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球磨機(jī)磨損的推斷模型
摘要：球磨機(jī)，典型礦物加工業(yè)，被用于將大小分布的礦石磨成別的。磨損與影響研磨性能的鋼球電荷方面的細(xì)碎機(jī)械學(xué)的建立聯(lián)系在一起。在本研究中，球磨機(jī)磨損與研磨工作參數(shù)有關(guān)，決定使用一種數(shù)學(xué)磨損模型。這種磨損模型聯(lián)合在壓碎過(guò)程中能量的消散和填裝層的磨損區(qū)段同粘著力和磨損描述。這種模型已經(jīng)被加到鋼球裝填層運(yùn)動(dòng)模型，球磨機(jī)磨損率的模擬實(shí)驗(yàn)以及球磨機(jī)的元件磨損和它的對(duì)研磨性能的影響。現(xiàn)有模擬結(jié)果顯示了在磨損和研磨性能之間的交互作用。更深入的研究是用工業(yè)日期確認(rèn)裝填層和磨損模型的答案。1997年，加拿大采礦和冶金學(xué)會(huì)。由Elsevier Science Ltd出版。
引言
粉碎，磨碎到微粒，磨成粉都是用于該礦產(chǎn)加工工業(yè)的水磨程序的同義名詞。與這些過(guò)程有關(guān)聯(lián)的是金屬磨損，加拿大和美國(guó)的年度消耗鋼鐵量達(dá)300 000噸。磨損也影響著研磨性能和品質(zhì)。在這一篇論文中，磨損的推斷模型的預(yù)測(cè)對(duì)減少工序磨損、保持研磨性能和品質(zhì)最理想的磨損狀況的決定是必需的。
磨損和它與磨損有關(guān)機(jī)械學(xué)原理已經(jīng)被廣泛的應(yīng)用在[2-4]期實(shí)驗(yàn)數(shù)據(jù)上，有用的模型對(duì)磨損現(xiàn)象的理解在[5-9]期實(shí)驗(yàn)數(shù)據(jù)上，理論研究在[10-12]期實(shí)驗(yàn)數(shù)據(jù)上。這篇論文的目的是根據(jù)球磨機(jī)水磨工序理論的發(fā)展而做出的磨損模型的推測(cè)。
背景
球磨機(jī)(圖1所示)是由許多相互聯(lián)系和相互作用的零件組合成的一個(gè)體系，這些零件組合起來(lái)是為了磨碎礦石。這種機(jī)械粉碎法的工序是由一些特殊的鋼球組成的球磨機(jī)的零件用來(lái)磨碎礦石的過(guò)程。同時(shí)，這些球在球磨機(jī)工作期間能夠很好地建成球磨機(jī)的填裝層的輪廓，如圖2所示。
填裝層的輪廓標(biāo)志描繪為三個(gè)有典型破碎特性的區(qū)域。研磨區(qū)段是由彼此滑動(dòng)球?qū)訑?shù)描述，磨碎它們之間的礦石；翻滾的區(qū)域是在低能量的沖擊下對(duì)鋼球相互的滾動(dòng)和磨碎礦石的描述。擊碎的區(qū)域是在高能量的沖擊下對(duì)鋼球的飛行和擊碎礦石的描述。
填裝層的輪廓的形成是直接地依靠存在填裝層和球磨機(jī)滾筒壁之間的摩擦力。通過(guò)用不同的襯板輪廓，摩擦力也可改變由此而生的影響球磨機(jī)輪廓的形式。
輪廓運(yùn)動(dòng)模型
如上述所提。球磨機(jī)磨損是能量轉(zhuǎn)移的作用在襯板和球輪廓之間以及在二個(gè)碰撞的球之間。因此，塑造輪廓運(yùn)動(dòng)模型是預(yù)測(cè)球磨機(jī)磨損和對(duì)研磨的影響的作用第一步。
模型的發(fā)展開始以單個(gè)的球的運(yùn)動(dòng)為對(duì)象(如圖4)。如所描述由Mclvor和Powell [15、16]，在球磨機(jī)中單個(gè)球飛行的問(wèn)題是由轉(zhuǎn)動(dòng)速度，滾筒半徑，靜摩擦因子和輪廓的角度決定：
(1)
無(wú)論如何，Hukki [17] 提及球輪廓的運(yùn)動(dòng)不是全部依靠一個(gè)的單點(diǎn)飛行如假設(shè)上述等式。它也是依靠描述球輪廓和襯板類型相互聯(lián)系的有效的摩擦因子是否和我的差不多。
所以，如果我們描述動(dòng)力傳遞損耗在二球?qū)訑?shù)之間作為靜態(tài)和運(yùn)動(dòng)摩擦因子[17]之間的]一個(gè)關(guān)系;
(2)
旋轉(zhuǎn)的動(dòng)力傳遞損耗速度為：
(3)
使用這個(gè)結(jié)果，我們可以區(qū)分球飛行和點(diǎn)穩(wěn)定的動(dòng)力傳遞損耗之間如下：
1. 飛行的問(wèn)題 (μ≥1.0)
(4)
2. 穩(wěn)定的動(dòng)力傳遞損耗問(wèn)題(μ<1.0)
(5)
有效的摩擦因子被定義為;
(6)
用這些關(guān)系與被描述的那些一起[18-20]和應(yīng)用于他們描述離散的球填充層的質(zhì)點(diǎn)系，它變得可能模仿球填充運(yùn)動(dòng)(如圖5所示)。
因而被定義為填充層運(yùn)動(dòng)，我們?cè)诟鞣N各樣的粉碎區(qū)域在填充外形可以通過(guò)確定被消耗的運(yùn)動(dòng)和分布能量(如圖2)，使用下式[18-21]：
(7)
(8)
(9)
在球磨機(jī)中這種能量輪廓(如圖6)現(xiàn)在被確定為Hardinge磨機(jī)(如圖1)。這個(gè)輪廓運(yùn)動(dòng)模型確定顯示能量怎么在研磨，擊碎和翻滾時(shí)被消耗后分布作為磨機(jī)轉(zhuǎn)動(dòng)速度、滾筒直徑、填充層和襯板表示法功能。
磨損率的估計(jì)
如前面所提到，在球磨機(jī)的球輪廓的運(yùn)動(dòng)外形有三個(gè)粉碎區(qū)域。雖然有別的磨損機(jī)械學(xué)的存在，只有膠粘劑和磨蝕是與這些粉碎區(qū)域有聯(lián)系。同翻筋斗和擊碎區(qū)域聯(lián)系在一起，當(dāng)球在這些區(qū)域碰撞，當(dāng)黏著性磨損聯(lián)系到研磨區(qū)域時(shí)，球滑過(guò)另一個(gè)球或越過(guò)另一個(gè)球。這些機(jī)制可以被表達(dá)，根據(jù)在[23-24]磨損的能量率而做黏著性磨損
(10)
腐蝕磨損
(11)
運(yùn)用這些磨損模型于球磨機(jī)的情況，我們可以得出：
(12)
(13)
(14)
比較最初和最后的襯板磨損外形，襯板磨損率可以被估計(jì)使用：
(15)
我們可以通過(guò)比較公式（15）和公式(12)確定磨蝕因素，因而得到：
(16)
進(jìn)一步，使用公式（13）和公式（14）以及(16)的結(jié)果，我們可以盡可能地確定黏著力的大?。?
(17)
以磨蝕因素并且為特定球磨機(jī)操作的上下文確定的黏附力可能性，現(xiàn)在我們從這個(gè)上下文可以確定球磨機(jī)的磨損率的影響是怎么改變的。保持和P為常數(shù)，變化的參量例如磨房轉(zhuǎn)動(dòng)速度和輪廓邊界，我們可以推出對(duì)球磨機(jī)磨損率[24的]有關(guān)的變動(dòng)。然而，推測(cè)球磨機(jī)磨損率是有限的，在球磨機(jī)研磨時(shí)，考慮到我們的確定球的磨損的影響的目標(biāo)時(shí)的表現(xiàn)。
襯板磨損
在球磨機(jī)研磨的表現(xiàn)取決于主要能量怎樣被分布在球輪廓外形的各種各樣的粉碎區(qū)域。
如被提及，輪廓外形的形式，和每個(gè)粉碎區(qū)域的重要特性，是直接地依靠存在球輪廓和球磨機(jī)桶壁之間的摩擦力。
不同的襯板輪廓類型(圖3) 在球輪廓和球磨機(jī)筒壁以及研磨的表現(xiàn)之間影響這中摩擦力。由于有特定的上下文，所以使用被認(rèn)為優(yōu)選的襯板類型的輪廓是可能的。然而，以時(shí)間，球磨機(jī)磨損將修改最初的襯板輪廓外形和隨后碾碎研磨。在球磨機(jī)輪廓的力量下塑造運(yùn)動(dòng)形式成為推斷球磨機(jī)磨損和它對(duì)研磨性能的影響的下一個(gè)步驟。
在球磨機(jī)運(yùn)轉(zhuǎn)期間，整個(gè)輪廓大廳由重心和離心力然后施加力場(chǎng)組成襯板輪廓（圖7）[23-25]。
用這種描述，平均的組成力可以被確定作為展示在圖8中；
(i)平均組成的離心力
(18)
(ii)平均組成的重力
(19)
進(jìn)一步，在球磨機(jī)襯板上面作為球磨機(jī)輪廓，球磨機(jī)輪廓的襯板部分的位移會(huì)產(chǎn)生一個(gè)壓縮力（圖9）。 這力量被定義如下：
(20)
這里
通常表示為：
(21)
在襯板表面總的平均作用力可以成為：
(22)
襯板磨損，作為球磨機(jī)球輪廓?jiǎng)?chuàng)造的力場(chǎng)的位置和強(qiáng)度功能以及磨蝕因素，成為：
(23)
那里
(24)
指出在襯板的滑動(dòng)速度由早先的公式（2）重新整理。
在襯板離散化到的區(qū)別，定時(shí)乘，這種仿真算法可以被開發(fā)之后成為[23-25]，包括襯板齒廓磨損的模擬實(shí)驗(yàn)。
舉例來(lái)說(shuō)，圖10中圖解的一種波形襯板的輪廓磨損的模擬實(shí)驗(yàn)和在圖11中現(xiàn)實(shí)的襯板的輪廓磨損差不多。
球磨機(jī)磨損和研磨性能
雖然工業(yè)上的研究需要更多這樣的磨損模型，但是想象出一種彈性襯板的進(jìn)化磨損的推斷模型是有可能的。當(dāng)然，這將計(jì)算怎樣的磨損影響對(duì)研磨性能的決定，這里解釋為在顆粒測(cè)定的生產(chǎn)能力的變化。由圖1.的Hardinge情況得，消逝在傾斜的襯板的現(xiàn)象由這里轉(zhuǎn)化為模擬如圖12所示。
進(jìn)一步，該圖6的能量率曲線是怎么隨這種襯板磨損而變化的，推斷球磨機(jī)的輸出同輸入的變化一樣，是有可能的。從表格1可以看出，使用一種已經(jīng)被開發(fā)的破碎模型，可以解釋襯板的壽命期限是有可能的。這里，球磨機(jī)生產(chǎn)能力將提高而襯板磨損下降。
與這種現(xiàn)象相聯(lián)系，球磨機(jī)能源消耗的減少如所顯示。這兩種現(xiàn)象說(shuō)明最佳化球磨機(jī)的性能作為一種推斷磨損影響的函數(shù)是可能的。
表1.球磨機(jī)的輸出量作為襯板曲線的一個(gè)函數(shù)。
顆粒大小
(μm)
最初通
過(guò)量%
1/2通過(guò)
量 %
最后通
過(guò)量%
74
100
150
300
830
1170
1650
58.08
68.25
78.54
92.12
99.45
99.96
100.00
58.25
68.44
78.68
92.15
99.45
99.96
100.00
60.13
70.38
80.25
93.96
99.61
99.97
100.00
討論
在結(jié)束這篇論文之前應(yīng)該對(duì)球磨機(jī)填裝層運(yùn)動(dòng)、襯板磨損和球磨機(jī)輸出的產(chǎn)品做幾點(diǎn)評(píng)論。
如所示，球磨機(jī)的填裝層運(yùn)動(dòng)依靠許多物理和操作因素；它也依靠礦漿的流變特性。當(dāng)然這些礦漿的流變特性是百分比固體的作用并且礦石的性質(zhì)。在填裝層運(yùn)動(dòng)的模型中，這些因素的影響被包括在（2）滑動(dòng)速度的關(guān)系中，如所描述的靜態(tài)和動(dòng)摩擦因素。摩擦因素的變化，如礦漿的流變特性的一個(gè)可能的變化造成的，可以增加或減少在鋼球?qū)又g和襯板磨損之間的相當(dāng)數(shù)量的動(dòng)力傳遞損耗。
在球磨機(jī)的磨損模型和兩種磨損機(jī)械學(xué)中，它只是一種假設(shè)，球磨機(jī)不是空轉(zhuǎn)的或者運(yùn)轉(zhuǎn)狀況不直接地把鋼球放到球磨機(jī)襯板里。在這樣的情況下，襯板磨損同磨損機(jī)械學(xué)一樣（表面，破裂，磨損），都增加了襯板的磨損率。
球磨機(jī)產(chǎn)品的變化的重要性與襯板的磨損有關(guān)，是從屬與一種特殊的礦石的破碎特性。同樣地，檢驗(yàn)使用的工業(yè)數(shù)據(jù)為了確定這個(gè)模型精確度是必要的。然而，假設(shè)，整體模型的一個(gè)充分檢驗(yàn)是可能的，這里被提出的初步結(jié)果表示，考慮使用這個(gè)模型為球磨機(jī)優(yōu)化作為能源消耗和球磨機(jī)襯板磨損功能變得可能。
結(jié)論
這篇論文根據(jù)球磨機(jī)的一個(gè)理論描述提出了一個(gè)有預(yù)測(cè)性的襯板磨損模型。當(dāng)包括一個(gè)簡(jiǎn)單的膠粘劑和磨損模型的應(yīng)用對(duì)一個(gè)復(fù)雜的鋼球填裝層模型時(shí)，不僅能確定必要的磨損參數(shù)，而且推斷襯板和研磨性能是可能的。
無(wú)論如何，進(jìn)一步的研究對(duì)模型的檢驗(yàn)和參數(shù)確定是必要的。盡管這是必要的，但在將來(lái)，球磨機(jī)在設(shè)計(jì)和操作方面的優(yōu)化也有了可能性。
鳴謝--本文的出版物由加拿大研究經(jīng)費(fèi)自然和工程研究委員會(huì)使成為可能。
PREDICTIVE MODEL FOR BALL MILL WEAR
Abstract-ball mills, characteristic of the mineral processing industry, are used to reduce ore from one size distribution to another. Wear is associated with comminution mechanisms found in the ball charge which in turn affects grinding performance. In this work, ball mill wear, as a function of mill operating variables, is determined using a mathematical wear model. The wear model incorporates the energy dissipated in crushing, tumbling and grinding zones of the charge profile with adhesive and abrasive wear descriptions. This model has been added to a ball charge motion model allowing the simulation of mill wear rates as well as ball mill element wear and its affect on grinding performance. Simulation results presented show the interaction between wear and grinding performance. Further work is necessary to validate charge and wear model results using industrial date.1997 Canadian Institute of Mining and Metallurgy. Published by Elsevier Science Ltd.
INTRODUCTION
To comminute, to reduce to minute particles, to pulverize, are all synonyms of grinding processes used in the mineral processing industry. Associated with these processes is metal wear which in Canada and the United States represents an annual consumption of some 300 000 tons of iron and steel [1]. Wear also affects grinding performance and quality. In such a context, predictive wear models become a necessity to determine optimal grinding conditions that reduce process wear while maintaining grinding performance and quality.
Wear and its mechanisms related to grinding has been studied extensively using experimental data [2-4], models useful to understanding of wear phenomena [5-9] and theoretical studies [10-12].The goal of this paper is the presentation of a predictive wear model based on a theoretical development for one such grinding process, the ball mill.
BACKGROUND
The ball mill (Fig.1) is a system composed of a number of interrelated and interactive elements that work together in order to grind a given ore. This comminution process is achieved by the individual balls which constitute the actual ball mill element that brings about ore breakage. Together, these balls form the mill ball charge which, during ball mill operation, typically has a charge profile as found in Fig.2.
Note that the charge profile shows three zones that are characterized by the type of breakage occurring there. The grinding zone is described by ball layers sliding over one another, breaking the material trapped between them; the tumbling zone is described by balls rolling over one another and breaking the material in low-energy impact; the crushing zone is described by balls in flight re-entering the ball charge and crushing the material in high-energy impact.
The form of the charge profile is directly dependant on the friction force existing between the charge and the ball mill wall. By the use of different liner profile (Fig.3), the friction force can be changed subsequently affecting the form of the mill charge as well.
Charge motion model
As mentioned. Mill wear is a function of the energy transferred between liner and ball charge as well as between two colliding balls. Therefore, modelling charge motion is a first step to predicting mill wear and its effect on grinding.
Model development starts with defining single ball motion (Fig.4). As described by Mclvor and Powell [15、16], the point of flight of a single ball in a ball mill can be determined as a function of rotation speed, mill radius, static friction factor and the liner lifer angle:
(1)
However, Hukki [17] mentions that ball charge motion is not entirely dependant on a single point of flight as assumed with the above equation. It is also dependant on whether the effective friction factor describing the interrelationship between ball charge and type of liner used is greater or less than I.
Therefore, if we describe slippage between two ball layers as a relationship between static and kinetic friction factors [17];
(2)
Rotational slippage speed becomes:
(3)
Using this result, we can differentiate between ball flight and the point of stable slippage as:
1. point of flight (μ≥1.0)
(4)
2. point of stable slippage (μ<1.0)
(5)
Where the effective friction factor is defined as;
(6)
Using these relationships along with those described in [18-20] and applying them to a system of particles that describe a discretized ball charge, it becomes possible to simulate ball charge motion (Fig. 5).
Having thus defined charge motion, we can further this development by determining energy consumed and distributed in the various comminution zones on the charge profile (Fig, 2) using the following equations [18, 21]:
(7)
(8)
(9)
The energy profile (Fig. 6) in a ball mill can now be determined as for the Hardinge mill of Fig. 1. Note that this charge motion model determines how energy is consumed and then distributed in grinding, crushing and tumbling as a function of mill rotation speed, mill diameter, ball charge and liner representation.
Wear rate estimation
As mentioned earlier, there are three comminution zones in the ball charge motion profile. Although other wear mechanisms exist, only adhesive and abrasive wear are associated here with these comminution zones. Adhesive wear is associated with the tumbling and crushing zone as balls in these zones collide while abrasive wear is associated to the grinding zone where balls slide pass one another or over the null liner. These mechanisms can be expressed in terms of energy rate used in wear as [23-24]:adhesive wear
(10)
abrasive wear
(11)
Applying these wear models to the ball mill case, we write;
(12)
(13)
(14)
Comparing initial and final liner wear profiles, liner wear rate can be estimated using:
(15)
We can determine the abrasion factor by equating eqn (15) with eqn (12), thus getting:
(16)
Further, using eqn (13) and eqn (14) with the result of eqn(16)，we can determine the adhesion probability:
(17)
With the abrasion factor and adhesion probability P determined for a given mill operating context, we can now determine how changes from this context affect mill wear rates. Keeping and P constant, and varying parameters such as mill rotation speed and charge column, we can predict the associated changes to mill wear rates [24]. However, predicting wear rates is only of limited use when considering our goal of determining the effect of wear on ball mill grinding performance.
Liner wear
Grinding performance in a ball mill is determined primarily by how energy is distributed into the various comminution zone found in the ball charge profile. As mentioned, the form of the charge profile, and consequently the importance of each comminution zone, is directly dependant on the friction force existing between the ball charge and the mill wall.
Different liner types (Fig.3) affect this friction force between the ball charge and the mill wall and the mill grinding performance. For a given grinding context, it is possible to use a liner type that is considered optimal. However, with time, mill wear will modify the initial liner profile and subsequently mill grinding. Modelling the forces acting on the mill liners becomes the next step to predicting mill wear and its effect on grinding.
During mill operation, the hall charge exerts a force field composed of gravitational and centrifugal components on the mil liner (Fig.7) [23, 25].
Using this description, normal force component can be determined as show in Fig.8, giving:
(i) centrifugal normal component
(18)
(ii) gravitational normal component
(19)
Further, as the ball charge slip over the mill liner, a compression force is created with the local displacement of the mill charge by the liner (Fig.9). This force is defined as:
(20)
Where
The normal compression component as:
(21)
The total normal force acting on the liner surface becomes:
(22)
Liner wear, as a function of the position and intensity of the force field created by the ball charge as well as the abrasion factor 0, become:
(23)
Where
(24)
Note that slippage speed on the liner is defined previously by rearrangement of eqn (2).
After liner discretization into differences ，and time into , a simulation algorithm can be developed [23, 25] which allows liner profile wear simulation.
As an example, Fig. 10 illustrates a wave liner profile wear simulation which is comparable to the real liner profile wear presented in Fig.11.
MILL WEAR AND GRINDING PERFORMANCE
Even though industrial studies are needed to further validate these wear models, it is possible envisage the prediction of wear evolution of a given liner type. This, of course, wou1d allow the determination of how wear affects grinding performance here defined as variations in output granulometry. For the Hardinge case of Fig.1, this translates into simulating the effect of wear on the bevel liner as shown in Fig.12.
Further, simulating how the energy rate profile of Fig.6 changes with this liner wear, it is possible to predict the changes in mill output granulometry for the same input granulometry. Table 1 shows how, using a breakage model developed in [22,23], it is possible to illustrate output variation over the life period of the liner. Here, mill output becomes finer with liner wear.
Associated with this phenomenon, mill energy consumption decreases as shown in Fig.13. Both these phenomena illustrate the possibility of optimizing ball mill performance as a function of the predetermined effect of wear.
Table 1. Ball mill output granulometries as a function of worn liner profile
Particle size
(μm)
Initial %
passing
1/2 life %
passing
Final %
passing
74
100
150
300
830
1170
1650
58.08
68.25
78.54
92.12
99.45
99.96
100.00
58.25
68.44
78.68
92.15
99.45
99.96
100.00
60.13
70.38
80.25
93.96
99.61
99.97
100.00
DISCUSSION
Before concluding this work a few remarks should be made concerning charge motion, liner wear and associated mill output product.
As shown, ba11 charge motion is dependant on a number of physical and operating factors; it is also dependant on the rheological characteristics of a given slurry. These rheological characteristics are of course a function of percentage solids as well as ore properties. In the model of charge motion the effect of these factors is included in the relationship (2) for slippage speed as escribed using static and kineticd friction factors. A variation in the friction
factors, as caused by a possible change in rheological characteristics of a given slurry, can increase or decrease the amount of slippage between ball layers and thus increase or decrease liner wear.
In modelling ball mill wear with only two wear mechanisms, it is assumed that a mill is not run empty or that operating conditions do not send mill balls crashing directly into the mill liner. Under such conditions, liner wear increases considerably with added wear mechanisms (surface fatigue, fracture, cratering) gaining importance.
The importance of ball mill product variations as a function of liner wear is dependant on the breakage characteristics of a particular ore. As such, validation using industrial data is necessary in order to determine the precision of this model. However, assuming that an adequate validation of the whole model is possible, the preliminary results presented here show that it becomes possible to consider using this model for ball mill optimization as a function of energy consumption and mill wear.
CONCLUSIONS
This work has presented a predictive wear model based on a theoretical description of the ball mill. While covering the application of a simple adhesive and abrasive wear model to a complex ball charge model, it was possible not only to identify the required wear parameters, but also to predict liner wear and grinding performance.
However, further work is required particularly in model validation and parameter identification. Notwithstanding this need, ball mill optimization, in design and operation, becomes a future possibility.
Acknowledgement--The publication of this paper has been made possible by a Natural and Engineering Research Council of Canada research grant.