Large-scale Homogenization of Bulk Materials in Mammoth Silos

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Large-scale Homogenization of Bulk Materials in

Mammoth Silos

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Large-scale Homogenization of Bulk Materials in

Mammoth Silos

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College van Promoties,

in het openbaar te verdedigen op maandag 28 juni 2004 om 15:30 uur

door

Dingena Lijntje SCHOTT werktuigkundig ingenieur geboren te Leeuwarden

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Prof. dr. ir. G. Lodewijks

Samenstelling Promotiecommissie:

Rector Magnificus voorzitter

Prof. ir. W.J. Vlasblom Technische Universiteit Delft, promotor Prof. dr. ir. G. Lodewijks Technische Universiteit Delft, promotor Prof. dr. A. Schmidt-Ott Technische Universiteit Delft

Prof. dr. ir. H.J. Glass University of Exeter (United Kingdom)

Prof. dr. J. Tomas Otto-von-Guericke-University Magdeburg (Germany) Dr. C. Kraaikamp Technische Universiteit Delft

Dr. ir. L.A. van Wijk ESI Eurosilo BV

Dr. ir. L.A. van Wijk heeft als begeleider in belangrijke mate bijgedragen aan de totstandkoming van dit proefschrift.

ISBN 90-9018116-4

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Summary

This thesis focuses on the applicability of large-scale homogenization in mammoth silos. Mammoth silos are frequently used for the storage of both free flowing and cohesive materials up to volumes of 100,000 m3. For the upgrading of free flowing bulk materials blending piles can be used. However, to date, an upgrading facility for cohesive bulk material is not available. Adding the functionality of homogenization to a mammoth silo means that both free flowing and cohesive materials can be stored and homogenized under climatological controlled circumstances.

The competitiveness of homogenizing in mammoth silos versus blending piles depends on the achievable rate of homogenization and the required homogenization efficiency for successive processes. Investment costs and available space are also relevant factors: there is an increasing market for compact and indoor storage facilities in densely populated areas.

In 1994 a research project was started at Delft University of Technology that resulted in several proposed methods to achieve homogenization in mammoth silos, each method having its own specific degree of homogenization. The methods derived from this research project are described in a patent (Gerstel and Van Seters, 1998). At the time, however, it was neither possible to indicate the homogenization efficiency of the different methods, nor to make a well founded choice of the method with the largest homogenization efficiency.

The main objective of this study was to determine the best stacking and reclaiming method for homogenization of bulk materials in mammoth silos. The determining criterion for the best method is the homogenization efficiency, i.e. the reduction of the standard deviation (homogenization effect) and the improvement of the auto correlation function (ACF) of the material properties. An increasing homogeneity of the material implies an increasing coherence of the material resulting in a decreasing homogenization effect and an increasing characteristic volume in the ACF.

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The input properties modeled in the literature on homogenization systems are assumed to be stochastic and weak-stationary. It was believed (Van der Mooren, 1965 and 1967) (Van der Grinten, 1973) (Gerstel, 1979) that these properties are sufficiently modeled using a first order autoregressive (AR(1)) model (Chapter 2). However, Schofield (1979 and 1980) used higher order models (Auto Regressive Moving Average models) based on his observation that modeling with AR(1) was not sufficient to characterize the input properties of the homogenization system he considered (Chapter 6).

From the literature (Van der Mooren, 1965 and 1967) (Gerstel, 1979) (Fu, 1995) it was also found that existing theories on blending and homogenization of raw materials are only applicable on homogenization systems when the volume or mass fluctuations of the different layers around the mean value are random and small (Chapter 2). In that case the volume distribution can be linearized around its mean value. It is shown in this thesis that homogenization in mammoth silos is characterized by a deterministic non-uniform volume distribution. Therefore, linearization is not applicable and hence a homogenization theory for mammoth silos had to be developed (Chapter 3).

The established homogenization theory of mammoth silos (Chapter 4) is a function of:

– The input properties of the material; its fluctuations are the main reason for application of homogenization.

– The volume distribution; the number of intersected layers and the volume of these layers inside a slice. Changes in silo geometry and stacking and reclaiming methods (including angles and thicknesses of layers and slices) affect the volume distribution and thus the homogenization performance.

In this thesis field data was fit with different order ARMA models to find the appropriate characterization of the material properties. It is shown that AR(1) models based on ARMA modeling criteria are not always adequate to account for the stochastic properties of the data. In addition, to predict the homogenization effect of a particular material the ACF of the ARMA model should be a good approximation of the ACF of the data set. This is more important for homogenization purposes than the determination of a perfect fit on the basis of ARMA criteria.

Since the volume distribution is discrete and different for all homogenization methods and silo geometries, a simulation program was developed to calculate the homogenization performance. An additional advantage of the simulations is that the attribution of the input properties is not restricted to stochastic properties.

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The simulation program was validated using:

– Process dynamics (PD). The impulse response functions of the part in the silo where both stacking and reclaiming angles are constant (the homogenization process is linear and time invariant in this part) correspond with the theoretical impulse response function derived by PD.

– Predefined situations for which the homogenization effect is known, such as blending piles.

The simulations were performed with the same set of input properties modeled by a first order autoregressive process with different characteristic volumes of the input. These simulations showed that (Chapter 7):

– A decreasing H/D ratio with equal storage capacity improves the homogenization effect (i.e. the reduction of the standard deviation);

– A decrease in the average layer thickness (in practice: 10 cm) does not lead to a substantially better homogenization effect;

– The number of slices has practically no effect on the homogenization;

– An increase in stacking and reclaiming angles up to the angle of repose of the material improves the homogenization effect;

– The stacking method based on controlled dumping homogenizes better than the use of an inclined conveyor when the other parameters are equal;

– The homogenization effect depends strongly on the characteristic volume of the input properties when it is larger than a third of the layer volume and smaller than the storage capacity. The homogenization performance in this area decreases for increasing characteristic volumes of the input properties.

Experiments were set up to investigate the suitability of a screw conveyor for inclined or declined stacking and reclaiming (Chapter 8). The main objective of these experiments was to compare the performance of the screw in an inclined position with that of the screw in a horizontal position. In conventional mammoth silos the screw conveyor operates horizontally. The experiments showed that for stacking and reclaiming downward a screw conveyor can be used to implement the stacking and reclaiming methods with an inclined conveyor.

The results of the simulations should be interpreted qualitatively instead of quantitatively when assessing the homogenization effect. The homogenization effect depends strongly on the input properties and therefore no conclusions can be drawn for estimating the homogenization effect for a particular material without knowing the characteristics of the input property. Therefore, the input properties have to be determined separately for each single batch of material.

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As a result of these simulations, it was found that the combination of the controlled dumping stacking method and the inclined conveying reclaiming method as shown in the following figure, gives the best homogenization performance compared to other methods in identical silo configurations.

stacking using controlled dump

reclaiming using an inclined conveyor

acd< 0 b > 0

The accuracy of the simulation program in predicting the homogenization performance is unknown since the simulation results were not validated in practice. To this date no mammoth silos with a large-scale homogenization function have yet been built, and therefore time series of the input and output properties of such a system were not available to validate the simulation results. When a mammoth silo with a large-scale homogenization function does become available, it is recommended to asses the results of the simulations on its quantitative merit.

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Samenvatting

Het onderzoek in dit proefschrift richt zich op de grootschalige homogenisering van stortgoed in mammoet silo’s. Deze silo’s hebben een opslagcapaciteit tot 100.000 m voor zowel vrijstromend als cohesief materiaal. Voor het uitvlakken van kwaliteitsfluctuaties (homogeniseren) en daarmee het verbeteren van de kwaliteit van vrijstromend stortgoed, wordt veelvuldig gebruik gemaakt van menghopen. Voor cohesief materiaal is een dergelijke grootschalige faciliteit nog niet beschikbaar. Wanneer een mammoet silo naast de opslag van het stortgoed ook de homogenisering kan realiseren, ontstaat een grootschalige homogeniseerinstallatie voor zowel vrijstromend als cohesief materiaal.

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De marktpositie van mammoet silo’s met homogeniseerfunctie ten opzichte van de menghopen is afhankelijk van de graad van homogenisering die behaald kan worden. Naast de eisen aan homogeniteit zijn investeringskosten en beschikbare ruimte ook relevante factoren, omdat er een toenemende vraag is naar compacte en overdekte opslagfaciliteiten in dichtbevolkte gebieden.

In 1994 is aan de TU Delft een onderzoek gestart dat heeft geresulteerd in diverse methoden om te homogeniseren in mammoet silo’s. Deze methoden zijn vastgelegd in een patent (Gerstel and Van Seters, 1998), maar destijds is de homogeniseringgraad van de verschillende methoden niet bepaald. Daarom kon geen gefundeerde keuze gemaakt worden voor de methode met de hoogste homogeniseringgraad.

Het hoofddoel van dit onderzoek is het bepalen van de inslag (vul) en uitslag (leeg) methode die de hoogste graad van homogenisering oplevert. Om dit te bepalen wordt de homogeniseringgraad gebruikt; de reductie van de standaard afwijking (het homogeniseringeffect) en de verbetering van de auto-correlatiefunctie van de stortgoed eigenschappen. Een toename in de homogeniteit van de eigenschappen wordt gekenmerkt door een afname van het homogeniseringeffect en een toename van de coherentie (toename van het karakteristieke volume in de auto correlatie functie).

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In de beschikbare literatuur over grootschalig homogeniseren wordt aangenomen dat de input eigenschappen stochastisch en zwakstationair zijn. Deze eigenschappen werden veelvuldig gemodelleerd door een eerste orde auto regressief proces (AR(1)), maar incidenteel werden ook hogere orde modellen (ARMA, Auto Regressive Moving Average modellen) toegepast wanneer de eerste orde auto regressieve modellen niet voldeden.

Vanuit de literatuur is ook bekend dat de bestaande theorieën voor het homogeniseren en van stortgoed alleen toepasbaar zijn als van de verschillende lagen de volume of massa schommelingen rond het gemiddelde klein en willekeurig zijn. In die gevallen kan de volume verdeling gelineariseerd worden om het gemiddelde. In dit proefschrift wordt aangetoond dat het homogeniseren van stortgoed in mammoet silo’s gekarakteriseerd wordt door een niet uniforme volume verdeling. Dit heeft tot gevolg dat linearisatie niet toepasbaar is en dat een homogeniseringtheorie voor mammoet silo’s vereist is. Deze theorie is in dit proefschrift uitgewerkt.

De ontwikkelde homogeniseringtheorie voor mammoet silo’s is een functie van:

– De kwaliteitseigenschappen van het materiaal; de fluctuaties hierin zijn de reden voor homogenisering.

– De volume verdeling; het aantal lagen dat doorsneden wordt en het volume van deze lagen in een snede. Verschillen in de silo geometrie en verschillen in de in- en uitslag methoden (inslag- en uitslaghoek, laag- en snededikte) beïnvloeden de volume verdeling en dus ook het homogeniseringresultaat.

In dit proefschrift zijn praktijkgegevens gebruikt om de materiaal eigenschappen te karakteriseren met behulp van verschillende ARMA modellen. Er kan geconcludeerd worden dat AR(1) modellen gebaseerd op ARMA criteria, de stochastische eigenschappen niet altijd juist representeren. Tevens kan geconcludeerd worden dat het voor de bepaling van de homogeniseringgraad belangrijker is om een goede benadering van de auto correlatie functie te hebben dan een perfecte fit op basis van ARMA criteria.

Er is een simulatie programma ontwikkeld voor het berekenen van de mate van homogenisering. Dit was nodig omdat de volume verdeling diskreet is en varieert voor elke homogeniseringmethode en silo geometrie. Een bijkomend voordeel van het simuleren is dat er geen beperking aan gegenereerde input eigenschappen is; naast stochastische signalen kunnen ook deterministische signalen gebruikt worden als input eigenschap.

Het simulatie programma is gevalideerd met:

– Proces dynamica (PD). De impuls respons functies van het gedeelte van de silo waar zowel de vul als leeghoeken constant zijn, komen overeen met de theoretische impuls respons functie volgens de proces dynamica.

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De simulaties zijn uitgevoerd met één en dezelfde set van input eigenschappen. Deze eigenschappen zijn gemodelleerd door een eerste orde auto regressief proces (AR(1)) met verschillende correlatie (verschillende karakteristieke volumes). De resultaten van de simulaties waren als volgt:

– Een afname van de hoogte-diameter verhouding bij gelijkblijvende opslagcapaciteit verbetert het homogeniseringeffect (reductie van de standaard afwijking).

– Een afname van de gemiddelde laagdikte (in mammoet silo’s van het Eurosilo type vaak 10 cm) leidt niet tot een substantieel betere homogenisering.

– Het aantal sneden heeft vrijwel geen invloed op de homogeniseringgraad.

– Een toename van de in- en uitslag hoeken tot een maximum van de hoek van inwendige wrijving van het materiaal leidt tot een verbetering van het homogeniseringeffect.

– De inslagmethode die gebaseerd is op gecontroleerd dumpen zorgt voor een betere homogenisering dan het gebruik van een schuinstaande transporteur voor het vullen van de silo.

– Het homogeniseringeffect hangt sterk af van het karakteristieke volume van de input eigenschappen wanneer het groter is dan een derde van het laagvolume en kleiner dan de opslagcapaciteit van de silo. In dit bereik neemt de homogeniseringgraad af voor een toename van het karakteristieke volume van de input eigenschappen.

Om het gebruik van een schuinstaande schroeftransporteur voor het homogeniseren te onderzoeken, zijn experimenten opgezet. Het doel van deze experimenten was om de prestatie van de schroef in schuinstand te vergelijken met de conventionele horizontale stand. De experimenten zijn uitgevoerd met zowel cohesief als vrijstromend materiaal.

Uit de resultaten van de experimenten kan geconcludeerd worden dat de schroeftransporteur geschikt is om de inslag en uitslag onder een hoek uit te voeren zolang het materiaal neerwaarts getransporteerd wordt.

De resultaten van de simulaties moeten kwalitatief geïnterpreteerd worden. Het homogeniseringeffect hangt namelijk sterk af van de input eigenschappen en daarom heeft de homogeniseringgraad geen kwantitatieve waarde wanneer van een specifiek materiaal de input eigenschappen onbekend zijn. In verband hiermee zouden de input eigenschappen van elke silo bekend moeten zijn om een redelijke absolute schatting te kunnen maken.

Met behulp van de simulaties is aangetoond dat de combinatie van inslag met behulp van gecontroleerd dumpen en uitslag met behulp van een schuinstaande transporteur de beste homogenisering oplevert (zie de figuur op de volgende pagina) wanneer de silo configuratie verder identiek is.

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inslag door gecontroleerd dumpen

uitslag met een schuinstaande transporteur

acd< 0 b > 0

De nauwkeurigheid van het simulatie programma voor het schatten van de homogeniseringgraad is onbekend omdat de resultaten niet zijn gevalideerd met praktijkwaarden. Tot nu toe zijn er namelijk geen mammoet silo’s met homogeniseringfunctie gebouwd, zodat geen gegevens van input en output eigenschappen van zo’n systeem beschikbaar waren. Op het moment dat er een mammoet silo met grootschalige homogeniseerfunctie beschikbaar komt, wordt aanbevolen om de resultaten van de simulaties met de praktijkwaarden te vergelijken. Op die manier kan het simulatie programma gevalideerd worden en kan ook de kwantitatieve waarde van de simulaties beoordeeld worden.

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Contents

2.4.4 Homogenization in end cones... 27

Introduction

Worldwide, industry uses bulk material as a source of energy or as a raw material for production processes. Since most production processes are continuous, intermediate storage facilities are required in order to uncouple the (discontinuous) supply of raw materials from the (continuous) production process. Depending on the scale of the production processes, large or small-scale storage is required.

1.1 Large-scale storage of bulk materials

A means of storage is considered to be small-scale if it contains anything from a few liters of material e.g. for use in the pharmaceutical industry, up to 2,000 m3 for applications e.g. in the agricultural industry as with grain silos. Large-scale storage can be achieved with stockpiles, heaps or large silos, e.g. dome silos and mammoth silos (Figure 1.1 and Figure 1.2). These storage facilities can hold between 5,000 and 100,000 m3 of bulk material. In general these systems require mechanical devices for stacking and/or reclaiming of the materials.

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(a) (b)

Figure 1.2: Large-scale storage systems: (a) dome silo, (b) mammoth silo of the Eurosilo type.

Piles and heaps have always been used for the storage of free flowing or slightly cohesive materials. Stacking takes place at the top and the material flows down naturally. The final angle of the material surface with the horizontal equals the angle of internal friction of the material. At mines and coal fired plants, piles are in the open air, implying that:

– only non- or slightly degradable materials can be stored, i.e. materials whose core properties are not affected by weather conditions;

– environmental pollution can occur, e.g. noise and dust due to strong winds.

The latter can be avoided by taking preventive measures, such as sprinkling with water or covering the pile. This, however, results in increased construction and operating costs.

An alternative to open air storage is the use of large silos, which, besides the environmental benefits, also offer the advantage of a higher storage capacity per m2 compared to stockpiles (Figure 1.3). A x103 [m2] V x103 [m3] stockpile mammoth silo 0 0.5 1 1.5 2 2.5 3 0 25 50 75 100 h=40[m] b=30[m], αr=45[°] h [m] V x103 [m3] D=40[m] D=50[m] D=30[m] D=20[m] 0 10 20 30 40 0 20 40 60 80 (a) (b)

Figure 1.3: (a) Storage capacities of mammoth silos vs. stockpiles as a function of the surface area A

(H = silo height [m], b = width of stockpile [m], α = angle of repose [˚]), and (b) storage capacities of mammoth silos as a function of the fill height H (V = storage capacity [m ],

D = silo diameter [m]).

r

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Dome silos are filled in the same way as piles; the material flows from the center at the top down towards the wall. Thus, a more or less free flowing material is required. The reclaiming process starts with gravitational flow until the angle of internal friction is reached, for the remaining part a screw reclaimer is used.

Mammoth silos are the only available large-scale storage facilities for cohesive materials such as gypsum and potato starch. In addition, these flat-bottomed silos can store free flowing materials as well, such as salt and coal (Van Wijk, 1993). A mammoth silo can store both cohesive and free flowing materials because stacking and reclaiming are controlled operations using screw conveyors to force the bulk material to move.

1.2 Storage in mammoth silos

The working principle of mammoth silos of the Eurosilo type can be characterized by the First In Last Out (FILO) principle and is illustrated in Figure 1.4.

Figure 1.4: Operational scheme of a mammoth silo of ESI Eurosilo BV (Schott and Van Wijk, 2001a).

Supply of material takes place over the roof of the mammoth silo. From there it is transported through a telescopic chute (1) to a screw conveyor. Consecutively, the material is transported to the outside (2) and spread on top of the material already in the silo by moving the auger frame in a clockwise direction (3). After each layer the conveyor is hoisted until the silo is totally filled. For reclaiming, the rotational directions are reversed (4 and 5). As a result, the material is transported (5) to the central column where it leaves the silo through a gravitational cone (6). Alternatively, in the case of cohesive material, a slotted column can be used through which the material is forced. Finally, the material is transported to processing facilities by a transport system (7).

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The flat bottom and the augers, which are always on top of the material, are a characteristic of a mammoth silo. These silos have a storage capacity of up to 100.000 m3. Typical characteristics of mammoth silos of the Eurosilo type are:

– height up to 40 m; – diameter up to 60 m;

– stacking capacity: 35 t/hr for potato starch up to 1,000 t/hr for potash; – reclaiming capacity: 50 t/hr for potato starch up to 2,500 t/hr for potash.

The residence time of the bulk material in the silo depends on the vertical position of the material in the silo in relation to the FILO principle, the supply to and the demand from the silo. For seasonal bulk material, e.g. potato starch, the residence time can be typically a few months up to a year. In the high capacity silos however, the residence time can be much shorter: typically a few days to weeks.

1.3 Upgrading and homogenizing raw materials

Large quality fluctuations of the material properties can occur due to the geographical origin of the raw materials. This means that processes using these raw materials have to deal with these fluctuations in order to produce products with a constant quality. The fluctuations can be dealt with in two ways:

– adapting the process, so that it can handle quality fluctuations of the raw material – adapting the raw material to the requirements of the process: upgrading.

Upgrading is defined as achieving a more constant quality throughout the material flow, where the mean quality cannot be influenced, but the frequency and amplitude of fluctuations around the mean value can.

y(t) upgrading

facility mx

x(t)

Figure 1.5: Systematic representation of upgrading, where x(t) represents the input property as a time

series with mean µx. The homogenized output property is represented by y(t) with mean µy = µ . x

Upgrading can very well be combined with the intermediate storage of raw materials as occurs in the mining industry, power plants, iron plants and cement plants where so-called blending piles are used. Adapting the process to the raw materials in these circumstances mainly is achieved by adding extra components to obtain a predefined recipe.

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storing

input output

material mixing material

blending homogenizing upgrading

Figure 1.6: Systematic representation of a storage facility.

The functionality of storage facilities can be expanded by upgrading as visualized in Figure 1.6 (Schott and Van Wijk, 2001a). From this figure it can be seen that upgrading can be divided into three functions: mixing, blending and homogenizing. The definitions are as follows:

– Mixing: defined as a random rearrangement of particles by means of mechanical energy, e.g. rotary devices in a fixed volume. Traces of individual components can still be located within a small quantity of the mixed material of two or more material types (Zador, 1991). Application: small-scale storage.

– Blending: defined as the integration of a number of raw materials with different physical or chemical properties in time in order to create a required specification or blend. The aim is to achieve a final product from, for example, two or more coal types, that has a well-defined chemical composition in which the elements are very evenly distributed and no large pockets of one type can be identified. When sampled, the average content and the standard deviation from the average are be the same (Carpenter, 1999). Application: e.g. using different types of coal for specific recipes.

– Homogenizing: defined as the systematic regrouping of the input flow in order to provide a more homogeneous output flow of one type of material so that inherent fluctuations of chemical or physical properties in time are evened out compared to the input flow. Application: e.g. one batch of limestone or coal, i.e. to homogenize the material in itself.

This thesis concentrates on large-scale homogenization. Small-scale systems and mixing will not be considered. Blending, as defined here, is also left out of consideration; however, the term ‘blending pile’ will frequently be used. This should be read as ‘homogenization within a blending pile’, i.e. upgrading one single material.

1.4 Problem domain

To date, no large-scale single upgrading facilities except for blending piles are available. As mentioned before, blending piles can only be used to store and homogenize free flowing materials and can therefore not be used for the storage of cohesive materials. Homogenization of cohesive materials requires a storage facility that can handle cohesive materials, e.g. mammoth silos.

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A mammoth silo has the following advantages compared to blending piles: – storage and homogenization of free flowing as well as cohesive material;

– lower surface - volume ratio: storage volume is independent of the angle of internal friction, but depends on the silo geometry;

– less segregation (i.e. involuntary separation of fractions of a bulk solid, which was previously more homogeneous), depending on the material handling and transportation methods used;

– degradable bulk solids can be stored, due the full enclosure of the material and the possibility to control the climatological conditions inside the silo.

However, the competitiveness of mammoth silos versus blending piles depends on the achievable rate of homogenization and the required homogenization efficiency for successive processes. Investment costs and available space are also relevant factors: there is an increasing market for compact storage facilities in densely populated areas.

In 1994 a research project was started at Delft University of Technology that resulted in several methods to achieve homogenization in mammoth silos, each method having its own specific degree of homogenization. At the time, however, it was neither possible to indicate the homogenization efficiency of the different methods, nor to make a well founded choice of the method with the largest homogenization efficiency.

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The methods derived from the research project are described in a patent (Gerstel and Van Seters, 1998) and as an example one of the methods is presented in Figure 1.7. It shows the current horizontally orientated stacking method of the silo combined with an alternative reclaiming method: reclaiming using an inclined conveyor. As can be seen in the figure multiple layers are intersected during reclaiming. This means that material in a reclaimed slice is a weighted average of the intersected layers, which is the principle of homogenization. The patent (Gerstel and Van Seters, 1998) contains relative indications of the homogenization performance of the methods. However, for use in industry the exact degree of homogenization should be explicitly known. After all, different industries use different raw materials and (may) have different requirements regarding the homogeneity and quality fluctuations of the material they are processing.

Several people (Van der Mooren, 1967), (Gerstel, 1979) have investigated homogenization and have set up theories to describe the homogenization efficiency of blending piles. There are two major problems associated with these theories.

First, they are simplified and do not take the geometry of the blending piles into account. Recently, Pavloudakis and Agioutantis (2003a, 2003b) developed a blending pile simulation program to investigate the effects of different operating parameters on the blending and homogenization performance, e.g. the storage capacity, the stacking and reclaiming method, and the number of layers.

Second, the results of the theories are based on simplified models of the input properties. These simplified stochastic models are useful to determine the homogenization efficiency of blending piles and multiple bins analytically. Nowadays, however, more complicated preprocessing stages may give cause for the use of more complex time series models for the input properties.

As will be explained in Chapter 2, the homogenization theory of blending piles cannot be used for mammoth silos. The consequence is that the existing homogenization theories must be extended and adapted so that the geometry of mammoth silos can be taken into account. Furthermore, the modeling of the input properties has to be investigated in order to determine the applicability of the simplified models used by Van der Mooren (1967) and Gerstel (1979). In Chapter 4 it will be shown that the extended theory for mammoth silos will have no general analytical solutions, so numerical calculations are necessary to:

– determine the homogenization efficiency for the different stacking and reclaiming methods,

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Summarizing, this thesis focuses on two main issues. First, the homogenization of the different patented stacking and reclaiming methods is unknown. These should be determined and compared so that the optimal homogenization system in mammoth silos can be established. Second, it will be shown that the available homogenization theory of blending piles does not apply to the homogenization process of mammoth silos. Therefore the homogenization theory must be extended in order to calculate the homogenization efficiency of mammoth silos.

1.5 Research objectives

The main objective of this study is to determine the best stacking and reclaiming method for homogenization of bulk materials in mammoth silos. The determining criterion for the best method is the homogenization efficiency, i.e. the reduction of the standard deviation (homogenization effect) and the improvement of the auto correlation function (ACF) of the material properties. An increasing homogeneity of the material implies an increasing coherence of the material resulting in a decreasing homogenization effect and an increasing characteristic volume in the ACF.

In this thesis available space and investment costs are no criteria for determining the best homogenization method. This research is further limited to the silo content including the stacking and reclaiming mechanisms, i.e. no attention will be paid to systems for supplying bulk solids to the silo and systems providing transport from the silo to processing facilities. Small-scale behavior of the material (particle-particle interactions and mixing of particles) will also be left out of consideration. After all, large-scale homogenization is involved in regrouping units of material, where these units can contain enormous amounts of particles. Changes in the positions of particles relative to each other are only important for the local state of the material. This small-scale behavior is not important for large-scale homogenization of the material.

In order to be able to compare the different stacking and reclaiming methods with each other a homogenization theory for mammoth silos must be developed. This theory can be based on existing homogenization theories of blending piles and multiple bins. Then, the development of a homogenization model enables to calculate the homogenization efficiency as a function of the different input parameters, such as silo diameter, silo height, stacking and reclaiming angles, number of layers, number of slices and raw material properties.

1.6 Outline of thesis

The outline of this thesis is visualized in Figure 1.8. The state of the art in homogenization is presented in Chapter 2. This chapter deals with various aspects of homogenization: assessing homogenization, homogenization theory of blending piles and multiple bins.

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2 The state of the art in large-scale homogenization

5 Homogenization indicators

4 Homogenization theory of mammoth silos

6 Modeling input properties

7 Results of simulations and performance of indicators

8 Practical aspects of homogenization in mammoth silos

9 Conclusions and Recommendations 3 Concepts for homogenization in

mammoth silos

Figure 1.8: Outline of this thesis.

The stacking and reclaiming methods to homogenize in mammoth silos presented by the patent of Gerstel and Van Seters (1998) are discussed in Chapter 3. From this chapter it follows that the existing homogenization theory has to be adapted for application to mammoth silos. This ‘Homogenization theory of mammoth silos’ is presented in Chapter 4. It will be shown that the theory has no general analytical solutions, and therefore numerical calculations are necessary. The results of these calculations are presented in Chapter 7.

In Chapter 5 the characteristic volume of a method (CVM) and process dynamics (PD) as indicators of homogenization efficiency are discussed. Chapter 6 focuses on modeling of the input properties, e.g. Auto Regressive Moving Average (ARMA) modeling, and field data will be presented with their characteristics.

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Chapter 7 provides results from the numerical calculations of the homogenization theory of mammoth silos. The influence of silo geometry, material properties, and stacking and reclaiming methods will be discussed, together with adequacy of the homogenization indicators: characteristic volume of a method (CVM) and process dynamics (PD).

In Chapter 8 the translation of the developed homogenization theory to practice is discussed and the following aspects are considered:

– compressibility of the material;

– small-scale homogenization in present-day operation; – feasibility of stacking and reclaiming methods;

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2 The state of the art in homogenization

Chapter 2

The state of the art in homogenization

As described in Chapter 1 homogenization systems for free flowing or slightly cohesive materials exist mainly on a large scale. The most important theories on and models of these homogenization systems were developed by Van der Mooren (1967) and Gerstel (1979). Van der Mooren mainly describes multiple bin systems (Figure 2.2) whereas Gerstel focuses on longitudinal and circular blending piles (see Figure 1.1, Figure 2.4 and Figure 2.6). Essential for the application of the models are the characteristics of the raw material properties to be homogenized (Chapter 6) and the way the degree of homogenization is assessed. If the characteristics of the material properties are known, the homogenization efficiency of mammoth silos can be investigated, as will be shown in Chapter 7. This study is restricted to single time series, i.e. the homogenization is assessed based on a single characteristic property of the material to be homogenized, e.g. calorific value of coal or dirt in potato starch.

The objective of this chapter is to describe the state of the art in large-scale homogenization. First, Section 2.1 deals briefly with the characteristics of time series. An extensive discussion on characteristics of time series is presented in Appendix A. The assessment of homogenization is discussed in Section 2.2. Section 2.3 explains the homogenization theory of multiple bins in detail. Subsequently, the homogenization theory of longitudinal discontinuous blending piles is discussed in Section 2.4. A more compact and continuous homogenization facility is a circular blending pile with simultaneous stacking and reclaiming operation (Section 2.5).

This chapter concludes with a discussion on the state of the art in homogenization. In Chapter 3 the concepts for homogenizing in mammoth silos will be introduced.

2.1 Characteristics of time signals

Roughly speaking, a stochastic process is a collection of random variables, indexed by either a discrete or a continuous time parameter. Obviously, a stochastic process cannot be fully described in a deterministic way because of the random variable. A measured result of a stochastic process during a certain time is called a stochastic signal. Such a signal can consist

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of a deterministic pattern due to the process and a random component. The properties of this random component can be described in statistical terms if the signal meets certain requirements regarding stationarity and ergodicity. Both terms will be discussed hereafter (Bendat and Piersol, 1986).

Stationarity

A signal is stationary when its statistical properties do not change over time. Consequently, the expected value and the variance are independent of the time parameter t. In addition, the auto covariance function (ACVF) depends only on the difference between ta and t and not on the absolute value of t, i.e. it is time-invariant.

b

In practice, a definition of stationarity called second-order stationarity or weak stationarity is employed. A time series x(t) is weak-stationary or second-order stationary if it satisfies the following conditions:

mean value: µ_x( )t = µ_x

2 2

variance: σ_x( )t = σ_x

auto covariance function: φ τ = φ_x( ) _x(t_a−t_b) is a function of τ =t t_a- only_b

In this thesis a stationary process will always refer to a second-order or weak-stationary process. By definition, stationarity implies that the process has a constant mean µ. Therefore, without loss of generality only zero-mean processes can be considered, since the statistical properties are relative to the mean value.

Ergodic properties

A signal can be considered as a drawing from an ensemble (i.e. a set of results due to repetition of a process, see Appendix A). Repetition of the experiment results in many signals used to determine the properties of the ensemble. In the stationary case absolute time is not a factor and therefore different observation periods can be chosen. If instead of many signals different drawings of one signal may be considered as independent drawings from the ensemble, the experiment needs to be carried out only once to determine the required properties of the ensemble. Signals meeting this requirement meet the so called ergodic property and are called ergodic signals.

A result of the ergodic property is the equality between the expected values over time and the expectations of the ensemble values. The statistical properties of a time series x(t) can then be written in a continuous form or a discrete form as shown in equations (2.1) to (2.3).

(2.1) (2.2) 0 0 1 [ ( )] lim ( ) ( ) T n x _T x i E x t x t dt x t T n →∞ ₌ µ = =

∫

µ = 1

∑

_i 2 2 2 2 0 0 1 1 [( ( ) ) ] lim ( ( ) ) ( ( ) ) T _n x x _T x x i i E x t x t dt x t T n →∞ ₌ σ = − µ =

_∫

− µ σ =

∑

− µ 2 x

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0 0 1 ( ) [( ( ) )( ( ) )] lim ( ( ) )( ( ) ) 1 ( ) ( ( ) )( ( ) ) T x x x _T x n x i x i x i E x t x t x t x t T k x t x t k n →∞ = φ τ = − µ + τ − µ = − µ + τ − µ φ = − µ + − µ

∫

∑

x dt (2.3)

Auto correlation function (ACF)

To indicate the coherence and the frequency of fluctuations of the signal, the auto covariance function (ACVF) φ is used, this being the expected value of the product between x(t) and

x(t+τ) relative to the mean value (equation (2.3)).

x

This function is a basic parameter for describing the dynamic properties of a signal and as explained before it depends on the time difference or lag τ for stationary continuous signals. For discrete signals k is used to indicate the lag.

The smaller the auto covariance, the smaller the coherence between the amplitudes of both times and the higher the frequency of fluctuations will be. For τ = 0 the ACVF turns into the variance of the signal. The ACF ρ (τ) is defined as the normalized ACVF by the variance: x

(2.4) The ACF and ACVF only differ from each other by a constant factor (the variance), and they

therefore have the same meaning. Since the ACF is normalized it can easily be used to compare ACFs.

2.2 Assessing

homogenization

2 ( ) ( ) x x x φ τ ρ τ = σ

The degree of homogenization obtained by any system (both small-scale or large-scale) can be evaluated in several different ways. Boss (1986) analyzed 37 different statistical expressions describing the state of a mixture based on a number of (arguable) criteria. The three most convenient of his two-parameter expressions of the homogenization effect use the reduction of the standard deviation in a material.

In large-scale homogenization the ACF is also used to evaluate homogenization systems (Gerstel, 1979).

Reduction of the standard deviation

The homogenization effect is judged by the ratio of the standard deviations of the input and the output in three slightly different ways.

The most obvious measure for the homogenization effect is the ratio of the standard deviations of the output and the input. That is, the decrease in intensity of the fluctuations in the output compared to the input, or:

(2.5) y H x σ η = σ

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This definition of the homogenization effect was first used by Michaels and Puzinauskas (1954) and was adopted by Gerstel (1979) for his homogenization theory of blending piles. Both Van der Mooren (1967) and Boss (1986) argued that this ratio has the disadvantage of resulting in a lower value corresponding with a better homogenization result. Therefore, Van der Mooren used the expression introduced by Rose (1959):

(2.6) Here ηH' equals zero if no homogenization occurs and 1 if no quality fluctuations remain in

the output flow.

1 1 x y y H H x x ′ σ − σ σ η = = − = − η σ σ

The third way of indicating the homogeneity effect, first used by Smith (1955), is the inverse of equation (2.5): (2.7) 1 _x H H y ′′ σ η = = η σ

This expression was used by, among others Hasler and Vollmin (1975), Schofield (1979 and 1980) and Zador (1991), and is often used in the cement industry. A very high value of ηH'' indicates excellent homogenization and low values (≥1) very poor homogenization. The main disadvantage of this relation is that the operational parameter is in the denominator, implying that the homogenization effect does not lie between the boundaries 0 and 1, but between 1 and ∞.

In the literature on homogenization and blending all three relations are frequently used. In this thesis relation (2.5) will be used since it is a direct ratio of the output and the input, and the results are therefore easier to visualize. The choice of this definition will not affect any theoretical discussions on homogenization effects, simply because all the other expressions can be derived directly from this one.

Use of correlation functions to assess homogenization

Very often, only the reduction of the standard deviation (ηH) is used as measure for homogenization and the ACVF is left out of consideration. The ACVF, however, stores information about the degree of coherence in a signal and can play a role in theory, simulations and assessment of homogenization in two ways.

Cochran (1946) and Stange (1958) used the covariance of the quality differences in their sampling theory. In homogenization, Van der Mooren (1965 and 1967) was the first to use the ACF of the input properties in order to take the coherence of the material quality into account. Dufour (1970), Van der Grinten (1973) and Gerstel (1979) also followed this approach. Lemke (1962), Larsen (1962), Denny and Harper (1962) and Bemelman (1963) modeled the input properties as being random fluctuations without any coherence between neighboring

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data points. However, Bemelman (1963) treated the situation without correlation between the elements as a special case and instead concentrated on the input properties modeled by a negative exponential ACF. He however did not use this ACF for his calculations and commented that its exponential behavior should be verified by experiments. Even without preprocessing stages it is a serious simplification to assume that the input properties can be modeled as white noise. Sarma (1974) performed a series of observations on the grade of gold ore just excavated from the mine. He found that the characteristics of the data set could very well be described by a negative exponential ACF.

Gerstel (1979) introduced the use of both the homogenization effect (ηH) and the ACF ρy of the output compared to the input to examine the homogenization performance. That is, a signal that is more evened out has less fluctuations and less intensity of fluctuations (Figure 2.1a). In that case the ACF of the homogenized material will be improved compared to the input (Figure 2.1b).

m mx= y x t y sx sy P(t) x t 1 0 y r(t) (a) (b)

Figure 2.1: Change in statistical properties after homogenization (a) the input function x and the (homogenized) output function y with their course in time and standard deviation, (b) ACF of x and y.

The procedure followed by Gerstel applied to blending piles will be discussed further in Section 2.4. First, the general applicability of the work of Van der Mooren on linear large-scale homogenization systems is discussed.

2.3 Homogenization with multiple bins

Van der Mooren (1967) considered in his dissertation two different ways of homogenizing in bins: systematic and preselected.

He uses ‘systematic homogenization’ to indicate that the input stream is divided into parts with equal lengths lip. These parts are in turn stored in a predefined number of bins (nbi). The output streams of the bins then come together with equal and constant flow rates in the output stream of the homogenization installation. So n input elements (or number of bins) at a constant distance l from each other come together in the output stream. This is visualized in Figure 2.2.

bi ip

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With ‘preselected homogenization’, the parts of the input stream are not stored in turn in one of the bins, but are separated on the basis of their quality. This results in a situation that can be described as blending rather than homogenizing and will not be discussed here.

i=1 i=2 i=nbi

lip input x

output y lip

lop

Figure 2.2: Homogenization using nbi bins according to Van der Mooren (1967).

2.3.1 Assumptions and homogenization model

The quality fluctuations of the input properties are assumed to be realizations of a stochastic process that at least meets the requirements of a weak-stationary process. Thus, the ACVF of the input parts is only a function of their distance and not of their mutual distance or quality. Furthermore it is assumed that the process is ergodic and that the mean values of the input and output properties are equal and zero (µx = µy = 0). This assumption does not influence the other statistical parameters like variance and ACF.

The input stream is modeled with constant flow, thus time and length scales are proportional.

Model of the input properties

Van der Mooren assumed throughout his dissertation that the ACVF of the input fluctuations can be described by a negative exponential function (see also Figure 2.3b):

2 ( ) V cx V x V xe τ − φ τ = σ (2.8) where:

τ V lag on volume scale [m ] 3

Vcx characteristic volume of the input property x [m3]

In addition, he defined an ACVF as the summation of two negative exponential functions:

1 1 2 ( ) V V cx cx V x V C e C e τ τ − − φ τ = + V 2 (2.9)

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V 0 e-1 Vcx V eVcx V

Figure 2.3: Negative exponential auto covariance function (ACVF) of the input properties.

Furthermore, Van der Mooren (1965) discussed the use of other functions such as a damped periodic function (equation (2.10)) with as a special case the exponential function (equation (2.8)). (2.10)

(

2 ( ) cos V cx V x V xe Vcxp V τ − φ τ = σ τ

)

in which Vcxp is the characteristic volume of the periodic input property x in m3.

However it has not been proved that these functions reflect reality.

Homogenization model

Homogenization in mammoth silos shows many similarities with the systematic homogenization described by Van der Mooren. However, as will be explained in detail in Section 3.4 and Chapter 4, due to the homogenization method in mammoth silos, the output stream consists of a weighted average of the input parts. Initially, Van der Mooren argued that the output stream is equal to a weighted average of the input stream (equation (2.11)).

(2.11)

where:

y property of output part s [-]

1 1 bi bi n i i i s n i i w x y w = = =

∑

s i input part [-] nbi number of bins [-]

xi property of input part i [-]

wi mass of input part i [kg]

But, in contrast to mammoth silos, this weighted average is not introduced because of the homogenization method but because of the density or volume fluctuations of the input stream.

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In order to obtain a linear function y in terms of the expectations of the random variables x and w , Van der Mooren used a Taylor approximation. This Taylor approximation of degree 1 is as follows: s i i

(

)

(

)

2 1 1 1 1 [ ] [ ] [ ] [ ] ( ) [ ] bi bi bi bi n n n i i i s s s n i i i i i i i i i E w E x y y y x E x w E x w E w = = = = ∂  ∂  = + _ − _+ _ − _+ ∂ ∂    

∑

wi O h (2.12)

According to the working rule of Davies (1958) the series can be terminated after the first order terms when the variation coefficients of xi and w are smaller than 0.2, when calculations of the statistical properties are required. The variation coefficient (VC) is defined as:

i

(2.13) Van der Mooren assumes that for practical purposes second and higher order terms are

negligible for variation coefficients smaller than 0.4. However, this working rule has not been found elsewhere and Van der Mooren omitted to prove that the variation coefficients indeed meet this requirement.

VC=σ µ

Elaboration of the first order terms of equation (2.12) gives:

(2.14)

Since the average value of the material was chosen to be zero, equation (2.12) transforms with equation (2.14) into: [ ] [ ] [ ][ ] 1 1 2 1 ₁ 1 and 0 bi bi bi _bi i i i i i i i i n n i i i i s i s i i n _n w E w w E w i bi i x E x _i x E x i i _i x w w x y w y x n w w _w = = = = = = = ₌ − ∂ ∂ = = = ∂ ∂ _ _    

∑

_∑

= 1 1 nbi s i i bi y n = =

∑

x (2.15)

2.3.2 Results of homogenization in bins

The general solution of the variance of the output presented by Van der Mooren after applying the Taylor approximation about the expectations is (by substitution of equation (2.15) in (2.2) with y = x(t)): s (2.16)

( )

2 ₁ 2 2 2 2 2 1 1 1 1 1 1 2 ( ) ( ) bi bi bi bi n n n n y i i i i i j i bi bi bi E x E x E x n n n − = = = = + _ _      σ = _ _ = _ _+    

∑



∑

∑ ∑

i j x

in which E[(x )2] = σ and the covariance on distance j-i (φ(j-i)) is indicated by E(x x ). Recall that E[x ]=0. Let d = j-i, so that n-d combinations occur within distance d⋅l and the

relationship between the quality in different parts is expressed by:

i x2 i j

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1 1 2 2 1 1 1 2 2 ( ) ( ) ( bi bi bi n n n i j bi x ip i j i d bi bi E x x n d dl n n − − = = + = = − φ

∑ ∑

∑

) (2.17)

in which l is the length of the input parts in m. ip

The standard deviation of the output can then be written as:

(2.18) 2 2 1 ( ) ( bi bi n y bi x d n bi n d dl n =− σ =

∑

− φ _ip)

This equation can also be written as a function on a volume scale due to the proportionality of the length and volume of the input parts if the flow is constant:

(2.19) in which V is the volume of the input parts in m .

2 2 1 ( ) ( bi bi n y bi x d n bi n d dV n =− σ =

∑

− φ _ip) ip 3

The homogenization effect then equals:

(2.20) When the input ACVF or ACF (φx or ρ ) of the quality fluctuations is known (e.g. in the form

of equations (2.8) - (2.10)) the homogenization efficiency can be calculated by substituting ρ in equation (2.18). 1 ( ) ( bi bi n y ₎ H bi x ip d n x bi n d dV n =− σ η = = − ρ σ

∑

x x

Van der Mooren (1967) presented results based on negative exponential ACVFs with varying characteristic volumes of the input properties (V ). Furthermore, Van der Mooren (1965) gives general considerations on results based on periodical ACVFs. However, results have been omitted due to the infinite combinations of parameters.

cx

2.4 Homogenization in longitudinal blending piles

A very frequently used theory for homogenization in longitudinal blending piles was presented by Gerstel (1979). In his thesis he considers both correlations in piles and correlations between piles. This thesis focuses on homogenization of the contents of a single mammoth silo and therefore leaves the correlation between piles and between silos out of consideration.

Gerstel also discussed circular blending piles. That approach differs strongly from the longitudinal blending piles and shows some similarities with mammoth silos. This will be discussed in Section 2.5.

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2.4.1 Assumptions and homogenization model

The longitudinal blending pile is built up using nbp layers with equal volumes. Figure 2.4 presents the rearrangement of the input material relative to the output material after reclaiming. The analysis of the pile is based on a time scale.

i=1 i=2 i=nbp

input x output y tbp nbp t2 nbp t1 nbp t1 t2 tbp tx0 ty0

Figure 2.4: Homogenization using a longitudinal blending pile with nbp layers (Gerstel, 1979).

Assumptions

In contrast to Van der Mooren, Gerstel assumes the input process to be continuous, i.e. he does not consider discrete elements within one layer. However, to arrive at a homogenization theory this assumption is redundant because of the statistical nature of the input properties. I.e. when a system is time invariant or the signal is stationary, the properties are completely described by the time difference or lag τ (or k). In that case the number of elements in between is irrelevant, the amount of material, however, is relevant.

Since the rearrangement of layers is a discrete process, the homogenization efficiency will be expressed in summation terms as by Van der Mooren.

Gerstel made the following assumptions to arrive at the homogenization efficiency: 1. The pile has no end cones.

2. Each layer is stacked in the same traveling direction of the stacker. 3. Each layer contains the same amount of material per meter pile length. 4. The mass flow of the input of the pile is constant.

5. The material is reclaimed simultaneously from all the layers.

6. The mass flow of the output is constant and equal to the mass flow of the input.

7. The fluctuations of a property in the input flow are assumed to be a realization of a weak-stationary and ergodic process.

As a result of these assumptions each slice contains an equal amount of all intersected layers, i.e. the property of the slice equals the average property of the layers. The significance of these assumptions is discussed in detail in the following:

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1 End cones (see also Section 2.4.4)

The presence of end cones means that the homogenization process is not stationary and that the overall result is less due to the worse homogenization efficiency of the cones. After all, in these cones only a selection of layers will be intersected; in an extreme situation the first slice will only intersect the last layer.

2 Stacker

The usual practice for the stacker is to travel in a zigzag course to deposit material. Due to this oscillating operation mode only in every second layer it is ensured that material is separated from adjacent material by a variable quantity of one to two layers. When every layer is stacked in the same direction, the material is separated by a quantity of exactly 1 layer.

3 and 4 Varying mass flow

Gerstel dealt with varying mass of the input flow in the same way as Van der Mooren, by linearization (Section 2.3.1) and came to the conclusion that it can be considered as a secondary effect in estimating the homogenization efficiency of blending piles. Fu (1995) proposed to use a stochastic variable for the mass flow so that better account can be taken of the conditions in a real blending bed. The results presented by Fu will be discussed in Section 2.4.2 together with the result presented by Gerstel.

5 Reclaimer

An ideal reclaimer meets the assumption of simultaneous reclaiming from all layers. Although this depends on the type of reclaimer there are some variations in arrangement within the slice. Nevertheless, the output property is still the average of the input properties of the layers, only disturbed on a smaller scale. This has no effect on the homogenization efficiency as determined in the following.

6 Output flow

This assumption concerning the constant mass flow is necessary in order to be able to use the same time scale for input flow, output flow and pile. Use of a length or volume scale would have made this assumption redundant.

7 Model of input properties

The input properties are modeled as by Van der Mooren (1967), i.e. the input fluctuations can be described by a negative exponential function (equation (2.8)):

(2.8) 2 ( ) V cx V x V xe τ − φ τ = σ

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Homogenization model

From Figure 2.4 it follows that the property content in the cross-section y1 equals:

1 1 0 1 1 1 1 ( ) ( ) ( ( 1) ) bp bp n n bp xi x i i bp bp bp bp t y t x t x t i n = n = n n τ =

∑

=

∑

+ + − (2.21) where:

nbp number of layers in the blending pile τbp duration of reclaiming the blending pile [s]

t x0 origin of the input flow [s]

An element in the output stream y(t ) equals the summation of the n input layers at a certain time t . Analogous, another output element in the same pile has property y(t ) equal to:

1 bp

xi 2

(2.22)

2.4.2 Results of homogenization in blending piles

The ACVF of the output can be calculated using expectations and equals:

2 2 0 1 1 ( ) ( ( 1) ) bp n bp x j bp bp bp t y t x t j n = n n τ =

∑

+ + − 2 1 1 2 1 ( , ) 1 ( ) bp bp n bp y x d n bp bp bp bp d t t t t d n =− n n   n τ − φ = _ − _φ +  

∑

(2.23)

So the variance is given by:

(2.24)

(

)

2 2 1 (0) ( ) bp bp n bp y y bp x d n bp bp n d d n =− n τ σ = φ =

∑

− φ

Where τbp/nbp equals the time to build up one input part or a layer. Due to linearization and the assumptions 3, 4 and 6, the time scale is proportional to the volume scale. Thus, the volume based expression of the variance then becomes:

(

)

2 2 1 ( bp bp n y bp x d n bp n d dV n =− σ =

∑

− φ _ip) (2.25)

The homogenization effect then equals:

(2.26) The homogenization efficiency of bins corresponds with the efficiency of a blending pile if

and only if the number of bins equals the number of layers in the pile (n = n ) and if the volume of the element stored in a bin and a layer (V = τ /n ) are equal (compare equation (2.20)). 1 ( ) ( bp bp n y ₎ H bp x ip d n x bp n d dV n =− σ η = = − ρ σ

∑

bp bi ip bp bp