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翻譯部分
1.1 中文譯文
輥式破碎機(jī)的運(yùn)轉(zhuǎn)模型
摘要
在這一篇論文中我們?yōu)檩伿狡扑闄C(jī)發(fā)展一個(gè)運(yùn)轉(zhuǎn)模型。產(chǎn)品尺寸分布被認(rèn)為是破碎機(jī)的轉(zhuǎn)子半徑、角速度、補(bǔ)給速度和補(bǔ)給尺寸分布聯(lián)合作用的結(jié)果。模型建立在含密級(jí)和破裂矩陣標(biāo)準(zhǔn)矩陣公式的基礎(chǔ)上。這個(gè)模型能被應(yīng)用到槌式和垂直軸式輥式破碎機(jī)，幫助估算每單位質(zhì)量需要的沖擊能。
在這里我們建議把破碎機(jī)的分類和破碎作用納入破碎特性的動(dòng)力學(xué)考慮范圍。密級(jí)功能有累積的Weibull分布的形狀并且合并一個(gè)取決于沖擊能和補(bǔ)給速度的多數(shù)最小的顆粒會(huì)破尺寸。破裂功能被做成二個(gè)Broadbent—Callcott分布合成的模型。它假定對(duì)產(chǎn)品的粒度破面的比例比速度起決定作用的沖擊能和補(bǔ)給速度通過(guò)被提議的表達(dá)式。模型的預(yù)測(cè)結(jié)果被和一個(gè)試驗(yàn)工場(chǎng)的槌式破碎機(jī)破碎石灰?guī)r的實(shí)驗(yàn)的數(shù)據(jù)相較。轉(zhuǎn)子速度和補(bǔ)給速度的改變引發(fā)產(chǎn)品尺寸分布的變化被研究。
1. 介紹
時(shí)下,輥式破碎機(jī)被廣泛地用于粉碎操作，因?yàn)樗麄兊母叱叽缱冃伪?，產(chǎn)品的容易修正和相對(duì)簡(jiǎn)單的設(shè)計(jì)。另一方面，作為一種可靠,節(jié)約時(shí)間和節(jié)省費(fèi)用的方法，預(yù)測(cè)礦物處理廠會(huì)通過(guò)越來(lái)越多的做模型和模擬來(lái)發(fā)展，分析和優(yōu)化起決定作用的電路。在這篇論文中,輥式破碎機(jī)的有關(guān)數(shù)學(xué)上的模型的實(shí)用性是對(duì)于一個(gè)這樣的工廠的成功模擬是非常重要的。
盡管它重要，然而,這個(gè)輥式破碎機(jī)粉碎行為的模型只受到文獻(xiàn)的小關(guān)注。最近已經(jīng)有一些人嘗試為這發(fā)展運(yùn)轉(zhuǎn)破碎機(jī)這種類型的模型,舉例來(lái)說(shuō)Csoke和Racz(1998)、Attou et al(1999),但是然而, 一些大量的重要工作還需要去做。除此之外，那為礦石的模擬處理的可得的商業(yè)代碼對(duì)于輥式破碎機(jī)仍然設(shè)置缺乏特性模型,它們明顯地減少了應(yīng)用字段。
在這一個(gè)工作中我們發(fā)展一個(gè)能被應(yīng)用到所有類型的輥式破碎機(jī)的運(yùn)轉(zhuǎn)模型。我們的目標(biāo)是通過(guò)提供了破碎機(jī)的轉(zhuǎn)子速度和半徑和補(bǔ)給速度和尺寸分布，在工作之前被預(yù)測(cè)產(chǎn)品尺寸分布。通過(guò)一些合理數(shù)目的可調(diào)整的參數(shù)考慮特定的礦石道具和破碎機(jī)的設(shè)計(jì)。
現(xiàn)在,Whiten和White(1979)發(fā)展的錐形和顎式破碎機(jī)的標(biāo)準(zhǔn)模型被認(rèn)為是出發(fā)點(diǎn)。因?yàn)闆_擊破裂的特殊性,在它的最初形態(tài)中，這一個(gè)模型不能夠作為輥式破碎機(jī)的模型。然而，在錐形和顎式破碎機(jī)的一般破裂加工方案(見(jiàn)到圖.1)在我們的情況中仍然可適用,密級(jí)和那破裂描述碎裂的功能過(guò)程從統(tǒng)計(jì)觀點(diǎn)應(yīng)該被再考慮。
錐形和顎式破碎機(jī)的碎裂過(guò)程相對(duì)地慢，破碎過(guò)程建立在壓縮應(yīng)力作用于顆粒一部分的表面的基礎(chǔ)上。二者擇一地，沖擊破裂發(fā)生在比較短時(shí)間暗示著一個(gè)動(dòng)態(tài)的裂痕擴(kuò)散導(dǎo)致非?？斓乃俣绕茐念w粒。依照奧斯汀(1984)的理論，沖擊發(fā)生時(shí)有壓縮力，而且拉的陡震波穿透顆粒。由于這個(gè)重要的陡震波的出現(xiàn),快速成長(zhǎng)的拉應(yīng)力幫助顆粒從里面破斷。除此之外，顆粒破裂理論被Oka和Majima(1970)提出，因?yàn)榇簏c(diǎn)的顆粒含比小的顆粒含有更多的微裂紋，所以它們應(yīng)該更容易破斷。為了要解釋動(dòng)力學(xué)沖擊破裂的特性,我們用一個(gè)決定于沖擊能的累積的Weibull分布代替標(biāo)準(zhǔn)的密級(jí)為。因此, 為輥式破碎機(jī)的運(yùn)轉(zhuǎn)的重要的參數(shù)如轉(zhuǎn)子半徑和速度和補(bǔ)給速度自然地在我們基于簡(jiǎn)單顆粒動(dòng)力學(xué)考慮的模型的基礎(chǔ)上組合成一體。
接下來(lái), Whiten和White(1979)發(fā)展的破碎機(jī)的破裂功能被二個(gè)Broadbent—Callcott分布合成一起代替，分別表現(xiàn)產(chǎn)品的粒度和產(chǎn)品的粗破面。細(xì)顆粒部分在產(chǎn)品中比例被假定隨著增加轉(zhuǎn)子速度增加和減補(bǔ)給速度而增加，這是符合實(shí)驗(yàn)的觀察的。模型的預(yù)測(cè)結(jié)果被和一個(gè)試驗(yàn)工場(chǎng)的槌式破碎機(jī)破碎石灰?guī)r的實(shí)驗(yàn)的數(shù)據(jù)相較。 在文章中，矢量(f)和矩陣(C)被在符號(hào)下面劃線指示。
2.模型發(fā)展
2.1. 質(zhì)量平衡
Whiten(1972)為發(fā)展了一個(gè)示意性的圖表模型來(lái)表示錐形和顎式破碎機(jī)的尺寸分布，稍后被Whiten和White(1979) 改良，模型在圖.1中被顯示。，顆粒被他們的尺寸分布，分別被矢量f(補(bǔ)給)和p(產(chǎn)品)表現(xiàn)的不連續(xù)形狀，表示的特色。密級(jí)算子C(一個(gè)對(duì)角線矩陣)為每個(gè)不同尺寸的顆粒計(jì)算破裂的可能性。破裂算子B(一個(gè)比較低次的三角形矩陣)在初步定義的尺寸等級(jí)管理破碎的顆粒的重新分配。
補(bǔ)給顆粒為被C選擇進(jìn)行破裂。沒(méi)有破裂的不改變形狀在產(chǎn)品中通過(guò)。破裂的顆粒碎片被B重新分配而且連同新的補(bǔ)給材料一起進(jìn)行碎裂。根據(jù)Whiten(1972)的理論,產(chǎn)品尺寸分布p能被公式表示為下面的樣子:
p=(I-C)·（I-B·C）-1·f (1)
I是特性矩陣
最近，Csoke和 Racz(1998)發(fā)展了一個(gè)槌式破碎機(jī)的破碎模型，建立在假設(shè)當(dāng)單一破裂在以槌桿擠入之后，槌式破碎機(jī)做模型受制于顆粒補(bǔ)給速度一個(gè)參數(shù)的基礎(chǔ)上。這造成下列的質(zhì)量平衡方程式:
p=B·C·f+(I-C)·f (2)
上述的方程式符合密級(jí)和破裂矩陣被沒(méi)有反饋的連接方案，在圖.1中被顯示。結(jié)論，方程式(2)沒(méi)有包含方程式(1)中的逆矩陣。Attou et al(1999)發(fā)展了Csoke 和 Racz 的方式而且考慮二種不同的破裂過(guò)程在槌式破碎機(jī)中處理。在他的模型里,由于轉(zhuǎn)子的槌桿顆粒在沖擊之前破斷或在槌桿和破碎機(jī)的內(nèi)壁沖擊之后。這個(gè)模型的質(zhì)量平衡方程式(2)的延伸，做不到從現(xiàn)在顆粒的破裂出發(fā)，進(jìn)一步進(jìn)行合并碎裂的可能性。我們用兩者的質(zhì)量運(yùn)行數(shù)字的模擬平衡定律 (1) 和 (2)進(jìn)行比較，發(fā)現(xiàn)方程式 (2) 是和破裂的清晰度和密級(jí)矩陣矛盾的。實(shí)際上，它預(yù)測(cè)那產(chǎn)品含有一個(gè)無(wú)補(bǔ)給顆粒的可以忽略部分，破碎機(jī)有100%的破裂速度,實(shí)際上物理上是不可能的，因?yàn)楸惶岢龅姆匠淌經(jīng)]有考慮顆粒聚結(jié)。因?yàn)檫@一個(gè)原因，我們使用標(biāo)準(zhǔn)的質(zhì)量平衡定律(1),意味著從破裂發(fā)行母顆粒破碎出來(lái)的碎片能進(jìn)行進(jìn)一步的碎裂。我們相信假定合并在方程式(1)中適用于在沖擊的情況，破碎機(jī)大部分的補(bǔ)給顆粒的確受制于超過(guò)一個(gè)碎裂，由于顆粒–鑄壁和顆粒–顆粒碰撞。
2.2單位質(zhì)量沖擊能
考慮在一個(gè)單一顆粒和一個(gè)槌式破碎機(jī)的破碎桿之間的沖擊。給予轉(zhuǎn)子的質(zhì)量比補(bǔ)給一個(gè)單一顆粒質(zhì)量重一些，破碎桿的線速度比微粒的速度重要些，單一顆粒動(dòng)能與轉(zhuǎn)子的動(dòng)能相比是可以忽略的?？紤]顆粒的動(dòng)量在沖擊前后的線性保護(hù)系統(tǒng)， Attou et al (1999)得出下面的單位質(zhì)量沖擊能的公式：
E=0.5·（R+0.5·Hb）2·ω2 (3)
式中R（m）是轉(zhuǎn)子半徑，Hb（m）是重要齒根破碎面的高度，ω（s-1）是轉(zhuǎn)動(dòng)角速度。
在垂直軸式破碎機(jī)中，顆粒被輸送到一個(gè)放射狀的水平轉(zhuǎn)動(dòng)的轉(zhuǎn)臺(tái) (轉(zhuǎn)子)而且通過(guò)離心力向破碎機(jī)的周壁拋射。 不像在槌式破碎機(jī)中,這里大部份的碎裂發(fā)生在破碎機(jī)的墻壁而不是在轉(zhuǎn)子的外圍。假定顆粒在飛向破碎機(jī)周壁的時(shí)候能量沒(méi)有改變，也就是說(shuō)忽略能顆粒–顆粒的作用。被疏忽在第一個(gè)逼近， Nikolov 和 Lucion (2002)為單位質(zhì)量沖擊能導(dǎo)出了下面的公式：
E=Rv2·ω2 （4）
式中Rv（m）和ω（s-1）代表轉(zhuǎn)子半徑和角速度，符號(hào)Rv被用于在槌式和垂直軸式破碎機(jī)在公式（3）和（4）中各自的區(qū)別。
有趣的是，我們注意到相同的轉(zhuǎn)子半徑下，槌式破碎機(jī)的單位質(zhì)量沖擊能比垂直軸式破碎機(jī)要低一些。這可以說(shuō)明垂直軸式破碎機(jī)比槌式破碎機(jī)產(chǎn)生更多的微粒和在微粒必須減小尺寸（沖擊能在這些機(jī)器中能達(dá)到更高的水平）的情況下工作更好的事實(shí)。
3. 結(jié)果
模型在早先的斷面中發(fā)展已經(jīng)在一個(gè)內(nèi)部的互傳式譯碼中實(shí)現(xiàn)。這個(gè)模型已經(jīng)與實(shí)驗(yàn)一起有效在一個(gè)槌式破碎機(jī)上運(yùn)行，它的轉(zhuǎn)子直徑和寬度分別為0.65m和0.45m。 轉(zhuǎn)子半徑是R=0.325 m;那轉(zhuǎn)子沖擊桿高度是Hb=0.1 m。用的材料是來(lái)自比利時(shí)的 Tournai 區(qū)域的石灰?guī)r,。 補(bǔ)給已經(jīng)由屏幕校正，使用材料的尺寸從 14 到 20 毫米不等。 最大的顆粒補(bǔ)給的尺寸是直徑為dmax=26 mm。 參考補(bǔ)給速度 Q0 和參考沖擊能 E0 是輪流分別地是 2t/h和 300 J/ kg。 其余方程式(8)的參數(shù)。依下列各項(xiàng)被取值:c0=1.4, c1=0.12,n=0.35。破裂動(dòng)作的參數(shù)( 方程式(11)和(12))，m，l和 c2 被分別地設(shè)定為0.74, 2.6 和 0.55。參數(shù)的值必需計(jì)算密級(jí)的形狀參數(shù)功能被調(diào)整到k0=1.35, k1=0.1。
我們已經(jīng)完成了以不同的補(bǔ)給速度補(bǔ)給的二個(gè)組進(jìn)行模擬實(shí)驗(yàn)的數(shù)據(jù),即 Q=2 和 7t/h。以固定的補(bǔ)給速度，產(chǎn)品獲得以三個(gè)不同的轉(zhuǎn)子速度(ω=540；720；900轉(zhuǎn)/分)已經(jīng)被分析。重要的是注意除了轉(zhuǎn)子速度和補(bǔ)給速度以外, 所有的其他模型參數(shù)已經(jīng)被保持不變地進(jìn)行完全部運(yùn)行模擬。
在2t/h的速度的實(shí)驗(yàn)下，以不同的轉(zhuǎn)子速度獲得的產(chǎn)品的模擬尺寸分布在圖.2 中被比較。在7t/h的速度的實(shí)驗(yàn)中對(duì)應(yīng)的產(chǎn)品的模擬尺寸分布在圖.3中被描述。
可以看到，模型能夠用一個(gè)合理的精度預(yù)測(cè)產(chǎn)品尺寸分布，即使當(dāng)重要的變化在轉(zhuǎn)子速度和那補(bǔ)給速度兩者之間被強(qiáng)加。以固定的補(bǔ)給速度,較高的轉(zhuǎn)子速度生產(chǎn)一個(gè)較好的產(chǎn)品尺寸分布。另外,補(bǔ)給速度的增加造成在固定的轉(zhuǎn)子速度下的產(chǎn)生較粗的產(chǎn)品。結(jié)果證實(shí)產(chǎn)品從打破得支離破碎的沖擊下獲得的尺寸分布比以錐形或顎式破碎機(jī)破碎獲得更寬廣。
轉(zhuǎn)子速度和補(bǔ)給速度的在顆粒遭受破裂的最小的尺寸dmin的影響力在圖 4 中被顯示。可以看到 dmin 強(qiáng)烈地依賴轉(zhuǎn)子速度和補(bǔ)給速度，在給定的操作條件下，它的范圍從3.8 到 7.8 毫米不等。
關(guān)于轉(zhuǎn)子速度和補(bǔ)給速度的粒度破面的比例的進(jìn)化在產(chǎn)品中被顯示在圖.5中.可以看出，隨著轉(zhuǎn)子速度的增大和補(bǔ)給速度的減小，它是增加的，在給定的操作條件下它的變化范圍是0.35 到 0.59。
通過(guò)比較,當(dāng)Whiten和White的模型被用來(lái)進(jìn)行短冒口錐形破碎機(jī)的仿真的話，參數(shù)φ、m和l是常常分別地固定在0.2，0.5和2.5。 一個(gè)比較有價(jià)值的發(fā)現(xiàn)是，通過(guò)我們的實(shí)驗(yàn)，我們得出來(lái)一個(gè)大家都知道的事實(shí)，就是輥式破碎機(jī)粒度φ比錐式和顎式破碎機(jī)有更高的粒度。
4. 結(jié)論
在結(jié)論中，我們已經(jīng)發(fā)展了一個(gè)輥式破碎機(jī)的運(yùn)轉(zhuǎn)模型，它可以在不變的操作條件和合理的一組參數(shù)下，預(yù)測(cè)產(chǎn)品的尺寸分布。。輥式破碎機(jī)的特殊特性是模型的兩個(gè)參數(shù)，密級(jí)和破裂功能，兩者都決定于轉(zhuǎn)子半徑、角速度和補(bǔ)給速度上。介紹破碎顆粒的最小可能尺寸和小粒度在產(chǎn)品中的比例對(duì)于一個(gè)成功的輥式破碎機(jī)模型是非常重要的。模擬結(jié)果很好的符合了實(shí)驗(yàn)，并且指出在補(bǔ)給速度和尺寸被保持不變的情況下，較高的轉(zhuǎn)子速度可以使產(chǎn)品尺寸分布變的較好。另外，轉(zhuǎn)子速度一定的情況下，比較高的補(bǔ)給速度容易產(chǎn)生較粗的產(chǎn)品。模型能容易地被實(shí)現(xiàn)在現(xiàn)有的礦物模擬處理的商業(yè)的代碼上。它可以應(yīng)用于預(yù)測(cè)運(yùn)轉(zhuǎn)速度不變的槌式或垂直軸式破碎機(jī)在復(fù)雜的程序表中整合。進(jìn)一步的工作是要求模型適應(yīng)不穩(wěn)定和瞬態(tài)操作。
1.2 英文原文
A performance model for impact crushers
S. Nikolov *
Centre Terre et Pierre, 55 Ch. d’Antoing, B-7500 Tournai, Belgium
Received 3 May 2002; accepted 17August 2002
Abstract
In this paper we develop a performance model for impact crushers. The product size distribution is obtained as a function of the Crusher’s rotor radius and angular velocity, the feed rate and the feed size distribution. The model is based on the standard matrix formulation that includes classification and breakage matrices. It can be applied to both hammer and vertical-axis impact crushers with the help of the corresponding estimations for the impact energy per unit mass.
Here we propose classification and breakage functions for impact crushers taking into account the dynamic character of the impact breakage. The classification function has the form of a cumulative Weibull distribution and incorporates a minimum breakable size of the particles depending on the impact energy and the feed rate. The breakage function is modelled as the sum of two Broadbent–Callcott distributions. It is assumed to depend on the impact energy and the feed rate through the proposed expression for the proportion of the fine fraction in the product.
The model predictions are compared with experimental data for limestone treated in a pilot-plant hammer crusher. The variations of the product size distribution resulting from changes in the rotor velocity and the feed rate are investigated.
Θ 2002 Elsevier Science Ltd. All rights reserved.
1.Introduction
Nowadays, impact crushers are widely used for comminution operations because of their high sizereduction ratio, easy modification of the product and a
relatively simple design. On the other hand, the prediction of the behaviour of mineral processing plants through modelling and simulations is the more and more employed as a reliable, time- and cost-saving approach for development, analysis and optimisation of crushing circuits. In this context, the availability of relevant mathematical models for impact crushers is important for a successful simulation of such plants.
Despite its importance, however, the modelling of the comminution behaviour of impact crushers received little attention in the literature. There have been some recent attempts to develop performance models for this type of crushers, for example by Csoke and Racz (1998) and Attou et al. (1999), but nevertheless, a significant amount of work remains to be done. In addition, the available commercial codes for simulation of ore processing plants still lack specific models for impact crushers, which obviously reduces their field of application.
In this work we develop a performance model that can be applied to all types of impact crushers. Our goal is to predict the product size distribution, provided that the crushers rotor velocity and radius as well as the feed rate and size distribution are known before hand. The specific ore properties and the crusher’s design are taken into account through a reasonable number of adjustable parameters.
Here, the standard model for cone and jaw crushers developed by Whiten and White (1979) is taken as a starting point. Because of the specificity of the impact breakage, this model cannot be used for impact crushers in its original form. While the general scheme of the breakage process in cone and jaw crushers (see Fig. 1) is still applicable in our case, the classification and the breakage functions that describe the fragmentation process from statistical point of view should be reconsidered.
The fragmentation process in cone and jaw crushers is relatively slow and is based on the application of a compression stress on a part of the particles surface.
Alternatively, the impact breakage takes place at a much shorter time scale and implies a dynamic crack propagation that leads to a much faster failure of the particles. According to Austin (1984), the impact generates compressive and tensile shock waves travelling throughout the particle. The presence of a significant, rapidly mgrowing tensile stress helps the particles to break from within. In addition, the particle breakage theory proposed by Oka and Majima (1970) states that larger particles should break more easily because they contain larger micro-cracks compared with the smaller ones.
In order to account for the dynamic character of the impact breakage, we replace the standard classification function for crushers with a cumulative Weibull distribution depending on the impact energy. Thus, important parameters for the performance of impact crushers such as the rotor radius and velocity as well as the feed rate are naturally incorporated in our model on the basis of simple particle dynamics considerations.
Next, the breakage function for crushers proposed by Whiten and White (1979) is replaced with the sum of two Broadbent–Callcott distributions representing the fine and the coarse fractions in the product. The proportion of the fine fraction in the product is assumed to increase with increasing the rotor velocity and to decrease with increasing the feed rate, which is in accord with the experimental observations.
The model predictions are compared with experimental data for limestone treated in a pilot-plant hammer crusher. Throughout the text, vectors (f) and matrices (C) are denoted by underlined symbols.
2. Model development
2.1. Mass balance
A schematic representation of the size-distribution model developed for cone and jaw crushers by Whiten (1972) and later improved by Whiten and White (1979) is shown in Fig. 1. The particles are characterized by their size distribution, which is represented in a discrete form by the vectors f (feed) and p (product) respectively.
The classification operator C (a diagonal matrix) computes the probability of breakage for each particle size. The breakage operator B (a lower triangular matrix) governs the redistribution of the broken particles in the preliminary defined size classes.
The feed particles are selected for breakage by C. Those that do not break pass unchanged in the product. The debris of the broken particles are redistributed by means of B and are eventually subjected to further fragmentation together with the new feed material. According to Whiten (1972), the product size distribution p can be expressed as follows:
(1)
where I is the identity matrix and denotes the inverse of a square matrix.
More recently, Csoke and Racz (1998) developed a model for hammer crushers with the basic assumption that the feed particles are subjected to a single breakage after impact with the hammer bars. This results in the following mass balance equation:
(2)
The above equation corresponds to a scheme where the classification and the breakage matrices are connected in series without the feedback shown in Fig. 1. Consequently, Eq. (2) does not contain the inverse matrix appearing in Eq. (1). Attou et al. (1999) extended the approach of Csoke and Racz and considered two different breakage processes in hammer crushers. In his model, the particles can break either after impact with the hammer bars of the rotor or after collision with the internal walls of the crushers chamber. The mass balance of this model is an extension of Eq. (2) and does not incorporate the possibility for further fragmentation of the debris issued from breakage of parent particles.
We performed numerical simulations with both mass balance laws (1) and (2) and found that Eq. (2) is incompatible with the definition of the breakage and the
classification matrices. Actually, it predicts that the product contains a non-negligible fraction of feed particles having a probability of breakage of 100%, which is physically impossible because the proposed equations do not account for particles agglomeration.
For this reason, we use the standard mass balance law (1), which implies that the debris issued from breakage of parent particles can be subjected to further fragmentation. We believe that the assumptions incorporated in Eq. (1) are applicable in the case of impact crushers with the argument that most of the feed particles are indeed subjected to more than one fragmentation due to the particle–wall and the particle–particle collisions.
2.2. Impact energy per unit mass
Consider the impact between a single particle and a crushing bar attached to the rotor of a hammer crusher. Given that the rotor mass is much greater than the mass of a single particle in the feed, and that before impact the linear velocity of the crushing bar is much more important than the particle velocity, the kinetic energy associated with a single particle is negligible compared with that of the rotor. Considering the conservation of linear momentum of the system particle-crushing bar before and after impact, Attou et al. (1999) derived the following expression for the impact energy per unit mass:
(3)
where R (m) is the rotor radius, Hb (m) is the height of the impact surface of the crushing bars and is the rotor angular velocity.
In vertical-axis crushers, the particles are fed to a horizontal turning table (rotor) with radially oriented guides and are projected towards the crushers walls by the centrifugal forces. Unlike in hammer crushers, here most of the fragmentation takes place at the crusher’s walls rather than at the rotors periphery. With the assumption that the particle energy does not change during its flight from the rotor periphery to the crushing walls, i.e., the particle–particle interactions are neglected in a first approximation, Nikolov and Lucion (2002) derived the following expression for the impact energy per unit mass:
(4)
where Rv (m) and are the rotor radius and angular velocity respectively. The notation Rv is used to distinguish between the impact energy for hammer and vertical-axis crushers given with Eqs. (3) and (4) respectively.
It is interesting to note that for the same rotor radius, the impact energy per unit mass provided by hammer crushers is lower than that provided by verticalaxis crushers. This could explain the fact that verticalaxis crushers produce more fines and perform better when finer granulate must be reduced in size, which is most probably due to the higher level of impact energy reached in these machines.
3. Results
The model developed in the previous section has been implemented in an in-house FORTRAN code. It has been validated with experiments performed on a hammer crusher with rotor diameter and width of 0.65 and0.45 m respectively. The rotor radius is R=0.325m; the height of the rotor’s impact bars is Hb=0.1m The material used is limestone from the region of Tournai, Belgium. The feed has been calibrated by screening and its size ranges from 14 to 20 mm. The maximum particle dimension in the feed is dmax=26mm The reference feed rate Q0 and the reference impact energy E0 are taken to be 2 t/h and 300 J/kg respectively. The rest of the parameters in Eq. (8) are identified as follows: c0=1.4,c1=0.12 and n=0.35. The parameters of the breakage function (Eqs. (11) and (12)) m, l and c2 are set to 0.74, 2.6 and 0.55 respectively. The values of the parameters necessary to compute the shape of the classification function are fixed to k0=1.35 and k1=0.1.
We have performed simulations for two sets of experimental data taken at different feed rates, namely Q=2 and 7t/h. At fixed feed rate, the products obtained with three different rotor velocities (=540; 720; 900 rpm) have been analysed. It is important to note that except for the rotor velocity and the feed rate, all other model parameters have been kept unchanged for all performed simulations.
The experimental and simulated size distributions of the products obtained with different rotor velocities at feed rate of 2 t/h are compared in Fig. 2. The corresponding size distributions obtained at feed rate of 7t/h are depicted in Fig3.
It is seen that the model is able to predict the product size distribution with a reasonable accuracy even when important variations in both the rotor velocity and the feed rate are imposed. At fixed feed rate, higher rotor velocity produces a finer product size distribution. Alternatively, an increase in the feed rate results in a coarser product at fixed rotor velocity. The results also confirm that the product size distribution issued from impact crushing is broader than that obtained with cone or jaw crushers. The influences of the rotor velocity and the feed rate on the minimum size of the particles that undergo breakage dmin are shown in Fig. 4. It is seen that dmin strongly depends on both the rotor velocity and the feed rate and ranges from 3.8 to 7.8 mm for the given operating conditions.
The evolution of the proportion of the fine fraction in the product with rotor velocity and feed rate is shown in Fig. 5. It increases with increasing the rotor velocity and decreases with increasing the feed rate as expected and ranges from 0.35 to 0.59 for the given operating conditions.
For comparison, when the model of Whiten and White (1979) is used for simulation of the behaviour of short-head cone crushers, the values for , m and l are often fixed to 0.2, 0.5 and 2.5 respectively. A greater value for the fine fraction in our case reflects the wellknown fact that the product issued from impact crushing contains more fines than that obtained with cone or jaw crushers.
4. Conclusions
In conclusion, we have developed a performance model for impact crushers that is able to predict the product size distribution at steady-state operating conditions and contains a reasonable number of parameters. The specific behaviour of impact crushers is modelled through classification and breakage functions that both depend on the rotor radius and angular velocity as well as on the feed rate. The introduction of variable minimum size of the breakable particles and proportion of fine fraction in the product seems to be very important for successful modelling of the impact crushing.
The simulation results are in a good agreement with the experiment and show that at higher rotor velocity the product size distribution becomes finer, provided that the feed rate and size are kept unchanged. Alternatively, higher feed rate at constant rotor velocity results in a coarser product.
The model can be easily implemented in the existing commercial codes for mineral processing simulations. It can be used for prediction of the steady-state performance of hammer and/or vertical-axis impact crushers integrated in complex flowsheets. Further work is required to adapt the model for unsteady and transient operating regimes.
Acknowledgements
The Belgian Walloon Region government and the European Community have jointly funded this research as an Objective 1 European project. The author thanks Dr. A. Attou for his contributions at the early stage of the project, Mr. Chr. Lucion for the useful discussions as well as Mr. R. Lemaire for providing the experimental results.
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