THE HOMOGENISATION OF BULK MATERIAL
IN BLENDING PILES
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IN BLENDING PILES
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR
MAGNIFICUS PROF. DR. IR. F.J. KIEVITS VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE
VAN DEKANEN, TE VERDEDIGEN OP WOENSDAG 14 NOVEMBER 1979 DES MIDDAGS
OM 16.00 UUR DOOR
ARNOLD WILLEM GERSTEL
werktuigkundig ingenieur geboren te Voorburg n r \ r ;^'
1378
124
4
BIBLIOTHEEK TU Delft P 1599 1241r
Dit proefschrift is goedgekeurd door de promotor PROF.IR. G. PRINS
en de co-promotor
opgedragen aan :
mijn moeder
mijn schoonmoeder mijn vrouw
SUMMARY
In this thesis the homogenisation of bulk material in three types of piles is dealt with.
The homogenisation implies that the fluctuations of a material prc-i^rty in the input flow of the pile are transformed into output flu^^^ations, whereby the latter ones are evened out.
Analyses are presented concerning the transformation of the auto correla :ion function and the standard deviation of the fluctuations. The results are presented in graphs.
Furtheriiore the similarity between the homogenisation in bins, divided in a number of compartments, and the homogenisation in the above piles is dealt with.
Finally various types of machines for the recovery of the material from the pile are described and evaluated.
SAMENVATTING
Tegenwoordig worden vaak menghopen toegepast in systemen van grondstoffen-voorbereiding, met name in de cement- en staalindustrie. Ook is echter te denken aan de behandeling van steenkolen.
In deze dissertatie wordt het homogeniseren van bulkgoed in drie typ.en menghopen behandeld. Daarbij wordt uitgegaan van stochastisch fluctuerende materiaaleigenschappen in de toevoerstroom naar de menghoop. Het
homogeniseren heeft tot gevolg dat de materiaaleigenschappen in de afvoer-stroom minder heftig en gemiddeld langzamer fluctueren dan in de toevoer-stroom. Dit betekent dat de fluctuaties in de afvoerstroom worden gekenmerkt door een verminderde standaardafwijking en een verbeterde autocorrelatie-functie.
Voor de drie typen menghopen wordt de transformatie van deze statistische kenmerken geanalyseerd als functie van de voor het homogeniseren relevante parameters. Als reden van deze analyse wordt het systeem van grondstoffen-voorbereiding in de cementindustrie beschouwd (hoofstuk 1; figuur 2-2). In een dergelijk systeem dient de standaardafwijking van de materiaal-eigenschappen voldoende te worden verkleind om een optimaal regelbaar ovenproces te verkrijgen met een minimaal energieverbruik.
De correlatie is met name van belang in de systeemfase volgend op het afgraven van de menghoop. In deze fase wordt de gewenste samenstelling gerealiseerd door materiaal van bijzondere samenstelling vanuit bunkers op de hoofdstroom te doseren. In de meeste gevallen gebeurt dit onder controle van een terugkoppeling en op basis van monsters die na de bunkers uit de stroon worden genomen. Een dergelijke regeling blijkt slechts effectief te werken bij voldoende correlatie van de materiaaleigenschappen.
De eertse van genoemde typen menghopen wordt behandeld in hoofdstuk 2 en is de conventionele menghoop die sedert 1905 wordt toegepast (figuren 2-1, 2-3 en 2-4). Deze menghoop wordt in lagen opgebouwd en vervolgens zodanig afgegraven dat het afgraafvlak steeds alle lagen doorsnijdt. Dit laatste is essentieel voor het homogeniseren maar wordt slechts gerealiseerd in het regelmatig opgebouwde middendeel van de menghoop. Dit middendeel wordt begrensd door eindkegels die door hun afwijkende configuratie in mindere mate bijdragen aan de verbetering van de materiaaleigenschappen.
De correlatie tussen twee doorsneden van eenzelfde menghoop blijkt
aanzienlijk verbeterd te kunnen worden door het aantal lagen te vergroten. De correlatie tussen doorsneden gelegen in verschillende menghopen is over het algemeen slecht en laat zich ook niet automatisch verbeteren door vergroting van het aantal lagen.
Voor wat betreft de voor de afvoerstroom geldende standaardafwijking worden twee gevallen onderscheiden. Het ene geval betreft de in formule gepresen-teerde standaardafwijking die bepalend is voor de mate waarin een materiaal-eigenschap afwijkt van menghoopgemiddelde van die materiaal-eigenschap. In het andere geval wordt de standaardafwijking beschouwd die gerelateerd is aan het lange termijn gemiddelde van die eigenschap en waarvoor reeds eerder door anderen formules werden opgesteld. Eerstgenoemde standaardafwijking wordt voorgesteld als een extra criterium voor het beoordelen van het homogeniseren. In hoofdstuk 3 wordt het tweede type menghoop behandeld. Dit is de sinds
kort toegepaste z.g. continue menghoop (figuren 3-1 en 3-2). Kenmerkend voor deze hoop is dat deze aan het kopeinde continu wordt afgegraven. Het toegevoerde materiaal wordt in lagen op het naar verhouding lange en flauw aflopende staarteinde gestort, dat daardoor langzaam verplaatst zonder dat de configuratie ervan wijzigt. Ook hier komt het homogeniseren tot stand doordat het afgraafvlak de lagen doorsnijdt.
Kenmerkend voor genoemde menghoop is dat af te graven eindkegels ontbreken en dat slechts een volgens een cirkel liggende of zo men wil, wandelende -menghoop in de praktijk bruikbaar is.
Bij meer dan 50 lagen wordt het homogeniseren in een continue menghoop nagenoeg bepaald door het proces van voortschrijdende middeling. In dat geval is er sprake van een lineair proces en vertoont de op de afvoerstroom betrokken autocorrelatiefunctie geen discontinuïteiten. Bij het homoge-niseren in achtereenvolgende conventionele menghopen zijn er wel discon-tinuïteiten en is er dus geen sprake van een lineair proces. Dit heeft met name consequenties voor eerder genoemde regeling met terugkoppeling.
Het derde type menghoop wordt behandeld in hoofdstuk M- en is een variant van de continue menghoop (figuren 4--1 en M--2). In plaats van een langgerekt staarteinde aan de opstortzijde wordt de hoop begrensd door een halve kegel, waardoor de homogeniserende werking minder is. Het opstorten van het
materiaal gaat gepaard met een langzame verplaatsing van het opstortpunt waardoor de kegelconfiguratie verschuift. Overigens behoeft in het onder-havige geval nooit een kegel zoals bij een conventionele menghoop te worden afgegraven. Het afgraafvlak blijft steeds het volle oppervlak houden. Het doorsnijdt daarbij wel een aantal "kegelschillen", wat in dit geval
essentieel voor het homogeniseren is.
In hoofdstuk 5 wordt in het kort aandacht besteed aan het homogeniseren in bunkers. De hier bedoelde bunkers zijn verdeeld in een aantal verticale compartimenten die volgens een bepaalde systematiek worden bijgevuld en leeggetrokken. Aannemende dat de zich in een compartiment voortbewegende materiaaldeeltjes een onveranderlijke positie ten opzichte van elkaar houden, blijkt er een volledige dan wel aanzienlijke overeenkomst te bestaan tussen de bunkers en genoemde menghopen voor wat betreft de beschrijving van het homogeniseren.
In hoofdstuk 6 wordt de z.g. machinehoeveelheid geïntroduceerd, doorgaans batch genoemd, die van belang is bij een heterogene verdeling van de materiaaleigenschappen over de afgraafdoorsnede. De batch is de kleinste door een afgraafmachine geproduceerde hoeveelheid materiaal die is samen-gesteld uit materiaal van het gehele afgraafvlak. Dit betekent dat het batchgemiddelde van een materiaaleigenschap gelijk is aan het afgraaf-vlakgemiddelde van die eigenschap.
Voorts worden enige typen afgraafmachines beschreven en beoordeeld aan de hand van een aantal criteria, waaronder genoemde batch. Het betreft hier de afgraafmachine met bakkenwiel ; de afgraafmachine met horizontaal langs het afgraafvlak voortbewogen schraapbladen ; de afgraafmachine met de van bakken voorziene draaibare trommel die over de gehele breedte van de menghoop werkzaam is ; en tenslotte de afgraafmachine met de van hark- en meeneemtanden voorziene draaibare schijf die het gehele afgraaf-vlak bedekt.
In hoofdstuk 7 wordt de literatuur betreffende het homogeniseren van bulkgoed besproken. Er blijken niet veel publicaties over dit onderwerp te bestaan. De meeste handelen over de reductie van de standaardafwijking in een menghoop of in een bunker, waarbij de autocorrelatiefunctie van de toevoerstroom niet altijd in de beschouwingen betrokken wordt.
De resultaten van de in de hoofdstukken 2, 3 en 4 gegeven analyses worden in grafieken gepresenteerd, waarbij het begrip effectiviteit van het homogeniseren (homogenizing efficiency) wordt gehanteerd.
Daaronder wordt verstaan de verhouding van de standaardafwijkingen betrokken op de toevoer- resp. afvoerstroom van de menghoop ; er
kan echter ook mee bedoeld worden de verbetering van de autocorrelatie-functie in de menghoop.
CONTENTS PAGE
SUMMARY 1 SAMENVATTING (DUTCH) 2
CONTENTS 4
1. INTRODUCTION 6
2. HOMOGENISATION IN CONVENTIONAL PILES 9
2.1. Introduction 9 2.2. Model of the homogenisation 12
2.3. Formula for the property content in a cross-section 13 of the output flow
2.4. Autocovariance function, variance and 15 autocorrelation function of the output
Piles with an "Infinite" number of layers 20 Covariance between cross-sections of the 23 output flow coming from the same pile
The exponential autocorrelation . function 23 Graphs of the autocorrelation function of the 24
input and output fluctuations
Fluctuating mass flow of the input 29 Reduction of the standard deviation in a 31
conventional p i l e , especially the evening out of the fluctuations of a property around its pile average
2.8.1. Formula for a and a 3 3
yp « p
2.8.2. Formulas for the reduction of the standard 35 deviation
2.8.3. Graphs 35 3. THE HOMOGENISATION IN A CONTINUOUS PILE 40
3.1. Introduction 40 3.2. Continuous circular pile versus conventional 42
(linear) pile
3.3. Model of the homogenisation 43 3.4. Formula for the property content in a 43
cross-section of the ouput flow
3.5. Autocovariance function, variance and 45 autocorrelation function of the output
3.6. Piles with an "infinite" number of layers 48 3.7. Graphs of the autocorrelation function of 49
the input and output fluctuations 2.4.1 2.4.2 2.5. 2.6. 2.7. 2.8.
PAGE
4. THE HOMOGENISATION IN AN ALTERNATIVE 53 CONTINUOUS PILE
4.1. Introduction 53 4.2. Model of the homogenisation 53
4.3. Analysis of the homogenisation 56 4.4. Autocorrelation function of the output ; 57
reduction of the standard deviation
4.5. Graphs 59
5. HOMOGENISATION IN BINS 62
5.1. Introduction 62 5.2. Homogenizing method 1 and 2 62
5.3. Homogenizing method 3 and 4 65
6. RECLAIMING OF PILES 66
6.1. Introduction 66 6.2. The batch 66 5.3. The reclaiming equipment 68
6.3.1. The bucket wheel reclaimer 68
5.3.2. The front scraper 71 5.3.3. The drum reclaimer 74 5.3.4. The disc reclaimer 76
7. 'LITERATURE INVESTIGATION 81
LITERATURE, Category 1 . 86 Category 2 88
LIST OF SYMBOLS 102
CHAPTER 1
INTRODUCTION
In the last decennia blending piles, especially the so-called conventional piles are increasingly applied in cement-, steel- and other raw material preparation systems (figures 2-1, 2-3 and 2-4).
Generally a conventional pile can perform three functions : Buffering
Composing Homogenizing
Buffering is the function of providing sufficient reserve of raw material
to guarantee a continuous operation of the processing plant under all normal circumstances.
Composing is the function of the integration of a number of raw materials
with different chemical and/or physical characteristics in such proportions that a completed pile represents the requisite composition.
Homogenizing is the function of a systematic (non-random) transformation
of the input flow of the pile into the output flow, so that the fluctuations of a property in the flow are evened out.
In this dissertation the homogenisation of bulk material in a conventional pile as well as in two other kinds of piles will be analysed.
Apart from this, the homogenisation in a bin consisting of a number of compartments will be dealt with.
The announced analyses are limited to the homogenisation of one property, for example CaO, moisture. Lime Saturation Factor, etc.
The homogenisation results in an improved autocorrelation function and a reduced standard deviation of the property fluctuations in the output flow of the pile(s). The analyses contain the development of the formulas for these statistical characteristics, whereby the composing and buffering functions of the pile are left out of consideration for the sake of clarity. Furthermore, the mass flow of the input and output of the pile will be
assumed to be constant. As pointed out in section 2.7., this does not imply a strong limitation.
It is desirable to explain here by means of an example, why it is of interest to have the above formulas available for the autocorrelation function and the standard deviation. The example concerns the preparation of cement raw material (figure 2-2).
In a cement plant the standard deviation of the relevant material properties must be reduced in favour of a regular kiln process and a' minimization of the energy consumption in the kiln /98/.
A stable composition is realised in the phase after the pile where material is drawn from additional bins, which correct the output of the pile
additively (figure 3-3).
The bins mainly contain homogeneous material of known composition, while the output flow of the pile usually has a fluctuating composition. The bins are located before the mill.
The composition adjustment is achieved by a feedback system on the basis of samples which are periodically taken from the output of the mill 758/, 751/, /58/.
The above procedure implies that a change in each of the capacities of the output of the pile and the bins is determined by samples taken 1-2 hours before.
This implies that the feedback system can only function in a proper way, if values of the several properties in the output flow of the pile can be predicted with sufficient reliability over a time interval of at least 1-2 hours.
The above reliability finds expression in the standard deviation of the prediction. A small deviation is realised by applying a blending system wherein, referring to the chapters 2, 3 and 4, the autocorrelation function of the material properties is sufficiently improved /42/.
Finally in this introduction a survey is presented of the subjects treated in the following chapters.
In chapter 2 the homogenisation in conventional piles will be analysed (figures 2-1, 2-3 and 2-4).
Concerning the standard deviation of the output of the piles a distinction will be made between two cases.
Firstly attention is paid to the standard deviation which may be assumed to indicate the extent in which the fluctuations of a property in the output flow of the pile(s) deviate from the long term average of the property. The formula for this standard deviation was already developed by Van Der Mooren and is needed to arrive at the auto correlation function of the output 715/, 7427.
Secondly the formula for an alternative standard deviation will be developed. In this standard deviation is expressed the extent in vzhich the output
fluctuations of a pile deviate from the pile average or short term average of the property.
The last standard deviation is introduced as an alternative norm to judge the homogenisation in the pile.
In chapter 3 the analysis of the homogenisation in a so-called continuous pile will be presented (figures 3-1 and 3-2).
The building up and reclaiming of this pile, which has a circular form can be done simultaneously. The material is layered on the sloping rear end of the pile and is reclaimed from the digging front at the other end. So the
pile behaves like a moving dune.
The continuous pile will appear to have a number of advantages in comparison with a conventional pile.
In chapter 4 a variance of the continuous pile will he analysed, the so-called alternative continuous pile (figures 4-1 and 4-2).
This pile has a cone-shaped rear end on which material is discharged while the discharging equipment is moving slowly.
Because the pile has a more sloping rear end than the continuous pile, the homogenisation is less.
In chapter 5 the homogenisation in bins will be dealt with (figures 5-1, 5-2, 5-3 and 5-4).
The bins in question consist of a number of compartments which are periodically provided with material.
The output of each compartment is a continuous (sub) flow having a constant mass.
Four homogenizing methods will be dealt with.
In principle two methods will appear to be equal to the methods presented in the chapters 2 and 3.
In chapter 6 the so-called hatch is introduced and dealt with. This is the smallest amount of material containing equal portions from all places of the digging front of the pile.
Various types of machines for the recovery of the material from a pile will
be described and evaluated.
Chapter 7 will deal with publications about the systematic homogenisation of bulk material.
A list of literature is presented at the end of chapter 7.
The results analysed in the chapters 2, 3 and 4 are presented in graphs. On the basis of these graphs conclusions are drawn concerning the
homogenizing efficiency of the pile. This is the ratio of the standard
deviations of the input and the output flow of the pile respectively. However it can also refer to the improvement of the autocorrelation function in the pile.
CHAPTER 2
THE HOMOGENISATION IN CONVENTIONAL PILES
2.1. Introduction
Figure 2-1 shows a conventional blending system.
Such a system consists at least of two piles, whereby the one pile is being reclaimed while the next one is being built up (figures 2-3 and 2-4). The building up is performed by a so-called stacker which is receiving the material from the side belt conveyor.
The material is discharged on the pile while the stacker is travelling to and fro along the side of the pile, which causes a zig-zag configuration of layers.
o reclaiming homogenizing composing - ~ \ discharging Dile
t/iniiii^nif, l i t nji i>r>>>>n>>irftr*f r )i //0>fn^l~i, > >i i}t>>>>>>>>>>'f'""*"""ff'" ' >'rm
pile (i + 7) bulk material
i i f
additional bins m i l l ^ raw meal samples **—-^ tendency | Z Z ~ i control I-analysis 1 capacity change (feedback) I fuel addition(s) cement meal mill0
L^i ""
kiln! t t
clinker cooling/\ A /
\i/
o 3 3 (t o c TO 3 / \ bulk material cyclic procedureti_Y_:i
/ pre-heatingC\
KJ
storage bin raw mealforward run 1 reverse run 'zig-zag' configuration reverse point full pile / ^ ^ / / ^ V C < \ \ cross-section
2|f^vv'{^'v^^^^^^^^^^^^^^^^v^^^^^^^^^^<^^^'vuu^^^u^^^^•^^^
input flow H L [property I I fluctuationsFigure 2-3 Building up of a conventional pile
digging front
'<k
^fSWSSWSS^v^ V^vv v^^vv^^jg
\:©
I11
j \ \ o u t p u t flow p r o p e r t y f l u c t u a t i o n sFigure 2-4 Reclaiming of a conventional pile
The side belt conveyor shown in figure 2-3 is carrying the input flow of the pile.
The reclaiming of the pile is done in such a manner that the digging front cuts all layers. The rearrangement of the material thus realised underlies the homogenisation.
The output flow of the pile is also carried by a side belt conveyor (figure 2-4). VJhether this is the above conveyor or another one depends on the lay-out of the blending system : the piles can be positioned in line or next to each other.
At both ends (cones) of the pile the rearrangement of the material is much more complicated than in the middle of the pile and can in fact not be described accurately.
Usually a pile contains,a material supply for one week production. An illustration is given :
reclaiming capacity 500 - 2000 t7h length of the pile 50 - 150 m width of the pile 20 - 35 m number of layers 300 - 700 amount of material 10000 - 75000 t
2.2. Model of the homogenisation
The analysis of the homogenisation will be based on the following assumptions.
a. Each layer is made in the same travelling direction of the stacker. b. Each layer contains the same amount of material per meter pile length. c. The digging front is perpendicular to the direction of the layers. d. The material is simultaneously reclaimed from all places of the digging
front.
e. The pile has no cones.
ƒ. The mass flow of the input of the pile is constant.
g. The mass flow of the output is constant and equal to the mass flow of the input.
h. The fluctuations of a property in the input flow are assumed to be one realisation of a weak-stationary and ergodic random process 7127,
7157-With "weak-stationary" in assumption h is meant that the average and the autocorrelation function of the input fluctuations of the property are independent or almost independent of the real time during a period being sufficiently long in comparison with the time which is relevant for the homogenizing process.
The conventional pile (figures 2-3 and 2-4) as well as a variant of this pile are covered by the above rules.
This variant has no cones while each layer is made in the same travelling direction of the stacker 7159-27. Consequently the rules a and e are not relevant.
According to the relevant rules both types of piles are continuously built up and reclaimed without interruptions, whereby as stated in chapter 1 the buffering and composing functions of the pile are left out of consideration. Of course in practice not all above assumptions will be completely satisfied, However, in many cases this only means an introduction of effects of second degree, e.g. the influence of a varying mass flow of the input analysed in section 2.7.
As far as the conventional pile is concerned assumption a needs special mention because it evidently does net apply. It is introduced for reasons of simplicity ; however one may expect that the fault made will be small if, as usual, the number of layers of the pile is large.
2.3. Formula for the property content in a cross-section of the output flow According to the rules presented in the previous section, figure 2-5 shows the diagram of the homogenisation in conventional piles.
The material of input part i, situated between the cross-sections x and x, of the input flow, is rearranged in pile i.
This results in output part i located between the cross-sections y and y-, of the output flow.
The parts {i - 1 ) , {i + 1 ) , {i + 2 ) , etc. are similarly transformed. All input and output parts have the same flow time r .
In this time a pile is built up and a pile is reclaimed.
Incidentally all flow times in figure 2-5 are taken proportionally to the lengths of the material parts because input and output flow have the same (uniform) speed.
Building up the pile in n layers, input part i breaks up in n equal fractions Ij . . . , j , . . . , n. Each fraction has a flow time : T ^ - r /n.
L p
As a result of the layering the fractions are stretched out n times so that they are found as longitudinal parts in output part i. Consequently these parts have a flow time r and are positioned between the cross-sections
y and w,. ^
''a '^b
The above implies that a cross-section z/- of output part i contains material of n cross-sections x. of input part t (j = 1, 2, ..., n ) . All these cross-sections have the same position in the successive fractions. Being t- the
flow time of output part y -J/,, the flow time of input part j of fraction -7 7^ci
j equals t / n . ^- cAud^t^ ""^ j ^^4^'k^'-^J'^''
To arrive at the formula for the property content in cross-section zy ^, ^ the fixed observation points 0 and 0 are introduced which have an
^ x y
arbitrary position along the input and output flow respectively (figure 2-5). It is assumed that cross-section x passes 0 at the time t ; y passes 0 at the time t ^.
y ya
The difference t - t - t consists of a part r and an arbitrary part. a ya xa P The part T is evident because the building up of a pile has to be finished before reclaiming may be started. The arbitrary part is determined by the locations of 0 and 0 .
X y
Other times as t -,, t ., etc. are similarly defined as t and t . With yl' xj ^ ya xa reference to figure 2-5 and using t = t - t , then :
Cc ya tXfCc
t . = t + tjn + ( j - 1).T /n ^ t - t + tjn + ij - D.r/n ( 2 . 1 . ) x;] xa 1 " p ya a 2 v , p/
1 p
The property content in the cross-sections y - , and x. is noted as y{t ^) and
xit .) respectively. "^ ^
X passes O at the time t
a X xa
y passes 0 at the time t
°a ^ y ya input part i^ flow time r
output flow
iiiiiiiilBlimiiirrmTTTnhiii
\y^%l^ --y^\a ' *3^
output p a r t i , flow time T
l l l l l l l l l l
Figure 2-5 Model of the homogenisation in conventional piles
This content may be defined as a mass portion in a cross-section, whereby this section is assumed to approach zero.
Applying formula (2.1.) and taking care that there is balance of mass, we have :
,(t^^)
-- iiM.lxit^^
0<t^<r^
*a + *;;/« + (j - D.T /n} (2.2.)
To arrive at the autocovariance function of the output, the formula for the property content yit „) found in a cross-section i/„ of part H + k) of the
y k. Ó
output flow has to be determined (figure 2-6). Using formula (2.2.) it can easily be seen : n yit^^) -- (l/n),^x{t^^ - t^ t k.r t t/n t (j - l).r^/n} J - 1 / I 0 < t^ < r ; k = 0, 1, 2, ... (2.3.) 'Of É^: •
2.4. Autocovariance function, variance and autocorrelation function of the output
The autocovariance function 0 it , t ,) of the property fluctuations in the
" y-^ _ y'^'
output flow of conventional piles is defined to be 7127, 7887 :
where E means expected value.
A' is the average on the long term of the property in the output flow.
y
Of course M equals M , the average of the property in the input flow.
y X
Without loss of generality fi and ju will be chosen equal to zero, so : 1 i { I
Having a weak-stationary process as described in 7127 and in section 2.2., the autocovariance function </> (...) of the input is for a time r equal to :
Using (2.2.) and (2.3.), formula (2.5.) becomes :
n
.{(I7n).2 x(t - t + k.T + tJn + {I - l).r /n)}] (2.7.)
7-I y^ a p ó p
Applying the well-known theorem that the expectation of a linear function of several random variables equals the linear function of the expectations, formula (2.7.) may be transformed into :
'^y^''yl'^y2^ = ^'^"'^^1 J , ^ ^ ^^^V " '« " V " ^ ^^' " l)-^p/'^)>- ^2.8.)
.{xit - t + k.T + tJn + il - D.T /n)}\
ya a p ^ p
Figure 2-5 O < t , j t „ < r ^ 1 2 p output flow ( i + 1) (i + k) a i- k + 1) k.r
h-h
F i g u r e 2-7 Case 1 : 0 < t ^ j t „ , ( t - t ^ ) < T j \ V-, ~ 1 ih
^output flow u.' ( i + 1) ' k.r P r " I 1 ( i + fe) 1 1 ik.D.r^ u . u . ^. *2. *i-*2 / (i+k+J)
r^
y
ƒ
\ Figure 2-8 Case 2 : 0 < tj,tJtj - t„) < Tn n
'^,/^,;7^*,.?) = (1/^^-^ 2 '^^{fe.'-„ + (tp - t.Vn + (^ - J).r /n} (2.9.)
y y-L y'^ j - - ^ 2,=i -^ p ^ -^ p
After some calculations we have that :
0 (t 7,t g) = (17n).2 (1 - lpl/n).0 {fe.r + it - t )/n + ip/n) .T } (2.10.)
y y y p--n ^ ^ 0<t^,t^<r^
Due to the non-linearity of the homogenizing system 0 depends on t ^ and t not only because of their difference. So, contrary to the input process, the output process is not-stationary. Therefore a distinction is made between two
cases, as cross-section y^ can be located in pile or part ii + k), or in pile
a + k + i) (figure 2-7 and 2-8). With regard to these cases T is defined as :
Case 1
This case is presented in figure 2-7 (see also figure 2-5). The cross-sections y-J and y^ have such a position in the output flow that : t^ < t^.
In combination with 0 < t^.t^ < r this implies :
r 2 p ^
0 < t^.t^At^ ~ *2^ ^ \ (2.12.)
Obviously the following is valid :
T - k.T + it - t^) , wherein
k - int(r77 ) = 0, 1, 2, ..., thus I (2.13.)
t^ - t- - T - k.T
2 1 p
With (2.13.) the time k.r + it^ - t^)/n in formula (2.10.) can be transformed into :
k.T + (t„ - t^)/n = k.T + (T - k.T )/n = T/n + (1 - 1/n).k.T
p 2 1 ' p p p
n 0 (T ) = (l/n).2 (1 - \p\/n).(t> {T/n + (1 - l/n) .k.T + ip/n) .T }
y
p^_„ ^
p
p
(2.14.)0^ W ( ^ 2 - * i ^ ^ ^ p
; fc = int(T/T ) = O, 1, 2, Case 2 O W In this case cross-section y^ has such a position in part i of the output fl that for the same value of T as in case 1, cross-section z/„ is located in part (t + fe + 1). This situation is presented in figure 2-8. Similar to (2.12.) and (2.13.) it appears from this figure, that :0^ W ^ * !
-^2^^'p
T - ik + 1).T + (t„ - t^) , wherein P Ó J-k - int(r/T ) = 0, 1, 2, ..., thus t - t - T - ik -IT 1) .T (2.15.) (2.16.)The altered position of cross-section y„ implies that k in formula (2.3.) has to be replaced by ik + 1). The same applies in formula (2,10.).
At this point it can easily be concluded from case 1 that k in formula (2,14.) has to be replaced by ik + l) to arrive at the formula for 0^ (T) which is
relevant in the present case. It is : y
<P iT) = (17n).2 (1 - \p\/n).(p {T/n + (1 - l/n) .ik + 1) .r + ip/n) .T }
y
p^_„
^ p p0 < t^.t^Atj - t^) <T^ ; fe = int(r/r^) = 0, 1, 2, ... . (2.17.)
(2.14.) and (2.17.) can be expressed in one formula, the final formula for the autocovariance function.
0 (r) = (17n).2 (1 - |p|/n).0 {T/n + (1 - l/n).ik + a) .T + ip/n) .T ]
y
p-_„
^ p p0 < t^,t^A2a - l)it^ - t^) <T ', fe = int(r/T ) = 0, 1, 2, ... (2,18.)
Case I : a = 0 Case 2 : a = 1
The formula for the variance a of the property fluctuations in the output flow, follows from (2.18.) for T = k - a = 0 icase 1).
2
Using <t> iT ) = 4> i-T ) and 0 (O) - o , being the variance of the input fluctuations, it follows :
^
2
2
^
°t = 0,(0) = (I7n).a^ + (27n).2 (i - p/n) .<!> {ip/n) .T } (2.19.) - ^
y y -^ v-i
This formula was developed for the first time by Van Der Mooren 7147» 7157. It will be further dealt with in chapter 7.
(2.19.) enables to calculate the reduction of the standard deviation o /a .
y X
The graphs of o /a are dealt with in section 2.8. (figure 2-16).
y "^
The autocorrelation function C (T) of the output fluctuations is defined as :
y C iT) ^ <j> iT)/a^ (2.20.) y y y 2 0 (0) = a , therefore : C (0) = 1. y y • y
As explained in chapter 1 the autocorrelation function can be used to predict values of the material property, whereby for instance in a cement raw material preparation system the accuracy of the feedback control is improved.
E.g. it can be proved that the value y it + T ) , defined by :
y it + r ) r A(, + C iT).{yit) - n } (2.21.)
^ y y y
is the best prediction of the expected value at the time (t + r ) , at least in the case that this prediction has to be based on yit) alone 7427.
In section 2.6, the improvement of the autocorrelation function in conventional piles will be judged on the basis of graphs. Before that
2
some variants of the formulas for 0 iT ) , a and C iT ) being valid under
y y y
special conditions will be developed and interpreted.lt concerns the cases that
a. the piles have an "infinite" number of layers
b. the covariance is considered between cross-sections of the output flow coming from the same pile (finite number of layers)
Case a is analysed in section 2.4.1, and enables to search in how far the homogenisation in a real pile is approximated by the homogenisation in a pile with an "infinite" number of layers.
Concerning case b, analysed in section 2,4.2., it will appear that the correlation within a pile is much stronger than between the piles.
^
I'
2.4.1/PtZes with an "infinite" nimher of layers
" The time T to build up a pile equals the product of the number of layers n, and the time A0 to deposit one layer :
n.A0 - T (2.22.)
P
In this section an analysis is made of the homogenisation in a hypothetical pile which satisfies the condition :
^ ^ oo, A0 ^ 0, T^ = constant. ^ ^^^ ^U
A0 ->- 0 means that the thickness of ttie layers is approaching zero. Using formula (2.22.) and putting p.A0 = 0 (0 = A0, 2.A0, ..., n.A©)
2
formula (2.19.) for a can be transformed into :
y
p P n.A©
o = o 1/n + (2/r ).S (l - 0/r ).0 (0).A0
y ^ p Q^^Q p ^
When n ->- °°, A0 ^ 0 and n.A© = r , this form approaches to the integral :
T
of^^ = (2/T ).ƒ (1 - © A ).0 (0)d0 (2.23.)
y p e-o P ^
On the basis of this function the ratio a /a can be calculated. Graphs of this ratio are discussed in section 2.8.
Applying the same procedure as above, the integral form of formula (2.18.) for 0 (T) becomes :
r P
'^,.J'^) = (l/^^)--'" il - \&\ /r ).<Pj(k + a).T + @}de (2.24.)
y P Qzz-T P -^ P p
0 < t^,t^,i2a - Dit^ - t^) < T ; k = intir/T ) = 0, 1, 2, ...
Case 1 : a = 0 Case 2 : a = 1
Before interpreting the formulas (2.23.) and (2.24.) the covariance (correlation) is considered within the output of one pile containing an infinite number of layers.
The formula for this covariance follows from (2.24.) whereby case 1 is relevant and k - 0 (figure 2-9) :
T
P
^ Jr) = (1/r ).ƒ (1 - |0|/r^).0^(©)d© (2.25.)
y p ©=_,- p X 0<t^,t^,r <r^ ; r -- t^ - t^
Considering that 0 (-0) = 0 (©), formula (2.25.) is identical to formula (2.23.) for a .
Consequently, using definition (2.20.) the autocorrelation function equals :
Cy^ir) = 1 (2.26.)
0 <
t^,t^,r <r^ • r -- t,^- t^
This formula implies that the content of a property in each cross-section of the pile equals the pile average of that property.
Referring to figure 2-5 and 2-9 this can also be understood from a
physical point of view. When building up a pile with an "infinite" number of layers, every part of the input flow, however small its length, is spread out over the full length of the pile.
Such a part being very short, it may be assumed that the content of a property is constant over the length of that part. Consequently the layer into which the part is transformed, has a constant content over its length. Since this is valid for all layers the pile average is realised in all cross-sections of the pile.
This implies that the function of the property in the output flow of conventional piles which have an "infinite" number of layers, is in fact a step function (figure 2-9). Each step represents the property average
in a pile. „ Proceeding from this explanation the variance o and the covariance
0 ^i'^ ) •, determined according to (2.23.) and (2.24.), can be interpreted
y
as follows :
0 ^{T ) indicates the extent in which two pile averages are correlated ;
y 2
in Ö ^ is expressed the extent in which the pile averages deviate from the y
long term average. „ With reference to the theorem of Shannon /12/ it can be shown that o ^ equals
the standard deviation of the process (figure 2-10) :
T
P
y^(t) = (1/r ).ƒ x(t - Q)dQ (2.27.)
y (.t) equals the average of x(t) between the times (t - T ) and t,
function of the propertv property average (i - 1) output flow part i
I
(t + 2) Figure 2-9 n -^ «> xit)' ym^'^ iM
j j i £ ! l _ >M
-xit - e) • de.
1 % .
t - e ^ -t xit)- V * )
timeFigure 2-10 Moving average
function of the p r o p e r t y
yit^^^ t tj) y ^ ' y ^ ^ ' l ' ^ ^
2.4.2. uovarianoe between aross-seations of the output flow aoming from the same pile
This case is concerned with piles containing a finite number of layers.
The formula for the covariance follows from (2.18.) for k '=• a - 0. {Case 2). It is (figure 2-11):
0 (r) = (l/n).r (1 - lp|/n).0 {T/n + (p/n) .r } (2.28.)
y X p
" p--n ^
0 < t^,t^,r < T p ; r -- t ^ - t^
It is remarkable that T in the argument of 0 is divided by n. This is inherent to the pile build up and recovery. The material of a layer is supplied to the pile during the time r /n but reclaimed during r , So all input fractions r /n which are transformed into layers are stretched out n
times. ^
This implies that the property fluctuations in the layers are "slowed down" n times. Consequently this effect is also found in the course of the property in the output flow of the pile.
If n approaches infinity, (2.28.) becomes identical to (2.25.). Then figure 2-9 is relevant.
2.5. The exponential autocorrelation function
Of course the formulas developed in the preceding sections can only be used when the autocorrelation function of the input is known.
In many cases this function may be approximated by the exponential function /12/, /15/, /18/.
T being the characteristic correlation time of the input fluctuations, s
we have that (figure 2-12) :
C (T) - 4> ij)/o^ - exp(-|T| /r ) (2.29.)
X X X ^ s
In practice one of the possibilities for estimating r is by determining the time T during which the input function xir ) cuts its average at least
2 = 60 times /42/ :
T «* 2T /H (2.30.) s e
In the next section, graphs of the autocorrelation function of the output will be dealt with. These are based on (2.29.).
2.6. Graphs of the autocorrelation function of the input and output
fluctuations
Graphs of C (r) and C {r ) are presented in figure 2-14. X y
j|^( The graphs are related to the homogenisation of one material and are only '^ drawn for values of r fr being larger than zero. The figure contains 5
kinds of curves c^ ... a^ which, anticipating further explanations are :
c, C (r) y i
^^2 v^ / ° ^
^*2
- 'i^ ^'p rf^
o^ ^ C^(Oj 0 < ( t ^ - t ^ ) < r p 3 n ^ « .
oo
For the graphical representation of C'^(T), determined by C^{r) = exp(-r/Tg) (2.29.), the relation was used:
r /r
' 's"p
C {T) depends on r/r ^ T /T and n.
y ^ ' p ' s' p
Cr, and c^ relate to a finite number of layers and result from the combination of the formulas (2.29.), (2.18.) and (2.19.), using C = 0 /a .
y y y
c. and Or follow similarly from the combination of (2.29.), (2.23.) and (2.24.).
n = 1 10 1 50 1 250 1 500 1 n i 1 1 1 T / r = 1 1 \ 4 \ 1 1 S P 1 1 1 1 ,
1 1 1 1 ^
0 , 1 i I 1 5 1 1 11 1 1
^
1
^
1
0 , 0 1 \ 2 \ 6 \ 9 \ 1 1 \-- 1 i ^1 0,001 1 3 1 7 \ 10 \ 12 \ 1 1 1 1 i 0,0001 1 \ 8 \ 11 \ 13 \Th e curves e ^ c. and a have not been presented if they coincide with the axes or with c and c^,
If a^ coincides with the axes this curve has not been presented either,
r
f) _-',,/ c and a are step functions, the first step of c. has the value 1. All curves c„j c„j c and c show a discontinuity for T/T - 1, 2, ... This relates to the change-over to another pile_^ / ' ^ If T /r approaches zero, C (r) becomes identical to the autocorrelation function of so-called white noise :
C (r) = 0 for r # 0 and C (7) = 1 for T = 0 /12/, /42/.
In reality white noise can not exist but is considered here as a kind of a border-line case of the real situation.
2 —
Using C - <t)/a the values of C in case of white noise can easily be ' derived from the combination of (2.18.), (2.19.) and (2.29.), as well as from the combination of (2.23.), (2.24.) and (2.29.).
For T /T -> 0 it appears : s p ^^ Formulas (2.18.) and (2.19.) Case 1 : a - Q ; for T = 0 1/n for T - T C^ir)- 1 C^ir ) = 1 0<t^,t^At^-t^)<r^ for T i= 0 P r ¥= T (2.31a.)
Case 2 a 1/n
0 <
t^,t^At^
for T = 0 for T i= 0h^^^p
(2.31b. ) Formulas (2.22.) and (2.24.) Case 1 : a = 0 ; 1 for 0 < T < T C ( T ) _y°° C (r ) = 0 y°° Case 2 ; C (r) = 00 < t ^ . t ^ . ( t ^ - t p < r p
for T > TP
0< W ( t ^ - V < r .
for 0 < T n ->- °° (2.31c.) (2.31d.)For the smaller values of T /r and n, figure 2-14 shows that the curves for & P
C approach the curves of white noise
y
As far as defined by (2.31a.) the last curves rise to the value (l-l/n) for ''A = 1, which for example can be seen in graph 8.
Referring to the model of the homogenisation in section 2.2., this pattern of the curves results from the configuration of the layers as presented in figure 2-13 (see also figure 2-5).
layering direction \ ^b n-1 n-1 flow time T F i g u r e 2-13
According to this configuration each layer is made in the same travelling direction of the stacker. This implies that the property content at the end b-, of a layer equals the content at the beginning a^ ^ of the next layer.
Only a^ and b^ do not satisfy this rule.
As a consequence the property content in cross-section y is almost equal to the content in cross-section y-,, as a result of which figure 2-14 shows a strong correlation for r/r = 1 .
"' ^ / / /
^^oyiclusions " ^ J>&fa^^-'^^'
A preferable result of the homogenisation would be a reasonable auto correlation function C over several piles.
y
In that case the feedback system used in the phase after the blending system to arrive at the requisite composition (see chapter 1 and section 3.2.), could be better adjusted to the qualities of the pile to be
reclaimed subsequently. / However a reasonable correlation over several piles is not possible, >- O.y^'^o.-i ^'^p' i
The correlation decreases considerably for r/r > 1 (figure 2-14),, With Jl^i^'y^.^' respect to a continuous pile this occurs less or not at all (figure 3-8).
A considerable correlation can only be realised within a pile. In how far this is effectuated depends on the parameters r /T and n.
5 P
When 7- /T is small, C, appears to be a "bad" function, even when the pile contains a large number of layers,
This is illustrated in graph 13 of figure 2-14, where T /r - 0,0001
^. P / and n - 500. Consequently the homogenisation efficiency is small. •^«
This can also be understood from a physical point of ''iaw. For n =• 500
and r/r - 0,0001, T = 0,057 /n is valid, where r /n equals the flow
s p " "^ s ^ p ' p
time of an input part of the pile that is transformed into a layer (figure 2-5). Because of the factor 0,05 a "reasonable" number of property
fluctuations may occur in each layer of the pile which results in a weak correlation within the pile.
The correlation can be improved by increasing the number of layers and/or the ratio r /r , the last of course by decreasing r because T generally
s p P- s can not be influenced.
Because of the layering equipment it is preferred to choose a number of layers not exceeding 700.
The time T is usually between 50 and 80 hours, being 5 days of 12-16 hours stocking (cycle of a week). There are no objections to make this period shorter for exam.ple 30-40 hours.
However when choosing r too short, for example one day stocking, one has to expect that the homogenisation is too much influenced by the end cones of the pile (figure 2-4).
In that case it may be better to apply a continuous or an alternative continuous pile, dealt with in chapter 3 and 4 respectively.
As T /T - 0,001 , the function C_ can be considerably improved by
s V y
increasing the number of layers, for example from 250 to 500 (figure 2-14, the graphs 10 and 12).
For T /T = 0 , 0 1 a rather "good" function C is already obtained for a
s p y smaller number of layers than above (graphs 5 and 9 with n = 50, and 250).
Figure 2-14
Autocorrelation function of the input and output
of conventional piles
T : characteristic time of the random input fluctuations ^ (relation 2.29. )
T : time to build up a pile
P ^ ^
r : correlation time n : number of layers
C : autocorrelation function of the input C : autocorrelation function of the output
c, C
1.^ 1 X
Jk.^ ,
.3
C^ 0<it^-tp<r^
(figure 2-8)
/
^ " V . j f -
^4 'y 0<U^-tp<r^,n
\ xV'^
^^^•é
• l ^, ^^ ^:D £ N ^ l/l ^ 11 II - * C Cl, < co1
/ 1 / / / ' 1 ƒ /' 1
\ 1 •j J' 1
/ ] 1 J' 1
/ J 1 ] j /'
1
/ 1 -^/ 1
/._l
i H / J / J ' ' ' ^ '' 1 4^
1 1
'' \ \ ^ i 1 ^ 1 -J^ i 1
' ' • • • • i! 1 L/l O II II ' T ~" I 1 I • • • 'I Sf £ 1 , c,^ o"" o J •I "^^•v.^^ \ ° ^ X J II II 1 -1 w ^ r J ^^y^ J • • • • " ^ ^ H :3s ^ o o o • r-\ O II II m K ÏX < co ^1
^ •1 J J , ^ °o c a, ^^The larger T /T , the smaller the number of layers needs to be to
s p
arrive at an acceptable function C .
For T /T = 0 , 1 and n = 50 the function is almost ideal (graph 5 ) .
s p
For T /T = 1 and n = 50 the correlation is ideal which means s p
C (T) = 1 for 0 < T A < 1 (graph 4 ) .
y p
In this connection it is remarked that the function C ( T ) is also improved by increasing T /T (graphs 8, 7, 6, 5 and 4 ) .
s p
In fact a representative impression of the homogenisation efficiency is obtained by comparing C with C . In this respect graph 9
y X
(r A n - 0,01 ; n = 250) is a much better picture than graph 4 s p
(^s/^r. = 1 ; n = 50).
b p
2.7. Fluctuating mass flow of the input
In the preceding sections the mass flow of the input of the pile was assumed to be constant.
In consideration of efficiency one will strive for this situation in practice. However, there are cases wherein a fluctuating mass flow occurs.-In this section the influence of such a flow on the analysis of the homogenisation will be examined.
To do so it will be assumed that the mass of the flow varies according to a function w{t) of the time, and that this function can be considered as the realisation of a weak-stationary and ergodic random process.
Break-downs in the material supply to the pile are not covered by the analysis.
When both a material property in the input flow of the pile and the mass of this flow fluctuate, formula (2.2.) for the property content y{t ^) in a
cross-section y^ of the output flow has to be replaced by a generalized variant. Taking care that there is balance of mass, it appears:
yit ) = { 2 w..x.}/{ 2 w.} (2.32.) y^ j = i J J j = i ^ X. - x{t - t + tJn + (j - 1).T /n} J ya a T P w. - w{t - t + t-,/n + {j - 1).T /n] J ya a T ^ '^ p 1 p
With reference to formula (2.3.), it follows similarly for the property content yit _) in a second cross-section yr, of the output flow :
n n
yit ) = { S w'..x'.}/{ E w'.} (2.33.)
» t
X . and Wj are defined in a similar way as a;^. and y • .
Substituting (2.32.) and (2.33.) in (2.5.), the formula for the auto covariance function of the output is obtained.
This formula contains the expected value of a non-linear combination of the
f t
variables x .. W ., x. and w .. This implies that a formula similar to (2.18.) 3 Ü 3 3
can not be developed.
However there is another method to draw conclusions on the influence of a fluctuating mass on the input flow.
It is well-known that a non-linear function of random variables may often be approximated by a linear function. This function can be found by
developing the non-linear function in a Taylor series about the
expectations, and terminating this series after the terms of the first
degree. . ^^ The approximation is known to be rather good when the coefficients of],^^i»^in-^'^' ^ variation of the random variables are smaller than 0,2 /45/. ^
The method will be applied here to formula (2.32.) but is also valid regarding formula (2.33.).
The procedure is partly identical with the one presented by Van Der Mooren /15/.
Ignoring the argument of y, formula (2.32.) is copied : n n
1/ = { 2 W..X .}/{ 2 w.} (2.34.)
j = l ^ ^ j = l ^
Since i/ is a function of 2n variables W. and x ., the formula can be noted as :
^ 3 3
n n
yix^, . . . , a: , M . . ., w ) = { 2 w ..x .}/{ 2 w .} (2.35.)
J n 1 n ^^^3 3 ^^^ 3
Using jU and M being the averages on the long term of the mass and
W X
property fluctuations respectively, the Taylor expansion gives :
yix^, . . . . x^, w^j, ..., w^) = yi^i^, . . . . M^, My, .,.., My) +
n g,
\^/^j -'^x^-a^V ••-
"x,
V ..., M ,) +
3 = 1 3 ^ Yl^^
(^j -'^«^•I^V ••- ^x^ V ••-
^'-w^
^
+ further terms (2.36 .) /The linearisation implies that the further terms are neglected. The remaining expression is reduced with the help of formula (2.35.) and the fact that M may be replaced by M :
x y
^^V • • - V V • • - "w^ --"y
9 y nB x / v ••- V V •••^ ^z.^ ^ V ^ ^ ' ^ j ' ='/"
t/ J - 1 9 V n n n ^ (M J . . . J M J M J . . . J M ) = { X . . 2 U . - 1.2 w ..X . } / { 2 w ,} ^w. X X w w 3.._^3 j._^3 3 j._,3 So f o r m u l a ( 2 . 3 6 . ) becomes n (y - IJ- ) - i 1/n) . 2 ( x . - M ) o r V = ( 1 / n ) . 2 x . y j = l <>^ ^ 3 = 1^ ( 2 . 3 7 . )Now the arguments of time joined to x and y in formula (2.32.) are introduced again :
yit ) = (1/n).2 x{t - t + tjn + (j - D.r /n}
t/J, ' - i y a u i p
^ p
(2.38.)
This relation is identical to (2.2,).
Manipulating formula (2.33.) in a similar way leads to a relation which is identical to (2.3.).
From the above it appears that a fluctuating mass flow of the input
represents an effect of second,degree if the variation coefficient is not larger than 0,2. [U^f, IUQ-<<• /ei^'-j. }
Therefore the m.ass flow of the input is assumed to be constant regarding the analysis of the homogenisation in the several piles.
2.8. Reduction of the standard deviation in a conventional pile,
especially the evening out of the fluctuations of a property around its pile average.
In section 2.4. formula (2.19.) was presented for the reduction of the standard deviation o /a in conventional piles.
y X
o may be assumed to indicate the extent in which the function of a property in the input flow of the piles deviates from its long term
average M
a refers similarly to the output flow whereby of course M = M •
y y X
The above implies that the ratio o /a which is generally used as the
y X
norm to judge the homogenisation efficiency, covers the transformation of the entire input flow. This flow is distributed over the successive piles. As concluded from figure 2-14, based on the long term average, a strong correlation can be realised within a pile.
Referring to chapter 1 and section 3.2., such a correlation is desirable for a proper functioning of the feedback system.
In fact one is also interested in the extent in which a property deviates from its pile average of which a /o does not give a direct indication. Consequently two alternative norms are introduced which enable us to judge the homogenisation within a pile. As a /o these norms are also quotients of standard deviations :
a /o
yp X
and a /a yp xp
T is the time to build up or to reclaim a pile.
p . o may be assumed to indicate the extent in which the function of a property
in the output flow of a pile deviates from the pile average of that property. This average will be called the short term average,
In the same way o indicates the extent in which the function of the xp
property in the input flow of a pile deviates from this average.
input part ^, flow time T
I function of the propertv
Figure 2-15
Dependent whether the output or the input flow is considered, this average will be noted as m or m (figure 2-15). Of courseyp xp ^ ^ yp xp ^ m = m .
Concerning the development of the formulas for o and a it will be yp xp
assumed
2 . 8 . 1 . Forrmxla for o and a
yp xp
with reference to section 2.3. and figure 2-5, the property average in the output of an arbitrary pile i is defined as :
T
P
m = (1/T ).ƒ it + t,)dt, (2.39.)
yp P ^ ^Q ya 1 1 2
In the variance s is expressed the extent in which the property function in the ouput of pile i deviates from m . Its definition is :
yp r
2 P 2
s = (1/7- ).ƒ {yit + t,) - m } dt, ,^ „„ .
yp p ^ -Q " ya 1 yp ^ (2.40.)
The value of the variance still depends on the time t . This dependence can be eliminated by taking the ensemble average over 3 . Thus the
within-2 yp
pile variance s becomes:
yp
T
4 = '^^'/'pVJ'^V"*i^" V'^'i^ ^2.41.)
To reduce this formula, it is divided in 3 parts.
' 2 2 By d e f i n i t i o n i t i s : E { y ( t + t , , ) } = o t c o n s e q u e n t l y : z/ci 1 y T T
E^--^[i-2/r^).m^^.fyit^^.t^)dt^]
U s i n g ( 2 . 3 9 . ) t h e f o l l o w i n g i s v a l i d : ff„ = -2E [m ] 2 2 yp'T
E^-Z[iUr^).s' {m^/dt^\ -^irn^/
Z^ -j — u
Combining the relations for E^, E and E , formula (2.41.) becomes :
2 2 „ r n 2
a = a - E l m ]
yp y yp
Being m = w , this formula may be written as :
^ yp xp ^
a^ = o^ - E [m ]2 (2.42.)
yp y xp'
m is determined in a similar way as m
xp yp With reference to figure 2-15 and the definitions presented in section 2.3., it is :
r P
m = il/T ).f xit + tMtn ( 2 . 4 3 . ) xp p xa d d
Using t h i s r e l a t i o n , formula ( 2 . 4 2 . ) goes over i n t o :
T
P
a^ = a^ - E{(l/T ).ƒ xit + t , ) d t , } ^ yp y ' p'_^ _^ xa d d
d
Replacing t-, by a and /? respectively, this formula may be transformed into :
r T
2 2 2 P P
a'' = a^ - (1/7 )^.E{ ƒ 5 xit + a).xit + /3 )dad^ } =
yp y P „,o ^ , 0 ^^ '^'^
T T 2 2 P P
= a^ - ( 1 / r y .S ƒ E{x(t t a ) . x ( t + /? )}dadj3
2^ P a = 0 P=0
Applying definition (2,6.) it follows : T r 2 2 9 P P a^ = a'' - (1/T y .i ƒ 0 (a - ^ )dad^ 2^P y P a=0 (3=0 ^ 2 a appears to be equal to :
yp
^^ ^
T 2 2 P a'' = a^ - (2/r ). ƒ (1 - ©/T ).0(©)d© (2.44.) yp y p 0 - 0 P ^'Of course there may be a difficulty in the derivations given as the output process is non-stationary.
However form.ula (2.44.) can also be obtained by using formula (2.2.) and (2.41.) whereby yit + t^) is substituted for yit ,,). In that case the formula""'
2 y "^ y
for CT is developed in terms of the input process and that is by
assumption a stationary and ergodic process. , J
With reference to formula (2.23.) in section 2.4.1. the integral in formula o
(2.44.) equals the variance a ^. If the number of layers n approaches
2 . — ^ -^ infinity, o becomes identical to this variance, implying that a
y • ! _ _ _ l _ _ _ yp -^ ^ approaches zero.
This result can be understood from a physical point of view. With | I [' I reference to the comments concerning formula (2.25.), the pile average / yjouyl of a property is realised in all cross-sections of a pile that consists
of an "infinite" number of layers.
In that case there are no deviations from this average, so or equals
zero. y"
2
The basic formula for a is similar to (2.42.). Therefore, referring to ( 2.44.), ^P
T
•"L = ''x " ^2/r ).ƒ (1 - 0/r ).0 (0)d© (2.45.)
xp J. P 0_Q P X
2.8.2. Formulas for the reduction of the standard deviation For reasons explained in section 2.5. the exponential function
2
0 (r) =.a .exp(-l rl /r ) (2.29. ) was substituted for 0 (...) in the
X X s X formulas (2.44.), (2.45.), (2.19.) and (2.23.). Then the formulas for the
ratios o /o , o /o and o /a 'were developed.
y x' yp X yp xp ^
Each ratio, to indicate as reduction of the standard deviation^is proposed here as a relevant norm to judge the homogenisation in the pile.
Simplified form.ulas for the ratios were developed as well. These approximate the exact ones in the case that r , r and n or combinations of these
p' s
variables have special values or are within special regions.
As a i^esult 3 groups of formulas v.'ere obtained, which are presented on the next page. The first formula in every group is the exact one, the other _ones are_the approximations .__By > i<) is meant : "much larger (smaller)
than". " ' •'*"
e - exp(-r A ) X ^ p s
e - exp{-r /in.r )}
Group 1
i^f
, / ^ . . , . . 2 . . _ .2. a /a = V (1/n) + (2/n ).{e / ( l - e ) } . { n . ( l - e ) - ( l - e ) } (2.46a.) y X n n n x%^"x-^'-
(1/3).(1- ! / " ' ) • V % ^
r >r
(2.46b.)
^ ^ s p
"y^"x
= V 1 - (l/3).(r A ) r > T , n > 1 (2.46c.)
"^ p s s p^/^x = V 1/n T
< T In
(2.46d.)
s p
a A = V 2 . ( T A ).{1 - (r A ).(1 - e ) n -^ ~ (2.46e.)
y X s p s p X Group 2L
o /o = y ( a A )2 - 2.(r A ).{1 - (r A ).(1 - e )} (2.47a.)yp X i/ X s p
«
'
^ P X
'^wp^'^x = (l/n).y (l/3).(r A ) r > i" (2.47b.)
'^'^ p s s p a /o = O T > T , n > 1 (2.47c.) yp X s p "up X - \l 1/n T < T /n (2.47d.) ^'^ s p Group Z a /a = A/B (2.48a.) yp xp^ = y ( a ^ A ^ ) 2 - 2 . ( r ^ A p ) . { l - ( r / r p ) . ( l - . ^ ) }
B = V l - 2 . ( r ^ A ^ ) . { l - (T/r^).(l-.^)}
a A = 1/n r > r (2.48b.)
yp xp s p a /a = V 1/n r <^ r A (2.48c.) yp xp . -s p 2.8.3. GraphsGraphs of o /a , a /a and a A ar^e shown on a logarithmic scale ^ y x' yp X yp xp
in figure 2-16.
These graphs are related to the homogenisation of one material. Conclusions drawn from the graphs are presented hereunder.
Group Ij a /a : formulas (2.46a.) (2.46e.) y "^
-Mostly the formula a /o = >,A/n (2.46d,) is used to judge the
y __x "
homogenisation efficiency of a pile.
With reference to the horizontal lines at the left side of figure 2-16, this is only correct if r is short in comparison vfith r /n. For the longer values of r the formula gives too favourable results, which means
O
that the ratio o /a to expect in reality will be larger than calculated
y X
according to (2.46d.).
In that case the general formula (2.46a.) has to be used, implying that correlation and so the time r to build up the pile is taken into account. This formula was for the first time developed by Van Der Mooren /14/, /15/. All curves for a /a tend to the curve being valid for an infinite number
y X ^
of layers, and determined by (2.46e.).
According to this formula, a indicates the extent in which the pile y
averages of a property deviate from the long term average (see also section 2.4.1.).
When r A increases, a /a approaches the value 1. In that case o /a is
s p ' y X ^^ y X
a less relevant norm to judge the homogenisation in a pile.
Group, 2 a /a : formulas (2.47a.) (2.47d.)
7 A being larger than 0,001 - 0,01 the standard deviation a is
s p ^ yp "recommended to be used. It indicates the extent in which the function of a
"property"within a pile deviates from its pile average.
A small deviation, just as a strong correlation may be considered to be an advantage to the functioning of the feedback system, frequently applied in the 'phase after the blending system to arrive at the requisite
composition (chapter 1 and section 3.2.).
As figure 2-16 shows, when r A 'increases the ratio a /a (2.47a.)
s p yp X
decreases. For T /r ->• °° this ratio approaches zero, because a
s p ^^ yp
approaches zero ((2.47b.) and (2.47c.)).
The decline of the curves for o /a is considerable. In this connection
, . yp X
an example may be given.
Van Der Grinten presents graphs of the autocorrelation function of several properties /12/. From the graph in which the correlation of the specific mass of coal is presented, it appears : r =^ 2 hours.
A usual value of the time r to build up the pile is 60 hours , so T A ** 0,03. P
s p
Assuming n = 400 it follows from figure 2-16 : a /a '^ 0,006 , which '~ may considered as relatively small. "^^
Incidentally for the same values of r A and n it appaears : a /a '^ 0,25. Group 2, a /a : formulas (2.48a.) (2.48c.)
yp xp
Figure 2-16 illustrates that for 7 A < 0,1 the curves for a /a _ ^ s p yp xp
(2.48a.) coincide with the curves for a /a (2.47a.). yp X
If r A approaches infinity, o approaches zero, just as o .In that case the ratio o /a approaches the value 1/n (2.48b.), and is
yp xp ^^ ' represented by the horizontal lines at the right side of figure- 2-16.
Both criteria o /a and o /a may give relevant information about the yp X yp xp ^ ^
evening out of the fluctuations of a property around its pile average.
If T < T /n, all ratios a /a , a /o and a /a are equal to the
s p y x' yp X yp xp ^
value l/\4r {(2.46d.), (2.47d.), (2.48c.)} and are represented by the horizontal lines at the left side of figure 2-16. Referring to the
explanations on page 27, this situation results from the weak correlation within a pile for : T < T /n.
^ s p
In /21/ are developed the generalized formulas for a and a , which cover
^ yp y
the combination of homogenizing, buffering and proportioning in a conventional pile.
