this PDF file - Archives of Mining Sciences

Arch. Min. Sci., Vol. 56 (2011), No 3, p. 441–450
441
Electronic version (in color) of this paper is available: http://mining.archives.pl
BOHDAN BOZHENKO*, ROBERT JUNGA*1, JANUSZ POSPOLITA**
MATHEMATICAL MODEL OF THE MILLING PROCESS ON THE RING-ROLLER’S TABLE
PART 1. MATHEMATICAL MODEL AND IT’S NUMERIC SOLUTION
MODEL MATEMATYCZNY PROCESU ROZDRABNIANIA NA MISIE MŁYNA
ROLKOWO-MISOWEGO
CZĘŚĆ 1. BUDOWA MODELU I ROZWIĄZANIE NUMERYCZNE
The first part of the article presents a balance model of grinding on the ring-roller mill’s table. The
model describes the grinding process by means of the milling parameters and selection, as well as the
grind material movement and the diffusion of its particular fraction in the layer. Also, this part contains
assumptions made for the examined model, stable form of differential equation system of the flow and
grinding material milling with the boundary and initial conditions. The system of equation was solved with
the use of meshfree finite point method according to the enclosed scheme with a calculation algorithm.
The second part presents identification of the above parameters on the basis of the test results; the test
having been performed on a test stand i.e. 1:4 model of the RP1043x ring-roller milling system. Finally,
realization of calculations is presented, i.e. the grinding effects across changes of the rotary velocity and
the feed stream. The model also, allows to define the grinding effect across changes in positioning bumpers
(slot width) and the grind material’s different fractional composition getting under the rollers.
Keywords: Process modeling, ring-roller mill, Finite Point Method (FPM)
W pierwszej części artykułu przedstawiono model bilansowy procesu mielenia na misie młyna rolkowo-misowego. Model opisuje proces mielenia poprzez parametry rozdrabniania i selekcji a także ruch
mieliwa oraz dyfuzję poszczególnych jego frakcji w warstwie na misie. Podano założenia przyjęte dla
opracowanego modelu, stabilną postać układu równań różniczkowych przepływu i rozdrabniania mieliwa
w układzie mielącym, z warunkami brzegowymi i początkowymi. Układ równań rozwiązano z zastosowaniem bezsiatkowej metody punktów skończonych, według załączonego schematu z algorytmem obliczeń.
W drugiej części artykułu przedstawiono identyfikację w/w parametrów w oparciu o wyniki badań
wykonanych na stanowisku badawczym – modelu układu mielącego młyna rolkowo-misowego RP1043x
*
**
1
OPOLE UNIVERSITY OF TECHNOLOGY, INSTITUTE OF MATHEMATICS AND PHYSICS, 45-271 OPOLE, UL. LUBOSZCZYKA 3, POLAND; e-mail: [email protected]
OPOLE UNIVERSITY OF TECHNOLOGY, FACULTY OF MECHANICAL ENGINEERING, 45-271 OPOLE, UL. STANISŁAWA
MIKOŁAJCZYKA 5, POLAND
Article co-financed by the European Union under the European Social Fund.
442
w skali 1:4. Ostatecznie przedstawiono realizację obliczeń tj. efekty mielenia osiągane przy zmianie
prędkości obrotowej i strumienia nadawy węgla. Model pozwala określić efekt mielenia również przy
zmianach ustawienia położenia zderzaków (szerokości szczeliny) oraz różnym składzie frakcyjnym
mieliwa wchodzącego pod rolki.
Słowa kluczowe: modelowanie procesów, młyn rolkowo-misowy, Metoda Punktów Skończonych
1. Introduction
Physical complexity of processes occurring in coal mills is the reason of problems with their
mathematical description. Grinding in a mill includes both the milling process and the material’s
movement in the mill which is connected with its drying and separation in the milling chamber
and the sifter. Every occurring process is difficult for mathematical description, mainly because
of limited possibilities of modelling parameters identification, particularly on the basis of real
object measurements.
Mathematical models found in literature concern mainly milling process in the table-roller
(ball) systems. Models describing the entire mill are relatively sparse. In this case, they are models describing individual processes in the mill in a concentrated way, i.e. defining connection
between input material, e.g. material on the table, and output by adequately defined transition
function (Kersting, 1984; Knittel & Ostertag, 1991; Pospolita, 1996; Otwinowski, 2000; Sato et
al., 1996; Shoji et al., 1998).
In case of mathematical description of the milling process in the table-grinders systems
different models and their classification are suggested. In the work (Górecka & Otwinowski,
2001) the following division of models is suggested: deterministic, where energy hypotheses and
thermodynamic models are included, and statistic, where approximation models and grain mass
balance are distinguished. In the work (Brożek et al., 1995) thermodynamic models are classified both as deterministic and statistic, whereas in the work (Pastucha & Mielczarek, 1994) they
constitute separate groups. Models of grinding kinetics were described from the deterministic and
probabilistic point of view in the work (Auer, 1978). Discrepancy of depictions and an analysis
of the works concerning modelling inclines to distinguish five model groups: -mechanics of the
elementary grinding processes, -thermodynamic, -statistic, -grinding hypotheses, and -balance.
Judging from the model description concerning medium speed coal mills the most frequently
used models are balance models. The models are based on material mass balance and its particular fractions. The balances concern both the grinding process and the material movement on
the mill’s table, in the milling chamber and the sifter. This model was also used in the presented
work for description of grinding process on the table of ring-roller mills.
2. Mathematic model equations – stable form
The following conditions were assumed for the formulated model:
– the material is fed onto the centre of the table creating a cone shaped bulk where the
grinding process does not occur
– the milling process takes place only in Ω space between the roll and the table, i.e..
Ω = {r : r ≥ rB , r ≤ rE}, according to Fig. 1,
443
Fig. 1. Simplified diagram of the grinding system
– the material moves on the table equally from the cone formed in the centre of the table
towards the ring. This assumption allows to consider the system as entirely symmetrical
and to examine the material movement regardless φ angle,
– the material taking part in the grinding process is divided into n ranges of fractions. Also,
the selection parameter Sj and the distribution parameter bij are introduced. The selection parameter defines the weight part of grains from the range j, which is actually being
milled. The distribution parameter defines the fractional part of grains, which goes from
the range j to i after grinding,
– analyzing the mass flow mi i-th material fraction on the grinding area, its radial velocity
vi and convection flow are introduced. The material stream transported by diffusion κm·
is taken into consideration, which according to Fick’s principle can be written as:
κm· = –Di grad m
(1)
where:
κ — coefficient corresponding to the area in the spatial issue,
Di — diffusion coefficient for the examined issue.
The numerical solution of the flow issue dominated by advection is influented by oscillations
within considerable gradients of change in the mass flow parts in the neighborhood of the boundary and internal layers (Liu, 2003; Zienkiewicz & Taylor, 2001). Existing stabilizing techniques
consist in modifying the system of differential equations of the model. We will formulate stable
form of the system of differential equations for the analyzed problem with the use of the finite
calculus (FIC) method.
For the above assumptions and for the random segment Ω∆r of ∆r lenght (Fig. 2) of Ω
region, the balance equation of the i-th fraction of the material in the process of grinding is the
following:
[FLOW CHANGE]i = [FLOW(r – ∆r)] i – [FLOW(r)]i + [GRINDING EFFECT]i
(2)
444
i 1
¦b S
ij
j
m j S i mi
j 1
Nm i r Nm i r 'r rB
Q i mi r 'r r 'r
'r
r
Q i mi r rE
Fig. 2. Finite balance domain Ω∆r
Taking into consideration the linear mass change of the fraction of the material in the subdomain Ω∆r and approximating mi (r – ∆r) by the values in the point r using the expansion in the
Taylor series of second order we obtain:
(3)
Acting in an identical way for
[GRINDING EFFECT] i =
æ i -1
æ
ç bij S j mj - Si miç dr =
ç
ç
è
r -D r è j =1
r
å
i -1
é i -1
ù
æ
Dr ¶ æ
Dr ¶
ç r bij Sj m j ( r )ç - Si mi (r ) +
rSi mi ( r ))ú
= D r ê bij S j mj ( r ) (
ç
2r ¶ r çè j =1
2 r ¶r
êë j =1
úû
è
å
(4)
å
The flow of the i-th material fraction in the point r is determined by the sum of its diffusive
and advective components
[FLOW (r)] i = κm· (r) + vi mi (r)
(5)
Let us assume that the changes in the advective and diffusive terms over the balance domain
Ω∆r are so important that a higher-order expansion is required for both terms
(6)
(7)
445
Taking into consideration the Fick’s principle (1) and the equations (3)÷(7), the balance
equation (2) of i-th fraction is written down:
¶mi
¶t
=
+
¶ mi æ 1 æ ¶
1 ¶ æ
æ
ç rDi
ç - ç (rvi mi )ç +
¶r è r è ¶ r
r ¶r è
è
D r 1 ¶ ìï é ¶ mi 1 ¶
ír ê
2 r ¶r ï ê ¶t r ¶r
ë
î
i -1
å bij Sj mj - Si mi
j =1
(8)
æ ¶mi æ 1 ¶
( rvi mi ) çrDi
ç+
¶r è r ¶ r
è
i -1
ùü
j =1
û ïþ
å bij Sj mj + Si miúúïý
where ∆r – characteristic length which plays a role of the stabilization parameter (Onate, 1998;
Onate et al., 1996, 2006).
For the simplification it is assumed that vi = const and Di = const and thus we get the final,
stable form of the system of differential equations for the flow and pulverized material in the
ring-roller milling system (Kersting, 1984).
(9)
The initial condition for the system of equation (9) is determined by the distribution of each
of the material fraction with Ω domain at the time t = t0
mi (r, t0) = f i S(r)
on Ω
(10)
In the boundary points the following considerations are postulated
mi (rB , t) = f i B(t),
κm· i (r E, t ) = 0
(11)
3. The method of solving the problem
The problems (9)÷(11) can be solved with the use of weighted residuum (Liu, 2003;
Zienkiewicz & Taylor, 2001; Onate, 1998; Onate et al., 1996, 2006; Belytschko et al., 1996). It
is assumed that within the domain Ω, functions mi can be described as a combination of locally
defined functions
(12)
where: N = [N1(r), N2(r), ... NkC (r)]T, ωi = [ωi1(t ), ωi2(t), ... ωikC (t)]T,
446
Nj (r) — known functions which satisfy the conditions Nj (r) ≠ 0, x ∈ Ω j; Nj (r) = 0, x ∉ Ω j,
Ω j — local subdomains Ω which satisfy the conditions
,
kC — total number of subdomains Ω j,
ωij (t ) — unknown time functions.
Inserting (12) in (9)÷(11) we obtain
Li{NT(r) ωi (t)} = Ri (r, t )
(13)
NT(rB ) ωi (t ) – fB = RiB(t )
(14)
(15)
where Li{.} — is a differential operator of the system of equations Ri (r, t), RiB (t ), RiE(t ) residual functions.
Writing the weighted sum of the residual functions we obtain a variation form for (9)÷(11)
(16)
where: Wj (r) — the known weigh functions.
Assuming that the weigh functions Wj = Wj' = Wj'' = δ(r – rj) (where δ is Dirac’s delta,
r1 = rB, r2, r3, ... rkC –1, rkC = rE are collocation points) we obtain a typical collocation method,
which implies a differential equation system (16) neglecting, for simplicity, the effect of the time
derivative in the stabilization term:
Cω· i – Hi ωi – Fi = 0,
—
i = 1, n
(17)
where (·) is a time derivative,
(18)
447
(19)
(20)
(21)
(22)
In the literature many function versions are discussed Nj (r) (Belytschko et al., 1996). What
is assumed here is the shape function obtained by approximation, using the method of moving
least-square (MLS) (Liu, 2003; Zienkiewicz & Taylor, 2001; Onate, 1998; Onate et al., 1996,
2006; Belytschko et al., 1996):
NT(r) = pT(r)A–1(r)B(r)
(23)
p(r) = [1, r, r 2, ..., r q ]T, pi = p(ri), P = [p1 p2 ... pkC ]
(24)
A(r) = B(r)PT
(25)
B(r) = PΦ(r)
(26)
where:
448
(27)
q — degree of the approxymizing polynominal;
φ (r) — weight function satisfying
φ (0) = 1; φ(r) ≠ 0, r ∈ < –rΩ, rΩ > ; φ(r) = 0, r ∉ < –rΩ, rΩ >
(28)
The equation system (17) can be solved using an iterative method „midpoint”. The unknown
values ωi (t) in the time point t l+γ can be approximated by the following expression
ωil+γ = γωil+1 + (1 – γ)ωil
(29)
whereas the first derivatives ω· i (t), by the expression:
(30)
where
t l+1 = tl + ∆t,
ωil = ωi (t l ),
0≤γ≤1
(31)
∆t — step through time t.
Calculating the equation (17) in the time point t l+γ and taking into consideration approximation (29) and (30) we obtain
(C – γ∆t Hi)ωin+1 = (C + (1 – γ)∆tH i)ωin + Fi
(32)
The diagonal structure matrix size C, Hi and vector Fi depend on the number of collocation points kc, whereas the number of elements in particular matrix lines C and Hi depend on
parameter rΩ.
Finally, the equation system (32) is solved in relation to the unknown values ωil+1. The
values ωi0 can be obtained from the initial condition (10).
4. Calculation algorithm
Fig. 3 represents a general diagram of calculating an algorythm of the advective-diffusion
model of the milling process in the system of a ring-roller mill.
449
Concentration distribution
of the material on the table
Literature data
Model
coefficients
Absolute point
method
Mass streams of the
fraction
Measurement data
DATA INITIATION
CALCULATIONS
RESULT RECORD
Fig. 3. General algorithm of the problem solving
The algorithm of absolute point method is realized in the following way:
[I] Value initiation for the initial condition – reading from the disc set of the grain composition of the material in front of the roll. Initiation of the collocation points numbers,
the weigh function type and its attributed parameters,
— and ∆r, ∆t.
[II] Loop through collocation points rk, where k = 1, kc.
[III] Calculating the weight function φ(rk) and its derivatives
[IV] Calculating the matrix pT(rk) and its derivatives
[V] Calculating the shape function matrix NT(rk) and its derivatives
[VI] End of the loop through collocation points
[VII] Loop through grain fractions
[VIII] Calculating the expressions (C – γ∆t Hi) and (C + (1 – γ)∆tH i)
[IX] Loop through time points
[X] Calculating the expressions (C + (1 – γ)∆tH i)ωin + Fi.
[XI] Solving the equation system (32) using the Gauss elimination method with the choice
of main element, in relation to the unknown values ωil+1.
[XII] End of the loop through time points
[XIII] End of the loop through the grain fractions
[XIV] Saving the solution in the disk set
[XV] Ending of the calculations
450
5. Summary
The elaborated mathematical model describes the mechanism of the milling process performed in a ring-roller mill’s grinding system in its advection-diffusion aspect. The milling process was expressed by the following parameters: selection and distribution. The advection was
characterized by radial velocities of the material on the rotating table whereas the diffusion – by
the respective diffusion parameters differenciated in accordance with the material grain size. In
order to solve the task the authors used meshless finite point method FPM.
On the basis of the above model for the selected constructional-operational parameters (e.g.
rotation velocity of the table, material feed onto the table and height of the pile-up ring, material
granulometry) the authors performed calculations of the material pulverization. The results of
the calculations will be presented in the second part of the article.
References
Auer A., 1978. Model i identyfikacja procesów klasyfikacji i mielenia. Zeszyty Naukowe WSI w Opolu, Nr 26, Opole.
Belytschko T., Krongauz Y., Organ D., Fleming M., Krysl P., 1996. Meshless methods: An overview and recent developments Comput. Methods Appl. Engrg. 139, 3-47.
Brożek M., Mączka W., Tumidajski T., 1996. Modele matematyczne procesów rozdrabniania. Monografia Akademii
Górniczo-Hutniczej, Kraków.
Górecka A., Otwinowski H., 2001. Teorie rozdrabniania i ich klasyfikacja. Część 1. Cement Wapno Beton 3, 96-98.
Kersting F.J., 1984. Mathematisches Model einer Walzenschüsselmühle zur Brennstoffzerkleinerung und –trocknung in
Kohlenkraftwerken. Aufbereitungs-Technik Nr. 10, 563-571.
Knittel D., Ostertag E., 1991. Matematisches Modell einer Kohlenmühle. VGB Kraftwerkstechnik 71, Heft 12, 11131117.
Liu G.R., 2003. Mesh free methods. Moving beyond the Finite Elements Method. CRC Press LLC.
Onate E., 1998. A mesh-free finite point method for advective-diffusive transport and fluid flow problems. Computational
Mechanics 21, 283-292.
Ońate E., Idelsohn S., Zienkiewicz O.C., Taylor R.L., Sacco C., 1996. A stabilized finite point metod for analysis of fluid
mechanics problems. Comput. Methods Appl. Engrg. 139, 315-346.
Onate E., Miquel J., Hauke G., 2006. Stabilized formulation for the advection-diffusion-absorption equation using finite
calculus and linear finite elements. Comput. Methods Appl. Mech. Engrg. 195, 3926-3946.
Otwinowski H., 2000. Statistical modelling of comminution process. Arch. Min. Sci., Vol. 45, No 3, p. 427-437
Pastucha L., Mielczarek E., 1994. Kinetyka i termodynamika rozdrabniania strumieniowego. Wyd. Politechniki Częstochowskiej, Częstochowa.
Pospolita J., 1996. Model matematyczny procesu mielenia w młynie węglowym rolowo-misowym. Prace IX Sympozjum
Symulacja procesów dynamicznych. Polana Chochołowska 10-14 czerwca 1996, 205-210.
Sato K., Meguri N., Shoji K., Kanemoto H., Hasegawa T., Maruyama T., 1996. Breakage of coals in ring-roller mills.
Part 1. The breakage properties of various coals and simulation model to predict steady-state mill performance.
Powder Technology 86, 275-283.
Shoji K., Meguri N., Sato K., Kanemoto H., Hasegawa T., Maruyama T., 1998. Breakage of coals in ring-roller mills.
Part 2. An unsteady-state simulation model. Powder Technology 99, 46-52.
Zienkiewicz O.C., Taylor R.L., 2001. The Finite Element Method. Fifth edition, Vol. 1: The Basis. Butterwort-Heinemann.
Received: 14 September 2010