers 119895 and 119903 ≪ 119895 we canassume that ℎ = 119895radic3119903 that is

119886 = 2119894119903 ℎ = 119895radic3119903 (1)

(b) Considering themodel of nonperfect close packing ofthe particles from the dispersion phase

119886 = 2119894119903 ℎ = 2119895119903 (2)

Therefore we assume the average values of 119886 and ℎ that is

119886 = 2119894119903 ℎ = (1 +radic3

2) 119895119903 (3)

3 Volume of the Sediment andthe Emergent in Liquid Dispersion

According to formula (3) for the volume 119881 = 119881119904(119881 = 119881

119891)

of the full ultimate sediment (emergent) in 119870 we have 119881 =

1198862ℎ = 2119894

2119895 (2 + radic3)119903

3 where 119903 is the radius of the Stokesparticles of the dispersion phase

On the other hand119881 = 1198811015840+11988110158401015840 where1198811015840 is the volume of

the dispersion phase in this sediment (emergent) and11988110158401015840 is thevolume of the bound dispersion medium of the sediment(emergent) The following equation is valid

1198811015840=4

3120587 11990331198942119895 (4)

Therefore 11988110158401015840 = 119881 minus 1198811015840 = 21198942119895(2 + radic3 minus (23)120587)1199033 Fromthe obtained expressions for1198811198811015840 and11988110158401015840 we determine that

1198811015840

119881=

2120587

3 (2 + radic3)

asymp 0 56

11988110158401015840

119881= 1 minus

2120587

3 (2 + radic3)

asymp 0 44

(5)

Then

1198811015840asymp 0 56119886

2ℎ 119881

10158401015840asymp 0 44119886

2ℎ (6)

From the equations in (6) it follows that 1198811015840 asymp 1 311988110158401015840Let 1205881015840 = 120588

1015840

119904(1205881015840= 1205881015840

119891) be the average volumetric mass

density of the particles from the dispersion phase in the ulti-mate sediment (emergent) In addition let 12058810158401015840 be the averagedensity of the dispersion medium (bound or free) in theLD Then according to formulae (6) for the masses 1198981015840 and11989810158401015840 respectively in the dispersion phase and the bound dis-

persion medium in the ultimate sedimentemergent in119870 wehave

1198981015840= 11988110158401205881015840asymp 0 56119886

2ℎ1205881015840 119898

10158401015840= 1198811015840101584012058810158401015840asymp 0 44119886

2ℎ12058810158401015840

(7)

where ℎ = ℎ119904(ℎ = ℎ

119891)

4 The Scientific World Journal

In (7) if 1205881015840 = 1205881015840

119904 then 1205881015840 gt 120588

10158401015840 and if 1205881015840 = 1205881015840

119891 then

1205881015840lt 12058810158401015840

In our studies we obtain approximated formulas in whichthe practically negligible distance between the particles isignored that is we assume that the particles of the dispersionphase by ultimate sedimentation are closely fitted to eachotherThis does not affect the calculations because we do notconsider only the case when the particles are perfectlyarranged closely to each other

4 Determination of the Average StokesRadii of the Particles in the Sedimentand in the Emergent

Using (3) we can find the numbers 119894 = 119894119904(119894 = 119894

119891) or 119895 =

119895119904(119895 = 119895

119891) and hence the average Stokes radius of the parti-

cles from the sediment (emergent) To this end we count thedispersion particles symmetrically around the center of themasses of the ultimate sediment (emergent) along the verti-cal in a chosen linear unit of distance (This is done becauseeven in monodisperse system the dispersion particles are notof absolutely the same size) Then the obtained number ismultiplied by the number of single distances contained in ℎThis is how we find 119895 and then according to (3)

119894 =

119886 (1 + (radic32))

2ℎ119895 (8)

Hence from (3) for the average Stokes radii of theparticles from the sediment or the emergent we have

119903 =ℎ

(1 + (radic32)) 119895

=119886

2119894 (9)

If the LD in 119870 is polydisperse the obtained radii 119903 of theparticles from the sediment or the emergent are determinedin average values

With this we offer a new method different from Stokesrsquofor determination of the average Stokes radii in dispersionparticles in LD The determination of the 119899 = 119894

2119895 of the dis-

persion particles from the ultimate sediment or emergent inLD in 119870 provides important information also for the foodmedical chemical and so forth industries

5 Algebraic Method for Determiningthe Average Densities and the Masses ofthe Components of a Liquid Dispersion

The densities of the ultimate sediment and emergent as wellas the free dispersionmedium can be definedwith the widelyused pycnometers as discussed by Westwood and Kabadi [4]But to this end these three components of the LD must bemechanically separated This requires the ultimate emergentto be removed from the container first and then pouringout the free dispersion medium and finally there is access tothe ultimate sediment After that the three substances sam-ples are taken which are placed in the pycnometer Thepycnometer method for measuring the densities is easy to

K1 K2 K3

m1

R R R

h1

m2h2

hf3

hf2

hf1

m3 h3

hs1ms1 ms2 ms3hs2 hs3

mf2

mf3

mf1

Figure 2 Vertical axial section of three equal straight cylindricalcontainers with separated components of liquid dispersions

apply and inexpensive but the measurement takes a lot oftime

The measurement can be also done in a noncontact waybymeans of ultrasound densitometers as discussed by Puttmeret al [5]These devices however use complex equipment andare expensive

We offer an algebraic method with significantly simplifiedmeasurements It employs containersmdashprismatic or cylindri-cal At the beginning themasses of the LDmust bemeasuredwhich need to be equal but taken in different proportionsfrom the dispersion phase The LD is placed in the respectivecontainers under constant temperature and external pressureAfter the formation of the ultimate sediment and emergentand the free dispersion medium between them in theconditions of gravitational or centrifugal field we performelementary linear measurements of the geometrical sizes ofthe separated particles in the containers with LD (Figure 2)The densities and the masses of the components are deter-mined with simple algebraic operations

As it is not necessary to count the particles in the ultimatesediment and emergent in this case we can use straightcylindrical containers Let119870

119894(119894 = 1 2 3) be such containers

which for convenience we assume are same with radii of themain planes of 119903 as they are presented in the vertical axialsections in Figure 2The three containers contain LDwith thesame components of the dispersion medium and the disper-sion phase taken in different proportions (These proportionscan be achieved for example by minimal dilution withdispersion medium of the given LD that is by increasingthe relative share of the dispersion medium in the differentcontainers)

Let us assume that at the initial moment 119905 = 0 the testedLD is in homogenized condition that is the dispersion phaseof this dispersion is evenly distributed in the whole volume ofthe containers Moreover the containers are placed underspecific constant temperature and external pressure

In the conditions of gravitational or centrifugal fieldcomes a moment of time 119905 gt 0 when in LD of119870

119894(119894 = 1 2 3)

appears a condition of ultimate sediment and emergent withconstant heights of their volumes Thus within time 119905 com-plete phase separation is achieved that is the volumes and themasses of the formed ultimate sediment and ultimate emer-gent in 119870

119894(Figure 2) no longer changemdashthe heavy particles

have settled and the light particles have emerged

The Scientific World Journal 5

Some cases of LD are possible when free dispersionmedium and only ultimate sediment or only ultimate emer-gent are formed Some more complicated cases are alsopossible when more than one sediment andor emergent areformed

The values of the average density of the substance in theultimate sediment 120588

119904of the average density of the substance in

the ultimate emergent 120588119891and the density of the dispersion

medium 120588 = 12058810158401015840 in the containers119870

119894(119894 = 1 2 3) are the same

Characteristics of the ultimate sediment and emergentare the lights of their heights ℎ

119904119894and ℎ

119891119894(119894 = 1 2 3) in 119870

119894

which are constant after certain moment of time 119905 Withℎ119894(119894 = 1 2 3) we designate the heights of the free dispersion

medium and with 119898119904119894 119898119891119894 119898119894(119894 = 1 2 3) we designate the

masses of the ultimate sediment and emergent and the freedispersion medium of LD in 119870

119894from Figure 2 Then we can

write

119898119904119894= 1205871198772ℎ119904119894120588119904 119898119891119894= 1205871198772ℎ119891119894120588119891 119898119894= 1205871198772ℎ119894120588

(119894 = 1 2 3)

(10)

where 119877 is the radius of the internal bases of the containersWith 119872

119894(119894 = 1 2 3) we designate the total masses of LD

which fill up 119870119894 We consider these masses to be known

(measured) According to the law on conservation of masseswe have the following system of three linear equations

119898119904119894+ 119898119891119894+ 1198980119894= 119872119894 119894 = 1 2 3 (11)

If 119867119894= ℎ119904119894+ ℎ119894+ ℎ119891119894 and 120588

119894is the average volumetric

mass density of LD in119881119894 then it is known that 119896

119894= 1198721198941205871198772=

119867119894120588119894(119894 = 1 2 3) Then from (10) and (11) the system follows

ℎ119904119894120588119904+ ℎ119891119894120588119891+ ℎ119894120588 = 119896119894 119894 = 1 2 3 (12)

By solving (12) the three densities are determinedmdashof theultimate sediment of the ultimate emergent and of thedispersion medium

120588119904=1198631

119863 120588

119891=1198632

119863 120588 =

1198633

119863 (13)

where11986311986311198632 and 119863

3 according to the Cramers rule are

determinants of third orderSince the proportions of the components in LD are differ-

ent then the equations in the system (11) are not equivalentand the system (12) is consistent But if we pour the sameLD even in three different in size and shape straight con-tainers then the equations in (11) will be proportional Thenthe system (12) will be inconsistent

From (10) and (13) we can determine themasses119898119904119894119898119891119894

119898119894 as well as the mass 119898

119901119894= 119898119904119894+ 119898119891119894

of the dispersionphase for any 119894 Then we also find the density 120588

119901of the

dispersion phase in the tested LD

120588119901=

ℎ119904119894120588119904+ ℎ119891119894120588119891

ℎ119904119894+ ℎ119891119894

(14)

If we need to determine only the density 120588119901of the

dispersion phase and the density 120588 of the dispersionmedium

two containers 1198701and 119870

2are enough and a system with two

linear equations

119898119901119894+ 1198981= 1198721lArrrArr (ℎ

119904119894+ ℎ119891119894) 120588119901+ ℎ119894120588 = 119896119894 119894 = 1 2

(15)

Thus

120588119901=1198631

119863 120588 =

1198632

119863 (16)

where 119863 1198631 and 119863

2 according to the Cramers rule are

determinants of second orderIf we know the density of one of the ingredients for

example 120588 for distilledwater then also two containers1198701and

1198702are enough Then 120588

119904and 120588

119891are also determined with a

system with two linear equations

119898119904119894+ 119898119891119894+ 119898119894= 119872119894lArrrArr ℎ

119904119894120588119904+ ℎ119891119894120588119891= 119896119894minus ℎ119894120588

119894 = 1 2

(17)

which is solved analogically to system (12)In case when the density 120588 of the dispersion medium is

known andwe have only ultimate sediment or ultimate emer-gent it is sufficient to use only one container 119870 = 119870

1 Then

we have only one linear equation

119898119904+ 119898 = 119872 or 119898

119891+ 119898 = 119872 (18)

that is

120588119901=119896 minus ℎ120588

ℎ119904

or 120588119901=119896 minus ℎ120588

ℎ119891

(19)

where 120588119901= 120588119904or 120588119901= 120588119891and 119896 = 119872120587119877

2Because119898

119904= 1198981015840+ 11989810158401015840(119898119891= 1198981015840+ 11989810158401015840) where1198981015840 is the

mass of the dispersion phase and11989810158401015840 is the bound dispersionmedium in the ultimate sediment (emergent) and accordingto formula (19) we can find the density 1205881015840 of the dispersionphase

1205881015840=

120588119901minus 0 44120588

0 56 (20)

In this case when the tested LD has sediments and emer-gents consisting of several components then the density ofeach component can be determined through a measurementwith more containers In the general case we use a systemconsisting of 119899 linear equations and 119899 unknowns which ispresent in the following way

119901

sum

119896=0

119898119904119896 119894+

119897

sum

119895=0

119898119891119895 119894+ 119898119894= 119872119894 119894 = 1 2 119899 (21)

where 119899 = 119901+119897+1 From (21) the unknown densities 120588119904119896 119894 120588119891119895 119894

and 120588 are determined

From (13) (14) (16) and (19) using the method shownin Kolikov et al [7] we can determined the maximumabsolute inaccuracies (errors) Δ120588

119904 Δ120588119891 Δ120588119901 and Δ120588 and the

maximum relative inaccuracies (errors) Δ120588119904120588119904 Δ120588119891120588119891

Δ120588119901120588119901 and Δ120588120588 discussed by Kolikov et al [8]

6 The Scientific World Journal

6 Characterization Scale forSedimentation Stability

There are different methods for evaluation of the sedimenta-tion stability 119878 of LD One of the most popular methods is thevisual methodThe visual methodmeasures the observed dis-placement over a period of time of the horizontal separationline between the clear and the muddy part of the tested LD asdiscussed elsewhere [9 10]Themethod employs transparentcontainer and a clock

From the instrumental methods most popular ones indetermining the sedimentation stability are the opticalabsorptionmethodsmdashthroughmeasuring the intensity of thelight passing through the LD as discussed elsewhere [11 12]

According to our method the evaluation of 119878 is donethrough the densities of the components of the LD and it isnot necessary to measure the time and optical transmissionof LD is no required Moreover we introduce scale forcharacterization of the sedimentation stability 119878

Let one particle of the dispersion phase with volume 119881and density 120588

119901be located in dispersion medium with density

120588 in the conditions of gravitational field Then the value ofthe resultant force of its weight and the Archimedean force is

119865 = 119881119892 (120588119901minus 120588) (22)

where 119892 is gravitational constant (In (22) we neglect the fric-tion force which is very little for small velocities of theparticle)

Therefore if the particle is heavy that is 120588119901gt 120588 hArr

120588120588119901lt 1 then119865 gt 0 andwe have straight sedimentation (set-

tling) And if the particle is light that is 120588119901lt 120588 hArr 120588

119901120588 lt

1 then 119865 lt 0 and we have reverse sedimentation (emer-gence)

Thus we suggest the evaluation of the sedimentationstability 119878 to be done through the following dimensionlessphysical variables 119868

119904= 120588120588

119904 119868119891= 120588119891120588 (120588119904 120588 = 0) and hence

through the variable

119868 = max (119868119904 119868119891) = max(

120588

120588119904

120588119891

120588) (23)

The number 119868 is the sedimentation stability index in adimensionless scale for characterization of 119878 Since 0 lt 119868

119904

119868119891le 1 then 0 lt 119868 le 1 that is the scale is the interval (0 1]

The more the index 119868 of an LD is closer to 1 the more stablethe LD is and the closer the index 119868 is to 0 the more unstablethe LD is Thus if 119868 = 1 the LD is absolutely sedimentarystable that is there is no phase separation

We shall notice that if the sediment is absent in the LDthen 119868 = 119868

119891isin (0 1] and if the emergent is absent then 119868 =

119868119904isin (0 1]

7 Determining the Percentage ofthe Substance in the Sediment andthe Emergent

The amount of the substance in the dispersion phase is amajor factor which determines the quality of many products

The drying is widely used method for testing the dry matterin different suspensions To this end a sample is taken fromthe homogenized LD which is placed in a suitable containerweighted with a scale and placed in a drier for drying untilconstant weight is obtained This weight is divided into theinitial (not dried) full weights of the sample and thenmultiplied by 100 The obtained value is called dry substancepercentage of LDThe disadvantages of thismethod are its rel-atively long duration and the spending of significant thermalenergy for drying a unit of weight from the tested substanceMoreover the heating usually to 105∘C required by thestandard results in destruction of the treated substance and itcan only be used for suspensions

We offer a different method which does not have theseflaws

If 119870 is a straight cylindrical container filled-up with LDwith weight 119872 and internal radius of the bases 119877 thenaccording to formula (7) the masses of the dispersion phaseand the bound dispersion medium in the ultimate sedimentor emergent are respectively 1198981015840 = 0 561205871198772ℎ1205881015840 and 11989810158401015840 = 0441205871198772ℎ12058810158401015840 Then knowing the geometrical parameters of the

used container as well as the heights masses and densitieswe have in percentage the amount of the substance 1198751015840 in theultimate sediment or emergent and of the bound dispersionmedium 119875

10158401015840 respectively

1198751015840= (

1198981015840

119872sdot 100) = (

0 561205871198772ℎ1205881015840

119872sdot 100) (24)

11987510158401015840= (

11989810158401015840

119872sdot 100) = (

0 441205871198772ℎ12058810158401015840

119872sdot 100) (25)

The percentage 119875 of the free dispersion medium can alsobe calculated with our method namely

119875 = (

1205871198772(119867 minus ℎ

119904minus ℎ119891) 12058810158401015840

119872sdot 100) (26)

Let us denote that unlike the method of drying until con-stant weight through formula (24) the percentage 1198751015840 of thetested substance is determined with the participation of thebound dispersion medium We take into account only theamount of the pure dispersion phase in the specific sedimentor emergentThrough drying complete dryness of the samplecannot be achieved in practicemdashin the sediment therealways remains some percentage bound dispersion mediumTherefore we cannot expect to have coinciding percentage ofdry substance determined according to the two methods

Let us consider a case of sediment If 1198751015840119904is the percentage

of dry substance in the sediment determined with the dryingmethod then for practical reasons we can write downmodification of formula (24) namely

1198751015840

119904= 1205761198751015840

119904asymp (120576

0 561205871198772ℎ1199041205881015840

119872sdot 100) (27)

Here the variable 120576 is a conversion factor which canchange within the limits 1 lt 120576 lt (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904 where

The Scientific World Journal 7

Table 1 Linear dimensions masses and densities of the components of liquid dispersions

Exp 119877 [cm] ℎ1199041

[cm] ℎ1[cm] ℎ

1199042[cm] ℎ

2[cm] 119872

1[g] 119872

2[g] 120588

119904[gsdotcmminus3] 120588 [gsdotcmminus3]

1 13 24 63 36 76 5607 7397 169 1032 13 24 61 36 76 5517 7433 173 1023 13 24 61 35 78 5509 7439 172 1024 13 24 62 36 77 5574 7495 174 1025 13 24 62 36 77 5551 7458 171 1026 13 24 62 36 76 5557 7383 163 106Avg 13 24 62 36 77 5553 7434 170 103

K1 K2

m1

R R

h1

m2h2

hs1ms1

ms2 hs2

Figure 3 Vertical axial sections of two identical straight cylindricalcontainers with ultimate sedimentation

1198981015840

119904+ 11989810158401015840

119904= 119881

1015840

1199041205881015840

119904+ 11988110158401015840

11990412058810158401015840 is the total weight of the

sediment (dispersion phase and bound dispersion medium)The unobtainable value 120576 = 1 corresponds to the drysubstance in the ultimate sediment in case of complete dryingof the sample in119870 that is without bounddispersionmediumas with formula (24) we offer If the ultimate sediment is notsubjected to drying then the value 120576 = (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904asymp

1 + (120588101584010158401205881015840

119904) lt 2 Thus for the conversion factor in formula

(27) we have 1 lt 120576 lt 2

8 Example

We present models of two dispersion system suspensionsfrom calcium carbonate powder respectively 700 g and1100 g measured with electronic scale with accuracy of 001 gand distilled water respectively 50mL and 65mL measuredwith graduated cylinders with accuracy of 05mL In order toevaluate the accuracy of our method we assume the densitiesof CaCO

3and the distilled H

2O to be unknown

In order to determine the densities respectively 120588119904of the

ultimate sediment and 120588 of the dispersion medium we usetwo identical cylindrical containers 119870

1and 119870

2 with radius

of the internal basis 119877 = 0 013m (Figure 3)In the two containers we pour the two homogenized sus-

pensions (their components are in different proportions)Weperform 6 observations with room temperature 20∘C

Table 1 presents the values of the measured variables119877 ℎ1199041 ℎ1 1198721(for container 119870

1) and 119877 ℎ

1199042 ℎ2 1198722(for

container1198702) during the 6 observations

For densities 120588119904and 120588 according to formula (15) we find

with the system of linear equations

10038161003816100381610038161003816100381610038161003816

1198981199041+ 1198981= 1198721

1198981199042+ 1198982= 1198722

lArrrArr

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1199041120588119904+ ℎ1120588 =

1198721

1205871198772

ℎ1199042120588119904+ ℎ2120588 =

1198722

1205871198772

(28)

Its solutions are

120588119904=1198631

119863=

1

1205871198772

1198721ℎ2minus ℎ11198722

ℎ1199041ℎ2minus ℎ1ℎ1199042

120588 =1198632

119863=

1

1205871198772

ℎ11990411198722minus1198721ℎ1199042

ℎ1199041ℎ2minus ℎ1ℎ1199042

(29)

Then for the densities of the ultimate sediment and thedispersion medium we have respectively the average values120588119904asymp 1 70 gsdotcmminus3 and 120588 asymp 1 03 gsdotcmminus3 which are also

presented in Table 1For the index of the sedimentation stability we find

119868 =120588

120588119904

asymp 0 61 (30)

The value of 119868 according to our scale (01] shows that thetested LD has a little over the average sedimentation stability

According to the model discussed by Kolikov et al [7]we shall determine the maximum absolute inaccuracies Δ120588

119904

and Δ120588 for the established densities 120588119904and 120588 as well as the

maximum relative inaccuraciesΔ120588119904120588119904andΔ120588120588 IfΔ119872

119894Δℎ119894

Δℎ119904119894(119894 = 1 2) andΔ119877 are absolute inaccuracies of119872

119894 ℎ119894 ℎ119904119894

and 119877 then from (29) it follows that

Δ120588119904

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(31)

8 The Scientific World Journal

Table 2 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198631|

10038161003816100381610038161003816100381610038161003816

1198631

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

1 444 75 169 1056 1393 119 143 785 65 1065 1284 11542 372 644 173 1039 1400 115 143 777 624 1055 1315 9903 263 457 174 1037 1401 115 147 792 619 1061 1357 7044 384 663 173 1050 1411 117 145 788 635 1073 1332 10205 384 664 173 1045 1405 117 145 782 630 1073 1332 10226 408 661 162 1047 1390 117 143 807 658 1004 1231 628Avg 376 640 171 1046 1400 117 144 789 636 1055 1309 920

Table 3 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198632|

10038161003816100381610038161003816100381610038161003816

1198632

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

1 444 459 103 1056 1393 045 068 610 407 247 371 1582 372 380 102 1039 1400 045 068 625 417 245 367 1573 263 268 102 1037 1401 045 068 605 415 245 357 1574 384 394 103 1050 1411 045 068 618 411 247 371 1585 384 390 102 1045 1405 045 068 620 413 245 367 1576 408 433 106 1047 1390 045 068 584 390 254 382 163Avg 376 387 103 1046 1400 045 068 610 409 247 369 158

Δ120588

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(32)

where Δℎ1= 0 056 cm Δℎ

2= 0 067 cm Δℎ

1199041= Δℎ1199042

=

Δ119877 = 0 cm Δ1198721= 0 27 g and Δ119872

2= 0 30 g

Considering the calculations fromTables 2 and 3 we havethe values of the maximum absolute inaccuracies Δ120588

119904asymp

0 43 gsdotcmminus3 and Δ120588 asymp 0 18 gsdotcmminus3From here we find the values of the maximum relative

inaccuracies

Δ120588119904

120588119904

=043

170asymp 025

Δ120588

120588=018

103asymp 017 (33)

Thus obtained value of 120588 at 20∘C temperature is withinthe error for the density of the distilledwater which shows theaccuracy of our model

Then we find the percentage 1198751015840 of the dry substance inthe sediment and the percentages 11987510158401015840 and 119875 of the bounddispersionmedium in the sediment and of the free dispersionmedium accordingly We use formulae (20) (24) (25) and(26) and the data from Table 1 The results are shown inTable 4

Because of the approximations in the calculations thesum of the values of 1198751015840 11987510158401015840 and 119875 is around 100

9 Discussion

Our model of close packing of the particles from the dis-persion phase of the studied liquid dispersion offers a newmethod for finding the volumes and the masses of the sedi-mented substances in LD as well as the average Stokes radiiof the particles of these substances Moreover this study isdistinguishedwith simplicity and its application is very cheap

Our simple algebraicmethod offers an increased accuracyin determining the average densities and weights of the ulti-mate sediment and emergent of the dispersion medium andof the dispersion phase in the liquid dispersionThefinding ofthese densities is achieved without the use of pycnometers orother more complex methods This prevents the destructiveeffects in the LD and the vibrations practically have no effecton the measurements results unlike the methods existinguntil now

Our method of determining the coefficient of the quanti-ties of substances in the components of LD does not requirepouring out the liquid from the container in order tomeasurethemasses of the ultimate sediment and of the free dispersionmedium (water) The known method of pouring out anddrying is destructive Using the drying method on LD it isgenerally even impossible to determine the percentage of thesubstance amount in the emulsion The method we proposeovercomes this disadvantage

A great advantage and particularly valuable for onemethod is that it has numerical characteristicmdashscale for eval-uation of a given variable The index 119868 which we introduce in(23) is a dimensionless scale for characterization of the sedi-mentation stability 119878 naturally following from the law on themovement of the dispersion particle from the dispersionphase of LD It provides a very easy way to evaluate 119878 of

The Scientific World Journal 9

Table 4

Exp1 2 3 4 5 6

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

119872 (g) 5607 7397 5517 7433 5509 7439 5574 7495 5551 7458 5557 7383120588119904(gcm3) 169 173 172 174 172 163

120588 = 12058810158401015840 (gcm3) 103 102 102 102 102 106

1205881015840 (gcm3) 221 229 227 23 227 2071198751015840 () dry matter 2811 3197 2973 3296 2939 3174 2942 3284 2917 3256 2656 300011987510158401015840 () bound dispersion

medium in the sediment 1029 1169 1037 1153 1038 1121 1026 1143 1030 1149 1069 1207

119875 () free dispersion medium 6146 5620 5985 5535 5994 5675 6021 5561 6046 5589 6275 5790

one LD in straight andor reverse sedimentation through thedensities 120588

119904 120588119891 and 120588 which are the solution of the system of

linear equations of type (11)The index 119868 isin (0 1] provides the possibility not only to

evaluate and to predict the sedimentation stability but alsoto compare different LDs according to their sedimentationstability Moreover the tests of the different LDs must beperformed in constant temperature and external pressure

10 Conclusion

The method which we offer does not require special equip-ment The phase separation of LDs can be achieved in theconditions not only of gravitational (earthrsquos) but also of cen-trifugal field In the case of centrifugal field the time interval 119905for determining the ultimate sediment and emergent can bemuch smaller compared with time for treating the LD withthe acceleration of gravity 119892

The advantage of our physicomathematical model forfinding the Stokes radii in the dispersion phase is that it doesnot use complicatedmathematical analysis for structuring thesedimentation curve

The three variables 1198751015840 11987510158401015840 and 119875 characterizing in per-centage the respective quantity of substance in the ultimatesediment or emergent bound dispersion medium in the sed-iment or the emergent and the free dispersion medium areintroduced in the literature for the first time with this paper

And with the sedimentation stability index 119868 whichwe introduce it is possible to demonstratively and withoutparticular difficultiesmake qualitative analysis of suspensionsand emulsionsmdashproducts of the food medical cosmeticchemical and so forth industries Such products are forexample different suspensions and emulsions such as foodsand drinks medications and cosmetic products paints andvarnishes and cement and lime solutionsThis is particularlyand technically useful in the creation of products withpredetermined sedimentation stability

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The obtained results are made with the support of Fund ldquoSci-entific Studiesrdquo of the Bulgarian Ministry of Education andScience under the Contract T 12 This paper is publishedwith the financial support of the Project BG051PO00133-05-001 ldquoScience and Businessrdquo financed by the HumanResources Development Operational Programme of theEuropean Social Fund

References

[1] D Dakova D Christozov and M Beleva ldquoBarycentric methodof determining the physical parameters of a single-phase parti-cle in liquid disperse systemsrdquo Journal of Colloid and InterfaceScience vol 256 no 2 pp 477ndash479 2002

[2] R Chanamai and D J McClements ldquoDepletion flocculation ofbeverage emulsions by gum arabic andmodified starchrdquo Journalof Food Science vol 66 no 3 pp 457ndash463 2001

[3] G S Hodakov andU P Udkin SedimentationAnalysis of HighlyDispersed Systems (in Russian) Chemistry Moscow Russia1981

[4] B M Westwood and V N Kabadi ldquoA novel pycnometer fordensity measurements of liquids at elevated temperaturesrdquoTheJournal of Chemical Thermodynamics vol 35 no 12 pp 1965ndash1974 2003

[5] A Puttmer R Lucklum B Henning and P HauptmannldquoImproved ultrasonic density sensor with reduced diffractioninfluencerdquo Sensors and Actuators A Physical vol 67 no 1ndash3 pp8ndash12 1998

[6] E D Shchukin A V Percov and E A Amelina ColloidalChemistry pp 306ndash309 Yurait Moscow Russia 6th edition2012 (Russian)

[7] K Kolikov G Krastev Y Epitropov and D Hristozov ldquoAna-lytically determining of the absolute inaccuracy (error) ofindirectly measurable variable and dimensionless scale char-acterising the quality of the experimentrdquo Chemometrics andIntelligent Laboratory Systems vol 102 no 1 pp 15ndash19 2010

[8] K Kolikov G Krastev Y Epitropov and A Corlat ldquoAnalyticallydetermining of the relative inaccuracy (error) of indirectlymeasurable variable and dimensionless scale characterising thequality of the experimentrdquoCSJM vol 20 no 1 pp 314ndash331 2012

[9] M H Ly M Aguedo S Goudot et al ldquoInteractions betweenbacterial surfaces and milk proteins impact on food emulsionsstabilityrdquo Food Hydrocolloids vol 22 no 5 pp 742ndash751 2008

10 The Scientific World Journal

[10] J P F Macedo L L Fernandes F R Formiga et al ldquoMicro-emultocrit technique a valuable tool for determination ofcritical HLB value of emulsionsrdquo AAPS PharmSciTech vol 7no 1 pp E1ndashE7 2006

[11] M Corredig and M Alexander ldquoFood emulsions studied byDWS recent advancesrdquo Trends in Food Science and Technologyvol 19 no 2 pp 67ndash75 2008

[12] J M Quintana A Califano and N Zaritzky ldquoMicrostructureand stability of non-protein stabilized oil-in-water food emul-sionsmeasured by opticalmethodsrdquo Journal of Food Science vol67 no 3 pp 1130ndash1135 2002

Submit your manuscripts athttpwwwhindawicom

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

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The Scientific World Journal 3

Udkin [3] Moreover as a result of the homogenizing of theLD and the conditions in which the sedimentation takesplace we can also assume sequential particle distributionalong the horizontals in the maximum sediment and emer-gent

The concentration gradient of the dispersion particlesduring the formation of the sediment and the emergentalso significantly contributes to this distribution along thehorizontals (As end result this gradient becomes 0)

After the ultimate separation a part of the dispersionmedium is located among the particles from the dispersionphase of the sediment and the emergent This dispersionmedium (in most cases water) is called bound (nonfree) Thedispersion medium separated outside the sediment and theemergent is called unbound (free) We shall emphasize thatunlike the free dispersion medium the bound dispersionmedium cannot be separated even during centrifugationbecause between the particles of themediumand the particlesof the dispersion phase there are huge quantum-mechanical(close) interactionsmdashmuch larger than any centrifugal

It should be taken into account that by approaching thesurfaces of two particles the forces of attraction between themgrow relatively slowmdashfor dispersion interactions by the law11205913 where 120591 is the distance between the particles Contrari-

wise the forces of repulsion are practically insignificant toenough short distance but by further reduction of 120591 theseforces are increasing rapidly by the low 1120591

119899 where 119899 asymp 10As a result by approaching the surfaces of the particles toequilibrium state of a close packing the majority of the workis for the forces of attraction and the depth of the minimumof the potential curve is close in absolute value to the workof the forces of attraction that is molecular component ofthe energy Furthermore the energy and the interaction forcebetween the particles are determined not only by the distancebetween the particles and the value of the complex constant ofHamaker but also by the size and the shape of the interactingparticles discussed by Shchukin et al [6]

Let the square of the basis 119870 have an internal side length119886 and let the height of the LD in119870 be ℎWith ℎ we designatethe height of the ultimate sediment and with ℎ

119891we designate

the height of the ultimate emergent (Figures 1(a) and 1(b))We shall find the number of the particles from the

dispersion phase We assume that an ultimate (maximal)sediment (emergent) in close packing along the length 119886 canbe applied 119894 number of dispersion particles and each of theseparticles has an average Stokes radius 119903 = 119903

119904(119903 = 119903

119891) We

also assume that the height ℎ = ℎ119904(ℎ = ℎ

119891) can be applied 119895

number of such particles

(a) Considering the model of perfect close packing of theparticles from the dispersion phase due to the largevalues of 119894 and 119895 we can ignore the difference in oneparticle through row or column

Then the number of the particles in the sediment (emer-gent) is 119899 = 119894

2119895 Moreover because of simple geometrical

considerations it follows that ℎ = (119895 minus 1)radic3 + 2119903 = 119895radic3119903 +

(2 minus radic3)119903

Due to the large values of numbers 119895 and 119903 ≪ 119895 we canassume that ℎ = 119895radic3119903 that is

119886 = 2119894119903 ℎ = 119895radic3119903 (1)

(b) Considering themodel of nonperfect close packing ofthe particles from the dispersion phase

119886 = 2119894119903 ℎ = 2119895119903 (2)

Therefore we assume the average values of 119886 and ℎ that is

119886 = 2119894119903 ℎ = (1 +radic3

2) 119895119903 (3)

3 Volume of the Sediment andthe Emergent in Liquid Dispersion

According to formula (3) for the volume 119881 = 119881119904(119881 = 119881

119891)

of the full ultimate sediment (emergent) in 119870 we have 119881 =

1198862ℎ = 2119894

2119895 (2 + radic3)119903

3 where 119903 is the radius of the Stokesparticles of the dispersion phase

On the other hand119881 = 1198811015840+11988110158401015840 where1198811015840 is the volume of

the dispersion phase in this sediment (emergent) and11988110158401015840 is thevolume of the bound dispersion medium of the sediment(emergent) The following equation is valid

1198811015840=4

3120587 11990331198942119895 (4)

Therefore 11988110158401015840 = 119881 minus 1198811015840 = 21198942119895(2 + radic3 minus (23)120587)1199033 Fromthe obtained expressions for1198811198811015840 and11988110158401015840 we determine that

1198811015840

119881=

2120587

3 (2 + radic3)

asymp 0 56

11988110158401015840

119881= 1 minus

2120587

3 (2 + radic3)

asymp 0 44

(5)

Then

1198811015840asymp 0 56119886

2ℎ 119881

10158401015840asymp 0 44119886

2ℎ (6)

From the equations in (6) it follows that 1198811015840 asymp 1 311988110158401015840Let 1205881015840 = 120588

1015840

119904(1205881015840= 1205881015840

119891) be the average volumetric mass

density of the particles from the dispersion phase in the ulti-mate sediment (emergent) In addition let 12058810158401015840 be the averagedensity of the dispersion medium (bound or free) in theLD Then according to formulae (6) for the masses 1198981015840 and11989810158401015840 respectively in the dispersion phase and the bound dis-

persion medium in the ultimate sedimentemergent in119870 wehave

1198981015840= 11988110158401205881015840asymp 0 56119886

2ℎ1205881015840 119898

10158401015840= 1198811015840101584012058810158401015840asymp 0 44119886

2ℎ12058810158401015840

(7)

where ℎ = ℎ119904(ℎ = ℎ

119891)

4 The Scientific World Journal

In (7) if 1205881015840 = 1205881015840

119904 then 1205881015840 gt 120588

10158401015840 and if 1205881015840 = 1205881015840

119891 then

1205881015840lt 12058810158401015840

In our studies we obtain approximated formulas in whichthe practically negligible distance between the particles isignored that is we assume that the particles of the dispersionphase by ultimate sedimentation are closely fitted to eachotherThis does not affect the calculations because we do notconsider only the case when the particles are perfectlyarranged closely to each other

4 Determination of the Average StokesRadii of the Particles in the Sedimentand in the Emergent

Using (3) we can find the numbers 119894 = 119894119904(119894 = 119894

119891) or 119895 =

119895119904(119895 = 119895

119891) and hence the average Stokes radius of the parti-

cles from the sediment (emergent) To this end we count thedispersion particles symmetrically around the center of themasses of the ultimate sediment (emergent) along the verti-cal in a chosen linear unit of distance (This is done becauseeven in monodisperse system the dispersion particles are notof absolutely the same size) Then the obtained number ismultiplied by the number of single distances contained in ℎThis is how we find 119895 and then according to (3)

119894 =

119886 (1 + (radic32))

2ℎ119895 (8)

Hence from (3) for the average Stokes radii of theparticles from the sediment or the emergent we have

119903 =ℎ

(1 + (radic32)) 119895

=119886

2119894 (9)

If the LD in 119870 is polydisperse the obtained radii 119903 of theparticles from the sediment or the emergent are determinedin average values

With this we offer a new method different from Stokesrsquofor determination of the average Stokes radii in dispersionparticles in LD The determination of the 119899 = 119894

2119895 of the dis-

persion particles from the ultimate sediment or emergent inLD in 119870 provides important information also for the foodmedical chemical and so forth industries

5 Algebraic Method for Determiningthe Average Densities and the Masses ofthe Components of a Liquid Dispersion

The densities of the ultimate sediment and emergent as wellas the free dispersionmedium can be definedwith the widelyused pycnometers as discussed by Westwood and Kabadi [4]But to this end these three components of the LD must bemechanically separated This requires the ultimate emergentto be removed from the container first and then pouringout the free dispersion medium and finally there is access tothe ultimate sediment After that the three substances sam-ples are taken which are placed in the pycnometer Thepycnometer method for measuring the densities is easy to

K1 K2 K3

m1

R R R

h1

m2h2

hf3

hf2

hf1

m3 h3

hs1ms1 ms2 ms3hs2 hs3

mf2

mf3

mf1

Figure 2 Vertical axial section of three equal straight cylindricalcontainers with separated components of liquid dispersions

apply and inexpensive but the measurement takes a lot oftime

The measurement can be also done in a noncontact waybymeans of ultrasound densitometers as discussed by Puttmeret al [5]These devices however use complex equipment andare expensive

We offer an algebraic method with significantly simplifiedmeasurements It employs containersmdashprismatic or cylindri-cal At the beginning themasses of the LDmust bemeasuredwhich need to be equal but taken in different proportionsfrom the dispersion phase The LD is placed in the respectivecontainers under constant temperature and external pressureAfter the formation of the ultimate sediment and emergentand the free dispersion medium between them in theconditions of gravitational or centrifugal field we performelementary linear measurements of the geometrical sizes ofthe separated particles in the containers with LD (Figure 2)The densities and the masses of the components are deter-mined with simple algebraic operations

As it is not necessary to count the particles in the ultimatesediment and emergent in this case we can use straightcylindrical containers Let119870

119894(119894 = 1 2 3) be such containers

which for convenience we assume are same with radii of themain planes of 119903 as they are presented in the vertical axialsections in Figure 2The three containers contain LDwith thesame components of the dispersion medium and the disper-sion phase taken in different proportions (These proportionscan be achieved for example by minimal dilution withdispersion medium of the given LD that is by increasingthe relative share of the dispersion medium in the differentcontainers)

Let us assume that at the initial moment 119905 = 0 the testedLD is in homogenized condition that is the dispersion phaseof this dispersion is evenly distributed in the whole volume ofthe containers Moreover the containers are placed underspecific constant temperature and external pressure

In the conditions of gravitational or centrifugal fieldcomes a moment of time 119905 gt 0 when in LD of119870

119894(119894 = 1 2 3)

appears a condition of ultimate sediment and emergent withconstant heights of their volumes Thus within time 119905 com-plete phase separation is achieved that is the volumes and themasses of the formed ultimate sediment and ultimate emer-gent in 119870

119894(Figure 2) no longer changemdashthe heavy particles

have settled and the light particles have emerged

The Scientific World Journal 5

Some cases of LD are possible when free dispersionmedium and only ultimate sediment or only ultimate emer-gent are formed Some more complicated cases are alsopossible when more than one sediment andor emergent areformed

The values of the average density of the substance in theultimate sediment 120588

119904of the average density of the substance in

the ultimate emergent 120588119891and the density of the dispersion

medium 120588 = 12058810158401015840 in the containers119870

119894(119894 = 1 2 3) are the same

Characteristics of the ultimate sediment and emergentare the lights of their heights ℎ

119904119894and ℎ

119891119894(119894 = 1 2 3) in 119870

119894

which are constant after certain moment of time 119905 Withℎ119894(119894 = 1 2 3) we designate the heights of the free dispersion

medium and with 119898119904119894 119898119891119894 119898119894(119894 = 1 2 3) we designate the

masses of the ultimate sediment and emergent and the freedispersion medium of LD in 119870

119894from Figure 2 Then we can

write

119898119904119894= 1205871198772ℎ119904119894120588119904 119898119891119894= 1205871198772ℎ119891119894120588119891 119898119894= 1205871198772ℎ119894120588

(119894 = 1 2 3)

(10)

where 119877 is the radius of the internal bases of the containersWith 119872

119894(119894 = 1 2 3) we designate the total masses of LD

which fill up 119870119894 We consider these masses to be known

(measured) According to the law on conservation of masseswe have the following system of three linear equations

119898119904119894+ 119898119891119894+ 1198980119894= 119872119894 119894 = 1 2 3 (11)

If 119867119894= ℎ119904119894+ ℎ119894+ ℎ119891119894 and 120588

119894is the average volumetric

mass density of LD in119881119894 then it is known that 119896

119894= 1198721198941205871198772=

119867119894120588119894(119894 = 1 2 3) Then from (10) and (11) the system follows

ℎ119904119894120588119904+ ℎ119891119894120588119891+ ℎ119894120588 = 119896119894 119894 = 1 2 3 (12)

By solving (12) the three densities are determinedmdashof theultimate sediment of the ultimate emergent and of thedispersion medium

120588119904=1198631

119863 120588

119891=1198632

119863 120588 =

1198633

119863 (13)

where11986311986311198632 and 119863

3 according to the Cramers rule are

determinants of third orderSince the proportions of the components in LD are differ-

ent then the equations in the system (11) are not equivalentand the system (12) is consistent But if we pour the sameLD even in three different in size and shape straight con-tainers then the equations in (11) will be proportional Thenthe system (12) will be inconsistent

From (10) and (13) we can determine themasses119898119904119894119898119891119894

119898119894 as well as the mass 119898

119901119894= 119898119904119894+ 119898119891119894

of the dispersionphase for any 119894 Then we also find the density 120588

119901of the

dispersion phase in the tested LD

120588119901=

ℎ119904119894120588119904+ ℎ119891119894120588119891

ℎ119904119894+ ℎ119891119894

(14)

If we need to determine only the density 120588119901of the

dispersion phase and the density 120588 of the dispersionmedium

two containers 1198701and 119870

2are enough and a system with two

linear equations

119898119901119894+ 1198981= 1198721lArrrArr (ℎ

119904119894+ ℎ119891119894) 120588119901+ ℎ119894120588 = 119896119894 119894 = 1 2

(15)

Thus

120588119901=1198631

119863 120588 =

1198632

119863 (16)

where 119863 1198631 and 119863

2 according to the Cramers rule are

determinants of second orderIf we know the density of one of the ingredients for

example 120588 for distilledwater then also two containers1198701and

1198702are enough Then 120588

119904and 120588

119891are also determined with a

system with two linear equations

119898119904119894+ 119898119891119894+ 119898119894= 119872119894lArrrArr ℎ

119904119894120588119904+ ℎ119891119894120588119891= 119896119894minus ℎ119894120588

119894 = 1 2

(17)

which is solved analogically to system (12)In case when the density 120588 of the dispersion medium is

known andwe have only ultimate sediment or ultimate emer-gent it is sufficient to use only one container 119870 = 119870

1 Then

we have only one linear equation

119898119904+ 119898 = 119872 or 119898

119891+ 119898 = 119872 (18)

that is

120588119901=119896 minus ℎ120588

ℎ119904

or 120588119901=119896 minus ℎ120588

ℎ119891

(19)

where 120588119901= 120588119904or 120588119901= 120588119891and 119896 = 119872120587119877

2Because119898

119904= 1198981015840+ 11989810158401015840(119898119891= 1198981015840+ 11989810158401015840) where1198981015840 is the

mass of the dispersion phase and11989810158401015840 is the bound dispersionmedium in the ultimate sediment (emergent) and accordingto formula (19) we can find the density 1205881015840 of the dispersionphase

1205881015840=

120588119901minus 0 44120588

0 56 (20)

In this case when the tested LD has sediments and emer-gents consisting of several components then the density ofeach component can be determined through a measurementwith more containers In the general case we use a systemconsisting of 119899 linear equations and 119899 unknowns which ispresent in the following way

119901

sum

119896=0

119898119904119896 119894+

119897

sum

119895=0

119898119891119895 119894+ 119898119894= 119872119894 119894 = 1 2 119899 (21)

where 119899 = 119901+119897+1 From (21) the unknown densities 120588119904119896 119894 120588119891119895 119894

and 120588 are determined

From (13) (14) (16) and (19) using the method shownin Kolikov et al [7] we can determined the maximumabsolute inaccuracies (errors) Δ120588

119904 Δ120588119891 Δ120588119901 and Δ120588 and the

maximum relative inaccuracies (errors) Δ120588119904120588119904 Δ120588119891120588119891

Δ120588119901120588119901 and Δ120588120588 discussed by Kolikov et al [8]

6 The Scientific World Journal

6 Characterization Scale forSedimentation Stability

There are different methods for evaluation of the sedimenta-tion stability 119878 of LD One of the most popular methods is thevisual methodThe visual methodmeasures the observed dis-placement over a period of time of the horizontal separationline between the clear and the muddy part of the tested LD asdiscussed elsewhere [9 10]Themethod employs transparentcontainer and a clock

From the instrumental methods most popular ones indetermining the sedimentation stability are the opticalabsorptionmethodsmdashthroughmeasuring the intensity of thelight passing through the LD as discussed elsewhere [11 12]

According to our method the evaluation of 119878 is donethrough the densities of the components of the LD and it isnot necessary to measure the time and optical transmissionof LD is no required Moreover we introduce scale forcharacterization of the sedimentation stability 119878

Let one particle of the dispersion phase with volume 119881and density 120588

119901be located in dispersion medium with density

120588 in the conditions of gravitational field Then the value ofthe resultant force of its weight and the Archimedean force is

119865 = 119881119892 (120588119901minus 120588) (22)

where 119892 is gravitational constant (In (22) we neglect the fric-tion force which is very little for small velocities of theparticle)

Therefore if the particle is heavy that is 120588119901gt 120588 hArr

120588120588119901lt 1 then119865 gt 0 andwe have straight sedimentation (set-

tling) And if the particle is light that is 120588119901lt 120588 hArr 120588

119901120588 lt

1 then 119865 lt 0 and we have reverse sedimentation (emer-gence)

Thus we suggest the evaluation of the sedimentationstability 119878 to be done through the following dimensionlessphysical variables 119868

119904= 120588120588

119904 119868119891= 120588119891120588 (120588119904 120588 = 0) and hence

through the variable

119868 = max (119868119904 119868119891) = max(

120588

120588119904

120588119891

120588) (23)

The number 119868 is the sedimentation stability index in adimensionless scale for characterization of 119878 Since 0 lt 119868

119904

119868119891le 1 then 0 lt 119868 le 1 that is the scale is the interval (0 1]

The more the index 119868 of an LD is closer to 1 the more stablethe LD is and the closer the index 119868 is to 0 the more unstablethe LD is Thus if 119868 = 1 the LD is absolutely sedimentarystable that is there is no phase separation

We shall notice that if the sediment is absent in the LDthen 119868 = 119868

119891isin (0 1] and if the emergent is absent then 119868 =

119868119904isin (0 1]

7 Determining the Percentage ofthe Substance in the Sediment andthe Emergent

The amount of the substance in the dispersion phase is amajor factor which determines the quality of many products

The drying is widely used method for testing the dry matterin different suspensions To this end a sample is taken fromthe homogenized LD which is placed in a suitable containerweighted with a scale and placed in a drier for drying untilconstant weight is obtained This weight is divided into theinitial (not dried) full weights of the sample and thenmultiplied by 100 The obtained value is called dry substancepercentage of LDThe disadvantages of thismethod are its rel-atively long duration and the spending of significant thermalenergy for drying a unit of weight from the tested substanceMoreover the heating usually to 105∘C required by thestandard results in destruction of the treated substance and itcan only be used for suspensions

We offer a different method which does not have theseflaws

If 119870 is a straight cylindrical container filled-up with LDwith weight 119872 and internal radius of the bases 119877 thenaccording to formula (7) the masses of the dispersion phaseand the bound dispersion medium in the ultimate sedimentor emergent are respectively 1198981015840 = 0 561205871198772ℎ1205881015840 and 11989810158401015840 = 0441205871198772ℎ12058810158401015840 Then knowing the geometrical parameters of the

used container as well as the heights masses and densitieswe have in percentage the amount of the substance 1198751015840 in theultimate sediment or emergent and of the bound dispersionmedium 119875

10158401015840 respectively

1198751015840= (

1198981015840

119872sdot 100) = (

0 561205871198772ℎ1205881015840

119872sdot 100) (24)

11987510158401015840= (

11989810158401015840

119872sdot 100) = (

0 441205871198772ℎ12058810158401015840

119872sdot 100) (25)

The percentage 119875 of the free dispersion medium can alsobe calculated with our method namely

119875 = (

1205871198772(119867 minus ℎ

119904minus ℎ119891) 12058810158401015840

119872sdot 100) (26)

Let us denote that unlike the method of drying until con-stant weight through formula (24) the percentage 1198751015840 of thetested substance is determined with the participation of thebound dispersion medium We take into account only theamount of the pure dispersion phase in the specific sedimentor emergentThrough drying complete dryness of the samplecannot be achieved in practicemdashin the sediment therealways remains some percentage bound dispersion mediumTherefore we cannot expect to have coinciding percentage ofdry substance determined according to the two methods

Let us consider a case of sediment If 1198751015840119904is the percentage

of dry substance in the sediment determined with the dryingmethod then for practical reasons we can write downmodification of formula (24) namely

1198751015840

119904= 1205761198751015840

119904asymp (120576

0 561205871198772ℎ1199041205881015840

119872sdot 100) (27)

Here the variable 120576 is a conversion factor which canchange within the limits 1 lt 120576 lt (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904 where

The Scientific World Journal 7

Table 1 Linear dimensions masses and densities of the components of liquid dispersions

Exp 119877 [cm] ℎ1199041

[cm] ℎ1[cm] ℎ

1199042[cm] ℎ

2[cm] 119872

1[g] 119872

2[g] 120588

119904[gsdotcmminus3] 120588 [gsdotcmminus3]

1 13 24 63 36 76 5607 7397 169 1032 13 24 61 36 76 5517 7433 173 1023 13 24 61 35 78 5509 7439 172 1024 13 24 62 36 77 5574 7495 174 1025 13 24 62 36 77 5551 7458 171 1026 13 24 62 36 76 5557 7383 163 106Avg 13 24 62 36 77 5553 7434 170 103

K1 K2

m1

R R

h1

m2h2

hs1ms1

ms2 hs2

Figure 3 Vertical axial sections of two identical straight cylindricalcontainers with ultimate sedimentation

1198981015840

119904+ 11989810158401015840

119904= 119881

1015840

1199041205881015840

119904+ 11988110158401015840

11990412058810158401015840 is the total weight of the

sediment (dispersion phase and bound dispersion medium)The unobtainable value 120576 = 1 corresponds to the drysubstance in the ultimate sediment in case of complete dryingof the sample in119870 that is without bounddispersionmediumas with formula (24) we offer If the ultimate sediment is notsubjected to drying then the value 120576 = (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904asymp

1 + (120588101584010158401205881015840

119904) lt 2 Thus for the conversion factor in formula

(27) we have 1 lt 120576 lt 2

8 Example

We present models of two dispersion system suspensionsfrom calcium carbonate powder respectively 700 g and1100 g measured with electronic scale with accuracy of 001 gand distilled water respectively 50mL and 65mL measuredwith graduated cylinders with accuracy of 05mL In order toevaluate the accuracy of our method we assume the densitiesof CaCO

3and the distilled H

2O to be unknown

In order to determine the densities respectively 120588119904of the

ultimate sediment and 120588 of the dispersion medium we usetwo identical cylindrical containers 119870

1and 119870

2 with radius

of the internal basis 119877 = 0 013m (Figure 3)In the two containers we pour the two homogenized sus-

pensions (their components are in different proportions)Weperform 6 observations with room temperature 20∘C

Table 1 presents the values of the measured variables119877 ℎ1199041 ℎ1 1198721(for container 119870

1) and 119877 ℎ

1199042 ℎ2 1198722(for

container1198702) during the 6 observations

For densities 120588119904and 120588 according to formula (15) we find

with the system of linear equations

10038161003816100381610038161003816100381610038161003816

1198981199041+ 1198981= 1198721

1198981199042+ 1198982= 1198722

lArrrArr

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1199041120588119904+ ℎ1120588 =

1198721

1205871198772

ℎ1199042120588119904+ ℎ2120588 =

1198722

1205871198772

(28)

Its solutions are

120588119904=1198631

119863=

1

1205871198772

1198721ℎ2minus ℎ11198722

ℎ1199041ℎ2minus ℎ1ℎ1199042

120588 =1198632

119863=

1

1205871198772

ℎ11990411198722minus1198721ℎ1199042

ℎ1199041ℎ2minus ℎ1ℎ1199042

(29)

Then for the densities of the ultimate sediment and thedispersion medium we have respectively the average values120588119904asymp 1 70 gsdotcmminus3 and 120588 asymp 1 03 gsdotcmminus3 which are also

presented in Table 1For the index of the sedimentation stability we find

119868 =120588

120588119904

asymp 0 61 (30)

The value of 119868 according to our scale (01] shows that thetested LD has a little over the average sedimentation stability

According to the model discussed by Kolikov et al [7]we shall determine the maximum absolute inaccuracies Δ120588

119904

and Δ120588 for the established densities 120588119904and 120588 as well as the

maximum relative inaccuraciesΔ120588119904120588119904andΔ120588120588 IfΔ119872

119894Δℎ119894

Δℎ119904119894(119894 = 1 2) andΔ119877 are absolute inaccuracies of119872

119894 ℎ119894 ℎ119904119894

and 119877 then from (29) it follows that

Δ120588119904

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(31)

8 The Scientific World Journal

Table 2 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198631|

10038161003816100381610038161003816100381610038161003816

1198631

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

1 444 75 169 1056 1393 119 143 785 65 1065 1284 11542 372 644 173 1039 1400 115 143 777 624 1055 1315 9903 263 457 174 1037 1401 115 147 792 619 1061 1357 7044 384 663 173 1050 1411 117 145 788 635 1073 1332 10205 384 664 173 1045 1405 117 145 782 630 1073 1332 10226 408 661 162 1047 1390 117 143 807 658 1004 1231 628Avg 376 640 171 1046 1400 117 144 789 636 1055 1309 920

Table 3 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198632|

10038161003816100381610038161003816100381610038161003816

1198632

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

1 444 459 103 1056 1393 045 068 610 407 247 371 1582 372 380 102 1039 1400 045 068 625 417 245 367 1573 263 268 102 1037 1401 045 068 605 415 245 357 1574 384 394 103 1050 1411 045 068 618 411 247 371 1585 384 390 102 1045 1405 045 068 620 413 245 367 1576 408 433 106 1047 1390 045 068 584 390 254 382 163Avg 376 387 103 1046 1400 045 068 610 409 247 369 158

Δ120588

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(32)

where Δℎ1= 0 056 cm Δℎ

2= 0 067 cm Δℎ

1199041= Δℎ1199042

=

Δ119877 = 0 cm Δ1198721= 0 27 g and Δ119872

2= 0 30 g

Considering the calculations fromTables 2 and 3 we havethe values of the maximum absolute inaccuracies Δ120588

119904asymp

0 43 gsdotcmminus3 and Δ120588 asymp 0 18 gsdotcmminus3From here we find the values of the maximum relative

inaccuracies

Δ120588119904

120588119904

=043

170asymp 025

Δ120588

120588=018

103asymp 017 (33)

Thus obtained value of 120588 at 20∘C temperature is withinthe error for the density of the distilledwater which shows theaccuracy of our model

Then we find the percentage 1198751015840 of the dry substance inthe sediment and the percentages 11987510158401015840 and 119875 of the bounddispersionmedium in the sediment and of the free dispersionmedium accordingly We use formulae (20) (24) (25) and(26) and the data from Table 1 The results are shown inTable 4

Because of the approximations in the calculations thesum of the values of 1198751015840 11987510158401015840 and 119875 is around 100

9 Discussion

Our model of close packing of the particles from the dis-persion phase of the studied liquid dispersion offers a newmethod for finding the volumes and the masses of the sedi-mented substances in LD as well as the average Stokes radiiof the particles of these substances Moreover this study isdistinguishedwith simplicity and its application is very cheap

Our simple algebraicmethod offers an increased accuracyin determining the average densities and weights of the ulti-mate sediment and emergent of the dispersion medium andof the dispersion phase in the liquid dispersionThefinding ofthese densities is achieved without the use of pycnometers orother more complex methods This prevents the destructiveeffects in the LD and the vibrations practically have no effecton the measurements results unlike the methods existinguntil now

Our method of determining the coefficient of the quanti-ties of substances in the components of LD does not requirepouring out the liquid from the container in order tomeasurethemasses of the ultimate sediment and of the free dispersionmedium (water) The known method of pouring out anddrying is destructive Using the drying method on LD it isgenerally even impossible to determine the percentage of thesubstance amount in the emulsion The method we proposeovercomes this disadvantage

A great advantage and particularly valuable for onemethod is that it has numerical characteristicmdashscale for eval-uation of a given variable The index 119868 which we introduce in(23) is a dimensionless scale for characterization of the sedi-mentation stability 119878 naturally following from the law on themovement of the dispersion particle from the dispersionphase of LD It provides a very easy way to evaluate 119878 of

The Scientific World Journal 9

Table 4

Exp1 2 3 4 5 6

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

119872 (g) 5607 7397 5517 7433 5509 7439 5574 7495 5551 7458 5557 7383120588119904(gcm3) 169 173 172 174 172 163

120588 = 12058810158401015840 (gcm3) 103 102 102 102 102 106

1205881015840 (gcm3) 221 229 227 23 227 2071198751015840 () dry matter 2811 3197 2973 3296 2939 3174 2942 3284 2917 3256 2656 300011987510158401015840 () bound dispersion

medium in the sediment 1029 1169 1037 1153 1038 1121 1026 1143 1030 1149 1069 1207

119875 () free dispersion medium 6146 5620 5985 5535 5994 5675 6021 5561 6046 5589 6275 5790

one LD in straight andor reverse sedimentation through thedensities 120588

119904 120588119891 and 120588 which are the solution of the system of

linear equations of type (11)The index 119868 isin (0 1] provides the possibility not only to

evaluate and to predict the sedimentation stability but alsoto compare different LDs according to their sedimentationstability Moreover the tests of the different LDs must beperformed in constant temperature and external pressure

10 Conclusion

The method which we offer does not require special equip-ment The phase separation of LDs can be achieved in theconditions not only of gravitational (earthrsquos) but also of cen-trifugal field In the case of centrifugal field the time interval 119905for determining the ultimate sediment and emergent can bemuch smaller compared with time for treating the LD withthe acceleration of gravity 119892

The advantage of our physicomathematical model forfinding the Stokes radii in the dispersion phase is that it doesnot use complicatedmathematical analysis for structuring thesedimentation curve

The three variables 1198751015840 11987510158401015840 and 119875 characterizing in per-centage the respective quantity of substance in the ultimatesediment or emergent bound dispersion medium in the sed-iment or the emergent and the free dispersion medium areintroduced in the literature for the first time with this paper

And with the sedimentation stability index 119868 whichwe introduce it is possible to demonstratively and withoutparticular difficultiesmake qualitative analysis of suspensionsand emulsionsmdashproducts of the food medical cosmeticchemical and so forth industries Such products are forexample different suspensions and emulsions such as foodsand drinks medications and cosmetic products paints andvarnishes and cement and lime solutionsThis is particularlyand technically useful in the creation of products withpredetermined sedimentation stability

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The obtained results are made with the support of Fund ldquoSci-entific Studiesrdquo of the Bulgarian Ministry of Education andScience under the Contract T 12 This paper is publishedwith the financial support of the Project BG051PO00133-05-001 ldquoScience and Businessrdquo financed by the HumanResources Development Operational Programme of theEuropean Social Fund

References

[1] D Dakova D Christozov and M Beleva ldquoBarycentric methodof determining the physical parameters of a single-phase parti-cle in liquid disperse systemsrdquo Journal of Colloid and InterfaceScience vol 256 no 2 pp 477ndash479 2002

[2] R Chanamai and D J McClements ldquoDepletion flocculation ofbeverage emulsions by gum arabic andmodified starchrdquo Journalof Food Science vol 66 no 3 pp 457ndash463 2001

[3] G S Hodakov andU P Udkin SedimentationAnalysis of HighlyDispersed Systems (in Russian) Chemistry Moscow Russia1981

[4] B M Westwood and V N Kabadi ldquoA novel pycnometer fordensity measurements of liquids at elevated temperaturesrdquoTheJournal of Chemical Thermodynamics vol 35 no 12 pp 1965ndash1974 2003

[5] A Puttmer R Lucklum B Henning and P HauptmannldquoImproved ultrasonic density sensor with reduced diffractioninfluencerdquo Sensors and Actuators A Physical vol 67 no 1ndash3 pp8ndash12 1998

[6] E D Shchukin A V Percov and E A Amelina ColloidalChemistry pp 306ndash309 Yurait Moscow Russia 6th edition2012 (Russian)

[7] K Kolikov G Krastev Y Epitropov and D Hristozov ldquoAna-lytically determining of the absolute inaccuracy (error) ofindirectly measurable variable and dimensionless scale char-acterising the quality of the experimentrdquo Chemometrics andIntelligent Laboratory Systems vol 102 no 1 pp 15ndash19 2010

[8] K Kolikov G Krastev Y Epitropov and A Corlat ldquoAnalyticallydetermining of the relative inaccuracy (error) of indirectlymeasurable variable and dimensionless scale characterising thequality of the experimentrdquoCSJM vol 20 no 1 pp 314ndash331 2012

[9] M H Ly M Aguedo S Goudot et al ldquoInteractions betweenbacterial surfaces and milk proteins impact on food emulsionsstabilityrdquo Food Hydrocolloids vol 22 no 5 pp 742ndash751 2008

10 The Scientific World Journal

[10] J P F Macedo L L Fernandes F R Formiga et al ldquoMicro-emultocrit technique a valuable tool for determination ofcritical HLB value of emulsionsrdquo AAPS PharmSciTech vol 7no 1 pp E1ndashE7 2006

[11] M Corredig and M Alexander ldquoFood emulsions studied byDWS recent advancesrdquo Trends in Food Science and Technologyvol 19 no 2 pp 67ndash75 2008

[12] J M Quintana A Califano and N Zaritzky ldquoMicrostructureand stability of non-protein stabilized oil-in-water food emul-sionsmeasured by opticalmethodsrdquo Journal of Food Science vol67 no 3 pp 1130ndash1135 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

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International Journal ofPhotoenergy

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Journal of

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

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Medicinal ChemistryInternational Journal of

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Chromatography Research International

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CatalystsJournal of

4 The Scientific World Journal

In (7) if 1205881015840 = 1205881015840

119904 then 1205881015840 gt 120588

10158401015840 and if 1205881015840 = 1205881015840

119891 then

1205881015840lt 12058810158401015840

In our studies we obtain approximated formulas in whichthe practically negligible distance between the particles isignored that is we assume that the particles of the dispersionphase by ultimate sedimentation are closely fitted to eachotherThis does not affect the calculations because we do notconsider only the case when the particles are perfectlyarranged closely to each other

4 Determination of the Average StokesRadii of the Particles in the Sedimentand in the Emergent

Using (3) we can find the numbers 119894 = 119894119904(119894 = 119894

119891) or 119895 =

119895119904(119895 = 119895

119891) and hence the average Stokes radius of the parti-

cles from the sediment (emergent) To this end we count thedispersion particles symmetrically around the center of themasses of the ultimate sediment (emergent) along the verti-cal in a chosen linear unit of distance (This is done becauseeven in monodisperse system the dispersion particles are notof absolutely the same size) Then the obtained number ismultiplied by the number of single distances contained in ℎThis is how we find 119895 and then according to (3)

119894 =

119886 (1 + (radic32))

2ℎ119895 (8)

Hence from (3) for the average Stokes radii of theparticles from the sediment or the emergent we have

119903 =ℎ

(1 + (radic32)) 119895

=119886

2119894 (9)

If the LD in 119870 is polydisperse the obtained radii 119903 of theparticles from the sediment or the emergent are determinedin average values

With this we offer a new method different from Stokesrsquofor determination of the average Stokes radii in dispersionparticles in LD The determination of the 119899 = 119894

2119895 of the dis-

persion particles from the ultimate sediment or emergent inLD in 119870 provides important information also for the foodmedical chemical and so forth industries

5 Algebraic Method for Determiningthe Average Densities and the Masses ofthe Components of a Liquid Dispersion

The densities of the ultimate sediment and emergent as wellas the free dispersionmedium can be definedwith the widelyused pycnometers as discussed by Westwood and Kabadi [4]But to this end these three components of the LD must bemechanically separated This requires the ultimate emergentto be removed from the container first and then pouringout the free dispersion medium and finally there is access tothe ultimate sediment After that the three substances sam-ples are taken which are placed in the pycnometer Thepycnometer method for measuring the densities is easy to

K1 K2 K3

m1

R R R

h1

m2h2

hf3

hf2

hf1

m3 h3

hs1ms1 ms2 ms3hs2 hs3

mf2

mf3

mf1

Figure 2 Vertical axial section of three equal straight cylindricalcontainers with separated components of liquid dispersions

apply and inexpensive but the measurement takes a lot oftime

The measurement can be also done in a noncontact waybymeans of ultrasound densitometers as discussed by Puttmeret al [5]These devices however use complex equipment andare expensive

We offer an algebraic method with significantly simplifiedmeasurements It employs containersmdashprismatic or cylindri-cal At the beginning themasses of the LDmust bemeasuredwhich need to be equal but taken in different proportionsfrom the dispersion phase The LD is placed in the respectivecontainers under constant temperature and external pressureAfter the formation of the ultimate sediment and emergentand the free dispersion medium between them in theconditions of gravitational or centrifugal field we performelementary linear measurements of the geometrical sizes ofthe separated particles in the containers with LD (Figure 2)The densities and the masses of the components are deter-mined with simple algebraic operations

As it is not necessary to count the particles in the ultimatesediment and emergent in this case we can use straightcylindrical containers Let119870

119894(119894 = 1 2 3) be such containers

which for convenience we assume are same with radii of themain planes of 119903 as they are presented in the vertical axialsections in Figure 2The three containers contain LDwith thesame components of the dispersion medium and the disper-sion phase taken in different proportions (These proportionscan be achieved for example by minimal dilution withdispersion medium of the given LD that is by increasingthe relative share of the dispersion medium in the differentcontainers)

Let us assume that at the initial moment 119905 = 0 the testedLD is in homogenized condition that is the dispersion phaseof this dispersion is evenly distributed in the whole volume ofthe containers Moreover the containers are placed underspecific constant temperature and external pressure

In the conditions of gravitational or centrifugal fieldcomes a moment of time 119905 gt 0 when in LD of119870

119894(119894 = 1 2 3)

appears a condition of ultimate sediment and emergent withconstant heights of their volumes Thus within time 119905 com-plete phase separation is achieved that is the volumes and themasses of the formed ultimate sediment and ultimate emer-gent in 119870

119894(Figure 2) no longer changemdashthe heavy particles

have settled and the light particles have emerged

The Scientific World Journal 5

Some cases of LD are possible when free dispersionmedium and only ultimate sediment or only ultimate emer-gent are formed Some more complicated cases are alsopossible when more than one sediment andor emergent areformed

The values of the average density of the substance in theultimate sediment 120588

119904of the average density of the substance in

the ultimate emergent 120588119891and the density of the dispersion

medium 120588 = 12058810158401015840 in the containers119870

119894(119894 = 1 2 3) are the same

Characteristics of the ultimate sediment and emergentare the lights of their heights ℎ

119904119894and ℎ

119891119894(119894 = 1 2 3) in 119870

119894

which are constant after certain moment of time 119905 Withℎ119894(119894 = 1 2 3) we designate the heights of the free dispersion

medium and with 119898119904119894 119898119891119894 119898119894(119894 = 1 2 3) we designate the

masses of the ultimate sediment and emergent and the freedispersion medium of LD in 119870

119894from Figure 2 Then we can

write

119898119904119894= 1205871198772ℎ119904119894120588119904 119898119891119894= 1205871198772ℎ119891119894120588119891 119898119894= 1205871198772ℎ119894120588

(119894 = 1 2 3)

(10)

where 119877 is the radius of the internal bases of the containersWith 119872

119894(119894 = 1 2 3) we designate the total masses of LD

which fill up 119870119894 We consider these masses to be known

(measured) According to the law on conservation of masseswe have the following system of three linear equations

119898119904119894+ 119898119891119894+ 1198980119894= 119872119894 119894 = 1 2 3 (11)

If 119867119894= ℎ119904119894+ ℎ119894+ ℎ119891119894 and 120588

119894is the average volumetric

mass density of LD in119881119894 then it is known that 119896

119894= 1198721198941205871198772=

119867119894120588119894(119894 = 1 2 3) Then from (10) and (11) the system follows

ℎ119904119894120588119904+ ℎ119891119894120588119891+ ℎ119894120588 = 119896119894 119894 = 1 2 3 (12)

By solving (12) the three densities are determinedmdashof theultimate sediment of the ultimate emergent and of thedispersion medium

120588119904=1198631

119863 120588

119891=1198632

119863 120588 =

1198633

119863 (13)

where11986311986311198632 and 119863

3 according to the Cramers rule are

determinants of third orderSince the proportions of the components in LD are differ-

ent then the equations in the system (11) are not equivalentand the system (12) is consistent But if we pour the sameLD even in three different in size and shape straight con-tainers then the equations in (11) will be proportional Thenthe system (12) will be inconsistent

From (10) and (13) we can determine themasses119898119904119894119898119891119894

119898119894 as well as the mass 119898

119901119894= 119898119904119894+ 119898119891119894

of the dispersionphase for any 119894 Then we also find the density 120588

119901of the

dispersion phase in the tested LD

120588119901=

ℎ119904119894120588119904+ ℎ119891119894120588119891

ℎ119904119894+ ℎ119891119894

(14)

If we need to determine only the density 120588119901of the

dispersion phase and the density 120588 of the dispersionmedium

two containers 1198701and 119870

2are enough and a system with two

linear equations

119898119901119894+ 1198981= 1198721lArrrArr (ℎ

119904119894+ ℎ119891119894) 120588119901+ ℎ119894120588 = 119896119894 119894 = 1 2

(15)

Thus

120588119901=1198631

119863 120588 =

1198632

119863 (16)

where 119863 1198631 and 119863

2 according to the Cramers rule are

determinants of second orderIf we know the density of one of the ingredients for

example 120588 for distilledwater then also two containers1198701and

1198702are enough Then 120588

119904and 120588

119891are also determined with a

system with two linear equations

119898119904119894+ 119898119891119894+ 119898119894= 119872119894lArrrArr ℎ

119904119894120588119904+ ℎ119891119894120588119891= 119896119894minus ℎ119894120588

119894 = 1 2

(17)

which is solved analogically to system (12)In case when the density 120588 of the dispersion medium is

known andwe have only ultimate sediment or ultimate emer-gent it is sufficient to use only one container 119870 = 119870

1 Then

we have only one linear equation

119898119904+ 119898 = 119872 or 119898

119891+ 119898 = 119872 (18)

that is

120588119901=119896 minus ℎ120588

ℎ119904

or 120588119901=119896 minus ℎ120588

ℎ119891

(19)

where 120588119901= 120588119904or 120588119901= 120588119891and 119896 = 119872120587119877

2Because119898

119904= 1198981015840+ 11989810158401015840(119898119891= 1198981015840+ 11989810158401015840) where1198981015840 is the

mass of the dispersion phase and11989810158401015840 is the bound dispersionmedium in the ultimate sediment (emergent) and accordingto formula (19) we can find the density 1205881015840 of the dispersionphase

1205881015840=

120588119901minus 0 44120588

0 56 (20)

In this case when the tested LD has sediments and emer-gents consisting of several components then the density ofeach component can be determined through a measurementwith more containers In the general case we use a systemconsisting of 119899 linear equations and 119899 unknowns which ispresent in the following way

119901

sum

119896=0

119898119904119896 119894+

119897

sum

119895=0

119898119891119895 119894+ 119898119894= 119872119894 119894 = 1 2 119899 (21)

where 119899 = 119901+119897+1 From (21) the unknown densities 120588119904119896 119894 120588119891119895 119894

and 120588 are determined

From (13) (14) (16) and (19) using the method shownin Kolikov et al [7] we can determined the maximumabsolute inaccuracies (errors) Δ120588

119904 Δ120588119891 Δ120588119901 and Δ120588 and the

maximum relative inaccuracies (errors) Δ120588119904120588119904 Δ120588119891120588119891

Δ120588119901120588119901 and Δ120588120588 discussed by Kolikov et al [8]

6 The Scientific World Journal

6 Characterization Scale forSedimentation Stability

There are different methods for evaluation of the sedimenta-tion stability 119878 of LD One of the most popular methods is thevisual methodThe visual methodmeasures the observed dis-placement over a period of time of the horizontal separationline between the clear and the muddy part of the tested LD asdiscussed elsewhere [9 10]Themethod employs transparentcontainer and a clock

From the instrumental methods most popular ones indetermining the sedimentation stability are the opticalabsorptionmethodsmdashthroughmeasuring the intensity of thelight passing through the LD as discussed elsewhere [11 12]

According to our method the evaluation of 119878 is donethrough the densities of the components of the LD and it isnot necessary to measure the time and optical transmissionof LD is no required Moreover we introduce scale forcharacterization of the sedimentation stability 119878

Let one particle of the dispersion phase with volume 119881and density 120588

119901be located in dispersion medium with density

120588 in the conditions of gravitational field Then the value ofthe resultant force of its weight and the Archimedean force is

119865 = 119881119892 (120588119901minus 120588) (22)

where 119892 is gravitational constant (In (22) we neglect the fric-tion force which is very little for small velocities of theparticle)

Therefore if the particle is heavy that is 120588119901gt 120588 hArr

120588120588119901lt 1 then119865 gt 0 andwe have straight sedimentation (set-

tling) And if the particle is light that is 120588119901lt 120588 hArr 120588

119901120588 lt

1 then 119865 lt 0 and we have reverse sedimentation (emer-gence)

Thus we suggest the evaluation of the sedimentationstability 119878 to be done through the following dimensionlessphysical variables 119868

119904= 120588120588

119904 119868119891= 120588119891120588 (120588119904 120588 = 0) and hence

through the variable

119868 = max (119868119904 119868119891) = max(

120588

120588119904

120588119891

120588) (23)

The number 119868 is the sedimentation stability index in adimensionless scale for characterization of 119878 Since 0 lt 119868

119904

119868119891le 1 then 0 lt 119868 le 1 that is the scale is the interval (0 1]

The more the index 119868 of an LD is closer to 1 the more stablethe LD is and the closer the index 119868 is to 0 the more unstablethe LD is Thus if 119868 = 1 the LD is absolutely sedimentarystable that is there is no phase separation

We shall notice that if the sediment is absent in the LDthen 119868 = 119868

119891isin (0 1] and if the emergent is absent then 119868 =

119868119904isin (0 1]

7 Determining the Percentage ofthe Substance in the Sediment andthe Emergent

The amount of the substance in the dispersion phase is amajor factor which determines the quality of many products

The drying is widely used method for testing the dry matterin different suspensions To this end a sample is taken fromthe homogenized LD which is placed in a suitable containerweighted with a scale and placed in a drier for drying untilconstant weight is obtained This weight is divided into theinitial (not dried) full weights of the sample and thenmultiplied by 100 The obtained value is called dry substancepercentage of LDThe disadvantages of thismethod are its rel-atively long duration and the spending of significant thermalenergy for drying a unit of weight from the tested substanceMoreover the heating usually to 105∘C required by thestandard results in destruction of the treated substance and itcan only be used for suspensions

We offer a different method which does not have theseflaws

If 119870 is a straight cylindrical container filled-up with LDwith weight 119872 and internal radius of the bases 119877 thenaccording to formula (7) the masses of the dispersion phaseand the bound dispersion medium in the ultimate sedimentor emergent are respectively 1198981015840 = 0 561205871198772ℎ1205881015840 and 11989810158401015840 = 0441205871198772ℎ12058810158401015840 Then knowing the geometrical parameters of the

used container as well as the heights masses and densitieswe have in percentage the amount of the substance 1198751015840 in theultimate sediment or emergent and of the bound dispersionmedium 119875

10158401015840 respectively

1198751015840= (

1198981015840

119872sdot 100) = (

0 561205871198772ℎ1205881015840

119872sdot 100) (24)

11987510158401015840= (

11989810158401015840

119872sdot 100) = (

0 441205871198772ℎ12058810158401015840

119872sdot 100) (25)

The percentage 119875 of the free dispersion medium can alsobe calculated with our method namely

119875 = (

1205871198772(119867 minus ℎ

119904minus ℎ119891) 12058810158401015840

119872sdot 100) (26)

Let us denote that unlike the method of drying until con-stant weight through formula (24) the percentage 1198751015840 of thetested substance is determined with the participation of thebound dispersion medium We take into account only theamount of the pure dispersion phase in the specific sedimentor emergentThrough drying complete dryness of the samplecannot be achieved in practicemdashin the sediment therealways remains some percentage bound dispersion mediumTherefore we cannot expect to have coinciding percentage ofdry substance determined according to the two methods

Let us consider a case of sediment If 1198751015840119904is the percentage

of dry substance in the sediment determined with the dryingmethod then for practical reasons we can write downmodification of formula (24) namely

1198751015840

119904= 1205761198751015840

119904asymp (120576

0 561205871198772ℎ1199041205881015840

119872sdot 100) (27)

Here the variable 120576 is a conversion factor which canchange within the limits 1 lt 120576 lt (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904 where

The Scientific World Journal 7

Table 1 Linear dimensions masses and densities of the components of liquid dispersions

Exp 119877 [cm] ℎ1199041

[cm] ℎ1[cm] ℎ

1199042[cm] ℎ

2[cm] 119872

1[g] 119872

2[g] 120588

119904[gsdotcmminus3] 120588 [gsdotcmminus3]

1 13 24 63 36 76 5607 7397 169 1032 13 24 61 36 76 5517 7433 173 1023 13 24 61 35 78 5509 7439 172 1024 13 24 62 36 77 5574 7495 174 1025 13 24 62 36 77 5551 7458 171 1026 13 24 62 36 76 5557 7383 163 106Avg 13 24 62 36 77 5553 7434 170 103

K1 K2

m1

R R

h1

m2h2

hs1ms1

ms2 hs2

Figure 3 Vertical axial sections of two identical straight cylindricalcontainers with ultimate sedimentation

1198981015840

119904+ 11989810158401015840

119904= 119881

1015840

1199041205881015840

119904+ 11988110158401015840

11990412058810158401015840 is the total weight of the

sediment (dispersion phase and bound dispersion medium)The unobtainable value 120576 = 1 corresponds to the drysubstance in the ultimate sediment in case of complete dryingof the sample in119870 that is without bounddispersionmediumas with formula (24) we offer If the ultimate sediment is notsubjected to drying then the value 120576 = (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904asymp

1 + (120588101584010158401205881015840

119904) lt 2 Thus for the conversion factor in formula

(27) we have 1 lt 120576 lt 2

8 Example

We present models of two dispersion system suspensionsfrom calcium carbonate powder respectively 700 g and1100 g measured with electronic scale with accuracy of 001 gand distilled water respectively 50mL and 65mL measuredwith graduated cylinders with accuracy of 05mL In order toevaluate the accuracy of our method we assume the densitiesof CaCO

3and the distilled H

2O to be unknown

In order to determine the densities respectively 120588119904of the

ultimate sediment and 120588 of the dispersion medium we usetwo identical cylindrical containers 119870

1and 119870

2 with radius

of the internal basis 119877 = 0 013m (Figure 3)In the two containers we pour the two homogenized sus-

pensions (their components are in different proportions)Weperform 6 observations with room temperature 20∘C

Table 1 presents the values of the measured variables119877 ℎ1199041 ℎ1 1198721(for container 119870

1) and 119877 ℎ

1199042 ℎ2 1198722(for

container1198702) during the 6 observations

For densities 120588119904and 120588 according to formula (15) we find

with the system of linear equations

10038161003816100381610038161003816100381610038161003816

1198981199041+ 1198981= 1198721

1198981199042+ 1198982= 1198722

lArrrArr

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1199041120588119904+ ℎ1120588 =

1198721

1205871198772

ℎ1199042120588119904+ ℎ2120588 =

1198722

1205871198772

(28)

Its solutions are

120588119904=1198631

119863=

1

1205871198772

1198721ℎ2minus ℎ11198722

ℎ1199041ℎ2minus ℎ1ℎ1199042

120588 =1198632

119863=

1

1205871198772

ℎ11990411198722minus1198721ℎ1199042

ℎ1199041ℎ2minus ℎ1ℎ1199042

(29)

Then for the densities of the ultimate sediment and thedispersion medium we have respectively the average values120588119904asymp 1 70 gsdotcmminus3 and 120588 asymp 1 03 gsdotcmminus3 which are also

presented in Table 1For the index of the sedimentation stability we find

119868 =120588

120588119904

asymp 0 61 (30)

The value of 119868 according to our scale (01] shows that thetested LD has a little over the average sedimentation stability

According to the model discussed by Kolikov et al [7]we shall determine the maximum absolute inaccuracies Δ120588

119904

and Δ120588 for the established densities 120588119904and 120588 as well as the

maximum relative inaccuraciesΔ120588119904120588119904andΔ120588120588 IfΔ119872

119894Δℎ119894

Δℎ119904119894(119894 = 1 2) andΔ119877 are absolute inaccuracies of119872

119894 ℎ119894 ℎ119904119894

and 119877 then from (29) it follows that

Δ120588119904

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(31)

8 The Scientific World Journal

Table 2 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198631|

10038161003816100381610038161003816100381610038161003816

1198631

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

1 444 75 169 1056 1393 119 143 785 65 1065 1284 11542 372 644 173 1039 1400 115 143 777 624 1055 1315 9903 263 457 174 1037 1401 115 147 792 619 1061 1357 7044 384 663 173 1050 1411 117 145 788 635 1073 1332 10205 384 664 173 1045 1405 117 145 782 630 1073 1332 10226 408 661 162 1047 1390 117 143 807 658 1004 1231 628Avg 376 640 171 1046 1400 117 144 789 636 1055 1309 920

Table 3 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198632|

10038161003816100381610038161003816100381610038161003816

1198632

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

1 444 459 103 1056 1393 045 068 610 407 247 371 1582 372 380 102 1039 1400 045 068 625 417 245 367 1573 263 268 102 1037 1401 045 068 605 415 245 357 1574 384 394 103 1050 1411 045 068 618 411 247 371 1585 384 390 102 1045 1405 045 068 620 413 245 367 1576 408 433 106 1047 1390 045 068 584 390 254 382 163Avg 376 387 103 1046 1400 045 068 610 409 247 369 158

Δ120588

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(32)

where Δℎ1= 0 056 cm Δℎ

2= 0 067 cm Δℎ

1199041= Δℎ1199042

=

Δ119877 = 0 cm Δ1198721= 0 27 g and Δ119872

2= 0 30 g

Considering the calculations fromTables 2 and 3 we havethe values of the maximum absolute inaccuracies Δ120588

119904asymp

0 43 gsdotcmminus3 and Δ120588 asymp 0 18 gsdotcmminus3From here we find the values of the maximum relative

inaccuracies

Δ120588119904

120588119904

=043

170asymp 025

Δ120588

120588=018

103asymp 017 (33)

Thus obtained value of 120588 at 20∘C temperature is withinthe error for the density of the distilledwater which shows theaccuracy of our model

Then we find the percentage 1198751015840 of the dry substance inthe sediment and the percentages 11987510158401015840 and 119875 of the bounddispersionmedium in the sediment and of the free dispersionmedium accordingly We use formulae (20) (24) (25) and(26) and the data from Table 1 The results are shown inTable 4

Because of the approximations in the calculations thesum of the values of 1198751015840 11987510158401015840 and 119875 is around 100

9 Discussion

Our model of close packing of the particles from the dis-persion phase of the studied liquid dispersion offers a newmethod for finding the volumes and the masses of the sedi-mented substances in LD as well as the average Stokes radiiof the particles of these substances Moreover this study isdistinguishedwith simplicity and its application is very cheap

Our simple algebraicmethod offers an increased accuracyin determining the average densities and weights of the ulti-mate sediment and emergent of the dispersion medium andof the dispersion phase in the liquid dispersionThefinding ofthese densities is achieved without the use of pycnometers orother more complex methods This prevents the destructiveeffects in the LD and the vibrations practically have no effecton the measurements results unlike the methods existinguntil now

Our method of determining the coefficient of the quanti-ties of substances in the components of LD does not requirepouring out the liquid from the container in order tomeasurethemasses of the ultimate sediment and of the free dispersionmedium (water) The known method of pouring out anddrying is destructive Using the drying method on LD it isgenerally even impossible to determine the percentage of thesubstance amount in the emulsion The method we proposeovercomes this disadvantage

A great advantage and particularly valuable for onemethod is that it has numerical characteristicmdashscale for eval-uation of a given variable The index 119868 which we introduce in(23) is a dimensionless scale for characterization of the sedi-mentation stability 119878 naturally following from the law on themovement of the dispersion particle from the dispersionphase of LD It provides a very easy way to evaluate 119878 of

The Scientific World Journal 9

Table 4

Exp1 2 3 4 5 6

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

119872 (g) 5607 7397 5517 7433 5509 7439 5574 7495 5551 7458 5557 7383120588119904(gcm3) 169 173 172 174 172 163

120588 = 12058810158401015840 (gcm3) 103 102 102 102 102 106

1205881015840 (gcm3) 221 229 227 23 227 2071198751015840 () dry matter 2811 3197 2973 3296 2939 3174 2942 3284 2917 3256 2656 300011987510158401015840 () bound dispersion

medium in the sediment 1029 1169 1037 1153 1038 1121 1026 1143 1030 1149 1069 1207

119875 () free dispersion medium 6146 5620 5985 5535 5994 5675 6021 5561 6046 5589 6275 5790

one LD in straight andor reverse sedimentation through thedensities 120588

119904 120588119891 and 120588 which are the solution of the system of

linear equations of type (11)The index 119868 isin (0 1] provides the possibility not only to

evaluate and to predict the sedimentation stability but alsoto compare different LDs according to their sedimentationstability Moreover the tests of the different LDs must beperformed in constant temperature and external pressure

10 Conclusion

The method which we offer does not require special equip-ment The phase separation of LDs can be achieved in theconditions not only of gravitational (earthrsquos) but also of cen-trifugal field In the case of centrifugal field the time interval 119905for determining the ultimate sediment and emergent can bemuch smaller compared with time for treating the LD withthe acceleration of gravity 119892

The advantage of our physicomathematical model forfinding the Stokes radii in the dispersion phase is that it doesnot use complicatedmathematical analysis for structuring thesedimentation curve

The three variables 1198751015840 11987510158401015840 and 119875 characterizing in per-centage the respective quantity of substance in the ultimatesediment or emergent bound dispersion medium in the sed-iment or the emergent and the free dispersion medium areintroduced in the literature for the first time with this paper

And with the sedimentation stability index 119868 whichwe introduce it is possible to demonstratively and withoutparticular difficultiesmake qualitative analysis of suspensionsand emulsionsmdashproducts of the food medical cosmeticchemical and so forth industries Such products are forexample different suspensions and emulsions such as foodsand drinks medications and cosmetic products paints andvarnishes and cement and lime solutionsThis is particularlyand technically useful in the creation of products withpredetermined sedimentation stability

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The obtained results are made with the support of Fund ldquoSci-entific Studiesrdquo of the Bulgarian Ministry of Education andScience under the Contract T 12 This paper is publishedwith the financial support of the Project BG051PO00133-05-001 ldquoScience and Businessrdquo financed by the HumanResources Development Operational Programme of theEuropean Social Fund

References

[1] D Dakova D Christozov and M Beleva ldquoBarycentric methodof determining the physical parameters of a single-phase parti-cle in liquid disperse systemsrdquo Journal of Colloid and InterfaceScience vol 256 no 2 pp 477ndash479 2002

[2] R Chanamai and D J McClements ldquoDepletion flocculation ofbeverage emulsions by gum arabic andmodified starchrdquo Journalof Food Science vol 66 no 3 pp 457ndash463 2001

[3] G S Hodakov andU P Udkin SedimentationAnalysis of HighlyDispersed Systems (in Russian) Chemistry Moscow Russia1981

[4] B M Westwood and V N Kabadi ldquoA novel pycnometer fordensity measurements of liquids at elevated temperaturesrdquoTheJournal of Chemical Thermodynamics vol 35 no 12 pp 1965ndash1974 2003

[5] A Puttmer R Lucklum B Henning and P HauptmannldquoImproved ultrasonic density sensor with reduced diffractioninfluencerdquo Sensors and Actuators A Physical vol 67 no 1ndash3 pp8ndash12 1998

[6] E D Shchukin A V Percov and E A Amelina ColloidalChemistry pp 306ndash309 Yurait Moscow Russia 6th edition2012 (Russian)

[7] K Kolikov G Krastev Y Epitropov and D Hristozov ldquoAna-lytically determining of the absolute inaccuracy (error) ofindirectly measurable variable and dimensionless scale char-acterising the quality of the experimentrdquo Chemometrics andIntelligent Laboratory Systems vol 102 no 1 pp 15ndash19 2010

[8] K Kolikov G Krastev Y Epitropov and A Corlat ldquoAnalyticallydetermining of the relative inaccuracy (error) of indirectlymeasurable variable and dimensionless scale characterising thequality of the experimentrdquoCSJM vol 20 no 1 pp 314ndash331 2012

[9] M H Ly M Aguedo S Goudot et al ldquoInteractions betweenbacterial surfaces and milk proteins impact on food emulsionsstabilityrdquo Food Hydrocolloids vol 22 no 5 pp 742ndash751 2008

10 The Scientific World Journal

[10] J P F Macedo L L Fernandes F R Formiga et al ldquoMicro-emultocrit technique a valuable tool for determination ofcritical HLB value of emulsionsrdquo AAPS PharmSciTech vol 7no 1 pp E1ndashE7 2006

[11] M Corredig and M Alexander ldquoFood emulsions studied byDWS recent advancesrdquo Trends in Food Science and Technologyvol 19 no 2 pp 67ndash75 2008

[12] J M Quintana A Califano and N Zaritzky ldquoMicrostructureand stability of non-protein stabilized oil-in-water food emul-sionsmeasured by opticalmethodsrdquo Journal of Food Science vol67 no 3 pp 1130ndash1135 2002

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CatalystsJournal of

The Scientific World Journal 5

Some cases of LD are possible when free dispersionmedium and only ultimate sediment or only ultimate emer-gent are formed Some more complicated cases are alsopossible when more than one sediment andor emergent areformed

The values of the average density of the substance in theultimate sediment 120588

119904of the average density of the substance in

the ultimate emergent 120588119891and the density of the dispersion

medium 120588 = 12058810158401015840 in the containers119870

119894(119894 = 1 2 3) are the same

Characteristics of the ultimate sediment and emergentare the lights of their heights ℎ

119904119894and ℎ

119891119894(119894 = 1 2 3) in 119870

119894

which are constant after certain moment of time 119905 Withℎ119894(119894 = 1 2 3) we designate the heights of the free dispersion

medium and with 119898119904119894 119898119891119894 119898119894(119894 = 1 2 3) we designate the

masses of the ultimate sediment and emergent and the freedispersion medium of LD in 119870

119894from Figure 2 Then we can

write

119898119904119894= 1205871198772ℎ119904119894120588119904 119898119891119894= 1205871198772ℎ119891119894120588119891 119898119894= 1205871198772ℎ119894120588

(119894 = 1 2 3)

(10)

where 119877 is the radius of the internal bases of the containersWith 119872

119894(119894 = 1 2 3) we designate the total masses of LD

which fill up 119870119894 We consider these masses to be known

(measured) According to the law on conservation of masseswe have the following system of three linear equations

119898119904119894+ 119898119891119894+ 1198980119894= 119872119894 119894 = 1 2 3 (11)

If 119867119894= ℎ119904119894+ ℎ119894+ ℎ119891119894 and 120588

119894is the average volumetric

mass density of LD in119881119894 then it is known that 119896

119894= 1198721198941205871198772=

119867119894120588119894(119894 = 1 2 3) Then from (10) and (11) the system follows

ℎ119904119894120588119904+ ℎ119891119894120588119891+ ℎ119894120588 = 119896119894 119894 = 1 2 3 (12)

By solving (12) the three densities are determinedmdashof theultimate sediment of the ultimate emergent and of thedispersion medium

120588119904=1198631

119863 120588

119891=1198632

119863 120588 =

1198633

119863 (13)

where11986311986311198632 and 119863

3 according to the Cramers rule are

determinants of third orderSince the proportions of the components in LD are differ-

ent then the equations in the system (11) are not equivalentand the system (12) is consistent But if we pour the sameLD even in three different in size and shape straight con-tainers then the equations in (11) will be proportional Thenthe system (12) will be inconsistent

From (10) and (13) we can determine themasses119898119904119894119898119891119894

119898119894 as well as the mass 119898

119901119894= 119898119904119894+ 119898119891119894

of the dispersionphase for any 119894 Then we also find the density 120588

119901of the

dispersion phase in the tested LD

120588119901=

ℎ119904119894120588119904+ ℎ119891119894120588119891

ℎ119904119894+ ℎ119891119894

(14)

If we need to determine only the density 120588119901of the

dispersion phase and the density 120588 of the dispersionmedium

two containers 1198701and 119870

2are enough and a system with two

linear equations

119898119901119894+ 1198981= 1198721lArrrArr (ℎ

119904119894+ ℎ119891119894) 120588119901+ ℎ119894120588 = 119896119894 119894 = 1 2

(15)

Thus

120588119901=1198631

119863 120588 =

1198632

119863 (16)

where 119863 1198631 and 119863

2 according to the Cramers rule are

determinants of second orderIf we know the density of one of the ingredients for

example 120588 for distilledwater then also two containers1198701and

1198702are enough Then 120588

119904and 120588

119891are also determined with a

system with two linear equations

119898119904119894+ 119898119891119894+ 119898119894= 119872119894lArrrArr ℎ

119904119894120588119904+ ℎ119891119894120588119891= 119896119894minus ℎ119894120588

119894 = 1 2

(17)

which is solved analogically to system (12)In case when the density 120588 of the dispersion medium is

known andwe have only ultimate sediment or ultimate emer-gent it is sufficient to use only one container 119870 = 119870

1 Then

we have only one linear equation

119898119904+ 119898 = 119872 or 119898

119891+ 119898 = 119872 (18)

that is

120588119901=119896 minus ℎ120588

ℎ119904

or 120588119901=119896 minus ℎ120588

ℎ119891

(19)

where 120588119901= 120588119904or 120588119901= 120588119891and 119896 = 119872120587119877

2Because119898

119904= 1198981015840+ 11989810158401015840(119898119891= 1198981015840+ 11989810158401015840) where1198981015840 is the

mass of the dispersion phase and11989810158401015840 is the bound dispersionmedium in the ultimate sediment (emergent) and accordingto formula (19) we can find the density 1205881015840 of the dispersionphase

1205881015840=

120588119901minus 0 44120588

0 56 (20)

In this case when the tested LD has sediments and emer-gents consisting of several components then the density ofeach component can be determined through a measurementwith more containers In the general case we use a systemconsisting of 119899 linear equations and 119899 unknowns which ispresent in the following way

119901

sum

119896=0

119898119904119896 119894+

119897

sum

119895=0

119898119891119895 119894+ 119898119894= 119872119894 119894 = 1 2 119899 (21)

where 119899 = 119901+119897+1 From (21) the unknown densities 120588119904119896 119894 120588119891119895 119894

and 120588 are determined

From (13) (14) (16) and (19) using the method shownin Kolikov et al [7] we can determined the maximumabsolute inaccuracies (errors) Δ120588

119904 Δ120588119891 Δ120588119901 and Δ120588 and the

maximum relative inaccuracies (errors) Δ120588119904120588119904 Δ120588119891120588119891

Δ120588119901120588119901 and Δ120588120588 discussed by Kolikov et al [8]

6 The Scientific World Journal

6 Characterization Scale forSedimentation Stability

There are different methods for evaluation of the sedimenta-tion stability 119878 of LD One of the most popular methods is thevisual methodThe visual methodmeasures the observed dis-placement over a period of time of the horizontal separationline between the clear and the muddy part of the tested LD asdiscussed elsewhere [9 10]Themethod employs transparentcontainer and a clock

From the instrumental methods most popular ones indetermining the sedimentation stability are the opticalabsorptionmethodsmdashthroughmeasuring the intensity of thelight passing through the LD as discussed elsewhere [11 12]

According to our method the evaluation of 119878 is donethrough the densities of the components of the LD and it isnot necessary to measure the time and optical transmissionof LD is no required Moreover we introduce scale forcharacterization of the sedimentation stability 119878

Let one particle of the dispersion phase with volume 119881and density 120588

119901be located in dispersion medium with density

120588 in the conditions of gravitational field Then the value ofthe resultant force of its weight and the Archimedean force is

119865 = 119881119892 (120588119901minus 120588) (22)

where 119892 is gravitational constant (In (22) we neglect the fric-tion force which is very little for small velocities of theparticle)

Therefore if the particle is heavy that is 120588119901gt 120588 hArr

120588120588119901lt 1 then119865 gt 0 andwe have straight sedimentation (set-

tling) And if the particle is light that is 120588119901lt 120588 hArr 120588

119901120588 lt

1 then 119865 lt 0 and we have reverse sedimentation (emer-gence)

Thus we suggest the evaluation of the sedimentationstability 119878 to be done through the following dimensionlessphysical variables 119868

119904= 120588120588

119904 119868119891= 120588119891120588 (120588119904 120588 = 0) and hence

through the variable

119868 = max (119868119904 119868119891) = max(

120588

120588119904

120588119891

120588) (23)

The number 119868 is the sedimentation stability index in adimensionless scale for characterization of 119878 Since 0 lt 119868

119904

119868119891le 1 then 0 lt 119868 le 1 that is the scale is the interval (0 1]

The more the index 119868 of an LD is closer to 1 the more stablethe LD is and the closer the index 119868 is to 0 the more unstablethe LD is Thus if 119868 = 1 the LD is absolutely sedimentarystable that is there is no phase separation

We shall notice that if the sediment is absent in the LDthen 119868 = 119868

119891isin (0 1] and if the emergent is absent then 119868 =

119868119904isin (0 1]

7 Determining the Percentage ofthe Substance in the Sediment andthe Emergent

The amount of the substance in the dispersion phase is amajor factor which determines the quality of many products

The drying is widely used method for testing the dry matterin different suspensions To this end a sample is taken fromthe homogenized LD which is placed in a suitable containerweighted with a scale and placed in a drier for drying untilconstant weight is obtained This weight is divided into theinitial (not dried) full weights of the sample and thenmultiplied by 100 The obtained value is called dry substancepercentage of LDThe disadvantages of thismethod are its rel-atively long duration and the spending of significant thermalenergy for drying a unit of weight from the tested substanceMoreover the heating usually to 105∘C required by thestandard results in destruction of the treated substance and itcan only be used for suspensions

We offer a different method which does not have theseflaws

If 119870 is a straight cylindrical container filled-up with LDwith weight 119872 and internal radius of the bases 119877 thenaccording to formula (7) the masses of the dispersion phaseand the bound dispersion medium in the ultimate sedimentor emergent are respectively 1198981015840 = 0 561205871198772ℎ1205881015840 and 11989810158401015840 = 0441205871198772ℎ12058810158401015840 Then knowing the geometrical parameters of the

used container as well as the heights masses and densitieswe have in percentage the amount of the substance 1198751015840 in theultimate sediment or emergent and of the bound dispersionmedium 119875

10158401015840 respectively

1198751015840= (

1198981015840

119872sdot 100) = (

0 561205871198772ℎ1205881015840

119872sdot 100) (24)

11987510158401015840= (

11989810158401015840

119872sdot 100) = (

0 441205871198772ℎ12058810158401015840

119872sdot 100) (25)

The percentage 119875 of the free dispersion medium can alsobe calculated with our method namely

119875 = (

1205871198772(119867 minus ℎ

119904minus ℎ119891) 12058810158401015840

119872sdot 100) (26)

Let us denote that unlike the method of drying until con-stant weight through formula (24) the percentage 1198751015840 of thetested substance is determined with the participation of thebound dispersion medium We take into account only theamount of the pure dispersion phase in the specific sedimentor emergentThrough drying complete dryness of the samplecannot be achieved in practicemdashin the sediment therealways remains some percentage bound dispersion mediumTherefore we cannot expect to have coinciding percentage ofdry substance determined according to the two methods

Let us consider a case of sediment If 1198751015840119904is the percentage

of dry substance in the sediment determined with the dryingmethod then for practical reasons we can write downmodification of formula (24) namely

1198751015840

119904= 1205761198751015840

119904asymp (120576

0 561205871198772ℎ1199041205881015840

119872sdot 100) (27)

Here the variable 120576 is a conversion factor which canchange within the limits 1 lt 120576 lt (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904 where

The Scientific World Journal 7

Table 1 Linear dimensions masses and densities of the components of liquid dispersions

Exp 119877 [cm] ℎ1199041

[cm] ℎ1[cm] ℎ

1199042[cm] ℎ

2[cm] 119872

1[g] 119872

2[g] 120588

119904[gsdotcmminus3] 120588 [gsdotcmminus3]

1 13 24 63 36 76 5607 7397 169 1032 13 24 61 36 76 5517 7433 173 1023 13 24 61 35 78 5509 7439 172 1024 13 24 62 36 77 5574 7495 174 1025 13 24 62 36 77 5551 7458 171 1026 13 24 62 36 76 5557 7383 163 106Avg 13 24 62 36 77 5553 7434 170 103

K1 K2

m1

R R

h1

m2h2

hs1ms1

ms2 hs2

Figure 3 Vertical axial sections of two identical straight cylindricalcontainers with ultimate sedimentation

1198981015840

119904+ 11989810158401015840

119904= 119881

1015840

1199041205881015840

119904+ 11988110158401015840

11990412058810158401015840 is the total weight of the

sediment (dispersion phase and bound dispersion medium)The unobtainable value 120576 = 1 corresponds to the drysubstance in the ultimate sediment in case of complete dryingof the sample in119870 that is without bounddispersionmediumas with formula (24) we offer If the ultimate sediment is notsubjected to drying then the value 120576 = (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904asymp

1 + (120588101584010158401205881015840

119904) lt 2 Thus for the conversion factor in formula

(27) we have 1 lt 120576 lt 2

8 Example

We present models of two dispersion system suspensionsfrom calcium carbonate powder respectively 700 g and1100 g measured with electronic scale with accuracy of 001 gand distilled water respectively 50mL and 65mL measuredwith graduated cylinders with accuracy of 05mL In order toevaluate the accuracy of our method we assume the densitiesof CaCO

3and the distilled H

2O to be unknown

In order to determine the densities respectively 120588119904of the

ultimate sediment and 120588 of the dispersion medium we usetwo identical cylindrical containers 119870

1and 119870

2 with radius

of the internal basis 119877 = 0 013m (Figure 3)In the two containers we pour the two homogenized sus-

pensions (their components are in different proportions)Weperform 6 observations with room temperature 20∘C

Table 1 presents the values of the measured variables119877 ℎ1199041 ℎ1 1198721(for container 119870

1) and 119877 ℎ

1199042 ℎ2 1198722(for

container1198702) during the 6 observations

For densities 120588119904and 120588 according to formula (15) we find

with the system of linear equations

10038161003816100381610038161003816100381610038161003816

1198981199041+ 1198981= 1198721

1198981199042+ 1198982= 1198722

lArrrArr

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1199041120588119904+ ℎ1120588 =

1198721

1205871198772

ℎ1199042120588119904+ ℎ2120588 =

1198722

1205871198772

(28)

Its solutions are

120588119904=1198631

119863=

1

1205871198772

1198721ℎ2minus ℎ11198722

ℎ1199041ℎ2minus ℎ1ℎ1199042

120588 =1198632

119863=

1

1205871198772

ℎ11990411198722minus1198721ℎ1199042

ℎ1199041ℎ2minus ℎ1ℎ1199042

(29)

Then for the densities of the ultimate sediment and thedispersion medium we have respectively the average values120588119904asymp 1 70 gsdotcmminus3 and 120588 asymp 1 03 gsdotcmminus3 which are also

presented in Table 1For the index of the sedimentation stability we find

119868 =120588

120588119904

asymp 0 61 (30)

The value of 119868 according to our scale (01] shows that thetested LD has a little over the average sedimentation stability

According to the model discussed by Kolikov et al [7]we shall determine the maximum absolute inaccuracies Δ120588

119904

and Δ120588 for the established densities 120588119904and 120588 as well as the

maximum relative inaccuraciesΔ120588119904120588119904andΔ120588120588 IfΔ119872

119894Δℎ119894

Δℎ119904119894(119894 = 1 2) andΔ119877 are absolute inaccuracies of119872

119894 ℎ119894 ℎ119904119894

and 119877 then from (29) it follows that

Δ120588119904

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(31)

8 The Scientific World Journal

Table 2 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198631|

10038161003816100381610038161003816100381610038161003816

1198631

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

1 444 75 169 1056 1393 119 143 785 65 1065 1284 11542 372 644 173 1039 1400 115 143 777 624 1055 1315 9903 263 457 174 1037 1401 115 147 792 619 1061 1357 7044 384 663 173 1050 1411 117 145 788 635 1073 1332 10205 384 664 173 1045 1405 117 145 782 630 1073 1332 10226 408 661 162 1047 1390 117 143 807 658 1004 1231 628Avg 376 640 171 1046 1400 117 144 789 636 1055 1309 920

Table 3 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198632|

10038161003816100381610038161003816100381610038161003816

1198632

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

1 444 459 103 1056 1393 045 068 610 407 247 371 1582 372 380 102 1039 1400 045 068 625 417 245 367 1573 263 268 102 1037 1401 045 068 605 415 245 357 1574 384 394 103 1050 1411 045 068 618 411 247 371 1585 384 390 102 1045 1405 045 068 620 413 245 367 1576 408 433 106 1047 1390 045 068 584 390 254 382 163Avg 376 387 103 1046 1400 045 068 610 409 247 369 158

Δ120588

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(32)

where Δℎ1= 0 056 cm Δℎ

2= 0 067 cm Δℎ

1199041= Δℎ1199042

=

Δ119877 = 0 cm Δ1198721= 0 27 g and Δ119872

2= 0 30 g

Considering the calculations fromTables 2 and 3 we havethe values of the maximum absolute inaccuracies Δ120588

119904asymp

0 43 gsdotcmminus3 and Δ120588 asymp 0 18 gsdotcmminus3From here we find the values of the maximum relative

inaccuracies

Δ120588119904

120588119904

=043

170asymp 025

Δ120588

120588=018

103asymp 017 (33)

Thus obtained value of 120588 at 20∘C temperature is withinthe error for the density of the distilledwater which shows theaccuracy of our model

Then we find the percentage 1198751015840 of the dry substance inthe sediment and the percentages 11987510158401015840 and 119875 of the bounddispersionmedium in the sediment and of the free dispersionmedium accordingly We use formulae (20) (24) (25) and(26) and the data from Table 1 The results are shown inTable 4

Because of the approximations in the calculations thesum of the values of 1198751015840 11987510158401015840 and 119875 is around 100

9 Discussion

Our model of close packing of the particles from the dis-persion phase of the studied liquid dispersion offers a newmethod for finding the volumes and the masses of the sedi-mented substances in LD as well as the average Stokes radiiof the particles of these substances Moreover this study isdistinguishedwith simplicity and its application is very cheap

Our simple algebraicmethod offers an increased accuracyin determining the average densities and weights of the ulti-mate sediment and emergent of the dispersion medium andof the dispersion phase in the liquid dispersionThefinding ofthese densities is achieved without the use of pycnometers orother more complex methods This prevents the destructiveeffects in the LD and the vibrations practically have no effecton the measurements results unlike the methods existinguntil now

Our method of determining the coefficient of the quanti-ties of substances in the components of LD does not requirepouring out the liquid from the container in order tomeasurethemasses of the ultimate sediment and of the free dispersionmedium (water) The known method of pouring out anddrying is destructive Using the drying method on LD it isgenerally even impossible to determine the percentage of thesubstance amount in the emulsion The method we proposeovercomes this disadvantage

A great advantage and particularly valuable for onemethod is that it has numerical characteristicmdashscale for eval-uation of a given variable The index 119868 which we introduce in(23) is a dimensionless scale for characterization of the sedi-mentation stability 119878 naturally following from the law on themovement of the dispersion particle from the dispersionphase of LD It provides a very easy way to evaluate 119878 of

The Scientific World Journal 9

Table 4

Exp1 2 3 4 5 6

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

119872 (g) 5607 7397 5517 7433 5509 7439 5574 7495 5551 7458 5557 7383120588119904(gcm3) 169 173 172 174 172 163

120588 = 12058810158401015840 (gcm3) 103 102 102 102 102 106

1205881015840 (gcm3) 221 229 227 23 227 2071198751015840 () dry matter 2811 3197 2973 3296 2939 3174 2942 3284 2917 3256 2656 300011987510158401015840 () bound dispersion

medium in the sediment 1029 1169 1037 1153 1038 1121 1026 1143 1030 1149 1069 1207

119875 () free dispersion medium 6146 5620 5985 5535 5994 5675 6021 5561 6046 5589 6275 5790

one LD in straight andor reverse sedimentation through thedensities 120588

119904 120588119891 and 120588 which are the solution of the system of

linear equations of type (11)The index 119868 isin (0 1] provides the possibility not only to

evaluate and to predict the sedimentation stability but alsoto compare different LDs according to their sedimentationstability Moreover the tests of the different LDs must beperformed in constant temperature and external pressure

10 Conclusion

The method which we offer does not require special equip-ment The phase separation of LDs can be achieved in theconditions not only of gravitational (earthrsquos) but also of cen-trifugal field In the case of centrifugal field the time interval 119905for determining the ultimate sediment and emergent can bemuch smaller compared with time for treating the LD withthe acceleration of gravity 119892

The advantage of our physicomathematical model forfinding the Stokes radii in the dispersion phase is that it doesnot use complicatedmathematical analysis for structuring thesedimentation curve

The three variables 1198751015840 11987510158401015840 and 119875 characterizing in per-centage the respective quantity of substance in the ultimatesediment or emergent bound dispersion medium in the sed-iment or the emergent and the free dispersion medium areintroduced in the literature for the first time with this paper

And with the sedimentation stability index 119868 whichwe introduce it is possible to demonstratively and withoutparticular difficultiesmake qualitative analysis of suspensionsand emulsionsmdashproducts of the food medical cosmeticchemical and so forth industries Such products are forexample different suspensions and emulsions such as foodsand drinks medications and cosmetic products paints andvarnishes and cement and lime solutionsThis is particularlyand technically useful in the creation of products withpredetermined sedimentation stability

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The obtained results are made with the support of Fund ldquoSci-entific Studiesrdquo of the Bulgarian Ministry of Education andScience under the Contract T 12 This paper is publishedwith the financial support of the Project BG051PO00133-05-001 ldquoScience and Businessrdquo financed by the HumanResources Development Operational Programme of theEuropean Social Fund

References

[1] D Dakova D Christozov and M Beleva ldquoBarycentric methodof determining the physical parameters of a single-phase parti-cle in liquid disperse systemsrdquo Journal of Colloid and InterfaceScience vol 256 no 2 pp 477ndash479 2002

[2] R Chanamai and D J McClements ldquoDepletion flocculation ofbeverage emulsions by gum arabic andmodified starchrdquo Journalof Food Science vol 66 no 3 pp 457ndash463 2001

[3] G S Hodakov andU P Udkin SedimentationAnalysis of HighlyDispersed Systems (in Russian) Chemistry Moscow Russia1981

[4] B M Westwood and V N Kabadi ldquoA novel pycnometer fordensity measurements of liquids at elevated temperaturesrdquoTheJournal of Chemical Thermodynamics vol 35 no 12 pp 1965ndash1974 2003

[5] A Puttmer R Lucklum B Henning and P HauptmannldquoImproved ultrasonic density sensor with reduced diffractioninfluencerdquo Sensors and Actuators A Physical vol 67 no 1ndash3 pp8ndash12 1998

[6] E D Shchukin A V Percov and E A Amelina ColloidalChemistry pp 306ndash309 Yurait Moscow Russia 6th edition2012 (Russian)

[7] K Kolikov G Krastev Y Epitropov and D Hristozov ldquoAna-lytically determining of the absolute inaccuracy (error) ofindirectly measurable variable and dimensionless scale char-acterising the quality of the experimentrdquo Chemometrics andIntelligent Laboratory Systems vol 102 no 1 pp 15ndash19 2010

[8] K Kolikov G Krastev Y Epitropov and A Corlat ldquoAnalyticallydetermining of the relative inaccuracy (error) of indirectlymeasurable variable and dimensionless scale characterising thequality of the experimentrdquoCSJM vol 20 no 1 pp 314ndash331 2012

[9] M H Ly M Aguedo S Goudot et al ldquoInteractions betweenbacterial surfaces and milk proteins impact on food emulsionsstabilityrdquo Food Hydrocolloids vol 22 no 5 pp 742ndash751 2008

10 The Scientific World Journal

[10] J P F Macedo L L Fernandes F R Formiga et al ldquoMicro-emultocrit technique a valuable tool for determination ofcritical HLB value of emulsionsrdquo AAPS PharmSciTech vol 7no 1 pp E1ndashE7 2006

[11] M Corredig and M Alexander ldquoFood emulsions studied byDWS recent advancesrdquo Trends in Food Science and Technologyvol 19 no 2 pp 67ndash75 2008

[12] J M Quintana A Califano and N Zaritzky ldquoMicrostructureand stability of non-protein stabilized oil-in-water food emul-sionsmeasured by opticalmethodsrdquo Journal of Food Science vol67 no 3 pp 1130ndash1135 2002

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CatalystsJournal of

6 The Scientific World Journal

6 Characterization Scale forSedimentation Stability

There are different methods for evaluation of the sedimenta-tion stability 119878 of LD One of the most popular methods is thevisual methodThe visual methodmeasures the observed dis-placement over a period of time of the horizontal separationline between the clear and the muddy part of the tested LD asdiscussed elsewhere [9 10]Themethod employs transparentcontainer and a clock

From the instrumental methods most popular ones indetermining the sedimentation stability are the opticalabsorptionmethodsmdashthroughmeasuring the intensity of thelight passing through the LD as discussed elsewhere [11 12]

According to our method the evaluation of 119878 is donethrough the densities of the components of the LD and it isnot necessary to measure the time and optical transmissionof LD is no required Moreover we introduce scale forcharacterization of the sedimentation stability 119878

Let one particle of the dispersion phase with volume 119881and density 120588

119901be located in dispersion medium with density

120588 in the conditions of gravitational field Then the value ofthe resultant force of its weight and the Archimedean force is

119865 = 119881119892 (120588119901minus 120588) (22)

where 119892 is gravitational constant (In (22) we neglect the fric-tion force which is very little for small velocities of theparticle)

Therefore if the particle is heavy that is 120588119901gt 120588 hArr

120588120588119901lt 1 then119865 gt 0 andwe have straight sedimentation (set-

tling) And if the particle is light that is 120588119901lt 120588 hArr 120588

119901120588 lt

1 then 119865 lt 0 and we have reverse sedimentation (emer-gence)

Thus we suggest the evaluation of the sedimentationstability 119878 to be done through the following dimensionlessphysical variables 119868

119904= 120588120588

119904 119868119891= 120588119891120588 (120588119904 120588 = 0) and hence

through the variable

119868 = max (119868119904 119868119891) = max(

120588

120588119904

120588119891

120588) (23)

The number 119868 is the sedimentation stability index in adimensionless scale for characterization of 119878 Since 0 lt 119868

119904

119868119891le 1 then 0 lt 119868 le 1 that is the scale is the interval (0 1]

The more the index 119868 of an LD is closer to 1 the more stablethe LD is and the closer the index 119868 is to 0 the more unstablethe LD is Thus if 119868 = 1 the LD is absolutely sedimentarystable that is there is no phase separation

We shall notice that if the sediment is absent in the LDthen 119868 = 119868

119891isin (0 1] and if the emergent is absent then 119868 =

119868119904isin (0 1]

7 Determining the Percentage ofthe Substance in the Sediment andthe Emergent

The amount of the substance in the dispersion phase is amajor factor which determines the quality of many products

The drying is widely used method for testing the dry matterin different suspensions To this end a sample is taken fromthe homogenized LD which is placed in a suitable containerweighted with a scale and placed in a drier for drying untilconstant weight is obtained This weight is divided into theinitial (not dried) full weights of the sample and thenmultiplied by 100 The obtained value is called dry substancepercentage of LDThe disadvantages of thismethod are its rel-atively long duration and the spending of significant thermalenergy for drying a unit of weight from the tested substanceMoreover the heating usually to 105∘C required by thestandard results in destruction of the treated substance and itcan only be used for suspensions

We offer a different method which does not have theseflaws

If 119870 is a straight cylindrical container filled-up with LDwith weight 119872 and internal radius of the bases 119877 thenaccording to formula (7) the masses of the dispersion phaseand the bound dispersion medium in the ultimate sedimentor emergent are respectively 1198981015840 = 0 561205871198772ℎ1205881015840 and 11989810158401015840 = 0441205871198772ℎ12058810158401015840 Then knowing the geometrical parameters of the

used container as well as the heights masses and densitieswe have in percentage the amount of the substance 1198751015840 in theultimate sediment or emergent and of the bound dispersionmedium 119875

10158401015840 respectively

1198751015840= (

1198981015840

119872sdot 100) = (

0 561205871198772ℎ1205881015840

119872sdot 100) (24)

11987510158401015840= (

11989810158401015840

119872sdot 100) = (

0 441205871198772ℎ12058810158401015840

119872sdot 100) (25)

The percentage 119875 of the free dispersion medium can alsobe calculated with our method namely

119875 = (

1205871198772(119867 minus ℎ

119904minus ℎ119891) 12058810158401015840

119872sdot 100) (26)

Let us denote that unlike the method of drying until con-stant weight through formula (24) the percentage 1198751015840 of thetested substance is determined with the participation of thebound dispersion medium We take into account only theamount of the pure dispersion phase in the specific sedimentor emergentThrough drying complete dryness of the samplecannot be achieved in practicemdashin the sediment therealways remains some percentage bound dispersion mediumTherefore we cannot expect to have coinciding percentage ofdry substance determined according to the two methods

Let us consider a case of sediment If 1198751015840119904is the percentage

of dry substance in the sediment determined with the dryingmethod then for practical reasons we can write downmodification of formula (24) namely

1198751015840

119904= 1205761198751015840

119904asymp (120576

0 561205871198772ℎ1199041205881015840

119872sdot 100) (27)

Here the variable 120576 is a conversion factor which canchange within the limits 1 lt 120576 lt (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904 where

The Scientific World Journal 7

Table 1 Linear dimensions masses and densities of the components of liquid dispersions

Exp 119877 [cm] ℎ1199041

[cm] ℎ1[cm] ℎ

1199042[cm] ℎ

2[cm] 119872

1[g] 119872

2[g] 120588

119904[gsdotcmminus3] 120588 [gsdotcmminus3]

1 13 24 63 36 76 5607 7397 169 1032 13 24 61 36 76 5517 7433 173 1023 13 24 61 35 78 5509 7439 172 1024 13 24 62 36 77 5574 7495 174 1025 13 24 62 36 77 5551 7458 171 1026 13 24 62 36 76 5557 7383 163 106Avg 13 24 62 36 77 5553 7434 170 103

K1 K2

m1

R R

h1

m2h2

hs1ms1

ms2 hs2

Figure 3 Vertical axial sections of two identical straight cylindricalcontainers with ultimate sedimentation

1198981015840

119904+ 11989810158401015840

119904= 119881

1015840

1199041205881015840

119904+ 11988110158401015840

11990412058810158401015840 is the total weight of the

sediment (dispersion phase and bound dispersion medium)The unobtainable value 120576 = 1 corresponds to the drysubstance in the ultimate sediment in case of complete dryingof the sample in119870 that is without bounddispersionmediumas with formula (24) we offer If the ultimate sediment is notsubjected to drying then the value 120576 = (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904asymp

1 + (120588101584010158401205881015840

119904) lt 2 Thus for the conversion factor in formula

(27) we have 1 lt 120576 lt 2

8 Example

We present models of two dispersion system suspensionsfrom calcium carbonate powder respectively 700 g and1100 g measured with electronic scale with accuracy of 001 gand distilled water respectively 50mL and 65mL measuredwith graduated cylinders with accuracy of 05mL In order toevaluate the accuracy of our method we assume the densitiesof CaCO

3and the distilled H

2O to be unknown

In order to determine the densities respectively 120588119904of the

ultimate sediment and 120588 of the dispersion medium we usetwo identical cylindrical containers 119870

1and 119870

2 with radius

of the internal basis 119877 = 0 013m (Figure 3)In the two containers we pour the two homogenized sus-

pensions (their components are in different proportions)Weperform 6 observations with room temperature 20∘C

Table 1 presents the values of the measured variables119877 ℎ1199041 ℎ1 1198721(for container 119870

1) and 119877 ℎ

1199042 ℎ2 1198722(for

container1198702) during the 6 observations

For densities 120588119904and 120588 according to formula (15) we find

with the system of linear equations

10038161003816100381610038161003816100381610038161003816

1198981199041+ 1198981= 1198721

1198981199042+ 1198982= 1198722

lArrrArr

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1199041120588119904+ ℎ1120588 =

1198721

1205871198772

ℎ1199042120588119904+ ℎ2120588 =

1198722

1205871198772

(28)

Its solutions are

120588119904=1198631

119863=

1

1205871198772

1198721ℎ2minus ℎ11198722

ℎ1199041ℎ2minus ℎ1ℎ1199042

120588 =1198632

119863=

1

1205871198772

ℎ11990411198722minus1198721ℎ1199042

ℎ1199041ℎ2minus ℎ1ℎ1199042

(29)

Then for the densities of the ultimate sediment and thedispersion medium we have respectively the average values120588119904asymp 1 70 gsdotcmminus3 and 120588 asymp 1 03 gsdotcmminus3 which are also

presented in Table 1For the index of the sedimentation stability we find

119868 =120588

120588119904

asymp 0 61 (30)

The value of 119868 according to our scale (01] shows that thetested LD has a little over the average sedimentation stability

According to the model discussed by Kolikov et al [7]we shall determine the maximum absolute inaccuracies Δ120588

119904

and Δ120588 for the established densities 120588119904and 120588 as well as the

maximum relative inaccuraciesΔ120588119904120588119904andΔ120588120588 IfΔ119872

119894Δℎ119894

Δℎ119904119894(119894 = 1 2) andΔ119877 are absolute inaccuracies of119872

119894 ℎ119894 ℎ119904119894

and 119877 then from (29) it follows that

Δ120588119904

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(31)

8 The Scientific World Journal

Table 2 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198631|

10038161003816100381610038161003816100381610038161003816

1198631

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

1 444 75 169 1056 1393 119 143 785 65 1065 1284 11542 372 644 173 1039 1400 115 143 777 624 1055 1315 9903 263 457 174 1037 1401 115 147 792 619 1061 1357 7044 384 663 173 1050 1411 117 145 788 635 1073 1332 10205 384 664 173 1045 1405 117 145 782 630 1073 1332 10226 408 661 162 1047 1390 117 143 807 658 1004 1231 628Avg 376 640 171 1046 1400 117 144 789 636 1055 1309 920

Table 3 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198632|

10038161003816100381610038161003816100381610038161003816

1198632

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

1 444 459 103 1056 1393 045 068 610 407 247 371 1582 372 380 102 1039 1400 045 068 625 417 245 367 1573 263 268 102 1037 1401 045 068 605 415 245 357 1574 384 394 103 1050 1411 045 068 618 411 247 371 1585 384 390 102 1045 1405 045 068 620 413 245 367 1576 408 433 106 1047 1390 045 068 584 390 254 382 163Avg 376 387 103 1046 1400 045 068 610 409 247 369 158

Δ120588

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(32)

where Δℎ1= 0 056 cm Δℎ

2= 0 067 cm Δℎ

1199041= Δℎ1199042

=

Δ119877 = 0 cm Δ1198721= 0 27 g and Δ119872

2= 0 30 g

Considering the calculations fromTables 2 and 3 we havethe values of the maximum absolute inaccuracies Δ120588

119904asymp

0 43 gsdotcmminus3 and Δ120588 asymp 0 18 gsdotcmminus3From here we find the values of the maximum relative

inaccuracies

Δ120588119904

120588119904

=043

170asymp 025

Δ120588

120588=018

103asymp 017 (33)

Thus obtained value of 120588 at 20∘C temperature is withinthe error for the density of the distilledwater which shows theaccuracy of our model

Then we find the percentage 1198751015840 of the dry substance inthe sediment and the percentages 11987510158401015840 and 119875 of the bounddispersionmedium in the sediment and of the free dispersionmedium accordingly We use formulae (20) (24) (25) and(26) and the data from Table 1 The results are shown inTable 4

Because of the approximations in the calculations thesum of the values of 1198751015840 11987510158401015840 and 119875 is around 100

9 Discussion

Our model of close packing of the particles from the dis-persion phase of the studied liquid dispersion offers a newmethod for finding the volumes and the masses of the sedi-mented substances in LD as well as the average Stokes radiiof the particles of these substances Moreover this study isdistinguishedwith simplicity and its application is very cheap

Our simple algebraicmethod offers an increased accuracyin determining the average densities and weights of the ulti-mate sediment and emergent of the dispersion medium andof the dispersion phase in the liquid dispersionThefinding ofthese densities is achieved without the use of pycnometers orother more complex methods This prevents the destructiveeffects in the LD and the vibrations practically have no effecton the measurements results unlike the methods existinguntil now

Our method of determining the coefficient of the quanti-ties of substances in the components of LD does not requirepouring out the liquid from the container in order tomeasurethemasses of the ultimate sediment and of the free dispersionmedium (water) The known method of pouring out anddrying is destructive Using the drying method on LD it isgenerally even impossible to determine the percentage of thesubstance amount in the emulsion The method we proposeovercomes this disadvantage

A great advantage and particularly valuable for onemethod is that it has numerical characteristicmdashscale for eval-uation of a given variable The index 119868 which we introduce in(23) is a dimensionless scale for characterization of the sedi-mentation stability 119878 naturally following from the law on themovement of the dispersion particle from the dispersionphase of LD It provides a very easy way to evaluate 119878 of

The Scientific World Journal 9

Table 4

Exp1 2 3 4 5 6

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

119872 (g) 5607 7397 5517 7433 5509 7439 5574 7495 5551 7458 5557 7383120588119904(gcm3) 169 173 172 174 172 163

120588 = 12058810158401015840 (gcm3) 103 102 102 102 102 106

1205881015840 (gcm3) 221 229 227 23 227 2071198751015840 () dry matter 2811 3197 2973 3296 2939 3174 2942 3284 2917 3256 2656 300011987510158401015840 () bound dispersion

medium in the sediment 1029 1169 1037 1153 1038 1121 1026 1143 1030 1149 1069 1207

119875 () free dispersion medium 6146 5620 5985 5535 5994 5675 6021 5561 6046 5589 6275 5790

one LD in straight andor reverse sedimentation through thedensities 120588

119904 120588119891 and 120588 which are the solution of the system of

linear equations of type (11)The index 119868 isin (0 1] provides the possibility not only to

evaluate and to predict the sedimentation stability but alsoto compare different LDs according to their sedimentationstability Moreover the tests of the different LDs must beperformed in constant temperature and external pressure

10 Conclusion

The method which we offer does not require special equip-ment The phase separation of LDs can be achieved in theconditions not only of gravitational (earthrsquos) but also of cen-trifugal field In the case of centrifugal field the time interval 119905for determining the ultimate sediment and emergent can bemuch smaller compared with time for treating the LD withthe acceleration of gravity 119892

The advantage of our physicomathematical model forfinding the Stokes radii in the dispersion phase is that it doesnot use complicatedmathematical analysis for structuring thesedimentation curve

The three variables 1198751015840 11987510158401015840 and 119875 characterizing in per-centage the respective quantity of substance in the ultimatesediment or emergent bound dispersion medium in the sed-iment or the emergent and the free dispersion medium areintroduced in the literature for the first time with this paper

And with the sedimentation stability index 119868 whichwe introduce it is possible to demonstratively and withoutparticular difficultiesmake qualitative analysis of suspensionsand emulsionsmdashproducts of the food medical cosmeticchemical and so forth industries Such products are forexample different suspensions and emulsions such as foodsand drinks medications and cosmetic products paints andvarnishes and cement and lime solutionsThis is particularlyand technically useful in the creation of products withpredetermined sedimentation stability

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The obtained results are made with the support of Fund ldquoSci-entific Studiesrdquo of the Bulgarian Ministry of Education andScience under the Contract T 12 This paper is publishedwith the financial support of the Project BG051PO00133-05-001 ldquoScience and Businessrdquo financed by the HumanResources Development Operational Programme of theEuropean Social Fund

References

[1] D Dakova D Christozov and M Beleva ldquoBarycentric methodof determining the physical parameters of a single-phase parti-cle in liquid disperse systemsrdquo Journal of Colloid and InterfaceScience vol 256 no 2 pp 477ndash479 2002

[2] R Chanamai and D J McClements ldquoDepletion flocculation ofbeverage emulsions by gum arabic andmodified starchrdquo Journalof Food Science vol 66 no 3 pp 457ndash463 2001

[3] G S Hodakov andU P Udkin SedimentationAnalysis of HighlyDispersed Systems (in Russian) Chemistry Moscow Russia1981

[4] B M Westwood and V N Kabadi ldquoA novel pycnometer fordensity measurements of liquids at elevated temperaturesrdquoTheJournal of Chemical Thermodynamics vol 35 no 12 pp 1965ndash1974 2003

[5] A Puttmer R Lucklum B Henning and P HauptmannldquoImproved ultrasonic density sensor with reduced diffractioninfluencerdquo Sensors and Actuators A Physical vol 67 no 1ndash3 pp8ndash12 1998

[6] E D Shchukin A V Percov and E A Amelina ColloidalChemistry pp 306ndash309 Yurait Moscow Russia 6th edition2012 (Russian)

[7] K Kolikov G Krastev Y Epitropov and D Hristozov ldquoAna-lytically determining of the absolute inaccuracy (error) ofindirectly measurable variable and dimensionless scale char-acterising the quality of the experimentrdquo Chemometrics andIntelligent Laboratory Systems vol 102 no 1 pp 15ndash19 2010

[8] K Kolikov G Krastev Y Epitropov and A Corlat ldquoAnalyticallydetermining of the relative inaccuracy (error) of indirectlymeasurable variable and dimensionless scale characterising thequality of the experimentrdquoCSJM vol 20 no 1 pp 314ndash331 2012

[9] M H Ly M Aguedo S Goudot et al ldquoInteractions betweenbacterial surfaces and milk proteins impact on food emulsionsstabilityrdquo Food Hydrocolloids vol 22 no 5 pp 742ndash751 2008

10 The Scientific World Journal

[10] J P F Macedo L L Fernandes F R Formiga et al ldquoMicro-emultocrit technique a valuable tool for determination ofcritical HLB value of emulsionsrdquo AAPS PharmSciTech vol 7no 1 pp E1ndashE7 2006

[11] M Corredig and M Alexander ldquoFood emulsions studied byDWS recent advancesrdquo Trends in Food Science and Technologyvol 19 no 2 pp 67ndash75 2008

[12] J M Quintana A Califano and N Zaritzky ldquoMicrostructureand stability of non-protein stabilized oil-in-water food emul-sionsmeasured by opticalmethodsrdquo Journal of Food Science vol67 no 3 pp 1130ndash1135 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

The Scientific World Journal 7

Table 1 Linear dimensions masses and densities of the components of liquid dispersions

Exp 119877 [cm] ℎ1199041

[cm] ℎ1[cm] ℎ

1199042[cm] ℎ

2[cm] 119872

1[g] 119872

2[g] 120588

119904[gsdotcmminus3] 120588 [gsdotcmminus3]

1 13 24 63 36 76 5607 7397 169 1032 13 24 61 36 76 5517 7433 173 1023 13 24 61 35 78 5509 7439 172 1024 13 24 62 36 77 5574 7495 174 1025 13 24 62 36 77 5551 7458 171 1026 13 24 62 36 76 5557 7383 163 106Avg 13 24 62 36 77 5553 7434 170 103

K1 K2

m1

R R

h1

m2h2

hs1ms1

ms2 hs2

Figure 3 Vertical axial sections of two identical straight cylindricalcontainers with ultimate sedimentation

1198981015840

119904+ 11989810158401015840

119904= 119881

1015840

1199041205881015840

119904+ 11988110158401015840

11990412058810158401015840 is the total weight of the

sediment (dispersion phase and bound dispersion medium)The unobtainable value 120576 = 1 corresponds to the drysubstance in the ultimate sediment in case of complete dryingof the sample in119870 that is without bounddispersionmediumas with formula (24) we offer If the ultimate sediment is notsubjected to drying then the value 120576 = (119898

1015840

119904+ 11989810158401015840

119904)1198981015840

119904asymp

1 + (120588101584010158401205881015840

119904) lt 2 Thus for the conversion factor in formula

(27) we have 1 lt 120576 lt 2

8 Example

We present models of two dispersion system suspensionsfrom calcium carbonate powder respectively 700 g and1100 g measured with electronic scale with accuracy of 001 gand distilled water respectively 50mL and 65mL measuredwith graduated cylinders with accuracy of 05mL In order toevaluate the accuracy of our method we assume the densitiesof CaCO

3and the distilled H

2O to be unknown

In order to determine the densities respectively 120588119904of the

ultimate sediment and 120588 of the dispersion medium we usetwo identical cylindrical containers 119870

1and 119870

2 with radius

of the internal basis 119877 = 0 013m (Figure 3)In the two containers we pour the two homogenized sus-

pensions (their components are in different proportions)Weperform 6 observations with room temperature 20∘C

Table 1 presents the values of the measured variables119877 ℎ1199041 ℎ1 1198721(for container 119870

1) and 119877 ℎ

1199042 ℎ2 1198722(for

container1198702) during the 6 observations

For densities 120588119904and 120588 according to formula (15) we find

with the system of linear equations

10038161003816100381610038161003816100381610038161003816

1198981199041+ 1198981= 1198721

1198981199042+ 1198982= 1198722

lArrrArr

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

ℎ1199041120588119904+ ℎ1120588 =

1198721

1205871198772

ℎ1199042120588119904+ ℎ2120588 =

1198722

1205871198772

(28)

Its solutions are

120588119904=1198631

119863=

1

1205871198772

1198721ℎ2minus ℎ11198722

ℎ1199041ℎ2minus ℎ1ℎ1199042

120588 =1198632

119863=

1

1205871198772

ℎ11990411198722minus1198721ℎ1199042

ℎ1199041ℎ2minus ℎ1ℎ1199042

(29)

Then for the densities of the ultimate sediment and thedispersion medium we have respectively the average values120588119904asymp 1 70 gsdotcmminus3 and 120588 asymp 1 03 gsdotcmminus3 which are also

presented in Table 1For the index of the sedimentation stability we find

119868 =120588

120588119904

asymp 0 61 (30)

The value of 119868 according to our scale (01] shows that thetested LD has a little over the average sedimentation stability

According to the model discussed by Kolikov et al [7]we shall determine the maximum absolute inaccuracies Δ120588

119904

and Δ120588 for the established densities 120588119904and 120588 as well as the

maximum relative inaccuraciesΔ120588119904120588119904andΔ120588120588 IfΔ119872

119894Δℎ119894

Δℎ119904119894(119894 = 1 2) andΔ119877 are absolute inaccuracies of119872

119894 ℎ119894 ℎ119904119894

and 119877 then from (29) it follows that

Δ120588119904

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(31)

8 The Scientific World Journal

Table 2 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198631|

10038161003816100381610038161003816100381610038161003816

1198631

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

1 444 75 169 1056 1393 119 143 785 65 1065 1284 11542 372 644 173 1039 1400 115 143 777 624 1055 1315 9903 263 457 174 1037 1401 115 147 792 619 1061 1357 7044 384 663 173 1050 1411 117 145 788 635 1073 1332 10205 384 664 173 1045 1405 117 145 782 630 1073 1332 10226 408 661 162 1047 1390 117 143 807 658 1004 1231 628Avg 376 640 171 1046 1400 117 144 789 636 1055 1309 920

Table 3 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198632|

10038161003816100381610038161003816100381610038161003816

1198632

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

1 444 459 103 1056 1393 045 068 610 407 247 371 1582 372 380 102 1039 1400 045 068 625 417 245 367 1573 263 268 102 1037 1401 045 068 605 415 245 357 1574 384 394 103 1050 1411 045 068 618 411 247 371 1585 384 390 102 1045 1405 045 068 620 413 245 367 1576 408 433 106 1047 1390 045 068 584 390 254 382 163Avg 376 387 103 1046 1400 045 068 610 409 247 369 158

Δ120588

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(32)

where Δℎ1= 0 056 cm Δℎ

2= 0 067 cm Δℎ

1199041= Δℎ1199042

=

Δ119877 = 0 cm Δ1198721= 0 27 g and Δ119872

2= 0 30 g

Considering the calculations fromTables 2 and 3 we havethe values of the maximum absolute inaccuracies Δ120588

119904asymp

0 43 gsdotcmminus3 and Δ120588 asymp 0 18 gsdotcmminus3From here we find the values of the maximum relative

inaccuracies

Δ120588119904

120588119904

=043

170asymp 025

Δ120588

120588=018

103asymp 017 (33)

Thus obtained value of 120588 at 20∘C temperature is withinthe error for the density of the distilledwater which shows theaccuracy of our model

Then we find the percentage 1198751015840 of the dry substance inthe sediment and the percentages 11987510158401015840 and 119875 of the bounddispersionmedium in the sediment and of the free dispersionmedium accordingly We use formulae (20) (24) (25) and(26) and the data from Table 1 The results are shown inTable 4

Because of the approximations in the calculations thesum of the values of 1198751015840 11987510158401015840 and 119875 is around 100

9 Discussion

Our model of close packing of the particles from the dis-persion phase of the studied liquid dispersion offers a newmethod for finding the volumes and the masses of the sedi-mented substances in LD as well as the average Stokes radiiof the particles of these substances Moreover this study isdistinguishedwith simplicity and its application is very cheap

Our simple algebraicmethod offers an increased accuracyin determining the average densities and weights of the ulti-mate sediment and emergent of the dispersion medium andof the dispersion phase in the liquid dispersionThefinding ofthese densities is achieved without the use of pycnometers orother more complex methods This prevents the destructiveeffects in the LD and the vibrations practically have no effecton the measurements results unlike the methods existinguntil now

Our method of determining the coefficient of the quanti-ties of substances in the components of LD does not requirepouring out the liquid from the container in order tomeasurethemasses of the ultimate sediment and of the free dispersionmedium (water) The known method of pouring out anddrying is destructive Using the drying method on LD it isgenerally even impossible to determine the percentage of thesubstance amount in the emulsion The method we proposeovercomes this disadvantage

A great advantage and particularly valuable for onemethod is that it has numerical characteristicmdashscale for eval-uation of a given variable The index 119868 which we introduce in(23) is a dimensionless scale for characterization of the sedi-mentation stability 119878 naturally following from the law on themovement of the dispersion particle from the dispersionphase of LD It provides a very easy way to evaluate 119878 of

The Scientific World Journal 9

Table 4

Exp1 2 3 4 5 6

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

119872 (g) 5607 7397 5517 7433 5509 7439 5574 7495 5551 7458 5557 7383120588119904(gcm3) 169 173 172 174 172 163

120588 = 12058810158401015840 (gcm3) 103 102 102 102 102 106

1205881015840 (gcm3) 221 229 227 23 227 2071198751015840 () dry matter 2811 3197 2973 3296 2939 3174 2942 3284 2917 3256 2656 300011987510158401015840 () bound dispersion

medium in the sediment 1029 1169 1037 1153 1038 1121 1026 1143 1030 1149 1069 1207

119875 () free dispersion medium 6146 5620 5985 5535 5994 5675 6021 5561 6046 5589 6275 5790

one LD in straight andor reverse sedimentation through thedensities 120588

119904 120588119891 and 120588 which are the solution of the system of

linear equations of type (11)The index 119868 isin (0 1] provides the possibility not only to

evaluate and to predict the sedimentation stability but alsoto compare different LDs according to their sedimentationstability Moreover the tests of the different LDs must beperformed in constant temperature and external pressure

10 Conclusion

The method which we offer does not require special equip-ment The phase separation of LDs can be achieved in theconditions not only of gravitational (earthrsquos) but also of cen-trifugal field In the case of centrifugal field the time interval 119905for determining the ultimate sediment and emergent can bemuch smaller compared with time for treating the LD withthe acceleration of gravity 119892

The advantage of our physicomathematical model forfinding the Stokes radii in the dispersion phase is that it doesnot use complicatedmathematical analysis for structuring thesedimentation curve

The three variables 1198751015840 11987510158401015840 and 119875 characterizing in per-centage the respective quantity of substance in the ultimatesediment or emergent bound dispersion medium in the sed-iment or the emergent and the free dispersion medium areintroduced in the literature for the first time with this paper

And with the sedimentation stability index 119868 whichwe introduce it is possible to demonstratively and withoutparticular difficultiesmake qualitative analysis of suspensionsand emulsionsmdashproducts of the food medical cosmeticchemical and so forth industries Such products are forexample different suspensions and emulsions such as foodsand drinks medications and cosmetic products paints andvarnishes and cement and lime solutionsThis is particularlyand technically useful in the creation of products withpredetermined sedimentation stability

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The obtained results are made with the support of Fund ldquoSci-entific Studiesrdquo of the Bulgarian Ministry of Education andScience under the Contract T 12 This paper is publishedwith the financial support of the Project BG051PO00133-05-001 ldquoScience and Businessrdquo financed by the HumanResources Development Operational Programme of theEuropean Social Fund

References

[1] D Dakova D Christozov and M Beleva ldquoBarycentric methodof determining the physical parameters of a single-phase parti-cle in liquid disperse systemsrdquo Journal of Colloid and InterfaceScience vol 256 no 2 pp 477ndash479 2002

[2] R Chanamai and D J McClements ldquoDepletion flocculation ofbeverage emulsions by gum arabic andmodified starchrdquo Journalof Food Science vol 66 no 3 pp 457ndash463 2001

[3] G S Hodakov andU P Udkin SedimentationAnalysis of HighlyDispersed Systems (in Russian) Chemistry Moscow Russia1981

[4] B M Westwood and V N Kabadi ldquoA novel pycnometer fordensity measurements of liquids at elevated temperaturesrdquoTheJournal of Chemical Thermodynamics vol 35 no 12 pp 1965ndash1974 2003

[5] A Puttmer R Lucklum B Henning and P HauptmannldquoImproved ultrasonic density sensor with reduced diffractioninfluencerdquo Sensors and Actuators A Physical vol 67 no 1ndash3 pp8ndash12 1998

[6] E D Shchukin A V Percov and E A Amelina ColloidalChemistry pp 306ndash309 Yurait Moscow Russia 6th edition2012 (Russian)

[7] K Kolikov G Krastev Y Epitropov and D Hristozov ldquoAna-lytically determining of the absolute inaccuracy (error) ofindirectly measurable variable and dimensionless scale char-acterising the quality of the experimentrdquo Chemometrics andIntelligent Laboratory Systems vol 102 no 1 pp 15ndash19 2010

[8] K Kolikov G Krastev Y Epitropov and A Corlat ldquoAnalyticallydetermining of the relative inaccuracy (error) of indirectlymeasurable variable and dimensionless scale characterising thequality of the experimentrdquoCSJM vol 20 no 1 pp 314ndash331 2012

[9] M H Ly M Aguedo S Goudot et al ldquoInteractions betweenbacterial surfaces and milk proteins impact on food emulsionsstabilityrdquo Food Hydrocolloids vol 22 no 5 pp 742ndash751 2008

10 The Scientific World Journal

[10] J P F Macedo L L Fernandes F R Formiga et al ldquoMicro-emultocrit technique a valuable tool for determination ofcritical HLB value of emulsionsrdquo AAPS PharmSciTech vol 7no 1 pp E1ndashE7 2006

[11] M Corredig and M Alexander ldquoFood emulsions studied byDWS recent advancesrdquo Trends in Food Science and Technologyvol 19 no 2 pp 67ndash75 2008

[12] J M Quintana A Califano and N Zaritzky ldquoMicrostructureand stability of non-protein stabilized oil-in-water food emul-sionsmeasured by opticalmethodsrdquo Journal of Food Science vol67 no 3 pp 1130ndash1135 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

8 The Scientific World Journal

Table 2 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198631|

10038161003816100381610038161003816100381610038161003816

1198631

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ2

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1199042minus1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772minus1198631

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ1

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198631

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198631

119877

10038161003816100381610038161003816100381610038161003816

1 444 75 169 1056 1393 119 143 785 65 1065 1284 11542 372 644 173 1039 1400 115 143 777 624 1055 1315 9903 263 457 174 1037 1401 115 147 792 619 1061 1357 7044 384 663 173 1050 1411 117 145 788 635 1073 1332 10205 384 664 173 1045 1405 117 145 782 630 1073 1332 10226 408 661 162 1047 1390 117 143 807 658 1004 1231 628Avg 376 640 171 1046 1400 117 144 789 636 1055 1309 920

Table 3 Coefficient values of the absolute inaccuracies Δ1198721 Δ1198722 Δℎ1 Δℎ2 Δℎ1199041 Δℎ1199042 Δ119877

Exp |119863| |1198632|

10038161003816100381610038161003816100381610038161003816

1198632

119863

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

1 444 459 103 1056 1393 045 068 610 407 247 371 1582 372 380 102 1039 1400 045 068 625 417 245 367 1573 263 268 102 1037 1401 045 068 605 415 245 357 1574 384 394 103 1050 1411 045 068 618 411 247 371 1585 384 390 102 1045 1405 045 068 620 413 245 367 1576 408 433 106 1047 1390 045 068 584 390 254 382 163Avg 376 387 103 1046 1400 045 068 610 409 247 369 158

Δ120588

=1

|119863|(

10038161003816100381610038161003816100381610038161003816

ℎ1199042

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987211003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

ℎ1199041

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δ11987221003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199042

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1199041

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ21003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

1198722

1205871198772minus1198632

119863ℎ2

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990411003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

1198632

119863ℎ1minus1198721

1205871198772

10038161003816100381610038161003816100381610038161003816

1003816100381610038161003816Δℎ11990421003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816

21198632

119877

10038161003816100381610038161003816100381610038161003816

|Δ119877|)

(32)

where Δℎ1= 0 056 cm Δℎ

2= 0 067 cm Δℎ

1199041= Δℎ1199042

=

Δ119877 = 0 cm Δ1198721= 0 27 g and Δ119872

2= 0 30 g

Considering the calculations fromTables 2 and 3 we havethe values of the maximum absolute inaccuracies Δ120588

119904asymp

0 43 gsdotcmminus3 and Δ120588 asymp 0 18 gsdotcmminus3From here we find the values of the maximum relative

inaccuracies

Δ120588119904

120588119904

=043

170asymp 025

Δ120588

120588=018

103asymp 017 (33)

Thus obtained value of 120588 at 20∘C temperature is withinthe error for the density of the distilledwater which shows theaccuracy of our model

Then we find the percentage 1198751015840 of the dry substance inthe sediment and the percentages 11987510158401015840 and 119875 of the bounddispersionmedium in the sediment and of the free dispersionmedium accordingly We use formulae (20) (24) (25) and(26) and the data from Table 1 The results are shown inTable 4

Because of the approximations in the calculations thesum of the values of 1198751015840 11987510158401015840 and 119875 is around 100

9 Discussion

Our model of close packing of the particles from the dis-persion phase of the studied liquid dispersion offers a newmethod for finding the volumes and the masses of the sedi-mented substances in LD as well as the average Stokes radiiof the particles of these substances Moreover this study isdistinguishedwith simplicity and its application is very cheap

Our simple algebraicmethod offers an increased accuracyin determining the average densities and weights of the ulti-mate sediment and emergent of the dispersion medium andof the dispersion phase in the liquid dispersionThefinding ofthese densities is achieved without the use of pycnometers orother more complex methods This prevents the destructiveeffects in the LD and the vibrations practically have no effecton the measurements results unlike the methods existinguntil now

Our method of determining the coefficient of the quanti-ties of substances in the components of LD does not requirepouring out the liquid from the container in order tomeasurethemasses of the ultimate sediment and of the free dispersionmedium (water) The known method of pouring out anddrying is destructive Using the drying method on LD it isgenerally even impossible to determine the percentage of thesubstance amount in the emulsion The method we proposeovercomes this disadvantage

A great advantage and particularly valuable for onemethod is that it has numerical characteristicmdashscale for eval-uation of a given variable The index 119868 which we introduce in(23) is a dimensionless scale for characterization of the sedi-mentation stability 119878 naturally following from the law on themovement of the dispersion particle from the dispersionphase of LD It provides a very easy way to evaluate 119878 of

The Scientific World Journal 9

Table 4

Exp1 2 3 4 5 6

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

119872 (g) 5607 7397 5517 7433 5509 7439 5574 7495 5551 7458 5557 7383120588119904(gcm3) 169 173 172 174 172 163

120588 = 12058810158401015840 (gcm3) 103 102 102 102 102 106

1205881015840 (gcm3) 221 229 227 23 227 2071198751015840 () dry matter 2811 3197 2973 3296 2939 3174 2942 3284 2917 3256 2656 300011987510158401015840 () bound dispersion

medium in the sediment 1029 1169 1037 1153 1038 1121 1026 1143 1030 1149 1069 1207

119875 () free dispersion medium 6146 5620 5985 5535 5994 5675 6021 5561 6046 5589 6275 5790

one LD in straight andor reverse sedimentation through thedensities 120588

119904 120588119891 and 120588 which are the solution of the system of

linear equations of type (11)The index 119868 isin (0 1] provides the possibility not only to

evaluate and to predict the sedimentation stability but alsoto compare different LDs according to their sedimentationstability Moreover the tests of the different LDs must beperformed in constant temperature and external pressure

10 Conclusion

The method which we offer does not require special equip-ment The phase separation of LDs can be achieved in theconditions not only of gravitational (earthrsquos) but also of cen-trifugal field In the case of centrifugal field the time interval 119905for determining the ultimate sediment and emergent can bemuch smaller compared with time for treating the LD withthe acceleration of gravity 119892

The advantage of our physicomathematical model forfinding the Stokes radii in the dispersion phase is that it doesnot use complicatedmathematical analysis for structuring thesedimentation curve

The three variables 1198751015840 11987510158401015840 and 119875 characterizing in per-centage the respective quantity of substance in the ultimatesediment or emergent bound dispersion medium in the sed-iment or the emergent and the free dispersion medium areintroduced in the literature for the first time with this paper

And with the sedimentation stability index 119868 whichwe introduce it is possible to demonstratively and withoutparticular difficultiesmake qualitative analysis of suspensionsand emulsionsmdashproducts of the food medical cosmeticchemical and so forth industries Such products are forexample different suspensions and emulsions such as foodsand drinks medications and cosmetic products paints andvarnishes and cement and lime solutionsThis is particularlyand technically useful in the creation of products withpredetermined sedimentation stability

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The obtained results are made with the support of Fund ldquoSci-entific Studiesrdquo of the Bulgarian Ministry of Education andScience under the Contract T 12 This paper is publishedwith the financial support of the Project BG051PO00133-05-001 ldquoScience and Businessrdquo financed by the HumanResources Development Operational Programme of theEuropean Social Fund

References

[1] D Dakova D Christozov and M Beleva ldquoBarycentric methodof determining the physical parameters of a single-phase parti-cle in liquid disperse systemsrdquo Journal of Colloid and InterfaceScience vol 256 no 2 pp 477ndash479 2002

[2] R Chanamai and D J McClements ldquoDepletion flocculation ofbeverage emulsions by gum arabic andmodified starchrdquo Journalof Food Science vol 66 no 3 pp 457ndash463 2001

[3] G S Hodakov andU P Udkin SedimentationAnalysis of HighlyDispersed Systems (in Russian) Chemistry Moscow Russia1981

[4] B M Westwood and V N Kabadi ldquoA novel pycnometer fordensity measurements of liquids at elevated temperaturesrdquoTheJournal of Chemical Thermodynamics vol 35 no 12 pp 1965ndash1974 2003

[5] A Puttmer R Lucklum B Henning and P HauptmannldquoImproved ultrasonic density sensor with reduced diffractioninfluencerdquo Sensors and Actuators A Physical vol 67 no 1ndash3 pp8ndash12 1998

[6] E D Shchukin A V Percov and E A Amelina ColloidalChemistry pp 306ndash309 Yurait Moscow Russia 6th edition2012 (Russian)

[7] K Kolikov G Krastev Y Epitropov and D Hristozov ldquoAna-lytically determining of the absolute inaccuracy (error) ofindirectly measurable variable and dimensionless scale char-acterising the quality of the experimentrdquo Chemometrics andIntelligent Laboratory Systems vol 102 no 1 pp 15ndash19 2010

[8] K Kolikov G Krastev Y Epitropov and A Corlat ldquoAnalyticallydetermining of the relative inaccuracy (error) of indirectlymeasurable variable and dimensionless scale characterising thequality of the experimentrdquoCSJM vol 20 no 1 pp 314ndash331 2012

[9] M H Ly M Aguedo S Goudot et al ldquoInteractions betweenbacterial surfaces and milk proteins impact on food emulsionsstabilityrdquo Food Hydrocolloids vol 22 no 5 pp 742ndash751 2008

10 The Scientific World Journal

[10] J P F Macedo L L Fernandes F R Formiga et al ldquoMicro-emultocrit technique a valuable tool for determination ofcritical HLB value of emulsionsrdquo AAPS PharmSciTech vol 7no 1 pp E1ndashE7 2006

[11] M Corredig and M Alexander ldquoFood emulsions studied byDWS recent advancesrdquo Trends in Food Science and Technologyvol 19 no 2 pp 67ndash75 2008

[12] J M Quintana A Califano and N Zaritzky ldquoMicrostructureand stability of non-protein stabilized oil-in-water food emul-sionsmeasured by opticalmethodsrdquo Journal of Food Science vol67 no 3 pp 1130ndash1135 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

The Scientific World Journal 9

Table 4

Exp1 2 3 4 5 6

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

1198701

1198702

119872 (g) 5607 7397 5517 7433 5509 7439 5574 7495 5551 7458 5557 7383120588119904(gcm3) 169 173 172 174 172 163

120588 = 12058810158401015840 (gcm3) 103 102 102 102 102 106

1205881015840 (gcm3) 221 229 227 23 227 2071198751015840 () dry matter 2811 3197 2973 3296 2939 3174 2942 3284 2917 3256 2656 300011987510158401015840 () bound dispersion

medium in the sediment 1029 1169 1037 1153 1038 1121 1026 1143 1030 1149 1069 1207

119875 () free dispersion medium 6146 5620 5985 5535 5994 5675 6021 5561 6046 5589 6275 5790

one LD in straight andor reverse sedimentation through thedensities 120588

119904 120588119891 and 120588 which are the solution of the system of

linear equations of type (11)The index 119868 isin (0 1] provides the possibility not only to

evaluate and to predict the sedimentation stability but alsoto compare different LDs according to their sedimentationstability Moreover the tests of the different LDs must beperformed in constant temperature and external pressure

10 Conclusion

The method which we offer does not require special equip-ment The phase separation of LDs can be achieved in theconditions not only of gravitational (earthrsquos) but also of cen-trifugal field In the case of centrifugal field the time interval 119905for determining the ultimate sediment and emergent can bemuch smaller compared with time for treating the LD withthe acceleration of gravity 119892

The advantage of our physicomathematical model forfinding the Stokes radii in the dispersion phase is that it doesnot use complicatedmathematical analysis for structuring thesedimentation curve

The three variables 1198751015840 11987510158401015840 and 119875 characterizing in per-centage the respective quantity of substance in the ultimatesediment or emergent bound dispersion medium in the sed-iment or the emergent and the free dispersion medium areintroduced in the literature for the first time with this paper

And with the sedimentation stability index 119868 whichwe introduce it is possible to demonstratively and withoutparticular difficultiesmake qualitative analysis of suspensionsand emulsionsmdashproducts of the food medical cosmeticchemical and so forth industries Such products are forexample different suspensions and emulsions such as foodsand drinks medications and cosmetic products paints andvarnishes and cement and lime solutionsThis is particularlyand technically useful in the creation of products withpredetermined sedimentation stability

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The obtained results are made with the support of Fund ldquoSci-entific Studiesrdquo of the Bulgarian Ministry of Education andScience under the Contract T 12 This paper is publishedwith the financial support of the Project BG051PO00133-05-001 ldquoScience and Businessrdquo financed by the HumanResources Development Operational Programme of theEuropean Social Fund

References

[1] D Dakova D Christozov and M Beleva ldquoBarycentric methodof determining the physical parameters of a single-phase parti-cle in liquid disperse systemsrdquo Journal of Colloid and InterfaceScience vol 256 no 2 pp 477ndash479 2002

[2] R Chanamai and D J McClements ldquoDepletion flocculation ofbeverage emulsions by gum arabic andmodified starchrdquo Journalof Food Science vol 66 no 3 pp 457ndash463 2001

[3] G S Hodakov andU P Udkin SedimentationAnalysis of HighlyDispersed Systems (in Russian) Chemistry Moscow Russia1981

[4] B M Westwood and V N Kabadi ldquoA novel pycnometer fordensity measurements of liquids at elevated temperaturesrdquoTheJournal of Chemical Thermodynamics vol 35 no 12 pp 1965ndash1974 2003

[5] A Puttmer R Lucklum B Henning and P HauptmannldquoImproved ultrasonic density sensor with reduced diffractioninfluencerdquo Sensors and Actuators A Physical vol 67 no 1ndash3 pp8ndash12 1998

[6] E D Shchukin A V Percov and E A Amelina ColloidalChemistry pp 306ndash309 Yurait Moscow Russia 6th edition2012 (Russian)

[7] K Kolikov G Krastev Y Epitropov and D Hristozov ldquoAna-lytically determining of the absolute inaccuracy (error) ofindirectly measurable variable and dimensionless scale char-acterising the quality of the experimentrdquo Chemometrics andIntelligent Laboratory Systems vol 102 no 1 pp 15ndash19 2010

[8] K Kolikov G Krastev Y Epitropov and A Corlat ldquoAnalyticallydetermining of the relative inaccuracy (error) of indirectlymeasurable variable and dimensionless scale characterising thequality of the experimentrdquoCSJM vol 20 no 1 pp 314ndash331 2012

[9] M H Ly M Aguedo S Goudot et al ldquoInteractions betweenbacterial surfaces and milk proteins impact on food emulsionsstabilityrdquo Food Hydrocolloids vol 22 no 5 pp 742ndash751 2008

10 The Scientific World Journal

[10] J P F Macedo L L Fernandes F R Formiga et al ldquoMicro-emultocrit technique a valuable tool for determination ofcritical HLB value of emulsionsrdquo AAPS PharmSciTech vol 7no 1 pp E1ndashE7 2006

[11] M Corredig and M Alexander ldquoFood emulsions studied byDWS recent advancesrdquo Trends in Food Science and Technologyvol 19 no 2 pp 67ndash75 2008

[12] J M Quintana A Califano and N Zaritzky ldquoMicrostructureand stability of non-protein stabilized oil-in-water food emul-sionsmeasured by opticalmethodsrdquo Journal of Food Science vol67 no 3 pp 1130ndash1135 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

10 The Scientific World Journal

[10] J P F Macedo L L Fernandes F R Formiga et al ldquoMicro-emultocrit technique a valuable tool for determination ofcritical HLB value of emulsionsrdquo AAPS PharmSciTech vol 7no 1 pp E1ndashE7 2006

[11] M Corredig and M Alexander ldquoFood emulsions studied byDWS recent advancesrdquo Trends in Food Science and Technologyvol 19 no 2 pp 67ndash75 2008

[12] J M Quintana A Califano and N Zaritzky ldquoMicrostructureand stability of non-protein stabilized oil-in-water food emul-sionsmeasured by opticalmethodsrdquo Journal of Food Science vol67 no 3 pp 1130ndash1135 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of