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7/31/2019 Math Model
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Boletim Paranaense de Geocincias, n. 50, p. 69-80, 2002. Editora UFPR 71
LESKO, M. et al. Development of processing technology and mathematical model of modification of magsenite raw material
- thermic processing of magnesite concentrates,- modification of magnesite sinters,- production of magnesite fireproof materials.
The listed technological units are presented by
technological operations:- grinding and milling,- briquetting and pressing of fittings,- phase separations:
sorting on sieves,hydraulic sorting,sorting in dense suspensions,flotation sorting,magnetic sorting,
- interphase sorting:unwatering
dust sedimentation, filtration- thermal operations:
burning of magnesite sinters,burning of magnesite building materials,
- chemical operations:production of pure MgO,production of chemically fixed buildingmaterials and fireproof concretes.
Complex of these operations in mutual connectioncreates a technological scheme: the operations arerealized in corresponding machines and equipments. Theresult of the technological method of modification isdetermined by:
- properties of raw material,- structure of technological scheme,- control level of the technological process.
The content of the contribution is not the analysisof the level of control of the technological process, waterand sludge management, circulation of operating anddiluted suspension, thermic processes either theprocesses of briquetting and pressing.
The properties of raw material are always qualifiedaccording to that characteristic that is used in thetechnological operation and that reflects theperformance of the raw material in this operation.
Properties of the raw material
The properties of raw material are qualifiedfollowing the contents of harmful materials (Fe
2O
3, CaO,
SiO2
etc.), the way to outgrow the useful and harmfulcompounds are given by the genesis of the deposit. Fromthe technological position for sorting operations theyare characterised:
- by the function of separation of mass accordingto the separation characteristic (k) . Thesymbol krepresents the characteristics that areused at separation. For example at sorting onsieves it is the size of particle d, at sorting indense suspensions it is the density etc.,
- by the function of separation of useful andharmful compounds according to the separationcharacteristic (k).
If there are more arguments, functions with two orthree arguments can be created.
Functions representing the properties of rawmaterials are detected by technological analysisand they are presented:
- in numerical form tables,- in graphical form granularity curves,
modifiability curves,
- in analytical form.The structure of the technological scheme
determines the separation characteristic of the objectand it is determined by:
- characteristics of elements creating the object machines and equipments,
- relations between the elements of the object mutual connection of machines and equipments.
The characteristics of each machine used aredescribed by the separation capability that can bepresented:
- in the worst case by technological parameter,e.g. partial weight yield of particular product andpartial recovery of the studied compound,
- distribution function e.g. Tromps curve innumerical or analytical form (k).
}
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Boletim Paranaense de Geocincias, n. 50, p. 69-80, 2002. Editora UFPR72
LESKO, M. et al. Development of processing technology and mathematical model of modification of magsenite raw material
In more complicated schemes the theory oftransmissive functions can be used for this purpose(Leko 1998).
Mathematical model of magnesite processing
Various modification processes are used whenprocessing magnesite from grinding to briquetting withconsequent thermal operations scheme in Fig.1. Wewill follow this scheme to discuss this problem.
Mathematical model of modification
of magnesite raw material
Vectorial expression will be used for themathematical description of the technological schemeof modification of magnesite because it enablescharacterization of technological relations respectingthe structure of technological process. The
technological scheme of modification is shown inFig.2. Mathematical models of particular operationswere presented before, e.g. models of sorting andgrinding (Leko 1996; Lynch 1981), models of sorting
(Leko 1998; Leko & Zelek 1994). Thecharacteristics of particular components of mathematicalmodels will be described in particular operations.Vectorial expression is characterised by brief expressionbecause each vectorial expression represents as manylinear equations as fractions or components we study.The following expression describes the flux of material solid state through a technological line. In similarway it is possible to create a model of flow of water,sludge, operating and diluted suspension.
Symbols used:Q
i vector representing material flux i in the system
column format (Kx1),i = 0, 1, ,I, whereI number of material fluxes
Fig. 1. Scheme of magnesite processing
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Boletim Paranaense de Geocincias, n. 50, p. 69-80, 2002. Editora UFPR 73
LESKO, M. et al. Development of processing technology and mathematical model of modification of magsenite raw material
Fig. 2. Technological scheme of processing of magnesite raw material
in the system, the listed vector can also bemultidimensional. For simpler calculation it is useful toexpress the amount of particular components, e.g.fractions, fractions of different density etc. by weightof these components, k= 1, 2, , K number of studiedcomponents.
VectorQ0 presents the input batch to the finishing
plant.In general the vector Q
ican be expressed:
(2)
where qik
are the elements of the vector.The elements can be calculated from the expression
(3)
where ik
expresses the amount of the kfraction in thematerial flux i [weight fraction]. If it is possible todetermine the granular structure by certain type ofdistribution (normal, lognormal, Weibull, exponentialetc.) or certain distribution function, the content ofparticular fractions can be calculated fromcorresponding functions and it is enough to set just the
parameters of these functions. Batch to the mill fromthe point of granularity has various compositions. If thematerial is grinded in the 1st and 2nd stage, the granularityof the material fluxes changes thus the amount offractions decisive for technology is changing. They arethe fractions 0(3)5 mm, (3)512(15) mm, (12)1540(60) mm that are in given order to be modified byflotation (operations 11, 12, 13), sorting in dynamic
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Boletim Paranaense de Geocincias, n. 50, p. 69-80, 2002. Editora UFPR74
LESKO, M. et al. Development of processing technology and mathematical model of modification of magsenite raw material
conditions (operation 8) and static conditions(operations 6, 7). If the batch is variable and it is grindedin existing grinders operations 1 and 2, if we havedifferent efficiency of wash and sorting operations 3,4, 5, portion of these fractions is different. Changeabilityof composition can be found out by simulation usingmathematical model of preparatory operations,
equations (1 to 9) from file 1.Operators:DR
j operator of grinding in operationj, represents
the principles of disintegration of existing and offormation of new fractions. Operator of grinding DR isrepresented as follows
DRj
= Bj
Sj
+ E Sj
The matrix Bj
presents disintegration (columns) andformation (rows) of fractions. It is a triangular matrix
format (KxK), its elements are determined by thestandard characteristics of the grinder.
The matrix Sj
(KxK) presents the conditions ofgrinding in gap in operationj.
E is a unit matrix with same format (KxK).The models of grinding and milling in further
technological operations are formed similarly (Leko
1998). Values of elements of matrix B and S are givenby the results of measurement.
PMj
operator of wash represents the distributionof solid state in consequence of wash. It is a diagonalmatrix format (KxK). Operator (E PM
j) represents
the transition of solid state to sludge.TR
j operator of sorting represents the distribution
of fractions to sifting products. It is a diagonal matrixformat (KxK). The element on the main diagonal tr
ii
(1)
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Boletim Paranaense de Geocincias, n. 50, p. 69-80, 2002. Editora UFPR 75
LESKO, M. et al. Development of processing technology and mathematical model of modification of magsenite raw material
represents the distribution of fraction to sifting product.For sorting on sieves and grate is valid the conditionthat grains with diameter d< d
Rpass to the sifting
product in certain proportion depending on effectiveness
of sorting r< 1. Grainsd