apelt2001

Pergamen

0892-6875(01)ooo52-4

Minerals Engineering, Vol. 14, No. 6; pp. 575-591,2OOI 0 2001 Elsevier Science L&d

All rights reserved 0892~6875/U/$ - see front matter

INFERENTIAL MEASUREMENT OF SAG MILL PARAMETERS*

T.A. APELT*“, S.P. ASPREY” and N.F. THORNHILL**

* Department of Chemical Engineering, University of Sydney NSW 2006, Australia $ Centre for Process Systems Engineering, Imperial College SW7 2BY, UK. Email: [email protected]

$ Department of Electronic & Electrical Engineering, University College London WClE 7.TE (Received 2 January 2001; accepted 20 February 2001)

ABSTRACT

Semi-autogenous (SAG) mill total load and ball load have a marked influence on mill performance. Direct measurement of these inventories is difsicult, hence inferential measurement is an attractive option to open the way to their control. Advances have been made in this area based on combined state and parameter estimation formulations, though there is still scope for further soft sensor development. This work investigates the use of mill powerdraw and weight simulation models combined with the corresponding measured variable (to form residual equations) to obtain estimates of total volumetric load fraction, J,, and ball charge volumetric fraction, Ju by the application of a constrained nonlinear optimisation technique. The sensitivity of the estimates to model parameters and the uncertainty in the estimates are investigated. Results show that, although all estimates illustrated good agreement with nominal conditions, the estimates from the weight- based models contained the least uncertainty. The inclusion of a mill liner wear model in combined state and parameter estimation formulations is recommended due to the notable contribution of the liner weight to the mill weight measured variable. 0 2001 Elsevier Science Ltd. All rights reserved.

Keywords SAG milling; comminution; modelling; simulation

INTRODUCTION

Problem definition

Semi-autogenous (SAG) mill ball charge and rock charge have a high impact on mill performance. Throughput and product quality are affected by the conditions inside the mill. The mill inventories are therefore key process variables and obtaining a “measurement” of these variables will provide means for their control. Improving their controllability, in turn, will provide means for improvement in process operations and subsequently cost-effectiveness.

Measurement options for these process variables include direct and indirect methods. Direct measurement involves designing a hardware sensor that allows immediate inference of the process variable of interest. Indirect measurement involves designing a software sensor that allows inference of the process variable of interest through the measurement of one or more alternative process variables, interrelated through an analytical process model. Direct measurement of the mill inventories is problematical for various reasons, such as the rotational motion of the mill and the destructive tumbling action of the charge. Recent

* Presented at Minerals Engineering 2000, Cape Town, South Africa, November 2000

575

576 T. A. Apeh et 01.

developments have been made in the direct measurement of the total load by use of conductivity probes situated within the mill lining (Moys et al., 1996). On the other hand, advances have also been made in the indirect measurement of the mill contents (Herbst and Pate, 1999; Schroder, 2000) using Kalman filters in a combined state and parameter estimation framework. These works use powerdraw and mill weight models as soft sensor measurement models.

As such, there is still considerable scope to further explore indirect measurement of the mill charge. As mentioned, the mill inventories affect mill weight and mill powerdraw. This feature allows the utilisation of a given model and its corresponding plant measurements to provide an inferential measurement of the mill total charge and ball charge. This paper studies a selection of four such model combinations, along with the associated “accuracy” and “dependability” of each. The paper begins with a description of the grinding circuit under consideration, followed by presentation of the inferential measurement models for mill powerdraw and mill weight, based on works by the Julius Kruttschnitt Mineral Research Centre and JA Herbst and Associates. Both sensitivity and error analyses are then carried out on the inferential measurements provided by the various models and are discussed in detail. Finally, some concluding remarks are made, followed by recommendations on which model combination gives the best estimation results, taking into consideration both accuracy and uncertainty.

Circuit description

This research focuses on the Module 1 grinding circuit at Northparkes Mines (NPM), a copper-gold mine and concentrator in New South Wales, Australia. The circuit treats approximately 250 tph of low-grade copper ore and consists of primary and secondary grinding circuits. The primary grinding circuit is illustrated in Figure 1. Stockpile feed ore is fed to the SAG mill for primary grinding. SAG mill discharge is screened with the oversize recycling via a gyratory cone crusher, while the undersize is diluted with water and fed to the primary cyclones for classification. Primary cyclone underflow is split between a small recycle stream to the SAG mill feed chute and a ball mill feed stream. The primary grinding circuit products are subjected to further size reduction (ball mill), classification (cyclones), and separation (flash flotation) in the secondary grinding circuit. The secondary cyclone overflow constitutes the flotation plant feed. Further details of the NPM surface operation can be found elsewhere (Apelt er al., 1998; Freeman er al., 2ooo).

SRW ,......................................................... . . . . .,....... . . . . . . . . ,

Ofi \!/I A -.

/ ‘! I. /&Q

L-2 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . * . . . . .

SMRC = SAG Mill Rock Charge; SMDC E SAG Mill Discharge; OSCF E Oversize Crusher Feed

SMBC = SAG Mill Ball Charge; PCFD = Primary Cyclone Feed

Fig. 1 The NPM primary grinding circuit.

Jnferential measurement of SAG mill parameters 517

INFERENTIAL MEASUREMENT MODELS

Several mill powerdraw models have been developed since the pioneering work of Bond (1961). These models have generally been developed from the refinement of the Bond models (van Neirop and Moys, 1997; Herbst and Pate, 1999), or from more detailed characterisation of the mill charge (JKTech, 1994; Valery Jr. and Morrell, 1995; Napier-Munn ef al., 1996; Valery Jr., 1998). Given that powerdraw is a function of mill load (mass and volume), the development of mill weight models has progressed accordingly. This paper further explores the indirect measurement of the mill inventories by utilising combinations of mill powerdraw and weight models and the corresponding plant measurements. As briefly mentioned, a selection of four such model combinations is studied, including estimates of the “accuracy” and “dependability” of each through use of sensitivity and error analyses, respectively.

Here, we classify the models by their prediction or response variable and model origin, i.e., the measured variable (process output) that is predicted by the model, and the research group that developed/utilised the model. The measured variable model classification used is (indicated by subscripts):

kW Mill powerdraw;

wt Mill weight (load cell).

While the research group model classification is:

JK

JAH

Julius Kruttschnitt Mineral Research Centre (Australia);

JA Herbst and Associates (USA).

Powerdraw models

The Julius Kruttschnitt Mineral Research Centre Model

The powerdraw model developed by the Julius Kruttschnitt Mineral Research Centre (Napier-Munn et al., 1996) can be written as:

P GrssJK = pNoLad +&huge (1)

In Eq (l), PG,~~~,JK is the mill powerdraw [kW], P N0 Ld is the no-load power of the mill (the-empty mill powerdraw) [kW], Pchg, is the mill powerdraw attributable to the entire contents of the mill [kW], and k is a lumped mill powerdraw parameter that accounts for heat losses due to internal friction, energy of attrition/abrasion breakage, rotation of the grinding media, inaccuracies in assumptions, and charge shape and motion measurements [dimensionless].

For purposes of completeness, the equations for each of the components in Eq (1) are presented here. With exception to Eq (6), Eq (1 l), and E!q (16), all equations Bqs (1 - 22) can be found in the monograph by the Julius Kruttschnitt Mineral Research Centre (Napier-Munn et al., 1996). The important characteristic of these equations is, by means of back-substitution, they reduce to functions of mill ball charge fraction, Jh, mill total charge fraction, J,, and a number of parameters, 8. This characteristic allows the formulation of an implicit inferential model that can be solved numerically for the key variables Jt and 36.

The no-load component of the mill powerdraw, PNO Load is:

P &,,Q& = 1.68(Di549, (0.667L,,, + Lm)pg2 (2)

where D, is the mill inside diameter [ml, +fcs is the mill fraction critical speed [fraction], L,,,, is the length of the conical section of the mill [ml, and L,,, is the length of the cylindrical section of the mill [ml.

578 T. A. Apelt et al.

The powerdraw component attributable to mill charge contents, Pchorgo comprises the components of powerdraw attributable to material in the conical feed end section of the mill and the material in the cylindrical section of the mill:

Ch q c = pNet + pCone (3)

In Eq (3), PN~~ is the mill powerdraw attributable to the contents of the cylindrical section of the mill [kW] and Pcone is the mill powerdraw attributable to the contents of the conical (feed) section of the mill [kW].

The powerdraws attributable to the cylindrical and conical sections of the mill are determined by Eq (4) and Eq (5), respectively:

pNet =

ngL, N, r,,, 2r; ( -3zr,2ri +$(3z-2))do,(sin(Qs)-sin@r)))

3r, -3?,ri

ngL, N, r,,, (2ri - 3Zri ri + rf (3.z - 2)b B (sin@r )-sin@&, )))

3r, -3Zri

+ Lmp,N$~n3 (r,,, -zri)4 -r~(.z-1)4

(rm - Zri)3

P ~gL,ne N, ri - 4r,,,r/ + 3ri Con.? =

4bc(si~(% )-sin(% )))

3(rnI - rr )

(4)

Here, g is the gravitational acceleration constant [9.81 m/s’]; N,,, is the actual mill speed [revolutions per second]; ri is the mill charge kidney inner radius [ml; r,,, is the mill radius [ml; r, is the mill trunnion radius [ml; z is the charge velocity profile parameter [dimensionless]; pC is the mill charge density (or specific gravity) [t/m3]; pP is the slurry pulp density (or specific gravity) [t/m3]; 8s is the charge shoulder angle [“I; or is the charge toe angle [“I; and Or0 is the slurry pool angle [“I.

The mill pulp density, pP, is assumed to be equal to the mill discharge pulp density. Mill cone length is determined from geometry as:

where %,,, is the mill cone angle [“I. The angle of the mill charge shoulder, es, is given by:

While the angle of the mill charge toe, Clr, is given by:

=2.5307(1.2796-J, (8)

Inferential measurement of SAG mill parameters 579

In consideration of the features of the SAG mill discharge grate (details precede Eq (30) below), the angle of the mill charge slurry toe, t3ro is equal to the charge toe angle, i.e.:

eT0 = eT

The charge velocity profile parameter, z, is given by:

(9)

z = (1 - J, )“.4532 (10)

Mill critical speed, RPM c,t,rcd, can be derived from a force balance (the rotational speed where angular .. acceleration is equal to gravitational acceleration) and is given by:

The actual mill speed, represented as a fraction of the critical speed, $fCa, is given by:

The actual mill speed , N,,,, in revolutions per second, is obtained from:

N,=E 60

The mean rotational rate, i , is given by:

The mill charge density, pC, is given by:

p = (JtPo(l-~+aij)+Jbbb -PoK-E)+Jtd-iiiJ~ c

Jt

(11)

(13)

(14)

(1%

Here, E is the porosity of the mill grinding charge [dimensionless]; S is the mill discharge volumetric solids content; p0 represents the ore density [t/m3]; pb is the’grinding ball density [t/m3]. The equation for U, the fraction of grinding media voidage occupied by the slurry, is not explicitly stated in the JK Monograph (Napier-Munn et al., 1996). However, the definition of U gives rise to the following expression:

(16)

where Jp,,, is the mill volumetric fraction of the slurry contained within the grinding charge [fraction]. From Eq (8), &, the experimentally determined fraction of critical mill speed at which centrifuging is fully established, is calculated by:

eC = 0.35(3.364 - J,) (17)

The mean travel time of material in the charge (from the charge toe to the charge shoulder), to is given by:


t, = 2x-8, +es

2nN (18)

The mean travel time of material in free fall (from the charge shoulder to the charge toe), $ is given by:

tf =

i

2T(sin(8S)-sin(8,)) O5

g 1

The fraction of charge that is active, p, is determined by:

The mean radial position of the mill charge, Y , is approximated by:

Finally, the radial position of the mill charge inner surface, ri, is given by:

ri =rm l- 2x-8, +e,

(19)

(21)

(22)

The JA Herbst and Associates model

The powerdraw model utilised by JA Herbst and Associates (Herbst and Pate, 1999; Herbst and Associates, 1996) is one of the measurement equations presented within a combined state and parameter estimation formulation used in comminution soft sensor design. Within this framework, mill powerdraw is calculated by:

P GrO.vs,JAH = C, sin(ol)Di3W, (3.2 - 3V* )N’ (23)

where PorOSs,,~ is the mill powerdraw [kW]; N’ is the fractional critical speed; WC is the charge mass [t] (see the JAH weight model below); v* is the mill fraction occupied by the charge; cx is the charge angle of repose [“I; and C3 is a constant.

Equation (23) is also described in the Julius Kruttschnitt Mineral Research Centre Monograph (Napier- Munn et al., 1996), where it is reported to be one of the correlations “published widely by Allis Chalmers” and is classified as a “conventional approach to ball mill power prediction that has been in use for some years.” This model is based on the work conducted by Bond (1961). Moys (1993) and van Neirop and Moys (1997) have made further developments to this model to account for partial load centrifuging, based on the use of conductivity probes in the mill lining to measure load behaviour. Recognising that:

N* =Qfcs and V*=J, and c~=(e~ -e,) (24)

allows Eq (23) to be rewritten using nomenclature consistent with that used in the equations from the Julius Kruttschnitt Mineral Research Centre model above, and thus enabling Eq (23) to be reduced to; a function of only Jb, J, and model parameters, 8 (the components comprising 8 are presented in Table 1):

Inferential measurement of SAG mill parame&s 581

P CrosS,,AH = C, sin(0s -e,)Di3Wc(3.2-3Jtkf, (25)

Mill weight models

The Julius Kruttschnitt Mineral Research Centre model

The mill weight model presented in Eq (26) is a generic model, but is referred to as the “JK” model due to the fact that the charge mass, Mcharge, calculation is based on the equations for ri, p. 0s and Or described earlier by Eq (22), Eq (15), Eq (7), and Eq (8), respectively:

LcJK = MCharge + Mdaell (26)

The mass of the charge, Mctigc, can be expressed as:

M Charge = Pcvkiaby + M pulp (27)

The volume of the charge (kidney), Vbh [m3], is given by:

(28)

The pulp mass, MpUlp [t], can be expressed as:

Mpuc = JpVJGp (29)

where V,,, is the mill volume (w~L, ) [m3]; SC,, is the pulp specific gravity [t/m3]; and J,, is the mill pulp

volumetric fraction.

Specifying the mill volumetric discharge, Q,, and assuming that no pooling is occurring within the mill, allows the calculation of the mill pulp (slurry) volumetric fraction, J,, (see Eq (3)). Due to the assumption that there is no pooling, the “p” subscript is augmented with an “m”, indicating that the flow is through the charge media. This assumption is reasonable in this case since the mill discharge grate has:

n a high fractional open area, fracoA (0.179);

H relative radial pOSitiOn of the open area, rvoA (0.8031), and;

n relative radial position of the outermost aperture, ru (0.972).

(30)

The JK weight model reduces to an implicit function of only Jb, J, and model parameters, 9, through the equations for p=, OS, and Or given in the JK powerdraw model section.

The JA Herbst and Associates model

The JAH weight model, Eq (31), is similar to the JK weight model, and can also be reduced to an implicit function of only Jb, J, and model parameters, 8. The main difference is the introduction of the “co-efficient” and “slope” terms, C1 and C,, respectively:

LcJAH =C2(H,+H,+Hw+H,+WL)+C, (31)

582

or, 44H = c,(w, +wJ+q

T. A. Apelt ef al.

(32)

where LC,AH is the mill load cell measurement [t]; Hi are the mill hold-up of component i (t), where i E {R, rocks; P, particles; W, water; B, balls}; IV, is the mill liner weight [t]; and Ci, C, are constants.

The first statement of the model is written in terms of the hold up of rocks, particles, water and balls, arising from the context of the utilisation of the model. A combined state and parameter estimation model problem formulation utilises dynamic models for each state (Hs, HP, Hw, Hs, and IV,) (Herbst and Pate, 1999). Neglecting mill liner wear in this case, i.e., assuming constant mill shell weight, Mshcll, which includes the weight of the mill lining, WL, leaves only ore, water and grinding balls to consider. These three components constitute the mill charge and sum to the mass of the charge, McharBE, regardless of how one classifies the mill charge constituents. The JAH model classifies the mill charge constituents into rock, particles, water and balls, whereas the JK weight model groups tine particles and water together into a “pulp” term and the coarse particles and the grinding media into a “media” term. Taking this into consideration, the JAH weight model can be simplified to the second statement of the model in Eq (32). Therefore, as for the JK weight model, the JAH weight model reduces to a function of only Jb, J,, and parameters, 0.

Inferential models

Throughout the previous discussion, it has been highlighted that the powerdraw and mill weight simulation models can be reduced to implicit functions of the variables J,, Jb, and 8. Equating the simulation models to the measured variables allows, after rearrangement of terms, the expression of residual equations that can be utilised as inferential models. Specifically, Eqs (33 - 36) are the inferential model equations under study in this research:

F kW,JK =“=MVkW -Pnnlnod(e)-kPcharge,JK(Jr,Jb,e) (33)

F kW,JAH =“=MVkW -Pcharge,JAHtJr~Jb~e) (34)

F wt,JK =’ = MVwt - Mshell - Mchex~e~Jt’ Jbd (35)

F wt,JAH =O=M”WI --cl -Cdbi,.(J,~Jd) (36)

where Fkw,JK is the powerdraw residual based on the JK model; F kW,JM is the powerdraw residual based on the JAH model; F,,,,,JK is the mill weight residual based on the JK model; Fw,Im is the mill weight residual based on the JAI-I model; MVkw is the powerdraw measured variable [kW]; MV,,,, is the mill weight measured variable (load cell) [t]; Pchrgr,,~ is the JK mill powerdraw due to the mill charge, Eq (3), [kW]; P cbrge,~~~ is the JAH mill powerdraw due to the mill charge, Eq (25), [kW]; Pm w is the JK no load mill powerdraw, Eq (21, WV; Mchnrge is the mill charge mass, Eq (27), [t]; M,Vtirl is the mill shell,mass [t]; k is the JK mill powerdraw lumped parameter, Eq (l), [d imensionless]; Ci, C,, and Cs are the JAH model parameters, Eq (3 1) and Eq (23); and 8 are the model parameters as given in Table 1.

The application of a constrained nonlinear optimisation technique’ - a gradient-based sequential quadratic programming (SQP) method - to Pq (33 - 36), respectively, provides a solution for Jb and Jt for each of the inference models. In Table 1, the parameter numbering scheme included assignment of No. 14 and No. 15 to J, and 36, respectively. Given that the treatment of each residual model, Eq (33 - 36), results in an estimated [J,, Jb] pair, the treatment of Jt and Jb was excluded from the subsequent parametric sensitivity and uncertainty analyses. The method of obtaining estimates for both J, and Jb, and the subsequent parametric sensitivity and error analyses presented here are the novel aspects of the work.

’ fiincon function in the MATLAB optimisation toolbox.


TABLE 1 Model parameters

1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 16. 17. 18. 19. 20. 21. 22.

Parameter, 8 1 k, JK mill powerdraw lumped parameter [dimensionless]

Dma, mill diameter [m] &ill, mill length [m] 8 c0m, mill cone angle [“I D,, mill trunnion diameter [m] RPM, mill speed [RPM] pO, ore specific gravity [t/m31 pb, ball specific gravity [t/m31 E, charge voidage [fraction] tph_s, mill solids throughput [tph] %sols (w/w), discharge density, [%sols w/w] rrpoA, relative radial position of open area [fraction] fracoA, discharge grate fractional open area [fraction] MV,,, mill weight measurement [t] M slvN, mill shell weight [t] MVkw, mill powerdraw measurement [kW] Ci, JAH weight parameter [t] CZ, JAH weight parameter [dimensionless] C’s, JAH powerdraw parameter [dimensionless] 172, relative radius of outermost grate [fraction]

RESULTS

Value Error % Error 1.39 0.15 10.79 7.12 0.15 2.10 3.53 0.15 4.24

15 2.00 13.33 1.6 0.10 6.25

12.014 0.50 4.16 2.65 0.10 3.77 7.80 0.20 2.56

0.4 0.015 3.75 252.1 10.00 3.96 75.93 3.00 3.94

0.8031 0.05 6.25 0.179 0.02 11.11

176 10.00 5.68 64 5.00 7.81

2800 200.00 7.14 64 5.00 7.81

1 0.057 5.68 9.9735 1.50 15.00 0.972 0.028 2.89

Solutions of the residual equations, Eq (33-36), are illustrated graphically in Figure 2. The estimates given by the constrained minimisation of the residual equations are indicated by the various symbols on the graph, and agree well with the actual (nominal) volumetric charge fractions indicated by the diamond (0). As can be seen in the graph, the estimates from the weight residuals (+ and x) coincide. This is to be expected due to the fact that the residual equations, Eq (35) and Eq (26), themselves are collinear (through adjustment of the “co-efficient” and “slope” terms, Ci and C,, respectively, in Eq (36)).

The estimates obtained for J, and Jb, corresponding to those presented in Figure 2, are listed in Table 2, along with their comparative error to the nominal conditions (Column 4). As indicated, the estimates show good accuracy, as all estimates are within a 7% error band of the nominal conditions, and a majority of the estimates are within a 2% error band of the nominal conditions.

TABLE 2 J, and Jb estimates

Inferential Measurement

(actual)

(0.298)

(O.:n42)

Model

Fkw,JK Fkw,JAn

F wt,JK

F Wt,JAH

Fkw,JK FkW.JAH

F wt,JK

F wt,JAH

Value (vol. Fraction)

0.2297 0.2300 0.2328 0.2328

0.1328 0.1377 0.1446 0.1446

Error wrt actual

(%)

0.04 0.09 1.31 1.31

6.48 3.03 1.83 1.83


0 = actual conditions [J,, Jb] = [0.229&O. 1421 V = Fkw,J~ estimate [J,, Jb] = [0.2297,0.1328] A = FkW,JAH estimate [J,, Jb] = [0.2300,0.1377] + = F,,,JK estimate [Jt, JJ = [0.2328,0.1446] x = FwlAH estimate [Jt, JJ = [0.2328,0.1446]

Fig.2 Inferential model contours.

Parametric sensitivity and error analyses

Good agreement with nominal conditions (or accuracy) is obviously a desirable characteristic for the estimates; however, the dependability (or uncertainty) of these point estimates requires further investigation in order to assign some degree of confidence to the estimates for their subsequent use in control applications. The contours in Figure 2 indicate that a range of plausible solutions for each residual equation is possible. To determine both the sensitivity of the equations to the parameters and the uncertainty of the estimates, parametric sensitivity and error analyses were conducted; the results of which are presented here.

Parametric sensitivity analysis

The sensitivity of the mill charge estimates to errors in the parameters, aJ/%, is determined by the application of the Differentiation of Composite Function rule or implicit differentiation (see Perry et al., 1984), which can be written as follows:

Given an implicit model equation F(J,, Jb, 9) = 0, then for cYFKIJ f 0:

dJ dF/Xl -=-_ 6% dF/aJ

(37)

Implicit differentiation (Eq (37)) was subsequently applied to each of the inferential models, Eq (33 - 36) for each of the mill inventories, J, and Jb, respectively. The results of this analysis, around the nominal conditions, are shown graphically in Figures 3 and 4.

Jnferential measurement of SAG mill parameters 58.5

4.5 r . . . . . i”““‘i”“” :...... ‘(“““T’-.T”““T”““T”“‘T”““~~~_.~~~ . . . . . . i”““‘:_ . . . . . r”““‘

: .’

2.5. ., .j. ..: : ..: : ‘. :... :..:. ..j. .j .,.. .

,.5_ I. ,, .; i. 1. I.

,_ : j. j j

: ..;. 1. .:.. ;

:

Fig.3 Total charge, J,, sensitivity analysis.

.. :” ” : : :

. . .

.()gj i . . . . . . i . . . . . . i . . . . . . i . . . . . . i . . . . . . i . . . . . . J. . . . . . . i . . . . . . . . . . . . . J. . . . . . . . . . . . . . . 1 . . . . . ..i . . . . . . . . . . . . . . . i.......; . . . . . . . i.......: . . . . . . . . . . . . . . . * . . . . . . . L . . . . . . . t . . . . . . . i . . . . . . I 0 1 2 3 4 5 6 7 9 9 10 11 12 13 14 15 16 17 19 19 20 21 22 23

Parameters, 0

Fig.4 Ball charge, Jb, sensitivity analysis.

There are a number of observations worth noting from the graphical results of the sensitivity analysis:

(9

(ii)

(iii)

The sensitivities of the ball charge estimates are an order of magnitude lower than for the total charge estimates;

The sensitivities of the ball charge estimates are generally more consistent across the different inferential models;

The sensitivities of the total charge estimates are generally higher for the powerdraw models.


Furthermore, Figures 3 and 4 indicate high sensitivity of the J, and Jb estimates to the following parameters:

1. k JK powerdraw lumped parameter

2. &ill mill diameter

3. Lmill mill length

9. E mill charge voidage

12. wOA relative radial position of the D/C grate open area

13. fracoA fractional D/C grate open area

20. CZ JAH weight parameter (“slope”)

These characteristics of the models revealed by the parametric sensitivity analysis require further investigation. In particular, the sensitivity analysis is somewhat incomplete without a complementary error analysis to determine the overall uncertainty in the estimate and the relative importance of the role that each of the parameters play in contributing to the overall uncertainty in the estimates.

Error analysis

From the sensitivities, dJh33, and estimates of the error in each of the parameters, 68, the overall uncertainty in the estimates, 6J can be calculated as follows (Taylor, 1982):

An error analysis (Eq (38)) was subsequently applied to each of the inferential models, Eq (33 - 36), for each of the mill inventories, Jt and Jb, respectively. Table 3 contains the results for the error analysis and shows that, despite good agreement between the estimates and the nominal conditions, some of the estimates contain a high degree of uncertainty. In particular, the Jt estimates from the powerdraw models FkwSJK and FkW,JAAH, though showing high accuracy, contain uncertainty in excess of 150%, thus generating a low degree of confidence in their subsequent use for purposes of model-based control.

TABLE 3 Uncertainty analysis summary

Inferential Measurement

(actual)

(0.2J;98)

(O.:h42)

Model

Fkw,JK

Fkw.JAfi F wt,JK

F w,JAH

Fkw,JK

Fkw,J. F wt,JK

F w,JAH

Value (vol. Fraction)

0.2297 0.2300 0.2328 0.2328

0.1328 0.1377 0.1446 0.1446

Error Uncertainty wrt actual Absolute Relative

(%) (vol frac) (%)

0.04 0.42 181 0.09 0.36 157 1.31 0.06 26 1.31 0.07 29

6.48 0.04 33 3.03 0.06 42 1.83 0.04 26 1.83 0.04 28

These results of the error analysis indicate the following:

a. Although displaying good agreement, the estimates of J, from the powerdraw models, Fkw,JK and FkW.JAff, contain a high degree of uncertainty;


b. The powerdraw models, Fkw,JK and F~w,JM, provide good estimates of Jb with reasonable uncertainty;

c. The weight models, Fwr.lK and Fw,MH, p rovide good estimates of both Jt and Jb with reasonable uncertainty;

d. The estimates (both J, and Jb) from the JK weight model, F,,,,,JK, contain the least uncertainty.

A detailed mathematical analysis of the high degree of uncertainty in the J, estimates is outside the scope of this paper. Briefly stated, the large uncertainty is a result of the relative complexity of the powerdraw equations and the number of times and the nature of the appearance of J, within them (i.e., the parameterisation of the model equations with respect to J,). This manifests itself as relatively large values for the c~J@CI terms in the summation in Eq (38), as was presented in the Sensitivity Analysis (refer to Figure 3).

The error contributions of the parameters to the estimates, %%I, are shown graphically in Figures 5 and 6

for J, and Jb, respectively.

0.1:

0.1

-6.2:

-0.:

1 i L ( I I I : : : : : :

! . : I ! I , I 1

: : : :

1.. I. . .I. . . . . . : ,, ., 1.. ,. ( .: ..I ..:

: : : : :

: :

. . :. .:. ,, ,,;,Y.,:,, 1,. ; i ;, i : i

‘. ‘. ” .ti’ j,,

‘. .

: : : :

ii : : : :

. . .:. ,, : .:. : : : :

: -. : : :j~~,JAH 1’ : :

1. :_ : _I. .: WV...., .,..: .,..... ., . . . . I . . . . . . . . . .: : : : :

. . . . . , : . : .I. .:. ..: :. :.. :.

1 2 3 4 5 6 7 6 9 1011121314151617161920212223 Parameters,0

Fig.5 Total charge, J,, error analysis.

The general findings of the parametric sensitivity analysis (Points (i) - (iii) above) hold here as well, and can be further expanded as follows:

(iv)

(v)

The combination of the parametric sensitivity and the errors in the parameters results in changes in the relative importance of the role that the parameters play in the overall uncertainty in the estimates;

The most influential parameters on the error in the estimates are:

JK powerdraw lumped parameter

mill diameter

588 T. A. Apelt er al.

3. L mill mill length

6. RPM mill speed

‘16. MV,, mill weight measurement (load cell)

17. M shell mill shell weight

18. MVkW mill powerdraw measurement)

19. Cl JAI-I weight parameter (“intercept”)

20. c2 JAH weight parameter (“slope”)

21. c3 JAH powerdraw parameter

()a_.. .:.. :. :. : ,. .: :. .: :.

~/

0.005 ._:..; . :..,. , ; .:.

4.05 I ..__. i . . .._.. _..... i . . . . . . J.._.A . . . . . . i . . . . . . . . . . . . . i . . . . . . J . . . . . . i . . . . . . J. . . . . . . t . . . . . . i . . . . . . . . . . . . . .i . . . . . . J . . . . . . . L . . . . . . i....: . . . . . ..i .._. J . . . . . . . L...... 1 0 1 2 3 4 5 6 7 8 Q l;sm'03 14 15 16 I? 18 19 2Q 21 22 23

Fig.6 Ball charge, Jb, error analysis.

The results of Figures 5 and 6 are presented in tabular form in Tables 4 and 5, respectively, and are reported in terms of the relative contribution to the estimate errors. The parameters listed above are those in Tables 4 and 5 that represent at least 10% of the error in the estimate. The parameters marked with an asterix (*) are those that represent between 20-308, namely k and Mv,. The lower levels of uncertainty in the estimates from the weight-based models imply that they be utilised in preference to powerdraw-based models. Having the lowest level of uncertainty, the JK weight-based model (F,,,,x) combined with its independence of the influential model parameters, k, Cl, CZ, and Cs, advocate that it be the favoured model overall.

Focussing now on the JK weight-based model, F,,,,,JK, inspection of Tables 4 and 5 reveals that the most influential parameters regarding uncertainty are:

2. &ill

3. Lill

16. MV,, 17. M dlell

mill diameter

mill length

mill weight measurement (load cell)

mill shell weight

Mill diameter, Dmiu, and mill length, Lmill, are dependent on the condition of the mill lining, which influences both the mill weight measurement, MV,, and the mill shell weight, MS,. These points give rise to the following possibilities for improved effectiveness in use of the model:

a. Justification of the installation of a load cell or bearing pressure measurement (in the absence of such a mill weight measurement);


b. Justification for the installation of a mill bearing pressure measurement to complement a load cell device, or vice versa, to improve the uncertainty in the MV,, measurement;

C. Dual weight measurement could result in decreased uncertainty in the MV,,, measurement. A 50% decrease in the MVw uncertainty would decrease the uncertainty in the J, and 56 estimates by = 15% (50% of 31.39%) and serve to bring the Jt and Jb uncertainties to = 22%;

d. Dynamic models of mill weight should include a mill liner weight model (as practised by JA Herbst and Associates (Herbst and Associates, 1996)).

TABLE 4 Jr sensitivity analysis

Ri Model Parameter, 8

1. K 2. Dmill 3. Lnill

4. bae

5. D, 6. RPM

7. P” 8. PI, 9. E

10. tph_s 11. %sols 12. rrpcA 13. fracOA 16. MV,,, 17. MShcli 18. MVK, 19. Cl 20. c, 21. c, 22. Rn

kW, JK kW, JAI-I 1 Jt (%I I wt, JK wt, JAH

27.62 0 0 0 16.74 12.99 12.97 10.89 10.28 9.80 14.72 12.36 2.01 0 0 0

0 0 0 0 13.81 11.02 0.61 0.52

1.75 1.65 2.48 2.08 4.99 4.10 6.16 5.17 5.89 4.84 7.26 6.10 0.40 0.74 1.11 0.93 0.04 0.08 0.11 0.10 1.59 2.91 4.37 3.67 1.13 2.07 3.11 2.61

0 0 31.39 26.36 0 0 15.70 0

13.73 16.24 0 0 0 0 0 13.18 0 0 0 16.04 0 33.58 0 0 0 0 0 0

tion to Error in

CONCLUSIONS AND RECOMMENDATIONS

In this work, through use of an SQP-based constrained optimisation technique, it has been shown that estimates of mill total volumetric charge, Jt, and volumetric ball charge, Jb, can be obtained from inferential models constructed from models of mill powerdraw and mill weight. In addition, estimates of J, and Jb that display good agreement with the nominal conditions can be obtained from both powerdraw and mill weight based models, although with a range of uncertainty: (i) estimates of J, from powerdraw-based models display a high degree of uncertainty (in excess of 150%); (ii) estimates of Jb from powerdraw-based models display a reasonable degree of uncertainty (3342%); (iii) estimates of J, and Jb from weight-based models display a reasonable degree of uncertainty (less than 30%). Finally, it has been shown that estimates of J, and Jb from the JK weight-based model, Fw,,JK, contain the least uncertainty (26%) or highest confidence.

Based on these conclusions, it is recommended that the JK weight-based model, Fw,JK, be used for estimates of J, and Jb with greatest achievable precision Furthermore, it is recommended that a mill liner weight model be incorporated when utilising a dynamic mill weight model for the estimates of J, and Jb (as practised by JA Herbst and Associates (Herbst and Associates, 1996)).


TABLE 5 Jb sensitivity analysis

Rela! Model Parameter, 8

1. K

2. Dmit/ 3. Lill

4. Rm?

5. D, 6. RPM

7. PO 8. Ph 9. E

10. rph_s 11. %sols 12. rvOA

13. fracOA

16. MV,,, 17. Mvhcll

18. MVkw 19. c, 20. c, 21. c, 22. Rn

- e Contribut kW, JK

27.66 0 16.73 12.99 10.28 9.80 2.01 0

0 0 13.81 11.02

1.75 1.65 4.99 4.10 5.89 4.84 0.40 0.74 0.04 0.08 1.59 2.91 1.13 2.07

0 0 0 0

13.73 16.24 0 0 0 0 0 33.58

OL 0 L

0 0 12.97 10.89 14.72 12.36

0 0 0 0

0.61 0.52 2.48 2.08 6.16 5.17 7.26 6.10 1.11 0.93 0.11 0.10 4.37 3.67 3.11 2.60

31.39 26.36 15.70 0

0 0 0 13.18 0 16.04 0 0 0 0

wt, JAH 1

REFERENCES

Apelt, T.A., Galan, 0. and Romagnoli, J.A., Dynamic Environment for Comminution Circuit Operation and Control. In CHEMECA ‘98, 26* Australian Chemical Engineering Conference. CHEMECA. Port Douglas QLD Australia, 1998

Bond, F.C., Crushing and Grinding Calculations Part I and II. British Chemical Engineering, 1961, 6, pp. 378-385 and 543-548.

Freeman, W.A., Newell, R., and Quast, K.B., Effect of Grinding Media and NaHS on Copper Recovery at Northparkes Mine. Minerals Engineering, 2000,13(13), pp. 1395-1403.

Herbst and Associates, J.A., Semi Autogenous Mill Filling Using a Model-based Methodology. Proposal submitted on North Limited - Northparkes Mines, 1996.

Herbst, J.A. and Pate, W.T., Object Components for Comminution System Softsensor Design. Powder Technology, 1999,105, pp. 424-429.

JKTech, JK SimMet Steady State Mineral Processing Simulator: User Manual Version 4, -1994, JKTech - JKMRC Commercial Division. Julius Kruttschnitt Mineral Research Centre, University of Queensland, Australia.

Moys, M.H., A Model of Mill Power as Affected by Mill Speed, Load Volume, and Liner Design. Journal of the South African Institute of Mining and Metallurgy, 1993,93(6), pp. 135-141.

Moys, M.H., van Nierop, M.A. and Smit, I., Progress in Measuring and Modelling Load Behaviour in Pilot and Industrial Mills. Minerals Engineering, 1996,9(12), pp. 1201-1214.

Napier-Munn, T.J., Morrell, S., Morrison, R.D. and Kojovic, T., Mineral Comminution Circuits, 1996, Julius Kruttschnitt Mineral Research Centre, Australia.

Perry, T. H., Green, D.W., and Maloney, J.O., eds., Perry’s Chemical Engineers’ Handbook. 6” edn, 1984, McGraw-Hill.

Schroder, A.J., Towards Automated Control of Milling Operations, In Crushing ana’ Grinding Technologiesfor Mineral Extraction. IIR Conference, Perth 18-19 May, 2000

Taylor, J.R., An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 1982, University Science Books.


Valery Jr., W., A-Model for Dynamic and Steady-state Simulation of Autogenous and Semi-autogenous Mills, 1998, PhD Thesis, University of Queensland (Department of Mining, Minerals, and Materials Engineering)

Valery Jr., W. and Morrell, S., The Development of a Dynamic Model for Autogenous and Semi- autogenous Grinding. In Minerals Engineering Conference, 1995, St. Ives, England.

van Neirop, M.A., and Moys, M.H., The Effect of Overloading and Premature Centrifuging on the Power of an Autogenous Mill. Journal of the South Africa Institute of Mining and Metallurgy, 1997, 97(7), pp. 313-317.

Correspondence on papers published in Minerals Engineering is invited by e-mail to bwills@min-engcom

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