滾筒混合機(jī)混合單元的設(shè)計(jì)-自落式混凝土攪拌混合機(jī)的設(shè)計(jì)研究含開(kāi)題及8張CAD圖

滾筒混合機(jī)混合單元的設(shè)計(jì)-自落式混凝土攪拌混合機(jī)的設(shè)計(jì)研究含開(kāi)題及8張CAD圖,滾筒,混合,單元,設(shè)計(jì),混凝土,攪拌,研究,鉆研,開(kāi)題,cad 基于斯托克斯的三維流動(dòng)混合：再次討論分區(qū)管混合器問(wèn)題 摘要——對(duì)速度場(chǎng)和所謂的分區(qū)管道混合器混合反應(yīng)進(jìn)行了研究。和以前使用的近似方案相比，一個(gè)從以前研究的具有相同物理模型入手的帶來(lái)更準(zhǔn)確的流量描述的精確分析方案正在發(fā)展中。另外，這些結(jié)果是根據(jù)更好的報(bào)道實(shí)驗(yàn)數(shù)據(jù)得到的。 斯托克斯流動(dòng)/層流分布混合/靜態(tài)混合器 朗讀 顯示對(duì)應(yīng)的拉丁字符的拼音 1.介紹 文章的目的是研究一個(gè)內(nèi)部無(wú)限長(zhǎng)，被內(nèi)壁劃分成一個(gè)順序排列的半圓形管道的圓管的三維蠕動(dòng)流。這樣一個(gè)系統(tǒng)，稱(chēng)為'分區(qū)管混頻器（PPM），是由Khakhar等人引進(jìn)的。 [1]作為樣機(jī)模型廣泛使用在Kenics靜態(tài)混合器。 在Kenics混合器中每個(gè)元素是一個(gè)螺旋，扭曲180?的金屬板;元素排列在圓管的軸向上，使元素的領(lǐng)先優(yōu)勢(shì)相對(duì)前一個(gè)是沿直角的。流體動(dòng)力學(xué)計(jì)算工具使這種三維流動(dòng)數(shù)值模擬簡(jiǎn)單可行（Avalosse，Crochet[3]，Hobbs和Muzzio[4]，Hobbs等人）。然而，這樣做需要大量的模擬計(jì)算資源，尤其是在研究不同的攪拌工藝參數(shù)的影響。因此，簡(jiǎn)化分析模型，即給出了模擬的過(guò)程快的可能性（或模仿其功能密切就夠了），也仍然是有用的。. 這種本質(zhì)上的三維流PPM模式是高度理想化，但保留了正在研究流動(dòng)的主要特征。在每一個(gè)半圓形軸向風(fēng)道，該模型包括兩個(gè)疊加，獨(dú)立，二維流場(chǎng)：一橫截面（旋轉(zhuǎn)）的速度場(chǎng)和一個(gè)全面發(fā)展的Poiseuille。在這里給了兩個(gè)獨(dú)立的二維邊界問(wèn)題，而不是三維問(wèn)題的。由Khakhar等人提出的解決方案。 [1]的橫截面速度場(chǎng)只是一個(gè)近似的。然而在一個(gè)封閉的形式下，存在一個(gè)'精確'分析解決方案。 在本論文中，我們利用這些精確的解決方案對(duì)這個(gè)三維混合機(jī)的混合性能就行審查。在一些混合模式下的重要區(qū)別已經(jīng)得到了，而且我們的結(jié)果更接近可用的Kusch和Ottino[6]的 實(shí)驗(yàn)結(jié)果。 2.PPM的速度領(lǐng)域 考慮一個(gè)0≤ r ≤a, 0≤θ≤2π, |z| < ∞內(nèi)部的無(wú)限缸體，，這里面包含一個(gè) 剛性矩形板的長(zhǎng)度為L(zhǎng)的序列（見(jiàn)，例如,Ottino [7] 圖6.2）。鄰近板塊相互正交放置，即0 ≤r ≤a,，θ= 0，π，2kL ≤ z ≤（2k+ 1）L和0≤r≤a, θ =π/2,3π/2,(2k+1)L≤z≤(2k+2)L,其中k = 0，± 1，± 2，。 所有半圓形管道流動(dòng)是由一個(gè)恒定的壓力梯度?p /?z和圓柱墻? =一內(nèi)壁保持不變恒定速度V的勻速轉(zhuǎn)動(dòng) 感應(yīng)得到的。遵循Khakhar [1]，我們假設(shè)在每一個(gè)橫截面軸向速度是充分發(fā)展的（忽略?xún)蓚€(gè)板塊之間的過(guò)渡效果）并且橫截面速度vr和V，正如他們將一個(gè)無(wú)限長(zhǎng)的半圓形管道。在斯托克斯近似零組件虛擬現(xiàn)實(shí)穩(wěn)定的速度場(chǎng)vr , V，和vz，是從兩個(gè)解耦合的獨(dú)立的兩維問(wèn)題定義的 在每個(gè)半圓形每一個(gè)領(lǐng)域。在這里，△代表拉普拉斯算子站極坐標(biāo)，μ代表流體粘度，而ψ（r，θ）代表相關(guān)的截面流流功能 我們認(rèn)為在下面的'基本'范圍 0≤r≤a, 0≤ θ≤π, 0 ≤z≤ L;其他領(lǐng)域的解決方案，可簡(jiǎn)單的從這個(gè)基本之一獲得。就無(wú)滑移邊界條件而言，ψ和Vz是 并且 對(duì)于式（1）和（2），是各自（獨(dú)立）的。 雙諧波問(wèn)題（1），（4）存在一個(gè)確切的解析解： 可通過(guò)以下方式獲得。 讓我們介紹下雙極坐標(biāo)系（ξ，η），這樣的坐標(biāo)的兩極都在位于x軸的點(diǎn)（± a，0）上： 三維Stokes流動(dòng)混合 785 這樣 當(dāng)a/J =coshη-cosξ， 1 / J的數(shù)量在這個(gè)直角坐標(biāo)系是第一個(gè)不同的Lame參數(shù)。這個(gè)支持二維雙調(diào)和方程式的系統(tǒng)首次采用在Joukowski[8];看到Joukowski和Chaplygin[9] 適用于 這個(gè)問(wèn)題的偏心Stokes流動(dòng)的精確解。 由極坐標(biāo)中的0≤r ≤a, 0≤θ≤π的半圓改變?yōu)殡p極坐標(biāo)的-∞≤η≤∞,π/2 ≤ξ≤π。雙調(diào)和方程式（1），必須符合在雙極坐標(biāo)中的流函數(shù)ψ，可寫(xiě)為 對(duì)于輔助函數(shù) Ψ=ψ/J 通過(guò)等式的方法 （其中nξ是指直線(xiàn)ξ=常數(shù)的外部標(biāo)準(zhǔn)），我們可以根據(jù)Ψ用形式表示邊界條件（3）為 通過(guò)選擇等式9的解決方案 我們能滿(mǎn)足所有邊界條件（11），并且假設(shè)常量的A,B,C,D的值是 通過(guò)以下等式將（12）中的Ψ(ξ)代回到流函數(shù)ψ（r，θ） 經(jīng)過(guò)一番簡(jiǎn)化，我們歸結(jié)到表達(dá)式（6）。 流函數(shù)ψ逼近點(diǎn)（a，0）的特征可以從等式6擴(kuò)張為現(xiàn)有的x =a-ρsinχ, y =ρcosχ的極坐標(biāo)（）中的泰勒級(jí)數(shù) 中獲得ρ的第一象限長(zhǎng)度是 圖1。（a）的流線(xiàn)圖案（流函數(shù)輪廓圖）分析解決方案（6）并且把（b）叫做一個(gè)近似的解決方案（16）。輪廓線(xiàn)與在兩個(gè)平面圖中相同的階段是等距離的。在（b）中的虛線(xiàn)代表輪廓，而在（a）中是不存在的。 對(duì)于ρ>0，0≤χ≤π/2的水平平面（古德?tīng)朳10]，泰勒[11]），隨著不變的常量速度- V應(yīng)用在平面χ=0上，類(lèi)似于刮的方案 圖1（a）顯示了流函數(shù)（6）外形的水平高度。截面流量在一個(gè)橢圓形停滯點(diǎn)（0.636a，π/ 2）處呈現(xiàn)出單渦旋細(xì)胞。 以往的研究（Khakhar等。[1]，Ottino[7]）暗示了近似一個(gè)條件的關(guān)于邊界問(wèn)題（1），（4）的解決辦法： 已經(jīng)靠一個(gè)變分法獲得它。然而這個(gè)表達(dá)式（16）不能同時(shí)滿(mǎn)足支配性雙調(diào)和方程（1）在移動(dòng)邊界無(wú)滑移條件！原來(lái)，在邊界r=a處的切向速度變化為（4 / 3）Vsin2θ,而不是恒定V。因此，在一些地區(qū)遠(yuǎn)離平面邊界，速度被高估了（高達(dá)33％，在圓形邊界），它是人為地平滑接近近角。根據(jù)一個(gè)長(zhǎng)遠(yuǎn)的解決辦法（16），流函數(shù)輪廓的劃分圖呈現(xiàn)在圖1（b）中。 充分成熟的關(guān)于一個(gè)半圓導(dǎo)管的軸向流邊界問(wèn)題（2），（5）顯示（Ottino[7]） 其中 是平均軸向速度。使用簡(jiǎn)單的轉(zhuǎn)換和無(wú)窮總結(jié)表（Prudnikov等人 [12]），我們可以提出在一個(gè)封閉形式下的表達(dá)（17）： 這對(duì)于在平流過(guò)程的數(shù)值模擬可取。值得一提的是，被Khakhar等人使用的條款[1]的第一個(gè)（17）的三極限無(wú)窮求和以及Ottino[7]提供的最大只有幾個(gè)百分點(diǎn)的精度的誤差（相對(duì)于'精確'表達(dá)式（18））。 圖2。等高線(xiàn)圖的軸向速度的Vz：實(shí)線(xiàn)對(duì)應(yīng)的確切表達(dá)式（18），虛線(xiàn)對(duì)應(yīng)于表達(dá)式（17）三個(gè)方面的近似值 在圖2中由（18）定義的Vz的輪廓線(xiàn)，顯示為實(shí)線(xiàn)，而三極限（17）近似相同的輪廓由虛線(xiàn)繪制。盡管這種近似的輪廓形狀很相似，但是平均流速vz差異量高達(dá)7%，其中一個(gè)最大速度vz離角點(diǎn)不遠(yuǎn)，所以vz是被低估了?？梢栽黾樱?7）的項(xiàng)數(shù)至一百，降低相對(duì)誤差到小于0.005％，但是，像這樣模擬被動(dòng)追蹤物的水平流動(dòng)會(huì)花掉更多的計(jì)算機(jī)時(shí)間。 3。PPM混沌混合 由平流方程描述的一個(gè)被動(dòng)的個(gè)體（拉格朗日）粒子的運(yùn)動(dòng)。 與右手邊由速度場(chǎng)（6）和（18）定義的（19）。當(dāng)t=0時(shí)初始條件為 r= R0的，θ=θ0和z=0。 這里定義變量φ顯然是 其中k=0，±1，±2，。 。 。 。 ? 系統(tǒng)（19）描述了沿每個(gè)流線(xiàn)區(qū)間的單個(gè)粒子穩(wěn)定的動(dòng)作。然而，由于流動(dòng)是三維和空間周期性的，它可以表現(xiàn)為混亂行為（Aref【5.4部分的13】）。在Khakhar[1]以及其他人中，單個(gè)無(wú)量綱參數(shù)，混合強(qiáng)度， 混合強(qiáng)度被提出給了完整描述這樣一個(gè)表現(xiàn)的系統(tǒng)。雖然參數(shù)γ沒(méi)有精確解（6）的特殊意義，但是β值是用來(lái)比較我們包含那些文獻(xiàn)的結(jié)果。 龐加萊映射是應(yīng)用于揭示規(guī)則與混沌運(yùn)動(dòng)區(qū)域的。龐加萊映射是通過(guò)采取在水平面Z = 0的初始點(diǎn)（r0，0）并記錄交叉坐標(biāo)常角軌道Zn= 2nL，n = 0，± 1，± 2。 。 。 。 幾個(gè)值為β的龐加萊映射是同時(shí)使用近似精確解計(jì)算和分析的。在這里，我們介紹的一個(gè)單一出發(fā)點(diǎn)的龐加萊映射被選在了混亂的區(qū)域（圖3）選擇的地圖。在平面圖的白色區(qū)域?qū)?yīng)于島狀物。這些島狀物的邊界是用細(xì)實(shí)線(xiàn)繪制的。 龐加萊映射島狀物相應(yīng)于流動(dòng)的Kolmogorov–Arnold–Moser (KAM) 管。在這樣一個(gè)管里捕獲的流體將只能在里面移動(dòng)，不與外管中的其余流體混合。這個(gè)混KAM管的影響，相比混合器的總流量，可以描繪為通過(guò)軟管的相對(duì)流量。因此，對(duì)于兩個(gè)島狀物的通過(guò)KAM管搬運(yùn)的面積和流量是可進(jìn)行評(píng)估的。流量可以計(jì)算看做在島狀物面積上v2的積分，或者作為一個(gè)島狀物的輪廓邊界通過(guò)斯托克斯定理積分。 圖3（a）和3（b）呈現(xiàn)了β= 4龐加萊映射。對(duì)于近似解的八個(gè)最大的島嶼是清晰可見(jiàn)（圖3（a））。他們占據(jù)了約49％橫截面積并且包含約55％的總流量。確切的解決方案提供了一個(gè)完全不同的群島系統(tǒng)（圖3（b））。他們的影響是相當(dāng)?shù)偷模驗(yàn)樗鼈冎徽技s13％的面積，并承擔(dān)總流量的18％。 對(duì)于混合強(qiáng)度為β的大量實(shí)用性的不同就開(kāi)始變得更強(qiáng)了。圖3（c）和3（d）代表β= 8時(shí)的情況。該近似解提供了兩個(gè)占據(jù)約13%截面（見(jiàn)圖3（c）），承擔(dān)總流量的18％的島狀物，同時(shí)群島的精確解（圖3（d））揭示了只占約0.7％的斷面面積。通過(guò)KAM管總數(shù),在這種情況下的相對(duì)流量約只有1％的總流量。 在這兩個(gè)例子中介紹了KAM管橫截面總面積明顯較小時(shí)使用的精確解。由于兩個(gè)近似和精確的解是以相同的簡(jiǎn)化模型的PPM為基礎(chǔ)，i.e.忽略了在混和器元素的結(jié)合處的過(guò)渡影響，計(jì)算出的KAM管的形狀，應(yīng)持一點(diǎn)保留態(tài)度。通過(guò)這些管道的相對(duì)的截面和相對(duì)流量是非常適當(dāng)?shù)?，并且他們可以給出一個(gè)實(shí)用的實(shí)際流動(dòng)價(jià)值標(biāo)準(zhǔn)的估計(jì)值。 條紋線(xiàn)可作為一種表征混合和可視化基礎(chǔ)的混合機(jī)械裝置的工具。Kusch和Ottino [6]指出，從創(chuàng)始的KAM管的橫截面計(jì)算條紋線(xiàn)，遠(yuǎn)比實(shí)驗(yàn)觀察得到的不同。對(duì)計(jì)算的β= 8.0的條紋線(xiàn)和實(shí)驗(yàn)獲得的β=10.0 ± 0.3結(jié)果進(jìn)行了比較至少有些相似。他們指出，PPM模式難以接近地模仿實(shí)驗(yàn)結(jié)果（由于劃分的板塊比小半徑管長(zhǎng)度要短）。然而，采用修正速度場(chǎng)進(jìn)行的數(shù)值模擬的結(jié)果（6），（18）和β值給出一個(gè)更好的一致。圖3（e）和3（f）顯示了β= 10的同時(shí)使用解決方案的龐加萊映射。在圖3（f）中周期2的兩島狀物的大致輪廓是用實(shí)線(xiàn)繪制的。這些輪廓被用來(lái)揭示相應(yīng)的KAM管的形狀（參見(jiàn)圖4（c））。封閉多邊形描繪的輪廓和通過(guò)4個(gè)混合元素的數(shù)字追蹤而來(lái)的這些多邊形的至高點(diǎn)，顯示了KAM管外邊界。另外兩張?jiān)趫D4中的圖像代表圖像的數(shù)值（a）和實(shí)驗(yàn)（b）分別來(lái)自于Kusch和Ottino [6]的成果。對(duì)于實(shí)驗(yàn)結(jié)果的實(shí)際混合強(qiáng)度的值，是β= 10.0 ± 0.3，我們計(jì)算了KAM管 圖3。龐加萊映射分別地作為4種不同攪拌強(qiáng)度，β值=（（a）和（b）），β= 8（（c）和（d）段），β= 10（（e）和（f）段）的結(jié)果。圖片中左邊的列（（a），（c），（e）段）通過(guò)方案（16），（17） 采用近似解獲得，而在右列的通過(guò)精確的方案（6），（18）獲得。 圖4。對(duì)比（a）計(jì)算的（β= 8）和（b）實(shí)驗(yàn)性的（β= 10.0 ± 0.3）庫(kù)施和Ottino [6]的實(shí)驗(yàn)性的條紋線(xiàn)，計(jì)算PPM模式下的KAM管的混合強(qiáng)度參數(shù)β= 10.0（c）（圖像（a）及（b）是來(lái)自于許可轉(zhuǎn)載的劍橋大學(xué)出版社的論文圖9。） 以及形狀限制值β= 9.7，β= 10.3。鋼管的整體造型變化不大，混合強(qiáng)度的變化主要是管壁厚度的影響：它是較大的β參數(shù)，反之也一樣。 Kusch和Ottino [6]沒(méi)有明確指定條紋線(xiàn)可視化染料注射的位置。然而在KAM管外面注入染料一點(diǎn)點(diǎn)很容易發(fā)現(xiàn)，因?yàn)槿玖祥_(kāi)始傳遍混合元素所以這是顯而易見(jiàn)的。為了說(shuō)明這一點(diǎn)，島的幾何中心周?chē)L制了圓（參見(jiàn)圖3（f））。標(biāo)記是均勻分布在每個(gè)圓邊界，并通過(guò)四要素混合的PPM（兩個(gè)空間時(shí)間）進(jìn)行跟蹤。在圖5（a）中圓的半徑為0.03a，因此所有的標(biāo)記都定位在KAM管里。在圖5（b）中圓（半徑0.062a）觸及到管邊界。這種條紋線(xiàn)可以稍微變形，但仍完全在管內(nèi)抓獲。在圖5（c）中最初的圓比在圖3（f）中的島狀物稍大，因而標(biāo)記包含在KAM管外?？梢郧宄乜吹剑灰卸潭趟膫€(gè)混合細(xì)胞，標(biāo)志物就可以在整個(gè)管道截面蔓延。 使用近似數(shù)值解（16），（17）指引Kusch和Ottino [6]得到一個(gè)偉大差異的對(duì)于10 <β<4的情況下的實(shí)驗(yàn)結(jié)果：實(shí)驗(yàn)顯示了非常穩(wěn)定KAM管，而計(jì)算顯示出了許多分支（比如，可見(jiàn)于他們的論文中的圖10（d））。但是使用相對(duì)簡(jiǎn)單穩(wěn)定結(jié)構(gòu)的精確方案（6），（18）卻被預(yù)測(cè)到了。例如，相對(duì)于一個(gè)比較大的β= 20混合強(qiáng)度，四個(gè)第一次序的KAM管被發(fā)現(xiàn)，但沒(méi)有發(fā)現(xiàn)2時(shí)期的KAM管。 圖5。這些標(biāo)記的痕跡，最初由規(guī)律地以時(shí)期2的到得幾何中心為中心空出不同半徑的圓。每個(gè)圓圈包含100個(gè)標(biāo)記。半徑是：（一）0.03a完全地再KAM管里面，（二）0.062a，觸摸到它的邊界，（三）0.08a – 劃定軟管邊界。 它們的管截面（并且因此，與他們相關(guān)的流量）相對(duì)較小。，不過(guò)這些周期性結(jié)構(gòu)穩(wěn)定。 4。結(jié)論 雖然正在研究的流動(dòng)僅僅是一個(gè)原型，但是它擁有廣泛使用的攪拌裝置流動(dòng)的一些重要特征。在同一模型框架獲得一個(gè)近似的比較和精確解，顯示了某些數(shù)學(xué)簡(jiǎn)化可能產(chǎn)生重大影響。這種簡(jiǎn)化在預(yù)測(cè)系統(tǒng)的行為上可能造成很大的差異，特別是對(duì)于那些應(yīng)該表現(xiàn)出的混沌性系統(tǒng)。在這里，由于使用（在以前的研究）了人為平滑的橫截面速度場(chǎng)一個(gè)長(zhǎng)期的近似解，在預(yù)測(cè)行為上就有差異。確切的解決方案顯示與已報(bào)道的實(shí)驗(yàn)結(jié)果能很好地一致。 當(dāng)然，存在的一個(gè)重要問(wèn)題就是在混合元件和忽視了這些過(guò)渡的發(fā)展流動(dòng)之間，有了突然的轉(zhuǎn)變。事實(shí)上最近的數(shù)值模擬結(jié)果（霍布斯等人[5]）表明，對(duì)于帶有螺紋的有限厚度的螺旋板的Kenics混合攪拌機(jī)，在入口和出口流動(dòng)的每個(gè)轉(zhuǎn)變?cè)囟紩?huì)強(qiáng)烈影響到一個(gè)以上元素長(zhǎng)度的速度場(chǎng)，這是一個(gè)重要的假設(shè)。 然而，從結(jié)果中呈現(xiàn)的——準(zhǔn)確描述了在混合流動(dòng)中哪怕是很小的變化也能顯著地改變系統(tǒng)整體混合反應(yīng)的速度區(qū)域的——重要性結(jié)論，仍然是適用真正的工業(yè)場(chǎng)合。 致謝 ? 筆者要感謝授予了編號(hào)為EWT44.3453的荷蘭科技基金（STW）的支持。我們也感謝其中一個(gè)發(fā)表“在閉塞或不了解情況下參數(shù)化的使用計(jì)算機(jī)從而產(chǎn)生一個(gè)荒謬結(jié)果”觀點(diǎn)的證明人。 Eur. J. Mech. B/Fluids 18 (1999) 783792 1999 ditions scientifiques et mdicales Elsevier SAS. All rights reserved Three-dimensional mixing in Stokes flow: the partitioned pipe mixer problem revisited V. V. M e l e s h ko a , O.S. Galaktionov a;b , G.W.M. Peters b; *, H.E.H. Meijer b a Institute of Hydromechanics, National Academy of Sciences, 252057 Kiev, Ukraine b Dutch Polymer Institute, Eindhoven Polymer Laboratories, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (Received 24 April 1998; revised 22 February 1999; accepted 7 March 1999) Abstract The velocity field and mixing behaviour in the so-called partitioned pipe mixer were studied. Starting with the same physical model as in previous studies, an exact analytical solution was developed which yields a more accurate description of the flow than the previously used approximate solution. Also, the results are in better accordance with the reported experimental data. 1999 ditions scientifiques et mdicales Elsevier SAS Stokes flow / laminar distributive mixing / static mixers 1. Introduction The aim of the present paper is to study the three-dimensional creeping flow in an infinitely long cylindrical pipe with internal walls, that divide the pipe into a sequence of semicircular ducts. Such a system, called the partitioned pipe mixer (PPM) was introduced by Khakhar et al. 1 as a prototype model for the widely used Kenics static mixer (Middleman 2). In the Kenics mixer each element is a helix, twisted on a 180 , plate; elements are arranged axially within a cylindrical tube so that the leading edge of an element is at right angles to the trailing edge of the previous one. Computational fluid dynamics tools make a straightforward numerical simulation of this kind of three- dimensional flow feasible (Avalosse and Crochet 3, Hobbs and Muzzio 4, Hobbs et al. 5). However, such simulations do require significant computational resources, especially when studying the effect of varying parameters on the mixing process. Therefore, simplified analytical models, that give the possibility of fast simulations of the process (or mimic its features closely enough), are still useful. The PPM model of the essentially three-dimensional flow was highly idealized, nevertheless retaining the main features of the flow under study. The model involves two superimposed, independent, two-dimensional flow fields: a cross-sectional (rotational) velocity field and a fully developed axial Poiseuille profile in every semicircular duct. This gives two independent two-dimensional boundary problems instead of the three- dimensional problem. The solution proposed by Khakhar et al. 1 for the cross-sectional velocity field was only an approximate one. There exists, however, exact analytical solutions in a closed form. In the present paper we use these exact solutions to examine the mixing properties in this three-dimensional mixer. Important differences in some mixing patterns were obtained, and our results resemble more closely the available experimental results of Kusch and Ottino 6. * Correspondence and reprints: Department of Mechanical Engineering, Building W.h. 0.119, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands; e-mail: gerritwfw.wtb.tue.nl 784 V. V. M el eshko et al . 2. Velocity field in PPM Consider the interior of an infinite cylinder 06r6a; 06 62 ; jzj0; 06 6 =2 with the constant tangential velocity V applied at the plane D0. Figure 1(a) shows contour levels of the stream function (6). The cross-sectional flow exhibits a single vortex cell with one elliptic stagnation point at (0:636a; =2). The previous studies (Khakhar et al. 1, Ottino 7) suggested the approximate one-term solution of the boundary problem (1), (4): 9 D 4Va 3 r a 2 1 r a sin 2 ; D.11=3/ 1=2 1 0:915; (16) which has been obtained by a variational method. This expression (16), however, does not satisfy both the governing biharmonic equation (1) and the no-slip condition at the moving boundary! It turns out that the tangential velocity at the boundary rDa varies as .4=3/V sin 2 instead of being constant V. Therefore, the velocity is overestimated (up to 33% at the circular boundary) in some zones far from the flat boundary, and it is artificially smoothed near corners. The contour plot of the stream function according to one-term solution (16) is presented in figure 1(b). The solution of boundary problem (2), (5) for the fully developed axial flow in a semicircular duct reads (Ottino 7): v z D 16 2 8 hv z i 1 X kD1 r a 2k 1 r a 2 sinT.2k 1/ U .2k 1/f4 .2k 1/ 2 g ; (17) where hv z iD 8 2 4 2 1 p z a 2 is the average axial velocity. Using straightforward transformations and tables of infinite sums (Prudnikov et al. 12), we can present expression (17) in a closed form: v z D 2 2 8 hv z i ( r 2 a 2 sin 2 C r a a r sin 1 4 r 2 a 2 a 2 r 2 sin.2 / ln r 2 C2arcos Ca 2 r 2 2arcos Ca 2 C 1 2 2 r 2 a 2 a 2 r 2 cos.2 / arctan 2arsin a 2 r 2 ) ; (18) which is preferable for numerical simulations of the advection process. It is worth mentioning that the first three terms of the infinite sum (17) used in Khakhar et al. 1 and Ottino 7 provide reasonable accuracy with EUROPEAN JOURNAL OF MECHANICS B/FLUIDS, VOL. 18,N 5, 1999 Three-dimensional mixing in Stokes flow 787 Figure 2. Contour plots of the axial velocity v z : solid lines correspond to the exact expression (18), dotted lines correspond to three-terms approximation of (17). maximum errors (compared to exact expression (18) that are within a few percent. In figure 2 the contour lines of v z , defined by (18) are shown as a solid lines, while the same contours for three-term approximation of (17) are plotted as dotted lines. Despite this approximation the shape of the contours is rather similar, the discrepancy amounts up to 7% of the average velocityhv z i, reaching a maximum not far from the corner points, where the velocity v z is underestimated. Increasing the number of terms in (17) to one hundred, reduces the relative error to less then 0:005%, but, it will take much more computer time to simulate the passive tracers advection. 3. Chaotic mixing in PPM The motion of a passive individual (Lagrangian) particle is described by the advection equations dr dt Dv r .r; /; r d dt Dv .r; /; dz dt Dv z .r; /; (19) with the velocity field on the right hand side of (19) defined by (6) and (18). The initial conditions are rDr 0 ; D 0 ;zD0attD0. Here the variable is obviously defined as D 8 : ; 2kL6z.2kC1/L; 06 6 ; ; 2kL6z.2kC1/L; 2 ; =2;.2kC1/L6z.2kC2/L; =26 63 =2; C =2;.2kC1/L6z.2kC2/L; 06 =2; 3 =2;.2kC1/L6z.2kC2/L; 3 =2 2 ; (20) where kD0; 1; 2;: System (19) describes a steady motion of an individual particle along the streamline in each compartment. However, as the flow is three-dimensional and spatially periodic, it can exhibit chaotic behaviour (Aref 13, Section 5.4). In Khakhar et al. 1 the single non-dimensional parameter , the mixing strength D 4VL 3 hv z ia ; (21) EUROPEAN JOURNAL OF MECHANICS B/FLUIDS, VOL. 18,N 5, 1999 788 V. V. M el eshko et al . was introduced to completely describe the behaviour of such a system. Although the parameter has no particular meaning for the exact solution (6), the value of is used to compare our results with those of the literature. Poincar mapping was applied to reveal the zones of regular and chaotic motion. The Poincar maps were constructed by taking an initial point .r 0 ; 0 /at the levelzD0 and recording the coordinates of the intersections of the trajectory with the planes z n D2nL; nD0; 1; 2;: The Poincar maps for several values of were computed and analysed using both the approximate and exact solution. Here we present the resulting Poincar maps for which one single starting point was chosen in the chaotic zone (figure 3). White regions in the plots correspond to islands. The boundaries of the islands are plotted as thin solid lines. Islands in Poincar maps correspond to the KolmogorovArnoldMoser (KAM) tubes in the flow. The fluid captured in such a tube will only travel inside, not mixing with the rest of the fluid outside the tube. The influence of the KAM tube on mixing can be characterized by the relative flux carried by the tube compared to the total flux through the mixer. So, for the islands both their area and the flux carried by corresponding KAM tubes are evaluated. The flux can be computed as the integral of v z over the islands area, or, by using Stokes theorem, as a contour integral over the boundary of islands. Figures 3(a) and 3(b) present the Poincar maps for D4. For the approximate solution the eight largest islands are clearly seen (figure 3(a). They occupy about 49% of the cross-section area and carry approximately 55% of the total flux. The exact solution provides a completely different system of islands (figure 3(b). Their influence is considerably lower since they occupy only about 13% of the area and bear 18% of the total flux. The difference becomes even stronger for larger values of the mixing strength . Figures 3(c) and 3(d) represent the case of D8. The approximate solution provides two large islands that occupy about 13% of the cross-section (see figure 3(c) and bear 18% of total flux, while the islands revealed by the exact solution (figure 3(d) occupy only about 0:7% of the cross-section area. The relative flux through KAM tubes amounts in this case to approximately only 1% of total flux. In both examples presented the total area of the cross-section of the KAM tubes is significantly smaller when the exact solution is used. As both the approximate and exact solutions are based on the same simplified model of the PPM, i.e. neglecting the transition effects at the joints of the mixer elements, the calculated shape of the KAM tubes should be considered with some reservations. The relative cross section of, and the relative flux through these tubes are of more relevance and they can give an useful estimation of these values for practical flows. Streaklines can serve as a tool to characterise the mixing and to visualise underlying mixing mechanisms. Kusch and Ottino 6 noted that computed streaklines, originating from a cross-section of a KAM tube, are much different from those experimentally observed. Computed streaklines for D8:0 and the experimental results obtained for D10:0 0:3 were compared to get, at least, some resemblance. They pointed out that the PPM model can hardly mimic closely the experimental results (due to the small length of dividing platesless than the pipe radius). However, the results of numerical simulations using the corrected velocity field (6), (18) and the right value for gives a much better agreement. Figures 3(e) and 3(f) show the Poincar maps for D10, using both solutions. In figure 3(f) the approximate contours of the two islands of period 2 are plotted with solid lines. These contours were used to reveal the shape of the correspondent KAM tubes (see figure 4(c). Contours were represented by closed polygons and the vertices of these polygons were then tracked numerically through four mixing elements, showing the outer boundary of the KAM tube. The other two images in figure 4 represent the numerical (a) and experimental (b) results from Kusch and Ottino 6, respectively. As for the experimental results the actual value of mixing strength was D10:0 0:3, we calculated the KAM tube EUROPEAN JOURNAL OF MECHANICS B/FLUIDS, VOL. 18,N 5, 1999 Three-dimensional mixing in Stokes flow 789 (a) (b) (c) (d) (e) (f) Figure 3. Poincar maps for different values of mixing strength D4 (a) and (b), D8 (c) and (d), D10 (e) and (f), respectively. Pictures in the left column (a), (c), (e) were obtained by using approximate solution (16), (17), while those in the right column were obtained by using the exact solution (6), (18). EUROPEAN JOURNAL OF MECHANICS B/FLUIDS, VOL. 18,N 5, 1999 790 V. V. M el eshko et al . (a) (b) (c) Figure 4. Computed KAM tubes for the PPM model with mixing strength parameter D10:0 (c) compared with (a) computed ( D8) and (b) experimental ( D10:0 0:3) streaklines from Kusch and Ottino 6. (Images (a) and (b) are taken from figure 9 of the cited paper, reproduced with permission from Cambridge University Press.) shapes for the limiting values D9:7and D10:3 as well. The overall shape of the tubes does not change much, variation of mixing strength influences mainly the tube thickness: it is thinner for larger parameter and vice versa. Kusch and Ottino 6 did not specify explicitly the location where the dye for streakline visualization was injected. However, it is easy to show that when the dye is injected just a little outside the KAM tube, this is clearly visible because the dye starts to spread over the mixing elements. To illustrate this, circles were drawn around the geometrical center of the island (see figure 3(f). Markers were evenly distributed on the boundary of every circle and tracked through four mixing elements (two spatial periods) of the PPM. In figure 5(a) the radius of the circle was 0:03a, thus all markers were positioned well inside the KAM tube. In figure 5(b) the circle (of radius 0:062a) touches the tube boundary. Such streaklines can be slightly deformed but are still captured completely within the tubes. In figure 5(c) the initial circle was slightly larger then the island shown in figure 3(f), and thus contains markers outside the KAM tube. It is clearly seen that within just four mixing cells the markers spread over the whole cross-section of the pipe. The use of approximate numerical solution (16), (17) led Kusch and Ottino 6 to a great discrepancy with experimental results for 10 40: experiments showed remarkably stable KAM tubes, while computations exhibited a lot of bifurcations (see, for example, figure 10(d) from their paper). However, using the exact solution (6), (18) relatively simple stable structures are predicted. For example, for a relatively large mixing strength of D20, four KAM tubes of first order were found but no KAM tubes of period 2 were detected. EUROPEAN JOURNAL OF MECHANICS B/FLUIDS, VOL. 18,N 5, 1999 Three-dimensional mixing in Stokes flow 791 (a) (b) (c) Figure 5. Traces of the markers, originally regularly spaced on circles of different radii, centered around the geometrical centers of the islands of period 2. Each circle contains 100 markers. The radii are: (a) 0:03awell inside the KAM tube, (b) 0:062atouching its boundary, (c) 0:08acircumscribing the tube boundary. The cross section of these tubes (and, consequently, the flux associated with them) is relatively small. These periodical structures are, nevertheless, stable. 4. Conclusions Although the flow under study is merely a prototype flow, it possesses some important features of flows in widely used mixing devices. The comparison of an approximate and an exact solution, obtained within the framework of the same model, shows the possible major consequences of some mathematical simplifications. Such simplifications can cause large differences in the predicted systems behaviour, especially for systems that are supposed to exhibit chaotic properties. Here, the difference in the predicted behaviour was caused by the use (in previous studies) of a one-term approximate solution that artificially smoothes the cross-sectional velocity field. The exact solution shows much better agreement with the reported experimental results. Of course, there exists an important problem regarding the abrupt transition between mixing elements and ignoring developing flows at these transitions. Results of recent numerical simulations (Hobbs et al. 5) show that, indeed, this is a major assumption: for the Kenics mixer with a finite thickness of helical screwed mixing plates, flow transitions at the abrupt entrance and exit of each element strongly affect the velocity field over up to one quarter of the element length. However, the conclusion from the results presented of the importance of an accurate description of the velocity field in mixing flows, where even small changes can significantly alter the overall mixing behaviour of the system, is still applicable for real industrial situations. EUROPEAN JOURNAL OF MECHANICS B/FLUIDS, VOL. 18,N 5, 1999 792 V. V. M el eshko et al . Acknowledgements The authors would like to acknowledge support by the Dutch Foundation of Technology (STW), grant no. EWT44.3453. 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