Analytical solution for the dynamic model of tumbling mills

Ping Yu1, Weiguo Xie1, Lian X. Liu2*, Malcolm S Powell1

1Julius Kruttschnitt Mineral Research Centre, Sustainable Minerals Institute, the University of Queensland, Indooroopilly, Brisbane, QLD 4068, Australia22Department of Chemical and Process EngineeringUniversity of Surrey, Guildford, Surrey, GU2 7JP, United Kindom

*Corresponding author. Email address: l.x.liu@surrey.ac.uk; Ph: +44 1483686594

Abstract

Optimisation of grinding circuits is invariably dependent on sound process models together with process simulators that can solve the process models accurately. Most of the process models are solved numerically because analytical solutions are not available, which can lead to errors in the results due to the numerical approximation of mathematical equations. Whiten [1], and Valery Jnr & Morrell [2, 3] have developed a dynamic model with numerical simulation for autogenous and semi-autogenous mills, and validated the model with dynamic response of mills in terms of power draw, grinding charge level, slurry level and product size distribution to changes in feed rate, feed size, feed hardness and water addition [2, 3]. In this work, an analytical solution for their dynamic model of tumbling mills has been developed based on the knowledge of solutions to the first-order nonhomogeneous linear differential equations. Two algorithms, Direct Single Time method (DST) and Direct Multiple Time method (DMT), were applied to obtain the analytical solutions respectively. It was found that analytical solutions are more accurate than the traditional finite difference numerical methods. However, the DST analytical method has a drawback of numerical instability due to the accumulation of round-off errors which are amplified by exponential functions, whilst the DMT method can provide stable solutions. To test the DMT analytical method, two cases of SAG mill dynamic operation were studied with both the traditional numerical method and the newly developed analytical method, further proving the robustness and feasibility of the analytical solutions.

Key words: dynamic model, tumbling mills, SAG mill, analytical solution, Direct Single Time method (DST), Direct Multiple Time method (DMT)

1. Introduction

In general, comminution processes require a relatively large amount of energy in the mineral processing industry. The energy consumption of comminution far exceeds the energy consumption of other processes. For example, in copper and gold mines in Australia, comminution processes consume 36% of the total energy utilised by the mines [4, 5]. The total energy consumed for

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the comminution of copper and gold ores accounts for at least 1.3% of Australia’s electricity consumption [5]. Comminution consumes up to 4% electrical energy globally and average comminution energy consumption can be approximately 6,700 kWh /kiloton for a single mine [6]. Grinding mills, which feature in most comminution circuits, take the biggest section of the energy consumption pie in comminution, arousing researchers’ interests in the optimisation of grinding circuits.

Tumbling mills are most commonly seen in mineral plants, such as Ball mill, AG (Autogenous Grinding) mill and SAG (Semi-Autogenous Grinding) mills. The installation of AG and SAG mills for comminution circuits has led to many economic advantages in mine site over the years [7], such as their high throughput and low maintenance. SAG mills are the largest tumbling mills used for primary grinding in mineral processing, using a coarse feed (typically up to 250 mm diameter rock) coming from a crushing stage of the grinding circuit [8]. One of the largest SAG mills (12m diameter), has a variable speed gearless mill drive (GMD) with a capability of 28 MW for refining copper- molybdenum-silver [6], and the average power draw is around 12.5 MW with a feed rate of up to 6,000t/h [9]. Due to the large size of the equipment and the high energy consumption, process simulation and optimisation become critically important. The grinding process optimisation is invariably dependent on sound process models together with process simulators that can solve the process models accurately. Most of the process models are solved numerically if analytical solutions can’t be obtained and numerical methods can often lead to errors on the results due to the numerical approximation of mathematical equations.

There are generally two categories of models for grinding mills, namely batch and continuous. In batch grinding models, the main focus in terms of analytical solutions is on the prediction of product size distribution. For example, Kapur [10] provided approximate analytical solutions for cumulative mass fraction, Reid [11] presented a fundamental integral-differential equation for batch grinding process and then developed an analytical solution for the product size distribution. Das, Khan and Pitchumani [12] developed an analytical solution for a cumulative batch grinding equation for first-order breakage. They discretised the conservation equation of mass fraction for particles of different size classes and found the analytical solution for mass fraction changing with time. Hoșten [13] also proposed an analytical solution for batch grinding equation in cumulative size distribution form. He developed an alternative analytical solution for the discretised equation for batch grinding. A simple selection function model proposed by Austin, Klimpel and Luckie [14] with size range less than 1mm was used.

Due to the demands for process control, dynamic models for AG/SAG and ball mills have attracted more attention in recent years. Valery Jnr and Morrell [2] proposed a conceptual model to provide accurate dynamic response of mills in terms of power draw, grinding charge level, slurry level and product size distribution to changes in feed rate, feed size, feed hardness and water addition. The model was solved numerically and was verified with plant data

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from a number of mills. Liu and Spencer [15] developed a powerful library of unit operation models for mineral processing circuit in the SIMULINK programming environment to describe dynamic response in grinding mills. Their AG/SAG mill and ball mill model were also based on the idea of Whiten [1] that the population balance modelling approach worked with the assumption that mill dynamics can be modelled by a number of perfect mixers in series. They compared their ball mill model to the corresponding model in DYNAMILL, originally developed by Raj Rajamani and John Herbst at University of Utah, and found that under the same conditions, the models produced simulation results that differed by less than 2% in product size distribution.

However, the analytical solutions for dynamic grinding models were not derived by the researchers. This paper looks into an analytical solution to dynamic models in tumbling mills. The flowrate, particle size distribution and content volume are obtained analytically and are compared with the traditional numerical solutions.

2. Methodology

2.1 Dynamic modelling

A dynamic or time-based grinding mill model is particularly useful for optimizing mill operation because it can provide the mill response to the change of operating conditions. Under the assumption of perfect mixing condition, the following equation describes the dynamic condition of a tumbling mill [1 - 3]:[Beginning of the document][Automatic section break][Automatic section

break] d si ( t )

dt=f i−p i+∑

j=1

i

aij r j si−r i si 1()

pi=di ∙ si 2()

d i=dmax ∙ ci 3()

where, si(t ) is the mass of material in size class i at time t in the mill; f i is the total flow rate of feed material in this size class; pi is the total flow rate of discharge material in this class; r i is the rate at which particles in size class i break; a ij is the breakage distribution or appearance function which describes the fraction of material breaking into size class i due to breakage of size class j; d i is the discharge rate of class size i. c i is the classification function value for size class i. The classification function is a continuous polyline in the logarithmic coordinates in this structure. dmax is the maximum discharge rate [16]. dmax is calculated iteratively using the mass transfer function which relates the slurry hold-up to the normalised volumetric flowrate of slurry discharged from the mill:

L=m1(F /V )m2 4()

3

where L is the fraction of mill volume occupied by below grate size material; F is the normalised flowrate through mill (mill fillings per minute in equivalent); V is the normalised volumetric flowrate of slurry. (F/V) means the volumetric discharge rate. m1is a parameter linked to grate design, grate open area and mill speed. m2=0.5.

2.2 Analytical solution

The model needs three input parameters: ri, a ij and d i.

Combining eq. 1() and eq. 2():

d si( t)

dt=f i−d i si+∑

j=1

i

aij r j si−ri si 5()

In matrix form:

(ds1/dtds2/dt

⋮dsn/dt )n × 1

=(f 1

f 2

⋮f n

)n ×1

−[d1 ¿d2 ¿⋱ ¿dn]n× n(

s1

s2

⋮sn

)n ×1

+(a11 0a21 a22

0 ⋯ 00 ⋯ 0

⋮ ⋱an 1 an 2

⋱ ⋱ ⋮⋯ ⋯ ann

)n× n

[r1 ¿r2 ¿⋱ ¿r n]n×n(

s1

s2

⋮sn)

n×1

−[r 1 ¿r 2 ¿⋱ ¿rn]n× n(

s1

s2

⋮sn

)n ×1

6()

dsdt

=f −Ds+ ARs−Rs 7()

dsdt

= ( AR−R−D ) s+f 8()

where A is the appearance function matrix, R is the breakage rate (or selection function) matrix, D is the discharge function matrix. If we let AR−R−D=Z, we can obtain first-order nonhomogeneous linear differential equations:

ds( t)dt

=Z (t )s(t )+ f ( t) 9()

4

The Z(t) matrix is related to appearance function, breakage rate and discharge rate. The appearance function is determined by the ore property and the breakage rate and discharge rate are linked to the mill load and power. Z(t) is a constant when the mill is operating at steady state with a constant feed, mill content, power, and throughput. As a simplification, Z(t) can be considered as a constant matrix during calculation inside a short time interval, which can be denoted as H(t) in the following sections. H(t) will be iteratively calculated for each time step. The general solution of first-order nonhomogeneous linear differential equations can be expressed as the sum of the general solution of the corresponding first-order homogeneous linear differential equations and the special solution of first-order nonhomogeneous linear differential equations [17].

According to the mathematical theory [17], the solution for eq. (9) is:

s ( t )=e tH ( t )[ N+N ( t )] 10()

s (t )=P etΛ P−1[ N+N (t )] 11()

where N is a constant vector, N(t) is a variable vector and is a function of time. P is an invertible matrix. Matrix H(t) can be diagonalized:

P−1 HP=Λ=diag (λ1 , λ2 ,⋯ , λn) 12()

If H(t) satisfies one of the following conditions, it can be diagonalized:1. H(t) is a real symmetric matrix2. The n eigenvalues of H(t) are mutually different , H(t) must be

diagonalizable.3. There are n linearly independent eigenvectors for H(t), so H(t) can be

diagonalized.4. The number of linearly independent eigenvectors for each of the

corresponding repeated eigenvalues of H(t) is equal to the repeated value of the repeated eigenvalues.

are the eigenvalues of the matrix H(t) which can be derived from the determinant of (H (t )−λI):

det ( H ( t )− λI )=0 13()

N(t) can be calculated by

N (t )=∫ ( P e−tΛ P−1 ) f (t ) dt 14()

Given an initial condition,s (t 0 )=[s1 (t 0 ) ,⋯ , sn (t 0 )]T substitutings (t 0 ) into eq. 11():

s (t 0 )=P e t0 Λ P−1[ N+N (t 0 )] 15()

Rearranging eq. (15), N can be obtained:

N= ( P e−t 0 Λ P−1 ) s (t 0)−N (t 0 ) 16()

Substituting eq. 16() into eq. 11(),

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s ( t )=P etΛ P−1 {[( P e−t 0 Λ P−1 ) s (t 0 )−N (t 0 ) ]+N (t)} 17()

s (t )=P etΛ P−1 ( N ( t )−N ( t0 ))+P e(t−t0 ) Λ P−1 s (t 0 ) 18()

if t0=0, s (t )=P etΛ P−1 ( N (t )−N (0 )+s (0)) 19()

Apparently, N (0 )=0; if the mill is empty at the initial time, s (0 )=0,

s ( t )=P etΛ P−1 N (t) 20()

Because

e−tΛ=diag (e−λ1 t , e−λ2 t ,⋯ , e− λn t) 21()

where

e−tΛ=[e− λ1 t ¿e− λ2 t ¿⋱ ¿e−λn t]

n× n

22()

P=[ p11 p12

p21 p22

⋯ p1 n

⋯ p2 n

⋮ ⋱pn1 pn 2

⋱ ⋮⋯ pnn

]n× n

23()

f (t)=(f 1(t)f 2(t)⋮

f n(t))

n×1

24()

Substituting eq. 16() and eq. 23() into eq. 11(), we have

P e−tΛ P−1=[ p11 p12

p21 p22

⋯ p1n

⋯ p2n

⋮ ⋱pn1 pn 2

⋱ ⋮⋯ pnn

]n ×n

[e− λ1 t ¿e− λ2 t ¿⋱ ¿e−λn t]

n× n[ p11' p12

'

p21' p22

'

⋯ p1 n'

⋯ p2 n'

⋮ ⋱pn 1

' pn 2'

⋱ ⋮⋯ pnn

' ]n × n

25()

P e−tΛ P−1=[ p11 e− λ1 t p12 e−λ2 t

p21 e−λ1 t p22 e−λ2t

⋯ p1 ne− λn t

⋯ p2 ne− λn t

⋮ ⋱pn 1 e−λ1 t pn2 e−λ 2 t

⋱ ⋮⋯ pnne− λn t ]

n ×n[ p11

' p12'

p21' p22

'⋯ p1n

'

⋯ p2n'

⋮ ⋱pn1

' pn2'

⋱ ⋮⋯ pnn

' ]n× n

26()

6

P e−tΛ P−1=[∑i=1

n

p1 i e−λi t pi 1

' ∑i=1

n

p1 i e−λ it pi 2

'

∑i=1

n

p2 i e−λi t pi 1

' ∑i=1

n

p2 i e−λ i t p i2

'

⋯ ∑i=1

n

p1 i e−λ i t p¿

'

⋯ ∑i=1

n

p2 i e−λ i t p¿

'

⋮ ⋱

∑i=1

n

p¿ e−λ it pi 1' ∑

i=1

n

p¿ e−λ it pi 2'

⋱ ⋮

⋯ ∑i=1

n

p¿e− λi t p¿' ]

n ×n

27()

P e tΛ P−1=[∑i=1

n

p1 ieλi t p i1

' ∑i=1

n

p1 ieλi t p i 2

'

∑i=1

n

p2 ieλi t p i1

' ∑i=1

n

p2 ieλi t p i 2

'

⋯ ∑i=1

n

p1i eλ i t p¿

'

⋯ ∑i=1

n

p2i eλ i t p¿

'

⋮ ⋱

∑i=1

n

p¿e λi t p i 1' ∑

i=1

n

p¿eλ it pi 2'

⋱ ⋮

⋯ ∑i=1

n

p¿ eλi t p¿' ]

n× n

28()

N ( t )=∫ ( P e−tΛ P−1 ) f ( t ) dt=∫ [∑i=1

n

p1 i e−λ it pi 1

' ∑i=1

n

p1 i e−λ it p i2

'

∑i=1

n

p2 i e−λ it pi 1

' ∑i=1

n

p2 i e−λ i t p i2

'

⋯ ∑i=1

n

p1 i e−λ i t p¿

'

⋯ ∑i=1

n

p2 i e−λ i t p¿

'

⋮ ⋱

∑i=1

n

p¿ e−λ it pi 1' ∑

i=1

n

p¿ e−λ it pi 2'

⋱ ⋮

⋯ ∑i=1

n

p¿e− λi t p¿' ]

n ×n

(f 1 (t )f 2 (t )⋮

f n (t ))

n ×1

dt

¿∫(∑j=1

n [∑i=1

n

p1 i e−λi t pij

' f j(t)]∑j=1

n [∑i=1

n

p2 i e−λ it pij

' f j(t)]⋮

∑j=1

n [∑i=1

n

p¿e− λi t pij' f j( t)] )dt 29()

Eqs. 18() 27() 29() form the analytical solutions for the dynamic model. Based on the above analytical solution, eq.20() provides a simple method to calculate mill contents s(t). This algorithm can be named as Direct Single Time method (DST). The time zero point is fixed and the expression is simple.

In order to avoid exceeding the computation power because of the rapid growth of exponential functions with time, another solving method was introduced. That is, the time zero point is not fixed and it moves forward

7

progressively. For every time interval, let t n−1=0, t n=t n−1+τ=τ, eq. 18() can be rewritten as:

s (τ )=P eτΛ P−1 ( N ( τ )−N (0 ) )+P e(τ −0) Λ P−1 s (0 ) 30()

Sinces (t n )=s (τ ), s (0 )=s (t n−1 ),

s (t n )=P eτΛ P−1 ( N (τ )−N (0 ) )+P eτΛ P−1 s (tn−1 ) 31()

From eq. 14(), N (0 )=0, eq. 31() can be rewritten as:

s (t n )=P eτΛ P−1 N (τ )+P eτΛ P−1 s ( tn−1 ) 32()

The algorithm based on eq. 32() is named as Direct Multiple Time method (DMT). Eq. 20() and eq. 32() are the analytical solutions for the dynamic model of SAG mills and will be tested in the following sections.

2.3 Sub-models for solving the dynamic equation

In order to calculate the dynamic equation of a tumbling mill (eq.1()), some sub-models, such as selection function, discharge function and appearance function are needed. In a dynamic situation, the breakage rate or selection function is dependent on the type of tumbling mills. For ball mills the selection or breakage function is generally not affected by mill holdup and can be treated as independent of time. For SAG/AG mills, both the mill holdup and the amount of coarse particles in the mill may affect the breakage rate and therefore the selection function is not a constant at different times. However, no relationship between the breakage rate and mill holdup is reported in literature and more work in this area is required for true dynamic control of mill operation. As long as the breakage rate at a particular time is known for SAG/AG mills, one can apply the time dependent selection function to the analytical solution. For demonstrating the analytical solution developed in this work, Austin’s time-independent selection function Seli (hour-1) (the specific rates of breakage) [18] is used here:

Seli=A1( xi

x0)

α 1

[ 1

1+( xi

μ x0 )β ]+ A2( x i

x0)

α2

33()

where, x i is the upper size of size class i, x0 is a standard size (usually 1mm ). A1, α 1, and β are parameters which are linked to the nipping breakage of smaller particles by larger grinding media and A2, α 2 are linked to the breakage of large lumps by collision with grinding media and lumps of similar size (self-breakage). The units if A1 and A2 are time-1(e.g. hour-1). α 1,α 2, and β are dimensionless.

8

Figure 1 shows a typical selection function curve used in the two test cases. The shape of this selection function is of typical features of a breakage rate curve for a SAG mill [19] and it is assumed that it does not change with time.

The mean energy E (kW/t) can be obtained from the following equation:

E= PowerTotal mass 34()

Normally, total mass throughput in one hour is used in Eq. 34(). It is assumed that the energy is not evenly distributed to all particles with different sizes. The selection function can regulate the size-specific energy level [20].

Ecsi=EcsSeli

35()

Ecsi is size-specific energy (kWh/t), Seli is the selection function of size class i.

As for the discharge function, it is closely related to the classification function. Typically, a classification curve (e.g. figure 2) can be divided into three regions according to the particle size [16, 21]. For particles smaller than Xm, it is assumed that they behave like water and are subject to no classification. Xg is the effective grate aperture size and Xp is the effective pebble port aperture size, the classification function ci is shown in Figure 2, where all the particles greater than Xp, cannot be discharged. The discharge rate and the classification function are related through relationship Eq.3(). The classification function c i is given by Eq.(36)

c i={1 x<Xm

lnx ∙ (1−cg )ln Xm−ln Xg

+cg−ln Xg ∙ (1−c g )ln Xm−ln X g

Xm<x<X g

lnx ∙ cg

ln X g−ln X p−

c g∙ ln X p

ln X g−ln X pX g< x< X p

0 x> X p

(36)

where: c g is a constant classification factor at the effective grate aperture size Xg.

The appearance function describes the breakage characteristics of the ore under certain comminution energy input. In this paper, the M-p-q t10-tn model [16, 22] was used (Eq.(37)):

t 10=M [1−e(−F mat ∙ X ∙ Ecs)] (37)

where, t 10 is defined as the percentage passing one tenth of the original mean particle size. M is the maximum possible value of t 10 in impact (percentage).

9

Fmat is material property (kgJ-1m-1) ¿ f (X , p ,q), in which p and q are ore-specific constants. q is ore size effect parameter. X is the feed ore particle size (mm).

The relationship of t10-tn is established by ore characterisation experiments [16], and the appearance function matrix A is obtained by spline function at any particle size of interest, in this case the chosen sieve size series.

3. Results and discussion

3.1 Comparison of the two analytical methods DST and DMTThe two analytical solution methods, namely DST and DMT methods were applied to obtain the analytical solutions respectively using MATLAB software. The mill is empty at time zero point and a constant feed rate of 150t/h is added into the mill and kept constant after time zero. The internal length of the SAG mill is 6.6m and the internal diameter is 5.0m. With eq. 20() and eq.32(), the mass fraction of each size for mill contents can be calculated. Adding each fraction of the size class together, the total mass and total content volume can be obtained. Figure 3 shows the mill content volume versus time using the two methods. It can be seen that the DST method will exhibit numerical instability when the time is increasing over 13 hours. This is because there is an absolute item “t” (time) on the exponente tΛ (eq. 20()). With time increasing, the value of exponent function goes up dramatically and exceeds the handling ability of the computer. If the exponential function doesn’t exceed the handling limitation of the computer, DST is stable and accurate. In contrast, the DMT method (in this case τ was chosen as 10 hours, which is less than the unstable time of 13 hours) is much more stable because it moves the time zero point ahead, avoiding the exponent function from

transgression. Besides time τ, also has an influence on the stability. Λ=diag(λ1 , λ2 ,⋯ , λn), which is related to the eigenvalues of matrix H. From eq. 8() and eq. 9(), H is linked to the selection function and discharge function. Thus the selection function, discharge function and time all have an impact on the numerical stability. As the former two factors are not adjusted, time τ can be changed by movement of time zero point (DMT method), resulting in a stable analytical solution.

Unlike DST, which might incur instability, DMT has the advantages of both high accuracy and strong robustness. Thus for the following cases, the DMT method is used to obtain the analytical solution, which is compared to the numerical solution.

3.2 Comparison between the analytical method and the numerical method

As mentioned previously, the mill products and contents, product size distribution can be solved numerically using the finite difference form of eq. 5():

S j , t+1−S j , t

∆ t=f j , t+1−p j , t+1+(AR−R)∙ S j , t+1 (38)

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Figure 4 shows the comparison between analytical solution and numerical solution. The feed rate has step changes from 150 t/h to 125 t/h and then to 180 t/h.

It is apparent that the DMT analytical method is more accurate since it has no other input computational parameters such as time step. With the numerical solution, if time step increases from 0.01 hour to 0.1 hour and to 1 hour, the numerical solution curve gradually moves away from the analytical solution, indicating a dramatically increasing error (Figure 4). Yet the choice of time step length is arbitrary and thus the error caused by the traditional numerical method is inevitable to some extent. In addition, iteration to converge the solution will always be involved in numerical techniques which lead to longer computation time. Through the use of an analytical solution, it is easier to analyse the response of various mill parameters to the change of operating conditions.

3.3 Case studies for analytical solutions

Two cases are studied to demonstrate the applicability of the analytical solution. The conditions for the two cases are shown in Table 1.

In Case 1, the feed rate step changes to 150t/h after time zero and is kept constant. In Case 2, the feed rate changes from 0 at time zero to 150t/h at 2.5 hours, keeps at this level for about 7.5 hours and then changes to 125t/h for around7.5 hours, and then changes to 195t/h. As for the classification function, they are also different. Case 2 has a finer classification function (with smaller Xm and Xg,) than Case 1, which makes the discharged products finer. For example, for size 10mm, the passing probability is 30% in Case 2, while it is 60% in Case 1.

The analytical solutions for the dynamic operation in both cases were obtained by the analytical DMT method (eq. 32()). The results are shown in Figs 5-7.

It can be seen from the above results that the analytical method can serve as a feasible way to get the solutions of the dynamic model for tumbling mills. In Fig. 6, when the feed changes from zero to 150t/h, and then changes to 125t/h, and finally changes to 195t/h after a period, the flow rate of the products and the volume of the slurry contents in the mill also change accordingly. If the feed goes up the product and the content also increase. However there is a dynamic response time (time lag) between the change of feed and change of product and content. The dynamic response time can be determined by the time period it takes for the mill to reach the new steady state after the change of feed. If the feed increases, the content also increases. Some of the contents will turn into product after grinding and those products will be discharged out of the mill. Thus product also increases until it reaches the new steady state where the flow rate of the product equals the feed flow rate, and vice versa.

In the above case, the dynamic response time for feed change from 0 to 150t/h is around 2 hours. In other words, it takes about 2 hours for this SAG mill to reach new steady state when the feed changes from 0t/h to 150 t/h. If the feed change range reduces, the dynamic response time also reduces. For example, when the feed decreases from 150t/h to 125t/h, the dynamic response time is around 1.25 hours.

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The product size will become a little bit coarser during the initial period when the product changes from 0 t/h to 150 t/h (Fig.8). The product size distribution of non-steady state at 4.08 hours is coarser than the non-steady state at 3.06 hours and it will get coarser until the steady state is reached. Thereafter, the product size distribution will be stabilized at the steady state. The product size distribution of steady state at 8.67 hours is the same as the steady state at 10 hours. This is because of the change of average residence time of the particles in the mill. At the non-steady state stage, the discharge flowrate is less than the feeds as the contents build up. This will result in a longer average residence time for the particles in the mill and hence finer products. At the steady state, the flowrate of the products equals the feeds, with the average residence time reduced compared with the non-steady stage. When the steady state is reached, the average residence time will not change any more, thus resulting in a constant product size distribution with time.

In Case 2, when the feed rate changes from 150t/h to 125t/h suddenly, as shown in Fig. 6, the product rate also changes. During this period, as shown in Fig. 9, the product size distribution (PSD) also changes slightly. Because there is a lag between the feed rate change and the product rate change, the mill content reduces accordingly, which results in a shorter average residence time. So the mill content of non-steady state at 10.46 hours when the feed is 125t/h is the coarsest (Fig. 9). Thereafter, the product rate gradually catches up with the pace of the feed rate reduction (Fig. 6). The product takes longer to be discharged than the beginning of sudden reduction in feed rate. Thus the average residence time is getting longer and the PSD is getting finer (see the amplified spot in Figure 9). When the steady state is reached, the PSD doesn’t change anymore. So there is an overlap between the line of steady state at 15.30 hours and the line of steady state at 17.85 hours.

Comparing steady states at feed rate 150t/h and 125t/h, the PSD at feed rate 125t/h is slightly finer than the feed rate at 150t/h (Fig. 9). With an increase in feed rate, the mill charge and power draw also increase, partially compensating for the increased feed rate and resulting in only a slightly coarser product. In real operation, the product size is mainly determined by the feed size distribution and ore hardness and feed rate does not change dramatically. If the feed rate is too high, the grinding performance will deteriorate sharply and the mill will eventually choke, or reach the motor power limit.

Comparing Case 1 and Case 2 with the same feed rate of 150t/h, because of the finer classification function in case 2, the contents are kept in the mill until they are ground finer and then discharged. Thus in Fig. 10, Case 2 has a higher content level than Case 1. In Case 1, with a coarser classification function, there is a wider size range that can be discharged. So the dynamic response time for Case 1 is shorter than Case 2 (Figure 10). The dynamic response time for Case 1 is about 1 hour (point C in Fig. 10) whilst it is about 2 hours in Case 2 (point F in Fig. 10). Such a conclusion can also be drawn from the observation of Figs 5 and 6.

Although a SAG mill is chosen as the case study in this work, it can be easily used for other types of tumbling mills such as ball mills. With ball mills, the selection function or breakage rate is less affected by mill holdup and therefore a constant selection function like what is used in this work can be

12

used. Future work needs to focus on the development of breakage rate function that includes the effect of mill holdup and the amount of coarse particles in SAG mills.

4. Conclusions

An analytical solution for the dynamic equation of tumbling mills has been presented. Two algorithms, Direct Single Time method (DST) and Direct Multiple Time method (DMT) were applied to get analytical solutions and were compared and analysed. Direct Multiple Time method (DMT) is proved to be a better method for obtaining a stable analytical solution while DST method leads to numerical instability because of overstepping computer handling ability caused by the exponential function. Two SAG mill operation cases were used to demonstrate the applicability of the DMT analytical solution method. It is proven that the DMT analytical method is robust, feasible and reliable, giving stable and straightforward results. It also does not need an iteration methodology, comparing favourably with the traditional numerical solution whose solution accuracy is dependent on the time step used. The case studies using the DMT analytical solution method showed that, as the classification function gets finer, the content in the mill increases because a wider size range of material is kept inside the SAG mill and the dynamic response time is also longer. In addition, the dynamic response time is linked to the feed change range. The greater the feed changes, the longer the dynamic response time.

The results showed the advantage of applying the analytical method. Future work will be aimed to compare the analytical solution against a case of industrial mill dynamic data for further validation and applications.

Acknowledgements

The author wishes to acknowledge the financial support of the Commonwealth Scholarship from the Australian Government and Scholarships from the University of Queensland. The author wishes to acknowledge the partial financial support on supervision from AMIRA P9P project.

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List of Figures

Figure 1 Selection function (specific rate of breakage)Figure 2 Typical classification function (linked to discharge function)Figure 3 Comparison between two analytical solution algorithms for calculation of mill content

Figure 4 Results of traditional numerical method compared with analytical methods

Figure 5. Dynamic response to feed rate in Case 1 by using the DMT methodFigure 6 Dynamic response to feed rate in Case 2 by using the DMT method

Figure 7 Product size distribution by using the DMT method (feed rate=195t/h Case 2)

Figure 8 Product size distribution variation when feed changes from 0 to 150 t/h in Case 2Figure 9 Product size distribution variation when feed changes from 150 to 125 t/hFigure 10 Comparison of content curves with different classification functions

(Feed changes from 0 to 150t/h, C and F represent the time to reach steady state in mill content volume)

16

10-2 10-1 100 101 102 103

Particle size (mm)

10-1

100

101

102

Sel

ectio

n fu

nctio

n (h

our-1

)

Figure 1 Selection function (specific rate of breakage)

17

10-2

10-1

100

101

102

103

0

20

40

60

80

100

Particle size Ln(size)

Cla

ssifi

catio

n fu

nctio

n %

Figure 2 Typical classification function (linked to discharge function)

18

Xm Xg Xp

0 2 4 6 8 10 12 14 160

5

10

15

20

25

30

35

40

Time (hour)

Vol

ume

(m3 )

Content volume variation curves in comparison among different algorithms

Direct Single Time method (DST)Direct Multiple Time method (DMT) = 10 hours

Figure 3 Comparison between two analytical solution algorithms for calculation of mill content

19

0 5 10 15Time(hour)

0

20

40

60

80

100

120

140

160

180

200

Flow

rate

(t/h

)

Comparison between Analytical products and Numerical products

FeedProduct of DMT Analytical MethodProduct of Numerical Method, Time Step=0.01 hourProduct of Numerical Method, Time Step=0.1 hourProduct of Numerical Method, Time Step=1 hour

Figure 4 Results of traditional numerical method compared with analytical methods

20

0 5 10 15 20 25 30Time (hour)

0

50

100

150

200

Flow

rate

(t/h

)

The dynamic response of product and slurry content to the feed

0

15

30

45

60

Volu

me

(m3 )

Feed t/hAnalytical Product t/hAnalytical Slurry Volume m3

Figure 5. Dynamic response to feed rate in Case 1 by using the DMT method

21

0 5 10 15 20 25 30Time (hour)

0

50

100

150

200Fl

owra

te (t

/h)

The dynamic response of product and slurry content to the feed

0

15

30

45

60

Volu

me

(m3 )

Feed t/hAnalytical Product t/hAnalytical Slurry Volume m3

Figure 6 Dynamic response to feed rate in Case 2 by using the DMT method

22

10-3 10-2 10-1 100 101 102 103

Ln(Size)

0

10

20

30

40

50

60

70

80

90

100

Cum

ulat

ive

perc

ent p

assi

ng %

Product size distribution @ constant feedrate 195 t/h

Product from Analytical methodFeed

Figure 7 Product size distribution by using the DMT method (feed rate=195t/h Case 2)

23

10-3 10-2 10-1 100 101 102 103

Ln(Size)(mm)

0

10

20

30

40

50

60

70

80

90

100C

umul

ativ

e pe

rcen

t pas

sing

%

Product size distribution for Analytical methods (feed=150t/h)

Non-steady state at 3.06 hoursNon-steady state at 4.08 hoursSteady state at 8.67 hoursSteady state at 10 hoursFeed @150t/h

Figure 8 Product size distribution variation when feed changes from 0 to 150 t/h in Case 2

24

10-3 10-2 10-1 100 101 102 103

Ln(Size)(mm)

0

10

20

30

40

50

60

70

80

90

100C

umul

ativ

e pe

rcen

t pas

sing

%

Product size distribution variation analysis (feed=125t/h)

Steady state at 10.21 hours, feed=150t/hNon-steady state at 10.46 hours, feed=125t/hNon-steady state at 10.97 hours, feed=125t/hSteady state at 15.30 hours, feed=125t/hSteady state at 17.85 hours,, feed=125t/h

Figure 9 Product size distribution variation when feed changes from 150 to 125 t/h

25

0 2 4 6 8Time(hours)

0

10

20

30

40

50C

onte

nt v

olum

e( m

3 )

Content volumes at different classification functions

Case 1 with coarser classificationCase 2 with finer classification

Figure 10 Comparison of content curves with different classification functions

(Feed changes from 0 to 150t/h, C and F represent the time to reach steady state in mill content volume)

26

C F

List of Tables

Table 1. Conditions for the case studies

Table 1. Conditions for the case studies

Feed Classification function

Case 1

0 5 10 15 20 25 30Time (hour)

0

50

100

150

200

Flow

rate

(t/h

)

Feed variation curve10-1 100 101 102 103

Particle size Ln(size)

0

20

40

60

80

100

Cla

ssifi

catio

n fu

nctio

n %

Case 2

0 5 10 15 20 25 30Time (hour)

0

50

100

150

200

Flow

rate

(t/h

)

Feed variation curve10-1 100 101 102 103

Particle size Ln(size)

0

20

40

60

80

100

Cla

ssifi

catio

n fu

nctio

n %

27